The cover picture shows JT-60, one of the largest operational tokamaks.
By courtesy of the Japan Atomic Energy Research Institute, Tokyo.

PLASMA PHYSICSAND CONTROLLED
NUCLEAR FUSION RESEARCH
1986
VOLUME 2

The following States are Members of the International Atomic Energy Agency:

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© IAEA, 1987

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to the International Atomic Energy Agency, Wagramerstrasse 5, P.O.Box 100, A-1400 Vienna, Austria.

Printed by the IAEA in Austria
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NUCLEAR FUSION, SUPPLEMENT 1987

PLASMA PHYSICSAND CONTROLLED
NUCLEAR FUSION RESEARCH
1986
PROCEEDINGS OF THE
ELEVENTH INTERNATIONAL CONFERENCE ON PLASMA PHYSICSAND CONTROLLED NUCLEAR FUSION RESEARCHHELD BY THE
INTERNATIONAL ATOMIC ENERGY AGENCYIN KYOTO, 13-20 NOVEMBER 1986

In three volumes

VOLUME 2

INTERNATIONAL ATOMIC ENERGY AGENCY
VIENNA, 1987

PLASMA PHYSICS AND CONTROLLED NUCLEAR FUSION RESEARCH 1986
IAEA, VIENNA, 1987
STI/PUB/723
ISBN 92-0-130187-1

FOREWORD
The IAEA Conferences on Plasma Physics and Controlled Nuclear Fusion
Research, organized biennially, are the largest and most important meetings in thefield. They are a scientific forum not only for the presentation of the best results
achieved in all laboratories performing fusion research but also for the co-ordinationof international fusion efforts.
The Eleventh Conference, which was organized in co-operation with theJapanese Atomic Energy Research Institute, was attended by about 650 participants
and observers representing 32 countries and two international organizations. About
200 papers were accepted for oral and poster presentations. The Conference opened
with the traditional Artsimovich Memorial Lecture.
The Eleventh Conference became an important event for fusion scientists andengineers all over the world, for two special reasons. First, on the basis of the results
achieved on several experimental devices, the plans for reaching plasma ignition
within the next two years were presented. Second, the Conference not only marked
25 years of international co-operation under the auspices of the IAEA but also
represented the starting point for a new international venture — an international
thermonuclear experimental reactor.
These Proceedings, which include all the technical papers and five Conference
summaries, are published in English as a supplement to the IAEA journal Nuclear
Fusion.
The Agency promotes close international co-operation among plasma and fusion
physicists and engineers of all countries by organizing these regular conferences oncontrolled nuclear fusion and by holding seminars, workshops and specialists
meetings on appropriate topics. It is hoped that the present publication, as part ofthese activities, will contribute to the rapid demonstration of fusion power as one of
the world's future energy resources.

EDITORIAL NOTE
The Proceedings have been edited by the editorial staff of the IAEA to the extent considerednecessary for the reader's assistance. The views expressed remain, however, the responsibility of thenamed authors or participants. In addition, the views are not necessarily those of the governments of
the nominating Member States or of the nominating organizations.
Although great care has been taken to maintain the accuracy of information contained in thispublication, neither the IAEA nor its Member States assume any responsibility for consequences which
may arise from its use.
The use of particular designations of countries or territories does not imply any judgement by the
publisher, the IAEA, as to the legal status of such countries or territories, of their authorities and institutions or of the delimitation of their boundaries.
The mention of names of specific companies or products (whether or not indicated as registered)does not imply any intention to infringe proprietary rights, nor should it be construed as an endorsement
or recommendation on the part of the IAEA.
The authors are responsible for having obtained the necessary permission for the IAEA toreproduce, translate or use material from sources already protected by copyrights.
Material prepared by authors who are in contractual relation with governments is copyrighted
by the IAEA, as publisher, only to the extent permitted by the appropriate national regulations.

CONTENTS OF VOLUME 2
MAGNETIC CONFINEMENT THEORY (Session E)
Sawtooth oscillations (IAEA-CN-47/E-I-l-l)
J.A. Wesson, P. Kirby, M.F. Nave
Theory of sawtooth disruption (IAEA-CN-47/E-I-1-2)
D. Biskamp, H. Welter
Sawteeth and ideal MHD (IAEA-CN-47/E-I-1-3)
M.N. Bussac, K. Lerbinger, R. Pellat, M. Tagger
Discussion on Papers IAEA-CN-47/E-I-1-1 to E-I-l-3
Second stability access in tokamak plasmas (IAEA-CN-47/E-I-2-1)
K. Yamazaki, H. Naitou, T. Amano, Y. Hamada, K. Matsuoka,
Y. Midzuno, Y. Nakayama, T. Sato, T. Tsunematsu, T. Tuda, S. Seki,
M. Azumi, S. Tokuda, T. Ozeki, T. Nemoto, Y. Kishimoto, T. Takeda
Stable tokamak access to, and operation in, the second stability region(IAEA-CN-47/E-I-2-2)
A.M.M. Todd, M. Phillips, M. Chance, J. Manickam, N. Pomphrey
Ellipsoidal shell tokamak (IAEA-CN-47/E-I-2-3)
F.J. Helton, Mingsheng Chu, J.M, Greene, R.W. Harvey, J.K. Lee,
T. Ohkawa, P. A. Politzer
MHD stability in low-aspect-ratio tokamaks (IAEA-CN-47/E-I-2-4)
B.A. Carreras, L.A. Charlton, J.T. Hogan, J.A. Holmes, E.A. Lazarus,
W.A. Cooper, T.C. Hender
Discussion on Papers IAEA-CN-47/E-I-2-1 to E-I-2-4
Simulation studies of tokamak dynamics (IAEA-CN-47/E-I-3)
M. Tanaka, T. Hayashi, K. Harafuji, Y. Nakayama, T. Sato
Stability conditions for helical MHD perturbations and disruptive
instability in tokamaks (IAEA-CN-47/E-I-4-1)
L.E. Zakharov
Bifurcation of plasma equilibrium in tokamaks (IAEA-CN-47/E-I-4-2)
Yu.N. Dnestrovskij, D.P. Kostomarov, A.V. Pedorenko, A.M. PopovDiscussion on Papers IAEA-CN-47/E-I-4-1 and E-I-4-2
Radiofrequency stabilization of ideal ballooning modes in atokamak plasma (IAEA-CN-47/E-II-l-l)
A. Sen, P.K. Kaw, A.K. Sundaram

3
11
17
25
27

37
45

53

63
65

75
89
99
101

Electromagnetic particle simulations of RF stabilization and ECRH ofover-density plasmas (IAEA-CN-47/E-II-1-2)
109
H. Abe, S. Nakajima, Y. Kadoya, H. Okada, R. Itatani, K. Sakai,
S. Takeuchi, M. Matsumoto, R. Sugihara
Discussion on Papers IAEA-CN-47/E-II-l-l and E-II-1-2
117
Theory of resistive magnetohydrodynamic instabilities excited by energetic
trapped particles in large-size tokamaks (IAEA-CN-47/E-II-2-1)
119
H. Biglary, L. Chen, R.B. White
Alfve'n wave phase mixing in a sheared magnetic field as a mechanism of'fishbone' instability damping and oscillation energy pumping into theregion of large k ± ~ wp/c (IAEA-CN-47/E-II-2-2)
127
A.D. Beklemishev, O.P. Pogutse
Effects of energetic particles on tokamak stability (IAEA-CN-47/E-II-2-3) . 135
J. Van Dam, M.N. Rosenbluth, H.L. Berk, N. Dominguez, G.Y. Fu,
X. Llobet, D.W. Ross, D.P. Stotler, D.A. Spong, W.A. Cooper,
D. Sigmar, D.E. Hastings, J.J. Ramos, H. Naito, T. Todoroki,
S. T. Tsai, S. G. Guo, J. W. Shen
Discussion on Papers IAEA-CN-47/E-II-2-1 to E-II-2-3
147
Nonlinear toroidal plasma dynamics by reduced fluid models(IAEA-CN-47/E-II-3-1)
149
M. Kotschenreuther, A. Aydemir, R. Carrera, R.D. Hazeltine,
J.D. Meiss, P.J. Morrison
Neoclassical MHD equations, instabilities and transport intokamaks (IAEA-CN-47/E-II-3-2)
157
J.D. Callen, W.X. Qu, K.D. Siebert, B.A. Carreras, K.C. Shaing,
D.A. SpongI. Plasma edge microinstability with up-down asymmetry.
II. Theory of collisional tearing modes for arbitrary Larmor radii(IAEA-CN-47/E-III-l)
167
A. Rogister, G. Hasselberg
Edge plasma oscillations in the H-mode: Effect or cause?
(IAEA-CN-47/E-III-3)
177
S.I. Krasheninnikov, A.S. Kukushkin, V.A. Pozharov
Magnetic islands and chaos induced by heat flow (IAEA-CN-47/E-III-4) ... 187
P.H. Rebut, M. Brusati, M. Hugon, P. Lallia
Incomplete thermalization and enhanced transport in tokamak plasmas(IAEA-CN-47/E-III-5)
197
S.-I. ltoh, K. Itoh
Effect of quasi-linear evolution of the minority ion distribution function
on the wave dispersion and absorption and on the thermonuclear
reaction rate (IAEA-CN-47/E-III-6)
205
A.V. Longinov, S.S. Pavlov, K.N. Stepanov

Effect of x-point location on edge confinement in diverted tokamaks(IAEA-CN-47/E-III-24)
T. Ohkawa, F.L. Hinton

221

OPEN CONFINEMENT SYSTEMS (Session C)
TMX-U tandem mirror thermal barrier experiments(IAEA-CN-47/C-I-1)
T.C. Simonen, S.L. Allen, D.E. Baldwin, J.D. Barter, L.V. Berzins,
M.R. Carter, T.A. Casper, J.F. Clauser, F.H. Coensgen,
D.L. Correll, W.F. Cummins, C.C. Damm, G. Dimonte, B.H. Failor,
S. Falabella, J.H. Foote, T.K. Fowler, A.H. Futch, R.K. Goodman,
D.P. Grubb, D. Hill, E.B. Hooper, R.S. Hornady, R.D. Horton,
W.L. Hsu, B. Huff, A.L. Hunt, R.A. James, C.J. Lasnier,
C.A. Leavitt, W.H. Meyer, J.M. Moller, A.W. Molvik, W.E. Nexsen,
D.E. Perkins, W.L. Pickles, P.A. Pincosy, G.D. Porter, P. Poulsen,
D.B. Ress, M. Shiho, E.H. Silver, B.W. Stallard, W.C. Turner,
R.D. Wood, T.L. Yu
Discussion
Studies of potential formation and transport in the tandem mirror
GAMMA 10 (IAEA-CN-47/C-I-2)
T. Cho, M. Ichimura, M. Inutake, K. Ishii, A. Itakura, I. Katanuma,
Y. Kiwamoto, A. Mase, S. Miyoshi, Y. Nakashima, T. Saito,
K. Sawada, D. Tsubouchi, N. Yamaguchi, K. Yatsu
Discussion
Recent results from the Tara tandem mirror and the Constance-B mirror(IAEA-CN-47/C-I-3)
R.S. Post, K. Brau, J. Casey, X. Chen, J. Coleman, H.R. Garner,
M. Gerver, S. Golovato, D. Goodman, W. Guss, S. Hiroe,
S. Hokin, S. Home, J. Irby, J. Kesner, B. Lane, T. Moran,
L. Pocs, E. Sevillano, D. Smatlak, D.K. Smith, J. Sullivan,
R.P. Torti, X.Z. Yao
Discussion
Stabilization of MHD modes in an axisymmetric magnetic mirror by
applied RF waves and initial results of Phaedrus-B(IAEA-CN-47/C-I-4-1)
R.A. Breun, P. Brooker, D. Brouchous, J. Browning, G. Butz,
J. Conrad, E. Dales, J. Ferron, R. Goulding, N. Hershkowitz,
T. Intrator, C. Litwin, R. Majeski, S. Meassick, B. Nelson,
L. Peranich, H. Persing, J. Radtke, D. Roberts, G. Severn, D. Sing,
Enyao Wang, D.A. D'Ippolito, J.R. Myra, G.L. Francis

231

241
243

250
251

262

263

Axisymmetric mirror plasma produced, heated and stabilized by
ICRF only (IAEA-CN-47/C-I-4-2)
R. Itatani, H. Takeno, Y. Yasaka
Discussion on Papers IAEA-CN-47/C-I-4-1 and C-I-4-2
A tandem mirror modelling code (IAEA-CN-47/C-II-1)
W.M. Nevins, B.M. Boghosian, R.H. Cohen, W.F. Cummins,
P.F. Dubois, A. Friedman, L.L. LoDestro, Y. Matsuda,
L.D. Pearlstein, G.D. Porter, M.E. Rensink, T.D. Rognlien,
G.R. Smith, J.J. Stewart, M. Phillips
Drift pumping of tandem mirror thermal barriers (IAEA-CN-47/C-II-2) ....
D.E. Baldwin, J.A. Byers, Y.J. Chen, T.B. Kaiser
Stabilization of flute instability in axially symmetric mirror machines(IAEA-CN-47/C-II-3)
I.A. Kotel'nikov, G.V. Roslyakov, D.D. Ryutov, G.V. Stupakov
Progress in plasma heating with powerful electron beams in open
systems (IAEA-CN-47/C-II-4)
A.V. Arzhannikov, V.T. Astrelin, A.P. Avrorov, B.N. Brejzman,
A.V. Burdakov, V.S. Burmasov, V.V. Chikunov, I.V. Kandaurov,
V.A. Kapitonov, A.D. Khil'chenko, B.A. Knyazev, V.S. Kojdan,
V.V. Konyukhov, Eh.P. Kruglyakov, S.V. Lebedev, K.I. Mekler,
O.I. Meshkov, V.V. Postupaev, D.D. Ryutov, A.L. Sanin,
M.A. Shcheglov, S.L. Sinitskij, S.G. Voropaev, L.N. Vyacheslavov
Enhancement of radial transport by collective processes(IAEA-CN-47/C-II-5)
D.L. Eggleston, J.D. Crawford, T.M. O'Neil, J.H. Malmberg
Radiofrequency plugging and tandem mirror experiments in a
cusp-anchored mirror device (IAEA-CN-47/C-II-6)
T. Sato, R. Kumazawa, S. Okamura, K. Adati, T. Aoki, H. Fujita,
S. Hidekuma, T. Kawamoto, H. Masumoto, K. Nishimura, T. Hatori,
K. Takayama, H.R. Garner, H.D. Price, A.M. Howald, B.J. Leikind,
H. Ikezi, P.B. Parks, P. Andrews

273
281
283

293

305

323

337

343

ALTERNATIVE AND SUPPORTING SYSTEMS (Session D)
Studies on currentless Heliotron E plasma (IAEA-CN-47/D-I-1)
K. Uo, A. Hyoshi, T. Obiki, S. Morimoto, O. Motojima, M. Sato,S. Sudo, F. Sano, T. Mutoh, K. Akaishi, M. Nakasuga, K. Kondo,
K. Hanatani, H. Zushi, S. Besshou, H. Kaneko, T. Mizuuchi,
H. Okada, Y. Takeiri, N. Noda, I. Ohtake, Y. Ijiri, M. lima,
T. Kawabata, T. Senju, K. Yaguchi, T. Baba, S. Kobayashi,
R. Takahashi, M. Murakami, H.C. Howe, D.L. Hillis,
P.K. Mioduszewski, G.H. Neilson, J. Rice, J. Terry, E. Marmar,

355

T. Yamashina, T. Hino, H. Minagawa, T. Oda, M. Hamamoto,
T. Ohgo, Y. Sakamoto, H. Amemiya, K. Okazaki, H. Oyama, S. Kato,
A. Amemiya, H. Sugai, S. Urano, K. Uchino, T. Koga, K. Muraoka,
M. Akazaki, M. Maeda, S. Morita
Discussion
Plasma confinement in the Wendelstein VII-A stellarator(IAEA-CN-47/D-I-2)
H. Wobig, H. Maassberg, H. Renner,
W VII-A Team, ECRH Group, NI Group
Discussion
Physics studies for advanced stellarators (IAEA-CN-47/D-I-3)
W. Dommaschk, F. Herrnegger, W. Lotz, P. Merkel, J. Niihrenberg,
A. Schluter, U. Schwenn, R. Zille
Discussion
Transport theory in stellarators (IAEA-CN-47/D-I-4)
K.C Shaing, S.P. Hirshman, CO. Beasley, Jr., E.C. Crume, Jr.,
W.I. VanRij
Results from HBTX with improved confinement (IAEA-CN-47/D-II-l)
B. Alper, V. Antoni, M.K. Bevir, H.A.B. Bodin, C.A. Bunting,
P.G. Carolan, J. Cunnane, D.E. Evans, A.R. Field, S.J. Gee,
C.G. Gimblett, R. Hoyden, T.R. Jarboe, P. Kirby, A. Manley,
A.A. Newton, P.G. Noonan, A. Patel, R.S. Pease, M.G. Rusbridge,
K.P. Schneider, D.P. Storey, H.Y.W. Tsui, S. Whitfield, P.D. Wilcock
Discussion
Experimental and theoretical studies of the STP-3(M) reversed field
pinch in the high current density regime (IAEA-CN-47/D-II-2)
K. Sato, T. Amano, Z.X. Chen, H. Arimoto, S. Yamada, A. Nagata,
K. Yokoyama, Y. Kamada, A.^Matsuoka, S. Masamune, H. Shindo,
K. Saito, H. Murata, H. Oshiyama, S. Shiina, T. Tamaru
Discussion
Reversed field pinch experiments in ZT-40M and ZT-P
(IAEA-CN-47/D-II-3)
K.F. Schoenberg, G.A. Wurden, P.G. Weber, J.C Ingraham,
C.P. Munson, D.A. Baker, C.J. Buchenauer, L.C. Burkhardt,
T.E. Cayton, J.N. DiMarco, J.N. Downing, R.M. Erickson,
P.R. Forman, A. Haberstich, R.B. Howell, F.C Jahoda, R.S. Massey,
G. Miller, R.W. Moses, Jr., R.A. Nebel, J.A. Phillips, M.M. Pickrell,
A.E. Schofield, R.G. Watt, DM. Weldon, K.A. Werley
Experimental studies of reversed field pinch in REPUTE-1 andultra-low-q discharge (IAEA-CN-47/D-II-4)
N. Asakura, A. Fujisawa, T. Fujita, Y. Fukuda, H. Hattori, N. Inoue,
S. Ishida, K. Itami, Y. Iwata, H. Ji, Y. Kamada, K. Kusano,S. Matsuzuka, K. Miyamoto, J. Morikawa, C.P. Munson, Y. Murakami,

368
369

381
383

390
391

399

410
413

422
423

433

Y. Nagayama, H. Nihei, S. Onodera, M. T. Saito, T. Sato,S. Shinohara, H. Toyama, Y. Ueda, M. Utumi, H. Yamada,K. Yamagishi, Z. Yoshida
Discussion
Heating and confinement studies in reversed field pinches(IAEA-CN-47/D-II-5)
V. Antoni, A. Buffa, L. Carraro, S. Costa, P. De Simone, G. Flor,
F. Flora, L. Gabellieri, M. Giubbilei, L. Giudicotti, P. Innocente,
K. Jain, P. Martin, S. Martini, D. Merlin, S. Ortolani,
R. Paccagnella, M.E. Puiatti, B. Sass, P. Scarin, H. Sorensen,
M. Valisa, K. Weisberg, S. Zago, E. Zilli
Discussion
Results from the reversed field pinch experiment on TPE-1RM15 with aprogrammed vertical field (IAEA-CN-47/D-II-6)
T. Shimada, Y. Hirano, Y. Yagi, A.A. Newton, K. Ogawa
Discussion
Plasma heating in stellarator/torsatrons and tokamaks by ion cyclotron
waves (IAEA-CN-47/D-III-l)
E.F. Jaeger, D.B. Batchelor, K. Imre, H. Weitzner
Discussion
ICRF heating of currentless plasma in Heliotron E(IAEA-CN-47/D-III-2)
T. Mutoh, O. Motojima, M. Sato, H. Okada, H. Zushi, K. Kondo,
N. Noda, H. Kaneko, T. Mizuuchi, S. Besshou, F. Sano, S. Sudo,
Y. Takeiri, K. Akaishi, T. Kawabata, Y. Ijiri, K. Itoh, S. Morimoto,
A. Iiyoshi, K. Uo
Discussion
I. Electron cyclotron resonance plasma heating by ordinary waves
in the L-2 stellarator.
II. Fast wave eigenmodes in the L-2 stellarator during ICR plasma heating(IAEA-CN-47/D-III-3)/. E.D. Andryukhina, D.K. Akulina, G.M. Batanov, V.A. Batyuk,
M.S. Berezhetskij, M.A. Blokh, G.S. Voronov, S.E. Grebenshchikov,
S.I. Dolgaev, N.P. Donskaya, K.S. Dyabilin, L.M. Kovrizhnykh,
L.V. Kolik, N.F. Larionova, K.M. Likin, A.I. Meshcheryakov,
P.E. Moroz, S.N. Popov, A.E. Petrov, V.I. Roshchin,
A.V. Sapozhnikov, K.A. Sarksyan, N.N. Skvortsova, I.S. Sbitnikova,
A.D. Smirnova, V.N. Sukhodol'skij, M.F. Ternovskaya, O.I. Fedyanin,
S.I. Fedorenko, Yu.V. Kholnov, M.G. Shats, IS. Shpigel'
II. V.A. Batyuk, S.E. Grebenshchikov, A.I. Meshcheryakov,
P.E. Moroz, I.S. Sbitnikova, I.S. Shpigel'
Discussion

440
441

450
453
461
463
472
473

479

481

487

Plasma potential and confinement in the Nagoya Bumpy Torus (NBT-1M)(IAEA-CN-47/D-III-4)
H. Ikegami, M. Hosokawa, H. Iguchi, T. Shoji, T. Kamimura,
H. Sanuki, K. Takasugi, F. Tsuboi, T. Takeuchi, A. Komori,K. Kadota, J. Fujita, C. Honda, M. Yokoo, M. Maeda, K. Muraoka,
T. Yuyanta, T. Michishita, H. Kubo, M. Fujiwara
Discussion
Effects on equilibrium and stability of high beta RF heated plasmas in the
URAGAN-3 torsatron (IAEA-CN-47/D-III-5)
V.L. Berezhnyj, N.T. Besedin, M.P. Vasil'ev, V.S. Vojtsenya,
E.D. Volkov, Ya.F. Volkov, S.P. Gubarev, Yu.V. Gutarev, A.G. Dikij,
V.M. Zaikind, G.V. Zelenin, V.K. Konovalov, V.I. Kononenko,
V.D. Kotsubanov, B.V. Kravchin, A.E. Kulaga, Yu.K. Kuznetsov,
G.G. Lesniyakov, A.P. Litvinov, Yu.K. Mironov, N.I. Nazarov,
O.S. Pavlichenko, V.K. Pashnev, N.F. Perepelkin, V.V. Plyusnin,
G.N. Polakova, N.P. Ponomarenko, T.Yu. Ranyuk, L.N. Remez,
K.S. Rubtsov, S.I. Solodovchenko, A.I. Skibenko, A.S. Slavnij,
I.S. Sushko, V.S. Taran, V.T. Tolok, T.O. Toryak, I.P. Fomin,
A.N. Shapoval, O.M. Shvets, A.F. Shtan'
Experimental and theoretical studies of toroidal heliacs(IAEA-CN-47/D-III-6)
B.D. Blackwell, R.L. Dewar, H.J. Gardner, S.M. Hamberger,
L.E. Sharp, X.H. Shi, C.F. Vance, D.F. Zhou
Discussion
Helicity conservation and energy confinement in CTX spheromaks(IAEA-CN-47/D-IV-l-l)
B.L. Wright, C.W. Barnes, J.C. Fernandez, I. Henins, H.W. Hoida,
T.R. Jarboe, S.O. Knox, G.J. Marklin, R.M. Mayo, D.A. Platts
Confinement experiments and simulation on the CTCC-I spheromak(IAEA-CN-47/D-IV-1-2)
N. Satomi, S. Goto, Y. Honda, Y. Kato, M. Nagata, M. Nishikawa,A. Ozaki, T. Takaishi, T. Uyama, K. Watanabe, T. Sato, T. Hayashi
Field reversed configuration transport: Theory and measurements of flux,
energy, and particle lifetimes (IAEA-CN-47/D-IV-2-1)
A.L. Hoffman, J.T. Slough, L.C. Steinhauer, N.A. Krall, S. Hamasaki
Slow formation and stabilization of FRC plasmas(IAEA-CN^7/D-IV-2-2)
S. Okada, Y. Ito, M. Kako, R.E. Chrien, S. Ohi, S. Goto,
T. Ishimura, H. ltd, T. Takahashi, M. Ohara, S. Shimamura,
M. Machida, Y. Nogi
Discussion on Papers IAEA-CN-47/D^V-2-1 and D-IV-2-2

489

495
497

511

518
519

529

541

551

561

High beta plasma confinement with near-neoclassical transport in TPE-2(IAEA-CN-47/D-IV-3)
563
K. Hayase, I. Hirota, H. Kiyama, S. Kiyama, Y. Maejima, Y. Sato,
S. Takeda, E. Yahagi, M. Kito, K. Koyama
Z-pinch experiments and theory (IAEA-CN-47/D-IV-4)
573
M.G. Haines, A.E. Dangor, A. Folkierski, P. Baldock, CD. Challis,
P. Choi, M. Coppins, C Deeney, M.B. Favre-Dominguez, E. Figura,
J.D. SethianDiscussion582
Investigation of plasma dynamics and emission in dense plasma focus(IAEA-CN-47/D-IV-5-1)
583
K. Hirano, T. Yamamoto, K. Shimoda, Y. Okabe, K. Wakabayashi,
M. Yokoyama, Y. Yamamoto, Y. Yamada, Y. Kitagawa, M. Yamanaka
Investigation of collapse, pinch dynamics and fast particle emission in aplasma focus discharge (IAEA-CN-47/D-IV-5-2)
593
S. Czekaj, S. Denus, A. Kasperczuk, R. Miklaszewski, M. Paduch,
T. Pisarczyk, S. Sledzinski, A. Szydiowski, Z. Wereszczynski
Discussion on Papers IAEA-CN-47/D-IV-5-1 and D-IV-5-2
601
Impurity behaviour in the Wendelstein VH-A stellarator(IAEA-CN-47/D-V-1)
603
H. Ringler, F. Sardei, A. Weller,
W VIl-A Team, ECRH Group, Nl Group
Stable access to second stability regime in large-aspect-ratio
torsatron/heliotron configurations (IAEA-CN-47/D-V-2)
615
B.A. Carreras, L. Garcia, LA. Charlton, J.A. Holmes,
V.E. Lynch, P.H. DiamondThree-dimensional MHD studies of ideal and resistive interchange modes
in heliotron configurations (IAEA-CN-47/D-V-3)
625
M. Wakatani, K. Ichiguchi, H. Sugama, K. Itoh, A. Hasegawa,J. Todoroki, H. Naitou
Self-consistent radial electric field evaluations for the TJ-II Heliac(IAEA-CN-47/D-V-4)
633
J. Guasp
Evolution, relaxation and transport in spheromaks and reversed field
pinches (IAEA-CN-47/D-V-5)
639
R.Y. Dagazian, J.P. Dahlburg, G. Doolen, G.J. Marklin, A. A. Mirin,
J.P. Mondt, D. Montgomery, A.G. Sgro, L. Turner, J. Weiland
Experiments with a fully toroidal Extrap Z-pinch (IAEA-CN-47/D-V-6) .... 647
J.R. Drake, J.E. Eninger, B.P. Lehnert
Characteristics of reversed field pinch plasmas with a resistive shell(IAEA-CN47/D-V-7)
655
F. Tamano, W.D. Bard, T. Carlstrom, C. Chu, R.R. Goforth, Y. Kondoh,
R.J. La Haye, P. Lee, E.J. Miles, M.T. Saito, M.J. Schaffer, P.L Taylor

Energy confinement and nonlinear tearing mode dynamics in a highcurrent reversed field pinch (IAEA-CN-47/D-V-8)
Z.G. An, P.H. Diamond, T.S. Hahm, G.S. Lee, B.D. Scott,
B.A. Carreras, L. Garcia, J.A. Holmes, V.E. Lynch
Theoretical and experimental studies of field reversed configurations(IAEA-CN-47/D-V-9)
D.C. Barnes, W.T. Armstrong, E.J. Caramana, R.E. Chrien,
W.N. Hugrass, H.R. Lewis, R.K. Linford, KM. Ling, K.F. McKenna,
D.J. Rej, J.L. Schwarzmeier, R.E. Siemon, M.M. Tuszewski,
K.A. Werley, D.S. Harned, S. Okada, C.E. Seyler, D.E. Shumaker,
H. Tuczek, G. Vlases, R.D. Brooks, Z.A. Pietrzyk, K.D. Hahn,
D. Lotz, R. Ramen, J. Wight
Experiment on relativistic electron beam ring core spherator(IAEA-CN-47/D-V-10)
K. Narihara, T. Tomita, S. Kubo, M. Hasegawa, M. Matsui,
T. Tsuzuki, A. Mohri
Three-dimensional MHD studies of the magnetic energy relaxation
processes (IAEA-CN-47/D-V-ll)
T. Sato, K. Kusano, R. Horiuchi
Modelling and transport simulations of oscillating field current drive(F-0 pumping) in a reversed field pinch (IAEA-CN-47/D-V-12)
R.A. Nebel, K.A. Werley, R.A. Scardovelli, G.H. Miley,
D.S. Earned, H.R. Strauss, D.D. Schnack, Z. Mikic
Relaxation phenomena in reversed field pinches and tokamaks(IAEA-CN-47/D-V-13)
A. Bhattacharjee, C.K. Chu, Y.-C. Kwok, E. Hameiri, H.R. Strauss,
A.H. Boozer
Physics considerations on a fusion plasma in a field reversed
configuration (IAEA-CN-47/D-V-14)
H. Momota, M. Okamoto, Y. Nomura, K. Yoshikawa, M. Ohnishi,
A. Ishida, K. Sato, S. Ohi, H.L. Berk, T. Tajima
Preliminary experiments in a compact toroid formed by four C-guns(IAEA-CN-47/D-V-15)
S. Sinman, A. Sinman
Linear, high beta, £ = 1 and heliac stellarator experiments(IAEA-CN-47/D-V-22)
F.L. Ribe, M.E. Koepke, E.R. Hedin, CM. Greenfield
Chairmen of Sessions and Secretariat of the Conference

663

673

687

693

701

711

719

731

739
747

Session £MAGNETIC CONFINEMENT THEORYChairmen of Sessions
Session E-I

D.B. NELSON

USAPapers E-I-l-1, E-I-l-2 and E-I-l-3
were presented by J.A. Wesson as Rapporteur
Papers E-I-2-1 to E-I-2-4
were presented by T. Amano as Rapporteur
Papers E-I-4-1 and E-I-4-2
were presented by B.B. Kadomtsev as Rapporteur
Session E-II

R.R. PARKER

Papers E-II-1-1 and E-II-1-2
were presented by A. Sen as Rapporteur
Papers E-II-2-1, E-II-2-2 and E-II-2-3
were presented by L. Chen as Rapporteur
Papers E-II-3-1 and E-II-3-2
were presented by M. Kotschenreuther as Rapporteur
Session E-IH (Posters)

USAIAEA-CN-47/E-M-1

SAWTOOTH OSCILLATIONS
J.A. WESSON, P. KIRBY1, M.F. NAVEJET Joint Undertaking,
Abingdon, Oxfordshire,
United Kingdom
Abstract
SAWTOOTH OSCILLATIONS.
The recent observations of sawtooth oscillations having a very fast collapse phase call for a reexamination of the theoretical description of sawtooth behaviour. One possibility is that the instability
involved is an ideal m = 1 "quasi-interchange" mode. A calculation of the stability of this mode showsthat flattening of the q profile reduces the critical j8 required for instability. Thus, contrary to previous
expectations, the ideal mode can be unstable even at the low values of /3 observed experimentally.
Non-linear simulations show the predicted cellular convection. The distinction between the behaviourof the quasi-interchange mode and a tearing mode is described and a comparison is made withexperimental results from JET. Finally, attention is drawn to a fundamental difficulty in explaining the
rapid collapse in precursorless sawtooth oscillations.

1.

INTRODUCTION

With the advent of larger tokamaks it was found that thesawtooth collapse time did not increase as expected from theKadomtsev scaling [1] and attention was drawn to this discrepancy
by Campbell et al. [2]. An explanation of the observed behaviour
was proposed by Wesson [3]. This predicted an mhd timescale forthe collapse, in reasonable agreement with the experimental
results.
The model follows from a simple calculation showingthat if the q profile is flat after the sawtooth collapse, then it
will remain quite flat during the resistive diffusion of the ramp
phase. Configurations of this type were shown to be unstable toan ideal m=1 mode.
A further prediction of this model was that the form of theinstability would not be a "rigid shift" of the plasma core, aspredicted by previous models, but would be a cellular convective
motion. This would then lead to the formation of a cold plasma
bubble at the centre of the plasma. The magnetic field would notreconnect on this fast timescale,but reconnection would occur on
a longer, resistive timescale after the collapse.
These predictions were confirmed by subsequent experimental
studies on JET. Tomographic analysis of the soft X-ray emission
from JET [t] indicated a broad convective motion and the formation of a plasma bubble. The existence of decaying "post-cursor"
1

Euratom-UKAEA Fusion Association, Culham Laboratory, Abingdon, Oxfordshire, UK.

WESSON et al.
(b)(a)
j Profile(^ Profile

0-3. , ^Analytic

02 N u m « f c 5 t \ \UNSTABLESTABLE

096

0-98

100

102

So
FIG. 1. (a) Form of assumed j and q profiles, (b) Stability diagram for critical &p against q0.

oscillations after the collapse seems to conform to the prediction of delayed reconnection.
In this paper we report further work on this model in thefollowing areas:i)

Mhd stability and its implications for the plasma
motion.
ii) Non-linear simulations of the sawtooth collapseiii) Distinction between ideal mode and tearing instability.
Finally, attention is drawn to a fundamental problem inunderstanding the trigger for the sawtooth collapse.

2.

STABILITY OF THE IDEAL m=1 MODEIn reference [3], numerical calculations were presented,
showing the crucial dependence of the stability of the ideal m=1
mode on the flatness of the q profile within the q=1 surface.
These results can also be obtained analytically. In the large
aspect ratio expansion the potential energy 6W given by Bussac etal. [5] may be written in the form
6W = a + bg_ +1(-dp/dr) r2dr

where

(1 )

1
r 1 is the radius of the q=1 surface and the coefficients are
determined from the solution of a differential equation which
gives the toroidal coupling to the m=2 component. Taking themodel illustrated in Fig.1a, for which
1 - I—

r > r-.

2B—1

1 - 2(1-qQ)Ir <where j c =(2BVR)(2-1/q o )(1-(r|/a 2 )) v , the coefficients of eq.(1)were calculated and the criterion for instability was found to be

IAEA-CN-47/E-I-l-l

tin ! inV)
j . 2. Flow pattern of quasi-interchange (q0 = 1.0) and "rigid shift" (q0 = 0.8).

where Aq=1-q Q and Aq*( = vr^2/2a2)
is the Aq which would haveoccurred without flattening.
It should be noted that in both cases there is a q=1 surface
at r=r 1 . The stability boundary for criterion (2) is shown in
Fig.1b for v=1 and r 1 /a = 0.3, together with the result of anumerical calculation. It is seen that the toroidal stabilising
effect is much reduced for a flattened q profile and furthermorethat if q Q > 1, while retaining the q=1 surface, the m=1 mode isunstable for arbitrarily small ($ .
It was pointed out in reference [3] that for flattened q
profiles with 1-qQ sufficiently small, the calculation becomes
invalid. The principal and important consequence of the changedordering of 1-q is that the plasma flow associated with the
instability is completely changed. The "rigid" displacement of
the conventional case [6| is replaced by a convective flow in the
reduced shear case. The two types of motion are illustrated in
Fig.2. The physical basis of this distinction is that, for the
more strongly sheared, unflattened q-profile the energy of theinstability is inadequate to bend the magnetic field^ whereas forsmall shear the magnetic field is carried with the fluid,preserving the flux surfaces but playing a negligible role in themechanics.
This type of behaviour could be called
quasi-interchange.

3.

SIMULATIONS OF SAWTOOTH COLLAPSE

A full simulation of the sawtooth collapse would require a
toroidal calculation. However, the essential dynamical featuresof the present theory should be seen in the simpler case of
cylindrical geometry.
To this end, calculations have been
carried out using a cylindrical, non-linear, incompressible,

WESSON et al.

FIG. 3. Development of flux surfaces for (a) q0 = 1.04 and (b) q0 = 1.005,
together with associated soft X-ray emission.

mhd code with various q profiles. The code has finite resistivity, but for the simulations reported here the unstable modes
are ideal and the resistivity has no significant effect.
To demonstrate the effect of a flattened q profile, simulations have been made using constant B z and a model current
profile given byJz =const,

r < r';(c-r 2 /a 2 ) 2
j 7 = j.;ro (c-r' 2 /a 2 ) 2 '> r1This means that q is constant for r < r f , and then rises
inonotonically for r > r 1 .
The runs described here had S=10 s ,r'/a=0.3 and c=1.26.
In these calculations the higher modes
m/n=2/2, 3/3... are unstable, with growth rates comparable to thatof m=1.
For a relatively large q 0 the plasma is displaced a small
amount by the m=1 instability, as shown in Fig. 3a for qQ=1.04. A
small bubble forms but does not penetrate fully because of themagnetic shear. This may represent the behaviour of a partial
sawtooth collapse.
As q 0 is reduced, the instability is more able to sweep thefluid into a full cellular pattern. Figure 3b gives the resultfor qQ=1.005, showing deeper penetration of the bubble.
The corresponding time development of the X-ray emission canbe calculated for a given initial profile on the assumption that(given the rapid collapse) the emission is convected with theplasma. The calculated profiles are shown in Fig.3 for the case

IAEA-CN-47/E-I-l-l

(b)

(a)

»

tearing

inlerchangerI i) quasi - interchange

liil tearing(iii) intermediate

FIG. 4. Eigenfitnctions %r(r) for (a) quasi-interchange mode and(b) tearing mode, together with diagrams showing thecontrasting non-linear topological development for these cases.

of an initially Gaussian emission profile. The general form of
the development is very similar to that observed in the JET
experiment [i]. A further comparison of theory and experiment ismade in the following section.

4.

DISTINCTION BETWEEN QUASI-INTERCHANGE AND RESISTIVEINSTABILITY

The characteristic form of the radial displacement eigenfunctions for an ideal quasi-interchange mode and a tearing modeare shown in Figs.la and to.
The general characteristics ofthese flows persist into the non-linear regime. The ideal modeleads to a convective flow on the scale of the radius of the q=1
surface.
The tearing mode leads to a Kadomtsev type reconneetion, the core of the q < 1 region moving almost rigidly to one
side and the return flow taking place in a very narrow resistive
layer.

WESSON et at.
c^uasiinterchange

tearing

r(m>

(b)

(C)

/

(d)

Fig. 5. Eigenfimction %e(r)for (a) quasi-interchange mode and(b) tearing mode, together with experimental forms for(c) %r(r) and (d) £e(r) from the JET experiment.

The flux surfaces which would develop in the two cases areillustrated in Figs 4(i) and 4(ii). The contours of equal temperature and equal X-ray emission would have a similar topology.
The distinction between the two cases is quite clear.
It ispossible that the type of behaviour will depend upon the plasma
parameters and size.
If there is a continuous transition fromquasi-interchange (4(1)) to tearing (4(ii)), then an intermediatecase would take the form shown in t(iii).
The distinction between the quasi-interchange and thetearing mode is brought out most clearly by considering theazimuthal displacement eigenfunction € e (r). For R/ri>>1, this isrelated to £;;r byby ££QQ == - ^^((rrSSpp)).. The
The functions
functions ??ee(r;(corresponding to the cases illustrated in Fig.1) are shown in Figs 5a and5b.
Experimental forms for £ p and 5 Q have been derived from the
X-ray emission at an early stage of a JET sawtooth collapse. The
radial displacement is calculated from the initial emission
profile and the profile after 50ns.
The function 5 r (r) and thederived £ e (r) are shown in Figs 5c and 5d. Comparison of Fig.5dwith Figs 5a and 5b gives clear evidence that the displacement
has the form of a quasi-interchange.

5.

A FUNDAMENTAL DIFFICULTY

In addition to the features described above, the model
proposed by Wesson [3] also offered an explanation of the trigger
for the rapid sawtooth collapse.
Further examination of thisproblem shows that not only is the proposed explanation
inadequate but also there is a fundamental difficulty for allmodels of "precursorless" sawteeth.
This difficulty, which is
associated with the rapidity of the onset of the collapse, isdescribed below.

IAEA-CN-47/E-I-l-l
If the collapse time is T C , it is necessary for a growthrate Y - T 0 " 1 to develop on a timescale T C . Let the change incentral q necessary to produce this Y be 6q. If this change in q
results from resistive diffusion, then(3)

6q - — Aq

where Aq i s the change in q during the ramp phase, the durationof which i s approximately the sawtooth period i . The magnitude
of t h i s Aq can be estimated from the r e s i s t i v e diffusion of thecurrent [3] and i s given by

where AT/T i s the fractional change in the temperature during thes a w t o o t h , and t h e r e s i s t i v e d i f f u s i o n time T R = vor-\ 2 /")n.
Relations (3) and (4) determine the expected change 6q during thecollapse time.
I t i s then necessary that the required i n s t a b i l i t y growth r a t e can be switched on by t h i s 6q.
Typical JET v a l u e s are AT/T - 10"1 and T S / T R - 10"1, giving
Aq - 10~ 2 .
Then, s i n c e T 3 - 100ms and T C - 100ps, r e l a t i o n (3)gives the required sharpness of the switch-on of the i n s t a b i l i t y ,<5q - 10"3x10~2 - 10" 5 .
Even allowing for the uncertainty in the estimation, t h i s <5q
seems to be improbably small.
We can make an estimate of 6q
required to switch on the quasi-interchange mode as follows. Let
the growth r a t e for zero shear be yQ, then the d e s t a b i l i s i n ge n e r g y d e n s i t y a v a i l a b l e for the i n s t a b i l i t y i s - %pY 2 £ 2 .
P u t t i n g d£/dr-£;/r the s t a b i l i s i n g energy d e n s i t y [3]is- ( 1 - q ) 2 ( B e z / 2 u o ) ( £ / r ) 2 . Thus the d i s p e r s i o n r e l a t i o n can becharacterised by
Y2 = Y2 - (1 - q ) 2 T A 2

where

TA = r^/^/f.^)^).

(5)

Although the i n i t i a l switch-on, dY/dq, at Y=0 is very rapid,i t is necessary for y to grow to give a value (in JET) of Y"1 100 vs.
Now, for the i d e a l m=1 mode, YQ - cB p /T A , where E=r1/R- 10" 1 , Bp - 10"1 and TA - 1 ps .
Thus, Y^1 - 100 v s , and t h i s i sof the order of the required growth time.
Consequently, thechange in q necessary to produce the required growth r a t e i sgiven by equation (5) to be

Hence the t h e o r e t i c a l switch-on width in q i s 6q - 10~ 2 , and t h i s
i s orders of magnitude g r e a t e r than the required experimental
value of 1 0 " s .

10

WESSON et al.

The problem is even more severe for tearing modes where thetheoretical value for the 6q necessary to provide a growth rate yis [7]

6q - W . xA

V

The typical JET values given above,with tp - 1 3, give 6q - 1, tobe compared with a required switch-on 6q - 10~ 5 .
It is clear therefore that a fundamental element is missingin our theoretical understanding of fast precursorless sawteeth.
Thus, although the experimental evidence on the flow behaviour
seems to support the quasi-interchange model, the validity ofthis model cannot be accepted while we still lack an explanationof the rapidity of the onset of the collapse.

6.

CONCLUSIONS(i)

A flattened
q-profileinside the q=1 surface allows
instability of the ideal m=1 mode which would otherwise bestable at low g .

(ii) Non-linear simulations of the instability illustrate thequasi-interchange behaviour.
The broad convective flow
takes the outer plasma into the centre to form a colder"bubble".
(iii)The behaviour of the quasi-interchange is quite distinct
from that of a tearing mode and is in better agreement withexperimental observations of sawteeth in JET.
(iv) The sudden onset of the collapse in preeursorless sawteeth
presents a fundamental problem for all existing models.

ACKNOWLEDGEMENTS
The authors would like to thank J.Hastie for discussions andR.Granetz for help in analysing the soft X-ray results.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]

KADOMTSEV, B.B., Sov. J. Plasma Phys. 1 (1975) 389.
CAMPBELL, D.J., et al., Nucl. Fusion 26 (1986) 1085.
WESSON, J.A., Plasma Phys. Controll. Fusion 28 (1986) 243.
EDWARDS, A.W., et al., Phys. Rev. Lett. 57 (1986) 210.
BUSSAC, M.N., et al., Phys. Rev. Lett. 35 (1975) 1638.
ROSENBLUTH, M.N., et al., Phys. Fluids 16 (1973) 1894.
COPPI, B., et al., Sov. J. Plasma Phys. 2 (1976) 583.

IAEA-CN-47/E-I-1-2

THEORY OF
SAWTOOTH DISRUPTION
D. BISKAMP, H. WELTER
Max-Planck-Institut fur Plasmaphysik,
Garching, Federal Republic of Germany
Abstract
THEORY OF SAWTOOTH DISRUPTION.
Numerical investigations of the MHD processes related to the sawtooth oscillation have been
performed, by using the full non-ideal MHD equations. Sawteeth are found to exist in cylindricalgeometry, but the amplitudes become too small for small ij. Toroidal effects are important, allowing
larger energy storage.

I. Introduction
The sawtooth oscillation is a well known relaxation phenomenon , observed in virtually
all tokamaks. It periodically flattens the temperature, density and probably also current
density profiles in the central part of the discharge column up to a certain radius n , thesawtooth inversion radius. While in Ohmic discharges or those with moderate amountsof auxiliary heating sawtooth amplitudes are small, ATe/Te ~ 0.05 typically, and seem tohave an overall beneficial effect in preventing central impurity accumulation, discharges
with strong additional power injection such as the ICRH experiments on JET exhibit sawteeth of much larger amplitudes, which strongly deteriorate the global confinement. Hencethe sawtooth disruption has become a problem of major concern in tokamak research.
The traditional theoretical interpretation of the sawtooth phenomenon that until recently had been generally accepted is based on a model by Kadomtsev 1 '. If the g-profilein the plasma center drops below 1, the resistive m = 1 kink mode becomes unstablegiving rise to a rapid convective transport of the central plasma toward the 9 = 1 surface and a flat current profile with q(0) > 1, from where the process restarts. In recent
years, however, observational results in particular from the large tokamak devices suchas TFTR and JET, indicate that the typical sawtooth oscillation shows features whichcannot be reconciled with this simple model. Since skin times for major changes in thecurrent profile far exceed the sawtooth period, the ^-profile inside the inversion radius
should remain flat close to unity. No m = 1 oscillations preceding the sawtooth crash areobserved, which should be present if the m = 1 mode starts growing slowly accelerating
gradually. The crash-time is very short, so that magnetic reconnection or other resistivity
related processes do not seem to play a major role. In addition smaller disruptions occur
at half period between main crashes, so called subordinate sawteeth, which are localized
close to the inversion radius and indicate some local ^-profile modification at this radius.
Recent numerical simulations2) using the reduced resistive MHD equations including theselfconsistent evolution of the resistivity have successfully reproduced many features ofthese observations. They show that in fact the ^-profile remains almost flat and that aweak skin current develops near the inversion radius, which under certain conditions gives
rise to the subordinate sawtooth disruption. (A "magnetic trigger" of this kind has also
been proposed in Ref. 3.)

11

12

BISKAMP and WELTER. m=1

Pp = 1

-

"

\

'•

PP
-

1

0.90

1

0.95

1.00

1

n

t

1

105

"A

0.90

095

1.00

n

4

105

0

FIG. I. Growth rates for cylindrical equilibria versus q0, for Aq = 0.

The main shortcoming of these simulations is the use of the reduced MHD equations
which neglect pressure driven effects. This is particularly restrictive for flat (/-profiles,
where shear effects become virtually negligible. In this case the m = 1 mode is marginallystable in the reduced equations (or possibly weakly unstable including resistivity gradienteffects), while it is known to be strongly unstable using the full equations, at least in
the cylindrical approximation. In view of this deficiency the qualitative agreement of theresults of Ref. 2 with experimental observations appears to be somewhat fortuitous. In thepresent investigation we therefore consider the full MHD equations with resistivity, heat
conductivity and viscosity included, which are solved numerically in cylindrical as well as
toroidal geometry. Section II outlines the numerical procedure and presents the results forcylindrical geometry: discussion of linear stability, nonlinear properties, and simulationsof the full sawtooth period. In Section III the modifications of these results due to toroidal
geometry are discussed. In Section IV the possibility of an alternative sawtooth model isinvestigated based on recent experimental observations'1' that the current profile is peaked
inside the inversion radius.
II. Cylindrical Geometry
A numerical code has been developed for the solution of the full nonlinear MHD equations
ft H—- = V x (v x B) - V x r;(V x B)
at(1)

— = -v-Vv-Vp-Bx
at

(2)

(VxB)

(3)
The density is assumed to be homogeneous, p = 1.

IAEA-CN-47/E-I-1-2

13

2 5 . to22-10

FIG. 2. Evolution of central pressure cylindrical geometry, for t\0 = 10 3.

To eliminate the fast compressional Alfven time step restriction a semi-implicit schemesuggested recently 5 ' is used, which is efficient and accurate. Finite difference approximation is applied in radial direction, while Fourier expansion is used in azimuthal and axial
directions. Linear mode properties for m = 1 as well as m > 1 have been investigated forvarious types of cylindrical equilibria by prescribing p'- and ^-profiles. Guided by previous
results 2 ' and the sawtooth simulations given below, the g-profile is chosen to consist of aflat part q = q0 inside TQ , a shallow well with depth Aq between r 0 and r j , and a parabolic
rise beyond ro, p' is characterized by a parameter fif The main results are:i) For small values of the transport coefficients r),K±,fi < 10~7 (in units of av^t) unstablemodes are essentially ideal modes. Choosing Ky ~ 102 instead of /cj| = 0 has a negligibly
small effect. Fig. 1 gives growth rates of the (m,n) = (1,1) and (2,2) modes for 0P = 1
and 0.25, with Aq = 0.
ii) Instability ranges have a sharp upper boundary in go and lower boundary in /3 P ,7 oc yfqc — q and 7 <x yjfip — Ppc respectively. Growth rates and mode structures are quite
insensitive to the detailed pressure profile depending only on some average p'{r), essentially
P(0)-p(r,).
iii) For <jo close to unity the instability is of the interchange type, i.e. many m-modes are
unstable, not only m = 1 as in the case of a peaked current profile with go < 1- Maximum
growth rates, which occur at q0 slightly below 1, increase with m and /?p, 7 ex yJmPy,, while
the unstable band shrinks with growing wave number and decreasing pressure gradient 0p.
iv) For q(J above the stability threshold as shown in Fig. 1 addition of a shallow well Aq
also leads to instability. While eigenmodes remain global, growth rates depend on thevalue of the pressure gradient at r =; r,. A local flattening can suppress the instability,
even if p(0) - p{r\) > 0.
Concerning the nonlinear behavior of these instabilities, two questions are addressed.
The first is the relationship between the linear growth rate *u, and the "disruption time"
defined by T = ri/vo, where UQ is the maximum value the velocity reaches in the center.

14

BISKAMP and WELTER
2.5»10

09

20«10

Oi.

FIG. 3. Evolution of q- and p-profiles during sawtooth period,(a), (b) and (c) refer to the times indicated in Fig. 2.

For interchange type mode, i.e. qo = 1, varying 11 by increasing 0P from 0.1 to 1 we find astrictly linear relationship r a 1.5"/^X- Since r can directly be observed experimentally andthe pure interchange mode probably gives the fasted nonlinear process, this relationshipindicates a minimum value of the linear growth rate required to explain the disruption time.
The second question concerns the transitionfrom small amplitude saturation to disruption.
We consider a sequence of equilibria with f}p — 0.25, go = 1-03, and Aq varying from 0.01
to 0.04. The saturation amplitude is defined by A / r i , where A is the off center shift ofthe pressure maximum. The transition is in fact very sharp, A/ri ~ 0.25 for Aq = 0.02
and A/rj ~ 0.8 for Aq = 0.03. It should be noted that the linear eigenmodes always havea global structure, also in cases saturating at small amplitudes.
We have performed simulations of the complete sawtooth cycle evolving selfconsistently
under the effects of Spitzer-type resistivity, Ohmic heating and heat conduction. Startingfrom a resistive equilibrium with qo < 1 the dynamical evolution has been followed over a
sufficiently long lime to reach dynamical behavior independent of the initial state. Steadysawtooth oscillations do in fact exist, as shown in Fig. 2, where the evolution of the centralpressure is given. This phenomenon, is by no means self-evident. Since for a flat g-profile
with 90 = 1 any 0P = p(0) — p(ri) > 0 gives rise to instability with 7 oc \fWf leading toa rapid interchange motion, a stationary convective cell seems to be the most likely final
state, as has recently been found under certain conditions in numerical simulations 3 ' using
the reduced MHD equations. The evolution of the p and q profiles during a sawtooth period
is illustrated in Fig. 3, with r\ = 10~ 5 ,/i = 10~6,/cj. = 3 x 10~ 7 , and a),b),c) referring to
the times indicated in Fig 2. The disruption is due to an m = 1 interchange-like instability
excited by a slightly inverted g-profile, a). Note that the instability sets in while g m l n >1 still. The rapid motion leads to an inverted p-profile (the actual dynamical process
resembles strongly the tomographic results from JET 5 )) and a monotonic ^-profile with
9(0) slightly below 1 temporarily, b). Due to the relatively low pressure and correspondinglarge r],q increases soon to q > 1, c).

15

IAEA-CN-47/E-I-1-2

10"'

10"°

1) = ^

10"*

FIG. 4. Growth rates for toroidal equilibria versus ij0 , q0 — 1, A q = 0.

Nevertheless, the cylindrical model cannot describe the sawtooth phenomenum in lokamaks, because of the inherent inefficiency of energy storage. This leads to a wrong r/scaling of the sawtooth amplitude. While the sawtooth period increases with decreasing r)
roughly as T a V~i/2, the amplitude decreases Ap/p oc r;1/2. In the case shown in Fig. 2
for r) = 10~ 6 , Ap/p ~ 0.03 which is already rather small. These computations have been
performed for pp ~ 1. We expect that in the limit 0P —» 0 and finite TJ the resistivity
driven effects dominate as described in Ref. 2. For a qualitatively correct modelling oftokamak sawtooth oscillation, in particular at small but finite /? p , toroidal effects have tobe taken into account.
III. Toroidal Geometry
A toroidal version of the code decribed in section II has been set up in quasi-flux coordinates u,0,ip, where u(r,6) = const are circles with the axis shifted in such a way thatthe coordinate axis coincides with the magnetic axis of the equilibrium. The advantage
as compared to numerically computed genuine flux coordinates is that all metric coefficients can be obtained analytically. For circular boundary to which the investigations havethus far been restricted, the deviation from exact flux surfaces is small. For nonlinear
computations the equilibrium flux coordinates would lose their advantage anyway.
Equilibria are obtained by choosing a cylindrical configuration, which is allowed to relax
into a toroidal equilibrium with the coordinate shift determined iteratively. Since r) = 0 in
the relaxation process, the ((-profile does not change and can thus be prescribed with highprecision. This allows a direct comparison of cylindrical and toroidal equilibria.
Fig. 3 shows growths rates of the n = 1 and n = 2 modes as functions of rj for a flat qprofile, q = 1 inside rt, for /3p = 1 and 0.25. Growth rates seem to be independent of ri forr\ < 3 x 10~ 7 , hence there is ideal instability also in the toroidal case, but i is proportional
1 ti

to 0,, instead of 0r. in the cylindrical case. In addition the unstable g-range is narrower.
These properties have a crucial effect on the sawtooth behavior, since they allow that freethermal energy may be accumulated much more easily than in the cylindrical model. On
the other hand a certain problem arises in explaining the experimentally observed fastcrash time scales. The toroidal configurations studies thus far yield growth rates thatare somewhat too small. Since growth rates depend rather sensitively on details of theconfiguration, while the plasma will automatically find the fastest way of energy release, a16

BISKAMP and WELTER

selfconsistent simulation of the full sawtooth cycle in toroidal geometry is required. Thoughsuch simulations have not yet performed, we expect that while much larger amplitudes Ap
may be build up with Ap independent of the resistivity, the basic mechanism of energy
release is essentially as in the cylindrical case, illustrated in Fig. 3.

IV. Alternate Model of the Sawtooth Oscillation
It has recently been observed experimentally4' that the current profile of a sawtoothing
tokamak discharge is peaked with go ~ 0.7 - 0.8. This result would be in contradiction to
the sawtooth model outlined in the previous sections where the current profile is flat inside
ri with </o — 1- Such a behavior is in principle conceivable. The sawtooth disruption would
be analogous to the major disruption, where the sudden outflow of thermal energy seems to
be caused by the destruction of the magnetic surfaces rather than a convective process andwhere the current profile does not change appreciably. The basic problem in this concept is
to explain the absence of the (m,n) = (1,1) mode. A previous theory of the internal kinkmode 7 ' in toroidal geometry predicts stability (of the ideal mode) for sufficiently small
Pr. Though the mode would still be resistively unstable, saturation at finite amplitude
could occur as in the case of m > 1 tearing modes, mainly due to a quasi-linear flatteningof the current profile close to the 9 = 1 surface. We have started to investigate thisquestion numerically using the cylindrical as well as the toroidal code for 0f, = 0.25 andan aspect ratio Rja = 5. In both cases no stabilization, not even a substantial reductionof the growth rates, due to a local current- (and pressure-) profile flattening could befound. Though the results are still preliminary and nonlinear computations concerning
this problem have not yet been performed, the indication is that a stabilization of theinternal kink mode is not possible within the resistive MHD approximation and, if further
corroborated experimentally, would require a more refined theoretical model.

Acknowledgements
The authors gratefully acknowledge valuable discussions with Dr. D. Harned and would
like to thank Mrs. M. Walter for her assistance in the numerical computations.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]

KADOMTSEV, B.B., Sov. J. Plasma Phys. 1 (1975) 389.
DENTON, R.E., DRAKE, F., KLEVA, R.G., BOVD, D.A., Phys. Rev. Lett. 56 (1986)
2477.
WESSON, J.A., Plasma Phys. 28 (1986) 243.
SOLTWICH, H., STODIEK, W., private communication, 1986.
HARNED, D.S., KERNER, W., J. Comput. Phys, 60 (1985) 62.
EDWARDS, A.E. et al., Phys. Rev. Lett. 57 (1986) 210.
BUSSAC, M.N., PELLAT, R., EDERY, D., SOULE, J.L., Phys. Rev. Lett. 35 (1975)
1638.

IAEA-CN-47/E-I-1-3

SAWTEETH AND IDEAL MHD
M.N. BUSSAC, K. LERBINGER, R. PELLAT
Centre de physique thdorique,
Ecole polytechnique,
Palaiseau
M. TAGGER
CEA, Centre d'e"tudes nucteaires,
Fontenay-aux-Roses
FranceAbstract
SAWTEETH AND IDEAL MHD.
Recent experimental results support the idea that in tokamaks the sawtooth crash should be linked
to an ideal MHD instability. Two possible interpretations, according to the current profile, arediscussed.

Recent results on JET, TEXT, TFR, ALCATOR, etc. [1-5] have shown thatthe sawtooth crash has an m = 1 structure with a growth rate much larger than theusual m = 1 precursor, which may or may not be present. This supports the idea thatthe crash should be linked to an ideal MHD instability [6, 7]. In the following, we
discuss two possible interpretations:(i) The first interpretation corresponds to a monotonically increasing q(r) profile
with q(0) < 1 (q(0) is the safety factor at the magnetic axis) and a partiallyreconnected topology with a finite magnetic island inside (q)r0 = 1.
(ii) The second interpretation corresponds to a non-monotonic q-profile with
q(0) > 1, but with a minimum q(r0) away from the magnetic axis, which
decreases in time and becomes equal to one, following the current penetration.

1.

OBSERVATIONS

Recent tomography of sawteeth in JET has revealed a lot of interesting features;
we shall discuss here the results of one of these experiments (shot 6320 with ICRH).
Although a great variety of sawteeth exist (with or without precursor, postcursor; single partial, giant sawteeth...) it is believed [5, 10] that the observed behaviour of thisshot is quite general.
17

18

BUSSAC et al.

According to soft-X ray reconstruction [1], the sawtooth collapse occurs essentially in two phases:
(a)

(b)

(c)

2.

Initially, the region of maximum emissivity (the hot core) is rapidly displaced
off axis. The volume of the hot core and the level of emissivity remain approximately constant. This phase, which lasts —100 ^s, has clearly an m = 1
character and appears as a solid translation of the hot core (Fig. 3 of Ref. [1]).
At the end of this first phase, an up-down asymmetric deformation is visible(Fig. 4 of Ref. [1]).
During the second phase (Fig. 4, C and D of Ref. [1]), this deformation
increases, giving rise to a bubble-like structure of cold plasma. As a result, thehot core is pressed against the 'q = 1' surface and surrounds the bubble in a
crescent (or island) shape. Its temperature, then, decreases rapidly (on a timescale of ~ 100 /*s), while the bubble appears as a hollow temperature region.
In a third, much slower phase ( - 1 . 5 ms), the central plasma develops a hollow,
poloidally symmetric profile (Fig. 3 of Ref. [1]).
DISCUSSION

Comparison with ECE signals [2] shows that the observed emissivity is here
mainly due to impurities (He-like nickel at 5090 A). The temperature profiles seem,
however, to follow the observed emissivity profiles.
The spatial resolution is ~ 7 cm and an island less than 10 cm wide is not visible
by tomography. Nevertheless, for a classical sawtooth with precursor, it has up to
now been believed that the topology of the central plasma core is partially reconnected as a consequence of the resistive m = 1 mode [17].
The detector configuration allows reconstructions with poloidal harmonics
m = 0, cos d, sin 6, cos 26. The outer 'q = 1' surface has an ellipticity(b - a)/(b + a) - 10%.

3.

FIRST INTERPRETATIONLet us first recall that all along the sawtooth cycle an inversion radius isobserved, and that its position does not noticeably change on a time-scale of many
sawteeth. Therefore, we favour the interpretation that a q = 1 surface always exists
in the plasma if one assumes a monotonically decreasing current profile. If q(0) < 1
and the shear is infinite on q(r0) = 1, the ideal MHD internal kink may be stable intoroidal geometry [11]. On the other hand, if unstable, this mode is expected tosaturate to a low amplitude for finite shear [12]. In any case, an unstable resistive
mode [18] allows magnetic field reconnection: a magnetic island grows on a very
long time-scale (~T/" I / 4 or r)~'A), so that the size of that island cannot change substantially during the sawtooth crash phase.

IAEA-CN-47/E-M-3

19

The presence of that island can, however, affect the potential energy of the ideal
MHD kink mode [6]:8W = 6WT,K + 5Wshape + 5Wisland(1)

where the first terms describes the internal kink in toroidal geometry [11], the second
term takes the shape of the plasma cross-section [8] into account, and the third termis due to the presence of the island causing the 'half-coalescence' instability [6] (wehave neglected cross-terms in the modulation of the poloidal magnetic field). On theassumption of low shear and low j8p inside q = 1 and finite shear outside, the toroidal contribution is of the order of

R J

\RJ

\

v

VR/

VR

where q(r0) = 1, and R is the major radius. B is the magnetic field; /3p is defined by
Eq. (3).
For an elliptical cross-section, 6Wshape takes the form:

Finally, the deformation of the hot core due to the presence of the island gives thecontribution

5WisIand = 2^2R (^fJ (1

qo)2

(£

where 6 is the island width and f is a measure of the poloidal magnetic field modulation along the separatrix. This factor can be positive or negative, depending on thedetailed island shape.
It is very likely that, in different experiments, various terms in Eq. (1) can be
dominant to drive (or stabilize) the mode, and that this should explain the different
types of behaviour observed during sawteeth. There is an important feature to thisMHD instability, whatever the origin of the potential energy. If 5W becomes negative, then the well-known non-linear stabilizing effect [12], which limits the plasma
displacement to the linear boundary layer, disappears: now the mode consists primarily of a solid displacement of the hot plasma core and a return motion in the island.
The maximum solid displacement is expected to be of the order of the island size.

20
4.

BUSSAC et al.

SECOND INTERPRETATION

According to [1, 2], it has been shown that JET sawtooth collapse may occur
without any precursor. These results and the observation of a hollow temperature
profile after the collapse [1, 19] led us to consider the possibility of skin currents
forming at a finite radius. As is shown in numerical simulations [20], discharges withgood central thermal confinement develop skin currents which grow stronger during
the rise of the sawtooth. They become more pronounced and cause the safety factor
q to fall below one, away from the magnetic axis, while q(0) remains larger than one.
Another source of local enhancement of the current is the bootstrap current due totrapped electrons, which is proportional to the local pressure gradient. In a tokamak,
when q(0) > 1, the ideal kink is linearly unstable if the safety factor q(r) has a minimum q(r0) = 1 away from the magnetic axis (as is shown in Fig. 1). In the limit ofsmall (1 - q(0)) ~ s (but Isl > (ro/R)2) and finite /3p, and keeping only terms up
to first order in s, the available energy may be written asf) B^r«)x(3 + Ae)- Ae + 8s-

5<3ps (3 +Ae)
1 - Ae13(2)where Bp(r) is the poloidal magnetic field:

Bpn = -

8ir

B>(r 0 )L r l drq(r)FIG. 1. q(r) profile of safety factor.

(3)

IAEA-CN-47/E-I-1-3

s =

- 1

—<

Jo r o

Ae =

21

dr

where £e is the solution of the Euler equation for the harmonic m = 2, n = 1 which
vanishes at the edge of the plasma and is equal to 1 on r = r0. For a monotonicallydecreasing current density outside r = r0) we obtain t - 3 < Ae < 0.
In the limit of vanishing j3p « s, the internal kink is stable for a monotonicallydecreasing current profile and unstable for a hollow current profile. Here, we noticethat this instability is a purely toroidal effect, since, in cylindrical geometry, we
obtain the opposite result if I3p — 0.
Moreover, the linear growth rate is larger than in the regime where q(r)increases everywhere and q(0) < 1. Indeed, if the safety factor has minimumq(r0) = 1 away from the magnetic axis,
1
q(r)

1 -

q"(r0)(r - r0)2

in the singular layer, and the lowest order in ro/R the growth rate 7 can be writtenas:

T = TA—

where 7A is the inverse poloidal Alfve'n time: y\ = B2P (ro)/(47rprg). The singularlayer A scales as r0 (7/7 A ) w . i-e. it is wider than for the usual kink with q(0) < 1,
where it scales as A<xr0 (7/7A).
More precisely, we obtain for a hollow current profile:1(q" (r0)

-5WT
p

D2

/r \

Therefore, in the non-linear regime, we may expect a saturation of the instability
which limits the plasma displacement to the singular layer width.

22

BUSSAC et al.

The feature common to these two interpretations is to allow a large plasma displacement. The essential difference is the following: in the first case, the available
potential energy becomes positive (5W < 0) in time, the necessary q-profile being
already settled. In the second case, the potential energy is available when the necessary current profile becomes accessible, i.e. when q(r) has a minimum equal to one,away from the magnetic axis where q(0) > 1.
We now discuss the non-linear evolution of this instability and its relevance tothe explanation of the sawtooth crash.
In toroidal geometry (with or without non-circular cross-section) neither analytical nor numerical calculations exist, and one can only discuss the non-linear evolution qualitatively. In the first interpretation, where q(0) < 1, the surface q = 1 isnow distorted, and since the destabilizing terms 5Wisland are now of the order of theexternal kink, one can expect a non-linear evolution similar to that of the externalkink which leads to bubble formation. In this process [14, 15], a configuration with
hot plasma and strong current, surrounded by a vacuum, evolves to a state of lower
energy where the hot plasma (and the current channel) surrounds the vacuum bubble.
This evolution for the external kink in an axisymmetric configuration requires very
low shear in the plasma. However, as was discussed in [6], this would be a natural
evolution in the presence of an m = 1 magnetic island, which would become the cold
bubble. We note that the island topology is very favourable to this mechanism since
(a)

(b)

the island and the hot core have the same topology (forming two interlacedrings) and thus can exchange their cross-section shapes by a continuous, ideal
MHD process;the separatrix remains intrinsically a q = 1 surface throughout this evolution,
which might hence be less insensitive to shear.

In situations without an island, where the mode is due to skin currents and toroidal geometry (at least with very low shear), bubbles may also exist, but only by
numerical simulations will we be able to discuss their extent, their importance andtheir sensitivity to shear.
In any of these situations, bubble formation leads to an inverted temperature
profile. This inverted profile creates large temperature, pressure and current gradients at the q = 1 surface, leading to very fast transport by diffusion, but, certainly,
also causing enhanced stochasticity and microturbulence [16].
The same mechanism, with the bubble now remaining at the periphery of theq = 1 region, could also explain the partial sawteeth observed in (large) tokamaks.

REFERENCES[1][2]

EDWARDS, A.W., et al., Phys. Rev. Lett. 57 (1986) 210.
CAMPBELL, D.J., et al., in Controlled Fusion and Plasma Physics (Proc. 12th Europ. Conf.
Budapest, 1985), European Physical Society (1985).

IAEA-CN-47/E-M-3

[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]

23

KIM, J.B., et al., paper presented at Am. Phys. Soc. Meeting Boston, 1984.
DUBOIS, M.A., et al., Nucl. Fusion 19 (1979) 703.
CAMACHO, R., GRANETZ, R., paper presented at Am. Phys. Soc. Meeting Boston, 1984.
BUSSAC, M.N., PELLAT, R., SOULE, J.L., TAGGER, M., Phys. Lett., A 105 (1984) 51;
109 (1985) 331.
WESSON, J.A., Plasma Phys. Contrail. Fusion 28 (1986) 243.
EDERY, D., LAVAL, G., PELLAT, R., SOULE, J.L., Phys. Fluids 19 (1976) 260.
ROBINSON, D., in Magnetic Reconnection and Turbulence, Editions de physique, Orsay (1985)
259.
SMEULDERS, P., ibid., p. 239.
BUSSAC, M.N., et al., Phys. Rev. Lett. 35 (1975) 1638.
ROSENBLUTH, M.N., et al., Phys. Fluids 16 (1973) 1984.
KADOMTSEV, B.B., Sov. J. Plasma Phys. 1 (1975) 389.
KADOMTSEV, B.B., POGUTSE, D.P., Sov. Phys. - JETP 38 (1974) 283.
ROSENBLUTH, M.N., Phys. Fluids 19 (1976) 1987.
DUBOIS, M.A., SAMAIN, A., Nucl. Fusion 20 (1980) 1101.
BUSSAC, M.N., in Plasma Physics and Controlled Nuclear Fusion Research 1976 (Proc. 6th
Int. Conf. Berchtesgaden, 1976), Vol. 1, IAEA, Vienna (1977) 607.
WADDELL, B., et al., Nucl. Fusion 18 (1978) 735.
BOYD, D.A., private communication, 1986.
DENTON, R.E., et al., Phys. Rev. Lett. 56 (1986) 2477.

DISCUSSION
ON PAPERS IAEA-CN-47/E-I-l-l TO E-I-l-3

A. GIBSON: At the beginning of your oral presentation, you mentioned
experimental results by the TEXTOR group at Jiilich, showing that q is less than 1
throughout the sawtoothing phase of some discharges. Cases have also been reported
where the sawtooth is stabilized by lower hybrid current drive but where an m = 1
oscillation remains; presumably, q = 1 is still within the plasma. Can you say how
these results affect the theoretical treatments which you discussed?
J.A. WESSON: I cannot answer this question in detail, but I would like to say
the following. Firstly, it should be made clear that I accept Soltwisch's result thatq0 < 1. If these q o < 1 profiles are smooth, they present a very serious problem for
MHD because they should be highly unstable with respect to tearing modes. If, on
the other hand, they are stable because of flattening around the q = 1 surface, then,because of the large shear, the sawteeth must be different from those in JET.
K. McGUIRE: The new features in connection with sawteeth discovered overthe past 4-5 years are the observed compound sawteeth, subordinate or partial
relaxations and a fast collapse time. In your paper you deal with the problem of afast sawtooth collapse time. Does your model also explain compound sawteeth, or
subordinate relaxations?
There is another point. On TFTR, the collapse time for sawteeth may vary from40 /xs to 10 ms and, even on one discharge, the crash time can vary by a factor offive. In your presentation you imply that all sawteeth crash in 100 /xs on JET.
Is this true?
J.A. WESSON: It seems possible that the partial bubble formation which we
see in the simulations for larger values of q0 can explain the partial relaxations.
Then, the full collapse would occur when there had been a further reduction in q0.
Regarding the collapse times on JET, I use 100 /xs as a typical value. Collapse
times which appear to be longer are sometimes observed, but full reconstruction
shows this to be a geometric effect. With neutral beam counter-injection, much
longer collapse times sometimes occur. Of course, I emphasize the fast collapses
because it is these which present the theoretical problem.
B. COPPI: What happens to the quasi-interchange mode when the theory for
it is applied to toroidal geometry, with finite Larmor radius effects?
J.A. WESSON: Toroidal calculations have been carried out and they showthat the quasi-interchange mode persists in toroidal geometry. Given the fast growthrate and the absence of any layer, I would not expect the quasi-interchange mode to
be significantly affected by finite Larmor radius effects.
M.A. DUBOIS: A major problem with sawtooth models is that they have to
explain not only the time-scale of the fast collapse but also the temperature profiles
observed after the collapse, which are sometimes flat and sometimes hollow; also,25

26

DISCUSSION

sometimes a very sharp temperature spike remains within the q = 1 surface, as
observed in TFR with ECRH. I feel that a sawtooth model should explain all observations. Do you think that with the models you presented a very sharp temperature spike
can be kept motionless in the plasma centre?
J.A. WESSON: It is difficult to comment on the result you mention without
knowing the details. As a general point, I think it quite possible that different typesof sawteeth occur, but I also hope that one model would suffice to explain all results.

IAEA-CN-47/E-I-2-1

SECOND STABILITY ACCESSIN TOKAMAK PLASMAS
K. YAMAZAKI, H. NAITOU, T. AMANO, Y. HAMADA,
K. MATSUOKA, Y. MIDZUNO, Y. NAKAYAMA 1 , T. SATO 1
Institute of Plasma Physics,
Nagoya University, Nagoya
T. TSUNEMATSU, T. TUDA, S. SEKI, M. AZUMI, S. TOKUDA,
T. OZEKI, T. NEMOTO 2 , Y. KISHIMOTO, T. TAKEDANaka Fusion Energy Research Establishment,
Japan Atomic Energy Research Institute,Naka-machi, Naka-gun, Ibaraki-ken
Japan

Abstract
SECOND STABILITY ACCESS IN TOKAMAK PLASMAS.
Methods of beta enhancement in a tokamak plasma are investigated. One is shaping
optimization for a slightly indented cross-section, and the other one is optimization of plasma
current profile for a circular or a dee-shaped cross-section. Access to the second stability
region of the ballooning modes is demonstrated for both cases. The stability of external kinkmodes is also studied. The kink instability is of crucial importance for the exotic shape orcurrent profile used in the analysis. The conducting shell placed at a w / a > 1.2 enhances thebeta limit more than twice the conventional value (a w , a are, respectively, the horizontal
minor radii of the conducting shell and the plasma). The non-linear evolution of the ballooning
modes is studied by using a two-fluid model, and the fluctuation level of the magnetic field isevaluated.

INTRODUCTION
One of the critical issues in tokamak fusion research is the improvementof the beta limit of a plasma (0 is the ratio of plasma to magnetic pressure).
We describe two different methods of beta limit enhancement in this paper.
Part A demonstrates enhancement by using an indented cross-section. Shaping
optimization is expected to reduce the indentation, and the pusher coil can be
simplified. The effect of triangularity in the indented configuration on kinkand ballooning beta limits in the first region of stability and on the accessibility
to the second region of stability is studied. In Part B, enhancement by using
1
2

Institute of Fusion Theory, Hiroshima University, Hiroshima, Japan.
On leave from Fujitsu Ltd., Tokyo, Japan.

27

28

YAMAZAKI et al.

current profile optimization is described for circular and dee-shaped crosssections. The accessibility to the second stability region of the ballooning
modes is investigated. The stability of external kink modes is also studied forthe equilibria that are stable against the ballooning modes. Beta enhancement
by using this method reduces the plasma current and the current in poloidal
coil systems, as well. In Part C, the non-linear evolution of the ballooning
modes is studied by using a two fluid model, and the dependence of the magnetic
field fluctuation level on the beta value is evaluated.

Part ASECOND STABILITY ACCESS IN CRESCENT SHAPING3
A 'crescent' configuration [ 1 ] is characterized by optimal shaping withsmall indentation ( i ^ O . l ) but large triangularity (5>0.5) and moderately highellipticity (/c ~ 2). This high beta configuration (|3> 10%) lies in the first regionof stability and enables a smooth transition to the second region. This is aneasily achievable extension from the conventional dee shaping of large tokamak
design.
Figure A-l shows a typical configuration related to the shaping effect onaccessibility to the second region of stability. In the dee-shaped case, thefirst stability beta limit is ~ 3 % , and the second stability region near thecentre appears at the toroidal beta value of ~ 2 5 % (Fig. A-l (a)). By increasing
the triangularity in addition to indentation (crescent shaping), the centre
region passes to second stability (Fig. A-l (b) and (c)), and, finally, the unstableregion is diminished by enhancing the crescent shaping. Indentation without
sufficient triangularity (kidney bean shaping) is not favourable for accessibility
to the second region (Fig. A-l (d)).
The change of local shear by adding triangularity is shown in Fig. A-2 for|3 ~ 13% configurations. In the bad curvature region, the local shear is increasedby raising triangularity, keeping the indentation constant.
As to first stability beta limits against n = °° ideal ballooning modes,
pressure optimized configurations are studied by using the FCT-BALLOON
code, and a new beta scaling for a wide range of shapings (ellipse, dee and crescent)is obtained [ 1 ]. Higher elongation or a lower aspect ratio does not give rise to j3enhancement unless sufficient triangularity is added. The effect of indentationtriangularity 5* is shown in Fig. A-3. Smoother but quicker transition to second
stability is obtained by an indented configuration with a sharp tip (crescent shape,
X = 1) than by shaping with a round tip (kidney bean shape, X = 2). In this3

K. Yamazaki, H. Naitou, T. Amano, Y. Hamada, K. Matsuoka, Y. Midzuno
Y. Nakayama.T. Sato.

IAEA-CN-47/E-I-2-1

• A-5
• A-3

29

I Balloon Unstable
Mercier Unstable

FIG. A-l. Effects of triangularity and indentation on accessibility to second regime ofballooning mode stability for A = 3.0, K = 2.0, q0 = 1.0 and qs = 3.0. The plasma shape isdefined by (X/a* + 5*(Z/K*a*)2)2+ (Z/K*a*)2X = 1.
(a) Dee: \ - 1, 8* = 0.5, i = 0.5, i = 0.0, 5 = 0.5(bj Weak crescent: X = 1, 8* = 0. 75, i = 0.04,6 = 0.68(c) Crescent: X = 1,8* = 1.0, i = 0.11, 8 = 0.76(d) Kidney bean: X = 2, 8* = 0.75, i = 0.11, 8 = 0.56.

( a ) \ = 1,6*=1.0( b ) X = 2,6*=075

Z 1-0

FIG. A-2. Contours of constant local shear for (a) crescent (X = 1) and kidney bean f\ = 2)
shaping at the same indentation of 0.11. Bad curvature region is on right hand side of\i^ = 0(zero principal normal) line.

30

YAMAZAKI et al.

00

FIG. A-3. Permissible beta values normalized by In = lp(MA)/(a(m)B(T)f for n=°°ballooning and (b) kink modes. In kink stability analysis, current profile is assumed asP'=cp2fJ - 4i2)ty and Tf = cR\(l - \pa)(t+ (1 - t)\p2) -RlP' (p2,aand Tare optimization parameters) for wall location ofa^/a = 2.0.

study, the q-profile is fixed at two positions only: at the centre, qg = 1.0,
and on the surface, q s = 0. When the same q-profile is assumed, a clearertransition to the second region of stability may be obtained.
The effectiveness of crescent shaping for low n kink modes is also clarified
by using the ERATO code. Several current profile models have been taken
into account to obtain optimal configurations for kink as well as Merrier andballooning modes (Fig. A-3(b)). The coefficient g of the Troyon type beta
scaling increases up to a 10 to 20% higher value for the X = 1 shape than for
the X = 2 shape, by raising the triangularity of the indented shape. Higher beta
values may be expected from a more accurate optimization of the profiles.
In connection with the achievement of these crescent configurations,axisymmetric mode stabilities are studied by using the linear ideal MHD code
ERATO; in addition, bifurcation problems are investigated by using a fullresistive MHD code [2], and transport processes accessible to the optimized
profile are studied with a 1.5-D equilibrium transport code [3]; thus, thefavourable stability features of crescent shapings in high beta regimes are
confirmed. Several engineering merits related to this crescent shaping in reaction
and ignition plasma experiments are also claimed [3,4].
It is concluded that this crescent shaping makes the device more compact
than normal dee shaping; it is supposed to be a more reliable extension,for beta enhancement, from conventional dee shaped design than is second
stability bean shaping.

31

IAEA-CN-47/E-I-2-1

J"TJ'.O

D.2

a.*

o.s

o.a

1.0

F / C B-l. Pressure gradient versus i// for A — 3, <?0 = -?.4, qs = 3.1, K = 1.6, 8 = 0.3 and(3 = £.i%. Z,arge pressure gradient is in low shear region.

PartB
SECOND STABILITY ACCESS IN CONVENTIONAL CROSS-SECTION4
When the region with negative local shear extends from the bad to thegood curvature region, the plasma enters the second stability region forballooning modes on magnetic surfaces with negative local shear. The local
shear becomes negative more easily in a low shear and a large pressure gradientregion [5]. By using this property, direct access to second stability, without
passing the unstable region, can be realized for a plasma with conventionalcross-section shape and q 0 > 1 (q 0 is the safety factor at the magnetic axis).
For fixed q-profile, equilibrium is obtained by solving the Grad-Shafranov andthe ballooning mode equations [6] iteratively. The ballooning mode equation
determines the marginal pressure gradient, dp^/di//, for a given equilibrium(\p is the normalized poloidal flux). The equilibrium at the next iteration step
is obtained by using the pressure gradient, Cdp^/di//. The constant C isadjusted so as to yield A/3 = 0.2%, where A/3 is the increment of the volume
averaged beta value. Figure B-l shows the resulting pressure gradient ofequilibria with a q-profile chosen as S w = (q(i// = 0.5) - q o )/(q s - qo) * 0.05,
where qs is the safety factor at the plasma surface. In the low shear region, thepressure gradient is unlimited for ballooning modes, and the beta value increases
4

T. Tsunematsu, S. Seki, M. Azumi, S. Tokuda, T. Ozeki, T. Nemoto, T. Takeda.

32

YAMAZAKI et al.

1.5
1.4
1.3

Access to Second Stability
Region of Ballooning Mode

1.2

1.0

I.I

1.2

1.3

1.4

1.5

1.6

FIG. B-2. Critical q0 versus K for the access to second stability region of ballooning modes.

A

1
v

Lo

'

o'.z

oU

FIG. B-3. Parallel current j)} = (f-B)/(B2).

'

Q.8

o.a

I.O

Equilibrium is same as in Fig. B-l.

up to the limit of numerical accuracy of the toroidal current in the equilibrium
calculation. Figure B-2 shows the critical value of qg as a function of elongation, K, for an aspect ratio of A = 3 and qs « 3 . 1 . Above the critical qo> directaccess to the second stability region is possible.
For such an equilibrium, the profile of the current parallel to the magnetic
field becomes hollow (Fig. B-3), which makes the stability of the external kinkmodes deteriorate. The stability of the n = 1 external kink modes is studied

33

IAEA-CN-47/E-I-2-1

(%)
1

1

i

i

10
Ballooning Mode <Unstable
/\

; ,Jl
1.0

| aw/a=1.2| a w •> ao

i

i

i

i

I.I

1.2

1.3

1.4

1.5

FIG. B-4. Beta limit of n = 1 external kink modes versus q0 for K = 1.6, 5 .= 0.3, A = 3,
and q as 3.1. In hatched region ballooning modes are unstable.

for the equilibria obtained in the analysis of the ballooing modes for the caseof K = 1.6 and 5 = 0.3, where 8 is the triangularity. Figure B-4 shows the beta
limit of the external kink modes. The upper and lower lines denote the casesof a w /a = 1 . 2 and aw = °°, where aw and a are the horizontal radii of the
conducting wall and the plasma, respectively. Because of the reduction of
global shear, the beta limit decreases with q 0 for the case where no conducting
walls are present. When a conducting wall is placed close to the plasma surface,a w /a = 1.2, 0 « 11% is possible for q 0 = 1.5. The beta limit due to the n = 1
external kink modes is close to Troyon's beta limit, 0 = gI p /aB, /3 « 4% for
g = 3 without conducting wall. A conducting wall placed at a w /a = 1.2
increases the beta limit up to /3>8% and Troyon's g factor becomes g = 6-8.
For the case of K < 1.6, the factor g takes almost the same value. Above thebeta limit, the dominant mode is m = 4, n = 1 for the case of no conducting
walls, and the mode is localized near the plasma surface. When the conducting
shell is placed at a w /a = 1.2, the dominant component becomes m = 1, n = 1.
By using current profile control, direct access to the second stability regionof the ballooning modes is realized for plasmas with circular and dee shaped
cross-sections and q 0 > 1. A plasma with q 0 > 1 was shown to be possible in
JT-60 experiments with clean plasmas [7] and in current drive experiments in
ASDEX [8]. The ballooning modes do not limit the beta value but low n
external kink modes determine the beta limit, and a conducting wall is required
to increase the limit.

34

YAMAZAKI et al.

10 '

5x10

-6

0.0

0.001

0.1

0.01

FIG. C-l. Time average of fluctuating magnetic energy versus beta value. Size of system
normalized by ion gyroradius is 16 X 64, density scale-length is 100.

PartC
SATURATION MECHANISM OF BALLOONING MODES5Previous studies of the non-linear behaviour of electrostatic ballooning
modes [9] are extended to the case of finite beta; the saturation mechanism andfluctuation level of the electromagnetic field are studied. We use a system of
two-fluid equations with Maxwell's equation:

if - -

A = -^C\, u) —

dadb

r-p

t

dadb

where L n denotes the density scale-length.
T. Tuda, Y. Kishimoto, T. Takeda.

Al

=

ay2

IAEA-CN-47/E-I-2-1

35

For beta values below the critical value as derived by MHD analysis, theelectrostatic ballooning modes are unstable, and moderate-m (m is the poloidal
mode number) eddies are formed. With increasing beta value, longer wavelength
modes grow, and, finally, the m = 1 mode dominates the other modes. Abovethe critical beta, magnetic fluctuations suddenly become large (Fig. C-l).
Saturation occurs when the order of magnitude of the m = 0 component thatis excited non-linearly by the unstable modes becomes sufficiently large tocancel out the background density gradient. In the vicinity of the critical beta
value, the ballooning modes degrade plasma confinement and induce a soft
beta limit, but have no other effect. At higher beta values, however, they may
cause a disruption of the current channel by large vortex formation.

ACKNOWLEDGEMENTS
The authors wish to thank Dr. Y. Shimomura for helpful suggestions.
They also express their gratitude to Drs M. Tanaka and M. Yoshikawa forcontinued encouragement.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]

YAMAZAKI, K., AMANO, T., NAITOU, H., HAMADA, Y., AZUMI, M., Nucl. Fusion
25(1985) 2543.
NAKAYAMA, Y., et al., Res. Rep. HIFT-125, Hiroshima University (1986).
KITAGAWA, S., et al., in Fusion Engineering (Proc. 1 lth Symp. Austin, TX, 1985),
Vol. 1, IEEE, New York (1986) 56.
YAMAZAKI, K., et al., Comments Plasma Phys. Controll. Fus. 10(1986) 69.
SEKI, S., et al., submitted to Nucl. Fusion.
CONNOR, J.W., et al., Phys. Rev. Lett. 40 (1979) 396.
JT-60 TEAM, in Controlled Fusion and Plasma Physics (Proc. 1 2th Europ. Conf.
Budapest, 1985), Vol. 1, European Physical Society (1985) 375.
McCORMICK, K., et al., in Controlled Fusion and Plasma Heating (Proc. 13th Europ.
Conf. Schliersee, 1986), Vol. 1, European Physical Society (1986) 253.
TUDA, T., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984
(Proc. 1 Oth Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1 985) 1 79.

IAEA-CN-47/E-I-2-2

STABLE TOKAMAK ACCESS TO,
AND OPERATION IN,
THE SECOND STABILITY REGION
A.M.M. TODD, M. PHILLIPS
Grumman Corporation
M. CHANCE, J. MANICKAM, N. POMPHREY
Princeton Plasma Physics Laboratory,
Princeton University
Princeton, New Jersey,
United States of America
Abstract
STABLE TOKAMAK ACCESS TO, AND OPERATION IN, THE SECOND STABILITY REGION.
Tokamak operation in the second region of stability to ballooning modes holds promise for
economic fusion reactor operation. A second region of stability is confirmed to exist in circular
tokamaks with a nearby conducting shell for ef3g > 1. In addition to ballooning instabilities with hightoroidal mode number n, the presence of low n internal pressure driven modes in low shear regions canpose serious problems for access to the second region. These modes can be present even when ballooning modes with infinite n are stable. Stable access to the second region for all ideal modes is demonstrated with current profile programming that raises the safety factor on axis above unity. Present
calculations for stable access using this technique indicate that a conducting wall must be located close
to the plasma edge ( — 0.1 plasma minor radius) to stabilize external modes. Pressure and shear profile
optimization could be used to increase this value. As e/3g is raised above unity, the stabilizing wall canbe moved to progressively larger major radii. This behavior is attributed to restabilization of thepressure driven component in low n kink modes. Finally, it is shown that poloidally discontinuous
conducting structures are effective in stabilizing low n external kink modes.

1.

INTRODUCTION

The economic advantage of high <8> fusion reactor
operation has long been recognized.
Here, <0> - 2ji <p>/B ,where <> denotes a volume average and BQ the nominal vacuum
toroidal f i e l d at R, the major radius of the plasma center.
I t has t r a d i t i o n a l l y been sought in Tokamak configurations by
maximizing the <(3> limit through cross-sectional shaping andminimizing the aspect r a t i o , A. The widely used c r i t e r i o n foet h i s ' f i r s t region 1 l i m i t , the Troyon l i m i t , i s <&T> = 3x10~8
I/aB Q [ 1 ] . I t has been shown [2,3] that high toroidal mode
number, n, ideal MHD pressure driven i n s t a b i l i t i e s canr e s t a b i l i z e at high values of e&e>1, giving r i s e t o a 'seconds t a b l e region' (e - a/R, Be-= 8ir p / y o l 2 ,a i s the minor
37

38

TODD et al.

radius, I the toroidal current, p the cross-sectional pressure
average - a l l u n i t s a r e S I ) . In g e n e r a l , a close conducting
shellis required to simultaneously stabilize parallelcurrent-driven low-n kink instabilities
in this second
stability region.
This paper explores the potential for high <0> second
stability region operation and access in moderate to large
aspect ratio circular Tokamaks.
Some of the benefits arethat, within standard constraints such as wall loading, thehigher power density second region reactor operates withsignificantly
reduced toroidal fieldand current, whencompared to firstregion Tokamak reactor concepts, oralternatively has the potential for advanced fuel operation.
Additionally, a large aspect ratio circular configuration may
have significant advantages from the important standpoint ofaccessibility
and hence maintainability.
The results
described here span a parameter space ranging from that of theCompact Ignition Tokamak (CIT) [1] to a conceptual large
aspect ratio second region experiment (SRX) [5].

2.

SECOND REGION PARAMETER SPACE

Our parameterization of the second region boundary is not
yet sufficiently comprehensive to permit the definition of anequivalent to the Troyon criterion. However, a few general
remarks can be made. When the safety factor on axis, qQ - 1, we
find a wall stabilized second stable region when e8e«. 1.2. As
qQ is allowed to rise above unity at fixed qe, the edge safety
factor, the first and second region eBe boundaries shiftdownward in approximate proportion to o^.
Figure 1 shows the lower bound on unstable toroidal mode
numbers, n , using the ballooning mode formalism. It alsoshows
the criticalconducting wall location, (b/a)(expressed in units of the minor radius) for n=1 stability.
For clarity, the stability of n=2,3,1< which has been
calculated, is not shown. The n=2 and 3 modes onset and begin
to restabilize at successively higher E B Q . The first andsecond region c0 e boundaries for the n=3 internal mode are-0.7 and ~1.1 respectively. The n=4 mode is stable with awall at infinity for all e8 e . A circular A=9 equilibrium
sequence with qQ => 1.01, qe •= M.1 and qa^1* was used for thesecalculations. The pressure profile was initially optimized tomarginal n = » first region ballooning stability on all flux
surfaces and then scaled to generate the E0Q variation. The
purpose of Figure 1 is to illustrate that conventional
ballooning theory is not a sufficient criterion for internal

39

IAEA-CN-47/E-I-2-2
4.0

1

j

1

1

1

1

l/i

1
1

050

-0.45

3.6-

\

I"
I'

3.2 --

-0.40

2.8-

-0.35

2.4 -

-0.30

120-

- 0.25n-0.201.6 1.2 --

-0.15

!

*)

-0.10

0.8-

0.4 -

y

i

-

1

-0.05

1

1

— ••n

i

i

i1

0.2

0.4

0.6

0.8

1.0

1.2

FIG. 1. Critical toroidal mode number, nc, for ballooning stability (n<nj; n — l mode growthrate, w2, with a wall at infinity; and n = l mode critical conformal wall location for stability, (b/a)c,
expressed as the displacement from the plasma edge in units of the minor radius, as Junctions o /mode s t a b i l i t y , since low n modes onset long before the lower
ballooning
thresholdand restabilize
after
the
upper
ballooning threshold. This also violates the ballooning mode
picture of decreasing s t a b i l i t y with n.
Conventional ballooning mode theory r e l i e s on an ordering
where the wavelength transverse to the magnetic field is shortin relation to other equilibrium variations.
However, when
the shear is weak, the ballooning mode ordering breaks down
and a new higher order theory is required [6]. At the largest
values of n >> (i{) dq/d\jj)~2 >> 1, the standard theory isrecovered, but when (\\> dq/diji)>> n >> 1, the new theory,
which shows an oscillatory dependence of the growth rate u
on 1/n, is required (here ty is the poloidal magnetic flux).
However, when the shear i s further weakened, we observe thateven this theory breaks down and there is no longer a simple

40

TODD et al.

behavior of the envelope of the o s c i l l a t i o n .
In p a r t i c u l a r ,i t i s possible to have bands of i n s t a b i l i t y , as 1/n i s varied,
which do not extend to high n. Consequently, high n modes canbe stabilized without s t a b i l i z i n g the low and intermediate ninternal ballooning modes, the "infernal" modes.
This
situation can be remedied by increasing the shear or shifting
the peak gradient in the pressure to regions of larger
shear.
Thus, the e x p l i c i t form of the shear and pressure
profiles i s crucial to access and operation in the seconds t a b l i t y region.

3.

STABLE SECOND REGION ACCESS

Several mechanisms(see for example [5])have been
proposed for reaching the second region of s t a b i l i t y . Here wewill only consider current profile programming [7]. As CJQ israised, n c r i s e s , the f i r s t and second region boundaries moveto lower values of ef5e, and the width of the unstable region,A(e($e), decreases significantly.
Peaking the pressure
profile, flattening the safety factor profile or decreasing A
will raise n c .
This dependence on A is similar to thatobserved when stable second region transition is obtained byindenting bean shaped plasmas. As A is increased in beans, thef i r s t and second region <g> boundaries decrease and the widthin <B> of the unstable gap is reduced.
However, theindentation required to access the second region in a stablemanner increases.
Figure 2 demonstrates that stable second regiontransition with respect to all ideal MHD modes i s possible, in
principle, with current programming.
A circular, A - 9
equilibrium sequencewith qQ = 3.1, q ^ ^ - l and q~i|> wasused.
The eJ e v a r i a t i o n was generated by scaling a pressure
profile - \ | r ' 5 .
Optimization of the plasma profiles in themanner suggested above could be used to reduce the high q0
value and relax the constraint on the low n mode stabilizingwall.
We have also performed accessibility studies where theform of the pressure profile is iteratively modified to remain
stable to n « • modes as <f}> is increased.
When the shear
near the axis is small and q^i.O, central surfaces in the low
shear region are observed to have a stable transition to thesecond region.
This leads to a centrally peaked two-region
pressure profile, where a second region core i s surrounded by
a f i r s t region mantle, whose transition
is inhibited by thepeaked pressure. Such cases can be stable at twice the Troyon
limit, for high and low n modes with (b/a)~ ~ 0.1. Resistive

41

IAEA-CN-47/E-I-2-2
40

—i

1

1

r

0.50

36

0.45

32

0.40

28

0.35

Ib/a)

0.30

0.25 ~

i 20

0.20

0.15

0.10

0.05

0.2

0.4

J

L

0.6

0.8

1.0

1.2

FIG. 2. n = l mode growth rate, o 2 , and critical wall location, (b/a)c,
resolved to within 0.05, as Junctions ofe&Q. n = oo ballooning stability exists for all £00.

Interchange instability near the first/second region boundary
is a potential problem, since, if a resistive interchange
stability
requirement
isincludedin the optimization
procedure, then the central surface transition is inhibited.
Alternatively, the shear profile can be adjusted to inhibit
central transition and promote edge transition. We find thatedge transition is difficult to achieve at high A, and atlower A i t tends to be localized to the outermost flux
surfaces.
The transitioning region of the plasma does notpropagate inwards towards the magnetic axis as <f}> isincreased.
Consequently, the edge pressure gradient grows
with <6> and leads to the development of unphysical profiles,
as, for instance, in the case of a large parallel skincurrent.
Similar localized edge transition is observed forq^i,and either high qe (~9) circular equilibria that
correspond to Tokamak Fusion Test Reactor (TFTR) supershot
like parameters [8] or conventional qe (~3) highly triangularTODD et al.

42

WALL ANGLE a(Degrees)

30

60

90

120

i

i

i

i

7"

150

Plosmoin second region

0.5
1.0
WALL DISTANCE, b/a

1.5

FIG. 3. n = l mode growth rate, co2 (solid curve), as a function of the angle, a, subtended .from theoutboard midplane by a circular wall. The dotted curve shows the dependence on wall distance (b/a).

and highly elongated CIT equilibria [*»]. Since the local formof the shear profile is critical for transition, othertransition scenarios will exist in addition to these examples.
1». POLOIDALLY DISCONTINUOUS HALL STABILIZATION OF KINK MODES
Whereas the consequences of internal pressure driven
modes can be minimized by tailoring the plasma profiles, low n
global kink modes with significant surface perturbations
require a conducting shell for stabilization.
Since physicalaccess to the plasma precludes a completely closed shell, we
have calculated the effect of the presence of an axisymmetricgap in the shell. Typical results for the n=1 mode are shownin Figure 3 for a bean-shaped plasma of elongation 1.38 andindentation 0.3 in the second region of stability.
We seethat, firstly, a partial wall at (b/a) =0.3 can stabilize then-1 mode if the angle a sustended by the wall is ~100°.
Secondly, the inner major radius side of the shell plays an
insignificant role in stabilization, so the hashed portions of
the wall may be removed.
Finally, a moderate gap on theouter major radius midplane can be tolerated.

IAEA-CN-47/E-I-2-2
5.

43

CONCLUSIONS

E q u i l i b r i a which a r e s t a b l e t o a l l i d e a l MHD modes i n t h esecond r e g i o n of s t a b i l i t y a r e observed with eg Q >1, provided anearby conducting s h e l l s t a b i l i z e s the low n modes.
Stablea c c e s s t o t h i s r e g i o n i s not e a s y , but programming t h e c u r r e n t
so t h a t q_>1, coupled with o t h e r a c c e s s t e c h n i q u e s [ 5 ] , shows
promise.
R e s t a b i l i z a t i o n of low n as well as high n modes i sobserved as eB e ia i n c r e a s e d above u n i t y .
The p o t e n t i a lbenefits of second region fusion reactor operation encourage
further study of these i s s u e s .

ACKNOWLEDGEMENTS
This work was supported by the United States Department of Energy,
under Contracts Nos. DE-FG02-86ER53226, DE-FG02-86ER53222 andDE-AC02-CHO-3073, and by the Grumman Corporation.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]

TROYON, F., GRUBER, R., SAURENMANN, H., SEMENZATO, S., SUCCI, S., Plasma
Phys. Contrail. Fusion 26 (1984) 209.
STRAUSS, H.R., PARK, W., MONTICELLO, D.A., WHITE, R.B., JARDIN, S.C.,
CHANCE, M.S., TODD, A.M.M., GLASSER, A.H., Nucl. Fusion 20 (1980) 638.
COPPI, B., FERREIRA, A., MARK, M.W.-K., RAMOS, J.J., Nucl. Fusion 19 (1979) 715.
SCHMIDT, 1., et al., Paper IAEA-CN^7/H-I-2, these Proceedings, Vol. 3.
NAVRATIL, G.A., et al., these Proceedings, Vol. 1, p. 299.
HASTIE, R.J., TAYLOR, J.B., Nucl. Fusion 21 (1981) 187.
SYKES, A., TURNER, M.F., in Controlled Fusion and Plasma Physics (Proc. 9th Europ. Conf.
Oxford, 1979), UKAEA, Culham Laboratory, Abingdon (1979) 161.
HAWRYLUK, R.J., et al., these Proceedings, Vol. 1, p. 51.

IAEA-CN-47/E-I-2-3

ELLIPSOIDAL SHELL TOKAMAK
F.J. HELTON, Mingsheng CHU,
J.M. GREENE, R.W. HARVEY, J.K. LEE,
T. OHKAWA, P.A. POLITZERGA Technologies Inc.,San Diego, California,
United States of America
Presented by T. Amano
Abstract
ELLIPSOIDAL SHELL TOKAMAK.
The paper describes a novel toroidal plasma configuration, taking the shape of an elongated ellipsoidal shell, and some features of a realizable tokamak design which has been developed from it. The
ellipsoidal shell tokamak (EST) has favourable high pressure MHD stability properties for a wide varietyof shell geometries. Highly elongated tokamaks also have the possibility of achieving ohmic ignition
at moderate toroidal field and moderate size. When the safety factor is held fixed, the current density
is proportional to elongation and so the required high heating power density can be obtained. Empiricaltransport scaling laws indicate that ignition may be achieved at a range of aspect ratios, with a moderate
plasma minor radius, using a feasible magnetic field. The considerations leading to the choice of anellipsoidal shell configuration — the equilibrium and ballooning stability properties, global MHD stability, thermal instability and possible mitigating factors — are discussed and the parameters of a sample
design for a modest proof-of-practicality EST are presented. The critical issues for the EST involve its
behavior under transient conditions. It is important to determine whether the plasma configuration canbe established quiescently and whether it can be sustained stably by active shape and position control.

INTRODUCTION
This paper describes a novel toroidal plasma configuration andsome features of a realizable tokamak design which has been developed
from it. In this configuration the plasma takes the shape of an elongatedellipsoidal shell. The ellipsoidal shell tokamak (EST) has favorable highpressure MHD stability properties for a wide variety of shell geometries.
Highly elongated tokamaks also have the possibility of achieving
ohmic ignition at moderate toroidal field and moderate size [l]. Whenthe safety factor is held fixed, the current density is proportional toelongation, so the required high heating power density can be obtained.
Empirical transport scaling laws indicate that ignition may be achieved
45

46

HELTON et al.

at a range of aspect ratios, 3 < l/e < 7, with a moderate plasma minor
radius, a = 0.25 m, using a feasible magnetic field, BT < 6 T.
We discuss the considerations leading to the choice of an ellipsoidal
shell configuration, the equilibrium and ballooning stability properties,
global MHD stability, thermal instability and possible mitigating factors, and present the parameters of a sample design for a modest proofof-practicality EST.

PLASMA SHAPING
The limitation on current density at the magnetic axis for which afavorable average magnetic well can be obtained (the Mercier criterion)is one of the constraints that suggests the geometry of this device. This
limitation is relaxed by increasing the triangularity of the plasma shape.
For the highly shaped EST devices, it is desirable to recast the triangularity in terms of the shape of the surface defined by the vanishing of
Bz. If the radius of curvature of this surface is denoted by RT, then in
the usual notation [2]where R is the major radius of the magnetic axis, and r is the triangularity. In terms of RT, the Mercier stability criterion isB%ql

1 + K2

Pp + « 2i +, l-i ~W« 2 ( « + l ) Mp
RT

where ^ ( 0 ) is the toroidal current density at the magnetic axis. Withhighly elongated shapes, qa can be as small as unity when RT = R- With
this condition,Bj vanishes on a section of the surface of a sphere, leading
naturally to a plasma which takes the shape of a section of a sphericalshell and a device in which the vacuum chamber and poloidal fieldshaping coil systems also conform to the spherical shell configuration.
In practice, the shape can be relaxed to a prolate ellipsoid (e.g. 3:2
axis ratio — RT = 9/4 R), without undue degradation of the overall
stability estimates.

IAEA-CN-47/E-I-2-3

47

FIG. 1. (a) \B\ contours for an EST plasma shape.
(b) EST equilibrium stable to ballooning and Mercier modes at 45%.

Ballooning mode instabilities are stabilized by strong magnetic
shear and by reduction of the geodesic curvature. For a thin sphericalshell, extending in latitude from - ^ to +#maxj geodesic curves are
great circles. However, the safety factor is restricted to q(a) < 1, theequality holding for the geodesic field line that just reaches the top(maximum latitude) of the plasma and passes from the outer surface ofthe shell to the inner surface. The generalization to a thin ellipsoidal
shell permits higher, tokamak-like values of q(a) — the maximum valueof q depends on the elongation of the ellipsoid and on the maximum
latitude of the shell plasma.
Minimizing geodesic curvature maintains nearly constant \B\ on
the outer magnetic surface. This partial omnigeneity arises becauseof the interplay between the variation of the toroidal field with major
radius and and the variation of the poloidal field with both major radiusand vertical position. The major variation in \B\ is at the top andbottom of the plasma, due to the finite thickness [see Fig. 1 (a)] of theshell. Using this property, the EST plasma shape should provide thepossibility of improved confinement.

48

HELTON et al.

FIG. 2. Sequence of asymmetric equilibria demonstrating
the startup of a K = 6 EST tokamak.

EQUILIBRIUM AND BALLOONING STABILITY
Analysis of EST equilibria indicates that they are stable to ballooning and Mercier modes at very high /?, without fine-tuning of theplasma shape. These equilibria can be obtained for a broad range of current profiles. Varying the width of the current profile from flat to ohmic
changes the limiting /? (= f3c) by less than a factor of 2. There are EST
equilibria with /3C as high as 45% [see Fig. 1 (b)]. When EST equilibria
are optimized, the shape becomes slightly pointed and there is a small
increase in (3C, in agreement with the work of Todd and Phillips [3]. The
equilibria are established with the limiters, flux loops, and field-shaping
coils lying on nested ellipsoids, with corresponding flux loops and coils
on orthogonal hyperboloids. The solutions are robust, that is they are
not sensitive to small changes in plasma current or in the control flux
values at the field-shaping coils.
We have obtained a sequence of asymmetric equilibria (see Fig. 2)
which demonstrates the startup of a highly elongated tokamak [4]. These
equilibria are limited by a physical limiter at the top of the resistive vacuum tank and a poloidalfieldnull (divertor) at the lower side. By proper
positioning of the the field null, the plasma evolves from low (« = 2) tohigh (K = 6) elongation on the time-scale given by the resistive tank.

IAEA-CN-47/E-I-2-3

49

We have compared the EST with equivalent racetrack and elliptical configurations. Racetrack equilibria have been obtained for K < 11,
but it is difficult to obtain any racetrack above K = 7, and very flat
current profiles are required as the elongation increases. The solutions
are not robust and can have multiple magnetic axes. The non-existenceof robust solutions is probably a consequence of the increased elongation of the central flux surfaces of the straight tokamak as compared to
the EST. The racetrack solutions cannot be made stable to ballooning
and Mercier modes at high (3 without fine-tuning the field-shaping coils.
When the elliptical and racetrack equilibria are optimized the shapes
become more EST-like. When elliptical and racetrack equilibria withthe same plasma current (1.0 MA), aspect ratio (6.0), area and current
half-width were optimized to ballooning and Mercier modes, the ballooning /? of the elliptical equilibrium exceeded that of the racetrackequilibrium by a factor of 5.0.

GLOBAL MHD STABILITY
Pressure and current driven global MHD instabilities for highly
elongated tokamaks were analyzed using the initial value code CART
[5] and the ideal stability code GATO [6].
The CART code employs the reduced tokamak equations in a
Cartesian coordinate system. The CART results show that for a «=6
EST example the n=0 mode can be stabilized with a conducting wallat 0.3 a while the n = l mode can be stabilized only with a very close
conducting wall. For a re=6 racetrack the n=0 mode can be stabilizedwith a conducting wall at 0.3 a, but the n = l and n=2 modes cannot
be stabilized even with a wall at the plasma boundary. For this fixedboundary case a double tearing mode was observed with a small growthrate. We have also analyzed the quasilinear evolution of the modes withn — 0, 1, and 2.
Various stabilizing schemes were considered using CART. A toroidal limiter covering 20% of the poloidal plasma circumference reduces
the growth rate by a factor of five for both n=0 and n = l modes. The
axisymmetric mode with n = 0 is a shifting mode (see Fig. 3), generating perturbed flow pointing vertically upward in the plasma central
region. It is more strongly stabilized by placing a conducting wall at theoutboard side than at the top-bottom side. The ideal mantle with infinite conductivity does not significantly affect the n=0 mode. It reduces

50

HELTON et al.

0

if
:;;!
iUl-

••i

\'

1
\\

i

FIG. 3. n = 0 shifting mode for a K = 6 racetrack equilibrium.

the growth rate of the n = l mode only slightly; the extent of the reduction depends on the open magnetic field configuration in the mantle
region.
Ideal MHD kink instabilities for highly elongated tokamaks wereanalyzed using G ATO. The n=0 results for the «=6 EST were consistent
with the CART results. The axisymmetric mode with n = 0 is also ashifting mode for the EST. The position of the wall necessary to stabilize
the n=0 kink mode for an EST with K = 6 is greater than 0.3 a,except
at low (3P and peaked current profile. We were not able to stabilize
the n=0 kink mode for a K=6 tokamak with an approximate racetrack
plasma shape. The racetrack results are in agreement with other results[7] obtained for «=3 racetracks. Our equilibrium calculation is freeboundary; it is possible to triangulate an approximate racetrack andstabilize the n=0 kink mode with a very close wall.
THERMAL INSTABILITY AND TRANSPORT
A major concern for EST plasmas is the ohmically driven thermal
instability which can lead to central peaking of the current profile. This,

IAEA-CN-47/E-I-2-3

51

in turn, may prevent the stationary maintenance of elongated equilibria.
We have carried out transport code simulations of the evolution of thecurrent, and the central peaking is inhibited by several mechanisms. As
usual in tokamaks, we assume a sawtoothing instability when <?(0) < 1,which flattens the profiles. This process is sufficient to prevent thermal instability if the equilibrium flux surfaces are assumed to be fixed
ellipses.
However, in the EST equilibria, the g-values are observed to actually rise near the magnetic axis, being dominated by the flux surface
shape rather than by the current profile. In the absence of the q < 1
sawtoothing instability, two further mechanisms are invoked to preventthe thermal instability. At low plasma density, MHD studies indicatethat the n = l and higher n modes can be located well inside the plasma
surface. These modes are taken to have current profile flattening effects
equivalent to the sawtoothing instability. In addition, at sufficiently
high density but still below the Murakami limit, there is a regime ofdominance of the ion neoclassical transport over the conventional empirical transport. The close coupling of the electrons and ions and the(1/r) 3 ' 2 variation of the ion diffusion coefficient give thermal stability.

SAMPLE DESIGN
Figure 4 illustrates an engineering design for an ellipsoidal shell
tokamak. The plasma shell is a portion of a prolate ellipsoid with a
3:2 ratio of axes. Thus, for the major radius of 0.69 m, the radius ofcurvature is 1.50 m.
A typical g-profile for an EST is inverted: it is maximum on axis,
decreases to the minimum close to the limiter and then increases toward
the limiter. When the current half-width of this equilibrium was varied
from flat to ohmic, j3c decreased by less than 50%.

SUMMARY
These analyses indicate that there are significant benefits inherent in the EST configuration. It provides the possibility of stable, highpressure plasmas, as well as the prospect of ignition with ohmic heating
alone at feasible magnetic field strengths. The critical issues for the52

HELTON et al.

Major radius, R

0.69 mShell curvature, RT

1.5 m

Halfwidth, a0.10 mElongation, K

6.0

BT

1.04 Th

1.0 MAPP

0.79

AxdJoon

37%
14%

Current halfwidth (S/a)

0.39

FIG. 4. Engineering design for an EST and typical q-profile for an EST equilibrium.

EST involve its behavior under transient conditions. It is important todetermine whether the plasma configuration can be established quiescently and if it can be sustained stably by active shape and position
control.

ACKNOWLEDGMENT
This is a report of work sponsored by the U.S. Department of Energy under Contract No. DE-AC03-84ER53158 and by GA Technologies
Inc.

REFERENCES[1] CHU, M.S., HARVEY, R.W., POLITZER, P.A., YAMAGISHI, T., Nucl. Fusion
25 (1985) 835.
[2] WEIMER, K., FRIEMAN, E., JOHNSON, J.L., Plasma Phys. 17 (1975) 645.
[3] TODD, A.M.M., M.W. PHILLIPS, Bull. Am. Phys. Soc. 30 (1985) 1424.
[4] MARCUS, F.B., JARDIN, S.C., HOFMAN, F., Phys. Rev. Lett. 21 (1985) 2289.
[5] LEE, J.K., Phys. Fluids 29 (1986) 1629; Nucl. Fusion 26 (1986) 955.
[6] BERNARD, L.C., HELTON, F.J., MOORE, R.W., Comp. Phys. Comm. 24(1980) 377.
[7] HOFMAN, F., MARCUS, F.B., TURNBULL, A.D., Plasma Phys. and Cont.
Fusion 28 (1986) 705.

IAEA-CN-47/E-I-2-4

MHD STABILITY
IN LOW-ASPECT-RATIO TOKAMAKS*
B.A. CARRERAS, L.A. CHARLTON1, J.T. HOGAN,
J.A. HOLMES1, E.A. LAZARUS
Fusion Energy Division,
Oak Ridge National Laboratory,
Oak Ridge, Tennessee,
United States of America
W.A. COOPER
Centre de recherches en physique des plasmas,
Ecole poly technique fe'de'rale de Lausanne,
Lausanne, Switzerland
T.C. HENDERUKAEA, Culham Laboratory,Abingdon, Oxfordshire,
United Kingdom

Abstract
MHD STABILITY IN LOW-ASPECT-RATIO TOKAMAKS.
Interest in tokamak configurations with high beta has led to renewed study of low-aspect-ratio
tokamaks. The paper presents a quantitative estimate of the MHD </3crit> predicted for such configurations, with emphasis on experimentally feasible cases. Ideal MHD stability calculations, extending
previous results to the regime with aspect ratio less than 2.5, show that stability follows the Troyon
semi-empirical scaling law. We find this scaling law to be a reliable theoretical guide in the search forhigh beta configurations at low aspect ratio. Ideally, stable equilibria with </3> - 40% are found at
an aspect ratio of 1.67. These calculations employ high-resolution equilibria to examine axisymmetric,
low-n (n < 3), and infinite-n ideal modes. Nonlinear resistive tearing mode calculations show that
decreasing aspect ratio also leads to a reduction in the predicted saturation levels of m = 2/n = 1,m = 3/n = 2 islands and in their overlap region. The m = 1/n = 1 reconnection time has been foundto increase as the aspect ratio is reduced, leading to the possibility of sawtooth stabilization at low aspectratio and high temperature.

Research sponsored by the Office of Fusion Energy, United States Department of Energy,
under Contract No. DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc.
Computing and Telecommunications Division, Martin Marietta Energy Systems, Inc.,

53

54

CARRERAS et al.

1. EQUILIBRIUM PROPERTIES OF LOW-ASPECT-RATIOCONFIGURATIONS
Many of the significant features of low-aspect-ratio equilibria havebeen pointed out by Peng and Strickler [l], who found a number of distinguishing properties:- There is a natural shaping effect, so that little additional explicit
external shaping current is needed to produce highly elongated shapes.
- The magnetic aspect ratio (defined as the ratio of poloidal to toroidal
magnetic field at the periphery) is characteristically much larger ina low (geometric) aspect ratio tokamak than it is in a conventional
tokamak with the same safety factor.
- The strongest differences in magnetic aspect ratio between low andhigh geometric aspect ratio tori are predicted to occur for A < 2.5(where the geometric aspect ratio A = R/a; R is the major and a thehorizontal minor radius of the tokamak).
These attributes have some immediate implications for MHD stability:- The absence of an externally applied quadrupole field for an elongatedlow-A plasma means that the field index is 0; hence, axistability will
be improved relative to conventional tokamaks, in which this index isnegative for elongated configurations.
- Since the magnetic aspect ratio is proportional to the well-known
Troyon /? scaling parameter Ip/aB^, both empirical scaling and existing calculations lead us to expect improved stability at low A.
- Existing stability calculations (covering only A > 2.5 in a systematicway) are not adequate to establish meaningful quantitative (/?) limits,
since the major differences in magnetic aspect ratio occur for lower
geometric aspect ratio.
The goal of our calculations is to make a quantitative estimate of thelimiting {j3} values in \ovt-A tokamaks, with some important constraints.
We wish to consider cases for which the marginal /? value does not lie at theextreme of tokamak operation; in such cases the confinement deteriorationand increased disruptivity associated with q or density limits can either
hinder the experimental study of the /?-limiting physics or even prevent
entirely the attainment of the theoretically expected /?Crit- Our emphasis
is thus placed on experimentally realizable cases: we require that theboundary safety factor lie in the range 3-4, we seek to minimize the central
safety factor to coincide with the experimental observation that discharges
with q(0) ~ 1 are often observed, and we require that the shapes becompatible with those which can be constructed from realistic external
shaping and OH coil sets. To support the last requirement and to^provide
a detailed basis for the stability analysis described here, we have extended
the equilibrium work reported in Ref. (lj.
A survey was carried out with the free-boundary EFIT code [2], including the effects of the Ohmic solenoid and of the finite size of the poloidal
coil set. The equilibrium survey shows that the "natural" elongation is

IAEA-CN-47/E-I-2-4

55

dependent on the value of A and on the higher order fields applied. For
example, /cnatUrai = 1-7 at A = 1.67, with zero applied quadrupole field.
Of relevance to non-axisymmetric stability, a large range of plasma shapes
has been produced, with elongation K up to 2.7, and triangularity 6 up to
0.5, for the specific case A = 1.67.

2. IDEAL MHD STABILITY
The results of a number of limiting 0 experiments are represented by
the Troyon semi-empirical formula [3], which is based on ideal MHD stability calculations and predicts a strong improvement in /? cr j t as aspectratio decreases. The improvement of ideal MHD /? limits with decreasing aspect ratio has been regarded as "simple and predictable" for sometime [4].
Our approach to investigating /? limits for low A is to follow the trends
suggested by the Troyon scaling law and to choose the direction for further
improvement from its predictions. Thus, we begin by performing an aspectratio scan, making the first systematic MHD stability study in the regime
with A < 2.5, while keeping both the shape and the profiles fixed. Then,
adopting an aspect ratio which is as low as possible, but still feasible forexperimental implementation, we fix both the aspect ratio and the profiles
to explore the detailed dependence of high-n stability on plasma shaping.
With this information, we proceed to our goal of estimating the maximum(/?} attainable with realistic values for both gedge and q(0).
Detailed calculations showing the general improvement of /3crit withdecreasing A have, for example, been presented by Todd et al. [4] for
A > 2.5. A summary of other recent work in low A stability is presented in the Proceedings of the IAEA Specialists' Meeting on Tokamak
Concept Improvements [5]. In particular, we note that Degtyarev et al.
report a degradation of pcTa compared with the Troyon parametrization
for g edge < 3.
Adopting the representation used by Troyon et al. for the JET
configuration (K = 1.68, 6 = 0.3) as a starting point, we have carriedout the aspect ratio survey to explore the regime A < 2.5, with fixed
shape, and pressure and q profiles. The aspect ratio scan includes an
examination of axisymmetric (n = 0) and low-n free-boundary modes(for n < 3), as well as fixed-boundary infinite-ra ballooning and Mercier
modes.
The pressure and rotational transform profiles used for this study arethose optimized by Troyon et al. for the JET analysis:p = p(0)(1 - 0.20 - 2.6V2 + 1.803)

111

g =

l,62

6

1 +

where 0 is the (normalized) poloidal flux and p is the effective plasma
radius (~ (volume) 1 / 2 ). Since such a parametrization would lead tolarge edge current densities at low aspect ratio if applied literally, a local
modification to q is made, which does not affect the central or edge q
values, to null the current density at the edge.

CARRERAS et al.

56
20
2<p>

' =3.5q

I (MA)

oJmlBJT)= 3.0

K-- 1.68
S =0.30.5

i.O

FIG. 1. Dependence o//3 cri , on inverse aspect ratio, following Troyon et al. [3].

We find that (Fig. 1.):a. The Troyon scaling law is a good extrapolation at low aspect ratio
for gedge = 3 - The predicted (3 scaling agrees well with the numericalresults. (The numerical coefficient is found to be somewhat larger
than the value in [4], 0.035 rather than 0.027.) As no optimization
was done at each A, the scaling law may be an underestimate.
b. The /3 cr ; t values increase sharply with decreasing aspect ratio, especially for A < 2.5 (in Fig. 1, e is the inverse horizontal aspect ratioa/R).
As described earlier, we now fix the aspect ratio corresponding to thehighest /? from this scan, at A = 1.67, and carry out a study of the effectsof plasma shape, with fixed profiles. This is intended to quantify theeffect of increased K and 6 on high-n ballooning stability at this aspectratio. Figure 2 shows the calculated dependence of /?(0)cr;t (for infinite-nmodes) on elongation for A = 1.67. For these profiles, (/?) ~ 1/2 x /?(0).
Two values of triangularity are shown: 6 = 0.3 (the previous JET case)and 6 — 0.13, near the "natural" shape at low aspect ratio. Two featuresof special significance for our goal are that:- pcT\t shows a weak optimum as a function of K. (It should be borne inmind that, since we have not optimised the profiles, the /3™ff versus
K curve could have a slightly different dependence than the one shown
here for fixed profiles.)- There is a decided improvement in /?crit with increasing 6 at a fixed
elongation.

IAEA-CN-47/E-I-2-4

57

1.7
1.8
ELONGATION

FIG. 2. Dependence of infinite-n /3 cr;i on elongation for 8 = 0.13 and 0.30. The scan is taken with
fixed aspect ratio and profiles. Curves are shown for the marginal case (0% unstable volume) and forthe 10% and 20% unstable volume cases.

Thus, following the scaling law predictions, the route to higher (3 liesin the direction of increasing the triangularity.
Figure 2 exhibits the marginal case (no unstable volume, /3 = /?Crit) as
well as cases with 10% and 20% ballooning unstable volume. According to
the Connor-Taylor-Turner soft-/? limit picture [6], the high-n unstable volume can be modelled as an effective reduction in energy confinement time,so that these curves could represent the f3cr\t attainable with, respectively,
10% and 20% more heating power than that required for the marginal case.
Although fixed-boundary equilibria have been used as input to these
infinite-n stability calculations, detailed free-boundary equilibria (including the effects of the OH solenoid and finite-size coils) have been foundfor configurations approximating those described here with A = 1.67.
The high-n ballooning results suggest increasing the triangularity.
Following this approach, the stable (/?) values at A = 1.67 have been
extended to the 40% range. We have used a ^-profile similar to that in
the original JET case, q = 1.61 + 1.89 • x 1 2 {q{0) = 1.61,<?edge = 3.5).
For a specific case the geometric parameters are K = 2.35, 6 = 0.55, and
A = 1.67, and (/?Crit) — 41%. A different parametrization of the pressure
profile was required to reach this /? value:
P = Po [ewhere a = 1.2 and x is the toroidal flux normalized to the value at theedge. This configuration is stable to ideal low-n modes with n < 3, toaxisymmetric modes, and to high-n ballooning modes. The equilibrium
was computed with the ISLAND free boundary code, and low-n stability
was studied with the ERATO code.
The ^-profile used for the cases thus far described, although widelyused in MHD optimization studies, falls somewhat short of our goal, inCARRERAS et al.

58
3.5

3.0

1

1

1

1

— 2.0

-j

1.8

J
\ —
—
\\, , , ,\
1.6

2.5

2.0

1.4 •«

1.2 I
c

1.0 1

r

c

1.5

0.8 CL
0.6

1.0
0.5
0

0.4
0.2

0

0.2

0.4

0.6

0.8

n
(.0

P/o
FIG. 3. Pressure and q-profiles for the

20% stable case, with A = 7.67.

that g(0) > 1.5. We have found, though, that this constraint can berelaxed by profile optimization. Stable (/?) = 20% cases have been foundwith 1.05 < g(0) < 1.5. Again, for a specific (/?Crit) = 20% case, thegeometric parameters are K = 2, 8 = 0.4 , and A = 1.67. The pressure
profile is broader, and the ^-profile satisfies the criteria for our study,
9edge = 3.5 and g(0) = 1.05:

q =9(0) - (?(0) The pressure and g-profiles for this case are shown in Fig. 3, and thecorresponding flux surfaces are presented in Fig. 4. The equilibrium used
was obtained from the Jardin-Monticello code [7], and ideal stability wasdetermined from PEST 2 [8].
Thus, following the Troyon scaling formula, we have been led to discover a very promising direction for enlarging the /3-stable tokamak operating regime with experimentally feasible configurations. The fact thatultra-high /?-values can be found by following Troyon scaling, using the
Troyon JET g-profile, means that there is substantial flexibility in therange of acceptable ^-profiles which will produce (/?Crit) in the range of 20
to 40% for low aspect ratio.
We further note that these configurations fill the region inside thetoroidal field coils relatively efficiently and thus do not provoke questions

59

IAEA-CN-47/E-I-2-4

0.50

-

-0.94

-0.71

-0.47

-0.24

0

0.24

0.47

0.71

0.94

FIG. 4. Poloidal flux contours for the case in Fig. 3, showing the compactness of the configuration.

concerning the relevant definition of the (ultimately economic) value of/?, as do more highly distorted shapes which do not effectively fill themagnetic volume.

3. AXISTABILITY
Attention is drawn to the fact that the stability studies described in
Section 2 all include n = 0 stability. The qualitative expectations notedin our discussion of equilibrium results in Section 1 have been borne out
by detailed calculation. That is, axistability is improved at low aspectratio.

4. TEARING STABILITY
The current density profiles used in the foregoing ideal stability calculations are relatively broad and, from the perspective of cylindrical calculations [9], might be considered liable to strong tearing instability andhence to disruption. Fully toroidal calculations show, however, that tearing stability actually improves in many respects with decreasing aspectratio.
Linear calculations of the tearing growth rate of an m = 2/n = 1 dominated eigenfunction show a weak equilibrium-induced stabilizing effect ofincreasing toroidicity on the current density profile. This favorable lowaspect-ratio effect competes with the potentially unfavorable developmentof coupling to toroidicity-induced satellite modes. Nonlinear studies havebeen carried out for a circular cross section using the full set of MHD
equations, in toroidal geometry, with a nonlinear version of the FAR

60

CARRERAS et al.

0.40 -

A—-CYLINDRICAL

0.30 _'€ = 0.4

1
m = 2/n = 1 MODE
6 = INVERSEASPECT
RATIO

0.20

/
It

/
t

€ = 0.7

0.10

0

U_

'i-""" ,

1

Q02

0.03

0.01

0.04

t/T, Hp
FIG. 5. Time evolution of the m = 2/n = 1 island width for various aspect ratios.
S — 10s for his case.

code [10]. The calculations employed an incompressible model. Figure 5
shows the variation with e (= A'1) of the m = 2/n = 1 island width.
We find a significant reduction in saturated island width with decreasing
aspect ratio. This reduction is actually accompanied by an increase in
the toroidally generated satellite activity. The 1OW-J4 improvement in the
single-island-dominated result for the m = 2/n = 1 mode is also found
for the m = 3/n = 2 behavior. Our nonlinear calculations show thatthe overlap volume is significantly reduced for lower aspect ratio. The
simultaneous reduction of the island extent for these modes suggests an
improved "disruption immunity" at low aspect ratio.
The deleterious effects of the m = 1/n = 1 "sawtooth" instability arefamiliar, and the behavior of this mode assumes greater importance at
low-,4 with the discovery of high ideal MHD /?crit values, with g(0) ~ 1.
At high temperature, as the resistive kink growth rate is slowed, recent
linear calculations [11] show that a transition to a tearing-dominated situation is expected to occur, as the Lundquist number (5 = rresjstive/rAifV6n)
increases, and as A decreases. In this case, the m = 1 tearing mode is therelevant instability. We have examined the nonlinear behavior of the resistive m = 1/n = 1 instability at different aspect ratios. The calculations
were done with the full resistive nonlinear MHD equations, for a zero-/?,incompressible toroidal model. The configuration is circular, with a conducting wall on the surface. The profile has q(0) = 0.9 and qedSe = 2.3.
The computed prediction for the time required for the m = 1/n = 1 mode
to reconnect increases by a factor ~ 7 as the aspect ratio in the calculation
is decreased from 10 to 1.4, for 5 = 10 s . The trend is to longer reconnection time for the tearing mode as aspect ratio is reduced. The nonlinear
calculations thus suggest that the m = 1/n = 1 tearing mode could actually be stabilized at low aspect ratio, hence producing complete sawtoothIAEA-CN-47/E-1-2-4

61

stabilization. It is important to note, however, that compressibility effects
could modify these results.
The overall trend for resistive stability is a broadening of the operational regime, decreasing the likelihood of disruptions, as the aspect ratio
is lowered.

5. S U M M A R Y
Quantitative estimates of the theoretically expected (/?Crit) in the lowaspect-ratio regime are quite favorable. Prospects for the experimental
exploration of this area are encouraging. Both ideal and resistive stability
are found to be improved for A < 2.5, and values of (/?Crit) in excess of
20% are indicated for tokamak operation with qedae (> 3).
The existence of ultra-high stable /? cr j t implies that the study of
confinement at low aspect ratio can be decoupled from the enhanced losses
thought to be associated both with marginal j3 behavior and with conventional current and density limits. The reduction in saturated island
widths suggests a concomitant reduction in disruptivity. Stabilization of
the resistive m = 1/n = 1 instability at high S and low A may provide
an intrinsic improvement in confinement. Improved axisymmetric stability should enable more efficient experimental exploration of the effectsof cross-section shaping on confinement and stability, with reduced disruptivity due to vertical instability. Recent experimental results withdiverted, shaped plasmas have suggested that operation at gedge ~ 2 may
be an unrealistic goal [12]. If these experimental results are borne out,
then the existence of high /?crit at fledge > 3 may mean that low A is
required to access high /? with the optimal (H-mode) confinement associated with divertor operation.
A new tokamak, the Spherical Torus Experiment (STX), has been
proposed by ORNL [13] to exploit this predicted expansion in the /?-stabletokamak operating regime.

REFERENCES[1][2]

PENG, Y-K.M., STRICKLER, D.J., Nucl. Fusion 26 (1986) 769.
LAO, L.L., ST. JOHN, H., STAMBAUGH, R., et al., Rep. GA-A17910, GA Technologies,
Inc., San Diego, CA (1985).
[3] TROYON, F., et al., Comments Plasma Phys. Controll. Fusion 26 (1984) 209.
[4] TODD, A.M.M., et al., Nucl. Fusion 19 (1979) 743.
[5] INTERNATIONAL ATOMIC ENERGY AGENCY, Tokamak Concept Innovations (Proc.
Specialists Meeting Vienna, 1986), IAEA-TECDOC-373, IAEA, Vienna (1986).
See specifically PENG, Y.-K.M., et al., p. 567-570; BESPOLUDENNOV, S.G., et al.,p. 571-574; TSUNEMATSU, T., et al., p. 575-579.
[6] CONNOR, J.W., TAYLOR, J.B., TURNER, M,F., Nucl. Fusion 24 (1984) 642.
[7] MONTICELLO, D., JARDIN, S., personal communication.
[8] GRIMM, R.C., DEWAR, R.L., MANICKAM, J., J. Comput. Phys. 49 (1983) 94.

62

CARRERAS et al.

[9] WADDELL, B.V., CARRERAS, B.A., HICKS, H.R., HOLMES, J.A., Phys. Fluids 22 (1981)
1131.
[10] CHARLTON, L.A., HOLMES, J.A., HICKS, H.R., LYNCH, V.E., CARRERAS, B.A.,
J. Comput. Phys. 63 (1986) 107.
[11] HASTIE, R.J., HENDER, T.C., CARRERAS, B.A., CHARLTON, L.A., HOLMES, J.A.,
Stability of Ideal and Resistive Internal Kink Modes in Toroidal Geometry, Rep. ORNLTM-10331, Oak Ridge National Lab., TN (1987).
[12] BURRELL, K.H., PRATER, R., EJIMA, S., et al., in Plasma Physics and Controlled Nuclear
Fusion Research 1984 (Proc. 10th Int. Conf. London, 1984), Vol. 1, IAEA, Vienna (1985) 131.
[13] LAZARUS, E.A., ATTENBERGER, S.E., BAYLOR, L.R., BOROWSKI, S.K.,
BROWN, R.L., et al., A Feasibility Study for the Spherical Torus Experiment, Rep.
ORNL-TM-9786, Oak Ridge National Lab., TN (1985).

DISCUSSION
ON PAPERS IAEA-CN-47/E-I-2-1 TO E-I-2-4

B.P. LEHNERT: Has the accuracy of MHD theory been examined in thepresent cases of high beta confinement, in which the ion Larmor radius does not
become entirely negligible as compared to the characteristic dimensions of the systemand of the relevant instability modes? In particular, kinetic effects may become significant for the fine structure — the high wave numbers — of such modes.
T. AMANO: None of the papers I have presented consider kinetic effects.
It is generally believed that kinetic effects have a stabilizing influence on such MHD
modes.
K. UO: The second stability region is frequently discussed at present and someof the designs for new fusion devices are based on concepts including this region.
Is there really experimental evidence for a second stability region?
T. AMANO: Yes, there seems to be some evidence for a second stability
regime in high beta torus experiments. Also, most of the systems I described havesufficiently high first stable critical beta ( > 10%).
S. ITOH: With regard to the MHD stability in low-aspect-ratio tokamaks,
described in paper E-I-2-4, the authors considered high n-modes as well as
n = 1, 2, 3 modes. Do you have any information on intermediate n-modes?
T. AMANO: Since this system has rather high shear, I assume that intermediate n-modes are stable.
K. ITOH: Could you comment on the compatibility of systems with high beta
configurations, such as a highly elongated system or a low-aspect-ratio system withthe separatrix at the surface?
T. AMANO: I think it is possible to install open divertors in these
configurations.

63IAEA-CN-47/E-I-3

SIMULATION STUDIES OF TOKAMAK DYNAMICS
M. TANAKA, T. HAYASHI, K. HARAFUJI,
Y. NAKAYAMA, T. SATO
Institute for Fusion Theory,
Hiroshima University,
Higashisenda-machi, Naka-ku,
Hiroshima, Japan
Abstract
SIMULATION STUDIES OF TOKAMAK DYNAMICS.
Three new simulation approaches for tokamak studies are proposed. First, a new particle simulation code applicable to kinetic phenomena of time- and spatial scales comparable to the magnetohydrodynamic (MHD) scales is developed and applied to tokamak current drive and sustainment.
Second, a full MHD simulation model that can treat a plasma-vacuum-wall system for arbitrary plasma
beta is developed and applied to a study of the bifurcation problem in a non-circular cross-section tokamak. Third, a full 3-D MHD simulation code is applied to the problem of suppression of major disruptions by a resonant helical field.

1.

INTRODUCTION

There are many problems to be overcome in demonstrating the usefulness andpracticability of computer simulations in fusion plasma research. Because of the subtleties of particle simulations, it has been believed that their application to real fusion
plasmas is rather limited. As one method of getting rid of this drawback of particle
simulations, we have developed a macroscale particle simulation code that can adequately deal with the MHD time- and spatial scales along with the kinetic aspects pertinent to the MHD scale phenomena. The code developed is successfully applied totokamak current drive and sustainment by electron beam injection.
One main concern in MHD simulation is its numerical accuracy. To increase
the accuracy it is necessary to develop a new code. On the other hand, it is also
important to know the limit of applicability of a conventional code. In this paper,attempts are made to attack two tokamak problems that seem to require some effort
to be solved by using the conventional Lax-Wendroff code. One is the bifurcation
problem of non-circular cross-section tokamaks in which some uncertainties must
always remain within the framework of the mathematical analysis. In this regard, twonumerical issues must be clarified. One is the treatment of the plasma-vacuum interface, and the other is that of an arbitrary beta plasma. The code developed has discovered an interesting new fact that would otherwise remain unpredicted. The second
problem is the suppression of the major disruption by application of a resonant helicalfield. Since this requires a three-dimensional, high resolution code, it was quite
65

66

TANAKA et al.

unclear whether the full 3-D MHD code based on the Lax-Wendroff method could
indeed solve the problem. The result has shown that a reasonably good solution caneven be obtained with the Lax-Wendroff code.
These three problems will be described in order.

2.

CURRENT DRIVE BY RELATIVISTIC ELECTRON BEAM INJECTION

Intense relativistic electron beam (REB) injection into tokamak plasmas has
recently been studied as an efficient means of quasi-stationary current generation andsustainment [1-4]. The current drive method by REB injection is considered to haveadvantages over the Ohmic current drive method in that the electron injection system
can be mechanically simple and efficient and the plasma particle and energy losses
arising from the pulse mode operation, hence the device wall damages, might be
reduced substantially.
A simulation study of the REB injection current drive has been performed byusing the macroscale particle simulation code [5]. This new class of particle code candeal with spatially large and slow time-scale (MHD scale) phenomena in which particle effects and the scalar potential electric field are non-negligible.
Two types of REB current drive simulation are studied here. In the first typeof simulation, the electron beam is continuously injected into a longitudinally magnetized plasma channel [6] to illustrate the physics process involved in the monoenergetic single REB injection. It is found that in an early stage of REB injection aplasma return current is excited by the electromagnetical beam induced electric field.
The primary electron beam is decelerated by the induced electric field as long as thebeam injection continues. The aforementioned plasma return current nearly cancels
the primary beam current before anomalous collisions participate. The beam
produced scalar potential electric field causes radially inward diffusion of the ions,which substantially reduces the radial expansion of the beam. When anomalous collisions take part in the process, further development of the plasma return current isprevented, and a net current, hence the aximuthal magnetic field, is created aroundthe beam path. The radial extension of the azimuthal field is confined to the vicinityof the beam path, because of the global cancellation of the net current. The net saturation current increases remarkably with beam relativisticity as Inet — (mc3/e)@y[Fig. 1], where |3 = vb/c, y = (1—/32)"1'2 and vb is the speed of the electron beam.
On the other hand, the net saturation current is independent of the injection current
intensity of the REB. When the beam relativistic factor is considerably large, a kinkinstability develops, and the net saturation current does not reach the previous scalinglevel (solid circle at y ~ 5 in Fig. 1). The instability is, however, easily suppressedby increasing the strength of the longitudinal magnetic field in such a manner thatthe q-value on the beam path does not fall below unity (open circle in Fig. 1). (Notethat the results shown above are applicable to the flat profile electron beam, where
the q-value on the beam path is comparable to unity.)

67

IAEA-CN-47/E-I-3

FIG. 1. Dependence of net saturation current, Inel, on relativistic factor <?> of electron beam injected
into longitudinally magnetized plasma channel. Ib: injection beam current; <7>: average relativistic
factor defined by <YVZ>/<VZ>.

(t)

-*- Y
FIG. 2. Beam electrons passing the same poloidal cross-section (x,y) at several different times;
t/rA = 0.5, 1.0, 2.0, 2.5, and 4.0 (from a tof) (poloidal magnetic surfaces present). TA: Alpin transit
time defined by using poloidal magnetic field at t = 4 TA.

68

TANAKA et al.

(a)

(b)

..

•

; / / / s~— ^s \ \ 1
1 I1 /"--WWI l l / I . - ' U I I
1 1 1 I 1 • • • I 1 1 t
1 l\\v

I III

i

1 /*r
III

- -

.\\\\
\ 1 t

1

'

• 1

t 1
| 1

1

1 •

• •

1

1 > • • • • ,

I

1

V V

1 1 1

•

•

•

• 1

i

I \ \-.'•-•/ I 1 l

FIG. 3. Poloidal magnetic field in the same cross-section as in Fig. 2 at times (a) t = 0 and(b) t = 4TA. Maximum field strengths: (a) for oice/ape = 0.17 and (b) for 0.28.

In the simulation of the second type, the REB is injected into a plasma where
poloidal magnetic surfaces are present (such as tokamak plasmas). A toroidal (longitudinal) magnetic field is applied in the z-direction. The initial q-value is q ~ 4.
In this simulation, the beam current is injected in a direction parallel to the initial
toroidal current. In the early stage of the injection, the beam electrons make manyround trips along the constant magnetic surface in the longitudinal (periodic) direction. This produces a current carrying thin annulus whose radius is comparable withthe minor radius (Fig. 2(a)). A modulation of the annulus with the poloidal mode
number m = 3 is observed (Fig. 2(b)). The modulation is due to the beam produced
poloidal magnetic field. The radius of the outer edge of the annulus in Fig. 2(b) is
more or less bounded by the magnetic surface where the beam injector is located, but
the inner edge lies distinctly inside.
With the passage of time, the beam electrons tend to gather at the inner edgeof Fig. 2(a) [see Fig. 2(c)]; eventually, the central region is to be filled with beamelectrons [Figs 2(e) and (f)]. The whole process occurs in a few Alfve"n transit times
when the magnetic pinch is taking place, since the electron population tends to be
more dense at the inner edge of the annulus. (We note that our MHD simulation
has shown that two current filaments with 200 kA placed 1 m apart merge
in a few Alfve'n transit times). The resultant poloidal magnetic field is shownin Figs 3 (a) and (b). The initial poloidal field strength is maximum near the wall(Fig. 3(a)). When the electron beam is injected, the poloidal magnetic field is intensified and the distribution corresponding to the time shown in Fig. 2(f) becomes hollow(Fig. 3(b)). The maximum of the poloidal field strength appears near the inner edgeof the beam electron annulus. There is a great possibility of a macroinstability suchas the tearing mode instability participating in this process. However, we have notconfirmed this possibility, so far.
From this study, it can be concluded that the electron beam injected from near
the wall can penetrate into the central region of the tokamak plasma and form thetoroidal current through the aforementioned process.

IAEA-CN-47/E-I-3

3.

69

HIGH BETA TOKAMAK WITH NON-CIRCULAR CROSS-SECTION

It is maintained that shaping the poloidal cross-section to assume a non-circularform can increase the beta limit of tokamak plasmas. Shaping coils, however, exert
additional external forces on confined plasmas, which may cause destruction of theplasma shape.
To study the stability of a non-circular cross-section tokamak, a compressible
non-linear MHD simulation model is developed which can treat a finite beta plasma
surrounded by a vacuum and a conducting wall [7].
The parameter range accessible to study depends strongly on the ability of theequilibrium code to obtain an initial equilibrium configuration of the non-circularcross-section. In our present study, the EQUCIR code was used to obtain initial
equilibria [8]. Since we found it rather difficult to obtain various bean shaped
equilibria in a systematic way, we started with the study of elliptically shaped
equilibria. Elliptically shaped equilibria were obtained in the range of K = 2.4 to 4.2(K is the ellipticity) and (3 = 1 to 20% (/3 is the toroidal beta) for a fixed toroidal
current.
It is found that the elliptical cross-section tokamak is always positionally unstable for the range surveyed and that the instability is of an ideal, current driven type.
We thus assume that the positional instability is feedback stabilized. We then proceed
to the bifurcation mode, i.e. the m = 2/n = 0 mode. It is found that there is a beta
limit for this bifurcation mode and that the beta limit is larger for smaller ellipticity.
The stability diagram in the 0-K plane is shown in Fig. 4. Along with the fact that
there is a beta limit for the instability, comparison of the induced flow pattern andthe pressure profile suggests that the instability is a pressure driven instability rather
than a current driven instability (see Fig. 5). Simulation runs for a wide range of theresistivity (magnetic Reynolds number in the range of 2000 to infinity) indicate thatthe instability is of an ideal type. Thus, the initial linear instability is concluded to
be an ideal, pressure driven instability.

20
UNSTABLE
X
xx

U 10
J
STABLEOL
2

J*°~"°""---..

.

3

<

I
I
I NO
EQUILIBRIUM!

ELLIPTICITY

FIG. 4. Stability diagram of m = 2/n = 0 (bifurcation) mode in @-K plane.

70

TANAKA et al.

T=22.6
P=9.5%
K=3.0
S=8000VMAX=0.008V*

FIG. 5. Induced flow vectors overlaying pressure contours at early phase of linear instability
development.

-1
C3
K UJ
UJ

KIN

z o -2
UJ
o 10
o
UJ o

K=3.0

0=9.6%
S=8000-3

f/G. 6. Temporal development of induced flow energy and poloidal flux contours at three typical bifurcation times: initial, last stage of ideal pressure driven instability, and development stage of non-linear
reconnection.

Another interesting finding is that when the ideal (pressure driven) instability
develops, the original elliptical current distribution deforms vertically, and a currentsheet-like distribution is formed. As a consequence of the current sheet development,
reconnection is non-linearly driven near the magnetic axis in the presence ofresistivity; this behaviour is shown in Fig. 6. In this figure, the solid line indicatesthe development of the induced flow energy; also the poloidal flux contours at threetimes of the evolution are shown. The first, gradual growth (0 < t < 45T A , TA

IAEA-CN-47/E-I-3

71

being the Alfven transit time) corresponds to the ideal pressure driven instability
during which only elongation of the elliptical shape occurs. As is seen in this figure,a burst-like increase occurs at about t = 45 rA. This burst-like phenomenon isclosely associated with the onset of non-linear reconnection. It is, therefore, concluded that bifurcation of the elliptical cross-section takes place in two steps: ideal
instability and subsequent non-linear reconnection.
Since a few bean shaped equilibria were obtained additionally by the EQUCIR
code, the positional instability and the bifurcation were also studied. It is likely thatthe growth rate of the positional instability is greatly reduced compared with the elliptical case although quantitative comparison is, in a strict sense, difficult. For theequilibria we have studied, no bifurcation instability was found, indicating that thebean shaped tokamak would be far more stable than the elliptical tokamak.

4.

SUPPRESSION OF TOKAMAK DISRUPTION BY RESONANT HELICAL
FIELD

It is widely believed that a tokamak major disruption can be prevented if we cansuppress the growth of the m = 2/n = 1 magnetic island [9, 10]. One way to suppress
the island growth is to apply an external helical field which resonates with the 2/1
magnetic surface [11]. However, complete control of the major disruption has not
yet been attained experimentally by this method. Making use of a 3-D full MHD
simulation code, we thus wish to investigate the stabilization effect of the resonant
helical field in the island growth, thus verifying the viability of this method.
The initial condition of the tokamak is such that q0 = 1.3, ^ = 4.0,
/30 = 0.5%, and A = 20 (aspect ratio). When the resonant helical field is absent,

FIG. 7. Poincare plots of magnetic field lines in poloidal plane observed at t = 3000 TA: (a) resonant
helical field (RHF) with AB/Bp = 0.3% and 180° out of phase from spontaneous 2/1 mode applied at
t = 2000 TA; (b) no resonant helical field applied (spontaneous development of 2/1 mode).

72

TANAKA et al.

10

tt|i|i|i|i|iii|i|imii<ii|i|imiii|iii|i|i|i|i|iii|Hi|>|i|i|i|ira

2000

3000

4000

5000

TIME (NORMALIZED)FIG. 8. Temporal development of total kinetic energy for (a) no RHF, (b) ABp/Bp = 0.3% and out ofphase, (c) ABp/Bp = 0.57% and out of phase, (d) AB/Bp = 1.1% and out of phase, and(e) ABp/Bp = 0.57% and in phase.

(a)FIG. 9. Temporal development of several dominant Fourier modes: (a) case (b) of Fig. 7, and(b) case (a) of Fig. 7.

an ergodic region of the magnetic surface develops, due, primarily, to the overlapping of the islands of the 2/1 and 3/2 (and 3/1) modes (see Fig. 7(b)). When,however, a 2/1 resonant field is applied at the instant when the island width of the
2/1 mode grows up to about one-tenth of the minor radius a, the 2/1 mode is almost
completely suppressed (see Fig. 7(a)). In this example, the width of the island createdby the applied resonant field was about a/10 (ABp/Bp = 0.3%), and the phase was
180° out of phase with the 2/1 mode ('out'). [Here, we assumed that the applied helical field was immersed instantaneously in the plasma.]

IAEA-CN-47/E-I-3

73

The temporal development of the total kinetic energy for different magnitudes
and phases of the applied resonant field is shown in Fig. 8(a). It is observed that when
the applied phase is in coincidence with that of the spontaneous one ('in'), theresonant field always accelerates the development of the instabilities. Even if thephase is 'out', it results in development of the 2/1 mode with the same phase as theresonant field when the resonant field is very strong.
The development of several dominant Fourier modes of the perturbed magnetic
field is shown for the case without applied resonant field in Fig. 9(a), and for the casewith applied resonant field in Fig. 9(b). The growth of the 2/1 mode is strongly suppressed by the resonant field. The growth rate of the 3/2 mode is also somewhat
reduced because no steep current gradient, necessary for the excitation of the 3/2
mode, is generated, owing to the suppression of the 2/1 mode. Furthermore, theonset of the rapid growth of the 5/3 mode, which would occur as soon as the 2/1 and3/2 islands overlap, disappears.
It is concluded that the resonant helical field can really act to stabilize the 2/1
mode. However, in the actual experiment, it might require special care to control themagnitude, phase and timing of the external helical field by which the 2/1 mode canbe suppressed.

5.

SUMMARY

Several investigations of the future prospects of numerical simulations, bothparticle and MHD, in fusion plasma research were attempted successfully. The firstexample, a macroscale particle simulation, has proven the usefulness of particle
simulations in the extended area of fusion plasma research. The other two examples
also indicate that the full MHD code can, to a greater extent, contribute to successthan had been thought probable before. In particular, the last example suggests thatby developing a new, slightly higher resolution MHD code, one can indeed solve
even geometrically complicated stellarator problems.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]

BAILEY, V.L., Steady State Current Drive in Tokamaks, Rep. DOE/ET-0077 US Departmentof Energy, Washington, D.C. (1979).
WONG, K.L., ONO, M., WURDEN, G.A., Rev. Sci. Instrum. S3 (1982) 409.
MOHRI, A., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1978 (Proc. 7th
Int. Conf. Innsbruck, 1978), Vol. 3, IAEA, Vienna (1979) 311.
TANAKA, H., et al., Jpn. J. Appl. Phys. 24 (1985) 1644.
TANAKA, M., to appear in J. Comput. Phys. (1986).
TANAKA, M., SATO, T., Phys. Fluids 29 (1986) 3823.
SATO, T., NAKAYAMA, Y., MATSUOKA, K., in Plasma Physics (Proc. Int. School Varenna,
Aug. 1985).

74

TANAKA et al.

[8] SHINYA, K., et al., An MHD Equilibrium Code 'EQUCIR Version 2' Applicable to Up-down
Asymmetric Toroidal Plasma, Rep. JAERI-M-9278, Japan Atomic Energy Research Institute(1981).
[9] CARRERAS, B., et al., Phys. Fluids 23 (1980) 1811.
[10] KLEVA, R.G., DRAKE, J.F., BONDESON, A., Phys. Fluids 28 (1985) 2478.
[11] KARGER, F., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1974 (Proc.
5th Int. Conf. Tokyo, 1974), Vol. 1, IAEA, Vienna (1975) 207.

IAEA-CN-47/E-I-4-1

STABILITY CONDITIONSFOR HELICAL MHD PERTURBATIONSAND DISRUPTIVE INSTABILITYIN TOKAMAKS
L.E. ZAKHAROV
I.V. Kurchatov Institute of Atomic Energy,
Moscow,
Union of Soviet Socialist Republics

Abstract
STABILITY CONDITIONS FOR HELICAL MHD PERTURBATIONS AND DISRUPTIVEINSTABILITY IN TOKAMAKS.
The effects of the current profile on the stability of kink and tearing modes are studied
on the basis of the successive current layer principle. The dependence of the stability against
disruptions on the global current profile parameters is analysed. Effects of local current
redistribution near the resonant surface on the tearing modes are demonstrated. A complete
model for quasi-linear saturation of the tearing modes is proposed. The stability of the edge
localized kink modes is analysed, and the optimum current profiles are given. It is proved thatq(a) = 2 is the asymptotic theoretical limit, ensuring the stability against all kink and tearing
modes. A characteristic exact solution for the m = 1 mode equilibrium in a cylinder is given.
The main reasons for the appearance of disruptions in a tokamak are mentioned.

1.

PRINCIPLE OF SUCCESSIVE CURRENT LAYERS (SCL PRINCIPLE)
AND STABILITY AGAINST DISRUPTIONS

The linear stability analysis of current driven MHD modes in the cylindricaltokamak model reduces to the study of a well-known equation and of thefunctional W:

dp dp

p

ju-n/m

'

J \
0

p

fi — n/m(1)where p is the minor radius, M(P) = 1/qO) is the rotational transform,
and m and n are the wavenumbers. The current density is given in dimensionlessunits, i.e. j -»• (47r/c)jR/Bs, so that q(0) = 1 corresponds to j(0) = 2.
75

ZAKHAROV

760

p P+6p0

FIG. 1. Successive current layers.

Equation (1) is actually a perturbation of the equilibrium equation(2)

where j m / n = 2nBg/(mR) = const, and * is the helical flux. The existence of asmooth solution of Eq. (1), A' = 0, means that a linear near-equilibrium with
continuous current density exists [1,2]. When A' > 0, the plasma column must
pass into this near-equilibrium state, with the formation of magnetic islands.
The fact that Eqs (1) and (2) are independent of the dissipative plasma properties
explains the good agreement between theoretical conclusions and experiment.
Equation (1) alone does not, however, say anything about the consequencesof the development of instabilities. Calculation of the instability growth rate
does not help either, in this case. The SCL principle, which will be set forth
below, clarifies the relationship between stability properties and the consequencesof the instabilities.
According to this principle, any current profile is considered to be the sumof nested cylindrical current layers disposed around the initial axis (Fig. 1). The
effect of each layering event on stability is analysed. Without conducting wall,
this effect is simply determined by the sign of the expression
6V = jnq - (m - 1)
(nq+-m)(nq_-m)

(3)

2mq
P

where j is the current density in the layer added, p and dp are the radius and thethickness of a layer, and q+ and q _ are the values of q(p) on the outside andinside of the layer. When 5 V > 0, stability deteriorates with an additional layer,
whereas when 5V < 0, it improves [2],Equation (3) is valid for the most general case, including discontinuities in
the current density, presence of a resonant surface inside the layer, etc. In thepresence of a casing, only the term ( m - 1) of Eq. (3) should be replaced by an
explicit, simple function nq m (p) [2].

77

IAEA-CN-47/E-I-4-1

nq

m-1

fd)
FIG. 2. Characteristic unstable current profiles: (a), (b): without supporting channel, subject
to major disruption: (c), (d), (e): with supporting channel, p° < a, subject to minor disruption.

In spite of its simplicity, the SCL approach gives a practically exhaustive
picture of the effect of current profile on the stability of kink and tearing modes
and yields criteria on the disruptive instability.
1.1. Kink instabilities, nq(a) < m
Only when the layer added includes the resonant surface, nq(p m ) = m, does
the sign of the dominator in Eq. (3) become negative. In all other cases, the signof 6 V is determined only by the signs of the numerator and j .
We easily see that current distributions with j(p) > 0 and nq(p) > m - 1(Fig. 2(a), (b)) are unstable against the m/n kink mode. This is obvious for amonotonic current distribution. The initial current channel, p < a, is certainly
unstable (uniform current, m - 1 < nq(0) < m). Further layer adding only makes
the stability worse because nq(p) > m — 1 yields 5V > 0 [ 1 ]. For a skin current
distribution including an additional internal resonant surface nq(p m ) = m(Fig. 2(b)), the current channel with radius p m + e is unstable (see Eq. (1)). An
outer current layer, p> pm+ e,\s destabilizing, which results in instability of thewhole profile.
For j(p) > 0, only a distribution having a region with nq(p) < m — 1 can bestable. For a monotonic current density, (Fig. 2(c)), the initial current channel
p < a is stable because nq(0) < m - 1. Additional current layers up to the point^m - 1' n 1 ^ m - 1 ) = m ~ 1 improve the stability. The next layers at p > p m _ jmake the stability worse, and it is possible that a neutral stability will be reached

78

ZAKHAROV

for a current cylinder with P - P°m- We shall call such a cylinder a supporting
current channel for the m/n kink mode. The exact value of p ^ is determinedby the relationshipmY;+ 1 —Lo= 0(4)

rnn — nwhere Y ; (p) is the internal solution to Eq. (1), Y^O) = 0.
The physical meaning of the supporting current channel consists in the factthat all bulk forces j X B inside it are exactly balanced for a small helicaldeformation. There are no reasons for this balance to be violated seriously, evenat a non-linear stage. Therefore, in the development of the m/n kink instability,
the supporting current channel will not be modified significantly. In the absenceof such a channel, the kink instability should completely remix the whole
distribution because it is unable to reach a new stable equilibrium [4]. The
absence of a supporting current channel means that the current profile is unstableagainst a major disruption.
For the most important mode, the 2/1 mode, the supporting channel
exists only if q(0) < 1. This has made it possible to draw the conclusion [ 1 ],based on the stable regimes with q(a) ^ 2 in T-10 and T-l 1, that q(0) < 1 is notforbidden in tokamaks and that internal relaxation oscillations are not related
to m = 1 reconnection. For regimes with q(a) < 2, any m = 1 reconnection should
lead to a major disruption.
For a skin profile, the supporting current channel may be hollow. In theprocess of SCL adding, the internal unstable cylinder (Fig. 2(d)), wherenq(p) > m— 1, is surrounded by a stabilizing layer with nq(p) < m— 1, and, thus,
neutral stability is first reached for p0 = p ^ j . The supporting current channel
is situated at p ^ j < p < p ^ 2 , where p ^ 2 is the radius where neutral stability isreached for the second time. The outer shell, P > P^ 2' destabilizes the wholecurrent profile. Owing to the presence of a supporting current channel, such a
linearly unstable current profile is, nevertheless, stable against a major disruption.
A similar picture with a hollow supporting channel is also possible in the presenceof a casing [2].
The addition of negative current density, j < 0, causes the stability todeteriorate if nq(p) < m - 1 and improves it if nq(p) > m - 1 . In this case,stabilization is possible even of a current profile with nq(p) > m - 1 everywhere [5](Fig. 2(e)). In particular, this situation occurs in the Kadomtsev reconnection
model, where q(p)> 1 is established after remixing. The 2/1 mode, nevertheless,remains stable [6], because a negative 5-function-type current sheet is generated
at p -\/2pl».
In this case, the supporting current channel for the 2/1 kink mode
becomes hollow but is not destroyed. Later, however, because of the decay ofthis negative current, the supporting channel will disappear. If q(a) ^ 2, this

IAEA-CN-47/E-I-4-1

79\

FIG. 3. (a) Stabilizing current layer embracing resonance surface; (bj unstable current
profile with internal supporting channel, p^, and with hollow supporting channel,
Pm < p < p ^ t < a, stabilizing the tearing mode.

should lead to the major disruption due to the kink mode 2/1. Note that, in
accordance with the SCL principle, the abovementioned negative current reduces
the stability of the 3/1 mode because, for m = 3 and n = 1, nq{y/2p^,^) < m - 1 .
The generation of a negative current at the plasma edge (Fig. 2(e)), where
nq > m - 1 , may create a hollow supporting channel which stabilizes the unstableinterior part [5]. In particular, such a current is always generated during thenon-linear stage of the kink instability [4], leading to its stabilization, a factwhich was observed in numerical simulations [7]. However, such a stabilization
cannot be stationary in a real plasma. Therefore, an unstable kink mode should
either lead to a major or a minor disruption.
The examples given here show how we can analyse the stability of any
conceivable current profile by using the SCL principle. This principle connects
the linear stability properties with the stability criteria referring to the disruptions.
A kink mode (as well as a tearing mode) is linearly stable only when a supporting
current channel exists and has an external radius exceeding the plasma radius,p£j > a. In the presence of a supporting channel with p ^ < a, the plasma islinearly unstable but the instability only leads to a minor disruption. The interiorof the supporting channel cannot be modified significantly by such a kinkinstability. In the absence of a supporting channel, the kink instability leads toa major disruption.
1.2. Tearing modes, nq(a) > m, Pm/n < aAdding a positive current layer of width 2e, which includes the resonantsurface p m (Fig. 3(a)), transforms a current distribution which is unstable against
the kink mode (a = P m — e) into a distribution stable against the tearing mode(a = p m + e). In this case, 6V ~ j/e is negative, because the denominator in
Eq. (3) is negative (nq+ > m, nq_< m), although nq > m - 1. Adding further
current layers only worsens the stability and may, finally, produce an unstablecurrent distribution (Fig. 3(b)).

80

ZAKHAROV

Thus, when the resonance surface is inside the current channel, p m < a,there is always an additional hollow supporting channel, pm < p < p ^ t , where
the radius p ^ t limits a neutrally stable cylinder (the subscript t denotes thetearing mode supporting channel). The radius p ^ t is determined by the same
rule — see Eq. (4) — as p ^ , except that Yj(p) intersects the resonance surface
with a continuous derivative.
The stability of tearing modes is thus ensured by a current layer adjacentto the resonance surface. Like the kink mode, the tearing mode is stable only
for p£j t > a. For p ^ t < a, the tearing mode may be transformed into a kinkmode or can pass into a new, quasi-linear equilibrium with magnetic islands [8],in which case it is not dangerous.

2.

LOCAL INFLUENCES AFFECTING THE STABILITY OF TEARINGMODES

The stability of tearing modes is sensitive to a number of local effects whichwere mentioned explicitly for the first time in Ref. [9].
One important example [2], given in Fig. 3(a), shows that a jump in thecurrent density beyond the resonance surface stabilizes the tearing mode.
Surprisingly, a similar case has been realized in H-regimes in tokamaks,
stabilizing the 2/1 mode. In the general case, a negative jump in the current
density just before the resonance surface destabilizes the tearing mode but, justbeyond the resonance surface, stabilizes it (Figs 4(a), (b)). In accordance with
the SCL principle, the addition of a positive current layer just at the resonantsurface (Fig. 4(c)) stabilizes the tearing mode.
These examples are useful for understanding the interaction of different
tearing modes by current redistribution during island formation. For example,
the development of the 2/1 tearing mode, at the beginning, produces a jumplike current distribution beyond the resonant surface p 3 / 2 and stabilizes the
3/2 mode. During the growth of the 2/1 island, this jump penetrates inside theradius p3,2» immediately destabilizing the 3/2 tearing mode. Some authors
consider this situation to be the start of a disruption. A non-linear effect of theinternal 3/2 tearing mode on the 2/1 mode is always destabilizing because the
3/2 mode produces a jump before the surface p2ji •The example presented in Fig. 4(c) fully clarifies the nature of tearing modestabilization by local current drive. An additional positive current layer willstabilize the tearing mode if it contains the resonant surface and will have adestabilizing effect if the resonant surface is outside this layer (Fig. 4(c)). The
addition of a negative current layer (Fig. 4(d)) has exactly the opposite effect.
In all the examples described, the effect of the current jump is proportional
to Aj/e, where e is the distance from the jump to the resonant surface. A weakereffect with 5W ~ Aj'ln e ~ Aj' is the formation of a break in the current profile

81

IAEA-CN-47/E-I-4-1

(a)

(bj

(O

(d)

FIG. 4. (a) Unstable and (b) stable current density drops near resonance surface; positively
directed (cj and negatively directed (d) stabilizing (solid line) and destabilizing (dashed line)
locally driven current.

[9, 2]. It is clear that a break of the type presented in Fig. 5(a) localized close to
the resonant surface stabilizes the tearing mode. This effect, which was pointed
out, first, only theoretically in Ref. [2], was later realized on T-10 [10] by local
ECR heating which produced a break in the Tg(p) and, consequently, the j(p)profiles in the heating zone. Although the experiment was initiated for otherreasons, the suppression of the 2/1 mode has completely confirmed the analysis
in Ref. [2]. Note that a break of the opposite sign, Aj' > 0, has a destabilizing
effect (Fig. 5(b)).
In the next section, we shall show that finite amplitude effects, which are
determined by the relationship between e and the island width w and by thepoloidal position of the current drive or heating zone with respect to the island,
do not significantly affect the active stabilization of the tearing modes. Therefore, linear stability considerations, indeed, yield complete information on thecontrol of tearing modes by local current redistribution.

ZAKHAROV

82

(a)

(b)

FIG. 5. (a) Stabilizing and (b) destabilizing breaks on the current profile.

3.

QUASI-LINEAR MODEL FOR SATURATION OF TEARING MODES

To describe the equilibrium (see Eq. (2)) of a configuration with magnetic
islands we have analysed the following model of current distribution near theislands:(5)

where x = 2 ( * - * S ) / ( J O - J m / n ). * s is the value o f * at the separatrix; d,, dr
are the widths of the left and right hand parts of the magnetic islands
d t + dr = 2w; J o , Jg r , J'g r j are constants determining the current
distribution on the left ($.) and on the right (r) of the separatrix and inside
the island (i). Unlike other models (e.g. [8]), the current distribution given by
Eq. (5) does not contradict the stationary transport equations and is the simplest
model for quasi-linear analysis. We have proved that the linear Eq. (1) can beused, with an accuracy to within ~ 1 % , up to the separatrix, permitting a maximum
reduced number of terms in Eq. (5) for the remaining 2-D zone. Hence, the model
given by Eq. (5) is practically exhaustive.
The parameters J o , Jg r , J j r are determined from the matching conditions
for total current, current density and its derivatives. The quantity J-' is a freeparameter also affecting the equilibrium conditions.

IAEA-CN-47/E-I-4-1

83

The equilibrium conditions can be written in. the following simplified form:
A t / e\\2.72J" w + 0.64(J'p + J'')w + 0.28(j; - J'p)
J

J

fi+Jr

o

Jm/n
Y + Y

i

e

(6)

Here, A^,(0) is the standard value of A'(0) calculated for a modified current
profile (the subscript w indicates the current modification). In this case, thelinear Eq. (1) is solved outside the separatrix, introducing the unperturbed value
X = Xo i n Eq. (5). Within the two-dimensional zone, Eq. (1) is solved with azero right hand side. In Eq. (6), Yj and Y^ on the right hand side are calculated
at the radii p m — d j and pm + d r , respectively. Note that A^(0) in Eq. (6) must
be distinguished from the quantity A'(w), which is often incorrectly used inquasi-linear analysis. The first term in Eq. (6) describes the stabilizing effectdue to negative second derivatives, J'n f ., while the second term is usuallydestabilizing.
From Eq. (6), it follows:(i) Local heating of the internal and external parts of the islands has a favourable
stabilizing effect. The position of the heating zone in the poloidal direction
determines only the efficiency.
(ii) A positive current drive inside the islands has a stabilizing effect, whereas
the same current drive outside the islands is destabilizing. It must be notedthat a positive current drive over both regions has a stabilizing effect, whichis, however, not so effective as in the case of a positive current inside theislands alone.
(iii) If w < e, the stabilization effects remain, because of the direct influence ofthe current redistribution on A^(0), as was analysed in the previous section.
These results, unlike the analysis in Ref. [6], show that synchronization ofan external action with island rotation is not very important and that the island
width, w, has virtually no influence on the stabilization effect.

4.

EQUILIBRIUM CONDITIONS FOR THE m = 1 PERTURBATION

For the main approximation, Bs > BQ , the equilibrium conditions of ahelical, m = 1 perturbation in a cylinder are elucidated on the basis of an exact
solution of Eq. (2). In regions I and III (Fig. 6), the cross-sections of themagnetic surfaces are circular. The total current in region I corresponds to84

ZAKHAROVfb)
FIG. 6. Exact solution of equilibrium equation (2) for m = 1 mode: (a) current distribution;fb) magnetic field geometry.

q = 1 at its boundary. Inside the island, the current density j;(U) = jj/j = const,
and * = const.
Note that, as a result of any inductive process splitting the magnetic surfaces,
the current density within the island cannot become greater than j 1/fj (in general,Jm/n)- A r t i f i c i a ' current drive, with jj >J~i/i> is also probably impossible, because
reconnection conditions would come about within the island. For jj(^) <jj/i>equilibrium is impossible because the lesser total current in the island cannot
compensate for the destabilizing force,]* X B s acting on the internal helicalcurrent channel, I. Thus, it will be clear that the Kadomtsev reconnection process
cannot be stopped at a non-linear stage. Therefore, we should invoke some
additional considerations to explain the existence, in a tokamak, of a zone withNote that for the m = 1 mode, in contrast to the m > 2 modes, stabilizationis possible only by current drive inside the island, provided jjOI') = j j . j .
Of course, this becomes more and more difficult with decreasing w.

5.

EDGE KINK MODES AND OPTIMUM CURRENT PROFILES

Optimum current profiles, which are stable against all the current driven
modes and have a minimum q(a) value [11], should satisfy the following requirements: a) q ( 0 ) > 1, b) q(a)> 2, c) the 2/1 mode should be stabilized by local
effects, such as a break in the current profile at p 2 n (Fig- 5 (a)), or by local
current drive at p 2 /i (Fig- 4(c)). If necessary, the 3/2 mode should bestabilized in a similar manner.
In this case, only the problem of higher mode stabilization remains, because
an unfavourable break in the j(p) profile at the plasma boundary p = a canIAEA-CN-47/E-I-4-1

-1

0

85

1

FIG. 7. Function FfzJ for the criterion (7j.

destabilize the edge localized modes with p m ,n «« a. For such edge modes, thestability criterion can be written in the form:aj < — mFl
a
q(a)
\

S

I
/

(7)

where }'a and S are the current density gradient and the shear at the current
channel boundary, and the function F(z) is shown in Fig. 7; z > 0 corresponds
to the edge kink modes and z < 0 to the edge tearing modes. When z -> 0, thefunction F(z) « l/|lnz| -+ 0.
From Eq. (7), it follows that there can be a stable current channel boundarywith a finite current density gradient: j'(a—0) = j ^ ¥= 0, j'(a + 0) = 0. In fact,
for an irrational q(a) value, the limitation on j ^ remains finite because the factor
m compensates a comparatively slow decrease in F(z) for z -* 0 when the resonancesurface approaches the plasma boundary. In this case, the limitations on j ^ are
imposed by the low m modes. The limiting j ^ , according to criterion (7), is shownin Fig. 8 as a function of q(a) for a tokamak of circular cross-section (S = 2). If
the 2/1 tearing mode is stabilized by the break of the current profile at p 2 /i > thediagram given in Fig. 8 permits the construction of completely stable current
profiles with j ^ ¥= 0 and q(a) = 2 + e, where e is an arbitrary, small quantity.
Therefore, the theoretical limit for q(a) is actually 2, instead of the previous
limit q(a) = 2.6 [11]. The point which corresponds to the current profile ofRef. [ 11 ] is presented in Fig. 8.
The diagram of Fig. 8 gives a natural limitation on the current density
gradient j ^ , due to the edge kink modes. When the q(a) value changes, thelimiting current density corresponds to the right hand edge of the teeth, jumping
from one to another with the variation in q(a). From the diagram presented inFig. 8 we see that there are some natural characteristic values of q(a) corresponding
to the gaps between the teeth, the most representative being located nearq(a) = 2.5 and q(a) = 3.
Let us note that the limitation on j ^ due to the edge MHD modes is proportional to the shear, S. In tokamaks with a separatrix, because of the increase

ZAKHAROV

86

3

1.

FIG. 8. Limiting f at plasma boundary. Dashed line illustrates limitation due to 3/1 mode,
as obtained from criterion (7), which is not well applicable to 3/1 mode. 2/1 tearing mode isassumed to be stabilized by a local effect in the same way as all m/n = 2/1 modes.

in shear at the edge of the current channel, S > 2, the limiting values of j ^ aregreater, making it more likely to stabilize the 2/1 mode by a fast drop in j(p)outside the resonance surface p 2 /i • It is natural to assume that the difference in S
permits an H regime only in tokamaks with divertors and not in tokamaks withlimiters.

6.

APPEARANCE OF DISRUPTIONS IN TOKAMAKSFrom our point of view [4] , the disruptive instability is a manifestationof a kink instability in a real plasma. This statement is based on the fact thatkink modes (nq(a) < m), being unstable, do not display saturation in a realplasma and cannot lead to a new equilibrium (with j(p) > 0). In the presenceof a supporting channel, p^/ n < a (see Eq. (4)) corresponding to the existenceof a region with nq(p) < m— 1, only the plasma outside p ^ , n will berearranged, which takes the form of a minor disruption. If there is no supporting
channel, m - 1 < nq(p) < m, the whole current profile will be rearranged. This
will appear as a major disruption.
Let us consider some examples of different disruption possibilities. During
the current rise, four characteristic cases can be identified when q(a) crosses theresonance value [12]:87

IAEA-CN-47/E-I-4-1

m-1

nq
\
m-1

FIG. 9. Typical regimes of passage through resonance q(a) values at current rise phase:(a) with major disruption (b) with minor disruption, both without precursor; (c) with minor
disruption preceded by Mirnov oscillations; (dj without MHD perturbations.

(a)nq(O) > m - 1, and an abrupt current density decrease beyond the resonancesurface p m . (Fig. 9(a)). In this case, the m/n tearing mode is stable. The
kink mode leading to a major disruption is excited when the surface p m ; ncrosses the plasma boundary.
(b) nq(0) < m - 1, and the same abrupt decrease in current density (Fig. 9(b)).
The kink mode only leads to a minor disruption, without precursor.
(c) nq(0) < m - 1, and smooth current density outside the surface pm ,n(Fig. 9(c)). The tearing mode is in a saturated state (Mirnov oscillations).
A slightly pronounced minor disruption arises from the Mirnov oscillations
when the surface p m . crosses the plasma boundary.
(d) nq(0) < m - 1, and the difference between nq(O) and nq(a) is sufficient tostabilize the m/n tearing mode. The resonance value of q(a) is crossed without MHD perturbations.
In the stationary state, the 2/1 tearing mode plays the main role. Whenq(a) ^ 2, any reconnection due to the 1/1 mode will always lead to a major
disruption. For q(a) > 2 and q(0) > 1, the 2/1 tearing mode will always be
unstable if no local mechanisms of stabilization are at work. There is a 2/1
magnetic island, which plays a basic role in current profile self-consistency [131.
In this case, disruption becomes possible when the separatrix of the island goesbeyond the boundary of the current channel [2]. In this situation, the width of
the islands can be estimated from quasi-linear theory (Eq. (6)). The last casecorresponds to the transformation of the tearing mode into a kink mode.

88

7.

ZAKHAROVCONCLUSIONSThe SCL principle is shown to be the most suitable method of analysing thestability of current driven modes. While the formal linear stability theory of the
1970s operated with badly defined concepts (such as 'peaked' or 'rounded'current profiles, etc.), the present approach elucidates the physics of the stability
properties for any current profile. In addition to a new representation of linear
stability it provides full information on current profile stability against disruption,even for the most exotic current distributions.
Let us also note that synchronization of the external impact with the island
rotation frequency is not essential for tearing mode suppression by local current
redistribution.

REFERENCES[1] ZAKHAROV, L.E., Fiz. Plazmy 7(1981) 1295.
[2] BOGOMOLOV, L.M., ZAKHAROV, L.E., BLEKHER, P.M., Nucl. Fusion 27 (1987) 241.
[3] BLEKHER, P.M., Fiz. Plazmy 10 (1984) 493.
[4] ZAKHAROV, L.E., Fiz. Plazmy 7 (1981) 18.
[5] DNESTROVSKIJ, Yu.N., KOSTOMAROV, D.P., POPOV, A.M., Zh. Tekh. Fiz. 42(1972) 182S.
[6] IGNAT, D.W., RUTHERFORD, P.H., HSUAN, H., in Course and Workshop on
Applications of RF Waves to Tokamak Plasmas (BEABEI, S., GASPARINO, U.,
SANDONI, E., Eds), Vol. 2, Monotypia Franchi (1985) 525.
[7] ROSENBLUTH, M.N., MONTICELLO, D.A., STRAUSS, H.R., WHITE, R.B., Phys.
Fluids 19(1976) 1987.
[8] WHITE, R.B., MONTICELLO, D.A., ROSENBLUTH, M.N., WADDELL, B.V., Phys.
Fluids 20 (1977) 800.
[9] BLEKHER, P.M., ZUEVA, N.M., YURCHENKO, E.I., Preprint, Keldysh Institute ofApplied Mathematics, Moscow (1982) 121.
[10] ALIKAEV, V.V., STEPANENKO, M.M., Pis'ma Zh. Ehksp. Teor. Fiz. 40 (1984) 327.
[11] GLASSER, A.H., FURTH, H.P., RUTHERFORD, P.H., Phys. Rev. Lett. 38(1977) 234.
[12] ZAKHAROV, L.E., PUTVINSKIJ, S.V., in Itogi Nauki i Tekhniki, Ser. Fizika
Plazmy (SHAFRANOV, V.D., Ed.), Vol. 7, VINITI, Moscow (1985)4.
[13] OHYABU, N., LEE, J.K., DE GRASSIE, J.S., Nucl. Fusion 26 (1986) 593.
[14] KURITA, G., AZUMI, M., TAKIZUKA, T., TUDA, T., TSUNEMATSU, T., TANAKA, Y.,
TAKEDA, T., Nucl. Fusion 26 (1986) 449.

IAEA-CN-47/E-I-4-2

BIFURCATION OF
PLASMA EQUILIBRIUM IN TOKAMAKSYu.N. DNESTROVSKU
I.V. Kurchatov Institute of Atomic Energy
D.P. KOSTOMAROV, A.V. PEDORENKO, A.M. POPOV
M.V. Lomonosov Moscow State University
Moscow, Union of Soviet Socialist Republics

Abstract
BIFURCATION OF PLASMA EQUILIBRIUM IN TOKAMAKS.
Helical equilibria for the m/n=2 mode in a plasma cylinder are studied by numericaland analytical methods. These helical equilibria are shown to be present within the subrange
Icr < Qo < 1 of the kink instability range q cr < qo. Transition across the stability boundary,qcr, corresponds to a bifurcation from axially symmetric equilibria towards helical equilibria
with elliptical cross-section.

1.

INTRODUCTION

Careful analysis of plasma behaviour in the presence of a linearly unstablehelical mode is necessary for understanding the mechanism of dischargetransition into a regime with low qa = q(a 0 ). The m/n = 2 mode is the mostdangerous one. The development of unstable motions for this mode, as has beenshown in Refs [ 1, 2] by a numerical solution to the set of MHD equations, canresult in stabilized states of. the plasma column in a non-linear stage. This behaviourof the solutions demands a study of the helical equilibria.
Existence and properties of some helical equilibria have been consideredin [3—6]. In particular, in [4], the bifurcation of a cylindrical equilibrium to ahelical one is shown to be subcritical, and there are no adjacent helical equilibria
for a homogeneous current. The bifurcation can be supercritical for an inhomogeneous current [3]. This implies that the evolution of unstable helical modes
will result in a transition of the plasma column from a cylindrical equilibrium
into an adjacent helical equilibrium without discharge destruction.
Helical equilibria in a plasma cylinder for the m/n=2 mode are studied bynumerical and analytical methods in this paper. Conditions for the bifurcationof axially symmetric equilibria to helical ones, the nature of bifurcation and its
dependence on plasma parameters and current profile are found. The case ofhomogeneous current, is, from the viewpoint of bifurcation, found to be89

90

DNESTROVSKU et al.

degenerate. The bifurcation is always supercritical for a monotonically decaying
inhomogeneous current. The bifurcations are studied by two methods, thefirst of which is based on the direct numerical solution of a 2-D helicalequilibrium problem [6]. The second technique reduces the problem of helicalequilibrium within an unlimited region to a one-dimensional integrodifferential
equation [4], which is also solved numerically. It is possible to expand thebifurcation equation in terms of ellipticity in the neighbourhood of thebifurcation points and to obtain some analytical expressions for the bifurcation
curve. The results for the two- and one-dimensional problems are in good agreement with each other.

2.

2-D PROBLEM OF HELICAL EQUILIBRIUM

Let us consider a plasma cylinder, with a length of L = 27rR, a radius ofa0 and identified ends. All physical parameters characterizing the configuration depend only on two variables, r and 6 = <p ~ aZ, in the presence of helicalsymmetry. Here, r, <£, Z are cylindrical co-ordinates with the Z-axis along thecylinder axis. The parameter a = 27rn/mL = n/mR determines the pitch,n and m are the numbers of starts along Z and v?, respectively. The equation
for a helical equilibrium has a form [5]:(1)cwhere i//* is the helical potential, JB = c (47r2aB z ),and

The boundary condition for Eq. (1) is set on an ideally conducting casingof circular cross-section with radius r w :=0(2)

The longitudinal current density, }($*), is given in 2-D calculations by thefollowing relationship:r|f J]

IAEA-CN-47/E-I-4-2

91

Here, \p$ and \p* are the potentials at the magnetic axis and the plasmaboundary, respectively. The parameter I//Q is an arbitrary quantity. 1//* isdetermined by the condition:

The parameter K allows the localization of the current distribution and, thus,
the ratio 7 = q a /qo to be varied. The quantity j 0 is found, in the process ofsolution, from the condition of total current conservation,
l

v=
Sp

where S p is the plasma column cross-section. It is convenient to use the relationv = I P / ( S P J B ) as the bifurcational parameter. For a circular plasma column
cross-sec"tion, we have v = 2/q a . If the parameter v is within the zone of linear
stability for a kink mode {v > vCT), the plasma cross-section and the crosssections of the internal magnetic surfaces will be a family of concentric circumferences for the equilibrium found. If the value of v passes into the zone of linear
instability, two possibilities will appear: either there is no solution to the
problem (1), (2), or the equilibrium plasma cross-section and the cross-sectionsof the internal magnetic surfaces are extended towards the fixed points (4), in
spite of the circular cross-section of the casing (2). The configuration of thesecross-sections is close to elliptical and can be characterized by the ratio X = b/a,
where b and a are the major and minor semiaxes of the magnetic surface crosssection. On the plasma surface, a = a 0 , b = b 0 , X = Xo = b o /a o . For the case ofhomogeneous current within the plasma, X = const = Xo. For an inhomogeneous
current, X is a function of the magnetic surface. The results of calculations to
be given below refer to a case with r w /b 0 =1.5; m/n = 2.
Later on, the dependence of Xo on v will be identified with a bifurcational
curve. Some bifurcational curves for different 7 are given in Fig. 1. The choiceof K, in (3), for a current profile is done to retain 7 in the process of solution.
The bifurcation points in Fig. 1 correspond to entering the zone of linear
instability with respect to v (transition across v = vCI). There are no helicalequilibria to the left of point vCI = 2 for 7 = 1 (homogeneous current, curve 1).
This corresponds to the results of Ref. [4], and to the analytical dependence
found there:(5)
For a parabolic current, when 7 = 2 (curve 2), a branch of the bifurcational
curve corresponding to an elliptic plasma cross-section emerges at the point of

92

DNESTROVSKIJ et al.

FIG, 1. Ratio of semi-axes, \0 = bo/ao , for elliptical plasma cross-section in helical equilibrium
versus v (bifurcation curves). Curves 1, 2 and 3 correspond to y~ 1, 2.0 and 2.1; they arecalculated by 2-D model. Curves 4 and 5 correspond toy = 2.0 and 2.5 and are calculatedby 1-D model.

entry into the linear instability zone. The bifurcation is, in this case, supercritical. Therefore, the evolutionary transition to a helical equilibrium should
be of a 'soft' nature. This behaviour is confirmed by a numerical integrationof the time dependent, non-linear MHD equations [ 1, 2].
When the current profiles are more peaked, 7 > 2 (curve 3), the bifurcation
curve does not tend to infinity, but has a maximum within the instability range
1 < v < uct. The values of Xo on this curve are not large. Therefore, the temporal
evolution of the unstable kink modes should only result in weak perturbations
within the plasma column.
3.

1-D MODEL OF HELICAL EQUILIBRIUM

The helical equilibrium problem of Eq. (1) can be solved by the methodof preset magnetic surfaces [4]. Let the cross-sections of the magnetic surfaces
be a family of nested ellipses:
x = a cos 6, y = X(a) a sin dwhere a is the radial co-ordinate (marker) of magnetic surfaces and X(a) is theratio of the semi-axes of the nested ellipses. An equation for ellipticity X(a)can be written in the following form:(6)

IAEA-CN-47/E-I-4-2

93

The following designations are introduced here:
A = 2(3X2 + 1) f0 + 2 f! -8X 2 f2
B = 2(1 - X4) f0 - 2 X2 + 1) f, + 4(X4 - 1) f2
C = 2(3 + X2)f0 X2 - 2X2 f, - 8X2 fa(7)

fo = X [j(a) — j B ]aa2

a a 4 x^ ~ ] ) da > f2 = ~ a^
T /J J J'(a)a Xda
J/ J'( )Equation (6) is a non-linear integrodifferential equation. To complete theposing of the problem, we have to prescribe the current distribution, j(a), andthe boundary condition on the plasma column edge, a = a0:
X(a o ) = Xo

(8)

Substituting condition (8) into (6), we find X' and the field B** at theplasma column boundary. As a result, we obtain Cauchy's problem for Poisson's
equation (1) for an external field. Using the method of separation of variables,
for an external quadrupole field, we have [4]:

.

x

° ~1

•

(Xo + 1)a0

- I/a? /j'a 2 Xda

0

Xp

(X

g

1 + X o ao XQ/XQ

94

DNESTROVSKU et al.

The helical equilibria in which we are interested satisfy the condition i//ext = 0, i.e.
F=0

(10)

which defines the permissible values of Xo for boundary condition (8).
For a homogeneous current, j'(a) = 0 and X'(a) = 0, and Eq. (10) leads to (5).
Solving Eq. (6), we use two different parametrizations for the current profile:•a'/aS)?" 1(11)(0<a<r0)(12)

J = Jo

1-

a r

~°

Equation (6), with current profiles (11) and (12) and boundary conditions (8) and (10), has been solved numerically. The same parameters, v and 7
as in the previous case have been used as variables. Curves 4 and 5 in Fig. 1
correspond to 7 = 2.0 and 7 = 2.5. Curves 2 and 4 are slightly shifted withrespect to each other. This shift is related to a different position of the casing
for the 2-D problem (r w /a 0 = 1.5) and for the 1-D problem (r w /a 0 = °°). From
Fig. 1, we also see that, with the casing approaching the plasma, the modulus of thederivative dX0/df|y=ycr is reduced, which implies a 'softer' bifurcation of theaxially symmetric equilibria towards the helical ones.

1.00

FIG. 2. Solution \(a) to problem (6) to (10) aty = 2.5, Xo = 1.02. Curve 1 corresponds to
v= 1.165, Curve 2 to e= 1.07.

95

IAEA-CN-47/E-I-4-2

stability/"boundary

FIG. 3. Curves \,fr0, v) = const for current distribution (12) at q0 = 0.9. Curve 1 corresponds
to Xo = 1.0; Curve 2 to\0 = 1.1; Curve 3 to\0 = 1.2.

When 7 > 2, i.e. the current profile is more pronouncedly peaked thanparabolic, each bifurcation curve has a maximum. In this case, there are twohelical equilibria with the same boundary extension, Xo, and with different
values of v. Two curves, X(a), for 7 = 2.5 with the same Xo = 1.02 and withdifferent values of v - v = 1.165 (Curve 1) and v = 1.07 (Curve 2) - are shownin Fig. 2. The shape of these curves shows that Ma) is close to unity at theplasma column centre, i.e. the magnetic surface cross-sections are almost
circular and the extension of the magnetic surfaces only occurs close to theboundary a = a0. The solutions to Eq. (8) are well approximated by the
relationshipy = y° -(13)

wherey = E/E 0 ,E = X(a)- 1,EO = Xo - 1, x = a/a0> n s [y1] - 1.
The results of calculations for current profile (12) are shown in Fig. 3. The
cross-sections of the bifurcational surface, Xo = S(q 0 , r 0 , v) with the planes
Xo = 1.0; 1.1; 1.2 at q0 = 0.9 are given in the figure. Xo = 1 is the stability
region boundary. It consists of bifurcation points.

4.

ANALYTICAL RELATIONSHIPS FOR SOLUTIONSIN THE NEIGHBOURHOOD OF BIFURCATION POINTS

Analytical representations for the equilibrium solutions in the neighbourhoodof the bifurcation points can be derived. Let us use a 1-D model and expand

DNESTROVSKU et al.

96

Eqs (6) and (10) in terms of e0- Terms of e 0 with powers higher than one should be
retained in order to derive the bifurcational relationships. As a result, we obtain:t (a 0 ) = £e0 + ^eg4F/j B = e0D(i^cr - v) + Se20 = 0

(14)

where
£ = 2 A 2 , ^ = 2(A2 - 1) (A, + A 2 ) + 3A3 - 2A2
4+£

Here,

k

5

f

(

£2

(15)

(K= 1,2,3)"l-7Qo/2
11

1, = J I(x2y)'dx, I, = J I(x4y)'dx, I, = J I (x4y2)'dx000

and y(x) is found as a solution to the non-linear integro-differential equationXxy = y1-(16)

with the boundary condition y( 1) = 1.
It can be shown that Eq. (16) coincides with an equation for the eigenfunctions of a linear stability problem of external modes.
It follows from (14) that bifurcation is supercritical for 6 =£ 0. When 7 = 1 ,which corresponds to a homogeneous current, 5 = 0, and terms of the order of
ejj should be taken into account in expansion H4). For a parabolic current,
relationships (15), with account of (13), yield deoldv \v=Va = D/8 = - 3 . 8 . The
numerical calculations, using the complete 1-D model, result in de o /d^ \v=v =~4.

5.

DISCUSSION

It is shown, by solving 2-D and 1-D problems, that bifurcation of axiallysymmetric equilibria towards helical ones occurs at the stability boundary forIAEA-CN-47/E-I-4-2

97

helical modes. The bifurcation is supercritical for an inhomogeneous current,
which results in a possible non-linear stabilization of external kink modes.
The region where adjacent equilibria are present, for infinitely removed casing,
qcr<q0<l(17)

is located within the instability range qcr < q 0 . The case of homogeneous current
is degenerate; in this case, bifurcation is subcritical, and no mode stabilization
occurs. A casing approaching the plasma increases the stability range and changes
the region where helical equilibria are present. For the most stable, piecewise
current, we have qcr = 1, and range (17) disappears.
Thus, the really dangerous zone, qo > 1, is determined by the most stablecurrent profile. A search for the instability range boundaries at different
current profiles is unnecessary since the results of such a search do not provide acomplete picture of real plasma column evolution anyway. The consequencesof the development of kink modes can be completely determined by non-linear
effects.
ACKNOWLEDGEMENT
The authors express their gratitude to V.D. Shafranov for useful discussions
and valuable remarks.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]

ROSENBLUTH, M.N., MONTICELLO, D.A., STRAUSS, H.R., WHITE, R.B., Phys.
Fluid 19(1976) 1987.
DNESTROVSKIJ, Yu.N., KOSTOMAROV, D.P., POPOV, A.M., SHAGIROV, E.A.,
Fiz. Plazmy 11(1985) 1080.
SYKES, A., WESSON, J.A.., in Plasma Physics and Controlled Nuclear Fusion Research
1980 (Proc. 8th Int. Conf. Brussels, 1980), Vol. 1, IAEA, Vienna (1980) 237.
ZAKHAROV, L.E., Fiz. Plazmy, 7(1981) 18.
ZAKHAROV, L.E., SHAFRANOV, V.D., Voprosy teorii plazmy, Vol. 11, Energoizdat,
Moscow (1982) 118.
DNESTROVSKIJ, Yu.N., KOSTOMAROV, D.P., PEDORENKO, A.V., POPOV, A.M.,
in Controlled Fusion and Plasma Physics (Proc. 1 lth Europ. Conf. Budapest, 1985),
Vol. 1, European Physical Society (1985) 102.

DISCUSSION
ON PAPERS IAEA-CN-47/E-I-4-1 AND E-I-4-2
D.B. NELSON: Several years ago I considered a model similar to the current
layer model described by you. I found the stability results to be very sensitive
to whether the layer boundary occurred at a mode rational surface, and to themagnitude of the jump in current density at a mode rational surface. I therefore
abandoned the model as being an unrealistic description of an actual current profile.
I wonder if you have encountered the same problem and, if so, what you
have done about it?
B.B. KADOMTSEV: Yes, I believe that this problem has been found, but
I would suggest that you discuss it with the authors of papers IAEA-CN-47/
E-III-15 andE-III-16.
H.P. FURTH: I would like to comment on the finding of A'-stable tokamakr(r) profiles. Ten years ago, the only known solutions had very close fitting
shells or very high values of q(a). Glasser et al. then found a r(r) model thatwas stable for q(a) = 2.6 without a shell. We did not carry the r(r) profile
optimization further at that time, because it seemed unlikely that such refinements could be realized experimentally. In recent experiments, however, thetokamak has turned out to be an excellent solver of the A' < 0 problem: this is
an important part of the 'profile consistency' phenomenon. Inspired by thisobservation, Dr. F. Cheng at Princeton has carried through a full optimization
study. Like Dr. Zakharov, he finds that q(a) -> 2 is consistent with A' stability,
but he identifies not just a single r(r) solution for this case, but a whole familyof them. They are all rather peculiar looking but perhaps the tokamak plasma
will find them all.
B.B. KADOMTSEV: Thank you for the comment.
L. CHEN: It seems to me in the case of the quasi-linear saturation of
ballooning modes that the choice of the trial function disregards the inertial
layer physics. Specificially, one would expect the formation of singular current
sheets which could affect the quasi-linear saturation process. Could you commenton this point?
B.B. KADOMTSEV: I don't know exactly, but quasi-linear saturation may
resemble near-equilibrium, so that inertia can be disregarded.

99

IAEA-CN-47/E-II-l-l

RADIOFREQUENCY STABILIZATIONOF IDEAL BALLOONING MODESIN A TOKAMAK PLASMA
A. SEN, P.K. KAW, A.K. SUNDARAM
Institute for Plasma Research,Bhat, Gandhinagar District,
India
Abstract
RADIOFREQUENCY STABILIZATION OF IDEAL BALLOONING MODES IN A TOKAMAKPLASMA.
A new method of stabilizing ideal ballooning modes in a tokamak plasma through the useof radiofrequency waves is suggested. Model calculations show that ponderomotive forces
produced by radial gradients in the RF energy can provide an effective stabilizing mechanism
and create a stable window to the second stability regime.

1.

INTRODUCTION

The maximum achievable beta (the ratio of plasma pressure to magnetic
field pressure) in a tokamak plasma is limited to rather low values by the onsetof the ideal ballooning instability. To improve the critical beta, several methods
have been proposed, such as the use of bean shapes or crescent shapes for theplasma cross-section and the introduction of energetic particles [1-3]. One
important aim of these investigations was to find a stabilization method permitting
access to the second stability regime without having to burn through an unstable
domain. In this paper, we propose the use of radiofrequency (RF) waves (close
to characteristic frequencies such as the lower hybrid frequency and the ion
cyclotron frequency) for stabilization of the ideal ballooning instability. We
show that for rather modest amplitudes of the RFfields a stable window to thesecond stability regime can be made available.
We take our cue from some recent experiments on magnetic mirrors [4, 51
demonstrating that Rayleigh-Taylor instabilities driven by bad magnetic curvature
can be stabilized by application of RF waves near the ion cyclotron frequency.
Theoretical calculations [6] indicate that the dominant physical effect responsible
for RF stabilization is the non-resonant ponderomotive force which counteracts
the effective gravitational force due to bad magnetic curvature; other competing
effects contributing to RF stabilization are related to resonant mode coupling
phenomena [7]. Our calculations reveal similar stabilizing effects on the ideal
ballooning modes in a tokamak plasma.
101

102
2.

SEN et al.
BALLOONING EQUATION WITH RF FIELDS

We derive a model equation for the ideal ballooning modes from thefollowing closed set of linear one-fluid equations:
3V!

-

_

7, X B o

V, XB 0E1 + - ^ - ^ = O

7 0 X B,

(2)

9P, _
^rV)Po=O(3)

3pj
T-+(Vr7)/>0=0
ot

(4)

ot

(5)
and Maxwell's equations. Here, VQE is the induced oscillating velocity of thefluid (ions) in the presence of an external RF field and the other symbols havetheir standard meanings. The term P]<(VoE-V) VQ E ) is the ponderomotive contribution due to the RF field and can be rewritten as

F P = P i <(V0E-V) Vfo> = ~ <ViV20El >for electrostatic RF waves.
For simplicity, we assume that the ponderomotive force is radially symmetric(with respect to the minor radius) and look for effects arising from radial
gradients in the RF energy. We also neglect for the moment other non-linear
RF effects (for instance side-band coupling or modification of the equilibrium
profile) and assume a model equilibrium with shifted circular magnetic surfaces.
The procedure for determining the marginal stability of such an equilibrium toballooning instabilities has been well developed and considerably simplified
through the use of ballooning co-ordinates [8-10]. This procedure involves
only the solution of an ordinary differential equation, which in our case isgiven by

IAEA-CN-47/E-n-l-l

103

9
3F
— [1 + ( s 7 ? - o ; s i n 7 ? ) 2 ] — or?
or)+ a [cos r? + sin r? (sr? - a sin r?)] F + P rad - F = 0
where s = (r/q)(dq/dr) is the shear parameter, a = -(2Rq 2 /B 2 ), (dp/dr) is ameasure of the maximum stable pressure gradient, q is the safety factor,
r? = s0 is the ballooning co-ordinate and F is the normalized potential.

'OE'

is an additional term due to the RF presence. Its physical origin is the leading
contribution from

CV- —— A \V| Vncl I I =T-Vjui A v | VQgi

For Pra(j = 0, Eq. (6) reduces to the standard model equation for the ideal
ballooning mode [8-10]. Our aim is to study the effect of the additional term
due to RF waves on the stability properties of the ballooning mode.

3.

NUMERICAL RESULTS AND ANALYTIC LIMITS

We solve Eq. (6) numerically, with the boundary conditions F-*0 as|T? I -»•«>. Numerically, this involves setting F(r? 0 ) = 0 at some arbitrarily large
T?o and iteratively obtaining the eigenvalue <x for a given s. For large s, the wave
function F has a narrow mode width and falls off quite rapidly at moderatevalues of TJ. However, in the region of small s, it is necessary to integrate out toa larger?, since the wave functions falls off rather slowly (~l/s). ForP ra( j = 0,
we recover the standard results obtained by others, namely that in the s-a plane
there is a zone of instability which extends all the way to the origin and separates
the first and second stability regimes (solid lines in Fig. 1). For P r a ( j>0, theinstability zone is found to broaden. This is consistent with the physical effectthat for P rac i> 0 the RF forces enhance the gravitational forces due to bad
curvature on the outside of the torus. For P r a d < 0, the instability zone shrinks
and also moves away from the origin (dotted curves in Fig. 1), thus providing
a stable window which connects the first and second stability regimes. This
fundamental change in the s-a diagram is brought about by finite values of

104

SEN et al.

O.5r

UNSTABLE

0.7

FIG. 1. Marginal stability curves for ideal ballooning modes in the s-a plane. The solid
curves are for P rad = 0 and the dashed curves are for various values of Pni.

Pra(j and is noticeable in our numerical solutions for even very small values of
P r a d (about 10~4). At large values of Pratj the window broadens considerably
and the stability region is enhanced. The stabilization effect persists even atlarge s-values and is quite appreciable for Pra(j «* 1CT1.
To understand these numerical results, we have considered analytic
approximations to Eq. (6) in two limits. In the limit of small s and a. - the'weak ballooning limit' - the extent of the mode is large compared with 2ir, anda multiple scale analysis can be applied in order to solve the ballooning equation[9-11 ]. Basically, we allow variation in F on two length scales, r\ <*> 1 and77 «s 1/s, to account for the rapid variation due to the periodicity of theequilibrium and to account for a lower secular decay due to shear. The slow
variation is given by
Z = S17and we expand the 17-derivatives as

9

d

a

dr}

y — +
3T?

9Z

•

.

IAEA-CN-47/E-II-l-l

105

and F asF = F 0 (Z,t?) + F1(Z,r?) + ...
To understand the behaviour near the origin, we also assume P r a d to be of ordera 4 . Then, in zero order, we find that F o = F o (Z) and, in first order,F0Fi =— (cos 7? + Z sin rj)
1+z

Likewise, second- and third-order equations for F 2 and F 3 satisfy periodicity
conditions and do not yield any constraint equation. In fourth order, however,periodicity of F 4 results in an equation determining F o :
9
S2

3F 0
(1+7?)VZ

a 2 ( 2 s - 3a 2 /8)

+

1+Z2

^T

F

0+PradF0-0(7)Equation (7) can be rewritten as a Schrodinger equation by setting

0.22)1/2

to obtain-s 2 r " | + V(Z) <>
/ = 0

(8)

where the potential V(Z) is

V(Z) =

,

.

s2

.

,2(3a2/8-2s)

1 +Z2

(9)

Equation (9) shows that if P rad > 0, the potential is negative at sufficiently
large Z and can lead to instability. This agrees with our numerical findings.
A simple sufficient condition for stability is V(Z) > 0 for all Z. This gives
the stability criteria as

(10)

106

SEN et al.

From Eq. (10) we can obtain the two stability boundaries as
5a 4

1/2
(11)

The two curves given by Eq. (11) approximately agree with the numerical s-a
curves in the vicinity of the origin. Note that in the absence of P r a d , s = a = 0
is a solution of Eq. (11), confirming the fact that the two curves emanate fromthe origin. However, such a solution is not possible for P rad ^ 0 . In addition,
Eq. (11) also indicates that there is a minimum a, (8P rad /5) 1/4 , below which no
instability is possible for any value of s. These features explain the moving
away of the instability zone from the origin and the consequent creation of astable window bridging the two regions.
Another interesting analytic limit is the high shear case or the strong
ballooning limit ( s ^ a > 1) where the mode is localized to a narrow range of TJ.
To analyse this situation, we convert Eq. (6) to a Schrodinger form by puttingF=

y/\ + (STJ —a sinTj)2

to obtain•V(T?)*=0

(12)

with
V(T?) =

( s - a. COST?)2(P rad + a COST?)[1 +(sT?-asinTj) 2 ] 2,
1 + (STJ - a sin TJ)2

(13)

Our numerical solutions show the wave function to be peaked at TJ = 0 with anarrow mode width. It is thus appropriate to expand V(T?) near T? = 0 and toretain terms up to second order in T?. This reduces Eq. (12) to a Weber equation
which can be studied analytically [12]. For our purpose, we can get a qualitative picture of the stabilizing influence of P rad by considering the expanded
potential
V(TJ) = a+ b 2 rj 2

(14)

a = P H~Q! + (S~oi) 2(15)

b 2 =a-[s-oc + i + ( s - a ) 2 ] - 2 ( s - a ) 4 + ( s - a ) 2 P r a d(16)

IAEA-CN-47/E-H-l-l

107

For stability, we require b2 < 0 . The stability boundary can again be determined(within the present approximation) by solving b 2 = 0. An approximate solution
is given by

• a ±

(17)

which agrees roughly with the numerical curves, although the agreement is poor
for the lower branch. For P r a d < 0 , Eq. (17) indicates lowering of the upper
branch and raising of the lower branch, whereby the instability regime shrinks.
This is in qualitative agreement with our numerical results.

4.

DISCUSSION

We have carried out a model calculation to study the influence of
externally imposed RF waves on the stability of ideal ballooning modes.
Retaining the ponderomotive forces produced by radial gradients in the RFenergy, we have obtained a modified marginal stability equation, which wehave solved numerically and studied analytically in a few limits. Our results
indicate significant stabilization effects and the creation of a stable window
linking the first and second stability regimes. Thus the method offers the possibilityof accessing the high beta regime of tokamak operation without having to burnthrough an unstable domain. The stabilization mechanism is significant at
modest RF powers, for instance P r a d *» a 4 , which corresponds to V Q E / V ^ J * a 2 < 1.
Some of the simplifying assumptions made for our calculation and thepossible limitations they may impose on our results are the following: A major
simplification is the neglect of other non-linear RF effects, notably side-band
coupling phenomena. These can become quite significant near wave resonancesand can affect the stability through parametric interactions. For interchange
modes in mirror plasmas, side-band coupling has been shown to have a stabilizing
influence [7], but for ballooning modes no such calculations have been performed.
Our preliminary investigations show that the side-band coupling contribution canbecome comparable to the non-resonant contribution in some realistic parameter
regimes (particularly in the edge plasma) and that side-band coupling should beincorporated into the stability calculations. Then, the model equation would
no longer be so simple as Eq. (6) but would involve higher derivative modifications. Work on such a modified equation is in progress. The P rad term in Eq. (6)is independent of r? because of the assumed symmetry of the RF ponderomotive
forces. Such an assumption is quite reasonable, judging from ray tracing andresonance cone results of lower hybrid waves, for example, which show that the
RF energy is uniform over magnetic surfaces and changes as a function of r.

108

SEN et al.

We have also ignored possible RF modifications of equilibrium profiles (e.g.
distortion of magnetic flux surfaces) and this restricts the validity of our results
at low RF power levels ( V Q / V ^ J ^ 1). In the ponderomotive force term wehave also neglected the electron contribution arising from the E X B motion.
This is justified in the limit cj2pj£l% < 1 for lower hybrid waves. It is also clearthat the present calculations are made with electrostatic RF waves in mind, but
an extension to electromagnetic waves is not difficult. The basic stabilization
mechanism arising from the radial ponderomotive force is, however, quite
unambiguously shown by our simple model. This is the principal result of ourcalculations and it is not expected to be seriously modified by any of the above
simplifications.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]

CHANCE, M.S., et al., Phys. Rev. Lett. 51(1983) 1963.
YAMAZAKI, K., et al., Nucl. Fusion 25 (1985) 1543.
ROSENBLUTH, M.N., et al., Phys. Rev. Lett. 51 (1983) 1967.
FERRON, J.R., et al., Phys. Rev. Lett. 51 (1983) 1955.
1NUTAKE, M., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1982
(Proc. 9th Int. Conf. Baltimore, 1982), Vol. 1, IAEA, Vienna (1983) 545.
MYRA, J.R., IPPOLITO, D.A., Phys. Rev. Lett. 53(1984)914.
McBRIDE, J.B., et al., Phys. Rev. Lett. 54 (1985) 42.
McBRIDE, J.B., Phys. Fluids 27 (1984) 324.
CONNOR, J.W., HASTIE, R.J., TAYLOR, J.B., Phys. Rev. Lett. 40 (1978) 396.
STRAUSS, H.R., Phys. Fluids 24 (1981) 2004.
HENDER, T.C., et al., Phys. Fluids 24 (1984) 1439.
LORTZ, D..NOHRENBERG, J., Nucl. Fusion 19 (1979) 1207.
HAZELTINE, R.D., MEISS, J.D., Phys. Rep. 121 (1985) 1.

IAEA-CN-47/E-II-1-2

ELECTROMAGNETIC PARTICLE SIMULATIONSOF RF STABILIZATIONAND ECRH OF OVER-DENSITY PLASMAS
H. ABE, S. NAKAJIMA, Y. KADOYA, H. OKADA, R. ITATANI
Department of Electronics,
Kyoto University,
Kyoto
K. SAKAI, S. TAKEUCHI, M. MATSUMOTO, R. SUGIHARA1
Department of Engineering,
Yamanashi University,
Kofu
Japan
Abstract
ELECTROMAGNETIC PARTICLE SIMULATIONS OF RF STABILIZATION AND ECRH OF
OVER-DENSITY PLASMAS.
Employing a slab model and using 2Vi-D, electromagnetic particle simulation codes developedin two institutes of the Kyoto and Yamanashi universities, the authors study two subjects: (1) RF stabilization of flute and drift modes and (2) heating of an over-density plasma (ODP) by injection of an electron cyclotron resonance wave (ECRW).

1.

RF STABILIZATION OF FLUTE AND DRIFT MODES

1.1. Model of the simulation
A slab plasma is immersed in a uniform magnetic field B0(llz) (Fig. l(a)). The
density is initially parabolic along the x-axis and uniform along the y-axis, except fora small perturbation. The RF is excited by sheet currents flowing parallel either tothe y-axis (perpendicular antenna) or to Bo (parallel antenna). The system is uniform
along Bo. So, the wave vector IT is always perpendicular to Bo (i.e. k|| = 0).
If no RF is applied, the flute mode grows as a result of the gravitational acceleration g (llx). The growth rate agrees with the theoretical value of 70 = (-glQ'AHere 4,= n(dn/dr)~' is the scalelength of the density gradient. The scalelength for
the RF intensity (E = E 2 /(dE 2 /dr)~' will also be used.

Institute of Plasma Physics, Nagoya University, Nagoya 464, Japan.

109

110

A B E et al.

antennaantenna(c)(d)

FIG. 1.

(a) Configurationof the system.

Contours show initial(b), (c), (d) Density contours at fir = 348;

specifically,(b) with no RF, (c) w/G = 1.25,

0.67.

(d) co/fi =density,eno" 'pump
FIG. 2. Frequency spectrum. Pump at u> = 1.25 0 mainly has x-component, while sidebands mainly
have y-component.

1.2. Results of the simulation
1.2.1.

Case 1: stabilization of drift mode by perpendicular antenna

Even if gravitation is absent, a slowly growing mode (a kind of drift mode) has
been observed (Fig. l(b)). This mode is stabilized when w < fi (ion cyclotron frequency), and destabilized when o > fi by RF with \(E\ - |^| excited by perpendicular antennas (Fig. l(c), (d)), The growth rate of the drift mode is so slow thatdestabilization of the plasma by RF is shown clearly (Fig. l(c)). In the case ofdestabilization, sidebands are observed in the frequency spectrum of the electric field(Fig. 2).

IAEA-CN-47/E-II-1-2

111

TABLE I. SUMMARY OF THE THEORETICAL AND SIMULATIONRESULTS141 > 141
141 *• 141

1.2.2.

4 >o4 <ooi > Q

Dominant mechanism

s(s)

u©

Sideband coupling

s
u

s
u

Ponderomotive force

Case 2: stabilization of flute mode by perpendicular antenna

In the presence of gravitation, theory predicts that the flute mode is stabilized
when o> < fi and destabilized when oi >Q by the RF with |l E | > | 4 | , where theeffect of the ponderomotive force is supposed to be negligible. Even though
141 - | 4 | , the abovementioned phenomenon may occur. Examples were alreadyreported [1]. When the mode was stabilized, the growth rate y decreased as theantenna current A increased, following the relation y2 = a - £A2 (a, & are positive constants). This indicates that the suppression is due to a non-linear process.
1.2.3.

Case 3: stabilization of drift and flute modes by parallel antenna

Parallel antennas excite parallel electric fields. These fields do not propagate but
are damped out at a relatively short distance. They were observed to stabilize both
MHD modes, independently of the sign of w - (2 in the simulations.
1.3. Interpretation of the results
A theory [2] has been presented on RF stabilization; it seems to be closely
related to the present simulation. We briefly summarize the results of the theory.
First, suppose that the high frequency waves are magnetosonic and the condition
Upj/kV < 1 holds. This corresponds to Case 2 (perpendicular antenna). The results
are shown in Table I, where S and U stand for 'stabilization' and 'destabilization'or 'enhancement of instability'. The simulation results are also denoted by encircled
S and U.
Let us compare simulation with theory. The simulation results for
141 > 141 in Case 2 are consistent with the theoretical prediction. Furthermore, thesimulation results for | 4 | ~ |4| imply that sideband coupling is also dominant inthis case. It should be noted that a case with | 4 | « 141 has not been realized in thepresent simulation. For Case 3 (i.e. parallel antenna), theory predicts that the effects

112

ABE et al.

of sideband coupling are negligible and the ponderomotive force acting on the electron fluid is dominant in the stabilization. The simulation results are consistent withthis notion.

1.4. Discussions and conclusions
For the case of |fE| ~ |4I and a magnetosonic pump wave, the theory does
not give an explicit indication because the results, in a subtle manner, depend on theparameters and the polarization of the waves used.
The simulation results in the present parameter region and under the condition
2 2a)pj/k c < 1 (or X (wavelength) •* c/210Vn) imply that sideband coupling effects
and dominant in flute mode suppression and consistent with the theory. Therefore,
it is concluded that the simulation results are successfully explained by the theory
although the parameter region covered by the simulation is limited.
Referring to Table I, we should understand the experimental results of Akimune
et al. [3] and part of the results of Yasaka et al. [4]: If some experimental condition
prevents sidebands, in addition to the condition lE > 0, which seems to be the casefor most of the experiments, the plasmas are stabilized, regardless of the sign ofoi — 0. In other words, polarization may offset the effect of the sign of o> — 0
included in the ponderomotive force term, in contrast to the simple picture that is stillwidely believed.

2.

HEATING OF OVER-DENSITY PLASMA BY INJECTION OF ELECTRONCYCLOTRON RESONANCE WAVE (ECRW)

2.1. Introduction
According to the linear theory, an O-mode of ECRW with w0 > fie injectedfrom the low field side at the optimum incident angle is converted to an electron
Bernstein wave (EBW) through the O-X-B mode conversion process [5].
On the other hand, a fast X-mode incident from outside the plasma tunnels
through the evanescent region between the cut-off (w0 = wR) and the upper hybrid
resonance (UHR) and is converted into both the slow X-mode and the EBW [6].
In either case, the EBW propagates from the UHR to high density regions
beyond the O-mode cut-off and X-mode cut-off (a>0 = a>L) and is expected to heatthe electrons in ODP (over-density plasma). wR and wL are the cut-off frequencies of
the right and left hand elliptically polarized wave.
We have first observed ODP heating by injection of a fast X-mode and an
O-mode, in the EM simulation results.

IAEA-CN-47/E-II-1-2

113

WG. 5. (a; Profiles of external magnetic field B0(\\y) and initial density distribution.
Initial electron temperature is 1.3 keV. (b) Interferogram of E^ and energy deposition to electrons&(ncTe)/he0Te0 (a/k0 = 2.06).

2.2. O-mode injection
A microwave with appropriate polarization is injected at the optimum incident
angle (0; = 46.7°). The initial plasma parameters are shown in Fig. 3(a). The EBW
excited through the O-X-B process propagates to higher density regions, is graduallydamped and heats the electrons between UHR and ECR (Fig. 3(b)).
Considering the results of ray tracing analyses [7], we know that the trajectoriesof the O-mode are distorted in the toroidal section near the cut-off (w0 = fl,*), andthat this three-dimensional effect prevents the O-mode from being converted to the
X-mode.
2.3. Dependence of conversion efficiency of the fast X-mode on density
gradients
The fast X-mode is observed to propagate up to the cut-off (a>0 = O>R)
(Fig. 4(a)) and to be converted to an EBW at UHR (Fig. 4(b), (c), (d)). A slow

114

(arb. unit;

ABE et al.

N

UJ

0.1 • * k0-0.1

Jfnu1ill) IIII1

-2

AftA

x/a

(a)
V

"n • u L0f/G. 4. (a) Interferogram of Ez. a/k0 = 8.24. (b), (c), (d) Interferogram of E^ and energy deposition
to electrons Af^TJ/fi^T^
specifically (b) a/ho = 2-06, (c) a/kg = 4.12, (d) a/kg = 8.24.
(e) Conversion efficiency to EBWoffast X-mode. Ordinate is TB = 1 - R, where R is power reflection
coefficient measured in vacuum region.

IAEA-CN-47/E-H-1-2

115

X-mode is also observed to be converted from the fast X-mode (Fig. 4(a)). The EBW
propagates, first, with small damping until finally it is damped quickly at ECR. In
accordance with wave damping, the electron energy increases near ECR, where thedensity is co^/wg = 2.72-2.93.
Figure 4(e) shows the total conversion efficiency TB versus JJ = koAx, where
ko is the wavenumber of the incident wave in vacuum and Ax is distance between
the cut-off (co0 = OR) and UHR. The parameter T? is proportional to n/(dn/dx) and,
therefore, to the minor radius in the case of a parabolic density profile.
The theoretical conversion efficiency from a fast X-mode to an EBW and a slow
X-mode is TFB = exp(-ir)j) - exp(-2xij) and TFS = exp(-irr)), respectively [8].
Here, we present a new idea on the ODP heating process referred to as the X-XB process. The slow X-mode converted from the fast X-mode is reflected at the cutoff (u0 = «L), partly converted to an EBW at UHR with a conversion efficiency of
TSB = 1 - exp(-irr;) (X-X-B process) and partly converted to a fast X-mode via
tunnel effect. Neglecting phase effects of the two EBW, the total conversion efficiency is calculated to be TB = 2exp(—mf) (1 — exp( — m/)), which is plotted as thesolid line in Fig. 4(e), which includes a broken curve, showing for comparison theusual theoretical value TFB.
As is seen in Fig. 4(e), the optimum density gradient is localized in a narrow
parameter space 77, and the maximum of TB is close to unity. These two results areat variance with the theoretical prediction.
2.4. Discussions and conclusions
The accessibility condition of the EBW to the centre of the plasma, including
three-dimensional effects [7], excludes the possibility of ODP heating by using the
O-mode. In the case of X-mode injection, normal injection is possible and may avoid
three-dimensional effects. Therefore, X-mode injection remains a candidate for ODP
heating.
If a plasma manages to adjust an optimum density gradient at the plasma surface
and to keep the electron temperature high enough to avoid collisional damping, theinjected X-mode is well converted to an EBW, and high efficiency electron heating
is expected in over-density regions, even for large scale plasmas, as is shown in thepresent simulations.

REFERENCES
[1]
[2]
[3]

SAKAl, K., et al., in Mirror Fusion Research (Proc. IAEA Tech. Comm. Meeting Tsukuba,1985).
D'IPPOLITO, D.A., MYRA, J.R., Rep. SAIC-85-1922/PRI-94, Science Applications, Inc.,
Pleasanton, CA (1986).
AKIMUNE, H., et al., J. Phys. Soc. Jpn. SO (1981) 2729.

116

ABEetal.

[4]
[5]
[6]
[7]
[8]

YASAKA, Y., ITATANI, R., Nucl. Fusion 24 (1984) 445.
PREINHAELTER, } . , KOPECKY, V., J. Plasma Phys. 10 (1973) 1.
SUGAI, H., Phys. Rev. Lett. 47 (1981) 1899.
TAKEUCHI, K., et al., J. Phys. Soc. Jpn. 54 (1985) 2915.
BUDDEN, K.G., Radio Waves in the Ionosphere, Cambridge Univ. Press, London (1961) 476.

DISCUSSION
ON PAPERS IAEA-CN-47/E-n-l-l AND E-II-1-2
S. ITOH: In your calculation, for the stabilization of the ballooning mode the
RF power must balance the excess energy of the unstable mode. How much power
is needed for stabilization in the case of typical large tokamak parameters? When one
considers an actual experiment, the dissipative effect on the RF power must surely
be seriously taken into account.
A. SEN: In answer to the first part of your question, I would like to emphasizethat the RF power ensures stabilization in a reactive manner and not by producing
an additional damping mechanism to balance the growth of the ballooning mode. The
magnitude of the RF power needed for access to the second stability regime can be
estimated from the scaling (VE/Vfod ~ a2. In the low shear region near the centreof large tokamaks, our estimates show that a few megawatts of RF power with thecorrect profile should be able to provide a stable window. I agree with you that inan actual experiment the dissipation of this RF power must be seriously taken into
account.
R. ITATANI: I would like to point out the importance of the profile of RF field
intensities in plasmas. When using standing waves or mixing two waves, there is ponderomotive force in the azimuthal direction which produces convective motion acrossa confining magnetic field, because of the inhomogeneity of the field strength. Twodimensional consideration is then always unavoidable for the RF field profile.

117

IAEA-CN-47/E-II-2-1

THEORY OF RESISTIVEMAGNETOHYDRODYNAMIC INSTABILITIESEXCITED BY ENERGETIC TRAPPED PARTICLESIN LARGE-SIZE TOKAMAKS
H. BIGLARY1, L. CHEN, R.B. WHITE
Princeton Plasma Physics Laboratory,
Princeton University,
Princeton, New Jersey,
United States of America

Abstract
THEORY OF RESISTIVE MAGNETOHYDRODYNAMIC INSTABILITIES EXCITED BY
ENERGETIC TRAPPED PARTICLES IN LARGE-SIZE TOKAMAKS.
It is shown that, in present-day large-size tokamaks, finite resistivity modifies qualitatively thestability properties of magnetohydrodynamic instabilities resonantly excited by the unfavorable precessional drift of energetic trapped particles, the so-called "fishbone"-type instabilities. Specifically, it isfound that (i) the n = 1 energetic trapped particle induced internal kink ("fishbone") instability isstrongly stabilized by resistive dissipation, and (ii) finite resistivity lowers considerably the thresholdconditions for resonant excitations of high-n ballooning/interchange modes. The possibility of exciting"fishbones" by alpha particles in ignition experiments is also considered.

I.

INTRODUCTION

In recent years, observations of magnetohydrodynamic (MHD)
instabilities and associated particle loss in tokamak experimentswith high-power near-perpendicular neutral beam injection » havestimulated
considerable
theoreticalinterest
in MHD modes
resonantly excited by the precessional drift of energetic
particles trapped in the region with unfavorable curvature, i.e.,the outside of a tokamak.
These so-called "fishbone"-type MHD
instabilities
thus
have, generically, mode frequencies, to,comparable to the trapped-particle precessional frequencies,u dh .3
Theoretical models based on this resonance excitation
mechanism have been rather successful inexplaining many
experimental features observed in both the n = 1 "fishbone"and high-n "precursor"-'' events in tokamaks the size of PDX.
Here, n is the toroidal mode number. To extend these theoretical
1

Institute for Fusion Studies, University of Texas, Austin, TX, USA.

119

120

BIGLARY et al.

results to current high-field large-size tokamaks such as TFTR,
JT-60 and JET, we must examine the validity of the core-plasma
ideal MHD approximation employed in the theoretical models. Notethat uidn •* E/BR, with E = v2/2 and R the major radius. Hence, inlarge devices, uL n may become comparable to or even smaller thanthe characteristic resistive frequency cuR = S^~'" u^,and finiteresistivity effects may be significant. Here, S^ is the magnetic
Reynold number, u»A = (1 + 2 q s 2 ) 1 / 2 V A /q s R is the toroidal Alfven
frequency, q s is the safety factor q at the singular surface r =r s , and Vft is the toroidal Alfven velocity.
In the next section, we first consider the effects of finite
resistivity on the n = 1 "fishbone" internal kink instability.
Thehigh-n"fishbone"-typeresistive
ballooning/interchange
instabilities are then considered in Sec. III.
Finally, inSec. IV, noting that fusion alpha particles will carry significant
8(= plasma pressure/magnetic pressure) values in future ignition
experiments, we also consider the possible excitations of aparticle induced "fishbone" instabilities.

II.

n=1 RESISTIVE "FISHBONE" INSTABILITIES'As in the ideal "fishbone" theory , we consider a largeaspect-ratio (e = a/R << 1) circular tokamak plasma consisting ofcore (c) and hot (h) components, and perturbations of predominantly
the m=1, n=1 internal kink mode. Here, m is the poloidal mode
number. We adopt the following formal orderings: 8 p c - 1, 8 p h e, T c /T h - e , and IUJ/OJA| - |5dh/u>A| - e . In addition, we orderto ~ (uR, i.e. S M~ e , which represents an optimal ordering
for the resistive effects. Consistent with these orderings, thehot component is described by the gyrokinetic equation, but thecore plasma must be treated as a resistive MHD fluid obeying the
Ohm's law
«B = V x ({; x B) - (nc2/4iriu)) V25B
where 5 is the (core-plasma) fluid displacement andresistivity.

(1)
n is theBy separating the regions outside and inside the singularlayer located at q(r ) = 1, we can derive the dispersion relation
variationally.
Since resistivity is ignorable in the outside
region, the only modification to the ideal "fishbone" theory is

IAEA-CN-47/E-II-2-1

121

the replacement of the i n e r t i a l - l a y e r Lagrangian by the following
r e s i s t i v e - l a y e r Lagrangian
D [5] = - 2TT R r s 3 J dx(3<o 2 P m |d5 r /dx| 2 + iBk! x ^ V s B / d x 2 ](2)where x = r-rs, k| = q (rs)/R and 6B r from Eq. (1) acts as theconstraint.
Q

Following the analyses of Coppi et al. , it is straightforward to solve the resistive-layer Euler-Lagrange equation andmatch the solutions in the two regions. The final dispersion
relation is then given by*fc

A « -1/3.-5/4 rfffl372 + 5lMi
Mr | (fl 3/2 _ i)/t} ]

xG
k

n

m

where n = - iu/ujR, 5W fc , defined in Ref. [4], is the normalized
core-plasma contribution to the ideal MHD 6W,«W.K =

2 3/2 M. ir2

1+r/R

-

Bo

1-r/R

o

f~ < ;Jiu-.- K 2d(«B) J dEE5/2 /- - S b w..
dh -uiohdh<y> = (2/r s 2 ) J^s y rdr, a = v, 2 /v 2 , u» h = -(!/<!>,,)(§„ x v In
F o h )-7, K b = # (d9/2ir)(1 - o B ) " 1 / S K 2 = £ (d9/2Tr)cose (1-aB)- 1 / 2 ,and only m = 1, n = 1 perturbations are kept in Eq. (4).
In the |n| >> 1 ideal limit, we recover from Eq. (3) the
ideal "fishbone" dispersion relation«W fc + «Wk - 1»/«A = 0(5)

and the corresponding threshold condition, using a model slowingdown distribution function as an example, is given by
<EB

P h,t

>

cr ' "dm /u A

(6)

122

BIGLARY et al.

where the subscript t denotes trapped particles and u>dm =iB dh (E=E in1 ).
Note that below the threshold, i.e. for B ni l i ti t d withith dampedd d
i d l "fishbone""fihb<B n t^cr' k ne singularity
associatedidealmodes is regularized by finite resistivity with a characteristic
width of S M ~ 1 / 3 .
In the |fl| << 1 regime of interest here, i.e.
limit, Eq. (3) reduces to
*u
fc +

*u
k=

a
s

i5ir/8the resistive<s " 1 / 3 T o r 5 / 4
M
*• '

/7\

where S = <o/5dm and A s = - 8 ( 5 d m A i R ) ~ 5 / 4 r(5/4)/r(-1/4) > 0. The
instability threshold condition then becomes, again, for a modelslowing-down distribution function,
29/4><e6 . . >R
ph,t crs\/3/ A
M
s

>

1

(8)

~

and 1/2 < flcr < 1, that is,R
p h . t cr

1 /*3
s M

R
*--

"dm

UJ
A

_

"dm

uu. i, MI-(9)Thus, in the resistive limit, the threshold S value of theenergetic trapped particles is raised dramatically over the ideallimit value.
Since physically the threshold condition can beunderstood in terms of balancing negative dissipation due to thetrapped-particle precession mode with positive dissipation of
Alfven wave resonance absorption at the singular layer, thisstabilization
then corresponds
to enhancement of
positive
dissipation due to finite resistivity.

IV.

HIGH-N RESISTIVE "FISHBONE"-TYPEBALLOONING/INTERCHANGE MODES9This study is in part motivated by observations of high-n(n ~ 5) "precursors" to "fishbones" in PDX experiments, which havebeen theoretically interpreted as high-n ideal MHD ballooning
modes resonantly excited by trapped beam-ion precessional drift,
i.e. similar to the n = 1 "fishbone" instability. In addition to
the above-mentioned issue concerning validity in large-size
devices, the applicability of the Ideal-MHD theoretical models

IAEA-CN-47/E-II-2-1

123

appears to be also severely restricted to either singular-type
energetic trapped-particle distribution functions or regimes nearideal-MHD ballooning-mode marginal stability. Specifically, notethat the dispersion relation of high-n ideal "fishbone"-typeballooning modes is of the same form as that of n _ : 1 ideal"fishbone", Eq. (5), with appropriate replacements of 6W f c and 5W k
in terms of surface quantities.
For e S p c ~ e <<^ (e6 p
^
~
p
p
bilit
W
1, i.e. away from ideal-MHD marginal stability, 6W f c ~ 1, due tofield-line bending and, hence, balancing the real part of Eq. (5)dictates Re(6W k ) - 0(e$ p h ( t ) ~ 1, except for non-analytic F o h suchas the model slowing-down distribution function which yields
logarithmically singular 6W k .
Since field-line bending and,hence, 6W f c can be minimized by finite resistivity, in this casehigh-n resistive "fishbone"-type instabilities could be excitedwith regular F Q nand e0 h t~ e 2 , more consistent withexperimental conditions.
?
1 f\
Employing two small parameters, e and X =(n £ /S M ) '^, theorderings adopted here are S p c - 1, 6p h - e, cuR = Xa>A, c3dh e 4 / ^u R , and, in a maximal scheme, X - e-*. The frequency orderingt|5dh/ioR| - e ^ < ]t corresponds to the resistive limit ofinterest in large-size tokamaks and also implies eigenmode
properties similar to those of the electrostatic resistive
interchange mode.
The dispersion
relation
can
then
be
straightforwardly derived and is given, in the incompressible
limit and for the lowest (most unstable) eigenstate, by
6W k - D R C + (-iu>A.R)3/2 = 0(10)where DDr. =* E + F ~ 0 (e'^-.J< 0 represents favorable average

KLr * APC

curvature.
We remark that inclusion of compressibility does notalter our final conclusions in any substantial way.
Stability
properties predicted by Eq. (10) have been analyzed via Nyquist
technique for both model slowing-down and Maxwellian F Q h and are
qualitatively similar.
In particular, the instability thresholdcondition is given byHere, <<>> denotes surface average.
Equation (11) indicates
that, with the field-line bending stabilizing term minimized byfiniteresistivity,
high-nresistive"fishbone"-typeballooning/interchange instabilities can be excited for 6 p c ~ 1

124

BIGLARY et al.

and 0 p n j. > e, which is much lower than the ideal prediction[e(6p C +S pn t ) - 0 (1) for a Maxwellian F Q n ] and, hence, appears toagree better with the PDX experiments. Also, we remark that,since
the eigenmodes are highly localized around the mode-rationalsurfaces,
finitebananawidth
may
assertastabilizing
influence.
In this case, it can be shown that the thresholdcondition is increased by a factor (PDt/pR) l n ^ p b h / p R ^ w n e r e Pfc>his the typical energetic trapped particle banana width and p R ~rs(nSM)~1/'^(aJdh/(DR)is the typical mode radial width. Finally,
we note that, by performing a three-region asymptotic matching,
the
usual
high-nresistive
ballooning/tearing
dispersion
relation 11 ' 12 can be generalized to include effects due toenergetic trapped particles such that both ideal and resistive"fishbone"-type modes are derivable in the appropriate limits.

IV. ALPHA PARTICLE-INDUCED "FISHBONE" INSTABILITY13
In proposed tokamak ignition experiments, fusion o-particles
generally contribute a significant fraction of the total 6-values,
and,hence,
thepossibilityofexciting"fishbone"-typeinstabilities and an associated anomalous loss of a-particles
needs to be seriously considered.
Assuming typical reference
parameters, R = 135 cm, a = 42 cm, B = 10 2 KG, and n Q = 10 15 /cm 3 ,the precessional frequency of the 3.5 MeV o-particles is then uido
- 105n and u>A - 2 x 10 7 . That is, 5da/o>A ~ n 5 x 10~ 3 > (n 2 /S M ) 1/ 3
for, typically, S M > 10^. We, therefore, need only to consider
ideal "fishbone"- type modes.
Furthermore, since 6-values areusually sufficiently below (SM^[))cri high-n modes are stable andonly the n = 1 "fishbone" instability may be excited.
From
Eq. (6), we find < B O ) t > £ r ~ 1-7 x 10" 3 or, noting n a f t / n o ~ 1/3
for isotropic a-particle distributions, <Ba>";r - 0.5$.
Taking6

ct/8total ~ °- 1 . 1 4 w e fchen derive (B t o t a l )J P - 5% as thecorresponding instability threshold condition, which may be
exceeded in some designs. Finally, it is instructive to note thescaling e a t/<8a t >£ r - n a /ng /2 ,and, hence, high-density ignition
experiments are more favorable with regard to the stability of the"fishbone".
ACKNOWLEDGMENT
This work was supported by United States Department of Energy
Contract No. DE-ACO2-76-CHO-3O73.

IAEA-CN-47/E-II-2-1

125

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]

McGUIRE, K., et al., Phys. Rev. Lett. 50 (1983) 891.
ARMENTROUT, C.J., STRAIT, E.J., Rep. GA-A-17891, GA Technologies, Inc., San Diego,
CA (1986).
CHEN, L., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc. 10th
Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 59.
CHEN, L., WHITE, R.B., ROSENBLUTH, M.N., Phys. Rev. Lett. 52 (1984) 1122.
WEILAND, J., CHEN, L., Phys. Fluids 28 (1985) 1359.
VAN DAM, J.W., ROSENBLUTH, M.N., Paper 1C-6, presented at Annual Controlled Fusion
Theory Conference (Apr. 1985, Madison, WI).
BIGLARI, H., CHEN, L., Phys. Fluids 29 (1986) 1760.
COPPI, B., et al., Fiz. Plazmy 2 (1976) 961; Sov. J. Plasma Phys. 2 (1976) 533.
BIGLARI, H., CHEN, L., Phys. Fluids 29 (1986) 2960.
GLASSER, A.H., GREENE, J.M., JOHNSON, J.L., Phys. Fluids 18 (1975) 875.
CONNOR, J.W., HASTIE, R.J., MARTIN, T.J., in Plasma Physics and Controlled Nuclear
Fusion Research 1982 (Proc. 9th Int. Conf. Baltimore, 1982), Vol. 3, IAEA, Vienna (1983) 403.
DRAKE, J.F., ANTONSEN, T.M., Phys. Fluids 28 (1985) 544.
WHITE, R.B., et al., Paper 2D-9, presented at Sherwood Controlled Fusion Theory Conference(Apr. 1986, New York).
HOULBERG, W.A., paper presented at Alpha Particle Effects Workshop, Courant Institute,
New York University, Dec. 1985.

IAEA-CN-47/E-II-2-2

ALFVEN WAVE PHASE MIXINGIN A SHEARED MAGNETIC FIELD
AS A MECHANISM OF'FISHBONE' INSTABILITY DAMPING
AND OSCILLATION ENERGY PUMPING
INTO THE REGION OF LARGE k ± ~top/c
A.D. BEKLEMISHEV, O.P. POGUTSE
I.V. Kurchatov Institute of Atomic Energy,
Moscow, Union of Soviet Socialist Republics
Presented by B.B. Kadomtsev
Abstract
ALFVEN WAVE PHASE MIXING IN A SHEARED MAGNETIC FIELD AS A MECHANISM
OF 'FISHBONE' INSTABILITY DAMPING AND OSCILLATION ENERGY PUMPING INTO
THE REGION OF LARGE kj_ ~ w /c.
New dynamic equations for the m = 1 mode in a tokamak are obtained; they may be
used to describe the 'fishbone' instability. Other types of Alfven waves in a sheared magnetic
field have an analogous property allowing them to transfer oscillation energy into the short wave
region. This is used to explain the pumping of energy into the region k^ ~ co /c and toconstruct a non-linear model.

Theoretical simulation of the 'fishbone' instability is based on allowancefor two processes: excitation of oscillations at the toroidal drift frequency offast particles and wave induced fast particle radial transport [1,2]. Experimental
observations [3] have shown that low ky Alfven modes, in particular the internalm = 1, n = 1 kink mode, are excited by fast particles.
The first phenomenological 'fishbone' oscillation model was proposed by
Gribkov et al. [4]. A more detailed study of m = 1, n = 1 mode excitation anddamping is presented in [2]. However, the ideal dispersion relation-iw/o; A + W = 0 obtained there has a limited range of validity.
In this paper, dynamic equations for the m = 1, n = 1 mode are obtained,
enabling us to take finite viscosity, conductivity and ion Larmor radius effects
into account.
The physical mechanism of phase mixing, studied in connection with'fishbone' oscillations, has another, very interesting application: it enables us
to explain the process of oscillation accumulation in the collisionless skin layer
region, kj_ ~ cop/c, which is very important in collective transport processes ina tokamak [5].
127

128

BEKLEMISHEV and POGUTSE

1. DYNAMIC EQUATIONS FOR THE m = 1, n = 1 MODE
Let us proceed from the dynamic equations for the m = 1, n = 1 ideal kinkmode in a tokamak, derived in [6]:i J ' - U n X , = M 12 x 2(1)i) 2 x 2 =M 2 I x,

(2)

Here, Xj and x2 are the poloidal harmonics of the radial component of plasma
displacement (x • Va = Xj cos(0 — ?) + x2(a)cos(20 — £)) in an equilibriumco-ordinate system (a, 8, f), in which the magnetic field lines are straight, /u(a) is
the rotational transformation, t is the time normalized to the poloidal Alfv6ntransit time, r$ = R/jucA, and R is the major radius. Concrete forms of thecoefficient Uu(a) < 1 and of the differential operators M 12 and M21 are given
in Ref. [6].
It is well known that the function x^a) is well approximated by a stairlike function with a jump near the n = 1 (a = aj) resonant surface for moderatefrequencies ( w 2 r | <(H(0)I) 2 ). Thus, the derivative of Xj(a) has a small,bounded support.
Using Eq. (1), we easily derive the integral representationa—3 / ( U n x ! +M 1 2 x 2 )da

a J

(3)

0

where the right hand side is a weak function of the quantity near the resonancea = a^ Hence, it can be calculated at this point by using a stair-like trial function:

Here,
a.

x 2 M 2 l X l + a 3 ( M - i ) 2 x ' 2 2 + 3a(Ai-4)2x22)da(5)This expression was evaluated in [7, 8] and represents the potential energyof perturbation.

IAEA-CN-47/E-n-2-2

129

Let us introduce a new independent variable, z = (a/a,) - 1, and consider
only the first non-vanishing term of an expansion ju - 1 = -sz near a = a,. Here,s = - / / ( a , ) . The quantity a, is the shear parameter.
Using the Fourier transformation
+ OO

f(k)=

f

x'.e^dz

we obtain

3k 2

3t 2

f=27rW6(k)f(6)

This equation governs the Alfven wave phase mixing process in a shearedmagnetic field.
Considering a wave of constant frequency, we can easily include the effectsof viscosity, resistivity and finite ion gyroradius (for the core ions), to obtain:
s2' — (1 +ik 2 /coSr 1 — + 2is 2 /wS (1 +ik 2 /coS)" 2ok
3k>+)F(z) + icoXk2 f=2;rW6(k)f(7)

Here, S = Tg/t^ > 1 is the magnetic Reynolds number, rg the resistive transit
time, X = VTg^2 the normalized viscosity coefficient, oj+ the drift frequency,
F(z) = (1 - I0(z)exp(—z))/z, z = p 2 a7 2 (l + k 2 ) and p ; the core ion gyroradius.
The first term in Eq. (7) characterizes the field line bending and the second termis the source of the tearing instability. The third and the fourth terms describe
inertial and viscous effects, respectively.

2.

PHASE MIXING AND SPECTRAL METHOD

Equation (6) describes the evolution of waves travelling along the k-axis andbeing scattered on the 5-function-like barrier at k = 0. Thus, any initial
perturbation will evolve to large values of k, and the maximum of the displacement will decrease roughly as x, ~ f/k. We shall call this process 'phase mixing'.
It is natural to assume the evolution of short, perpendicular wavelengths
to result in a further decay, due to the dissipation. This is why Eq. (6) must be
supplemented by the 'outgoing wave' boundary conditions, i.e. for |k| -*• °°,130

BEKLEMISHEV and POGUTSE

there will be only waves travelling away from the region of generation at smallvalues of k towards still larger values of this parameter.
By matching the constant frequency solution around k = 0, we obtain thedispersion relation obtained previously in Ref. [2]:-iw

+ TTW/S* = 0(8)In a more realistic situation, however, small effects may become importantat large values of k and partially reflect the outgoing wave back into the regionof generation or absorb it. If the return time of the reflected wave, t r = k r /s,
is shorter than, or comparable to, the time of instability evolution, Eq. (8) andthe assumption of an outgoing wave are no longer applicable. In other words,
the small effects may discretize the continuous Alfven spectrum and, if thegrowth rate or the decay rate is smaller than, or comparable to, the difference
between successive eigenfrequencies, it is only the spectral theory that is accurate.
Equation (7), with the usual boundary conditions Jirr^ f(k) = 0, predicts
a tearing instability with a growth rate of 7 T = s2 W"4/5 S"3/s for W > 0. With theinclusion of fast particles trapped in the region of unfavourable field curvature,a corresponding contribution to W is obtained and the excitation of an instability
at their toroidal drift frequency becomes possible [2]. For this instability to bevisible against a tearing background, it is necessary that co^S; y > 7 T . For Eq. (8)to be applicable, still higher frequencies are necessary: 7 2 w d h > s2/4S and^dh ^ co*- Therefore, the spectral method (7) is apparently more adequate fordescribing the 'fishbone' instability in large tokamaks.
3.

ALFVEN OSCILLATIONS WITH m = 1 AND ENERGY TRANSFER
INTO THE REGION OF LARGE kModern theory of collective transport processes in a current carrying
plasma (tokamaks) predicts a special role for oscillations in the region of wavelengths of the order of the collisionless skin layer width c/cope [5, 9]. However,there are no accepted mechanisms of energy pumping into the small scale
fluctuations. In this paper, Alfven wave phase mixing is shown to be a possible
candidate for this role.
Oscillations with an m = 1 poloidal wavenumber in a tokamak are of a
specific nature since the eigenfunction is similar to a stair-like function. However,for m = 1 it is also possible to obtain a dynamic equation in the Fourier space,and it will be a wave equation, although one of a more general form than thatgiven by expression (6).
In the limit of long wavelengths, the Alfven waves obey the well known
equation^

0(9)

IAEA-CN-47/E-II-2-2

131

In a sheared magnetic field, the parallel wavenumber is a function of theradius. Expanding k|| near the resonance point kj|(as) = 0: ky = k'yAx,
Ax = a — as and carrying out a Fourier transformation, we obtain:

where k2 = k2 + k2 k2, = m 2 a7 2 ss const.
The wave equation (10) describes the shift of any perturbation into theregion |k x | -»• °°, the velocity of this shift being constant and equal to cAk'||.
As the wave approaches k x ~ pr 1 , its dispersion properties change. We shallimmediately consider the limit k x p s > 1, where, instead of Eq. (10), we obtain:^ ^ ^ - k

i

A - ,

k

= 0

(.1)

This equation describes the so-called kinetic Alfven waves with a dispersion
relation co2 = 2 c ^ k | kjc 2 w~ 2 ; all perturbations propagate into the large k region,
with increasing velocity: k x ~ k x .
No energy change occurs during the spectral shift although there are certainchanges in amplitude. These may be expressed as A ~ (k2 + k2 )~'' 2 fork x p i < 1 and A ~ (k2 + k* r 1 ' 4 for k x p; > 1.
Let us assume that at the very short wave end of the spectrum (wherek x ~ io jc), the level of oscillations reaches the maximum, strongly turbulent
regime e^/T ~ 1/k^a ~ c/(w a). What amplitude of oscillations in the region
k x ~ I/a is necessary to justify this assumption? Matching the asymptotic
spectra at k x p 4 = 1, we easily obtain:
n/n 0 ££ etp/T s (m/M/3)l/2 < 1(12)This result yields fi/n0 = 0.2 for typical tokamak parameters (/3 ~ 1%),
which is virtually always observed in experiments. It should be noted that (12)holds only for j3 > m/M, i.e. pi > c/co . In the opposite limit, px < c/cj pe , the^k ~ k x : s P e c t r u m should be between I/a < k x < w p e /c, and the fluctuations
at k x a ~ 1 should be of the order of unity for the strongly turbulent regime.
4.

NON-LINEAR MODEL

Full dynamics include transverse (drift) vortices and the Alfven (or tearing)oscillations. For their general description we shall use the well known set of
equations governing plasma behaviour in a strong magnetic field [10]:
^

(13)

132

BEKLEMISHEV and POGUTSE

(14)
cB -*
dn
= - - — ( b , V ) A xi A
47re
dt

(15)

where
d
dt

3

_v

c

_^

X V 4 B = B o e z - [ e[ez X V A ] , b = B / B ,T == const
These equations differ from the original ones [10] by the inclusion of an
equation for the density [ 11 ], as was done in Refs [11,12]. In particular, inRef. [11] they were used to describe anomalous transport processes due toflutter. Here, we shall rewrite the system (13) to (15) in order to show thephysics more clearly, To do this, we write for the density:n = no(x) + ^- + 5n(16)

where 6n describes the non-Boltzmann part of the distribution n. Subtracting
Eq. (13) from Eq. (15), we obtain:
^

g

(17)

where
T d

Ii7

6nP

Mc2T

-7F~

For the variable g we can obtain an equation by combining (13) and (15):c\(b,V)A1(b,V)- — iThis equation is only an approximate one since small insignificant corrections
were omitted in its derivation. Equations (17) and (18) may be rewritten as asingle equation, but in the present form they are clearer; the vortices
governed by Eq. (17) generate AlfVen perturbations (18), which in turn (through g)
feed the vortices.

IAEA-CN-47/E-II-2-2

133

The linear dispersion relation, following from (17) and (18), is wellknown [5]:( w - w , + cok2p2)(u-k2cA) = w3k2p2(19)Strong interaction between Alfven and drift waves occurs in the region <»;„ se k||CA.
To the right of the interaction point k a c A > cot and for the drift branch, weobtain, as an estimate from (18): g ~ Gj;Jp2(k||CA)-2Aj_^. In this case, thecorrection with g on the right hand side of Eq. (17) is small, and we get a normal
Hasegawa-Mima equation [13]. In the opposite limit, k|C A < co,, for the drift
branch, we have g = -2(d/dt)Ajy> and the chief non-linear terms in (17) vanish.
This can lead to an additional concentration of oscillations in the region
Ax < p y/^ji e ju. In general, Eqs (17) and (18) describe the pumping ofoscillations into the region of large k.
To demonstrate this statement, let us consider the case where ^ is governedby the Hasegawa-Mima equation (d/dt)(^-p 2 AJ^J) = 0. Let us take the turbulent
limit to describe the evolution in detail. It is well known that the Hasegawa-Mima
equation gives a spectrum with a flux of the form <p2 ~ k ~ a , where 2 < a < 4 [14].
In the case <p2 ~ k"4, (18) has a solution in the k-space, independent of k:g~co,,p 2 k 2 <£~ co.p 2 (let us recall that (b,V) 2 ~ 9 2 /dk 2 ). The variable g, whichis independent of k, leads to a 8-function-type source (in real space), on theright hand side of Eq. (17). Taking multiple resonances into account, we write= )S is a sum over all resonances, y?r is the potential averaged over a cell with given
helicity. It is a source of this form that describes the generation of small scale
oscillations.
It is conceivable that this result, obtained for the <p2 ~ k~4 model, has amuch broader applicability since, in the general case, the operator on the left
hand side of (18) provides an efficient averaging of the right hand side over k x .
The appearance of a source on the right hand side of (17) of the form25(x-x r )i^ r will lead to a slower drop in the spectrum for large values of k x ,i.e. to the very effect that we discussed above.
These considerations suggest the possibility of two-dimensional simulationof a three-dimensional spectral problem in the large k region. In general, Alfven
oscillations may be replaced by a source of given magnitude localized near the
resonant flux surface.

134

BEKLEMISHEV and POGUTSE

REFERENCES
[1]
[2]
[3]
[4]

WHITE, R.B., GOLDSTON, R.J., et al., Phys. Fluids 26 (1983) 2958.
CHEN, L., WHITE, R.B., ROSENBLUTH, M.N., Phys. Rev. Lett. 52 (1984) 1122.
McGUIRE, K., GOLDSTON, R.J., et al., Phys. Rev. Lett. 50(1983)891.
GRIBKOV, W.M., KADOMTSEV, B.B., POGUTSE, O.P., Pis'ma Zh. Ehksp. Teor. Fiz.
38(1983)14.
[5] KADOMTSEV, B.B., in Voprosy teorii plazmy (LEONTOVICH, M.A., Ed.), Vol. 4,
Atomizdat, Moscow (1964) 183.
[6] MIK.HAILOVSKIJ, A.B., ibid., Vol. 9(1979) 3.
[7] BUSSAC, M.N., PELLAT, R., et al., Phys. Rev. Lett. 35 (1975) 1638.
[8] ZAKHAROV, L.E., Fiz. plazmy 4 (1978) 898.
[9] OHKAWA, T., Phys. Rev., A 67 (1978) 35.
[10] KADOMTSEV, B.B., POGUTSE, O.P., Zh. Ehksp. Teor. Fiz. 65 (1973) 575.
[11] KADOMTSEV, B.B., POGUTSE, O.P., Pis'ma Zh. Ehksp. Teor. Fiz. 39 (1984) 225.
[12] COZZANI, F., HORTON, W., Preprint IFSR-214, Univ. of Texas at Austin (1985).
[13] HASEGAWA, A., MIMA, K., Phys. Fluids 21 (1978) 87.
[14] HASEGAWA, A., MacLENNAN, C.G., KODAMA, Y., Phys. Fluids 22 (1979) 2122.

IAEA-CN-47/E-II-2-3

EFFECTS OF ENERGETIC PARTICLESON TOKAMAK STABILITY
J. VAN DAM1, M.N. ROSENBLUTH1, H.L. BERK1,
N. DOMINGUEZ1, G.Y. F U \ X. LLOBET1,
D.W. ROSS2, D.P. STOTLER1
1
Institute for Fusion Studies,
2
Fusion Research Center,
University of Texas at Austin,
Austin, Texas,
United States of America
D.A. SPONG, W.A. COOPER
Oak Ridge National Laboratory,
Oak Ridge, Tennessee,
United States of America
D. SIGMAR, D.E. HASTINGS, J.J. RAMOS
Plasma Fusion Center,
Massachusetts Institute of Technology,
Cambridge, Massachusetts,
United States of America
H. NAITO, T. TODOROKI
Institute of Plasma Physics,
Nagoya University,
Nagoya, Japan
S.T. TSAI, S.G. GUO, J.W. SHEN
Institute of Physics,
Academia Sinica,
Beijing, China

Abstract
EFFECTS OF ENERGETIC PARTICLES ON TOKAMAK STABILITY.
Energetic particles (for instance fusion product alphas or beam-injected/rf-heated ions) can excite
MHD ballooning modes through wave-particle interaction at their large magnetic drift frequency. The
use of a "sloshing" distribution avoids resonant destabilization, but tends to excite a new type of continuum instability. At higher energies, the usual MHD unstable modes are suppressed, although subtler
instabilities related to shear Alfve'n gap modes and global Alfve'n waves occur. When these low growthrate modes are neglibible, access to the high-beta regime of second stability can be attained with sufficiently hot energetic particles.

135

136

VAN DAM et al.

1. INTRODUCTION
Energetic particle species may be produced in tokamaks by neutral beam
heating, by rf heating, or, in ignition devices, by alpha particle production.
Due to their large magnetic curvature drift velocity and possibly superAlfvenic parallel velocity, these hot particles can significantly affect the low
frequency MHD and also the higher frequency Alfvenic stability propertiesof tokamaks, depending on their average energy. In this paper, we discussand present detailed results on a number of such issues.
2. MHD BALLOONING MODES
2.1. Moderately Energetic Regime
A moderately energetic hot particle species whose typical magnetic curvature and gradient-i? drift frequency uJdh is comparable to MHD frequencies can excite new resonant instabilities such as "fishbones" and their highfrequency precursor oscillations. These modes were observed in the PDX
tokamak and have been associated, respectively, with trapped particleinduced internal kink and ballooning instabilities [1]. Here we consider
the latter.
The behavior of high-toroidal-mode-number ballooning modes interacting with an anisotropic population of energetic trapped particles can be
described with the vorticity equation+ e • K (e- VP,| + - e • VP X ) $ + oBe- K ( Q , | - °-Be • K $ ) = 0 (1)and the perturbed pressure balance equation

Here, $ is related to the magnitude of the field line displacement; Q\\ isrelated to the parallel component of the perturbed (Lagrangian) magnetic
field; angle brackets denote bounce averages: perturbed quantities are assumed to vary as exp(iS -iut), where B • VS = 0, with B the equilibrium
magnetic field; and PXl\\ = Pe(i/>)+ Px,||/»(^S-S) are the total pressure components of the isotropic core plasma and anisotropic [P||/,/PJL/, ~ a/R <C 1J
hot species. Also, we havecr = l + 47r(P L -P ) |)/2? 2 , r = 1 + 4*- (dP±/BdB),vA = (5 2 /4rriV i M i ) 1/2 , K = [(B/B) • V] (B/B), e = VS x B/B2. V =V - {VB)(d/dB), ft = Mv\/2B, and Mv\ = 2(E - pB). The hot particle

137

IAEA-CN-47/E-II-2-3

0

0.2

0,1 0.6

0.6

I.O

i.2

(.4

I.6 4.8

0.2

0.4
aH/ac

0.6

FIG. 1. (a) Dependence of (s, aj stability boundaries on Tefor a,/a c = 0.3, q = 2, kLpL = 0.28,
and ep = R\d In PJdr\ = 6.
(b) Ratio of $cril = (ep/q2)accri, with alphas to that without alphas, at s = 0.6, versus ah/ac forvarious Te.

diamagnetic frequency is w*ft = -(c/ehZh)(e- VFh)(dFh/dE)-x
and theirmagnetic drift frequency is u>dh = (c/ehZh)e- (MI)|JK + fiVB^. Also, the total ion diamagnetic frequency, including both core plasma and energetic ion
species, is w*,- = (B/NiMi) J2 (e • VP±i) /Clj, with ft,- = ejZjB/Mjc. Equa3

tions (1) and (2) were solved numerically to obtain the stability boundaries
for these resonant modes, with various models for the energetic particle
distribution function Fh. The low-beta, large-aspect-ratio model tokamak
equilibrium with shifted circular flux surfaces and steepened pressure gradients was adopted.
Figure l(a) shows the results [2] for a collisionally slowing-down distribution of alpha particles with a deeply trapped pitch angle dependence:
Fh(E,(i) oc {E*!2 + Ell2)'1 6((xB0/E - 1), where Ec is proportional to thecore plasma electron temperature Te. Increasing Te is strongly destabilizing, which can be understood as increasing Re(u>) through w*,- (if T) = Te)and thus maintaining the destabilizing resonance condition Re(w) ~ Udh
at higher energies. In particular, the unstable region at low shear, s =d(£n q)/d(£n r),and high values of the normalized core pressure gradient,ac = -(8irq2R/B$)(dPc/dr)>is
enlarged; q is the safety factor. This typeof flat q, peaked pressure profile will be characteristic of low q(a) ignited
plasmas, especially those with large alpha-driven sawtoothing. Figure l(b)indicates the Te effect in lowering the critical beta value. Results similar

138

VAN DAM et al.

unstablestableunstable

0.5

1.0

1.5

2.0

FIG. 2. Stability boundaries in cth versus wdi/wAfor the low-frequency MHD branch (lower curve) andthe high-frequency precessional branch (upper curve).

to those of Fig. l(a) have also been obtained for an isotropic pitch angle
dependence: the ballooning mode is somewhat more stable at large shear.
However, at low shear (s ~ 0.2), both types of pitch angle distributions
prevent entry into the second stability regime if Te ^ 40 keV. This feature
may provide an automatic burn control mechanism for ignited tokamaks.
As the hot particle transverse energy Txh is raised such that wj/, becomes comparable to the Alfven frequency U>A = VA/QR, the threshold forthese resonant instabilities increases. Figure 2 shows that a stable operating window exists for intermediate values of the energetic particle pressure
gradient ah = -(8Trq2R/ B$)(dPlk/dr) when &dh/u>A Z 0.6. For PDX parameters (B - 2 T, Ni = 3 x 1013 cm" 3 , a = 25 cm, q - 2) and toroidal
mode number n = 2, this would require T±h > 500 keV. Below the lower
marginal stability boundary, which is independent of Tj_fc, the instability isa nearly purely growing MHD mode, whereas above the upper boundary,given approximately by a/, > 3(a7d/l/w>i), the unstable mode is characterized by a finite oscillation frequency Re(u>) RS Udh- The results of Fig. 2
were obtained [3] with the neglect of w*;/w effects and for a Maxwellian
distribution function of the form
- E) exp{-E/Txh)

E/Bc

(3)

where Bc = Bo f 1 — — cos i?0) is the field strength at the angle i90 thatmeasures the poloidal extent of the trapped particles and where AA controls
the peaking of the pressure profile.

139

IAEA-CN-47/E-II-2-3

FIG. 3. Maximum growth rate ymw. (curve i) and frequency Re u> (curve ii), normalized to the Alfven
frequency uA, versus ah for a sloshing distribution.

A "sloshing" population of energetic particles strongly pressure-weighted
in the favorable curvature region, which may be created by off-normal neutral beam injection or by cyclotron heating on the small-major-radius side,
docs not cause resonant destabilization, at least for those particles whosebounce-averaged magnetic precessional drift is reversed: W*A {<^dk) < 0.
However, a study [4] of such a sloshing distribution, given by Eq. (3) with
AA <C 1 and i90 close to TT, reveals a new type of continuum instability. A circulating anisotropic component was added to the distribution of
Eq. (3) to satisfy the mirror mode constraint, r > 0. For large i9, where
the extended poloidal angle d of the ballooning representation describes thevariation along a field line, the wavenumber is secular, |V5| 2 —* (si?)2; also,
the geodesic curvature dominates, e- K —> itfsintf. Thus, the asymptotic
form of Eqs. (1) and (2) isrd3y(?-VF
W—

J

1-

= 0

B
1

-

(4)

(udh)

The wavefunction $(#) = d • $(i9) in Eq. (4) can be expressed in Floquet
form: $($) = ^'('d)exp(ikd), where $ is periodic. Equation (4) is found tohave a nearly purely growing solution, whose growth rate 7 = lm(uj)/u>A,
shown in Fig. 3, is maximum when k = 1/2. An analytical estimate with adelta-function pressure model, Pxh($) <* £(i? + i?o)+^($~-i?o)> confirms this

VAN DAM et al.

140
0.15

0.10 -

0.05

-

0.00

40

FIG. 4. Stability diagram in core plasma beta on axis, (30, versus flux surface \p for a fixed q profile.
Unstable surfaces in the absence of hot particles (H = 0) are shown byx; the small unstable region (U)shown for H = 0.3, where H <= (t3Li,/t3c, disappears for H a 0.5. Drift reversal (V < 0) occurs
above the upper curve.

zero-frequency instability above an Oh threshold. At small a/, values, however, as shown in Fig. 3, the mode converts to a finite-frequency resonantly
unstable Alfven gap mode, to be discussed in Sec. 3.1.
2.2. Highly Energetic Regime
On the other hand, energetic particles that are trapped in the largemajor-radius, unfavorable curvature region of a tokamak and whose transverse energy is high enough to preclude resonant interactions can provide
enhanced ballooning stabilization and access to the high-beta second regimeof stability. This has been confirmed by 2D numerical computations [5] ofballooning stability for finite aspect ratio (R/a = 3) anisotropic equilibria.
Figure 4 shows that the number of unstable flux surfaces if> decreases as H,
which is approximately the ratio of the peak perpendicular pressure of the
hot particles to the core plasma pressure on axis, increases; for H > 0.5, allsurfaces are stable. Current programming has been assumed to raise qm (onaxis) above unity in order to reduce the second stability threshold below
the drift reversal curve, where V = u*h (wdh) > 0 and where the analysis
yields a sufficient condition for stability.
Access to second stability has also been explored [6] by means of atime-evolved solution for the radial profile of the plasma pressure in a ballooning mode transport formalism. The hot-particle-enhanced marginal
stability relationship between ac oc (r/Bp)2(dPc/dr) and the shear s =Bpd(r/Bp)/dr can be specified, where the poloidal field Bp is evolved self-

IAEA-CN-47/E-II-2-3

141

consistently with the pressure Pc(^), and the thermal conductivity (normalized to INTOR scaling) is taken to be unity in the stable region andanomalously large but finite in the unstable region. Fairly high auxiliary
heating power is found to be required for the plasma to penetrate the unstable zone and attain a second-stable steady state.
The detailed structure of the second stability regime for ideal ballooning
modes was also investigated [7] for large pressure gradients, without hot particles. The threshold for first stability is most stringent for modes that peakon the tokamak midplane, whereas second stability is degraded for off-angle
modes. The numerical results yield the empirical scaling ac ~ 4.2s1'25 for
2 < s < 100 as the most stringent second stability boundary. At low shear,
the computed boundary joints onto the known relationship a = 2.13s 05 .
Noting that the behavior of the eigenfunction is primarily determined bychanges in slope at points where the local shear or the integrated local shear
vanishes, one can analytically derive a = 4.6s 125 (1 — 0.3J~°l13) ' for s ~S> 1
by taking into account only one such pair of neighboring points.
Finally, even if the ideal MHD modes and the alpha particle resonantly
excited modes can be stabilized, resistive ballooning instabilities may persist. The A' parameter that controls their stability was obtained [8] numerically and, in a low beta limit (ac ~ ah ~ s < 1), analytically from the solution in the ideal region with the presence of highly energetic, nonresonantparticles. The dispersion relation A'(w) was then derived by asymptoticallymatching to the solutions in the resistive and deeply resistive regions, with
the inclusion of parallel compressibility and cross-field transport effects.
The critical value for A' indicates that the hot species in the nonresonant
limit tends to stabilize the resistive modes, at least for ac values below theideal threshold.

3. ALFVEN MODES
3.1. Continuum Gap ModesAt the high energies,such that UJdh is comparable to or larger than u>A,interaction between the hot particles and shear Alfven modes occurs. In
particular, a new mechanism for continuum gap eigenmodes is found [9].
Note that the third term in Eq. (4), which is proportional to o^, introduces
a Mathieu equation type of modulation. For a square-top hot pressure
profile (ofA=const. for |?9| < i?0, zero elsewhere), the spectrum for the characteristic exponent A; = Ar(w) can be derived. Frequencies within gaps ofwidth proportional to a/,, which are located in the MHD stable continuum
near the points where u/u)A = 0.5,1.0,1.5,2.0,.., lead to spatially amplifying solutions at |i?| —> 00. Within each such gap, however, there exists
a discrete eigenmode that is spatially evanescent. This high-mode-number

142

VAN DAM et al.

localized gap mode can be resonantly excited, without threshold, by the
hot particle-wave magnetic drift resonance. Its growth rate is proportional
to ah but decreases inversely with UJdh/^A', usually its magnitude is largest
when W/U>A ~ 1- For the parameters used in Fig. 3, one observes the
IJJ/IJJA = 0-5 shear Alfven gap mode instability at low ah values.
3.2. Global Alfven Waves
Energetic particles whose parallel motion is super-Alfvenic can resonantly destabilize the global Alfven eigenmodes. These MIID-like modes
have discrete frequencies lying just below the Alfven continuum, i.e., u>2 <min(u;^), where U>A — Vj\(nq + m)/qR and {n,m) are the toroidal andpoloidal mode numbers. They are very weakly damped by electron Landau damping and are therefore more strongly destabilized than the kinetic
Alfven eigenmodes [10]. The instability is driven by the alpha particle
density gradient. Coupling to sidebands occurs through the curvature andgradient-Z? drift of the alphas. A necessary condition for instability is
u*Jto > (2TJma) (dFa/dv*)± / (Fa)±(5)

with {.. .) ± = J <Pv (u 2 /2 + vj|) 6(u- %m±i«||) ..., To = j
d3vFamav2/3,
w*a = Tac/eB0LQ, and La = —n~ldna/dr. For low m axisymmetric modes{n = 0) and typical CIT parameters, we find instability when La/a < 0.5
for a slowing-down distribution. Since the birth distribution of the alphas is
expected to be narrower (roughly La/a = 0.4) than this critical value, themodes are likely to be unstable, leading to possible radial transport. (Studies employing Maxwellian alphas yield somewhat more optimistic stability
estimates.)Figure 5 shows the damping effect of increasing La for the (0,-2) mode,
as obtained from a full numerical calculation. A simpler analytic calculation
gives nearly identical results. For the CIT parameters, we assume R =1.4 m, a = 0.67 m [representing (ab)1^2 for a shaped tokamak], Bo = 10 T,
Te(0) = Ti(0) = 30 keV [volume average 10 keV], and ne(0) = n;(0) =1021 m~3 [volume average 5 x 1O20 m~ 3 ]. The alpha particles are describedby a gaussian radial distribution with n a (0) = 2 X 1019 m~3. The slowingdown distribution yields T a « 1 MeV, with the Maxwellian Ta taken to
be the same for comparison. The real mode frequency is W(Oi_2) = 4.05 x
106sec~1. The (0,-1) mode has a much weaker growth rate than the (0,-2)
mode. However, the inclusion of toroidicity or noncircularity effects may
reduce the (0,-2) growth rate by introducing an Alfven resonance throughcoupling to the continuum of the (0,-1) mode.
Toroidicity can also induce a new class of global low-ra discrete eigenmodes, with frequencies that lie within gaps in the shear Alfven continuum

IAEA-CN-47/E-H-2-3

143

slowing down
distribution-4X10 c

•8X10 J
0.20

0.30

0.40

0.50

0.60

FIG. 5. Growth rate y of axisymmetric (0, -2) global Alfven eigenmode for CITparameters versus thedensity gradient scale length La, for 1 MeV slowing-down and Maxwellian alpha particles.

[11]. These toroidicity-induced shear Alfven eigenmodes can be destabilized by inverse Landau interaction with energetic particles. The growthrate for such a toroidal gap mode due to coupling between the (-1,-1) and(-1,-2) modes was estimated to be approximately an order of magnitudelarger than that for the most unstable cylindrical global Alfven eigenmode,(0,-2), described previously.

4.

CONCLUSIONS

The following picture emerges of the effects of energetic particles on
tokamak stability. At moderately high energies, characteristic of neutralbeam-injected ions or medium energy alpha particles (for example [11],Tia/Ti < 150), resonant destabilization of macroscopic MHD modes such as
ballooning is to be expected when the beta value of the hot particle trappedcomponent exceeds a modest threshold. In particular, flat q and peakedpressure profiles that will be characteristic of low edge q ignited plasmas
with alpha particle driven sawtoothing are susceptible to instability; hence,
sawtooth-free operation is desirable if the trapped alpha resonance effects
cannot be stabilized. Increasing the plasma electron temperature (Te >40 keV) can also degrade stability; this feature could provide an automaticburn control mechanism. The destabilizing effects of moderately energetic
particles can be avoided with the use of "sloshing" distributions that arepressure-weighted on the small major radius side; however, these driftreversed hot particles tend to drive a purely growing continuum instability,with growth rates comparable to MHD rates.

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VAN DAM et al.

At higher energies - typically 500 keV for PDX-Iike parameters or afew MeV for reactor parameters - the usual MHD modes are stabilized.
The energetic particles can permit access to the high-beta second regimeof ballooning stability, although fairly high auxiliary heating power may
be required during the dynamic process. Also, second stability is more
stringent for off-midplane modes. In the nonresonant limit, the energetic
particles do tend to stabilize low-beta resistive ballooning modes.
At these high energies, however, the energetic particles interact withthe shear Alfven continuum to produce a new unstable discrete gap modewith no threshold, as well as to resonantly destabilize the low-n toroidicityinduced gap eigenmode. Moreover, for the parameters of the proposed
Compact Ignition Tokamak, the birth distribution of alpha particles is expected to be narrow enough to lead to destabilization of the global Alfven
modes. Thus, although the MHD modes can be suppressed at high energies,
subtler instabilities related to Alfven waves are predicted, whose nonlinear
and experimental consequences warrant scrutiny.

ACKNOWLEDGMENTS
This work was supported in part by the U.S. Department of Energy
under Contract No. DE-FG05-80ET-53088 with The University of Texas
and Contract No. DE-AC05-84OR-21400 with Martin Marietta Energy Systems, Inc., and Massachusetts Institute of Technology.

REFERENCES
[1]

[2]

[3]
[4]
[5]

[6]

CHEN, L., WHITE, R.B., CHENG, C.Z., ROMANELLI, F., WEILAND, J., HAY, R.,
VAN DAM, J.W., ROSENBLUTH, M.N., TSAI, S T . , BARNES, D.C., in Plasma Physics andControlled Nuclear Fusion Research 1984 (Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA,
Vienna (1985) 59.
SPONG, D.A., SIGMAR, D.J., TSANG, K.T., RAMOS, J.J., HASTINGS, D.E.,
COOPER, W.A., Effects of Isotropic Alpha Populations on Tokamak Ballooning Stability,
Rep. ORNL-TM-10200, Oak Ridge National Lab., TN (1986).
STOTLER, D.P., The Effect of Energetic Particles on Stability of Mirror and Tokamak Plasmas(PhD Thesis), Rep. IFSR-238, Inst. for Fusion Studies, Texas Univ., Austin (1986).
STOTLER, D.P., BERK, H.L., The Effects of Sloshing Energetic Particles on Ballooning
Modes in Tokamaks, Rep. IFSR-259, Inst. for Fusion Studies, Texas Univ., Austin (1986).
NAITOU, H., VAN DAM, J.W., BARNES, D.C., Energetic Particle Stabilization of Ballooning
Modes in a Finite-Aspect-Ratio Tokamak, Rep. IPPJ-797, Inst. of Plasma Physics, Nagoya Univ.
(1986).
FU, G.Y., VAN DAM, J.W., ROSENBLUTH, M.N., Effects of Ballooning Instability on
Tokamak Confinement, Rep. IFSR-254, Inst. for Fusion Studies, Texas Univ., Austin (1986).

IAEA-CN-47/E-n-2-3
[7]

145

LLOBET, X., BERK, H.L., ROSENBLUTH, M.N., Finite Pressure Ballooning Mode Stabilityin Toroidal Equilibrium, Rep. IFSR-263, Inst. for Fusion Studies, Texas Univ., Austin (1986).
[8] DOMINGUEZ, N., Stability of Ballooning Modes in Tokamaks with Energetic Particles (PhD
Thesis), Rep. IFSR-245, Inst. for Fusion Studies, Texas Univ., Austin (1986).
[9] VAN DAM, J.W., ROSENBLUTH, M.N., in Sherwood Annual Controlled Fusion Theory Conference (Proc. Conf. Madison, WI, 1985), Wisconsin Univ., Madison (1985) Paper 1C6.
[10] LI, Y.M., MAHAJAN, S.M., ROSS, D.W., Destabilization of Global Alfen Eigenmodes andKinetic Alfven Waves by Alpha Particles in a Tokamak Plasma, Rep. FRCR-276, Texas Univ.,
Austin (1985)[11] CHENG, C.Z., CHANCE, M.S., Phys. Fluids 29 (1986) 3695.
[12] SPONG, D.A., SIGMAR, D.J., COOPER, W.A., HASTINGS, D.E., TSANG, K.T., Phys.
Fluids 28 (1985) 2494.

DISCUSSION
ON PAPERS IAEA-CN-47/E-II-2-1 TO E-II-2-3
J. A. WESSON: Do you still accept your original interpretation of the fishbone
activity observed on PDX?
L. CHEN: Since for PDX parameters, wdh ^ coR, ideal MHD approximation
remains valid.
J. A. WESSON: Further, do you have any understanding of the observation offishbone instabilities on JET which should be stable according to theory?
L. CHEN: No. Our prediction is based on the assumption that the equilibria
in JET are more or less similar to those in PDX. If this is not the case, as suggested
by your sawtooth theory, we need to re-examine the situation for JET.
J.A. WESSON: A final point — would quasi-interchange m = 1 behaviour
have fishbone stability different from the m = 1 kink behaviour which you assumed?
L. CHEN: We have not carried out detailed analyses and so I can only speculate. Since the eigenmode structures and the singular layer properties are different,
I would say that this is the case.
Mingsheng CHU: Can the fishbone excitation process which you described be
operative at the q = 2 surface? If so, what is the signature of such a mode in the nonlinear regime?
L. CHEN: If I stretch the argument, I could regard modes localized at about
m/n = q = 2 as either ideal ballooning modes or resistive interchange modes. Since
these modes are radially localized and break the third invariant of energetic trapped
particles, the instabilities would definitely lead to local flattening of hot particles
through instability induced outward diffusion.
K. ITOH: When the stability of the modes in the presence of alpha particles
is estimated, the result depends on the assumption regarding the source distribution
in real space and in velocity space. What assumption is made and which model is used
for the conclusion that there is alpha driven instability?
My second question concerns the excitation of high k± ~ «p/c modes. The
convection to the high k-axis in the k-space corresponds to the condensation near therational surface. Can global modes excite k ± ~ <op/c modes on every magnetic surface, and do you need many high m-modes?
L. CHEN: For alpha induced fishbones, the alpha velocity space distribution
is taken to be sufficiently smooth and the modes are therefore essentially driven by
the real space gradients. Our results are estimated on the basis of transport code simulations of alpha particle distributions.
Regarding your second question, strictly speaking, propagation to high k corresponds to mode behaviours as exp (itlr-r s l) (i.e. k a t), and condensation
around mode rational surfaces occurs if dissipations are introduced.

147

IAEA-CN-47/E-II-3-1

NONLINEAR TOROIDAL PLASMA DYNAMICS
BY REDUCED FLUID MODELS
M. KOTSCHENREUTHER, A. AYDEMIR, R. CARRERA,
R.D. HAZELTINE, J.D. MEISS, P.J. MORRISON
Institute for Fusion Studies,
The University of Texas at Austin,
Austin, Texas,
United States of America

AbstractNONLINEAR TOROIDAL PLASMA DYNAMICS BY REDUCED FLUID MODELS.
Fluid models are presented which generalize reduced MHD by allowing for compressibility,
Finite Larmor Radius, and long mean free path in toroidal geometry. The Hamiltonian structure of themodels leads to a generalized energy principle for determining linear and nonlinear MHD stability ofequilibria with flows and Finite Larmor Radius effects. Neoclassical effects from the long mean freepath lead to new dissipative terms giving rotation damping, bootstrap currents and the Ware pinch. Rotation damping from non-ambipolar transport in stellarators can lead to self-consistent plasma currents
which strongly reduce or "heal" steady-state magnetic stochasticity (e.g., from coil errors or
Pfirsch-Schliiter currents). The bootstrap current in tokamaks causes the growth of nonlinear magnetic
islands in the Rutherford regime. Thus, otherwise stable moderate mode number islands can potentiallyoverlap, with serious detriment to confinement.

INTRODUCTION
We present models that generalize reduced MHD by allowing for compressibility, Finite Larmor Radius, and long mean free path. The Hamiltonian
structure of the models leads to a significantly generalized energy principle.
Applications include linear and nonlinear instabilities and results regarding
self-consistent magnetic stochasticity.
1. MORE REALISTIC FLUID MODELS
A. Conventional high /3 reduced MHD[l] contains three fields: the poloidal
flux function tj>, a velocity stream function <f>, and pressure p. To allow forcompressible flows, we add parallel velocity. Previous Alfvenic normalizations are used[l]. FLR terms enter proportional to a parameter 6, definedso that the normalized diamagnetic velocity (cT/neB)z x Vn/ev^ is equal
to 6z x Vp. We also define [Qi,Q2] = z • VQi x VQ 2 , V\\Q = f ? ~ [$,Q]->
F=<t> + k6p, W = V]_F, J = V2±ip, with k = Tt/Te. The conserved energy
is

149

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KOTSCHENREUTHER et al.

We therefore write the four-field model with the terms needed for energy conservation on the left, and higher order FLR terms needed to be
Hamiltonian on the right.
W + [F, W]-6kV • [p, VF] + Vh J + (1 + k)[h,p]={k6fi/2){ - 26(1 + k)V3±[h,p] - V* V,|(« + 26J)
(2)=0[F,p] + /?V,|(t» + 2SJ)-2/3[h,F~(l(3)+ k)6p] = U20{P+ 20h, W] (4)+ [F - kp,v] + ^(1 + fc)V,|P = -fc6 [v,2/3/> + ki2i3V\p - 26/JVV]^(5)

where h gives the effects of curvature; h is the normalized distance in the
direction outward from the torus.
B. In modern tokamaks the collisional mean free path exceeds the device
size. Nonetheless, fluid equations can be rigorously derived to describe theregion near a rational surface. The calculation uses kinetic theory together
with a systematic two-scale expansion in the parallel gradients, and is similar in spirit to previous MHD analysis by Glasser et al. and Kotschenreuther
et al. For simplicity we use an aspect ratio expansion and plateau collisionality. For maximum generality, we include FLR effects, parallel compressibility,
resistive flux diffusion in the inner layer, semicollisionality, rotation dampingof parallel velocity, nonlinear convection and nonlinear modifications of theequilibrium field. A maximal ordering to systematically include all the above,in which all quantities are ordered in terms of e, is: Ar ~ p p , pp/r ~ e2,u) ~ e2Vi/R, f$ ~ e2, BpJB ~ e, veR/ve ~ e, Jme/rrii ~ e, where Ar, r, Vi,and pp are the layer width, minor radius, ion thermal velocity and poloidal
gyroradius.
The final fluid equations have non-dissipative and dissipative terms usually found in slab models, but also additional dissipative terms giving rotation damping, bootstrap currents, and the Ware pinch; similar effects werepreviously found by Callen and Shaing[2]. The dissipation-free terms are,in fact, the same as those on the left sides of Eqs. (2)-(5), but with h = 0(also, the terms on the right do not occur—they are higher order). The newneoclassical terms in Eqs. (2) and (5) are(6)

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151

(7)
For an axisymmetric tokamak with nearly circular surfaces,!/ — i/nfliq, 6 =8' — the ratio of poloidal to toroidal field. We have general expressions for
non-circular non-symmetric geometries, which are omitted due to space.
The vorticity is damped at a rate i/, and the parallel velocity is damped
at a rate 08'u ~ e2u. For moderate /?, the rotation damping term is muchlarger than the usual inertia] term in Eq. (2). For axisymmetry the damping
terms tend to equilibrate both F and v to satisfy v — -O'^dF/dr, as in equilibrium neoclassical theory. The toroidal angular momentum is conserved by
the damping terms for axisymmetry. In a stellarator, \8'\ > |0|. Then thedamping terms tend to make both F and v vanish. This also is the condition in equilibrium neoclassical theory, where the ions are electrostaticallyconfined.
The Ohms' Law and pressure evolution, Eqs. (3) and (4), have neoclassical dissipative terms

For equilibrated ion flows, these give a bootstrap current and Ware pinch
in agreement with equilibrium neoclassical theory. For plateau collisionality,a ~ eve/'Rve.

2. ENERGY PRINCIPLES FOR NONLINEAR STABILITY
Here we present generalizations of the ideal MHD energy principle, 6W, forfluid and kinetic models, that can handle equilibrium flow and FLR effects.
Consider a noncanonical Hamiltonian system with field variable f(z,t).
Functionals called Casimirs exist for such systems, which are constants of the
motion. (For example, in the Vlasov equation with distribution function / ,f dx dvG(f) is constant for any function G.) Denote such a Casimir by C[f],and denote the Hamiltonian by H[f]. It has previously been shown[3,4] thatfor each Casimir C, an equilibrium can be found by solving A' — A ',where 6/6f means functional differentiation with respect to / . Consider
perturbations 6f away from /o- We have— ^ / 2 +Higher order terms

(10)

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KOTSCHENREUTHER et al.

H[fo + Sf]-H[fo]=Jdz — Sf
2

i r s H+ - I dz —— 6f2 + Higher order terms
^J
°J

(11)

Now we can obtain the change in energy from a perturbation 6f. Since thesystem is constrained by C(f) — constant, the only accessible perturbations
are those where the left side of Eq. (10) vanishes. Then by the equilibrium
condition we have that the change in H at constant Casimir C, AH | , is
AH\c = - / dz- K—'-6f2 •+ Higher order terms

and we define F = H — C to be the free energy for accessible perturbations.
As expected, the perturbed energy is second order in 6f. Also note that theenergy released or absorbed to create an allowable perturbation 6f depends
on the equilibrium through C.
If 62F is positive definite, then all accessible small perturbations 6f increase H. Since H is conserved, no small perturbation can grow indefinitely,
and we have both linear and nonlinear stability for small bf. (Technicallythis only implies formal stability, but the step from formal to nonlinear stability is usually straightforward[3,4].)Note that we have not considered the linearized equations of motion tocome to this conclusion. The linearized equations depend on both H and
the Poisson bracket. The latter can be quite complicated for noncanonicalsystems.
If 62F is indefinite, then either there is a linear instability or a linearlystable negative energy wave (or direction in function space). It is not possible
to determine which without considering the Poisson bracket. Nonetheless,note that a linearly stable negative energy wave can lead to serious nonlinear
instabilities. For example, explosive instabilities are possible in Hamiltoniansystems with three interacting waves with resonant frequencies. More complicated nonlinear phenomena are also possible in continuous media; these
are currently under investigation. The important point is that when 62F isindefinite, an instability of some kind is likely.
Negative energy waves can also be found in simple cases by applying
the Brillouin-Laue condition wde/du> < 0. We can show that this condition
must agree with 62F. However, e can only be easily defined in uniform
homogeneous media. Dielectric response theory in non-homogeneous media
is very lengthy or intractable. The free energy functional 62F is far morepractical in such cases.
Thus, we believe that 62F is a better tool than linear spectral theory tofind the total instability potential of many configurations.

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153

As an example we consider reduced low (3 MHD with poloidal flow.
Suppose we wish to consider stability of cylindrically symmetric equilibria
to perturbations with a given helicity. Let x be the relevant helical flux,
X — V> - r2/2q0. The Casimirs are / H(ip)dx and JUG(^), for arbitrary
functions H and G. The free energy F is
(12)

Equilibria are given by V~_ (X + r2/dq0) = H'(X) + V\GG', 4> = G(X). To
examine stability, take the second variation and rearrange terms to get

62F = I ( \+ SX2 [G"V2G + H" + G'V • G"VX] )dx(13)

An instability or negative energy wave exists if 62F can be made negative.
To minimize 62F, choose 6<f> to make the first term vanish. The minimizing 6x can be found by standard techniques, e.g., solve the relevant Eiders
equation, or for an approximate answer, insert trial functions and vary theirparameters, etc.
For equilibria without flow, G = 0, the Eulers equation is equivalent tothe small aspect, ratio 8w result for ideal kinks; the second term gives thestabilizing influence of line bending, and the H" term gives the destabilizing
influence of current gradients. We see that the nonlinear stability with flow
can be found by a similar Eulers equation; no qualitatively new features arisein the mathematical analysis given Eq. (13).
3. APPLICATIONS
A. Large self-consistent plasma currents can arise in magnetic stochasticity and islands when the neoclassical transport is not intrinsically ambipolar. Here we consider only steady-state stochasticity (from non-axisymmetricequilibrium Pfirsch-Schliiter currents or coils).
Let us estimate the size of the self-consistent parallel currents and fields
from radial electron diffusion in stochasticity. Resonant magnetic perturbations 6Br — 6Bm<i + 6Bext arise on an integrable background field B,
due to both an external agent (i.e. field coils or Pfirsch-Schliiter currents)
and the induced plasma response. The flux average radial electron current
from parallel motion in the stochasticity is (j*) = (6j\\6BT/B). Breaking
this into the various helicities, which we label by perpendicular wavenumberfcThej Or) = Y,k (tiJ\\h&Br/B)motion of electrons in a stochastic field
is only correlated with the resonant harmonic near its respective rationalsurface; call the distance Axc. Thus, at a given radius, only terms from

154

KOTSCHENREUTHER et al.

nearly rational surfaces contribute. Denote the typical spacing between therational surfaces by A; then Axc/X harmonics contribute at a given point,
and if all resonant harmonics have roughly the same amplitude we have{j') ~ (Axc/X) 6Bk/B6j^k.
The induced magnetic perturbation can be
simply computed using the "constant rp" approximation, which here implies(A'/k)6B^d= (4n/c)fdx6j^~ {Air/c)8j^Axc. Self-consistency is impor-tant when l>Bmd ~ &Br, which gives the criterion

A neoclassical ion current arises because the magnetic stochasticity
changes the radial electric field. Equal radial electron and ion current occurs.
For typical reactor parameters and ion transport levels consistent with theLawson criterion, self-consistency is important for 6B/B ~ 10~ 2 , which cancause very large overlapping islands.
More detailed analysis of both the single helicity and multihelicity cases
reveals that the self-consistent currents cause the resonant perturbation to
be shielded out, and thus islands and stochasticity are "healed" in steady state
if the inequality in (13) is satisfied.
B. The bootstrap current term in Ohm's Law can cause magnetic islands
to grow in the Rutherford regime. For sufficiently large islands, the vorticity
equation implies V|| J = 0, and J is a flux function, so as in previous analysis
the current, can be found by taking the flux average of Ohm's Law. Also,
the pressure becomes a flux function because of parallel convection by soundwaves. The pressure profile is found as in Appendix A of Ref. 6.
We obtain the following evolution equation for Ax, the island width_„ + *,£
rj

at

Ax ax
2

where K: ~ 1.6. k2 ~ 6, and 5 ' = q /r(dq/dr).
For small islands thebootstrap current term dominates and the island grows. For A' < 0, thesteady-state island width is given by Ax — 2' A, '.
This is similar to previous results for unstable interchanges[5,6]. Thus,
as in that case, island overlap always occurs if islands up to sufficiently highmode numbers are considered [6]. For typical parameters and j3 ~ 1%, island
overlap occurs for islands with m ~ 5 — 10.
One must keep in mind that the above calculation is only valid forsingle helicity islands; the dynamics may change considerably as overlap isapproached. If islands significantly overlap, the stochastic diffusion can be
crudely estimated from the quasilinear formula; for /3 ^ l%,the confinement

IAEA-CN-47/E-II-3-1

155

deterioration is significant, for current and future devices and rapidly worsens
with increasing (3.
C. We find that the shear Alfven resonant mode is destabilized by gradientsin the equilibrium u>,-profile. Significantly, its growth rate dominates thetearing mode for either large w. or low shear and depends only weakly on
the resistivity, 7 oc 771/4. Thus, it should predominate in reactors. The
mode's stability is independent of A'.
D. Numerical studies of the m = 1 tearing mode show that it continues
exponential growth well into the nonlinear regime. We have an analytic
theory explaining this. Space does not allow a full discussion of this, but the
primary reason for the rapid growth of the m. = 1 mode is thai its island size
depends linearly on the magnetic flux perturbation.

ACKNOWLEDGEMENTS
This work was supported by USDOE, Contract No. DE-FG05-80ET53088.
REFERENCES
[1]
[2]
[3]
[4]

[5]
[6]

STRAUSS, H.R., Phys. Fluids 19 (1976) 134.
CALLEN, J.D., SHAING, K.C., Bull. Am. Phys. Soc. 30 (1985) 1424.
HOLM, D.D., MARSDEN, J.E., RATIU, T., WEINSTEIN, A., Phys. Rep. 123 (1985) 1.
HAZELTINE, R.D., HOLM, D.D., MARSDEN, J.E., MORRISON, P.J., in Plasma
Physics (Proc. Int. Conf. Lausanne, 1984), Centre de recherches en physique des plasmas,
Ecole polytechnique fe'de'rale de Lausanne, Lausanne, Switzerland (1984).
KOTSCHENREUTER, M., HAZELTINE, R.D., MORRISON, P.J., Phys. Fluids 28 (1985)
294.
CARY, J.R., KOTSCHENREUTHER, M., Phys. Fluids 28 (1985) 1392.

IAEA-CN-47/E-II-3-2

NEOCLASSICAL MHD EQUATIONS,INSTABILITIES AND TRANSPORTIN TOKAMAKS*
J.D. CALLEN, W.X. QU 1 , K.D. SIEBERT2
University of Wisconsin,
Madison, Wisconsin
B.A. CARRERAS, K.C. SHAING, D.A. SPONGOak Ridge National Laboratory,
Oak Ridge, Tennessee
United States of America

Abstract
NEOCLASSICAL MHD EQUATIONS, INSTABILITIES AND TRANSPORT IN TOKAMAKS.
The reduced equations of resistive MHD are extended for tokamaks into the experimentally relevant banana-plateau collisionality regime by adding parallel viscous stress effects. The resultant neoclassical MHD equations include: (1) rapid (~v) viscous damping of poloidal ion flows; (2) the trapped
particle effects on the electrical conductivity and bootstrap current contributions to Ohm's law; (3) anenhanced (by B 2 /B|) perpendicular dielectric constant; and (4) all neoclassical transport effects. The
types of neoclassical MHD instabilities that occur are analogous to the resistive-g, tearing and rippling
modes of resistive MHD, but more virulent. Low mode number tearing type modes evolve into the usual
nonlinear stage of resistive tearing modes. Turbulent transport due to medium mode number neoclassical
MHD instabilities is estimated and shown to lead to energy confinement scalings similar to experimental
results. Finally, the total transport in neoclassical MHD is shown to be comprised of equilibrium andtransient Coulomb collisional effects (classical and neoclassical), convection-driven turbulent transportand magnetic flutter effects.

1.

INTRODUCTION

Resistive magnetohydrodynamic (MHD) models have proven to bevery successful [i] in describing many macroscopic phenomena intokamak discharges (e.g., major disruptions, Mirnov oscillations
and anomalous skin penetration).
Also, a resistive MHD basedmodel of anomalous transport due to medium mode number (n > 10)
resistive ballooning modes correlates well with the finite g* Research supported by the United States Department of Energy underContract No. DE-FGO2-86ER5318 (Univ. Wisconsin) and Contract
No. DE-AC05-84OR21400 (ORNL/Martin Marietta Energy Systems, Inc.).
12

Permanent address: Southwestern Institute of Physics, Leshan, Sichuan, China.
Present address: Mission Research Corporation, Santa Barbara, CA, USA.

157

158

CALLEN et al.

ISX-B experiments [2]. Further, the energy confinement scaling
in tokamaks has been found [3] to be best described in terms of
dimensionless parameters derived from resistive MHD [*(].
The resistive MHD model is, however, only valid in acollisional or short mean-free path (X << R Q q, Pfirsch-Schluter)regime, whereas tokamaks operate in a long mean-free path (A >>R Q q, banana-plateau) regime. In this work we extend the reduced
equations [5] of resistive MHD into the tokamak-relevant bananaplateau collisionality regime by adding the parallel viscousstress effects (Section 2 ) . The resultant "neoclassical MHD"
equations (Section 3) are the appropriate Braginskii [6] typeequations for a tokamak plasma. The neoclassical MHD equations
include all neoclassical transport effects in equilibrium(Section 4) and an enhanced dielectric constant (Section 5 ) .
Neoclassical MHD instabilities (Section 6) and the transport they
induce (Section 7 )> together with the implied energy confinement
scaling (Section 8) and the overall transport fluxes (Section 9),
are also discussed.
Section 10 summarizes the impact of neoclassical MHD on our understanding of resistive MHD typephenomena in tokamaks.

2.

VISCOUS FORCES IN A TOKAMAKThe viscous stress tensor IT has parallel [TTM- noVV], crossor gyroviscous [TVX ~ nat* BxVv] and perpendicular~[irx- ru , 2(VV)jJ
components in a "magnetized plasma [6]. In analogy~with the heatflow cj, the parallel stress (no - vA 2 ) dominates the cross-stress(n3,i - vpX - pv T ) by one order in the gyroradius expansion andthe perpendicular stress (111,2 - v p 2 ) by two orders. The
dominant, parallel stress can be written in the usual
anisotropic-pressure (CGL) form for all collisionality regimes:
211 = (pii - Px)(.bt> ~ 1/3) in which b = B/B is a unit vector along
The magnetic field direction.
The pressure anisotropy caused by flows in the plasma has
been derived previously in the collisional regime [7]. For
incompressible flows it can be written as p,, - p x = -3n°(V-VI!,nB),
which indicates the magnetic pumping rather than diffusive
character of the parallel viscous stress [8]. For arbitrary
collisionality it can be written, to lowest order in an aspectratio expansion, as
p,, - p„ - "inu<B2> (V.V&nB),<(b-7B)*>tr,, E (P.. - Pa.)(bb - 1/3) (1)=—=in which < > denotes the flux surface average. Here, p is thefrequency for viscous damping of flows in the VB direction. Its
static value for electron and ion plasma components in a Z = 1
plasma is [9] 3
3However, the normalization of p. is different; there, /* has the units of Braginskii's ij 0 [6],whereas here, n has units of frequency.

IAEA-CN-47/E-II-3-2
2.3

159

c\o

(1+1.07v»e* + 1.02v* e )(1+l.07v» e e 3/2 )0.66 e^Vj(1+1.OSv*^ + 0.
Here, v* = ve" 3 ^ 2 RQq/v.j. is the collisionality parameter that is
unity at the transition from the banana to plateau collisionality
regime. In the banana collisionality regime (v#<<1 ), \i ~ e v,
which reflects the viscous damping of parallel flows carried bycirculating particles through collisions with the e fraction of
immobile trapped particles. The time-varying form of the viscousdamping frequency in the banana collisionality regime is derived
in reference [1O].
The dominant viscous force V_']t_u and its parallel (to B)
component are given by~"
7-TT.. = (p,,-pi)[(b'7)b-b(b-7)JlnB]+b(b-7)(pI1-pj.)-V(p,1-px)/3(3)

B-V-TTn = - (Pn-PxXb'V)B + (2/3)(B.7)(p,,-Px)(^)

ue<B2>, u e = v-ve/(B'Ve) = v e /B e

3.

(5)

NEOCLASSICAL MHD EQUATIONSThe neoclassical MHD equations [8] result from adding theparallel and cross viscous stress effects to the usual reduced
equations of resistive MHD [5]. The usual assumptions are made:T = constant (but T e * T i)i a small gyroradius expansion (6 p/i << 1 ) ; a twistable but incompressible magnetic field (3B_/3t *0, but 3B/3t = 0 — valid for low 6 ) ; and axisymmetric magnetic
field geometry B = IV? + V? x i?i|> with I(I|J) = RB , |V?[ = 1/R and
B x V41/B2 = R27? - IB/B 2 . In these plasma approximations anelectron-ion plasma is governed by density and momentum consei—
vation equations for each plasma species:
3n/3t + V«nV = 0(6)mn dV/dt = nq(E+c~iVxB) - Vp - R - V«ir

(7)

where £ = nq(Jx/oj,+J,t/Oii) is the interspecies frictional force.
The perpendicular components of Eq. (7) can be solved
iteratively in the small gyroradius expansion to yield (E = - V<J>
- c"1 3A/3t with A =)Vx = 2BxfVU+lfcnn) + — ( m n — + R + V-TT)] + i=lLv* ^ (8)
2
2
B

- ~

q

nq

dt

~

~ =

B

~ 3t

160

CALLEN et al.

Jj. = (c/B2) B x [VP + p m dVj./dt + I V.^_3](9)Here, the perpendicular flow V x has first order (in 6) Bxy^ anddiamagnetic flows, and second order polarization drift, friction
and viscous force driven flows. The last (3i|j/3t) term is a small(~62)"grid velocity" contribution arising from the vector
potential part of jE_xj3.
Since the lowest order flows arederivable from a potential, V is incompressible to lowest order[^•V - 6 2 ] . Also, the poloidal flow velocity in Eq. (5) can now
be written as U Q = V,,/B + (CI/B 2 )O/3I|J) [$ + (T/qHnn] + &(.62).
The perpendicular current £j. has similar contributions, but ismissing the _Exj5 and friction driven terms because these flows areambipolar. The viscous force effects contribute to Jj. because,
while they lead (to order 6 2 ) to only ambipolar flows inequilibrium, they are in general nonambipolar [11J.
The parallel flows are determined from the sum and electroncomponent (with d/dt << v g ) of the parallel momentum balances[8J:
pm(3/3t +• VR-V)(V,MB) = - (B-V)P - B'V^ir,,.
III

—n

—

l

S=

(10)

a

0 = - nee(E-B) - (B«V)pe + neeJ,,B/o,, - B^V«ir,,e(11)

H e r e , VE = ( c / B 2 )Bx7((> i s t h e p o t e n t i a l p a r t of t h e ExB flowv e l o c i t y . I t i s the only order 6 flow which e n t e r s the convective d e r i v a t i v e on the l e f t of Eq. (10) because the diamagnetic
p a r t of t h i s convective d e r i v a t i v e i s cancelled by B « V / I T X [ 8 , 1 2 ] .
The n e o c l a s s i c a l MHD equations are time evolution~equations
for the d e n s i t y , perpendicular and p a r a l l e l plasma flows and themagnetic flux. They are obtained from the preceding equations asfollows.
The density conservation equation for e l e c t r o n s i s
u t i l i z e d with V«nV1Ie = (B.y_)(nV,,e/B) = (B«7)(nV M1 /B - J,,/eB) and
V_xe a s d e f i n e d in Eq. ( 8 ) . The p e r p e n d i c u l a r flow evolution
e q u a t i o n r e s u l t s from 0 = V»J = (B-V)(J,,/B) + V-J^ with J A as
given in Eq. ( 9 ) . The p a r a l l e l flow evolution equation i s given
in Eq. (10).
F i n a l l y , the magnetic flux evolution equation i sobtained from the Ohm's law with E = - 7<j> + c" 1 ( 3 i J ) / 3 t ) ^ . Then e o c l a s s i c a l MHD equations can thus be w r i t t e n as
3n/3t + 7«nV 4e + (B»V)(nV ni /B - J,,/eB) = 0(12)- V'(c P m /B 2 )BxdV i ,/dt = (B'7)(J,,/B)+V.[(c/B 2 )Bx(VP+V.ZTr ltQ )]
—

in

—

pm d ( V , , . B ) / d t
in
I
-

_L_

3]k

cR

3t

2

= -

— x

—

= -

(B«V)P -

T R
J!_
-

(B«7)(*

a,,

~

—

—

— 3—

B'7'ZTT,,.
a
3 =
-

T
-i

e

X,n

(13)

o

(11)

n)

+ i-

ne

B-V-TT,,.

(15)-

where d/dt = 3/9t + Vg-V and J,,/B = - (cR2/4irI)V» (Vip/R2 )+CT(e2).
The terms involving the divergence of the parallel viscous stress

IAEA-CN-47/E-H-3-2

161

in Eqs ( 1 2 , 8 ) and (13) can be s i m p l i f i e d using 7- (B~ 2 J3X7-JT,,) =7 . ( B - 2 ( P , , - P l ) b x V B + ( 2 / 3 B ) bxV(p,,-p A ) - ( P i I - P l ) ( W c B 2 ) ? , , ] =7/ [B~ 2 (p,,-p A )j>x7B] , where in t h e l a s t ap P roximatlon f i n i t e g andsmall p r e s s u r e a n i s o t r o p y terms a r e n e g l e c t e d compared t o s i m i l a r
i s o t r o P i c p r e s s u r e terms in Eqs (12) and ( 1 3 ) .
If d e s i r e d ,p e r p e n d i c u l a r v i s c o s i t y e f f e c t s can be added t o t h e s e e q u a t i o n s
by r e p l a c i n g *,, by TT_,, + £ i «The n e o c l a s s i c a l MHD e q u a t i o n s simplify t o t h e u s u a l reducedr e s i s t i v e MHD e q u a t i o n s [ 5 ] in t h e l i m i t of small p a r a l l e lv i s c o u s s t r e s s ir,,(u •» 0 for \ « R Q q), s i n c e then t h e l a s t termin each of Eqs 1 8 ) , ( 1 3 ) - ( 1 5 ) becomes n e g l i g i b l e .
They can beshown t o s a t i s f y an energy c o n s e r v a t i o n r e l a t i o n .
This i so b t a i n e d by m u l t i p l y i n g Eqs ( 1 2 ) - ( 1 5 ) by t h e f a c t o r s (Tg+T^Hn n ,(<p + (Ti/e)lnn ) , V n i / B and J , , / B , i n t e g r a t i n g o v e r a l l s p a c e ,n e g l e c t i n g s u r f a c e terms f o r s i m p l i c i t y and summing t h e r e s u l t i n gequations t o yield (correct to order 62)
|

^

i

1

P£nn]

/ d x ^^

^ V ^ , ^ ] (16)

Neoclassical MHD adds to the comparable energy conservationr e l a t i o n for r e s i s t i v e MHD [13] the p a r a l l e l flow energy termpm V " i / 2 a n d t h e P a r a l l e l viscous dissipation Z V g ^ ' l n g which,for t h e form of <B«7«iTn> given in Eq. (5), becomls ranp U2<B2> andindicates the viscous~dissipation due to poloidal flow against
the p a r a l l e l viscosity — neoclassical t r a n s p o r t . The crossv i s c o s i t y irx i s d i s s i p a t i o n l e s s [ 6 , 1 t ] and so does not enter
Eq.(i6).
TTie various dissipation terms on the r i g h t of Eq.(i6)could be removed [13] by adding temperature evolution equations
for electrons and ions.

4.

NEOCLASSICAL MHD EQUILIBRIUM

In e q u i l i b r i u m ( d / d t « p , ) P=P(i|i), Ue=Ue(i^) and t h e t o t a l
p a r a l l e l momentum balance E q . ( i 4 ) i s 0=<^«V^«ZTT,,3>=minipiU9i<B2>.
Thus, t h e p a r a l l e l v i s c o u s s t r e s s damps t h e p o l o i d a l ion flow t oz e r o [ 9 ] , which c a u s e s t h e i o n flow t o be t o r o i d a l : U e l =0,
V l t i o B= - c I O / 3 ^ ) [ 4 , 0 + ( T i / e ) J l n n 0 ] , VQ = n o R 2 7e, flo(i|>) = - c?3/3^)l(j>o+(Ti/e)Jlnno].
( I f 7T i e f f e c t s were included, t h e r e would be asmall,
residualpoloidalion flow.)
The predominance ofelectron-ion friction over electron viscosity causes the lowest
order Ohm's law to be <J,,B>/o,, = <E^B> and hence a poloidal
electron flow Uee = [-(cI/ne)(dP/dij)) + (e/meve)<EAB>]/(1+lie/ve)<B2>,
where E* i s the transformer-induced E f i e l d .
The p a r a l l e lviscous force <B«7-TT,,e> on t h i s poloidal electron flow induces
the bootstrap current and t r a p p e d - p a r t i c l e
conductivity
modification [ o,, e f f = o,,/(1 +y e /v e ) ] contributions to the parallel
Ohm's law [9].
Also, the radial flux driven by the electron
viscous force, <n7^«(c/eB2)Bx7«Trlle>, yields [8] the neoclassical162

CALLEN et al.

d i f f u s i v e f l u x e s (D n c -v e p|q 2 +MaPg|) a n d t n e neoclassical pinchv e l o c i t y , - ( u e / v e ) c<E^B>I (i|>) / [v<p[<B2>. Thus, the equilibrium
limit of the neoclassical MHD equations includes a l l known
neoclassical e f f e c t s .

5.

DIELECTRIC PROPERTIESThe perpendicular d i e l e c t r i c constant can be inferred bynotingfromthedivergenceofAmpere'slaw
that
V«JX = -(1 /<JTr)V«£ x «OE i /3t).
In r e s i s t i v e MHD the J^ induced bythepolarizatTon
drift,
orperpendicularflowinertia[cf. Eqs ( 9 ) , ( 1 3 ) ] l e a d s t o the d i e l e c t r i c constant £j.=1+cz/vj[,
with vA = B/(4irp m ) .
In n e o c l a s s i c a l MHD the J^ induced by theviscous force [cf.
Eqs ( 9 ) , ( 1 3 ) ] introduces a p a r a l l e l flow
i n e r t i a term [ 8 ] , which i s effectively an additional polarizationd r i f t and leads to a perpendicular d i e l e c t r i c constant for r a d i a l(or V^) charge imbalances given by e,|.= 1 + (c 2 /v^g) (u^/(uj-iiii)), withv A e = Bg/(4iTpm) .
(There i s a l s o a s m a l l e r , P f i r s c h - S c h l U t e r
c o n t r i b u t i o n , - c 2 q 2 / v | , which we n e g l e c t h e r e . )For slowp r o c e s s e s ( - i u = d / d t < < u i ) t h i s d i e l e c t r i c constant i s B 2 /BQ~10 2 ,larger than the r e s i s t i v e MHD one and r e f l e c t s the additional
i n e r t i a l effects due to the poloidal flow damping.

6.

NEOCLASSICAL MHD INSTABILITIESThe types of i n s t a b i l i t i e s obtained from the n e o c l a s s i c a l
MHD equations are analogous to the standard r e s i s t i v e MHD
instabilities[15]: resistive-g,t e a r i n g and r i p p l i n g .
Thep r e s s u r e - g r a d i e n t - d r i v e n n e o c l a s s i c a l MHD i n s t a b i l i t i e s [10,16]
a r e found to have growth r a t e s Y and r e s i s t i v e l a y e r widths 6 :

T

Aebe

v

e

dr

v

2 ^

SQ

where S e = 8irP0/B|, 1/r =rdX,nP0/dr, s e = T R / T A e > tR=i)Tro,,r2/c2, andT
Ae= ^1|7r/)m^ RoC)/'B6* T n e s e n e w modes are analogous to resistive-gmodes [T5], except that they are more virulent in that they[16]: (l)depend on the parallel electron viscous damping rate u e ,which in neoclassical MHD predominates over the magnetic fieldcurvature in providing access to the expansion free energy, and(2) have growth rates and resistive layer widths that are larger
by ( M e / v e ) l 7 3 ( B / B Q ) 2 / 3 - 5 and depend only on the poloidal
magnetic field B Q , because of the larger, B g dependent dielectric
constant in neoclassical MHD. Including the modifications due todiamagnetic drift effects [10,16] yields a growth rate Y Y /(1+o)|/Y2) and a resistive layer width 6 - 6 /(1+<4/Y2) •Neoclassical MHD tearing instabilities [17] have access toboth the current profile free energy through a' and to the

IAEA-CN-47/E-II-3-2

163

expansion f r e e energy through t h e b o o t s t r a p c u r r e n t .
For 6 e u e / v e> > S ~ 2 / s , t h e y h a v e g r o w t h r a t e s and r e s i s t i v e l a y e r w i d t h s
approximately equal t o those given in Eq.(17) above ( i . e . the a 1
dependence i 3 very weak in t h i s l i m i t ) and so a r e p r i m a r i l y
p r e s s u r e - g r a d i e n t - d r i v e n modes in tokamaks.
In t h e i r n o n l i n e a r
e v o l u t i o n [ 1 8 ] , the n e o c l a s s i c a l MHD t e a r i n g modes f i r f s t growe x p o n e n t i a l l y in t i m e .
They n e x t e n t e r a w - iji^ - t * growth
regime where the b o o t s t r a p c u r r e n t c o n t r i b u t i o n i s important[18,19].
For A*>0, they u l t i m a t e l y e n t e r t h e Rutherford growth
regime [ 2 0 ] where t h e b o o t s t r a p c u r r e n t c o n t r i b u t i o n
isnegligible,
w-t and ultimate s a t u r a t i o n
i s achieved
ifA'(w m a x )=0, or d i s r u p t i o n occurs if two i s l a n d s of different
helicity overlap [ 1 ] .
Thus, low mode number tearing type
neoclassical MHD i n s t a b i l i t i e s with A'>0 eventually evolve
according to standard r e s i s t i v e MHD nonlinear theory. For higher
mode numbers with &'<0, individual neoclassical MHD modes woulds a t u r a t e at the end of a w - t
growth regime, but then theplasma becomes turbulent.
New, rippling-type
neoclassical
MHD i n s t a b i l i t i e sarepossible for v*e~1 through density perturbation effects in thee f f e c t i v e p a r a l l e l e l e c t r i c a l conductivity ait eff »0n/(1 +u e /v e ).
The neoclassical MHD i n s t a b i l i t i e s are a l l purely growing ina t o r o i d a l E «B r e s t frame (UQio=0) and have a laboratory frame
frequency [i6J,u E = - nc dif>0/dij> = - (nq/r)(c/B)(d<(>_/dr). Since intokamak experiments, one often finds [21 ] (nee/Te)(d<j>0/dr) - d n g / d r , these modes would usually be observed experimentally to
be moving in the electron diamagnetic drift direction.

7.

INSTABILITY - INDUCED TRANSPORT

The turbulent transport induced by medium mode number(n-3-30) neoclassical MHD instabilities can be estimated frommixing length type turbulence arguments.
(This procedure isanalogousto, but more physical than, scale length invariancearguments [H]. While it properly predicts the dominant scalings,
more precise estimates [22] of the turbulence and transportcoefficients are larger by effective Reynolds number type factors- 10.) The radial flow, density and magnetic field fluctuationlevels estimated from mixing length arguments arev l x ~ V p , n - f n £2o. !* . ( L d t t l ^ ,
1X

vv

nn0

0

dr

B

Hi,a,,/c2

In the Direct-Interaction-Approximation, D - I ^ v ^ ^ + k x D ^' w n i c n
in the strong turbulence limit (Y,. < k?.D) yields (for kv - 1/6 )D - (EV x /k 2 )* - coYu(5u* i n w n i c n c o i s a dimensionless spectrum
sum.
Similarly,
the magnetic field
diffusioncoefficient
Dm - <AX2>/Ll ~ E(B x /B) 2 /k x D m , which yields [2] Dm - C,6 (Bx/Bi
with C1 being another spectrum sum. To calculate the spectrum
coefficients CQ,C^, one must go beyond the mixing length arguments

164

CALLEN et al.

and do a two-point correlation theory (cf. [22,23]) to determine
the spectrum.
While this has not yet been done, we cananticipate CO,C1>>1.
The anomalous electron heat conduction induced by themagnetic field diffusion can be estimated [24] as xe ~ vTeDm*
The resultant estimates for the particle and electron heat
conduction coefficients are similar to those derived by Cameras,
Diamond et a l . [ 2 ] for resistive ballooning modes, but are
somewhat larger and are, in contrast, valid for a r b i t r a r i l y smallvalues of BQ. They can be written as

in which| r / r | = |dX,n PQ/dJln ql and D =c2/tiroif=riti/p0 ( i n mks) i s

the c o e f f i c i e n t for magnetic f i e l d l i n e diffusion in a r e s i s t i v eplasma.
[Since in the present theory the temperature i s held
constant, the pressure gradient scale length in Eqs (17.19) i s
rigorously j u s t the density gradient s c a l e l e n g t h . ]The p a r t i c l ediffusioninduced by n e o c l a s s i c a l MHD
turbulence i s proportional t o n e o c l a s s i c a l diffusion (D - C Q p e pg|- C 0 D n c ) , but exceeds i t by the factor CQ. The induced impurity
diffusion and ion heat conduction c o e f f i c i e n t s should a l s o becomparable [ D i m p - D, x i ~ ( 5 / 2 ) D ] .
The t u r b u l e n c e - i n d u c e de l e c t r o n h e a t c o n d u c t i o n i s , however, much l a r g e r [ x e /D (C-i/CQMngBQ/vgXnij/mg)^ ~ 5~10], and comparable t o t h a t observed
experimentally.

8.

CONFINEMENT SCALING

For ohmically heated plasmas, since t h e x e given in Eq.(19)i s p r o p o r t i o n a l t o D J 9 1 t h e ohmic power b a l a n c e l e a d s t o aunique p r e d i c t i o n for f?Q of ( m e / m i ) i / 6 ( u e / v e r 2 / 3 < 0 . 3 , which i ssimilarto experimental
observations.
For the t r a n s p o r t
d e t e r m i n i n g r / a > & r e g i o n of o h m i c a l l y h e a t e d plasmas wherev* e > 1, we have u e / v e - £ ^ / v # e .
Under t h e s e c o n d i t i o n s t h eohmic power balance y i e l d s (with T E - a z / 4 x e ) the p r e d i c t i o n
TE - n e ' 3 / " t a ' l / 7 R 0 1 5 / 7 q 5 / 7 B e " ' 7(20)

in which we used t h e often observed experimental s c a l i n g Z e f f . ~1/n g .
This energy confinement scaling (and magnitude for C1 ~10) is reasonably close to the neo-Alcator scaling xE /For hotter,, auxiliaryy heated plagmas
which have
p g
e j q
v* e <<<< 1
1 overover most
most ofof thethe plasma,
plasma, \i\iQ/v/ve - e
and
the
v*
e
and
the powere
Q eb a l a n c e y i e l d s t h e u l t i m a t e ( P a u x >> P OH^ s c a l i n g
E
P
>>
^
P
aux~i/3ne"1/3Be8/3>\ t h e t r a n s i i tat iuioxn f r o m ohmic to auxiliary
p
5/7#
The3e
heating, where ve/ve- e /v* e , one finds t E - a u x ~various scalings are in reasonable agreement with experimental
observations.

IAEA-CN-47/E-H-3-2
9.

165

OVERALL TRANSPORT BEHAVIOUR
The f l u x - s u r f a c e - a v e r a g e d
1^ = <nV«Vi|i> = <n 0 V2«y>radial+ <n,VfViJ)>

particle

flux(~62) i s

+ <(B, •Vi)j)noV,ll/B>(21)

Here, t h e t h r e e terms i n d i c a t e , r e s p e c t i v e l y , t h e t r a n s p o r t due
t o Coulomb c o l l i s i o n s( c l a s s i c a l and n e o c l a s s i c a l ) ,j£ixj3 oc o n v e c t i o n - d r i v e n t u r b u l e n c e and "magnetic f l u t t e r " [ 2 5 ] . (The\[s term can conceivably include f l u c t u a t i o n driven terms as
well.)
The analogous radial heat flow q^= <£»^ijj> also has thesethree types of contributions.
The neoclassical contributions are usually derived
inequilibrium.
For rapid transients (d/dt > v^), the poloidal ion
flow must be determined from Eqs (5),(11) and leads to additionalt r a n s p o r t f l u x e s with v e l o c i t y V^ - ( n e / f t 6 e ) ( U Q i B ) .
These
effects are nonambipolar and lead to transient (at - v^1) changes
in the e q u i l i b r i u m p o t e n t i a l <J>o(iJi) and e l e c t r o n and ion heat
conduction.
They should be taken into account in modellingtransient effects and could be important for "fast time-scale"transport effects
in tokamaks.
The <niVi«7iji> contribution is predominantly the usual EixB0
convection driven
transport.
Since
the neoclassical MHD
i n s t a b i l i t i e s are purely growing, n and ?1 . are in phase and theturbulent transport can be calculated as indicated in Section 7.
The l a s t contribution to the fluxes i s caused by magnetic
fluctuations.
To lowest order, i t yields ambipolar p a r t i c l efluxes since [ 2 6 ] , from Ampere's law, 3,, = (c/4ir) b 0 «7x8,so that<(EJi-Vi|i) U,,/B> •» 0.
However, i t does lead to a heat flux
contribution which is predominantly that due to electrons andwhich has been estimated in Eq.(19).

10.

SUMMARY

The fluidlike resistive MHD theory has been extended fortokamaks into the experimentally relevant banana-plateau
collisionality regime. The resultant neoclassical MHD equations
include the reduced equations of resistive MHD and all
equilibrium and transient neoclassical effects, including aperpendicular dielectric constant that depends on the poloidal
magnetic field B Q . Neoclassical MHD instabilities are similar toresistive MHD ones, but more virulent.
Low mode number
tearing-type modes evolve into the usual nonlinear regime
governed by regular resistive MHD. Thus, the previous goodcorrelation of these fluidlike theories with tokamak discharge
phenomenology [1] is preserved.
Higher mode number (n - 3-30)
neoclassical MHD instabilities lead to anomalous transportestimates which correlate reasonably well with tokamak
experimental results — primarily because they depend on only thepoloidal magnetic field B Q . A neoclassical MHD computer code

166

CALLEN et al.

similar to those in references [2,5,22,23] has been written. The
code works, has been debugged, and is beginning to be used to
explore the linear and nonlinear behavior of the neoclassical MHD
equations more comprehensively and precisely.

REFERENCES
[1]

[2]
[3]
[4]
[4]
[5]
[6]
[7]
[8]
19]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]

CALLEN, J.D., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1978
(Proc. 7th Int. Conf. Innsbruck, 1978), Vol. 1, IAEA, Vienna (1979) 425;
FURTH, H.P., Phys. Fluids 28 (1985) 1595.
CARRERAS, B.A., et al., Phys. Rev. Lett. 50 (1983) 503.
CORDEY, J.G., et al., these Proceedings, Vol. 1, p. 99.
THOMAS, P.R., in Controlled Fusion and Plasma Heating (Proc. 13th Europ. Conf. Schliersee,1986), European Physical Society (1986).
CONNOR, J.W., TAYLOR, J.B., Phys. Fluids 27 (1984) 2676.
STRAUSS, H.R., Phys. Fluids 19 (1976) 134;
WADDELL, B.V., et al., Phys. Fluids 22 (1979) 896.
BRAGINSKII, S.I., "Transport processes in a plasma", Reviews of Plasma Physics(LEONTOVICH, M.A., Ed.), Vol. 1, Consultants Bureau, New York (1965) 205.
GRIMM, R.C., JOHNSON, J.L., Plasma Phys. 14 (1972) 617.
CALLEN, J.D., SHAING, K.C., Rep. UWPR-85-8, University of Wisconsin (1986).
HIRSHMAN, S.P., SIGMAR, D.J., Nucl. Fusion 21 (1981) 1079;
HIRSHMAN, S.P., Phys. Fluids 21 (1978) 1295.
SHAING, K.C., CALLEN, J.D., Phys. Fluids 28 (1985) 1859.
HIRSHMAN, S.P., Nucl. Fusion 18 (1978) 917.
HSU, C.T., et al., Phys. Fluids 29 (1986) 1480.
DRAKE, J.F., ANTONSEN, T.M., Jr., Phys. Fluids 27 (1984) 898.
SIEBERT, K.D., Ph.D. Thesis, Univ. of Wisconsin, 1986.
FURTH, H.P., et al., Phys. Fluids 6 (1963) 459.
CALLEN, J.D., SHAING, K.C., Phys. Fluids 28 (1985) 1845;
CONNOR, J.W., CHEN, L., Phys. Fluids 28 (1985) 2001.
QU, W.X., CALLEN, J.D., Rep. UWPR-85-2, University of Wisconsin (1985).
QU, W.X., CALLEN, J.D., Rep. UWPR-85-5, University of Wisconsin (1985).
CARRERA, R., et al., Phys. Fluids 29 (1986) 899.
RUTHERFORD, P.H., Phys. Fluids 16 (1971) 1903.
HALLOCK, G.A., et al., Phys. Rev. Lett. 56 (1986) 1248.
CARRERAS, B.A., et al., Theory of resistive pressure-gradient-driven turbulence,
to be published.
TERRY, P.W., et al., Phys. Fluids 29 (1986) 2501.
RECHESTER, A.B., ROSENBLUTH, M.N., Phys. Rev. Lett. 40 (1978) 38.
CALLEN, J.D., Phys. Rev. Lett. 39 (1977) 1540.
MANHEIMER, W.M., COOK, I., Comments Plasma Phys. 5 (1979) 9;
WALTZ, R.E., Phys. Fluids 25 (1982) 1269;
KOTSCHENREUTHER, M., Ph.D. Thesis, Princeton Univ., 1982.

IAEA-CN-47/E-ni-l

I. PLASMA EDGE MICROINSTABILITY
WITH UP-DOWN ASYMMETRY
II. THEORY OF COLLISIONAL TEARING MODES
FOR ARBITRARY LARMOR RADII
A. ROGISTER, G. HASSELBERG
Euratom-KFA Association,
Institut fiir Plasmaphysik,
Kernforschungsanlage Julich GmbH,
Julich, Federal Republic of Germany
Abstract
I. PLASMA EDGE MICROINSTABILITY WITH UP-DOWN ASYMMETRY.
II. THEORY OF COLLISIONAL TEARING MODES FOR ARBITRARY LARMOR RADII.
The linear theory of the rippling instability in a torus permits the occurrence, at the edge oftokamaks, of density fluctuation spectra with up-down asymmetries to be anticipated; for low kg, thewaves propagate in the electron diamagnetic direction with a common phase velocity
vph = (1 + 1.71 VC)VDC- These predictions are supported by experimental results from TEXT. The
limitations of the present theory are discussed and additional experiments are proposed. A simple
differential formulation is given of the collisional tearing problem for arbitrary values of the ratio widthof the resistive layer to the Ion Larmor radius without having recourse to any approximation (e.g. the'constant \p approximation'). The technique used is that of a Fourier transform of generalized functions;the most significant mathematical step has been to transpose the boundary conditions from geometricalspace to k-space.

I. PLASMA EDGE MICROINSTABILITY
WITH UP-DOWN ASYMMETRY

1.

INTRODUCTION

The rippling instability [1, 2] is efficiently suppressed, in the cylindrical model,
by the diamagnetic drifts [3, 4]. In toroidal geometry, we have found that an instability window reappears owing to the curvature and to the field gradient. The generalmathematical analysis has been presented elsewhere [5, 6]. Our purpose here is (1) tooutline the conclusions, but also the approximations, of the calculation; (2) to compare the theoretical and experimental results; (3) to identify the necessary extensionsof the theory; (4) to suggest complementary experimental investigations.
167

168

2.

ROGISTER and HASSELBERG

OUTLINE OF THEORY, COMPARISON WITH EXPERIMENTAL
RESULTS AND DISCUSSION

2.1. Theoretical results and their limitation
The theoretical results simplify appreciably, although they remain essentiallycorrect, if we consider the parameter -q = (d In T/dr)/(d In N/dr) to be large compared to unity. The results may be summarized as follows:(1) The mode frequency is given bycoo = (1 + 1.71 7,e)We*(1-1)

where w* = -kecT e /|e|BL N is the electron diamagnetic frequency (ke = m/r is thepoloidal mode number and LN = (d In N/d r ) " ' is the density gradient length scale).
(2) The growth rate is given by
7re= 0.24 ^ - ' " I t f V L r l 2 0 - k2x,,ere/1.5N(i-2)

where TC is Braginskii's collision time, /x = me/mj, £s = Un/c, is the streaming
parameter (q = VTj/m; is the ion thermal velocity), Ls is the shear length,
Lr = Ln/7/, kg = lT-B/B is the parallel component of the mode number, and xi,e isthe parallel electron thermal conductivity. The instability only occurs in a window
defined approximately by
0.13/q2 < X2 < 0.30

(1-3)

within which the quantity A = -1.59 + 12.25 q2X2/(l - 3.25X2) is positive.
X = Cj/qRwo is the ratio of the distance travelled by a thermal ion in one wave
period to the connection length qR; the first term originates from ion inertia andgyroviscosity, and the second term from the radial component of the magnetic drifts(the mode is, thus, stable in a cylinder).
(3) The density fluctuation spectrum,,/. „ krL

I nke,kr|2 « (1 + sin 0 - gN -J

3.17 + 1.95 X2 \\ 2i

_ 3 ^ x2

, ,,

,,„

J exp (-kr2 a2/S)(1-4)

is to be contrasted with the potential fluctuation spectrum:l<A ke ,k r l 2 «exp(-k 2 a 2 /S)
Here, S = 0.41A~2/3|£sLi./Ls|2/3; a; = c^fy is the ion Larmor radius.

(1-5)

IAEA-CN-47/E-III-l

169

The following assumptions have been made (in addition to 77 > 1) in deriving
these results: (i) kr t> ke and, more precisely, kr LN ~ keR — as appropriate torippling modes; (ii) co0 > 7, i.e. 'weak instability', (iii) Tj(r) = Te(r) and /3 — 0 —i.e. negligible kinetic pressure compared to magnetic pressure — as appropriate to
the edge plasma.
2.2. Comparison with experimental results
An up-down asymmetry of the density fluctuation spectrum has been observedin TEXT, the Texas Experimental Tokamak [7]. Both the experimental and thetheoretical asymmetries reverse (even if this does not appear clearly from formula(1-4)) with the plasma current. We consider the parameters of the TEXT discharge
described in [8] at the normalized radius r/a = 0.8: q = 1.0 X 10 7 cm-s~';ai = 0.052 cm; £s = 0.67 (derived by assuming q = 1 on axis and Spitzer
resistivity); N = 1.2 x 1013 cm" 3 ; Te = T; = 100 eV; rj = 1, but the theoreticalresults remain essentially valid; Ls = 116 cm; LN = 6.5 cm; we* = 3.7 x 103 m
cycles-s"'; r e = 4.2 X 10" 7 s (assuming Z ^ = 4); q = 2.33. We conclude:(1) The phase velocity of the modes isvph = 2.71 VDe(1-6)

where v ^ = we*/ke is the electron diamagnetic velocity, in close agreement with theexperimental result vph « 3vDe (we note that plasma rotation can probably be ruledout in these experiments [7]).
(2) The instability window, Eq. (1-3), implies that the linearly unstable mode
numbers are in the range
8 < m < 28 or

0.37 cm" 1 < ke ^ 1-30 cm" 1(1-7)

Although the destabilizing term in Eq. (1-2) vanishes at the above lower limit (whereA — 00) and becomes infinite at the upper limit (where A — 0, see Section 2.3), theresults of the linear theory disagree mildly with the experimental finding that the
spectrum maximum occurs between 2 and 4.5 cm" 1 . This, as well as the fact thatasymmetric modes have been observed at poloidal wavenumbers as high as
13 cm" 1 , could result from a cascading process from the long to the shortwavelengths of the spectrum.
2.3. Discussion and necessary theoretical improvement
The divergence of the destabilizing term at the upper limit of the instability
window (Eq. (1-7)) could not be anticipated. It is, of course, unphysical and implies

170

ROGISTER and HASSELBERG

the breakdown of the approximation coo > 7 (Section 2.1), but suggests a probably
very large growth rate 7 ~w 0 ~ 3 x 105 s"1! This quite strong instability will,
however, be limited to a narrow range of mode numbers for the following reason.
The second and the first terms in Eq. (1-2) have a ratio of 1.19 A-m 2 (we use theexpression for E in evaluating kj); although it can be shown that parallel electron
heat conduction does not stabilize the rippling modes but only reduces their growth
rates [9] — as in the resistive MHD approximation [2] — this large value implies a
rapid fall-off of 7 away from that value of m for which A = 0.
2.4. Proposed additional experiments(1) Equation (1-5) shows that, within the limits of the theoretical approximations, the spectrum of the potential fluctuations is symmetric. This point could be
investigated with Langmuir probes.
(2) If the source of edge turbulence has such a narrow bandwidth as discussed
in the preceding section, it might be possible to use selective stabilizing means. Doesa magnetic separatrix perhaps play such a role? Is there possibly a connection
between edge localized modes [10], which grow out of a quiet background level inless than 20 ps, and the toroidal rippling instability which could have growth rates
as large a s 3 x l 0 5 s " ' i n a narrow kg bandwidth?

REFERENCES[1][2]

FURTH, H.P., KILLEEN, J., ROSENBLUTH, M.N., Phys. Fluids 6 (1963) 459.
CARRERAS, B.A., GAFFNEY, P.W., HICKS, H.R., CALLEN, J.D., Phys. Fluids 25 (1982)
1231.
[3] COPPI, B., Phys. Fluids 8 (1965) 2273.
[4] ROGISTER, A., Plasma Phys. Controll. Fusion 26 (1984) 1063.
[5] ROGISTER, A., Plasma Phys. Controll. Fusion 28 (1986) 547.
[6] ROGISTER, A., Nucl. Fusion 27 (1987) 325.
[7] BROWN, D.L., PEEBLES, W.A., LUHMANN, N.C., Jr., SAVAGE, R.L., Jr., Phys. Rev.
Lett. 54 (1985) 689.
[8] RITZ, C.P., BRAVENEC, R.V., BENGTSON, R.D., et al., Edge Diagnostics and Transport
in the TEXT Tokamak, Rep. Univ. of Texas (1986).
[9] AMEIN, W.H., ROGISTER, A., Plasma Phys. Controll. Fusion 28 (1986) 1511.
[10] KEILHACKER, M., et al., Plasma Phys. Controll. Fusion 26 (1984) 49.

IAEA-CN-47/E-III-l

171

II. THEORY OF COLLISIONAL TEARING MODES
FOR ARBITRARY LARMOR RADII

1.

INTRODUCTION

With the parameters achieved in present day tokamak plasmas where the widthof the resistive layer of tearing modes (A) is of the order of — and often less than
— the ion Larmor radius (a;),(i) the electrons can effectively be described by the fluid equations;(ii) only a Vlasov description is justified for the ions since the relevant radial
length scales are not larger than the particle Larmor radius of the particles and theirperiodic drift (~ qai) away from the magnetic surfaces; the parallel fluid equations
also break down because Xmfp > c/w.
In the cylindrical limit, the introduction of the Fourier transform and of thenotion of generalized functions [1] leads to an elegant differential formulation of thetearing problem — directly amenable to numerical treatment — for arbitrary valuesof the ratio A/aj without — which is worth noting — having to introduce the 'constant ip approximation'. The theory thus applies to the 1/1 mode as well and, in particular, to the neutral MHD mode; it has, in this respect, a broader scope than previous
works by Crew et al. [2] and Cowley et al. [3] and is easier to handle than theirintegral formulation. The present work is, however, restricted to the regime which
Drake et al. [4] call 'collisional' (C > 1) but we hope that it can be extended to otherregimes. In TEXTOR, tearing modes are just at the edge of collisionality withC « 1/m.

2.

ANALYTICAL FORMULATION OF THE PROBLEM

2.1. The ion and electron responses
Straightforward solution of the ion gyrokinetic equation in the parameter range
of interest yields the following results:(zn - <fr)k = [(rz + 1) (r 0 - 1 ) + i?ik2r0T <t>k - zf(k) <f>k(II-l)The Fourier transform (F.T.) is defined by
dk exp (ik£)«kwhere £ is the normalized distance (= x/a{) from the mode rational surface; in addition to the definitions introduced in (I), we have set To = exp(-k 2 ) Io(k2),172

ROGISTER and HASSELBERG

T = Te/Ti and z = w/we*, where w is the wave frequency; r is the radius of therational surface where the resistive layer is localized.
The electron fluid equations, taking proper account of the projection of thethermal force and of the electron heat flux along the total (static plus perturbed)
magnetic field, are given byzn - 0 = -i(0$/Dz) (£<£ + z«i0 [(t-z) + l.O7i0£ 2 (l-z)/z](II-2)

Here,
D = 1 + 4.03i (0£2/z) - l.O7(0$2/z)2d = (k|faiCe)2Te/0.51we* (=• 1/Cr)
a = ftLs2/L&, k,| = m^i/rLj, ft = 4irNTj/B2 and t = 1 + 1.71ije
The electrostatic potential $ and the parallel component of the electromagnetic potential An associated with the wave have been normalized according to<t> = (e^/T,), I = (A,LN/B a, L sft)(II-3)In the limit of high collisionality (C P- 1), the F.T. of Eq. (II-2) is easily combinedwith Eq. (II-l) to yield:
f(k)0k = id (t-z) z- 2 M- izo^')

where the prime stands for d/dk. Upon F.T., Ampere's law becomes:

Thus, since f(k) <x k2 for small k's and we look for odd <j>, even ^, solutions,^ = (i/ra) [f-'(k) Ik' + C,«'(k)](II-4)

Ik" - (f'/f) Ik' - z2 [(ra/k 2 ) - i/0(t-z)] flk = -C,f"(0)5(k)(n-5)

where Ik = k2^k is the F.T. of the normalized parallel current density
k,' aiJ,/«itNei.
2.2. The boundary conditions
For the purpose of this analysis, the hydromagnetic solution which holds in theouter layers can, in the neighbourhood of the resistive layer, be cast in thesymmetric/antisymmetric form:

IAEA-CN-47/E-HI-lt>H = - a z ^ H / ?

173

(H-6)

where Ai/% is the jump of ^H across the resistive layer. We have

J -co

J-

dk 5(k) Ik = -2ir lim Ik(II-7)

m [(k/f) Ik -Solving Eq. (II-5) in the neighbourhood of k = 0 shows that lim (k/f) Ik- a r z 2 lim Ik. Thus C, = 0. We have
k-0

*

'

dk e ik{ (kI k /0"-co

The last expression where one eliminates Ik7f through Eq. (II-5) is non-zero only if
Ik"" behaves like a Dirac function as k — 0; Ik ' ' has thus a discontinuity. Since Ik
is even, we finally obtain:
tf. = 2 (I k " ')o+/arz2f"(O)(U-9)

The problem has thus been reduced to solving the differential equation (II-5) subject
to the boundary conditions (II-7, 9). To illustrate the procedure leading to (II-9), one
can consider the function 0(£) = tan~'£).

3.

ANALYTICAL AND NUMERICAL SOLUTIONS

3.1. The limiting case A/a, — oo
Here f(k) = f"(0) k2/2 = -z~'(s + rz)k2, where s = 1 + rj,. Equation (II-5) can be reduced to Kummer's differential equation for the confluent hypergeometric function [5] w(a, - V4, Ak2) = exp (Ak2/2) Ik, where the parameters A
and a are defined by
A2 = iz(s + Tz)/0(t-z), ReA > 0(H-10)
a = - [ 1 + arz(s + re)/A]/4

174

ROGISTER and HASSELBERG\ \ \\\ \10-.•••""^--^IA?"TJV—•"tj\. . - ••' °FIG. 1. Growth rates and frequencies of the 1/1 tearing mode as Junctions of i/z^/Aip^and...:
exact results; —: approximate ones from the finite Larmor radius expansion;(1) with (LN, LT) = (58, 29) cm, i.e. w* = 4.7 x 103 cycles-s'1; (2) with (LN, LT) = (oo, <x).

Convergence of Ik as k — oo selects the solution w(a, — 'A, Ak2) = U(a, - 1 / 2 ,Ak2). The ratio (I k '' '/I^)o+ is, thus, obtained from Eq. (13.1.3) of Ref. [5]. There,
it follows that
= [8A3/2/xaTz(s + TZ)] [r(a + 3/2)/r(a)](H-ll)

where ReA1'5 > 0 since the principal branch of U is given by - ir < arg A k2 < TT
and we consider positive k's. T is the gamma function. Introducing
X/e = i 0 ar (t-z), X2 = - a r z (s + TZ), (thus A2 = Xe), £ = X/e%, and
XH = — ir}pJA\p'm we recover the dispersion relation by Coppi et al. [6]. For the
MHD neutral mode (^» = 0) we obtain a = 0, i.e. z(s + TZ) (t-z) = i / a V 6.
3.2. The case of arbitrary A/a,
We consider a typical TEXTOR discharge (shot 19730, B = 2 T, Teo = Ti0= 850 eV, No = 4.92 X 1013 cm" 3 , Z = 2, Ip = 334 kA, R = 175 cm,
a = 46 cm, qa = 3.6) for which the current density profile and thus Ls(r) is
known [7]: J ~ Jo [1 - (r/a) 2 ] 2 , Jo = 220 A-cm- 2 , q0 = 0.83. We calculate theeigenvalues of the 1/1 tearing mode (r = 16 cm) for different values of ^^/A^i fortwo sets (LN, Lj). The results are given in Fig. 1. We note that, with the finite Larmor radius expansion the unstable solution becomes unbounded for \{/JA\l/'m « 0.5

IAEA-CN-47/E-HI-l

175

if we* = 4.7 x 103 cycles-s" 1 . The curves obviously show that the finiteness of theratio A/a, cannot explain an eventual stabilization of the mode following temperaturecrashes in TEXTOR. If such a stabilization occurs, it must result from the changein ^ / A ^ i , A simple calculation which expands on the 'g' term in the MHD
cylindrical equation and assumes a pressure and a (frozen) current profileoc [1 - (r 2 /a 2 )] p( ' ) indicates that this ratio changes sign when p = — j/6j6e(q = 1),
i.e. a moderately hollow pressure profile stabilizes the mode.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]

LIGHTHILL, M.J., Fourier Analysis and Generalised Functions, Cambridge University Press(1962).
CREW, G.B., ANTONSEN, T.M., Jr., COPPI, B., Nucl. Fusion 22 (1982) 41.
COWLEY, S.C., KULSRUD, R.M., HAHM, T.S., Linear Stability of Tearing Modes, Rep.
PPPL-2262, Plasma Physics Lab., Princeton Univ., NJ (1986).
DRAKE, J.F., ANTONSEN, T.M., Jr., HASSAM, A.B., GLADD, N.T., Phys. Fluids 26(1983) 2509.
ABRAMOWITZ, M., STEGUN, I.A. (Eds), Handbook of Mathematical Functions, Dover, NewYork (1965) 504.
COPPI, B., GALVAO, R., PELLAT, R., ROSENBLUTH, M., RUTHERFORD, P., Fiz.
Plazmy 2 (1976) 961.
SOLTWISCH, H., STODIEK, W. MANICKAM, J., SCHLUTER, J., these Proceedings,
Vol. 1, p. 263.

IAEA-CN-47/E-III-3

EDGE PLASMA OSCILLATIONSIN THE H-MODE:
EFFECT OR CAUSE?
S.I. KRASHENINNIKOV, A.S. KUKUSHKIN,
V.A. POZHAROV
I.V. Kurchatov Institute of Atomic Energy,
Moscow, Union of Soviet Socialist Republics
Abstract
EDGE PLASMA OSCILLATIONS IN THE H-MODE: EFFECT OR CAUSE?
A model for self-sustained oscillations in a dense divertor plasma which provides a unified
explanation for both improved plasma confinement and non-steady edge phenomena in the
H-mode is further developed.

INTRODUCTION
This paper discusses the behaviour of a dense divertor plasma. The recently
emerging interest in this subject is connected both with experiments on existing
tokamaks, which led to the discovery of the H-modes and with divertor studies
aiming at tokamak reactor design. Recently, it was shown [1,2] that undercertain conditions in the divertor plasma, self-sustained oscillations due topeculiarities of the interaction between the plasma and the neutral gas formed
upon its neutralization on the divertor target may develop. The results of further
advancing the theoretical model of the oscillations are given, and good agreement
between the 2-D numerical simulation and a simple analytical model is shown, in
Section 1 of the paper. The effect of the self-sustained oscillations on the bulk
plasma parameters is addressed in Section 2. The oscillations are shown to lead,for ASDEX parameters, to a strong modulation of the neutral flow into the mainchamber, F N . In view of the fact that the edge plasma parameters, whichapparently exert a strong influence on the global energy confinement time r E ,are to a great extent determined by the flow F N , it may be expected that themodulations F£ will cause variations in r E and trigger the H-mode. In this way,
the non-steady edge phenomena accompanying the H-mode [3, 4] are also
provided with a natural explanation.

1.

SELF-SUSTAINED OSCILLATIONS IN A DIVERTOR PLASMA

According to experimental data and numerical results [5], the energy andparticle fluxes flowing within the scrape-off layer towards the target are localized
177

178

KRASHENINNIKOV et al.

FIG. 1. Geometry of the problem.

in a rather narrow hot 'core', which is surrounded by a relatively cold and rarefied plasma (Fig. 1). We shall particularly be concerned with conditions where
the ionization length for neutral particles leaving the target, # d , is less than thecore width, A k , i.e. the neutrals are localized in the core [1,2]. In this case,the plasma flow entering the scrape-off layer, IL, becomes much weaker thanthe plasma flow onto the target, P d , and plays no significant part. An estimateof the particle localization factor, £k = £ d /A k , can be obtained from the following
equations of balance of particles and energy in the recycling zone (the D-zone,
Fig. 1):

D d N d /C d -K,(T d )n d N d £ d = 0(2)

lo = J d (0T d + E d )som d c d siniKf3T d + E d )(3)where n d , N d and Td are the plasma and neutral density and temperature in the
D-zone; q0 is the energy flux entering the D-zone; \p < 1 is the angle between thethe magnetic field lines and the target plane; D d = T d /MK cx (T d ) n d ; E d is theenergy loss per ionized atom in the D-zone; M is the mass of the atom; K cx , Kj
are charge exchange and ionization rates, respectively; a and j3 are factors
correcting for the deviation of plasma fluid velocity in the D-zone from thesound speed c d , and for the conditions of convective energy transport to thetarget, respectively.
The q0 value is related to the power, Wo, entering the scrape-off layer of atokamak with a ko-null divertor: Wo = 47rk o q o RA k , R being the major radius.
Then, we find from Eqs ( l ) - ( 3 ) := 47rok 0 v d c d e d R sin ^/W 0 (K^Kf) 1 / 2(4)

IAEA-CN-47/E-III-3

179

where e d = /3Td + E d ; vd = (TJM)1/2; K^x l = Kcx ,(T d ). The dependence
Sj(T d ) = £ k W 0 M^7(Rak 0 ) is shown in Fig. 2 (here,W 0 is in MW, R being in m).
It is easily seen that a heavier filling gas and a single-null divertor do facilitate
the localization (£k < 1) condition Wo > 14 ak 0 sin $ R/Mjf (5 eV < T d < 15 eV).
This yields the following estimate for ASDEX: Wo > 0.7 MW (k 0 = 1, M H = 2,sin^ * 0.1,a ^ 0.5). For £k < 1, particle exchange between the core and thesurrounding cool plasma is reduced and can be neglected in a first approximation.
Then the basic parameter, besides q 0 , determining the steady state distributionof the core plasma parameters along the magnetic field is the core averaged numberof particles N (ions plus neutrals). It has already been shown [1,2] that thedependence of the plasma parameters on N for sufficiently high q0 values may
be ambiguous and discontinuous, and that particle exchange between core andcool plasma causes variations in N and leads to the self-sustained oscillations,involving transitions between the stable branches of the one-dimensional steadystates (OSS). A quasi-stationary theory of the variations in N, developed inRef. [2], does not allow us to consider the transitions between the different
OSS branches which are accompanied by particle redistribution between therecycling region (the D-zone) and the rest of the core (the T-zone, Fig. 1)
because that theory assumes equality of the plasma pressures, P o and P T , in theD- and T-zones. Provided that the pressure difference between the two zones issmall, the plasma fluid velocity, v0, in the transition zone between the D- andT-zones depends on the longitudinal viscosity i?(Td) of the plasma:
T

D

d

(5)

where
Zo = [qo(7 e /M T d) + Ti/Ki(T d ))/2T d sin M" 1is the characteristic scale-length for the variation in the plasma fluid velocity;« e and Kj are the electron and ion heat conductivities along the magnetic field;
7 e and yi are the relative amounts of total power as transported by electrons
and ions, respectively.
The pressure P T can be expressed as P T = (5/7)N T T s [2], where N T is a
T-zone contribution to N; Ts = (7/2)(q 0 LT d /2 sin i//)2/7 [(7 e /Ke(T d )) 2/7+ ( T J / K ^ T J ) ) 2 / 7 ] is the total (ion and electron) temperature at the separatrix,
L being the length of the magnetic field line in the core. The pressure P D can befound together with N D - a D-zone contribution to the N - from the steadystate equations (1) to (3) since the relaxation times of the plasma parametersin the D-zone are roughly proportional to ( K ^ ) " 1 and small:P

D = l o V ^ d «fd sin 4,

(6)

180

KRASHENINNIKOV et al.

75

50

25

10

20

Td(eV)

FIG. 2. Reduced parameter of neutral localization in the core, %*, versus plasma temperature
at the target.

L

(7)

sin

We use the diffusion approach to treat the particle exchange between core andcool plasma, which is mainly caused by neutral component transport [2]. Taking
the N(x,z) dependence to be N = N d (x) exp(-z/£ d (x)) and using the
T-approximation along the x-coordinate, we obtain from Eqs (5—7):
(8)

dt
dN H
— S - ( N T + N D (T d )) =(9)

where T is the cool plasma temperature at the target, andsin= (5/7) 2 T^zo (T d )*(T d )/T d 7 7 (T d )and K*=
Note that Gj^ > 0 for T d > T# and G i < 0 for Td < T# (we shall not consider
the range of extremely high temperatures). The system of Eqs (8) and (9)IAEA-CN-47/E-ni-3

181

FIG. 3. 'Phase portrait' of self-sustained oscillations.

allows us to study all phases of self-sustained oscillations. Linearizing
Eqs (8) and (9) around the steady-state solution, Td = T , N T = 3>(T ), we
obtain the following instability criterion:
dF(T )

v,

dlnTd

P,

d
—f--—

„

NDD(Tdd)

>0

(10)

T =T

where F(T d ) = *(T d ) + N D (T d ) and vL = BGj^/Sln^ > 0. Note that theinequality (10) is somewhat more stringent than the instability condition for
the OSS branch, dF/dT d > 0 [1 ]. The behaviour of the solution to Eqs (8) and(9) in the (N, T d ) phase space obeys the equationg D (T d )N D (T d )dT,,

N-N M (T d )

(11)

where
= -dlnND/dlnTd>0(T d ) = F(T d ) - N D (T d )G i (T d ,In case of an unstable equilibrium, the phase trajectories tend to a stable
limiting cycle (Fig. 3). Figure 4 shows the results of a numerical solution to
Eqs (8) and (9) and of a 2-D numerical simulation of the INTOR divertor
plasma. The simulation was carried out in conditions similar to those used in
Refs [1,2], but with a fluid description of the neutral component [6]. The
sharp drops in T d (t) correspond to transitions between the different OSS branches

KRASHENINNIKOV et al.

182

t (ms)

FIG. 4. Temporal evolution of plasma temperature at the sheath during oscillations obtained
from analytical model (a) and 2-D computation (b).

(,vjv\\ < 1). The good agreement between patterns (a) and (b) in Fig. 4 justifies
the application of this simple model to an analysis of the effect of the processes
in the divertor upon the bulk plasma.

2.

SELF-SUSTAINED OSCILLATIONS AND THE OCCURRENCE OF
H-MODESAs follows from the theory of tearing modes, the profiles of current
density j(r), and hence of the electron temperature Te(r) (assuming j a T^ 2 ),are of a rather peculiar kind [7-9]. Thus, it is desirable to have a j(r) profile
with a sharp drop just beyond the q = 2 resonant surface in order to stabilize
the most dangerous m/n = 2/1 mode [8]. The T-10 experiments on electron
cyclotron heating [10] apparently confirm this conclusion. An analysis of TFTR
experiments also leads the author of Ref. [9] to infer a strong dependence ofr E on the T e (r) profile, at the column periphery.
The temperature profile satisfies the heat conduction equation,

IJL
r 3r

(12)

where Q h , Q r , Q ei are the power densities of heating, radiative losses and electronion energy exchange, respectively; K^_ is the cross-field thermal conductivity ofthe plasma. It follows from Eq. (12) that, for low values of Q r and Qe}, theIAEA-CN-47/E-in-3

o
3

183

1

10

20
t (ms)

FIG. 5. Modulation of neutral influx into main plasma by self-sustained oscillations in thedivertor (2-D modelling, ASDEX, single-null divertor, deuterium).

Te(r) profiles necessary for 2/1 mode stabilization and hence leading to high T E
could be realized. However, an increase in Q r and Qej increases the second
derivative, d 2 T e /dr 2 , and finally leads to 2/1 mode destabilization and deteriorationof T E . Q r is determined by the level of impurity content, wall sputtering by
charge exchange neutrals being one of the main channels of impurity input into
diverted discharges [11]. Apart from this effect, the neutrals give rise to charge
exchange energy losses from the ion component, lowering the ion temperatureand increasing Q ei . Thus, r E may depend strongly on the neutral influx, F N ,into the main tokamak chamber. In steady state operation, the flow F ^ isconstant and F ^ —Fs. If the plasma confinement is poor, V* is high (we assume
r p oc T E ) so that the high energy losses from electrons make the optimum Te(r)profile impossible.
The self-sustained oscillations in the divertor plasma modulate rjjj. Figure 5
shows the pattern of F^(t) variation as obtained from 2-D modelling, making
use of a Monte Carlo technique for the treatment of neutrals, for ASDEX
parameters (single-null divertor, deuterium, Wo = 2.5 MW). Note that, for
Wo = 2.5 MW, double-null divertor and hydrogen as filling gas, no oscillations
were observed in the simulation. During time intervals when F ^ is reduced, theimpurity influx and charge exchange losses are small, and the Te(r) profile could
shift to optimum, thus improving the confinement.
Let us now see to what extent the dependences of the self-sustained
oscillation thresholds, of power Wo and plasma density at the separatrix, n s ,on the sort of filling gas (H, D) and on the divertor geometry (single-null,
double-null: k 0 = 1, 2) match the corresponding dependences of the conditions
for the occurrence of the H-modes. Let us first find the threshold values of q 0

184

KRASHENINNIKOV et al.

from Eq. (10) and then relate them to the values of Wo and n s . Uvjv^ > 1,
then, considering that q 0 A k « W 0 /k 0 and L « l/k 0 , the inequality (10) can be
written as
/2

(13)

where K|| = (1 to M^1/2), depending on the relative contributions of electrons
and ions to the energy flow along the magnetic field in the T-region; M H isthe atomic mass in amu, Wo is a normalization constant. For vjv^ < 1, theinequality (10) can be represented in the following form:

q0 > q? « koftf/Mft/*°(14)

Note that, for q0 « q*, the temperature T d is about 10 eV. To relate q0 to Wo,
we should examine the energy balance in the T-zone:
V K

i l V1T +V K

II II V tl T = °(15)Hence, assuming KL = k^J0
and taking account of the pressure constancy along
a field line and of the boundary conditions at the sheath (3) and at the separatrix,
2
T l = W0(27r) a R (a is the minor radius), we obtain, after extensive
calculations, the following condition for the onset of self-sustained oscillation:
.b'+2

W
~
Wo

~ l^l

i/c

. . i . . 5b + 6c + 9(16)

The minimum plasma density at the separatrix is only determined by the thresholdvalue of q0 and is independent of the K1 scaling:<;/7/M^/4(17)

For various scalings, such as T-l 1, Alcator, Bohm-like and n ^ n scalings,
Eq. (16) yields results that are rather close to each other:
Wo/Wo > 4 1 1 0 US)/M$*to

»•">

(18)

Thus, the functional dependences (13), (17).and (18) of the oscillation thresholdson the atomic mass of the filling gas and on the divertor configuration are inagreement with the corresponding conditions for the onset of the H-modes [12].

IAEA-CN-47/E-III-3

185

The set of the three main factors — (a) the effect of edge plasma parameters
on global plasma confinement; (b) the important role of neutral influx into themain plasma, F ^ , in the energy balance of the plasma edge; (c) buildup, in certain
conditions, of self-sustained oscillations modulating P^ - enables us to propose
the following model for H-mode onset in a diverted tokamak discharge: in the
L-mode, the power Wo is below the threshold value, (13) and (18), and the divertor
acts steadily. Plasma confinement is poor; the strong flow F|j cools the plasmaedge down, inhibiting the formation of optimum plasma parameter profiles, inparticular around the q = 2 surface. With Wo increasing beyond the thresholdvalue, self-sustained oscillations are triggered in the divertor. In the appropriate
phase of the oscillations, when the F|j flow decreases, the energy losses from theplasma edge are reduced, the plasma profiles are rearranged, and T E rises. Further
development of the oscillation leads to an increase in F ^ , causing edge coolingand lowering of r E . The power Wo released into the scrape-off layer increases
sharply, ensuring ionization of the neutrals and limiting further increase in F^
and edge cooling. In the next oscillation phase, F^ decreases again, etc. As aresult, the good plasma confinement in the H-mode is accompanied by regularbursts of Hft line radiation, together with MHD activity, reducing r E and giving
rise to an enhanced power release into the scrape-off layer. Note that, in thismodel, the edge plasma oscillations are the primary cause of the onset, not a
feature, of the H-mode.
The proposed model is, of course, still of a qualitative character; further
studies are necessary to refine and check the model. In particular, the effect ofenhanced particle and energy release into the scrape-off layer upon the oscillation
dynamics is not quite clear. Nevertheless, the theory provides a unified
explanation for both the improvement of plasma confinement and the non-steadyedge phenomena in the H-mode. A number of coincidences are hardly attributable
to simple chance, such as: (a) the closeness of the threshold Wo values for
onset of the H-mode [12] and the buildup of self-sustained oscillations (the above
presented example of modelling ASDEX with SN and DN divertor configurations);(b) the similar types of dependence of these thresholds on the sort of filling gas
and on the divertor geometry, [12] as well as Eqs (13) and (18); (c) the agreement
between the period of the oscillations (Fig. 5) and the characteristic time interval
between Hft spikes in the H-mode [3, 4]; (d) the important role played by neutral
recycling at the plasma edge in the H-mode onset, which is pointed out in the bulkof the experimental papers (remember the difficulties in producing the H-mode
in open divertor geometry [4], where the neutrals can penetrate more effectively
into the main plasma).

ACKNOWLEDGEMENT
The authors are grateful to Dr. A.M. Stefanovskij for discussion of the subject.

186

KRASHENINNIKOV et al.
REFERENCES[1] KRASHENINNIKOV, S.I., et al., in Controlled Fusion and Plasma Physics (Proc. 12th
Europ. Conf. Budapest, 1985), Part II, European Physical Society (1985) 500.
[2] KRASHENINNIKOV, S.I., et al., Self-Sustained Oscillations in the Divertor Plasma,
Preprint IAE-4235/6, Moscow, 1985.
[3] KEILHACKER, M., et al., Nucl. Fusion 25 (1985) 1045.
[4] KAYE, S.M., et al., J. Nucl. Mater. 121 (1984) 115.
[5] INTOR GROUP, International Tokamak Reactor, Phase Two A, Part II (Rep. Int.
Workshop Vienna, 1984 and 1985), IAEA, Vienna (1986).
[6] KRASHENINNIKOV, S.I., POZHAROV, V.A., Fluid Description of Neutral Component
in Divertor Plasmas, Preprint IAE-4173/6, Moscow, 1985 (in Russian).
[7] GLASSER, A.H., et al., Phys. Rev. Lett. 38 (1977) 234.
[8] BOGOMOLOV, L.M., et al., Kink- and Tearing-Mode Stability Conditions in theTokamak, Preprint IAE-3827/6, Moscow, 1983 (in Russian).
[9] FURTH, H.P., in Controlled Fusion and Plasma Physics (Proc. 12th Europ. Conf.
Budapest, 1985), Part II, European Physical Society (1985) 980.
[10] ALIKAEV, V.V., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984
(Proc. 10th Int. Conf. London, 1984), Vol. 1, IAEA, Vienna (1985)419.
[11] STAUDENMAIER, G.J., Vac. Sci. Technol., A 3 (1985) 1091.
[12] WAGNER, F., et al., Nucl. Fusion 25 (1985) 1490.

IAEA-CN-47/E-IH-4

MAGNETIC ISLANDS ANDCHAOS INDUCED BY HEAT FLOW
P.H. REBUT, M. BRUSATI, M. HUGON, P. LALLIAJET Joint Undertaking,
Abingdon, Oxfordshire,
United Kingdom

Abstract

MAGNETIC ISLANDS AND CHAOS INDUCED BY HEAT FLOW.
A theoretical model is presented which treats the problem of confinement degradation in tokamaks with increasing input power. It is shown that, when full account is taken of the magnetic field
line topology, the adiabatic invariance conditions and the particle motion equations in the guiding centre
approximation permit the heat flow to be related with the level of magnetic perturbation associated withlarge (M ~ 15) poloidal mode numbers. It is shown that substantial power can be transported acrossthe plasma by low level magnetic fluctuations. One possible mechanism for the self-sustainment of themagnetic islands is investigated.

1.

INTRODUCTION

The degradation of confinement in tokamaks when asubstantial amount of heating power is applied is generally
ascribed to anomalously high electron losses.
In this paper we show how low-level broad-band magnetic
perturbations (B/B - 3x10"") associated with large poloidal
mode numbers (M - 15) are sufficient to account for the losses
observed experimentally. To achieve this result it is necessary
to take full account of the topological properties of themagnetic field lines.
The calculations are carried out for low collisionality
plasmas in steady state regime, neglecting curvature and finite
Larmor radius effects. The low collisionality assumption isjustified by present experimental conditions; for typical JET
parameters [1] the electron mean free path (< i C m ) is much
longer than the plasma toroidal length (- 20m).

187

188

REBUT et al.
yf

1

N-1

N

N+1

No

28
FIG. 1. Schematic diagram of the slab model.

2.

MAGNETIC TOPOLOGY
The magnetic field in a slab geometry (Fig 1) is describedby the components of the vector potential _A(x,y,z):
Ax= 0!°1 x 2 + A ?° cos k (y + M )

(1)

2

=

R=o

kR

1x A
Y = ilBo'

In toroidal geometry:

where q1 is the shear term, e the virtual island half-width and
Y the overlapping parameter of the different chains of magnetic
islands; — and 2nR are the periodicities in the y and z
directions, respectively. Eq. (1) corresponds to N o chains ofmagnetic islands [2].

189

IAEA-CN-47/E-III-4
Force Field Lines

3.0

J9HB

2.0

1.0

X
<

0

I
w

few

i1 Hi1
1

-1.0
-2.0

I

-3.0

-2.0

1.0

-1.0

2.0

X-AXIS
FIG. 2. Poincare map computed for the field distribution of Eq. (1), with y = 1.0.
Satellites (secondary islands) are visible around the main magnetic island structures.

The different chains of islands can be mapped into eachother through a translation by A (distance between two chains ofislands) and a rotation due to the shear. Consequently, they are
all equivalent and the full slab topology can be deduced fromthe detailed knowledge of the properties of a single chain ofislands. In Fig. 2 the (x,y) Poincare map is shown for fivechains of islands with an overlapping parameter 7 = 1.0. The
onset of stochasticity affecting neighbouring chains of islands
occurs at values of Y £ Y c = .8. For .8 < Y < 1.5, islands withclosed magnetic surfaces are embedded in a chaotic region.
With reference to Fig-1, the normalised cross-sectional
area o of the tube of flux associated with a chaotic field line
and crossing one chain of islands is given by:(2)

where Hii(Y)H(B x )H cr03S dy dz
z

cross

dy dz

f 1 when the field line crosses the island chain
0 otherwise.

Similarly, H(B X ) is the Heaviside function for B x .
Following the same formalism, we can define the fraction
ct(Y) of the flux tube which crosses the next chain of islands

190

REBUT et al.

a (o)

FH
3

1.0
0.8
0.6
0.4
0.2

O8 / 1.0

1.2

1.4

1.6

1.8

2.0

FIG. 3. Plots of a, a,a/(l - a) and F as a Junction of
the overlapping parameter y.

and the normalised cross-section of the island. Results ofnumerical computations are shown in Fig. 3 for a and the quantitya o

The behaviour of the field lines in the chaotic zone
corresponds to a diffusion process through neighbouring chainsof islands according to an equation of the type:
2

(3)

3L

= 1 A2 3 F
2

SCCY)
o(Y)

(a =1)
Mq'A

where F(L,x) is the probability of finding a field line at x
after a path length L,and L o is the required path length for afield line to cross an island chain; and where S C (Y) is the
cross-sectional area of the chaotic region (normalised).

3.

PARTICLE MOTION
The following equations are written in a reference frame inwhich the magnetic islands are at rest and where <j> is theelectric potential which ensures the plasma quasi-neutrality.
The electric field satisfies the symmetry invariance of themagnetic field.

IAEA-CN-47/E-HI-4

191

Taking into account the constancy of the magnetic field in
the chosen frame, the particle adiabatic invariants are:U.

= 1 m wf

JL

p

U,, =

1

~

(4)
m w,2, + q<j>where subscripts 1, u refer to the field lines. The motion
equations in the guiding centre approximation are:
(5)

dr
-=
dt

=

w . B
=
=
| B| 2B + v0~~

They correspond to a motion along the field lines and a perpenE x B
dicular drift with velocity v 0 =-, where E = -Vd> is the
~
IBl2
~
~
electric field.
From conservation of the number of guiding centres, thetotal current j associated with them satisfies the equation:(6) V . j g = V.(2e g + i i g ) = V.[j ||g e M+ (q^+ qenjvo] = 0

when perpendicular diffusion is neglected.
In conditions of quasi-neutrality and taking into accountthat the parallel plasma current jequals j M , we have:
(7)

V.(j

e

Defining a scalar A by:

JM= A 13£M) = 0

we get:(8)

B.V A = 0

Eq.(8) shows that the parameter A is constant on every fieldline. In particular, A is constant on a magnetic surface.
When the parallel resistivity n M is taken into account, A
is defined by:
"

(9)

A /L

n

dl = L

1

JL

J o
"
B
2TTR
where L is the field line path and V is the loop voltage. Eq.(9)can be solved using Eq.(3).
A particular case arises from Eq.(8), when all field lines
behave similarly; in this case, A is single valued over the whole
region [3].
In the chaotic region,field lines with different paths havedifferent values of Y, which depend also on the boundary
conditions. At a given location, an average value A Q h can be

192

REBUT et al.

defined statistically over three classes of field lines: fieldlines connected to each boundary (X lf X 2 ) and linking bothboundaries ( X 3 ) . On the contrary, the field lines in themagnetic islands remain localised and have in general adifferent \^ (Fig. 1). The guiding centre trajectories can becalculated simply by referring to pseudo field line equations.
In the particular case when <j> = <f>(x) = -E o x,the guiding
centre equations become:
dX
dt
(10)
dY
dt
where X

x + B

Y = y.

o' w nFor a given w(| the guiding centre

trajectories have the same behaviour as the field lines and the
topological properties summarised in section 2 apply to theparticle motion.
The effect of the electric field is smallest for fast
particles; when E 0 /B o 'w M < A,particles follow closely the fieldlines, while drift effects dominate when E 0 /B 0 'w,, > A (slowparticles).

1.

PARTICLE AND ENERGY FLOW
To maintain temperature and density gradients as particles
are crossing the various chains of islands, suitable sources andsinks have to be introduced at the boundaries of the slab.
These will be schematically represented by hot and cold "seas"of particles with Maxwellian velocity distributions F^ and F C J
respectively, at temperatures and densities T^, n H and T Q , nQ
(Fig-1).
Taking into account the symmetry properties of the magnetic
islands and the adiabatic invariants (Eq-(4)), it can be shown byrecurrence analysis that the electron flow ¥ through N o chainsof islands is given by:
(11)

a o

2KT
'H

- exp(TTITK- nc

1M 3L A 2

where <f>0 (<j>0 = E 0 N 0 A) is the potential difference between the
hot and the cold sea and 1 is the slab dimension in the ydirection.
In order to maintain quasi-neutrality conditions,theelectron flow has to equal the ion flow. Assuming similar
temperatures for electrons and ions we get:

IAEA-CN-47/E-III-4
qp<f>
(12)

exp

(-

n

-*1JL)r

KTHYt_ £ ( _ £ ) +

nH

TH

m, 'i
(-£
mi

(nig/m^) 2 = 0 :In the approximation

KTH

(13)

T

193

,V Y nE o = —Jl
%

,

VYT,

(JL. + 1 J L )
n
2
T
H
H

which shows that the velocity v 0 of the chosen reference frame
follows the diamagnetic velocity of the electrons, a result
already obtained for microtearing modes [1],Similarly, the heat flows are given by:
2

(lib)

PHpat . =
Heat

i+

2

n

a a
2_2

J

/8KT
9

(KTHH - KT_)K Tc J ) 1M S 2ii A
(KT
A2
q
q'
1

/ — - ii
Hll VKT
T
IM q
n
A2
.n
1 H -/
iH 1M H A
i H

N 0 (l-a) + 1

y

lH

•nmi

q2

The electron distribution function in the chaotic region at
the Nth chain of islands is:which shows a linear combination of the Maxwellian distributionfunctions describing the hot and cold "seas".

5.

SELF-SUSTAINMENT OF THE PERTURBATIONThe necessary condition for self-sustainment is foundthrough the determination of the relative differencep = U c h -A 1 )/<A> between the values of A associated to thechaotic zone and to the islands respectively. In the presence of
perturbations witli a range (AM) £f poloidal mode numbers aroundan average value M, such as (AM)/M = \i, the mean separation A
between chains of islands in toroidal geometry is:(16)

A = L- . 1_ . 31 ; q =
2p

M2

q'

"ll
RB p

Ampere's equation and the definition of the overlapping
parameter Y leads to the relation:
B(17)

2U . M . p=

F(Y) .

Uo-JR

where j is the average current parallel to the magnetic field at
the minor radius r and where F(7) depends on the shape and194

REBUT et al.

cross-section of the islands. As shown in Fig. 3, F(Y) has aminimum close to 0.5 for Y - 1.1 - 1.2.
Then,by using the safety factor q0 and current density j 0
on the axis of a Tokamak, we get the necessary condition ofself-sustainment of the perturbation:

(18)

8p

. 5L . M
Qo>1

6j = j c h - Ji

Jo

where <Sj is the default of current inside the islands withrespect to the chaotic region.
Substituting Eq.(i6) into Eq.( 1 4a), the electron heat flow can be
written as:
P

Heat e e"

2|i M s q'

1-a

V v m ,eEq.(19) shows that the heat flow is strongly dependent on the
average poloidal mode number M which defines the number N o ofchains to be crossed, and on the temperature and density of thecold "sea". Around the separatrix in X-point discharges, q andconsequently R go to infinity, implying that P H e a t _ go to zero.
However, the transport in this region could be explained byother phenomena (neoclassical limit, MHD instabilities).
Combining Eq.(17) with Eq.(19) leads to:
2/s

3/s _

(20) p -l(l-) (1)
2 2p

3

P

l/s

F(Y).(i2i) . i^o.(q q "
a o

l/s

Hpa1-

jq

2

)

2-rrr nQ vQ VKTThe minimum value of p occurs at Y = 1.2.
By reference to Section 3, the average electricalresistivity n along a field line depends upon the relative
length of its path in the different plasma regions including thetwo "seas" and on the local distribution function given by
Eq.(15). In case of "infinite seas" the current density in the
chaotic zone can be written as:

(2D
(21)
As n

i _ Jc ( . ) J H _
Jch
z
a

T

v
2irR

n

ch

, j ht is larger than the current density of a plasma-"

N T * (N0~N)TH

with temperature T c h =". The determination of
j , , the current density in the°islands, requires to calculate
the local distribution function by integrating the particle
trajectories including the E x B drift.
For small temperature and density differences between the
hot and the cold sea, it can be shown that

IAEA-CN-47/E-III-4

(22)

195

p =v

where v is a numerical coefficient, the value of which depends
on the details of the collision and on the relative value of theradial electric field. Using Spitzer conductivity and a Krook
collisional model, v is found to be in the range 0.1 - 0.3. A
rough estimate of (T H "T C )/T c n based on the relative plasma
volumes at high and low temperatures leads to a self-consistentp value of a few percent.
It must be noted that a full treatment should take into
account the ion Larmor radius when it is only marginally smaller
than the island size. We should also remark that the crash of asawteeth can also contribute to p as the current expelled fromthe central region of the plasma should flow preferably in the
chaotic zone.
Preliminary comparison with JET experimental results has
been carried out. Inserting JET data into Eq.(20) for q = 1.5 we
get p = M > 2.5 x 10~ 2 and a band of M numbers extending fromj ~10 to 20. The corresponding magnetic perturbation B/B has anamplitude of 3-10"" and the chains of islands are separated by adistance A - 10" 2 m, i.e. only a few times larger than the ion
Larmor radius. From Eq.(22) with v = 0.3, (T H -T c )/T c n ~ 0.3 andthe number of chains of islands between the two "seas" would bein the range 15 to 20.

6.

CONCLUSION
The topology of magnetic field lines is a dominant feature
affecting the properties of magnetically confined plasmas. An
equilibrium could exist (Y - 1.2) between undestroyed magnetic
islands and chaotic regions.
This topology may be the key feature which explains the"anomalous" confinement in tokamaks, by allowing transport along
field lines linking different regions of the plasma. Thistransport is affected by the plasma density and temperature at
the edge. The reverse field pinch configuration could represent
an extreme case of this model.
The current perturbation and then the topology may be
self-sustained by the difference of resistivity between chaotic
zone and islands, resulting from the field line path and thepresence of hot electrons in the chaotic zone. The islands havea phase velocity close to the electron diamagnetic velocity.
Other effects, not taken into account so far, like finite
ion Larmor radius, perpendicular diffusion and field curvature,
could play an important role in the self-sustainment of such a
topology.

196

REBUT et al.

When the model is applied to JET, the observed heat fluxes
can be accounted for by perturbation levels in the magnetic
field B/B
3-10"" and in the cm-rent 6j/j - 3%, corresponding
to average poloidal mode number M - 15.

ACKNOWLEDGEMENTS
We acknowledge the contribution of A. Samain, who is
working along similar lines at Cadarache (CEA).

REFERENCES[1] JET TEAM (presented by P.H. Rebut), these Proceedings, Vol. 1, p. 31.
CORDEY, J.G., et al., these Proceedings, Vol. 1, p. 99.
[2] REBUT, P.H., HUGON, M., in Plasma Physics and Controlled Nuclear Fusion Research 1984
(Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 197.
[3] TAYLOR, J.B., Phys. Rev. Lett. 33 (1974) 19.
[4] SAMAIN, A., Plasma Phys. Controll. Fusion 26 (1984) 5.

IAEA-CN-47/E-HI-5

INCOMPLETE THERMALIZATIONAND ENHANCED TRANSPORTIN TOKAMAK PLASMAS
S.-I. ITOH
Institute for Fusion Theory,
Hiroshima University,
Hiroshima
K. ITOH
Plasma Physics Laboratory,
Kyoto University,
Kyoto
Japan

Abstract
INCOMPLETE THERMALIZATION AND ENHANCED TRANSPORT IN TOKAMAKPLASMAS.
Plasma transport in the presence of unthermalized particles is analysed. During intensive
heating a high energy tail develops and the spatial loss of fast particles affects the global energybalance. The paper examines the anomalous transport of electrons in the tail current operationof Ohmic discharges, the effect of the spatial loss on RF current drive, and the ion confinement
in ICRF heating plasmas. Experimental results show that the tail current operation of Ohmic
plasmas reduces the anomalous loss. On the other hand, the fast particle losses put an upper
limit to the efficient heating and current drive. The small amount of fast particle loss (such as
ripple loss) annihilates the RF enhancement of the fusion reactivity. In this case, the order ofmagnitude of the global confinement time is between that of the bulk particle confinement
time and that of the slowing-down time. Effects of incomplete thermalization in real space
are also discussed.

1.

INTRODUCTION

Recently, alternative operation in tokamaks has attracted attention. For
instance, enhancement of the fusion reaction rate by taking advantage of the
non-Maxwellian ion component has been discussed. Various methods ofintensive heating, current drive, current profile control, and slide-away dischargeusing a highly conductive plasma have been applied in order to extend the attainable parameter regime. The formation and sustainment of the tail components
are key issues in the realization of these operations.
If spatial loss of tail components occurs, the thermalization of the tailbecomes incomplete. Mechanisms of incomplete thermalization put a limit
to the extension of plasma parameters and affect global confinement. In an

197

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S.-I. ITOH and K. ITOH

investigation of the conditions of alternative operation methods and theachievable plasma regime, we have examined mechanisms of the limitation of
the tail development.
We analyse electron tail operation of Ohmic discharges in the low density
regime and discuss the possibility of reduction of anomalous transport.
Concerning RF-induced ion tail development, we report on serious constraints
due to direct tail loss for the input power and the reaction enhancement. The
reduction of the current drive efficiency is also examined.

2.

TAIL OPERATION OF OHMIC PLASMA

A model of magnetic braiding [1-3] is employed for the electron loss
mechanism, using the velocity dependent diffusion coefficient D = IUJDQ(u = v/ve, D o = Cv e 5 2 /qR, C being a numerical coefficient, q the safety factor,
ve the electron thermal velocity and R the major radius). For application to
low density plasma, a correction to the collisionless skin depth is taken into
account, i.e. the smaller quantity of c/cjp and 1/RkJ is indicated by 6; we
simply have 82 = c2/o>p (1 + n c /n). To determine the velocity distributionfunction, we first solve the one-dimensional Fokker-Planck equation forsteady state:•_99u
where E is the DC electric field, v is the collision frequency at v e , g = \/(3y/W+ |u| 3 ),TQ = a2 /Do and S is the source of cold electrons. We have f = f^^ + ftail, andf t a i ] (u)s 10>/2e exp(-l/4e-u 2 /2et>T 0 ) f (0) in the small e limit (e = eE/vmve).
The tail current is given by Jtai] = -ev e / uftaii du.
The spatial profiles are determined by the radial transport equation
V-q + E-J = 0, where q = —nD()VTe and E = const. The boundary conditionsa r e q ( 0 ) = 1, J' = 0 for r < f l (q( r i ) = l ) a n d J = 0 for r > r 2 (q(r 2 ) = 2) [4, 5].
The density profile is assumed to be parabolic. This model has been usedsuccessfully to explain the results for ordinary Ohmic discharges. Our analysis
includes the tail component following the bulk distribution according to Eq. (1).
The plasma current profile is obtained from the transport equation
2oB

2aB

J0(x)Y0(x2)-Y0(x)J0(x2)(2a)

J 0 (x 1 )Y 0 (x 2 )-Y 0 (x 1 )J 0 (x 2 )199

IAEA-CN-47/E-IH-5

(b)

3
\
—o-er^ ._--°- —201n e (1O"in3)o2FIG. 1. Low density OH discharge for PLT parameters: R = 1.35 m, a- 0.4 m, B = 3.1 T,/ p = 0.25 MA, Zeft = 2 and nc=10iam-3.
(a) One-turn voltage (solid line for theory, dashed line for experiment; V% is given in volts)
and tail current normalized to I versus n(bj TJO) versus h~e. The solid line indicates the present model and the open circles areexperimental values. For explanation of dashed line and dashed-dotted line see text.

2B (1 - a) e

(2b)where x = per, M2 = 7 7 . 4 ^ Re 0 R 2 /m e Z eff In Ac 2 , e = e(O)h 2/3 ,a(0) E = 2aB/noR, a ' s ^le electrical conductivity at T e , x 1 2 = A""]^ anda = J^-^/JCO). The eigenvalues E and a are obtained numerically. Figure 1
shows the density dependence of the loop voltage, Vg= 27rRE, Itajj/Ip and T e (0).
A recent PLT experiment has been simulated [6]. The numerical coefficient
C for D o is given by adjusting Vg to the experiment at n = 2 X 1019 m" 3 . The
solid lines indicate the results of our analysis. Figure 1 (b) also shows the resultsof a model which neglects the tail (dashed line) and of a model without
6-correction (n c = 0, dashed-dotted line). These two models do not explain
the n-dependence of T e ; the confinement time is longer than that expectedfrom Alcator scaling. Our model seems to fit the experimental results in therange o f i T > 5 X 1018 m" 3 . However, for n < 5 X 1018 m" 3 there is stilla discrepancy; theory predicts Vg a y/n, but the experiment shows Vg °c n.
It is not only the magnetic braiding model which gives this proportionality of Vj;
other models, such as that for the collisionless trapped particle mode [4], also
give Vg cc y/~n j n the low density limit. Experimental observation shows afurther reduction of Vg and thermal conductivity [6, 7]. This fact raises
questions regarding further experiments, for instance whether anomalous
electron loss can be reduced by sustainment of the beam current (to make thebulk electrons current-free as in the Astron-Spherator configuration [8, 9]).

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3.

S.-I. ITOH and K. ITOH

ION TAIL GENERATION BY RF WAVES

We investigate the effect of the high energy particle loss by examining theminority ion cyclotron resonance heating. The Fokker-Planck equation, withRF heating and particle loss, may be written as— = C(f) + K ( f ) - X f + S
ot

where C(f) is the collision term and X is the spatial loss, assumed to have theform X = |UJ S /7Q(U = v/vj, v T being the thermal velocity at backgroundtemperature). For simplicity, we use the linearized collision operator, assume
isotropic distribution, and write [10, 11]:9(KvK(f) =

3v-v2£)

The absorbed RF power is given by P^ = 3 mKn.
In the high heating limit, £ = P rf T s /3nT e > 1 (with r s the slowing-downtime by electrons), the ion tail develops and the characteristic tail temperatureis given by £Te. In this limit, the tail distribution is approximately given by2oftail * e *P I " "
tml

V

if

3 = uS/2+1
s+2 v

^

If the index s is greater than 2, the tail distribution decays much faster than theGaussian distribution for u > u c . The cut-off velocity u c is approximately given?). The loss rate T? is calculated as
n = Pioss/Prf = / Xfwdv / / wK(0 dv" (w = mv 2 /2)
In the weak loss limit, we have(if s is odd and s = 2£ + 1). The loss becomes appreciable for £ > (T O /T S ) 2 / S .
We solve the Fokker-Planck equation numerically. Considering ripple
loss [ 12], we take s = 7. The power contributing to bulk heating is given aspbulk = /C(f)wdv > . In the range of ^ O( 10 2 ) and T - 0 / V - S ^ O ( 1 0 4 ) , the bulk

201

IAEA-CN-47/E-in-S

100 -

a 0.5

200

100

FIG. 2, Power partition and power amplification factor for s = 7.
(a) Ploss (dashed line) and i>bulk (solid line) for TJTS = 10s (I) and TO/TS = 103 (II). The
dashed-dotted line is for P - Prt.
(b) Qr{ versus % for TO/TS = 10s (I) and TO/TS = 103 (II). The dashed line is for T S /T 0 = 0(no spatial loss).

heating rate, Pbulk/Prf = 1 — "i?, is approximately given by 1 -rj^ 3£ °- 86 (T 0 /r s ) 0 - 12
for s = 7. Figure 2 (a) shows the partition of the absorbed power as a functionof the heating density for the cases of TQITS - 10s and TQ/TS = 10 3 . Although
the spatial loss is negligibly small for thermal particles, P )oss exceeds P bu]k for$ > £c- £c =s 20 for TO/TS = 103 and £c ^ 35 for TO/TS = 10 s . It is also notedthat if % exceeds £ c the power transferred to bulk particles saturates and theincrement of the heating power enhances only the fast ion loss.
The deterioration of the tail confinement affects the fusion reaction rate.
The fusion reaction with the background tritons can be calculated assumingthat the minority deuterons are heated. For the estimate we employ the Duaneformula for D-T reactions [13]. In the calculation of the average reactivity <av>
we use the power amplification factor, Q rf = PfUSion/Prf = w f n T n D <av>/P rf(wf = 22.4 MeV). Figure 2(b) illustrates Q rf as a function of % forT e = 6 keV;
TJTQ is chosen to be 0, 10"5 and 10" 3 . The enhancement of the reactivity isannihilated by the small amount of the fast particle loss.
We note that the estimated loss rate ?? is the upper limit (see Fig. 2). The
spatial loss \ starts to decrease in the high energy limit because the VB drift
becomes large for energetic particles. For particles w > w H (w H satisfies therelation £2H (w H ) = Peff(wH)> ^ H being the poloidal rotation by VB drift),s = -0.5 holds. In this case, 77 saturates for large £, and P^ss/Prf scales as(r,/T 0 ) 0 - 7 .

4.

EFFECT ON RF CURRENT DRIVE EFFICIENCY

The efficiency of the RF current drive is limited by the spatial loss. If wewrite the loss term as |u| f/r 0 (see Section 2), Eq. (1) reduces to

C(f)+K(f)- |u|f/T0

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S.-I. ITOH and K. ITOH

where K(f) = V-RVf (R being the driving term due to the RF momentum
absorption). For high energy particles, the RF force compensates the spatial
loss of the momentum.
If the RF driving term extends to the region of high phase velocity andK(f) is written as K 3* R3 2 f/du 2 , the asymptotic form of the tail distribution
is given as* exp(^ e /Ru - 2u
The tail distribution decays exponentially in the presence of spatial loss. Theenergy limit of the tail electrons is estimated as u c « (p e /R) 0 - 2 (v e rQ) 0 - 2 . Foru > u c , f decays quickly. As the power level increases, the critical velocity slows
down and the sustained current normalized to the RF power decreases.
When u c approaches unity, the current drive efficiency becomes very low.
This process imposes an upper limit on the sustained current for a given value
ofverQ.

5.

SUMMARY AND DISCUSSION

The confinement of a tokamak plasma with unthermalized particles has been
investigated. The role of the tail component appears to be different for OH plasma
and ICH plasma. In OH plasma, the generation of highly conductive and high energy
electrons reduces the electric field. The confinement is not deteriorated by selective
loss of fast electrons; the experimental results show a reduction of bulk transport.
In contrast, in ICH plasma the high energy ion mode (if> 10) is largely affectedby selective loss of energetic particles. In the density limit of the high heating
regime the order of magnitude of the global confinement time is between
the order of magnitude of the collision time and the order of magnitude of thebulk diffusion time. The lowering of the current drive efficiency is also discussed.
The processes discussed in this paper are the phenomena associated withincomplete thermalization in velocity space. Incomplete thermalization also
occurs in real space where it is accompanied by thermoelectric effects. The
electron temperature is usually considered to be constant on the magnetic
surface. Temperature inhomogeneity along the field lines appears in the
disruption phase [14] and/or near the plasma periphery where local coolingtakes place owing to interaction with external materials. In this situation
the heat flux along the field lines completes the thermalization, and thisflux can enhance the cross-field plasma losses. These processes illustrate theimportance of incomplete thermalization in global confinement.

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203

ACKNOWLEDGEMENTS
The authors wish to thank Dr. A. Fukuyama for many useful comments
and discussions. Continuous encouragement by Profs. K. Nishikawa, A. Iiyoshi
and K. Uo is also acknowledged.
This work was partly supported by a Grant in Aid for Scientific Research
and a Grant in Aid for Fusion Research of the Ministry of Education of Japan.

REFERENCES[1] OHKAWA, T., Phys. Lett., A 67 (1978) 35.
[2] RECHESTER, A.B., ROSENBLUTH, M.N., Phys. Rev. Lett. 40 (1978) 38.
[3] KADOMTSEV, B.B., POGUTSE, O.P., in Plasma Physics and Controlled Nuclear Fusion
Research 1984(Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 69.
[4] PERKINS, F.W., in Heating in Toroidal Plasmas (Proc. 4th Int. Symp. Rome, 1984),
Vol. 2, Int. School of Plasma Physics, Varenna( 1984) 977.
[5] ITOH, K., 1TOH, S.-I., PERKINS, F.W., Res. Rep. HIFT-121, Hiroshima University(1986); Comments Plasma Phys. Controll. Fusion 10 (1986) 103.
[6] CHU, T.K., et al., Plasma Confinement in Toroidal Discharges Sustained by a Highly
Conducting Plasma Current, preprint, Princeton University, 1985.
[7] FUSSMANN, G., BECKER, G., BEHRINGER, K., et al., in Plasma Physics and Controlled
Nuclear Fusion Research 1982 (Proc. 9th Int. Conf. Baltimore, 1982), Vol. 3, IAEA,
Vienna (1983) 295.
[8] YOSHIKAWA, S., CHRISTOFILOS, N.C., in Plasma Physics and Controlled Nuclear
Fusion Research 1971 (Proc. 4th Int. Conf. Madison. 1971), Vol. 2, IAEA, Vienna(1971)357.
[9] MOHRI, A., NARIHARA, K., YOMITA, Y., et al., in Plasma Physics and Controlled
Nuclear Fusion Research 1984 (Proc. 10th Int. Conf. London, 1984), Vol. 3, IAEA, Vienna(1985)395.
[10] STIX, T.H., Nucl. Fusion 15(1975)737.
[11] FUKUYAMA, A., et al., these Proceedings, Vol. l,p.627.
[12] KOVRIZHNYKH, L.M., Nucl. Fusion 24 (1984) 851.
[13] BOOK, D.L., NRL Plasma Formulary, Office of Naval Research (1978) 36.
[14] ITOH, K., ITOH, S.-I., Phys. Lett., A 109(1985) 31.

IAEA-CN-47/E-IIKi

EFFECT OF QUASI-LINEAR EVOLUTION OF
THE MINORITY ION DISTRIBUTION FUNCTION
ON THE WAVE DISPERSION AND ABSORPTIONAND ON THE
THERMONUCLEAR REACTION RATE
A.V. LONGINOV, S.S. PAVLOV, K.N. STEPANOV
Kharkov Institute of Physics and Technology,
Ukrainian SSR Academy of Sciences,
Kharkov,
Union of Soviet Socialist Republics

Abstract
EFFECT OF QUASI-LINEAR EVOLUTION OF THE MINORITY ION DISTRIBUTION
FUNCTION ON THE WAVE DISPERSION AND ABSORPTION AND ON THE
THERMONUCLEAR REACTION RATE.
The effect of the quasi-linear (QL) evolution of the minority ion distribution function
caused by slow (ion cyclotron) wave absorption at minority ion resonances, CO^CJ^ or w ! «2wj ; ,on the dispersion and damping of the slow waves is considered. The conversion of the slow
waves (SW) propagating from the low magnetic field side into shorter wavelength SW and theopacity zone found between conversion and ion cyclotron resonance points existing forminority ion densities in excess of the critical density are shown to disappear at large SW
amplitudes. This is due to the high energy tail 'cut-off for minority ions as a result of quasilinear diffusion of resonant ions in the velocity space, perpendicular to the magnetic field.
If the opacity zone disappears and the zone o^^Za>'c is accessible, it is possible to heat theplasma in powerful regimes with the help of SW excited on the low magnetic field side, either
directly or by using fast wave conversion into SW. — It is also shown that SW accessibility of
the cyclotron resonance zone for minority ions allows one to reach the two-component
reactor regime (resonance w = w cd for the deuteron minority in the tritium plasma andresonance co = 2w ct for the triton minority in the deuterium plasma). In the former, somewhat
more efficient regime (d-minority in the t-plasma), the Q-factor reaches values of Q « 1.5-2.5
at T = 5-10 keV; moreover, the energy confinement time can be reduced by more than anorder of magnitude, and the specific power of the d-t reaction increases three to five times
over a conventional regime with nd = n,.

1.

INTRODUCTION

In the region w ~ o j c i , along with long wavelength ( k i P i ~ v T i / v A ^ 1)
fast magnetosonic waves (FMSW), short wavelength (k^Pj ~ 1) ion cyclotron(slow) waves (SW) are employed for plasma heating and current drive [1-13].
The waves can be excited either directly by an external antenna [2, 3] or by
conversion effects [2, 4-12]. SW can be absorbed by minority ions at co = 2o>'c

205

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LONGINOV et al.

[2, 7-15] if their density does not exceed a critical value, ri <n'CT [1,7]. Forn' ^n'c,., a SW propagating from the low magnetic field side converts into awave with a shorter wavelength that propagates backwards, i.e. down in themagnetic field. In this case, the opacity zone appears between the conversion
point and the minority ion cyclotron resonance (MICR) zone, and, hence, theSW cannot reach the MICR zone.
At the same time, at high SW power levels, quasi-linear deformation of theminority ion distribution function (MIDF) due to SW cyclotron absorption
by these ions may cause a substantially decreased ion contribution to the RF
current as compared with the linear case, because of the sharp tail cutoff of the
distribution function [8. 11]
This paper shows that the strong influence of quasi-linear effects on SW dispersion may take the opacity region disappear and the SW reach the MICR zone
co = S.u>'c (£ = 1 or £ = 2), even for n' > n'CT. This enables one to employ theresonance for SW in powerful regimes. These effects may also be used to createa two-component reactor. An analysis of the parameters of such an RF reactor
is presented in this paper.

2.

EFFECT OF MINORITY ION HEATING ON SW PROPAGATION

2.1. Formulation of the problem
Consider a dispersion equation for SW in the range u> * £u'c. The distribution functions of the bulk ions and electrons are assumed to be Maxwellian, with
Te = Tj = T. We determine the MIDF f'g (v^,v,), averaged over the magnetic
surface r = const, from the quasi-linear kinetic equation accounting for thecollisions with bulk ions and electrons. The dispersion equation relating theperpendicular and the parallel refractive indices N_L = kj_c/a; and Nj = kgc/co
has the form [1]:
=0

(l)

(2)

IAEA-CN-47/E-III-6

207

where e a , m a , n a and e', m', n' are the charges, masses and densities of the bulk
electrons (a = e) and ions (a = i) and the minority ions 'The MIDF is approximated by a productwhere(in the following, the difference between T|| and T is not significant). Averaging
the kinetic equation over the magnetic surface, we assume the SW electric field
Er and kj_ to be slightly varying in the range CJ » £co^ (kx = k<[ = kx (Zg = 0)
= const), which corresponds to weak damping of SW, Imk^ < k^, /Imk^dR < 1.
We also assume that the QL diffusion changes only vj^ so that (v\) > iv]\) andneglected the angular scattering of minority ions. (This is true for heavy ions,m'> m^, for m ' ~ m j the scattering may change fp to a certain extent.) Then, inthe steady case, the kinetic equation for fo(v^) reduces toi +(F^F)f' =0

(4)

208

LONGINOV et al.

(5)Here, L = 15 is the Coulomb logarithm, and P R F is the SW power flux density.
The solution to Eq. (4) is)clll)(7)(8)

where C is determined from the condition

Equation (7) is used to derive the specific RF power (averaged over themagnetic surface) which is absorbed by minority ions, the fraction of the energy
transferred from these minority ions to ions p; and to electrons p e (p e + Pj = 1),
the ratio of the average transverse energy of the minority ions to the transverse
energy of bulk ions, W^, and the contribution of resonance ions to the dielectricpermeability e':0r . , 3O

IAEA-CN-47/E-HI-6n.

„-...,,

209:

i-DlA)t

JeJ(P(-_f/Uu.)2)ffF

The damping coefficient is proportional to exp(~Zg2 )<!>; it differs from zeroin the narrow range where |Zg| < 1. The resonance surface co = Hu'c is assumed
to be located at R = R o so that Zg = (ui'k\\v'r) (R - RQ)/RO-

2.2. Accessibility of co = &CJ'C zone at high n' values
Consider the accessibility condition of the zone a> = CcoJ. at n ' > n'a for|e 3 i > N\ when Eq. (1) reduces to

8, + £'-A/,f=0(15)where ex can be taken at the surface co = iw'c and e' is determined by formula (13).
As an example, let us take a 7Li impurity in a deuterium plasma (N| = 9,n e = n d = 7 X 1 0 1 3 c n T 3 , T = 1 keV, u = 2u'c, co/cocd = 1.7, Xo = 0 . 9 ^ 2 ,Xo being weakly dependent on N | , rij and T, if N^ = Wpi/a>cj>Njj,' ] +ajpe/c0ge).
It can be seen from Fig. 1 that for high powers (large £ 2 ) a distribution function
is formed with a pronounced effect of tail cutoff in the range where D R F -> 0.
Figure 2 shows that, as £2 grows, the function 4> and, hence, the values of e' forsufficiently large k ^ become very small.
Figure 3 demonstrates a graphic solution to Eq. (15). Here, for £2 - 0,
T' = T(j, Z£ = 1 the function e' is represented by Curve 1 for n' = n^ < n^r,
Curve 2 for n' = n 2 = n^r and Curve 3 for n' = n 3 > n'cr. As is seen in Fig. 3,
for n' = nj at Z2 > 0 (Re e ' < 0), we have positive roots kj and k2 for all Z^> 0;
for n' = ri2i the roots kj and kj merge, producing the conversion region at
Z2 «* 1; for n' = n 3 , positive roots kl and k 2 for Z 2 = 1 are absent, which is

210

LONGINOV et al.

= 500

FIG. 1. D R F and igf^ versus u2 for nLi minority ions at w = 2w\ for various i

0.06
0.05
0.04
0.03
0.02
0.01

F/G. 2.

k\p\.

indicative of the presence of an opacity zone (obviously, positive roots appear
for Z2 ^* 1 )• Thus, the MICR zone can be reached for T' = T^ only at n' < n'CT,
where the values of r\'CT may be very small, especially for large differencesw
c ~ w cd anc^ small N | (see also [2, 8]).
It can be shown that in the case of n' = 113 > n^r and a strongly deformed
Li ion distribution function (£2 = 500, Fig. 1) due to a substantial decrease in
the function (see Fig. 2) for k°p d = XQC^/GJ =1.17 (the same value as in apure d-plasma) at Z2 «* 1 (and hence, throughout the range Z 2 > 0), we have
two roots of Eq. (15), kj « k ] 0 and k 2 , even for T | = T (see Fig. 3, broken
curves).
In Fig. 4, the solutions to Eq. (1) for £2 = 0 (T' = T) and £2 = 500 are
presented, showing that for £2 = 500 and n ' > n'cr, the solutions to the dispersion
equation are close to each other for n'<^n;.r, £2 = 0 and n ' ^ n ^ , £2 = 500.
The opacity zone for SW propagating from the low magnetic field side
appears also for the resonance a; = oo'c, e.g. in a d-plasma with a 3He impurity.
For a; = coc3He (cj/oj cd = 4/3) one has k°p d = 2,3; Xo = 1.7. Since Xo is high,a significant transport of 3He ions through the zeros of D R F occurs due to the
diffusion in deuterium, but still the tail cutoff caused by decreased D R F (u) is
distinct (Fig. 5). The function ^(k^pj) for 3 He is shown in Fig. 6, and the211

IAEA-CN-47/E-III-6• (K,PJ 2

FIG. 3. Graphical solution of Eq. (15) at co^2ca'c (Z'^ ~ 1) for nLi minority ions in d-plasma.
The root kl0 corresponds to pure d-plasma (w/cocd = 12/7, k1pd = 0.77).

1- d-d+7Li

FIG. 4. fe±pd versus Z'2 obtained from Eq. (1); 1Li minority ions in d-plasma, co = 2o)'c,
ne = 7 X 10li cm'3, T= 1 keV, Nn = 3,B0= 3T. Curves 2 (T' = T) and 3 ff =500)
correspond to d + nLi (6%)-plasma, Curve 1 refers to pure d-plasma. Shaded area shows
maximum cyclotron SW absorption.

16

FIG. 5. DRF and 2gf^ versus u2 for d-plasma with 3He minority ions at co = co'Q for £at \ = 1. 75.

solution to Eq. (1) is in Fig. 7 for Ny = 3, ne = 7 X 10 1 3 cm" 3 , B o = 3T, T = 1 keV(£2 = 500 for Curve 3, £2 = 0 for Curves 1 (n' < n^.) and 2 ( n ' > n'ct). The highvalue of XQ = 1.7 limits the 3He ion energies and;consequently, the 3He ionpressure. This effect can be employed in high RF power experiments withbulk ion heating.

212

LONGINOV et al.

0.5
0.4
0.3
0.2
0.1

£ 2 =500i\

2
10 (K.ft)
I'd1

FIG. 6. <&(kLpd) versus k2Lp2A (d-plasma with 2He minority ions at o; = o>'c, Ao = 1.75).

To achieve the regime where the resonance co = 9.w'c is accessible at n ' > n^.
it is necessary for the SW specific power flux to exceed the value(16)
For P R F < P m i n , a stable steady state with accessible MICR zone is not possible.
When condition (16) is satisfied, two stable steady solutions to the above
set of equations are possible, corresponding to Curve 2 (A regime) and to
Curve 3 (B regime) of Fig. 4.
Which of the two regimes will materialize depends on the initial conditions.
As an example, two temporal diagrams are presented in Fig. 7, showing how
different regimes are realized. If the density n' ceases to increase before thevalue n' = n"a is reached where condition (16) holds, the B regime may well
exist as a steady state. For n ' > n"CT, the RF power flux is hot sufficient tomaintain the B regime, and return to the A regime will occur. (Note that sucha scheme of 22Ne isotope puffing was realized in ICF heating in T-10 [10].

3.

TWO-COMPONENT THERMONUCLEAR REACTOR USINGICR HEATING WITH SLOW WAVES

3.1. Formulation of the problem
In a two-component reactor [13-15], where in the d(t)-plasma, the nonequilibrium distribution function of minority ions t(d) is maintained with thehelp of external power sources, an important problem is to achieve a high power
multiplication factor Q. In this context, it is most useful to employ 5-functiontype distributions, fo<x6(E - E o ), with E = E o corresponding to maximum Q.

IAEA-CN-47/E-III-6

213

(a)

n e r —•/-(b)

FIG. 7. Time behaviour of ion impurity puffing showing possibility of achieving A regime(cyclotron resonance zone for minority ions is not accessible to SW) and B regime (cyclotron
resonance zone is accessible to SW).

For example, for Tj< 1 keV and T e >40 keV,

,.

4T

47re

L

E

A = 22A

MeV

>

a

=

/J = d(t)whence it follows that, for a = d, Q m a x = 8.2 and for a = t, we obtain Q m a x = 5.5.
The distribution function of the d-ions for deuterium atoms injected into thet-plasma is similar to a 5-function [14, 15]. In ICR heating of d-ions in thet-plasma with FMSW, fg « exp(-aE) differs from the 6-function essentially;
therefore, the Q-factor is substantially lower [13].
Plateau formation due to high power SW and the MIDF tail cutoff provides
a rather good approximation to the 6-function, enabling high enough Q-values
to be reached. Tail cutoff leads to a limitation of the impurity ion transverse
pressure, permitting an increase of the specific thermonuclear power.
Regimes with absorption at the fundamental cyclotron frequency of
deuterium minority ions in the t-plasma (ldt regime) and with absorption at
the second harmonic of tritium minority ions in the d-plasma (2td regime) canbe implemented with the help of high power SW.
Consider the following basic parameters Q, p T N and rg.
(i) The Q-factor is the ratio of the specific power of the thermonuclear reaction
accounting for neutron capture in the lithium blanket, p T N , to the specificpower required to maintain the non-Maxwellian distribution of minority
ions and to heat the bulk plasma (P>RF,Q = Pr*/<p> ftF

(17)
(18)

214

LONGINOV et al.

<crv> = ^ ^- l^dii^^l^/ij^A^U'i^)(19)

where y = (u' + u - |u' - u|t + |u - u'|, a is the d + t reaction cross-section
at relative velocity, <orv)RF is the value of av averaged over the distributionfunctions of bulk particles with a Maxwellian distribution of temperature T
and of the resonant minority ions. With (19), we obtain(}(ii) The maximum value pj^ = p m a x , which is reached for nj and n' corresponding
to the limiting plasma pressure pj/ = max(n e + n^ + n'W^)T = |3 max Bo/8
determined by the equilibrium and stability conditions

H - £ , n!-^(21)

is given by
OdF EA

J. _ • 2(22)
In a conventional reactor (n t = n d = 0.5 n e ):(Hi) The energy confinement time T£ in the steady regime is given by

Pu
where p a = 3.5 P X N / 2 2 . 4 = 0.156 P T N is the specific power heating theplasma by a-particle deceleration. Assuming n' to be optimum (21) andCoulomb collisions to be the main source of energy losses for fast ions,
we obtain

215

IAEA-CN-47/E-III-6

50 1.00.8
30 0.60.4
10 0.2

0

5

10

15

20

T

FIG. 8. Maximum Q (g, X0A W.L and X o p t versus T for Idt and 2dt regimes.

(25)

We now introduce a parameter(26)
where Tg = 12.6 n e T / p j N is the energy confinement time in a conventional
reactor (n d = n t ; Tt = Td = T) and a parameter a m i n = r E m i n / r E wherer
Emin = 12-6 neTopt/PTN is the minimum value of r°E for given p m a x (T = T o p t
corresponds to the maximum <CTV>0/T2 ). The parameter a describes the decreasein T E compared to r E at the same values of T, and the parameter a m i n corresponds
to the case where the thermal conductivity does not depend on T and the value of
T = T o p t for a conventional reactor regime is chosen for minimum T E at fixed pj_.
From (24) and (25), it follows that
(TL

(27)

(28)
Q

where T' is the plasma temperature in a two-component reactor.
3.2. Q-factor optimization
In accordance with (20), the Q-factor depends on XQ and £ 2 . Figure 8
shows Q as a function of T'; for given T' the values of Xo and £2 are chosen

216

LONGINOV et al.

-4-2

0

2

4

- 4 - 2 0 2 4 6

G

Z.

Z2

FIG. 9. kLp{ versus Z'% in the resonance region w = wj, for 1 dt regime (Curve 1: pure d-plasma,
Curve 2: (t + d 15%)-plasma (A regime, £2 = 150), Curve 3: ft+d 15%)-plasma (B regime,| 2 = 150) and for 2td regime (Curve 1: pure d-plasma, Curves 2 and 3: (d +1, 25%)-plasma,%2 = 500);«; = iO 1 4 cm'3, BQ = 5T, Nl{ = 3.

such as to maximize Q which has maximum at Xo = X opt (Fig. 8). As is seenin Fig. 8, the Q-values for the ldt regime are higher than for the 2td regime.
Moreover, the latter has higher Wj_ values. Xopt increases as T rises. It follows
from (21) and the curve for Wj/T) that the optimum minority ion concentrationis relatively high (10-30%).
3.3. Accessibility of cyclotron resonance region for d-ions in t-plasma
and for t-ions in d-plasma
Although, to achieve the maximum Q-factor, one has to choose theoptimum value for Xo> the possibility of changing Xo is, in reality, rather limited.
(This can, e.g., be done by varying N | or using an additional ion impurity (3Hein the ldt regime or H+ in the 2td regime)). Therefore, we shall below consider
the simple case where the parameter XQ corresponds to a value in the zone
w = cocd in pure t-plasma or to a value in the resonant zone CJ = 2w ct in pured-plasma. Obviously, such a situation prevails for low impurity concentrations(n' < n'cr) or high impurity concentrations in the B regime.
In Figure 9, k | p d versus Zg is plotted for pure d-plasma in the regionco = |-CJC(I (resonance co = 2<^'c for t) and for pure t-plasma in the range
co = 7 coct (resonance a> = oo^. for d). As can be seen in Fig. 9, the values
kjpj = 2.3 (Xo = 3.45) and kjpj = 1.76 (Xo = 1.17) are well in excess of the
optimum values, especially for the 2td regime (see Fig. 8). A numericalanalysis of the dispersion equation (1) shows that the MICR zone is only accessible at £2 -»• 0 for low impurity ion concentrations ( n ' / n ^ 1%).

217

IAEA-CN-47/E-III-6

o

faj

-2
-4
20 40 60 80
0.05

0.03

(b)

0.01

FIG. 10. (a) igf^ versus u7, (b) 4> versus k\p\for various £2 (solid line: Idt regime, = 3.45;
dotted line: 2td regime, \ 0 = 1.14).

Let us discuss the possibility of realizing the B regime at high impurity ion
concentrations. As an example, Fig. 10 shows distribution functions fo(u) andfunctions •SCkjPj) for Xo = 1.17(1 dt) and Xo = 3.45 (2dt). It can be seen from
Fig. 10 that the impurity ion contribution to the RF current is reduced by afactor of 200, only because of the decrease in ^(kj^Pi) at k^Pj = \QCOJ<^C\ (theadditional decrease in the minority ion RF current in the range Zj * 1 is due tothe increase in Tj). The behaviour of kj^Pj in the B regime (see Fig. 9, brokenline) is obtained from the solution to Eq. (1) taking into account 3>(k|pj) (Fig. 10).
Thus, because of a significant QL evolution of the MIDF, the Zg-dependence ofkj^pj is close to the Zg dependence of k^Pj for n' -*• 0, even for n' > n'a.
3.4. Analysis of reactor parameters
Idt regime. Temperature dependences of reactor parameters are shownin Fig. 11 for \ 0 = 1.17. The parameter £2 is chosen for maximum Q; therefore,
the values of T?, a and a m i n in Fig. 11 are somewhat lower than the corresponding
optimized values. The Q-factor is now lower than at Xo = XOpr F ° r T > 10 keV,
Q = 3-4, but in this region p j ^ «< p^N' anc* w e n a v e lower ei*£ ( a ^ a , , ^ ^ 3 - 1 0 ) .
The principal advantages of this two-component regime are found at T < 10 keV
where TJ/T70 ~ 3 - 5 a n d a > a m i n > 10 though Q is not very high here, Q = 1.5-2.4.
This region is of interest in view of its application in a hybrid reactor where the
Q-factor can be increased five to ten times owing to the energy release in theuranium blanket [16] as well as in the neutron sources. In the region T = 5-10 keV,
the optimum concentration of deuterons is 14-25%.

LONGINOV et al.

218

1dt<\0=1.14)
Q<av) 0 .(ov) RF

w,

4
Q

3
2

Pe

20 0.80.65xid 1 O^

10 0.4

1

0.2
P(

a

\(<TV> 0
10

5
-°min

15

20 T(k8t
no.r<

100 \l
a

80

\

60

V^

40

\

20

J^.

3.5X10 6n1.75X10"6

—
10

15

20

T (keV)

FIG. 11. <cw R F , Q, WL, pe, a, a m i n and -q versus temperature for Idt regime (\0 = 1.14) and<<7V>Q and T?o versus temperature for conventional regime.

2td regime. In this case, the parameters of the two-component reactor,
e.g. Q-factor, are somewhat lower than for the 1 td regime. This is, in particular,
due to the high value of the parameter XQ = 3.45.

REFERENCES[1] LONGINOV, A.V., STEPANOV, K.N., in Proc. Ail-Union Meeting Gorky, 1982, Inst.
Prikl. Fiz. AN SSSR, Gorky (1983) 105.
[2] LONGINOV, A.V., Zh. Tekh. Fiz. 42(1972) 1591.
[3] ONO, M., STIX, T.H., WONG, K.L., HORTON, R., in Heating in Toroidal Plasmas(Proc. 2nd Joint Varenna-Grenoble Int. Symp. Como, 1980), Vol. 1, CEC, Brussels(1980)593.
[4] KLIMA, R., LONGINOV, A.V., STEPANOV, K.N., Nucl. Fusion IS (1975) 1157.
[5] PERKINS, F., Nucl. Fusion 17(1977) 1197.
[6] SCHARER, J.E., MacVEY, B.D., MAU, T.K., in Theoretical and Experimental
Aspects of Heating in Toroidal Plasmas (Proc. 3rd Int. Meeting Grenoble, 1976),
Vol. 1 (1976) 79.
[7] LONGINOV, A.V., STEPANOV, K.N., in Plasma Heating in Toroidal Systems(Proc. 4th Int. Symp. Rome, 1984), Vol. 1, (1984) 486.
[8] LONGINOV, A.V., PAVLOV, S.S., STEPANOV, K.N., in Controlled Fusion andPlasma Physics (Proc. 12th Europ. Conf. Budapest, 1985), Part II, European PhysicalSociety (1985) 132.

IAEA-CN-47/E-HI-6[9] ALIKAEV, V.V., BEREZOVSKIJ, E.L., VASIN, N.A., et al., ibid., p. 156.
[10] TFR GROUP, ibid., p. 112.
[11] LONGINOV, A.V., PAVLOV, S.S., STEPANOV, K.N., in Controlled Fusion andPlasma Heating (Proc. 13th Europ. Conf. Schliersee, 1986), European PhysicalSociety (1986) 107.
[12] LONGINOV, A.V., PAVLOV, S.S., STEPANOV, K.N., ibid, p. 141.
[13] STIX, T.H., Nucl. Fusion 15(1975) 737.
[14] DAWSON, J.M., FURTH, H.P., TENNEY, F.H., Phys. Rev. Lett. 26 (1971) 1156.
[15] PISTUNOVICH, V.I., At. Ehnerg. 3 5 ( 1 9 7 3 ) 1 1 .
[16] KADOMTSEV, B.B., ORLOV, V.V., Itogi Nauki i Tekhniki, Fizika Plazmy, Vol. 6,
Moscow (1985) 5 (in Russian).

219

IAEA-CN-47/E-HI-24

EFFECT OF x-POINT LOCATION ON
EDGE CONFINEMENT IN DIVERTED TOKAMAKS
T. OHKAWA, F.L. HINTONGA Technologies Inc.,San Diego, California,
United States of America
Abstract
EFFECT OF x-POINT LOCATION ON EDGE CONFINEMENT IN DIVERTED TOKAMAKS.
The edge mirror physics mode in a diverted tokamak is shown to have good confinement
properties. This mode is possible only if the x-point is located away from the outboard median planeso that a trapped ion population can exist near the separatrix. The H-mode has been found experimentally only in this case.

1.

EDGE MIRROR PHYSICS

When the ion. collisionality near the separatrix is low, the confinement properties of the edge region are essentially the same as in a mirror
device [1,2]. Loss regions in velocity space exist, analogous to loss conesin mirrors, for ions inside the separatrix, as well as outside. These loss
regions correspond to collisionless drift orbits which cross the separatrix, allowing ions to escape into the divertor. Small angle Coulomb
scattering into these loss regions is the primary loss mechanism whenthe effective collision frequency is smaller than the bounce frequency
and when turbulence is suppressed. All energy leaving the central region must pass through the edge, when charge exchange losses in thecore are negligible. The edge region, within a few ion banana widths ofthe separatrix, is thus in series with the core or central region for ion
energy transport. The edge can therefore form an effective barrier toenergy flow in the ion channel.
The energy transport out of the edge region is purely convective,
since energy is lost only when ions escape. Confinement is then simply
particle confinement, just as when a plasma is surrounded by vacuum.
Since electrons cannot escape in this manner from as far inside theseparatrix as ions can, a negative radial electric field must be present,
to partially hold the ions back. For ambipolarity, an anomalous valueof the electron mobility is required, which is related, via the Einstein
relation, to a diffusivity comparable to the ion thermal diffusivity, whichis therefore comparable to the empirical electron thermal diffusivity, on
the order of 1 m?sec~1.
221

222

OHKAWA and HINTON

When ions and electrons diffuse through the electric field in thisregion, the ions do work against the field, which does work on the electrons, so that thermal energy is transferred from ions to electrons. Thus,
the electron thermal confinement is coupled to that of the ions. The
global energy confinement time is proportional to the edge temperature, which can be kept high by keeping the neutral density low, as weshall now show.
The edge confinement time r 0 is the time for an ion to scatter
into a loss region, or roughly To = T^e, where TH is the ion collisiontime and e is the inverse aspect ratio. Since the thickness of the edgelayer is roughly an ion banana width, the volume of the edge region is
Vo = Se1/2p-ip, where 5 is the plasma surface area and p-ip is the ion
poloidal gyroradius. Assuming that the edge layer thickness is muchsmaller than the plasma radius, and assuming no confinement in thecore, the confinement time is given byrr0

where V ~ Sa/2 is the plasma volume and T is the particle flux. Hence,aor, using e = 1/3,
nr = 1.5 x 109 (Tj/e) /For a given density, the confinement time is proportional to theedge temperature and to the current. For T» = 1.5 x 103 eV and I= 5 x 105 Amp, for example, we havenr ~ 1018 m~3sec
which is roughly the total confinement observed in H-mode experiments[3,4]. For an ignition device, with / = 107 Amp and Ti = 2 x 104 eV,
we obtain

IAEA-CN-47/E-UI-24

223

n r ~ 3 x 1020 m~3sec
Note that ignition occurs even without the benefit of core confinement.
The power throughput isor
P = 4.3 x 10" 28 n?V/I
which is independent of edge temperature.
= 4 X 105 Amp, we have

For V = 4 m 3 and I

P = 4.3 x 10s (n x 1(T 19 ) 2 watt
which is roughly the heating power in the H-mode experiments. The
heating power determines the density and not the temperature.
The particle throughput determines the temperature. Since themirror loss is convective, the power throughput is related to the particle
throughput F by

If the beam fueling is insignificant, the particle flux is given by
F = Snovo
where no is the neutral density outside the plasma, vo is the averagevelocity of the neutral atoms, and it is assumed that the neutral atoms
are ionized and heated before they are lost. The edge temperature isthen given by

Inovn

224

OHKAWA and HINTON

For a given minor radius, plasma density, current, and neutral velocity,
the edge temperature is inversely proportional to the neutral density.
For a = 0.4 m, / = 4x 105 Amp, n = 3x 1019 m~ 3 , and vo = 103 m-sec"1
we have T/e = 4 X 1020 n" 1 . An edge temperature of lkeV corresponds
to about 1 x 10~5 torr of neutral atom pressure.
If there is fueling only from neutral beams, the temperature mustrise to 1/3 the beam energy. This would be the ideal situation,withbeam fueling, and only a small neutral density at the edge, so that theedge could be heated to increase the confinement time.
The edge temperature can be kept high if there is negligible coolingof ions by neutral atoms in the recycling process. One condition for thisis that charge exchange does not dominate the ion energy losses in theedge. The charge exchange time r ^ is given by

where n* is the neutral density in the plasma, <rcx is the charge exchange
cross section, and Vi is the ion thermal velocity. The condition that the
charge exchange time is longer than the mirror loss time, r ^ > To, is

n"Jn

<

1.85 x 10~16 , _ . , 2(T/e)

For o-c, = 10" 19 m 2 , and T/e = 103 v, we haven*o/n < 1.85 x 10~3
For n = 2 X 1019 m" 3 and n* < 3.7 x 1018 m~ 3 , for example, thiscondition is satisfied.
Another condition for no cooling of ions in the edge by recycling isthat the cold ions created by ionization in the edge are scattered into theloss regions before they are heated by the hot ions. The neutral atoms
coming into the plasma edge are ionized by electron impact within amean free path

IAEA-CN-47/E-OI-24

225

where <7ion is the ionization cross-section and ve is the electron, thermalvelocity. The cold ions produced by ionization are typically heated toa velocity vc before they are scattered into the loss regions, where ve issuch that the ion banana width at that velocity equals the distance between their birth location and the separatrix. We estimate this distanceto be about one ionization mean free path and obtain

where fl-ip is the ion poloidal gyrofrequency. For ion energy losses to besmall, we want the velocity ve to be less than the ion thermal velocity,
vc < V{. This is the same as the condition that the mean free path beshorter than the ion banana width,

For Bp = 0.2 T, T = 103 eV, vo = 103 m - sec" 1 , and cr-iaa = 10~21 m2
we have n > 6 X 1018 m~ 3 . For the experimental parameters, therefore,
the neutral density is attenuated in the edge region, and the cold ionsare lost before they cool the edge.
Good mirror confinement at the edge is obtained in a window ofthe edge density. A high density shortens the mirror confinement, whileat low density, the cold ions produced by the recycling process cool theedge. At sufficiently low density, charge exchange becomes the dominantloss.
The position of the x-point on the separatrix plays an importantrole in the mirror confinement. (We assume single-null operation, forsimplicity.) It is necessary that a trapped-ion population exist, near
the separatrix. But this requires that the effective collision frequency
divided by bounce frequency, i>»i = ^/(wje), be small compared withunity. The bounce frequency u>& becomes very small for ions which passnear the x-point, since for them,where q$ is the MHD safety factor, which becomes infinite at the separatrix. Such ions cannot be considered to be trapped long enough tocontribute to the mirror confinement at the edge. Only those ions withturning points sufficiently close to the median plane, which are never

226

OHKAWA and fflNTON

close to the x-point, play a role in the mirror confinement process. This
restriction is effectively eliminated if the x-point is near the minimum
major radius side of the separatrix, as in a "D" shape. In the otherextreme case, when the x-point is on the outboard median plane, as in
the JT-60 design, the trapped ion population is effectively eliminatednear the separatrix. The beneficial effects of the mirror confined edge
region would not be expected to be present in the latter case.

2.

THE TRANSITION FROM A COLLISIONAL
EDGE PLASMA TO A COLLISIONLESS ONE

The foregoing assumes a high enough edge ion temperature forthe collisionality to be low, I/,J < 1. A threshold amount of power
was needed to cause a transition from the L-mode to the H-mode in
ASDEX [3], which was also a transition from utl > 1 to v%\ < 1. In
single null operation, the threshold power depended upon the locationof the x-point, and was lower when the x-point was in the direction of
the ion grad-B drift. A theoretical model which attempted to explain
this was given previously [5], and the results will now be summarized.
We begin by considering the high-collisionality regime, since that ismore appropriate for the L-mode. With single-null divertor operation, if
the ion grad-B drift is toward the x-point, the Pfirsch-Schluter parallel
ion heat flow is away from the x-point. This heat flow adds to theparallel heat flow due to the lower temperature in the divertor, to givethe net heat flow. The net heatflowinto the divertor must equal the ionheat flow out of the plasma edge, assuming there are no other energy
losses. The Pfirsch-Schluter heatflowthus reduces the net heat flow into
the divertor chamber, for the same ion temperature gradient dTi/dr.
Or, if the heat flow is given, the ion temperature gradient must belarger, due to the toroidal effect. We assume that a threshold dTi/dr isrequired for the H-mode, which may be understood in terms of the needfor the ions to be hot enough in the edge to be in the low collisionality
regime. Then the threshold heating power for the H-mode is lower withthis x-point location, than when the ion grad-B drift is away from thex-point. This effect has been clearly seen experimentally in ASDEX [6].
With double-null operation, the ion grad-B drift direction should
not matter, because the effects of the two nulls cancel. Also, at higher
heating power, the ion grad-B drift direction should not matter, because the ion loss occurs through banana orbit excursions, and is not

IAEA-CN-47/E-ffl-24

227

sensitive to null location. These predictions are also in agreement withexperiment [6],The location of the x-point thus plays an important role in the
L- to H-mode transition. The favorable locations are above the medianplane when the ion grad-B drift is up, and below the median plane,
when the ion grad-B drift is down. An x-point on the median plane, as
in the JT-60 design, would be expected to have no effect, favorable orunfavorable, just as with symmetric double null operation. In this case,the power threshold for the H-mode would be expected to be higher
than with the favorable location.
3.

SUMMARY

When the edge plasma is governed by the mirror physics we havedescribed, confinement in a divertor tokamak has the following properties.
(1) The global energy confinement time is proportional to the edge
temperature and to the current, for a given density.
(2) The edge density is determined by the heating power.
(3) The edge temperature is determined by the particle throughput.
(4) If the edge density is not too low, excessive cooling of the edge
plasma by recycling can be avoided.
The edge mirror physics mode is possible only if the x-point islocated away from the outboard median plane, so that a trapped ion
population can exist near the separatrix. The transition from a collisional edge plasma to a collisionless one requires a lower heating power
when the x-point is located away from the median plane, in the directionof the ion grad-B drift.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]

HINTON, F.L., CHU, M.S., Nucl. Fusion 25 (1985) 345.
OHKAWA, T., Combined confinement system applied to tokamaks (to be publishedin Kakuyugo Kenkyu.
WAGNER, F., et al., Phys. Rev. Lett. 49 (1982) 1408.
FONCK, R.J., et al., in Heating in Toroidal Plasmas (Proc. 4th Int. Symp. Rome, 1984),
Vol. 1 (1984)37.
HINTON, F.L., Nucl. Fusion 25 (1985) 1457.
WAGNER, F., et al., Nucl. Fusion 25 (1985) 1490.

Session C
OPEN CONFINEMENT SYSTEMSChairman of Session
Session C-I

D.D. RYUTOVPapers C-I-4-1 and C-I-4-2
were presented by R. Itatani as Rapporteur
Session C-II (Posters)

USSR

IAEA-CN-47/C-I-1

TMX-U TANDEM MIRROR
THERMAL BARRIER EXPERIMENTS*
T.C. SIMONEN, S.L. ALLEN, D.E. BALDWIN, J.D. BARTER1,

L.V. BERZINS2, M.R. CARTER2, T.A. CASPER,
J.F. CLAUSER, F.H. COENSGEN, D.L. CORRELL,
W.F. CUMMINS, C.C. DAMM, G. DIMONTE1, B.H. FAILOR2,
S. FALABELLA2, J.H. FOOTE, T.K. FOWLER, A.H. FUTCH,
R.K. GOODMAN, D.P. GRUBB, D. HILL, E.B. HOOPER,
R.S. HORNADY, R.D. HORTON1, W.L. HSU3, B. HUFF 1 ,A.L. HUNT, R.A. JAMES, C.J. LASNIER4, C.A. LEAVITT1,
W.H. MEYER, J.M. MOLLER, A.W. MOLVIK, W.E. NEXSEN,
D.E. PERKINS, W.L. PICKLES, P.A. PINCOSY1,
G.D. PORTER, P. POULSEN, D.B. RESS, M. SHIHO5,
E.H. SILVER, B.W. STALLARD, W.C. TURNER,
R.D. WOOD, T.L. YU 6

Lawrence Livermore National Laboratory,
University of California,
Livermore, California,
United States of America
Abstract
TMX-U TANDEM MIRROR THERMAL BARRIER EXPERIMENTS.
Thermal barrier experiments have been carried out in the Tandem Mirror Experiment-Upgrade(TMX-U). Measurements of nonambipolar and ambipolar radial transport show that these transport
processes, as well as end losses, can be controlled at modest densities and durations. Central cell heating
methods using ion cyclotron heating and neutral beam injection have been demonstrated. Potential
measurements with recently developed methods indicate that deep thermal barriers can be established.

1.

INTRODUCTION

This paper describes thermal barrier experiments from theTandem Mirror Experiment-Upgrade (TMX-U) using new heating
systems and improved diagnostics aimed at improving ourunderstanding of how tandem mirrors work. We can operate TMX-U* Work performed under the auspices of the United States Department of Energy by the Lawrence
Livermore National Laboratory, under Contract No. W-7405-ENG-48.
1

TRW, Inc., Redondo Beach, CA, USA.
University of California, Davis, CA, USA.
3
Sandia National Laboratories, Livermore, CA, USA.
4
University of Maryland, College Park, MD, USA.
5
Japan Atomic Energy Research Institute, Naka-machi, Naka-gun, Ibaraki-ken,
Japan.
5
Johns Hopkins University, Baltimore, MD, USA.
2

231

232

SIMONEN et al.

as a standard tandem mirror at densities near 10 13 cm~3 and Ion
temperatures of 2 keV with H ms energy confinement time.
However, we have concentrated our experiments on the thermal
barrier mode which could scale to a reactor. In this mode,
three processes limit the central-cell density. First isnonambipolar radial transport, second is ambipolar transport,
and third is the loss of end plugging. When those three loss
processes are controlled, the central-cell density increases at
the ionization rate up to a maximum line averaged value of
1 to 3 x 10 12 cm""3 .
The purpose of the program of measurements presented here
has been to improve our understanding of the above three
processes in order to continue increasing the density toward the
10 l3 cm~ 3 theoretical maximum. Because of changes in funding
priorities, for the present we will not be able to complete thisprogram on TMX-U, but the work will be continued on the othertandem mirror facilities elsewhere,such as GAMMA-10, TARA and
Phaedrus. However, we have made considerable progress,
especially in understanding and controlling radial transport,
which was of considerable concern in the past. Section 2
describes this work. The remaining sections deal with two
topics related to understanding and controlling end losses.
Section 3 summarizes our progress on central-cell heating andfueling by ICH and neutral beams, which ultimately control
barrier filling. Section 1 describes our recent measurements ofthermal barrier axial potential profiles.
TMX-U employs minimum-B, thermal barrier end plugs [1,2].
The 0.3 T nonaxisymmetric central cell is 8 m long, and iscurrently operated with a 2H cm radius limiter. The heating
systems operate for a 75 ms duration. Ten of the 2H neutral
injectors are located in the central cell and operate at either
2 or 20 kV. A total of fourteen 20 kV beams are used in the end
plug as sloshing beams (^7° and 10°) and as pump beams (18° and24°). Two 200 kW, 28 GHz gyrotrons are used in each end cell.
Two 200 kW transmitters drive separate ICH antennas at each endof the central cell, between which D 2 gas ia Injected to provide
fueling. The end-wall plasma potential control (PPC) plates [3]can operate either grounded or floating.

2.

RADIAL TRANSPORT AND PARTICLE ACCOUNTABILITY

In TMX-U, nonambipolar radial transport is driven by thecentral-cell potential [1], Recent experiments with grounded
end walls indicate that the nonambipolar radial current depends
on the plug sloshing beam current in addition to the plasma
potential. (The two terms are comparable.) A portion of thesloshing ion loss is radial. To compare the central-celltransport with resonant neoclassical theory, we evaluate its
nonambipolar confinement time by using regression analysis to233

IAEA-CN-47/C-M
300

I

(a)

Central-cell global radial transport (rc < 2 0 c m l .

\r \. •// - Resonant neoclassicaltheory

X

100 -

-

£
0

-

10 -

1

i

0.05

0.1

1.6
(b)

0.5

I

r

1

5.0

1.0

I

-Sloshing-beams fired;plugging starts
Plates floated

1.2

Plugging fails
5-kG
ECRH off

5 0.8

0.4

FIG. 1, (a) Variation of radial average central-cell nonambipolar radial transport time with plasma
potential shows agreement with neoclassical theory, (b) Thallium ion beam probe measurements of thecentral-cell potential show a sharp decrease when the end walls are floated.

subtract the Inferred radial sloshing ion current from the total
measured net current. In Fig. 1a, the variation of TLftk withpotential is compared with the theoretical prediction atr » 20 cm, assuming Tic - <J»/-U. Good agreement is found in thisradially averaged confinement time.
Nonambipolar transport has been significantly decreased byelectrically floating segmented end PPC plates C3]« During
plasma operation, the 26 plates on each end charge to a negative
potential of 1 to 3 kV negative through 10 4 ohm resistors togrounds. Potential measurements have now demonstrated that the

SIMONEN et al.

234
(a)

i

'
JL

0.8

£

u
M

; u o.4

Gas feed off

1~^^T\K

9/11/86
Shot 14

Loss of / f
plugging—//

'A
(b)

1 (A)

0
10

i

ill
— Density buildup
during end plugging
1
1

1

dn J
/n = •II + iNA
-idt

'

3

+

i
30

-4
20

1

1

I I

40

50

60

70

t (ms)

FIG. 2. Measurements of (a) central-cell density and (b) ionization current ls and total measured
loss current IT. These measurements show that TMX-U can operate with small ambipolar losses.

end-wall plates reduce the central-cell plasma potential,
confirming the conclusion deduced from the effect of the plates
on the nonambipolar radial ion transport [1,3]. Figure 1b shows
the time history of the central-cell potential during adischarge in which the plates were floated during plugging. The
potential on-axis dropped from 1.1 to 0.5 kV with the plates
floating. The plates usually drop twice as much below ground.
In this shot, plugging failed 3 ms later and the potential
partially recovered.
Since radial transport can occur independently in thecentral cell and the end cells, a low PPC plate net current does
not guarantee small nonambipolar radial transport in the central
cell
nor does it indicate low ambipolar transport.
Consequently, particle accountability is used to evaluate theamount of ambipolar radial transport in the plasma; here we
consider the core of the plasma (r < 10 cm). The ion particle
balance equation is:

dt

s

II

i

i

where Is is the core ionization source current determined froman absolutely calibrated imaging high-speed H-alpha camera
system [5], Ii is the axial ion loss current obtained from anarray of Faraday cups, I^ A is the nonambipolar loss current
measured by the end-wall PPC plates, and I A is the residual(ambipolar) current required to balance the equation. Resultsof our measurements are shown in Fig. 2, which shows (a) the235

IAEA-CN-47/C-I-1

(a)

(b)

O(kV)

/

-1.5

\ "

I

\ t = 42 ms

7/22/86-23
il

yj
Jut =

42 ms

t = 22ms\l US-25 cm25 cm$
I-25 cm

• 0.5

22 ms

\

1

I 1II
25 cm

FIG. 3. Thallium ion-beam probe central-cell radial profiles of (a) secondary current (proportional tone (ov)) and (b) plasma potential. At 22 ms the plasma is plugged, but with large ambipolar transport,
and at 42 ms the plasma is not plugged and ambipolar transport is small.

central-cell density and (b) a comparison between the measured
core source Is and the sum of the terms: I T = (qdN/dt + I^ A
+ In). Between 25 to 30 ms, we have achieved plasma conditions
with small axial losses and the PPC plates were floated(10 4 ohms) to reduce nonambipolar radial losses. The total
particle confinement time under these conditions is 10 ms.
During this plugged period (25-30 ms) the density rises and is
the main contribution to IT- Later in this shot, we observe a
discrepancy indicative of some ambipolar transport, which is
reduced when the 2uce electron cyclotron heating (ECH) is turned
off. Under usual operating conditions, Is and If agree tobetter than a factor of two, implying that ambipolar losses are
no greater than the sum of the other losses. However, undersome conditions of central-cell fueling and heating rates, weobserve larger amounts of ambipolar transport usuallyaccompanied by 2 to 7 kHz fluctuations.
Figure 3 shows the central-cell radial "density" (thallium
ionization current) and potential profiles with and without
significant ambipolar radial transport. At 22 ms, the measured(H-alpha) ambipolar transport is large and the radial "density"
profile is flat. During this time the axial ion end losses areplugged and the radial potential profile is peaked. Later, at
t2 ms, the measured ambipolar radial transport is small and the"density" profile is strongly peaked. At this time, axial
plugging is weak and the potential profile is nearly constant in
the plasma core. Such peaked density profiles are also measured
during plugging, when the central-cell density increases.

236
3.

SIMONEN et al.
CENTRAL-CELL HEATING AND FUELING

End plugging requires that the passing-ion thermal barrier
collisional filling rate be reduced below the pumping rate. The
filling rate should be reduced by ICH of the passing ions and by
reducing the passing-ion density at the plug mirrors throughanisotropic trapping in the central cell. Our previous
experiments used gas fueling 2.25 m west of the central-cell
mldplane and a double half-turn loop 1.26 m west of thetnidplane. These resulted in efficient perpendicular heating of
mirror-trapped ions to 2 keV but did not heat ions entering thewest plug. Present experiments use a more symmetricalarrangement, with gas fueling near the central-cell midplane(0.25 m east) and double half-turn antennas toward both ends ofthe central cell. The ICH frequencies are selected forresonance between each antenna and the gas box so that newly
ionized particles must pass through an ICH resonance before
reaching an end cell. The increase in the parallel ion
temperature with ICH power is similar at both ends, as shown in
Fig. la. Using this ICH configuration, we increased thecentral-cell density to 3 * 10 12 cm"3 as shown in Fig. Ut>. In
this shot, Tec - 90 eV and Tic= 160 eV. During thisoperation, we achieved long axial and nonambipolar confinement
times; however, low-frequency fluctuations driven by the ICH
limited the ambipolar confinement time to 2 ms.
With 20 kV extraction voltage, the ten central-cell
neutral beams inject 2.5 MW of power for heating and beta
studies. These experiments were carried out in the standard
tandem mirror operating mode rather than the thermal-barriermode. At densities above 10 12 cm" 3 , the beam coupling to thelow-energy ions increases, as indicated by the increase In Tj(
shown in Fig. 5a. The average ion perpendicular energy for thisoperation was 2 keV. The average plasma beta increases to 6% at
higher density and is limited by beam power.
With 2 kV extraction, low-energy neutral injection (LEND,
the ten injectors are designed to fuel the central-cell plasmaat densities above 3 x 10 12 cm"3 without gas fueling. Buildup
calculations using the TREQ code predict that density buildup
will require 170 atom-amp incident and 10 ms particle
confinement time. In the experiments to date, the incident LENI
beam current and particle confinement time are each a factor of
2 to 3 less than these levels, except for periods shorter than
the required 20 ms buildup time. As a consequence, the densityof the trapped beam Ions increases to only 1 x 1 0 u cm"3 withLENI beam Injection. Thallium beam probe density measurements(Fig. 5b) and diamagnetic loop measurements (Fig. 5c)
Indicate that this could be a promising method of fueling theplasma core with modest energy ions. The emphasis of ourpresent work Is to understand and improve confinement while
increasing the LENI beam current so that a higher plasma density
can be sustained with the LENI beam system.

237

IAEA-CN-47/C-I-1
500

(a)
•
-

Diamagnetic T

O East ELIS> T

Q West ELIS I

1-

o D D

o
a

O Y
a •
I
200

400

Radiated ICRF power (kW)

5
»

I

r

I

I

S 3
o

2

c'

1
0

I

I

12/5/85
Shot 35
Time = 22 mi

4

•

-

I
-6

-4

East end
plug

I
I

-

2

0

2

Central cell

46IWest end
plug

Axial position, z (m)FIG. 4. (a) Measurements of ICRF heating with diamagnetic loop and end-loss spectrometer (ELIS).
(b) Axial density profile during high density plugging using ICRF.

A very important question for tandem mirrors concerns theelectron temperature that can be achieved with open field line
geometry. Central-cell electron temperatures have reached
280 eV. As shown in Fig. 6, this temperature can be strongly
correlated with the stored end-cell diamagnetic energy. The
achievement of these electron temperatures is significant
because it indicates the absence of very strong electron power
losses along open field lines.
4. AXIAL CONFINEMENT AND PLASMA POTENTIAL MEASUREMENTS
Past measurements of TMX-U axial plasma potential profiles
were obtained during experiments with single-ended plugging.

SIMONEN et al.

238

na (10 12 cm"3)«tr

10.000

0

10

20

30

40

50

60

70

80

t (ms)

1'\'

3/12/86
Shot 22-100

FIG. 5. Results of central-cell neutral-beam injection: (a) Average beta <0>, parallel ion temperature Tj and diamagnetic energy confinement time TE, as a Junction of on-axis electron density;
20 kV beams without strong end plugging are used, (b) With 2 kV LENl beams, the thallium ion-beamprobe measures a small but encouraging increase in density, (c) The diamagnetic loop indicates anincrease in the central-cell energy.

More recently, we have developed several new methods using
particle beam probing to measure plasma potential in different
axial locations with improved energy and temporal resolution andwith redundancy. We conclude that thermal barriers as large as
0.75 kV are present in TMX-U.
We are currently carrying out measurements under various
TMX-U operating conditions. An example of our recent thermal
barrier region potential measurements is shown in Fig. 7.
Figure 7a shows the thermal barrier potential measurements as afunction of time. These are made by measuring the reflectionenergy of a 24° hydrogen pump beam with an end-loss ion
spectrometer (ELIS) [6], In this shot, the thermal barrier

239

IAEA-CN-47/C-M
3UU

300

I

I

I

I

I I II

Unplugged operation
1 1 / 2 5 / 8 5 - 4 5 ms

—

1

200

—•

100 —

^

I

(eV)

A
•A

A

- A

50

:

A

A
A

A

A

20 —

m

0.1

I

A
A

1
0.2

1

1

1 1 1 1 1
0.5
1.0

2.0

End cell diamagnetic energy (kJ/ml

FIG. 6. High central-cell electron temperature measured by Thomson scattering, correlated with endcell diamagnetic energy (or ECH power).

depth reaches 0.75 kV at 44 ms. The PPC end plates are groundedin this operation. Note that the thermal barrier depth rapidly
decreases when the 2 u ce ECH is turned off. These measurements
are localized along magnetic field lines, as indicated by thepump neutral beam footprints shown in Fig. 7b. Using thesefootprints to estimate spatial locations of the potential
measurements, we have constructed the axial potential profile
shown in Fig. 7c. The solid circle is measured with a thalliumion beam probe at z •= 0. The solid square is from the centroidof the transmitted 6F-18 0 beam measurements and the open squares
are from the energy spread in the transmitted signal assuming amonotonic decreasing potential. The solid diamond is from thecentroid of the reflected 2A-24° pump beam and is arbitrarily
located at the end-cell midplane. The barrier could be evendeeper, since atoms ionized at that location would be trappedand not detected. The open diamonds are obtained from thecentroid of the transmitted signal and are plotted symmetricallyaround the potential minimum. The bar indicates the ELIS energy
range between the minimum energy detected and the energy of the
maximum current. The open triangle is the calculated potentialthat provides the measured 13 ms axial confinement time. The
measured barrier depth is 0.75 kV, much larger than the ±0.1 kV
uncertainties of this data.

240

SIMONEN et al.
8/13/86-16• Thallium beam
• 6F transmitted
centroid• 2A reflected
centroidiiCentral cell,[

J D+ ELISV Calculated
D 6F energy spread
0 2A transmitted
centroid
1
1

,

2 (m)

FIG. 7. (a) Simultaneous plasma potential measurements in the TMX-U thermal barrier region,(b) Axial locations of pump beam intersections with plasma and measured hot electron lie diamagnetic
length, (c) Axial potential profile constructed at 44 ms. The rise in the potential at 6.8 m is notmeasured directly by the ELIS but is indicated by the lack of density at the outer sloshing-ion
turning point.

5.

CONCLUSIONS

Thermal barrier experiments have been carried out in the
TMX-U tandem mirror experiment. Measurements of nonambipolar
and ambipolar radial transport show that these processes, aswell as end losses, can be controlled at modest densities anddurations. Central-cell heating methods using ICH and neutral-

IAEA-CN-47/C-I-1

241

beam Injection have been demonstrated. Potential measurementswith several recently developed methods indicate that deep
(0.75 kV) thermal barriers can be established. These results
provide encouragement to the prospect that the tandem mirror
could be developed into an attractive reactor concept.

REFERENCES[1]SIMONEN, T.C., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984
(Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 255.
[2] GRUBB, D.P., et al., Phys. Rev. Lett. S3 (1984) 783.
[3] HOOPER, E.B., et al., Phys. Fluids 27 (1984) 2264.
[4] HOOPER, E.B., et al., Phys. Fluids 28 (1985) 3609.
[5] ALLEN, S., et al., Rev. Sci. Instrura. 57 (1986) 2072.
[6] FOOTE, J.H., Rev. Sci. Instrum. 57 (1986) 1786.

DISCUSSION
T. CHO: With regard to the thermal barrier operation with 3 x 10 n cm" 3 inthe central cell, I would like to ask the following. Is the ratio of the hot mirror trappedelectron density to the total density consistent with the previous theory of thermal barrier potential formation?
T.C. SIMONEN: The ratio of mirror trapped electrons to total electrons was
not measured during this operation. Under similar conditions we have determined
this ratio to be about 40%. However, since the determination of it is very modeldependent, the uncertainties are as large as a factor of two. We cannot say with certainty whether this ratio is less than in the theory of thermal barrier potential
formation.
T.N. TODD: You referred to central cell heating by ICH and 20 kV neutralbeam heating. What is the incremental energy confinement time, considering only thecentral cell energy and power input?
T.C. SIMONEN: The incremental central cell energy increase depends on thelocation of the gas puffing. For off-midplane gas puffing the increase is about
10 eV/kW. For midplane gas puffing the increase is about 1 eV/kW.
R.J. HAWRYLUK: What is the incremental confinement time with the application of auxiliary heating to the central cell?
T.C. SIMONEN: Depending on the location of gas puffing, the incremental
central cell energy confinement time is 3 ms for off-midplane gas puffing, and
0.3 ms for midplane gas puffing.

IAEA-CN-47/C-I-2

STUDIES OFPOTENTIAL FORMATION AND TRANSPORT
IN THE TANDEM MIRROR GAMMA 10
T. CHO, M. ICHIMURA, M. INUTAKE, K. ISHII,
A. ITAKURA, I. KATANUMA, Y. KIWAMOTO, A. MASE,
S. MIYOSHI, Y. NAKASHIMA, T. SAITO, K. SAWADA,
D. TSUBOUCHI, N. YAMAGUCHI, K. YATSU
Plasma Research Centre,
University of Tsukuba,
Ibaraki-ken, Japan

Abstract
STUDIES OF POTENTIAL FORMATION AND TRANSPORT IN THE TANDEM MIRRORGAMMA 10.
Experimental studies have been carried out on axial and radial transport in the tandem mirror
GAMMA 10. It has been demonstrated that contributions of the thermal barrier enhance the selective
heating of plug electrons and therefore increase the attainable ion confining potential, supporting strong
plugging at both ends. The axial confinement time has reached 0.4 s for a central cell density of
3 x 1012 cm" 3 . The barrier decay mechanism has been found to be classical, so that the pumping
requirements for steady state operation can be reduced by decreasing the collisionality of passing ionsand local ionizations. Non-ambipolar radial transport has been observed to be consistent with neoclassical theory. The associated confinement time T^ can be increased above 1 s by properly terminating
the end plates.

1.

THE GAMMA 10 DEVICEGAMMA 10 is a thermal barrier tandem mirror with inboard minimum-B
anchors flanking the central solenoid (see Fig. 1). At each end of the axisymmetricmirror cell an electron repelling thermal barrier potential dip and an ion confining
plug potential hill are created by a combination of sloshing ion and hot and warm
electron populations. To reduce resonant radial transport, the magnetic configuration
is effectively axisymmetrized so that the drift orbits of confined ions form concentric
circles in the central cell midplane [1-3]. At each end of the machine, radially andazimuthally segmented plates with resistors of 1 Mfi are provided which terminate
the loss fluxes of the plasma. The base pressure is kept below 7 x 10 ~6 Pa and thepressure rise is limited to less than 7 X 10 ~5 Pa during the plasma shot.
Key diagnostics in tandem mirrors are measurements of the potentials at cardinal points and of losses to the ends. The potentials 4>c at the central cell and $8
at the thermal barrier midplane are directly measured with Au° beam probes [4].
The amount of end losses and the plug potential $ p are determined with radiallyscannable end loss analysers of a multigridded structure.
243

244

CHO et al.
lEastl Plug/Barrier Anchor Central

ELA*f

FIG. 1. GAMMA 10: Axial profiles of magnetic field strength and locations of heating sources andkey diagnostics (top); potential distribution (middle); density and temperature profiles (bottom).
ELA is end loss analysis.

Time (ms)FIG. 2. Time evolution of central cell line density n lcc, barrier depth <j>b, confining potential <j>c andend loss flux.

245

IAEA-CN-47/C-I-2*SX ABSORPTION"SX EMISSIVITY% 0.5 * 2co r B EMISSION

0.4

PLUG

ffi 0.3

0.2

CENTRAL -

0.1

0.2

0.4

BARRIER DEPTH
FIG. 3.

Electron temperatureplug is measured

2.

of thermal barrier depth <t>b. The temperature

with the soft-X-ray absorption

cyclotron emission at 2u>ce (•).
emission at 2oiceas a function

0.6
0.8
(kV)

b

The temperature

method (A),the soft-X-ray

Tep at theemissivity method (A) and

Tec at the central cell is measured

with 2oice

cyclotron(o).

AXIAL TRANSPORT

2.1. Potential formation and plugging
The axially confining potential profile is created by a combination of neutral
beams injected at 41° (25 kV, 70 Adrain, two modules) and 28 GHz gyrotron power( < 140 kW, four modules). The fast ion population forms the basis of the density distribution, with pitch-angle distribution peaking at 41°. The temperature of the hot
electrons at the thermal barrier cells has been successfully controlled to below about
50 keV, as theoretically required by the flute stability criterion. The technique isbased on relativistic detuning of overheated electrons according to the 2coce ECH
resonance condition; a spatially limited microwave beam power is injected
dominantly in the x-mode [5],On application of « ce ECH at the 1 T point of the plug, the potentials $ c , $ B
and $ p increase until the thermal barrier distribution $ B < $c < * P is formed,
while the potential of each end plate decreases. Figure 2 shows the time evolutionof the potentials, the end losses and the line density at the central cell. A substantial
reduction of the end losses (i.e. plugging) is observed in connection with the formation of the ion confining potential, <f>c = * p - # c . A large confining potential iscreated when the thermal barrier depth, 4>b = * C -* B , is large. The plugging
becomes weaker when 4>c decreases as </>b decays.

246

CHO et al.

0.2
0.4
0.6
0.8
THERMAL BARRIER DEPTH 0b(kV)

1.0

kSX ABSORPTIONiSX EMI55IVITY

-H

0.5

fb)
T ep (keV)

0.5

FIG. 4. (a) Confining potential <j>c as a Junction of<t>bfor <o« ECHpowers of 20 kW(&), 40 kW(a) and
70 kW (o), with other parameters kept unchanged and with n lcc ~ 1 x I013 cm~2. The plots
marked by • and © are those obtained with slightly lower density and in better vacuum conditions,(b) Confining potential <t>c as a Junction of plug electron temperature Tep, determined from cyclotron
emission (•), the X-ray absorption method (ij and the X-ray emissivity method (A).

Among the heating sources applied to the GAMMA 10 plasma, coce ECH has
a dominant effect on the formation of the electron velocity distribution and therefore
on the potential profile. The x-mode microwave beam, obliquely injected from thehigh field side, illuminates the 1 T resonant surface over a half-power width of about
5 cm. More than 90% absorption is estimated for one pass through the resonant surface according to microwave attenuation measurements. The electron temperature
Tep, determined at the plug from the soft-X-ray spectrum and the cyclotron emission, increases with increasing power of wce ECH. At the same time, Tep is significantly influenced by the thermal barrier depth, as shown in Fig. 3. The electron
temperature T^ at the central cell does not increase so much as Tep during the formation of the thermal barrier. This feature represents a reduction of thermal conduction across the thermal barrier.

IAEA-CN-47/C-I-2

247

The maximum attainable confining potential <j>c increases with increasing <£b,as shown in Fig. 4(a). Here the w« ECH power is varied, with the plasma density
kept nearly constant for the plots indicated by A, • and O . The increase of 0C is
well correlated with the increase in Tep, as shown in Fig. 4(b). An advantage of theinstallation of the thermal barrier is the efficient creation of a confining potential. In
the presence of the thermal barrier, the confining potential can be established witha plug density np lower than the central cell density n<.. It has been observed that theaxial particle confinement time T\, defined as the total number of ions confined in acore flux tube (rc < 10 cm at the central cell midplane) divided by the end loss
fluxes, is in agreement with the theoretical prediction of the Pastukhov scaling formula [3]. For an ICRF heated plasma of T i± = 1.7 keV and Tj, = (0.2-0.3) keV,
7|| = 0.4 s has been obtained at n^ = 3 X 1012 cm" 3 .
The observed depth 0 b of the thermal barrier potential is normally larger thanT^ m [Hc/(nb-nbh)]. One possible explanation for this, derived from our FokkerPlanck code simulation [6], is that the velocity distribution of low energy electrons
at the barrier is distorted from Maxwellian under strong 2wce ECH fields so that thedensity of the mirror trapped electrons is larger than the measured hot electron
density nbh. The simulation predicts an effective reduction of the thermal electron
density by a factor of 1.5-1.8. Experimental evidence from the energy analysis ofend loss electrons suggests the production of an abundant population of mirror
trapped warm electrons.
2.2. Thermal barrier duration
In examinations of the feasibility of steady state operation of tandem mirror
reactors, one of the fundamental problems is the evaluation of the decay rate of thethermal barrier potential. Collisional trapping of passing ions from the central cell

1.0

2.0

n c (xi0 12 cm" 3 )FIG. 5. Duration of thermal barrier, TU, plotted as a junction of central cell density nc. The dottedlines are best fits to the data of two cases, with Tit = 0.05 keV, nb = (3-6) X 10" cm'3, andTa =0.1 keV, nb= ~ 7 x 10" cm'3.

248

CHO et al.

and local ionizations are conceivable classical processes which tend to weaken thepotential dip. Without pumping, the decay rate, or equivalently the required pumping
rate for steady state operation, is related to the duration of the barrier potential. In
Fig. 5, the inverse of the observed duration of the thermal barrier, T^1, is plotted asa function of the central cell density n<. for different passing ion temperatures,
Tn = 0.05 keV and 0.1 keV, and barrier densities, nb = (3-6) X 10" cm" 3 andabout 7 X 10" cm" 3 . These values can be represented by a fitting equation
T M ' = 6 . 0 x lO"

12

32

n.Til '

This expression is in very good agreement with the theoretically predicted collisional
barrier filling rate of passing ions,vb = (3.2-6.4) x 10 "12derived from the classical Futch-LoDestro formula. Also, it has been observed thatrbV increases linearly with nbT^3'2 for relatively low nb if n. is kept nearly constant.
These observations indicate that the decay of the thermal barrier is mainly due to classical processes. The required rate for barrier pumping can be reduced by decreasing
the collisionality of the passing ions and by reducing the local neutral density.

3.

RADIAL TRANSPORT

Radial transport inherent in an open system is of a non-ambipolar nature, aspredicted by neoclassical theory, which means that the quadrupole magnetic field
causes enhanced radial transport of ions in relation to electrons. The charge balance
condition requires an excess flow of electrons to the ends. Since the non-ambipolar
particle flux is measurable in terms of a net current flowing into the end plates, theassociated confinement time T^A can be determined in a similar manner as Tj [7].
When the end plates are forced to be short-circuited to the machine wall, anempirical relation of T^A « *<r' is obtained [3]. The observed * c "' dependence isconsistent with that predicted by neoclassical theory, but the value of r^A is smaller
by a factor of two to three.
On the other hand, in the normal mode of operation of GAMMA 10, the end
plates are terminated by resistors of 1 MO. Measurements of T"A have been madefor different resistances REP grounding each segment of the plates to the machine(see Fig. 6). As REP is increased from 1 fl to 1.135 Mfl, TiA increases from about
10 ms to above 1 s, and the central cell potential $ c decreases from 2 kV to 0.6 kV,
allowing for a change in n<. of within a factor of three. In the relatively narrow rangeof <i>c observed, T™A is proportional to $~a, with a varing from 1 to 2 for
REP < 135 kO. When REP = 1.135 MQ, r^A is poorly correlated with $ c , but aclear correlation between T^A and n2 is observed. A full theoretical explanation is

249

IAEA-CN-47/C-I-2
lltllllj

1U

10'

r

hull

si

t

10°
10"'

r

<

luliltIO-1

yt
Illlllll 1 illlllll

in"'

10

j y
'T ^ \•NEOCLASSICAL -.
THEORY:1

i 111 mil 1 1 itlltlt 1 lllml

I tilti.J

I ( llnnl

101 101010
10' 1010
END PLATE RESISTANCE lohm)t Mint10"

FIG. 6. Non-ambipolar confinement time, T^A, as a function of termination resistance of each end
plate segment. The black circles represent the experimental data and the solid line indicates the resultof neoclassical calculations.

not available yet, but it can be said that this correlation is favourable for increasing
the plasma density. A neoclassical calculation has been carried out with a simplified
model in which the end plates form one plate which is grounded to the machine
through a variable resistance. The radial profile of the potential is self-consistently
calculated by using previously determined neoclassical transport coefficients. The
result of the calculation is shown by the solid line in Fig. 6. The good agreement
between the theoretical and experimental results indicates the validity of the neoclassical treatment of radial transport in GAMMA 10. These results also indicate that
non-ambipolar radial transport can be effectively suppressed in GAMMA 10 byincreasing the terminating impedance of the end plates.

4.

CONCLUSION

Efficient formation of a confining potential in connection with a thermal barrierhas been realized. The confinement time is improved by more than two orders ofmagnitude over that of a single mirror. It has been confirmed that the axial particle
confinement time is in good agreement with the Pastukhov scaling formula. The
observed duration of the thermal barrier indicates that the barrier pumping requirements for steady state operation can be determined from classical scaling, including
collisional and ionization processes. The non-ambipolar radial transport is observed
to be consistent with neoclassical theory.
REFERENCES
11] M1YOSHI, S., et al., Selected Lectures Presented at Spring College on Fusion Energy, Trieste,
1981, International Centre for Theoretical Physics, Trieste (1982) 239.

250

CHO et al.

[2] CHO, T., ICHIMURA, M., INUTAKE, M., et al., in Plasma Physics and Controlled Nuclear
Fusion Research 1984 (Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 275.
[3] INUTAKE, M., et al., Phys. Rev. Lett. 55 (1985) 939.
[4] ISHn, K., et al., Rev. Sci. Instrum. 56 (1985) 1053.
[5] KIWAMOTO, Y., et al., Phys. Fluids 29 (1986).
[6] KATANUMA, I., et al., Phys. Fluids 29 (1986) 4138.
[7] HOOPER, E.B., et al., Phys. Fluids 28 (1985) 3609.

DISCUSSION
R.S. POST: I would like to ask two questions. First, referring to what you said
about confinement times, could you say whether r = 0.4 s corresponds to particle
confinement or energy confinement? Second, if you take the quoted values for TPH
and TP± together, does that give you an energy confinement time consistent with themeasured values?
Y. KIWAMOTO: The quoted value of T = 0.4 s corresponds to the particle
confinement time.
The energy confinement time rEi was estimated from the decay time, T^ a y , ofthe central cell diamagnetism after turning off the low power ICRH. We subtracted
the charge exchange rate 1/TCX from the decay rate l / r ^ y , since the charge
exchange process ultimately poses minimal technical problems.
The neutral density, which was obtained from both the Ha measurement andthe neutral pressure measurement, has been consistently used in the evaluations ofrp and rEi.
T.C. SIMONEN: You describe new results at a density of 8 x 1012 cm" 3 .
What was the duration of the strong axial confinement and how did you start andoperate these plasma experiments?
Y. KIWAMOTO: The startup and operation sequence is somewhat different
from the standard mode. The magnitude of the central cell magnetic field is 55% of
the standard value and the central cell plasma is sustained by ICRF. The duration ofstrong plugging is 10 ms, which corresponds to the width of the plug ECH pulse.

IAEA-CN-47/C-I-3

RECENT RESULTS FROM
THE TARA TANDEM MIRROR
AND THE CONSTANCE-B MIRROR*
R.S. POST, K. BRAU, J. CASEY, X. CHEN,
J. COLEMAN, H.R. GARNER, M. GERVER,
S. GOLOVATO, D. GOODMAN, W. GUSS, S. HIROE1,
S. HOKIN, S. HORNE, J. IRBY, J. KESNER, B. LANE,
T. MORAN, L. POCS2, E. SEVILLANO, D. SMATLAK,
D.K. SMITH, J. SULLIVAN, R.P. TORTI, X.Z. YAO3
Plasma Fusion Center,
Massachusetts Institute of Technology,
Cambridge, Massachusetts,
United States of. America

Abstracc
RECENT RESULTS FROM THE TARA TANDEM MIRROR AND THE CONSTANCE-B
MIRROR.
The Tara tandem mirror program has studied anchor and ponderomotive stabilization, axicell
plugging with ECH and ICRF, sloshing ion buildup in the axicells, and halo formation and stabilization
by an axisymmetric divertor. Central cell plasma parameters achieved by midplane fueling and slow
wave ICRF heating from a local magnetic hill are /3 = 1.2%, n,, = 3 x 1012 cm" 3 . The plasma isstabilized both by anchor ion /3 and by ponderomotive stabilization with the central cell ICRF in combination with a magnetic divertor, realizing a completely axisymmetric configuration. Anchor ICRF creates non-Boltzmann potential plugging of central cell ions. Neutral beam injection establishes a sloshingion distribution for a cold dense cell stream; the hot ion confinement is classical and dominated by electron drag. Axicell ECH plugging experiments lead to near total reduction in endloss, but also to adecrease in the central cell density, indicating increased radial losses. Single ended ECH plugging showsno increase in opposite endloss. Single-ended plugging with axicell ICRF produces 50% reduction in
ion endloss, with about half of the reflected ions observed in the opposite endloss. In the Constance-B
quadrupole mirror the hot electron pressure profile is peaked off-axis and has the shape of a baseballseam.

* Funded by the United States Department of Energy, under Contract
No. DE-AC02-78ET51013.
1
2
3

Oak Ridge National Laboratory, Oak Ridge, TN, USA.
Central Research Institute for Physics, Budapest, Hungary.
Institute of Physics, Beijing, China.

251

252

POST et al.

Central Cell Startup
ICRF healing and midplane gas fueling on a magnetic hill are used
to create the startup plasma1'2 (fig. 1). Figure 2 is a plot of the axial
density profile with slot antenna heating only indicating the presence of
mirror-trapped ions in the two central cell magnetic wells. Assuming flat
temperature profiles, and measured parabolic density profiles typical onaxis parameters in the magnetic well closest to the slot antenna are ne0 =3 - 4 x 10 12 cm- 3 , Ti± = 700 eV, Tin = 150 eV, and 7; = 60 - 80 eV.
Stability of the central cell plasma to flute-like fluctuations is obtainedby creating average minimum-B with an ICRF heated anchor plasma, orby the use of high power central cell ICRF heating. In discharges where
the anchor ICRF is terminated well before the slot antenna, the central
cell plasma remains stable at sufficient slot antenna power, which suggests
ponderomotive stabilization. The anchor ICRF creates non-Boltzmannpotentials 3 ' 4 that plug the central cell stream.
In fig. 3 the central cell diamagnetism and line density, and total
endloss currents are plotted for a discharge where the anchor ICRF is
off at 20 msec and back on at 40 msec. During anchor ICRF the total
ion endloss is reduced, while the total net current indicates increased ion
radial transport. Assuming the rise in central cell density to be caused byreflected ions, the reflected current is about 10 A, in agreement with thedrop in ion endloss in the core (r < a/2) plasma. Ions reflected near theedge are subject to substantial radial losses and thus do not contribute to
the central cell density rise. Pressure rises are observed in the transition
region between the axicell and anchor, where the ion radial drifts arelargest. These losses maintain a low density in the transition and are an
example of passive pumping.5

Axicell Sloshing Ion Formation by Neutral Beam Injection
Three Berkeley-type injectors per axicell together provide 70 A atomic
on target at about 11 kV average energy. The 40° injection angle results
in a sloshing ion distribution peaked at a mirror ratio of about 2.4. A
doubling of the density can be observed throughout the machine underconditions of low central cell ICRF power and high gas feed, which produces a cold, dense plasma stream to the plug. Figure 4 shows the data

253

IAEA-CN-47/C-I-3
SLOT ANTENNAPLUG ECH •HALO GAS BOX

AXI-CELL ANTENNA-GAS BOXBARRIER ECH-

-DOUBLE HALF-TURN ANTENNAa£piniiiM
600

300
z (cm)

FIG. I. Schematic ofTara magnetic geometry, heating and fueling systems for north central cell andaxicell, and part of south central cell (anchors not shown). Tara is symmetric about midplane divertor
except for central gas box and slot antenna.

3r i r i i

Mirror Wei
t=l7msMirror Well/
Transition"Plug..Transition
i

-10

t

i

I

i

i

-5
Axial

i i

I

i

i t

i

I

0

5

location

(m)

i

i

i

i

10

FIG. 2. Axial profile of line averaged central cell density during slot antenna heating.

from a particular shot. The diamagnetism and line density at the plug
midplane increase; the mirror peak line density also increases because of
the higher ionization rate due to increased Tt. An energetic (> 600 eV)
ion component is observed in the endloss current. The ion confinement canbe calculated from the interferometer and endloss data. From the decay

POST et al.

254
i.o

SOUTH MAGNETIC WELL

0.0
I.S

SOUTH MAGNETIC WELL

nl

i

0.0
ZOO

W

I:(tot)
<A)
100

0
0
«(to1)

(A)

-50

-100
200

Ie(«ot)
(A)

100

,
6

1—I—'—I ' I
'—t
10
20
30
40
Time(ms)

'

50

60

FIG. 3. Central celt 0 and density, total ion, net, and electron endloss currents for a discharge withcentral cell slot antenna heating. Anchor RF on from 0 to 20 ms and 40 to 55 ms.

of the diamagneiic loop signal after beam turnoff and endloss measurements of the potential we conclude that Tt ~ 20eV and that sloshing ion
confinement is nearly classical and is dominated by electron drag. Underconditions given above of higher central cell heating power and lower gas
feed the sloshing ion buildup is much lower than expected based on T«,
the measured edge neutral gas pressure and target density, and assumingclassical confinement.

End Plugging with Gyrotron Heating
Each axicell is equipped with two gyrotron ECH systems at 28 GHz,
for fundamental and second harmonic heating. Each antenna typically

IAEA-CN-47/C-I-3

255

*NB
(A) 100

20

30
40Time (ms)FIG. 4. (Bottom to top traces) Injected neutral beam current in south axicell, line density in south
axicell midplane and south mirror throat, $ at south axicell midplane. The part of fluctuation in line
density signal due to bit noise can be seen from signal at 60 ms.

radiates 50 kW for pulse lengths up to 75 ms. In the following discussion ofthermal and potential barrier generation by ECH, end plugging is defined
as an increase in the ratio of the central cell mirror throat density to
the ion endloss current. Figure 5 is a plot of experimental data for a"single-ended" plugging shot, with ICRF in one anchor to provide MHD
stability, and the barrier (2w«) and plug (u>ce) gyrotrons in the opposite
end on at 10 and 25 ms, respectively. Immediately after initiation of theplug gyrotron there is a sharp reduction of the north endloss current, anda rise in TJ|. The reflected ion current (about 50 A for the core) is notobserved either in the opposite endloss current or as a rise in the central
cell density. This result is consistent with radial losses in either the axicellor the central cell. The plug potential, as measured by endloss analyzers,
increases on both the plugged and unplugged ends of the machine from200 V before the ECH to 275 V during the plugging. This potential is

POST et al.

256

0
60
40
lies
(A) 20060lien 40(A)
20-'"W*) I
0200
Pp(kW)100
0200
Pb(kW)
100-

-10

20
30
Time (ms)

40

50

60

70

FIG. 5. (Bottom to top traces) Barrier and plug gyrotron powers, north mirror throat density, south
and north ion core endloss currents, and core parallel confinement time, during single-ended ECH
plugging experiment.

non-Boltzmann since the axicell density is below the central cell density.
The rapid drop in axicell /? at barrier turnoff implies an electron energyof ~ 5 keV, assuming classical scattering.
Plugging occurs independently of neutral beam injection. The axicell midplane density is typically 30% of the mirror peak, indicating somemechanism which removes the low energy ions trapped in the negative
potential at the midplane. The density at the 10 kG point drops sharply
during plugging, which is consistent with a Boltzmann-factor reductionof the outgoing stream at the high potential region. The potential formation is consistent with a theoretical analysis of strong ECH diffusion,
similar to the treatment by Cohen6 assuming an anomalous radial ion
transport at the barrier. To achieve plugging normally requires both wce
and 2u)ct heating. Plugging is observed for on-axis central cell densitiesof 1 - 3 x 1012cm~3. Also characteristic of plugging is a flattening of

IAEA-CN-47/C-I-3

257

the endloss profile. This transport is ambipolar, in contrast to TMX-U
results. 7 In a typical double-ended plugging experiment (data not shown),with barrier and plug gyrotrons on at 20 and 25 ms, respectively, the central cell north throat interferometer rises by about 20% during the first
ms after application of the plug ECH, then drops abruptly to 60% of its
initial value. The central cell density drops by a factor of two. In thiscase the increased radial particle loss rate more than compensates the improvement in axial confinement. The initial rise in the north mirror throat
density indicates plugging by the north axicell ECH; then as the central
cell stream to the anchor is cut off, the anchor ion 0 drops nearly to zero,and a loss of MHD stability ensues.

End Plugging with Axicell ICRF
ICRF heating has been used in the north axicell to study the trapping and heating of the central cell stream in order to improve the neutralbeam target plasma, as well as to provide plugging by potential formation.
Unlike ICRF plugging in the quadrupole anchors plugging in the axicell
avoids the problems of neoclassical radial transport. The antenna is located near the R=2 point on the outboard side of the R=4 mirror. The
RF is normally excited at the midplane ion cyclotron frequency, which
produces the largest increase in density and diamagnetism. The potential
plugging by the ICRF is produced over the, range 0.9 < w/u;,.,- < 1.2 at the
midplane. For constant central cell conditions, the plugging, as measured
by the reduction in ion losses out of the north end, increases roughly as
the square root of axicell ICRF power.
In a shot with no ICRF plasma in the anchors (fig. 6) the ion loss
out of the north end is reduced by 50% across the entire profile during
application of axicell ICRF. The line density in the north central cell
mirror peak increases by 42%, showing that the ions are reflected fromthe north axicell. The plug potential (not shown) increases by 100 V
during the ICRF, with the pre-ICRF parallel ion temperature, T;|| = 120
eV. This would be expected to reduce the ion end loss by approximately
exp( — A$/Tj||) = 0.43, which is consistent with the observed end loss
reduction of 50%. About 55% of the reflected ion current is unaccounted
for in the axial losses out the other end, indicating that additional radiallosses are occuring when the axicell ICRF is applied.

POST et al.

258" * NMPSO
I s (lot)(A)

I N (tot)
(A)
50

400
P R F (kW)
200

10

20

30

40

SO

60

70

80

Time (ms)FIG. 6. (Bottom to top traces) North axicell ICRF power, mirror throat density, north and south ion
endloss for single-ended ICRF plugging discharge.

Magnetic Divertor
A magnetic flux divertor was recently installed at the central cell
midplane to provide stability to m=l fluctuations8 and to create a dense"halo" plasma.9 Up to 20% of the magnetic flux can be diverted around aninternal coil.10 During end plugged operation the gas fueling requirementsof the well confined core plasma should drop by an order of magnitudewith respect to the flow-confined edge. By diverting this edge plasma into
a separate chamber the two regions can be fueled independently. Gas is
injected at four azimuthal positions in the divertor using a system identicalto the central cell.11 Plasma which intercepts the internal ring supports
and current leads (total area 4.3 cm2) causes recycling of ~ 6% of thecalculated classical flow loss, and provides an additional fueling source.

259

IAEA-CN-47/C-I-3
i.owith diver tor(IOl4crn2>

without divertor

/^Sm

/wm

(IOMcm-2)
0
150

100Pp(kW)500
150

100Pb(kW))
50
O

20

40
Time (ms)6080FIG. 7. Comparison of axicell ECH induced fluctuations in central cell density for cases with andwithout axisymmetric divertor.

The halo power is supplied by edge heating from the ICRF antennas.
In the divertor the density peaks at ne ~ 3 x 10 n cm~ 3 with a half-widthof 0.5 cm, which maps to a 2 cm thick annulus in the magnetic wells. The
electron temperature is 30-40 eV across the profile. About 25% of the gas
injected in the divertor appears as ionization of the core, indicating good
plasma shielding. Stabilization of m=l fluctuations by the divertor duringaxicell ECH has been observed. Figure 7 illustrates the response of thecentral cell plasma density to the axicell ECH both with and without thedivertor, at a central cell fueling rate of 28 torr-1/sec.
Constance-B
In the Constance-B experiment the equilibrium, stability, and confinement of hot electrons in a minimum-B mirror with a large radial welldepth have been studied. Up to 4 kW of 10.5 GHz ECRH power is used

260

POST et al.

FIG. 8. Visible light photographs of Constance-B hot electron distribution.

IAEA-CN-47/C-I-3

261

to create plasmas with /? > 30%, Tth - 450 keV, neh = 2 x 10 n cnT 3 ,and rj ec = 2 x lO^cm" 3 . Visible light and x-ray pinhole photographs haverevealed that the hot electron pressure profile is peaked off-axis and has
the shape of a baseball seam. This curve coincides with the drift surfaceof the deeply-trapped electrons (fig. 8). The plasma is MHD stable bothduring ECRH and in the afterglow when only hot electrons are present.
The stability of the afterglow plasma contrasts with EBT results,12 thatstabilizing mechanisms other than cold plasma are important.
RF diffusion of hot (E > 50 keV) electrons has been studied by meansof x-ray and endloss measurements. The RF-induced endloss exceeds collisional losses by factors of 10-1000. The ratio of the two losses scalesas P^f. This is due in part to the increase in field strength, for which a
linear scaling with P ^ has been measured, and in part to the change inplasma parameters produced by the larger powers. These hot electrons
are heated up to well above the adiabatic barrier to diffusion, which is
calculated to be at E=20 keV for the electric field strength of 15 V/cm
measured outside the plasma. The rf-induced endloss is maximum when
the nonrelativistic ECRH resonance is near the mirror peak.

REFERENCES
[1]

[2]
[3]
[4]
[5]
[6]
[7]
[8]

[9]

POST, R.S., GERVER, M., KESNER, J., IRBY, J.H., LANE, B.G., et ah, in Plasma Physics
and Controlled Nuclear Fusion Research, 1984 (Proc. 10th Int. Conf. London, 1984), Vol. 2,
IAEA, Vienna (1983) 285.
POST, R.S., BRAU, K., GOLOVATO, S., SEVILLANO, E., SMITH, D.K., et al., to be published in Nucl. Fusion.
SMITH, D.K., BRAU, K., GOODRICH P., IRBY, J., MAUEL, M., et al., Phys. Fluids 29(1986) 902.
HERSHKOWTTZ, N., NELSON, B.A., JOHNSON, J., FERRON, J.R., PERSING, H., et al.,Phys. Rev. Lett. 55 (1985) 947.
KESNER, J., Comments Plasma Phys. Controll. Fusion 4 (1979) 123.
COHEN, R.H., Phys. Fluids 2<S (1983) 2774.
HOOPER, E.B., Jr., COHEN, R.H., CORRELL, D.L., GILMORE, J.M., GRUBB, D.P.,
Phys. Fluids 28 (1985) 3609.
KESNER, J., LANE, B., POST, R.S., Stabilization of MHD Modes in an Axisymmetric Plasma
Column Through the Use of a Magnetic Divertor, Rep. MIT/PFC/JA-86-53, Massachusetts Inst.
of Tech., Cambridge (1986); to be published in Nucl. Fusion.
POST, R.S., HORNE, S., BRAU, K., CASEY, J., GOLOVATO, S., et al., presented at 1986
AVS Meeting, Baltimore, MD, USA.

262

POST et al.

[10]

POST, R.S., BRAU, K., CASEY J., COLEMAN, J., GOLOVATO, S., et al., "The Tara
Tandem Mirror Experiment", in Plasma-Surface Interactions in Controlled Fusion Devices(Proc. 7th Int. Conf. Princeton, NJ, 1986).
[11] BRAU, K., GOODRICH, P., POST, R.S., SEVILLANO, E., J. Vac. Sci. Technol., A 4 (1986)
1079.
[12] HIROE, S., WILGEN, J.B., BAITY, F.W., BERRY, L.A., COLCHIN, R.J., et al., Phys.
Fluids 27 (1984) 1019.

DISCUSSION
T. KAWABE: You showed us a new scheme for stabilizing the MHD mode ina pure axisymmetric mirror by use of a divertor coil with ponderomotive force. I
think it is important to use a mirror confined plasma or small power reactor as a neutron source. My question is whether this new concept is scalable to the fusion condition or not.
R.S. POST: The answer to that rests in the precise position of the separatrix
in the pressure profile. Presently, we believe that the separatrix is located at ~ 10%of the peak pressure. Rough estimates indicate that associate support losses would be
acceptable. For a neutron source, one could consider a superconducting ring, as in
the Levitron or FM-1. In that case there would be no support losses, but adequate
shielding would be required.
N. HERSHKOWITZ: Could you compare the plasma parameters with andwithout this magnetic divertor and perhaps comment on possible changes inpoderomotive stability when the divertor is turned on.
R.S. POST: When the divertor is switched on, we can operate at - 4 0 % lower
gas feed rate. The electron and ion temperature rise by about the same factor, allother conditions being the same.
The ponderomotive stabilization may well be affected by the divertor due tochanges in the edge profile. This is an issue we are currently studying, but we do
not have any results yet.
T. CHO: With regard to thermal barrier operation in TARA with plugging but
without NBI, do you have any idea how one can explain the sustainment mechanismof the thermal barrier against barrier filling by passing ions?
R.S. POST: We assume that the loss or barrier pumping is by anomalous radialtransport. The rates required to pump the barrier can be consistent with the measured
rates.

IAEA-CN-47/C-I-4-1

STABILIZATION OF MHD MODESIN AN AXISYMMETRIC MAGNETIC MIRRORBY APPLIED RF WAVES
AND INITIAL RESULTS OF PHAEDRUS-B
R.A. BREUN, P. BROOKER, D. BROUCHOUS, J. BROWNING,
G. BUTZ, J. CONRAD, E. DALES, J. FERRON, R. GOULDING,
N. HERSHKOWITZ, T. INTRATOR, C. LITWIN, R. MAJESKI,
S. MEASSICK, B. NELSON, L. PERANICH, H. PERSING,
J. RADTKE, D. ROBERTS, G. SEVERN, D. SING, Enyao WANG
Nuclear Engineering Department,
University of Wisconsin,
Madison, Wisconsin
D.A. D'IPPOLITO, J.R. MYRA, G.L. FRANCIS
Science Applications International,
Boulder, Colorado
United States of America

Abstract
STABILIZATION OF MHD MODES IN AN AXISYMMETRIC MAGNETIC MIRROR BY
APPLIED RF WAVES AND INITIAL RESULTS OF PHAEDRUS-B.
Phaedrus was operated as a three-cell axisymmetric tandem mirror with dual half-turn end cellantennas at the end cell midplanes. The device was converted to a five-cell tandem mirror (Phaedrus-B)
with quadrupole end cells by adding choke coils at the ends of the central cell and dual half-turn antennas
throughout the device. The axisymmetric Phaedrus operation achieved enhanced axial plugging which
improved the central cell density to n = 1 x 1013 cm" 3 , the peak plasma beta to 13% and nrp to3 x 1010 cm" 3 . The MHD stability was enhanced by radial ponderomotive force using ICRF at 1.3,
1.7 and 4 Ur A kinetic theory of RF stabilization has been developed which includes the effects of RF
sideband modes. The results suggest that RF far fields 'lean' on the device walls and RF near fields'lean' on the antennas. The results were found to be sensitive to the antenna kz spectrum. Thermal barrier potential dips were identified without active barrier pumping. Low frequency fluctuations were
found to be sensitive to the phasing of the central cell antennas. Plugging was enhanced by the presenceof end cell RF which was resonant only in the central cell.

1.

INTRODUCTION

The Phaedrus Tandem mirror uses RF near the ion cyclotron
frequency (ICRF) to achieve: MHD stability by radial
ponderomotive force, electrostatic ion plugging in the end
cells, end cell fueling, and ion and electron heating[l,2,3]•
Phaedrus can be operated with quadrupole end cells or with

263

264

BREUN et al.

axisymmetric end cells and so has many operating modes. Here
we report Phaedrus operation as an axisymmetric three-cell
device,with MHD stability provided entirely by externallyapplied RF electric fieldsfl]. Our original experiments[1,2]employed RF frequencies slightly larger than the local ion
cyclotron frequency (i.e. « = 52. ). This paper presents resultsof recent experiments which demonstrate that stabilization does
not require » to be nearly equal to S2j.. Stability has beenenhanced by RF at 1.3, 1.7 or 4 52. . In addition, the use ofaxisymmetric end cell antennas resulted in stable plasmas withcentral cell densities as high as 10 cm" .
A theoretical treatment has been developed which showsthat the self-consistent interaction of the RF with the plasma
results in sideband mode coupling terms which tend to reduce
the stabilizing effects of the RF.
During the first half of 1985, Phaedrus was upgraded froma three-cell device to a five-cell device by inserting 1 T
choke coils at each end of the central cell. New central cellantennas were added to allow control of the spectrum of the
ICRF axial wavenumber k and the azimuthal mode number.
Thisdevice, Phaedrus-B, waszdesigned to provide thermal barriers in
the choke coil end cell transitions without the presence of hot
ECH produced electrons and to further improve the understandingof ICRF. This paper presents the first results of Phaedrus-B
which demonstrate the presence of thermal barrier potential
wells and axial density dips in the transition, achieved
without the presence of active barrier pumping. The newantenna set has resulted in increased plasma density. Effects
of antenna phasing on plasma stability and of operation of
additional end cell antennas at lower frequencies are alsopresented.

2.

RF STABILIZATION EXPERIMENTS IN THE AXISYMMETRICCONFIGURATIONThe stabilizing effects in these experiments are due
primarily to the radial ponderomotive force F of the ICRF onp
the plasma. F can be written:p

VE2

where E is the left circularly polarized field (which rotateswith the ions), E_ is the right circularly polarized field, andE is the field parallel to the magnetic field. Eq.(l) is
appropriate for a cold plasma, i.e. « - S » k v and » >>k v ,where v. and v are the ion and electron tnermal velocity.,
respectively* If w e = 52., the term which depends on E is265

IAEA-CN-47/C-I-4-1
Axisymmetric

300

-300
AXIAL POSITION (cm)FIG. 1. Schematic of magnetic field coils and axial magnetic field profile for axisymmetric three-cell
operation, illustrating the placement of the dual half-turn (DHT) antennas in the end cells and of thehalf-turn antenna (HT) in the central cell.

important. This term comes from the force on the ions. Itchanges sign with <o - 8. and may be responsible for the abruptchange in stability reported previously[1]. The terms which
depend on E_ and E are due primarily to the force onelectrons. In Phaldrus the E_ term becomes the largest
contributor to F P when w is not close tofl..The sign of Fpdepends on the direction of the radial electric field
gradients. Data from all experiments are consistent with thegradient directions required for stabilization. The only
propagating wave for the Phaedrus plasma parameters is theazimuthal mode number C = 1 fast magnetosonic wave. For thismode, radial E and E profiles peak on the edge of the plasma
while E_ profiles peak on axis. Measurements of the RF
magnetic field profiles in the plasma are consistent with thisdescription.
The axisymmetric configuration of the Phaedrus tandem
mirror is given schematically in Fig. 1. Unlike in previous
experiments, which used line antennas in the end cells, dual
half-turn antennas were operated at the end cell midplanes. As
shown in Fig. 2, application of RF to the end cell antennas
resulted in enhanced axial confinement and a factor of 3
increase in the central cell density. In the central cell theion temperature T = 40 eV, the electron temperature T = 20eV, the peak plasma density was n = 1 x 10 1 3 cm" 3 , theepeak
plasma beta was 13 percent and nt = 3 x 10 1 0 cm" -s wasmeasured,where x is the axial particle confinement time. (Inother experiment! a plasma beta of 11 percent was achieved at
n = 1 x 10 1 2 cm"3 and T, = 550 eV.) An £ = +1 fast wave at the266

BREUN et al.

4

8

12

lend cell Rl-I

Time (ms)Time (ms)FIG. 2. Plasma density (a) and energy density (b) versus time in each cell for axisymmetric three-cell
operation. End cell RF is on from 5 to 12 ms; central cell RF is on from 0 to 15 ms.

end cell RF frequency (approximately 4 times the central cellion cyclotron frequency) was observed in the central cell.
This wave was observed to enhance central cell stability. Fig.3
gives line density as a function of time. Note the reductionin the fluctuation level when the end cell RF was turned on in
one end cell at 5 ms. Application of RF to only one end cell
does not make significant changes in central cell plasma
parameters. In other experiments[4] the stability of plasma
produced by RF with » < S. was enhanced by application of RF ateither 1.3 S. or 1.7 S. in the central cell.

3. THEORY
A theory describing the nonlinear interaction of RF waves
with low frequency HHD modes has been developed[5,6] andapplied to model the recent RF stabilization experiments in
Phaedrus. The nonlinear RF coupling term which enters theballooning equation has been calculated,including the effect ofboth the RF wave driven by the antenna and the RF sidebands
generated by beating with the MHD wave. An important analytic
result[6] was obtained for m = 1 rigid shift interchange modes:
the ICRF stabilization vanishes as the antenna and conducting
walls are removed from the proximity of the plasma. Thissuggests that the plasma cannot "lean" on the high-frequency
fields themselves but only on the rigid structures which
support them.
In a numerical study the fast wave eigenmodes of a diffuse
cylindrical plasma column and the near-field patterncorresponding to the 9. = 1 Phaedrus antenna are calculated for267

IAEA-CN-47/C-I-4-1

8
12Time (ms)
I end cell RF I

16

20

FIG. 3. Probe ion saturation current versus time. One end cell RF system
is turned on at 5 ms and turned off at 13 ms.

typical experimental parameters. For simplicity, in the
numerical work we assume E = 0 (valid everywhere except within
an electron skin depth of ?he plasma edge where E is highly
evanescent), and the model is applied only in thezregime where
the RF frequency <o is greater than the ion cyclotron frequency
S. , in order to avoid the shear-Alfven resonance. The critical
RF field (or antenna current) for m = 1 interchange stability
is calculated using a rigid shift MHD trial function and thenumerically determined RF wave and sideband wave profiles.
For far-field fast wave stabilization (eigenmode "leaning"
on a conducting wall), the analytic prediction that thestabilizing effect vanishes as the wall moves out to infinity
has been verified numerically. It has been found that thedegree of stabilization is sensitive to the conducting wall
position and that an optimal wall position exists. Even at the
optimal position the ponderomotive and sideband terms tend tocancel to a large degree for m = 1 MHD stability, thereby
diminishing the effect of the far field compared with estimates
based on balancing the equilibrium ponderomotive force with thecurvature drive. For near-field stabilization (fields"leaning" on an antenna), the results depend sensitively on theradial profile of the applied waves and on the k spectrum of
the antenna (due to the presence of main wave ana" sideband
resonances). Our results suggest that an antenna design whichprimarily drives the near-field amplitudes just below the krequired for fast wave propagation is advantageous forstability. Applied to Phaedrus, the model predicts critical RF
fields for both far-and near-field stabilization on the orderof 20 V/cm (peak fields) and antenna currents consistent withthose used in the experiments.
Further generalizations of the model would be useful toobtain a more detailed understanding of the experiment. Firstis the inclusion of the E terms. Order of magnitude estimates
based on the full theoretical model[6], with E estimated from
McVey's antenna coupling code[7], indicate that the nonlinear
terms due to this component can be comparable to theBREUN et al.

268

Phaednis-B

11.2m
DHT

-400

DHT

-200

0

200

400

axial position (cm)FIG. 4. Schematic of magnetic field coils and axial magnetic field profile for Phaedrus-B, illustrating
the placement of the four double half-turn (DHT) antennas in the end cells and the two double dual halfturn (DDHT) antennas in the central cell.

contribution of the other components at the plasma edge.
Further work is in progress on this problem. Second, theobserved transition to instability[l] as w drops below S2. is
not addressed in the present calculation. In order to treat
this problem rigorously, a physical mechanism to resolve theshear-Alfve'n resonance must be introduced into the model.
Despite these omissions, the results obtained here indicatethat basic consistency between theory and experiment has
already been achieved, and that RF stabilization is a promising
tool for enhancing the performance of future experiments.
4.

PHAEDRUS-B

Recently, Phaedrus was upgraded to Phaedrus-B, a five-cell
device, by elongating the machine 1.2 m and inserting 1 T chokecoils at the ends of the central cell (see Fig. 4 ) . Overall,magnetic fields were increased throughout the machine. In
addition, two double dual half-turn antennas were installed in
the central cell (at z = + 50 cm) and two dual half-turn
antennas were installed in each end cell (at z = ±300 cm andz = ±322 cm). The antennas have normally been operated with w
close to the local ion cyclotron frequency, with the exceptionof the off-midplane antennas in the end cells. Use of the newantenna set resulted in n as high as 8 x 10 cm" without end

-0.50

0
-1

0.00

0.50

1.00

ANTENNA PHASE (rad/»r)
-0.5

0

0.5

ANTENNA PHASE (rad/77)

FIG. 5. Langmuir probe ion saturation current fluctuations versus relative antenna phase.
MHD activity is a strong junction of antenna phasing at (a) to/fi, = 0.8 and (b) CD/QJ = 1.0.

plugging. T increased from 20 eV to > 60 eV. Excitation ofslow waves in the west end cell, at half the end cell Q, ,resulted in a significant reduction in the net west ion1end
loss current.
The central cell antennas (at +50 cm) allow control of thespectrum of the ICRF wavenumber and the azimuthal mode number,through the relative antenna phase. The dominant propagating
wave (with « > Q. ) is a magnetosonic £ = +1 fast wave. By
adjusting the relative phase of the two dual half-turn antennas(located at ±50 cm) it is possible to approximately cancel out
this mode (because the central cell antenna separation is 1 m,comparable to the central cell parallel wavelength of the 2. =+1 mode). As shown in Fig. 5, the relative density fluctuationlevel depends on the antenna phase and varies from an on-axis
value.as high as 60 percent to as low as 5 percent.
Antenna phasing affects stability because RF stabilization issensitive to the radial electric field profiles. The antennaphase difference determines the axial wavenumber and theazimuthal mode number spectrum of the antenna set. Thisaffects the radial profile of the RF fields through the plasma
response to the antenna spectrum. Computations of the RF
fields show that the average radial E_ field is most altered by
the antenna phasing. For <j> = 0.75 n in Fig. 5, the gradientsprovide a stabilizing contribution,while for <j> = -0.75 n thecontribution is approximately neutral. It should also be notedthat the contribution from side-band mode coupling is also
sensitive to the k spectrum.
When the dual"half-turn antennas at the midplane of eachend cell are excited, the central cell density (n ) can
increase by a factor of two. The axial density profile isshown in Fig. 6. The maximum end cell density resulting from

270

BREUN et al.

_ 4
n

'g 3
N

*o 2
~e 1

-

4

-

2

0
Z(m)

2

4

F/G. 6. Axial density profile in Phaedrus-B for operation with resonant end cell and central cell RF.
Only a single dual half-turn antenna at z = —50 cm is excited in the central cell.

RF trapping in the end cells is never more than 60 percent of
n . This is a feature of RF enhanced end cell plugging[8].
Also, the density in the thermal barrier region is as low as
0.5 x 10 1 2 cm"3 or approximately 0.1 n . Direct measurementsof the barrier and central cell plasma°potential indicate that
thermal barrier potentials are present in the choke coil end
cell transition cell. The barrier potential is measured to beapproximately 80 V more negative than the central cell value of
110 V. The reduction in density in the thermal barrier (shownin Fig. 6) is then consistent with a Boltzmann relation,i.e.
eA<f> ~ T ln(n /n ), where A<f> is the change in plasma potential
and n is the barrier density.
Studies of the trapping and pumping mechanisms occurringin the thermal barrier cell indicate that RF trapping is thelargest contributor to barrier fueling. Barrier fueling by
neutrals is not found to be as large. The total neutralpressure in the region is low (< 1 x 10~ 6 Torr), and an order ofmagnitude increase in hydrogen pressure, produced by local gas
puffing, only doubles the local plasma density. In addition,
the increased rate of fueling of the trapped ion population
with the addition of gas into the thermal barrier is found tobe accompanied by an increase in barrier limiter current. This
indicates that a passive pumping mechanism, which produces aradial loss of locally trapped ions, is present. The thermal
barrier cell in Phaedrus-B is circular at one end andelliptical at the other, so it is a non-omnigenous mirror cellhaving trapped ion drift surfaces which differ from themagnetic flux surfaces. Transport associated with
non-omnigenous cells has been proposed as a passive pumping
technique to pump thermal barriers[9].
Good plugging of central cell end loss has been observedon operation of a second RF ("beach") antenna in the west end
cell (z = 300) at approximately half the end cell midplane 9..
An ion cyclotron resonance at this frequency exists only near
the end of the central cell, at R = 2. Slow waves launched by

271

IAEA-CN-47/C-I-4-1

Cent, cell midplane diamagnetism

Cent, cell line density

1

Off

1 '

1 '

1

'

_

,

I

Beach RF On

H

•

i

•

1

-Off

j

.

/

-

1

20

1

(b)

Plug RF On
.

6

8

10

12

14

IS

19

I

.

I

.

20

Time Cms)

I

1 .

I

10

.

I

.

I

12

14

. .1.

IS

19

20

Time (ms)FIG. 7. Symmetric use of the beach antennas produces large increases in(a) central cell density and (b) central cell midplane diamagnetism.
Peak density in (a) is (6-7) x 1012 cm"3. Peak beta in (b) is 1.5%.

the low frequency antenna propagate through the thermal barrier
region to the resonance where the wave is strongly damped.
Symmetric use of such antennas can produce a 300 percent
increase in central cell density, with a 250 percent increasein diamagnetism, for a 40 percent increase in total ICRF power(see Fig. 7 ) .

ACKNOWLEDGEMENTS
This work was supported by the United States Department of Energy, underContracts Nos DE-AC02-78ET51015 and DE-AC03-76ET53057.

REFERENCES[1] FERRON, J.R., et al., Phys. Rev. Lett. 51 (1983) 1955.
[2] HERSHKOWITZ, N., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984
(Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 265.
[3] PHAEDRUS GROUP, Nucl. Fusion 25 (1985) 1213.
[4] FERRON, J.R., et al., Interchange stabilization of an axisymmetric single cell mirror using highfrequency electric fields, to be published in Phys. Fluids.
[5] MYRA, J.R., D'IPPOLITO, D.A., Nucl. Fusion 26 (1986) 361.
[6] D'IPPOLITO, D.A., MYRA, J.R., Phys. Fluids 29 (1986) 2594.
[7] McVEY, B., Rep. PFC RR-84-12, Massachusetts Inst. of Tech., Cambridge (1984).
[8] HERSHKOWITZ, N., et al., Phys. Rev. Lett. 55 9 (1985) 947.
[9] KESNER, J., Comments Plasma Phys. Controll. Fusion 5 (1979) 123.

IAEA-CN-47/C-I-4-2

AXISYMMETRIC MIRROR PLASMAPRODUCED, HEATED AND STABILIZEDBY ICRF ONLY
R. ITATANI, H. TAKENO, Y. YASAKA
Department of Electronics,
Kyoto University,
Kyoto, Japan
Abstract
AXISYMMETRIC MIRROR PLASMA PRODUCED, HEATED AND STABILIZED BY ICRF
ONLY.
The HIEI device is an axisymmetric single mirror machine in which radiofrequency (RF) stabilization and heating have been investigated. It is shown that the MHD stable plasma can be started up
and sustained by selecting the azimuthal mode of the applied ICRF. The mechanism of stabilization is
identified as the RF ponderomotive force due to the radial gradient of the RF field. With a combinationof RF production and stabilization by the m = ± 2 RF mode and heating by the m = ± 1 mode, a stable
plasma with ion temperatures above 100 eV can be produced and sustained.

1.

INTRODUCTION

Non-axisymmetric MHD anchors have been predicted to give rise to resonantradial diffusion. Radiofrequency (RF) stabilization of MHD instabilities offers thepossibility of operating a tandem mirror in a purely axisymmetric mode, i.e. with asimpler and more compact configuration. The HIEI device is an axisymmetric single
mirror machine in which the effectiveness of RF stabilization has been demonstrated[1-4]. This paper describes a method by which high temperature plasma is started
up, heated, stabilized and sustained by ICRF only, with RF mode control. The
mechanism of RF stabilization is also discussed.
2.

RF PLASMA PRODUCTION AND STABILIZATIONThe HIEI device has a mirror length of 1.2 m, a maximum magnetic field BM
of 1.1 T, a mirror ratio R of 1.4 to 4, and a plasma radius of =4.5 cm. Two RF
antenna systems are installed; one system is located at the midplane, for plasma
production and stabilization, and the other is placed at one of the throats, for heating.
Each antenna consists of four elements aligned axially with 90° spacing in theazimuthal direction. By adjusting the phase and amplitude of the RF current in eachantenna element, the RF fields of the m = 0, + 1 , - 1 , ±1 and ±2 azimuthal modes
can be selectively established.
273

274

ITATANI et al.
I.U

X
UJ
>
UJ

0.8

0

m =± ,
m=• 2

X
XX

0.6

X

o
c
<

o

0.4

X

—XX'"

X

X "Xv
\

V °

0.2
o
I

cjj-^.—
1

1

10

15
RF

20

25

(arb.)

FIG. 1. Fluctuation level of the flute instabilities as a Junction of the squared RF current in the midplane antenna for the m = ±2 (circles) and m — ±1 (crosses) RF modes. R = 2.3, the radius of themagnetic field line curvature Rc = 4.3 m, and a>/un- = 2.3.

The plasma is produced by hydrogen gas puffing from a gas box; an RF pulse
from the midplane antenna follows. In order to eliminate the line tying effect, nostartup gun is used. The end-free plasma is sustained for the duration of the RF pulseof up to 10 ms. Figure 1 shows the fluctuation level of the plasma versus the midplane antenna current IRF for the m = ±1 and m = ±2 RF modes. The ratio ofradiofrequency to the local ion cyclotron frequency is w/wd = 2.3-2.6. For low
IRF the fluctuation level is nearly the same in both RF modes. With a correlation
measurement, the fluctuation has been identified as the flute mode, with a frequencyof 6-10 kHz, and the azimuthal mode m = 1. For higher IRF the flute mode is further destabilized when the IRF of the m = ±1 mode increases. In some shots theplasma density decreases to near zero, even during the RF pulse. In contrast, whenthe IRF of the m = ±2 mode is increased, the flute instability can be stabilized andthe plasma with a fluctuation level of less than 10% can be sustained for the whole
duration of the RF pulse. The plasma density for the m = ±2 RF mode isn = 1.2 x 1012 cm" 3 at r = 2 cm, which is twice as large as that for the m = ±1
RF mode. Since gas ionization is mainly caused by the axial electric field of theantenna near-field, the plasma production rate is insensitive to the RF mode number
m. This is supported by the fact that with low values of IRF both the fluctuation level
and the plasma density are almost the same for the m = ±1 and m = ±2 RF modes.
From a measurement of the profile of the RF magnetic field, it is found that the radial
gradient of the RF field for the m = ±2 mode is much larger than that for them = ± 1 mode. The large difference in stability for the two RF modes observed forhigher IRF is attributed to the difference in the radial gradient scale length lB of theperpendicular RF magnetic field B±.

275

IAEA-CN-47/C-I-4-2
1.2

mN 0
^ 3

-*—"*

"*----—— 4
2

^-

4
r(cm)

n

L—l
-0.1

0

0.1

Oi
1/I B

03 0.4(I/cm}F / C 2. Measured radial profiles of the perpendicular (Bx) and axial (BJ components of RF magnetic
fields for (a) the m = ±2 RF mode and (b) the m = ±1 RFmode. (c) Fluctation level of the flute instability versus the reciprocal of the radial gradient scale length lB of B±.

The midplane antenna is then used to generate a mixture of the m = ±1 andm = ±2 RF modes. The change in the ratio of the modes results in a change in 4The measured radial profiles of B ± and Bz are shown in Fig. 2, with (a) them = ±2 RF mode dominant and (b) the m = ±1 RF mode dominant. The fluctuation level of the flute instability is plotted in (c) as a function of the reciprocal of (Bfor B x = 5 G. The values of 4 corresponding to (a) and (b) are indicated byarrows. For lower values of i£l, the m = ±1 mode component is larger than them = ±2 mode component, and vice versa for higher values of t£l. It can be seenthat the fluctuation level decreases smoothly with a shortening of lB.
The radial ponderomotive force for ions, due to linearly polarized RF fields andin adiabatic conditions, is roughly given by

F p = -(e 2 /2 m j ) E\/(E(1)

where mj is the ion mass, Ej. is the perpendicular RF electric field and 4 is theradial scale length of E±. For stabilization of the flute mode, it is required that thisforce oppose the effective gravity G, which is equal to e(Tj + T^/R,., with R<. being
the radius of the field line curvature. Since the functional dependences of Br and
Bs on r are, respectively, the same as those of Ee and Er, provided Ez is small,

276

ITATANI et al.

FIG. 3. Radial density profiles for different lB values of RF fields
corresponding to points (a) and (b) in Fig. 2.

we can use the value of 4 as the radial scale length of E ± (4 = (B). For
4"1 £ 0.15 cm" 1 in Fig. 2(c), Fp/G a 2.1 is obtained from Eq. (1). Thus, theresult presented in Fig. 2(c) is in good agreement with Eq. (1). In previous experiments [2, 4] it was observed that the fluctuation level of the flute mode decreased
when B x was increased. These results indicate that the stabilization mechanism canbe attributed to the RF ponderomotive force for ions.
The ponderomotive effect on electrons is negligible because E ± is estimated to
be almost linearly polarized and the electrons are collisional. Side-band coupling
effects [5] are also calculated to be small compared with the ponderomotive effectbecause w is far from wci and the plasma radius is small enough for side-band waves
to be cut off [6].
Figure 3 shows a comparison of the density profiles for different 4 values corresponding to points (a) and (b) in Fig. 2. The density profile of curve (a) in Fig. 3
is almost flat over the radius of the gas box (made of insulator material). When theflute fluctuation level is high, the density decreases rapidly towards the outside, as
shown by curve (b). In the steady state, the ionization current I; is equal to I x + Ig
(radial and parallel total loss current), provided non-ambipolar loss is negligible [7].
The particle confinement time is TP = eN/Ij, with TP~' = T^ 1 + T{\ where
T ± = eN/I x , 7« = eN/In, and N is the total particle number. The value of Ij is considered to be almost the same for cases (a) and (b) in Fig. 3 because the production
rate is insensitive to the RF mode number, as discussed before. The value of N over
the radius of the gas box for (a) is 1.9 times larger than that for (b). This means that
TP for (a) is larger than rp for (b) by a factor of two. End-loss measurements of Inshow that T| for (a) is smaller than TB for (b). Therefore, the increase of TP for (a)should be due to the improvement of T X . We see that the RF mode with a larger
radial gradient stabilizes the flute instability, which results in an improvement ofradial confinement.

277

IAEA-CN-47/C-I-4-2(b)production
healingTime (ms)(c)
heating2
24Time (ms)-2

FIG. 4. Results of slow wave heating, (a) Perpendicular ion temperature at r = 2 cm as a junctionof net RF power of the m = ±1 mode from the throat antenna, (b) Time evolution of the ion saturation
current to a Langmuir probe at r = 3 cm. (c) AC components of the probe current for the midplaneantenna currents of 58 A (upper trace) and 67 A (lower trace).

3.

HEATING OF RF PRODUCED PLASMA

We performed additional ion heating with the throat antenna which excited slow
waves o f t h e m = + l , m = - l o r m = ±1 modes. The throat antenna was energized 2 ms after switching on the midplane antenna. The plasma produced by them = ± 2 RF mode from the midplane antenna was stable and had parameters ofn = 0.5 X 1012 cm" 3 , Te = 17 eV, T i x = 20 eV and Tj, = 10 eV. T i± and TU|
were measured with a movable multigrid energy analyser of 6 mm dia and an endloss analyser, respectively. The value of w/wci varied from 1.0 to 0.7 along thethroat antenna element. Figure 4(a) shows T i± at r = 2 cm versus the coupled RF
power from the throat antenna. The value of T i± increases from 20 eV to about
120 eV; Ta averaged over the radius is about 80 eV. At these temperatures the
charge exchange time is estimated to be about 20 /is because of the small volume andthe low pumping speed of HIEI. This results in an apparent saturation of T i ± .
Calculations of power balance equations combined with neutral pressure balance
equations are consistent with the results presented in Fig. 4(a).
During the application of heating, the fluctuation level is sometimes increased,
as shown in Fig. 4(b). It is likely that the flute mode becomes unstable owing to theincrease of Tj. This may occur when the effective gravity, increased by ion heating,exceeds the stabilizing ponderomotive force created by the m = ±2 RF at the midplane. In fact, this fluctuation is stabilized when the RF current in the midplane

278

I T A T A N I et a l .

2OxKT3

FIG. 5. Extension of the empirical scaling law. Previous results are indicated by black circles and theresults for higher Tt values by whitecircles.

antenna is increased. This is demonstrated in Fig. 4(c), where IRP for the lower
trace is larger than IRF for the upper trace.
As found previously with an empirical scaling law, the IRF required for flute
stabilization is proportional to n a 3 ± 0 0 5 G°- 4±01 for plasmas with T, up to 25 eV [4].
The scaling is shown by the straight line in Fig. 5, together with previous experimental points (black circles). The results for plasmas with higher T: (white circles) lie
on this line, showing the extension of the scaling law for Tj as high as 100 eV.

4.

CONCLUSIONSIn the axisymmetric mirror HIEI it has been shown that the MHD stable plasma
can be started up and sustained by selecting the azimuthal mode number of the applied
ICRF. The mechanism of stabilization is identified as the RF ponderomotive force
due to the radial gradient of the RF field.
With a combination of RF production and stabilization by the midplane antennaand heating by the throat antenna, a plasma with T| £: 100 eV has been started up,stabilized, heated and sustained by ICRF only.

REFERENCES[1]

YASAKA, Y., et al., in Plasma Physics (Proc. Int. Conf. Nagoya, 1980), Vol. 1, Fusion
Research Association of Japan, Nagoya (1980) 327.
[2] INUTAKE, M., ITATANI, R., et al., in Plasma Physics and Controlled Nuclear Fusion
Research 1982 (Proc. 9th Int. Conf. Baltimore, 1982), Vol. 1, IAEA, Vienna (1983) 545.

IAEA-CN-47/C-I-4-2

279[3] OKAMURA, S., ITATANI, R., et al., in Plasma Physics and Controlled Nuclear Fusion
Reserach 1984 (Proc. 10th Int. Conf. London, 1984), Vol. 1, IAEA, Vienna (1985) 337.
[4] YASAKA, Y., ITATANI, R., Nucl. Fusion 24 (1984) 445; 25 (1985) 29.
[5] D'IPPOLITO, D., MYRA, J., Rep. SAIC-85/1922, Science Applications, Inc., Pleasanton, CA(1986).
[6] YASAKA, Y., ITATANI, R., Phys. Rev. Lett. 56 (1986) 2811.
[7] INUTAKE, M., et al., Phys. Rev. Lett. 55 (1985) 939.

DISCUSSION
ON PAPERS IAEA-CN-47/C-I-4-1 AND C-I-4-2

T. KAWABE: I think your work on the stabilization of the flute mode by RF
ponderomotive forces is important for the creation of a neutron source by burning
D-T plasma confined in an axisymmetric mirror. At present, the plasma temperature
is of the order of 100 eV. What action should we take, in your opinion, to acquire
more confidence in the use of this method for fusion conditions as in the Fusion
Engineering Facility?
R. ITATANI: The scaling law we derived shows that the required RF current
is proportional to approximately nOi3(Tn/R)°i4. We have already performed acheck-up to 4 x 1014 cm""3 in plasma density. However, the effective gravity(which is proportional to TJ/R) should be increased so as to obtain a confirmation of
the scaling law. The radius of curvature of the magnetic field lines is already of thesame order as that in the FEF. It is necessary to test the scaling law, using plasmasof higher temperature, as high as 1 keV or 10 keV. Also, studies of the energy flowof applied RF should be performed so as to enable a proper choice of the RF mode
to be made.

281

IAEA-CN-47/C-II-l

A TANDEM MIRROR MODELLING CODE*
W.M. NEVINS, B.M. BOGHOSIAN, R.H. COHEN,
W.F. CUMMINS, P.F. DUBOIS, A. FRIEDMAN,
L.L. LoDESTRO, Y. MATSUDA, L.D. PEARLSTEIN,
G.D. PORTER, M.E. RENSINK, T.D. ROGNLIEN,
G.R. SMITH, J.J. STEWART
Lawrence Livermore National Laboratory,
University of California,
Livermore, California
M. PHILLIPS
Grumman Aerospace Corp.,
Princeton, New Jersey
United States of America
Abstract
A TANDEM MIRROR MODELLING CODE.
The paper describes a computer code that models the time evolution of tandem mirror plasmaand power deposition by radiofrequency (rf) fields within the plasma. Rate equations for seven groupsof particles are integrated in time. The effects of axial variations in the potential are included. An analytic model for rf-heated electron populations is described. Code predictions are compared with datafrom the TMX-U experiment. Power deposition profiles for electron cyclotron resonant heating in
TMX-U are shown. A full-wave solution for ion cyclotron heating fields is displayed, and the ponderomotive forces are evaluated.

1. I N T R O D U C T I O NThis paper describes comprehensive modeling of tandem mirrors, focusing on
our code MERTH (Mirror Equilibrium Radial Transport and Heating). MERTH
includes physics packages together with a command language interpreter thatprovides a convenient user interface. The MERTH physics packages describedhere include a transport package, a ray-tracing package, and an ion cyclotron
heating package. MERTH also contains a mapping package that provides selfconsistent axial profiles of density and temperature, and a MHD equilibrium
package that describes the magnetic field geometry. MERTH physics packages
can easily exchange information and use consistent models of the magnetic field
and the axial and radial profiles of density and temperature.

Work performed under the auspices of the United States Department of Energy,
under Contract No. W-7405-ENG-48.

283

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NEVINS et al.

2. RATE EQUATIONS
The transport package in MERTH, called TREQ (Tandem mirror Rate
EQuations), solves for the potential in five axial regions (central cell, inboard
mirror, thermal barrier, plug, and end wall) and solves rate equations for sevengroups of particles (thermal electrons and ions, central-cell and end-cell beamions, barrier-trapped ions, magnetically trapped electrons in the end cell, andplug-trapped electrons). For each group of particles in a thermal barrier tandem
mirror, we describe particle and energy balance with a system of zero-dimensional
rate equations. Axial coupling exists between particle groups because the bounce
orbits of two or more particle groups may pass through any one axial region.
The axial profiles of the particle and energy density for thermal electrons andions depend on the potential in each axial region. Rate equations for the thermal
electrons and ions that employ these self-consistent axial profiles are formulated
in section 2.1. An analytic model of the effect of electron cyclotron resonant
heating (ECRH) on the electrons in the plug cell of a thermal barrier tandem
mirror is described in section 2.2.
Calculations with this transport package indicate a strong coupling of theplasma densities in the various axial regions of the device. In section 2.3 this coupling is shown to be in qualitative agreement with Tandem Mirror ExperimentUpgrade (TMX-U) observations.

2.1. Thermal Particle Rate Equations
On the most fundamental level, particles in a tandem mirror obey a bounceaveraged Fokker-Planck equation. To reduce this equation for each group ofparticles into a system of rate equations for the number of particles, Ngt and thetotal energy, Eg ,we assume a functional form for the distribution of each particle
group that involves two parameters, and take the particle and energy moments ofthe bounce-averaged kinetic equation. For example, we assume that the thermal
central-cell electrons have a nearly Maxwellian distribution in e and /i,where ffoisM ' s t n e ' o s s boundary for the thermal central-cell electrons. The
two parameters used to fit the total particle and energy content of this group ofparticles are then nl°' and Te. A third parameter, ac, which measures the widthof the boundary layer near the loss cone over which the distribution function
departs significantly from a Maxwellian, is held fixed with a value of order unity.
The thermal barrier tandem mirror is modeled as a sequence of connectedsquare wells. The particle and energy content of group g may then be expressedin terms of n j ' ' and Tq as

IAEA-CN-47/C-II-l

285
(2.2)

where j labels regions, Lj is the length of the j t l h region, Bj is the magnetic induction in t h e j " 1 region. The symbol {<£} is used to indicate that a}9 {qg{4>}/Tg)
and c\s\qg{4>}/Tg) depend on the values of the potential <j> in all regions. For particle groups with nearly Maxwellian distributions, the functions af {qg{<f>}/Tg)
and Cj(qg{<f>}/Tg)—which involve moments of the distribution function over a
known region of phase space—may be expressed in terms of error functions andDawson's integral [I].
The particle and energy moments of the kinetic equation yield equations that
describe the rate of change for the number and energy content of each group ofparticles,
N9=

Ng

(s)
1gno
9

with a similar equation for the rate of change of the energy. The final term in
Eq. (2.3) describes the transfer of particles from one group to another becauseof the motion of separatrices between groups of particles. The term Ng\cl represents sources and sinks of particles that are due to classical processes such as
ionization, charge exchange, collisional coupling between groups of particles, andradial transport. Radial transport is modeled by a phenomenological loss rate
proportional to the central-cell potential.
Evaluating the source/sink term in Eq. (2.3), Ng\ct, requires knowing thedensity of each group of particles in each axial region. These densities may be
expressed in terms of nig\ Tg, and {</>} by taking the appropriate moment of the
distribution function defined in Eq. (2.1), Equation (2.3) for Ng is integrated
in time by the TREQ package, while the parameters n[g) and Tg, as well asg,the potentials {<£}, are obtained by enforcing Eqs. (2.2), and quasineutrality ineach region as constraints at each time level. The entire set is solved using thedifferential-algebraic equation package DASSL [2].
2.2 Hot-Electron Physics
Electron cyclotron resonant heating (ECRH) is used in the plug cell of athermal barrier tandem mirror to heat electrons, and to assist in the formationof the barrier and plugging potentials. ECRH is applied at both the plug andbarrier locations. The ECRH is typically strong enough to render the electron
distribution function far from Maxwellian.
We have developed approximate forms for the hot-electron distribution function in the strong-ECRH limit by assuming that the coefficients of collisional andrf diffusion are slowly varying. For fundamental heating at the plug.with weaker

286

NEVINS et al.

second-harmonic heating at the barrier, the non-relativistic Fokker-Planck equation for plug electrons can be written approximately in the form
d df
d - df—v—- H
Dp— = 0dx dx
dydy
where x and y are the parallel energies at the plug and the inner mirror, respectively, normalized to the central-cell electron temperature Tt\ Dp(x, y) is the diffusion coefficient for plug ECRH in these variables; and v{x,y) is a sum of collisionaland barrier-ECRH diffusion coefficients. The separatrix dividing passing frommagnetically trapped electrons is a portion of the line y — 0; in the limit where
passing (from the central cell) electrons are constrained to be Maxwellian (longsolenoid approximation), / must vary as exp(-px), where p = Rmp/(Rmp - 1)
on that separatrix. Here, Rmp is the mirror ratio from the inner mirror to theplug. One may verify a posteriori or observe from relativistic Fokker-Planck code
results that / varies most rapidly near the intersection of the passing, magnetically trapped, and potential-trapped regions, over an energy scale that is shortcompared with the variation of Dp and v. Treating vjDp as slowly varying andrescaling variables to X = p(x- XQ) and Y — p{i'/Dp)x^i[y — yo)t where XQ and yo
are the coordinates of the aforementioned intersection, reduces the Fokker-Planck
equation locally to Laplace's equation in X and Y. Extending the domain to all
X,Y and requiring that / be finite at infinity (validity requires that the hotelectron flux into the vj. axis in the plug is small compared with its maximum
value on the passing separatrix), we find / ex lm{Z\(X + iY)1/2}}, where Z is theplasma dispersion function. A similar result is obtained for electrons confined tothe barrier region. Here the boundary condition on the line vy = 0 at the plug
is obtained from the plug solution. The overall solution compares well to resultsfrom the bounce-averaged Fokker-Planck code SMOKE as shown in Fig. 1, wherelevel surfaces of the distribution function are plotted. We are developing a model
for TREQ based on this analytic solution.
The remainder of this section describes a number of features common to
the analytic and Fokker-Planck solutions that are qualitatively incorporated in a
cruder end-cell electron model that is now operational in TREQ. The combinationof rf diffusion and collisions provides an efficient means of transport from low tohigh parallel velocity for passing electrons. This process leads to heating ofcentral-cell electrons, and to enhanced electron end loss and thus high overallmachine potentials. The enhancement of end loss over the purely collisional level(from the entire machine) is modest when the ECRH is weak enough, so thepassing electrons are Maxwellian, but can become substantial when the ECRH isstrong enough to dominate over collisions for passing electrons. In the latter case,for a fixed potential profile, the hot-electron end-loss rate is limited by the rate
at which barrier electrons are replenished by collisional scattering of central-cellelectrons into the passing region. Thus TREQ and Fokker-Planck calculations
both show that as the ECRH level is increased, the overall electron end-loss ratevaries from the collisional (Pastukhov [3]) rate for climbing the potential hill tothe end wall, to the collisional rate for merely climbing the solenoid-to-barrier
potential.

287

IAEA-CN-47/C-n-lv (10 cm/sec)/mo(1O cm/sec)

FIG. 1. Comparison of (a) analytic and (b) Fokker-Planck distributions at the plug (no freeparameters). Inserts are enlargements of the region near the passing-trapped separatrix.

ECRH heating provides a large inventory of hot electrons, the size of which
depends on the strength of the plug and barrier ECRH; hence, TREQ calculates
barrier depths and plugging potentials that are sensitive to the ECRH levels.
The distribution function contains two potential drives for instability. The
first arises because a distribution function, which is monotonically decreasing up
the v± axis in the barrier, maps in the plug to a boundary condition that isinverted (increasing with ii||) on the separatrix bounding the potential-trapped
region. For strong plug ECRH, the plug distribution then has a peak near the
maximum t>|| point on the separatrix. This distribution yields a density that canbe a nonmonotone function of the plug-to-barrier potential. This nonmonotonic
density can result in multiple roots for the potential, leading to root jumping andhysteresis in TREQ solutions as other parameters (e.g., ECRH power, density)vary. This behavior is suggestive of the loss of plugging observed in the TMXU experiment. We also speculate that this bump-on-tail distribution can be
micro-unstable. An option in the TREQ model that fills in the plug distribution,
simulating but overestimating the effect of quasilinear diffusion, leads to a lower
plugging potential. However, the potential can still be a locally sensitive functionof parameters.
The second possible instability drive comes from the increasing of the distribution function with vj. at finite vy in the barrier. Instability can arise because of
the inverted energy population. For these cases, the absorption of power from thebarrier ECRH is found to be negative in both TREQ and Fokker-Planck studies.
2.3. Comparison of TREQ modeling with TMX-U
An important feature of the transport package within MERTH is the ability
to model the coupling of the plasma densities in various axial regions of a thermalbarrier tandem mirror. The ionization current that is injected in the central cell

288

NEV1NS et al.

n

1.0

I

(a)

i

o
CM

1.0 -

i
•D

/
0.5 —

/

/

n

/A
-

A

I

0

0.2

0.4

0.6

|

0

Central-eel ldensity(10 12 cm" 3 )55

Central-cell density (10 12 cm"3)

0.2

0.4

0.6

FIG. 2. Variation of (a) the total density at the midplane of the barrier, and (b) the sloshing ion density
at the turning point with the central-cell density. The triangles are data from the TMX-U experimentin which the central-cell density was varied by varying the fueling rate, and the lines are the result ofa TREQ simulation of these experiments. These experiments were done under conditions of good axial
confinement, so the end-loss currents were small compared with other loss mechanisms.

of a tandem mirror creates a plasma in that cell. Some of these ions, the passing
ions, lie in the region of phase space that permits their transit into the anchor cellsof the device. Scattering of these passing ions produces a warm ion populationthat is trapped in the thermal barrier. In turn, these cold locally trapped ions area target for the neutral beams that provide the sloshing ion population. Hencethese three ion populations—central cell, barrier-trapped, and sloshing—are all
coupled. Figure 2(a) shows the dependence of the total density at the thermalbarrier midplane; Fig. 2(b) shows the dependence of the sloshing-ion density
on the central-cell density. The solid triangles are data obtained in TMX-U byvarying the central-cell gas-fueling rate and keeping all other machine parameters
fixed. The solid lines are the results of the TREQ modeling of these experiments.
Good agreement exists between the experimental results and the model. The
data presented in Fig. 2 were obtained in the strong plugging mode in which theaxial ion losses were small compared with the radial losses. Hence, the results
are insensitive to the plugging model, i.e., the anchor electron model.
3. E L E C T R O N C Y C L O T R O N H E A T I N G
Propagation and absorption of waves in the electron-cyclotron frequency
range are modeled by tracing bundles of rays and calculating the power depositedin the plasma. We describe here some features of our computational model, andwe show results for one of the microwave-launching systems in TMX-U.
The equilibrium magnetic field is derived from a vacuum field that is calculated from a realistic coil set, together with a paraxial, low-/? approximation ofplasma diamagnetism. Inclusion of diamagnetism leads to cyclotron resonanceshifts that affect both ray trajectories and power absorption. The electron den-

289

IAEA-CN-47/C-II-l
30

Cyclotron
resonance

I

Horn\ A \\:*-' >' •'• •''10
Radial position, r (cm)

280

290

300

310

320

20

330

Axial position, z (cm)FIG. 3. Selected rays, launched by an 18 GHz horn in TMX-U. This zx-plane shows projections of 13
rays and cross-sections of both the plasma (bounded by vacuum) and the cyclotron resonance surface.
The numbers 1, 2, 3, 4 and 5 along the rays indicate, respectively, where 20, 40, 60, 80 and 99% of
the power has been absorbed. The peak values of density and temperature (on the magnetic axis
x = y = 0) aren = 1012 cm'3and te = 0.1 keV.

sity and temperature are assumed to depend only on the radial magnetic flux
coordinate and are therefore uniform along field lines and around the azimuth ofthe elliptical flux surfaces.
The power absorption calculation assumes a Maxwellian electron distribution and yields a power deposition profile that is a function of both the radial
and azimuthal flux coordinates. We can thus assess the azimuthal asymmetryof the deposition that is due to launching microwaves toward one side of theplasma. Averaging the deposition profile over azimuth, which presumably occurs
in experiments after an E x B rotation time, yields a radial deposition profile for
one-dimensional calculations of axial and radial transport.
We report here the ray trajectories and radial deposition profile for the 18GHz heating system in the central cell of TMX-U. We model one of the fourhorns included in the system. The reported values of power deposition are proportional to the 10 kW of incident power that we assume enters the plasma inthe extraordinary mode.
Figure 3 shows a view of the ray bundle impinging on the plasma and being
absorbed. Only 13 of the 121 rays are shown, the central ray and 12 rays withthe maximum divergence angle of 20°. Note the strong refraction that tends
to trap the power and guide it down the field until absorption occurs near the

290

NEVINS et al.

Antenna

2

0

-

2

-6

Axial position z (m)Antenna

IE.I-6
Axial position z (m)FIG. 4. Contours of the absolute values of the electric field strength in the E+ and E_ components in
TMX-U, when the east antenna is excited. Contour levels have linear spacing and are separated by
1.2 Vim (for \E+\)and0.6
Vim (for \E_\) per ampere of total current in the two drive elements. Notethe greatly enlarged scales used on the radial (vertical) axes compared with the axial (horizontal) axes.
The location of the antenna current elements is indicated. The dashed curve is a contour of constant plasma density at 0.1 A. We model only the central cell (between the inner mirrors at z = ±4 m)
and the east end cell (—7m<z<—4 m). Note that z increases from right to left.

IAEA-CN-47/C-II-l

291

cyclotron resonance. The inset shows the corresponding radial deposition profile.
The deposition near radial position r = 0 is four times that given by a uniformdeposition estimate, because the horn illuminates only a narrow swath near they = 0 plane. Integrating over r, we find that over 80% of the incident power isabsorbed in the plasma and that over 30% is absorbed in the core (r < 10 cm).

4. I O N C Y C L O T R O N H E A T I N G
Propagation and absorption of applied ion-cyclotron waves are investigatedby calculating the global wave structure of ion-cyclotron frequency fields in axisymmetric mirrors. Simpler, eikonal methods are not applicable to the mirror
machines of interest because wavelengths in the plasma are comparable to density and magnetic-field scale lengths. We use the GARFIELD code to solve forthe electromagnetic field in an axisymmetric region containing a plasma andsurrounded by conducting walls. The plasma is described by the cold-plasma
dispersion relation with a small collision term to simulate absorption. Further
details on GARFIELD can be found in Ref. [4].
We apply GARFIELD to the TMX-U experiment. The axial magnetic field
profile generated by the actual coil set is modeled by a number of filamentary
current loops of appropriate radii and axial locations. Radial density profiles
have exponential forms suggested by measurements, and axial ion density profiles model a mirror-trapped population plus a (constant) passing population.
Antennas are modeled by one or more azimuthal current loops. For instance,
opposed half-turn loop antennas with a shield are simulated by two drive currents for the main straps and six image-current elements representing the shield.
These latter currents are estimated using a finite-element antenna code.
Calculated contours of the left and right circularly polarized components ofthe electromagnetic field, |J5 + | and \E-\ respectively, are shown in Fig. 4 for the
TMX-U central cell and the east end cell,with excitation of the east antenna at
w — 1.03 fi.o, where Cli0 = Q,-(« = 0). The cyclotron frequency fi; at the antenna
is such that w < ft,-. The peak density is n = 2 x 1012 cm" 3 . Of the spectrum of
azimuthal modes excited by the antennas, only m = - 1 for E+ and m = +1 forE- are large in the plasma core near r = 0.
In TMX-U, the E+ component propagates where Q{ > a> and is absorbed at
the cyclotron resonance near z = — 1 m. Reflection of this component from theinner mirror causes a standing-wave pattern between the antenna and mirror.
The £L component is not absorbed by ions, but it is nevertheless absent forz > - I r a because it is cut off there; however, at higher density, n = 5 x l 0 1 2 cm" 3 ,this component propagates into the other half of the central cell. The wave fields
external to the plasma are probably related to modes discussed by Paoloni [5].
Those fields, rather than the fields in the plasma, excite standing waves in theend cell, which might heat electrons by transit-time pumping or cause anomalous
ion diffusion.
Ponderomotive forces of sufficient magnitude and appropriate spatial variation can affect plasma stability. Figure 5 shows the radial variation of these forces

292

NEVINS et al.
1

1

*
-

nes)

0.4 -

•o

•

itive force 1

b 0.2

tfs •/

a)
|
a.

\

)

••

Destabilizing^=-%
Stabilizing-0.2

Antenna10

2040Radius mapped to z = 0 (cm)60

FIG. 5. Magnitude and radial variation of the ponderomotive force in the TMX-U central cell, witheast and west antennas each excited at oi = 1.5 fi, 0 with a 2 kA current. The force is plotted against
a normalized radius rfBfy/BfO)]1*. The three axial locations are at the midplane (z = 0), whereuA)j = 1.5 (solid curve); at the west antenna (z = 1.1 m), where OJ/D, = 1.385 (dashed curve); andat the east antenna (z = -1.85 m), where wAl,- = 1.21 (dotted curve).

at three axial locations in the central cell of TMX-U. The east and west antennas(located at z — —1.85 m and 1.1 m) were both driven at w = 1.5f2io with a current of 2 kA. Differences in the radial force profiles at the two antenna locations
are due to the different values of w/fi, at those locations. As expected, the forces
are generally negative, i. e. stabilizing, inside the antenna current loops.

REFERENCES
[1]
[2]
[3]
[4]
[5]

GAUTSCHI, M., in Handbook of Mathematical Functions (ABRAMOWITZ, M.,
STEGUN, I.A., Eds), U.S. Government Printing Office, Washington, D.C. (1972) 295.
PETZOLD, L.R., in Scientific Computing (Proc. 10th World Congress on Systems Simulationand Scientific Computing, 1982), North Holland, Amsterdam (1982) 65.
PASTUKHOV, V.P., Nucl. Fusion 14 (1974) 3.
PHILLIPS, M.W., TODD, A.M.M., Comput. Phys. Commun. 40 (1986) 65.
PAOLONI, F.J., Phys. Fluids 18 (1975) 640.

IAEA-CN-47/C-II-2

DRIFT PUMPING OFTANDEM MIRROR THERMAL BARRIERS*
D.E. BALDWIN, J.A. BYERS, Y.J. CHEN, T.B. KAISER
Lawrence Livermore National Laboratory,
University of California,
Livermore, California,
United States of America
Abstract
DRIFT PUMPING OF TANDEM MIRROR THERMAL BARRIERS.
Drift pumping is a means of selectively removing low energy ions from the thermal barrier ofa tandem mirror by induced radial transport. The paper determines the conditions for, and the propertiesof, transport induced by externally applied low frequency waves resonating with the trapped-ion
azimuthal drift frequency. Considerations that are discussed include conditions necessary forstochasticity, derivation of the local diffusion coefficient, and computation of the perturbed fields
throughout the plasma for a model based on TMX-U profiles with and without a throttle-coil geometry.

1. INTRODUCTION AND OVERVIEW
A thermal barrier in a tandem mirror is a negative dip in
potential between the central cell and the confining plugpotential [1], as illustrated in Fig. 1. The barrierenergetically separates plug electrons from central-cell thermal
electrons, permitting a higher plug temperature and potential
with a reduced plug-to-central-cell density ratio. The
potential profile is generated by the space charge ofmagnetically trapped, energetic ion and electron plug
populations. Maintenance of the profile requires control ofthese particle distributions and of the local thermal densities.
The thermal ion accumulation in the barrier must be
prevented. The process of removing thermal ions as they aretrapped by collisions and ionization is called "pumping" of thethermal barrier. There are two possible channels for pumping:heating out of the barrier, and radial transport to the surface.
Current experiments use a version of the first: i.e. charge
exchange on injected neutral beams converts barrier-trapped ionsto energetic untrapped ions or to magnetically trapped ions—a
process prohibitively inefficient at reactor energies. As analternative, we describe a method for inducing the radialtransport of barrier-trapped ions, using appropriate externally* Work performed under the auspices of the United States Department of Energy,
under Contract No. W-7405-ENG-48.

293

294

BALDWIN et al.

A B (z)

A

n

JV
Z

T

\

>

/ ii
/ ^^

*b
\
\

/

/ 1
\
\

7. Typical thermal-barrier-plug axial profiles of magnetic field B(z) and potential <j>(z).

applied radio-frequency (rf) fields. At the plasma surface, theions would then be lost by axial flow. To minimize enhancementof the ion barrier-trapping rate, we adopt drift frequencies
that preserve the cyclotron and bounce adiabatic invariants,suggesting the name "drift pumping ".
The idea of expulsion of a certain class of ions by waves
resonant with their drift frequency is reminiscent of theexpulsion of energetic beam-injected ions from a tokamak byfishbone oscillations [2]. The difference here is that the rf
fields responsible would be externally driven.
A suitable rf drift-pump field must satisfy several
constraints. The perturbed fields must cause particle
displacements off the density-level surfaces as the latterare convected by the moving flux. In the Lagrangian frame, thismotion requires perturbations having either parallel electricfields, magnetic compression, or line curvature. The motion
must either be irreversible for diffusion to occur, or must
convect ions to the surface. The former condition implies astochastic motion of trapped ions in the combined equilibrium
and rf fields; either motion sets restrictions on the applied
spectrum, dependent on the equilibrium. The pump fields musthave low azimuthal mode number and adequate penetration to
induce transport more or less uniformly across the column. Thisrequirement is fairly well satisfied, given the low frequencies
required and the short AlfveVi penetration time. Although drift
frequencies in the central cell will in general differ fromthose in the plug, the pump fields must not penetrate toostrongly into the plug cell in order not to excessively enhance
ion transport there. The degree of penetration depends in somedetail on the magnetic geometry, and this issue could be adetermining one in selecting one pump configuration overanother.

IAEA-CN-47/C-H-2

295

Drift pumping also offers the possibility of pumping
impurities—a process very difficult to accomplish with charge
exchange. To the degree that drift frequencies are determinedby E x B drifts, impurities would be as easily pumped as ions.
Were magnetic drifts to dominate, the impurity drift frequencies
would differ owing to their charge-to-mass ratios; for narrow
bandwidth, resonance with a fuel-ion pump would be lost. It
would not be practical to impose a separate set of frequencies
for each impurity. Therefore, the preferred situation forpumping impurities is one in which the drift frequency ofparticles trapped in the barrier region is dominated by electricfields.
To the extent that E x B drifts do dominate, thermal
electrons would be transported along with ions. This is inpotential conflict with the requirements of a gridless directconvertor in a reactor [3] for which the total electrontransport must be small. However, in the barrier itself, thelarge (>80?) hot electron density is dominated by magnetic
drifts having quite different frequency (and possibly sign).
These electrons would not transport in drift-pump fields
resonating with the E x B frequency, and the charge separationin the particles lost to the direct convertor is thereby
preserved.

2.

LINEARIZED PARTICLE DRIFT EQUATIONS

We choose a gauge such that the electric field can be
writtenE - - c"1 1 x B - Vx

(1)where the dot denotes differentiation with respect to time,t"B = 0, andB - V x (| x B)

(2)

In this gauge, because V-(5 x B ) * 0, x is generally not theelectrostatic potential. Linearization of Eqs (1 and 2) isdirect, with jj, x * I, X- I is the velocity frame in which <|i, 9
do not change, i.e. iji + £'Vt|> - 9 + £«V9 » 0.
With this choice of gauge, the adiabatic invariant forbounce motion of a particle of energy e and magnetic moment \i(per unit mass) is given by

ds[2(e - pB - 2 x ) ] 1 / 2

(3)

296

BALDWIN et al.

where s measures distance along a field line. J is conserved
for the frequencies assumed (below the bounce frequency),
leading to Hamiltonian equations for the particle flux
coordinates

at - - ~ IS
,6,J,t) is defined by the constancy in J in time-dependent
fields
1/2
1/2
J = j> ds {2[K(i|»,e,J,t) - V(i|j,e,s,t)]f» constant

(5)

where
V - yB(i|),e,s,t) + £ x (t|>,e,s,t)
is the effective potential for parallel bounce motion.
The fields can be linearized into equilibrium and small
time-varying components. The Hamiltonian can likewise belinearized by linearizing Eq. (5), holding J constant. We
generate a Lagrangian linearization, e.g. V -» V Q + V , where V
is related to its Eulerian counterpart V by
VL - V(6)by adding to the displacement £ a component parallel to b,

51| " J ds £•£<7)

where K - b«Vb is the equilibrium line curvature. This gives(8)

where x n - d> r^- and VM - [2(e - V.)]D

J V ||". A feature of the gaugeUand displacement chosen is that the parallel electric field isgiven by b«Vx- In particular, when E,, - 0, owing to high~"~Lparallel conductivity,
x is independent of s. We

IAEA-CN-47/C-H-2

297

subsequently discard the zero subscript on equilibrium
quantities. Equation (8) is equivalent to that obtained byAntonsen et al. [M] for particle motion in low-frequency waves.
For equilibria independent of 6, the linearized equationsof motion are
dip _ medt " " q

3K
38

d9
dt

me TT7
3K
q 3^

,0,
*'

(

and
,,„,
(10;

-rr » Up. + —

D

where
,,.

me 3K

is the bounce-averaged equilibrium drift frequency.

3. CONDITIONS FOR DIFFUSIVE TRANSPORT
Drift pumping employs rf fields resonating with w_.
Convective loss in a small-amplitude, single-frequency pumpfield requires an uv that is nearly linear in i|>, which we assume
generally not to occur. Diffusive loss relies on an
irreversible element in Eqs (9 through 11). This could arise
from collisions, which in practice are too infrequent, or fromproperties of the particle motion in more complex applied
spectra. A rf source having sufficient continuous bandwidth tocover a practical range of u D is less electrically efficientthan one of the same spread but having a number of discrete
frequencies, each generated by high-Q resonant circuits.
Particle motion in these latter fields is stochastic only when
the resonance overlap condition is satisfied. Under thiscondition, using Eq. (8), the resulting diffusion coefficient isgiven byf
(12)
If we limit the rf field to azimuthal mode number X, - 1 withmultiple frequencies,

BALDWIN et al.

298

K = cos 8 Jia K n GOS
n

(13)

then

5
When Kand u

are slowly varying with n,K2
8 lq '««„

Following the analysis of Ref. [5], the condition forresonance overlap requires that the amplitude of particle motionin the vicinity of one resonance, Aiji , satisfy

where 6i)iAip

is the separation between adjacent resonances.

Whenis expressed in terms of the locally resonant component of

the perturbed Hamiltonian, this becomes

mo

(14)

If the number of applied frequencies N is increased by reducing6u) and K , while preserving the total bandwidth -N6u> and the
n
n
n
~ p
total power -NK , the diffusion coefficient given by Eq. (12)is unchanged, while the stochasticity condition given by
Eq. (11) becomes better satisfied.

CHARACTERISTICS OF CONSTANT-J PUMPING
Transporting a particle to the surface by drift pumping
requires that successive frequency components be resonant witha) • uD(i|0, where u is to be computed at constant J. Whiletransporting, a particle's kinetic energy and pitch angle willchange.

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IAEA-CN-47/C-II-2

FIG. 2. Typical separatrices between passing andtrapped ions at fixed \//.

FIG. 3. Typical separatrices between passing andtrapped ions at fixed /i.

To picture the process, assume a throttle-coll magnetic
configuration having negatively biased endwalls, giving rise to
the on-axis profiles illustrated in Fig. 1. Off-axis, assume
that the potential retains a similar shape but decreases inmagnitude to zero at the surface, i.e. <)>(<|J,Z) » <f>. (i|>) 4>2(z).
First, construct a familiar e-y diagram for the barrier region
at a fixed i|>, as shown in Fig. 2. Note the separatrices definedT, b, p, describe ionsby v
B + qx /m. Respectively, oof vanishing parallel velocity at the throttle coil z™, thebarrierz , and the plug location z .

For e > V (except atvery large y ) , ions are unconflned and the distribution vanishes
along e = V . For V > e > V™, ions pass from the central celland are confined by the plug potential; their distribution is
Maxwellian at central-cell density and temperature for that
ijj-value. For e < V T , ions are trapped between the throttle coiland the plug potential, and the trapped distribution must equal
the passing distribution along E = V™. Such a figure exists foreach ip, with the contribution of potential smaller in magnitude
at larger if).
Because y is nearly constant (affected only by collisions),
we construct e - if) diagrams at constant y, as shown in Fig. 3.
The separatrices here have variation due to x(f) an<3 also B(iji)(which we neglect). The central-cell distribution fc vanishes
along e » V (41); for E < V , f decreases with both increasingppce a n d i|i.
The trapped-region distribution ft is to be obtained bysolution of a diffusion equation competing drift pumping withionization and collisional trapping across e
V™ (as well as

300

BALDWIN et al.

FIG. 4. Typical variation ofoiD (\p) at fixed n, J.

FIG. 5. Typicalat fixed \p.

level

surfacesof o>Dfrom adjacent u-values). As a diffusion problem, f^ isprescribed along the broken boundary formed by the two
separatrices e » V-p and e - Vp: ft - f c along the V T portion,and ft » 0 along the Vp portion. Owing to J conservation, therf diffusion reduces to one dimension, i.e. the constant-J
surfaces in Fig. 3. which must be constructed numerically for
general B and x- J surfaces intersecting the e » V p boundary
lead to true loss of trapped particles; those that intersect thee - V T boundary enhance trapping. In the strong-pumping limit
in which drift diffusion dominates collisions, there is alimiting form to f t : on the former J surfaces, f t vanishes; on
the latter, f. has its value on the surface. This limiting form
will give the minimum attainable trapped-ion density for the
profiles assumed; collisions will only increase it.
Whether the motion along the J-surfaces is irreversible
depends on the profile and the applied spectrum. Figure 4
illustrates profiles of U Q at constant J; and Fig. 5 contoursof (oD in e - u for fixed iji. A selected band of rf frequencies
covers a band in <dD and in e - u» e.g. as shown shaded in Figs
4 and 5. The band of J-values that will be resonant can be read
off of Fig. k, giving the surfaces in Fig. 3 along which
diffusion will occur, provided the stochasticity condition[Eq. (1H)] is satisfied.

5.

EXAMPLE:

PUMP FIELDS IN PARAXIAL QUADRIPOLE GEOMETRY

As an example, we calculate the pump fields in a low beta
quadrupole configuration for which paraxial field lines are
given by

IAEA-CN-47/C-II-2

301

c(z) cos 8(15)
y = /2~ijj T ( Z ) sin 6
where O,T are the fanning factors. We limit ourselves toincompressible perturbations generated by a coil set external tothe plasma and we neglect the effects of potentials. For thiscase it is convenient in Eq. (Ha) to write

where K3 = B h_ x V^-jc is the geodesic curvature- Then, in theparaxial approximation, linearization gives

6

For rigid I » 1 displacements,j,ra==ll 3x-II 3yx 9? + 5 y 36

S

where the prime denotes differentiation with respect to z, andthe nth frequency component is given byn[o(ou n )" sin 6 + T (xv n )" cos e](19)

The expression for the diffusion coefficient becomes

2

2

r

% = ? ( f f 6^ [^T ^B).(aun)« + 7v}+UBMxvn)ll ] (20)where the overbar denotes the bounce average. For symmetricallyarranged pump coils, we have a(z) = T ( - Z ) . Then, withappropriate source terms, the equation for line-bending
perturbations becomes£- " f i [ i x ( o + T >]where <p> » <p^ + Pir>/2 i s the flux-averaged pressure.

(21)

BALDWIN et al.

302
Without throttle coil

-600-400-200

0

0.12

With throttle coil

200 400 600-600-400-200

0200 400 600

I'T

0.08
0.04
0.00

T

TT

-600-400-200 0

200 400 600

Z(cm)-600 -400 -200 0 200 400 600
Z(cm)FIG. 6. Typical axial profiles of B, a and 8 = -K/4, for TMX-U geometry with and without throttle
coils. The perturbation coils generating Bx and By are at z = 390 cm and z = -390 cm without
throttle coil, and at z = 300 cm and z = -300 cm with throttle coil. Vertical dotted lines in the lower
figures mark extrema of B.

Generally, even though B is localized in z, the function u
has sizable amplitude throughout the plasma column. However,when there is a choke coil or a thin fan between the perturbing
coils and the central cell, so that o = T - 1 //§ in the central
cell, H3 is small there. Figure 6 shows profiles of fo for the
TMX-U geometry without and with a throttle coil. In each casethe perturbation coil has been placed at the location of thethin fan, as illustrated in the figure.

6.

OTHER TOPICS

We have also considered creation of electrostatic
perturbations by using modulation of biased segments of theendwall, and by using modulation of power inputs to the plasma,such as by ECRH. The analysis of these perturbations is similar
to that of trapped particle modes [6]. Although somelocalization to the barrier is possible, we find considerable
penetration of perturbations into the central cell. Suchtechniques would generate effective pumps only if the ions in
the central cell did not exhibit the same drift resonances as
those in the barrier.
Drift pumping of an axisymmetric geometry is found to be
more difficult than one having equilibrium geodesic line
curvature. First, the geodesic curvature increases the flux

IAEA-CN-47/C-I1-2

303

displacements of particles relative to an axisymmetric geometry
with the same B(z) profile. Second, the thin fans of aquadrupole geometry permit tight coupling between perturbationcoils and the plasma. An experimental test of drift pumping
must compete the induced diffusion with collisional trapping.
Therefore, a near-term test in a relatively collisional machine
would be more easily carried out if the device had an asymmetricgeometry.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]

BALDWIN, D.E., LOGAN, B.G., Phys. Rev. Lett. 43 (1979) 131.
McGUIRE, K., et al., Phys. Rev. Lett. 50 (1983) 891.
LOGAN, B.G., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984
(Proc. 10th Int. Conf. London, 1984), Vol. 3, IAEA, Vienna (1985) 335.
ANTONSEN, T.M., Jr., LEE, Y.C., Phys. Fluids 25 (1982) 132.
LICHTENBERG, A.J., LIEBERMAN, M.A., Regular and Stochastic Motion, SpringerVerlag, New York (1983) Ch.4.
BERK, H.L., ROSENBLUTH, M.N., WONG, H.V., ANTONSEN, T.M., Jr.,BALDWIN, D.E., Fiz. Plazmy 9 (1983) 176; Sov. J. Plasma Phys. 9 (1983) 108.

IAEA-CN-47/C-H-3

STABILIZATION OF FLUTE INSTABILITYIN AXIALLY SYMMETRIC MIRROR MACHINES
LA. KOTEL'NIKOV, G.V. ROSLYAKOV,
D.D. RYUTOV, G.V. STUPAKOV
Institute of Nuclear Physics,
Siberian Branch of the
USSR Academy of Sciences,
Novosibirsk, Union of Soviet Socialist Republics
Abstract
STABILIZATION OF FLUTE INSTABILITY IN AXIALLY SYMMETRIC MIRROR MACHINES.
Axially symmetric tandem mirrors have a number of important advantages over axially nonsymmetric ones, but attempts to create them have foundered on the serious difficulty of stabilizing theflute oscillations excited in them because of the — on average — unfavourable curvature of the magnetic
field lines. The paper describes new results of studies on two methods of achieving stabilization inaxisymmetric mirror machines by using, first, a non-paraxial magnetic field and, second, sloshing ionsin a paraxial mirror machine. The overall conclusion of the paper is that there is some prospect of
creating an MHD stable axisymmetric mirror machine with 'natural' geometry (i.e. with a singleconnected plasma which does not contain conductors within it).

1.

INTRODUCTION

At present the main method of stabilizing plasma flute instability in open traps
is to create a magnetic well using multipole windings. However, the disruption of theaxial symmetry of the plasma which inevitably follows from this procedure produces
a number of undesirable effects, notably an increase in transverse transport [1] anddistortion of plasma equilibrium on long sections of the homogeneous field [2].
Consequently, a number of suggestions have been made recently regarding thepossibility of stabilizing flute perturbations within axisymmetric magnetic configurations [3-10]'. Most of these suggestions are based on the use of systems with nonsingle-connected plasma [4, 6, 9] and cusps [5, 7], The transition to a non-singleconnected plasma is, however, accompanied by certain technical difficulties, themain one being a reduction in plasma thickness (which carries with it the danger ofmicro-instability excitation); as far as the use of cusps is concerned, it should be
noted that their operation will depend largely on the behaviour of the hot plasma near
the magnetic field zero point, a question which has so far been the subject of little
experimental study.
1This paper considers devices with a long plasma particle free path, concentrating mainly on
tandem mirrors; for systems with a short free path, an effective mechanism for stabilization (namely,
the plasma flow beyond the mirror) has long been known [11, 12].

305

306

KOTEL'NIKOV et al.

References [3, 8, 10] dealt with plasma stabilization within the framework of'natural' open trap geometry. Timofeev [3] suggested stabilizing the plasma entirely
by kinetic effects similar to finite Lamor radius (FLR) effects. This approach seems
promising, but the possible role of fluctuations with negative energy which canappear in systems with fast ion drift is still not fully understood. In Ref. [8] Hinton
and Rosenbluth made a detailed study of a possibility which had frequently beenraised earlier, namely that of stabilizing a plasma by creating in it a population of'sloshing' ions having a turning point in a region of favourable curvature of the fieldlines; however, the technical requirements for the injection system (injectionenergy ~ 40 MeV) for the magnetic field configurations examined in Ref. [8] turned
out to be extremely demanding. In Ref. [10] Kesner suggested installing a conducting
casing near the plasma surface to achieve stability, but in this case stabilization —if achieved at all — occurs only at high plasma pressure and when the plasma has
a high degree of anisotropy.
The above short summary of work on MHD stabilization of axisymmetric traps
indicates that, although there are a number of interesting proposals, the problem isfar from resolved. The present paper examines a further two possibilities for creating
an axisymmetric MHD stabilizer for open (mainly tandem) traps.
The first possibility is the use of a stabilizer in the form of a non-paraxial mirror
machine in which the transverse dimension of the plasma is comparable with the
distance between the mirrors. We show that large scale flute oscillations can be madestable in such a device, while small scale fluctuations can be stabilized by the FLR
effect. It may even be conceivable that complete stabilization is not necessary sincea certain level of MHD activity at high modes will not result in excessively fastplasma losses. The essential point is that stabilization should occur within the'natural' open trap geometry and should not require transition to a non-singleconnected plasma.
The second possibility lies in the use of sloshing ions. We demonstrate that, by
optimizing the shape of the magnetic field and using drift pumping [13], the injectionenergy requirements can be lowered significantly (by comparison with the predictions in Ref. [8]), which greatly improves the prospects for this stabilization method.

2.

MHD STABILITY IN A NON-PARAXIAL MIRROR MACHINE

2.1. Model of plasma as an ideal gas
We shall consider the mirror machine to be part of a tandem mirror with a longcentral cell whose presence ensures the stability of all flute modes (except the global
mode m = 1) because of FLR effects [10].
FLR effects in the long homogeneous section of a central mirror machine exert
no influence on the rigid body displacement of the plasma in this section. SuchIAEA-CN-47/C-II-3

307

displacement is due to the homogeneous electric field to which the potential perturbation of the following form corresponds:<p = const Vi^ cos 0(1)where 0 is the azimuthal angle. In a flute perturbation the potential <p is constant along
the field lines, and so in a finite mirror machine the potential will also be given by
Eq. (1). In this case the potential perturbation energy is defined by the equation:

J

r

J L

where

u

p'U' + 7 —U'

L

-|

(2)

2J

J

U = 1~(3)is the specific volume of the flux tube (integration is performed with respect to thefield line between infinitely strong mirrors), \p is the magnetic flux inside a given
magnetic surface, p is the pressure, A > 0 is a constant which does not need to beconsidered further, and the prime indicates differentiation with respect to \p.
If the plasma transverse dimension is small compared with the scale of thechange in the magnetic field, then we have the inequality Ip'l/p > U7U and the lastterm in expression (2) can be neglected. Performing an integration by parts in the firstterm and taking into account that ^U" — 0, in the paraxial region, we obtain thefollowing for a 'thin' plasma:
5W = - A U '

< 0

(4)

It is well known that on the magnetic axis (at ^ = 0) the derivative U' is positive(see Ref. [14], for example), in other words that a 'thin' plasma proves to be universally — and independently of the pressure profile p(\p) — unstable in relation to theglobal mode [4],The last term in Eq. (2), which plays a stabilizing role, becomes crucial at
distances from the axis of the order of the length of a finite mirror machine. Thus,
if it is possible to stabilize the global mode at all, this can only be done if the plasma
is 'thick' (and the paraxial approximation is no longer valid).
Integrating the first term in Eq. (2) by parts we obtain

5W = Ap Fty) W

(5)

308

KOTEL'NIKOV et al.

FIG. 1. Magnetic field lines of two 'point" mirror coils located at points A and B. Bold line denotes
separatrix passing through the magnetic field zero point. Point 0 is equidistant from points A and B.

where

"jj

(6)

To prove the existence of stable pressure profiles it is enough to show that arange of values of ^ exists in which F(\p) > 0. For small \j/, ¥(\p) = — U'(0) and,in accordance with what was said above, F(\{/) > 0. As the distance from the magnetic axis increases, the plasma confinement region is usually bounded by theseparatrix passing through the magnetic field zero point (points) (see Fig. 1 for an
example). It is a simple matter to demonstrate that, as the distance to the separatrix
decreases, U increases according to the law
U a In1(7)where \j/s is the magnetic flux enclosed within the separatrix. Using Eq. (7), we findthat, for \p — &, F (f) = - a G H W 2 . with a > 0, in other words ¥($) < 0
here also. Hence, steady states can only exist at intermediate \p values, in a certain
ring region. It is worth noting that if such a region (stability ring) exists, then anypressure distribution inside it will be stable (with respect to the global mode).
Magnetic field configurations in which a stability ring exists are far from
common. The magnetic field of the two point coils whose field lines are shown in
Fig. 1 does not possess this property, for example; the function F(^) for this field

IAEA-CN-47/C-II-3

309

F
1

0.5

/
1

0.5
\

-0.5

A

FIG. 2. Graph ofjunction F(i) (in arbitrary units) given by expression (6). Curve (I) corresponds tofield of the point coils, and curve (2) to superposition of this field and the homogeneous field,
AB = 0.06B..

FIG. 3. Force lines of superposition of field of the two point coils and the homogeneous field,
AB = 0.06B.. Separatrix is indicated by bold line.

is shown in Fig. 2. If a homogeneous field AB directed along the coil axis is superimposed on this field, then a stability ring occurs in the range of values
0.023 B, < AB < 0.20 B.
where B. is the field at the point 0 in Fig. 1. For AB = 6 X 10~2 B., a graph ofthe function F(\t) and the field lines are illustrated in Figs 2 and 3, respectively.
Figure 4 presents graphs of the functions U(i/<) and Bo(^) (Bo is the field in theequatorial plane of the mirror machine) for this field.

310

KOTEL'NIKOV et al.

u

P, B

1.0

100

V

• 80

0.8

\
0.6

• 60

0.4

• 40

\

/
• 20

0.2

0
0.2

0.4

0.6

0.8

1.0

FIG. 4. Graphs of the dependences U(i) (curve (1)), B(i) (curve (2)) and p(<l>) <x U'087 (curve (3))
for the field shown in Fig. 3. The dependences p(i) and B($) are normalized to their values on the trap
axis, U($) is given in arbitrary units.

By creating a plasma in the stability ring, we ensure a certain MHD stability
margin which enables a certain quantity of plasma to be placed also in the paraxial
region of the outermost mirror cells. Thus for the magnetic configuration illustrated
in Fig. 3, a numerical calculation indicates that for pressure distributions of the form

(8)0,xj, > ,/,*

where ^* is the magnetic flux corresponding to the outer boundary of the stability
ring, and the index 'a' relates to the values p and U on the magnetic axis, there isstability at a < 0.87 (the maximum pressure profile, corresponding to a = 0.87, isshown in Fig. 4).
2.2. Kinetic energy principle
Above we have used the energy principle in an ideal magnetohydrodynamic
model in order to study stability. More appropriate for collisionless plasma is the
Kruskal-Oberman energy principle [15], which is not limited by the isotropic

IAEA-CN-47/C-II-3

311

pressure approximation and enables the equilibrium stability of anisotropic plasma to
be studied also.
Let us first consider the new contribution that the Kruskal-Oberman energy
principle makes to the problem of isotropic plasma stability examined in the preceding section. Using the Kruskal-Oberman energy expression changes the shape of thefunction F(^) entering into integral (5). It is possible to show, however, that thisfunction is, as usual, negative both for small \p and for \p —• ^ s . Consequently, as
before, the steady states — if they exist — form a stability ring, and this ring will
be wider than the one that results from magnetohydrodynamics, for the familiar
reason that the Kruskal-Oberman energy principle predicts better plasma stability
than magnetohydrodynamics with the adiabatic index 7 = 5/3. However, as thenumerical calculations show, the widening of the stability ring that results fromswitching to the Kruskal-Oberman energy principle is not large.
Let us now look at the question of the stability of a plasma with a strongly
anisotropic distribution function. This situation is of less interest from the practicalpoint of view for two reasons: first, the creation of strongly anisotropic distributions
in characteristic thermonuclear reactor conditions requires extremely high injectionenergies; and, secondly, such distributions are more susceptible to microinstabilities. We shall none the less consider such distributions also, since they canbe created in existing small scale devices with low electron temperature2.
Let us begin with a plasma with pj = 0. It occupies the equatorial plane of thetrap (z = 0). For the plasma to be stable in relation to displacements along the field
line, the magnetic field should have a 'longitudinal' minimum at z = 0. It can beshown that the condition for the existence of this minimum has the following
form [16]:

dr /

r

dr

dr

where B0(r) = B z | 2=0 is the magnetic field in the equatorial plane.
Let us clarify the question of whether it is possible to achieve stability in sucha system in relation to all flute modes. The expression for the flute perturbationenergy in a disc plasma has the following form [21]:5W = - Ad^r 2 ? 2

B£ W'± - 2 - ^ B^2(10)

where W ± is the energy per unit plasma area, £ is the radial displacement component in the flute perturbation and A is the positive constant. It can be seen fromexpression (10) that the outer plasma boundary can only be stable if it is located in2

A low electron temperature enables the injection energy Einj to be reduced since the condition
Einj s> (m/mj)" 3 T e has to be satisfied in order to obtain narrow angular distributions.

312

KOTEL'NIKOV et al.

the region where dB0/dr > 0. Two possibilities should now be considered: (1) Bo
increases monotonically from the axis itself to the outer plasma boundary; (2) there
is a minimum (minima) in the intermediate space. In the first case inequality (15)definitely ceases to be valid near the axis (at r — 0) 3 , in other words the paraxial
region is empty and the plasma has an inner boundary where 3W ± /3r < 0. But
because of the condition dBo/dr > 0 the inner boundary is unstable, and hence
absolute stability is impossible in the first case. Turning to the second case, let us
look at the vicinity of the Bo minimum nearest to the outer plasma boundary. In
the vicinity of the minimum, where the dB0/dr derivative is sufficiently small,inequality (9) clearly does not hold, in other words, the region around the minimum
is empty and the plasma has an inner boundary located outside the minimum.
However, this boundary is then unstable. We have thus demonstrated that absolutestability is impossible for a disc plasma.
Let us check, therefore, whether it is not possible for such a plasma to be stable
to the global mode. This condition is obtained if the displacement i- corresponding
to the disruption of the potential (1) is substituted in Eq. (10), thus producing:<5W = const

f

W,
d
dBa
<ty — j - —— yp -r—(11)

Consequently, it is essential for stability that the following inequality be satisfied at
the same time as Eq. (9) at one point at least on the radius.
d

dr

*
rB0

dB

" > 0
dr

(12)

It is easy to verify that inequalities (9) and (12) cannot be reconciled in the paraxial
region. However, they may be satisfied simultaneously at finite distances from theaxis. In the magnetic field shown in Fig. 3, for example, this happens in the interval
0.49 rs < r < 0.83 rs, where rs is the radius of the separatrix.
We can also examine the opposite extreme case where the transverse componentof the particle velocity in the equatorial plane is very small (extreme case of tangential
injection). The mirror ratio in this case is considered to be so large that these particles
are still retained within the trap, stopping in the region of the very strong field in the
mirror. For global mode stability it is necessary for the following condition to besatisfied on at least one magnetic surface [17]:e3
It should be recalled that near the axis Bo = C, + C2r2.

IAEA-CN-47/C-II-3

313

where £ is the length of the field line between the mirrors. This condition is not
fulfilled anywhere for the magnetic field shown in Fig. 3, so that this case offers noadvantages over that of the isotropic plasma where there is a wide stability ring inthe same field.

3.

STABILIZATION OF FLUTE PERTURBATIONS IN A PARAXIAL
MIRROR MACHINE BY MEANS OF SLOSHING IONSMHD stability of a plasma in a paraxial axisymmetric mirror machine can beachieved if a group of sloshing ions is introduced into the system and the magnetic
field profile is selected in such a way that the turning point and, consequently, thepressure peak of such ions are located in the region where the curvature of the fieldlines is favourable. This method of stabilizing flute fluctuations by means of sloshingions is fairly self-evident but the evaluations conducted earlier did not reveal all its
capabilities. It was found in Ref. [8] that to stabilize the system it is necessary to form
a population of sloshing ions with an extremely small angle straggling similar to thatof helium ions of 50 MeV energy during injection into a plasma with an electron
temperature Te = 50 keV (ions with so much energy are scattered much more
slowly than they are slowed down by electrons).
Below we show that optimizing the magnetic field profile can make MHD
stabilization by means of sloshing ions a more realistic possibility.
3.1. Stability condition
To determine whether a given system is stable with respect to flute perturbations, it is sufficient in a paraxial approximation to examine the case of a plasma witha sharp boundary, when the pressure is constant over its cross-section. Then, in
accordance with the Rosenbluth-Longmire criterion [18], an axisymmetricconfiguration is stable if1= L ^ -£- > 0(14)where a = a(z) is the boundary field line equation and d2a/dz2 is its curvature. The
sum of the longitudinal and transverse pressure P = pB + p ± is a function of themagnetic field induction B (see Ref. [19], for example) which in a paraxial approximation is proportional to a"2; hence it can be considered that P = P(a). Integrationin Eq. (14) is performed along the z-axis of the trap for the region occupied by theplasma. At the boundary of this region z = ±L, the function P(a) goes to zero while
the magnetic field, which can be considered symmetrical relative to the z = 0 plane

314

KOTEL'NIKOV et al.

FIG, 5. Qualitative form of the Junction Pa3 when criterion (16) is satisfied. Dashed line depicts
contributions of isotropic plasma (1) and sloshing ions (2).

without limiting its generality, reaches a maximum. Having performed an integration
by parts in Eq. (1):
I = -

dz

dz

da

[Pa3]

(15)

it is not difficult to establish that, if the function
g(a) =da[Pa3]

is of fixed sign and, as in the case of isotropic plasma, positive, then I > 0. The
system may be stable, I > 0, only if g(a) is negative in a certain interval. In this casea local minimum appears in the graph of the function Pa3 (Fig. 5). This is precisely
the situation that arises if a population of sloshing ions with a sufficiently small angle
straggling is created. The points a = a, and a = a2, where g(a) = 0, isolate on thefield line the section a! < a(z) < a2, in which g(a) < 0 and which makes astabilizing (positive) contribution to the integral (15). By reducing the length of thissection t = z2 - Z\ at fixed values a(z,) = a.x and a(z2) = a2 and thereby increasing
here the angle of inclination of the field line Ida/dz I (but none the less remaining
within the framework of the paraxial approximation), it is possible to satisfy thestability criterion I > 0 even at a very small (in absolute terms) negative value ofthe minimum g(a). Thus the question of MHD stabilization by means of sloshing ionsboils down to ascertaining the conditions at which
min g(a) < 0aIt should be noted that the stabilization boundary
min g(a) = 0(16)

IAEA-CN-47/C-II-3

315

corresponds to such a small peak value on the Pa3 function graph that the local
maximum and minimum run together, and at the point of convergence a! = a2 aninflection forms, i.e.
g(a,) = 0

dg

= 0

da

(17)

With regard to the stabilization boundary found in Ref. [8], it corresponds to a large
peak value such that P(a,)ai ~ P(ao)ao, where ao is the plasma radius at themagnetic field minimum.
3.2. Fast ion distribution function
A population of sloshing ions is formed by injecting monoenergetic particles
with a very small initial angle straggling and then slowing them down on electrons;
because of the large energy Eo of the injected particles this slowing down takes
place much more quickly than slowing down and scattering due to plasma ions.
Assuming that a major part of the plasma volume V is allotted to the longhomogeneous central part of the trap, we can write the kinetic equation for thesloshing ion distribution function f(V,0) in this part of the trap in the following form:v^ 0\vT»

VQ)

rst//ia \i

Z/Q I a \ i/im

4ir V sin 0O v 2

where the penultimate term describes the reduction in sloshing ions as a result ofeither external pumping (see below) or of charge exchange with the neutral atoms
present in the plasma; 60 is the injection angle (reckoned from the system axis);
Q is the injection power;
*

M

T 3/2
Ae 4 nis the slowing down time.of ions with charge Z and mass M due to electrons:

Tit=

2ir Z 2 A e4n

is the scattering time (due to ions), A is the Coulomb logarithm; v0 = (2Eo/M)1/2
and vc = (3ir1/2 me/4M)1/3 (2Te/me)1/2 are the initial and critical velocities of injected ions respectively (at v = vc the velocities of their scattering and slowing down

316

KOTEL'NIKOV et al.

due to electrons are comparable); Eo is the injection energy. Assuming for the sakeof simplicity that T* = const, it is a simple matter to find a steady state solution for
Eq. (18):f =Qr ei (v 3 + v 3 r'

4ir3/2 V Eo sin 60 A0+ e~ ^ *» ' I ( ^ ^ l ) ^ 7

(19)

where
1

.

V3/V3

\I2

3.3. Stabilization condition at T > rei
Let us first examine the case where the lifetime of the fast ions is not longenough to allow them to be converted by charge exchange into neutral atoms, i.e.
rei << 7*. In such a case, as we shall see, the minimum injection energy at which it
is possible to achieve stabilization by sloshing ions is several times higher than theenergy4?^2(where mH is the proton mass) at which slowing down due to electrons and ions iscomparable.
Having performed calculations similar to those in Ref. [20] (see also Ref. [8]),it is possible to obtain the function Pa3 and, using correlation (17), to determine theinjection energy and power at which flute perturbation stability can be ensured.
As has already been shown, the system is stable if the function Pa3 is nonmonotonic and if there is a marked contribution from the spike created by sloshingions against a background of a monotonically increasing contribution from theisotropic plasma 2p0a3 (see the dashed line in Fig. 5). Of course, the greater theisotropic plasma pressure is, the greater must be the power Q and (or) the injectionenergy EQ in order to ensure stability (the spike becomes higher as Q increases andnarrower as EQ increases). If the injection energy is not high enough, the function

317

IAEA-CN-47/C-H-3

FIG. 6. Graph of dependence of injection power on initial velocity of particles injected at angles:
1 - r . > rei, for the optimal magnetic field;
2 - T.> rei, result ofRef. [8];3 - T. = 0.5 rei, pumping of sloshing ions from additional short mirror cell;
4 - T. = 0.5 rei, pumping from long mirror cell.

Pa 3 becomes monotonic even if the isotropic plasma contribution is ignored. Hencestabilization can only be achieved if the injection energy is greater than a certain
critical value E».
Calculations (the details of which can be found in Ref. [21]) show that thedependence of the injection power required for stabilization on the injected particle
velocity has the form:
Q =

Vpo

33.2 sin1/2-2flo(vc/vo)9/4
1 - 11.8ctg0 o (Vc/vo) 3/2(20)

To compare Eq. (20) with the results of Ref. [8], which are for helium injection
into an isothermal deuterium plasma, we shall assume that dQ = arcsin V677 as inRef. [8] and will express Q in dimensionless form in terms of Q/Qc
_ 3

3/2

Po V

Te

4mH
M

HI;

2mH

lgR

where, neglecting ambipolar effects, the multiplier in front of the braces is, with anaccuracy up to the numerical coefficient, equal to the energy flow through themirrors, the combination in the braces is equal to unity for helium injection into an
isothermal deuterium plasma, mH is the proton mass,

JL

3/2

(2mime) 1/2
M

lg R

318

KOTEL'NIKOV et al.

is the scattering time into the loss cone for ions of the main plasma component whichhave a unit charge and mass m;. As a result, we obtain
94

0.86 (v c /v 0 ) '

1 - (2.85 v c /v 0 ) 3/2

Qc

It is very important that at a v 0 value 1.5 to 2 times greater than the minimum
injected ion velocity v» = V2E»/M = 2 . 8 5 v c , at which stabilization by sloshingions is still possible (Q — « at v —• v.), the injection power derived from relation (13) decreases much more quickly (Q oc vo"9/4) as v 0 increases than follows
from the calculation performed in Ref. [8] (Q oc vo~3/4). Equation (21) and thecorresponding dependence
0.022

Q
Qc

[8]

(Vo/3.81 v c ) 3/4 - 1(22)

obtained in Ref. [8] are plotted in Fig. 6. A comparison of Eqs (21) and (22) showsthat the minimum velocity v, of injected ions can be reduced by a factor of 1.34,
and hence the injection energy E» by a factor of 1.8.

4.

FURTHER REDUCTION OF MINIMUM INJECTION ENERGY

Since the minimum injection energy E. is still very high, it would be advantageous to find a way of reducing it further. This can be achieved by changing to asystem of two mirror cells — one long and the other short (but such that the paraxial
approximation in it still applies). We shall consider that tangential injection is only
performed into the long mirror cell, the injection angle 60 corresponding to theturning point of the fast ions at the magnetic field maximum in the mirror which
separates the two mirror cells (1 in Fig. 7). Under these conditions only those fastions whose pitch angle has decreased compared with 60 (because of scattering) can
penetrate the short mirror cell. In this case the fast particle angular distribution

FIG. 7. Two-mirror cell system. Outer mirrors 3 are stronger than intermediate mirror 1. Arrow
indicates injection of fast particles. Angle of inclination of field lines to system axis is not given to scale(in reality, it is less) and in the stabilizing section 2 the inclination is much greater than in the remaining
part of trap.

IAEA-CN-47/C-II-3

319

f(0)

FIG. 8. Fast ion angular distribution Junction in short mirror cell reduced to magnetic field in centreof long mirror cell. Curve 1 corresponds to the absence of pumping (T. > Tej), curve 2 to pumping
whose time T, satisfies conditions (23) and (24).

function in the short mirror cell will have the form shown by the solid line in Fig. 8,
which ensures the existence of a section with negative values of the function g(a) evenat large A0O values (i.e. at low injection energies). The appearance of this segment
makes it possible to stabilize the flute instability by creating a section with a sharply
increasing magnetic field in the outer mirror of the short mirror cell (2 in Fig. 7).
Collisions in the short mirror cell will, of course, cause blurring of the righthand boundary of the fast particle distribution function in Fig. 8 and a reduction in
the stabilizing effect. To prevent this from happening, some mechanism for pumpingout trapped fast particles must be introduced in the short mirror cell. The mostsuitable one is the drift pumping mechanism [13] associated with the use of a small
axially non-symmetric perturbation of the magnetic field oscillating at a frequency
close to that of the drift revolution of the fast particles around the magnetic axis. The
advantage of this pumping method is that, because of the considerable difference
between the drift frequencies of the three populations of ions — fast trapped, fastpassing (i.e. fast ions with 0 < 0O) and slow — it is possible to achieve selective
pumping of the first population without losing the second and third.
The substantial (of the order of A0O) time for the blurring of the right handboundary of the fast particle distribution function is equal by order of magnitude to
the scattering time for such particles at the angle AdQ (the scattering corresponds byorder of magnitude to their slowing-down time due to electrons r ei ). For this reason,
the characteristic 'pumping time' should satisfy the condition
7ei

(23)

Given condition (23) the width of the right hand boundary of the peak in Fig. 8 canbe calculated from the equation
Ad* ~ A0O V7./7ei

320

KOTEL'NIKOV et al.

while the power Q* lost from the system together with the pumped out particles canbe evaluated from the equation
Q.

Q

where V, is the volume of the short mirror cell. To ensure that pumping particles
out of the short mirror cell does not lead to a significant reduction in the number offast particles in the system, it is necessary for the condition Q. < Q to be satisfied
as this limits the permissible pumping time to below:

Y

(24)

Hence, for the minimum attainable A0. value, we have: Ad, ~ A0oV»/V.
Thus, when the volume of the short mirror cell is sufficiently small, it really is possible to increase substantially the steepness of the right hand slope of the fast particle
angular distribution and thereby to reduce injection energy.
The results of quantitative calculations of the dependence Q(v0) (for details see
Ref. [21]) are given in Fig. 6 (curve 3).
Another method of reducing E» is to pump sloshing ions out of the long mirror
cell. A natural mechanism for such pumping may be losses due to the charge
exchange of fast ions in the residual gas penetrating the plasma. Naturally, becauseof the pumping of ions directly out of the long mirror cell, the injection power has
to be increased in order to achieve stabilization, but under these conditions it is nolonger necessary to have an additional mirror cell. The dependence Q(v0) for sucha case is shown by curve 4 in Fig. 6.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]

RYUTOV, D.D., STUPAKOV, G.V., Pis'ma Zh. Ehksp. Teor. Fiz. 26 (1977) 186.
STUPAKOV, G.V., Fiz. Plazmy 5 (1979) 871.
TIMOFEEV, A.V., Pis'ma Zh. Ehksp. Teor. Fiz. 29 (1979) 277.
ARSENIN, V.V., Fiz. Plazmy 5 (1979) 1293.
LOGAN, B.G., Comments Plasma Phys. Contrail. Fusion 6 (1980) 199.
ARSENIN, V.V., Fiz. Plazmy 8 (1982) 484.
DIMOV, G.I., Prepr. No. 82-150, Nuclear Physics Institute, Siberian Dept. USSR Acad. Sci.
Novosibirsk (1982).
HINTON, F.L., ROSENBLUTH, M.N., Nucl. Fusion 22 (1982) 1547.
ARSENIN, V.V., Pis'ma Zh. Ehksp. Teor. Fiz. 37 (1983) 534.
KESNER, J., Nucl. Fusion 25 (1985) 275.
MIRNOV, V.V., RYUTOV, D.D., Pis'ma Zh. Tekh. Fiz. 5 (1979) 678.
NAGORNYJ, V.P., RYUTOV, D.D., STUPAKOV, G.V., Nucl. Fusion 24 (1984) 1421.

IAEA-CN-47/C-II-3

321113] BALDWIN, D., in Mirror-Based and Field-Reversed Approaches to Magnetic Fusion (Proc.
Workshop Varenna, 1983), Vol. 1, International School of Plasma Physics (1984) 109.
[14] MOROZOV, A.I., SOLOV'EV, L.S., Voprosy Teorii Plazmy, Vol. 2 (LEONTOVICH, M.A.,
Ed.), Gosatomizdat, Moscow (1963) 3.
[15] KRUSKAL, M.O., OBERMAN, C.R., Phys. Fluids 1 (1958) 275.
[16] ARSENIN, V.V., in Plasma Physics (Proc. Int. Conf. Lausanne, 1984), Vol. 1, CEC, Brussels(1985) 206.
[17] RYUTOV, D.D., STUPAKOV, G.V., Prepr. No. 85-146, Nuclear Physics Institute, Siberian
Dept. USSR Acad. Sci., Novosibirsk (1985).
[18] ROSENBLUTH, M.N., LONGMIRE, C , Ann. Phys. 1 (1957) 120.
[19] RYUTOV, D.D.,
STUPAKOV, G.V.,
in
Voprosy
TeoriiPlazmy,Vol. 13(KADOMTSEV, B.B., Ed.), Atomizdat, Moscow (1983) 74.
[20] KOTEL'NIKOV, I.A., RYUTOV, D.D., Fiz. Plazmy 11 (1985) 1155.
[21] KOTEL'NIKOV, LA., ROSLYAKOV, G.V., RYUTOV, D.D., Prepr. No. 86-31, Nuclear
Physics Institute, Siberian Dept. USSR Acad. Sci., Novosibirsk (1986).

IAEA-CN-47/C-II-4

PROGRESS IN PLASMA HEATINGWITH POWERFUL ELECTRON BEAMSIN OPEN SYSTEMS
A.V. ARZHANNIKOV, V.T. ASTRELIN, A.P. AVROROV,
B.N. BREJZMAN, A.V. BURDAKOV, V.S. BURMASOV,
V.V. CHIKUNOV, I.V. KANDAUROV, V.A. KAPITONOV,
A.D. KHIL'CHENKO, B.A. KNYAZEV, V.S. KOJDAN,
V.V. KONYUKHOV, Eh.P. KRUGLYAKOV, S.V. LEBEDEV,
K.I. MEKLER, O.I. MESHKOV, V.V. POSTUPAEV,
D.D. RYUTOV, A.L. SANIN, M.A. SHCHEGLOV,
S.L. SINITSKIJ, S.G. VOROPAEV, L.N. VYACHESLAVOVInstitute of Nuclear Physics,
Siberian Branch of the
USSR Academy of Sciences,
Novosibirsk, Union of Soviet Socialist Republics
Abstract
PROGRESS IN PLASMA HEATING WITH POWERFUL ELECTRON BEAMS IN OPEN
SYSTEMS.
Investigations of the effects of plasma heating with REB are reported. The plasma heatingproblem is an important part of the research programme designed to investigate the possibility ofbuilding a multimirror fusion reactor. Asymmetry of the distribution function along the system axis isobserved and investigated on the INAR machine. It was shown that high energy electrons are createdeven at large distances ( - 7 m) from the point of REB injection into the plasma (GOL-1). A newmachine for investigating REB interaction with plasma (GOL-M) is described, and the first experimental
results are reported. It is shown that the interaction efficiency increases with the magnetic field strength
up to the largest fields obtained in experiments (4.5 T on GOL-M, 7 T on INAR). The practical results
obtained in generating REB of microsecond duration (3-4 us) on a plane diode (U-l machine) is
discussed. A beam energy content of 130 kJ is obtained. After compression (in a 10 T field) the REB
energy content is 100 kJ. In the final section of the paper, a method is proposed for increasing theduration and energy content of the beam.

1.

INTRODUCTION

The plasma heating experiments with powerful relativistic electron
beams (REB) carried out at the Institute of Nuclear Physics of the USSR Academyof Sciences (Siberian Division) are directed towards the development of a method for
rapid heating of dense plasma in elongated open systems with magnetic confinement.
The heating is designed to take place in two stages whereby the beam gives up thelarger part of its energy in a plasma with a density of 1015 cm"3 in the central partof the trap, after which the energy is transmitted through two-particle collisions to
323

324

ARZHANMKOV et al.

a denser plasma (n ~ 1017 cm"3) adjacent to the ends. A multimirror magnetic
system is to be used to confine the dense plasma. The ultimate purpose of the present
stage is to build the GOL-3 machine [1]. Our paper presents the results of work
carried out under this programme quite recently.
Section 2 reports the results of experiments on the INAR device. Here, attention
is concentrated on phenomena occurring in the immediate neighbourhood of the beam
injection point (z < 1 m), and in a magnetic field up to 7 T. Section 3 presents theresults of experiments on the GOL-1 machine. Thanks to the comparatively great
length of this device (L = 7.5 m), it is possible to distinguish between effects
occurring at the entrance to the machine and those distributed over its length. In
Section 4 we describe the modernized GOL-M device and give the results of initial
experiments on it with magnetic fields of up to 4.5 T.
The final two sections (Sections 5 and 6) are devoted to work aimed at developing a technology for generating REB of microsecond duration for heating plasma in
machines of the next generation. Section 5 deals with beam generation on a plane
diode followed by magnetic compression of the beam. Section 6 describes a possible
scheme for enhancing energy content and beam duration by successive reduction of
a number of band-shaped beams from different generators onto a single ' 'magnetic
path", followed by transformation of the beam from the band shape to a circularcross-section.

2.

EXPERIMENTS ON THE INAR DEVICE

A new series of experiments has been carried out on INAR to investigate collective heating of plasma electrons by the stream instability, with the accent on
phenomena occurring close to the point of beam injection into the plasma. In earlier
experiments [1-5] it was shown that over the first metre from the injection point the
diamagnetism falls off sharply (by several factors) and then continues to drop more
slowly; the beam energy is imparted largely to the high-energy "tails" of the electron
distribution function; and with time the diamagnetism tends to even out over thelength of the machine.
In the experiments described here we obtained a direct recording of the directional movement of "hot" plasma electrons from the entrance to the outlet of themachine. The geometry of the experiment set up to determine the directional motionof hot electrons by small angle laser scattering is shown in Fig. 1. The relativistic
beam and the laser radiation propagate along the x and z axes respectively, and theradiation scattered at an angle of (j> = 8° is recorded in a solid angle
0 = sin <j> d<t> i/. Since the azimuthal angle region is symmetrical relative to the yaxis, a contribution to the useful signal is made by electrons having both positive andnegative velocity projections onto the direction of REB motion. The signals fromsuch electrons can be separated by measuring separately the power of the radiation
scattered in directions bounded by the angles \pi and \j/2- (The spectral range of the

325

IAEA-CN-47/C-H-4

X

FIG. 1. Geometry of experiment designed to record anisotropy ofelectron distribution function on INAR by Thomson scattering method.
fl = sin <t>Air is the solid angle of measurement;+, + h = 135°.

scattered radiation is localized on one side of the centre of the laser line.) Measurements of this kind for the 0.3-1.5 keV electron energy region gave a ratio of radiation intensities I+/I_ = 3 ± 1 for radiation scattered by electrons moving along the
REB and in the reverse direction. Thus, we observe a stream of fast electrons fromthe point of REB entry into the plasma in the direction of beam propagation.
The undoubted fact that the hot electron distribution function is strongly
asymmetrical (showing a "forward-backward" shape) indicates that heat transporttakes place at a velocity of the same order as the thermal velocity of these electrons.
The "secondary" instabilities which in theory might develop in a flux of this kind(see Ref. [6], for example) do not lead to any substantial slowing-down of the electrons, as predicted in Ref. [6].
A special experiment was set up to test the asymmetrical nature of the electron
distribution function. Earlier the plasma had been produced by a discharge ignited
between an annular electrode located in the centre of the discharge chamber and theend foils of the magnetic mirror trap. In the present experiment, on the other hand,the annular electrode was surrounded by a thin foil (normally 9 /tm of aluminium).
In this way the preliminary plasma was produced by a direct discharge between thecentral and the two end electrodes. The plasma parameters were the same in bothparts of the discharge. The plasma density, as before, was 1.5 X 1015 cm" 3 and thethickness of the foil was such that the REB passed through it without any appreciablechange in the angular and energy characteristics; at the same time the foil was opaqueto "hot" plasma electrons.

326

ARZHANNIKOV et al.

In these circumstances the diamagnetic signals in the region from the beam inlet
to the "internal" foil remained virtually unchanged whereas in the second part of themachine they were considerably reduced. This form of signal behaviour points to thepresence of an electron flux from the trap inlet.
In another series of experiments we considered how the plasma energy content
is affected by increasing the REB energy content and enhancing the longitudinal magnetic field. In order to increase the energy content and the beam duration, the beam
generator was modernized. As a result the REB duration increased from 70 ns at thebase and 30 ns at half-height to 100 ns and 60 ns, respectively. The current generated
by the accelerator also increased and the energy content of the beam was doubled.
Where this beam was injected into a plasma with a density of 1 x 1015 cm" 3 weobserved an increase in the plasma pressure measured by diamagnetic probes which
was proportional to the growth of the REB energy content — i.e. the plasma pressure
was approximately doubled. Preliminary laser measurements have indicated both anincrease in the temperature of the main plasma electron component and an increased
energy content in the "tails" of the distribution function. In these experiments, incontrast to earlier ones, the signals from laser scattering could be observed directly
during the time of beam passage through the plasma. The experiments suggest thatthe increase in REB energy content (duration and current) does in fact lead to anincrease in plasma energy content.
In order to examine the processes which occur when the REB interacts withplasma in a strong magnetic field (up to 7 T), we set up a magnetic discharge systemwith new geometry. A magnetic field up to 8 T at the mirrors and 7 T in thehomogeneous part of the system was produced by a solenoid 1 m in length. A preplasma 3 cm in diameter was formed by a pulsed Penning discharge. The plasma
parameters were measured with a Michelson interferometer (X = 3.30 jttm), a
system for scattering the rays emitted by a ruby laser at 90°, and diamagnetic probes.
The plasma density varied from 1015 cm" 3 to 1016 cm" 3 . The beam current rose to
35 kA and the current density was 10 kA-cm~ 2 .
In one series of experiments with plasma density of (3-5) x 1015 cm" 3 wefound an increase in the diamagnetic signals (nT^S) by factors of 2 to 2.5 when theexternal magnetic field strength was increased from 1 to 7 T.

3.

EXPERIMENTS ON THE GOL-1 DEVICE

The great length of the GOL-1 machine (7.5 m) makes it possible to elucidate
certain characteristics of the beam-plasma interaction which are difficult to investigate on machines of shorter length. The length factor becomes particularly important when we wish to study fine interaction effects such as non-equilibrium of theplasma electron distribution function following REB heating of the plasma.

IAEA-CN-47/C-II-4

327

In order to gain a better understanding of the physics of the beam-plasma interaction, we must not only establish the fact of non-equilibrium of the distribution function, but also find an answer to the question where the electrons of the " t a i l " areproduced and determine whether " h o t " electron generation is an effect distributed
over the length of the trap or whether these electrons are produced near the point of
REB injection into the plasma. The pulsed target method [7] was used to study REB
interaction with the plasma in GOL-1, the essence of which consists in determining
the " h o t " electron parameters from the characteristic radiation excited by these electrons in the plasma, with pulsed injection into the plasma of various gases (in theexperiments we used Ne, Ar, Kr and Xe, which were injected into the vacuum
chamber at a distance z = 650 cm from the point of REB injection into the plasma).
For a plasma density of r^ = 4.5 x 10 14 cm~ 3 and an REB current density ofj b = 2 kA-cm~ 2 (total current I b = 60 kA, electron energy E e = 0.5 MeV, initial
angular scatter VFQ = 18°,beam duration at half-height r b - 40 ns), we havesucceeded in establishing the following by means of the pulsed target method:
1.

2.

3.

At z = 650 cm from the point of REB injection into the plasma, the ' 'hot" electrons of the " t a i l " have a density of n* = (0.7 ± 0.2) X 10 12 c m " 3 and an
average energy e* = 45 ± 20 keV);Electrons with the above parameters were formed directly at the observation
point, i.e. z = 650 cm from the REB injection point.
This conclusion is based on the good correlation between the times of beam
arrival, and the appearance of the diamagnetic signal and the characteristic
X-radiation;
As the angular scatter of the REB increases, the energy content of the " h o t "electron group decreases; moreover, the increase in the angular scatter leads on
the whole to a decrease in the density of the ' 'tail" electrons (roughly according
to the law n* ~ 1/F), leaving the average energy of these electrons virtuallyunchanged.

In a theoretical paper which appeared quite recently [8] we find evaluations forthe effective energy e* of the " t a i l " electrons:(where T is the temperature of the plasma " c o r e " , and t is the REB duration), andfor the density of the electrons responsible for the absorption of Langmuir waves:*Tbn,, ~ n,.

me 2
~ n b •me 2 + E

328

ARZHANNIKOV et al.

The parameters of the "tails" obtained experimentally are in reasonable agreementwith theoretical evaluations.
For the conditions we have described, the distribution of diamagnetism over thelength of the system is strongly inhomogeneous and is in reasonable agreement witha relationship of the form

W ± (z) ~where 60 is the initial angular scatter of the beam, z is the distance between theanode foil and the point of REB injection, and I is the relaxation length (seeRefs [9,10] for more details).
As we have already indicated, the theoretical evaluations of Brejzman and
Erofeev [8] predict an increase in the density of "hot" electrons with decreasing
angular scatter. This has also been confirmed experimentally [11]. Thus we may
assume that the inhomogeneous distribution of diamagnetism is most likely due toinhomogeneity in the density distribution of "hot" electrons over the length of thesystem.

4.

THE GOL-M DEVICE

Since the beginning of 1986 a new machine has been operating at the Instituteof Nuclear Physics: the GOL-M, designed for continued research into the interactionof powerful REB with plasma in strong magnetic fields (up to B = 6 T). This device
can also be used for engineering experiments on various components of the large
multimirror system called GOL-3, which is being built at the Institute, and fordeveloping new methods of diagnostics which can be used in strong pulsed magnetic
fields. Figure 2 shows a diagram of this machine, which consists of the following
elements:(1)(2)

An REB generator (1) with a beam current of Ib = 100 kA, electron energy
Eb = 0.5 MeV, beam duration (at half-height) rb = 120 ns and energy of thebeam injected into the plasma Q = 3.5-4 U;
A system for producing a magnetic field, consisting of a solenoid (2) with a total
length of 7.5 m and maximum magnetic field strength 6 T and two autonomous
end mirrors (3) with maximum magnetic field strength up to 12 T. The periodof the magnetic field, both at the mirrors and at the solenoid, is about 11 ms.
The transverse component of the magnetic field at the solenoid is suppressed
to a level B ± /B = 10~3 while high transparency of the solenoid is conserved;

With these new parameters, then, we carried out experiments on magnetic beamcompression which were in fact a continuation of the work described in Refs [11, 13,

329

IAEA-CN-47/C-O-4

GOL-M
nD=1Q|Z|-10l6cm-3

E b =D.5MeV
I b >1DDkA
db=Bcmt 1/2) =12Dns
FIG. 2. Diagram of GOL-M device:
1 — diode ofREB generator
2 — solenoid section
3 — end magnetic mirrors
4 — current lead-in
5 — ruby laser and system for recording Thomson scattering
6 — nine-channel Michelson interferometer (Kg = 3,39. y.m)7 — single-channel interferometer
8 — cooled interferometer detectors (Ge-Au, T = 77 K)
9 — four-channel X-ray crystal spectrometer
10 — valve for pulse target shaping

Boor Q(J)

I

I

B(kG)
1O

SO

30

<4Q

BO

FIG. 3. Energy content of plasma after REB heating as a function of magnetic field (GOL-M):
ne - 10" cm'3, j b = 2.7 ± 0.3 kA-cm~2, db = 6 cm.

330

ARZHANNIKOV et al.

(3)

A vacuum chamber constructed of quartz tubes 108 mm in diameter, the endsof which are pneumatically sealed. Two methods of producing the pre-plasma
have been tried: an oscillating discharge of the Penning type and a direct discharge. In the density region n,. ~ 1015 cm" 3 , with a column diameter of
8 cm, the direct discharge gives better reproducibility of plasma parameters.

The results of the preliminary experiments suggest that the effectiveness of theinteraction is enhanced by increasing the magnetic field strength. Experiments werecarried out at ne ~ 10 15 cm~ 3 , Teo = 1 eV, transverse plasma dimension
d = 8 cm, and beam current density j b = 2.7 ± 0.3 kA-cm~2. The initial angular
scatter of the REB in the plasma is V?o = 20°. Figure 3 shows experimental
results for the efficiency of the REB-plasma interaction as a function of magnetic
field strength. Each point in this figure represents a calculation of total plasma energy
content:

3 fl

dZ

The measured temperature of the plasma electrons after heating is ~ 10 eV, whichis in agreement with the Te evaluation for the case of plasma heating by reverse current. An evaluation of Te from diamagnetic measurements gives a considerably
higher value. Thus, we find the results obtained earlier on INAR and GOL-1 confirmed in stronger magnetic fields as well: when a high current REB interacts withplasma, a non-equilibrium distribution function prevails. A substantial part of theenergy imparted by the beam to the plasma is given up in the "tails".

5.

NEW EXPERIMENTS DESIGNED TO PRODUCE A MICROSECONDELECTRON BEAM IN THE U-l MACHINEThe U-l electron accelerator is an LC-generator which produces a high voltage
pulse; this passes through the main discharge gap to a plane graphite cathode (1)21.5 cm in diameter, located in the diode chamber (Fig. 4). At the diode a longitudinal magnetic field of 0.5 T is set up. The electron beam is extracted from the diodethrough an anode foil (2).
The beam parameters were higher than those quoted in Ref. [11], firstly
because an extra LC-generator was installed parallel to the one already in place [12]and secondly because of the new design of the accelerator diode chamber which madeit possible to reduce current leakage. As a result, the energy content of the beam
extracted from the diode — which in this case was recorded by a graphite
calorimeter [9] located just after the anode foil (2) — was raised to 130 kJ. The
voltage on the diode was 1 MV, the current 60 kA, and the duration at the base 4 fis(Fig. 5(a)).

331

IAEA-CN-47/C-II-4-20

z (cm)

FIG. 4. (a) Experimental layout: I — cathode; 2 — anode foil; 3, 4, 5 and 6 — magnetic system coils;
7 — foil; 8 — cathode holder,(b) Distribution of magnetic induction along device axis.

14]. In these experiments the energy content of the initial beam Q1( received by thecollector after foil 2, was 110 kJ (Fig. 5(b)). The distance between the cathode andthe anode foil was 9 cm. The anode foil was dacron a few tens of microns thickcovered with aluminium on both sides. The beam compression was done with amagnetic field of mirror configuration [15] (0.5 T at the anode foil and 10 T at themirror). The distance between foils 2 and 7 was 55 cm. The pressure in the compression chamber could be varied from 10 ~5 to 1 torr. The Rogowski coils P2 and P5
measured the beam current at the inlet, I2, and at the outlet, I5, of the compression
chamber. Coils P3 and P4 recorded the total current made up of the beam current
and the inverse current in the plasma being formed. Coil P, recorded the diode current. The energy content of the beam, Q2, after passing through the magnetic mirror
was measured by the graphite calorimeter (9), and also from the integral J UI5dt.
The experiments showed that the beam is transported most effectively(v — Q2/Q] =90%) through the magnetic mirror when the gas pressure in the compression chamber is 10 ~4 torr. If this condition is satisfied, the beam in thecompression chamber is neutralized in relation to the space charge, but no current

332

ARZHANNIKOV et al.

1 (MV),

80 (kAl

FIG. 5. Oscillograms of diode voltage U and current I:(a) Beam is received by calorimeter 9 installed in place of the compression chamber: Ig — generator
current, Ip — beam current;(b) Beam is received by collector placed immediately after anode foil 2; I2 is current recorded bycoil P2;(c)
Bam passes through compression system: Is is the current recorded by coil P5.

K (%)

y.

(MV)
0.8
0.6

60

- \j

0.4 -

40

I \
/

*

20

0.2

/
/

0
Ik A)
40 -

80

t (ps)
i

i

i

0.5

1.0

1.5

„•".,

30 •
20
10

FIG. 6. Diode voltage U and current at inlet (I2) and outlet (Is) of compression chamber when the
beam is injected into compression chamber through 30 urn Al foil. Calculated values of the beam reflection coefficient K and of the current which should be passing through mirror are indicated by crosses(d = 8.5 cm).

IAEA-CN-47/C-II-4

333

compensation has yet been observed. As a result of these experiments, we obtained
a compressed beam with a diameter of 4 cm, energy content Q2 = 100 kJ andcurrent density 5 kA-cm" 2 . The corresponding oscillograms are shown in
Fig. 5(c). There is some reduction of energy content with beam compression owing
to the earlier shorting out of the diode.
Experiments were carried out to investigate how the angular scatter of the initial
beam affects the operation of the diode with REV injection into a magnetic mirror.
Replacement of the dacron anode foil by an aluminium layer with a thickness of
10 fim increases the number of electrons reflected from the mirror. These electrons
oscillate through the anode foil, losing energy in it, which results in earlier plasma
formation at the anode and accelerates the shorting of the diode. Increasing the thickness of the foil to 30 fim of aluminium puts the diode into an "impedance collapse"
regime in which the voltage on the diode falls off very sharply (Fig. 6) but the rateof current rise increases [16]. At the beginning of the pulse the diode is cut off by
the space charge from the reflected and oscillating electrons, and then, after theformation of plasma at the anode, moves into a collapse regime which also explains
the observed five-fold decrease of diode impedance. With a thin anode foil, when thenumber of reflected oscillating electrons is small, there is practically no influence ofthe compression system on the diode and the beam can be compressed with great
efficiency.

6.

BAND BEAMS: THE U-2 DEVICE

In order to increase the energy content of the plasma, Voropaev and coworkers [13] suggested injecting through the ends of the machine a series of electron
beams generated successively in a number of separate diodes. These are initiallygenerated as band beams and are cast onto a single "magnetic track"; subsequently,
they are converted into ordinary cylindrical beams by an appropriate manipulationof the magnetic flux. The scheme used for successive injection of two beams into theplasma is shown in Fig. 7. In the vacuum magneto-insulated diodes Dl and D2(Bz = 0.5 T), band beams with a 4 x 140 cm cross-section are generated in turn at
an electron current of j ~ 0.1 kA-cm~ 2 . The thin band beams are transported in
narrow channels Kl, K2 and K3 with conducting walls even without neutralization,
because with this type of beam and channel cross-section the potential sag due to thecharacteristic space charge of the beam electrons is not very large. In cases where
the switching field Bx = 0, the magnetic flux in channel K3 contains about half ofthe magnetic fluxes in channels Kl and K2. By switching on the magnetic field Bx
we can transfer virtually all the magnetic flux from transport channel Kl to channel
K3 (in one specific calculation variant Bx = 0.1 Bz). When the sign of Bx isreversed, the flux from channel K2 is transferred to channel K3.

334

ARZHANNIKOV et al.

oYFIG. 7. Layout for successive injection of band shaped REB into the plasma. D, and D2 are thevacuum magneto-insulated diodes, K,, K2 and K} are the beam transport channels.

U-2 GENERATORHIGH-VOLTAGE
GENERATOR

FIG. 8. Layout of the U-2 generator: I — high voltage generator, 2 and 4 — insulators, 3 — spark
gap switch, 5 — cathode.

IAEA-CN-47/C-II-4

335

Thus, transport channel K3 will receive first the beam from diode Dl and thenthe beam from diode D2. With further propagation of the beam in the transportchannel K3, the flux cross-section of the guiding magnetic field undergoes a twentyfold area compression as it approaches the plasma trap, and at the same time the shapechanges to circular from being strongly elongated along the y co-ordinate. Similarchanges also occur in the beam cross-section. It is assumed that charge neutralizationof the REB occurs as an effect of the ions.
One of the main difficulties in providing for the transport of a band beam
without axial symmetry is that the transverse cross-section shape of the beam canchange as a result of the drift movement of the electrons in crossed electrical andmagnetic fields. In order to keep the shape of the beam cross-section unchanged
during its movement in a homogeneous magnetic field, we must satisfy the condition
that the constant density level lines of the beam current should coincide with the lines
of electron motion in a plane perpendicular to the vector of the guiding magnetic
field [17]. Analytical and numerical calculations have made it possible to find a broadclass of such equilibria. It has emerged that, for the case where the beam issymmetrical in the channel and uniform in cross-section, the ratio of the beam width
to the channel width can — in an equilibrium configuration — be unambiguously
established at 3/4. The gap between the beam and the conducting wall, which accordingly, for this case, has to be 1/8 of the channel width, can be considered acceptablefrom the standpoint of insulating the electron flux from the metal in magnetic fields
with Bz S; 0.5 T.
The U-2 facility is being built at the Institute of Nuclear Physics forexperimental investigations of the generation, switching and transport of large band
beams. The structure of one large band-shaped REB generator is shown in Fig. 8.
LC-generator (1) generates a voltage pulse with a rise time of 12 ps and a no-load
amplitude of up to 1.7 MV. This pulse passes via insulator (2) to the controlled gas
discharge gap (3). When this operates, the voltage from the next insulator (4) istransferred to vacuum diode (5). The beam cross-section has a length of about
140 cm and its width can be selected in the range from 4 to 12 cm.
At the present time the experiments being carried out on the U-2 devices aredesigned to study generator operation at equivalent active load with a resistance ofabout 20 ohms. The tests have indicated great reliability of generator operation withan on-load energy output of more than 100 kJ within about 10 ms.

7.

SUMMARY OF MAIN RESULTS(1)

Direct measurements have shown (on INAR) that heat transport by hot plasma
electrons takes place at a velocity of the same order as the thermal velocity ofthe electrons. This is an important fact to know if we wish to devise a methodof "two-stage" plasma heating.

336

ARZHANNIKOV et al.

(2)

It has been established (on GOL-1) that in the REB-plasma interaction hot
electrons are formed even at large distances (~ 7 m) from the point of REB
injection into the plasma.
(3) The tendency observed earlier towards increased plasma heating efficiency withincreasing field strength continues right up to the largest fields obtained inpresent experimental work (4.5 T on GOL-M and 7 T on INAR).
(4) On the U-l device, REB parameters have reached the following values: electronenergy 1 MeV, pulse length 3-4 p.s, energy content before beam compression
130 kJ. After compression in a 10 T field the energy content is 100 kJ and thecurrent density 5 kA-cm"2.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]

[12]
[13]

[14]
[15]

[16]
[17]

KOIDAN, V.S., KRUGLYAKOV, E.P., RYUTOV, D.D., in High Power Electron and Ion
Beam Research and Technology (Proc. 4th Int. Conf. Palaiseau, France, 1981) 531.
ARZHANNIKOV, A.V., BURDAKOV, A.V., KOJDAN, V.S. et al., Pis'ma Zh. Ehksp. Teor.
Fiz. 27 (1978) 173.
ARZHANNIKOV, A.V., BURDAKOV, A.V., BURMASOV, V.S., etal., in High Power Electron and Ion Beam Research and Technology (Proc. 3rd Int. Conf. Novosibirsk, 1979) 29.
ARZHANNIKOV, A.V., BURDAKOV, A.V., KOIDAN, V.S., VYACHKSLAVOV, L.N.,
Phys. Scr. T2/2 (1982) 303.
ARZHANNIKOV, A.V., BURDAKOV, A.V., KOIDAN, V.S., et al., in Plasma Physics (Proc.
Int. Conf. Lausanne, 1984), Vol. 1, Invited Papers (1984) 285.
RYUTOV, D.D., SAGDEEV, R.Z., Zh. Ehksp. Teor. Fiz. 58 (1970) 739.
KRUGLYAKOV, Eh.P., LUK'YANOV, V.N., PODYMINOGIN, A.A., in Diagnostika
Plazmy, No. 5, Ehnergoatomizdat, Moscow (1986) 85.
BREJZMAN, B.N., EROFEEV, V.I., Fiz. Plazmy 11 (1985) 387.
BREIZMAN, B.N., KRUGLYAKOV, E.P., in Phenomena in Ionized Gases (Proc. 1st Int.
Conf. Minsk, 1981), Vol. 1, Invited Papers (1981) 93.
BURMASOV, V.S., KHILCHENKO, A.D., KORNILOV, V.A., et al., in Controlled Fusion
and Plasma Physics (Proc. 10th Europ. Conf. Moscow, 1981), Vol. 1 (1981) paper C-2.
ARZHANNIKOV, A.V., BREIZMAN, B.N., BURDAKOV, A.V., et al., in Plasma Physics
and Controlled Nuclear Fusion Research 1984 (Proc. 10th Int. Conf. London, 1984), Vol. 2,
IAEA, Vienna (1985) 347.
VOROPAEV, S.G., LEBEDEV, S.V., CHIKUNOV, V.V., SHCHEGLOV, M.A., Pis'ma Zh.
Ehksp. Teor. Fiz. 11 (1985) 267.
VOROPAEV, S.G., KNYAZEV, B.A., KOJDAN, V.S. et al., in Engineering Problems of
Fusion Reactors (Proc. 3rd Ail-Union Conf. Leningrad, 1984), Vol. 1, Moscow (1984) 298 (inRussian).
CHIKUNOV, V.V., KNYAZEV, B.A., KOIDAN, V.S., et al., in Controlled Fusion and Plasma
Physics (Proc. 12th Europ. Conf. Budapest, 1985), Vol. 9F (1985) 508.
VOROPAEV, S.G., GORBOVSKIJ, A.I., KNYAZEV, B.A. et al., Voprosy atomnoj nauki itekhniki (Problems of Atomic Science and Technology), Ser. termoyad. sintez (Nuclear Fusion
Series), Vol. 2 (1986) 25.
VOROPAEV, S.G., KNYAZEV, B.A., KOJDAN, V.S., et al., Preprint 86-82, Institute of
Nuclear Physics, Academy of Sciences of the USSR, Siberian Branch, Novosibirsk (1986).
ASTRELIN, V.T., EROFEEV, V.I., in Summaries of Papers Presented at 6th All-Union Symp.
High-Current Electronics), Vol. 1, Tomsk (1986) 160.

IAEA-CN-47/C-II-5

ENHANCEMENT OF RADIAL TRANSPORTBY COLLECTIVE PROCESSES
D.L. EGGLESTON, J.D. CRAWFORD,
T.M. O'NEIL, J.H. MALMBERG
Physics Department,
University of California, San Diego,
La Jolla, California,
United States of America
Abstract
ENHANCEMENT OF RADIAL TRANSPORT BY COLLECTIVE PROCESSES.
The paper shows how collective effects can enhance the radial transport produced when nonaxisymmetric fields are applied to a nominally cylindrical pure electron plasma. Both linear and nonlinear
collective effects are considered and experimental demonstrations of each effect given. In the linear
case, the field asymmetry drives an asymmetric plasma mode and the mode field produces additionaltransport. In the nonlinear case, the field asymmetry can either excite daughter modes via a decay
instability or drive a mode by induced particle scattering. Since the physical basis for these processes
is quite general, they may be relevant for understanding nonambipolar transport in other rotating plasma
systems such as tandem mirrors.

1. Introduction
This paper shows how collective effects can enhance the radial transport
produced when nonaxisymmetric fields are applied to a cylindrical pure
electron plasma. This enhancement may be produced by either linear or
nonlinear collective effects and both cases are observed experimentally.
Because the physical basis for these processes is quite general, they may be
relevant for understanding nonambipolar transport in other rotating plasma
systems such as tandem mirrors.
It is well known that the presence of nonaxisymmetric fields produces
radial ion transport in tandem mirrors [1]. Such fields also produce radialtransport in pure electron plasmas. The transport problem may be viewed ashaving two parts. The first part is the determination of the self-consistent
fields in the plasma and the second part is the determination of the transportof particles in those fields. The point of this paper is that the first part of the
problem is nontrivial. In a rotating plasma the field asymmetry may act as apump for various collective processes and thus excite large asymmetric mode
fields in the plasma. This collective enhancement of the field asymmetry
results in enhanced radial transport.

2. Theory
The simplest case is that of a linear resonance between an external fieldasymmetry with a Fourier component e »('*+**-"') a n d a standing wave in the
337

338

EGGLESTON et al.

plasma column. This requires that w, k , and / of the field asymmetry satisfy
the dispersion relation for the mode of interest. For confinement
considerations, it is of particular interest to note that a mode can occur which
propagates backwards on the rotating column so that it has zero frequency inthe laboratory frame. Such a mode can be driven to large amplitude by astatic field asymmetry and the large mode field can then produce enhanced
transport [2],It is important to consider nonlinear generalizations [3,4] of this linear
process, since a zero frequency eigenmode will not generally appear in abounded plasma for arbitrarily chosen values of density, magnetic fieldstrength, column length, etc. The simplest generalization merely replaces thelinear resonance with a resonant three-wave interaction in which the fieldasymmetry acts as a pump which drives two daughter modes via the decay
instability. For a static field asymmetry, the daughter modes must havefrequencies that satisfy the resonance condition o^+o^—0.
In a second generalization, one of the daughter modes is in effect replacedby resonant particles. The field asymmetry nonlinearly excites the remaining
mode through induced scattering off of these particles (i.e. nonlinear Landau
growth). For finite columns, which necessarily have discrete frequencies, thissecond mechanism can occur more generally than the decay instability because
the resonance condition <x>i+u)^Q is not required. These two processes are notexhaustive (higher order analogues are obvious), but are illustrative of thekind of nonlinear effects that can degrade plasma confinement.
3. Experiment
We have experimentally studied the collective enhancement of radialtransport in a pure electron plasma device [5]. The extremely long
confinement times [6] of these devices make them especially suitable forcontrolled studies of induced radial transport. Radial confinement is provided
by a uniform axial magnetic field while the axial confinement is provided byelectrostatic potentials. Typical parameters are nt f^lO7 cm"3, Tc f«l eV,
B0=350 G, and a plasma lifetime exceeding 105 axial bounce times.
Our device is shown schematically in figure la and our basic experimental
sequence is a modification of techniques described in detail elsewhere [5,6]. The
cylindrical conducting wall of the device is divided into rings of various
lengths and two of the rings (Si, S2) are divided azimuthally as well. Rings
Gl and G3 are normally held at a large negative potential and the remaining
rings are grounded. To start a cycle, ring Gl is switched to ground potential,
allowing electrons from a negatively biased spiral filament to fill the device.
Ring Gl is then returned to a negative potential, thus producing a confinedplasma sample. After the procedures described below are completed, theelectrons are dumped by grounding ring G3. Electrons are collected either by
a radially moveable probe (giving the line integrated density at a particularradius) or by a plate (giving the total number of remaining electrons).
To demonstrate the collective enhancement of transport by a linear
resonance we proceed as follows [7]. After the electrons are confined we apply
a nonaxisymmetric electrostatic perturbation by placing a signal of frequency
w on the two 180° azimuthal sectors of ring SI in a push-pull fashion. Ring
S2 is used to monitor the plasma for nonaxisymmetric modes. By applying

339

IAEA-CN-47/C-II-5(a)
SOVJf!CE!:!

Gl

G2 SI Rl

L1

12 1

(!

L3

((

R2

U

((

S2 R3 G3G4COLLECTORS

100

200

300

400

500

600

700

800

Perturbolion Frequency (kHz)FIG. 1. (a) Schematic of experimental device.
(b) Normalized central density change due to the application of an electrostatic field asymmetryof variable frequency. The radial transport is enhanced at normal mode frequencies.

this perturbation only on alternate machine cycles and analyzing the change insignal on the collectors, one may obtain, for example, the density change
6n ( r ) due to the applied field asymmetry. Figure lb shows the perturbationinduced density change at the center of the plasma as a function of appliedfrequency. At the frequencies where normal modes are excited (as indicatedby the detector on S2) we also find an enhanced level of radial transport asreflected in the increase of the magnitude of 6n .
We have also observed enhancement of transport by nonlinear collective
effects. Here we present an example of an induced scattering instability driven
by a static field asymmetry. For such an instability a single growing mode
should be observed and in this case the mode is an / = 0 , k =ir/L plasma
wave, where L is the length of the plasma column. The wave becomes
unstable when a threshold condition is exceeded and eventually saturates after
growing 40-50 dB. Three features of this instability identify it as due toinduced scattering: 1) The threshold condition, in addition to depending on

340

EGGLESTON et al.

the mode damping, depends on the amplitude of the applied asymmetry. 2)
The presence of the instability is associated with an enhanced level of radialtransport. This is significant since an azimuthally symmetric wave (/ = 0 ) does
not by itself produce a radial drift of particles. 3) The instability does not
require any particular tuning of conditions but appears over a broad range ofplasma parameters.
According to theory [4], the time evolution of the observed mode isgoverned by the equation

[

]m

(1)

where 7 m is the linear mode damping rate, Am and Ae are the mode and fieldasymmetry amplitudes, and F is the beat-resonant coupling coefficient.1 Whenthe bracketed term is positive, the mode is unstable and grows exponentially.
Since any real plasma device will have small asymmetries due to construction
imperfections, At is always nonzero. In our experiments, these small
asymmetries are sometimes large enough to exceed the threshold condition andthe instability appears as soon as the plasma is captured. Such a case isshown by curve a in figure 2. The wave starts to grow immediately after the
injection gate is closed ( t « l m s ) . 2 We may prevent the mode growth, in
accordance with equation (1), by increasing the value of | 7 m | • One way to
do this is by placing a resistive load between a ring within the containment
volume (e.g. L3) and ground. Wall currents induced by the wave flow throughthis load and dissipate the wave energy. This creates additional mode
damping beyond that produced by wave-particle effects. The stabilizing effectof such an addition is shown by curves 6 and c in figure 2. In curve 6 theplasma remains stable until the resistive load is removed at the time indicatedby the lower arrow. In curve c the instability is quenched by the addition of
the resistive load at the time indicated by the upper arrow.
Equation (1) also predicts that a stable plasma will be destabilized byincreasing the magnitude of the field asymmetry. To demonstrate this effectwe adjust the injection conditions to create a stable plasma (i.e. a plasma
where the mode Landau damping is large enough to prevent the spontaneous
growth shown in figure 2). We drive the / = 0 mode by applying a short (50/is) tone burst at the mode frequency to ring R3. At the same time we apply
a static field asymmetry to the plasma by placing voltages on the 180° sector
probes of SI and S2. Figure 3 shows the mode amplitude versus time,with thevoltage applied to the sector probes as a parameter. As the asymmetry
amplitude increases, the mode becomes more lightly damped and finallybecomes unstable.
The rate of loss of plasma particles due to radial transport is increased
by as much as a factor of twelve when the instability is present. Preliminary
measurements show that the loss rate increases with the amplitude of both theunstable mode and the field asymmetry. These results agree with theoretical
expectations.
1

2

In what follows we assume T is positive. The requirements for this to be true are
discussed in reference [4].
The large signal observed from t *» 0-1 ms is broadband noise due to the plasma injection
and should be ignored in this context.

341

IAEA-CN-47/C-O-S

1 2

FIG. 2.

3

4

5

6

7

Control of induced scattering instability by varying the mode damping rate.

50 r

t(ms)
FIG. 3. Time evolution of a launched t = 0 mode with field asymmetry magnitude as a parameter. The
mode is destabilized as the field asymmetry magnitude is increased.

342

EGGLESTON et al.

As stated above, this instability occurs over a wide range of plasma
parameters. This is consistent with expectations for an induced scattering
instability. In contrast, the linear resonance and the decay instability bothhave frequency matching conditions which, in general, would only be satisfied
for certain sets of plasma parameters. Because of this, the induced scattering
process may be more important for enhancing radial transport than either the
linear resonance or the decay instability.

4. Conclusion
We have shown how linear and nonlinear collective effects can enhanceradial transport in a rotating plasma subject to nonaxisymmetric fields.
Although our work involves transport in a pure electron plasma, similar
processes may affect the transport in other confinement devices. Processes
analogous to our linear collective enhancement have been considered for both
toroidal devices [8] and tandem mirrors [9]. To our knowledge, however, there
have been no studies of nonlinear collective enhancement in such devices.

Acknowledgments
This work was supported by DOE Grant DEFG 03-86ER53199 and NSF
Grant PHY83-06077.

REFERENCES
[]]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]

RYUTOV, D.D., STUPAKOV, G.V., Sov. Phys. Dokl. 23 (1978) 412.
KEINIGS, R., Phys. Fluids 24 (1981) 860.
CRAWFORD, J.D., O'NEIL, T.M., MALMBERG, J.H., Phys. Rev. Lett. 54 (1985) 697.
CRAWFORD, J.D., O'NEIL, T.M., Nonlinear collective processes and the confinement of apure electron plasma, submitted to Phys. Fluids.
MALMBERG, J.H., DeGRASSIE, J.S., Phys. Rev. Lett. 35 (1975) 577.
DRISCOLL, C.F., MALMBERG, J.H., Phys. Rev. Lett. 50 (1983) 167.
EGGLESTON, D.L., O'NEIL, T.M., MALMBERG, J.H., Phys. Rev. Lett. 53 (1984) 982.
STRINGER, T.E., Phys. Rev. Lett. 22 (1969) 770.
LEE, X.S., MYRA, J.R., CATTO, P.J., Phys. Fluids 27 (1984) 2248.

IAEA-CN-47/C-H-6

RADIOFREQUENCY PLUGGING AND
TANDEM MIRROR EXPERIMENTS
IN A CUSP-ANCHORED MIRROR DEVICE
T. SATO, R. KUMAZAWA, S. OKAMURA, K. ADATI,T. AOKI, H. FUJITA, S. HIDEKUMA, T. KAWAMOTO,
H. MASUMOTO, K. NISHIMURA, T. HATORI, K. TAKAYAMA1
Institute of Plasma Physics,
Nagoya University,
Nagoya, Japan
H.R. GARNER, H.D. PRICE2, A.M. HOWALD, B.J. LEIKIND,
H. IKEZI, P.B. PARKS, P. ANDREWSGA Technologies, Inc.,San Diego, California,
United States of America
Abstract
RADIOFREQUENCY PLUGGING AND TANDEM MIRROR EXPERIMENTS IN A CUSPANCHORED MIRROR DEVICE.
The paper covers three experimental topics: (1) An empirical scaling relation for the radiofrequency plug potential (ponderomotive potential) has been obtained and the maximum plug potential
measured is 1 kV. (2) A negative potential mode produced with ICRF power only has been discovered
and its equilibrium is investigated. (3) The MHD stability condition in a cusp-anchored mirror plasma
has been obtained experimentally.

1.

INTRODUCTION

The RFC-XX-M device has a linear, axisymmetric, MHD stable configuration,
which is a simple mirror (central section) terminated by spindle cusps with radiofrequency (RF) plugging. RF plugging is one of the important possible means of suppressing end losses in open-ended systems. Preferential RF plugging [1] is especially
useful for keeping the confinement region free of impurities. MHD stability in
RFC-XX-M is ensured by the axisymmetric cusp anchor which can prevent
anomalous perpendicular transport. It is valuable to compare the ambipolar potential
plugging in this axisymmetric device with other, non-axisymmetric tandem mirror
devices. Experiments on ambipolar potential plugging have been carried out since
1982, in a Japan-USA co-operative effort.
1
2

Tokai University, Hiratsuka 259-12, Japan.
Physics International, San Francisco, CA, USA.

343

344

SATO et al.

I
1

345

IAEA-CN-47/C-H-6(b)10

ij (10 cm-3)

RF electcode
55 60 65 70 75 80
r (cm)

55 60 65 70 75 80
i-

(cm)

FIG. 2. Ion density distribution along the magnetic lines of force in the plug section,(a) a/o>ci = ].35, (wp/wci)2 = 14, Vj=4kV,} = 0.8 kV. (b) a/o>ci = 0.89, (ap/aci)2 = 12,V^ = 4 kV, \p = 0 kV. o — without RF plugging, • — with RF plugging.

2.

RADIOFREQUENCY PLUGGINGIn 1984, at the London Conference [2] we reported that the energy confinement
time due to the end loss of ICRH-sustained plasma is improved to about 10 ms by
RF plugging and that the RF plug potential is produced for frequencies above thelocal ion cyclotron frequency at the line cusp. In this paper, we confirm the effectof RF plugging by measuring the ion density distribution along the magnetic lines offorce in the plug section, and we obtain an empirical scaling relation for the RF plugpotential in a broad range of plasma parameters.
A schematic diagram of RFC-XX-M is shown in Fig. 1. The distance between
field-null points is 3 m. The magnetic field strength at full excitation is 2.1 T at theline cusp, 0.35 T at the central mirror midplane, and 0.97 T at the mirror throat. A
rotating (phase) Nagoya type-Ill antenna is located in the mirror throat for plasma
production and is driven by two 0.4 MW RF oscillators (f = 7 MHz). For RF
plugging, a pair of ring electrodes is installed in each line cusp and connected to a
1 MW RF oscillator (f = 24.5 MHz).
The ion density distribution along the magnetic field lines in the plug section
is measured by a movable Langmuir probe. Figure 2(a) shows that the ion density
distribution has a very steep gradient under the RF electrodes when the applied RF
frequency, w, is greater than the local ion cyclotron frequency, « ci , under the RF
electrodes. This suggests that an RF plug potential exists under the RF electrodes.
In fact, the RF plug potential, \(/, measured by a method discussed later, is 0.8 kV.

346

SATO et al.

(a)f (kV)
1.0

10

20

50

100 200

YJkV)

0.5

160
Hh1

1.2

1.3

1.4

^T
1.5

1.6

10

20
50 100 200<Wpi/a>ci> 2

FIG. 3. (a) RF plug potential ^ versus oi/oicjfor V^= 4 kV, as a parameter of (o)pi/o>ci)2.
(b) Dependence of the optimum frequency wj<aci on (oipi/o>ci)2.
(c) ^ versus (upi/aci)2 when u = co^.

In the case of «/<oci = 0.89 ( < 1), however, no RF plug potential is produced andconsequently the ion density distribution is almost the same as in the case without RF
plugging, as shown in Fig. 2(b).
The RF plug potential is measured by means of multi-grid energy analysers.
(MGAs) located on the two end walls of the line cusp. In this experiment, the RF
plugging is applied to either the LS line cusp or the LB line cusp of RFC-XX-M. In
the ion energy analysis the RF plug potential is determined from the difference
between the plug-side MGA and the non-plug-side MGA. Details are described inRef. [2].
The RF plug potential is plotted in Fig. 3(a) as a function of co/coci. The
parameter is the normalized plasma density, (upi/wci)2, where o>pi is the ion plasma
frequency evaluated by using the density under the RF electrodes. The optimum
value of w/a)ci, where ^ reaches its maximum value ^!|t, varies with (a>pi/a)ci)2. This
optimum frequency, referred to as w^/a^;, is plotted in Fig. 3(b) as a function of(copi/wci)2. It shifts from 1.35 to 1.5 as (a>pi/uci)2 increases from 10 to 160. This isconsistent with the dispersion relation of the eigenmode of the ion Bernstein wave
in the plug region [3]. Figure 3(c) shows the dependence of ^ on (wpi/wci)2 in thecase of the applied RF voltage Vrf = 4 kV. The value of ^ reaches 1 kV for(upi/o>ci)2 = 10 and is proportional to (wpi/o.>ci)~'-6. This result is similar to thatobtained in the TPD-III device [4] and agrees with theoretical expectations [5]. The
dependence of ^ on the plug RF voltage is shown in Fig. 4 for co/wci = 1.41 and347

IAEA-CN-47/C-II-6

1 2

5
Vrf (kV)

10

FIG. 4. Dependence of0^ on V^foro>/ad - 1.41 and (upi/o>ci)2 = 40.

(wpi/wci)2 = 40. The results indicate that ^ is approximately proportional to thesquare of Vrf. These relations can be unified and expressed in the following equation, with the frequency chosen to be equal to w^[kV] = 4.7 E?f [kV/cm]

1-1.6

(1)

where Erf = Vrf/d and d is the separation of the RF electrodes.
We can conclude from these results that: (i) the RF field penetrates into the line
cusp plasma by coupling with the eigenmode of the ion Bernstein wave, (ii) the
penetrating RF electric field produces an RF plug potential (ponderomotive potential)
as expressed by Eq. (1), and (iii) the potential operates to suppress the ion loss flux
through the line cusp.

3.

NEGATIVE POTENTIAL MODE

In 1984, a new plasma equilibrium was discovered in RFC-XX-M. We call thisthe negative potential mode (n-mode) because end-loss analysers indicate the plasma
potential to be near zero, and because measurements of the negative potential and theprobe floating potential show a deep (negative) barrier between the central cell andthe cusp end cell. Heavy ion beam probe measurements of the central cell plasma
potential show it to be much lower than in the normal mode, and measurements ofplasma rotation using a carbon tracer indicate that the central cell plasma potential

348

SATO et al.

central cell nltem"2)
LS cusp nl (cm"2)
central cell V,,oat (V)LS throat V,,oat (V)LS cusp V, lMt (V)51015
time (ms)FIG. 5. Line density and floating potential as Junctions of time at various axial locations.
A small gas puff in the central cell is responsible for the changes at 14 ms.

-2.0

-1.0
0.0
1.0
axial position (m)2.0

FIG. 6. Axial floating potential profile and line density profile, showing the presence of a potential
barrier between the central cell and the cusp end-cells.

is near zero. The n-mode is a tandem mirror operating mode which is sustained by
ICRF only, for up to 150 ms (limited by the ICRF power supply).
The line density and floating potential as functions of time are shown for a typical shot in Fig. 5. For this shot the plasma starts out slightly negative and is driven
further negative by a small gas puff in the central cell at 14 ms. It is also possible
to adjust the RF phasing and fuelling so as to achieve a transition from a normal mode
plasma to the n-mode plasma during a single shot. The two modes have very different
axial and radial profiles and confinement properties.

IAEA-CN-47/C-II-6

349

The axial density profile and the floating potential profile in the n-mode areshown in Fig. .6. The cusp end-cell densities are high because of the fuelling there,
the central cell is fuelled by ions trapped by ICRF or collisions, and the density inthe barrier is low because it is located on a magnetic anti-well. Floating potentials
as large as —500 V have been measured on the anti-well. The radial density profile
is highly peaked on axis, and the electron temperature profile is flat at about 20 eV.
The potential barrier is largest near the plasma edge where the density is low and theE] fields are high.
The global particle and energy confinement time is less than 1 ms. The
measured central cell confinement time exceeds 1 ms and is possibly longer than the
shot duration (up to 150 ms).
Scaling studies indicate that the mode is sensitive to internal RF fields (i.e. thephasing of the rotating Nagoya type-Ill antenna), the gas puff location and timing,and the wall conditions. The potentials of the central cell and the barrier are insensitive to the RF power (less than Pft5). The mode can be formed as long as the fast
wave can propagate (high plasma density), and a region of strong E| exists between
the midplane and the field maximum at the end of the central cell. Gas fuelling is done
primarily in the cusp end-cells to prevent cold ions from filling in the barrier regions.
We have used the computer code GARFIELD [6] to calculate the E| RF fields
in the plasma. Using this code together with a new analytic model, a relationshipbetween the density variation and the potential variation is obtained which corresponds to the experimentally measured profiles. Model ion distribution functions
have been constructed, with the main features of the expected actual distributionfunctions in the strong ICRH limit [7]. Since these are functions of the invariant
energy and magnetic moment, parallel pressure balance is guaranteed. The Boltzmann equation fails under these conditions and the anisotropic electron pressure
driven by the E| RF fields must be taken into account. This model predicts barrier
potentials much greater than the perpendicular electron temperature, as have been
observed experimentally.

4.

MHD STABILIZATION BY THE CUSP ANCHOR

The mirror plasma in RFC-XX-M is quite stable in the usual operational modes,
such as the normal mode, the RF plug mode, the normal tandem mode and the negative potential mode [8]. To verify the effect of the cusp anchor, an MHD instability
is intentionally generated by reducing the cusp plasma pressure with respect to themirror plasma pressure. In the normal mode, gas is puffed into the central section
only. By adjusting the gas pressure, the ratio of the density in the cusp region to thedensity in the central section can be changed. When the ratio is less than a criticalvalue, MHD instability occurs, as shown in Fig. 7(a). During the instability, thediamagnetic loop signal and the microwave interferometer signal show a sawtoothtype relaxation oscillation, the repetition time of which is about 0.5 ms. The signal

35G

SATO et al.

(a)

(b)

fc«w

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4 1

central mirr Dr beta valueomagnetic probeodiamagnetismo"0

0.1 0.2 0.3 0.4 0.5
density ratio (n() c /(nl) M

FIG. 7. (a) Typical example of MHD instability, (b) Stable and unstable shots for various settings ofthe beta value of the central mirror plasma and the ratio of the cusp plasma line density to the mirror
plasma line density, (nt)c/(nt)M.

4
6
time (ms)4
6time (ms)FIG. 8. Stabilization of MHD instability by gas injection into the cusp region.
The traces are for (]) central mirror diamagnetism, (2) magnetic probe signal,(3) central mirror line density and (4) cusp line density.

of the MHD fluctuation is detected by a magnetic probe placed at the outer boundaryand by the crown-of-thorns probe shown in Fig. 1. The fluctuation shows an
exponentially growing oscillation, with a frequency of about 30 kHz, and rotates inthe direction of the ion diamagnetic current with an azimuthal mode number 1. Whenthe amplitude reaches a threshold level, the diamagnetic loop signal and the microwave interferometer signal begin to decay sharply and reach their minimum values
within 100 /*s. A that time, the MHD fluctuation has its maximum amplitude and thendisappears.

IAEA-CN-47/C-II-6

351

In Fig. 7(b), stable and unstable shots are plotted for various settings of the beta
value of the central mirror plasma and the ratio of the cusp plasma line density (nQc
to the mirror plasma line density (nQM. This figure shows that the mirror plasma isstable when the line density ratio is larger than 0.27. In this experiment, the ion temperature of the mirror plasma and the cusp plasma is measured. From this measurement the critical pressure ratio corresponding to the line density ratio can be
estimated as 0.13, which means that the cusp plasma, whose pressure is only 1/8 ofthe mirror plasma pressure, ensures MHD stability of the mirror plasma.
The plasma pressure in the cusp region is sustained by the outgoing ICRF wave
launched at the mirror throat and thus it is possible to raise the cusp pressure simply
by additional fuelling in the cusp region. An example of stabilization of MHD instabilities is shown in Fig. 8. During the unstable state, a small amount of gas is injected
into the cusp region. The cusp plasma density increases and the instability is stabilized when the density ratio exceeds the above-mentioned critical value.
A cusp anchor is very attractive from the point of view of axisymmetry, small
plasma volume and strong good curvature.

ACKNOWLEDGEMENTS
The authors wish to express their gratitude to Professor H. Kakihana, theformer director of the Institute of Plasma Physics of the Nagoya University, to Drs
T. Ohkawa and R.L. Freeman from GA Technologies Inc., and to Dr. R.E. Pricefrom the United States Department of Energy, for their encouragement.
This work was supported in part by the United States Department of Energy,
under Contract No. DE-AT03-83SF11713.

REFERENCES[1] HIDEKUMA, S., et al., Phys. Rev. Lett. 33 (1974) 1537.
[2] OKAMURA, S., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984
(Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 337.
[3] CHAUDRY, M.B., et al., J. Phys. Soc. Jpn. 51 (1982) 2666.
[4] HIROE, S., et al., Phys. Fluids 21 (1978) 676.
[5] HATORI, T., Phys. Fluids 28 (1985) 219.
[6] PHILLIPS, M.W., TODD, A.M.M., Comput. Phys. Commun. 40 (1986) 65.
[7] PARKS, P.B., et al., in Sherwood Controlled Fusion Theory Conference (Proc. Conf.
New York, 1986), Paper 3D-2 (1986).
[8] PRICE, H., et al., Bull Am. Phys. Soc. 30 (1985) 1493.

Session D
ALTERNATIVE AND SUPPORTING SYSTEMSChairmen of Sessions
Session D-I

L.M. KOVRIZHNYKH

USSR

Session D-II

T. UCfflDA

Japan

Session D-III

G. GRIEGER

Germany, Fed. Rep.

Session D-IV

F.L. RIBE

USAPapers D-IV-1-1 and D-IV-1-2
were presented by B.L. Wright as Rapporteur
Papers D-IV-2-1 and D-IV-2-2
were presented by T. Ishimura as Rapporteur
Papers D-IV-5-1 and D-IV-5-2
were presented by S. Denus as Rapporteur
Session D-V (Posters)

IAEA-CN-47/D-I-1

STUDIES ON CURRENTLESSHELIOTRON E PLASMA
K. UO, A. IIYOSHI, T. OBIK1, S. MORIMOTO, O. MOTOJIMA, M. SATO,
S. SUDO, F. SANO, T. MUTOH, K. AKAISHI, M. NAKASUGA, K. KONDO,
K. HANATANI, H. ZUSHI, S. BESSHOU, H. KANEKO, T. MIZUUCHI,
H. OKADA, Y. TAKEIRI, N. NODA, I. OHTAKE, Y. UIRI, M. IIMA,
T. KAWABATA, T. SENJU, K. YAGUCHI, T. BABA, S. KOBAYASHI,
R. TAKAHASHI,
M.MURAKAMI 1 ,H.C.HOWE 1 ,D.L. HILLIS 1 ,1
1
2
P.K. MIODUSZEWSKI ,G.H. NEILSON ,J. RICE ,J. TERRY 2 ,23
33
E. MARMAR , T. YAMASHINA , T. HINO , H. MINAGAWA , T. ODA 4 ,M. HAMAMOTO5,
T. OHGO 6 ,Y. SAKAMOTO7,
H. AMEMIYA7,77
78K. OKAZAKI , H. OYAMA , S. KATO , A. AMEMIYA , H. SUGAI9,
S. URANO9, K. UCHINO 10 , T. KOGA10, K. MURAOKA10, M. AKAZAKI10,
M. MAEDA", S. MORITA12

Plasma Physics Laboratory,
Kyoto University,
Gokasho, Uji, Kyoto,Japan

Abstract
STUDIES ON CURRENTLESS HELIOTRON E PLASMA.
Three kinds of heating, i.e. electron cyclotron resonance heating (ECRH), ion cyclotron frequency range heating (ICRF), and neutral beam injection heating (NBI) have been applied to heat the
Heliotron E plasma. There has been particular interest in studies of confinement scaling, particle andimpurity control, high beta properties, and high energy drift orbit dynamics. At a magnetic field of
Bh = 1.9 T, the central chord averaged ion temperature reached more than 1.2 keV with NBI and morethan 1.6 keV with ICRF at low densities (H e =2xl0 1 3 cm" 3 in NBI and ^ = 0 . 5 X10 13 cm" 3 in ICRF).
The central electron temperature Te(0) of ECRH plasmas scales as T e (0)a(P rf IHJ('M, where Prf is the
ECRH power injected. The observed rE(a) scaling of NBI plasmas is: rE(a) oc P^mM Re"'54. w h i c nshows a positive density dependence of the energy confinement time. This density dependence partially
offsets the negative power dependence. In the case of long pulse ECRH + NBI produced plasmas, a1

Oak Ridge National Laboratory, Oak Ridge, TN, USA.
Plasma Fusion Center, Massachusetts Inst. of Tech., Cambridge, MA, USA.
3
Department of Nuclear Engineering, Hokkaido University, Sapporo, Japan.
4
Department of Applied Physics, Hiroshima University, Higashi-Hiroshima, Japan.
5
Department of Applied Physics, Oita University, Oita, Japan.
6
Physics Department, Fukuoka University of Education, Munakata, Japan.
7
Institute of Physical and Chemical Research, Saitama, Japan.
Department of Nuclear Engineering, Nagoya University, Nagoya, Japan.
9Department of Electrical Engineering, Nagoya University, Nagoya, Japan.
10
Department of Energy Conversion, Kyushu University, Fukuoka, Japan.
11
Department of Electrical Engineering, Kyushu University, Fukuoka, Japan.
12
Institute of Plasma Physics, Nagoya University, Nagoya, Japan.
2

355

356

UO et al.

small ECRH power added to the NBI plasma was found to pump out impurities. A wall carbonization
technique was also found effective in reducing iron impurities and thus contributed to raising the stored
energy with extended pulse length. In addition, the topology of drift orbits modified by the presenceof a radial electric field was examined numerically.

1. INTRODUCTION
The recent Heliotron E experiments have emphasized studiesof heating and confinement of currentless plasmas, and are also
concentrated on particle and impurity control, aiming at longpulse operation. The currentless plasma is first initiated by
ECRH (53.2 GHz, <500 kW) and is additionally heated by NBI (3beam-lines, <4 MW through the ports) [1] or ICRF (26 MHz, 2 MW)[2],This paper discusses the behavior of ECRH and ECRH+NBI
heated plasmas with special reference to the mechanisms that
control the energy transport. Results of ICRF are reported inanother paper of this IAEA conference [3],2. ECRH EXPERIMENTS
The 53«2 GHz microwave power was injected into the Heliotron
E vacuum vessel with the T E Q 2 and H E ^ modes. The resonant
magnetic field is 1.9 T. The resonance position is adjusted tothe position of the plasma center. The microwave (rf) power
injected was varied from 160 kW to 500 kW by increasing thenumber of gyrotrons (up to 3 gyrotrons) [2]. Figure 1 shows atypical example of the temporal development of line-averaged
electron density, electron and ion temperatures, bolometric loss,
and soft X-ray intensity. These parameters become constant fromthe early phase of the rf pulse. The flat top time is much
longer than the confinement time estimated from the decay ofelectron and ion temperatures. This result shows that a steadystate plasma is maintained by ECRH. A temporal increase of
impurities has never been observed. The relevant studies of
impurity behavior are described in section A- The gross energy
confinement time estimated from the electron decay time is U ms.
Figure 2 shows the profiles of T e (r), n Q (r), T i (r), theabsorbed rf power density P a b s (r), and the integrated rf power
P i n t (r).
The density is n"e=2.5x1O13 cm"^. The electron density

357

IAEA-CN-47/D-M

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FIG. 1. Temporal behavior ofne, TJO), Tfh (9=0), PM, S.X. at Bh =1.9 T.
Pf = 500 kW with D2 gas.

profile is flat or slightly hollow at this density. However, inthe lower density plasmas, the hollow profile becomes more
typical. The rf energy-deposition profile, Pai3sM> w a s measured
from the electron energy decay time. I t was assumed that theenergy confinement time did not change immediately after the rfpower had been turned off.
The total rf absorption efficiency,P^ n ^(a)/P r £, was about 50 to 70%. I t depends on the electron
density and temperature. From one third to half of the absorbed
rf power was deposited inside the a/2 radius. The deposition
profile was found to be flatter than expected from a single-pass,
ray-tracing calculation for the TEQ^ mode injection.
The central electron temperature Te(0) scaling is shown in
Fig. 3. Te(0) was measured with a laser-Thomson-scatteringsystem. I t scales as Te(0) a (P r f /n e ) 0 # ^ 1 ± 0 ' 1 , where P r f is the
ECRH power injected. The heating rate becomes smaller as P r f /n e
increases. The c o l l i s i o n a l i t y of electrons near the plasma
center is in the plateau regime. On the other hand, i t is in thelocalized particle (ripple-trapped-particle) regime in the region
from around r =0.35 to f=0.8a.

UO et al.

358

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0J0

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cm'3 at Bh=1.9 T. P4=500kW with D2 gas.

0

FIG. 3. Central temperaturen =1.0-2.5XIO'3
cm'3.

0.1

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is shown for

0.4
0.5 0.6
0.7
Pe (MW/io'3cm3)

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T,

kW,

and

3 . NBI EXPERIMENTS

The f i r s t phase of the NBI experiment in the Heliotron E
device, which was reported previously [ 1 ] , focused on the ionheating efficiency achievable for the medium i n j e c t i o n power
level of Pin-s<2 MW. The encouraging results led to the secondphase experiment (from 1984) using much higher injection power up

359

IAEA-CN-47/D-I-1
t

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FIG. 4. Dependence of Tfh (6=0) on Pio^/ne.
injected through the ports.

Pimz is the initially ionized neutral beam power

to 4. MW. One of three beam lines was replaced. I t s injection
angle was changed from 62° to 90°. The second-phase study now inprogress i s also marked by an evolving understanding obtainedwith a variety of operational tools such as impurity injection[5]» p e l l e t fueling [6], pump-limiter [7], and i n - s i t u carbon
coating [8, 9]« The ion heating efficiency of the second-phase
experiment (13^=1.9 T) i s shown in Fig. 4-.
Within theexperimental error bars, i t i s the same as that of the f i r s t phase experiment (Fig. 1 of the previous report [1]), though theinjection angle was changed to perpendicular. With an injected
power of 3-5 MW, Heliotron E was able to achieve ion and electron
temperatures of Tich(6=0)>1.2 keV and Te(0)>0.7 keV at the lineaveraged densities of iie=2-3x101^ cm~^. Here T i ch (9=0) i s thecentral-chord averaged ion temperature measured by a neutral
particle analyzer. The central ion temperature is estimated tobe higher by a factor of 1.2 to 1.3 than t h i s value. The database obtained from 1985 to 1986 includes the whole operating
regime in density (5Q=1.5-9x10cm~^) and injection power
Pir.=O-3.5 MW at two different magnetic fields «(Bh=1.9 and0.94 T). The global energy confinement time T E (a), defined as
TE(a)=W ( a ) / P n e a t ( a ) , i s 8-20 ms. W (a) i s the t o t a l plasmainternal energy and Pnea-t(a) ^"s ^ n e n e a ^i n S power calculated by
6an orbit following Monte-Carlo-code [10]UO et al.

360

0

0.5

1.0
P

1.5
he 8 . <

2.0
MW

2.5

3.0

»

FIG. 5. Dependence of energy confinement time rE(a) on NBI heating power.

rg

As shown inthe confinement time was observed, to scale as T g(a)heatn j 3 , where a =-0.64+0.15 and 6=0.54+0.15. Although the T £ (a)dependence on the heating power P ^ e a t ^ i s a i m i l a r t o thatobserved in tokamaks (L-mode), the positive density dependence
partially offsets the negative power dependence. Since nQ <* ?^_nA
experimentally, the value of T- E ( a ) appears roughly constant athigh injection power. For this data base, the local energybalance has been studied using the ORNL-developed transport/data
analysis code, PROCTR [11]. A relatively large uncertainty
exists in these studies, which is the level of thermal neutrals
and the resulting loss of fast ions due to charge exchange
process.
To improve the accuracy of the neutral-density
estimation, the laser-induced fluorescence using Balmer alphatransition is measured [12]. In the case of low power and low
density, the convective loss can be dominant in the ion power
balance at r=2a/3. As the power and density are increased, theelectron conductive loss becomes dominant in the over-all energybalance. A comparison was made between the experimentallyinferred energy transport loss rate and the corresponding
neoclassical rate (including the helical ripple and radialpotential [13]). The inferred experimental electron energy
transport coefficient is close to the neoclassical value at very
low heating power. However, it increases to a substantiallygreater value than the neoclassical rate at high heating power.

361

IAEA-CN-47/D-I-1

25
to

20

5«

m=3/n=2
INTERNALDISRUPTION

-

Ul

o
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FIG. 6. Cosine coil signal versus density profile peaking parameter ?/.
i|«ii, (central chord)/ne (peripheral chord).

The ion loss rate is in the neoclassical range within a factor of
3 at all heating powers [22],In addition to the transport study, MHD activity has been
analyzed to explore the stability map for several operationalmodes (Bh=0.94 T). Figure 6 shows a plot of a cosine-coil signal
against the density-profile peaking parameter n (=ne(center
chord)/n e (peripheral chord)).
The cosine-coil signal is ameasure of the beta value and the pressure profile. The NBI
plasma initially lies in the stable-plasma region of low beta andflat density profile. With a temporal increase in beta followedby an increase in injection power, the plasma treads two
different paths. The first path can be called the stable path (Qmode) [ K ] , since no MHD instabilities have been observed. Alongthis path, the profile was kept broad by additional gas fueling.
The MHD beta limit is not reached yet. The second path (S mode)is chosen by a condition where the pressure profile becomespeaked with less or no gas puffing. Pellet injection usuallycauses the density profile to be more peaked. This produces the
S mode plasma. The initially observed instability of the S modeis the m=1/n=1 mode, and thereafter as predicted from MHD theory
the m=3/n=2 mode dominates with increases in beta and peaking
parametern C15 ] • The m=3/n=2 mode leads to an internal
disruption.
In this moment, the density (pressure) profilechanges from a peaked one (unstable) to a broader one (stable).
These observations suggest that the MHD beta limit in S mode isobserved in the case of peaked profiles.

UO et al.

362

30

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4

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250

X = 4cm

300

350

400
450
TIME (ms)

500

550

FIG. 7. Electron density nc and gas puff program: before activation of the limiter pumps (OFF) andafter activation (ON). The NBI power of3.2 MWwas injected through the ports. The limiter was inserted
4 cm inside the last closed flux surface.

LONG P 0 L S E

PLASMA PRODUCTION

AND RELATED PARTICLE AND

IMPURITY CONTROL

Long-pulse ECH+NBI plasma production has been achieved so
far with careful gas-puffing control. Gas-puffing control isclosely related to the particle- and impurity-transport behavior.
One approach to active particle control is pellet fueling [6]. A
sharp r i s e in density due to p e l l e t fueling provides anencouraging prospect for particle control and a
resultant

improvement in particle confinement. Both the electron and ion
temperatures recovered to the levels of the pre-pellet phase in
about 30 ms. Since the injecting power does not change, t h i smeans the improvement of energy confinement as density increases.
Another approach for particle control i s the u t i l i z a t i o n of a
pump-limiter [7]. Figure 7 shows the global effects of the pumplimiter on the core-plasma density. The gas flow rate, cj>[torr1/s], and neutral beam power were identical both before and after
the activation of the gettering-purap of the pump l i m i t e r . A
clear density drop was observed after the gettering-pump was
activated, when the limiter was located 4 cm inside the l a s t
closed flux surface.
For long-pulse operation, impurity contamination of theplasma is a large problem due to the enhanced radiation loss. The
main impurity species in neutral beam heated Heliotron E plasmas
are iron, chromium, nickel (the material of the wall, YUS 170
stainless steel), titanium (the gettering material), oxygen, and363

IAEA-CN-47/D-I-1

(a)

2000

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AVERAGED DENSITY (10'Vcm 3 )20

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TlME(ms)10
TlME(ms)

60

FIG. 8. (a) Transport properties analysed by simulation fitting. Diffusion coefficient D and flow
velocity V as a junction of ne. Dashed lines show the density dependence observed in well controlled
shots. The cases (b) and (c) below are pointed, (b) In NBI plasma (Te(0) — 400 eV), the linear increase
of SX at the central chord shows impurity accumulation. The code predicts the probable values of
D-800
cm2-s~' and V-400 cm-s~' (fine lines), (c) NBI plasma with overlapped ECRH
(TJ0) ~ 450 eV) shows shorter impurity confinement time. The SX signal increases more rapidly due toenhanced transport (D~ 2500 cm -s ', V ~ 500 cm-s ) than the previous case (b), but it decays.

carbon.
At high densities (ne>6x10 ^ cm ^), a substantial
contribution to the radiation loss from the iron impurity wasobserved in the spectroscopic measurements. The impurity problem
is solved essentially by suppressing the impurity influx, e.g.,
by wall carbonization. However, the impurity contamination
process is closely related to the impurity transport. In orderto study these transport mechanisms, the laser blow-off technique
has been used to inject trace impurities such as silicon into the
ECRH+NBI plasmas [5].
Figure 8(a) shows the values of D
(diffusion coefficient) and V (inward flow velocity at the plasma
surface), evaluated by a 1-D numerical simulation code. The
horizontal axis i s the line-averaged density.
With wellcontrolled operation, gas puffing was successfully applied toprovide a steady-state, long-pulse NBI plasma. This aims at

364

UO et al.

achieving a temporally constant density phase with adequate gas
puffing.
The comparison between ECRH and NBI plasmas yielded a
new aspect of impurity t r a n s p o r t .
In NBI plasmas with a densityof 1.7x10 -^ cm" , where t h e c o n t r o l of t h e gas p u f f i n g i s
d i f f i c u l t , impurity accumulation was observed a s shown i n F i g .
8(b) (see t h e t r a c e f o r SX). The c o r r e s p o n d i n g d a t a p o i n t s a r e
shown in Fig. 8(a). The confinement time of injected s i l i c o n was
900 u s . When t h e ECH p u l s e was a p p l i e d t o t h e NBI plasma, t h eimpurity density showed a simple decay with a time constant of
60 ms (Fig. 8(c)). This means t h a t the i m p u r i t i e s in NBI plasmas
were pumped out by applying the ECRH power.
A local measurement of iron atom d e n s i t i e s was made using a
c o a x i a l l a s e r - i n d u c e d f l u o r e s c e n c e s p e c t r o s c o p i c system [ 1 6 ] .
The t e m p o r a l and s p a t i a l measurements f o r NBI plasmas haverevealed t h a t the locations of the iron sources are broader than
the width of t h e d i v e r t o r t r a c e of t h e chamber [ 2 1 ] . Thissuggests t h a t a c o r r e l a t i o n i s p r e s e n t between charge exchange
neutral fluxes and fluorescence i n t e n s i t i e s for low-density
plasmas (ne=2-3x10 ^ cm"-'). The obtained results reveal that therelease of iron atoms from the surface of the chamber can beinterpreted as being due to sputtering by charge-exchange
neutrals.
The impurity fluxes in the plasma edge region were also
examined using a surface analysis station with an "in-situ" AUGER
electron spectroscopic system [17]. The dominant impurity
species deposited on the a-S^ probe was iron when the probesurface was d i r e c t l y exposed to the plasma (ne=2-3x10 ^ cm"-').
The amount of deposited iron atoms increased with increasing
neutral beam power. There was a correlation between this amountand the bolometer signal as well as with the brightness ofmultiply ionized iron ions. Since the low-temperature edge
plasma (Te<10 eV) can not assume a principal role in sputteringof iron atoms, the higher-energy p a r t i c l e s such as charge
exchange neutrals (and fast ions on loss orbits) are considered
to be responsible for releasing the iron atoms. This result wasconsistent with the observation obtained from the laser-induced
fluorescence measurement. However, i t should be noted that those
two measurements have been made locally in poloidal and toroidal
positions.
I t i s necessary to continue the experiments toinvestigate whether the impurity sources are local or not.
The carbonization ( i n - s i t u carbon coating) technique has
been established in tokamaks to reduce metal-impurity contents.

365

IAEA-CN-47/D-M
WITHOUT CARBONIZATIONWITH CARBONIZATION^BI 1.8r MW

5>

3.5 4.0
MW MW

\

\

390
TIME ( m s )

290

390

490

TIME (ms)

FIG, 9. Time behavior of ne , bolometric signal iand a diamagnetic loop signal in NBI heatedplasmas with and without carbonization. Heating scheme is shown at the top.

This is a method to make thin carbon films on the surfaces of a l l
in-vessel components, thus providing an environment for metalimpurity free plasma [8,9]. In Heliotron E, the carbonization
has been conducted with a DC glow discharge in a methane/hydrogenmixture with wall at room temperature. The coating rate was
200 A/hour. After the carbonization (300-500 A), the NBI pulse
length (at densities over ne=10 ^ cm~^) could be extended while
keeping temperatures almost constant. The radiation intensityfrom the iron impurity (e.g., Fe XVI) was reduced by more than afactor of 10 compared to the intensity before carbonization. Thet o t a l bolometric loss [18] was also reduced by a factor of 2-3.
This was a primary reason that the long-pulse operation became
possible with high-power NBI and RF. Figure 9 shows typical time
behavior of n e , P^o]_ and a diamagnetic-loop signal in NBI heatedplasmas with and without carbonization.
Without carbonization,
the radiation loss, especially in low-field operation, increased
continuously during the heating pulse, which was typical in lowfield operation. The stored energy decreased in the later halfof the pulse. With carbonization, the radiation intensity became
stationary at the end of the heating pulse. The stored energy
measured increased continuously. A higher averaged-beta value
was obtained. An additional r e s u l t with the carbonization was
the shift of achievable density to higher values due to thestrongly enhanced recycling on the carbonized surfaces. The

UO et al.

366

(a)

(b)

FIG. 10. Two different types oj the resonant orbits confirmed numerically in HeliotronE. (a) Resonant
superbanana orbit, (b) Resonant banana orbit.

highest density so far i s iie=1.8x101^ cm~^ at Bh=1.9 T in pelletaided gas fueled operation.
5 . DRIFT ORBIT CALCULATION

Recent neutral beam injection experiments on the WVII-As t e l l a r a t o r suggested that the e l e c t r i c potential <>j plays key
roles in both confinement and heating of the plasma [19]» It isinteresting to know under what conditions (sign, magnitude andprofile of (j)) the confinement of the ion orbits in Heliotron E isimproved or deteriorated by the Er field.
To answer thisquestion, calculations have been carried out by using a newly
developed orbit following code. In order to reduce the computing
time, this code uses the "field splitting" algorithm [20], which
decomposes the fully 3D field quantities into helically symmetric(2D) and symmetry-breaking (3D) p a r t s . The E r f i e l d i s
calculated from the potential which is constant on 3D magnetic
surface;
E=- V<>j (ty), 2 TT^ = /BdS being the toroidal flux. Two
different types of resonance were confirmed numerically. One isthe "helical resonance" (ft h +fi ExB =0) and the other is the"toroidal resonance" (-tvii/R+ ft EXJJ=O). Near these conditions,resonant super banana (Fig. 10(a)) and resonant banana orbits(Fig. 10(b)), which cannot be found without an Er field, wereobtained. If the Er field is strong enough (off resonance), theIAEA-CN-47/D-I-1

367

electric potential improves the confinement of helically trapped
particles. Another notable effect of the Er field is that i t
induces additional collisionless detrapping.
ACKNOWLEDGMENTS
I t i s a pleasure to acknowledge the other members of theLaboratory whose dedicated efforts have enabled us to conduct
these experiments. We are grateful to Drs. M. Wakatani and K.
Itoh for stimulating discussions. The US participants were
supported by the US Department of Energy Contact No. DE-ACO5840R2U00 and DE-AC02-78ET51013.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]

UO, K., HEUOTRON E GROUP, in Plasma Physics and Controlled Nuclear Fusion Research
1984 (Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 383.
IIYOSHI, A., HELIOTRON E GROUP, ibid., Vol. 1, p. 453.
MUTOH, T., HELIOTRON E GROUP, Paper IAEA-CN-47/D-III-2, these Proceedings, Vol. 2.
UO, K., HELIOTRON E GROUP, Nucl. Fusion 24 (1984) 1551.
RICE, J.E., TERRY, J.L., MARMAR, E.S., MOTOJIMA, O., HELIOTRON E GROUP,
Nucl. Fusion 24 (1984) 1205.
SUDO, S., HELIOTRON E GROUP, Nucl. Fusion 25 (1985) 95.
HILLIS, D.L., et al., in Plasma-Surface Interactions in Controlled Fusion Devices (Proc. 7th
Int. Conf. Princeton, NJ, 1986), to appear in J. Nucl. Mater.
WINTER, J., ibid.
NODA, N., HORI, Y., MASAI, K., et al., ibid.
HANATANI, K., et al., Nucl. Fusion 25 (1985) 259.
HOWE, H.C., Physics Models in Tokamak Transport Code PROCTR, Rep. ORNL/TM-9537,
Oak Ridge National Laboratory (1985).
OKANO, D., et al., in Plasma-Surface Interactions in Controlled Fusion Devices (Proc. 7th Int.
Conf. Princeton, NJ, 1986), to appear in J. Nucl. Mater.
HASTINGS, D.E., HOULBERG, W.A., SHAING, K.C., Nucl. Fusion 25 (1985) 445.
MOTOJIMA, O., HELIOTRON E GROUP, Nucl. Fusion 25 (1985) 1783.
WAKATANI, M., et al., Paper IAEA-CN-47/D-V-3, these Proceedings, Vol. 2.
KONDO, K., HELIOTRON E GROUP, in Plasma-Surface Interactions in Controlled Fusion
Devices (Proc. 7th Int. Conf. Princeton, NJ, 1986), to appear in J. Nucl. Mater.
MIZUUCHI, T., HELIOTRON E GROUP, ibid.
BESSHOU, S., MORIMOTO, S., MOTOJIMA, O., et al., Nucl. Fusion 26 (1986) 114.
GRIEGER, G., WVII-A TEAM, Plasma Phys. Controll. Fusion 28 (1986) 43.
HANATANI, K., UO, K., in 3-D MHD Studies (Proc. US-Japan Workshop Oak Ridge,TN,
1984) (1984) 7.
MOTOJIMA, O., MIZUUCHI, T., BESSHOU, S., IIYOSHI, A., UO, K., et al., Nucl. Mater.
128&129 (1984) 524.
SANO, F., OBIKI, T., MOTOJIMA, O., SASAKI, A., HANATANI, K., et al., Nucl. Fusion
24 (1984) 1103.

368

UO et al.

DISCUSSION
K. LACKNER: In your ECRH heating experiments, did you change the power
deposition profile by varying Bo? If so, how did the Te profile respond? For
example, did you observe 'profile consistency', such as in tokamaks, or adjustmentof the Te to the power deposition profile, as in the Wendelstein VII-A stellarator?
K. UO: Yes, we did change Bo. The central temperature Te(0) is about
1 keV, if coce is on the magnetic axis (Bo = 1.90 T). It decreases to 0.6 keV when
Bo is reduced on 1.84 T, while the temperature at the half-radius Te(a/2) increases
from 0.7 keV to 0.85 keV. The 'profile consistency' is now under study by measurement of the electron energy transport, and more details will be available later on.
O.S. PAVLICHENKO: Did you try to measure any unidirectional current
during your NBI heating experiments, or do you consider that your plasma was reallycurrentless?
K. UO: In the cases of unidirectional NBI and ECRH, we observed a small
amount of current — less than 1 kA.

IAEA-CN-47/D-I-2

PLASMA CONFINEMENT
IN THE WENDELSTEIN VII-A STELLARATOR
H. WOBIG, H. MAASSBERG, H. RENNER,
W VII-A TEAM1, ECRH GROUP2, NI GROUP3
Euratom-IPP Association,
Max-Planck-Institut fiir Plasmaphysik,
Garching, Federal Republic of Germany
Abstract
PLASMA CONFINEMENT IN THE WENDELSTEIN VII-A STELLARATOR.
The confinement behaviour of the W VII-A stellarator depends on the magnetic field structure,
the heating method as well as the plasma properties. At low (m,n) rational numbers of the rotationaltransform, t = m/n, strongly deteriorated confinement is observed which is explained by island formation and ergodization of magnetic surfaces. It is only in the close vicinity of these fundamental
resonances, t = xh and lh, where the most stable magnetic surfaces are expected, that optimum confinement is found. It is sensitively dependent on both the edge value of t and on small additional shear.
It is only for these optimum confinement conditions that the particle and energy transport is analysedin the paper. — The ion energy transport in W VII-A is found to be in rather good agreement withneoclassical theory. In detail, in case of NBI heating, large radial electric fields are built up by the fastions, strongly improving the confinement of thermal ions. Collisional slowing down of the ionsoriginating from the NBI is found. For ECF heated discharges, the observed radial electric fields arecomparable with neoclassical estimates (by means of the ambipolarity condition). For the central partof the plasma, the particle fluxes are explained in terms of neoclassical transport; at outer radii, strongly
enhanced particle transport must be concluded. ECF heated discharges, especially with low density,
indicate neoclassical electron heat conduction in the central part; in the edge region, however, the electron energy confinement must be described by anomalous transport.

I.

Introduction

Experiments in the low shear stellarator W VII-A exhibit a critical dependence ofthe plasma confinement on the structure of the magnetic field. At particular valuesof the external rotational transform ((,, = ~ and ~), the confinement is deteriorated
whereas in the close neighbourhood of these values confinement time and energy content
are maximum (Figure ]). Roth energy and particle balance at optimum confinement
have been studied in "currentless" plasmas generated by NBI and EORH and will be
discussed in the subsequent sections. The residual current (1-2 kA) observed under these
conditions does not contribute to the heating power.
1

G. Cattanei, D. Dorst, A. Eisner, V. Erckmann, U. Gasparino, G. Grieger, P. Grigull,
H. Hacker, H.J. Hartfufl, J. Hackel, R. Jaenicke, J. Junker, M. Kick, H. Kroiss,
G. Kuehner, H. MaaBberg, C. Mahn, G. Miiller, W. Ohlendorf, F. Rau, H. Renner,
H. Ringler, F. Sardei, M. Tutter, A. Weller, H. Wobig, E. Wursching, M. Zippe.
2
W. Kasparek, G.A. Muller, P.G. Schuller, M. Thumm, R. Wilhelm (Institut fur Plasmaforschung, Univ. Stuttgart).
3
R. Freudenberger, W. Ott, F.P. Penningsfeld, E. Speth.

369

370

WOBIG et al.

w
[fcj]
3.0

WVIIA

BO-3.2T #35982-36046
B0-3.0T #39914-39936
t>2

/I
20-

Ip - 1 - 2 k A

uL-ov

!\ o

nh N
Mi'

1.0-

IT
II

u

v

Nl/Ti
H2
# 2.4,9\i[160-180 ms]

M

M
Q30

0.25

lltl

V*pla
0.35

0.40

0.45

050

0.55

fit II lit II

FIG. 1. Total plasma energy in W VIl-A as a Junction of rotational transform at plasma edge,ta = <0 + tp|a. Heating by neutral beam injection. O indicates regions of optimum confinement, M arestates of minimum confinement.

In NBI heated plasmas (Section II), the main topics are ion energy balance, the roleof I ho radial electric field and the heating mechanism, where in contrast to earlier assumptions / I / the classical slowing down mechanism could be verified. Paricle transport,
is discussed in Section 111. ECR heated plasmas offer a better chance to analyse the electron energy balance (Section IV) than NEil heated plasmas. In ECU heated plasmas, theelectron transport in the bulk of the plasma can be explained by neoclassical effects, inthe boundary regions, however, anomalous transport has to be assumed. In the minimaof confinement (c() — i, and ±), the most plausible explanation for the enhanced losses
is island formation in the magnetic field structure and stochastir.ity caused by external
perturbations. This effect will be discussed in Section V.

II.

NBI heating

The slowing down of the fast ions originating from the nearly perpendicular (84")
neutral beam injection had been experimentally proved to be collisional both for stationary conditions and after switching off the injectors j'lj. This was demonstrated byneutron flux measurements of D-D reactions in case of D" injection into a D + target
plasma as well as by the charge exchange diagnostic. Both the absolute value of thestationary neutron production as well as the transient neutron flux after the D° injection
switched off were found to be in rather good agreement with simplified Fokker-Planck
estimates based on the assumption of only collisiojial interaction (Coulomb scattering)/ 3 / . Additionally, the measured time decay of high energetic CX neutrals after switching

371

IAEA-CN-47/D-I-2
ON|OFF|ON
Measured flux ofhigh energetic
particles°- 8

Calculated
density of highly•0-8 energetic particles

0.6

0.6

^

-1

27 £ 1 keV

27.1 ±4.0 keV

0.4

0.8
21:1 keV

21.4 + 2.4 keV

0.6
• 1

0.8 17 = 1 keV

17.6 ± 2.0 keV °- 8

O.6
Ion cyclotron

0.4

radiation from47 (o 44 MHz

156 158 160 162
Time [ms]156 158 160 162
Time [ms]FIG. 2. Charge exchange fluxes measured (on the left) and simulated (on the right) versus time forphase of all neutral beam injectors switched off.

AT

o

feVl

o

ECE electron temperature

150 -

n
O

CXRS
CX

.
.
ion temperature# 52254 -52274
4lh injector

O

o
3—-

*>

—a—a-

o

100-

a

AT;/

a

•

/

r

50-

/ °
/°

ATe

0 -

160

165

170

t tms)

FIG. 3. Ion and electron temperature increase, AT, versus time after fourth neutral beam injector isswitched on (7] = 650 eV and Te « 500 eV).

372

WOBIG et al.

off the NBI could be reproduced by Monte Carlo simulation on the basis of only collisional slowing down (see Figure 2). The ion distributions were estimated to be highly
anisotropic / 4 / due to the high power NBI. By means of linear stability analysis / 5 / ,these ion distributions were found to be stable against low harmonic ion cyclotron instabilities (ion Bernstein modes) for stationary conditions. Unstable lower hybrid modes
are expected to smooth the ion distribution function near the injection velocities buthaving negligible effect on the energy transfer.
Only in the transient phase after switching on the injectors, the radiation close to
the low ion cyclotron harmonics (ICE) measured by a probe in the limiter shadow wasstrongly increased. From the stability analysis of the transient ion distributions afterswitching on the NBI, low harmonic ion Bernstein modes were found to be strongly
unstable. Although these distributions were calculated on the basis of collision al slowing
down, strong instability is expected leading to preferential ion heating. This prediction
is consistent both with a measured delay of the neutron flux increase and with the strong
increase of the ion temperature immediately after switch on of an additional beam.
This temperature increase, AT, (see Figure 3), is in rather good agreement with theassumption that the absorbed NBI power of the additional beam given by the diamagnetic
signal is only transferred to the ions; direct electron heating is rather small (AT, <g AT,).
For collisional slowing down, however, preferential electron heating should be expected
for the initial phase since high energetic ions are mainly slowed down by electrons.
From the measured Doppler shift of impurity lines connected with poloidal plasma
rotation, radial electric fields up to 1 kV/c.m were deduced. The toroidal plasma rotation,t\,,, was found to be not larger than th« poloid.il one: vg » tv^.jA where A is the aspect,ratio (A —- 20 for W VII-A); this result is quit« different, to tokamak cases /<>, 7/. Without
radial electric fields, the fast ioj] Joss rate is quite high due to the nearly perpendicular
injection and the small plasma radius in W Vll-A (r,, <10 cm) leading to only poor
heating efficiency. It was clearly shown by Monte Carlo calculations (ODIN code, /8/)
that the fraction of fast ion losses can be significantly reduced by radial electric fields
which, on the other hand, art: built up by the loss of such fast ions /O/. Due to thestrong radial electric fields deduced from the measured poloidal plasma rotation theheating efficiencies for both co- and counter-injection (which are analogously definedwith respect to the rotational transform r) are strongly improved. For both injection
cases which are experimentally realized by changing all magnetic field directions, similar
confinement properties were found. However, in case of counter-injection, higher electric,
fields, Er, were observed than for co-injection. This is consistent with the condition
ET -t- tv\\Br/R ~ const which can be derived for strong radial electric fields from thekinetic equation (poloidal component of (Tfy +T^£>) • V / =? const) describing the partial
compensation of the V5-drift effects.
The connection of fast ion losses with the global confinement properties was demonstrated by measuring the Fe sputtering by means of laser induced fluorescence (LIF).
By Monte Carlo calculations with electric fields included, the fast ion flux distribution
onto the stainless steel vacuum vessel was estimated leading to Fe densities which arewithin a factor of 2 in agreement with the LIF measurements. Figure 4 shows the time
behaviour of the Fe sputtering (upper plot) which is equivalent to the orbit loss rate for adischarge in which the rotational transform was transiently shifted over the deteriorated
confinement regime from the optimum confinement below to the one above t =; i . This
shift of the external rotational transform leads to a strong decrease in the energy content, W (middle plot), within the deteriorated confinement regime. The corresponding

IAEA-CN-47/D-I-2

373

WVII-A
Iron
density

B o = 3.2 T,NI,-fc0 varied
Resonance
fluorescence

o.t

0.15

O2

FIG. 4. Iron sputtering, total energy content and line density versus time. For these discharges, external rotational transform was shifted from optimum confinement below (line on left) to the one abovet = 'A (line on right).

decrease of line density, fndl (lower plot), is partially compensated by higher neutral
gas influx due to (lie increased recycling. The stronger neutral gas production is confirmed by an increase of the H a signal. Then, both the energy as well as (he particle
confinement times are significantly reduced close to the rational value c = ~ which isconnected with an increase of fast ion losses leading to a reduced NBI heating efficiency
and the strong decrease of the energy content. For degraded confinement properties, thisflux of fast ions onto the vacuum vessel is up to 20% of the total influx of NBI neutrals(the fraction of transmitted neutrals is typically between 30% and 60% depending online density). Consequently, fast ion losses can contribute significantly to the particle
transport. Furthermore, the poloidal plasma rotation was found to be smaller within thet-range of deteriorated confinement. These experimental findings confirm the correlation
between radial electric fields and the confinement properties of the fast ions. Although
the fraction of fast ions is relative small (typically <10%), the fast ion loss contributes
significantly to the particle fluxes. Consequently, the radial electric fields are mainly
determined by the fast ion loss rate to satisfy the ambipolarity condition of the particle
fluxes.

374
HI.

WOBIG et al.
Particle transport

Since in general detailed information about (lie particle sources is not available(especially the neutral gas distribution within the torus is largely unknown), the analysis
of particle transport lias to be based on some reasonable assumptions. This situation isdifferent to the energy analysis where profiles of heating power deposition and radiatedpower are known, at least to some extent. However, both particle transport and heat
conduction are strongly correlated in all transport theories of interest.
In the early phase of W Vll-A operation with only ohmic healing, detailed energy transport studies were carried out leading to the electron thermal conductivity / 9 / :X®u <x n^'1 Te7(see Section IV). Only for the case of ohmically heated helium discharges the particle diffusion coefficient could be derived from the measured ionization rates/10/. The radial behaviour was found to be similar to the electron thermal conductivity,
but the absolute value was about one order of magnitude smaller: Doli or 0.] • x ? " The flattening of the density profiles at outer radii which is typical for all discharges
in W VII-A and the increased particle sources due to neutral gas ionization lead tostrongly enhanced particle transport as it is described by the n and T dependence of
the anomalous diffusion coefficient DOH. Contrary to the electron heat conduction, theanomalous flux, fAN ~ —D°a n'., is less than or of the same order of magnitude as theneoclassical particle flux in the central part, of the plasma. The transport coefficients ofthe thermal ions are mainly determined by the values of the plateau collisionatit.y regime
which without radial electric, fields strongly exceed the electron transport coefficients.
In case of the strong neutral beam heating in W VII-A, the beam particle depositionis the main particle source in the central region of the plasma column leading to the nearlylinear density increase during the duration of the NBI heating (< 200 ms) for good
confinement conditions. The beam particle deposition, however, is not known accurately
enough since the ionizat ion profiles of the charge exchange generated thermal neutrals
cannot be determined up to now; the transport of these neutrals is mainly determined by
CX collisions quite similar to the problem of neutral gas influx. Assuming that all theseneutrals are ionized, a lower limit of the particle confinement time can be estimated.
In doing this, particle confinement times of typically more than 300 ms were found forthe central plasma region. These very good confinement properties were confirmed bypellet injection experiments where all the additional particles were confined for the restof the discharge duration (< 100 ms). Only in the case of an additional EOF pulse,
theses particle confinement properties seem to be degraded / l l / . For the other cases,
the experimentally deduced particle fluxes are of the same order of magnitude as theneoclassical electron transport in the bulk part of the plasma within the uncertainty ofthe particle sources. However, already in the maximum gradient region the anomaloustransport becomes dominant.
EOF sustained discharges at higher densities where the ions are effectively heatedby electron collisions can be analyzed on the basis of the neoclassical formalism since
the ion distribution functions are close to Maxwellians. As the ions are in the plateau
collisonality regime, their transport coefficients are reduced by a suitable radial electricfield in order to satisfy the ambipolarity condition. On the basis of measured profilesof density and temperatures, the local radial electric field is calculated by means of theambipolarity condition of only the neoclassical fluxes Vfc = P ^ ° consisting of PfirschSchliiter and plateau regime contributions / 1 2 / . The calculated radial electric fields arein good agreement with the values deduced from the Doppler shift of impurity lines for375

IAEA-CN-47/D-I-2
ECF heated discharges

NBI heated discharges#61003-61069#60547-60605

r [cm]

r [cm]

FIG. 5. Radial electric fields measured (dots) and calculated versus plasma radius for ECF and NBI
heated discharges, plots (a) and (b), respectively.

the case of ECF heated discharges (s. Figure 5a); however, it must be mentioned thatthe measured values are obtained at radii where the anomalous transport, is dominant.
Furthermore, on the basis of measured profiles it is impossible to satisfy the continuity
equation with only neoclassical fluxes at outer radii.
In order to get a consistent description, the total particle flux density is modeled by:
P = T N C + F A N where F N G is the neoclassical term consisting of Pfirsch-Schliiter andplateau regime contributions and T AN the anomalous term derived from the ohmicallyheated discharges (F A N ^ - D0H • n'.). In the plasma boundary region, the anomalous
term exceed.* the neoclassical term by orders of magnitude. Up to now, no provenpicture exists for these anomalous fluxes, however, it is believed that both the anomalous
particle transport and the anomalous electron heat conduction are caused by convective
transport. This hypothesis of convective transport based on ExB-drift is in agreementwith the experimental experience. Then, the a«oiiia/on.« transport of electrons and ionsis ;imbipolar on their own: F(AN - T A N . With this assumption, the radial electricfields have to be determined by the balance of the neoclassical fluxes (mainly of theplateau regime contributions) with the fast ion losses included in case of neutral beam
heating. Then, neoclassical theory may be used to estimate the radial electric fields andthe ion heat conduction. The basis of the neoclassical formalism is the expansion of
the distribution functions starting from Maxwellians. For estimating the radial electricfields, experimental values of density and temperature have to be used to define the local
Maxwellians. On this basis, the analysis of the EOF heated discharges supports the
hypothesis of tho .unbipolarity of the anomalous (luxes.

376

WOBIG et al.

In case of strong neutral beam heating, however, the highly anisotropic ion distributions violate the bask assumption of distribution functions being close to Maxwellians.
For tliis case, the observed radial electric fields are much larger than those estimatedwith the assumption of only thermal ions (s. Figure 5b) supporting the conclusion thatthe radial electric fields are mainly generated by the fast ions. The thermal ion heat conduction is strongly reduced by the electric fields in both cases. Only with this reductionof the ion heat conductivity coefficients, the ion energy balance could be satisfied forboth the high density EOF as well as the NBI heated discharges / 1 3 / . However, as theneutral gas density and thus the OX losses are largely unknown, the ion energy balance
is not conclusive at outer radii.
Finally, the influence of the anomalous particle transport on the ion energy balance isdiscussed shortly. At outer radii, the dominant anomalous flux is related to the ionizationof neutral gas: div PAN w {av)cncn,, where {av},. is the ionization rate due to electron
collisions. The contribution to the ion heat conduction (J, is given by % T{ F^ N and lessthan (he charge exchange ion energy loss by the factor (<?v)t / {av)cx where {crv)cx >s
the charge exchange rate coefficient; this factor is typically of the order of 0.3 at outerradii. Consequently, anomalous transport is of minor importance for the ion energybalance which may be significantly influenced by CX losses at outer radii.

IV.

Electron h e a t conduction

Local analysis of electron thermal conduction was made for OH- discharges, NBIheated discharges and ECRH-discharges. The transport coefficient was obtained fromthe measured profiles with the power deposition profile taken from the current density
profile (OH) or from numerical calculations (NBI).
In the analysis of ohmically heated discharges one has to avoid strong effects ofcurrent driven MHD instabilities which significantly deteriorate the confinement. In theregion outside the q = 1 surface, when strong tearing modes are absent, the local thermal
conductivity, Xet n a s been evaluated from local measurements of density, temperatureand radiative power. Within the main field range Bt, = 2.5 - 3.5 T, an experimental
thermal conductivity x ° H ~ 3.8 • 1018 • n~J • T*21* \cmr/s\ was found depending on thedensity n e [cm'3] and Te \eV\, see / 9 / .
The analysis of electron heat losses during currentless operation with NBI was moredifficult since the separation of electron heat conduction is hampered by the uncertainties
of the heating power deposition, the strong coupling of electrons and ions at densitiesn, ~ 1014 cm"0, the radiative losses, and the nonstationary conditions due to the strong
particle input flux. However, there are strong indications that neoclassical thermal conductivity alone is not. sufficient to model the discharge.
ECF sustained discharges seem to be more suited for analysing the electron energy
transport. The wave energy is transfered directly to the electrons. Other loss channels
such as electron ion coupling and radiation play a minor role. Consequently, the dependence of the heat conduction on various parameters (B, n c , t, Tt) could be studied.
Launching the rf power with different modes of irradiation (TE02, HE11 at. o-niode or
x-mode) and for off-axis resonance conditions, strong variation of the power deposition
profiles and the electron temperature profiles have been obtained /14/. Furthermore,
a power modulation technique has been applied. Analysing the generated heat waves
across the plasma, the local conductivity coefficients could be derived from the timedependent temperature profiles / 1 5 / .

IAEA-CN-47/D-I-2

377

A coupled system of equations for the energy balance of electrons and ions was solved
to fit the experimental density and temperature profiles / 1 6 / . The steady state TEMPL
code models the electron thermal conductivity by a sum of the axisymmetric neoclassicalpart, the magnetic ripple dependent neoclassical part and the anomalous part. Especiallyin the boundary, enhanced election energy transport seems to be necessary to fit the flat
temperature profiles. The anomalous transport, coefficient was assumed to follow thesame scaling as in ohmically healed discharges.
So far, at t-values with optimum confinement good agreement of experimental datawith the neoclassical model is concluded for the central jiart of the plasma. Especially athigher temperature (low collisionality) and large magnetic ripple at higher t, ripple effects
become dominant. In the boundary region where the density and temperature are small,enhanced losses have to be assumed. Atomic processes, perturbations of the magnetic
surfaces or drift, instabilities may be responsible for the enhanced loss. However, thedependence of the anomalous heat conduction XgH ~ C'-n~ l Te
restricts its influence
more and more to the boundary the higher the energy density of the plasma grows. The
anomalous transport used to fit the currentless EOF heated discharges is following theempirical scaling for OH discharges assuming the coefficient C to be proportional to B. If
in both cases the same loss process occurs, this suggests that current driven instabilities
arc not responsible for the anomaly.

V.

D e s t r u c t i o n of magnetic surfaces

At particular values of the rotational transform, c = | , | , | , ..., confinement isdeteriorated, which indicates that around these low-order rational values of t magnetic
surfaces may be destroyed by symmetry breaking perturbation fields. This effect occurs
in currentless operation obtained by various heating methods (NB1, ECRH) and seems
to appear independently on the plasma pressure. The experimental results suggest thatthe perturbation breaks the five fold symmetry of the configuration and therefore canbe explained by field errors coming from the coil system. Numerical calculations haveindeed shown that current leads to the helix create a local perturbation field of 10 - 20 G,
which gives rise to island formation on rational surfaces. The largest islands occur at
e = k, three smaller islands at t = j and five islands at t = | . Islands are also foundat. c = 2 and A = ? . Other perturbations which are caused by misalignment of the roils
and magnetic material may also exist and lead to larger islands as calculated above.
The plasma confinement shows pronounced minima around t = ^, j , \ and | ; inclose proximity of c = £ and t — | , however, the confinement is maximum (Figure 1).
This effect can be understood on the basis of island formation by a general external, time
independent field B i . The Fourier spectrum of this field may contain several harmonics
with decreasing amplitude. Then, every harmonic generates a chain of primary islands ona rational surface with t = ™ (n = toroidal mode number, m = poloidal mode number).
In the neighbourhood t = ^ or ^ or any other low [m,n) rational surface only large(m, n) rational surfaces exist, which is a general result of number theory. Therefore only
high harmonics of B i , or nonlinear interaction of low harmonics can create islands inthe vicinity of t — \ or j . The size of these islands, in general, will be small.
In order to study the behaviour of small shear stellarators under the influence ofan external perturbation B i , the mapping technique developed by J. Greene / ] 7 / wasapplied / 1 8 / . In general the perturbation B i leads to a perturbed twist map T of the

WOBIG et al.

378

FIG. 6. Poincare plots of magnetic surfaces for the same perturbation function with nearly undisturbed
magnetic surfaces (0.47 < t < 0.49, left) and island formation (0.45 < * < 0.47, right).

A
a

I

10-

0.5-

• -M0.3

0.5

0.4

0.6

FIG. 7. Effective plasma radius, A, as a function of external rotational transform, t(a). K and y areparameters describing perturbation.

poloidal plane onl.o itself. Written in the magnetic coordinates i',6 of the unperturbed
held Bo the twist map can be derived from a generating function
V'<

/

t

and yield* T : (V>..A) —> WiJi)Iteration of T generates the desired Poincare plot.
The cfloct of the perturbation field B ^ s described by the function MV'»A) an<1 v a n o l i s
analytical perturbation functions fc^utf,,) ]>ave been investigated. Variation of the rotational transform t() + «, * with a fixed perturbation fc(*i.M a l w : i V S s l i o w s *ie f " l epattern : The vicinity of t = \ or i ((O) in Fig. l) is the most stable against island formation and destruction. The feature is independent of the parLcular choice of M V i . ^ l With increasing amplitude of h the neighbourhood of * - \ or A is the last one to be
destroyed. Figure 0 compares the region at t = 0.47 - 0.49 with c = 0.45 - 0.47 under
the same perturbation.

IAEA-CN-47/D-I-2

379

The size of the islands lias been computed / 1 8 / from the residues and the sum of allisland widths was subtracted from the plasma radius a, thus defining an effective plasma
radius, A. This effective plasma radius is plotted vs t{a) in Figure 7. One sees that theeffective plasma radius vs i(a) exhibits a similar pattern as the measured plasma energy.
These considerations are of pure geometric nature, the plasma may react in different
ways on the presence of islands. Plasma currents and electric fields will be modified byislands leading to enhanced convective losses. But also neoclassical particle losses will
be enhanced as has been shown by Boozer and White /19/ for tokamaks. It has to be
expected that due to islands and field ergodizntion also the energy transport of electrons
will be strongly enhanced, therefore, the confinement in W VII-A based on neoclassicaleffects will exhibit a similar dependence on t(l as the effective plasma radius, A. A Monte
Carlo code addressing this effect is in preparation.
VI.

The effect of shear

Experimentally the influence of shear on confinement was extensively studied starting from a nearly shearless magnetic configuration / 2 0 / with St/e EJ 10~ 2 and varying
the shear up to />c/t = d.0.2. Shear was introduced externally by operating the 2 setsof the helical windings with unbalanced currents (Torsatron mode /21/) and internallyby a small induced toroidal current and the inhomogeneous distribution of the residual
plasma current including effects of the plasma pressure. Mainly around t = | the effectof shear was investigated.
In the ECR-heated plasmas (B = 2.5 T, n ss 10 13 cm'3) the negative shear resulting
from 4 kA induced toroidal current improves the confinement at t = | but leads to nofurther improvement in the region of optimum confinement (O). At higher plasma density(n = 2 - 4xlO 1 3 cm"3), however, 2 kA plasma current provides optimum conditions,whereas 4 kA reduce the confinement everywhere to the level of t = J . At this density
the bootstrap current may already modify the t-profile, therefore the exact shear is notknown. Numerical studies indicate that strong shear reduces the island widths but due
to islands overlapping ergodization of the region in between islands occurs. Consequently
moderate shear may reduce the influence of a single island at the fundamental resonancebut a strong shear deteriorates confinement by island overlapping.
A small positive external shear seems to be more favourable for high ft plasmas(/'(O) < 1%). The residual plasma current which roughly grows proportional with plasmapressure and which probably is related to the bootstrap current leads to further positive
shear. Starting from negative external shear the current density profile of the residual
current which in case of the bootstrap current is peaked in the maximum gradient region
may compensate the shear locally to zero, which could lead to large islands. After a rapid
loss of plasma the build-up of the plasma pressure and the pressure driven current would
start again and thus lead to relaxation oscillations. Consistent with this idea, strongly
unstable conditions were observed for the negative external shear case during increaseof the plasma energy. These experiments support the picture of island formation andergodization of the magnetic surfaces as the reason for deteriorating the confinement.
VII.

S u m m a r y and conclusions

As demonstrated, plasma confinement in W VII-A depends very sensitively on the
structure of the magnetic field which is determined by the particular value of the rotational transform and the magnitude of the shear. The effect, appears with all heating

380

WOBIG et al.

methods if the shear is small enough. The sl.roug dependence of the confinement on thevalues of t0 and shear can be understood on the basis of magnetic surface destruction by
external symmetry breaking field errors among which some helix joints could be identified. Close to the fundamental resonance e — ^ and t — ^ only high Fourier harmonics
of the perturbation field may create small islands. This explains why this region is themost stable against any kind of external perturbations. Since external perturbations
are maximum at the plasma boundary, this also explains the experimental finding that
confinement is deteriorated in particular if the resonant t = ^ - surface is close to theplasma boundary. Small variations of the t-profile during the discharge, either caused bypressure effects or by small induced currents, may shift the plasma from optimum confinement (()) to minimum confinement (M). Therefore in the experiment, a careful controlof the edge values of the rotational transform and the shear is necessary to achieve optimum plasma parameters. The conclusion for future stellarator experiments with low
shear is that modifications of the t-profile with increasing plasma pressure be as small
as possible and external symmetry breaking field errors be avoided or minimized.
For optimum conditions of EC'R-heated currentless plasmas the electron thermal
conductivity in the central part of the plasma column can be described by neoclassicaltheory including effects of helical ripple. In the edge region an anomalous transportcoefficient, assumed to be the same as in ohmically heated discharges, is necessary to fit
the experimental data. Because of the uncertainties of the exact power deposition profile,
this hypothesis has to be considered with caution, so far, however, no contradiction
did arise. A conclusion from these results is that the anomalous thermal conductivity,
which exists in all tokamaks, is not caused by the olmiic heating current. Due to theTe2^3- scaling of the anomalous conductivity its influence in future high temperatureexperiments is expected to be small. The effect of the anomaly will be shifted more andmore towards the plasma boundary. A temperature independent thermal conductivity
Xc ~ 1/n did not provide a good fit. to the experimental data.
In ECR-heated plasmas neoclassical ambipolarity condition explains the measured£"-field. Anomalous particle transport - if ambipolar - does not change this result. In
NBI-heat.ed plasmas high energetic particle losses drive the observed radial electric field
which improves the confinement of the thermal ions in the plateau regime. This beneficial
effect is predicted by neoclassical theory and it is particularly effective in large aspectratio devices as W VII-A (A = 20).
Particle transport in the plasma center is comparable with neoclassical predictions,
if one lakes into accout the uncertainties of the particles sources. In the boundary region,
however, strongly enhanced particle losses (possibly by convective mechanisms) have to
be concluded.

REFERENCES[1] W VII-A TEAM, NI GROUP, in Plasma Physics and Controlled Nuclear Fusion Research 1980
(Proc. 8th Int. Conf. Brussels, 1980), Vol. 1, IAEA, Vienna (1981) 185.
[2] W VII-A TEAM, NI GROUP, in Plasma Physics and Controlled Nuclear Fusion Research 1984
(Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 371.
[3] WELLER, A., MAASSBERG, H., Neutron Flux Measurements at the Wendelstein VII-A Stellarator, Rep. IPP-2/278, Max-Planck-Institut fur Plasmaphysik, Garching (1985).
[4] MAASSBERG, H., Rep. IPP-2/281, Max-Planck-Institut fur Plasmaphysik, Garching (1986).

IAEA-CN-47/D-I-2
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]

[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]

381

MAASSBERG, H., HELLBERG, M.A., Rep. IPP-2/282, Max-Planck-Institut fiir Plasmaphysik, Garching (1985).
ISLER, R.C., et al., Nucl. Fusion 26 (1986) 391.
STACEY, W.M., Jr., SIGMAR, D.J., Phys. Fluids 27 (1984) 2076.
PENNINGSFELD, F.P., OTT, W., SPETH, E., in Controlled Fusion and Plasma Physics (Proc.
12th Europ. Conf. Budapest, 1985), Vol. 9 F, Pt. 1, European Physical Society (1985) 397.
GRIEGER, G., et al., Plasma Phys. Controll. Fusion 28 (1986) 43.
W VH-A TEAM, in Controlled Fusion and Plasma Physics (Proc. 9th Europ. Conf. Oxford,1979) paper A2.4.
RINGLER, H., et al., IAEA-CN-47/D-V-1, these Proceedings, Vol. 2.
MAASSBERG, H., et al., Controlled Fusion and Plasma Physics (Proc. 12 Europ. Conf.
Budapest, 1985), Vol. 9F, Pt. 1, European Physical Society (1985) 389.
W Vn-A TEAM, NI TEAM, in Fifth International Workshop on Stellarators (Proc. IAEA Tech.
Comm. Mtg on Plasma Confinement and Heating in Stellarators, Schloss Ringberg, 1984),
Vol. 1, EUR-9618-EN, CEC, Brussels (1984) 219.
ERCKMANN, V., et al., Plasma Phys. Controll. Fusion 28 (1986) 1277.
HARTFUSS, J., et al. Nucl. Fusion 26 (1986) 678.
WOBIG, H., W VD-A TEAM in Controlled Fuision and Plasma Heating (Proc. 13th Europ.
Conf. Schliersee, 1986), Vol. 1, European Physical Society (1986) 291.
GREENE, J., J. Math. Phys. 20 (1973) 1183.
WOBIG, H., Magnetic Surfaces and Localized Perturbations in the Wendelstein VII-A Stellarator, Rep. IPP-2/283, Max-Planck-Institut fur Plasmaphysik, Garching (1986).
BOOZER, A.H., WHITE, R.B., Phys. Rev. Lett. 49 (1982) 786.
RENNER, H., W VII-A TEAM, in Controlled Fusion and Plasma Heating (Proc. 13th Europ.
Conf. Schliersee, 1986), Vol. 1, European Physical Society (1986) 287.
HARMEYER, E., et al., Wendelstein VII-A in Torsatron Mode, Rep. IPP-2/275, Max-PlanckInstitut fur Plasmaphysik, Garching (1985).

DISCUSSION
K. ITOH: If the improvement in the confinement time in NBI plasma achievedby eliminating OH current is attributed to the stabilization of the tearing mode, whyis this improvement not observed in the ECH plasma? Since the reduction in the drift
parameter, \ <x I/n, is prominent in the low density case, one might expect thedecrease in the anomaly to be greater in the ECH plasma.
H. RENNER: The improvement in confinement in NBI heated plasmas wasobserved at high density. ERC heated plasmas operate at lower density whereas in
OH plasmas no strong effect by tearing modes was observed. Improvement in confinement of ECR heated as compared with low density OH plasmas is only marginal.
It is not as large as expected from the drift parameter scaling.
K. ITOH: I have-another question. Did you observe that the electron temperature profile was peaked, indicating the islands on the periphery in the discharges, therotational transform of which correspond to the minima of TE?
H. RENNER: Yes, the observed electron temperature profile is more peaked
under these conditions. The effect, however, depends on the amount of shear.

382

WOBIG et al.

S.M. HAMBERGER: Do you have any views on the origin or nature of theinstability responsible for the anomalous heat transport during ECRH operation of the
W VII-A stellarator?
H. RENNER: Fluctuations in density have been observed between 50 kHz and
1 MHz. These could be drift wave oscillations, but a detailed comparison of theory
and experiment has not been made, since the diagnostic equipment does not enable
us to investigate an extended k-range.

IAEA-CN-47/D-I-3

PHYSICS STUDIESFOR ADVANCED STELLARATORS
W. DOMMASCHK, F. HERRNEGGER, W. LOTZ, P. MERKEL,J. NUHRENBERG, A. SCHLUTER, U. SCHWENN, R. ZILLE
IPP-Euratom Association,
Max-Planck Institut fiir Plasmaphysik,
Garching, Federal Republic of Germany
Abstract
PHYSICS STUDIES FOR ADVANCED STELLARATORS.
In the Helias class of stellarators, a value of </3> = 0.09 can be achieved which is stable withrespect to resistive interchange modes. Accordingly, the parallel current density is strongly reduced,<j j/j2± > < lA, the shear of the configuration is virtually unaffected by /3, and the change in rotationaltransform is moderate.The self-consistent reduction of 13 due to resonances at rational values of the rotational transform is taken into account: the pressure driven current density varies smoothly through zeroat rational i. — Helias vacuum fields are studied in two ways: i) the NESTOR code obtains the vacuum
field which has the Helias boundary as the outermost flux surface; this procedure allows island sizes
and basic features of modular coils to be studied; ii) representation of Helias configurations with
Dommaschk potentials leads to more realistic coil studies. — Neoclassical energy confinement times
of Helias configurations are obtained by Monte Carlo simulation of a monoenergetic guiding centre test
distribution subject to pitch angle scattering. Preliminary results indicate that, with an electric potentialof the order of the particle energy, ion confinement in the long mean free path region improves to about
102 collision times.

1. INTRODUCTION
Advanced stellarators, exploiting favourable properties of stellarator
configurational space, are candidates for viable stellarator fusion reactors.
Two important confinement questions in stellarators are the /?-value which
can be achieved in a stable way and the neoclassical energy transport.
Results pertaining to these issues are obtained for Helias [l] configurations.
Three aspects of the equilibrium properties of stellarators are well described by today's 3D flux-variable based codes [2, 3, 4] supplemented byevaluation of the parallel current density [5]. The Shafranov shift yields
the simple equilibrium-/? estimate L2/A (L rotational transform, A aspectratio). In advanced stellarators this limit is overcome by reduction of
383

384

DOMMASCHK et al.

the parallel current density; W VTI-AS [6] and Helias rely on this principle. The rotational transform profile depends on /? for genuine stellarator
equilibria with identically vanishing net toroidal current; for advanced
stellarators this dependence is smaller than for stellarators of the 1 = 2
type [1, 7]. Near a resonant surface, i.e. a rational value of rotationaltransform, a finite pressure gradient drives a diverging parallel current
density which is of particular importance in devices with sizeable rotational transform per period, ip, as e.g. Heliacs [8], and can be eliminated bypressure profile flattening near the resonance [9]. A fourth aspect is thesize of the islands occurring at rational i in finite /? equilibria and constitutes the main risk in the assessment of equilibrium properties because its
computational treatment has still to be developed [10].
The MHD stability behaviour of stellarators without net toroidal current through each magnetic surface is completely different from that foundin tokamaks. The modes are purely pressure-driven and probably only theresonant ones are dangerous. Fixed- and free-boundary modes are of thesame nature and the growth rates of unstable modes increase with increasing node number [11, 12]. This behaviour leads to the conjecturethat local ideal modes are formally the most unstable ones. Evaluationof Mercier modes and ballooning modes has shown that Mercier modes
are observed first as /? is increased [13]. An even more stringent condition on /? is obtained from the resistive interchange criterion [14] which
endangers not only the second stability regime of tokamaks [15 ] but also
those stellarators which rely on shear and strong Shafranov shift as, e.g.
ATF [16]. This criterion has been used while finding the Helias class ofstellarators and provides a natural means for pressure profile flattening[9]. The resistive interchange criterion is, for the case of vanishing net
toroidal current, identical with the applicability condition of a sufficientstability criterion [17] and with the stability condition of peeling modes[18], which adds significance to its use for selecting finite-^ stellarators.
Moreover, it has been shown [19] that island growth (as a function of ff)
is connected with resistive instability.
Neoclassical energy transport in stellarators with long mean free pathhas been successfully simulated by Monte Carlo methods [20-22]. A calculation of an energy confinement time with the help of a monoenergetic
guiding centre test distribution subject to pitch angle scattering and energy relaxation in a prescribed electric potential is performed for Helias
configurations.

IAEA-CN-47/D-I-3

385

FIG. 1. Flux surface cross-sections atV = 0, w/2, r of a Helias equilibrium with N = 5, A = 11.5,
Roj = 0,8, ZOil = 0.4, Ali0 = 0.1, Ao = 0.07, A2i0 = 0.05, b.,_, = 0.39, A2>_, = 0.24,
A2.-2 = 0.07.

2. /?-VALUES IN HELIAS
In continuation of Helias stability studies [l], a configuration withfive periods and aspect ratio 11.5 has been identified in which a /?-valueof {/?} w 0.09 is marginally stable against resistive interchange modes.
Figure 1 shows the flux surface geometry of this configuration.
Figure 2a shows the rotational transform profile of the vacuum and thefinite-/? case. Figure 2b shows the corresponding pressure profile, —dp/ds,
which exhibits the flattening around the tp = \ resonance resulting in a
P value which is about 10% smaller than it would be without taking thisresonance into account.
Figure 3a shows the vacuum field solution obtained with the NESTOR
code [23]. Comparison of island thickness and the width of the pressure
profile flattening yields comparable numbers and clearly shows the needfor refined 3D codes which can compute the island size as a function of 0.
Given this lack of knowledge one would prefer configurations in which boththe size of the vacuum field islands and the width of the pressure profile
flattening are small. This is the case for Helias configurations which stay
below LP = j , as is shown in Fig.3. An alternative to these moderate-shear
cases would be to avoid resonances of this order by considering finite-/?
equilibria with very small shear. Such Helias equilibria have also beenfound and will be discussed elsewhere.

386

DOMMASCHK et al.
0.30-

o o

0.2

0.4

i.o

o.e

fb)

-p'

0.2

0.4

FIG. 2. (a) Three {.-profiles: circles are obtained from result of Fig. 3 (a), upper line is result obtainedwith MOMCON [4] for ji = 0 case, lower line is result for <0> = 0.09; (b) corresponding profile—p'(s). Radial label s is square root of normalized toroidal flux.

(b)

(c)

FIG. 3. Poincari plots ofHelias vacuum fields with surface parameters as given in Fig. 1, except for
A,,., = 0.39, 0.3336, 0.284.

IAEA-CN-47/D-I-3

387

(a)

FIG. 4. (a) A Helias boundary (at V = 0) with parameters of Fig. 1, except for A = 12,
A2,_; = 0.12, A2 _2 = 0.1, magnetic surfaces of approximate solution to boundary value problem
with Dommaschk potentials, and outer surface on which modular coils are calculated; (b) line currents,
ordinate being poloidal parameter, abscissa 0 < V < 2ir.

3. MODULAR COIL STUDIES FOR HELIASA Helias version with less indentation than that of Fig.l, but stillallowing (/?) « 0.05, was used to solve a boundary value problem with
Dommaschk potentials [24]. The result allows the computation of modular coils for this configuration on a surrounding surface, see Fig.4a. Fig.4bshows line currents which do not look more complicated than the corresponding result for WVII-AS [6].

4. MONTE CARLO SIMULATION OF ENERGYCONFINEMENT IN HELIAS
Monte Carlo methods developed previously [20, 21] for calculating
neoclassical heat conductivity in stellarators have been generalized to thecalculation of energy confinement times [22]. This generalization allows
one to treat not only the cases of large ratio Qp of plasma radius a to gyroradius p in which a local transport coefficient can be calculated (Qp ~ 500),
but also cases in which the gyroradius is too large for a strictly local transport coefficient to exist (Qp < 500). An electric field F is included in the388

DOMMASCHK et al.
10'

?

/

0/s)

o

o

o

+

°/
D

+

*
"o"""

o

y

10

I

o

*

+

+

+

o

_

*

+
o

+

o
*

o

"

o

•

o
o

a

.1
10'

a

'

—i—

10'

10

.1

FIG. 5. Loss rates of ions and electrons as Junctions of normalized mean free path in Helias,
parameters as in Fig. 1, except for A, _, = 0.29, (3 = 0. o Ion loss without electric field; +,- ionlosses with e<j>/E = ±1; a electron loss.

form of an electric potential <f> which is constant on magnetic surfaces,
F = <f>o2r/a2, and a monoenergetic test particle distribution with energy
E is used. With the help of Lc = nRo/i {Ro major torus radius), sothat Lc is half the connection length, and of the plateau transport value Sp = 3.6v/Q2pmRo [v = (2J£/m)5] the results can be described in adimensionless S'(L*) diagram, where S* = S/SP, L* = A / i c and S(A)is the actual loss rate as a function of mean free path A. For electrons
5 is obtained from the relation 5 = £>/(a/2.4)2 and D is the computed
local transport coefficient. Results which were obtained for Helias [25]show that for Qp = 100 and \e<j>o/E\ = 1 the ion loss rate is only a weak
function of mean free path in the range 1 < L* < 103, while the electrons
are in the f~1-regime, see Fig.5 where the ordinate corresponds to Bo =3 T, Ro = 5 m, and 4 keV deuterons and electrons. These results can beinterpreted as energy losses corresponding to T& ~ 200 msec. The mean
free paths of the two species are approximately equal.
5. CONCLUSION
The MHD, modular coil, and transport properties of the Helias configurations discussed above are promising so that further research is justified. With respect to the MHD problems, the finite-/? structure of islated
regions and the analysis of global MHD modes appear to be the most

IAEA-CN-47/D-I-3

389

important issues; the coil calculations have to be subjected to engineering
constraints; the transport calculations need to be refined, in particularwith respect to the quasineutrality problem. Thus, for an experimental
program, substantial improvement of the computational tools is needed.

ACKNOWLEDGEMENT
We are indebted to our colleagues at the Oak Ridge National Laboratory, in particular to S.P. Hirshman, R.H. Fowler, J.A. Rome, and J.F.
Lyon for providing us with the MOMCON code and the initial version of
their Monte Carlo transport code.

REFERENCES[1] NUHRENBERG, J., ZILLE, R., Phys. Letters 114A (1986) 129.
|2] BAUER, F., BETANCOURT, O., GARABEDIAN, P., A Computational Method
in Plasma Physics (Springer, Berlin, 1978); Magnetohydrodynamic Equilibrium andStability of Stellarators (Springer, Berlin 1984).
[3] SCHWENN, U., 5th Int. Workshop on Stellarators, SchloB Ringberg 1984, Comm.
EC, EUR 9618 EN, I, 377.
[4| HIRSHMAN, S.P., LEE, D.K., Comput. Phys. Comm. 39 (1986) 161.
[5] GROSSMANN, R., NUHRENBERG, J., Proc. ICPP 1984,1, 117.
[6] BROSSMANN, U. et al., Plasma Physics and Contr. Nucl. Fusion Research 1982,
Ninth Conf. Proc, Nucl. Fusion, Supplement 1983, III, 141.
[7] HENDER, T.C. et al., Nucl. Fusion 25 (1985) 1463.
[8] BOOZER, A.H. et al., Plasma Physics and Contr. Nucl. Fusion Research 1982,
Ninth Conf. Proc., Nucl. Fusion, Supplement 1983, III, 129.
[9] HIRSHMAN, S.P., NUHRENBERG, J., ZILLE, R., 8th Europhysics Conf.
Comp. Physics 1986, ECA Vol. 10D (1986) 157.

on[10] REIMAN, A., GREENSIDE, H., Proc. 8th Europhysics Conf. on Comp. Physics
1986, Comput. Phys. Comm. 1986.
[11] MERKEL, P., NUHRENBERG, J., GRUBER, R., TROYON, F., Nucl. Fusion 23(1983) 1061.
[12] HERRNEGGER, F., 8th Europhysics Conf. on Comp. Physics 1986, ECA Vol.
10D (1986) 153.

390

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[13] NUHRENBERG, J., ZILLE, R., Proc. 12th Eur. Conf. on Contr. Fusion andPLasma Physics (Budapest, 1985), EPS, Budapest 1985, Vol. 9F, I, 445.
[14] GLASSER, A.H., GREENE, J.M., JOHNSON, J.L., Phys. Fluids 18 (1975) 875.
[15] CORREA-RESTREPO, D., PI. Phys. and Contr. Fusion 27 (1985) 565.
[16] NUHRENBERG, J., ZILLE, R., Proc. 5th Int. Workshop on Stellarators (Schlofi
Ringberg, 1984), Comm. EC, EUR 9618 EN, I, 339.
[17) LORTZ, W., REBHAN, E., SPIES, G., Nucl. Fusion 11 (1971) 583.
[18] LORTZ, D., Nucl. Fusion 15 (1975) 49.
[19] CARY, J.R., KOTSCHENREUTHER, M., Phys. Fluids 28 (1985) 1392.
[20] BOOZER, A.H., KUO-PETRAVIC, G., Phys. Fluids 24 (1981) 851.
[21] FOWLER, R.H., ROME, J.A., LYON, J.F., Phys. Fluids 28 (1985) 338.
[22] DOMMASCHK, W., LOTZ, W., NUHRENBERG, J., Nucl. Fusion 24 (1984) 794;
LOTZ, W., NUHRENBERG, J., SCHLUTER, A., J. Comput. Physics (1986).
[23] MERKEL, P., J. Comput. Physics 65 (1986).
[24] DOMMASCHK, W., Comput. Phya. Comm. 40 (1986) 203.
[25] LOTZ, W., 8th Europhysics Conf. on Comp. Physics 1986, ECA Vol. 10D (1986)
105; IPP Annual Report 1985, p. 142.

DISCUSSION
B. COPPI: Why did you devote special attention to resistive interchange modes,
given the fact that they tend to become stable at high temperature as a result of theeffects of ion-ion collisions?
J. NUHRENBERG: Previous work has shown that the stability limit for ballooning modes proper is higher than the Mercier limit, which is close to the resistive
interchange limits for HELIAS. Thus, the resistive interchange criterion may be used
as a simple means to achieve complete stability with respect to fixed boundary MHD
modes. In addition, j8-iteration towards marginal stability with respect to thiscriterion is very convenient for elimination of the pressure driven singular currents
near rational values of rotational transform. Finally, this criterion is well suited forminimizing the parallel current density, which is the origin of a deterioration in thequality of the flux surfaces with increasing j3.

IAEA-CN^»7/D-I-4

TRANSPORT THEORYIN STELLARATORS*
K.C. SHAING, S.P. HIRSHMAN,
CO. BEASLEY, Jr., E.C. CRUME, Jr.
Oak Ridge National Laboratory
W.I. VAN RIJ
Computing and Telecommunications Division,
Martin Marietta Energy Systems, Inc.
Oak Ridge, Tennessee,
United States of America
Abstract
TRANSPORT THEORY IN STELLARATORS.
A comprehensive numerical treatment has been developed for analysing neoclassical transport ingeneral three-dimensional plasmas. It is used to study the assumptions underlying previous analytic
approximations and to develop accurate formulae for transport coefficients in stellarators. In the plateau
regime, a resonant enhancement factor for diffusion that agrees with analytic results is found. In thelower collisionality regimes, the structure of the particle distribution function in phase space is computed
to examine the validity of boundary conditions employed in analytic calculations. A diffusion
coefficient D in the collisionless detrapping regime is computed and found to have the form
D = eef'2 G(eh, e j f / t M W f l j ) . Here, G(eh, e,) is a weak function of eh and «„ in contrast withprevious analytic results. A resonance between parallel and E x B motions, which is neglected inbounce-averaged theories, is found to increase (decrease) diffusion in the v (1/v or plateau) regime. The
improved transport properties of configurations with helical wells localized at the inside of the torus,
which have been previously noted for the \lv regime, are found to persist into the lower collisionality
regimes. This tendency is confirmed by comparison with Monte Carlo results.

1. INTRODUCTION
The parameters such as rotational transform t, number of field periods m, inverse aspect ratio e<, and helical modulation eh for different
types of stellarators (e.g., torsatron, heliac, heliotron) can vary widely.
The stellarator transport coefficients obtained by analytically solving thebounce-averaged drift kinetic equation [1] may be invalid for realistic values of these parameters. To relax the approximations associated withbounce-averaging (e.g., J-conservation), a numerical code is used to solve* Research sponsored by the Office of Fusion Energy, United States Department of Energy,
under Contract No. DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc., andunder Contract No. DE-FGO5-8OET-53O88 with the University of Texas.

391

392

SHAING et al.

the linearized drift kinetic equation for the particle distribution function[2]. Transport fluxes are then calculated by taking appropriate momentsof the particle distribution. We use the code to study the assumptions
underlying previous analytic approximations and to develop accurate formulas for transport coefficients in stellarators.
2. DRIFT KINETIC EQUATION SOLVER (DKES)The DKES code solves the linearized drift kinetic equation [2]

^

=5

(1)

da

where / i is the deviation of the particle distribution from the MaxwellianfM)

v = v cos ah + EpVp X h/{B2),n = B/B,v\\/v = cos a, a =-(v/2)(sina)B • V ( l / 5 ) , S = fM [-v<* • V p(Ar + KA2) - Bvcos aA3],
Ai = n'/n - 3T'/2T - eEp/T, A2 = T'/T, Az = -e(E • B)/T(B 2 ),vd = -h x [eE - (Afw5./2)V in B] /Mfi + (vjj/n) [ V x n - ( ^ V x n)ft],
Ep = -d$/dp with $ the electrostatic potential, p is the radial flux coordinate, the angle brackets denote the flux surface average, and K =Mv2/2T. To obtain Eq. (1), we have neglected VB and curvature drift
terms in v • V / i , so the transport phenomena associated with resonant
superbananaorbits are ignored. However, we do include the poloidal E x B
drift to describe the effect of the collisionless detrapping due to helicallytrapped particle orbits. In this code a pitch angle scattering operator isused for Cifi). Equation (1) is solved in terms of Fourier-Legendre series
for / i at a fixed value of normalized energy K, and neoclassical fluxes arecalculated by taking the appropriate moments of f\. The DKES code canbe employed to calculate neoclassical fluxes for any toroidal device once
the Fourier decomposition of B is known. A numerical code has beendeveloped to Fourier analyze B in Boozer coordinates for realistic finite-/?
toroidal equilibria [3].
3. RESULTS AND DISCUSSION
Numerical calculations for stellarators are carried out by using a truncated magnetic field spectrum B = Bo [1 - et cosO - e^cos (£6 - m^)],where JBO is the magnetic field strength on the axis, et (eh) is the dominant toroidal (helical) magnetic field harmonic, and t (m) is the poloidal(toroidal) mode number. One should not use the values of diffusion coefficients shown in the paper to estimate the confinement time without determining electric field and flow velocity. In Fig. 1, we show the diffusion
coefficient D versus v/u>t, where v is the collision frequency and u>t is thetransit frequency for an ATF torsatron (Fig. la) with et = 0.12, eh = 0.14,£ = 2, m — 12, and v = 0.68 and for a stellarator (Fig. lb) with et = 0.1,
eh = 0.075, £ = 2, m = 5, and t = 1.8. Also shown is D for the equivalent symmetric configuration (with e^ = 0 for the same et, t, and Bo).

393

IAEA-CN-47/D-I-4
10V

e*/T
0.5
10-'

5^0
g
SYMMETRIC f

10"2

io-3

10-2

10-'
10°
u/u.

10'

via,

FIG. 1. Diffusion coefficient D versus v^/oi, for two torsatron devices and their equivalent
symmetric configurations, (a) ATF, ant = 2, m = 12, t = 0.68 torsatron, with no significant resonant
enhancement of D in the plateau and Pfirsch-Schltiter regimes, (b) A model ( = 2, m = 5,i = 1.8 stellarator, with resonant enhancement of D in the plateau and Pfirsch-Schliiter regimes.

CONTOURS:-

->0,

=0,

<0,

BOUNDARIES

1.2

(a)
1.0

^.HELICALLY/ ! TRAPPEDPARTICLES
A

A

rt

M

« '" M '", '". A •V- 0.6

0.6

CIRCULATING PARTICLES

8{v)FIG. 2. Contours off, in (n,d) space for a fixed magnetic field line: typical characteristics for(a) 1/v and (b) v regimes.

Figure la demonstrates the well-known fact that the electric field can reduce the diffusion coefficient in the low collisionality regime. The ratio ofthe diffusion coefficients for the stellarator and its equivalent symmetricconfiguration is DsjDe ~ 5 in the Pfirsch-Schluter regime and Ds/De ~ 4in the plateau regime for the device shown in Fig. lb. However, there isno significant resonant enhancement in the plateau and Pfirsch-Schluter
regimes in ATF. The results shown in Fig. lb agree with the analytic
calculations of Ds/De = 1 + I2 (r £ m /r 1 0 ) 2 (eh/et)2 ^ 4.7 in the PfirschSchluter regime [4] and D3/De = 1 + (? ( ^ m / n 0 ) {eh/et)2 =: 3.8 in theplateau regime [5], with r«m(E= wf 1 ^ - m/tj" 1 ) the resonant transit time.

394

SHAING et al.

10.0

0.1
0.01

0.1

1.0

0.2

0.3

FIG. 3. (a) Electric field and collision frequency and (b) goemetric scaling of the diffusion coefficientin the v regime. Q^ = fl£(e<!>/T = 1).

The ratio Ds/De can be greater than unity if r^ m /rio > 1 and/or e^ > e<.
In the case of Fig. lb, r£ m /rio = 1.28, so that there is a resonant enhancement of transport due to a finite value of c/m. This shows that
resonances should be avoided to minimize the stellarator transport. We
note that the helical contribution to the diffusion coefficient cannot be
obtained by solving the bounce-averaged drift kinetic equation.
In the low collisionality regime (uh = v/tk •C oJth, the transit frequency of helically trapped particles), analytic calculations assume thatthe particle distribution function / i is constant along the magnetic field
line to lowest order in Vhl^th- Figure 2 shows the contours of f\ in {n,0)space for a fixed magnetic field line in an I = 2,m = 12 torsatron, where
H is the magnetic moment and 6 measures the distance along the fieldline. In both the \ju (Fig. 2a) and v (Fig. 2b) regimes, the numericallycomputed f\ is nearly constant along the field line for fixed fj, within individual helical wells. Note that fx vanishes at the helical trapping boundary
in the l/u regime, but not in the v regime, as is evidenced by nonzero
contours in the toroidal well. This shows that the boundary conditions
employed in some analytic calculations [1] in the v regime are inappropriate. Note also that in both regimes ) \ varies slowly from one helicalwell to the next for helically trapped particles, since particles trapped in
one helical well cannot communicate with those trapped in the next well.
These characteristics of / i are not sensitive to the geometric parameters.
It is difficult to calculate transport coefficients analytically in thev regime because of the complicated boundary conditions for the kinetic
equation. In particular, one needs to impose ^ JidfT/dfi = Y,Ji^ff/dp
at the boundary (J, = nu between helically and toroidally trapped distributions. Here, J, is the longitudinal invariant in the t'th helical well,and / r ( / / 1 ) is the toroidally (helically) trapped distribution. In Fig. 3,
we show the diffusion coefficient scaling obtained using the DKES code

IAEA-CN-47/D-I-4

E

395

10
o 0.0001
D 0.0010
A 0.0100
0 0. 1000
V 1.000010o*/T

FIG. 4. Diffusion coefficient D versus ei/Tforvarious collision frequencies.

for an I = 2, m — 12 stellarator. From Fig. 3a, which shows the electric field and collision frequency scalings, we conclude that D oc y / f l t ,where n # is the E x B drift frequency. The geometric scaling shown in
Fig. 3b is obtained by systematically varying ej and e^. We find that D
is not very sensitive to the value of eh within the accuracy of the codeand that D is roughly proportional to e / . The weak eh dependence
is also observed in results obtained from a Monte Carlo code |6]. Thisscaling can be understood heuristically from the following random walkargument. In the v regime, the step size Ar is determined by the helically trapped particles and is Ar ~ Vdr/^E — ttT'/ (MCtrCls), where vjr
is the radial drift velocity and r is the local minor radius. The diffusion
coefficient D ~ f(Ar)2/At~ i / ( A r ) 2 / / , where At ~ (ujf2)'1
is assumed and / is the fraction of particles that participate in the transport process. The scaling obtained from the DKES results indicates that/ ~ >/£tG~l (eh, et) where G is a weak function of eh and et. Thus we obtain D ~ veZt/2G(eh,et)T2l(M2Q2r2n2E). Note that this scaling differsfrom those given in Refs. [l, 7, 8] for the v regime, showing a weakerdependence on eh than found previously.
We find a resonance effect for the diffusion coefficient in large aspectratio stellarators when the condition V|| — Ep/B$ ~ 0 is satisfied, where Be
is the poloidal magnetic field strength. A similar effect was also observedin [6,9].. In^Fig. 4, we show D versus e $ / T at various collision frequencies
for an i - 0.35, et = 0.037, eh = 0.022, t = 2, m = 12 stellarator. For theseparameters the effect of the resonance becomes important for e$/T ~ 5.5
where D has a local maximum. For the high collision frequency case, D
is not independent of $, as it would be in the absence of the resonance,
but decreases substantially for e $ / T > 5.5.

396

SHAING et al.

_E

io- 2

FIG. 5. Diffusion coefficient D versus vfor a a-optimized configuration. Both DKES results and Monte
Carlo results are shown. The solid lines are asymptotic limits ofllv diffusion coefficients for a = ±0.4.
The dashed line is the diffusion coefficient of the equivalent tokamak.

The DKES code can be used to calculate diffusion coefficients fora stellarator with a complicated magnetic field spectrum. As an example, we show in Fig. 5 the results obtained for cr-optimized configurations[10] with B = JBO[1 - ft cos 0 + (aefc/2) cos(0 - 12?) - 6jtcos(20 - 12?) +(o-efc/2)cos(30-12?)] = B 0 [l - ttcos9-th{\- a c o s ? ) cos(20- 12?)]. The
a = 0.4 configuration for e $ / T — 1, which reduces the helical ripple on
the outside of the stellarator, has better confinement than the a = —0.4
configuration, in agreement with other calculations [10]. We also compare
DKES results with Monte Carlo results [6] in Fig. 5. In the low collisionality regime, agreement between the two computations is good. The
discrepancy between the results in the plateau regime for a = —0.4 is notunderstood.
CONCLUSIONS
We find that analytic results obtained using the bounce-averaged kinetic equation are generally inappropriate for stellarator configurations.
For example, resonant enhancement of diffusion coefficients in the plateau
regime cannot be treated using the averaged kinetic equation. From thenumerically computed contours of the particle distribution f\ in phasespace, it is found that f\ does not vanish at the helical trapping boundary{K2 = 1) in the v regime. That casts doubt on the validity of certain analytic results obtained by assuming /i = 0 for /c2 > 1, where K2 ~ (cos2 a)
is the well-depth parameter. A transport scaling for an t = 2, m = 12
stellarator is inferred from the DKES code results in the u regime by systematically varying geometric and physical parameters. It has the form

IAEA-CN-47/D-I-4

397

D ~ ve3t/2G{eh,et)T2/ (M2n2r2n%). The function G depends weakly
on £t and eh- If the resonance condition v\\ — Ep/B$ ~ 0 is satisfied, thediffusion coefficient is modified significantly by the electric field.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]

[10]
[11]

KOVRIZHNYKH, L.M., Nucl. Fusion 24 (1984) 851.
HIRSHMAN, S.P., SHAING, K.C., VAN RU, W.I., BEASLEY, C O . , Jr.,CRUME, E.C., Jr., Phys. Fluids 29 (1986) 2951.
BEASLEY, C O . , Jr., ROME, J.A., ATTENBERGER, S.E., HIRSHMAN, S.P., Rep.
ORNL/TM-10222, Oak Ridge National Lab. (1986).
BOOZER, A.H., Phys. Fluids 24 (1981) 1999.
SHAING, K.C., HIRSHMAN, S.P., CALLEN, J.D., Phys. Fluids 29 (1986) 521;
SOLANO, E., SHAING, K.C., Phys. Fluids 30 (1987) 462.
BEIDLER, C D . , personal communication, 1986.
GALEEV, A.A., SAGDEEV, R.Z., Usp. Fiz. Nauk 99 (1969) 528.
MYNICK, H.E., Phys. Fluids 26 (1983) 2609.
MAASSBERG, H., in Fifth International Workshop on Stellarators (Proc. IAEA Tech. Comm.
Mtg on Plasma Confinement and Heating in Stellarators, Schloss Ringberg), EUR-9618-EN,
Vol. 2, CEC, Directorate-General XII — Fusion Programme, Brussels (1984) 507.
MYNICK, H.E., CHU, T.K., BOOZER, A.H. , Phys. Rev. Lett. 48 (1982) 322.
BEIDLER, C D . , HITCHON, W.N.G., personal communication, 1986.

IAEA-CN-47/D-II-l

RESULTS FROM HBTXWITH IMPROVED CONFINEMENT
B. ALPER, V. ANTONI1, M.K. BEVIR, H.A.B. BODIN,
C.A. BUNTING, P.G. CAROLAN, J. CUNNANE2, D.E. EVANS,
A.R. FIELD3, S.J. GEE4, C.G. GIMBLETT, R. HAYDEN2,
T.R. JARBOE5, P. KIRBY, A. MANLEY3, A.A. NEWTON,
P.G. NOONAN, A. PATEL3, R.S. PEASE, M.G. RUSBRIDGE4,
K.P. SCHNEIDER4, D.P. STOREY6, H.Y.W. TSUI,
S. WHITFIELD7, P.D. WILCOCK
Euratom-UKAEA Fusion Association,
Culham Laboratory,Abingdon, Oxfordshire, United Kingdom
Abstract
RESULTS FROM HBTX WITH IMPROVED CONFINEMENT.
In the HBTX Reversed Field Pinch (R/a = 0.8 m/0.25 m), the resistance has been decreased by
reducing field errors and, by using wall loading to increase the density, there is a fivefold increase in
T E . The resistance decreases with the size of the edge region, defined by field errors and Hmiters, andthe resistivity anomaly might be explained in terms of helicity transport and dissipation, where field lines
intersect material objects. From 200 to 450 eV, T;~T e ; plasma rotation in 0 and 9 is observed but thiscannot account for the anomalous ion heating. The on axis value of Te increases with L, up to 430 kA,
but slower than linearly and /39 falls with L,, from - 2 0 % at 80 kA to - 1 0 % at 220 kA and more
slowly thereafter, possibly saturating. With deep reversal a rotating m = 1, n = 6 helical structure isinterpreted as Taylor's second relaxed state.

1.

INTRODUCTION

This paper describes results on plasma resistivity andconfinement, radiation, ion heating, scaling, and fluctuations
from HBTX1B, an improved version of HBTX1A [1], and simulations
with a 3D single fluid incompressible MHD code.

1

Istituto Gas Ionizzati, Consiglio Nazionale delle Ricerche, Padua, Italy.
University College, Cork, Ireland.
3
Royal Holloway and Bedford New College, University of London, UK.
4
UMIST, Manchester, UK.
5
Los Alamos National Laboratory, NM, USA.
6
University of Hull, UK.
7
Imperial College of Science and Technology, University of London, UK.
2

399

400

ALPER et al.
a. A
a HBTXIBO HBTX1A
0.3

T

E
Ims)

oB
0.2

D
Q

0.1

10

15

20

25

FIG. 1. Energy confinement time as a junction of UN in HBTXIB with improved magnetic geometry
and density control using wall loading, compared with that observed previously in HBTX1A. Current
varies from 190 to 230 kA. (TE is calculated from profiles similar to those of Fig. 3.)

2.

FIELD ERRORS AND EQUILIBRIUM

Field errors due to windings ripple and port distortions
have been reduced from 0.2% to 0.01% (equivalent error flux area
normalised to liner surface area) using a new windings former
with one toroidal gap. This gives a reduction in resistance bya factor ~1.5-2. The equilibrium displacement, typically ~3 cm,
contributes an equivalent error of 1.5%, compensated by a steady
B up to 8 piT, to give a further twofold reduction in resistanceto 1 150 nohm at 200 kA. Neither T nor p vary significantlywith B or field error reduction. In conditions where bothcurrent and density (using wall loading) are sustained there isan overall improvement in T - by up to 5 to 0.35 ms. Figure 1
shows x E vs I/N for HBTX1A and HBTXIB.

3.

RESISTANCE ANOMALY

The resistivity calculated from the global resistance
Q = V /I (where V is the loop volts) using helicity balanceexceeds the Spitzer value for the observed T and spectroscopic
Z ff by a factor ~ 3 at high density (I/N -JxlO-1* A.m), rising

401

IAEA-CN-47/D-n-l
125n Scaling Datax Rail Limiterv B v -Scan* HBTX1A

100

75

50

2 5 ••

0

0-1

0-2

0-3

04

Volume of edge region (m3)

FIG. 2. Loop voltage as a function of edge region size, determined by field errors, equilibrium displacement, or limiters. Data from HBTX1A and HBTX1B.

to more than ten as n is reduced. This analysis takes accountof experimentally estimated radial profiles of T , n , ion
species and Z ... Moreover, Q is almost independent of T , eg
between 200 and 450 eV at I = 220 kA.
As seen in Fig. 2, V increases with the size of the edge
region, which was varied by reducing field errors, by changing
the equilibrium using B and, most strikingly, when a rail
limiter was inserted 5 cm into the plasma. In the latter caseV
increased from 30 V to 115 V at constant I [2]. Theseobservations have been interpreted in terms of helicity
dissipation in the edge region either close to the walls or at
the actual boundary where field lines intersect physicalsurfaces.
The helicity balance equation, 1n the steady state, can be
written in the formV • - / £."? d3 x + /
Bulk
Edge
piasmapiasma402

ALPER et al.

\- Vsp + Eedge

The injection term (LHS) is balanced by ohmic dissipation in thebulk plasma («I 4 Q S D h together with additional dissipation in
the edge region («=E?d Ved ); Ved
is the volume of the edge
region defined by f i e l d errors and litniters and E . is theequivalent time and volume averaged electric f i e l d in thisregion. I t is found that E e d q e is approximately constant at
27±7 volts-nr 1 for all the conditions shown in Fig.2.
When helicity is dissipated at physical boundaries where
iJ'TT+O, the helicity balance equation isV^ =/ ?.Bd3X + /plasma£.Tt d 2 X - I Q •+ AXBed A

surfaceorV^ = I^Qsp
tne
Here, Ax iselectric potential difference between thephysical surfaces where field lines enter and leave the plasma
and A is the projected area of such surfaces normal to the fieldlines. Additional power input, which appears as the resistanceanomaly, is required to compensate for the energy loss
associated with the helicity transport needed to balance its
dissipation at the boundary.
Studies with a small carbon limiter (5 cm wide) inserted
progressively up to 6 cm into the plasma confirm that theincrease in loop voltage A V is proportional to the flux
intercepted and AV scales linearly to the rail limiter data. No
increase in oxygen or metal impurities was observed. The risein Z ff from the measured increase in carbon makes an estimated
10-20X contribution to AV.
It is the helicity transport, by fluctuations of the form
<TuB*>, which couples the edge region to the bulk plasma.
Because the system is a relaxed state the loop voltage
associated with the edge region acts In addition to the Spitzer
loop voltage and appears as an anomalous resistance.
According to both of these models, when field errors are
eliminated, the resistance would be that derived using helicity
balance from Spitzer resistivity including impurities.
Probe measurements indicate that the carbon tile limiters
together (8% surface coverage) must draw 10-40 kA electron

IAEA-CN-47/D-II-l403

current and that the edge is active with 100 amp, 10 ^sec
spikes.
4.

FIELD CONTROLIn matched mode start-up over 7 ms, the planned RFX [3]value scaled for HBTX, current termination occurs before
reversal, but constant-e (ramping) yields standard plasmas.
From a comparison at risetimes at which both modes could be
used, the risetime dependent term in the volt seconds using
ramping is a factor of two smaller, indicating a useful
reduction for long risetimes. Controlled rundown at constant-e[4] forestalls abrupt termination until the current falls to
10-20% of its maximum value.
The voltages on the plasma surface during sustainment were
scanned through those calculated to minimise dynamo activity (V~ -10 volts, V ~ -6 volts). Fluctuations were reduced as Vvaried from +50 volts (flat top) to -50 volts, most markedly"
from +50 volts to 0 (which yielded only a modest reduction inxv. A significant effect was observed using a multichord
spectrometer which views the whole plasma volume, with atypically two fold decrease in %/B from edge coils. Previouslyf5] a similar effect was strongest when V < 0.
5.

IMPURITIESThe impurity concentration, typically 1-2% low Z and 0.1%
metals, is as in HBTX1A [1]. Impurity transport was studied byfitting C V profiles calculated with a 10 code (see [6]) for
different transport coefficients to observations from a 20 chordsurface barrier diode (SBO) array. Profiles of T (r) were
determined from CV line Intensity ratios from the cluster near
40A and the 2271A line, measured by the SBD array and a 20 chordvisible spectrometer respectively, which view the same volume.
A collisional-radiative computation shows that in HBTX thisratio is almost linear in T and virtually independent of n ,whose profile is measured with a 3-chord interferometer.
Computed CV distributions with diffusion at 100 and 150
are compared with measurements in Fig.3, for the T (r) and n (r)profiles indicated. A value of n t .
=« 1016 m - | s j s deduced,
comparable with n ^ f o r the plasma as a whole.

404

ALPER et al.

0-75 •

S
01

•o

I
0-50-

0-25

0-9
Ir/a)

FIG. 3. Relative density ofCVasa
function of normalised radius at 200 kA; the experimental curveis compared with simulation (dashed line) for two values of diffusion coefficient. Density andtemperature profiles estimated from experiment were used as follows:». ={ne0-nJ [1 - (r/a)2] as + nea, Te = <Teo - Tta) [1 - (r/a)2] 02! + Tca.

The total radiation during sustainment, measured by a threechord absolutely calibrated bolometer, 1s typically ~ 4£ p o h m ^ c «Radiation is insignificant in the power balance for the plasma
as a whole and for the edge, although 1t rises towards the walls
and can show poloidal asymmetries. There is no radiation cooledmantle controlling the edge temperature as In the Tokamak.
6.

ION HEATING

The value of T . from NPA data tracks T g within 20* from 200
eV to 450 eV. The ions are heated anomalously since In an ion

405

IAEA-CN-47/D-II-l

I

U0

460

500

480

10.052A/channel)

520

X-

F/G. 4. Experimental spectrum of the C V 2271 k line viewed through opposing tangential ports inhorizontal toroidal plane. Separation is due to red (view co-I^) and blue (view counter-I^j. Doppler
shifts correspond to an ion velocity in Indirection. Lower curves are two Gaussians whose sum, convolved with instrument Junction, fits experimental red shifted spectrum.

energy confinement time « 0.3 ms electron-ion collisions, taking
into account Impurities, only raise T. to less than 1/10 T .
A visible Doppler spectrometer with OMA and twin quartz
fibre inputs, viewing the plasma simultaneously in opposing
tangential directions, measures time integrated deuterium andimpurity ion temperatures and rotational velocities. Spectra ofa CV (2271) line viewed through opposing tangential ports in thehorizontal toroidal plane are shown in Fig.4. They appear to becomposite and can be fitted with 1% accuracy by the weighted sumof a pair of gaussians, one having T. = 455 ±20 eV and shiftedby v = 1.4 ±1 x 10* ms" 1 , the other having T. = 115 ±10 eV andv = 4 ±1 x 1 0 3 m-s" 1 . The ratio of the hot to the cold
components is ~ 1.7 ±0.2. The velocities are in the directionof the toroidal ion current.
These results suggest theexistence of rotational shear. Temperatures measured in thepoloidal plane tend to be 10-20% lower than those measured
toroidally; the velocity, also in the direction of the poloidal
ion current, is 2-3 times smaller. Although values of v up toALPER et al.

406

6»10 "

FIG. 5. Electron temperature Te as a Junction offynjor
Best fit curve is shown through points.

;

currents between 80 and 430 kA.

4X10"1 ms"1 have been observed and the carbon ions can get
significant energy directly from the applied electric f i e l d thiseffectappears i n s u f f i c i e n t
to explain the deuterontemperatures.
7.

SCALINGThe variation of T , p and T E with I from 80 kA to 435
kA (F = -0.1 and e = 1.4) has been studied. The density iscontrolled by preloading the carbon tiles, to give a five fold
Increase to ~ 5 x 1 0 ^ m -3 sustained for 5 ms; typically I/N ~ 5
x lO"11* A.m.
The observed T vs I dependence might be interpreted on thebasis of a linear 61 eV/kA variation (see [7][8]) but in thiswork it is more complicated with evidence for a linear variation
at lower current and some flattening off at higher values. At
220 kA a density scan yielded T e Z n p" 0 < 6 « A maximum likelihood
fit to all the data yields 1QZ I o « 7 ^ n e " ° . 5 5 .

IAEA-CN-47/D-H-1

407

This fit implies some reduction of p with current which
can be deduced from Fig.5 in which T is plotted against I2/n.
A constant p , i.e. T=I at fixed I/N, corresponds to a line ofconstant slope; a better fit is obtained by the curve. The
variation is most pronounced at low currents, and p falls from >20% at 80 kA to ~ 10% at 220 kA and more slowly thereafterT(Profiles similar to those in Fig.3 are used). When the data iscorrected for the observed profile broadening with increasing
current and replotted vs. Lundquist number there is someindication for saturation.
The value of tr rises with current at low values but
thereafter it is hard to identify a clear trend and there is awide spread in the data; at high currents the optimum B could
not be used and there was only limited conditioning. The Lawson
parameter increases strongly with I as observed elsewhere.
According to theories of transport due to resistive fluid
turbulence, when g-modes dominate [9] p ~ constant and n T F «I 2 « 5 . If tearing modes dominate [10], p = I "1/3 and n t E «I . 1 * 6 7 . Experimental uncertainty makes it difficult to say
wnich, if either, model fits HBTX1B data. The value of T £ isbased on the global resistance, and therefore depends on the nonohmic contribution due to the edge effects (Section 3) which
involves factors not taken into account in the theories. It
might be appropriate to deduct this part which would increase
Trto > 1 ms.

8.

FLUCTUATIONSFluctuations are detected by internal poloidal and toroidal
arrays of edge coils. With e = 1.4, F = -0.1, HBTX1B is similar
to HBTX1A with B/B = 1-2*, dominant poloidal modes m=0,lresonant inside the reversal surface, and a wide range oftoroidal modes with Inl = 8-17 peaking at 10-12. These modes
rotate in the same sense as the plasma ion current with toroidal
speed v = SxlO1* ms-i and poloidal speed v =3xl0"»ms-l, which may
be compared with the CV rotation (Section 6 ) .
With deep reversal, F< -0.35 and e>1.6, S/B = 5-10%, andthere is a dominant single rotating m=l structure resonant
outside the reversal surface, always with n=6, and v ~ 2-7 x
101* me* 1 which has been interpreted as Taylor's (second) m=l

408

ALPER et al.

helical minimum energy state (n=4 for HBTX1B). Since e has not
saturated the plasma is evidently not fully relaxed.
With deep reversal, sawtooth-like fluctuations are seen in
the m=0 B signals and in the soft X-ray emission detected by
the SBOs. Toroidal flux and I are observed to oscillate andthere are associated field profile changes seen by means of an
insertable magnetic probe. The global parameters such as n and
Tj are more influenced by changes in F (which characterises theplasma edge) than by e, which is an average quantity, indicatingthat the outer region plays an important role.
With deep
reversal both n and TJ are higher and more prone to variation.
Fine-scale fluctuations are observed on the SBD that have atransverse correlation length of the order of 0.1 minor radius
and are consistent with density fluctuations due to motion ofparticles along magnetic field lines.

9. 3D SIMULATIONS
Numerical simulation has been performed using asingle-fluid, incompressible MHD code in cylindricalgeometry [11]. Plasma parameters are specified in terms of the
Lundquist number S and the normalised radial profile of theresistivity. As it evolves the fluid exhibits both spontaneous
field reversal and sustainment. If S = 101* and resistivity
increases towards the wall, as in HBTX, the F-e variation is ingood agreement with curves modelled by the MBFM to fit
experiment. The code also displays field fluctuations, definedas departures from axisymmetry, which are about 10% for S=102
and fall as S is increased to about 4% at S^O 1 *. For uniform
resistivity and S=103 the energy spectrum of fluctuations inradial field B is dominated by two low order toroidal m=l modes
during sustainment, similar to experiment.
The relative strengths of the individual terms in theaxisymmetric part of the poloidal component of Ohm's law, (VxT)- vrB<t,-v(JIBr responsible for sustaining the reversal have been
evaluated for uniform resistivity and S=10 3 ; <v B > provides thedominant emf balancing resistive diffusion.
The computed
profile of p.=J /B corresponds well with experiment.

IAEA-CN-47/D-II-l

409

10. THIN SHELL OPERATION
A new shell assembly has been constructed with a B time
constant of 2 ms (compared with 80 ms at present). MHD
instabilities with a resistive wall have been studied
theoretically [12] and tearing modes, which can lead to magnetic
islands, are stabilised by toroidal rotation at ~ 102-10 3 me" 1 ,smaller than observed.
Ideal modes are not stabilised byrotation and lock to the wall, but parallel viscosity andnon-linear relaxation may play a role. Qualitatively, if thewall time constant exceeds the relaxation time continuously
growing modes may not occur. The centre limb of iron core will
be encased in a conducting bandage to minimise the possible
effects of shift and tilt modes.
11. CONCLUSIONS(i) In HBTX1B improved magnetic geometry and density control
have yielded a fivefold increase in x_ compared withHBTX1A in sustained density and current conditions.
(ii) The resistance increases with the size of the edge
region, varied by changing field errors, B and thelimiter configuration; it is postulated that anomalous
resistivity can be explained in terms of an additional
contribution to the loop voltage from helicity transport
in the plasma and dissipation where field lines intersect
physical obstructions at the edge.
(iii) Radiation is unimportant in the energy balance even at
the edge although it increases towards the walls.
(iv) From 200 to 450 eV T 1 = T & ±201, but the ions are notheated by collisions. Tangential CV measurements show
T(C V) ~ T. and rotation velocities v , v in the
direction of the ion current up to 7 4 x 101* ms~r.
(v) T increases as I n with 0.5 < n < 1; pfl decreases with I,
from 20% at 80 kA to ~ 10S at 220 kA and slowly
thereafter, possibly saturating with increasing S.
(vi) For F=-0.1 modes with m=l, In 1=8-17 rotate with v_ ~ v. =
II

(vii)

8 < 9

5xlO"» me"1; at deep reversal a rotating helical m=l
structure with n~6 is interpreted as Taylor's second
relaxed state.
Several aspects of RFP behaviour can be simulated by a
3-D single fluid incompressible code.

410

ALPERetal.

REFERENCES[1]

CAROLAN, P.G., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984
(Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 449.
[2] JARBOE, T.R., ALPER, B., submitted to Phys. Fluids.
[3] MALESANI, G., ROSTAGNI, G., paper presented at The RFX Experiment (14th Symp.
Fusion Technology, Avignon, 1986).
[4] NEWTON, A.A., NOONAN, P.G., Nucl. Instrum. Methods Plasma Res. A 425 (1986)
167.
[5] ROBINSON, D.C., KING, R.E., in Plasma Physics and Controlled Nuclear Fusion Research(Proc. 3rd Int. Conf. Novosibirsk, 1968), Vol. 1, IAEA, Vienna (1969) 263.
[6] BEHRINGER, K.H., et al., Nucl. Fusion 26 (1986) 751.
[7] TAMANO, T., in Plasma Physics and Controlled Nuclear Fusion Research (Proc 10th Int.
Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 43.
[8] BAKER, D.A., et al., ibid., p. 439.
[9] CONNOR, J.W., TAYLOR, J.B., Phys. Fluids 27 (1984) 2676.
[10] AN, Z.G., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc.
10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 231.
[11] KIRBY, P., in Computational Physics (Proc. 8th Europ. Conf. Bad Eibsee, 1986),
Vol. 10D, European Physical Society (1986) 23.
[12] GIMBLETT, C.G., Nucl. Fusion 26 (1986) 617.

DISCUSSION
H. TOYAMA: Is there any difference between the /3 scaling of the deep
reversal and that of the standard reversal?
H.A.B. BODIN: We did not make any scaling studies with deep reversal since
there was increased plasma-wall interaction and some risks to the integrity of theliner at high currents.
M. HAINES: How can you explain ion heating by rotation? Surely only
sheared rotation will lead to viscous heating.
H.A.B. BODIN: The rotation is sheared. The two fitted Gaussians for C V
contained in the figures with T = 455 eV and T = 115 eV have different rotation
speeds — 14 x 103 m-s" 1 and 4 x 103 m-s"1, respectively.
T. OHKAWA: What is your definition of the 'edge region'? After all, inserting the limiter further or retracting the conducting shell further both increase the edge
volume.
H.A.B. BODIN: The edge region is defined by the radial distance from theinner surface of the rail limiter or standard carbon tiles to the inner surface of theliner. For field errors and equilibrium cases, an equivalent 'edge volume' representing the size of the region where flux tubes due to field errors or displacement intersect
the liner is used. The position of the shell does not enter into it when the shell andliner are separate structures.
It should be noted that two ways of looking at this are described in my paper.
The one I discussed orally is in terms of the edge volume (above). The second, now

IAEA-CN-47/D-n-l

411

preferred, is in terms of the projected area of inserted objects normal to the field lines(approximately j86 near the wall). It should be mentioned as well that one of twotheories based on this second approach predicts a value of the change in loop voltage
when objects are inserted in rough agreement with observation without fitted
parameters.

IAEA-CN-47/D-n-2

EXPERIMENTAL AND THEORETICAL STUDIESOF THE STP-3(M) REVERSED FIELD PINCHIN THE HIGH CURRENT DENSITY REGIME
K. SATO, T. AMANO, Z.X. CHEN 1 , H. ARIMOTO,
S. YAMADA, A. NAGATA 2 , K. YOKOYAMA 3 , Y. KAMADA 4 ,A. MATSUOKA 5 , S. MASAMUNE 6 , H. SHINDO 2 , K. SAITO 3 ,H. MURATA 5 , H. OSHIYAMA, S. SHIINA, T. TAMARU
Institute of Plasma Physics,
Nagoya University,
Nagoya, Japan

Abstract
EXPERIMENTAL AND THEORETICAL STUDIES OF THE STP-3(M) REVERSED FIELDPINCH IN THE HIGH CURRENT DENSITY REGIME.
Reversed field pinch experiments are carried out, with a high current density of up
to 7 MAm" 2 . The electron temperature is nearly proportional to the current density. The
self-organizing action and its effect on plasma confinement are investigated experimentallyand theoretically. The typical RFP behaviour is essentially the same in the entire current
density region of 2 to 7 MA-m"2. In the high current density regime, the magnetic fluctuation level decreases slightly and plasma-wall interaction caused by the dynamo action sets in.

1.

INTRODUCTION

In this paper, the results of experimental and theoretical studies on the
self-organizing action of an RFP plasma [ 1 ] and its effect in the high current
density regime are presented. These studies are very important in realizing acompact and simple RFP reactor [2] with high power density.
The experiments are carried out in STP-3(M) [3], which has a stainlesssteel liner (R/a = 0.5/0.1 m) with molybdenum limiters. Its load assemblies
and electric power source are designed so that the plasma current can be driven
up to 350 kA, which corresponds to a current density of 15 MA-m~2. So far,the maximum plasma current density and plasma lifetime attained are 7 MA- m"2(170 kA) and 3.1 ms, respectively. STP-3(M) has no vertical field coils for
1

Institute of Plasma Physics, Hefei, China.
Faculty of Engineering, Hiroshima University, Higashi-Hiroshima 724, Japan.
3
Department of Physics, College of Science and Technology, Ninon University,
Tokyo 101, Japan.
4
Department of Nuclear Engineering, University of Tokyo, Tokyo 113, Japan.
5College of Technology, Gunma University, Gunma 376, Japan.
Department of Electrical Engineering, Kyoto Institute of Technology,
Kyoto 606, Japan.
2

413

414

SATO et al.

plasma position control. Equilibrium is maintained by a conducting shell,
which is a double shell structure for the purpose of reducing the field errors.
The outward displacement of the plasma column is kept within 6 mm during
the entire plasma lifetime. In the quiescent phase, the fluctuation level 5B/B is0.5-2% and the electron temperature is 120-300 eV, with a plasma current of
50-lOOkA.
To simulate MHD phenomena in STP-3(M), a three-dimensional semiimplicit compressible code has been developed. By using this code, themechanism of the self-organizing effect and its effect on the diffusion arestudied.

2.

SELF-ORGANIZING EFFECT

Typical waveforms of the characteristic parameters of the STP-3 (M) plasma
are shown in Fig. 1. The electron temperature (Te) and the electron density(n e ) are measured by Thomson scattering, and the conductivity temperature(T ea ) is estimated by assuming Zgff = 1 and the geometrical factor to be equal to 4.
The fluctuation level is high (5-10%) up to the end of the current rise phase.
In this phase, the toroidal flux increases. In the current decay, or flat-top, phase,the fluctuation level decreases to 0.5-1%, and the electron temperature increases
above 100 eV, which indicates improved confinement in this phase.
The field reversal ratio (F) and the pinch parameter (d) are kept constantafter the setting up of the RFP configuration. This is the result of the selforganization which accompanies the generation of toroidal flux. The self-organizing
effect is sensitive to the magnetic Reynolds number (S). In the low S case, theF and 6 values move along wide trajectories in the F-0 diagram. On the other
hand, in the high S case, the trajectory tends to stay, for a long time, at a particularpoint in the F-0 diagram, as is shown in Fig. 1 (e). The 0-value in STP-3(M) is
about 1.7-2.0. The F-0 diagram for high input power is shown in Fig. 2. The
ratio of the amplitude of the helical fluctuation to that of the symmetric fluctuation does not depend on the volt-seconds stored in the plasma. These experimental
results imply that the helical equilibrium of an RFP plasma with finite pressure is
not always described by Taylor's theory.
To investigate the mechanism of the self-organizing effect, magnetic probesof various kinds and flux loops located at toroidal positions are used. A poloidal
array of coils is installed behind the limiters at four poloidal locations in themeridional planes. A toroidal array of coils is set up between the liner and theshell.
The fluctuation propagates toroidally in the anti-plasma-current direction.
Poloidally, it propagates in the direction of the electron diamagnetic drift withrespect to the outer toroidal magnetic field. The screw of the observed fluctuation always coincides with that of the magnetic line of force inside the reversal

415

IAEA-CN-47/D-II-2
(kA)

(a)
(b)
(c)

(eV)

(d)

< F >

:

:

(e)

F=8.5/DIU

THETO=B.5'DIU<

FIG. 1. Typical waveforms of characteristic STP-3(M) plasma parameters:(a) plasma current and its time derivative; (b) toroidal magnetic field at wall; (c) averaged
toroidal magnetic field; (d) electron temperature, conductivity temperature and electron
density; (e) F-8 diagram.

surface, and the fluctuations always propagate in the anti-plasma-current direction.
Furthermore, the poloidal plasma flow inside the reversal surface is found bymeasuring the Doppler shifts of the impurity lines [4]. All the observed fluctuations, such as those of the toroidal flux, visible light emission and density at theedge propagate in the direction of the electron current flow [5, 6]. From theseobservations, we conclude that plasma flow along the magnetic field lines exists
and propagates in the same direction as the fluctuation.
The results of mode analysis made by using FFT show that the dominantpoloidal mode number (m) is m = 1; in = 0 and m = 2 modes are also observed.
The dominant toroidal mode numbers (n) are 9 to 13. The time evolution of
the toroidal flux in the quiescent phase is shown in Fig. 3, together with those416

SATO et al.

(a)
Y= 20 KfVDIU

X= .3 MS/DIU< FLUXd) >(b)
Y= 200 G'DIU

X= .3 tiS/DIU

< F >

(c)

F=0.5/DIU

nETfl=0.5'DIU< THHTB >FIG. 2. F-8 diagram for high input power: (a) plasma current; fb) toroidal magnetic flux;(c) F-d diagram.

0,4

0.6

0.8

t(ms)

FIG. 3. Time evolutions of toroidal flux and poloidal magnetic fluctuationin quiescent phase.

417

IAEA-CN-47/D-II-2
Ee2 -- 1.000

50

150

T.H

0.095

FIG. 4. Ee at reverse field surface and toroidal flux inside reverse field surface versus time.
EB is negative in shaded region.

of the poloidal magnetic fluctuation. Toroidal flux enhancement is observed
as a discrete event, and a growing sinusoidal oscillation usually appears in the
Bfl signal, which is composed of m = 0, 1,2 modes. These experimental results
indicate that the mechanism of self-organizing action is due to coupling withhelical modes having m = 1, n = 9 to 13.
To simulate the non-linear evolution of MHD fluctuations, two resistive MHD
codes (incompressible and compressible) solving primitive MHD equations incylindrical co-ordinates (y,8 ,Z) have been developed. Both codes use finite
difference in r and spectral representation in 6 and z. Fifty to one hundred(m,n) modes with multi-helicity are taken into account. The compressible codeuses a semi-implicit method in order to avoid the short time steps due tocompressional Alfven modes. As to tokamak disruption simulation, the twocodes give essentially the same results as the reduced MHD model. By comparing
the results of five different codes, Aydemir et al. [7] have shown that compressible codes predict field reversal maintenance while incompressible codes do not,for a configuration with an aspect ratio of R/a = 1 and S = 10 3 . Using a forcefree paramagnetic field model as initial configuration, our codes lead to the sameconclusion: the compressible code predicts appearance and maintenance offield reversal while the incompressible code does not.
By considering a more realistic configuration with R/a = 5, and startingfrom an initial finite beta Robinson equilibrium, a series of simulations has been
done in which the S number ranges from 104 to 10 6 . Axial flux conservationand constant toroidal current were used as boundary conditions. It was observedthat field reversal was maintained, even in the incompressible case, in contrast
to the R/a = 1 case.

418

SATO et al.

100

150

TH

FIG. 5. Magnetic field energy of ten dominant modes versus time. The (m,nj numbers ofeach mode are (1,-10), (1,-11), (1,-12), (1,-13), (1,-9), (1, -14), (1,-15), (1, -16),(0, -1), (1, -17) in the order of decreasing maximum amplitudes.

Figure 4 shows the time history of toroidal field flux <f> (inside the fieldreversed surface) and poloidal electric field Eg measured at the field reversal
surface for the case of S = 10 s , j3 = 8%. The F and 9 values are 2 and -0.23,
respectively. Beta was kept constant by introducing a diffusion term into thepressure equation. 58 modes were considered. A reverse field configuration
was maintained throughout the simulation, up to t = 200 T^. Negative Eg(shaded region) corresponds to toroidal flux enhancement. For other S values(10 4 , 10 6 ) and Rvalues (5%), similar trends were observed. Figure 5 shows
the magnetic energy often dominant modes: (1, —10), (1 ,—11), (1,-12), (1,-13),(1,-9), (1,-14), (1,-15), (1,-16), (0,-1), (1,-17) in order of decreasing maximum
amplitudes. During the simulation, the central safety factor, q(0), was less than 0.1
for most of the time; therefore, all modes except (1,-9) are resonant modes.
The total fluctuating magnetic field energy was about 1% of the equilibrium fieldenergy.

3.

EFFECT OF SELF-ORGANIZATION ON PLASMA-WALL INTERACTION

To study the relationship between plasma-wall interaction and selforganization, the electron saturation current (l's) in the edge region measured bydouble probes, visible line radiation of metal impurities and the toroidal flux aremeasured in the same plane. The results of the measurements are shown in
Fig. 6. Except for the initial phase before setting up the RFP configuration,
the large pulsed peaks in the line radiation and in I~ were synchronous with

419

IAEA-CN-47/D-II-2

1.2

t( ms

FIG. 6. Time evolutions of electron saturation current fls) in edge region, visible line
radiation (Mol) and toroidal magnetic flux.

0.2

0.3

0.4

0.5 t(ms)FIG. 7. Time histories ofl~, ion saturation current (fj and floating potential (VfJ measured
at nearly the same position.

420

SATO et al.

200 p s / d i v
0-5V/ div

FIG. 8. Typical waveform of soft X-ray measured by surface barrier diode (SBD).

toroidal flux enhancement. These facts show that self-organization enhances theplasma-wall interaction. As the current density increases, the plasma-wall interaction due to self-organization becomes more and more conspicuous.
To study the origin of the metal impurities, Ig, the ion saturation current(I*) and the floating potential (Vf) are measured at nearly the same positions
by triple probes installed behind the limiters. The time evolutions of I~, Ij
and Vf are shown in Fig. 7. I" and I* start increasing simultaneously at thetime of flux enhancement, but Ij varies more rapidly than l+s. As a result wefound that the floating potential assumes increasingly negative values first andthen returns to zero. The peak value of I* is nearly equal to that of I". These
facts indicate that electrons and ions are emitted from the plasma towards thewall, by the self-organizing action.
The soft X-rays measured by a surface barrier diode (SBD) are shown in
Fig. 8. There is a slowly changing and a pulsive signal. The occurrence of thelatter coincides with toroidal flux enhancement. By using collimators, it isconfirmed that almost the entire pulsive soft X-ray activity occurs at the wall.
This is consistent with the results obtained from the triple probes. Furthermore,
the electron energy is estimated by an absorber method using Be foils. Theenergy of the slowly changing signal is about 150-300 eV, which is in good
agreement with the results of Thomson scattering. The latter signal has an energyof 2 to 8 keV, on the assumption that the electrons collide with the vacuum vessel.
This shows that high energy electrons and ions are emitted from the plasma
towards the wall by the self-organizing effect and metal impurity influx is causedby these impinging high energy ions.

4.

HIGH CURRENT DENSITY OPERATIONHigh current density operation ( > 4 MA-rrT 2 , £100 kA) is carried out in
STP-3(M). The typical RFP behaviour for high current density is basically the

421

IAEA-CN-47/D-H-2

700

FIG. 9. Relation between electron temperatures (Te, Tea) and plasma current
for round top operation.

same as in the low current density regime. With increasing plasma current, theself-organizing action sets in, and the corresponding plasma-wall interaction isintensified.
The relationship between electron temperature (T e , T e a ) and plasma current
in the case of round top operation is shown in Fig. 9, together with Z e ff(T e /T ea ).
The electron temperature, T e , is nearly proportional to the plasma current. On
the other hand, the conductivity temperature (T ea ) shows a tendency to saturate
above a plasma current of about 100 kA (3.7 MA-m" 2 ). Accordingly, Zeff
increases gradually as the plasma current increases. Since the magnetic fluctuationlevel decreases slightly, the increment in Zeff seems to be caused by the influxof metal impurities due to the self-organizing effect. Preliminary bolometermeasurements of the radiation power emitted by the plasma support thesestatements.

5.

CONCLUSIONSThe self-organizing action and its effect on RFP plasma confinement werestudied experimentally and theoretically. It was found that self-organization

422

SATO et al.

is caused by coupling with helical instabilities having m = 1, n = 9-13 modes
inside the reversal surface and that it enhances plasma-wall interaction. It was
confirmed that the dominant process of metal impurity release is caused byion sputtering. The typical RFP behaviour for high current density is basicallythe same as that for low current density, except for the increase in plasma-wallinteraction. To attain a high current density in RFP, it is important to suppress
the plasma-wall interaction caused by the self-organizing action.

REFERENCES
[1]
[2]

[3]
[4]
[5]
[6]
[7]

TAYLOR, J.B.,Phys. Rev. Lett. 33(1974) 139.
HAGENSON, R.L., KRAKOWSKI, R.A., Compact Reversed Field Pinch Reactors(CRFPR): Sensitivity Study and Design-Point Determination, Rep. LA-9389-MS,
Los Alamos National Laboratory, NM (1982).
SATO, K.I., Nucl. Fusion 25(1985) 1301.
ARIMOTO, H., et al., submitted to Nucl. Fusion.
ARIMOTO, H., et al., J. Phys. Soc. Jpn. 54 (1985) 1232.
MATSUOKA, A., et al., to be published in J. Phys. Soc. Jpn.
AYDEMIR, A.Y., et al., Phys. Fluids 28 (1985) 898.

DISCUSSION
R.S. PEASE: How do you propose to reduce the undesirable plasma-wallinteraction?
K. SATO: The plasma-wall interaction (PWI) due to the dynamo process
seems to depend on I/N, where I is the plasma current and N is the line density.
With increasing I/N, PWI increases in reversed field pinches. In STP-3(M), theoperating region of I/N is above 1CT13 A-m, which is ten times larger than inother machines. In order to decrease the PWI, it is necessary to operate in a
regime of low I/N. We will therefore try to decrease the PWI using carbonization.

IAEA-CN-47/D-II-3

REVERSED FIELD PINCH EXPERIMENTS
IN ZT-40M AND ZT-P
K.F. SCHOENBERG, G.A. WURDEN, P.G. WEBER, J.C. INGRAHAM,
C.P. MUNSON, D.A. BAKER, C.J. BUCHENAUER, L.C. BURKHARDT,
T.E. CAYTON, J.N. DiMARCO, J.N. DOWNING, R.M. ERICKSON,
P.R. FORMAN, A. HABERSTICH, R.B. HOWELL, F.C. JAHODA,
R.S. MASSEY, G. MILLER, R.W. MOSES, Jr., R.A. NEBEL,
J.A. PHILLIPS, M.M. PICKRELL, A.E. SCHOFIELD, R.G. WATT,
D.M. WELDON, K.A. WERLEY
Los Alamos National Laboratory,
Los Alamos, New Mexico,
United States of America

Abstract
REVERSED FIELD PINCH EXPERIMENTS IN ZT-40M AND ZT-P.
Key experimental research on the Los Alamos ZT-40M and ZT-P Reversed Field Pinch (RFP)
experiments is summarized. Fuelling of an RFP by pellet injection has been successfully demonstrated
on ZT-40M. When employed with programmed gas injection, the fuelling effectiveness noticeably
increases and demonstrates the potential for density control in next-generation RFP devices. The
exploration of Oscillating Field Current Drive (OFCD), as a mechanism to provide steady-state RFP
operation by low frequency field modulation, is under experimental investigation. Initial results haveshown that the amplitude of the plasma current is dependent on the field modulation phase, as predictedby theory. However, the data suggest that the current drive response is presently offset by increased
plasma resistance due to modulation enhanced plasma-wall interactions. The use of wall carbonization
has been tested for controlling metal impurity influx during OFCD and standard discharges. During normal operation, significant reductions in metallic and oxygen impurities were noted, with a concomitantdecrease in the plasma Zeff. However, wall carbonization was less effective during OFCD discharges
because of its shortened life under the high power load conditions. For certain modes of ZT-40M operation, energetic ion plasmas are obtained, and the observed increase in plasma ion energy can significantly contribute to the total plasma beta. Investigation of ramped current start-up has allowed RFP
formation at reduced voltages, and scaling results project acceptable volt-second consumption for future,larger devices. The ZT-P experiment is being used to study design issues, such as dynamic equilibrium
control, field error effects, and high current density scaling, for the Los Alamos ZT-H experiment.

INTRODUCTION
In the last two years, the Reversed Field Pinch (RFP)
effort at the Los Alamos National Laboratory has been devoted to
conducting research on the ZT-40M[l] and ZT-P[2] experiments,
and to developing the design of the high current Los Alamos
ZT-H[3] experiment. This paper summarizes key experimental work
associated with ZT-40M and ZT-P. It is organized into six
423

424

SCHOENBERG et al.

sections: 1. Density control by programmed gas and pellet
injection; 2. Steady-state current sustainment by Oscillating
Field Current Drive; 3. Impurity control by wall carbonization;
4. Energetic ion plasma experiments; 5. RFP start-up; and6. ZT-P operations.

1. DENSITY CONTROL
Recent experiments on ZT-40M have successfully demonstrated
the first pellet refueling of a reversed field pinch (RFP). The
four-barrel pneumatic pellet injector, used on Alcator C [4],was modified for ZT-40M operation to inject cylindrical hydrogenor deuterium pellets with radii up to 0.7 mm at velocities of
150 to 800 m-s . Injection along the major
radius at
velocities exceeding 400
w-s
into
discharges
withL= 100 to250 kA, n e = 1 to 8xl0 19 m" 3 and T (0) = 1 0 0 to 300 eV, typicallyresulted in a density increase or 2x10 m . Still photographs
taken from above show full penetration for
low density
discharges, with some toroidal curvature of the trajectory.
Data from an array of D a monitors and from side photographs
show poloidal curvature of the trajectory, with the pellet
typically approaching only to 0.1 m ofthe minor
axis(a = 0.2 m ) . These deflections are attributed to preferential
ablation of the pellet by electrons moving along field lines.
As a result of the deflections, density profile peaking has notbeen observed. Also, because the particle confinement time iscomparable to the ablation time, only 40 to 60% of the pellet isseen in the plasma particle inventory.
Figure l(a) shows data from a two-chord FIR interferometer(impact parameters 0 and 0.12 m) for a pellet which curved
strongly and remained at r > 0.12 m. This is to be compared toa pellet which penetrated close to the major radius[Fig. l(b)].
Both pellets initially traveled at 500 m-s" 1 into 230 kA
discharges. The electron temperature on axis, measured byThomson scattering, and ion temperatures, measured by Doppler
broadening of impurity lines and a time-of-flight neutral
particle spectrometer, decrease sharply at the density rise, andrecover as the density decays so that the plasma P is maintainedroughly constant. The largest absolute density increase yetobserved is shown in Fig. l(c). Four almost simultaneouspellets were injected into a discharge where the current was
ramped from 100 kA to 180 kA in 10 milliseconds. The injection
resulted in a six-fold density increase of 8x10 V .
Noticeable improvements in the effectiveness of both pellet
injection and gas puffing have been observed when the twotechniques are combined. Figure l(f) shows the density traces
for an I. = 200 kA discharge with four pellets, and a sustained
total gas puff of 180 torr-1/sec from three piezo-electric

425

IAEA-CN-47/D-H-3(a):1^=230 kA :1 PELLET
-

0

2

4

6

8

10

,. NO PUFF, NO PELLETS

10

0

2

4

6

8

10

(f)
210 kAx10

x10 1 912 14

(nr 3 )
4 PELLETS +GAS PUFF-180 torr • i/s
0

2

4

6

8

10

12 14

TIME (ms)

0

2

4

6

8

10

12 14

TIME (ms)FIG. 1. Mean chordal density as a junction oftime for density control discharges. The dashed and solid
curves denote the r = 0 and r = 0.12 m impact parameters, respectively.

valves around the torus. Figure l(d) and l(e) show a comparable
discharge without and with gas puffing, respectively. The
improved performance with combined gas puffing and pellet
injection is hypothesized to result from an increased neutral
density in the plasma edge due to strong gas puffing, which
suppresses the energetic plasma electrons responsible for the
preferential ablation. The data demonstrate the ability to
produce a substantial density increase for many
particleconfinement times over a considerable discharge duration, anddemonstrate the potential for density control in future longpulse RFP devices.
2.

OSCILLATING FIELD CURRENT DRIVE

Steady-state current sustainment by Oscillating
Field
Current Drive (OFCD) employs a technique in which the toroidal
and poloidal magnetic fields at the plasma surface are modulated

SCHOENBERG et al.

426
200

(kA)
i

r

..

PUMPING't . . .

f . . .

i . . .

DUMPING^i . . .

i . . .

i

10

12

14

100

(mT)
0

2

4

6
8
TIME (ms)FIG. 2. Plasma current (I^J and toroidal magnetic field at the plasma edge (B^(a)) for one
non-modulated and two OFCD discharges.

-9-mFIG. 3. Time dependent, spatial toroidal magnetic field profile during an OFCD pumping discharge.

in quadrature
at audiofrequencies [5,6].
The non-linear
coupling of these modulations by the plasma is predicted toprovide steady-state maintenance of the mean magnetic fields andhence
the plasma current.
Becausethis
technique istechnologically simple, non-invasive, and scales well to hightemperature, high density plasmas[6], the possibility exists fora relatively simple current drive system in future large RFP
devices. In addition, OFCD may be applicable to other toroidal
confinement
systems,
providedappropriate"relaxation"mechanisms exist[7,8].
An experimental program was initiated on ZT-40M to assess
thiscurrent drive technique. Figure 2 illustrates threecomparative discharges. The non-modulated curve displays a

IAEA-CN-47/D-II-3

427

nominally flat-topped 180 kA discharge. The modulated traces
display the results of applying sinusoidal voltage components tothe toroidal and poloidal field circuits, but with different
relative phases. For the "pumping" trace, the applied V Q and V±
measured at the plasma edge were driven in quadrature. The"dumping" trace was obtained by changing the driving phase of V Q
with respect to V^ by n.
The experimental results were compared with 0-D simulations
that use the F-9 trajectory (F = B(j)(a)/<B(j)>, 0 = B e (a)/<B.>) andplasma inductance calculated from the Modified Bessel Function
Model (MBFM)[6]. The general dependence of discharge behavior
on driving phase is in agreement with the 0-D simulations. In
addition, internal magnetic field probes have been employed tomeasure the RFP magnetic field profiles during modulation. The
time dependent toroidal field profile for a 50 kA discharge isshown in Fig. 3.
The 15% oscillation of B ^ o ) on axis is inphase with the plasma current, and 180° out of phase withrespect to B. at the plasma edge. This phase relationship is
also in agreement with the MBFM calculation when the model is
suppliedwiththeexternallymeasured magnetic fields.
Comparisons of the data with 1-D MHD simulations are discussed
in Ref. [9].
The strong
agreement
betweenthe
MBFM
simulation
predictions and the data is encouraging in that it indicates aplasma response necessary for current drive. The data, however,suggest that the failure to increase the current under highpower "pumping" conditions is due to an increased plasma
resistanceresultingfrom modulation enhanced plasma-wallinteractions.
For example, during thepumpingdischarge
illustrated in Fig. 2, the 0-D model calculated that a 10 to 15
per cent increase in the bulk plasma resistivity and resistanceis necessary to match the experimental results. This is of thesame order as the expected OFCD current enhancement, assumingtheresistance was not directly affected by the applied
modulations. Although investigation of techniques to reduce thewall interactions on ZT-40M is continuing, an unambiguous
demonstration of OFCD may require the substantially improved
performance characteristics, in terms of plasma resistivity andwall protection, anticipated for ZT-H.

3. IMPURITY CONTROL
ZT-40M has typically operated without limiters, or withlimiters whose scrape-off lengths are short compared to theconnection lengths between the limiters.
As a result, thedegree of metal contamination in the plasma has been high enoughtoaccount for most of the radiated power and for the
spectroscopically determined Z e £f[10]. Because the metal influx

428

SCHOENBERG et al.

from the Inconel vall[ll] increases strongly with the ratio ofplasma current to electron inventory (I/N), a number of methods,
includingpulseddischarge cleaning, gas puffing, pellet
injection, and wall carbonization, have been used to maintain an
appropriately high density at a given current.
Wall carbonization[12] entails depositing thin layers ofcarbon on the Inconel vacuum liner with theintentofsubstituting a low-Z wall material for some higher-Z elements.
Depending on the wall carbonization method, one can either
generate a metal carbide layer
or a layer of carbon whose
structure is a function of the deposition procedure[12].
With
the method utilized on ZT-40M[13], approximately 40 monolayers ofcarbon were deposited. The carbon layer was presumably not the
very
hard
structurethatcan be obtained at elevatedtemperatures[12].
Reversed field pinch discharges were obtained
almostimmediately
after
carbonization
andshowed significant
differences in impurity content. Specifically, C III emission
implied an increase in carbon from 0.06 % with a metal wall to
1-2 % after carbonization. The reduction in metal impurities
wasapproximately
a factor of two, assuming self-similar
profiles, and both metals and oxygen were at the low end of
their historical range. Soft x-ray fluxes, which are sensitive
to metal impurities, were also lower.
Due to the increased
density, electron temperatures measured on axis by Thomson
scattering showed a reduction from 360 to 200 eV for discharges
with carbonization compared to the metal wall case. However,the plasma resistivity was nearly the same[13], implying a
reduction in Z e ff assuming the resistivity behaves classically.
The beneficial impurity control effects of carbonization on
ZT-40M are relatively short-lived.
Spectroscopy indicates an
e-folding time of 30-60 discharges for the carbon level todecrease and the metal influx to increase.
This effect was
further accelerated on the liner segment that lies under the gap
in the shell and during OFCD discharges, implying that a large
power flux will remove the layer more rapidly. However, in
future RFP devices such as ZT-H, a combination of an effective
limiter set and carbonization at higher wall temperatures is
expected to improve the effectiveness of this impurity control
technique.

4.

ENERGETIC ION PLASMAS

Normally, ZT-40M operates in a regime where the inferred
ion temperatures are approximately equal to theelectrontemperature.
However,duringlow density ramped current
discharges, OFCD discharges, or high-6 flat-topped discharges,

IAEA-CN-47/D-II-3

429

enhanced ion heating during the discharge is apparent. For
example, in discharges where the plasma current was ramped from100 to 200 kA with T e (o) ~ 300 to 400 eV and n £ = 1.8xl0 19 nr 3 ,inferred ion temperatures exceeded T g by at least a factor oftwo.
Measurements of ion energies in ZT-40M were made using
Doppler broadening of light impurity ions (CV, NVI, 0VII)[14], aTime-Of-Flight(TOF) charge
exchange neutral
particle
spectrometer, measured deposition profiles in silicone samples
exposed to the plasma[ll], and neutron production. Absolute
neutron yield was obtained from a post-shot arsenic activation
detector[15], while the time resolved neutron rate was obtained
from a r-insensitive ZnS(Ag) proton recoil detector, as well as
a e LiI scintillator.
A
TOF neutron spectrometer, from Phillips Petroleum
Company!16], has indicated that the measured neutron flux
corresponds to 2.5 MeV D-D neutrons, but with insufficient
statistics to resolve a spectral width or shift. For the ion
impurity inferred temperatures, the thermal characteristics ofthe high energy deuterons are also unresolved.
However, athermal deuterium ion distribution with T^ > T is consistent
with neutral energy spectra (< 2 keV) measured by the TOF charge
exchange spectrometer. The enhanced ion heating observed,
during energetic ion n discharges,
may be sufficient to maintain aconstant &JOTAL' * s P^- te °f tne slow decrease of g g with I/Nwhich was previously reported[17,18].
5. RFP START-UP
Studies on ZT-40M have addressed start-up methods in order
to assesstheir useinthe design of high current(multi-mega-ampere) RFP devices. The standard matched mode
start-up, in which the toroidal flux is kept constant, has beensuccessful for RFP experiments, but presents problems for futurehigh current devices. For example, a fast matched mode start-up
requires the use of high loop voltage, while a slow start
suffers from a long duration before field reversal, resulting inincreased energy losses and resistive V-s consumption.
The ramped mode[l,19] uses a fast, low-current, matchedmode RFP formation followed by a slow ramp to the final highcurrent. These ramped discharges have allowed successful RFP
start-up at reduced voltages, and their scaling results project
acceptable V-s consumption for
initiating
discharges
in
ZT-H[3,19], the next-generation RFP experiment at Los Alamos.
A survey of ramped mode start-up as a function of ramping
rate was made to determine its physical characteristics for use
in scaling to high currents. It was found that the bulk plasma

430

SCHOENBERG et al.

G
a
tr

200

FIG. 4. Calculated bulk plasma resistance (Rp) as a junction of the instantaneous current during aramped discharge. Each curve corresponds to a different ramping rate for a duration of 11 ms.
The curves are labelled by the peak value of 1^ measured at the end of the ramp.

resistance R , as deduced using the 0-D model[6], increased withthe ramping rate (Fig. 4.). This result may be attributed to anincreased Z g ££, due to enhanced wall loading and impurity
release accompanying the faster ramping rates. The increased
resistance may also be due to the increased dynamo activity
needed for the large rate of toroidal flux generation during thefaster ramps. Since the dynamo plays a prominent role during
ramped start-up, ramping provides a valuable experimental basis
for improving the understanding of RFP discharges[20,21,22].

6.

ZT-P OPERATIONSThe ZT-P experimental was designed as a prototype for the
ZT-H experiment.
ZT-P utilizes an air-core poloidal fieldsystem to drive the plasma current
and provide
plasmaequilibrium with intrinsically low magnetic field errors. ZT-P
began operation in October 1984 and obtained its first RFP
plasmadischarge
in January
1985. To date, ZT-P has
investigated ramped RFP plasma start—up and air-core RFP
equilibrium control by passive field programming, using bothflat-topped and decaying current
discharges.
For these
discharges, which have been run up to 60 kA (<j> ~ 5 MA/m ) for
durations up to 1.2 ms, spectroscopic measurements indicate ahot (T g > 70 eV) and relatively low~Z plasma. Ongoing ZT-P
experiments are continuing to investigate ZT-H design issues,

IAEA-CN-47/D-H-3

431

including dynamic RFP equilibrium control in an air-core system,
as well as the effects of imposed field errors on performance.
High current density RFP scaling is also being studied using
recently fielded 2-color interferometry and Thomson scatteringsystems.

ACKNOWLEDGMENTS
The authors gratefully acknowledge the contributions tothis work, by visiting staff members, including Dr. Zensho
Yoshida,Dr. Shunjiro
Shinohara,
Dr. Sadao
Masamune,
Ms. Elizabeth Nilles, and Dr. Klaus Buchl. We are also gratefulto the Los Alamos RFP operations team for their assistance andcooperation. This work was supported by the U.S. Department ofEnergy.

REFERENCES[1] BAKER, D.A., BUCHENAUER, C.J., BURKHARDT, L.C., et al., in Plasma Physics andControlled Nuclear Fusion Research 1984 (Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA,
Vienna (1985) 439.
[2] SCHOENBERG, K.F., et al., ZT-P: An Advanced Air Core Reversed Field Pinch Prototype,
Rep. LA-10593-MS, Los Alamos National Lab. (1986).
[3] THULLEN, P., SCHOENBERG, K.F. (Eds), ZT-H Reversed Field Pinch Experiment, Rep.
LA-UR-84-2602, Los Alamos National Lab. (1985).
[4] GREENWALD, M., GWINN, D., MILORA, S., et al., Phys. Rev. Lett. S3 (1984) 352.
[5] BEVIR, M.K., GRAY, J.W., Proceedings of the Reversed Field Pinch Theory Workshop, Rep.
LA-8944-C, Los Alamos National Lab. (1981) 176.
[6] SCHOENBERG, K.F., GRIBBLE, R.F., BAKER, D.A., J. Appl. Phys. 56 (1984) 2519;
Rep. LA-9161-MS, Los Alamos National Lab. (1984).
[7] BELLAN, P.M., Tokamak applications of helicity injection, Los Alamos Current Drive
Workshop 5/86, personal communication.
[8] HOGAN, J.T., OFCD studies at ORNL, Los Alamos Current Drive Workshop 5/86, personalcommunication.
[9] NEBEL, R.A., et al., Paper IAEA-CN-47/D-V-12, these Proceedings, Vol. 2.
[10] WEBER, P.G., Phys. Fluids 28 (1985) 3136.
[11] BASTASZ, R.R., CAYTON, T.E., J. Nucl. Mater., to be published.
[12] WINTER, J., WAELBROECK, F., WIENHOLD, P., et al., J. Nucl. Mater. 122&123 (1984)
1187.
[13] CAYTON, T.E., DOWNING, J.N., WEBER, P.G., et al., J. Nucl. Mater., to be published.
[14] HOWELL, R.B., NAGAYAMA, Y., Phys. Fluids 28 (1985) 743.
[15] JACOBS, E.L., BONAPARTE, S.D., THATCHER, P.D., Nucl. Instrum. Methods 213 (1983)
387.

[16]
[17]
[18]

WALKER, S.E., PRESZLER, A.M., MILLARD, W.A., Rev. Sci. Instrum. 57 (1986) 1740.
WURDEN, G.A., WELDON, D.M., WEBER, P.G., et al., Bull. Am. Phys. Soc. 30 (1985)
1402.
ALPER, B., EVANS, D.E., STOREY, D.P., TSUI, H., Bull. Am. Phys. Soc. 30 (1985) 1406.

432

SCHOENBERG et al.

[19] PHILLIPS, J.A., et al., Observations on Ramped Current Operations in ZT-40M, Rep.
LA-10060-MS, Los Alamos National Lab. (1984).
[20] CARAMANA, E.J., BAKER, D.A., Nucl. Fusion 24 (1984) 423.
[21] CARAMANA, E.J., SCHNACK, D.D., Phys. Fluids 29 (1986) 3023.
[22] HOWELL, R.B., INGRAHAM, J.C., WURDEN, G.A., WEBER, P.G.,
BUCHENAUER, C.J., Rep. LA-UR-86-1727, Los Alamos National Lab. (1986),
submitted to Phys. Fluids.

IAEA-CN-47/D-II-4

EXPERIMENTAL STUDIES OF
REVERSED FIELD PINCH IN REPUTE-1
AND ULTRA-LOW-q DISCHARGE
N. ASAKURA1, A. FUJISAWA1, T. FUJITA, Y. FUKUDA,
H. HATTORI, N. INOUE, S. ISHIDA, K. ITAMI, Y. IWATA, H. J I \Y. KAMADA, K. KUSANO2, S. MATSUZUKA1, K. MIYAMOTO1,
J. MORIKAWA, C.P. MUNSON3, Y. MURAKAMI, Y. NAGAYAMA1,
H. NIHEI, S. ONODERA1, M.T. SAITO, T. SATO2, S. SHINOHARA1,
H. TOYAMA1, Y. UEDA1, M. UTUMI, H. YAMADA,
K. YAMAGISHI1, Z. YOSHIDA
Department of Nuclear Engineering,
Faculty of Engineering,
University of Tokyo,
Tokyo, Japan
Abstract
EXPERIMENTAL STUDIES OF REVERSED FIELD PINCH IN REPUTE-1 AND ULTRA-LOW-q
DISCHARGE.
The general behaviour and the relaxation process of the RFP plasma in the REPUTE-1 experiment are described. The RFP plasma is formed in the resistive shell, with a time constant of 1 ms; themaximum discharge duration is 2.5 ms, at present. The dynamics of the relaxation phenomena is studied
by inserting magnetic probes. It is observed that the current density profile and the X(r) = jtjpB/B 2
profile become peaked and the safety factor q0 on axis decreases during the current sustainment phase.
Then, q0 recovery takes place in a time of the order of several tens of the Alfven transit time and thetoroidal current density flattens; this process is repeated. These observations and the results of resistive
MHD simulation correspond to each other satisfactorily. Stable toroidal discharges in an ultra-low-q(ULQ) regime, i.e. 1/2 < o^ < 1, have been realized and sustained for a much longer time than thecharacteristic MHD time. An ULQ equilibrium is set up through MHD relaxation, promoted by an
m = 1 global kink instability and characterized by a dq/dr<0 profile. A class of dq/dr<0 equilibria
is shown to be stable against low n kink modes, by using an energy principle.

1.

DEVICEREPUTE-1 (Reversed Field Pinch, University of Tokyo Experiment) is areversed field pinch (RFP) device with a major radius R of 82 cm, a minor radius
ap of 20 cm and a flux swing of the iron core transformer <t> of 1.6 V-s [1, 2]. The
vacuum chamber of Inconel bellows is surrounded by a 5 mm thick stainless steel
Department of Physics, Faculty of Science, University of Tokyo, Tokyo, Japan.
Institute for Fusion Theory, Hiroshima University, Hiroshima, Japan.
3
Los Alamos National Laboratory, Los Alamos, NM, USA.
2

433

434

ASAKURA et al.

250

t kA)
FIG. 1. Dependence of nt and 1% (under the assumption ofZeg = / and modified Bessel Junction
model) on plasma current Ip (Q = 1.6-1.8) for optimized conditions. Filling pressure is about
3.2 mtorr.

shell with a time constant of 1 ms for vertical field penetration. The equilibrium field
is applied externally by vertical field coils wound outside the toroidal field coils. The
rise time of the plasma current is 0.5 or 1 ms. This device is also used for ultra-low-q
discharge experiments.

2.

RFP EXPERIMENT IN REPUTE-1

2.1. Parameters of RFP plasma in REPUTE-1
Experiments have been carried out in the I p <240 kA range of plasma current.
The RFP plasma is formed in the resistive shell with a 1 ms time constant; themaximum discharge duration is 2.5 ms, at present. An enhancement of them = 1 MHD perturbation and a deviation of the externally applied vertical fieldfrom the value required for equilibrium are observed before discharge termination.
The line averaged electron density, fig, measured by a CO2 laser interferometer at
the current flat-top, and the conductivity electron temperature T^ on axis increase
approximately linearly with the plasma current in optimized conditions(ii,.- 1.0 x 1014 cm" 3 , T^ ~ 80 eV at Ip =200 kA, on the assumption of Zeff = 1;
see Fig. 1). The O V line is burnt through during current flat-top. The ion temperature T i; measured by a time of flight method, is about 80 eV at 200 kA. The time
variation of the plasma parameters of a typical RFP discharge, in the case of a highpinch parameter 0 , is shown in Fig. 2.

435

IAEA-CN-47/D-II-4-0.4 -

'£

i.o •

X
VI

3-0

FIG. 2. Time variation of loop voltage VL, plasma current lp, pinch parameter 0 , reversal ratio F, line
averaged electron density n€ , relative intensity of soft X-ray SX along horizontal line measured by surface barrier diode ORTEC CR-017-050-100 with 4 urn thick polypropylene filter, ion temperature T,
measured by time of flight method (closed circles) and by Doppler broadening of O Vline (278 nm, open
circles) in typical RFP discharges.

2.2. Relaxation process in RFP plasma
The internal distribution of the magnetic fields and field fluctuations of thepoloidal and toroidal fields during current flat-top is measured in detail by using
inserted magnetic probes in order to study the dynamics of the relaxation phenomenain the range of I p < 130 kA. The magnetic fields in the toroidal and poloidal directions are measured simultaneously at six radial points 3.4 cm apart from each other.

ASAKURA et al.

436

0.1

30.

I20
t-t 3
0

5

10 15
r(ctn)

20

FIG. 3. Time evolution of toroidal and poloidal current densities, j , (r) and jp (r), safety factor q(r)and \(r) parameter estimated from magnetic probe measurements at different times tt = 0.56 ms,
t2 = 0.66 ms and t3 = 0.70 ms for 6 "1.9 operation.

External magnetic probes and an array of surface barrier diodes without filter for
VUV measurement are also used. A large amplitude oscillation of the poloidal fieldwith mode numbers m = 0 and m = 1 is observed when the pinch parameter 0
exceeds about 1.75. Figure 3 shows the time evolution of the toroidal and poloidal
current densities, j t (r), j p (r), the safety factor q(r) and X(r) parameter defined byMr) = fiof-B/B2, estimated from magnetic probe measurement and plotted at different times tj = 0.56 ms, t2 = 0.66 ms and t3 = 0.70 ms from the start of the discharge (©=1.9). The safety factor on axis, q0, becomes lower, because of thepeaking of the current density profile (t, = 0.56 ms — t2 = 0.66 ms); the m = 1
fluctuation of the VUV signal becomes large. Then, q0 recovery takes place in atime (<40 (is) of several tens of the Alfve"n transit time, rA and the toroidal current
density flattens. The RFP plasma reaches a more relaxed state as is seen from theX(r) profile at t3 = 0.70 ms as the q0 value recovers (increases); this process isrepeated (Fig. 4). While the q0 value recovers, toroidal flux is generated. Theseobservations and the results of a resistive MHD simulation [3-5] show good correspondence to each other, and these experimental results could be explained by nonlinear, driven reconnection of a global (non-resonant) helical mode. A lower limit of

437

IAEA-CN-47/D-II-4

0.12
0.11
c? 0.10

0.09

0.4

0.5

0.6 0.7
t(ms)

0.8

0.9

FIG. 4. Time variation of safety factor on axis, q0 , and toroidal flux <£„• calculated from magnetic
field profile.

the q0 value exists at -0.095, and the q0 value after recovery exceeds 0.11-0.12,
in most cases. The observed dominant mode is (m,n)=(l,9). These results are consistent with the stability limit of an ideal internal m = 1 kink mode. We observe an
increase in the ion saturation current of the Langmuir probe at the plasma edge, atthe time of q0 recovery. This suggests that the plasma energy loss is enhanced by
the reconnection process.

3.

ULQ DISCHARGE

We shall now describe the observation of stable ultra-low-q equilibria. The
ultra-low-q (ULQ) regime, i.e. 0 < q a < 1, studied here lies in the parameter region
intermediate between tokamak and RFP. Z-pinch discharges as were studied extensively in the old days belong to the ULQ discharge. Their equilibria were, however,destabilized by kink instabilities in the characteristic MHD time-scale (about the
Alfve'n time) [6].
We have realized stable toroidal discharges against a global kink mode withq a ~ l / 2 , through the relaxation phenomena. The basic idea of the production of
ULQ equilibria has been proven experimentally in REPUTE-1 [7]; TORIUT-6(R/a = 0.33/0.05 m) was constructed for the specific purpose of ULQ experiments.
Figure 5 shows typical ULQ discharges on TORIUT-6 in different two operational
schemes. In the discharge of Fig. 5(a), the plasma is raised to the MHD excited stateby the current spike in the startup phase. Then, it experiences an MHD relaxation
associated with a global kink mode ((m,n) = (1,3)) and enters a quasi-stable phaseanalogous to the quiescent phase of an RFP, where the amplitude of the poloidal field
fluctuation at the wall is less than 2% of the equilibrium field. An ion density
pumpout phenomenon is also observed simultaneously with the relaxation. The magnetic Reynolds number, S, is estimated to be around 5 x 1013, during ULQ438

ASAKURA et al.

(b)

(a)
Plasma Current (KA)

#2375
^-1/2

«2369

20 z
10
PoLoidaL

FieldLine Averaged0

a t Wall(A.U.)

IonDensity

0.5 1. 0 1.5
Time (ms)0( 1 0 m)

0.5 1.0 1.5
Time (ms)FIG. 5. Plasma current, poloidal field at wall and line averaged ion density measured by a neutralbeam probe for ULQ discharges on TORIUT-6: (a) set up through MHD relaxation, (b) without
relaxation.

quiescence. After a period of a quasi-stable state for 600 jtts, another global kinkinstability ((m,n) = (1,2)) is excited and degrades the plasma. Obvious differencesfrom the other discharge without current spike can be seen as is shown in Fig. 5(b).
The fluctuation of the magnetic field is much larger than in the former case, and the(1,2) kink instability appears in an early phase of the discharge. The plasma is notstable against MHD modes throughout the discharge.
The toroidal current density and the q-profiles derived from the local fieldmeasurements by inserted probes are shown in Fig. 6 for the same discharges as areillustrated in Fig. 5. When the stable ULQ state is set up at t = 466 us, the current
profile is hollow in the central region, and dq/dr is negative and q lies between two
rational values, i.e. 1/3 and 1/2 (see Fig. 6(a)). The poloidal field decreases aroundthe edge of the plasma and in the vacuum region, which indicates that the volume
current does not flow behind the limiter. The dq/dr < 0 state is maintained for around
600 /is and no instabilities are observed to degrade the plasma within that scale.
However, the magnetic field diffusion ( T R = 2 0 0 US in these experiments) slowly
deforms the field structure to the state with dq/dr >0, where the global modes aredestabilized. On the other hand, while the dq/dr is a little negative in the initial phaseby the skin effect of the current penetration, the stable ULQ equilibrium cannot be
formed without the spire of the current (see Figs 5(b) and 6(b)). We can conclude
from the comparison between two shots that whether the relaxation takes place or notmakes a fundamental difference in equilibria.
Linear stability against ideal MHD modes is studied numerically by the energy
principle. A class of dq/dr < 0 ULQ equilibria is shown to be stable against low-n

439

IAEA-CN-47/D-II-4

(a)

Jt
lr

'D 20
M I nor(b)

4Q 60
Radius

0 20 40 60
M 1 n o r Rad t us

Jt
#2375
1 .0

i. 5

tu

0 20 40 60
tt Inor R a d l u s

0 20 40 60
MI nor Rad1 us

FIG. 6. Toroidal current density and q-profile evolution. Figures (a) and (b) correspond to discharges(a) and (b) of Fig. 5, respectively. Unit of minor radius is mm.

kink modes. The maximum beta limited by kink mode stability is about 5%. A
preliminary estimate of the average toroidal beta is typically around 2% in
TORIUT-6 experiments. The global mode stability is robust enough to permit a certain vacuum region between the plasma and the wall (typically, the radiusratio ~ 1.1). Local modes are destabilized in the ULQ equilibrium because of theexistence of the pitch minimum. They are, however, fairly localized in the pitch minimum region, and their growth rate is by an order of magnitude smaller than thoseof the internal kink modes.
The setting up of ULQ equilibrium is simulated and analysed by a 3-D MHD
code and contrasted with the self-reversal mechanism of RFP and the sawtooth
activity in a tokamak. It is concluded that the non-linear growth of the global helicalkink instability and the driven reconnection lead to a stable dq/dr<0 state in the ULQregion. This relaxation process occurs in the relatively high S regime (>10 3 ). In
the low S regime, the evolution is dominated by magnetic field diffusion, and nodq/dr<0 state is attained. Experiments and MHD simulation have led to a consistentand comprehensible picture of ULQ physics in terms of MHD relaxation.

REFERENCES[1][2]

MIYAMOTO, K., INOUE, N., Nucl. Fusion 25 (1985) 1303.
ASAKURA, N., FUJITA, T., HATTORI, K., INOUE, N., ISHIDA, S., et al., Plasma Phys.
Controll. Fusion 28 (1985) 805.

440

ASAKURA et al.

[3]
[4]
[5]
[6]
[7]

SCHNACK, D.D., CANAMARA, E.J., NEBEL, R.A., Phys. Fluids 28 (1985) 321.
SATO, T., KUSANO, K., Phys. Rev. Lett. 54 (1985) 808.
KUSANO, K., SATO, T., Nucl. Fusion 26 (1986) 1051.
FURTH, H.P., Phys. Fluids 28 (1985) 1595.
YOSHIDA, Z., et al., J. Phys. Soc. Jpn. 55 (1986) 389.

DISCUSSION
T. TAMANO: I should like to ask two questions. First, ultra-low-q discharges
last longer than RFP discharges. Do you have any explanation for that? Second, did
you observe that t h e m = 1, n = 10 mode appeared and disappeared according to
the change in q(0)?
K. MIYAMOTO: RFP plasmas are likely to be unstable for high n modes andare sensitive to the effect of local error fields. Ultra-low-q plasmas have a strong
toroidal field compared to the poloidal field and may withstand the local
perturbations.
In reply to your second question, although n = q is the dominant mode, there
are several different n modes. We did not study the differences in mode spectrum forthe change in q(0) in detail, but in any case we did not observe any big differencein fluctuation modes for the q(10) change during a discharge.

IAEA-CN-47/D-II-5

HEATING AND CONFINEMENT STUDIESIN REVERSED FIELD PINCHES
V. ANTONI, A. BUFFA, L. CARRARO, S. COSTA,
P. De SIMONE, G. FLOR, F. FLORA, L. GABELLIERI,
M. GIUBBILEI, L. GIUDICOTTI, P. INNOCENTE,
K. JAIN1, P. MARTIN, S. MARTINI, D. MERLIN,
S. ORTOLANI, R. PACCAGNELLA, M.E. PUIATTI,
B. SASS2, P. SCARIN, H. SORENSEN2, M. VALISA,
K. WEISBERG2, S. ZAGO, E. ZILLI
Associazione Euratom-ENEA-CNR,
Istituto Gas Ionizzati,
Consiglio Nazionale delle Ricerche,
Padua, Italy
Abstract
HEATING AND CONFINEMENT STUDIES IN REVERSED FIELD PINCHES.
A possible interpretation of the large resitivity anomaly factors characteristic of the low density
regime is related with an increase of Zeff in the plasma edge region, leading to an enhanced dynamo
contribution to the apparent on-axis resistivity. The natural tendency of the discharge to evolve towards
a low density, high I/N regime is counteracted by pellet injection, resulting in higher beta discharges.
The sustainment of the RFP distribution can be interpreted as the outcome of periodical relaxation
processes which counteract the resistive diffusion. Resistive m = 1 instabilities are a probable mechanism
underlying relaxation; they set a limit to q on axis of about % a/R. The expected plasma performancein the RFX experiment is discussed.

1.

INTRODUCTION

During 1985-1986, the experiments on Reversed Field Pinches (RFP) in
ETA-BETA II have been aimed at a more detailed study of the physics phenomena
underlying plasma heating and confinement as well as sustainment of theconfiguration. Since the 1984 IAEA conference [1], more detailed measurements
have been carried out, with spatial resolution of plasma density, temperature, impurities, fluctuations and mean magnetic field [2, 3], From these measurements it has

1
2

Institute for Plasma Research, Gandhinagar, Ahmedabad, India.
Rise National Laboratory, Roskilde, Denmark.

441

442

ANTONI et al.

become clear that the RFP behaviour is strongly related with the continuous relaxation process and the consequent sustainment of the magnetic field profiles. In particular, distinct progress has been made in the following areas:(a)Density dependence of heating and confinement: The confinement degradation
observed in the low density regime [4] has been interpreted as being due to theenhanced dissipation associated with the RFP profile sustainment when larger
Zeff values are present in the plasma edge region [5]. These studies havestressed even more the importance of density sustainment. First results of pellet
injection are reported in this paper.
(b) MHD stability and q-limits of the mean magnetic field profiles. During thesustainment phase, regular oscillations of the on-axis safety factor q(0) of about% a/R have been measured [3] and the MHD stability analysis shows that thislimit on q(0) is related with m = 1 tearing modes resonant inside the region of
toroidal field reversal [6].
Finally, beta limits and confinement scaling are briefly discussed and the
expected parameters of the RFX experiment [7, 8] under construction at Padua
are given.

2.

DENSITY DEPENDENCE OF HEATING AND CONFINEMENT

2.1. Resistivity anomaly factor
On ETA-BETA II, it has been observed [2, 4] that the low density regime (withan I/N ratio of > 10~14 A*m, where N is the cross-section integrated density) is
associated with a degraded confinement time, although it is characterized by highelectron temperatures. This confinement degradation is associated with a decrease ofbeta and with a strong increase of the on-axis resistivity anomaly factor, defined aswhere?j*(0)

istheresistivity derived from Ohm's law on axis,
The values of Z*ff(0) may exceed 15 in some cases andshow a clear correlation with the I/N parameter [4]. With the assumption of a uniform
effective ion charge Zeff, the highest Z*ff(0) can be accounted for only marginallyand by assuming the highest possible estimates for the ion charge Zeff [4], even bycomputing the on-axis resistivity from a helicity balance as rjk(O) = $V, /J ijJ'Bdv(where rj(i) is the resistivity normalized to the on-axis value).
On the other hand, we have recently considered the effect of a non-uniform
Zeff and we find that large resistivity anomaly factors can be associated with increased dissipation in the outer plasma region connected with a hollow profile of theion effective charge [5]. From the line emission profile of the dominant impurity ions
TJ*(O) = V]oo f(2irRJJP)).

443

IAEA-CN-47/D-II-5-0.1 -0.05

00.05. 0.1 r[m]FIG. 1. Experimental profile of ion effective charge.

- / T?K

15-

10-

5-

0

2

4

6

f

FIG. 2. Comparison of the ti'(0)/r)k(0) values corresponding to the experimental 2eff profile and thosewith 2eff = I (ZeB(r) = Zeff(r)/Ze/0)) for several J and Te profiles.

(oxygen) measured in ETA-BETA II [5] the radial profile of the ion effective charge
can be derived and is indeed found to be hollow, as shown in Fig. 1. The corresponding values of the 7j*(0)/»jk(0) ratio are shown for several current density and electron
temperature profiles in Fig. 2. The current density profiles are described by/i = JUO(J,/B), with jt//x(O) = 1 - (r/a)" (see Section 3), and the expression
T/T e (0) = 1 - (r/a)7 is used to study the effect of different T, profiles. It can be

444

ANTONI et al.

seen that for typical values of a and y (between 2 and 4 as deduced from theexperimental profiles [2, 3]), the Tj*(O)/ijk ratio ranges between 3 and 10. To highlight the effect of a non-uniform Zeff, the results obtained with Zeff = const are also
reported. The apparent resistivity on axis, T;*(0), is higher by a factor of about twofor the experimental Zeff profile. Since the corresponding average ion charge issmall (about 1.2), the difference found is practically only due to the profile shape.
Thus the i]*(0)/ri\0) enhancement due to a hollow Zeff profile could explain theobserved discrepancies between Z'^0) and the estimated global ion charge, particularly at high I/N [2, 4]. In fact, to sustain the current density distribution in a pinch
characterized by a relatively high ion charge (i.e. a highly resistive plasma) in theouter region, a higher loop voltage must be applied so that the enhanced dissipationof helicity can be balanced by a higher external input via the dynamo process. This
results in an increase of the apparent resistivity on axis.
It can therefore be concluded that, in general, the impurity content in an RFP
is of fundamental importance in the determination of the plasma behaviour, both in
the high density regime and in the low density regime [9]. At high density the impurities have a direct influence on the upper limit for the electron density (or, more precisely, the lower limit for I/N) beyond which radiation losses dominate the plasma
power balance [9, 10]. Although radiation is much lower in the low density regime,
the ion effective charge increases, owing to the density decrease and probably theincrease of impurities, and enhances the dissipation in the outer region of the plasma.
Hence the sustainment of the RFP requires a stronger dynamo activity, which in turnis associated with enhanced transport and confinement degradation. This interpretation clearly stresses the importance of operating at high density. First results withpellet injection are presented in the following section.

1.5
I—1
p>

V

«:•

"E 1.58
o 1IC

0.5Qi

t[ms]

2.5

WITH
PELLET

WI1HOUTPELLET0o'.5f

i 1 i!5

-'

,

2.5

FIG. 3. Line averaged density versus time for various discharges with and without pellet injection.
The arrows indicate the time at which the pellet is injected.

445

IAEA-CN-47/D-II-5
l/N& WITHOUT PELLET20-• W I T H PELLET

15-

10-

i

•

••

5—i—

1.5

0.5

t[ms]

Te[eV]

200-

•

•

100-

T-

0.5

1.5

t[ms]

1.5

0.1-

•V

.
a

^

•

•

0.05-

0.5

1

1.5

t[ms]

FIG. 4. Time evolution of I/N, Te from SXR pulse height analysis and 0e for three typical sets ofdischarges, with and without pellet injection. Each point is averaged over many discharges and over
200 us time intervals.

446

ANTONI et al.

2.2. Pellet injection results
Single deuterium pellets with velocities of about 100 m/s and a mass of about
1019 molecules have been injected in ETA-BETA II for the first time, with an injector system developed at Ris0 National Laboratory [11]. The pellet is completely
ablated during the pulse. Figure 3 shows some examples of the line integrated density
versus time, with and without pellet injection. It is seen that the time evolution of
the density is significantly altered by this technique. In some cases this leads todensities higher than 5 X 1019 m~3 which are maintained throughout the pulse. In
this way the natural tendency of the discharge to evolve towards a high I/N regime
can be somewhat controlled and counteracted (see Fig. 4). The change in density
behaviour is coupled with a change in the electron temperature; this is also shownin Fig. 4 for two typical sets of discharges, with and without pellet injection. Of
course, higher densities correspond to lower temperatures. However, as shown in
Fig. 4, when the density is sustained, beta is also sustained. Thus, these preliminary
results are a further proof of the interdependence between I/N, |3 and TE previously
found [2, 4]. Moreover, it appears now possible to escape from the high I/N regime.

3.

MHD STABILITY AND q-LIMITS OF THE
MEAN MAGNETIC FIELD PROFILES

The processes underlying the sustainment of the RFP distribution ('dynamo'
mechanism) are not well understood at present. However, under consideration is theeffect of the magnetic fluctuations and in particular of that portion of the spectrum
which is interpreted as being due to MHD activity. Actually, it has been foundrecently [3] that a periodical activity comes from the turbulent background fluctuations that are always present in RFPs with amplitudes of about 1 % in present experiments. This process is related with a substantial redistribution of currents in theplasma, resulting in a cyclic flattening of the current density profile which would
otherwise tend to peak on axis. It has also been found that the experimental profiles
can be well described by the expression ju(r) = ^(0)[l-(r/a)"] [12] and that theprocess results in a periodical oscillation of m(0) or equivalently of the on-axis safety
factor q(0) = 2/(Rjt(0)) [3].
An MHD stability analysis for current driven modes of these profiles has been
performed over a wide range of a and jt(O) [6]. The most severe boundaries forstability are due to m= 1 resistive modes; they are plotted in Fig. 5 in terms of thevalues of q on axis and q at the wall. Also shown are experimental values of q(a) andq(0) for typical RFP distributions. The mean field experimental distributions lie in thestable region below the uniform /t-curve corresponding to Taylor's fully relaxedstates; they are close to the m= 1 resistive internal (i.e. resonant inside the toroidal
field reversal surface) tearing mode boundary. Hence a lower limit on q(0),associated with the destabilization of these m = l modes, can be identified as

447

IAEA-CN-47/D-II-5

q(0)-§-

FIG. 5. Stability diagram for m = l modes in the plane q(a)R/a - q(0)R/a; the dots indicate experimental configurations. Also shown are the curve for the fully relaxed \L—const states with 6 in the range
1.2-1.56, and the curve for our model with a=8.

0.10

FIG. 6.

Time evolution of the experimental value of q on axis.

q(0) > % a/R. This limit is slightly dependent on the fi-profile (varying between
about 2A a/R for a = 3.9 and about Vi a/R for a = 2.7) and imposes an upper limiton the on-axis current density J(0) when J(0) < 3B(0)//iQa [12]. Generally, theexperimental magnetic field profiles are stable for current driven modes, but resistive
diffusion tends to destabilize the configuration by peaking the current distribution on
axis and thus leading to lower q(0). The excited internal current driven instabilities
lead to a redistribution of the current density profile and thus the stable configurationof the initial tearing mode is recovered; this process is repeated in a cyclic way [13].
In this sense the experimental mean field profiles, although they are not fully relaxed,
represent the outcome of periodical relaxation processes which counteract the diffusion process. This mechanism can explain quite well the periodical oscillations ofq(0) experimentally found in ETA-BETA II [3] around 0.11 (~ % a/R), as shown in
Fig. 6.

448

ANTONI et al.

TABLE I. PLASMA PARAMETERSEXPECTED FOR RFXa
(a)

(b)

a

(m)

0.5

0.5

R

(m)

2

2

I

(MA)

2

2

4

4

4

4

I/N

(10~

14

A-m)

z;ff
2O

3

<n>

(10 m~ )

0.6

0.6

ft

(%)

10

4.6

T(0)

(keV)

1.8

0.9

Te

(ms)

47

7

3

0.5

4.7

0.3

<n>rE (1018

3m- -s)(ms)" Scaling with 0 = const (a) and 0 = I~" 3 (b)iooooiboo T(eV)1014nT E (cm' 3 -s)

FIG. 7. Three-dimensional version of Lawson's diagram, including the beta values achieved in thepresent experiment and those expected in RFX.

IAEA-CN-47/D-II-S

4.

449

BETA LIMITS AND CONFINEMENT SCALING TO RFX

A simple estimate of the ideal MHD beta limits can be obtained by computing
the pressure gradient which marginally satisfies the Suydam's criterion for the^-profiles considered in Section 3. it is found that the tearing mode stable profiles
can support average beta values of the order of 30%.
However, in present experiments, beta values of only about 10% (well below
the ideal MHD limit) are achieved. Thus, the experimental beta values can be interpreted as the outcome of competing processes of relaxation and transport acting
against Ohmic heating. Two sets of plasma parameters as expected for RFX are listed
in Table I, according to two different scalings [7]. The value of (3gTE, which can beof the order of 1 ms in RFX, is of particular interest. Indeed, the beta values should
be considered when comparing various approaches to magnetic fusion. Figure 7
shows a three-dimensional version of the usual Lawson's diagram. A vertical axis isadded to represent beta, and a beta value of 5 % is set as the minimum for a fusion
reactor to be of practical interest.
The main goal of RFP research and in particular of the RFX experiment is todemonstrate at higher currents the possibility of confinement of a relatively high beta
plasma produced solely by Ohmic heating.

ACKNOWLEDGEMENTS
The authors wish to acknowledge the essential contribution of E. Baseggio in
the operation of the pellet injector and of ETA-BETA II. The valuable help of
G. Mella with the vacuum system and the technical support of I. Dalla Libera,
I. Molon, E. Perdon and A. Tamiazzo is also gratefully acknowledged.

REFERENCES[1]

ANTONI, V., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc.
10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 487.
[2] BUFFA, A., et al., in Controlled Fusion and Plasma Heating (Proc. 13th Europ. Conf.
Schliersee, 1986), Vol. 1, European Physical Society (1986) 349.
[3] ANTONI, V., et al., ibid., p. 337.
[4] ALPER, B., MARTINI, S., ORTOLANI, S., Nucl. Fusion 26 (1986) 1256.
[5] GABELLIERI, L., et al., Enhanced Dynamo Contribution to the Resistivity of a RFP due to Nonuniform Effective Charge Profiles, Internal Rep. 86/03, Istituto Gas Ionizzati, Padua (1986).
[6] ANTONI, V., MERLIN, D., ORTOLANI, S., PACCAGNELLA, R., Nucl. Fusion 26 (1986)
1711.
[7] ANTONI, V., MARTINI, S., ORTOLANI, S., in Controlled Fusion and Plasma Heating (Proc.
13th Europ. Conf. Schliersee, 1986), Vol. 1, European Physical Scoiety (1986) 385.
[8] MALESANI, G., ROSTAGNI, G., "The RFX experiment", Fusion Technology (Proc. 14th
Symp. Avignon, 1986), Pergamon Press, Oxford (1987).

450
[9]
[10]
[11]

[12]
[13]

ANTONI et al.
ORTOLANI, S., ROSAGNI, G., Nucl. Instrum. Methods 207 (1983) 35.
COSTA, S., De ANGELIS, R., ORTOLANI, S., PUIATTI, M.E., Nucl. Fusion 22 (1982)
1301.
SORENSEN, H., et al., "D 2 pellet injector for the Reversed Field Pinch Eta-Beta II in
Padova", Fusion Technology (Proc. 14th Symp. Avignon, 1986), Pergamon Press, Oxford(1987).
ORTOLANI, S., in Mirror-Based and Field-Reversed Approaches to Magnetic Fusion (Proc.
Workshop Varenna, 1983), Int. School of Plasma Physics (1984) 513.
ORTOLANI, S., in Twenty Years of Plasma Physics (Proc. ICTP Commemorative Meeting
Trieste, 1984), World Scientific Publishing, Singapore (1985) 75.

DISCUSSION
H. A.B. BODIN: I am puzzled by the curve in Fig. 1 showing that Zeff rises atthe edge. This seems to imply high impurity confinement times or nr values at theedge, so that the ions reach stages where they can contribute to Zefr; this also suggests slow impurity transport, otherwise the highly excited ions would be dominant
in the middle, giving higher Zeff there.
S. ORTOLANI: Actually, the hollow Zeff profile is due to the short particle
confinement time, which prevents higher ionization stages being reached and which
results in Zeff values close to one in the centre and relatively higher in the outerregion where the plasma electron density is lower.
A.A. NEWTON: You drew attention to the constancy of beta with time andunder a variety of conditions. Several experiments have shown beta to be about 10%,including the pellet injection in ZT-40M, and this now extends over a wide range ofparameters and Lundquist numbers. Does your cyclic model of diffusion, instability
and relaxation lead to a beta prediction?
S. ORTOLANI: The cyclic model of relaxation enables us to view the beta
values as the result of the counteracting actions of relaxation (and associated transport) and diffusion (heating). The model therefore suggests saturation at the beta
value at which this balance is achieved. So far, however, there is no quantitative
prediction for the beta values.
K. MIYAMOTO: If I understood you correctly, them = 1 resistive modes are
responsible for the MHD associated with the relaxation process. However, thegrowth time for resistive instability is of the order of 1 ms. The fluctuation periodof q(0) is shorter than the growth time. I would think that the ideal internal kink mode
could be responsible for the MHD activity associated with the relaxation process.
S. ORTOLANI: The MHD stability analysis shows that as q(0) decreases andthe current density profile becomes more peaked, the first modes to be destabilized
are the m = 1 tearing modes. However, for lower q(0) and more peaked current density distributions — as obtained at higher 6 values — ideal internal kink modes arealso found, as described in Ref. [6]. A quantitative comparison with the experimental

IAEA-CN-47/D-n-S

451

time-scales of the observed oscillations depends upon the relevant layer thickness
which enters into the process.
B. COPPI: Do you have a theoretical explanation for the observed nearconstancy of beta as n is increased?
S. ORTOLANI: When the density is sustained by pellet injection, the temperature decreases, but the resulting beta values are higher than those obtained withoutpellet injection. More precisely, beta remains high and does not decrease with time.
This is an experimental result and I do not have an exact theoretical explanation.

IAEA-CN-47/D-II-6

RESULTS FROM THE
REVERSED FIELD PINCH EXPERIMENTON TPE-1RM15 WITH A
PROGRAMMED VERTICAL FIELD
T. SHIMADA, Y. HIRANO, Y. YAGI,
A.A. NEWTON1, K. OGAWA
Electrotechnical Laboratory,
Sakura-mura, Niihari-gun, Ibaraki-ken,
Japan

Abstract
RESULTS FROM THE REVERSED FIELD PINCH EXPERIMENT ON TPE-1RM15 WITH A
PROGRAMMED VERTICAL FIELD.
The TPE-1RM15 is a new moderate size machine with R = 0.7 m and a = 0.135 m, operating
as a Reversed Field Pinch (RFP) at a current of up to 200 kA and a pulse duration of 8 ms. A DC
vertical field, B vdc , is used to centre the plasma toroidal equilibrium in a thick shell, and plasma
formation is assisted by a pulse cancellation field produced by coils inside the shell. During the flat-top
phase at I p = 135 kA, there is an optimum in Bvdc equal to 7.7 mT, and T^, ~ 600 eV andn,o ~ 2 x 1 0 " m~ 3 are obtained by Thomson scattering. The temperature is confirmed by analysisof the soft-X-ray spectrum measured with a SiLi detector. In these conditions the helicity resistivity
on axis shows an anomaly factor of four, Vloop = 30 V, /3p ~ 10%, T E — 0.2 ms and
1/N - 25 x 10" 14 A-m. The DC vertical field is varied over a wide range and T^ ~ 400 eV at
B vdc = 0.

1.

INTRODUCTION

The TPE-1RM15, which succeeds TPE-1RM [1], is a new moderate size
machine with R = 0.7 m and a = 0.135 m, operating as a Reversed Field Pinch(RFP) at a current of up to 200 kA and a pulse duration of 8 ms. Recent RFP experiments on various machines [2-4], including TPE-1RM, show that reduction ofmagnetic field errors and control of plasma equilibrium are essential for good RFP
discharges.
Our objectives in the TPE-1RM15 experiment are as follows:(a)To establish a technological and physical basis for obtaining and maintaining astable RFP configuration;
1

Euratom-UKAEA Fusion Association, Culham Laboratory, Abingdon, Oxfordshire, UK.

453

454
(b)

(c)

SHIMADA et al.
To demonstrate that temperatures in the range of Te = 0.5-1 keV and confinement times of up to 0.5 ms can be realized at high beta by Ohmic heating
alone;
To determine the scaling laws of temperature, beta and confinement time withcurrent and density; and to establish a benchmark at a moderate radius.

To realize these objectives, the following items were included in the design ofthe experiment:— Minimization of error fields arising from coils, current feeders, cuts in the conducting shell and supporting structures;— Control of the plasma equilibrium by two kinds of vertical fields, namely theDC vertical field, Bvdc, for aligning the outer magnetic flux surface with thelimiters during the flat top, and the control vertical field, Bvcv, for reducing theerror field at the shell gap;
— Transient cancellation of the DC vertical field at the set-up in order to facilitate
plasma formation;— Possibility of access for profile measurement of various plasma parameters.
The experiment has been successfully performed with vertical field control, andhigh electron temperatures (T^ ~ 600 eV) have been obtained at Ip = 135 kA.
The loop voltage has been reduced by a factor of two. The results show the importance of plasma positioning by the DC vertical field.
2.

THE DEVICE

The heart of the new device [5] is an air core solenoid of 0.8 V s , pulsed by
a 1.0 MJ capacitor bank so that the plasma current and the magnetizing current rise
simultaneously. The experiment starts with a slow current rise (2 ms) and earlyreversal of the toroidal field (0.5 ms). A power crowbar bank maintains a flat top
for 3-4 ms, followed by slow decay. A sudden termination of the discharge usuallyoccurs at 7-8 ms.
Equilibrium and stability are achieved with a 25 mm thick copper shell of
0.162 m inner radius. The vacuum vessel and the limiters (0.135 m inner radius) aremade of stainless steel. Because of the surface currents the poloidal field outside theshell would be that of a ring, and therefore a control vertical field, Bvcv, produced
by a pair of coils, is used to suppress the vertical component at the two shell gaps.
The control vertical field is driven by start and power crowbar banks to match thetime dependence of the plasma current. The same coils produce a DC vertical field,
Bvdc, which is applied 2 s before the discharge to centre the plasma during the flat
top. The shell gap fields, measured by coils across the gap on an inner surface of theshell, are < 5 mT during the flat top; the normalized gap flux, 4>%l\, is< 2 mWb/MAm, which is the flux used in medium-sized machines.

455

IAEA-CN-47/D-H-6

(a)

7.5
Urns)

(b)

2

1

1/

0

V
6

\\ 75
Kms)

0.5
F/G. 7. fa> Typical waveforms with an optimum DC vertical field (7.7 mT) and with optimum cancellation of the error field in the flat-top case with Ip = 135 kA. Filling gas D2, pressure Po = 4 mtorr,
initial toroidal field Bo = 100 mT.
(b) Time variations of the pinch parameter 8 and the reversal parameter F, under the same conditions
as in {a).

Since the DC vertical field would inhibit the growth of ionization, a pulse cancellation field, Bvpc, produced by coils inside the shell, is applied for 2 ms shortly
before the start of the discharge. In this way, the plasma can be formed with fillingpressures, P o , of > 4 mtorr and with Bvdc < 10 mT.
In this early period, the pulse cancellation field produces an additional verticalfield in the shell gap; this is reduced to < 30 mT by pulsed saddle coils placed acrossthe gaps.
3.

PLASMA PARAMETERS

Typical discharge waveforms, the pinch parameter 6 and the reversal parameter
F are shown in Fig. 1. The 0-value stays nearly constant (1.6-1.5) and the F-value

456

SHIMADA et al.

(o) density

7 0 0 •' * ' ' e m P e r a t u r e
*

4.5

•

3.0 2 O

plosma current

1.5

1.5

3.0

4.5
t(ms)

6.0

7.5

FIG. 2. Time variations of electron temperature and density measured by Thomson scattering, under
the same conditions as in Fig. I {a).

decays slowly from - 0 . 2 to - 0 . 1 . A low loop voltage (20-30 V) is obtained during
the flat-top phase at Ip = 135 kA when the optimum DC vertical field is applied.
The electron temperature and density are measured by Thomson scattering(Fig. 2). The temperature rises 1 ms after the current rise and reaches 500-600 eV,
where it remains constant for 3 ms near current maximum; the decay is slow at firstand more rapid near current termination. Simultaneous soft-X-ray measurementswith a SiLi detector between 2.4 ms and 5.4 ms also give a high electron temperature(about 570 eV); this agrees fairly well with the above measurements.
In the current rise and flat-top phases the electron velocity distribution is wellfitted by a Maxwellian, but there are about a factor of two more electrons of 2 keVin the current decay phase.
After the current rise phase the plasma density is constant at
n^ ~ 2 x 1019 m" 3 . The reproducibility of the line radiation of iron (225.9 run)and soft X-rays measured with a surface barrier diode is fairly good.
If we make the usual assumption of equal ion and electron temperatures and aparabolic pressure profile, j3p ~ 10% and TE ~ 0.2 ms. The estimated helicity
resistivity on axis, TJJ, using the modified Bessel function model (MBFM), gives ananomaly factor of four when calculated with the measured electron temperature andZ = l.
The value of I/N is about 25 x 10"14 A-m, assuming a parabolic density profile, and corresponds to a high Lundquist number S.

457

IAEA-CN-47/D-II-6

Ip(kA)
200
VCv=,0kV

3kV

3.5 kV4kV

4.5 kV

Kms)
FIG. 3. Variation of plasma current waveforms with different control vertical fields cancelling theerror field at the shell gap, from no cancellation (0 kV), through optimum cancellation (4 kV), to overcancellation (4.5 kV). Bo = 50 mT, p0 = 3.5 mtorr (D2), with the same bank conditions, except forthe control vertical field.

4.

SHELL GAP ERROR FIELD

A problem in connection with the conducting shell is the fact that the induced
surface currents inside and outside of the shell produce an error field at the gap (cut).
This error field is compensated by the control vertical field as described above. The
plasma behaviour is very sensitive to the measured Br signal at the gaps; this can beseen in Fig. 3, which shows the plasma current waveforms with different control vertical fields. If the cancellation of the error field is insufficient, the discharge pulse
is considerably shortened. With optimum cancellation, the discharge pulse lasts
8 ms.

5.

VARIATION OF THE DC VERTICAL FIELD

When plasma equilibrium is maintained by the conducting shell alone, the minor
radius, rp, of the outward toroidal shift of the plasma, A, is reduced. Since the conductivity of the shell is finite, the effective distance between the plasma and the shell
grows with time; this causes an additional shift of the plasma and a further reductionof rp. For positioning of the plasma column, a DC vertical field is applied which cancontrol the extent of toroidal shift. Figures 4 and 5 show the dependence of the electron temperature, density, plasma resistance (Rp) and energy confinement time on
the DC vertical field in the current flat-top phase. In these experiments the plasma
current is maintained in a flat top at around 135 kA for 3-5 ms.

458

SHIMADA et al.

700 . (• 1 temperature( o ) density
•

600 -

1

* 1

500

f 400

I

.

8

•

-4.5

*

*

• •

O

o

^300

"O

0
0
U

-3.0

O

A

0

200

OD

°

1

o

o

X

o

J

e

-1.5

100

1011
FIG. 4. Dependence of electron density and temperature on the DC vertical field. Flat-top cases with
Ip - 135 kA, 3.45 ms after start of plasma current.

o;Rp

"

• ; rE

t
;

O
0

:

• 8
•

•

o
*

8

*

-0.5

0

1

8

8

e

1

s

#
-

1

0

Bvoc

10

FIG. 5. Dependence of energy confinement time and plasma resistance on the DC vertical field.
Flat-top cases with I ~ 135 kA, 3.45 ms after start of plasma current.

The plasma resistance is at a minimum and the electron temperature is at a maximum around Bvdc .= 7.7 mT. When the effect of the penetration of the surface current into the shell is considered, this value agrees well with that calculated from
Shafranov's formula for A ~ 0. The ratio of the mean energy resistivity to the valuer)Q on axis, calculated using Spitzer's formula ( Z = l ) , decreases gradually with the459

IAEA-CN-47/D-H-6

0.7

•

1

0.5
t

'

!

»

•

•

•

0.3

0.1
I

1

1

1

i

.. I

Bvoc

1

J

1

1

10

FIG. 6. Dependence ofX, (the point where n in the MBFM begins to decrease) on the DC verticalfield. Flat-top cases with Ip ~ 135 kA, 3.45 ms after start of plasma current.

vertical field from 20 to 15 at the optimum Bvdc. The energy confinement time rE
peaks at the optimum Bvdc and the plasma resistance Rp decreases by more than afactor of two. The value of X,, which is the point where fi in the MBFM begins todecrease, increases from 0.35 to 0.55 (see Fig. 6), showing that the volume ofrelaxed plasma increases as the optimum Bvdc is approached.

6.

DISCUSSION

The improvement in the plasma properties, especially in the energy confinement
time, when a vertical field is used for control of plasma equilibrium is striking. Thishas been observed on HBTX1A [2] and HBTX1B [6], as well as on TPE-1RM15,
even though the aspect ratio of TPE-1RM15 is larger (5.2 compared with 3.2 on
HBTX1). It is interesting to compare the results for TPE-1RM15 (Bvdc = 0) withthose for ETA-BETA-II which has similar dimensions (R = 0.65 m, a = 0.125 m)
and a similar current range (100 - 180 kA). The value of I/N is a factor of two to
ten higher in TPE-1RM15, but /3p and rE are essentially the same [7]. The value ofZ*ff is about three-quarters of that expected from the trend with I/N as reported for
ETA-BETA-II [8]. When a vertical field is used, /?p is only marginally increased,
which indicates a strong tendency for /3p to remain constant as I/N is varied by anorder of magnitude. In TPE-1RM15, with the use of a vertical field, TE is increasedby more than a factor of two, mainly because of the low loop voltage, although Z*ff

460

SHIMADA et al.

is unchanged. The fact that r)\lriQ — 4 is also unchanged implies that the contribution due to impurities is also constant. According to the observed FE II line radiation,
the optimum positioning of the plasma column by the vertical field corresponds toless plasma-wall interaction, which may result in a low recycling level and low
plasma density (see Fig. 4). With /3p constant, the low density gives a high electron
temperature, which could be responsible for the observed low loop voltage in the
optimum condition.
An alternative explanation is that the dissipation of helicity in the outer plasma
regions is increased when the plasma is not centred and that more poloidal flux intersects the wall [6]. The estimated variation of the anomalous loop voltage with the DC
vertical field closely follows the observed loop voltage. The variation of X, with
Bvdc indicates that more helicity dissipation can be expected without a DC verticalfield.
The increase in the number of electrons with an energy three to four times larger
than T^ suggests that energetic particles are better contained with a slowly decaying
current.
7.

CONCLUSIONS

Reversed field pinches were obtained in the new TPE-1RM15 machine when
the toroidal equilibrium was corrected by a DC vertical field. Cancellation of thisfield by coils inside the shell for a short time (2 ms) permits plasma formation overa wide range of filling pressures ( > 4 mtorr) and vertical field ( < 10 mT).
In all cases the pulse duration is very sensitive to the shell gap error. When thisis minimized, the pulse length is about 8 ms.
During the flat-top phase at Tp = 135 kA, the electron temperature remains
constant at T^ ~ 600 eV for 3-5 ms, n ( 0 ~ 2 x 1019 m~3, Pp ~ 10% andrE ~ 0.2 ms, with the usual assumptions and with an optimum DC vertical field of
7.7 mT. With no DC vertical field, T^ and rE are reduced to about 400 eV and
0.08 ms, respectively. The variation of the loop voltage with the DC vertical field
follows that of the electron temperature, consistent with a resistance anomaly factor
of four.
The TPE-1RM15 is operated at very large I/N (25xlO~ 14 A-m), but significant degradation of j3p and rE is not observed.
The importance of positioning the RFP plasma in the vacuum vessel and ofcancellation of the shell gap error field, even for a machine with a large aspect ratio,
is clearly shown by the results presented.
ACKNOWLEDGEMENTS
This work was supported by the Atomic Energy Bureau, Science andTechnology Agency (STA), Japan. One of the authors is grateful for the hospitalityof the Electrotechnical Laboratory and for the support of STA.

IAEA-CN-47/D-II-6

461

REFERENCES[1]HIRANO, Y., MAEJIMA, Y., OGAWA, K., et al., in Plasma Physics and Controlled Nuclear
Fusion Research 1984 (Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 475.
[2] CAROLAN, P.G., ALPER, B., BODIN, H.A.B., et al., ibid., p. 449.
[3] BAKER, D.A., BUCHENAUER, C.J., BURKHARDT, L.C., et al., ibid., p. 439.
[4] NOONAN, P.G., TSUI, H., NEWTON, A.A., Plasma Phys. Contrail. Fusion 27 (1985) 1307.
[5] SHIMADA, T., HIRANO, Y., YAGI, Y., et al., in Fusion Technology (Proc. 14th Symp.
Avignon, 1986), Pergamon Press, Oxford, to be published.
[6] ALPER, B., ANTONI, V., BEVIR, M.K., et al., Paper IAEA-CN-47/D-II-l, these
Proceedings, Vol. 2.
[7] ALPER, B., MARTINI, S., Electron Temperature Studies on ETA-BETA-2, Rep.IGI-85/04,
Istituto Gas Ionizzati, Consiglio Nazionale delle Ricerche, Padua (1985).
[8] ALPER, A., ANTONI, V., BASSAM, M., et al., in Controlled Fusion and Plasma Physics(Proc. 12th Europ. Conf. Budapest, 1985), Vol. 9F, Part 1, European Physical Society (1985)
578.

DISCUSSION
T. TAMANO: Allow me to congratulate you on the good initial results. You
have constructed a scaled-up device with a design philosophy very similar to theprevious one, and, further, you have obtained almost the same temperature and density with an improved confinement time. Could you comment on the size scaling?
T. SHIMADA: Thank you. The equilibrium control reduces the outer volume
and this may lead to a reduction of the anomalous component of the loop voltage,as suggested by another RFP experiment. One consequence of this reduction is ahigher energy confinement time. However, it is too early to comment on the size
scaling.
S. ORTOLANI: As you have stressed, the densities at which you operate arerather low (N^ - 2 x 1019 m~3). Have you tried to operate at higher densities oris there something that prevents it?
T. SHIMADA: The density is determined by recycling. In our device it is low
because of the all-metal vessel. In principle, there is nothing to prevent higher densities being obtained with, say, gas puffing or pellet injection.
The good magnetic field configuration allows us to operate at a higher I/N ratio
than in any previous experiment and so we obtain high temperatures.

IAEA-CN-47/D-m-l

PLASMA HEATING IN
STELLARATOR/TORSATRONS AND TOKAMAKSBY ION CYCLOTRON WAVES
E.F. JAEGER, D.B. BATCHELOROak Ridge National Laboratory,
Oak Ridge, Tennessee
K. IMRE, H. WEITZNER
Courant Institute of Mathematical Sciences,
New York University,
New York, New York
United States of America
Abstract
PLASMA HEATING IN STELLARATOR/TORSATRONS AND TOKAMAKS BY IONCYCLOTRON WAVES.
High power ion cyclotron heating experiments are currently planned for the Heliotron-E device
in Kyoto and for the ATF torsatron at Oak Ridge National Laboratory. Reliable solutions for theradiofrequency fields and wave absorption are needed to optimize antenna designs and heating scenarios
as well as to determine heating profiles and tail formation for the interpretation of confinement experiments. The complicated field structure in torsatrons makes a number of approximations commonly usedin studying ICRH in tokamaks of questionable validity. For example, in tokamaks the cyclotron
resonance and ion-ion hybrid resonance layers are, to a good approximation, vertical sections and smallchanges of frequency or field B result in small shifts in their locations. However, in a torsatron a saddle
point in B occurs near the magnetic axis so that resonance and cut-off layers occur on two sheets oriented
either horizontally or vertically. Small changes of frequency therefore can cause radical changes in theresonance cut-off topology. This phenomenon has been investigated by performing two-dimensional calculations of the wave fields and heating profiles in a cold plasma model. The effects of finite temperaturein this geometry have also been investigated by calculating cyclotron damping and mode conversion ina slab having a non-monotonic B profile. Resonances and cut-offs therefore exist in pairs, and by adjusting the wave frequency they can be made to merge together and to annihilate in pairs. In large tokamaks
and stellarator/torsatrons the well known geometrical optics approximation may apply and in certain
regimes of configuration space the system is approximately perpendicularly stratified. In such regions
near cyclotron harmonic resonances, geometrical optics fails and boundary layer expansion techniques
are employed to obtain a wave equation whose solutions match the geometrical optics solutions. In thismanner, energy absorption, wave reflection, and mode conversion for fundamental, second and third
harmonic and ion-ion hybrid resonances are studied for incident fast waves and incident Bernstein
waves.

1.

FULL WAVE CALCULATIONS IN 2 DIMENSIONS

We consider a model torsatron equilibrium which isstraight and helically symmetric. That is,all equilibrium
quantities depend on r and <f> = 6 - kz.
We introduce axial

463

464

JAEGER et al.

and poloidal flux functions iKr,<(>)x(r,<t>) and use the Bessel
function model for i|>. The density is assumed to be afunction of ty. In these coordinates the equilibrium is
independent of z so that the wave fields can be Fourier
analyzed with respect to z. We take the limit of zero
electron mass and set EI
I 0. Then the projection of thevector wave equation in the ViJ) direction and in the
directions gives two coupled pde's for E e and E p . The fullequations are given in Ref. [1].
Two models have been used for the plasma current
response.
Most calculations were done using the cold plasma
conductivity with an ad hoc collision frequency used toresolve the cyclotron resonance. To test the sensitivity of
the results to the use of collisions to broaden the cyclotron
resonance we have also done calculations using the warm
plasma conductivity tensor. For these, the k ( p i = 0 limit was
taken and where kg appears in the plasma dispersion function,
k z was used.
Figure 1 shows contours of equal power deposition forthree different frequencies, giving different topologies ofthe minority cyclotron, ion-ion hybrid and cut-off layers.
Figure 2 shows the radial heating profiles for the threefrequencies of Fig. 1. The plasma parameters were chosen
relevant to ATF: 1 = 2, m = 12, a c o i l = 51* cm,
n e (0)3
cm" B(0)20 kG« nscrape-off = 4 x 1 ° 1 1

cm

~3.

FIG. 1. Two-dimensionalfull wave calculation for ATF, showing contours of constant power deposition
W(r, 4>) for (a) f = 28 MHz. (b) f = 30 MHz, and (c) f = 31 MHz.

465

IAEA-CN-47/D-HI-lo.io
0.08 ".f=30 MHz

0.375

FIG. 2. Flux surface average of the power absorbed (W)^for

0.500

the cases shown in Fig. 1.

k z =• NT/R~» where R T = 210 cm and the toroidal mode number
N™ » 8. The antenna current was an approximately 1/4 turncircular loop in the 6-direction at r = 42.4 cm. For the
calculations in Fig. 1 the fluctuating plasma wave current
was obtained from the warm plasma conductivity model with
T e = T^ = 1 keV. As expected when using the cold plasma
model with collisional broadening, the total power absorbed
is independent of v/u> for sufficiently small v. Similarly, we
find that in the warm plasma conductivity model at small T i thetotal absorption or antenna loading is independent of T^ andagrees with that predicted by the collisional model. The
power deposition patterns are also very similar for the two
conductivity models, except that collisional absorption awayfrom resonance layers (for example due to cavity modes) isabsent in the warm plasma conductivity model. In Fig. 1a(f - 28 MHz) a vertical chord through the center of theantenna intersects the minority cyclotron resonance and theion-ion hybrid resonance twice. In Fig. 1b (f = 30 MHz) the
topology has changed such that the hybrid resonances have
moved to the side and a vertical chord intersects theminority cyclotron resonance twice but does not cross the
hybrid resonances. In Fig. 1c (f = 31 MHz), both resonancelayers are at the side so that neither is directly in frontof the antenna. From Fig. 2 we see that the heating is muchgreater for the 30 MHz case and the profile is much morecenter peaked. Thus a frequency variation of 3% causes aradical change in loading and power deposition profile.
Obviously,the coupling to Bernstein modes will be muchdifferent for the three cases.

466

JAEGER et al.

A striking feature of these 2-D calculations is that thepower deposition lies on localized island structures lying on
flux surfaces rather than being smoothly spread over the
resonance surfaces. This phenomenon has been seen in a
number of previous 2-D calculations for tokamaks [2,3,4],
Hellsten et al. [5] have developed a theory of thesestructures based on a Frobenius type expansion of the wave
equation in the vicinity of possible singular layers. Othersuggestions have been made that these structures along the ^surfaces are in fact the result of cavity modes or reflect
the constancy of n e on ty or are perhaps merely numericalartifacts. We have studied these heating structures indetail in tokamak geometry and find them to be quite
insensitive to the wall and antenna location and also to thefineness of the numerical grid.
In addition, the islands
persist if the density is taken constant. This strongly
suggests that they are not due to cavity modes, density
contours or numerics. The islands do however depend on thequantities
appearing in Hellsten1s eigenvalue equation,
namely f, n e , B kz, i. However, having solved the eigenvalue
equation, we are not able to obtain any systematic agreementon the location and shape of the islands observed in the 2-D
calculations.
The theory of Hellsten et al. is an ideal
theory (v » 0) which relies on the self-adjointness of the 6
eigenvalue problem.
Indeed, the system has no periodic
solutions for real r when v * 0.
We conjecture that byintroducing v sufficient to resolve the resonance in ourmesh, we eliminate the singularity revealed by the Frobenius
expansion and thereby modify the problem so that detailed
agreement is no longer obtained.

2.

ONE-DIMENSIONAL FULL WAVE SOLUTIONS IN WARM PLASMA

Some insight into the effect of finite temperature forthis system can be obtained from an idealized model in whicha wave propagates across a magnetized plasma slab [§ = B(x)z]
where the plasma parameters n e (x), T e (x), T^Cx), B(x) arethose of a vertical chord through the axis of the 2-D model
above(the profiles are approximately parabolic).
Ourcalculation is similar to Ref. [6] in that the unperturbed
distribution function includes diamagnetic drifts and theassociated anisotropy to second order in Larmor radius. The
Vlasov equation is expanded to second order in gyroradiusretaining all x-dependence of B and p = n e T. This gives aplasma current operator of the form:

IAEA-CN-47/D-ni-l

467

->
1

2

(O°+P +T )-E

1

2+ [o + p ] u • ||Full expressions for the a, p and T tensors can be found in
Refs[6-73.
The electric field parallel to § is retained
to include collisionless damping of the ion Bernstein wave.
The resulting sixth order wave equation is solved as a 2point boundary value problem by finite differencing.
Avacuum region is included at the edge of the plasma andvanishing of transverse E is imposed as a boundary conditionat the wall.
In order to interpret the solutions with respect tolocal power deposition, it is necessary to use Poynting's
theorem and separate Re {£*•?} into a part P representing
local dissipation and a part v"«<5 where § is the kinetic wave
energy flux. There is no unique way to make this separation
and a number of different choices appear in the literature.
We use the definitions:

These forms for P_ and Q reduce to the correct forms
for local power dissipation and kinetic flux in the WKB limit
when a single wave is present in the system. Also the total
absorbed power as determined by integrating EPS over theplasma volume agrees to high accuracy with that obtained byintegrating
Re (E"*«3 fc} overtheantennacurrent
distribution.
When using the definitions given in Ref. [6]we find local regions of negative P e associated with transit
time magnetic pumping on the fast wave and Landau damping on
the Bernstein wave. With our forms the regions of negative
TTMP are eliminated and negative Landau damping is reduced.
At small kii we do see negative oscillations on Pa associatedwith cross-terms between fast wave and Bernstein wave (e.g.,^fast*°"^Bernstein^* O u r definition is similar to that given
in Ref. [8], although in that work the terms associated withdiamagnetic drifts are not retained in J nor in P S> Q S .

JAEGER eta!.

468

0.27
27.9

lefT2T4Mm^<cif=S0MH,

F l g u r e

3pCalculations: n

4"

\

B(0)

u on thetheused. 1.5 keV,GT"IIn Fig. ^a,

469

IAEA-CN-47/D-m-l

(a)

(b)
1

\

\

\

•

\

\ •

\
\

-20

T
0
LW/C= 0.95

\

-

\

1

•

1

'

= 20 keV

•

J

30-

-

=

•nll

m

_

ow

•40

1

_

^—

'

1

/ -

x

/

fastmooV>1

ujfl

1001 0

-i

'

1

1

10000
x

-1

U

20000

o

FIG. 4. (a) Dispersion relation for the fast and slow modes near the fundamental ion resonance,(b) Absorption contours for the slow mode incidence in the plane of X = u^Ao2 and n, = k, w/c.
For instance, a deuterium plasma with Bo = 3 T, T = 5 keV, and the characteristic equilibrium gradient length L = 100 cm.

modes generated which are trapped and must be absorbed
between the two resonance pairs. The Bernstein modes do not
propagate at the plasma edge where n o i < u>. In Fig. 1b,
f - 28.1 MHz, the hybrid resonances havemoved
closertogether and the separation between the hybrid and minority
cyclotron resonance has increased.
This frequency, whichcorresponds to one fast wavelength between hybrid resonances,
gives maximum absorption except when high Q cavity resonances
are present. In Fig. 1c, f - 30 MHz, the hybrid resonances
have come together at the center and disappeared.
Twominority cyclotron resonances do remain,however. One sees
center peaked minority heating and also somemajorityheating.
However, electron heating associated with Bernsteinmodes has disappeared.

3.

GEOMETRICAL OPTICS APPLIED TO CYCLOTRON RESONANCES IN A
PERPENDICULARLY STRATIFIED MEDIUM

In an approximately perpendicularly stratified plasma inwhich the geometrical optics approximation generally applies,
special techniques are needed near
cyclotron
harmonic
resonances.
We apply a boundary layer expansion method andfind a wave equation valid there whose solutions connect tothe geometrical optics solution away from resonance. In thismanner we study pure fundamental cyclotron resonance, puresecond harmonic and pure third harmonic resonances, andion-ion hybrid resonance of a minority fundamental resonance

470

FIG. 5. Absorption contours for (a) fast wave incident on the high field side, (b) fast wave incident
on the low field side, and (c) Bernstein mode incidence for the H-D hybrid resonance, where
TH - TD = 5 keV, XD = 500 and Lu/c = 1, corresponding to a plasma with nD = 4 x 10M cm'3,
B0 = 4TandL= 40 cm.

species with majority second harmonic resonant species. In
all but the pure fundamental resonance case, incident waves
may be either fast wave from the high field side or fast wave
from the low field side, or an ion Bernstein wave from theappropriate side.
In the pure fundamental resonance a fast
wave may be incident from either side or a slow wave may be
incident from the high field side. We characterize the wave
propagation in terms of energy flux
transmission
andreflection coefficients, energy flux absorption, and energy
flux mode conversion and/or reflected wave mode conversion
coefficients.
We can obtain a wide class of exact results. The fast
wave propagates on both sides of the resonance and thetransmission
coefficientsfor low or high field side
incidence are equal and are given by the geometrical optics
result and by an explicit formula. No fast wave is reflected
when a fast wave is incident from the high field side.
We
extend the reciprocity theorems of Chiu and Mau [9] and showthat the mode conversion transmisson coefficients and themode conversion reflection coefficients are independent ofincident wave type.
We may use the explicit formulas for the transmission
coefficient to isolate those cases for which the coefficientis much less than one and consequently plasma heating ispossible. By numerical calculation we may determine therange of values of n n in which significant energy absorption

471

IAEA-CN-47/D-IH-l

/

10

-I

VJ _y

:

HIGH FIELD

n

1

1

10

20

FIG. 6. Absorption contours for the third harmonic resonance for XD = 2000 and Luslc = 1, in the(n,, T) plane, corresponding to a deuterium plasma with nD = 5 x 1014 cm'3, Bo — 3 T and
L = 40 cm.

occurs.
Generally, the heating is more efficient as thedensity and/or temperature increase. We show in Fig. 1 wave
propagation properties at fundamental resonance. In Fig. 4a
the dispersion relation is given with both the fast waves andthe slow waves present. The slow wave is clearly accessible
from the high field side. In Fig. 1b the strong absorptionof the slow wave is evident. In fusion plasmas the fast wave
is only very weakly absorbed. Figure 5 considers pure second
harmonic resonance as well as the effect of the addition ofsmall concentrations of a fundamental harmonic resonant
species.
The fundamental resonant species reduces thetransmission coefficient of the fast wave, and a moderatevalue of n||, leads to effective energy absorption. Incident
ion Bernstein waves are also typicaly well absorbed for suchn ]f . Finally, a pure third harmonic resonant fast wave has
an extremely small transmission coefficient so that additionof a moderate n (| yields a substantial energy absorption.
We show such results in Fig. 6.

H. CONCLUSIONS
Because of the presence of the saddle point in B foundin torsatrons the detailed behavior of ICRH is expected to bequite different than in tokamaks. This makes calculations
and interpretation of experiments more difficult but may also
give flexibility in tailoring the heating profiles or theeffects on different species. The long VB scale length at

472

JAEGER et al.

the center may be used to advantage. Note the large value ofcentral ion heating in Fig.
3c even though no hybrid
resonance is present. A very critical unresolved question isthe confinement of energetic tails produced by ICRH.
Whengeometrical optics applies in a perpendicularly stratified
plasma we exhibit a wide range of effective wave energy
absorption
atfundamental, second
and third harmonicresonances.
ACKNOWLEDGMENTS
Research sponsored by the Office of Fusion Energy, U.S.
Department of Energy, under Contract No. DE-AC05-8^0R21^00
with Martin Marietta Energy Systems, Inc., and by U.S.
Departmentof Energy Grant Nos. DE-FGO2-86ER53223 andDE-FG02-86ER53233.

REFERENCES[1] JAEGER, E.F., WEITZNER, H., BATCHELOR, D.B., ICRF Heating in a Straight Helically
Symmetric Stellarator, Rep. ORNL-TM-10223, Oak Ridge National Lab. (1986).
[2] VILLARD, L., APPERT, K., GRUBER, R. ( VACLAVIK, J., Comput. Phys. Rep. 4 (1986) 95.
[3] FUKUYAMA, A., OKAZAKI, N., GOTO, A., ITOH, S.-I., ITOH, K., Nucl. Fusion26 (1986)
151.
[4] ITOH, K., ITOH, S., FUKUYAMA, A., Nucl. Fusion 24 (1984) 13.
[5] HELLSTEN, T., TENNFORS, E., in Controlled Fusion and Plasma Physics (Proc. 9thEurop. Conf. Oxford, 1979), UKAEA, Culham Lab., Abingdon, Oxfordshire (1979) 355;
Phys. Scr. 30 (1984) 341.
[6] MARTIN, T., VACLAVIK, J., Dielectric Tensor Operator of a Nonuniformly Magnetized
Inhomogeneous Plasma, Rep. LRP-259/85, Ecole polytechnique f&le'rale de Lausanne (1985).
[7] JAEGER, E.F., BATCHELOR, D.B., WEITZNER, H., Global Ion Cyclotron Waves in a
Parallelly Stratified One-Dimensional Warm Plasma, Rep. ORNL-TM-0224, Oak Ridge National
Lab. (1987).
[8] FUKUYAMA, A., ITOH, K., ITOH, S.I., Comput. Phys. Rep. 4 (1986) 137.
[9] CHIU, S.C., MAU, T.K., Nucl. Fusion 23 (1983) 1613.

DISCUSSION
F.L. RIBE: Could you describe the various types of calculations that you used?
I should also like to ask about the RF eigenmodes in the stellarator system.
H. WEITZNER: Three types of codes are described in the paper: they relate
to a full-wave, three-dimensionsal cold plasma model; a perpendicularly stratified
medium, including a Vlasov equation expanded through Qa±p{)2; and a boundary
layer expansion of the Vlasov equation near the resonance layers.
Regarding your second point, eigenmodes associated with the plasma and cavity
have been found, but they are not the origin of the effect shown in Fig. 1 of my paper.

IAEA-CN-47/D-DI-2

ICRF HEATING OF CURRENTLESS PLASMAIN HELIOTRON E
T. MUTOH, O. MOTOJIMA, M. SATO, H. OKADA,
H. ZUSHI, K. KONDO, N. NODA, H. KANEKO,
T. MIZUUCHI, S. BESSHOU, F. SANO, S. SUDO,
Y. TAKEIRI, K. AKAISHI, T. KAWABATA, Y. IJIRI,K. ITOH, S. MORIMOTO, A. IIYOSHI, K. UO
Plasma Physics Laboratory,
Kyoto University,
Gokasho, Uji, Kyoto, Japan
Abstract
ICRF HEATING OF CURRENTLESS PLASMA IN HELIOTRON E.
High power and long pulse ICRF heating of a currentless plasma is carried out in Heliotron E.
Three kinds of wave mode are studied in a wide range of plasma parameters. Fast wave (minority
heating), ion Bernstein wave and slow wave (ion cyclotron wave) heatings are performed by using thesame antenna system, while gas species, electron density and magnetic field strength are varied. With
the fast wave heating, the ion temperature increases from 200 eV to 650 eV at an electron density of
2.2 x 1019 m~3. The duration of effective heating is, however, limited to less than 50 ms, because of
the rapid increase in radiation loss power. Carbonization of the vacuum chamber reduces the radiation
loss power to about 30%, and then the plasma which has been initiated by an ECH pulse is effectively
heated and sustained by ICRF pulse for 100 ms only. The behaviour of the antenna loading resistanceis studied in comparison with a theoretical analysis. The fundamental cavity mode is clearly
identified. — With ion Bernstein wave heating, the ion energy spectrum has a two component form.
The effective ion temperature in the perpendicular direction increases quickly during the ICRF pulseof a frequency of 4 o)cD. With slow wave heating, the ion temperature increases up to 1.6 keV at anelectron density of 0.6 x 1019 m" 3 . The heating efficiency depends strongly on the electron density.

1.

INTRODUCTION

The ICRF method of ion and electron heating has been investigated in
Heliotron E since 1984 [1,2]. The previous, first phase experiment [3] was the firsttrial of ICRF fast wave heating in a fully toroidal device with current-free operation.
Heating characteristics and cavity mode excitation were studied, and agreement withtheoretical predictions was observed [3]. The available heating pulse length andpower were, however, restricted to less than 15 ms and 1 MW, because of the limitation of the RF power source. In the present, second phase experiment, the available
pulse length was extended to 200 ms, and the number of installed antenna loops andRF generators was doubled.
There were three modes of ICRF heating in Heliotron E: Fast wave
(minority) heating has been extensively carried out in a medium density
473

474

MUTOH et al.

: 1.5 x 1019 m 3) D2 plasma. Ion Bernstein wave heating was carried out at
the fourth harmonic ion cyclotron frequency in a low density(ne = 1.0 x 1019 m 3) D2 plasma, and slow wave (ion cyclotron wave) heating
was carried out in a low density (iie < 1.0 x 1019 m"3) H2 plasma. Direct excitation of an electrostatic wave (IBW) may have occurred, because of the incompleteness of the electrostatic shielding on the plasma boundary in the direction of themagnetic field.

2.

FAST WAVE HEATING AND CAVITY MODE OBSERVATIONThe eight antenna loops were fed by two generators at different frequencies. All
antenna loops were installed on the high field side inside the vacuum chamber.
Figure 1 shows a picture of the minor cross-section of the toroid in the feedthroughsection. The shapes of the cyclotron resonance layers are quite different from thoseof usual tokamaks. Normally, the magnetic field strength on axis was 1.9 T, which
was determined by the ECH gyrotron frequency (53.2 GHz). The ICRF frequencies(26.7 and 28.2 MHz) correspond to the proton cyclotron frequencies at 1.75 T and1.85 T, respectively. For the case at 26.7 MHz, the left hand cut-off and the two-ion
hybrid resonance layers are located near the axis as shown in Fig. 1, when theminority proton fraction is around 10—15%.
Figure 2(a) shows typical time traces of plasma parameters without carbonization. The target plasma was produced and heated by an ECH pulse of about 300 kW,
and, after that, the plasma was heated by an ECH pulse (150 kW) and an ICRF pulse(1.5 MW). The electron density increased from 1.5 to 2.2 x 1019 m""3 during the

FARADAY SHIELDLOOP ANTENNAFEEDTHROUGH

OUTERMOST MAGNETIC
SURBACEHELICAL COIL

n,f=R

FIG. 1. Geometries of proton cyclotron resonance surface (<a = aCH = 2<JJCD), left hand cut-off(n\ = L) surface, two-ion hybrid resonance (n\ = S) surface, and right hand cut-off (n\ = R) surfaceof the fast wave in heliotron minor cross-section. Antennas are placed on high field side of plasma.
(\B\r=0 = 1.9T, nH/nH + nD = 0.15, nJO) = 4 x 1&9 m~3, frequency = 26.7 MHz.)475

IAEA-CN-47/D-HI-2

0
500
(kW)

TIME (ms)

TIME |ims|

FIG. 2. Temporal evolution of electron density, electron temperature measured by electron cyclotron
emission, central chord ion temperature measured by charge exchange neutral particle analyser,
impurity line intensity ofFe XVI and bolometric loss power. ECU and ICRF pulses are also shown.
(a) without carbonization in ECH+ICRF mode (nH/nD + nH = 0.2-0.3, Pgf = 1.5 MW),(b) with carbonization in ICRF sustained mode. (n^nD+nH = 0.1-0.2, Pw = 700 kW).

20
19

OJ=OJCH
Ion Bemsteln I9 _Wave Heating
184-

Slow Wavee Heating(Ion Cyclo
lotron Wave)
WITH ICRF
1.6 keV

Fast Wave
Heating(Ilimlnorlty)

a

b
c

B(0)
0 .91T.
l .9 T.
l .9 T.

ne(10 1 9 nf 3 )

1.0
0.6
2 ,0
(H2

gas
D2

112
D2

m lnorlty)keV
T D = \ 6 0 0 eV

9
12 15
Energy (keV)

18

21

FIG. 3. Ion energy distributions for perpendicular component: (a) ion Bernstein wave heating mode
(Pgjr = 400 kW), (b) slow wave heating mode (Pgf = 400 kW), (c) fast wave heating mode(P^ = 1.5 MW)476

MUTOH et al.

0.1
0

5
tie

(10 1 9 m- 3 )FIG. 4. Comparison between measured loading resistance of an antenna loop (b)and theoretical result [4] (c); (a) shows temporal evolution of electron density.

ICRF pulse without additional gas puffing. The ion temperature of the majority deuterium increased from 200 eV to 650 eV. During the ICRF pulse, the bolometric signal and the impurity spectrum line intensities for Fe, O and Ti increased rapidly, andthe electron temperature decreased gradually. The duration of effective heating was
limited to less than 50 ms, and the heating efficiency of the ICRF pulse was also
reduced. The ion energy spectra which were measured by a perpendicular neutral
particle analyser (NPA) are shown in Fig. 3(c). The temperature of the high energy
tail of minority protons increased almost linearly with the ICRF power, and no particular deterioration of the tail confinement was observed in the range of E < 15 keV.
These phenomena are similar to those seen in minority heating experiments in usual
tokamaks.
After carbonization of the vacuum chamber, the bolometric loss decreased toabout 30%, and the impurity line intensities were also reduced. Figure 2(b) showstime tracers of the ICRF sustained plasma parameters. In this case, the ECH pulse
produced a seed plasma, and then the plasma was heated by the ICRF pulse only.
The electron density was not controlled by additional gas puffing during the ICRF
pulse. It increased from 1.5 to 3.1 x 1019 m~3 by enhanced influx from the wall
surface. Ion and electron temperatures were sustained at a level of 300 to 350 eV.
From the time traces of these temperatures, it seems that the ICRF power initiallyheated the ions and then transfer of the energy to the electrons occurred by collisions.
The bolometric signal increased slowly, and, finally, radiation loss terminated theplasma sustainment. The main improvement brought about by carbonization was that

IAEA-CN-47/D-m-2

477

the duration of effective heating increased from 50 to 100 ms as is shown in Figs 2(a)and (b).
Recently, the structures of the excited wave were analysed theoretically andnumerically in a 2-D straight heliotron configuration model by Fukuyama et al. [4].
Figure 4 shows a comparison between the measured antenna loading resistance andthe analytical result of Ref. [4]. In this case, the density increased almost linearlywith time; therefore, the measured loading resistance could easily be compared withthe analytical results. The first sharp peak coincides well with the calculated positionof the fundamental cavity mode (m = 0), with X, equal to half the helical pitch
length. Theory had predicted that the half-width of the peak is inversely proportional
to the damping rate of the wave. In the case of pure deuterium plasma, a higher andnarrower peak appeared, and it split into three or four sharper fine structure lines.
On the other hand, in the case of a high proton ratio ( > 5%) plasma, this peak became
lower and wider as the proton ratio increased. This observation of the antenna loading
peak suggests that the fast wave propagates to the centre of the plasma forfle > 1.0 x 1019 m~3 and that the absorption rate is enhanced by the introductionof minority protons.

3.

ION BERNSTEIN WAVE HEATINGAND SLOW WAVE (ION CYCLOTRON WAVE) HEATING

Recently, ion Bernstein wave heating has been tested in several tokamaks [5].
In Heliotron E, ion Bernstein wave heating was applied for the first time. By using
the same antenna system, ICRF power coupled to the plasma when the frequency was
the fourth harmonic of the central ion cyclotron frequency. The target plasma wasproduced by a second harmonic ECH pulse at IB I = 0.94 T, and the electron density was about 1 X 1013 cm" 3 . The ion temperature, which was measured by theperpendicular NPA, quickly increased from 100 eV to 700 eV during less than 5 ms.
The ion energy spectrum is shown in Fig. 3(a). The spectrum shows a two component form, which implies that the heated ions are not well thermalized.
Slow wave heating has been an efficient ion heating method for stellarators [6]and heliotrons [3, 7]. In Fig. 5(a), time traces of various plasma parameters for slow
wave heating are shown. The electron density was kept to a low level of less than
6 X 1018 m~3 during the ICRF pulse of 400 kW. The ion temperature, which wasmeasured by the perpendicular NPA, increased from 300 eV to 1.6 keV. The ion
energy spectrum is shown in Fig. 3(b). The electron temperature stayed around
1.0 keV and varied almost independently from that of the ions since the energy transfer between electrons and ions was negligibly small in this parameter range. The radiation loss had an effect on the electrons only.
In this heating mode, the ion temperature strongly correlated with the electron
density. The most effective heating was obtained at an electron density of(5—6) x 1018 m~3. Even a slight increase in density, for example, up to478

MUTOH et al.

2.5
1(10 1 9 '-

1

1

1

1

1

n~e

(b)

(a) :

wave
propagating
region

o
2000

(eV)
1000 - /-\j\J/"TTNPA

f

hi.

0
200
(mV)

0
-, 3

%2

JFe X V I :

o
500
(kW)
—t——r~^i

i

i

i

EC H
0

I CH

150

n e (I0 19 m" 3 )TIME (ms)FIG. 5. (a) Temporal evolution of plasma parameters in slow wave heating mode. Parameters are thesame as in Fig. 2 (frequency = 26.7 MHz, IBIr=o = 1ST, 100% H2); (b) positional shift of slow
wave propagating region on short axis of minor cross-section versus electron density (njr): parabolic,&i = 26 m~'); (c) dependence of ion energy increments, normalized by RF power, on line averaged
density. Data for fast and slow wave heating are shown.

ne = 8 x 1018 m 3, could reduce the heating efficiency for the central ion temperature by about 50%. This phenomenon can be explained by a position change of theslow wave propagating region as the density increases [3]. In Figs 5(b) and (c), theion heating efficiency and the position of the slow wave propagating region are shown
versus electron density. For slow wave heating, it was very essential to heat thecentral region in order to obtain high ion heating efficiency.

4.

SUMMARY

Fast, ion Bernstein and slow wave heating experiments were carried out on
Heliotron E. For fast wave heating, the ion temperature increased up to 650 eV at
an electron density of 2,2 x 10" m~3 and an input ICRF power of 1.5 MW. The
duration of effective heating was limited to about 50 ms by radiation loss. With carbonization of the chamber wall, the effective heating time increased to longer than
100 ms.Thismay be due to the reduction of the iron impurity influx. Then, the ICRF
pulse became able to sustain the plasma for 100 ms without an ECH pulse. The

IAEA-CN-47/D-III-2

479

behaviour of the antenna loading resistance with changing plasma density was studied
by comparison with theoretical predictions. The good agreement confirms that acavity mode was excited in the heliotron plasma.
These results suggest the effectiveness of the fast wave heating for heliotron
plasmas.
With ion Bernstein wave heating, the perpendicular ion temperature increased
from 100 eV to 700 eV at an electron density of 1.0 x 1019 m" 3 and an ICRF powerof 400 kW, but the ions were not well thermalized.
With slow wave heating, the ion temperature increased up to 1.6 keV, as determined by the perpendicular NPA measurement at an electron density of
6 x 1018 m""3 and an ICRF power of 400 kW. The heating characteristics depended
strongly on the electron density and were explained by a positional shift of the slow
wave propagating region. The ion heating efficiencies of fast and slow wave heating
are shown in Fig. 5(c). The dependences on the electron density are consistent withtheoretical predictions.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]

UO, K., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc. 10th Int.
Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 383.
IIYOSHI, A., ibid., Vol. 1, p. 453.
MUTOH, T., et al., Nucl. Fusion 24 (1984) 1003.
FUKUYAMA, A., et al., Nucl. Fusion 26 (1986) 151.
ONO, M., et al., Phys. Rev. Lett. 54 (1985) 2239.
BROWN, I.G., et al., Phys. Fluids 12 (1969) 1318.
IIYOSHI, A., et al., Nucl. Fusion 14 (1974) 789.

DISCUSSION
L. KOVRIZHNYKH: According to your Fig. 2, the ion temperature increases
monotonically during the HF pulse. Does this mean that the ion energy lifetime ismore than the duration of the HF pulse?
T. MUTOH: The ion energy confinement time was of the order of 10 ms.
hence the time-scale for the ion temperature variation for the slow wave mode was
longer than that time. Our understanding is that the power deposition profile gradually changed during the ICRF pulse. The changes in ne(r) and temperatures probably
relate to this mechanism.
R.J. GOLDSTON: You can now support similar plasmas with ICH and NBI
on Heliotron E. Would you compare the energy confinement time or the heating efficiency in the two cases?
T. MUTOH: We have not yet estimated the absorbed power of ICRF heating
so we cannot make an accurate comparison. Roughly speaking, heating efficiencies

480

MUTOH et al.

were almost the same for NBI and ICRF within a factor of 1.5. The power dependences of the gross energy confinement time in both cases were around <* p~ 0 5 ,given the same electron density.
S. SHINOHARA: In the case of the ion Bernstein wave, is there any possibility that other waves such as the fast magnetosonic wave can propagate in the plasmaand heat the ions? And what in your opinion is the fraction of the wave power carriedby the ion Bernstein wave? It seems to me that in the Heliotron E device, becauseof the magnetic field geometry and antenna current direction, it is rather difficult for
the ion Bernstein wave to propagate, as compared with the case of the tokamak withlow field side excitation, in which the antenna current is parallel to the toroidal magnetic field.
T. MUTOH: At fourth harmonic frequency, fast wave damping was negligibly small. The absorbed power in the plasma column was therefore mainly due tothe ion Bernstein wave. But the fast wave was excited simultaneously, so the excitedwave structure was probably a combination of the fast wave and the ion Bernstein
wave.

IAEA-CN-47/D-III-3

I. ELECTRON CYCLOTRON RESONANCE
PLASMA HEATING BY ORDINARY WAVESIN THE L-2 STELLARATOR
E.D. ANDRYUKHINA, D.K. AKULINA, G.M. BATANOV,
V.A. BATYUK, M.S. BEREZHETSKIJ, M.A. BLOKH,
G.S. VORONOV, S.E. GREBENSHCHIKOV, S.I. DOLGAEV,
N.P. DONSKAYA, K.S. DYABILIN, L.M. KOVRIZHNYKH,
L.V. KOLIK, N.F. LARIONOVA, K.M. LIKIN,
A.I. MESHCHERYAKOV, P.E. MOROZ, S.N. POPOV,
A.E. PETROV, V.I. ROSHCHIN, A.V. SAPOZHNIKOV,
K.A. SARKSYAN, N.N. SKVORTSOVA, I.S. SBITNIKOVA,
A.D. SMIRNOVA, V.N. SUKHODOL'SKIJ,
M.F. TERNOVSKAYA, O.I. FEDYANIN, S.I. FEDORENKO,
Yu.V. KHOLNOV, M.G. SHATS, I.S. SHPIGEL'
Institute of General Physics,
Academy of Sciences of the USSR

II. FAST WAVE EIGENMODESIN THE L-2 STELLARATOR
DURING ICR PLASMA HEATING
V.A. BATYUK, S.E. GREBENSHCHIKOV,
A.I. MESHCHERYAKOV, P.E. MOROZ,
I.S. SBITNIKOVA, I.S. SHPIGEL'
Institute of General Physics,
Academy of Sciences of the USSR
Moscow, Union of Soviet Socialist Republics
Abstract
I. ELECTRON CYCLOTRON RESONANCE PLASMA HEATING BY ORDINARY WAVES IN
THE L-2 STELLARATOR.
H. FAST WAVE EIGENMODES IN THE L-2 STELLARATOR DURING ICR PLASMA HEATING.
Experimental results on electron cyclotron resonance plasma heating (ECRH) by ordinary waves(ElIB) in the L-2 stellarator are presented. Microwave radiation (f = 37.5 GHz, P = 100 kW) is
launched into the plasma in the form of a Gaussian beam from the weak field side. The heating isobserved to be in resonance with the magnetic field. The electron temperature increases by a factor oftwo (up to 0.6 to 0.7 keV) at a plasma density of 1013 cm" 3 , the heating efficiency being 5 to
6eV-kW"'. The ion energy distributions are non-thermal during ECR heating. — Experimental
results on propagation and damping of waves and excitation of eigenmodes during ICR plasma heatingin the L-2 stellarator are presented and explained theoretically. In the theoretical model, plasma andmagnetic field inhomogeneities and the presence of an Alfv6n resonance are included. The experimental
results are in good agreement with the calculations.

481

482

ANDRYUKHINA et al.

I. ELECTRON CYCLOTRON RESONANCE
PLASMA HEATING BY ORDINARY WAVESIN THE L-2 STELLARATOR
1.

Introduction

The experiments were carried out in the L-2 stellarator [1] (R = 100 cm,
ap = 11.5 cm, I = 2, m = 14). Microwave power taken from a gyrotron with
f = 37.5 GHz, Po = 100 kW and a pulse duration of < 7 X 10~3 s, was launched
into the machine through a system of two lenses in the form of a linearly polarized
Gaussian beam withfsllB, a beam divergence of 0.02, and a beam diameter of about
6 cm, the ordinary wave containing 99% of the total power. The target plasma wasproduced by Ohmic discharges at relatively low levels of plasma current and Ohmic
power(Ip = 12-13 kA,
P 0H = 20-25 kW,n, « 7.5 x 1012 cnT 3 ,Te(0) = 0.35 keV, Tj(O) = 70 eV, hydrogen). To reduce the impurity content andthe radiative losses (RL), a graphite limiter was placed at the magnetic surface with
r = 9 cm.
2.

Results

Figure 1 shows the time histories of plasma current Ip, resistive component ofloop voltage U, radiative loss power Prad and mean electron density fie during a heating pulse when the resonance condition w0 = wce is fulfilled in the plasma centre.
When the ECR power is switched on, the plasma current starts decreasing more
slowly than before, and the loop voltage has a characteristic drop, i.e. it decreases
by a factor of about two at the end of the pulse, and the plasma conductivity increases
by about 2.5 times. During this process,ftj.changes by no more than 10%, and Pra<1
increases by 5 to 8 kW, reaching 18 to 20 kW at the end of the pulse.
The electron temperature at the centre, Te(0), as measured by Thomson scattering, soft X-ray PHA spectrometry and foils, increases from 0.35 keV to
0.6-0.7 keV, and the main temperature variation occurs during 1.5 to 2 ms afterswitching on the microwave power. The electron distribution function is Maxwellian.
Figure 2 shows the energy spectra of the charge exchange atoms for Ohmic and ECR
heating. In contrast to the case of Ohmic heating, non-Maxwellian ion distributions
with a tail of high energy particles are observed during ECRH. If we, conventionally,
divide the available energy range into two parts, E, = 0.2 - 0.5 keV andE2 = 0.5 — 1.5 keV, we can derive the effective temperatures for these ranges:
T[° (bulk ions) and T[2) (tail). The time histories of T[l) and Tf2) are presented inFig. 3. When we switch on the ECR power, the energy distribution is deformed
rapidly during about 1 ms, i.e. Tj1' rises from 70 to 90 eV and Tj2) increases toabout 150 eV, while T\n = T}2) before ECR heating. The spatial profile of T|" isrelatively flat in both cases.

483

IAEA-CN-47/D-ID-3

ECRH

i

1
IplkA)

U(V)• ^ - ^

3

10

u

—, v

—•

5 -

P

P

rad ( k W l
15 „

10 --

"nT"

- 0.5

0.5

/

:y

n e (10 1 3 cm- 3 )

**

1.0

0.6

0.3 - 1 - 1-——_,——%*^"',''T a (0).(keVl

0.4

rad .

2

"T""
,*
1
/
•

1

i

t

—•<•'"

lx(a-u.)

• < ^

0.3
0.2
\

"

Te(0) - ^ V
1

1

0.1

1

t (ms)

FIG. 1. Time dependences of Ip, V, Prad and ne during ECR heating pulse(
OH, — — OH + ECRH) and of central electron temperature, Te(0): (*: Thomson
scattering, + : soft X-ray spectra, 1^ ratio of X-ray fluxes passing through Be filters of thickness
25 fim and 50 tan.

200

600

1000

E(eV)

FIG. 2. Energy spectra of charge exchange neutral atoms (1: OH, 2: OH + ECRH).

ANDRYUKHINA et al.

484
200

150

r<2>

D

,_.- 100

50
EC H

-I
82

^

84

86

88

90

92

t (mslFIG. 3.

Time dependence of ion temperatures T\ and 7? during ECRH pulse.

To determine the efficiency of ECR heating, we measured the absorption of
microwave power by using two methods: measuring the power passing through theplasma near the launching point and analysing the plasma energy loss channels
related to radiation and thermal conductivity, the main part of the thermoconductive
flux being deposited at the limiter. These methods yield approximately the samevalue of absorbed power, i.e. 0.5 to 0.6 of Po.

3.

Discussion(a) The Ohmic heating power during ECRH is low (P 0H « 10 kW «0.15-0.2 Po) and is not essential in the plasma energy balance.
(b) The plasma conductivity increases continuously during the heating pulse
while Te(0) rises for 1.5 to 2 ms only. This fact indicates that the electron temperature profile broadens during heating since no increase in the effective plasma charge
was observed spectroscopically. The increase in conductivity could be due to current
drive or to an increase in the number of runaway electrons. When, however, thedirections of current and magnetic field were reversed, no changes in Ip(t) and U(t)were observed. On the other hand, according to the X-ray spectra in the range of 2
to 40 keV and from the estimates carried out, no increase in the number of runaways
takes place.
(c) The increase in ion temperature at Te/Tj = 7 - 8 and the non-thermal ion
energy distributions indicate that ion heating is due to a non-linear mechanism rather
than to Coulomb collisions.
(d) Microwave absorption was numerically calculated by using ray tracing (31rays), in geometrical optics approximation, in the real geometry of the L-2 magnetic
field. It was shown that the maximum absorption of about 40% takes place forIAEA-CN-47/D-III-3

485

FIG. 4. Structure of magnetic field module \B\ (3), ray traces (dashed lines) and absorption regions(4) at cross-section where microwave power is launched, vacuum vessel (1), and the last closed magnetic
surface (2); torus major axis is at left hand side.

fle(O) = 1O13 cm 3, Te(0) = 0.4 keV, and with resonance conditions at the plasma
centre. Figure 4 shows the ray traces calculated under the above mentioned conditions; also shown are the absorption regions and the structure of IB t. Microwave
absorption is localized within the saddle shaped region of IB I, where d In B/dr is
minimum. The results of the calculations are in good agreement with the experimental ones.
(e) Just as in the previous experiments [2, 3], the heating efficiency decreases
by a factor of three when the magnetic field changes by ± 5 % , which is due to thespecific structure of the magnetic field in the L-2 stellarator.
(f) The other important parameters are: plasma energy W ~ 120 J, global
energy lifetime « 2 ms, (3(0) « 0.16%, and ECR heating efficiency 5 - 6 eV-kW"1
per 1013 cm" 3 .

REFERENCES
[1]
[2]
[3]

ANDRYUKHINA, E.D., et al., Preprint No. 154, Lebedev Institute, Moscow (1977).
ANDRYUKHINA, E.D., in Plasma Physics and Controlled Nuclear Fusion Research 1984
(Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 409.
ANDRYUKHINA, E.D., et al., Pis'ma Zh. Ehksp. Teor. Fiz. 40 (1984) 377.

ANDRYUKHINA et al.

486

II. FAST WAVE EIGENMODESIN THE L-2 STELLARATOR
DURING ICR PLASMA HEATING
In the experiments on ICR heating (co = Q-t of hydrogen plasmas in the L-2
stellarator fl], the heating efficiency was rather high. The present experiments,
studying wave propagation in the ICR heating regime of a hydrogen plasma, werecarried out for plasma densities of no = n«,(0) :S 4 X 1013 cm" 3 and a magnetic
field of Bo « 12 kG at the axis. The antenna was a bent loop whose central part —without Faraday screen — was moved close to the plasma. The longitudinal component of the radiofrequency magnetic field was measured by magnetic probes placedalong the torus at different distances from the antenna. The eigenmodes of the fastmagnetosonic waves corresponding to waves with k| = f/Rj,, where t is an integer
and Ro is the major radius of the torus, were observed as spikes of the radiofrequency amplitude when the plasma density was changed. The experimental dependence of k| on no for the eigenmodes is shown in Fig. 5(a). The character of wave
propagation changed essentially (Fig. 5(b)) when the magnetic field Bo becameclose to Bc, which corresponds to the resonance condition w = Q, at the plasma
centre, i.e. when the longitudinal damping length, XD, was less than the torus(cm"1)

^

*

jar

0.1

if*.

7

0.05 .

(a)

•

i

•0*

n

13
3
0 <10 cnT )

i

I

1.2

B 0 /B c

L
1.0

0.5

0.9

1.0

1.1

FIG. 5. Theoretical and experimental dependence of (a) kf on n0 and (b) relative damping length,
L , on Bo/Bc.

IAEA-CN-47/D-III-3

487

perimeter, L = 2-n-Ro. These dependences can neither be explained by collisional
nor by cyclotron absorption of fast magnetosonic waves. Calculations allowing forthe presence of a resonance surface with e, = nf in the plasma show that the strong
damping and the observed wave dispersion may be due to Alfve'n resonance.
To determine the quality factor Q and the damping length XD, we have calculated the dependences of the loading resistance of the antenna R ^ on B0/Bc for various ky (no = const) for the m = 1 mode. The values of Q and XD were estimatedfrom Q = (26)"' and XD = 2Qvg/a>. Here, 6 is the ratio of the half-width-at-halfmaximum of the peak of R ^ to the value of B0/Bc at the maximum, and vg is thelongitudinal component of the mean group velocity. Using the dependences of R ^on kj for various no (Bo = Bc), we could find the kj that corresponds to the maximum of R^m, by analogy with the eigenmodes. The results of the calculations areshown by the solid lines in Figs 5(a) and (b). If the Alfve'n resonance is taken into
account, the results of the calculations differ essentially from those of Ref. [2] (dashdotted line in Fig. 5(a)), especially for low plasma densities, the results of [2] having
been obtained for a homogeneous magnetic field. The calculations are in good agreement with the experimental results.

REFERENCES[1][2]

BATYUK, V.A., et al., in Controlled Fusion and Plasma Physics (Proc. 11th Europ. Conf.
Aachen, 1983), Vol. 7D, Part 1, European Physical Society (1983).
KOVRIZHNYKH, L.M., MOROZ, P.E., Zh. Tekh. Fiz. 54 (1984) 649.

DISCUSSION
H. WOBIG: You reported a confinement time of 2 ms in ECR heated plasmas.
How did the confinement time change when you applied the ECRH power to theohmically heated target?
L.M. KOVRIZHNYKH: When the high microwave frequency power isapplied, the confinement time decreases slightly. But the reason for this may not only
be the degradation of confinement due to input of the power; it may also be a direct
result of an increase in electron temperature.
A.C. RIVIERE: You have described some anomalous ion heating during
ECRH. Do you believe that this is due to the connection of the EC wave to the electron Bernstein wave at the upper hybrid resonance, leading to a parametric decay
instability?
L.M. KOVRIZHNYKH: Yes, it is possible; but the exact mechanism underlying the ion heating is not yet known.

IAEA-CN-47/D-DI-4

PLASMA POTENTIAL AND CONFINEMENT
IN THE NAGOYA BUMPY TORUS (NBT-1M)
H. IKEGAMI, M. HOSOKAWA, H. IGUCHI, T. SHOJI,
T. KAMIMURA, H. SANUKI, K. TAKASUGI, F. TSUBOI,T. TAKEUCHI, A. KOMORI1, K. KADOTA, J. FUJITA,
C. HONDA2, M. YOKOO2, M. MAEDA2, K. MURAOICA1,
T. YUYAMA3, T. MICHISHITA3, H. KUBO3, M. FUJIWARA
Institute of Plasma Physics,
Nagoya University,
Nagoya, Japan
Abstract
PLASMA POTENTIAL AND CONFINEMENT IN THE-NAGOYA BUMPY TORUS (NBT-1M).
There are two critical issues to be confirmed for the bumpy torus concept to be established. One
is the average minimum-B configuration produced by the hot electron rings that stabilizes MHD instabilities to which bumpy torus plasmas are susceptible. The other is confinement scaling, especially confinement enhanced with the radial electric field. — With increased microwave power input of 90 kW into
one mirror sector, which is equivalent to a total input of 2.16 MW to the bumpy torus NBT-1M, gradient B reversal is not achieved. The hot electron ring beta is observed to remain at a level of several
per cent, i.e. almost at the same level as in the case of 30 kW total input power, although more than
one order of magnitude increase is detected in the diamagnetic loop signal and the hard X-ray intensity.
— The plasma space potential is found to be controlled by energetic electrons generated in the second
harmonic electron cyclotron resonance zone and by their loss mechanism. The potential profile is little
influenced by the transport of the core plasma for the present NBT-1M plasma parameters, but becauseof the potential produced by the energetic electrons, neoclassical transport models can be applied to thecore plasma. — As to confinement scaling, the effect of the radial electric field is studied with the useof high power ECH and ICH, for the purpose of controlling the polarity of the radial electric field andthe collisionality. No drastic improvement is observed, regardless of the sign of the electrostatic potential (well or hill) with respect to the plasma edge. The observed energy confinement times agree,
however, with the prediction from neoclassical transport theory.

1.

INTRODUCTION

The Nagoya Bumpy Torus (NBT-1M) [1] is composed of 24 simple mirrors
connected end to end into a torus with a major radius of 1.4 m and an average aspectratio of approximately 10. The plasmas are produced and heated by a combinationof microwave power (18 GHz and 28 GHz) for ECH and RF power (7-9 MHz) for
ICH. The MHD instabilities to which simple mirror plasmas are susceptible, could
1

Graduate School of Engineering Sciences, Kyushu University, Fukuoka, Japan.
Faculty of Engineering, Kyushu University, Fukuoka, Japan.
3
College of General Education, Kyoto University, Kyoto, Japan.
2

489

490

IKEGAMI et al.

be stabilized by the average minimum-B configuration produced by a hot electron
ring, which is generated by electron cyclotron heating (ECH) in the midplane of eachmirror sector; in addition, the hot electron ring beta is considered to control the maximum beta of the toroidal core plasma.
Neoclassical theory [2] predicts an electron energy confinement time TE
given byrEhei = 1.3(R/a)2 (1 + v2/ Q2) (e<£0/Te)2(1)

where Q is the poloidal precessional frequency dominated by E X B drift over VB
drift, 0O is the electrostatic potential at the plasma centre with respect to the plasmaedge, and 7ei, R, a are the electron collision time, the major radius and the averageplasma radius, respectively.

2.

GRADIENT B REVERSAL BY HOT ELECTRON RING

The most critical issue for the bumpy torus concept is whether the hot electron
ring can produce the average minimum-B configuration for the stabilization of MHD
instabilities. The ring beta must then be large enough to push out the local magnetic
field to reverse V,B in the outer edge of the ring.
With a microwave power of 3 kW at 18 GHz fed into each mirror sector, theplasma magnetic field at the ring location was measured by the Zeeman splitting oflithium beam atoms excited by plasma electrons, and the ring beta was observed toreach several per cent which was too small to reverse V r B. So the microwave
power was increased up to 90 kW at 28 GHz fed into a single mirror sector. For themeasurement, to enhance the fluorescence signal intensity, sodium beam atoms were
excited by a scanning dye laser as shown in Fig. l(a), and Zeeman splittings weredetected [3]. The microwave power input of 90 kW per mirror sector was equivalent
to a total power input of 2.16 MW, and, according to the 30 times greater input
power compared with the previous cases, both the ring energy detected by the
diamagnetic loops and the line integrated hard X-ray signals were observed toincrease more than one order of magnitude. However, as is seen in Fig. l(b), thechange in the magnetic field intensity was detected to remain always at the level ofseveral per cent and no reversal of VrB was ever observed.
It shoud be noted that even in this case the NBT plasma was observed to be
MHD stable. Several explanations for the observed stability [4], such as shear of
E X B poloidal drift, line tying, charge uncovering, etc., are proposed.

3.

EFFECT OF RADIAL ELECTRIC FIELD ON CONFINEMENT

The confinement properties and their relationship with the plasma potential in
an ECH plasma have been studied by using a high power 28 GHz pulse gyrotron

491

IAEA-CN-47/D-III-4

Fibre

NBT-1Ma>=6.6 kG
fjj =90 kW

(b)
}

\j.

.'A

7

0.1 •

\

/
/
/

J

-12

i

i

//// /

If
if
It

If
it

—•--wilh Plasma—o-wilhoul Plasma ' 0 2

/

-10

-8

-0.1

-6
-4
-2
RADIUS { cm )FIG. 1. (a) Schematic diagram of local magnetic field measurement with sodium beam, Zeeman
fluorescence spectroscopy induced by rapid frequency scanning dye laser.
(b) Magnetic field modification by hot electron ring generated with 90 kW (28 GHz, 50 ms) microwave
power. Measurement was made 5 ms after 28 GHz power turnoff.

(200 kW/75 ms). The initial plasma is produced by 18 GHz microwave power(4-26 kW), and a 28 GHz pulse is superimposed, when the initial plasma reaches
almost a steady state.
Figure 2(a) shows the typical time evolution of several plasma parameters,
together with the time sequence of the superimposed microwave power. Both plasma
density and ring stored energy increase gradually with the applied 28 GHz power,
but the nef(Te) signal shows a sharp rise, which may indicate that the electrons in thecore plasma are heated rapidly. From a comparison of n<. and nef(Te), the core electron temperature is estimated to remain nearly constant during the duration of theIKEGAMI et al.

492

(a)
28GHz i80itm

nt I SI-10

FIG. 2. (a) Time evolution of plasma parameters associated with superposition of 110 kW (28 GHz)
microwave pulse on plasma produced by 26 kW (18 GHz) microwave source. Line integrated density,(nt),from 75 GHz interferometer; central electron density, nt, from lithium neutral beam probe; ring
stored energy, Wx, from diamagnetic loop; sum of doubly ionized caesium beam currents on split plate
detectors, nJ(Tt); central plasma space potential, <j>, determined by heavy ion beam probe,(b) Time evolution of radial potential profile after 160 kW (28 GHz) pulse front superposed on plasma
produced by 4 kW (18 GHz) source in magnetic field of 6.6 kG.

28 GHz power. Thomson scattering revealed that the electron distribution was nonMaxwellian and the average energy was 100 to 150 eV. An additional slow increase
and decay of the nf(Te) signals after the rapid change, at 'on' and 'off of the
28 GHz pulse, will be due to ionization by slowly produced energetic electrons.
It is also found in Fig. 2(a) that the plasma potential is controlled by the energetic electrons rather than by the core plasma. The time evolution of the radial potential profiles is shown in Fig. 2(b), where it takes 20 ms before the potential forms

IAEA-CN-47/D-H1-4

493

600
400

— 200

-e-200-15 -10

-5

0
X

5
(cm ]

10

15

FIG. 3. Electron cyclotron resonance configuration in mirror sector, where solid and dashed lines
indicate fundamental and second harmonic electron cyclotron resonance zones. Corresponding to threecases [A (5.2 IcG), B (5.9 kG), C (6.9 kG)], each with different magnetic field intensity, space potential
profiles are shown. Plasmas are produced with combination of 160 kW (28 GHz) and 4 kW (18 GHz)for preionization.

a well depth of 200 to 250 V; thereafter it remains constant until the 28 GHz poweris turned off. The characteristic time-scale of this potential well formation is much
longer than the core plasma confinement time, indicating that the potential well formation is related to the energetic electrons rather than to the core plasma electrons.
The mechanism of potential formation was further examined by varying theelectron cyclotron resonance zone (the energy deposition profiles), i.e. by varying
the magnetic field strength. In Fig. 3, the radial profiles of the space potential are
shown in three cases with different magnetic field strengths. In the case of a lower
magnetic field, where the second electron cyclotron resonance zone is located in thecore plasma region, the potential profile is peaked at the centre, which indicates thatenergetic electrons are efficiently generated in the central area where the second ECRzone gets closer to the lower magnetic field. With increasing magnetic field, the peak
splits apart in radial direction, and between the two separate peaks there is a potential
well. When the second harmonic ECR zone is properly separated in the midplane,
a hot electron ring is most effectively generated there, encircling the core plasma.
The energetic electrons not only control the core plasma stability, but also have a profound effect on the core plasma placed in the potential produced in this way.
It is important to recognize that if we assume the potential profile to be the resultof ambipolar diffusion of the core plasma, the observed direction of the radial electricfield is contrary to the prediction of neoclassical transport theory. Because of thepotential produced by the energetic electrons, the neoclassical transport model canbe applied to the core plasma, but the core plasma itself will have little effect on thepotential, for the present core plasma parameters.

IKEGAMI et al.

494

10'

10"'

1

10

101

COLLISIONALITY PARAMETER ( f / f l )FIG. 4. Electron energy confinement time normalized by electron-ion collision time versus collisionality e/Q. Closed and open circles are for RF and ECH plasmas, respectively. Solid curve is given byneoclassical transport theory [SJ.

Studies were made on the confinement scaling of the core plasma by using ECH
and ICH, varying the value of collisionality v/Q over a wide range. The energy confinement time was estimated from density, temperature, and volume of the coreplasma, divided by the input power, on the assumption that the fraction of theabsorbed power is 30% and constant.
In an ECH plasma (n,, ~ 1012 cm" 3 , Te ~ 100 eV) of standard MBT-1M
operation, the collisionality was in the intermediate regime 1 < v/U < 10, and theenergy confinement was estimated to be in the range of three to ten times theelectron-ion collision time. The potential profile was observed to have a typical negative well.
In a high density ICH plasma (n,. ~ 1013 cm" 3 , Te ~ 10 eV), the electrons
were collisional, 10 < e/Q < 100, and the ratio T^JT^ ranged between 30 and 300,
in a peaked positive potential (0O ~ 500 V). In a low density RF plasma(n,, ~ 1010 cm" 3 , Te ~ 30 eV) and low magnetic field, to which no ECH wasapplied, no energetic electrons were supposed to be generated. The collisionality wasin the range 0.02 < v/Q < 1, with a positive hill potential generated; however, noclear improvement of confinement was observed.
The comparison with neoclassical transport theory [5] is summarized in Fig. 4,
where the experimental values show reasonable agreement with theory.

4.

CONCLUSIONSWith an increased microwave power input of 90 kW into one mirror sector,
which is equivalent to a total input of 2.16 MW into the bumpy torus NBT-1M, no
gradient B reversal has ever been achieved. The hot electron ring beta was always
observed to remain at a level of several per cent, i.e. at almost the same level as inIAEA-CN-47/D-m-4

495

the case of 30 kW total input power, although a more than one order of magnitude
increase was detected in the diamagnetic loop signal and the hard X-ray intensity.
There must be some mechanism at work that levels off the ring beta to such a low
level as several per cent.
The plasma space potential is determined by the generation of energetic electrons in the second harmonic electron cyclotron resonance zone and by their loss
mechanism. The potential profile may be little influenced by the transport of the coreplasma under the present NBT-1M plasma parameters, but the potential produced inthis way may have a profound effect on core plasma transport.
On the assumption that the absorbed power is a constant fraction of 30% of the
input power, the volume averaged energy confinement time shows reasonable agreement with the prediction of neoclassical transport theory, regardless of the sign ofthe potential, and is estimated to be three to ten times the electron-ion collision time,
for a collisionality of vlQ = 0.02-10.

REFERENCES
[1]

[2]
[3]
[4]
[5]

FUJIWARA, M., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1980 (Proc.
8th Int. Conf. Brussels, 1980), Vol. 2, IAEA, Vienna (1981) 845; in Plasma Physics and Controlled Nuclear Fusion Research 1982 (Proc. 9th Int. Conf. Baltimore, 1982), Vol. 2, IAEA,
Vienna (1983) 197; in Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc. 10th
Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 551.
JAEGER, E.F., et al., Phys. Rev. Lett. 40 (1978) 866.
HONDA, C , et al., Jpn. J. Appl. Phys. 24 (1985) 1402.
SANUKI, H., CHEN, F., Phys. Fluids 28 (1985) 3567.
HASTINGS, D., KAMIMURA, T., J. Comput. Phys. 61 (1985) 286.

DISCUSSION
H. WEITZNER: How would you compare your observation of the nonreversal of V ^ with the EBT results?
H. IKEGAMI: I don't think that the reversal of V r B has ever been achieved
with EBT at Oak Ridge. Whether the non-reversal of V r B is due to the canted mirror asymmetry, or to the nature of the hot electron ring itself, is not yet clear andrequires further research.
J.D. CALLEN: If I understand correctly, you increased the ECH power input
in one cavity from 3 to 90 kW, and the stored energy in the hot electron ring went
up by about a factor often. Theoretically, the AB would increase by a similar factor
unless the ring increased in size by a corresponding factor. Since AB apparently did
not increase, could you say whether the ring increased in its radial or axial extent?

496

IKEGAMI et al.

H. IKEGAMI: When the ECH power was changed from 3 to 90 kW, themicrowave frequency was also changed, from 18 to 28 GHz, so it is for the magnetic
field intensity to provide the ECH resonance, and in order to give the same AB/B,
the hot electron energy density must have increased by a factor of 2.4, at least. The
rest of the increment factor may be attributed to some expansion and spread of thering in a radial direction, especially towards the wall.
K. ITOH: The asymmetry of the potential profile persists during the change
in the sign of the potential. Does this mean the existence of the large convective cells,
and does it degrade the confinement?
H. IKEGAMI: In a system like the bumpy torus, there are several
unfavourable mechanisms, besides the asymmetry, causing deterioration of theconfinement. For example, the resonant cancellation in the poloidal precession,
Q, B + fiEXB « 0, may change the bumpy torus to almost a simple torus, as you
may have observed in Fig. 2(b). The convective cells you mention may develop fromany discrepancy between the equipotential surface and the constant IB I surface.

IAEA-CN-47/D-ffl-S

EFFECTS ON EQUILIBRIUM AND STABILITY
OF HIGH BETA RF HEATED PLASMAS
IN THE URAGAN-3 TORSATRON
V.L. BEREZHNYJ, N T . BESEDIN, M.P. VASIL'EV, V.S. VOJTSENYA,
E.D. VOLKOV,
Ya.F. VOLKOV,
S.P. GUBAREV,
Yu.V. GUTAREV,
A.G. DIKU,
V.M. ZAIKIND,
G.V. ZELENIN,
V.K. KONOVALOV,
V.I. KONONENKO, V.D. KOTSUBANOV, B.V. KRAVCHIN, A.E. KULAGA,
Yu.K. KUZNETSOV, G.G. LESNIYAKOV, A.P. LITVINOV, Yu.K. MIRONOV,
N.I. NAZAROV, O.S. PAVLICHENKO, V.K. PASHNEV, N.F. PEREPELKIN,
V.V. PLYUSNIN, G.N. POLAKOVA, N.P. PONOMARENKO, T.Yu. RANYUK,
L.N. REMEZ, K.S. RUBTSOV, S.I. SOLODOVCHENKO, A.I. SKIBENKO,
A.S. SLAVNU, I.S. SUSHKO, V.S. TARAN, V.T. TOLOK, T.O. TORYAK,
I.P. FOMIN, A.N. SHAPOVAL, O.M. SHVETS, A.F. SHTAN'

Institute of Physics and Technology,
Ukrainian Academy of Sciences,
Kharkov,
Union of Soviet Socialist Republics

Abstract
EFFECTS ON EQUILIBRIUM AND STABILITY OF HIGH BETA RF HEATED PLASMAS IN THE
URAGAN-3 TORSATRON.
Systematic studies of currentless plasma produced in the URAGAN-3 torsatron by RF heating
at w £cj H i have led to the discovery of three types of discharge. The first type (called D-discharges)
was encountered on working with bare, stainless steel, helical frame antennas. They were characterized
by rather high densities (n^O) <, 2 x 1013 cm" 3 ), moderate temperatures (T^O) < 0.3 keV,
Te(0) < 0.23 keV) and a short pulse duration (At = 3 to 4 ms) due to progressive degradation of theelectron and ion temperatures. The second type (called Q-discharges) was obtained when the antenna
was coated by TiN. These discharges were characterized by long pulse duration (At = 50 ms), Lower
electron densities (ne(0) < 4 x 1012 cm" 3 ) and higher temperatures (T, < 1.1 keV, Te < 0.3 keV).
The third type of discharge (called QR-discharges) was encountered when the magnetic configuration
axis was shifted outwards by the transverse magnetic field. These discharges were characterized byrelaxation oscillations of the plasma parameters, similar to those of the high-/3 S-regime of Heliotron E.

1.

INTRODUCTION

Currentless plasma production and heating by RF waves in the ion cyclotron
and Alfve'n resonance regions have been studied in the URAGAN-3 torsatron
during the last four years [1-5]. RF waves are excited in a plasma by helical frame
antennas (without Faraday screen) with a specially chosen spectrum of toroidal andpoloidal wavenumbers. Such antennas provide gas ionization and subsequent plasma
heating via a 'mode relay race' mechanism [2]. Plasma with an electron density ofne(0) < 2 x 1013 cm" 3 , an ion temperature of T;(0) < 300 eV and an electron
497

498

BEREZHNYJ et al.

temperature of Te(0) < 270 eV was obtained in URAGAN-3 for a confining fieldof Bo » 0.92 T with an RF radiated power of 0.6 MW [4]. Such parameters were
sustained for no longer than 3-4 ms, because of progressive electron and ion
temperature degradation [4, 5]. Plasma radiative cooling by metal impurities from astainless steel (SS) RF antenna was considered to be a probable reason for thetemperature degradation.
To study the role of the antenna material, a titanium nitride (TiN) coatedantenna was installed in URAGAN-3 in autumn 1985. Quasi-stationary discharges ofa duration determined by the RF pulse length (t < 50 ms) were obtained with thisantenna, and a plasma with a j8-value close to the critical one, from the view of
equilibrium, was produced at moderate RF power ( < 0.6 MW).
In this paper, results of studies of the effects of a plasma with maximum/8 = 0.6% on equilibrium and stability in URAGAN-3 are presented.

2.

EXPERIMENTAL TECHNIQUESThe URAGAN-3 torsatron design is described in Ref. [1]. It consists of amagnetic system installed inside a large (volume « 70 m3) vacuum chamber withdiagnostic ports.
The magnetic configuration is produced by one ( = 3, m = 9 helical winding
set wound on the torus with a major radius of R = 100 cm and a minor radius of
ah = 26 cm, according to the winding law<>/ = - — (6 - 0.2 sin B - 0.01 sin 20)
and by two circular coils (R = 150 cm, Az = ±50 cm) containing compensating
and correcting windings which produce a vertical magnetic field (Bx).
Two frame antennas (length = 70 cm, width « 20 cm) were placed near thetoroidal surface with a minor radius ra = 10-12 cm inside the helical winding at the
low field side. Antennas were fed by separate RF generators (5.3 MHz, 2 MW,
50 ms) with a two step RF pulse. Most experiments were performed at a constantgas (H2) inlet into the evacuated vacuum chamber and after extensive inside surface
cleaning by RF discharges at a low magnetic field [3].

3.

MAGNETIC FIELD CONFIGURATION

Magnetic field configuration studies performed since 1981 by the electron beam
mapping technique showed that the region of nested magnetic surfaces is smaller than
was predicted by calculations. Substantial improvement in the configuration took
place after introducing a 9% imbalance in the vertical field coil currents. Thissuggests the possible presence of an additional radial error field.

499

IAEA-CN-47/D-III-5- - 1 5 cm

15 cm

- --15 cm

FIG. 1. Flux surfaces in URAGAN-3 measured by stationary electron emitter and grid. Surfaces withdifferent magnetic field line transits N have the following numbers: (1): N = 10±2, (3): N = 20±5,(4): N = 30±5, (5): N = 40±5, (6): N = 50±5, (8): N = 70±5, (9): N = 100±10. Regime
0 (BJB0 = +0.',

2

5

3

a (cml

6

10

lira

1.0

0.75

0.5 -

t(a)

- 5

0.25

v
0
0.25

0.5

y 0.751.0

FIG. 2. URAGAN-3 magnetic configuration parameters as functions of vertical magnetic field Bx(results of numerical calculation). Numbers near arrows are the numbers of regimes used.

Construction and alignment errors in the massive magnetic system werepossible reasons for this situation. The magnetic system alignment for URAGAN-3
was verified by a novel technique developed in co-operation with ORNL [6].
Measurements by this technique showed that the vertical field coil was shifted byabout 1 cm relative to the axis of the helical coil set. This shift was eliminated bydisplacing the vertical field coil set. After coil alignment, the flux surfaces in
URAGAN-3 were studied by a novel technique [7]. These studies showed that coil

500

BEREZHNYJ et al.

alignment resulted in some improvement of the configuration although the radius of
the last closed surface remained smaller than the calculated value. Helical field coilshape studies showed local misalignment within the helical coil set itself. Flux surface
calculations including the data of helical coil shape studies showed that these local
misalignments probably were the reason for the observed flux surface destruction in
URAGAN-3.
Experimentally mapped flux surfaces in the optimum regime of URAGAN-3
after coil alignment are shown in Fig. 1. The general diagnosis of the magnetic field
configuration in URAGAN-3 is as follows: the nested flux surface region (averageradius about 9 cm) is surrounded by an ergodic magnetic layer (radial width 3 to4 cm), where the magnetic field lines go round the torus one to ten times. The
magnetic field configuration in URAGAN-3 can be changed by varying the verticalmagnetic field, B ± . The main effects of B x variation are a shift of the magnetic axis
and a change in the central rotational transform,-*(0), the outer transform-t(a) remaining practically unchanged (Fig. 2). This possibility of machine design was used inequilibrium and stability studies and proved to be very useful.

4.

PLASMA BEHAVIOUR IN Q-DISCHARGES

RF discharges with quasi-stationary behaviour of plasma parameters(Q-discharges) were encountered when one of the two installed SS antennas was
replaced by a TiN coated antenna. With this RF activated TiN antenna and after theusual RF discharge cleaning, discharges without temperature degradation during atime of up to 50 ms were obtained and studied.

600

pR
P

RF

(arb.
units)

f . , .,

(kW)

(arb.
units)

\

I

2
(kA)
0
A

(arb.
units)SXR, V
bolometer(arb.
units)

A
Cff

"e

(arb.
units)

1012cm 3 )
0
0

10

20

0

10

20

t (ms)

FIG. 3. Plasma parameter behaviour in Q-discharge (Bo = 0.43 T, BJB0

= +0.36%).

501

IAEA-CN-47/D-m-5

0

100

200

300

400

500

FIG. 4. Dependence of unidirectional (It) and bidirectional (Iu) currents on RF power,(Bo = 0.43 T, B±/Bg = +0.77%).

0

1

2

3

4

n e (10 12 crTT 3 >FIG. 5. Dependence of plasma pressure on electron density (Bo = 0.43 T, BJB0P*RF = 450 kW.

=+0.36%,

Figure 3 shows the time behaviour of the plasma parameters in a Q-discharge.
Here, PRF is the RF radiated power, I t l is the bidirectional current measured by asaddle-type magnetic winding [3], I, is the unidirectional current measured by a
Rogowski coil, n,. is the electron density averaged over the vertical chord andmeasured by a 2 mm interferometer, SXR is the soft X-ray diode detector signal,'bolometer' refers to the pyroelectric detector signal, C V is the intensity of the C V
spectral line, l2W(x is the intensity of the plasma radiation at 20;,*. The importantfeatures of these 'currentless' discharges were quasi-stationary, longitudinal
unidirectional (It) and bidirectional (dipole, I t |) currents. Both current directions
changed with the poloidal magnetic field direction of the helical winding. The current
increased with increasing RF power (Fig. 4). It was shown earlier [3] that the value

502

BEREZHNYJ et al.

TABLE I. RESULTS OF MEASUREMENTS
Regime 3 (PgF = 500 kW), t = 12.5 ms.
Bo(T)B x /B 0 (%)(cm'3)

P(eV •cm"3)

(eV)

(eV)

(eV)

(eV)

n (Te + Ti)(eV-cm" 3 )

0.43
+0.48

2 x 10'2

2.5 x 10"

980

930

250

270

2.4 x 10"of the bidirectional current in RF produced plasmas in URAGAN-3 is in good agreement with the current value as calculated from a simple analytical model of
MHD plasma equilibrium in a stellarator (Pfirsch-Schluter current). This conclusion
allowed us to use measurements of the dipole current poloidal magnetic flux
for estimating the average plasma pressure p = n (Tj + Te) and the parameter/3 = 8TT p/Bg.
For a given value of antenna radiated RF power, the plasma pressure depended
on a few parameters: confining and vertical magnetic field values and hydrogenpressure. A decrease in hydrogen pressure resulted in a decrease of the electron
density and a simultaneous increase in plasma pressure (Fig. 5), showing a strong
increase in plasma temperature.
In these experiments with a confining magnetic field, Bo = 0.43 T, and one
active antenna (PR F < 0.55 MW), a plasma with p = 2.6 X 1015 eV-cm""3 and/3 = 0.6% was obtained at an electron density of He = 2.1 x 1012 cm" 3 . The factthat the keV range of the average plasma temperature Te + Tj was reached wasconfirmed by independent measurements of the plasma temperatures. The ion
temperature was measured by two charge exchange neutral energy analysers measuring the longitudinal and transverse neutral fluxes in the equatorial plane of thetorsatron. The electron temperature was measured by laser scattering at R = 103 cmof the equatorial plane and by second harmonic electron cyclotron radiation. Verticalscanning of the transverse neutral flux showed that Tj did not change very much.
Measurement of the 2wcc spectrum showed that the Te(r) profile was close to theparabolic profile. Table I shows the results of these measurements. The averageplasma pressure n (Te + Tj) was calculated on the assumption that the electron
density and temperature distributions were parabolic and the ion temperature profile
was homogeneous; it was found to be 2.4 x 1015 eV-cm~ 3 . Comparisons of thiskind allowed us to conclude that the description of the dipole equilibrium current
chosen represents the experimental situation rather well up to values of J3 = 0.6%,
thereby permitting the use of dipole current measurements in order to measurep and J3.
The theory of finite /3 plasma equilibrium in a stellarator [8] predicts two main
effects: (1) shift of magnetic surfaces and (2) destruction of magnetic surfaces in the503

IAEA-CN-47/D-IU-5

>

0.3

0.2

0.1

200

400

P

(kWI

RF

600

FIG. 6. Dependences of plasma pressure and global energy confinement time on RF power, P^p, fordifferent
verticalfields(Bo = 0.43 T,
0: BJB0 = +0.77%,
3:BJB0=+0.<
5: BJB0 = +0.36%).

pre-separatrix region. The shift of the magnetic surfaces can be eliminated by using
a transverse magnetic field so that destruction of magnetic surfaces and related confinement degradation should be the main effect of high /3 plasma equilibrium [9]. A
conventional rough estimate of the critical beta, /3C = *2(a) a/R, for the URAGAN-3
yields a value of 1.4%.
MHD calculations for URAGAN-3 show that, for /3 > 0.2&, we may expect
a pronounced shift of the magnetic surfaces and a plasma pressure effect due to thetransverse magnetic field. These expectations were confirmed by measurements ofthe plasma pressure dependence on the transverse magnetic field Be (Fig. 6): we seethat the maximum plasma energy content is obtained for the inward shifted vacuum
magnetic configuration.
It was interesting to find out whether energy confinement degradation would
occur at beta values attained in URAGAN-3. As a confinement factor of merit, one
can use the value of the global energy confinement time:

PRRF

504

BEREZHNYJ et al.

0.1

0.2 0.3

0.4 0.5

0.6

FIG. 7. Energy decay time, T£ , and global energy confinement time, T £ , versus |8 (same conditions
as for Fig. 6).

where PRP is the absorbed RF power and V the plasma volume. Assuming that the
total antenna radiated power is absorbed by the plasma (PRF = PR>) andV = 1.7 x 105 cm" 3 (a = 9 cm), we calculated T% as shown in Fig. 6. We see
saturation and degradation of r% with increasing radiated RF power. Energy
confinement degradation in experiments with additional RF plasma heating is acommon feature in tokamaks. Therefore, to check this option, we measured theenergy decay time T"E :w
with

W = - f n <Te
0.2 ms after switching off the RF power. As to the RF power, it was not a parameterof importance for T{? ; we have plotted the dependence of T° on average beta at the
moment of the T° calculation (Fig. 7). This figure also shows r% at the end of the
RF pulse. From this figure, we see a qualitative similarity between T ° = i\(ff) andT{? = hW) and a strong quantitative difference between their values which could be505

IAEA-CN-47/D-III-5

E

2.7

>

) no1

in

-*«-—

2.3
/

1.9

I

/
/

c

1.5

•
c
.O

,2
U.
CC DC

a.
1

10

15

20

25

30

t (ms>

F/G. 5. Plasma pressure increase after step-like RF power increase
BL/B0 = +0.36%, P%r = 350 kW, AP^, = 70 kW, T | = 2.7ms, r f = j . 5 w j .

considered to be evidence for the negative influence RF power has on energy confinement. To verify this conclusion, we measured the plasma energy buildup time, T\,
which describes the time behaviour of the plasma energy increase AW(t) with a
step-like increase of the absorbed power
AW(t) =

r\ | 1 - exp ( -Figure 8 shows typical data for AW(t)/V, confirming the time behaviour as predicted
by Eq. (1). The general conclusion resulting from the available data on T\ and T|=
is:This means that the transition times, T\ and T%, represent the plasma energy
confinement time TED. The observed strong difference between T% and T | ~ T ° is
the result of a large difference between radiated and absorbed RF powers(PR F > PRP). This conclusion will need some explanation, in the future. At any
rate, a beta soft saturation at J8 = 0.5j8c, which results from the energy confinement
degradation appearing for ft > 0.3, is evident for URAGAN-3 plasmas.

BEREZHNYJ et al.

506
r

600

RF

bolometer(arb.
units)

(kW)
0

CV

'•I
(arb.
units)(arb.
units)(arb.
units)SXR(10 12 cm- 3 )S^rvWyV\,arb.
units)(arb.
units)

10

20

10

20

t (ms)

FIG. 9. Plasma parameter behaviour in QR-discharge (Bo = 0.43 T, B JB0 = +0.

1.0
CV

0.5

o

•

1.0
C III

1 0.5
d <

insity

J2

li

f

c

g 0.5

o
1.0

Cll

[/. . A
Til

0.5
0
0

5

10

15

20

t (ms)

FIG. 10. Time behaviour of carbon ion and titanium atom spectral lines
BJB0 = +0.77%, Pk = 400kW).

(Bo = 0.43 T,

507

IAEA-CN-47/D-III-5

(a)

•n

1
1
1
1

2
1
0
2
1
0

1

•

1

i

|

|

I
t

w
•

i

;
i

* . . .V

VV
i
t

1

•

'.^
C V

I

1

10

10
t Imsl

15

FIG. 11. Time behaviour of longitudinal neutral flux Tn, for different neutral energies shown byarrows in typical energy spectrum. Oscillograms of C V (plasma centre) and TiN (near antenna) arealso shown.

t (ms)

FIG. 12. Time behaviour of plasma pressure for different vertical field values (Bo = 0.43 T,
P
RF - 450 kW). Regime numbers from Fig. 2.

508
5.

BEREZHNYJ et al.
RELAXATION OSCILLATIONS IN A HIGH 0 PLASMAAll Q-discharges showed pronounced (~ 10 %) plasma parameter fluctuations in
the frequency range of 1 to 30 kHz. In regimes with the vertical field
B,/Bo > +0.5% and /3 > 0.3%, new types of discharge, called the QR-discharges
and characterized by the occurrence of relaxation oscillations, were encountered(Fig. 9). For most values of the plasma parameters, these oscillations were of disruptive nature, but for plasma parameters revealing radial localization (impurity ions),
the oscillation shape changed (Fig. 10). On some signals related to the periphery
localized parameters (Ti I emission near the antenna, H^ emission in the divertedflux) positive spikes were observed before the disruptions. This implies that the
perturbation propagated from the centre to the periphery of the discharge. The change
in shape of the relaxation oscillation for energetic neutrals with neutral energy variation (Fig. 11) and its correlation with the shape of spatially localized parameters(C V, Ti I) allowed us to conclude that the high energy neutrals of the energy distribution presented in this figure characterize the plasma central zone and its temperature. Low energy neutrals yield information on the ion temperature in the ergodic
magnetic layer.
The pattern of relaxation oscillations changed with the value of the verticalmagnetic field B ± : discharges with lower B± values were more stable, in spite of
the increase in plasma pressure (Fig. 12).

6.

CONCLUSIONS

After coating the RF antenna with a TiN layer, a quasi-stationary, high temperature (T, > 1 keV) plasma with /? = 0.5& has been obtained in URAGAN-3. In thisplasma a bidirectional current with all features of the Pfirsch-Schluter equilibrium
current and a unidirectional current proportional to the plasma pressure are flowing.
The observed relaxation oscillations appear similar to those of the high /?
S-regime of Heliotron E [10]. However, in URAGAN-3, in addition to the pressure
gradient, the unidirectional current can also be a driving force for the tearing mode
instability.
The observed soft saturation of j8 with the increase of the RF power results fromenergy confinement degradation for 0 > O.3/3c. To which of the effects —equilibrium, stability or peculiarity of transport — the observed TE degradation isrelated is a question to be elucidated in our future experiments on the URAGAN-3
torsatron.

REFERENCES[1]

ARSEN'EV, A.V., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1982
(Proc. 9th Int. Conf. Baltimore, 1982), Vol. 2, IAEA, Vienna (1983) 227.

IAEA-CN-47/D-ni-S
[2]

509

SHVETS, O.M., et al., in Heating in Toroidal Plasmas (Proc. 4th Int. Symp. Rome, 1984),
Vol. 1, CEC (1984) 513.
[3] BAKAEV, V.V., et al., in Plasma Physics and Controlled Nuclear Fusion Research (Proc. 10th
Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 397.
[4] PAVLICHENKO, O.S., paper presented at US-USSR Stellarator Workshop, Oak Ridge (1985).
[5] NAZAROV, N.I., et al., in Controlled Fusion and Plasma Heating (Proc. 13th Europ. Conf.
Schliersee, 1986), Vol. 1, European Physical Society (1986) 192.
[6] COLCHIN, R.J., et al., Rev. Sci. Instrum. 57 (1986) 1233.
[7] GEORGIEVSKU, A.V., et al., in Controlled Fusion and Plasma Physics (Proc. 12th Europ.
Conf. Budapest, 1985), Vol. 1, European Physical Society (1985) 68.
[8] SHAFRANOV, V.D., Voprosy teorii plazmy (Problems of Plasma Theory), Vol. 2 (1963) 92(in Russian).
[9] GORBACHEV, N.S., KUZNETSOV, Yu.K., PINOS, I.B., Nucl. Fusion 20 (1980) 341.
[10] HARRIS, J.H., et al., Phys. Rev. Lett. 53 (1984) 2242.

IAEA-CN-47/D-III-6

EXPERIMENTAL AND THEORETICAL STUDIESOF TOROIDAL HELIACS
B.D. BLACKWELL, R.L. DEWAR, H.J. GARDNER,
S.M. HAMBERGER, L.E. SHARP, X.H. SHI,
C.F. VANCE, D.F. ZHOU1Australian National University,
Canberra, Australia
Abstract
EXPERIMENTAL AND THEORETICAL STUDIES OF TOROIDAL HELIACS.
Experimental results are given for the small heliac SHEILA whose rotational transform profile
can be varied by an I = 1 helical current around the central ring. Introducing low order resonances
adversely affects the confinement, as predicted. The mode structure of an electrostatic fluctuation isshown to resonate with the helical field geometry. Computations presented for a large heliac demonstrate how both | B | ripple and toroidal effects can be greatly reduced by judicious choice of coil geometry. Results are presented for stability against ideal interchange and ballooning modes of freeboundary helically symmetric equilibria obtained from a new equilibrium code. Plasma shifts are foundto be stabilizing.

1. Experiments on Prototype Heliac SHEILA
Further detailed work has confirmed the results of initial experiments on
SHEILA (Table I) reported earlier [1] which demonstrated the existenceof well-defined, low-/?, long mean-free-path equilibria via the coincidenceof the experimentally measured isobaric surfaces (neTe = const) and thecomputed helical flux surfaces. For typical plasma conditions (n e ~ 1 — 5 x
10 1 8 /m 3 ,T e ~ 10eV,5 ~ 0.2 T), 0 < 10~4, and only small oscillatory
currents are used in plasma production. Thus the flux surfaces are, toa good approximation, those of the vacuum field computed from actual
conductor geometries and currents.
The subsequent addition of an Z = 1 helical winding around the ring,as suggested by Harris et al. [2], has allowed a wide measure of control
over rotational transform i(ip), shear Ae/i(0) and magnetic well W{if>) =1 - V ( ^ ) / V ( 0 ) . Fig. 1 shows some examples for various ratios of helix
to ring current C = /h// c . Notice that for C < - 0 . 1 the magnetic wellis very small, while the shear is large, and that for C ~ 0.125 the shear
vanishes while the well deepens. Another important feature is the presenceof the resonance t/JV = 1/2, (t = 1.5) for 0.09 < C < 0.125,which leads toisland formation.
On leave from the Institute of Plasma Physics, Hefei, China.

511

512

BLACKWELL et al:

TABLE I. COMPARISON OF EXPERIMENTAL RESULTS
FOR SHEILA AND H-1
No. of field periods N
No. of toroidal coils NT
Mean radius of toroidal coils r c (m)Helical excursion /?(m)Major radius Ro(m)Mean minor plasma radius (a)Max. field B(T)
Duration (s)Max. ring current Jc(kAT)Max. vertical field BV(T)Radius of control helix rvt(m)t(O)Ka)Well depth W
Dia. of vacuum enclosure(m)SHEILA
3
24
0.065
0.025
0.19
0.03
0.4

0.04
28

0.04
0.014
1.2|1.35
3%
0.6

H-1
3
36
0.383
0.22
1.0
0.20
0.2|1.0
oo|l
500
0.1
0.10
1.1|1.18
3%
3.8

(a)

t/H
1.8 3/5- ^~~~
1/7~~~~

k/k-0.25

U•—0.15

——.

~~~

U
3/7-

1.2
3/1-

--—

1.0 1/1-V

2/7-

0.8 l/t-y-O.IS
0.6 I/S-

^

-0.1
1

0

1

0.01

1

1

- -

0.02

flux surface parameter X Im)

0

0.01

0.02

flux surface parameter X (m)FIG. 1. Rotational transform t (a) and well depth (b) for different values ofC= Ih/Ic on the SHEILA
heliac. The flux surfaces are identified by the parameter X, defined as the distance between the magnetic
axis and the surface at <t> = 0, z = 0, for increasing R.

IAEA-CN-47/D-III-6

513

ne(o)

-0.2

-0.1

FIG. 2. Dependence of central electron density on helical winding current.
Representative vacuum magnetic surfaces at <J> = 0 are shown for some values of C.

Initial experimental observations obtained when the configuration isprogressively changed clearly show the adverse effects of the resonanceson particle confinement. For example, Fig. 2 shows the dependence on Cof central electron density for the same r.f. input and gas conditions, withvertical field adjusted for optimum particle confinement time.
At C = 0.1, computation shows that the 1/2 resonance causes the helical surfaces to bifurcate, while experimentally the particle confinement
time is clearly reduced. As C —» —0.07, where the i/N = 1/3 resonance islocated near mid-radius, the confinement again falls. For conditions corresponding to the shaded region, no plasma can be formed. We attribute
this to the effect of stray fields on the 1/3 resonance.
For some conditions (cg/w^ J$ o) a single mode of quasi-electrostaticoscillation frequency at ~ 10kHz and amplitude ne/ne ~ 10" 1 appears,
provisionally identified as a dissipative drift wave. Although it has only
slight effect on confinement, its structure (obtained using large arrays ofelectrostatic probes located on a given flux surface) can be used to provide
an independent check on the magnetic geometry in the helical plasma
frame. For example, Fig. 3 shows (for C = 0) the dependence of its phaseon position in the magnetic field-line coordinate system (tp,i!),<j>) [3,5],
suggesting a wave field of the form £ = £o cos(mi? — n<f>).
The propagation clearly shows m = 3, n = 5, which is resonant with thehelical rotational transform (ih = N — t) of 5/3 in the region of maximum

514

BLACKWELL et al.

-1120°

0

40°

80°

120° 160° 200° 240° 280°

320°

360°

BOOZER POLOIDAL COORDINATE 3-1120°(b)-960°

5 -640°g -480°a
S -320°4(P80°120° 16<J° 200° 240°
TOROIDAL ANGLE <t>

280°

320°

360°

FIG. 3. Phase of the electrostatic mode versus (a) the magnetic poloidal coordinate, i?,and (b) the toroidal coordinate <t>.

wave-field, where t c^ 1.33. Varying the ion mass confirms that the toroidal
phase velocity « c», while varying t produces the expected changes in bothmode number and frequency.
2.

Optimization of the Magnetic Structure of H-lExtensive calculations made during the process of optimization of the design of the much larger heliac, H-l (under construction, see Table I) have

515

IAEA-CN-47/D-m-6

0.3

(a

o.oi ;•2 0.20.07 0 a=0

g.30
C20

§•
" 0.1

cos

0.05

o.r

flux surface parameter

X (ml

flux

0.1

surface parameter

X (m)FIG. 4. (a) Amplitudes &mn of the Jacobian harmonics versus X (as defined in Fig. 1).
The case illustrated has a = 0.3, y = 1, Bv = 0.1 T.
(b) Ripple amplitude ((B,^ - BmiJ/(Bmax + BmJ) for different values of vertical field and index.
The azimuthally equispaced toroidal coil case (a = 0) is shown for comparison.

0*3

H.A. US

0.05

1.30
Radius Im)

0.1

flux surface parameter

X (m)FIG. 5. Rotational transform (a) and well depth (b) for various values of C.
Except where noted, a = 0.3, Bv = 0.07 and y = 0. M.A. is magnetic axis.

confirmed the suggestion [4] that strong toroidal effects associated with a
low aspect ratio device can be greatly reduced by azimuthal redistributionof the coils. Using the prescription
9

Mi - \)

j = l,2.JV

with a a constant (taken here as 0.3), we have been able to markedly
reduce the global \/R variation in |B|. This has the consequential advantage of reducing both the variation in surface cross-sectional shape withtoroidal angle and the total magnetic ripple. The Jacobian harmonic £3|0
which governs the Pfirsch-Schluter currents due to toroidicity [5] is also
considerably reduced, e.g. by a factor ~ 5 at mid-radius (see Fig. 4a).
The choice of rotational transform per period t/N ~ 0.379 (a "noble
ratio") and low shear (Fig. 5a) conveniently avoids the worst low order

516

BLACKWELL et al.

resonances, particularly those like (3,2) which can be pressure driven directly via £3i2. An additional improvement in magnetic ripple can beachieved if the applied vertical field Bv is made to vary with major radius
as BV(R) oc (RQ/R)"1, and its magnitude adjusted to centre the magnetic surfaces with respect to the minor axis. For an index 7 = 1 and
BV(RQ) = O.IBT, the |BJ ripple on the magnetic axis is reduced (Fig. 4b)from 10% to 4% and, at an outer surface, from 55% to 32%, at the expenseof a reduction in well depth and an increase in the poloidal variation insurface shape. The harmonics of the Jacobian for this case (Fig. 4a) show
the component S3t0, which corresponds to the 1/R variation of 1/B2 over
one helical period, to be everywhere less than 7.5%. The local 1/R variation, which contributes to 5o,i> is more difficult to control as it is linked tothe toroidal curvature (in a current-free region) by V X B = 0. By judicious adjustment of the vertical field index and magnitude, a compromise
can be achieved between these two extremes.
Excitation of the helical control winding gives access to a wide range ofmagnetic configurations: Fig. 5 shows its effect on t and W profiles, once
again demonstrating the possibility of comparing low shear, deep well configurations with moderate shear, antiwell configurations.

3- Stability Studies of Free-Boundary Equilibria
As a complement to the 3-D field-line-tracing studies, we have studied thefinite-/? stability properties of SHEILA in the helically symmetric limit. A
new code, FEQ2.5(FR), which models free-boundary heliac equilibria, hasbeen developed. The fixed-boundary inverse-equilibrium code FEQ2.5 [6]has been combined with an iterative technique using the Green's function,
G, for the helical Grad-Shafranov equation:
V . (KVG) = ^-6(r- ro)8(C - Co)(1)

where C = <f> — hz, his the helical pitch, K = h?/(l + h2r2) and (r,</>, z)

are the usual cylindrical coordinates. One solution for G is

^

\

(2)

- 4 r r 0 f ) lfn(nfcr<)Ji:;(nfcr>)coB[n(C - Co)]n=l

with r< = min(r,r 0 ), r> = max(r,r 0 ).
The code FEQ2.5(FR), which is described in detail in [7], assumes
that the vacuum helical flux function of the external coil system may be
expanded as a Fourier-Bessel series:

IAEA-CN-47/D-III-6

-0.05

517

STABLE

a-2

FIG. 6. Critical (@)for stability against ideal interchange and ballooning modes for helically symmetric, net-current-free SHEILA fixed-boundary (solid lines) and free-boundary (broken lines) equilibria.
The pressure profiles have the form p oc ^ ° . The plasma is stable above the curves and unstable below.

= ai ln(r) + a2r2/2

T

S KI'J,nhr) cos(nC)

(3)

oo

n=l

For a typical helically symmetric heliac, consisting of a helical solenoid
which carries a wholly azimuthal current, which is up-down symmetricand which encloses the z-axis and a number of helical windings, the series of Eq. 3 is certainly valid inside the inner tangent cylinder to thesolenoid, and outside the largest radius, n,, of the helical windings. For
circular cross-section solenoids the series can also be shown to converge
throughout the region interior to the solenoid, but outside r = 7*i, when thedisplacement of the solenoid satisfies certain conditions [8].
It is possible to construct a stability diagram for the SHEILA heliac
using the current in the extra helical coil as a parameter with which to deform the shape of the outer flux surface. This is shown in Fig. 6,where theplasma elongation and indentation increases moving up the vertical axis.
When the plasma boundary is fixed to its vacuum shape, we confirm thehigh critical (/?) to Mercier and ballooning instabilities predicted by others[6,9] with a stronger dependence on the pressure profile than that notedby [6]. The limiting point at zero (ft) corresponds to the appearance (withdecreasing elongation and indentation) of a magnetic hill close to, but not

518

BLACKWELL et al.

exactly at, the magnetic axis. When the plasma boundary is allowed to
move, it shifts away from the z-axis and broadens. The increase in helicalaspect ratio is sufficient to ensure Mercier stability for all those configurations for which there is a transition to instability at finite pressure. In
other words, any zero-pressure configuration with a uniform magnetic wellacross the plasma will remain Mercier-stable as {/?) is increased.
REFERENCES[1]BLACKWELL, B.D., HAMBERGER, S.M., SHARP, L.E., SHI, X.H., Nucl. Fusion 25(1985) 1485.
[2] HARRIS, J.H., CANTRELL, J.L., HENDER, T.C., CARRERAS, B.A., MORRIS, R.N.,
Nucl. Fusion 25 (1985) 623.
[3] BOOZER, A.H., Phys. Fluids 23 (1980) 904.
[4] CARY, J., KOTSCHENREUTHER, M., in Fifth International Workshop on Stellarators (Proc.
IAEA Tech. Comm. Mtg on Plasma Confinement and Heating in Stellarators, Schloss Ringberg,
1984), Vol. 1, CEC, Directorate-General XII — Fusion Programme, Brussels (1984) 291.
[5] REIMAN, A.H., BOOZER, A.H., Phys. Fluids 27 (1984) 2446.
[6] MONTICELLO, D.A., et al., Phys. Fluids 27 (1984) 1248.
[7] GARDNER, H.J., The free boundary equilibrium problem for helically symmetric plasmas,
PhD Thesis, Australian National University, Canberra, 1986.
GARDNER, H.J., DEWAR, R.L., SY, W.N.-C, The Free Boundary Equilibrium Problem for
Helically Symmetric Plasmas, Rep. ANU/TPP-86/02, Australian National University, Canberra(1986); submitted to J. Comput. Phys.
[8] DEWAR, R.L., GARDNER, H.J., The Eigenfunction Expansion for the Helical Solenoid,
Rep. ANU/TPP-86/03, Australian National University, Canberra (1986).
[9] MERKEL, P., NUHRENBERG, J., GRUBER, R., TROYON, F., Nucl. Fusion23 (1983) 1061.

DISCUSSION
F.L. RIBE: I would like to ask three questions. First, can you control theresonance with the helical windings; second, why do you not choose N = 4 insteadof N = 3; and third, what are your plans for heating your device?
S.M. HAMBERGER: The helical winding current controls the position of the
low order resonances: When one is inside the confinement region, the plasma density
falls. Notice that the shear is low where the one-half resonance is important and fairly
high for the one-third resonance.
The answer to your second question is that we prefer N = 3 for several reasons.
Firstly, the larger number of toroidal coils per period reduces magnetic ripple near
the outside and improves the flux surfaces. Secondly, N = 3 provides a more convenient experimental layout. The suggestion that, with N = 3, resonances cannot be
avoided is not correct, provided careful design is used.
Finally, it is intended to use RF heating only, using frequencies in the ICR
region. The actual method has not yet been decided. ECRH will be used if suitable
gyrotrons become available to us.

IAEA-CN-47/D-IV-l-l

HELICITY CONSERVATIONAND ENERGY CONFINEMENT
IN CTX SPHEROMAKS*
B.L. WRIGHT, C.W. BARNES, J.C. FERNANDEZ,
I. HENINS, H.W. HOIDA, T.R. JARBOE, S.O. KNOX,
G.J. MARKLIN, R.M. MAYO1, D.A. PLATTS
Los Alamos National Laboratory,
University of California,
Los Alamos, New Mexico,
United States of America
Abstract
HELICITY CONSERVATION AND ENERGY CONFINEMENT IN CTX SPHEROMAKS.
Results are reported for the CTX spheromak experiment with a flux conserver of 0.67 m radius.
The measured currents in the mesh allow a non-perturbing determination of the equilibrium current profile. Analysis of magnetic data shows that injected magnetic helicity is conserved. A mode of operation
has been encountered in which periodic decreases in the helicity content of sustained spheromaks arecorrelated with the quiescence of current-driven instabilities. Increase of the spheromak radius from
0.40 m to 0.67 m has not led to an increase in energy confinement. Attempts to resolve this issue,
including the use of a conducting core and applied bias fields, are discussed.

1.

INTRODUCTION

In the CTX experiment at Los Alamos, magnetized coaxial
electrodes are used for the formation and sustainment of
spheromak plasmas that are stably confined in an oblate
conducting shell called a flux conserver [1J. The electrodeconfiguration is that of a Marshall gun with a solenoid mounted
inside the inner electrode to provide an axial magnetic field.
Because this applied flux is linked with the injected azimuthal
flux, the gun is a source of magnetic helicity. The source andflux conserver are on a common axis and are separated by a shortcylindrical section with a conducting wall. Plasma contact withsurfaces is reduced and diagnostic access is improved byconstructing the flux conserver from a mesh of 12.7 mm diametercopper bars at an approximately 50 mm spacing. As reported
previously [1,2], the spheromak field configuration can besustained in CTX over many resistive decay times by thecontinuous injection of magnetic helicity from the coaxial
source. This prior work was performed in a 0.40 m radius flux
conserver and resulted in the achievement of a core temperaturein excess of 100 eV [3]. Ve now report on helicity balance andconfinement
results using a larger, 0.67 m radius, flux
conserver.
* Work performed under the auspices of the United States Department of Energy.
1

Department of Nuclear Engineering, Purdue University, West Lafayette, IN, USA.

519

520
2.

WRIGHT et al.
DIAGNOSTIC TECHNIQUESThe principal diagnostics in use on the CTX experiment are:over 35 Rogowski loops that measure currents flowing in the meshof the flux conserver, an 8-chord quadrature interferometer for
midplane density profiles, multipoint Thomson scattering for T gnear the magnetic axis, two steerable gold foil bolometers, a
Lyman-a monitor, a selection of VUV instruments for impurity
concentration studies, two polychromators for impurity ion
Doppler broadening, and a double Langmuir probe to measureconditions at the plasma edge.
An important feature of recent CTX work is our ability to
use Rogowski loop measurements of the toroidal equilibrium
currents flowing in the hoops of the mesh flux conserver toinfer the internal profiles of J and tf [4]. This technique,which eliminates the need for perturbing internal magnetic
probes, is based on fitting the hoop current data to computed
axisymmetric force-free equilibria defined by VxS m Xt, where thescalar X is considered to be a linear function of the normalized
poloidal flux, \J/: X(t) = X[l + a(2i^-l)]. Figure 1 shows thecurrent density profiles of three such equilibria. The fitted
parameter a. determines whether j/B increases (a > 0) or
decreases (a < 0) toward the magnetic axis.
The evolution of a typical CTX spheromak discharge has twophases: the sustainment phase when magnetic helicity is injected
to build up and maintain the spheromak fields, and the resistive
decay phase that begins when the helicity source is turned off.

20

40
Radius (cm)60

FIG. 1. Magnitude of the plasma current density versus radius in the midplane of the 0.67 m radiusflux conserver for three values of the profile parameter a. Solid diamonds indicate the radius of themagnetic axis.

IAEA-CN-47/D-IV-l-l

521

During sustainraent, the enhancement of current on the outer flux
surfaces, which are connected with the coaxial electrodes,
results in an observed value of a in the vicinity of -0.3. The
spheromak poloidal current is typically 5 times the source
current. During decay, the spheromak is isolated from theelectrodes and the lower resistivity of the core plasma near themagnetic axis results in peaking of the current profile. The
measured a becomes increasingly positive, reaching 0.6 to
1.0 toward the end of the decay. Our analysis of the Rogowski
loop data is supported by the appearance of saturated rotating
magnetic distortions under conditions for which the inferredq(ij/) profile predicts the instability of nonresonant internalkink modes [2,4].
3. HELICITY BALANCE AND RELAXATION IN SUSTAINED SPHEROMAKS
A recent analysis of magnetic helicity balance in CTX [5]gives compelling evidence for the existence and importance ofrelaxation processes that minimize magnetic energy on timescales much shorter than the resistive diffusion time scale forloss of the magnetic helicity. The significance of suchprocesses for the formation and evolution of RFP and spheromak
equilibria has long been recognized [6]. Though equilibria thatare close to the fully relaxed state have been observed in many
experiments, a key issue is also whether the relatively rapid
relaxation process is accompanied by any helicity loss. Thisissue has been examined by comparing CTX data with the following
model:at " "^ + 2 V *where K is the helicity of the spheromak equilibrium, T«r is its
characteristic resistive decay time, $ is the applied source
axial magnetic flux, and V is the elecnric potential across the
coaxial source electrodes measuring the rate of generation oflinking azimuthal flux. The term 2 V #gives the rate ofmagnetic helicity injection for our source. Thus the model
tests whether helicity injected by the source is conserved as it
is incorporated into the spheromak equilibrium. One result of
this study is shown in Fig. 2 where the increase in helicity ofthe spheromak equilibrium, AK, after time At = 50 us,is comparedwith the injected helicity, J^ 2V • dt, over a range of valuesof V_ and • . These data represenr nie early phase of sustained
CTX oischarges during which K is growing and resistive loss ofhelicity is unimportant (At « T K ) . They show that, within the± 12% uncertainty of the data, the injected helicity isconserved during the formation of a relaxed spheromak
equilibrium. The full model, with T K included as an unknown,
constant parameter, gives good fits to both sustained anddecaying phases of CTX discharges [5]. The values of TV sodetermined agree with estimates of the plasma resistivity.

522

WRIGHT et al.
l

o.oto

i

1

i

f

i

i

1

i

-

1 ' /

/

/ 1
—
-

0.005

-

ft'
1

n

f

•

•

r

,

0.005

1 ,
0.010

2VQ0adt (Wb2)-/ oF/G. 2. Comparison of the magnetic helicity content of the CTX spheromak with the magnetic helicity
injected by the source after a short time At.

During sustainment, the model is valid in spite of strong
relaxation processes that produce a sustained spheromak field
profile (a = -0.3) departing from the fully relaxed state(a = 0) by only 3% in magnetic energy.
The study of helicity conservation on CTX referred to above
did not identify the specific relaxation mechanism thatdissipates excess injected magnetic energy and maintains a
constant ratio of poloidal and toroidal spheromak. fluxes during
the sustainment phase. This phase is characterized by prominent
oscillations (AB/B ~ 0.1) associated with rotating saturated
n = 1 kink modes. The modes arise because of the unrelaxed
condition of the driven q-profile, with q «s 1 on the magnetic
axis.
Evidence for a connection between n = 1 modes andhelicity injection on CTX has now appeared with theimplementation of circuitry to achieve constant-current
discharges with I t o r as high as 700 kA sustained for up to
2.0 ms. In this mode of operation we have observed cyclic
variations in the plasma toroidal current, as illustrated inFig. 3a. Corresponding variations in the relative amplitude of
the n = 1 oscillations (as seen in %Doi at the wall) are shownin Fig. 3b. We note that although riie helicity injection rateis constant, I t o r and the spheromak helicity decrease during thequiescent intervals (when &B -^ is small) and increase when the
n = 1 oscillations are activfe. These findings support the viewthat n = 1 modes play a role in the assimilation of injected

IAEA-CN-47/D-IV-l-l

523

500 -

Time

(ms)

FIG. 3. (a) Cyclic variations in the toroidal plasma current during sustainment. The rate of helicity
injection was constant within ±20% until the source was turned off at 2.5 ms.
(b) Corresponding variations in the rms level of n = 1 perturbations of the poloidal magnetic fieldat the wall normalized to the equilibrium value.
Both (a) and (b) have been filtered to average over the individual n = 1 oscillations.

helicity by the spheromak. In terms of the helicity balance
model with 100% conservation, the observed variations in K would
imply a spheromak resistivity (as represented by 1 / T R ) that isperiodically increased by a factor sufficiently large that its
time-average value is about twice that measured for decaying
spheromaks (V_ = 0) under the same conditions. Because thisconclusion seems unlikely, we are looking into otherexplanations. For example, it could be that the assumption of alinear relation between X and ^ is not valid during thisprocess, but this possibility alone cannot account for theobserved variation of K. It is therefore likely that thetransfer of helicity from the source to the spheromak isperiodically interrupted by decreases in n = 1 mode activity.
One expects that such a process would be accompanied bynoticeable profile changes. However, as determined from ouranalysis, the parameter a does not show large variations. In
general, a current distribution that is somewhat more peaked
toward the outside is indicated during quiescent periods.
4.

CTX CONFINEMENT

Although encouraging results had been obtained in CTX with
the 0.40 m diameter flux conserver, a 0-D energy balance
analysis [7] showed that ionization of the background fill gas
to replace rapid plasma particle losses (T p e 0.17 ms) was thedominant electron energy loss mechanism of decaying spheromaks.

524

WRIGHT et al.
T A B L E I. C T X S P H E R O M A K P A R A M E T E R S F O R T W O SIZESOF THE FLUX CONSERVERR £ c <m)

O.4O

0.67

R

2.9 ± 0.3

4.9 ± 0.6

p63K(\cC* ^

I t o r <kA)

191 ±

20

471 ± 58

nfi (10 13 cm" 3 )

6.0 ±

1.2

5.5 ± 1.0

j/ne (10- 14 A'm)

2.0 ±

.2

2.1 ± .2Tcore < eV >90 ± 1080 ± 10<P>vol « >

7.0 ±

1.5

1.8 ± 0.3

TB2

(US)

240 ± 60840 ± 100

TE

(US)

24 ±

Alfven < us >

0.19

0.17

s (io4)

2.7

7.3

T

4

22 ± 3

Indeed, low impurity levels had been achieved by rapid particle
pumpout rather than by direct burnthrough. With the aim of
exploring a more relevant regime for confinement studies withhigher T g , T and S, the 0.67 m radius flux conserver was
installed and the helicity source radius was increased from
0.15 m to 0.28 m. Table I shows a comparison of the parameters
achieved for decaying spheromaks at both sizes.
The data
represent averages of a selection of shots chosen for best
simultaneous confinement and temperature. The temperature T C Q r e
is the value of T g obtained from Thomson scattering in the coreregion near the magnetic axis. The other parameters listed
correspond to the time of the Thomson scattering pulse. Valuesof 0 are based1 on the assumption T- = T g . For the Alfve*n time,T>, we use X~ as a characteristic length. The data are fordecaying spheromaks, but T g from Thomson scattering and thelevel of impurity radiation show no significant difference
between sustained and decaying phases. 2Although the magnetic
energy loss time, T B 2 , has increased as R (indicating no change
in average plasma resistivity), no significant change has
occurred in the plasma energy loss time, Tg, and the achievable
plasma 0 has correspondingly decreased. The achievement ofrecord spheromak currents (Ij-or = \fl&) an<* of values of spaceaveraged j/n
as high as 3 x 10"
A'm (sufficient to burnthrough low-Z impurities) has not succeeded in raising T e

IAEA-CN-47/D-IV-l-l

o.o

1

525

2

Time

(ms)

FIG. 4. Toroidal plasma current waveforms for a high-current mode of CTX operation.

significantly. Examples of high current CTX operation are shovnin Fig. 4. Because T has increased with size, ve may be facedwith a new energy loss process involving strong thermal
conduction or convection.
In order to reach such a conclusion, it is first necessary
to characterize energy losses from impurity radiation. A
steerable gold foil bolometer was initially used to look, fromthe source end of the experiment along various paths. It showedthat there exists a large region, including the magnetic axis,
from which, during the decaying phase, the radiated energy
density can be as low as 20% of the local initial magnetic
energy density at the start of the decay. However, relaxation
dynamics may be transferring magnetic energy outward without
local dissipation. Ve have recently extended the range of ourbolometer scans (with a second steerable instrument) to include
the outer flux surfaces near the geometric axis so that the
total radiative loss can be determined. Though the conditions
obtained during this study have not been our best in terms of
decaying spheromak. lifetime, we find that, during the decaying
phase, an upper bound for the total radiated energy is 45X ofthe total available magnetic energy. Our analysis is verified
by the observation of nearly complete conversion of field energy
to radiation under conditions of high impurity concentration(e.g. just after a vacuum opening or with the introduction oflarge amounts of argon). The observed general trend, both forsustained and decaying phases, is a reduction in the radiatedenergy fraction with increased time-averaged j/n. However, we
recognize that impurity radiation can dominate the global
dynamics of spheromak. evolution through its local effects on theplasma resistivity even though the net radiated power is small.
Ion heating above the classical rate may also be significant in
the energy balance. Doppler broadening measurements of C-III,
0-IV
and 0-V radiation show ion temperatures of theseWRIGHT et al.

526
—I

I

1

1

1

1

1

1—1—1—1—1—

1.4%

-

1.2

-

1.0

/

<£>

0.8
0.6
x

—

0.4

-

-

0.2 -

n

'

'
-0.2

I

•

I

1

0.2

1

0.4

1

1

0.6

<

•

1

OB

a
FIG. 5. Volume-averaged plasma beta at marginal stability (Mercier criterion) versus the CTX profile
parameter a.

impurities that are 5 x T g during sustainment and 2 x T during
the decay phase.
We are looking for a current-driven
instability (e.g. the electrostatic ion cyclotron instability)
which could produce these temperature differences.
Because of the low shear of spheromak equilibria, thepredicted threshold in 0 for the onset of pressure-driven modes
is low. Figure 5 shows the result of recent calculations using
the Mercier criterion for the marginal stability of CTX
equilibria described by the parameter a. It appears likely thatthese ideal modes are active in all spheromak discharges. In
order to see whether pressure-driven modes were determining Tg,
we attempted to raise the critical 0 for instability bymodifying the q-profile. We inserted a 0.15 m radius copper
cylinder, with a hemispherical cap, along the geometric axis of
CTX. The resultant configuration was like that used at Osaka
for the stabilization of disruptive instabilities on CTCC-1 [8].
Figure 6 shows theoretically how the volume-averaged (3 at the
instability threshold is raised with increasing insertion, L.
At L = 0.50 m in CTX, the axial current measured in the cylinder
was only 5X of the spheromak poloidal current. However, evenwith peak j/ng as high as 3 x 10" A"m, the electron
temperature of decaying spheromaks as measured by Thomson
scattering was lower than before. Insertion of the cylinder
also increased the radiated energy density near the magnetic
axis to 50% of the local magnetic energy density. The global

IAEA-CN-47/D-IV-l-l

527

FIG. 6. Volume-averaged plasma beta versus the insertion length of a conducting cylinder of
0.15 m radius in CTXfor three values of a.

energy fraction radiated during decay was 10%. Thus, changes in
the contribution of impurity radiation to the energy balance may
have obscured the effects of modifying the q-profile.
5.

STUDIES WITH APPLIED BIAS FIELD

During CTX discharges the flux conserver acts as the plasma
limiter. In addition, as magnetic flux is lost through thecopper hoops because of their resistivity, an increasing
fraction of the spheromak's magnetic field passes outside themesh boundary [7]. As a result, the edge plasma is cold, with
T = 5 to 10 eV, as measured by the Langmuir probe. To vary theedge conditions and thus study their effect on confinement,
coils located outside the CTX vacuum tank were used to apply an
external bias field, parallel to the spheromak geometricaxis [9]. With the sign of the applied field such as to push
the spheromak flux surfaces inward, the plasma density wasincreased and Lyman-a emission was reduced. A "global" particle
confinement time inferred from the ratio of these chord-averaged
measurements thus increased with the application of bias field.
It decreased for bias of the opposite sign. These results show
the sensitivity of particle confinement to separatrix geometry.
The effects of bias field on global energy confinement were
found to be more complex. Under some conditions with appliedbias, a quiescent phase occurred during spheromak decay
accompanied by reduced current flow to the flux conserver andincreased magnetic field decay time. The inferred increase inenergy confinement time from this reduction in edge effects
could be as large as a factor of 2. We are now testing theeffectiveness of a strongly mirrored bias field using a pair of

528

WRIGHT et al.

coils inside the CTX tank. This configuration can more
completely disengage the equilibrium fields of decaying
spheromaks from the flux conserver, but, as observed on CTCC-1[10], it also lovers the applied flux at which the spheromak
undergoes a 180° rotation of its axis during formation.

REFERENCES[1] JARBOE, T.R., et al., Phys. Rev. Lett. 51 (1983) 39.
[2] JARBOE, T.R., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc.
10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 501.
[3] JARBOE, T.R., et al., Phys. Fluids 27 (1984) 13.
[4] KNOX, S.O., et al., Phys. Rev. Lett. 56 (1986) 842.
[5] BARNES, C.W., et al., Phys. Fluids 29 (1986) 3415.
[6] TAYLOR, J.B., Rev. Mod. Phys. 58 (1986) 741.
[7] BARNES, C.W., et al., Nucl. Fusion 25 (1985) 1657.
[8] NAGATA, N., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc.
10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 655.
[9] BARNES, C.W., et al., Phys. Fluids 28 (1985) 3443.
[10] SATOMI, N., et al., Paper IAEA-CN-47/D-IV-1-2, these Proceedings, Vol. 2.

IAEA-CN-47/D-IV-1-2

CONFINEMENT EXPERIMENTS AND SIMULATIONON THE CTCC-I SPHEROMAK
N. SATOMI, S. GOTO, Y. HONDA, Y. KATO,
M. NAGATA, M. NISHIKAWA, A. OZAKI,T. TAKAISHI, T. UYAMA, K. WATANABE
Faculty of Engineering,Osaka University,
Suita, Osaka
T. SATO, T. HAYASHI
Institute for Fusion Theory,
Hiroshima University,
Higashisenda-cho, Naka-ku, HiroshimaJapan

Abstract
CONFINEMENT EXPERIMENTS AND SIMULATION ON THE CTCC-I SPHEROMAK.
The CTCC-I spheromak is confined in a metal flux conserver so that external global(tilt and/or shift) instabilities can be prevented. However, the plasma in close contact withthe solid wall, hence, the plasma-wall interaction, which is a drawback of the flux conserver
method, directly affects the confinement. The formation of a detached spheromak is studied
both experimentally and numerically to find a method of improving the present spheromak
confinement. The internal instabilities, especially the tilting instability, of the detachedspheromak are studied by a full 3-D MHD simulation code. The experiment of spheromak
detachment from the solid wall is attempted in two ways, by applying the external magnetic
field through conductive and through resistive walls. Numerically, a fairly stable detachedconfiguration for the tilting instability is found for L/a = 1.0, where L is the height of theflux conserver and a is the radius. No tilting mode is excited, even if the external flux applied
is more than 60% of the internal flux. Experimentally, the formation of the detachedspheromak is demonstrated, although still in the initial phase.

1.

INTRODUCTION

Basically there are two experimental approaches in spheromak research.
One is to confine a spheromak plasma by an external field pressure (verticalfield), notably the Princeton S-l, and the other by a conducting wall, notably
the Osaka CTCC-I and the LANL CTX. The former spheromak has an isolated
space between the separatrix and the rigid wall while it is always globally (tiltand/or shift) unstable. In contrast, the latter can be globally stable while theplasma is in close contact with the rigid wall. From the standpoint of MHD
confinement, globally unstable configurations do not satisfy the necessary

529

530

SATOMI et al.

condition for the confinement vessel. From the standpoint of particle confinement, however, the closeness of plasma and the wall is, in several ways, detrimental to confinement.
The Osaka group has, for many years, been conducting extensive research
into the flux conserver (FC) confined spheromak. The main results ofprevious work are described in the literature [1,2]. The experiments canbe roughly divided into three phases: In the first phase, efforts are primarily
focused on the creation of a well shaped, long lived spheromak by using, forexample, titanium coating, spheroidal vessel and highly conductive material.
The typical lifetime is 1 ms (the present record is 1.3 ms), withn = (3-5) X 1013 cm" 3 and Te = 20-30 eV. The second phase is moreclosely related to an understanding of physical processes, namely, the internal
MHD instability and associated MHD relaxation.
With these achievements, we attempt to detach the plasma (separatrix)
from the wall so that the plasma-wall interaction, which is a real drawback of
the flux conserver method, can be reduced. The attempt is carried out bothexperimentally and numerically. First, by means of a 3-D MHD simulation, theinternal MHD instabilities including the n = 1 (tilt), n = 2 and n = 3 instabilities,
are studied in detail. Then, a simulation model is developed which is capableof studying the detachment of the separatrix by externally applying a magnetic
flux through the FC wall. By this model the internal instabilities, in particular
the tilting instability, of the detached spheromak are studied.
Secondly, experiments of spheromak detachment from the FC wall areattempted in two ways: The first attempt is to apply an external flux througha wall of relatively high conductivity from external coils surrounding the flux
conserver before the co-axial gun is fired. The second method is to apply an
external flux through a wall of relatively low conductivity, together with thefiring of the co-axial gun.
In the following section, we describe some typical experimental data which
may show the difference in lifetime, depending on the vessel conditions and theexperimental devices used for the present detachment experiment. In Section 3,
the simulation model and results are described, and in Section 4 the present
status of the experimental results is presented. Finally, conclusions are given
in Section 5.

2.

DESCRIPTION OF CTCC-I SPHEROMAK FEATURES AND
DETACHED MODE EXPERIMENTIn the CTCC-I experiments, the spheromaks are produced by a magnetizedco-axial plasma gun and confined in a metal FC. Typical poloidal magnetic fieldmeasured at the symmetry axis is 2 kG. Assuming the forcefree minimum energy
state, the total toroidal and poloidal currents are estimated to be 90 kA andIAEA-CN-47/D-IV-1-2

531

(a)

1.0

as
0
n

Q5

1

To

t(ms)
FIG. 1. Temporal evolution of magnetic field in CTCC-I spheromak obtained with various
flux conservers.

140 kA, respectively. The initial plasma produced without inpurity control(titanium coating on the FC inner wall) is radiation dominated, and the plasma
exhibits a quiet exponential decay of 0.2 ms.
The cylindrical FC (1.5 mm thick copper) has been used from the start of ourspheromak experiment. By using this FC, the maximum configuration lifetimeof 0.8 ms was attained when the impurity control, the titanium coating, andthe central conductor were applied as is shown in Fig. 1 (a). The effects of FC
geometry, shape and wall thickness on plasma confinement were studied. The
FCs shown in Figs 1 (b) and 1 (c) both have oblate spheroidal shapes; the first
FC consists of 1.5 mm thick copper, and the second FCof 15 mm thick aluminium.
As is shown in Fig. 1 (b), plasma confinement was improved by using the spheroidal
FC to attain a maximum lifetime of 1.0 ms; this indicates that the vessel shapeplays an important role in the confinement. The wall thickness is also thought tobe significant for the confinement since the magnetic field lines in the plasma
would escape through an FC wall of finite conductivity, i.e. flux loss would take

532

SATOMI et al.

Entrance Region( stainless"sl«.M

Entrance Region
Flux Conserver(copper UV5)

Flux Conserver/ (Stainless Steel t=a5)

External Coll

FIG. 2. Flux conservers used in detached mode experiment.

place during long confinement. This was confirmed in the experiment whichused a thick wall FC, where the lifetime was extended up to 1.3 ms as is shownin Fig. 1 (c). It was in the long lived CTCC-I spheromak, a stepwise MHD
activity of the magnetic field in association with the internal MHD instability
during resistive plasma decay 11,2] was observed.
To study the detached spheromak, two types of FC with external coil asshown in Fig. 2 are used. The first type is made of highly conductive metal(1.5 mm thick copper); the major and minor semi-axes are 0.4 m and 0.24 m
long. As external coils, fourteen copper conductors (8 mm diameter) connected
in series are installed on the FC, spaced 3 cm from it by ceramic insulators. By
using this FC, thu spheromak is formed in the presence of an external bias flux
so as to detach the separatrix. The second type consists of 0.5 mm thickstainless steel, where the major and minor semi-axes are 0.35 m and 0.21 m long.
In this case, eight conductors (10 mm diameter) connected in series are used asexternal coils. They are supported on the frame surrounding the FC. The risetime of the coil current is about 60 MS, and the decay time can be varied in therange of 0.5 to 2.0 ms. The maximum coil current is 10 kA. In the second
experimental scheme, a pulsed vertical field is applied through the resistive FC
to detach the separatrix, after a spheromak has been produced in the FC without
external field.

533

IAEA-CN-47/D-IV-1-2

3.

SIMULATION STUDY OF DETACHED SPHEROMAK
FOR MHD INSTABILITIESThe principal aim of the present simulation study is to provide guidance
to planning and designing the detached mode experiment.
Before carrying out the detached analysis, let us examine the internal MHD
instabilities of spheromaks in an FC. To extract the general features of the
problem as thoroughly as possible, we adopt a simple geometry for the FC. The
FC is assumed to be drum shaped, with radius a and height L. The spheromak
equilibria to be studied are obtained by solving the Grad-Shafranov equations,where the toroidal current and the plasma pressure are assumed to be proportional to t/A (\p is the poloidal flux) and 0, respectively, and the total toroidal
current is fixed to a certain value. Thus, the initial equilibria are represented
by two parameters only, L/a and X. Plugging the equilibria obtained in this way
into the full 3-D MHD simulation code as initial conditions [3], we shall study
the behaviour of the MHD instabilities.
The growth rates of the three most dangerous modes, i.e. m/n = 1 / 1 , 1/2
and 1/3, are examined as functions of L/a and X. The general tendency for allthese modes is to become more unstable for larger L/a and larger X. There is a
threshold for L/a below which a mode becomes stable. Since we are here mainly
concerned with the tilt mode (m/n = 1/1), we show the growth rates of the 1/1
mode for three X values (X = 1.5, 2.0 and 3.0) as functions of L/a in Fig. 3.
Although not shown here, all modes are stable for L/a = 1.0. As can be seenfrom this figure, the threshold values lie in the range L/a = 1.0 to 1.3 for

1.0
UJ
Q

o

O A=1.5
A A=2.0
D A=2.5

u.

o

D
0.5

A

§
a
-1_O0.0

1.0

2.0

3.0

L/
FIG. 3. Growth rates of tilting mode, m/n = 1/-1, for X = 1.5, 2.0, 2.0, 2.5 (I<* \pK) as afunction of elongation, L/a, where L and a are height and radius of flux conserver.

534

SATOMI et al.
0.3• (1,-1) I L/a=1.3
A (1,2) f
O (1,-1) L/a=1.0

0.2

O

g 0.1
• 0.0

_L_O_1

1
0.5

Lo-J-

FIG. 4. Growth rate of tilting mode and helical kink mode as a function of external flux
normalized by internal plasma flux.

X — 1.5 to 3.0. A discrepancy is observed between our numerical and theanalytical result [4], which may, however, be attributed to the difference inthe equilibria. The equilibrium in [4] is the Bessel function model, whileours assumes ^ .
With this result in mind, we proceed to the detached mode. As an initialcondition, we use an equilibrium obtained by solving the Grad-Shafranov
equations. On choosing the equilibrium, we dynamically supply an external
flux through the FC wall. In the actual simulation run, the magnitude of the
on-wall magnetic field parallel to the wall is changed with time whereby themagnetic pressure on the wall pushes the plasma inwards and, subsequently,
the external field is brought inside the FC. When a certain amount of external
flux has been brought inside and the separatrix is detached from the wall, we
impose a tilting perturbation on the system and make the 3-D simulation
code run.
The growth rate of the tilt mode as a function of the external flux relative
to the spheromak flux is plotted in Fig. 4. The filled circles correspond to the
X = 1.5 and L/a = 1.3 case, and the dot-in-open circles refer to the X = 1.5 andL/a = 1.0 case. The filled circles indicate that the growth rate increases drastically as the external flux increases. Considering that the original spheromak istilt stable, it is evident that the external flux destabilizes the tilt mode. For
comparison, the growth rate of the 1/2 mode is shown by the triangles, showing
a slightly increasing tendency as the external flux increases, which is, however,substantially smaller than for the tilt mode. The slight increase may be attributed
to the change in the spheromak configuration due to the intrusion of external
flux. Incidentally, the 1/3 mode was always stable in this example. In Fig. 5 weshow an example of a spheromak that is non-linear]y deformed by n = 1 andn = 2 instabilities.

IAEA-CN-47/D-IV-1-2

535

FIG. 5. Magnetic field line trajectories in initial and developed (n = 1 and n= 2 mode)
spheromak phases.

Interestingly enough, the tilt mode was not excited in the L/a = 1.0 case,even if the intrusion of external flux exceeded 60% of the internal flux. Fromthese simulation results we can, therefore, conclude that certainly the external
flux contributes to destabilization but that there exists a realistic, detachedmode condition that can absolutely prevent the tilting instability. The conditionis that the shape of the FC should be as oblate as possible, most favourably
L/a = 1.0. The encouraging results of this simulation study led to planning thedetached mode experiment. This will be discussed in the following section.

4.

EXPERIMENTAL RESULTS

4.1. Bias flux experiment
The experiments presented here were carried out by the bias flux mode,
i.e. the spheromaks were formed in the presence of bias flux [5, 6]. The spheromak
plasma becomes tilt unstable within 0.1 ms if the bias flux exceeds a certain criticalvalue, 0.75 mWb, as is shown in Fig. 6 (a). Here, the bias flux inside the FC isestimated by using the measured bias magnetic field without spheromak formation.
The magnetic field shows rapid decay after the onset of tilting, and during thisdecay the density increases slightly, thus indicating increased plasma-wall interaction. The O V (630 A) line radiation is only observed in the early phase while thisradiation increases gradually in the period of 0.2 to 0.3 ms in the case of thelong lived spheromak obtained without bias flux. When external flux of theamount of this critical bias flux is applied, the spheromak suffers from tilting
instabilities in 30% of all the discharge shots. When a bias flux of 3 mWb isused, tilting is always observed. The tilting growth rate depends on the536

SATOMI et al.

it=0.75 mWb

Q8i

r=35cm

mWb

0.81
0.5

=0.13 mWb

N
03

=0.3 mWb

t=0.5 mWb

0.5t(tns)(b)
FIG. 6. Typical data in bias flux experiment:(a) Time evolutions of magnetic field, O V line radiation and density without central conductor;(b) time evolutions of magnetic field for various bias fluxes with centra! conductor (d = 18 cm).

02
04
0.6
Ofl
External Field UymWb)

FIG. 7. Internal plasma flux at termination onset time of discharge
as a function of external flux.

IAEA-CN-47/D-IV-1-2

537

amount of bias flux; for a bias flux of 0.75 mWb, it takes 50 to 60 jus to complete
the tilting. This feature is, at least qualitatively, in agreement with the previous
numerical result.
To stabilize this tilting as observed in the formation phase, we used a central
conductor. Figure 6(b) shows the results of the experiment with an 18 cm diameter
central conductor for various applied bias fluxes: 0.13, 0.3 and 0.5 mWb. In thesmall bias flux case (0.13 mWb), the plasma is not, within the reproducibility ofthe discharge, affected by the external field. These results indicate that the initial
tilting is not observed for an external field of less than 0.5 mWb, but terminationof the discharge is always observed and the time after which termination sets inbecomes shorter as the amount of the bias flux increases. When a bias flux higher
than 0.5 mWb was applied, the plasma could not be injected into the FC, and nospheromak formation took place, because of the small annular space between thecentral conductor and the entrance tube of the FC. The internal flux of the plasma
at the time when the termination of the discharge is observed is shown in Fig. 7
as a function of the amount of applied bias flux. This indicates that the flux
ratio at the onset time of the termination is about 10%.
4.2. Pulsed vertical field experiment
In this experiment, an external vertical field with a rise-time of about 60 us
was applied through the resistive wall after gun firing. The typical delay timeof the external coil discharge described here is 5 ^s. The experiments presented
here were carried out without titanium coating. So, the plasma is radiation
dominated, and the lifetime of the configuration is 100 to 150 us in the present
experiment, while the lifetime of the previous plasma in the experiment withcylindrical FC was, typically, 200 to 300 (is.
Figure 8 shows typical poloidal and toroidal magnetic fields, measured
simultaneously at five radial positions in the midplane, without and withexternal field. The radial position of the magnetic axis is estimated to ber = 22 cm if force-free minimum energy is assumed. In Fig. 8 (a), the poloidal
magnetic field at r = 23 cm changes from positive to negative, gradually in time,i.e. the magnetic axis of this plasma shifts outwards, probably because of flux
loss through the resistive wall. In most of the discharges, the poloidal magnetic
field at r = 23 cm is rather large throughout the discharge; thus, the spheromak
configuration in the resistive FC is far away from the force-free minimum energy
state. The global behaviour of the measured magnetic field does not change
markedly even if an external field is applied. It is, however, remarkable that,when the external field is applied, the toroidal magnetic field near the FC, forexample at r = 32 cm, becomes very small as is shown in Fig. 8(b). The B tsignal at r = 35 cm is an error field caused by the effect of the observation holeon the FC. Figure 9 shows the toroidal magnetic field at r = 32 cm and theratio of the toroidal fields at two positions, r = 32 cm and r = 23 cm, as a

538

SATOMI et al.

I
"a
c
in

o

in

u
«

o

in

o

in

s

I

o

o

E
I/)

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1
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I

I

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I

I <

t-t-

ino

ino

mo

w>

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2

(9>0 a

>-t-«

in

o

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if)

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.I

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IAEA-CN-47/D-IV-1-2

1

539

2
3
EF VOLTAGE (kV)

FIG. 9. Toroidal field Bt(r = 32 cm) and toroidal field ratio Bto(r = 32 cmj/Bt-Jr = 23 cm)
as a function of charging voltage of external field coil source.

function of the charging voltage of the power source for the external field.
This indicates that the separatrix of the spheromak is detached by about 2
to 3 cm from the FC wall when a rather high external field is applied. Thereis no evidence of tilting in the present detached spheromak.
5.

CONCLUSIONSThe numerical simulations have shown that the external magnetic flux
brought inside the FC acts to destabilize the tilt mode, but that if the shape of
the FC is sufficiently oblate (e.g. L/a » 1.0), we can realize a tilt stable, welldetached spheromak. Although the results reached so far are still only preliminary, the experiments are in good agreement with the simulations, for example,
as far as the tendency of tilt destabilization in proportion to the external flux
is concerned.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]

NAGATA, M., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984
(Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 655.
SATOMI, N., et al., in Advances in Compact Torus Research (Proc. Tech. Comm. Mtg.
Sydney, 1985) (1985) 203.
HAYASHI, T., SATO, T., Phys. Fluids 28 (1985)3654.
FIN, J.M., MANHEIMER, W.M., Phys. Fluids 24 (1981) 1336;
BONDESON, A., et al., Phys. Fluids 24 (1981) 1682.
BARNES, C.W., et al., in Compact Toroids (Proc. 7th Int. Conf. Santa Fe, NM, 1985),
Los Alamos National Lab., to be published.
SATOMI, N., et al., in Compact Toroids (Proc. US-Japan Workshop, Seattle, 1985)(1985)41.

IAEA-CN-47/D-IV-2-1

FIELD REVERSED CONFIGURATION TRANSPORT
Theory and measurements offlux, energy, and particle lifetimes*
A.L. HOFFMAN, J.T. SLOUGH, L.C. STEINHAUER
Spectra Technology, Inc.,
Bellevue, Washington
N.A. KRALL, S. HAMASAKIJAYCOR,San Diego, California
United States of America
Abstract
FIELD REVERSED CONFIGURATION TRANSPORT: THEORY AND MEASUREMENTS OF
FLUX, ENERGY, AND PARTICLE LIFETIMES.
Experimental flux, energy, and particle lifetimes are reported for a wide range of conditions inthe slow risetime TRX-2 field reversed theta pinch. The measurements have been compared with theoretical estimates based on low frequency drift waves and trapped particle waves in the peculiar FRC
geometry. These estimates give reasonable agreement with the measured lifetimes and, for the first time,model the anomalous flux loss rates.

1.

INTRODUCTION

The first detailed description of field reversed configuration (FRC) transport
was given by Hamasaki and Krall [1], and subsequently by Tuszewski and
Linford [2], who modelled particle transport across the separatrix and subsequent
flow along open field lines. The model was based on classical transport plus localized
lower hybrid drift (LHD) microturbulence, and it successfully predicted the R2/pi0
particle lifetime scaling seen between the FRX-B [3] and FRX-C [4] experiments(R is the radius of the magnetic field null and pi0 is the ion gyroradius in the external field Be; the ion temperature is assumed to be uniform throughout the FRC).
Localized LHD microturbulence is weak near the field null, however, and the LHDbased model cannot account for the fact that the flux loss rates are more rapid thanthe classical ones [5]. Comparisons with later experiments also showed deviations
from the LHD-predicted scaling dependences, even for particle lifetimes [4, 6],
In this paper we describe flux, energy and particle lifetime measurements in the
TRX series of experiments, particularly in TRX-2, which is a slow, multiturn field* Work supported by the United States Department of Energy.

541

HOFFMAN et ai.

542

2 u

-e

19 "

!f
s s

s
2
g
8
O

§
U

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m w i oOO O O O v i O Q Q ' r t V lo * o o < n r - > n Q Q c ^ f N

inin

intn»n

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*l

o « n o»-—CMr ^ ° ? r - o 7 o o < s < s « n
VO

^

H

^

^

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^

-

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^

-

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I
(X)
00
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IAEA-CN-47/D-IV-2-1

543

reversed theta pinch utilizing programmed formation [7, 8]. TRX-2 has been operated over a wide range of experimental conditions (fill pressures varying from 3 to
20 mtorr D2 and magnetic fields varying from 5 to 15 kG) and provides scaling
information for ion temperatures varying between 75 eV and 800 eV, and electron
densities varying from 2 X 1015 cm" 3 to 8 X 1015 cm" 3 . The measured lifetime
scaling dependences show the same deviations from LHD-based calculations as noted
previously. The new computations which are presented here and which are based on
low frequency drift and trapped particle waves give better agreement with themeasured lifetimes.

2.

EXPERIMENTAL METHODOLOGY

TRX-2 is a multiturn version of the previous TRX-1 device [9], with coils ofradius rc = 12 cm. Trigger and plug coils are used to achieve programmed formation [7, 10], and microwave pre-preionization is utilized, together with a z-discharge,
to permit operation at fill pressures as low as 3 mtorr. The results presented here are
only for 'well formed' FRCs, by which we mean symmetrical and untorn FRCs
(monotonically decreasing diamagnetism from the centre to the ends) well separated
from the end mirrors. Even with this selection, there was a scatter of about a factorof two in the lifetime data for shots run under nominally identical conditions [11].
The experimental lifetimes were determined from the decay rates of </>p, Ep
and N. An exponential fit was made for a time period at least as long as 0.4 of themeasured lifetime. Details of the data reduction procedure are given in Ref. [11].

3.

LIFETIME MEASUREMENTS

Most of the lifetime measurements on TRX-2 were obtained for one of theeleven conditions listed in Table I. Each point represents the average over at least
three shots. The changes in separatrix radius rs for shots with the same fill pressurep0 and external magnetic field reflect either different initial bias fields or different
amounts of flux retention. 4 is the full separatrix length and T, is the total temperature obtained from pressure balance and the measured electron density and magnetic
field. On the basis of previous FRC measurements, it was assumed that Te equalled
Ti for total temperatures below 200 eV and that Te scaled up to 150 eV as the total
temperature increased.
Several experimental averages from TRX-1 plus one condition from FRX-B andtwo conditions from FRX-C are also listed in Table I [4]. We have examined scaling
dependences for the TRX data and found correlations with rs and p io . To a first
approximation, all lifetime data appeared to scale with the parameter rjp'£. Thisscaling is shown in Figs 1-3 for the points listed in Table I. These averaged data
generally show a closer fit than the individual TRX measurements.

HOFFMAN et at.

544
500

100-

?
50-

• TRX-1o TRX-2* FRX-B* FRX-C

20-

10

—i

1—

5

10

FIG. 1. Flux lifetime measurements.

500

100• TRX-1
o TRX-2
a FRX-B* FRX-C
50-

20-

10

10

FIG. 2. Energy lifetime measurements.

545

IAEA-CN-47/D-IV-2-1
500

100D TRX-1

z

50-

o
TRX-2
a
FRX-B
* FRX-0

20-

10
10

FIG. 3. Particle lifetime measurements.

The flux lifetime measurements indicate best the average FRC cross-field-line
transport since they do not depend directly on open field line effects or bulk lossesof energy. The data for the small machines (TRX and FRX-B) show very strong
rs/pjo dependence,= 0.012

where the dimensions are given in centimetres. On the basis of a constant plasma
resistivity 77 and a calculated flux lifetime of the form T^RR = W ^ a 2 ( a is acharacteristic length equal to rs/4) [5], the effective flux diffusion coefficient can be
written as

43

For typical values of a/pio = 5, the flux diffusion coefficient has a value of about
4m2-s-'.

546

HOFFMAN et al.

The flux lifetime measurements from the larger FRX-C machine show somedeviation from the TRX scaling. The 5 mtorr data point is above the TRX scaling
curve; this may be indicative of a flattening of the magnetic field profile near the field
null, which is expected to occur when the flux lifetime exceeds the particle lifetime.
A low electron temperature could contribute to the 20 mtorr data point being below
the TRX scaling.
The energy lifetime T ^ is seen to scale approximately as (rjp'fy27. Theenergy lifetime is determined by several loss processes, including particle losses, ion
and electron thermal losses, and radiation. The TRX energy lifetime scaling shownin Fig. 2 is weaker than the corresponding particle lifetime scaling shown in Fig. 3.
This may be indicative of radiative losses becoming relatively more important as the
FRC size increases and the relative thermal and particle loss rates are lowered.
The TRX particle lifetime data show the same scaling tendency as the flux lifetime, although there is considerably more scatter. However, the scaling from therc = 12.5 cm FRX-B machine to the rc = 25 cm FRX-C machine indicates an additional dependence on rc. This is indicative of the fact that xs = rs/rc is, independentof rs alone, an important parameter for particle loss. Particle loss rates are strongly
influenced by the density gradients near the separatrix, and hence by the density gradients on xs. The rs portion of the TRX scaling shown in Fig. 3 could be written
equally well as xsL6rs, which would better fit the FRX-B and FRX-C measurements
where xs was approximately 0.4. On TRX, xs was varied from 0.3 to about 0.6.
4.

LOW FREQUENCY DRIFT WAVE THEORY

The experimental measurements indicate a resistivity dependence on the effective drift parameter (vE/v; <x pja), but the detailed form of this dependence isinconsistent with localized LHD predictions. Thus, low frequency electromagnetic
drift waves and trapped particle waves have been investigated theoretically with theaim of finding a possible explanation for the observed FRC transport behaviour.

TABLE II. COMPARISON OF PARAMETERS FOR FRCs AND TOKAMAKS
FRX-C(5 mtorr)

FRX-C(20 mtorr)

TRX-1

TFTR/JET

82 cm

10 cm

10 cm

6.2 cm0.45 cm

0.22 cm

0.33 cm

0.1 cm0 (r = 0)

oo

oo

oo

0.2 x 10~2

/3 (r =a p )

0.54

0.66

0.66

0.17 x 10 " 2

3/kai

100/ka,-15/ka,

0.07/ka,Pi

IAEA-CN-47/D-IV-2-1

547
^ .

Region 1
Region 2\

Region 1

Region 3

FIG. 4. Stability regions of field reversed configurations.

Low frequency stability has been explored for a variety of systems, and basic
equations have been derived for a wide range of parameters and geometry [12, 13].
These equations have been adapted in order to derive a dispersion relation for low
frequency waves in a slab plasma with parameters relevant to current FRCs. There
are substantial differences between this case and previous applications, including thewide variation of internal beta in the FRC (from < 1 at the separatrix to infinite at
the magnetic null) as well as the similarity of the characteristic distances along andacross the magnetic field, which can make w/kj comparable to or smaller than theion thermal velocity. Besides the geometry, key parameters such as 0, ap/pi (plasma
size divided by local ion gyroradius) and collisionality determine the parameter rangeof theoretical studies. The parameters of the FRC are compared with those of more
widely studied tokamak experiments in Table II, where ap is the plasma radius (rsfor FRCs), Pe,. is the electron-electron collision frequency (detrapping frequency),
and co* is the diamagnetic drift frequency.
Solution of the dispersion relation shows that three separate regions of instability exist, which are indicated in Fig. 4. A plasma with straight magnetic field lines
and a length (parallel to Bo) much greater than 4 = (dn/ndx)"1 is similar to thestraight region near the FRC separatrix (region 1 in Fig. 4) and can be unstable toa high beta, electromagnetic version of the familiar drift dissipative instabilities, with
cor = -co*, COJ = Wj8, and 5B > 8E, depending on the length of the region. A
plasma with magnetic mirrors and a length (along Bo) not much greater than 4 issimilar to the curved end of the FRC (region 2) and can be unstable to an electrostatic
trapped particle instability, cor = o>j = CJ*(?'/CO*)'A. The interior high beta ( > 1) end
region (region 3) can be unstable to an electromagnetic trapped particle mode with
cor = coL = 0.2cc*/k2pf and SB > 5E. This mode is distinct from any mode studiedin previous applications. Because of the electromagnetic nature of the internal trapped
particle mode, differential ion-electron energy loss is possible. Although the slab
model lacks some specific features of FRC geometry, these results give strong indications of high beta trapped particle instabilities in FRCs.

548

HOFFMAN et al.

Estimates have been made of the transport due to these instabilities. Two ofthem have 6B > 5E and are dominated by stochastic effects, which can be describedusing the usual theories for magnetic field line diffusion [14]. The mean free path
is assumed to be longer than L s kf ! and the saturated field is small, such thatb = 6B/B is less than L divided by the perpendicular correlation distance. For quasilinear flow (nvr) = 5 (nv,) 5B/B, the anomalous resistivity is tj = (vr) B/cJ, andwith the usual saturation fluctuation levels the magnitude and scaling of T 0 , T^,
and rN can be estimated. A fairly crude point model analysis, based only on low
frequency drift waves, gives the scaling as TEp = a3b2/T3/2, T$ = TN — a2B/T,
where a weak dependence on L is suppressed.
The above scaling is normalized to the conditions for TRX-2 (12 mtorr, 10 kG,
rs = 4.7 cm) and the scaling results are compared in Table I with the experimental
measurements for flux and energy lifetimes. The proper transport in regions 1, 2
and 3 is also included in a pseudo 2-D transport model, and the calculated flux lifetimes are indicated in Table I for several conditions.

5.

SUMMARY AND CONCLUSIONSThe theoretical predictions for low frequency drift waves agree well with allmeasurements, except for the 5 mtorr flux lifetime and the 20 mtorr energy lifetimeof FRX-C. However, as mentioned previously, the 20 mtorr FRX-C plasma is relatively cold, and classical loss processes plus radiation may be dominant. The 5 mtorr
flux loss is anomalously high, even compared with the TRX empirical scaling, andwarrants further experimental and theoretical attention.
The scaling results obtained empirically and those based on the LHD model andthe low frequency drift wave model are summarized in Table III. Because of the highbeta nature of the FRC the pi0 <x T^/B,, portion of the scaling has been written as
n~'A. The parameter s, given in the LHD particle scaling, is defined in Ref. [7] asapproximately the number of ion gyroradii between R and rs. Because of small s andinfluences of the open field line loss rate on the FRC pressure profile, it is difficult

TABLE III. SUMMARY OF EMPIRICAL AND THEORETICAL SCALING

T0

Experiment

LHD (classical)
model

Low frequency
drift wave model

n°-'r 3 - 5

Te3/2r*n 0 5 T" 0 ' 5 r 2

0 7 2.7
TB

TNn„ 1 T ~ 0 - 5 -3

rs—

n°-'x s '- 6 ^T"-"2 [1 + s/3] 3

n

i

rs

IAEA-CN-47/D-IV-2-1

549

to evaluate the appropriateness of new resistivity models. The low frequency drift
wave model, however, is the first explicit theoretical model which reproduces boththe flux and the energy/particle lifetimes in FRCs. This strongly suggests that low
frequency drift and trapped particle instabilities are an important component of FRC
transport physics. A new FRC experiment with large s (LSX) is being constructed,
which should provide additional transport information over a much wider and morereactor relevant range.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]

[11]
[12]
[13]
[14]

HAMASAKI, S., KRALL, N.A., in Plasma Science (Proc. IEEE Int. Conf. Montreal, 1979),
IEEE, New York (1979) 143.
TUSZEWSKI, M., LINFORD, R.K., Phys. Fluids 25 (1982) 765.
ARMSTRONG, W.T., et al., Phys. Fluids 24 (1981) 2068.
SIEMON, R.E., et al., Fusion Technol. 9 (1986) 13.
HOFFMAN, A.L., et al., Appl. Phys. Lett. 42 (1982) 31.
STEINHAUER, L.C., et al., Phys. Fluids 28 (1984) 888.
SLOUGH, J.T., et al., Nucl. Fusion 24 (1984) 1537.
HOFFMAN, A.L., et al., Fusion Technol. 9 (1986) 48.
ARMSTRONG, W.T., et al., Phys. Fluids 25 (1982) 2121.
BELDCOV, V.V., et al., "Suppression of losses in a compact torus with programmed shapingof the magnetic structure", Plasma Physics and Controlled Nucler Fusion Research 1982 (Proc.
9th Int. Conf. Baltimore, 1982), Vol. 2, IAEA, Vienna (1983) 343.
HOFFMAN, A.L., SLOUGH, J.T., to be published.
tfRALL, N.A., Advances in Plasma Physics, Vol. 1 (SIMON, A., THOMPSON, W.B., Eds),
Interscience, New York (1968) 153.
MANHEIMER, W.M., An Introduction to Trapped Particle Instability in Tokamaks, Rep.
T1D-27157, National Technical Information Service, Springfield, VA (1977).
LIEWER, P.C., Nucl. Fusion 25 (1985) 543.

IAEA-CN-47/D-IV-2-2

SLOW FORMATION AND STABILIZATIONOF FRC PLASMAS
S. OKADA, Y. ITO, M. KAKO, R.E. CHRIEN1, S. OHI,S. GOTO, T. ISHIMURA, H. ITO
Plasma Physics Laboratory,
Osaka University,
Osaka
T. TAKAHASHI, M. OHARA, S. SHIMAMURA,
M. MACHIDA2, Y. NOGICollege of Science and Technology,
Nihon University, Tokyo
Japan

Abstract
SLOW FORMATION AND STABILIZATION OF FRC PLASMAS.
In preliminary experiments of slow FRC plasma formation, the effects of an auxiliary
axial current on the formation and containment of FRC plasma in the NUCTE device at
Nihon University were studied. By applying the axial current before the main compression,
the bias field is amplified so as to satisfy the forcefree equilibrium condition. For larger
values of the axial current ( > 4 kA), the lifetime of the plasma becomes short. When theaxial current is applied to the FRC plasma after the main compression, the upper limit of
the current not destroying the FRC was found to be given by the condition that the safety factor q
on the separatrix is larger than unity. The onset time of the n = 2 rotational instability wasdelayed by the application of the axial current. This suggests that the axial current is effectivein improving the plasma lifetime.

FRC plasma stabilization experiments were done on the OCT device at
Osaka University. The modification of the multipole field from a straight toa helical configuration resulted in a drastic decrease of the threshold strengthfor suppressing the n = 2 rotational instability. This agrees qualitatively withprevious experiments on the NUCTE device but the dependence of the thresholdon the pitch angle of the helical field seems to be different from the previous value.
The rotational angular velocity of the FRC plasma was determined from theDoppler shift of the H^ line emitted from charge exchange neutrals producedby the injection of an intense, pulsed neutral beam. The measured angularvelocity of 4 X 10s rad-s"1 is lower than the real frequency of 7 X 105 rad-s"1of the n = 2 mode deduced from the interferometry signal. If the ideal MHD
1
2

On leave from Los Alamos National Laboratory, USA.
On leave from Universidade Estadual de Campinas, Brazil.

551

552

OKADA et al.

approximation were valid, the two values should agree with one another.
Therefore, the above discrepancy implies that ion kinetic effects are important
for the n = 2 rotational instability.

1.

INTRODUCTION

To construct a large scale machine producing FRC plasmas with a long
lifetime, many improvements are necessary in the plasma formation technique
Slow formation with axial current is considered to be one of the possible
improvements. In the NUCTE experiments at Nihon University, effects of theaxial current on the formation and containment of FRC plasma were investigated.
Suppression of the n = 2 rotational instability excited in FRC plasmas
has already been achieved by applying multipole fields; there remain, however,unsolved problems on the details of the suppression mechanisms. In the OCT
experiments at Osaka University, these problems were investigated.

2.

EFFECTS OF AXIAL CURRENT ON THE FORMATION AND
CONTAINMENT OF FRC PLASMA

The application of an axial current is proposed for a successful formationof the FRC. In the formation phase, three effects are expected: heating of theembedded plasma in the bias field, amplification of the initial bias flux, andcontrol of the plasma expansion to the wall by the azimuthal field. In theconfinement phase, the current heats the cold plasma near the separatrix sothat an improvement of the plasma confinement properties is expected.
The FRC plasma is generated by using the theta pinch device NUCTE II,
which has a 1.7 m long, 16 cm bore coil. Deuterium gas is filled into a 14 cm
bore discharge tube at 12 mtorr. When the negative bias field reaches —630 G,
the compression field is applied to the plasma preheated by the axial current I z i
as is shown in Fig. 1. The rise-time, maximum strength and e-folding time of thecompression field are 2 /JLS, 12 kG and 65 /JIS, respectively. Another current IZ2
is applied to the plasma in order to investigate its effects on the FRC plasma.
The parameters of the produced plasma without application of the IZ2 aredescribed elsewhere [1].
2.1. Amplification of the negative bias flux
When the axial current IZ2 is applied before the main compression, it flows
through the well ionized plasma immersed in the bias field and induces paramagnetic effects so as to increase the axial bias field. The axial field is measured

553

IAEA-CN-47/D-IV-2-2

10 Main Field

\
5 -

n

\
Bias Field

l_
0

—\—
140

160

l\ 80 1001 J 120 >140.
0 l 60(

160

—1—
20

60

40

80

100

120

40 30 -

Izi

20 -

|Z2

l\

10 0

1

1
1
0

20

i

V

1

i

(ps)

FIG. 1. Time sequence of discharge.

by a magnetic probe located at the centre of the discharge tube. As is seen in
Fig. 2, the bias field just before the main compression increases with the axial
current. The solid line in the figure shows the bias field as a function of I z 2
which is calculated on the assumption of the forcefree current profile having
boundary conditions B z = -630 G and Bg = HQ\Jliix^ at the tube wall r w .
The experimental results agree well with the theoretical prediction. This meansthat the bias flux is amplified by the axial current.
2.2. Disturbance in the FRC formation
It is an interesting problem whether the axial current disturbs the
FRC or not. A similar problem of spheromak formation was alreadyexamined in MS-1 experiments at the University of Maryland [2]. The
FRC is formed in our experiment; the lifetime is, however, strongly affectedby the magnitude of the current. The lifetime which is obtained from thee-folding time of nrl (rs is the separatrix radius of the FRC) becomes shortfor a current above 4 kA. On the other hand, when the current is higher than
10 kA, the number of trapped particles in the plasma is found to decrease.
Here, we have used the method of estimation of line mass density by analysing
the radial oscillation of the plasma [3].

OKADA et al.

554

-1.0

"5. -0.5

I
1020

Current (kA)

FIG. 2. Bias field strength versus axial current.

2.3. Effect of current flowing on the separatrix surface
In this experiment, the axial current is applied to the FRC after the formation
to avoid a disturbance in the formation phase. We find that there is a limit for thecurrent not to destroy the FRC. This limit is given by the following condition:

>1

(1)

where £p and B e are the separatrix length and the axial field strength at theoutside of the separatrix, respectively. The destruction of FRC can be seenfrom the end of the trajectory in the B e -I z diagram of Fig. 3, where 2 p is assumed
to be equal to half the coil length. The ratio of Joule energy generated by thecurrent to the loss of internal plasma energy can be obtained from the equation|R

0 = W/r

I|ep77s/27rrs5e(3/2)7rrs2J2p<p)/re(2)

where r e , t]s, 5 and <p> are the energy confinement time of the plasma, theresistivity of the plasma near rs, the depth of the current path on the separatrix,
and the average pressure of the plasma, respectively. The ratio calculated by

555

IAEA-CN-47/D-IV-2-2
U10"

m

1.5
2
(xlO"' Wb-m'M

F/G. 3. Be-Iz diagram. Termination of trajectory means that the cross-section S of theinitial plasma column decreases to S/e. Times: 21 (is [shot No. 6648), 24 \is (shot No. 6687),
and 31 ys (shot No. 6652).

using the plasma parameters at 25 /us after the formation of the FRC is 0.1-0.2,
in optimum current conditions. The onset time of the n = 2 instability, which
is determined by inspecting the photodiode signals, is delayed by the applicationof the current. We consider this to be one indication for the improvement ofconfinement because there is a prospect for a rotational instability which occurs
when some portion of the confined particles is lost [4, 5].

3.

ROTATIONAL INSTABILITY AND MULTIPOLE FIELD SUPPRESSION

Recent experimental studies on FRC plasmas in many laboratories haveestablished that the n = 2 rotational instability (n is the toroidal mode number)can be completely suppressed by applying a multipole magnetic field to the556

OKADA et al.

plasma [6]. On the basis of ideal MHD theory, the threshold field strength B s cnecessary to suppress the instability was obtained as- 1\Sl\/i(3)where 2m is the order of the multipole field, JLIQ the permeability of the vacuum,and n, rs and p the rotational angular velocity, the radius, and the mass densityof the plasma column, respectively [7]. In many cases, the reported experimental
thresholds have amounted to 25-50% of the theoretical predictions. Furthermore,
it has been reported that a helical multipole suppresses the instability more
effectively than a straight multipole [1]. However, experiments carried out byanother group have yielded results there are contradictory to those noted above[8]. Later, the helical multipole field stabilization experiments were repeatedto verify that helical stabilization was more effective. Next, we addressed oneof the most important issues when we compared theoretical and experimental
stabilization thresholds. The experimental measurements of the rotational
angular velocity £2 were regarded as ambiguous because the velocities were notmeasured directly but estimated from the observed real frequency of the instability or from the Doppler shift of an impurity line (usually, the C V line).
Since the rotational angular velocity £2 is the most important physical quantity
affecting the rotational instability, direct measurements of £2 are of vital
importance. In our experiment, we measured the ion velocity distribution todetermine the rotational angular velocity SI, together with the ion temperature
Tj, by using the beam probe spectroscopic technique which we had developed
previously [9].
3.1. Suppression of n = 2 instability by a helical quadrupole field
In this experiment, the hydrogen FRC plasma is translated into a stainlesssteel vessel with an inner diameter of 30 cm and is confined for ~ 100 jus between
mirror fields separated by a distance of ~ 130 cm. The plasma parameters in theconfinement region are typically as follows: ion temperature Tj < 80 eV, plasma
density n = 4 X 1014 cm" 3 , rs«* 8 cm, and plasma length fip « 90 cm. The
axial magnetic field outside the separatrix radius is about 1.5 kG at rs =» 7.5 cm.
The straight or helical winding coil was set just outside the vessel between the
mirror coils. The distortion Ar of the plasma column caused by the instability
was determined from the cyclic deviation of the interferometry signal, whichmeasured the integrated density, /nd£, along the diameter at the midplane.
From many interferograms, each of which shows the time history of /ndfi forone shot, we obtained a plot of the relative distortion Ar/rs versus the multipole field strength. The results are shown in Fig. 4 for straight (a = 0) and557

IAEA-CN-47/D-IV-2-2
0r=4.3rad/m

a=oiai/m
0.4

ir/r

a:
n:
o:
A:

A

55-65(lS
65-75 |JS75-85[IS85-95(IS

0.4

ir/r

<r:t=45-55pS
a: 55-65pS
a: 65-75 (ISo: 75-85(15A: 85-95 (IS

0.3

0.3
AA
O

0.2

0.2

0.1 i

0.1

°<zl
* oa
100

200

300
B a CG]

50

100

150
B 3 [G]

FIG. 4. Relative distortion Ar/rs of plasma column plotted against quadrupole fieldstrength 5 a on plasma surface for straight (ot = 0) and helical (ex. — 4.3 rad-m'1) cases.

helical (a = 4.3 rad/m) quadrupole fields, where B a is the maximum multipole
field strength on the plasma surface, which is equal to twice the field strength B s ,in the absence of plasma. In the straight multipole field case, the experimental
stability threshold B a;th is 250 G, while the theoretical threshold 2B SC isestimated to be 275 G, by using the value of £2 measured by beam probe
spectroscopy. The threshold in the helical multipole field case is 100 G. This
result agrees with the previous observation on the NUCTE device that thethreshold of the helical field is smaller than that of the straight field. However,in the previous experiment, the straight multipole field was only able to partly
suppress the instability so that a modification from straight to helical multipoles was definitely important. This suggests that the effect of the helicalmultipole field depends not only on the pitch a but on the other plasma
parameters, as well.
3.2. Rotational angular velocity of the FRC plasma
To generate charge exchange neutrals, intense pulsed hydrogen beams
produced by a Z-pinch type beam source were injected into the plasma along
the diameter of the midplane. The parameters of the probing beam at theobservation point (r = 7 cm) are: energy E^ s»400 eV, density n^, « 3 X 1012 cm' 3 ,and pulse width Atj, < 2 /us. The rotational angular velocity £2 and ion tempera-

558
too

OKADA et al.

L TjCeV]

Ca]

80

80

60

60

40

40

20

fl

0

20

20

40

60
C

80

0

120

40

60

• I

40

• 11l
20

• }

"J

20

40

60

Q C«10 tad/s]

60
80
C(is]

CO

80

5

fl [«10 <ad/s3_L

100 . Tj CeV]

80

0

20

1

1

1

40

60

80

cm

s

0

Cb]

100

0

20

40

s

Cb]

fiC«10 rad/s]

80

40

60

CO

60

80

CMS]

FIG. 5. Time evolution of ion temperature T{ and rotational angular velocitv Q. for
quadrupole field strength Ba = 0 (a), Bi^83 G (b), and 5 a «134 G fcj
atr= 7.5 cm.

ture Tj were estimated from the respective Doppler shift and the broadening of
the Hp line emitted by the charge exchange neutrals. In Fig. 5, the time development of Tj and £2 is shown for the cases of Ba = 0 (a), Ba « 83 G at rs = 7.5 cm (b),and B j * 134 G at rs = 7.5 cm (c), where B a is the strength of the helical quadrupole field of a = 4.3 rad-m" 1 used to suppress the instability. The temperature
increase from 30 eV to 80 eV up to 40 /us probably results from the relaxationof the kinetic to thermal energy of the plasma translated by reflection from themirrors because the macroscopic motion of the plasma ceases around 40 us.
The observed value £1 at t > 40 us. is ~4 X 10s rad • s"1 (a) and ~(2 -4) X 10s rad -s"1(b and c), where the direction of the rotation agrees with the ion diamagnetic drift.
The angular frequency core of the instability is estimated to be ~7 X 10s rad-s" 1
from the time history of /nd£. If the ideal MHD approximation were valid, £1 ando) re should agree with each other. Therefore, the apparent discrepancy between
the two values indicates the importance of the ion kinetic effects on the n = 2
mode. The ratio £2re/cj obtained here agrees well with computer simulation
results by Harned using a hybrid code, which includes both resonant ion andfinite ion Larmor radius effects [10]. The ion diamagnetic angular frequency
Cl* (= |(Tj/eBr)(l/n)(dn/dr)j) was estimated to be ~4 X 10s rad-s" 1 , on the

IAEA-CN-47/D-IV-2-2

559

assumption of a rigid rotor equilibrium. Thus, the ratio J2/^* is about unity at
the onset time of the instability (t «» 40 us). When a quadrupole field is applied
to the plasma (b and c), the ions continue rotating, although Q, is slightly
reduced to ~(2-4) X 10s rad • s"1.

4.

CONCLUSIONSThe effects of an axial current oh formation and confinement of FRC plasmas
were investigated. Preliminary results showed an amplification of the bias flux
and a delay of the onset time of the rotational instability. These effects aredesirable for the formation of the FRC, using a slowly rising compression field.
However, the lifetime becomes short when the axial current is too high.
Further investigation is necessary to utilize the flux amplification effect in
the formation phase in order to increase the trapped flux.
The helical multipole field was shown to be more effective in suppressing
the n = 2 rotational instability than the straight field. This agrees with previous
results; the dependence of the threshold field strength on the pitch angle a. seems,
however, to be different from the previous one. The rotational angular velocity,which was estimated from the Doppler shift of the H^ line emission from thecharge exchange neutrals, was smaller than the frequency of the n = 2 mode.
This suggests the importance of the ion kinetic effects on the n = 2 mode.

REFERENCES[ 1 ] SHIMAMURA, S., NOGI, Y., Fusion Technol. 9(1986)69.
[2] GOLDENBAUM, G., IRBY, J., CHONG, Y., HART, G., Phys. Rev. Lett. 44 (1980) 393.
[3] NIBLETT, G., GREEN, T., Proc. Phys. Soc. 74 (1959) 737.
[4] BARNES, D., SEYLER, C, in Compact Toruses and Energetic Particle Injection(Proc. US-Japan Joint Symp. Princeton, 1979), Plasma Physics Lab., Princeton Univ.,
NJ(1979) 110.
[5] HARNED, D., HEWETT, D., Nucl. Fusion 24(1984) 201.
[6] OHI, S., et al., Phys. Rev. Lett. 51 (1983) 1042.
[7] ISHIMURA, T., Phys. Fluids 27 (1984) 2139.
[8] REJ, D.J., et al., Phys. Fluids 29 (1986) 2648.
[9] ITO, Y., et al., J. Appl. Phys. 58 (1985) 1752.
[10] HARNED, D.S., Phys. Fluids 26 (1983)1320.

DISCUSSION
ON PAPERS IAEA-CN-47/D-IV-2-1 AND D-IV-2-2
M.G. HAINES: Can you tell us more about the physics of the suppressionof rotation in the FRC?
T. ISHIMURA: By using multipole fields we can suppress the deformationof the plasma column due to rotation. The multipole field exerts magnetic
pressure on the surface of the plasma column and, if the column deforms, thepressure is stronger at the part of it which bulges, and further deformation isthen suppressed.

561

IAEA-CN-47/D-IV-3

HIGH BETA PLASMA CONFINEMENT WITH
NEAR-NEOCLASSICAL TRANSPORT IN TPE-2*
K. HAYASE, I. HIROTA, H. KIYAMA, S. KIYAMA, Y. MAEJIMA,
Y. SATO, S. TAKEDA, E. YAHAGI, M. KITO, K. KOYAMAElectrotechnical Laboratory,
Sakura-mura, Niihari-gun,Ibaraki-ken, Japan
Abstract
HIGH BETA PLASMA CONFINEMENT WITH NEAR-NEOCLASSICAL TRANSPORT IN TPE-2.
The stability and the energy confinement of a high /3 tokamak-like plasma are studied experimentally at /3, = 0.02-0.1 and q, = 1-4 in ETL-TPE-2. Plasmas with ft * 0.1, 0p » 1 and ^ = 1.3
are stably confined at f = 30-400 eV and rie = 1 x 1020 to 1021 m' 3 . The volume averaged toroidal
beta, /5,, obtained so far increases with decreasing qj. The maximum value of #t tends to exceed the
Troyon limit. The global energy confinement time, r E , is a few milliseconds at q,, < 1.5 and /3p « 1.
The scaling with electron density and poloidal field is found to be close to neoclassical. The evolutionof the q^, profile shows that it may be self-organized by the relaxation process. The current density profile is nearly flat. A flat current profile is essential to good, stable confinement of a high 0 plasma withlow q,.

1.

INTRODUCTION

High /? confinement and stability of a screw pinch plasma are studied
experimentally in ETL-TPE-2. TPE-2 is a toroidal screw pinch device with elongatedcross-section (b/a = 21 cm/13 cm, K = 1.6) and small aspect ratio (A = 3.0)
[1, 2]. The device is capable of producing and setting up a high /? tokamak-like
plasma by a fast pulsed discharge. The operating parameters are flexible so that we
can vary the initial values of/3t, qa, n,,, etc., over a wide range. We can also choose
the plasma current operation mode: constant, ramping or decaying. Stable confinement of a plasma with /3, = 0.1 was reported in the previous conference [3]. The
maximum value of /3t that can be stably confined in the tokamak is suggested by ballooning mode theory [4, 5]. In tokamaks, high power auxiliary heating is necessary
for operation in a high jS region. In a screw pinch with high power implosion heating,however, high /3 operation is much easier to obtain than in a tokamak.
The main objective of the present experimental study is to explore the stability
and the energy transport of high /3 plasmas around the theoretical limit. The impurity* Supported by the Atomic Energy Bureau, Science and Technology Agency, Prime Minister's
Office, Tokyo, Japan.

563

564

HAYASE et al.

TABLE I. TPE-2 PARAMETER RANGE
Toroidal field, B, (T)0.6-0.75 (flat-top: - 0 . 6 ms)

Plasma current, Ip (kA)

40-175 (programmable)

Safety factor, c^

1.3-4.0

D 2 filling pressure, p (Pa, mtorr)

0.16-1.13 Pa (1.2-8.5 mtorr)

Electron density", n,. (m"3)

1 X 1020 to 1021

Temperature11, f (eV)30-400ab

CO2 and He-Ne laser interferometers.
T e : conductivity, Thomson scattering and soft X-rays.
Tji C V Doppler broadening.

reduction and the optimum setting-up of a confining configuration enabled us to operate it with qa « 1.3 and /3P « 1 (/3, « 0.1), where qa is the safety factor at the
periphery of the plasma column in cylindrical approximation. We show that highiS (/3t < 0.1) plasma can be stably confined at qa (1.3 < qa < 2) with nearneoclassical transport. We also discuss the characteristic features of the q* profile(the safety factor defined on each flux surface), which is formed by the self-relaxation
process in the screw pinch configuration.

2.

SETTING-UP

The initial /3t and j3p values are controlled by preheating and the filling pressure, p. We can easily achieve that j8p > 1 is fulfilled in this device. The fast
implosion heated plasma (T; l> Te) is further heated by Ohmic current in the confinement stage. When the Ohmic input overcomes the conduction and radiation
losses, the total plasma energy increases.
The operating range of parameters in the experiment is listed in Table I. We
investigate the initial parameter (qa, /3P and li,.) dependences of the discharge. So far,a discharge with an initial qa lower than 2 is not stable and is characterized by large
fluctuations of the magnetic and radiation signals and by a high toroidal one-turn voltage, Vp. A stable low qa discharge (qa < 2) can only be attained by ramping theplasma current over a long time, where qa starts at ~2.5 and decreases as low as
1.3. A discharge with a qa higher than 3 is very quiescent . A highly diamagnetic(/3p > 2) plasma is produced in the initial stage. However, the following Ohmic
heating is not sufficient to maintain it, and /3P relaxes to ~ 1. The high density operation with p > 8 mtorr is accompanied by strong radiation and conduction losses
and high Vp, even for qa > 2. Experiments are carried out mainly with an initial qa
of ~2.5 and p < 5 mtorr, both in constant and ramp current modes.

565

IAEA-CN-47/D-IV-3!p(kA)^ 100

Ip(kA)Vp(V)600

FIG. 1. Typical discharges. Upper and lower diagrams show B,, Ip, Vp and qa in the constant andramp operations of current, respectively.

In Fig. 1, we show typical waveforms of discharges in constant and ramp current modes with an initial qa of ~2.5 and /3P less than 1, at 2.5 mtorr. The plasma
current, Ip, is sustained for 600 j*s in the constant current mode. In the ramp current
mode, the plasma current builds up from 60 to 140 kA within 400 us, and qa
reaches 1.3. When qa passes through 2, some spikes in the Vp signal and a small dip
in the Ip signal appear. The low qa discharge (qa < 2) has been operated in tokamaks with unusual discharge conditions, for example in Ref. [6]. /3, is low(ft < 0.01) with a low electron density (lie ~ 1019 nr 3 ).
The ion temperature, T i; is 200 to 400 eV, from C V Doppler broadening in
the initial stage. It decreases gradually by equipartition among ions and electrons.
The radius of the high T; core is one-half to one-third of the vessel radius. On theother hand, the initially low electron temperature continues increasing by energy
transfer from the ions and by Ohmic heating and, finally, is nearly equal to the ion
temperature. The electron temperature on axis, Te, obtained from conductivity andThomson scattering at 400 to 450 (is after the start of the main discharge at
Ip « 120 kA is shown as a function of the filling pressure in Fig. 2. In deriving Tefrom the plasma conductivity, we have assumed Zeff = 1 and classical resistivity as
well as an appropriate radial distribution of the temperature. The electron density,rief, from the interferometer measurement is proportional to the filling gas pressure
and is roughly constant during the discharge.

566

HAYASE et al.

oTtion. Scott.
• Conductivity

300 -

.

o . §
•

— 100

1

o
0
o

I

50

:

30
t

1 i i i i

35
p (mtorr))0

FIG. 2. Electron temperature Tt on axis from plasma conductivity and Thomson scattering as aJunction of filling pressure p at 400-450 us (Ip = 120 kA).

3.

BETA

The /3, value is measured by magnetic analysis [7, 8]. The initial /3t is determined by preheating and implosion heating. In the same conditions, the value islarger at a higher filling pressure. It does not depend on the plasma current. The maximum value of the initial (S, can be varied up to 0.08. The corresponding j8p is given
by ~6/3, qj;. In Fig. 3, we show the temporal variation of /3, in the constant current(qa « 2.5) and ramp current (qa = 2.5-1.3) modes as a function of the fillingpressure. The dependence on the filling pressure is weak except for the values in theearly stage. The energy of the plasma increases when the Ohmic input is larger thanthe loss. This is more significant when the plasma current is ramped up. In this case,the final j8t reaches —0.1 at 500 /ts when the toroidal field begins to decay. The
value of /Sp increases to — 1 and is kept constant even if it is initially lower than 1,
regardless of the qa value.
By changing the preheating condition, the initial /3P and /3t are raised to higher
than 2 and 0.07, respectively. The temporal behaviour of j3, and /?p in the constantand decaying current modes is shown in Fig. 4. In both cases, /3P relaxes to ~ 1, as
the high 0 p state cannot be sustained by Ohmic heating alone. In the decaying current case, /3P > 1 is kept for more than 300 us. The value /3, decays with a time
constant of ~ 100 us in both cases and then reaches the same level as that of the low
initial $, case. Hence, the energy confinement time may be ~ 100 /is in the earlystage. The relatively poor energy confinement is not due to high /3P since it is thesame in the low initial /3p (J3P< 1) case.
The experimental values of /3t are compared with the values from ballooning
mode theory [4, 5]. The operating region for Figs 3 and 4 is shown in the /3t and567

IAEA-CN-47/D-IV-3

0.05 •

FIG. 3. Temporal variations of (3, in constant (left, qa » 2.5) and ramp (right, qafrom 2.5 to 1.3)
operations of current.

O.t

5.0

const. l 0

0.05

0.1

decay.

hv

5.0
>P
*P

0.05

•

^^^^.^
250

()

500

°C

250

2.5

*s.:

500 U

tUs)

FIG. 4. Temporal variations of $, and 0p with high initial value of $p (~2). Left hand figure showscase of constant current (qa ** 2.7); right hand figure is case of decaying current (qafrom 2.3 to 3.4).

l/qa space, compared to the Troyon limit in Fig. 5. In the ramp mode, /3, increases
from below to above the limit. In the constant current mode, /3t lies just below or
around the limit. In the high initial /3p case, however, the initial $t is far above thelimit, and then the value decreases to below or around that value.

4.

HIGH BETA STABILITY

To investigate the MHD stability of a high /3 discharge, we observe the fluctuations in the magnetic loops, the one turn voltage loops, visible radiation, etc. The
kink mode is observed around the integer a^ values, and a certain amount of plasmaenergy is lost when qa crosses ~ 2 transiently in the ramp current mode.
Two 3-D plots of the poloidal field on the wall surface in the outer half of thepoloidal plane are shown in Fig. 6. The distortion of the surface field is not verylarge, except for qa larger than 2.5. A low level fluctuation is observed, and it islarger for smaller qa. The small fluctuations for low qa are not correlated well on theplasma surface, in the poloidal and toroidal directions [9]. There are less temporal

568

HAYASE et al.

FIG. 5. Operating region of $,. (a) and (b) correspond to left and right hand parts of Fig. 3,
respectively, (c) and (d) correspond to left and right hand parts of Fig. 4, respectively. Troyon limit
is shown by two lines, C = 4 and C = 2.8.

U/is)
FIG. 6. 3-D plots of poloidal field in outer half-plane ofpoloidal surface. Left and right hand figures
are cases of slowly decaying and ramp modes, respectively. qa starts at 2.5 in both cases. Position
number indicates top (1) to bottom (9). Plasma is strongly unstable (qa £ 1) after 400 ps in ramp
mode.

(—10 /AS) and spatial (~ir/4) correlations in such magnetic fluctuations on the surface. As the surface mode is stabilized by the conducting shell, a kink may occur inthe interior. A significant change in the internal q* profile is only seen for qa « 2or < 1, as will be shown in the next section. This behaviour does not depend on /3P,
even if it is large (/3P > 1). Fluctuation of the soft X-ray radiation from the interior
is also observed. No fluctuation characteristic of high /3p is, however, detected.
Then, these fluctuations are considered to originate from an internal kink or a tearingof higher toroidal mode number in the edge plasma region. No ballooning mode has
been observed, as yet.
Therefore, plasma with /3p > 1 is grossly stable if qa is chosen appropriately.
The level of fluctuation changes with the discharge. If it is low, the confinement is

IAEA-CN-47/D-IV-3

569

FIG. 7. Time evolution of q? profile in ramp mode. It changes rapidly at 50-150 us by crossing
qa « 2. Profiles at 50 JXS and after 250 us are similar to one another.

better and /3t increases more with time by Ohmic heating. Hence, it is important for
confinement of high /3t plasma to choose qa and to control the equilibrium configuration in a precise manner.
5.

RELAXATION OF THE q* PROFILE

The shape of the q* profile is essential for the stability of a high /3, and low qa
plasma. In TPE-2, the flux profile and, hence, the q* profile are calculated from thepoloidal field distribution on the wall [7]. The temporal evolution of the q* profileis shown for the ramp mode in Fig. 7. We see that the profiles are independent of
qa (i.e. the plasma current) in the quasi-steady state. The profile is disturbed by an
MHD instability during the transition through qa « 2. The profiles are similar toeach other before and after the transition. For the constant current mode, the profile
remains unchanged except in the early stage. We can say that the current density distribution relaxes to a self-organized state. The. ratio of axial to peripheral q* isaround 1.8. The toroidal current density profile calculated from the q* profile isnearly flat over the plasma cross-section. By assuming a parabolic plasma pressure
profile, the poloidal current distribution is calculated for the case of /3P ~ 1. The
plasma is diamagnetic near the axis.
This type of profile seems essential for the stability of a high /3 plasma. If theconfiguration deviates substantially from this, the transition through qa » 2 is not
successful. In this case, the discharge is accompanied by a high one turn voltage, witha spike, and large fluctuations in the peripheral magnetic field, the visible and the
X-ray radiations. Hence, setting up the configuration in the production stage, which
is nearly the same as that in the equilibrium stage, is important for the good confinement of a high /? plasma.

HAYASE et al.

570

3
p (mtorr)

5

FIG. 8. Global energy confinement time TE as a Junction of filling pressure. Values change greatly
shot by shot, because of irreproducible electron density, (a) and (b) show values at 100 fis and~ 300 us, respectively, in constant mode (Ip = 70 kA). (c) shows the case at - 300 /is in ramp mode
(I ~ 120 kA). Lines (b)' and (c)' show results of numerical simulations for (b) and (c), respectively.

6.

ENERGY CONFINEMENT

The global energy confinement time TE is evaluated from TE = Wp/(Vp I p -dWp/dt). The plasma energy Wp is calculated from J3V The radiation loss has
been greatly reduced to less than one-third of the total loss although the older screw
pinch and belt pinch were dominated by the radiation loss [10].
The dependence of rE on the filling pressure and plasma current is measured(see Fig. 8). In the early stage, TE is ~ 100 /*s but increases gradually to more thana few milliseconds. The enhancement of energy confinement in the later stage may
be due to the self-relaxation and improvement of the equilibrium configuration as
described in the previous section. Following the establishment of the configuration,
the plasma energy is considered to be better confined and increased. If the ideal configuration is set up in the production stage, then, 100 jts later, rE is already high.
The value TE tends to be larger for higher /3t.
As to scaling, rE decreases with increasing filling pressure and with decreasingplasma current. As the electron density is approximately proportional to the fillingpressure, the dependence is similar to neoclassical, in the collision dominated regime.
To check this in greater detail, a numerical simulation is carried out, where neoAlcator scaling is assumed for the electron transport [11]. The assumption of neoclassical ion transport can explain the experimental results. We indicate the energy confinement time from the simulations in the same figure. The experimental values areclose to the neoclassical ones in the well confined discharges.

IAEA-CN-47/D-IV-3
7.

571

CONCLUSIONSThe experimental results are summarized as follows:
A stable discharge at /3p « 1, qa « 1.3 and /3, = 0.1 is established. Energy
transport here is near-neoclassical. A high @p (~2) discharge can be realized
without noticeable instability but is not sustained by Ohmic heating alone. No ballooning mode has been clearly observed in the high /3p operation. Relaxation andself-organization of the q + profile are observed. The current density is nearly flatover the cross-section, and the plasma is diamagnetic in the central region. This profile is essential to good, stable confinement in a high & tokamak configuration.

ACKNOWLEDGEMENTS
The authors are grateful to Drs H. Ashida and H. Sugimoto for useful
discussions on high fi theory.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]

TAKEDA, S., et al., in Fusion Technology (Proc. 9th Symp. Garmisch-Partenkirchen, 1976),
Pergamon Press, Oxford (1976) 1943.
SATO, Y., et al., in Fusion Technology (Proc. 11th Symp. London, 1980), Pergamon Press,
Oxford (1980) 1563.
KIYAMA, S., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc.
10th Int. Conf. London, 1984), Vol. 1, IAEA, Vienna (1985) 393.
SYKES, A., et al., in Controlled Fusion and Plasma Physics (Proc. 1 lth Europ. Conf. Aachen,1983), Vol. 2, European Physical Society (1983) 363.
TROYON, F., et al., Plasma Phys. Controll. Fusion 26 (1984) 209.
TFR GROUP, Nucl. Fusion 24 (1984) 784.
KIYAMA, S., et al., Bull. Electrotechn. Lab. 50 (1986) 95.
HAYASE, K., et al., Bull. Electrotechn. Lab. 47 (1983) 129.
HAYASE, K., et al., Bull. Electrotechn. Lab. 50 (1986) 137.
BECKER, G., et al., Nucl. Fusion 18 (1978) 1653.
KIYAMA, H., et al., Bull. Electrotechn. Lab. 50 (1986) 122.

IAEA-CN-47/D-IV-4

Z-PINCH EXPERIMENTS AND THEORY
M.G. HAINES, A.E. DANGOR, A. FOLKIERSKI,
P. BALDOCK, C D . CHALLIS1, P. CHOI, M. COPPINS,
C. DEENEY, M.B. FAVRE-DOMINGUEZ2, E. FIGURA,
J.D. SETHIAN3
The Blackett Laboratory,
Imperial College of Science and Technology,
London, United Kingdom
Abstract
Z-PINCH EXPERIMENTS AND THEORY.
The paper presents a report of density and temperature measurements performed in three Z-pinch
experiments. In the compressional pinch, a stable pinch is formed by shock compression. The density
profile and line density (about 1019 m~') are maintained even after current reversal. The temperature(about 130 eV) agrees with the Bennett relation, but the decay is more rapid. In the gas-embedded pinch,pressure equilibrium is obtained by preheating a laser-initiated filament in hydrogen at about 1 bar.
During the preheat phase, expansion of the plasma column generates a shock in the surrounding neutral
gas. The equilibrium is destroyed by a deformation of the plasma column into a radially expanding
spiral. The results from two experiments (f - 1.5 x 1012 A-s~' and 4.5 x 10 1 2 A-s~') show
different scalings of the duration of equilibrium with pressure. In the gas puff pinch, where an annular
gas jet is ionized and imploded, short lived (about 5 ns) 'hot spots' (ne = 1027 m~ 3 , Te ~ 600 eV) of
submillimetre dimensions are observed to occur at intervals of 50 ns. Simultaneously, a plasma of hightemperature (about 5 keV) and low density is detected. The 'hot spots' and the surrounding plasma areobserved only when a longer lived (about 50 ns) electron beam (about 10 kA and 10 kV) is present.
Self-similar profiles for an ohmically heated pinch in pressure equilibrium have been calculated.
Numerical solutions of the linear MHD eigenvalue equation are obtained for a variety of equilibria. A
minimum energy state corresponding to a helical current flow is found to exist. Initial results for thestabilization due to a large ion Larmor radius are presented.

Three types of Z-pinch experiments, and analytical and computational investigations in the Z-pinch area have been performed at the Imperial College of Scienceand Technology.

1.

COMPRESSIONAL PINCH

The experimental details of the apparatus used are described in Ref. [1], The
Marx generator and the pulse-forming line provide a current pulse of about 130 kA,
1

JET Joint Undertaking, Abingdon, Oxfordshire, UK.
Pontificia Universidad de Chile, Santiago, Chile.
3
Naval Research Laboratory, Washington, D.C., USA.
2

573

574

HAINES et al.

150 -

200

300

TIME(ns)

-50

-100

BOns

1= 5

r.

d

r. r.

r. r.

ELECTRON NUMBER DENSITY PROFILESCONSERVATION OF LINE DENSITY

100'J

r

80-

o

"loo

T

RECONSTRUCTION OF HOLOGRAM

n.

200 55o 2)6 560
TIME(ns)

FIG. 1. Current waveform and number density profiles in the compressional pinch (1 torr hydrogen).

of some 200 ns duration and 50 ns rise time. The quartz pinch tube is 50 mm longand 20 mm in diameter. Filling pressures of 300 mtorr to 2 torr of hydrogen areused. The discharge collapses to about one-third of the tube radius, in agreement withthe slug model [2], and then remains generally stable for some 500 ns.
Figure 1 shows the current waveform and the time resolved number density
profiles obtained by end-on pulsed ruby laser interferometry at 1 torr pressure. The
measured line density, N = 2 X 1019 m" 1 , in the central column remains constantfor more than 500 ns and agrees with the initial filling pressure. At t ~ 200 ns, additional plasma is generated at the wall; at about the same time there is a considerable
drop in the current measured in the external circuit. This behaviour is compatible
with the assumption of an internal crowbar occurring at this instant. Conservation of

IAEA-CN-47/D-IV-4

575

TIME(ns)

30

|

230

r—|

I

|

430

1

•I

\

h

•

1OOnm ALUMINIUM+ 2.5pm KIMFOIL

6mm

lOOnm ALUMINIUM
5 y m KIMFOIL

FIG. 2. Time-resolved X-ray images, end-on, from two pinholes (1 ton H2, 132 kA).

the magnetic flux after this period can explain the continuation of equilibrium after
current reversal. A similar behaviour and a constant line density are observed
at 2 torr.
Figure 2 shows streak photographs of two end-on X-ray images of the discharge
obtained simultaneously, using two pinhole cameras with different absorbers. The
hot phase lasts about 100 ns and the measured temperature is 250 ± 200 eV.
Experiments with different absorbers will improve the precision of this measurement.
The Bennett relation with Te = T i; I = 130 kA and N = 2 x 1019 m" 1
yields Te = 130 eV. The ions are magnetized with WTU — 1.8, and the ratio of theion Larmor radius to the pinch radius is a/a = 0.13. At 130 eV, the MHD
instability growth time (TMHD = radius/ion thermal speed) is 25 ns and the thermal
ion transit time from centre to electrode along the axis is 225 ns. The axial heat loss
calculated according to Haines [3] is about 42 MJ-s" 1 ; with a heat content of 62 J,
this yields a decay time for temperature of about 1 us. If the observed durationof the stable phase, > 500 ns, is interpreted as an inductive decay with
T = L/R = 200 ns, then the resistance R is 0.06 Q. With a thickness of 2 mm,
this gives for the wall plasma a temperature of more than 4 eV.

2.

GAS-EMBEDDED PINCH

In the compressional pinch a high temperature is achieved by dynamic compression . With the gas-embedded configuration the aim is to start with a high density
plasma and to heat it under conditions of pressure balance. Equating the rate ofincrease of internal energy to the rate of Joule heating, using the Bennett relation andthe transverse Spitzer resistivity, we obtain [4]:dIdt

= 2.7 x 10" 3 — log e A
aT(A-s" 1 )

576

HAINES et al.
8.0

6.0

4.0
electrons

2.0

0.0

0.2

0.4

0.6

0.8

PINCH RADIUS (mm)FIG. 3. Number density profiles during preheating in the gas-embedded pinch at p = 0.3 bar.

10.0

0.0
0.2

0.4
PINCH RADIUS

0.6

0.8

1.0

(mm)

FIG. 4. Number density profiles during the equilibrium phase at p = 0.3 bar.

IAEA-CN-47/D-IV-4

577

where n is the number density in m~3, T is the temperature in eV, and a is theradius in m. With the available dl/dt of 2 x 1012 A-s" 1 , the plasma channel must
be preheated if pressure balance is to be achieved.
The earlier low current experiment was described previously [1, 4]. First, anarrow channel of ionization is formed by a laser beam in hydrogen at p = 1 bar.
A current of 6 kA is used to preheat the plasma for about 60 ns and then the maincurrent is switched on. In the low current apparatus, dl/dt is limited to
1.5 X 1012 A-s" 1 and Ip^ is 50 kA. In the high current apparatus the corresponding figures are 4.5 x 10 1 2 A-s~' and 150 kA. Side-on pulsed laser Schlieren
photography and interferometry are used to examine the plasma.
Three phases are distinguished. In the first phase, the preheating phase, theplasma column expands from an initial radius of 100 fim with a radial velocity of
about 1.7 x 104 m - s ~ \ which drives a cylindrical shock wave in the neutral gas.
Figure 3 shows density profiles obtained by Abel inversion from an interferogram
taken during preheating. In the outward propagating shock, the initial step in neutral
gas density is a factor of four, as expected for a strong shock. The electron number
density profile is flat.
The second phase, the equilibrium phase, begins when the main current isswitched on. During this time, r = 400 ^m and the column is heated in radialequilibrium; no further expansion occurs. Figure 4 shows that the electron density
becomes peaked on axis. The equilibrium phase terminates when a spiral configuration rapidly develops. The pressure dependence of the duration of the equilibriumphase is:Low current apparatus
High current apparatus

P = 0 .3 bar

P = 1.0 bar

40 ns
12 ns

15 ns
27 ns

The behaviour in the low current case does not agree with MHD expectations. It
should be mentioned that the length of the discharge is 40 mm in the low current caseand 13 mm in the high current case. During the equilibrium phase the electron line
density increases by a factor of two, which indicates an accretion of plasma resulting
from the radial diffusion of neutrals into the plasma. In the low current apparatus at
0.33 bar the electron temperature was measured by X-ray absorber techniques [5].
A value of 70 ± 5 eV was obtained, while the Bennett relation with Te = T; gives
for 20 kA and N = 2 x 1012 m" 1 a value of 30 eV.
The third phase has the configuration of a spiral. Optical framing photographs
and Schlieren exposures indicate that the spiral grows from the axis. Interferometricmeasurements clearly show that the perturbation begins on the axis (Fig. 5). Whenthe perturbation reaches the surface, the overall diameter of the plasma increases
rapidly. The growth time r/fr in the low and high current cases is measured as 12 ns
and 9 ns at 1 bar, and as 14 ns and 5 ns at 0.3 bar. The ratio of the pitch of the spiral

578

HAINESetaJ.

0.3 mmFIG. 5. Interferogram showing how the spiral begins to grow from the axis (p = 1 bar).

to the radius of the pinch (in equilibrium), X/a, is 4 in the low current case and2 in the high current case. A single-turn coil positioned coaxially with the pinch andclose to the ground electrode is used to measure the axial magnetic field due to theazimuthal component of the helical current. This is done on the high current
apparatus only. It is found that the onset of Bz at p = 0.3 bar occurs at I = 25 kA,
while at p = 1 bar it occurs at I = 75 kA. The direction of Bz is always from
cathode to anode, but it has not been possible to reverse the polarity of the discharge.
The helical configuration, which terminates the attempt to heat the pinch underpressure equilibrium, is itself relatively stable, and the spiral remains inside an
overall cylindrical envelope.

3.

GAS PUFF Z-PINCH

A hollow cylindrical shell of gas is formed by supersonic gas injection. Whena large current (about 300 kA) is applied, the hollow shell collapses onto the axis anda hot plasma is formed. Normally, argon is used, but some experiments have been
carried out with neon and other gases. The apparatus is described in Refs [1, 6].
Optical streak measurements of velocity give the kinetic energy as 250-300 J
with a mass of 400 /ig.
Time-resolved streak photographs of two pinhole X-ray cameras, taken simultaneously with different absorbers, show short-lived 'hot spots' on the axis, as wellas anode emission of longer duration which is ascribed to the incidence of electron
beams. Figure 6 shows on a common time axis the AllAt trace, the hard-X-ray signal
from the PIN diode looking at the anode, and the soft-X-ray signal from the X-ray

IAEA-CN-47/D-IV-4

579

V

dl
dT

(1

I

HARD
X-RAYS

1

SOFT
X-RAYS
0 .

0L

200 WO 600
TIME (ns)FIG. 6. Time-correlated signals, showing dl/dt (I), hard X-rays from the anode and soft X-rays fromthe plasma in the gas puff pinch.

diode looking at the body of the discharge. The 'hot spots' are about 100 /un indiameter and 300-500 jum long, and are observed both in time-resolved and timeintegrated X-ray images. They occur randomly along the axis and last < 5 ns; theyoccur in bursts at intervals of 30-100 ns. Spectra taken with a PET crystal spectrometer (X = 3.7-4.3 A) are consistent with the hot spots having n,. = 1027 m~3 andTe = 600 eV. Measurements using collimated PIN diodes with appropriate
absorbers show that the hot spots are surrounded by plasma of higher temperature.
If the radiation is due to bremsstrahlung, then the temperature of this hot plasma is
about 5 keV. The hot spots are always preceded by electron beams, but not all beams
are followed by hot spots. The intensity and angular distribution of the X-rays fromthe anode indicate electron beams with I = 10 kA and electrons of about 10 keV.
There are two classes of beams. Beams occurring early in the discharge arealways associated with peaks in the I signal. These beams may be due to large inductive fields producing runaways, and they are not followed by hot spots. Later in the
discharge, beams occur when the 1 signal is fairly smooth, presumably owing to aninternal crowbar, and these beams are followed by hot spots.
4.

THEORETICAL INVESTIGATIONS

4.1. Self-similar profiles
To preserve pressure balance during the heating of a Z-pinch, the current mustrise according to a particular prescription, the 'Haines-Hammel curve' [7, 8]. It isfound that this imposes a restriction on the possible equilibrium profiles. Figure 7
shows self-similar profiles computed under this restriction. The inclusion of radial
ion thermal conduction gives rise to a small finite pressure at the edge.

580

HAINES et al.

'E
O

CM

O

LU

z o
a —
UJ

C/5
CO
UJ

INJ

u5

F o

1.5

2-9 4-4 5.8 7.3
R A D I U S (1(T< m)

1.5

2-9 4.4 5-8 7.3
RADIUS (10- 4 m)

FIG. 7. Computed profiles of pressure and current density for a self-similar casewith ion thermal conductions.

6.0

15.0
FIG. 8. MHD growth rates normalized in terms of thermal ion radial transit time
(k is axial wavelength, a is pinch radius).

<
O

o

0.04
RADIUS (10'2m)

0.08

FIG. 9. Current density (A -m'2) profile for an equilibrium marginally stable to the m = 0 mode.

IAEA-CN-47/D-IV-4

581

4.2. Ideal MHD stability
The 1-D shooting code developed by C. Davies at the Imperial College of
Science and Technology has been used to obtain numerical solutions of the ideal
MHD linear eigenvalue equation for a variety of equilibria satisfying pressurebalance. Figure 8 shows the growth rates of the m = 0 and m = 1 modes when auniform current distribution is assumed. The m = 0 mode grows faster over almost
the whole range of ka. Figure 9 shows an equilibrium current profile peaked on axis,
which is stable to m > 2 modes, marginally stable to the m = 0 mode [9], but
unstable to the m = 1 mode. The m = 1 eigenmode is peaked on axis and has a
predicted growth time of 7.3 ns, which agrees with the experiment. It is to be notedthat the experimental electron density peaks on axis before the development of the
spiral, but the current density could not be measured.
4.3. Minimum energy states
The gas-embedded pinches transform into helical structures with \/a ~ 4.
Linear MHD predicts the highest growth rates in the limit of X — 0. Minimum
energy states with helical configurations have been sought. If one assumes:(i) a cylindrical bulk plasma, (ii) a helical current path on the surface, (iii) a vacuum
surrounding the plasma, (iv) a perfectly conducting wall, and (v) conservation ofmagnetic flux and entropy, then a minimum energy state is found with
9.9 > X/a > 6.5, the precise value depending on the wall radius. In the more
realistic case of a uniform current density and with the wavelength of the helicalcurrent paths being independent of radius, a value of X/a s 9.0 is found. A lower
value is obtained if a centrally peaked current distribution is assumed.
4.4. Large ion Larmor radius effects
Our linear initial value code for Hall fluid instabilities [10] has been modified
to treat ions as an ensemble of particles, as described earlier [1]. Preliminary results
for m = 0 internal modes have been obtained for various values of e (the ratio of
average ion Larmor radius to pinch radius). For e = 0.3, a considerable reductionin the growth rate by up to a factor of five has been found, but definitive results could
not be obtained so far because of numerical problems in the scheme.

REFERENCES
[1]

[2]
[3]

HAINES, M.G., DANGOR, A.E., FOLKIERSKI, A., et al., "Z-pinch experiments andtheory", Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc. 10th Int. Conf.
London, 1984), Vol. 2, IAEA, Vienna (1985) 647.
POTTER, D.E., Nucl. Fusion 18 (1978) 813.
HAINES, M.G., Proc. Phys. Soc. 77 (1961) 643.

582

HA1NES et al.

[4] DANGOR, A.E., et al., Phys. Rev., A 27 (1983) 2751.
[5] FAVRE-DOMINGUEZ, M.B., Experimental investigation of gas-embedded Z-pinch,
PhD Thesis, Imperial College, Univ. of London, 1985.
[6) CHALLIS, C D . , X-ray observations of an annular gas-puff Z-pinch, PhD Thesis, Imperial
College, Univ. of London, 1986.
[7] HAINES, M.G., Proc. Phys. Soc. 76 (1960) 249.
[8] HAMMEL, J.E., Rep. LA-6203-MS, Los Alamos Scientific Lab., NM (1976).
[9] KADOMTSEV, B.B., in Reviews of Plasma Physics, Vol. 2, Consultants Bureau, New York
(1966).
[10] COPPINS, M., et al., Phys. Fluids 27 (1984) 2886.

DISCUSSION
B.P. LEHNERT: Just a short comment. In the theoretical analysis of our
EXTRAP pinch experments we also found that large Larmor radius effects1
contribute significantly to the stability of the Z-pinch; this is in agreement with yourresults. I think that a relevant analysis of high beta systems cannot be based on MHD
theory alone but must, at least in part, be extended to include kinetic effects.
A. SEN: In you compressional Z-pinch experiments did you estimate the sizeof betatron orbits and can you say something about their role in the stabilization of
the pinch?
M.G. HAINES: The size of the betatron orbits, or what we call singular orbits,
is approximately V(a|a) , where ai is the mean ion Larmor radius and a the pinch
radius. For our compressional Z-pinch, where a/a is 0.13, the singular orbits
occupy a region of radius 0.36 a from the axis. We believe that they play an importantrole in the stabilization of the pinch. As stated in the paper, a considerable reductionin the growth rate of the m = 0 mode has been found in a kinetic simulation witha(/a up to 0.3.

LEHNERT, B.P., Comments Plasma Phys. Contrail. Fusion 9 (1985) 91.

IAEA-CN-47/D-IV-5-1

INVESTIGATION OF
PLASMA DYNAMICS AND EMISSION
IN A DENSE PLASMA FOCUS
K. HIRANO, T. YAMAMOTO, K. SHIMODA,
Y. OKABE, K. WAKABAYASHI
Department of Electronic Engineering,
Gunma University,
Kiryu, Gunma
M. YOKOYAMA, Y. YAMAMOTO, Y. YAMADA,
Y. KITAGAWA, M. YAMANAKA
Institute of Laser Engineering,Osaka University,
Suita, OsakaJapan

Abstract
INVESTIGATION OF PLASMA DYNAMICS AND EMISSION IN A DENSE PLASMA
FOCUS.
The heating mechanism of plasma in a plasma focus discharge is investigated through correlation between plasma dynamics and emission of neutrons, X-rays and charged particles. Various
measurements are carried out by using a Mach-Zehnder interferometer, a time resolved soft
X-ray camera, and a Thomson parabola ion analyser. High energy deuterons are measured by
nuclear activation techniques. The target used in the measurement, consisting of two semicircles (aluminium and carbon), determines the mean energy of the deuterons, from the
13 28
N/ A1 ratio. The ion temperature and the kinetic energy, from the D-D/D-3He yield ratio,
are obtained by using an equimolecular Dj - 3 He mixture. The neutron emission region andthe neutron intensity distribution are determined by using a neutron pinhole camera. The
experimental results show that the plasma emits soft X- rays in three successive phases: themaximum compression, plasma disruption and decay phases. Immediately after disruption,
emission of neutrons, soft X-rays and charged particles is observed. The energy distributionof the deuteron beam produced by the plasma focus is obtained.

1.

INTRODUCTION

The plasma focus is considered to be one of the most efficient pulsed
neutron, X-ray and charged particle sources [ 1 ]. Moreover, recent developments
in plasma focus research suggest the possibility of a breakeven reactor [2].
However, the mechanism of these emissions is not fully understood, in spiteof investigations performed from various points of view.

583

584

HIRANO et al.

After the initial breakdown along the insulator surface, the filling gas D2
is snowploughed by the current sheet which is driven by the electromagnetic
force. When the current reaches the end of the inner electrode, the current
collapses radially and forms a plasma column. The length of the plasma column
extends axially in time. Then, the column is disrupted by the growth of them = 0 instabilities. After the disruption, neutron, X-ray and ion emissions areobserved.
The aim of the present study is to precisely investigate the correlation
between the dynamic behaviour of a focused plasma and the emission and toclarify the heating mechanism of the plasma.

2.

APPARATUS

The experiments were carried out by using two Mather type plasma focus
devices with which capacitor banks of 4.24 yF, 65 kV and 40 uF, 30 kV were
employed [3].
The macroscopic behaviour of the plasma was observed by a Mach-Zehnderinterferometer and a time resolved soft X-ray pinhole camera. The light source
for the interferometer was an N2 laser whose light pulse width (FWHM) was lessthan 1 ns. A microchannel plate (MCP) (Hamamatsu Photonics F2223-1 IP)
was employed as an image sensor of the pinhole camera. The effective exposure
time of the camera was 1.5 ns [4]. The soft X-ray emission was also recordedwith a PIN photodiode.
A Thomson parabola analyser was used to measure the charged particle
beam generated by the focused plasma in the downstream direction with respect
to the discharge current. Deflection fields of up to 5 kG and 5 kVcm" 1 , respectively, can be applied. A time resolved measurement of the ion beam was carriedout by developing a detector consisting of MCP with multianode. A method of
calibrating the electron multiplication factor of the MCP for charged particles
was developed [5].
We obtained the ion temperature and the kinetic energy from the D-D/D- 3 He
yield ratio by using a D 2 - 3 He mixture. The D(d,n) 3 He reaction yield was determined from the neutron yield measured by a silver activation counter, while the3He(d,n) 4 He reaction yield was determined by an analysis of the 7-decay spectrum
induced by the 63 Cu(p,n) 63 Zn reaction (the threshold is 4.21 MeV). Such a
mixture gas method was reported by Gullickson et al. [6].
Moreover, we used a neutron time of flight detector to obtain a maximum
neutron yield of 1.2 X 109 in 60 ns and a silicon PIN (250 jum thick) detector todetermine the ion generation time, which agrees with the 10 ns pulse duration of
hard X-rays. The spatial distribution of neutron emission was also measured bya neutron pinhole camera, consisting of a paraffin pinhole and a detector [7].

585

IAEA-CN-47/D-IV-5-1

20 ns/div— I.E.

— I.E.
1st peak

2nd peak

3rd peak

FIG. 1. Time correlation between emission of soft X-rays and macroscopic plasma behaviour.
Soft X-ray signal is obtained by the PINphotodiode detector (a), framing interferograms (b),time resolved soft X-ray images (c). Inner electrode face is designated by I.E.

3.

EXPERIMENTAL RESULTS AND DISCUSSION

A typical example for soft X-ray signal obtained by one of the PIN photodiodes is shown in Fig. 1 (a). Obviously, the soft X-ray signal has three peaks
with good reproducibility. The peaks of hard X-rays and neutrons are pointed
out by HX and N in this figure, respectively. Figures 1 (b) and (c) show interferograms and soft X-ray images taken at each peak of the soft X-ray signalin (a). It is observed that soft X-ray emission starts before maximum compression and continues for a relatively long time (60-70 ns), whereas hard X-rays
and neutrons are emitted in single pulses. The first peak corresponds approximately to maximum compression as is seen from Figs 1 (b) and (c). In thisphase, the soft X-rays are emitted nearly uniformly from the collapsed plasma
column. The second peak coincides with the time when the plasma column is
disrupted by the growth of the m = 0 instabilities. At this instant, it isrecognized that the plasma column is partly disrupted as is shown in Fig. 1 (b),HIRANO et al.

586
(cm)

3

0 •—
(a)

FIG. 2. Interferogram (a) and schematic drawing of soft X-ray images overlapping interferogram (b). Inner electrode face is labelled I.E.

and dense plasmas which are confirmed by the numbers of shifted fringes areseen at both sides of the disrupted region. At the third peak, we cannot recognize
the plasma column up to 1 cm above the inner electrode face. In this phase, soft
X-rays are only emitted from the inner electrode face. The peaks of the hard
X-ray and neutron signals appear immediately after the second peak of the soft
X-ray signal. This fact suggests that the second peak is closely related to themost interesting phenomena in the plasma focus discharge. Soft X-ray images,
with sufficient intensity, could only be taken in the vicinity of each peak.
Figure 2 shows an example for the interferograms and soft X-ray images
immediately before disruption. In this interferogram, extremely developedm = 0 instabilities appear in the plasma column. A filamentary structure anda high intensity spot can be seen on the electrode axis in the soft X-ray image.
The fringe shifts in the interferogram are proportional to the electron density, n e .
On the other hand, the intensity of the soft X-rays produced by the bremsstrahlung
corresponds to n | T 1 / 2 , where Te is electron temperature. The region has a higher
Te, if intense soft X-rays are generated from the region with lower electron density.
In Fig. 2(b), region <D shows a rather high n e , and so the intense X-rays are
ascribed to the higher n e . There are two or more fringes between the lateral face andthe axis of the plasma column at ® . It is, therefore, assumed that an extremely
intense spot of the soft X-rays at © is produced by high temperature electrons

IAEA-CN-47/D-IV-5-1

587

since the plasma has a low density in this region. The regions ® and © may beorigins of anomalous resistivity.
A deuteron beam produced by a single shot of the plasma focus was recorded
after energy analysis with the Thomson parabola analyser. An example for
deuteron tracks registered on a CN film after etching is shown in Fig. 3 (a).
Tracks of the same width as the multianode were counted by using a microscope.
The deuterons collected on the CN film are shown in Fig. 3 (b). The width of eachcurve on the energy axis (AE<j) is calculated by taking energy borne by an equivalent width of the multianodes into account. The analog signals which wereobtained simultaneously by the multianode are shown in Fig. 3(c). The signals
are caused by ions which are produced at the same time and the same location,
but have different energy, and so we can determine the energies which correspondto each signal. Integrating the curves in Fig. 3(b) across the beam width, weobtain the number of ions collected in each area, i.e. an energy distributionfunction as is shown in Fig. 4. Two vertical axes are used, corresponding to an
ion count per anode of an area 0.5 mm X 8 mm and an ion count per keV andsteradian.
From the ratio of the activated products 13N and 28A1 obtained from thetarget, we calculated the mean deuteron energies as 1.5MeVat 1.5 torr and
1.4 MeV at 5 torr, which shows that the mean energy depends weakly on thefilling pressure although the beam intensity is very sensitive to it. No copper
activation was observed, which indicates that deuterons above 4 MeV are very few.
To study the correlation between neutrons and energetic deuterons, wemeasured the deuteron temperature and kinetic energy by using a D2 - 3 He mixture
gas, where the pressure dependence of the neutron yields was the same as in pure
deuterium gas. The results are listed in Table I, where we estimated the deuterontemperature Tj from a pure thermonuclear model and the deuteron kinetic energy
E d from a pure beam-target model. Tj = 11 keV at 1.5 torr is much higher than
the neutron time of flight (3 keV), which implies that the thermonuclear model
is not applicable at the low pressure.
In the low pressure regime, optical and ion pinhole images indicate that thedeuterons are produced by the plasma diode. This plasma diode plays the roleof an opening switch interrupting the current with a very short rise-time, which
may induce a high accelerating voltage across the plasma diode. We estimate
the accelerating voltage Vp given by
Vp=V-LI
where V is the voltage across the electrodes, L the inductance in the discharge,
and I the discharge current. The inductance is assumed to be constant (40 nH
in our device), during the interruption. We assume AI equal to three-quartersof the total current, and to be 480 kA, dropping off in 5 ns, as is determinedfrom the hard X-ray emission time, so that

588

HIRANO et al.

1
1
c

•a

•§

'43
3

^1

IS

589

IAEA-CN-47/D-IV-5-1

10
1000

Ed

1500

(keV)

FIG. 4. Energy distribution ofdeuterons obtained by integrating tracks in Fig. 3 fb).

This voltage is consistent with the observed deuteron energy.
The plasma diode was formed above the anode surface, but no dense pinch
was formed in this regime. This suggests the possibility of current interruption
due to the enhancement of radiative loss caused by high Z impurities such as
tantalum from the anode tip [8].
To determine the location of the neutron emission region in the focus
pinch column, we first measured the axial intensity distribution of the neutron
emission by using a horizontal paraffin slit (1 cm X 10 cm X 5 cm) as neutron
collimator. Then, we measured the radial intensity distribution of the neutron
emission by using a vertical paraffin slit (10 cm X 0.5 X 50 cm).
As is seen in Fig. 5, the highest density of neutrons is located up to 2 cm
above the anode tip with a maximum radius of 0.5 cm; emission takes place
within a conical angle of 44° ± 10°, where the top of the cone is located at theanode tip [9].

590

HIRANO et al.

TABLE I. ION TEMPERATURE Tj AND DEUTERON KINETIC ENERGY Ed
Second column refers to equiatomic case, whose data are not so different fromequimolecular case (first column)
D-D neutrons
D-3He protons

T; (keV)(thermonuclear model)

Ed (keV)(beam acceleration model)

4torr(l:l)
4 torr(1:2)

14
13

5.0
5.2

42
43

1.5 torr(l:l)

3

11

74

2

4.

3

e

CONCLUSIONS
From the experimental results we conclude the following:(1) Soft X-rays are emitted for a rather long period from the collapse phase
up to the disruption of the plasma column, whereas the hard X-rays whose
appearance coincides with the emission of the electron beam consist of a narrowpulse.
(2) Typically, the soft X-ray signal had three successive peaks. According to
the interferometric observation, three soft X-ray signal peaks correspond to theinstant of (i) maximum compression, (ii) disruption and (iii) the decay phase of
the plasma column. This was also confirmed by the time resolved soft X-ray
pinhole camera.
(3) The soft X-rays emitted from the disrupted region were less intense thanthose of the unstable phase. The peak of the voltage spike appeared in thisphase. According to a previous paper, the generation of the voltage spike canbe ascribed to a rapid increase in plasma resistance rather than to a rapid
increase in plasma inductance [10]. This means that the electrons in the
disrupted region can be heated further, for example, by turbulent heating.
The plasma has, however, a low electron density in this region, as is shownby interferometric observation, and, thus, emission of the soft X-rays is notintense in spite of the higher electron temperature. Therefore, the high resistivity
generated in this region provides an intense electric field and the field accelerates ions up to a sufficient energy to produce neutrons.
(4) A Thomson ion analyser in which an MCP was employed as an ion detector
was developed. A method of calibrating the electron multiplication factor of
the MCP for charged particles was established. By using the Thomson ion
analyser, the energy distribution of the deuteron beam produced by the plasma
focus device was obtained.

591

IAEA-CN-47/D-IV-5-1

z(cm)

(cm)

(cm)

2.0

1.5

1.0

05

0

(cm)
(b)

(a)

FIG. 5. Isointensity contour display of neutron source by using (a) single collimator and(b) double collimators.

(5) We estimate the neutron yield using a pure beam target model as
Yb = n d n a vAL t
where n d and n are the deuteron beam and target densities, a is a reaction crosssection, v is the beam velocity, A and L are the target plasma cross-section andlength, and t is the production duration. For 74 keV deuterons directed intoa cold gas target of 1.5 torr, cr = 10"26 cm 2 ,v = 2.7 X 108 c m s " 1 ,n d £= 8.8 X 1015 cm" 3 (from experimental results), A « ir X 10~2 cm 2 ,t as 60 ns, and L = vt = 16 cm, which agrees with the results of the neutron
pinhole camera measurement. The neutron yield is then 5.1 X 10 8 . Since
the measured average neutron yield was 7.0 X 10 8 , the neutron production
mechanism at low pressure is explained by the beam target model. Note that
not the deuterons above 330 keV, but the deuterons < 74 keV, contribute to
the neutron yield.
(6) The highest density of neutrons is located up to 2 cm above the anodetip with maximum radius of 0.5 cm and emission is within a cone angle of
44°± 10°, where the top of the cone is located at the anode tip.

REFERENCES[1] GULLICKSON, R.L., SAHLIN, H.L., J. Appl. Phys. 49(1978) 1099.
[2] HEROLD, H., HAYD, A., in Plasma Focus Research (Proc. 3rd Int. Workshop Stuttgart,1983), IPF-83-6, Institut fur Plasmaforshung der Universitat Stuttgart (1983) 131.

592

HIRANO et al.

[3] HIRANO, K., YAMAMOTO, T., SHIMODA, K., et al., in Plasma Physics and Controlled
Nuclear Fusion Research 1984 (Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA,
Vienna (1985) 569.
[4] HIRANO, K., YAMAMOTO, T., SHIMODA, K., TAGAYA, Y., Jpn. J. Appl. Phys. 24(1985) 1514.
[5] HIRANO, K., YAMAMOTO, T., OKABE, Y., SHIMODA, K., to be published inRev. Sci. Instrum.
[6] GULLICKSON, R.L., LUCE, J.S., SAHLIN, H.L., J. Appl. Phys. 48 (1977) 3718.
[7] YAMAMOTO, Y., KISODA, A., YAMADA, A., NIKI, H., KITAGAWA, Y.,
YAMANAKA, M., YOKOYAMA, M., YAMANAKA, C, Jpn. J. Appl. Phys. 23 (1984)
1410.
[8] YAMADA, Y., KITAGAWA, Y., YOKOYAMA, M., J. Appl. Phys. 58 (1985) 188.
[9] YAMAMOTO, Y., KISODA, A., YAMADA, A., NIKI, H., KITAGAWA, Y.,
YAMANAKA, M., YOKOYAMA, M., YAMANAKA, C, J. High Temp. Soc. 11 (1985) 113.
[10] YAMAMOTO, T., SHIMODA, K., HIRANO, K., Jpn. J. Appl. Phys. 24 (1985) 324.

IAEA-CN-47/D-IV-5-2

INVESTIGATION OF COLLAPSE,PINCH DYNAMICS AND
FAST PARTICLE EMISSION
IN A PLASMA FOCUS DISCHARGE
S. CZEKAJ, S. DENUS, A. KASPERCZUK,R. MIKLASZEWSKI, M. PADUCH, T. PISARCZYK,S. SLEDZINSKI, A. SZYDLOWSKI, Z. WERESZCZYNSKI
Institute of Plasma Physics and Laser Microfusion,
Warsaw, Poland
Abstract
INVESTIGATION OF COLLAPSE, PINCH DYNAMICS AND FAST PARTICLE EMISSION IN A
PLASMA FOCUS DISCHARGE.
The experimental results of investigations into the plasma focus (PF) phenomenon are presented.
Interferometry was used on two devices, PF-360 and PF-150. The emission of fast particles was studied
on the PF-150 device only. The dynamics of the collapse and the pinch works differently in the two
devices. It seems that it is due to different geometry and circuit parameters. It was found in the PF-150
device that the emission characteristics of relatively slow D ions (within the energy range of
50-1000 keV) and of very fast ones (of energy above 1 MeV) are quite different.

1.

THE COLLAPSE AND THE PINCH DYNAMICS

Experimental investigations were carried out on the PF-150 (60 kJ) and thePF-360 (100-170 kJ) plasma focus devices by means of laser interferometry [1 —3].
Thus, we could compare two facilities with energies differing by a factor of two to
three. Unfortunately, the characteristics of the condenser bank used (lower inductance of the larger device) does not allow a simple scaling of the plasma dynamics
to be performed because of the difference in the dimensionless parameters relating
to the plasma and circuit characteristic times [4].
Essential differences in plasma sheath dynamics and pinch behaviour have been
observed [2 — 3]. In this paper, we present an attempt to explain the nature of thesedifferences on the basis of classical MHD theory. The interesting question is whether
these differences are due to a difference in energy in the condenser bank or whether
they are the consequence of different electrode geometries and circuit parameters. A
characteristic feature of plasma sheath dynamics in the collapse phase are the MHD
instabilities, which are not observed in the PF-150 device. Some explanations for thisphenomenon were presented in Refs [2, 3]. The second difference which has important consequences is the curvature radius of the plasma sheath just before the sheath
593

594

CZEKAJ et al.

— PF-15O— PF-36O
CE.
edge-

FIG. 1. Shape of plasma sheath.

8 [mm]
FIG. 2. Structure of plasma sheath (ne is the electron density).

arrives at the axis. This radius in the PF-360 device is about eight to nine times longer
than that in the PF-150 device (Fig. 1).
The greater thickness of the plasma sheath is an immediate consequence of thisfact. It agrees well with theoretical prediction:

Rkr

where A is the thickness of the sheath, R is the curvature radius, r the actual radiusof the sheath, and p/p0'the compression ratio. This formula is based on Bernoulli's
law; it is valid in the late, collapse phase and differs essentially from Gureev's formula [5].
Greater sheath thickness with virtually the same plasma density (Fig. 2) implies
a greater amount of plasma inside and, then, a greater radius of the plasma column
in the PF-360 device.

IAEA-CN-47/D-IV-5-2

595

77773

FIG. 3. Pinch formation geometry.

The difference in the curvature radii of the plasma sheath also affects thecharacter of the MHD instabilities in the pinch phase. Let us estimate the axial plasmapressure gradient during the process of pinch formation. Let the pressure rise-timein the z = zo plane be given by (Fig. 3):v cos aDuring this period, the point z moves towards ZQ, leading to
ZQ-ZQ =d=

TpU

where u is the rate of pinch formation. From simple geometrical considerations, we
obtain:u =sin aThen, the characteristic length of the pressure gradient is as follows:
a, =»

A

v

v cos a

sin a

=

2A
sin 2 aThus, the pressure gradient along the axis may be estimated to bev

1dsin 2 aA

When the sheath has a small curvature radius (PF-150 device), the thickness A
of the plasma sheath is also small, and the angle a is large; high pressure gradients

596

CZEKAJ et al.

FIG. 4. MHD pinch instability in PF-150.

FIG. 5. MHD pinch instability in PF-360.

develop during pinch formation, and an intense plasma outflow from the pinch isobserved. As a consequence of this outflow, there is a fast developing flute in thepinch which leads to a break of the column, as is shown in Fig. 4. In the PF-360
device, where the plasma sheath curvature radius is greater, weak plasma outflow
along the pinch axis and classical MHD instabilities were observed (Fig. 5). The
important role of the sheath curvature radius was confirmed in an experiment withan additional axial magnetic field B2 applied in the PF-150 device [6]. The rise in B2
caused an increase in curvature radius and sheath thickness and, in addition, changed
the character of the MHD instabilities.

597

IAEA-CN-47/D-IV-S-2

l\
/

8M0 6

6-10 6

4-10 6I/\/ lLiLivermore

/:

\

20

40

©d

FIG. 6. Angular distribution of ions in PF-150 obtained from track number on pinhole images.

2.

FAST PARTICLE EMISSION

Two methods were applied to obtain the angular distribution of the fast D +ions. The first method was based on counting the numbers of tracks in the pinhole
images (along the whole area). This method is time consuming and very cumbersome, but yields the angular distribution of relatively slow deuterons (Fig. 6). In thesecond method, the track detector plates were located inside the discharge chamber
along a semicircle (22 cm radius) around the inner electrode face. To prevent thedetector plates from being damaged by the plasma and the waves in the ionized gas,a metallic shield with drilled holes (1 mm in diameter) was placed in front of thedetector plates so that the detectors registered only deuterons which passed throughthe holes. To avoid track saturation effects, we covered the detectors with either
mylar or Al foils of different thicknesses. Up to now, only angular distributions ofvery fast deuterons which penetrated 9 /xm thick Al foils (Ed > 900 keV) have beenobtained (Fig. 7). A comparison of the angular distributions as obtained by the firstand by the second method seems to show that deuterons of lower energy have abroader angular spectrum. The angular distributions achieved by the second method
are very similar to those obtained with a carbon target activation technique in otherlaboratories [7—8]. It should, however, be noted that, by using an activation technique, one can only register D + ions with energies above 700 keV.
The energetic distributions of deuterons were studied by means of a Thomson
spectrometer and an ion collector. In certain spectra (Fig. 8) two components can be
distinguished: within the energy range 0.15 to 1 MeV, the shape of the spectrum is598

CZEKAJ et al.

150

II

\
1/
n
\
>

IDOf

-20

20

ed

FIG. 7. Angular ion distribution obtained with metal semi-ring located in front of detector.

dN—1-Vr_...
\

5

S

C - F

V 1

0.5

1

ION ENERGY

FIG. 8. Deuteron spectra registered by Thomson spectrometer in PF-150.

well approximated by the formula f{E^) « E^3, while above 1 MeV this function is
given by f(E<|) « E<[2. It was confirmed that the apparent maximum of the spectrum
in the low energy range ( < 300 keV) is due to the effect of an interaction between
the registered ions and molecules of the D2 gas filling the discharge chamber. The
maxima of the spectra obtained with the ion collector were more and more shiftedIAEA-CN-47/D-IV-5-2

599

FIG. 9. Pinhole images of ion source in PF-150 device: (a) ions passing through pinholestraightforward hitting the detector, (b) track detector covered with 1 \tm mylar foil, (c) track detector
covered with 9 iim Al foil.

towards the low energy side as the thickness of the gas layer along the ion path
became smaller.
The ion emission source was determined with the set of three pinhole cameras[9-10] located at angles of 0°, 9° and 11° with respect to the electrode axis. Ion
source images created by pinholes were recorded by the plates of a dielectric track
detector. By covering the pinholes with foils of various thicknesses, we could obtain
the images of ions with various energies. In this way, we found that very slow ions
(Ed > 50 keV) are emitted from a large area around the inner electrode and, probably, also from the converging plasma sheath. Very fast ions (Ed > 900 keV) areemitted from hot spots which are located near the central electrode axis (Fig. 9).
A schematic of view of the experimental set-up for measurements of the fastelectron energy spectrum is presented in Fig. 10. A magnetic spectrometer was used
to measure the fast electron energies. The electron beam deflected by the B field wasrecorded on a KODAK X-ray film of RAR type. A photograph of the exposed film
and its densitogram is shown in Fig. 10, as well. The maximum energy of the electrons was about 1 MeV. The fine structure of the energy spectrum is evident. One
also sees several distinctly marked components of the electron beam. In the bulk ofelectrons of continuous energy distribution, intense groups of monoenergetic electrons seem to appear.

3.

SUMMARY

Essential differences in the pinch behaviour in PF-150 (50 kJ) and PF-360 (100to 170 kJ) were observed. It was shown that these differences are connected with adifference in the plasma sheath curvature radius in a late phase of the collapse. Since
the plasma sheath curvature radius depends on the electrode geometry, one canCZEKAJ et al.

600

limators
fillingpressure
2to3[torrQ

Bfieldlotographic
film
470 215

127

fa)

68

47

E,keV

(b)

FIG. 10. Scheme of experimental arrangement for electron beam registration (a)and electron beam spectrum (b).

change the pinch properties by changing this geometry. The emission characteristics
of relatively slow D + ions (Ed > 50 keV) are quite different from the characteristics of very fast ions (Ed > 900 keV). Fast electrons with energies of up to 1 MeV
were registered. The fine structure of the fast electron spectrum was observed.

REFERENCES[1] JERZYKIEWICZ, A., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984
(Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 591.
[2] CZEKAJ, S., et al., in Controlled Fusion and Plasma Physics (Proc. 12th Europ. Conf.
Budapest, 1985), Vol. 9F, Part I, European Physical Society (1985) 550.
[3] CZEKAJ, S., et al., in Plasma Focus and Z-Pinch Research (Proc. 4th Int. Workshop Warsaw,
1985) 122.
[4] IMSHENNIK, V.S., FILIPPOV, N.V., FILIPPOVA, T.I., Nucl. Fusion 13 (1973) 929.
[5] CZEKAJ, S., et al., Rep. No. 4/81, Institute of Plasma Physics and Laser Microfusion, Warsaw(1981).
[6] GUREEV, K.G., Fiz. Plasmy 4 (1978) 304.
[7] GULLICKSON, R.L., SAHLIN, H.L., J. Appl. Phys. 49 (1978) 1099.
[8] STEINMETZ, K., Rep. Plasmabericht 1/8 (Nov. 1980).
[9] CZEKAJ, S., DENUS, S., SZYDLOWSKI, A., in Controlled Fusion and Plasma Physics (Proc.
11th Europ. Conf. Aachen, 1983), Vol. 1, European Physical Society (1983) 539.
[10] CZEKAJ, S., DENUS, S., SZYDLOWSKI, A., Atomkemenerg. Kerntech. 44 (1984) 189.

DISCUSSION
ON PAPERS IAEA-CN-47/D-IV-5-1 AND D-IV-5-2
K. SATO: Could you comment on future prospects for this kind of plasma
focus device, especially in terms of a future neutron source as well as a plasma fusion
machine?
S. DENUS: Plasma focus has an optimistic scaling law for thermonuclear neutron production, according to which the neutron yield per shot is proportional to thecondenser bank energy when squared or to the current when cubed (or even raised
to a higher power). This scaling law has now been confirmed by experiments up toa bank energy of about 400 to 500 kJ. An answer to your question can be given when
this scaling law, or a modified version of it, is confirmed by plama focus experiments
performed at higher energy levels equal to one to two or even more megajoules. As
far as I know, these experiments are being planned, including in Poland.

601

IAEA-CN-47/D-V-1

IMPURITY BEHAVIOUR
IN THE WENDELSTEIN VH-A STELLARATOR
H. RINGLER, F. SARDEI, A. WELLER,
W VII-ATEAM 1 ,ECRH GROUP2, NI GROUP3
Euratom-IPP Association,
Max-Planck-Institut fiir Plasmaphysik,
Garching, Federal Republic of Germany
Abstract
IMPURITY BEHAVIOUR IN THE WENDELSTEIN VD-A STELLARATOR.
Plasma buildup and heating of net-current-free plasmas by ECRH (70 GHz) and further heating
by NBI are successfully demonstrated at B o = 2.5 T. Impurity transport studies are carried out. In
NBI sustained discharges, neoclassical behaviour is found for impurities, similar to that in discharges
starting from OH target plasmas reported earlier. During combined application of ECRH and NBI heating aluminium introduced by laser ablation is lost much faster than would be expected neoclassically.
The electron line density, which increases linearly with time during NBI, levels off during the additional
ECRH pulse and then remains constant. A similar degradation in impurity confinement is observed inpure ECRH discharges.

Impurity Confinement
In the W VII-A stellarator high radiation losses due to impurity
accumulation were observed for NBI-heated, "currentless" plasma
discharges created by ohmic heating /I/. The impurity transport
study presented here refers to "currentless" plasmas created by
ECRH with a 70 GHz gyrotron at B o = 2.5 T with -€o <• 0.5. Threemain types of discharges were investigated:a) NBI-sustained discharges at an input power level of 750 kW,
n e 2! 3.7 - 5.5 x 1 0 1 3 cm"3, T e (o) ^ 4 0 0 eV, T ± (o) ^ 600 eV.
b) EC-sustained discharges at 70 GHz with a power ofn e ^ 1.7 - 2.7 x 1 0 1 3 cm"3, T e (o) ^ 900 - 1400 eV,
T ± (o) ££100 -300 eV.

2

3

150 kW ,G. Cattanei, D. Dorst, A. Eisner, V. Erckmann, U. Gasparino, G. Grieger, P. Grigull,
H. Hacker, H.J. HartfuB, H. Jackel, R. Jaenicke, J. Junker, M. Kick, H. Kroiss,
G. Kuehner, H. MaaBberg, C. Mann, G. Muller, W. Ohlendorf, F. Rau, H. Renner,
H. Ringler, F. Sardei, M. Tutter, A. Weller, H. Wobig, E. Wursching, M. Zippe.
W. Kasparek, G.A. Muller, P.G. Schiiller, M. Thumm, R. Wilhelm (Institut fur
Plasmaforschung, Univ. Stuttgart).
K. Freudenberger, W. Ott, F.P. Penningsfeld, E. Speth.

603

604

RINGLER et al.

c) NBI-sustained discharges with additional ECR heating during
part of the discharge, n e ^ 4 x lO 1 ^ cm~3, (n e (o) close to thecutoff density), T e (o) X 500 eV, T ^ o ) - 6 0 0 eV.
As in previous transport studies, both Al trace impurities
injected by the laser blow-off technique and intrinsic impurities
were considered. The simulations were done with the SITAR code/I/.
a) For NBI-sustained discharges starting from an ECRH target
plasma, the measured Al radiation is reproduced reasonably
well by neoclassical transport simulations using experimental
profiles for n e and T e . In particular, good agreement is found
for the time evolution and the absolute intensity of thecentral Al XII + Al XIII radiation from soft X-ray measurements and of line-averaged intensities from six Al ionization
states. According to neoclassical expectations, the measurements show Al accumulation at the plasma center and no evidence of appreciable losses during the time of the discharge.
For the intrinsic impurities (predominantly oxygen in W VII-A)
good agreement of soft X-ray measurements and neoclassicalcalculations was obtained by assuming in the calculations, inaddition to a 1 % oxygen contamination of the neutral beam,
which constitutes an oxygen deposition o f - 2 x 10l8 s-1, an
oxygen influx from the walls increasing from 1 x 1018 to
7 x 10*8 s -l during the discharge.
b) In contrast to the NBI-sustained discharges, for pure ECRH
discharges the Al XII + Al XIII radiation from soft X-ray
measurements (Fig. la) and the single line radiation from six
ionization states Al VIII - Al XIII could not be reproduced by
the neoclassical model (Fig. lb). The main discrepancy concerns the fast loss of Al from the plasma center after 80 ms
as indicated by the decrease of the central Al XII + Al XIII
radiation and of the corresponding line radiation, which cannot be explained by atomic processes (n e , T e — constant intime). Furthermore, the time scale for the Al penetration into
the plasma center is much shorter than in the neoclassicalsimulations. The fast loss of Al could be simulated, however,by an additional diffusivity D a n o m = 3 x 10^ cm^.s"!, constantin space and time. An additional inward velocityvanom = 3 x 10^ x r/a cm-s~l, linearly decreasing to zero
during the ECRH pulse, was used to accelerate the Al penetration and to fit the absolute level of the central radiation
approximately (Fig. lc).
c) The discharges heated jointly by NBI and ECR differ from thepurely NBI-heated discharges in that for the former an

IAEA-CN-47/D-V-1

605

exp.
A L - 1 2 . 1 3 INTENSIT
SHOT 6IO23 61026
So
MICRON 8E

FIG. 1. (a) Evolution of Al XII + Al XIII radiation power from soft X-ray measurements after
Al injection into an EC sustained discharge;(b) code simulation with neoclassical transport fluxes;(c) code simulation with additional anomalous diffusivity and inward velocity.

606

RINGLER et al.

FIG. 2. (a) Evolution of soft X-radiation for an NBl + EC heated discharge;(b) code simulation with oxygen impurities and neoclassical + anomalous transport;(c) same as (b), but with additional Fe impurities.

IAEA-CN-47/D-V-1

607

additional 150 kW ECRH pulse was switched on for - 50 ms
during the discharge. During this time interval a flatteningof the density evolution is typically observed (see Fig. 5 ) .
This flattening may be explained by a deterioration of theparticle confinement during the ECRH pulse or by a reduced
penetration of recycled neutrals due to edge deposition of
ECRH power, which is discussed later in this paper. Theinterpretation of reduced particle confinement is supported by
the observed evolution of the Al radiation from soft X andsingle-line radiation measurements, which indicate a fast lossof Al from the plasma center, in contradiction to neoclassical theory. The time scale for the decay of the radiation andthe values of D a n o m required to simulate it with the code wereabout the same as for pure ECRH discharges, with the anomalous
impurity behaviour extending throughout the gradient region.
The soft X-radiation from the intrinsic impurities for thisdischarge type is shown in Fig.2a. In a previous paper /2/ it
was shown that the increase of the central radiation could be
reproduced reasonably well by pure neoclassical transport withoxygen impurities from 1 % beam contamination and from wall
influx rising from 2 x 10*8 to 5 x lO 1 ^ s"1 during the discharge. However, if the same anomalous behaviour is assumed
for the intrinsic impurities as for the ablated Al during its
escaping phase from the plasma center, i.e. if the same D a nom
is added while keeping the wall influx unaltered, then thecentral oxygen radiation drops to unacceptably small
intensities. In this case, the measured radiation could only
be reproduced if the oxygen influx at the plasma edge wasraised by almost an order of magnitude with respect to thevalues assumed for the neoclassical simulations. However, only
about 10 % of this impurity influx contributes to the increaseof the central oxygen density. The remaining 90 % circulates,
after ionization, back to the wall, thus keeping the impurity
gradients at the plasma edge higher than in the neoclassicalcase.
In this picture of enhanced transport the typical neoclassicalimpurity accumulation is strongly reduced. That is, for agiven central radiation level the impurity density at the edge
can be maintained at a substantially higher level than in the
neoclassical picture. This may facilitate impurity removalfrom the plasma edge. A further consequence of this model isthat the plasma particle outflux to the walls, which is linearly related to the oxygen desorption rate, would also have to
be increased by about an order of magnitude with respect tothe neoclassical transport case.
Unfortunately, the experimental information; available on theedge plasma of these discharges was not sufficient to decide

RINGLER et al.

608

exp.(CXRS)#61678-61688

60

80

\

\l

100

Time (ms)FIG. 3. Central O density (activity units) versus time from active CXRS measurements in comparison with code calculations.

whether the high oxygen influx is a reasonable assumption. On
the other hand, the code calculations show that the higher
oxygen density at the plasma edge resulting from the increased
oxygen influx would still be too small to give an observable
increase of the bolometer signal. Uncertainties would also
arise in checking the simulations with spectroscopic
measurements of the lower oxygen ionization states. In fact,
the enhanced transport would lead to an outward shift of thehigher ionization states, thus decreasing the density of thelower states with respect to their values at coronalequilibrium. This would compensate to a large extent thedensity increase due to the higher fluxes at the edge.
Concerning the possibility of an impurity flow reversal
mechanism associated with the ECR heating process /3/, it is
not yet clear at present whether or not the required additional poloidal electric field was sufficiently high for the
W VII-A discharges discussed here to create outward drift contributions compensating to a large extent the neoclassicalinward drift. However, even if this were the case, the outward
drift would have essentially the same consequences as theassumed enhanced diffusion with respect to the high impurity
sources required to maintain the central radiation level.
The evolution of the 0VII + 0VIII radiation resulting from thecode calculations discussed is shown in Fig. 2b. At the latestage of the discharge, a drop of the measured soft X-radiation is observed (Fig. 2a), in evident contrast to thesimulation, which shows a steep increase of the 0VII + 0VIII
radiation. Such an increase of oxygen radiation is essentiallya consequence of the T e drop (&Te = 140 eV within 10 ms)
observed after switching off the ECRH pulse. In order toreduce the discrepancy, additional high-Z material was
included in the simulations for this discharge type (Fig. 2c).

609

IAEA-CN-47/D-V-1(a.u.)

code

collisional excitation)#61678-61688

60

80

100

120

Time (ms)FIG. 4. Flux of O VIII (2976 A) line intensity (activity units) versus time from electron excitation incomparison with code calculations.

Such impurities could originate from, for example, sputteringof plasma ions at the vacuum vessel (Fe, Cr) and sputtering at
the limiter and NI beam dump plates (Mo)/4,5,6/. By assuming a
Fe wall influx of 4 % of the oxygen influx, the measured
central radiation could be qualitatively reproduced.
For the same discharge type, the evolution of the 0%+ density
has been measured by CX recombination spectroscopy (active
signal) / 2 / . On the other hand, the passive signal, obtainedwith the diagnostic beam off, yields a flux which is proportional to the 0 7 + density / 2 / . The time histories of bothsignals are reproduced reasonably well by the simulations(Figs. 3,4). In particular, the decrease of the 08+ density
related to the T e drop observed after switching off the ECRH
pulse is clearly confirmed by the CX recombination measurements.

Electron Confinement
As shown above, Al introduced by laser ablation into a combined
NBI and ECRH discharge is lost faster than neoclassically.
(Similar results are reported for Si from the Heliotron E
experiment /7/.) Using the same transport coefficients for theoxygen transport to model the total radiation requires anincreased oxygen influx from the walls.
In this section the confinement behaviour of the backgroundplasma, protons and electrons during this phase will be
discussed.
Starting from ECRH target plasmas NBI leads to density increasesof N e =*• 0.5 - 2 x 1 0 2 0 s" 1 , as in NBI discharges starting from OH
target plasmas, which can be explained by the total particle

610

RINGLER et al.

Influx from the neutral beams (ECRH switched off after transition
phase). There is no saturation of the density increase during theinjection phase, which leads to particle confinement times in the'Cp^SlOO ms range. Impurity transport and particle confinementshow no difference from those measured earlier / I / .
With an additional ECRH pulse during NBI, however, the line
density increase levels off and stays constant, with asignificant increase in the He* signal. An influence of ECRH on
the time evolution of the density has also been observed intokamaks /8,9/. At electron densities n e o well below n e cr it.»core heating is observed, while for n e o approaching n e c r i f »edge heating is seen from ECE /10/, but even for n eo r» n e c r i t .
during ECRH /n e dl stays constant. Two possible explanations forthis behaviour have been given / 8 / :a) In addition to impurity confinement, the electron confinement
is also decreased, orb) the gas flux from the walls is ionized by ECRH power outside
the last closed flux surface, reducing the gas flux
penetrating through the scrape off layer and thus allowing
constant density with electron confinement still unchanged.
So far, no clear-cut experimental proof for either one of these
explanations can be given. In this position a discussion of both
possibilities in a more or less phenomenological way seems
appropriate.
a) - Looking at the time history of the H^ signal there isincreased recycling (or loss of particles) even in the early
ECRH target phase, when NBI starts and external fluxes are
switched off (Fig. 5 ) , with no density increase. After ECRH
shut-off the density increases strongly with a sharp drop in
the H<^ signal. With the second ECRH pulse applied later inthe discharge jn e dl stays constant and H^ increases, thisbeing followed by a drop and a corresponding density
increase after the ECRH is switched off again.
- At the line density under consideration (5.7 x 1014 C m ~ 2 ) alarge fraction of the beam particles 20 x 1 0 1 9 s"1 (63 % of
the NB power) hit the molybdenum beam dump. Owing to asurface temperature of C? 400°C all beam particles are
expected to be released from the beam dump during the 100 ms
interval of injection /ll/ and used to refuel the plasma.
The source of fast ions, produced mainly by CX, amounts to
12 x 10!9 B " 1 . Only a source of S e l e c t r o n s ~ 7 x 1 0 1 9 s"1 isestimated to lead to particle production, with 1/3 of thehalo neutrals assumed to be reionized in addition to 1/3 of611

IAEA-CN-47/D-V-1
WVII A

ECRH 70GHz + NBI
O-ModeBo= 2.5 T#61364 - 61385

15o

2oo

FIG. 5. Ha signal and line integrated electron density versus time. Ha signal taken at position of
ICRH antenna.

the flux of fast ions which is created by proton andelectron collision rather than by charge exchange.
A flux of 10 x 1()19 s"1 fast ions is lost via orbit losses(ODIN code calculations, including electric fields), is
deposited in the SS wall of the vacuum vessel and is notreleased during the injection interval /9/. The reflux fromthe wall and limiter due to plasma losses with an expectedparticle confinement time of TpCslOO ms during the pure NBI
phase and a recycling coefficient of 0.7(N e> total = 8 « 4 x 1 0 1 8 ) is also small compared with the flux
from the beam dump, -which thus appears to be the main
particle source.
Now, if during the N e CT 0 phase with ECRH applied a particle
confinement time of T^ cr 100 ms were to be maintained, a612

RINGLER et al.
flux of 2.5 x 1 0 1 9 s"1 particles would be sufficient, which
is of the order of the flux of particles directly producedby the beam.
The ECRH power would therefore have to decouple the largehydrogen flux out of the beam dump from the plasma almost
completely.
Density profiles during and after ECRH, however, show littlechange and thus do not support this explanation. In addition, the ionlzation length by electron collisions forthermal hydrogen atoms released at the beam dump is longenough to penetrate into the non-ergodic region, whereparticles are confined on flux surfaces. This would holdeven for electron densities up to lO* 2 ^,-3 w i t n i n theergodic region, because <<? v> is only weakly dependent on
temperature.
It is therefore not very likely that the neutral flux fromthe outside is strongly reduced by ECRH.

b) - Poloidal plasma rotation shows no change during the ECRH
pulse, so radial electric fields which are associated withthe rotation still indicate good confinement.
- By application of an additional external gas flux Jn e dl canbe made to increase again during ECRH, which, however, would
also be true with reduced particle confinement.

Conclusion
The technique of an additional ECRH pulse applied to a NBIsustained discharge has some very interesting aspects:- Stationary discharges can be maintained with good energy
confinement at tolerable radiation levels, whereas without
additional ECRH pulse the density increases linearly with timealong with high radiation losses due to impurity accumulation.
- There is experimental evidence that the electron particle
confinement also deteriorates.
- For the ablated Al a faster loss of impurities from the central
region is observed.

IAEA-CN-47/D-V-1

613

REFERENCES[1] W v n - A TEAM, NI GROUP, Nucl. Fusion 25 (1985) 1593.
[2] KICK, M., RINGLER, H., SARDEI, F., WELLLER, A., et al., in Controlled Fusion andPlasma Heating (Proc. 13th Europ. Conf. Schliersee, 1986), Vol. 10c, Part I, European PhysicalSociety (1986) 319.
[3] STACEY, W.M., The Effect of Poloidal Electric Fields on Impurity Asymmetries and Transportin Tokamaks, Rep. GTFR 63, Georgia Institute of Technology Fusion Research (1986).
[4] W VD-A TEAM, NI GROUP, in Fifth International Workshop on Stellarators (Proc. IAEA
Tech. Comm. Mtg on Plasma Confinement and Heating in Stellarators, Schloss Ringberg),
Vol. 1, CEC, Directorate-General XII — Fusion Programme, Brussels (1984) 259.
[5] BOGEN, P., SCHWEER, B., RINGLER, H., OTT, W., L Nucl. Mater. 111&112 (1982) 67.
[6] WELLER, A., Measurements of the Radiation Energy Distribution in the Soft X-Ray Region in
the Wendelstein VII-A Stellarator, Rep. IPP 2/277, Max-Planck-Institut fur Plasmaphysik,
Garching (1985).
[7] UO, K., paper IAEA-CN-47/D-M, these Proceedings, Vol. 2.
[8] ALIKAEV, V.V., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc.
10th Int. Conf. London, 1984), Vol. 1, IAEA, Vienna (1985) 419.
[9] FOM ECRH TEAM, TFR GROUP, in Controlled Fusion and Plasma Physics (Proc. 12th Europ.
Conf. Budapest, 1985), Vol. 12, European Physical Society (1985) 60.
[10] ERCKMANN, V., et al., Plasma Phys. 28 (1986) 1277.
[11] MOELLER, W., SCHERZER, B., private communication, 1986.

IAEA-CN-47/D-V-2

STABLE ACCESSTO SECOND STABILITY REGIMEIN LARGE-ASPECT-RATIOTORSATRON/HELIOTRON CONFIGURATIONS*
B.A. CARRERAS, L. GARCIA
Fusion Energy Division,
Oak Ridge National Laboratory,
Oak Ridge, Tennessee
L.A. CHARLTON, J.A. HOLMES, V.E. LYNCH
Computing and Telecommunications Division,
Martin Marietta Energy Systems, Inc.,Oak Ridge, Tennessee
P.H. DIAMOND
Institute for Fusion Studies,
The University of Texas at Austin,
Austin, Texas
United States of America
Abstract
STABLE ACCESS TO SECOND STABILITY REGIME IN LARGE-ASPECT-RATIOTORSATRON/HELIOTRON CONFIGURATIONS.
Shear stabilization caused by beta-induced modification of the transform profile can give stable
access to the second stability regime for Heliotron configurations to which an external toroidal field is
added. This stabilization mechanism is effective for free boundary and fixed boundary modes. The
existence of resistive instabilities and their implications for anomalous transport are also discussed.
Saturated resistive pressure-gradient-driven turbulence is studied, and an accurate saturation criterion
is proposed. The resulting predicted pressure diffusivity scales in a similar way as the mixing length
estimate, but it is significantly larger in magnitude.

1. EFFECT OF TOROIDAL FIELD COILS ON THE STABILITY OF
HELIOTRON CONFIGURATIONS
Two alternative routes to the second stability regime are possible for
torsatron devices. One, for small-aspect-ratio configurations, relies on the
deepening of the magnetic well with increasing /?. This /? self-stabilization
mechanism [l] has been studied in detail for some configurations [2|, andResearch sponsored by the Office of Fusion Energy, United States Department of Energy,
under Contract No. DE-AC05-85OR21400 with Martin Marietta Energy Systems, Inc., andunder Contract No. DE-FGO5-80ET-53O88 with the University of Texas.

615

616

CARRERAS et al.

the ATF device [3,4] has been designed to explore it. The second alternative, for higher aspect ratio configurations, relies on the increase of shearwith increasing /?. This stabilization mechanism should be able to eliminate the most unstable modes in Heliotron-type configurations and permit
the achievement of high-/? plasmas.
If one relies on the increase of shear with /?, it is necessary to allow theequilibrium to relax to a true stellarator (zero-current) equilibrium andlet the transform profile change accordingly. Therefore, slow heating andlong pulses are essential in gaining access to the second stability regime
for this type of configuration. It is also important to select the transformprofile in a way that places the most unstable surfaces in the high-shear
region. This can be done for the Heliotron E configuration [5] by using
toroidal field coils [6).
The studies presented in this paper are based on the stellarator expansion approach [7] to the equilibrium and stability of three-dimensional(3-D) configurations. For planar axis configurations, the stellarator expansion compares favorably with 3-D calculations [2,8]. Therefore, it is a
useful approach for magnetohydrodynamic (MHD) studies. Details of theequations and numerical methods used for the calculations presented here
can be found in Ref. [9]. For fixed-boundary calculations, the equilibrium
equations are solved by the RSTEQ code and the linear stability equations
by the full implicit FAR code. The latter solves the MHD equations forstellarator configurations derived by extending the stellarator expansion
to the dynamical evolution problem [10]. Free-boundary stability has beenstudied using the STEP code described in Ref. [11]. This code uses a 8W
approach, with an optional conducting wall that can be placed at any distance from the plasma. This free-boundary code is a modified version of
the tokamak MHD equilibrium and stability code PEST [12], which uses
the stellarator expansion.
To study the effect of an additional toroidal field (BT), we have added
19 circular toroidal field coils to the standard helical coil Heliotron E model
J6]. They are located at the beginning of each field period, and their radius
is ax = 0.59 m. The addition of this toroidal field to that generated by
the helical coils (Bo) reduces the rotational transform v and increases theplasma minor radius. In general, we have assumed that a limiter is used
to keep the plasma aspect ratio constant. The reduction of *• results in
an increase in the magnetic axis shift at finite (3. The larger shift gives a
deeper magnetic well. This affects the plasma core stability but not thestability at the i- — 1 surface.
The stability properties of the n = 1 internal mode are very sensitive
to the magnitude of the added toroidal field. When the added toroidal field
is 15% of the total field, the n = 1 internal mode is stable for all 0 values
considered. As in the case of the standard Heliotron E configuration, thedominant component of the n = 1 mode is the (m = 1; n = 1). This
component, for 0 values near the threshold (/?o ~ 3% for BT/Bo < 10%),is very localized radially. It is then reasonable to assume that the stability
properties of the n = 1 mode depend mostly on local quantities at the v —1 surface. For a constant /? value and increasing BT/BO, the linear growthrate decreases, the shear increases, and V" becomes increasingly positive.
Therefore, since the overall effect is stabilizing, the shear stabilization isthe dominant effect. As the n = 1 internal mode is stabilized by the effectIAEA-CN-47/D-V-2

617

0.25

0.20

0.15
o
01

0.10

0.05

FIG. 1. Fixed boundary stability:
Constant linear growth rate contour plots.

of the added toroidal field, the critical stability /? for this mode becomes
higher. All these effects are summarized in Fig. 1, where lines of constantn = 1 linear growth rate in the /? - BT plane have been plotted. This
figure also shows the critical equilibrium /3 contour (dotted line). For
BT/BO ^ 0.15, the /3 limitation is due only to equilibrium failure. In thissituation, peak 0 well above 10% could be achieved in the Heliotron E
device.
The results shown in Fig. 1 are for a strongly unstable pressure profile,p oc xj)2. For more favorable profiles, the toroidal field required to stabilize
the n = 1 mode is much smaller. For instance, for the p <x rp3 pressure
profile.only a 5% increase of the toroidal field is required. In this case,even higher values of j3 could be attained in the Heliotron E device.
Internal pressure-driven modes, when they are unstable, have growth
rates that increase with n. Therefore, higher n modes have higher growth
rates than the n = 1 mode. However, the numerical results show that
their linear instability threshold is close to the n = 1 mode threshold. We
have studied the stability of Heliotron E plasmas to n = 2 modes withvarying toroidal fields. The marginal stability contours in the /? - BT
plane are nearly identical to those for the n = 1 mode. Therefore, thecalculated marginal stability contours for the n = 1 mode can be taken as
the marginal stability contours for all n modes.
We can also see from Fig. 1 that the self-stabilization effect can bestrong enough to totally stabilize the n — \ mode and not merely reduce

618

CARRERAS et al.
0.08

0.05

0.10

0.16

0.20

0.25

FIG. 2. Free boundary stability:
Linear growth rates for fixed $0 = 8%.

its linear growth rate. The cause for the /? self-stabilization effect is thesame as for the added toroidal field. As /? increases, the transform changes
to maintain zero current in each flux surface. In doing so, the transformat the magnetic axis increases, that at the plasma edge decreases, andthe whole profile changes. This change produces an increase of the shear
at the * = 1 surface. As in the case of the addition of a toroidal field,
V" stays positive at the * = 1 surface and increases with /?. Therefore,
the second stability region in Heliotron-type configurations is again due
to shear stabilization and not to the deepening of the magnetic well.
Let us now consider stability to external modes. As the conductive wall
is moved away from the plasma, the n = 1 linear growth rate increases,
particularly at high 0 (/?o > 6%). The eigenmode structure also changes,
becoming less localized and more like a kink mode. The qualitative effectof removing the conducting wall is very similar to that observed for ashifted-in ATF-type configuration [13].
The effect of the added toroidal field on the free-boundary modes isanalogous to the effect on the internal modes (Fig. 2). There is a stabilization of the internal modes with increasing BT/BQ.
However, somewhat
larger values of BT/BO are required for full stabilization when the conducting wall is absent. In Fig. 2, the result for the linear growth rate
without the conducting wall (open circles) is contrasted with the casewith the conducting wall at a distance of half the plasma radius (opensquares). The marginal points for the latter are very close to those for theinternal modes. This situation is probably the most relevant one for theexperiment. Therefore, the conclusions drawn from internal mode stability are not modified by the most realistic assumptions on the position of
the conducting wall.

IAEA-CN-47/D-V-2

619

2. EFFECT ON CONFINEMENT OF THE RESISTIVE PRESSUREGRADIENT-DRIVEN TURBULENCEFor configurations relying on shear stabilization to access high /?, aquestion remains: resistive mode stability and its implications for confinement. The resistive interchange in particular is believed to be a potential
cause of anomalous transport [14] in devices with average unfavorable curvature, which is the case for the torsatron near the plasma edge. Thus,
it is important to consider the effects of resistive pressure-gradient-driven
turbulence.
Previous work on the theory of resistive pressure-gradient-driven turbulence has consisted primarily of the application of mixing length arguments [15] and dimensional analysis techniques [16]. Mixing length arguments, such as D = 7/fcj, attempt to relate the properties of fully evolved
turbulence to the characteristic scales of the underlying linear instability
by semiquantitatively balancing nonlinearity with the linear drive. Dimensional analysis techniques use the scale transformation symmetries ofthe basic, nonlinear equations. Both yield similar results for predictedfluctuation levels, diffusivities, etc. These predictions are in quantitative
agreement with the results of numerical simulations performed for relatively high resistivity and high velocity. The agreement was particularly
good for predicted scalings with /?p and dpo/dr. However, when compareddirectly with experimental results, the predicted thermal diffusivities were,in general, too small to explain the observed phenomena.
There are several deficiencies in the theoretical underpinnings of themixing length and dimensional analysis approaches. First, mixing length
predictions are derived from a heuristic procedure of balancing nonlinearity with linear drive. In particular, no genuine saturation criterion basedon considerations of energetics is satisfied or even established. Second,the dimensional analysis approach, as implemented in Ref. [16], omits
the effects of dissipation and the role of scales dominated by dissipative
damping. The omission of damping, which is analogous to the lack of awell-defined saturation criterion, explains why the (zero damping) mixing
length predictions agree with those of dimensional analysis.
The simple model of the resistive interchange instability used in thispaper contains the main physics properties of the instability, while taking
the average pressure gradient as the only source of free energy. The model
consists of two equations:^ v ^ V
at

t]p

»

+

ifvnx^J+MV^p= X J _Vip

dt

X±

l P

L

(i)

J

(2)
r 80 drK'

Here V, ' means that only the equilibrium magnetic field is included in
the parallel derivative (electrostatic approximation). A detailed analysisof this modela r eis given in Ref. [17]. The fluid viscosity fj, and thermal diffusivity x x
retained in the analysis and calculations. The viscosity
and thermal diffusivity guarantee the existence of a stable large-m dissipation range, which serves as an energy sink. In the numerical calculation

620

CARRERAS et al.

the electromagnetic model is considered, and the pressure gradient is held
fixed, thus avoiding possible confusion of quasi-linear and nonlinear effects.
Sufficient large-m resolution is retained, and the existence of a saturated
state is demonstrated by well-converged numerical calculations.
The resistive pressure-gradient-driven turbulence saturates by themechanism of nonlinear transfer of energy from large to small scales. An
analytical theory of such a process requires a renormalized two-point theory of the fluctuation spectrum to describe the nonlinear coupling of thelow-m energy-containing region to the large-m dissipation range. In thecase when a few low-m modes dominate the energy spectrum, an alternative, more tractable calculation is to determine the energy outflow fromthe low-m modes required for saturation. Such a tactic is implemented
by deriving low-m renormalized response equations and then determining
the level of diffusion, representative of the spectrum-averaged rate of coupling to smaller scales, required for low-m saturation. The latter method
is used in this paper.
The calculation of energy outflow from low-m modes is only meaningful if there is sufficient large-m dissipation so that a saturated state isattainable. Dissipation in both the vorticity and pressure evolution equations is required for a nontrivial stationary state (dpo/dr ^ 0). A related
requirement is that viscous dissipation and resistive diffusion are requiredin the vorticity evolution equation. The need for viscous dissipation is aconsequence of the fact that nonlinear transfer to large m ultimately couples to modes with m numbers for which mjr > Wfn'. Such m numbers
correspond to modes in the fast interchange regime, with (ideal) growth
rates for which resistive field line diffusion is dynamically irrelevant. Thus,
substantial viscous dissipation is required for the necessary stabilizationof short-wavelength interchange modes.
Renormalized response equations, which describe the nonlinear dynamics of resistive pressure-gradient-driven modes in the presence of turbulence, can be derived (I7j. For long-wavelength modes, a saturation
condition is established by setting the time derivative terms to zero. Then,

i

VPm

A ^

Pm <"

= -i&kytt

fc

(3)
(4)

where the nonlinear diffusion and viscosity coefficients are

DXX = £( £ ) <^,r-

(5)

m',n'

Here, T^,, and F2£,, are the propagators for the "beat mode" k". They
can be approximated by the decorrelation rate, which corresponds to the621

IAEA-CN-47/D-V-2o
D

NUMERICAL RESULTANALYTICAL RESULTMIXING LENGTH THEORY

0.005

0.010

0.015

FIG. 3. Nonlinear diffusion coefficient.

characteristic rate of nonlinear interaction for resistive pressure-gradientdriven turbulence.
Assuming that there is a known relationship between (ixx and Dxx,
Eqs. (3) and (4) define an eigenvalue problem. An approximate analyticalsolution valid for low m and low resistivity is(7)
where 7™ is the linear growth rate andthe linear mode width inabsence of collisional dissipation. The factor A,

3TT

Lvr

(-Y-

(8)

is an enhancement factor over the mixing length theory prediction.
The main properties of the nonlinear diffusion coefficient Dxx are as
follows. (1) The scaling with /? is very close to linear, as in the mixing
length prediction (Fig. 3). (2) The nonlinear Dxx is about an orderof magnitude larger than the mixing length result due to the A2 factor(Fig. 3). (3) The correction factor gives a weak dependence on m, close
to m" 1 . (4) The dependence on fj. and Xj. is weak for realistic values ofthese parameters. Once the relation between the coefficients Dxx and (j,xx622

CARRERAS et al.
(00"I

1—I I I I 11

1.0

J.

I'

0.1

i'f

la.

O NUMERICAL RESULT

tv r : • ANALYTICAL RESULTO NUMERICAL RESULTANALYTICAL RESULT
1
O.OCM

i

i

i

i

i

i

i i \

0.01
c,2

0.01

FIG. 4. Comparison between analytic andnumerical results for saturation levels.

is known, we can estimate the turbulence level at saturation and the main
physics parameters characterizing the turbulent state. In this way, we findthat the mean square radial velocity and the rms pressure fluctuation are
(9)
dpp
dr

(10)

The characteristic nonlinear width of the velocity fluctuation is A m =Wm 'A», and the decorrelation rate is Au m = ^ ' A i .
The nonlinear stability of a sequence of equilibria with different valuesof /?o/2e2 has been numerically studied. The values of the Vr and p fluctuations at saturation are given in Fig. 4. These values have been calculated
by averaging over time, and the error bar corresponds to one standard
deviation. The analytical results are calculated with Eqs. (9) and (10).
The values for (m2) used in those equations are taken from the numerical results. The scaling with 0 is very well reproduced by the analyticalmodel, and the agreement shown in Fig. 4 is good. The importance of thelogarithmic factor modifying the mixing length is clear.

IAEA-CN-47/D-V-2

623

The losses caused by the induced electron heat conductivity can be
estimated by modifying the result of Ref. [14]. Using the nonlinear width
A and the decorrelation rate Awm instead of the linear width and linear
growth rate, the calculatedL" ^ 0hea^(11)

The enhancement of Xe at finite /? can be up to an order of magnitude.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]

MIKHAILOVSKII, A.B., SHAFRANOV, V.D., Zh. Ehksp. Teor. Fiz. 66 (1984) 190;
Sov. Phys. - JETP 66 (1974) 88.
CARRERAS, B.A., et al., Phys. Fluids 26 (1983) 3569.
LYON, J.F., CARRERAS, B.A., et al., Fusion Technol. 10 (1986) 179.
CARRERAS, B.A., HICKS, H.R., HOLMES, J.A., LYNCH, V.E., NEILSON, G.H., Nucl.
Fusion 24 (1984) 1347.
UO, K., Plasma Phys. 13 (1971) 243.
CARRERAS, B.A., GARCIA, L., LYNCH, V.E., Phys. Fluids 29 (1986) 3356.
GREENE, J.M., JOHNSON, J.L., Phys. Fluids 4 (1961) 875.
HERRNEGGER, F., MERKEL, P., JOHNSON, J.L., J. Comput. Phys. 66 (1986) 445.
LYNCH, V.E., CARRERAS, B.A., CHARLTON, L.A., HENDER, T.C., GARCIA, L., et al.,J. Comput. Phys. 66 (1986) 411.
STRAUSS, H.R., Plasma Phys. 22 (1980) 733.
ANANIA, G., JOHNSON, J.L., Phys. Fluids 26 (1983) 3070.
JOHNSON, J.L, et al., J. Comput. Phys. 32 (1979) 212.
HENDER, T.C., et al., Nucl. Fusion 25 (1985) 1463.
SHAING, K.C., CARRERAS, B.A., Phys. Fluids 28 (1985) 2027.
AN, Z.G., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc. 10th
Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 231.
CONNOR, J.W., TAYLOR, J.B., ibid., p. 13.
CARRERAS, B.A., GARCIA, L., DIAMOND, P.H., Theory of Resistive Pressure-GradientDriven Turbulence, Rep. ORNL/TM-10150, Oak Ridge National Lab. (1986).

1AEA-CN-47/D-V-3

THREE-DIMENSIONAL MHD STUDIES OFIDEAL AND RESISTIVE INTERCHANGE MODESIN HELIOTRON CONFIGURATIONS
M. WAKATANI, K. ICHIGUCHI, H. SUGAMA, K. ITOH
Plasma Physics Laboratory,
Kyoto University,
Kyoto, Japan
A. HASEGAWA1
AT&T Bell Laboratories,
Murray Hill, New Jersey,
United States of America
J. TODOROKI, H. NAITOU
Institute of Plasma Physics,
Nagoya University,
Nagoya, Japan

Abstract
THREE-DIMENSIONAL MHD STUDIES OF IDEAL AND RESISTIVE INTERCHANGE MODESIN HELIOTRON CONFIGURATIONS.
In Heliotron E high beta experiments, the highest average beta value, # = 2%, is obtained fora flat profile, /3(0)/$ £ 1.5, while pressure driven instabilities are observed for peaked profiles. The
beta value at the magnetic axis is denoted by /3(0). By measuring density fluctuations and soft X-ray
fluctuations, sawtooth oscillations accompanied by an m = 1/n = 1 mode are observed for13(0)/$ = 2.5—3; an internal disruption accompanied by an m = 3/n = 2 mode is observed for0(O)//3 > 3—4. These fluctuations are well explained in terms of non-linear evolution of the resistive
interchange mode (G mode) under the assumption of multi-helicity. — Transport analysis of Heliotron
E predicts anomalous transport in the edge region; a candidate for explaining the transport is G mode
turbulence. Model equations including the electron temperature Larmor radius effect are derived. The
linear growth of the G mode with the electron diamagnetic frequency is examined, and threedimensional non-linear calculations of density and potential fluctuations are performed to obtain a saturated state for a fixed background density profile in the electrostatic approximation. — To improve thebeta limit of heliotron configurations in the context of ideal MHD, the effect of an additional verticalfield applied to shift the magnetic axis, on the ideal interchange instabilities, is studied by two
approaches: three-dimensional MHD equilibrium and stability code (BETA code) calculations and HERATO code calculations, which are a modification of the ERATO code for tokamaks, based on theassumption of the averaging method for stellarator fields. According to the BETA code, them = 1/n = 1 internal mode is completely stabilized by the outward shift of the magnetic axis for/3 •< 5 -6% in heliotron configurations. The MHD equilibria for the H-ERATO code are calculated bya code which is able to optimize the pressure profile according to the Mercier criterion and the local

1

Co-operative research supported by the United States Department of Energy, underContract No. DE-F4O580ET53O88.

625

626

WAKATANI et al.

ballooning stability. This study shows that stability against the low mode number interchange instability
is correlated with the local mode stability in the Heliotron E case. Improvement of the beta limit by
the additional vertical field is also confirmed by the H-ERATO code.

1.

m = 3/n = 2 RESISTIVE INTERCHANGE MODE DRIVEN INTERNALDISRUPTION

A recent high beta experiment in Heliotron E reveals that the density fluctuations as measured by the FIR laser and correlated with the internal disruption havea spatial structure which is not explained by the m = 1/n = 1 mode only [1]. In thepaper discussing the first high beta plasma confinement, both the pressure driven
sawtooth oscillation and the internal disruption in the soft X-ray signals were related
to the m = 1/n = 1 mode, where /3(O)//3 * 2 . 5 - 3 and 3.6% > 0(0) > 2% [2].
After the neutral beam injection power has increased from 2 MW to 4 MW, the density profile peaking is enhanced in the case of weak or no gas puffing. When density
fluctuations with m = 3/n = 2 mode structure were observed for j3(O)//3 > 3 - 4
and /3(0) > 2%, internal disruption or flattening of the density profile occurred successively. To examine the relationship between m = 3/n = 2 and m = 1/n = 1
modes, we extend the numerical studies of the non-linear evolution of them = 1/n = 1 resistive interchange mode in Ref. [3] to multi-helicity calculations.
Line integrals of the pressure profile are compared with the experimental observationof the density fluctuations, by assuming uniform temperature.
For a peaked profile, the linear growth of the m = 3/n = 2 mode is faster thanthat of the m = 1/n = 1 mode [4]. Thus, the non-linear behaviour of them = 3/n = 2 mode becomes dominant for the pressure profile p = p0 (1 - r 2 ) 4 and/3(0) = 3%, as is shown in Fig. 1. Here, the magnetic Reynolds number S is
assumed to be 5 x 103; twenty modes given by the mode selection rule introducedin the multi-helicity non-linear tearing mode calculation are included [5]. Figure 2
demonstrates the relative density fluctuations at T = 288, normalized by the poloidal
Alfv6n transit time, and the density profile along the 6 = 0° line, where 6 is a poloidal angle. The peaked density profile is dominantly flattened by the non-linear evolution of the m = 3/n = 2 mode. Figure 2 agrees with the experimental data ofRef. [1].
Thus, the internal disruption caused by the m = 3/n = 2 resistive interchange
mode resonant at the i = 2/3 surface limits the central beta value for highly peakedpressure profiles in Heliotron E. The beta limit is, however, improved by tailoring
the pressure profile to obtain broad profiles [6].
2.

FINITE LARMOR RADIUS (FLR) EFFECTS ON G MODE

In setting up the FLR-MHD equations, the FLR terms are introduced throughboth the ion pressure and the electron pressure gradients [7]. The former FLR effect627

IAEA-CN-47/D-V-3

0.00

80.CO

tSO.OO

2U0.00320.00400.00

ISO.00560.00TIME
FIG. 1. Evolution of kinetic energy of multi-helicity G mode for strongly peakedpressure profile in H-E.

was examined in the case of an ideal interchange mode in Heliotron E; a destabilizing
effect of FLR was pointed out [8]. Here, we are interested in the latter FLR effecton the G mode. Reduced MHD equations including the parallel electron pressure gradient in Ohm's law, which introduces the electron diamagnetic frequency, can bederived on the basis of the stellarator expansion and an averaging method [9],In the electrostatic limit, Ohm's law yields
ji = — V, (<p-n)v(1)where j j , <p and n are the parallel current normalized by the sound velocity,
jj/noeCs, the normalized electrostatic potential e<p/T and the normalized density
n/no, respectively. The temperature T is assumed to be constant, v is the normalized
collision frequency, and e = a/R. By using Eq. (1), the continuity equation is
written ase _dt= — V]

v

,

(n-<

(2)

628

WAKATANI et al.

(a)

(b)

FIG. 2. (a) Density profile along 9 = 0° line and (b) line density fluctuation profile at T = 288.

and the equation of motion is reduced to

-

? (n-

[V

(3)

where the perpendicular length is normalized by a, the parallel length is normalized
by R, and the time is normalized by o)ci(ft/a)h. Here, p is Larmor radius measured
by the electron temperature and wci is the ion cyclotron frequency. The brackets are
Poisson brackets. The curvature term, Q, is given by(2 = Ne/l \—2 r2t(r) + I

Jort(r)drwhere t is the pole number and N the pitch number. When the curvature term isneglected, Eqs (2) and (3) become the model equation derived by Hasegawa andWakatani [10, 11] for resistive drift wave turbulence. A cylindrical plasma model is
employed and the rotational transform is assumed as t = 0.51 + 1.69r25. Figure 3
shows a linear mode structure of the G mode of m = 1/n = 1, including the electron
diamagnetic effect for v = 10' 3 and p/a = 0.02. The potential and density pertur-

629

IAEA-CN-47/D-V-3

•i.a

o.a
RADIUS

FIG. 3. Linear mode structure of G mode including FLR effect with m =l/n = 1.

bations become close to the Boltzmann relation n « <p and show diamagnetic rotation
with an angular frequency of cot = 0.6445. The growth rate is (a-, = 2.222 andapproximately scales like vm. Non-linear calculations for the G mode turbulence
described by Eqs (2) and (3) are performed under the assumption of a fixed background density profile. When low mode resonant surfaces such as i = 1 and i = 2/3
exist inside the plasma, no saturation of density and potential fluctuations was
obtained without quasi-linear flattening of the background density profile in thepresent calculation using 59 modes. For a configuration with t = 0.68 + 0.27r2
and n(r) = no (0.7exp(-2r 2 ) 4- 0.3}, a saturated turbulent state was obtained byadding a small diffusion term on the right hand side of Eq. (2) for the fixed background density profile. The wavenumber spectrum is broad, and the average fluctuation level is 5/% « 0.06.

3.

BETA LIMIT IMPROVEMENT OF HELIOTRONS

The problem of MHD equilibrium and stability in fully three-dimensional
geometry can be treated numerically by applying the classical variational principleof MHD. This is implemented in the BETA code [12]. Stability against them = 1/n = 1 internal mode is investigated in an H-E model configuration for currentless equilibria [13]. The beta limit in the first stability region is /3 » 2%, whichis consistent with the maximum beta value obtained for H-E. For i3 > 5 %, a second

630

WAKATANI et al.

0.16-

0.08-

-0.08

FIG. 4. Eigenvalue of m = lln = / mode versus average beta in H-DR. Solid line shows case ofstandard configuration; dashed line shows case of outward magnetic axis shift.

0.16-

"b

0.08-1-0.08
FIG. 5. Eigenvalue of m = lln = 1 mode versus average beta in H-H. Solid line shows case ofstandard configuration; dashed line shows case of outward magnetic axis shift.

stability region appears. Stability against the same mode was examined for H-E currentless equilibria modified by an additional vertical field. By introducing about 15%
shift of magnetic axis with respect to the averaged radius, the m = 1/n = 1 internal
mode is completely stabilized. The same stability examination was also tried for otherheliotrons p = p 0 (1 - r 2 ) 2 in the straight approximation other heliotrons: HeliotronDR(H-DR) and Heliotron H(H-H). H-DR has R/a = 14 and ( = 2/N = 15, wherei(0) « 0.8 and t(a) « 1.3. The first stability beta limit of 0.8% is smaller than thatof H-E (see Fig. 4). Here, negative - w 2 corresponds to instability. By applying an
additional vertical field to shift the magnetic axis outside, the m = 1/n = 1 internal
mode disappears (as is shown in Fig. 4).
It is well known that the m = 1/n = 1 internal mode becomes dangerous when
the t = 1 surface is inside the plasma. When it is removed into the outer region, an

631

IAEA-CN-47/D-V-3

rv1

M

0.10

s

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K

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0.0 0.13 027 0.40 0.53 0.67 0.80 033
S

0.00-0.02 x
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O
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SFIG. 6. Distributions of destabilizing potential energy (K), dominant stabilizing potential energy (S)and total potential energy (W) for n = I mode (left) and n = 2 mode (right). M designates regions
violating Mercier criterion.

example of an 2 = 2 torsatron is the ATF. On the other hand, an example of a heliotron with rotational transform larger than unity, i(0) « 1.2andt(a) = 1.6, isH-H;
it also has £ = 2/N = 15 and R/a = 14. In Fig. 5, the first stability beta limit is/3 = 3.2%, which is determined by the non-resonant mode of m = 1/n = 1. H-H
is more stable than H-E. This may be due to the removal of the i = 1 surface. The
stabilization by the outward shift of the magnetic axis also works in H-H, as is shownby the dashed line in Fig. 5.
Thus, the BETA code predicts that the most dangerous mode in heliotron configurations can be suppressed by appropriately shifting the magnetic axis outside. The
stabilizing mechanism is magnetic well formation by the finite beta effect.
We have developed a new formulation of the averaging method on the basis ofan asymptotic expansion of the MHD equations with a co-ordinate transformation
technique [14]. The H-APPOLO and H-ERATO codes have been developed forequilibrium and stability analyses, respectively. H-APPOLO calculates free boundary currentless equilibria, tests the ballooning and Mercier criteria on each magnetic
surface, and is able to produce an optimized pressure profile for local modes. HERATO is the helical torus version of the ERATO stability code for tokamaks
designed to test low n global mode stability [15].
First, the low n mode stability of H-E free boundary currentless equilibria is
examined. A beta limit smaller than $ = 2% is given by the n = 1 mode localizedaround the i = 1 surface, when the pressure gradient on this surface is substantial.
When the pressure profile becomes flatter or the peak of the pressure gradient is
moved outside, the m = 2/n = 3 internal mode or the m = 1/n = 2 free boundarymode becomes unstable so as to limit the beta value. In H-E, it seems that the pressure profile favourable to Mercier and ballooning stabilities is also favourable to the
low n global modes. Figure 6 shows that the Mercier stability is violated near the

632

WAKATANI et al.

unstable region of the global modes with n = 1 and n = 2 for a peaked pressure profile of J3 = 2%.
Therefore, first, pressure profile optimization with respect to the local mode iscarried out in the H-APPOLO code; then the result is put into the H-ERATO code
to check the stability against n = 1, 2, 3 modes. When the additional vertical field
is chosen so as to shift the magnetic axis outward as is done in the BETA code study,
self-stabilization occurs, or the magnetic axis is shifted significantly by the finite beta
effect, and a magnetic well is created around the central region. The pressure gradient in the central region increases as a result of pressure profile optimization, anda plasma with J3 = 2.5% is easily obtained; it is stable against the n = 1, 2, 3
modes. As was discussed in Section 1, this kind of peaked pressure profile is,however, unstable with respect to the resistive interchange mode; internal disruption
may occur in H-E before self-stabilization is expected to take place. To enter the stable region with a highly peaked profile, a high beta plasma of high temperature will
be required.

REFERENCES[1] ZUSHI, H., et al., submitted to Nucl. Fusion.
[2] HARRIS, J.H., et al., Phys. Rev. Lett. S3 (1984) 2242.
[3] WAKATANI, M., SHIRAI, H., YAMAGIWA, M., Nucl. Fusion 24 (1984) 1407.
[4] YAMAGIWA, M., WAKATANI, M., NAKAMURA, Y., J. Phys. Soc. Jpn. 54 (1985) 15.
[5] HICKS, H.R., et al., 3-D Nonlinear Calculations of Resistive Tearing Modes,
Rep. ORNL/TM-7132, Oak Ridge National Laboratory (1981).
[6] WAKATANI, M., IEEE Trans. Plasma Sci. PS-9 (1981) 243.
[7] HASEGAWA, A., WAKATANI, M., Phys. Fluids 26 (1983) 2770.
[8] WAKATANI, M., SUGAMA, H., HASEGAWA, A., Phys. Fluids 29 (1986) 905.
[9] YAGI, M., HASEGAWA, A., WAKATANI, M., submitted to J. Phys. Soc. Jpn.
[10] HASEGAWA, A., WAKATANI, M., Phys. Rev. Lett. 50 (1983) 682.
[11] WAKATANI, M., HASEGAWA, A., Phys. Fluids 27 (1984) 305.
[12] BAUER, F., BETANCOURT, O., GARABEDIAN, P., Magnetohydrodynamic Equilibrium andStability of Stellarators, Springer, New York (1984).
[13] WAKATANI, M., ICHIGUCHI, K., BAUER, F., BETANCOURT, O., GARABEDIAN, P.R.,
to be published in Nucl. Fusion.
[14] TODOROKI, J., to be published in J. Phys. Soc. Jpn.
[15] GRUBER, R., RAPPAZ, J., Finite Element Methods in Linear Ideal Magnetohydrodynamics,
Springer, New York (1985).

IAEA-CN-47/D-V-4

SELF-CONSISTENTRADIAL ELECTRIC FIELD EVALUATIONSFOR THE TJ-H HELIAC
J. GUASP
Associaci6n Euratom-JEN,
Junta de Energia Nuclear,
Madrid, Spain
Abstract
SELF-CONSISTENT RADIAL ELECTRIC FIELD EVALUATIONS FOR THE TJ-II HELIAC.
Confinement times and self-consistent ambipolar radial electric field values have been calculated
by using the model introduced by K. Shaing et al. for helically trapped fluxes, together with the contribution of the 'corner' ripple term that, with a hypothesis of additivity, is taken from the banana plateau
rippled expressions. The behaviour of the confinement times and of the self-consistent radial electricfields for the expected TJ-II conditions is discussed, as well as the three root situations appearing forthe electric field.

TJ-II is a four-period helical axis stellarator of heliac type to be built at JEN[1, 2]; it has 1.5 m major radius, 0.20 m average plasma radius and an average
toroidal magnetic field of 1 T.
Former confinement time and transport coefficient evaluations were done byusing the Kovrizhnykh model [3, 4] with a simplified treatment for the corner ripple
term; the results were presented elsewhere [5], In this study, a different model basedon a model by K. Shaing et al. [6, 7] is used together with a more realistic treatment
for the corner ripple term [8].
To apply the model, we approximate the magnetic field intensity for TJ-II by
the expression:
B = Bo [1 — eT cos 6 — eh cos (0 + m<p) — 6 cos m<p](1)where m = 4 and eT and eh are toroidal and helical magnetic field modulations and6 is the ripple due to the variation in position of the helically placed TF coil centres(the so-called 'corner' ripple) [1], which reflects one of the main differences between
TJ-IJ and other stellarators, from the transport point of view.
For this work we have evaluated the parameters eT, eh, 6 and the radial dependence by means of a series expansion around the helical magnetic axis and by adjusting the resulting coefficients with the real surface average field intensity for TJ-II.
This gives, for the usual references case (circular coil current 95 kA, helical coil current 219 kA, rotational transform at axis -t0 = 1-46), eT = 0.104, eh = 0.109,
633

634

GUASP

Bo = 0.866 and 6 = 0.107 for the case of no TF coil current modulation, while fora modulation factor of 15% the ripple reduces to 8 — 0.01. Other current settings
produce different values for eT, eh and 5; in this study we have limited the calculations to the 'reference' case quoted.
More exact studies considering a greater number of Fourier harmonics for thefield in Boozer co-ordinates and solving the drift kinetic equation are also underway [9, 10].
To apply the quoted models to field expression (1), we suppose the additivitityof the fluxes to the 'corner' ripple terms and the remaining terms. In this way, thetotal particle flux could be divided into three parts:

r* = prc + m, + r^

(2)

the index a corresponding to electrons and ions and similar expressions governing theheat fluxes. The first part in (2), r nc , corresponds to the usual complete axisymmetric neoclassical flux given by Hinton and Hazeltine [11].
The second term, r bt , is the helically trapped contribution for a stellarator fieldwith the magnetic field given by the three first terms of Eq. (1), and we use for thisflux contribution the expressions given by the Shaing model (formulas (7) of Ref. [6]or (4) to (12) of [7]), where a strong dependence on the electric field appears.
The last term in the flux expression (2), r ns , corresponds to the nonaxisymmetric corner ripple, whose expression has been taken from Ref. [8] (formulas (40) and (41), valid for the banana plateau regime, retaining only the ripple
dependent part, both for the electric field dependent term and for the ambipolar part.
The self-consistent radial electric field can be obtained by setting equal the total
electron and ion particle fluxes:

re($') = r'(*')(3)

which, as will be confirmed later, may lead to multiple simultaneous roots for theelectric potential $, in many cases [7].
Once the fluxes are obtained, different confinement times [3] and 'effective'
diffusion or conductivity coefficients may be evaluated. In all cases, we calculate thefluxes at a radial point where the electron density is half the central value and suppose
parabolic profiles of density and temperatures for the dependence on the averageradius.
In this way, Fig. 1 gives the dependence of the global energy confinement time,TE, on the central electron temperature for a central density of no = 2 x 1019 m" 3
and an electron to ion temperature ratio of Te(0)/Ti(0) = 3.33, which corresponds
to the estimated energy balance situation for 400 kW ECRH injection in TJ-II. In thisfigure, three different values for the radial electric field are plotted, corresponding
to a ratio between the electric potential and the central electronic temperature ofe$/Te(0) = + 1 , 0 , - 1 . The influence of the electric field is very important at high

IAEA-CN-47/D-V-4

635

T o (eV)

FIG. 1. Global energy confinement time versus central electron temperature for typical TJ-IIECRH
situation (n0 = 2 x 1019 m'3, TJOf/TJO) = 3.33). Electric potential * normalized to TJO).

electron temperatures and more sensitive to the potential value than to the sign; thus,
for Te(0) = 770 eV, TE can change by 43 % for the quoted potential variation, but
the alteration decreases to 10% at 300 eV.
Figure 2 shows the temperature dependence of TE for the same case; but for aself-consistent radial electric field as determined by Eq. (3), it is clear that a threefold
root appears above 625 eV and remains up to 1020 eV; the root corresponding to alower rE value has a moderate negative electric potential (Fig. 3). It is the ion root
accessible from the low temperature region, while the upper TE value root corresponds to a high positive electric potential. It is the electron root accessible fromthe high temperature region; as usualy, the intermediate root corresponds to a low
field, generally positive, and represents an unstable situation. When the temperature
increases, the first and last roots approach each other, and finally, at 1020 eV,
coalesce and disappear, leaving only the electron root with high field and TE values(8.7 ms, e$/Te(0) = +2.15).
A rough global energy balance, taking in account transport, bremsstrahlung
losses and electron to ion energy transfer, yields an equilibrium central electron temperature of about 750 eV with a TE value, for the ion root (that which is accessible
from low temperature), of 4.85 ms and an electric potential of e0/Te(O) = -0.175.
These confinement time values are of the same order of magnitude as those calculated
previously by means of the Kovrizhnykh model [5].
The behaviour, as a function of Te, of the three self-consistent electric potential roots is shown in Fig. 3. In addition, the dependence of the electron and ion
fluxes on the electric potential for the quoted case and a Te(0) value of 770 eV isrepresented by Fig. 4, showing the pronouncedly peaked shape at null field of the

636

GUASP

10
T e <eV)

FIG. 2. Same as Fig. 1, but with self-consistent electric potential.

T e (eVI

FIG. 3. Dependence of self-consistent electric potential on TJO) (same case as Fig. I).

ion flux and the smoother behaviour of F e ; in this case, the two fluxes intersect atthree points, corresponding to the three roots discussed previously. A change in temperature shifts the curves so that two of the roots coalesce and disappear, restoringa single root situation [7].
The dependence on 'corner' ripple has also been studied, showing an increase
in TE with a modulation factor which is only appreciable for high temperatures,
because of the presence of the ripple term in the banana plateau regime. For a TF
coil current modulation factor of 15%, the increase for the former case at

637

IAEA-CN-47/D-V-4

s'-S.OO -TOO -3.00 -2.00 -1.09 0.00

i.00

2.00 3.00 4.00

5.00

*/Te

FIG. 4. Variation of electron and ion particle fluxes with electric potential (case of Fig. 1 for
7,(0) = 770 eV). Fluxes in
10l9rn'2-s-'.

Te(0) = 750 eV is only by 27 O o, while at 1100 eV where the electron root is alone
the increase is almost by a factor of 3 (23.5 against 8.1 ms without modulation); in
addition, the region of three roots shits slightly towards lower temperatures.
The dependence on density and the electron to ion temperature ratio has also
been studied, giving the usual increase in density for the ion root, which is lessimportant, but a visible increase for the electron root and a shift of the three root
region towards higher temperatures. On the other hand, an increase in the Te/Ti
ratio can totally alter the root topology, with a disappearance of the multiple root
region for values greater than 3.6 [3]; in this case, the single electric field root shifts
smoothly from the ion to the electron value, with small changes in the rE values. In
contrast, a decrease in the temperature ratio produces smooth changes in the root
topology and very slight changes in the TE values for each individual root, but
extends the width of the three root region without modification of the starting bifurcation temperature. For example, a decrease in Te/Tj from 3.3 to 2.7 brings the upper
limit of the three root region from 1020 eV up to 1945 eV, maintaining the starting
point at around 625 eV.
Calculations with this model for other TJ-II configurations and heating
scenarios (NB injection, in particular) are now under way, as is a more detailed,global energy balance analysis.

REFERENCES[1][2]

TJ-n GROUP, ATF DESIGN TEAM, in Controlled Fusion and Plasma Physics (Proc. 12th
Europ. Conf. Budapest, 1985), Vol. 9F, European Physical Society (1985) 441.
TMI GROUP, ATF DESIGN TEAM, ibid., p. 433.

638
[3]
[4]
[5]

GUASP

KOVRIZHNYKH, L.M., Nucl. Fusion 24 (1984) 435.
KOVRIZHNYKH, L.M., Nucl. Fusion 24 (1984) 851.
GUASP, J., in Controlled Fusion and Plasma Heating (Proc. 13th Europ. Conf. Schliersee,
1985), Vol. 10F, Part I, European Physical Society (1986) 299.
[6] SHA1NG, K.C., Phys. Fluids 27 (1984) 1567,[7] HASTINGS, D.E., HOULBERG, W.A., SHAING, K.C., Nucl. Fusion 25 (1985) 445.
[8] SHAING, K.C., HIRSHMAN, S.P., CALLEN, J.D., Phys. Fluids 29 (1986) 521.
[9] SOLANO, E.R., ROME, J.J., LYON, J.F., HIRSHMAN, S.P., Paper 7S9, presented at
American Physical Society Meeting San Diego, CA, Nov. 1985.
[10] SOLANO, E.R., SHAING, K.C., submitted to Phys. Fluids.
[11] HINTON, F.L., HAZELTINE, R.D., Rev. Mod. Phys. 48 (1976) 239.

IAEA-CN-47/D-V-5

EVOLUTION, RELAXATION AND
TRANSPORT IN SPHEROMAKS AND
REVERSED FIELD PINCHES*
R.Y. DAGAZIAN, J.P. DAHLBURG1, G. DOOLEN,
G.J. MARKLIN, A.A. MIRIN2, J.P. MONDT,
D. MONTGOMERY3, A.G. SGRO, L. TURNER, J. WEILAND4
Los Alamos National Laboratory,
University of California,
Los Alamos, New Mexico,
United States of America
Abstract
EVOLUTION, RELAXATION AND TRANSPORT IN SPHEROMAKS AND REVERSED FIELDPINCHES.
The strongly nonlinear dynamics present in spheromaks and reversed field pinches, which offer
the promise of compact, ohmically heated reactors, have been studied by various computational and analytical methods. This has resulted in new insight into the significance of minimum energy states andof transport mechanisms that limit the confinement. The paper reports on the following investigations:(1) Exploration of the tendency of plasmas to oscillate about or to approach relaxed states, using simulations of a decaying spheromak and of a low-S, magnetized, self-reversing Z-pinch. (2) Finding of finitebeta relaxed MHD states with Hall terms. (3) Presenting details of the structure of three-dimensional
states relevant to spheromak helicity injection. (4) Investigation of transport due to either thermal convection or steep temperature gradients.

The resistive evolution of a decaying spheromak initially ina constant J/B (minimum-energy) equilibrium is calculated. A
spatially varying resistivity h(t) [for which r) (magnetic axis)< ri (wall)] causes evolution away from the minimum energy statethat concentrates the current near the magnetic axis in a manner
similar to the RFP and other confinement schemes. Aftersufficient current peaking (when q falls below 1/2 everywhere),
the plasma becomes unstable to a mode having toroidal mode
number n=2 (Fig. la), the plasma distorts into a nonaxisymmetric
n=2 configuration, and the unstable mode saturates (Fig. lb,c).
* Work performed under the auspices of the United States Department of Energy.
!

United States Naval Research Laboratory, Washington, D.C. 20375, USA.
National Magnetic Fusion Energy Computer Center, Lawrence Livertnore National
Laboratory, Livermore, CA 94550, USA.
3
Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA.
4
Chalmers Institute of Technology, Goteborg, Sweden.

2

639

640

DAGAZIAN et al.

The shift away from axisymmetry is large. The value of 6B/B at
the boundary is about 10%, in agreement with the measurement of
this quantity on the CTX experiment, and the magnetic axis moves
about halfway to the wall. The nonaxisymmetric equilibrium is
initially stable and the plasma profiles then change, due todiffusion.
Eventually, the plasma becomes unstable again(Fig. Id), a new magnetic axis is formed (Fig. le), the old one
is pushed out toward the wall, and the plasma approaches theaxisymmetric (n=0) minimum energy state again (Fig. If). Since
the expulsion of the old axis also involves expulsion of theplasma near the old axis, this would presumably have the effectof a convective cooling of the plasma. Thus, superimposed on
the overall decay of the spheromak is an oscillation between an
axisymmetric equilibrium and an equilibrium with n=2 structure.
In Fig. 2a, T X / T « 2 » the ratio of the oscillation time to theconfiguration lifetime, is plotted as a function of S and three
regions are indicated.
In the large S region (region I ) ,T X ~ 1.5 x B 2. The lifetime of the decaying phase of theexperiment is typically several magnetic decay times, so that
more than one or two complete relaxation oscillations areunlikely. It is possible that only the first part of one
oscillation will be observed. In fact, the reconnection which
drives the magnetic axis to the wall may terminate
thedischarge.
Forintermediate
S (region II), x x can beconsiderably smaller than Tg2»so several oscillations may occur.
For very small S (region III), diffusion will dominate, resulting
in a smooth decay and no oscillations. It is not clear how
small S would have to be before the oscillations become
undetectable. For high S, the energy loss concomitant with themagnetic axis expulsion would be considerably reduced, since
there would be at most one oscillation. Only n = 0 and n = 2
activity is exhibited by 3D code simulations. When n = 1, 3 or
4 modes are present, they decay while n = 2 grows and saturates.
The predominance of closed flux surfaces minimizes possible heatloss along stochastic field lines. The numerical results
qualitatively agree with experimental data.
The nature of 3D minimum energy states is analyzed byexamining a configuration in which an unstable magnetizedZ-pinch is used to supply helicity through a long cylindricalentrance region (D = L = 40 cm) to sustain a spheromak in a flux
conserver (D = 80 cm, L •= 40 cm), as illustrated in Fig. 2c. In
the entrance region, which has no net magnetic flux, the minimum
energystate
has a helical m *1 1 structure varying as1
exp[i(9 + kz)lwith
k = 6.25
m"and
X E R = 15.6m"
(X = u j||/B). The minimum energy state in the flux conserver isthe usual axisymmetric (m = 0) spheromak configuration with X $ =13.2 m . The X value of the z-pinch
is the ratio of theIfcurrent to the external flux, X^ = V Q 1 ^ '\ p > *RR thez-pinch will become kink unstable and will feed helicity into
the entrance region and on into the spheromak since helicity
flows towards regions of lower X, and the spheromak can be
maintained in steady state against resistive dissipation.
This

641

IAEA-CN-47/D-V-5

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642

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DAGAZIAN et al.

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FIG. 2. (a) Plot of TJTB1 versus S. For large S (region I), TX is comparable to the plasma lifetime
and at most one or two complete oscillations (and possibly the first part of one oscillation) will beobserved. For intermediate S (region II), several oscillations are likely. For small S (region HI),the plasma is diffusion dominated and never evolves far from the minimum energy state, so that no
oscillations occur.
(b) Character of the eigenvalues, (i, as a function of the values ofKj and %.3. In the shaded regions,labelled I and II, the eigenvalues are real. In the clear regions, the eigenvalues are complex. In particular, the eigenvalues are imaginary on the hyperbolas labelled C. They are degenerate on the hyperbolas
labelled A or B. Previously studied force-free magnetic field configurations correspond to the domainof the X2 = 0 axis.
(c) Shape of the flux conserver.
(d) Eigenvalues of the m = 0 and m = 1 minimum energy states plotted as a function ofa/R, the radiusof the entrance region over the radius of the flux conserver.
(e) Three-dimensional perspective plots of the alignment cosine and for a relaxed Z-pinch state [I],(f) Same as (e) for X ^jB

1AEA-CN-47/D-V-5

643

equilibrium is difficult to calculate because there is no
overall symmetry and the fields are fully 3-dimensional.
Ve
attack this problem by separately computing global solutions to
VxB m = ^ B for the m = 0 and m = 1 modes in a conducting
boundary which includes the flux conserver and the entrance
region, but not the z-pinch. The eigenvalues ^ are computed asfunctions of a/R, the radius of the entrance region over theradius of the flux conserver, and are plotted in Fig. 2d. At
a/R ~.5 the m = 0 and m = 1 eigenvalues are equal, so the twosolutions B and B, can be added in any linear combination,
resulting "in a "fully 3-dimensional solution to VxB = XB. The
structure of the fields can then be studied by examining theproperties of this 3D minimum energy state. Flux surface plotsobtained by field line tracing show a helical structure in theentrance region and a toroidal structure in the flux conserver,
as expected. If the amplitude of the m = 1 component is small
compared to the m = 0, most of the volume of the flux conserver
has good magnetic surfaces which resemble a slightly tilted
spheromak. Vhen the amplitudes are equal, however, as theyusually are in experiments, the field is highly stochastic andthere are only a few good surfaces near the magnetic axis of thetilted and distorted spheromak.
A conservative rule of thumb for MHD computation is that one
grid point (or expansion coefficient, or finite element) is
required per unit Lundquist number S for each spatial dimension,in order to guarantee well-resolved dissipation scales for MHD
computation. This estimate probably cannot be exceeded by morethan an order of magnitude. Thus, an affordable (on the Cray I
or Cray II) 643 3D code probably cannot fully resolve nonlinear
Z-pinchdynamics at S > 1000. This limitation restricts
computations to S of a few hundred. Fortunately, it has turned
out that several experimentally significant effects can be seenfrom the equations of incompressible, resistive MHD at such S
values. Among those are relaxation toward a force-free stateand spontaneous reversal of the toroidal component of themagnetic field at the wall. Ve are currently performing 3D MHD
computations with a pseudospectral code in which the equationsof resistive, incompressible, non-viscous MHD are solved inside
a 3D region with a square (x,y) poloidal boundary and periodic
boundary conditions in the toroidal (Z) direction. The poloidal
boundary is rigid, perfectly conducting, and free-slip. A net
magnetic
flux
crossesevery
plane ofconstantZ.
Non-equilibrium initial Z current profiles evolve
throughseveral tens of poloidal AlfveVi transit times; typical runs
require 10 hours of CRAY-II time. Considerable small-scale
turbulence results (low "m", high "n") and the dissipation
spectra are not always fully resolved. The ratio of the totalenergy to the magnetic helicity decreases monotonically. The
kinetic energy rises from nearly zero to a few percent of themagnetic energy and then falls again. Since the current density
and the magnetic field obey different boundary conditions, a

644

DAGAZIAN et al.

turbulent boundary layer persists near the wall. Over theinterior of the plasma, a nearly force-free state results. The
flatter profile of j'B/jB = cosG, as compared with that of
X=j-B/B2 in a poloidal cross-section (Fig. 2e,f) suggests |1]that" the relaxed state is better characterized as "force free"
than as "minimum energy" (for which X as well as cos © would beconstant).
We have studied magnetic relaxation derived from a newvariational principle obtained from incompressible MHD, modified
to include the Hall term in Ohm's law [2] (referred to as HHHD).
The Hall term breaks the symmetry of ideal HHD under the gaugetransformation associated with the cross-helicity invariant( J y B d 3 r ) . However,we found that in ideal HMHD, a new quantity,
the hybrid helicity, is conserved instead. This hybrid helicity
represents the self-linkage or knottedness of a generalized
vorticity, 0 = eB/mc + V x y. The conservation results from g's
being"frozen"" tothe" fluid
motionaccording"to3P./3t = 7 x (y x Q).
The aforementioned variational principle
is a consequence of the hypothesis that when dissipative effects
are present,the total (kinetic plus magnetic) energy, V, decays
more readily than do the (ideally conserved) magnetic and hybrid
helicities, K and X, respectively. We then suppose that W
decays to the minimum value compatible with the initial valuesof the two helicities, the magnetic flux and the fluid vorticity
flux. Hence we arrive at the principle
8KX1K + X£X - W) = 0.
Thus the Lagrange multiplier parameter space is expanded to 2-D(namely, Xj and A 2 ) . The two Euler equations that result couple
the behavior of the magnetic field to that of the fluid
vorticity field. The solution for each field can be expressedas the sum of two eigenvectors of the curl operator witheigenvalues, y, that are possibly complex (Fig. 2b). As aresultofthiscoupling, we
foundthat the relaxedmagnetic field configuration need not be force-free if the fluid
vorticity is nonzero, leading to the existence of relaxedmagnetic field configurations with finite plasma beta. Our
resultsare enticing
forfurther
explorations.
Canfusion-worthy relaxed magnetic field configurations that confine
significant plasma pressure be achieved in the Hall regime? Byvirtue of the relaxation process that couples magnetic field to
fluid vorticity, are there new methods of current drive,
possibly mechanical in origin, that can be used to sustain afusion plasma? Can vorticity be used to stabilize such a plasma?
When Hall effects are present, is the concept of magnetic
reconnection
appropriately
replacedbythe concept ofreconnection of the generalized vorticity?
The theory of Rayleigh-Benard thermal convection in ordinary
fluids is employed as a prototype for studying plasma convection
resulting from small-scale g-mode fluctuations. For well
behavedstrongly magnetized plasmas, situations of strong
turbulence are unlikely and a quasi-linear theory as expoundedIAEA-CN-47/D-V-5

645

here may be adequate in describing basic physical processes.
The physical model that ve employ is that of one-fluid resistive
MHD augmented by collisional tensor viscosity and simple thermal
conductivity. The existence of stationary states for resistive
modes in this model has been proven [3]. The basic situation isthat of a diffuse cylindrical pinch with weak density variation
over the bulk of the plasma so that temperature convection isthe dominant nonlinear saturation mechanism for the
fastresistive interchange. This mode is chosen to work with becausea) it is the usual resistive g-mode of RFP's and underlies
resistive ballooning; b) its stationary eigenfunctions die off
quickly with distance from the resonant surface so that theanalysis is relatively simple.
The anomalous confinement time
can be written as T = 4/2a2p0(me/m^)1/2/(lipN)i where a is theplasma characteristic dimension, h. the plasma revistity, 0 the
kinetic to magnetic energy ratio, N the Nusselt number, and m g
and m^ the electron and ion mass, respectively.
Edge
plasmasarecharacterized by steep temperature
gradients and, in discharges with gas puffing, much smaller
density gradients.
In reversed field pinches the large plasma
current prompts an investigation of enhanced transport due tocurrent-convective instabilities.
The mode structure of suchlinear instabilities which are resonant in different regions may
overlap [4]. The present work consists of a two-fluid treatmentof the resultant second-order mode coupling and transport.
Inview of the short kj_, time-scales slower than the resistive
diffusion rate across 2 a perpendicularwave
lengthareconsidered, and kjj X ^ « k? p| is assumed, where p g is thethermal ion Larmor radius at trie electron temperature and \,f D
is the electron mean free path parallel to the magnetic field.
Then, magnetic islands are short-lived on time-scales ofinterest, whereas the perturbation in the parallel current
density dominates all other electromagnetic effects. The model
consists of the parallel component of the inertialess Ohm's law
including electron pressure and parallel thermal force effects,
and the electron energy equation including parallel electron
thermal conductivity and compressibility, the drift heat flux
and parallel thermal forces. Ohmic heating rates are typicallyslow compared with the relevant nonlinear rates. The parallel
component of Ohm's law eliminates the current perturbation,whilst the electron temperature disturbance is coupled back to
the electrostatic potential through the ion vorticity equation,
obtained from ion force balance with lowest-order finite Larmor
radius effects and quasi-neutrality in the usual manner. The
resultantheat
diffusioncoefficientisroughlyD = (cTe/eB) K V, where K»p g /1 T and Vsve||/C_ are the normalized
inverse temperature gradient scale lengtn and parallel electron
drift velocity, respectively.
It was also found that on timescales short compared with viscous damping, thermal solitaryvortices can exist even in the presence of shear and parallel
thermal diffusion. If such vortices are attractors of the

646

DAGAZIAN et al.

nonlinear dynamics, nonlinear steepening balancing dispersion
mayfurtherenhancethe heatloss
associatedwithcurrent-carrying edge plasmas.
Althoughthe thermal convective mode and the current
convective mode may effect the confinement properties of someregions, it is not implied that they determine the global
confinement time. Rather, the global confinement time will be
determined by regions having good flux surfaces and nearlyclassical cross-field thermal conduction.

REFERENCES[1] DAHLBURG, J.P., MONTGOMERY, D., DOOLEN, G.D., TURNER, L., Phys. Rev. Lett.
57 (1986) 428.
[2] TURNER, L., IEEE Trans. Plasma Sci. (Dec. 1986).
[3] DAGAZIAN, R.Y., PARIS, R.B., Phys. Fluids 29 (1986) 3.
[4] MONDT, J.P., GOEDERT, J., Phys. Fluids 28 (1985) 1816.

IAEA-CN-47/D-V-6

EXPERIMENTS WITH A FULLYTOROIDAL EXTRAP Z-PINCH
J.R. DRAKE, J.E. ENINGER, B.P. LEHNERTDepartment of Plasma Physics and Fusion Research,
Royal Institute of Technology,
Stockholm, Sweden
Abstract
EXPERIMENTS WITH A FULLY TOROIDAL EXTRAP Z-PINCH.
In the Extrap plasma confinement scheme, a Z-pinch is produced along the null of an octupole
field generated by currents in external conductors. In the paper, studies of the discharge startup process
in a fully toroidal configuration are described. Startup involves first breaking down a toroidal discharge
and then driving up the current in order to reach the pinch parameter regime. Current densities of2 x 106 A-m~2 have been achieved. The estimated plasma density is 6 x 1020 m~3, and the temperature is about 4 eV. These parameters correspond to pinch conditions.

1. INTRODUCTION
An Extrap Z-pinch is a high beta discharge where the current channel has
a characteristic non-circular cross-section achieved by bounding the discharge
by a magnetic separatrix [1-3]. The configuration is produced by generating
the discharge along the magnetic field null line of an octupole magnetic field.
In a toroidal configuration, the vacuum octupole field is produced by parallel
currents in four rings. The discharge current is located in the vicinity of theoctupole null and is antiparallel to the ring current. A schematic representationof the configuration is shown in Fig. 1.
A series of linear and toroidal sector experiments have shown that a
globally stable pinch equilibrium can be achieved. Theoretical studies indicatethat the stability of the configuration can depend on the imposed octupole field[1,4,5], the effects of the separatrix on the pressure profile [6],finite
Larmor radius effects [1,6] and large Lanmor radius effects [7].
The next step in the investigation of this scheme is a fully toroidal
experiment and the device Tl has been constructed for these studies. The
goals of the experiments are; 1) development of discharge startup procedures,
2) achievement of equilibrium and 3) studies of stability and confinement In
this report we present the results of studies of discharge startup.

2. DESCRIPTION OF THE EXTRAP Tl EXPERIMENTAL APPARATUS
The Extrap Tl apparatus, shown schematically in Fig. 2, consists of astainless steel toroidal vacuum vessel with a 0.20 m x 0.25 m rectangular
647

648

DRAKE et al.

-support
low beta scrape-off layer
x-point

separatnx
high beta discharge

OCTUPOLE FIELD

Z-PINCHEXTRAP

FIG. 1. Extrap Z-pinch configuration results when Z-pinch is generated along vacuum octupole magnetic field axis produced by currents in four conductors. Self-field of discharge, together with octupole
field, generates configuration where pinch discharge region is surrounded by separatrix field line segments defined by four X-point field nulls. High beta pinch is located inside region defined by nulls; lowbeta plasma, contained by vacuum field, is located outside this region.

cross-section. The vessel is divided mechanically and electrically into two
halves; this allows for easy assembly and disassembly as well as flexibility inthe circuit configuration. Each half is further divided into four sections which
are separated by polyethylene insulated gaps. In the present experiments thegaps are shorted to provide a return path for the ring current so that the ring
circuit, which produces the octupole field, is decoupled from the Ohmic
heating discharge circuit. The four rings, located at (R, 0, z) = (450 ±30mm, 6 , 0 ) and (R,6,z) = (450 mm, 6, ±30 mm), are made of 10-mm
diameter chrome-copper. The rings are supported mechanically at 50-mm
intervals by radial spokes machined into 4-mm thick stainless steel plates.
These support frames also connect the rings electrically around thecircumference. The rings are driven in parallel by a 12-mF, 8-kV capacitor
bank through a current limiting 20-|iH inductance which can be shorted out
by an ignitron switch during an experimental pulse in order to increase therate of rise of the ring current. The ring system is designed for currents up to
100 kA in each ring. The 2 \iH inductance connected in the gap at thevacuum vessel is essential for achieving a uniform plasma breakdown and aclosed toroidal current channel.
The plasma current is induced through a 0.5-Vs iron-core transformer
driven by two 24-(iF, 30-kV capacitor banks feeding a common two-turnprimary. Two separate banks having different resistors in series permits a soft
start up of a low current (< 1 kA) predischarge. The time constant of thepredischarge is 200 fj.s. The predischarge is followed by a second highcurrent, 60-|J.s pulse to build up the discharge.
The rings are uninsulated from the plasma except at the two breaks inthe vacuum vessel where there are 100-mm long, 45-mm diameter ceramic
tubes inserted inside the rings to provide insulation of the ring gaps.

IAEA-CN-47/D-V-6

649

14mF

FIG. 2. Schematic diagram of toroidal Extrap experiment.

3. DISCHARGE STARTUP
Experience has shown that it is difficult to achieve toroidal Extrap
discharges. This is of course expected as the desired discharge is _> _„perpendicular to the vacuum octupole magnetic field. The particle ExB drifts
hinder breakdown when the toroidal electric field is applied.
In linear Extrap experiments, discharges have been produced using
electrodes placed on the linear axis where the octupole field is zero. With
electrodes, breakdown is easily achieved. Initially, the current channel iscentered on the axis and is very thin. Breakdown occurs in the region where
the magnitude of the magnetic field is sufficiently small so that the ExB drift
has no effect on the electron trajectory and the breakdown conditions are thensimilar to the well known Townsend conditions [3].
In principle, breakdown could be achieved in the toroidal configuration
in a similar fashion. Ionization would be limited to a small region close to thetoroidal axis where the octupole field is small. When the ionization rate
exceeds the loss rate, breakdown would result [8].
However, it is difficult to achieve breakdown in this fashion on the fasttime scale desired for the Extrap discharge. In the linear experiments, fully
developed Extrap discharges exhibited good global stability but the initial thin

650

DRAKE et al.

discharge located in the weak octupole field region was observed to be
globally unstable. With electrodes it was possible to drive the discharge
current up to a level where full ionization was achieved in a few
microseconds. The current channel expanded into the region where theoctupole field was stronger and those stabilizing features, which characterize
the Extrap configuration, became effective before the global instabilties
destroyed the young pinch.
Two approaches to discharge startup in the toroidal experiment Tl havebeen studied. The primary approach, to be discussed in detail later in thisreport, incorporates a weak toroidal field which expedites breakdown byimproving the electron containment in the region of the octupole field null.
3.1 STARTUP WITHOUT A TOROIDAL B FIELD
A second approach, which has also been the subject of preliminary
investigations, involves using the ring currents which produce the octupole
field as the primary circuit for inducing the discharge. First the discharge isbroken down down at a gap in the rings. At this stage the plasma current is
not fully toroidal and this discharge current is driven directly by the voltage
applied at the gap. The plasma circuit does not include an induced EMF.
However the JxB force on the plasma current drives the discharge away fromthe gap, on both sides of the gap, so that the plasma discharge sector becomesa larger and larger fraction of the full torus. When the two ends of thedischarge meet at the antigap, 180° from the gap, the plasma discharge
becomes fully toroidal forming a closed loop secondary and the plasma
current is then driven by the induced EMF applied by driving the ring current
at the gap.
This approach to achieving breakdown is not as clean as the approach
where breakdown is achieved along the toroidal magnetic null without contact
with the rings. When the rings also function as electrodes during breakdown,
impurities are introduced which can of course affect the energy balancethroughout the discharge. The advantages of this approach are that no toroidal
field is required and very high power inputs can be achieved giving a fully
toroidal, fully ionized discharge on a fast time scale.
In the experiments performed to date, the discharge has propagated tothe antigap when ring gap voltages of the order of 2 kV have been applied.
The propagation velocity of the ionization front is about 5xl(T m s ' ' which isconsistent with earlier studies of moving ionization fronts [9]. After about 30
jo.s the two ends of the discharge meet at the antigap, 180° from the gap.
However,induced toroidal discharges have not been achieved. The problem isthat the current magnitude necessary to drive the discharge to the antigap
exceeds the stability limit and the discharge goes globally unstable before
forming a toroidal pinch discharge.
3.2 STARTUP WITH A TOROIDAL B FIELD
We now turn to the discharge startup studies that have been carried out
incorporating a toroidal magnetic field. The toroidal discharge is produced

IAEA-CN-47/D-V-6

651

directly along the octupole field null. Discharge startup involves first breaking
down the neutral fill gas near the toroidal axis thus forming a secondary for
the Ohmic heating primary circuit. The discharge starts on the axis because
the octupole field hinders breakdown elsewhere. An ECR source provides
necessary preionization. A loop voltage of 2 kV is needed for breakdown.
The current is then driven up thus increasing the Ohmic power input so thatthe plasma pressure builds up. The startup phase is complete when the
discharge parameters are in the pinch regime. In general this means that thepinch pressure is balanced by the magnetic field produced by the discharge
current.
The two basic requirements for achieving a pinch discharge aresufficient discharge current for confinement and sufficient power input forburning through die radiation barrier. The confinement requirement isexpressed in the Bennett relation which is a global expression for the force
balance in the form
I2 = 16rcNekT/no(1)

where N e is the electron line density, T is the temperature and I is thedischarge current.
During the buildup phase the dominant power loss is radiation and thepower balance becomes2(2)

where T](T) is the resistivity, Npj is the neutral hydrogen line density andR J J ( T ) is the radiation power factor. If light impurities are present, the power
balance can be dominated by impurity radiation. In reversed field pinch
discharges, which operate in a similar regime, burn-through of light
impurities requires discharge currents such that [10]I / N > lxlO" 14 Am(3)

These expressions can be used to establish operational regime curves forfilling pressure and discharge current. We assume that the discharge ispermeable to neutral gas so that NJJ is determined by the filling pressure andthat the plasma density is also equal to the filling density. The cross-sectional
area of the discharge is determined by the aspect ratio of the octupole rings
and the ratio of the discharge current to the ring current. However, thedependence on this current ratio is weak so a constant cross-sectional area
can be used for calculational purposes. In Fig. 3 we show curves of thefilling pressure versus discharge current derived from the Bennett relation andthe power balance for several temperatures.The operational regime of thebuildup studies carried out to date is also shown in Fig. 3. For these studies,
the toroidal field strength was 0.1 T. In a tokamak mode, a requirement of

DRAKE et al.

652

Operational regime.
at present

so

01

Ol

05
1
2
5
10
DISCHARGE CURRENT (kA)

20

50

FIG. 3. Operational regime curves of filling pressure and discharge current. Limits imposed by Bennett
relation and radiation losses are shown.

q = 1 would limit the discharge current to 440 A for the given aspect ratio. As
can be seen, discharge currents of up to about 4 kA have been achieved sofar. In the present device the desired operational regime requires currents
exceeding 10 kA. The limitation in the present series of experiments isdifficulty in raising the octupole field, in the presence of the discharge, due to
breakdown at the ring gap.
Magnetic piobes have been used to measure the magnetic field profiles inthe discharge current channel. In Fig. 4 we show a measured profile of thepoloidal field due to the discharge current alone. The vacuum octupole fieldhas been subtracted. This profile has been used to calculate a current
distribution which is also shown. Characteristic parameters for this discharge
are given in Table I.
4. SUMMARY
The discharge parameters shown in Fig. 3 and Table I indicate that thedischarge is in the pinch regime. Current densities of 2x 10" Am"2 have beenachieved. The estimated plasma density is 6X1020 m"3 and the temperature isabout 4 eV. These parameters correspond to pinch conditions.The pinch iscold and the power balance can be assumed to be dominated by radiation
losses. However,the pinch is well developed and well positioned near theminor axis. In Fig. 5 we show an open shutter photograph of the discharge
showing that the pinch boundary is well defined.

653

IAEA-CN-47/D-V-6

TABLE I. CHARACTERISTIC DISCHARGE
PARAMETERS

0.45

Major radius (m)Total ring current (kA)10

Toroidal magnetic field (T)0.1

Filling density, nH (1020 m~3)

6

Discharge cross-section (10~3 m2)

1

Total discharge current (kA)3

~15 A-m)

5

I/NH

Loop voltage (V)700

Calculated electron temperature (eV)4

Calculated electron density (1020 m~3)

6

AlfVen transit time, a/vA (jis)1Discharge pulse duration (us)

60

0

10

20

30

40

8 (mm)
FIG. 4. Measured profiles ofpoloidal magnetic field strength produced by discharge current. Coordinate p is radial distance from discharge axis which is about 5 nun outside geometric minor axis forthe four rings. With given vacuum octupole field, X-point is at p = 18 mm. Profile of corresponding
current density is also shown.

In the next series of experiments, the discharge current and octupole fieldstrength will be increased. A vertical magnetic field necessary for controllingthe equilibrium position of the pinch at the higher current levels will be
implemented, the goal is to maintain equilibrium at the higher currents andachieve higher temperatures.

654

DRAKE et al.

RINGS

upper
outer
lower

RING SUPPORT'' B Q COIL

FIG. 5. Open shutter photograph of discharge taken through a 100 mm diameter window in vacuum
vessel outer wall.

ACKNOWLEDGEMENTS
The authors wish to thank the academic and technical staffs of the
Department for their contributions and the effort that has been made to design,
construct and run the Extrap experiment.
This work has been supported by the European Communities under anassociation contract between Euratom and Sweden through the Swedish
Energy Research Commission, Project Number 4210014/2.

REFERENCES
[1]
[2]

[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]

LEHNERT, B., Phys. Scr. 10 (1974) 139; 16 (1977) 147.
DRAKE, J.R., HELLSTEN, T., LANDBERG, R., LEHNERT, B., WILNER, B., in Plasma
Physics and Controlled Nuclear Fusion Research 1980 (Proc. 8th Int. Conf. Brussels, 1980),
Vol. 2, IAEA, Vienna (1981) 717.
DRAKE, J.R., Plasma Phys. 26 (1984) 387.
WAHLBERG, C , ARORA, A.K., RING, R., Plasma Phys. Controll. Fusion 26 (1984) 1261.
BRYNOLF, J., RING, R., WAHLBERG, C , Plasma Phys. Controll. Fusion 27 (1985) 1255.
HELLSTEN, T., Nucl. Fusion 22 (1982) 565.
LEHNERT, B., Comments Plasma Phys. Controll. Fusion 9 (1985) 91.
TENDLER, M., Peculiarities in the Start-up Process in a Toroidal Extrap, Rep. TRITAPFU-86-08, Royal Institute of Technology, Stockholm (to be published in Europhys. Lett.).
ENINGER, J.E., in Phenomena in Ionized Gases (Proc. 7th Int. Conf. Belgrade, 1966), Vol. 1(1966) 520.
COSTA, S., De ANGELIS, R., ORTOLANI, S., PUIATTI, M.E., Nucl. Fusion 22 (1982)
1301.

IAEA-CN-47/D-V-7

CHARACTERISTICS OF
REVERSED FIELD PINCH PLASMASWITH A RESISTIVE SHELL
T. TAMANO, W.D. BARD, T. CARLSTROM, C. CHU,
R.R. GOFORTH, Y. KONDOH1, R.J. La HAYE, P. LEE,
E.J. NILLES2, M.T. SAITO, M.J. SCHAFFER, P.L. TAYLORGA Technologies Inc.,San Diego, California,
United States of America
Abstract
CHARACTERISTICS OF REVERSED HELD PINCH PLASMAS WITH A RESISTIVE SHELL.
Reversed field pinch plasmas in a resistive shell were investigated in the OHTE device. The discharges were sustained over 10 ms — much longer than the resistive shell time of 1.5 ms. Global plasma
characteristics, such as plasma temperature, density, resistance and magnetic fluctuations, were very
similar to those obtained with the previous highly conducting shell. A new type of toroidally localized
kink instability, called the 'slinky' mode, was found. It grew initially with a growth time of approximately the resistive shell time, as predicted by the linear MHD theories, but self-healing was usuallyobserved after a few milliseconds. This indicates that a highly conducting shell may not be required fora reversed field pinch and suggests that linear plasma stability is not always necessary in a confinement
system.

Remarkable progress has been made in recent years in thefield of reversed field pinch (RFP) and OHTE helical pinch [15]. Discharges have been sustained for over 10 ms and improvement of both the plasma temperature and the Lawson parameterhas been observed as the plasma current increased*
The pinch
devices have been operated with a conducting shell surrounding
the plasma that has a time constant much longer than the duration of the plasma discharges.
The linear MHD theories predictthat RFP plasmas are unstable with a resistive shell and that
unstable modes grow on the resistive shell time - scale T 8 =MgObw/2, where o, b and w are the electrical conductivity of theshell, the minor radius and the thickness of the shell, respectively [6-8]. In the future devices, the discharge duration will
be much longer than the resistive shell time.
Therefore, it isimportant to study plasma behavior in a resistive shell. For
this reason, the OHTE conducting shell was replaced by the resistive shell in the early spring of 1985 and experiments havebeen carried out with the resistive shell since then [9].
1

Present address: Department of Electronic Engineering, Gunma University, Kiryu, Gunma,
Japan.
2
Present address: Department of Physics, University of Wisconsin, Madison, WI 53706, USA.

655

TAMANO et al.

656

200
Itf(kA)I

400

A

rp(cm)
8^ (G)
1000
0(10 1 4 cm"3]

1

Z

4

6

10

thm)
10
t(ms)

FIG. I. Waveforms of plasma current 1$, loop voltage V^, average toroidal field B$ inside the liner,line-average electron density ne, pinch parameter 6, reversal parameter F, plasma radius rp from thetruncated Bessel function model, and plasma resistance R.

The OHTE device has a major radius of 1.24 m and a minor
radius of 0.183 m to the inside surface of graphite tiles thatcompletely cover the inside of the stainless-steel bellows plasma chamber.
The resistive shell has a minor radius of 0.200 m.
The resistive shell is made of brass with a thickness of 0.8 mm,bonded to a 13.2 mm thick plastic composite.
The electricalresistivity of the brass is 6 x 10" 8 52 • m.
The shell has asingle gap in the toroidal direction and is poloidally continuous. The resistive shell time is 1.5 ms while the shell time ofthe old aluminum shell (14 mm thick) was 25 ms.
The characteristic time of the plasma chamber for vertical field penetration
is 0.1 ms.
The copper helical windings occupy half the volume
between the minor radii of 0.215 m and 0.255 m.
This OHTE device with the resistive shell was operated in
the reversed field pinch mode.
Reversed field pinch discharges
were routinely produced for durations much longer than the resistive shell time.
The longest discharge produced had a reversed field duration of 11.3 ms, corresponding to a factor 7.5
times the shell time constant.

657

IAEA-CN-47/D-V-7
Current (kA)
300
(a)
i

l

0

i

i

i

i

i

i

r

1 . 2 3 4 5 6 7 B 9Time (ms)10

Pickup signals (±5 T/s)
. . .

...i^

t

i

(b)
i

i

i

(c)
1

l

i

t

i

Id)

(e)
I

1

i

i

i

1

1

i

i

i

(9)
i

1

i

t

t

(h)

3

4

5

6
7
Time(ms)

89FIG. 2. Typical plasma current waveforms (a) and magnetic pickup loop signals Br: top 78.50° (b),top 146.25" (c), top 157.50° (d), outside 140.625" (e), outside 151.875° (fl, bottom 135.00" (g)and bottom 146.25° (h).

An example of waveforms from a single discharge is shown in
Fig. 1.
The plasma current, IA, has a broad peak with a valueof 185 kA at 5.7 us.
The one-turn voltage, V,j,, at the peak is
123 V.
The average toroidal field, B A , inside the plasma chamber is about 1 kG.
The average electron density, n e , measured
with an infra-red laser interferometer, is nearly steady and is
4.A x lO1^ cm"3 at the current peak. The magnitude and time behavior of the line-average density are similar to those previously obtained with the conducting shell.
The central electron temperature measured at 4 ms is 178 ± 45 eV, which is also
in good agreement with the results of 1 eV/kA obtained in the
conducting shell experiments.

TAMANO et al.

658

Power Spectrum (t=2.1 - 2.6 ms).:....;...:...•..;..;-

10-'- ,-.m,-.,i».i-..-.-. i ^.;,,,,,^.
v

1O-2:

"-|

71

;:::l!:::::i:::::::s:::!|!:

3

o- -

;!i!l!l!;;;:!;i!:!««*;;s!:!:: : :: :10"'-: ; • . • . : : : • . : : • . " . ' . : • . - . :•.;•.•. • . : : • .

t

•(••

5

io- -

1

10

100

Power Spectrum (t=3.35 - 3.85 ms)k
P::*::;::::HJ:::::::::::l::i::::::::::l::::::!::r:::::::!::::^!::::^:'l:|JH:•=—=—r—=""=•"£""•.•:•:•;!-.::( :•.!•>. t-.ti*.

10"•; !;.-;J.•;•!;? J.-jJ.": i i i •" {•" i«\i i i £; •"; \\ *'!' v. *!"I::::::::::::!::::::::!:::::

10
Frequency (kHz)••:"'ipt::-!:S;

100

FIG. 3. Examples of frequency power spectra before the development of the slinky mode(t = 2.1-2.6 ms) and near its maximum amplitude (t = 3.35-3.85 ms).

The pinch parameter 0 - B0(a)/BA, the reversal parameterF » B ^ a ) / ! ^ , the equivalent plasma radius r p derived by use of
a truncated Bessel function model and the plasma resistance arealso very close to those with the conducting shell, and the resistance follows the previous I*" 3 / 2 scaling. The rapid rise in
the plasma resistance late in time is correlated with the mismatch in the equilibrium vertical field provided by the preprogrammed external coil currents, and the lack of precise equilibrium control appears to be a primary cause for the plasma
termination.
High frequency magnetic fluctuations measured just inside
the resistive shell for components of B A with m = 0 and m = -1
generally have frequency spectra and amplitudes very similar toIAEA-CN-47/D-V-7(m = - l659

10 — IS

FIG. 4. Time behavior of average power spectra, 8^, for the m -1 mode with
5kHz<f<
7.5 kHz and 10 kHz < / < 15 kHz.

those observed in the conducting shell. It appears that the replacement of the conducting shell by the resistive shell does
not seem to affect the internal resistive kink mode (tearingmode) with high toroidal mode numbers n, as expected by the linear MHD stability theories.
On the other hand, kink instabilities with low n predicted by the linear theories for the resistive shell case are not observed.
In order to investigate detailed MHD behavior of the plasmas with the resistive shell, an array of 128 pickup loops, 32
toroidally by 4 poloidally, was installed over the outer surfaceof the helical coils at a minor radius of 0.255 m.
Each pickup
loop spans 11.25° toroidally by 90° poloidally, and together
they cover the entire torus.
The pickup loops are sensitive toradial magnetic fluxes.
Figure 2 shows a typical plasma current waveform and themagnetic loop signals at about 180 kA.
Most of the pickup loop
signals are reasonably quiet, as represented by the top pickup
loop signal B r at the toroidal location of 78.50° (Fig. 2b).

660

TAMANO et al.

(a)Mode Amplitude (orbitrory units)

m=+1

m=0

n=-1

m=±2
0

2

4 6 8 10 12 14 16
Toroidal Mode Number n(b)Normalized Mode Plot (

m=-1
0

45

90 135 180 225 270 315 360
Toroidal Angle (°)

FIG. 5. (a) Mode amplitudes for m = +1,0, —1 and ±2, versus toroidal mode number,(b) Phases of the m = — I modes with 10 < n < 16, as a function of toroidal angle <j>.

Figures 2c through 2h show the B r signals of the top pickup
loops at 146.25°
and 157.50°; the outside pickup loops at
140.625° and 151.875°; and the bottom pickup loops at 135.00°
and 146.25°, respectively. These exhibit a local kink instability starting at about 4 ms into the discharge with a growth timeof about 1 ms (i.e., resistive shell time). Its toroidal extent
is limited to approximately one period and its toroidal period(~ 0.5 m long) corresponds to n ~ 16.
It has the same helicalhandedness as internal magnetic field lines.
The plasma shapewith this mode is analogous to the kink when the spring toy"slinky" is twisted. Thus, we have named it the "slinky" mode.
An interesting and important feature of the 6linky mode isthat it suddenly stops growing, as seen at 6 ms in the example,
and self-heals. Apparently, slinky modes arise randomly in time
and toroidal position.
As its amplitude increases, the mode
also expands spatially in the toroidal direction, but a sudden
rearrangement of the plasma prevents it from growing further beyond one or two periods, and the slinky mode decays away after
the rearrangement.

IAEA-CN-47/D-V-7

661

The plasma high frequency oscillations seen also have characteristic mode numbers n ~ 16. The spectra of m = -1 high frequency oscillations measured by a pickup loop inside the shellindicate development of a peak between 10 ~ 20 kHz when a slinkymode is growing nearby.
Figures 3a and 3b show the spectra obtained from another shot before the development of the slinkymode (2.1 — 2.6 ms) and near its maximum amplitude (3.35 ~ 3.85
ms).
Figure 4 illustrates the time behavior of average power
spectra, B|, for the m =• -1 mode with 5 kHz 4 f «; 7.5 kHz and 10
kHz s; f < 15 kHz. It indicates the power in 10-15 kHz started
decreasing soon after the plasma rearrangement, which occurred
at 3.6 ms in this case.
The Fourier analysis of the 128 pickup coil signals showsthat m = -1 is the dominant mode when a slinky mode exists, as
shown in Fig. 5a. The analysis of n mode numbers indicates thatthe dominant modes are 10 < n < 16. However, because of the finite number of pickup coils, modes with 16 < n <; 22 could also
have contributed. It turns out that the several m = -1 modes
with adjacent n-mode numbers around 16 form a slinky mode if
they all add up at one location.
Indeed, Fig. 5b confirms that
all the m =• -1 with n values mentioned above are adding up near<}> » 35°, where the slinky mode is located.
This leads to thefollowing explanation: Typical RFP plasmas are unstable againstm » -1 resistive internal kink modes with n > 10. The modes
with very high n numbers may be stable because of the magnetic
shear.
Thus, the m = -1 modes with 10 < n < 22 may become all
unstable and grow with the resistive shell time-scale.
Under acertain condition, they all lock together and add up at one location, forming a slinky mode.
However, the phase locking issuddenly lost and after that the modes no longer add up. Thisis likely due to mixing of field lines through the slinky region,
so the plasma finds a way to get to the relaxed state.
In summary, the results with the resistive shell are very
similar to those with the conducting shell, except for the observation of the slinky mode, which grows with the resistive
shell time-scale and then self-heals.
This indicates that ahighly conducting shell may not be required for a reversed field
pinch and suggests that linear plasma stability is not always
necessary in a confinement system.

ACKNOWLEDGEMENT
This work was supported by the United States Department of Energy, underContract No. DE-FGO3-86ER53228.

662

TAMANO et al.
REFERENCES

[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8J
[9]

TAMANO, T., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc.
10th Int. Conf. London, 1984), Vol. 1, IAEA, Vienna (1985) 431.
BAKER, D.A., et al., ibid., p. 439.
CAROLAN, P.G., et al., ibid., p. 449.
HIRANO, Y., et al., ibid., p. 475.
ANTONI, V., et al., ibid., p. 487.
GOEDBLOED, J.P., et al., Nucl. Fusion 12 (1972) 649.
TANAKA, M., et al., Nucl. Fusion 12 (1972) 119.
NALESSO, G.F., COSTA, S., Nucl. Fusion 20 (1980) 443.
GOFORTH, R.R., et al., Nucl. Fusion 26 (1986) 515.

IAEA-CN-47/D-V-8

ENERGY CONFINEMENT ANDNONLINEAR TEARING MODE DYNAMICS
IN A HIGH CURRENTREVERSED FIELD PINCH
Z.G. AN1, P.H. DIAMOND, T.S. HAHM,
G.S. LEE2, B.D. SCOTT
Institute for Fusion Studies,
The University of Texas at Austin,
Austin, Texas
B.A. CARRERAS, L. GARCIA,
J.A. HOLMES, V.E. LYNCH
Oak Ridge National Laboratory,
Oak Ridge, Tennessee
United States of America

Abstract
ENERGY CONFINEMENT AND NONLINEAR TEARING MODE DYNAMICS IN A HIGH
CURRENT REVERSED FIELD PINCH.
Energy confinement and nonlinear tearing mode dynamics in RFP plasmas are discussed. The
observed broadening of the m = 1 mode spectrum and the role of m = 1 interaction with the m = 0
mode in this process are examined. Magnetic fluctuation levels are found to be insensitive to resistivity.
Developments in the theory of resistive pressure gradient driven turbulence are reported, and ion temperature gradient driven modes are identified as a possible loss mechanism at higher current. Pellet
injection is proposed as a means of improving energy confinement in RFP plasmas.

1. NONLINEAR TEARING MODE DYNAMICS
The dynamics of m = 1 tearing modes has been linked to relaxationand configuration maintenance in Reversed Field Pinch (RFP) plasmas.
[l] In previous work [2,3], the basic theory of MHD turbulence in currentcarrying plasma was developed and used to describe the nonlinear interaction of m = 1 tearing modes in RFP. In particular, a new, nonlinear
saturation mechanism based on 'cascading' to small scales initiated by thegeneration of m — 2 current sheets was proposed and used to estimate (Bz)evolution (i.e. field-reversal) rates using quasilinear theory. Here, further
1

Present address: Center for Nonlinear Studies, Los Alamos National Laboratory, NM, USA.
Permanent address: Department of Physics, Beijing University, Beijing, China.
2
Present address: Oak Ridge National Laboratory, TN, USA.

663

664

ANetal.
10 -1

—i

r~

D t = 500 rfl
a t = 600 TAo t=650

I 10-2
o

10

15

20

25

30

35

FIG. 1. Evolution of the n-spectrum of' m = 1 modes in time.

detailed studies of the saturation and relaxation mechanisms are summarized. Attention is focussed on the broadening of the m = 1 spectrum intoroidal mode-number n and radius, the role of nonlinear interaction withthe m = 0, n = 1 mode in this process, and on what the nonlinear dynamics
imply about magnetic fluctuation levels. Throughout this paper, frequent
reference is made to the results of nonlinear, numerical calculations. These
computations, which utilize the reduced MHD (RMHD) equations for RFP[4] unless otherwise indicated, should be viewed as a means for elucidationof basic physical processes and not as 'simulations' of RFP plasmas.
Initially, tearing modes with m = 1, 10 < n < 20 are destabilized byresistive diffusion away from the Taylor state. However, in the course of
nonlinear evolution, the m = 1 spectrum broadens, as m = 1 modes with20 < n < 40 are generated. At the same time, the radial extent of the
turbulent zone broadens to encompass the region from the axis to reversal
surface. Figures 1 and 2 illustrate these features of m = 1 mode evolution.
The broadening of the m = 1 spectrum discussed above is a consequence of quasilinear 'mode competition' and nonlinear interaction. However, comparison of quasilinear numerical calculations, which retain m = 1
and m = 0, n = 0 evolution only, with corresponding full nonlinear calculations indicates that nonlinear coupling processes are dominant (Fig. 3).

665

IAEA-CN-47/D-V-8<n•^ - 2 --UJ

t = 500 rA
t = 65O rA

-

o

O

1

10,-4 -z-10',-8

0.2

1Il
V//.

0.3

0.4

0.5
r/o

-

0.6

0.7

0.8

FIG. 2. Evolution of the m = 1 mode spectrum in radius.

0 NONLINEAR CALCULATION
I = 6 5 0 TA
10

DOUASILINEAR CALCULATIONt = 700

FIG. 3. Comparison of quasilinear and nonlinear effects
in the evolution of the m — 1 mode spectrum.

666

AN et al.

Indeed, the m = 1 spectrum is strongly affected by nonlinear interaction
with the 77? = 0, n — 1 mode. This is because neighboring m = 1, n = no;7n = l , n = n o + l modes beat together to drive m = 0, n = 1 and m = 2,
n = 2n0 -f 1 modes. The m = 2, n = 2n0 + 1 mode is a localized, stable,
driven current sheet which transfers m = 1 energy to small scale dissipation. In contrast, the driven m = 0, n = 1 mode is global, with stability
maintained by strong magnetic shear at the reversal surface. It back-reacts
with the 7n = 1, n = n 0 -f 1 to drive the m. — 1, n = n0 -f- 2. As a result, them — 1 spectrum broadens. The radial spreading of the turbulent zone is,in turn, a consequence of the generation of m = 1 modes with progressively
lower resonant q(r,) values. This process of nonlinear mode competition
can thus be represented as:y
J 77i = 1, n = no(increasing\
TO = 1, n = no+\/m — 2, n = 2no + 1 —> dissipation1/\ m = 0, n = 1

m = 1, n = n 0 + 2

and has been observed experimentally. Indeed, the m = 1 spectrum evolution indicated by the nonlinear calculations (Fig. 4) is in good, qualitative
agreement with results from the HBTXl-A RFP [5].
Theoretical insight into the role of the m = 0 mode in m = 1 dynamics
may be gained by utilizing the previously derived energy spectrum equations [3] for MHD turbulence in current-carrying plasma. In particular, them = 0, n = 1 is driven by coherent scattering from 77? = 1, n = n0 off
m = l , n = n o + l - Similarly, the m = 1, n = no + 2 is driven by incoherent emission from the beating of m = 1,TO= no + 1 with m = 0, n = 1.
These two nonlinear interaction processes appear in the mode energy (E)evolution equations as:(1)

n=no+2
.

m'\n"

(2)

667

IAEA-CN-47/D-V-8
700

675

40

FIG. 4. Time evolution of the m — 1 spectrum.
(Compare with Fig. 19 of Ref. [5].)respectively, where7(r,s) =Here, A' is the MHD stability parameter, Au>ic is the nonlinear decorrelation
rate, <j> is the electrostatic potential and A\\ is the parallel component of thevector potential (i.e. A\\ ~ A. in the core, A\\ ~ A$ at the reversal surface).
It is important to note that A| n=0 < 0, A ^ = , < 0 and A^ = ] < 0 for 20 <n < 40, i.e. the m = 1 modes with higher n's are nonlinearly driven. As aconsequence of the global character of the m = 0 mode (i.e. Fig. 5), m = 1
interaction with m = 0 is stronger than with m = 2, discussed previously.
However, it is clear from Eqs. (1,2) that the nonlinear interaction terms areof opposite sign, so that cancellation would result if an n-summation were
performed. This result, which follows from energy conservation, indicatesthat the principal role of interaction with the TO = 0, ra = 1 mode is toredistribute m — 1 mode energy among different n's. However, since them = 1, n = n 0 spectrum broadens, the m" = 2, n" — 2nQ + 1 driven
mode spectrum also broadens, thus facilitating net m = 1 damping via
nonlinear coupling to smaller scales. Hence, the m = 0, n = 1 interaction

668

ANet al.

-1.0_

1.0 0.5 -

J
-0.5

0.5
r/a

1.0

FIG. 5, Comparison of the m = 0 and m= 2 radial eigenfiutclions.

also serves to accelerate the true nonlinear saturation mechanism. Indeed,
the slow decay of the E(m) spectrum (Fig. 6) is consistent with the notionof cascade to small scales. Finally, it should be noted that reconnection
associated with the driven m = 0, n = 1 mode serves to 'lock-in' thereversed (Bz(r)) quasilinearly generated by m = 1 kink modes [3].
The scaling of saturated m = 1 mode magnetic fluctuation levels withmagnetic Reynolds number 5 = (TR/TA) is of considerable interest. Previous work [3], which predicted a weak dependence of (B/B)m=i on S, wasbased on an assumed balance of linear m — \ growth with nonlinear coupling to m = 2. However, the results of the numerical calculations indicatethat {Br/B)m=l is virtually independent of S (Fig. 7)! Hence, despite therobust character of m = 1 modes, the linear growth rate is apparently
not a good approximation for the instability drive in the nonlinear regime.
Recently [6], the closure theory for MHD turbulence in current-carrying
plasma was extended to include a treatment of the driving term

669

IAEA-CN-47/D-V-8-i

10

D t = 500 rA
t = 600 rA
01 = 650 T.

FIG. 6. Mode energy spectrum as a Junction ofm.

This energy source term depends on the cross-correlationis, in turn, determined by the closure equation:
-

which

(a
(akk + S- k,kx «if) k -

(3)

where
(4)

Equation (3) represents an extension of mean-field-electrodynamics [7] toinhomogeneous, current-carrying plasmas. During nonlinear evolution,°k > ||Vj_/S|| and thus the solution for the driving term does not scalewith S. Alternatively, nonlinear interaction rather than resistive diffusion
determines the phase-shift between <$> and VxA||. These conclusions aresupported by scale transformation analysis of the basic nonlinear equations, in the limit of negligible inertia (i.e. B • V.7|| « 0). A further assumption, required to properly treat the multiplicity of scales inherent intearing modes, is that J|| ~ (A'/w)A§, where w is the radial scale factor
ultimately determined by dimensional analysis.

670

ANetal.

FIG. 7. m = J, n = 10 mode island width (oij) plotted versus S .

Investigations of high temperature tearing mode dynamics utilizing
RMHD with additional temperature and density evolution equations indicate that the nonlinear evolution of even a single tearing mode may depart significantly from'expectations based on linear theory. In particular,
finite-sized islands were initialized and allowed to evolve. Results indicatethat even if the underlying mode is linearly stable [8], the island need only
be weakly nonlinear (Wf > L*v+e/x\\k\, where Wj is the island width,u>*e — k±p3c,/Lx, X\\ — vTe/t/e,') m o r c ' e r that flattening of temperature
along field lines causes the islands to grow under the influence of A' [9].
Thus, the nonlinear interaction mechanisms and their consequences discussed above may be expected to persist into higher temperature regimes,
even if linear theory predicts stability there.

2. Energy Confinement
Energy confinement in RFP plasmas is determined by a combinationof tearing mode and microinstability induced loss mechanisms. Foremost amongst the latter is resistive pressure gradient, driven turbulence(RPGDT) due to resistive interchange modes. Previous work [3] predicted that magnetic fluctuations associated with RPGDT drive stochastic magnetic field induced transport with thermal diffusivity XE ~(e/aq')a/3l/2VTe<i/S. Balancing such losses with Ohmic heating yielded
the scaling relations Tc ~ 1%/N, where N is the line-averaged density, andAs ~ (7*g'A)(rTle/Tnt)1''6 "• 10%. These results were derived using heuristic
mixing-length arguments. However, recent theoretical and computational

IAEA-CN-47/D-V-8

671

studies of R P G D T [10,11] indicate that while heuristic approaches may beadequate for predicting scalings, they do not accurately predict fluctuationlevels. In particular, the thermal diffusivity is enhanced beyond the previous result according to XE = XE A 3 / 2 , where A [11] is a logarithmic functionof f3#, S, etc. For typical R F P parameters, XE/XE~ 10! As a consequence,
the new results predict, a slow degradation of (3# as Ip increases. Finally,
when the diamagnetic frequency u;*e exceeds the characteristic MHD frequency (i.e. growth rate) for R P G D T , XE is modified according to [12]:(5)where the bar denotes a spectrum average. Thus, steep density gradientstend to quench RPGDT losses.
The RPGDT model is, of course, based on resistive MHD. However, attemperatures expected in future high current RFP experiments, the conditions required for the applicability of resistive MHD will be violated. In
particular, the rather fiat density profiles and strong ion heating (via equilibration, proportional to n 2 , and anomalous processes) present in RFP
plasmas make it quite possible that rn = dtnTijdtnn exceeds the threshold
f?i,crit ~ 1.5 for the onset of another type of pressure gradient driven turbulence, namely that produced by ion temperature gradient driven (rj^-modes.
Ion temperature gradient driven modes [13] are insensitive to electron collisionality, with growth rates 7 ~ [ ( 1 + ^ ) / T ] \nq'\ psc,/r increasing with shear
and mode widths Ax ~ [(1 + T ^ / r ^ V . - Here r = Te/T{ and p, = c,/n,-.
Moreover, inclusion of non-adiabatic electron dynamics allows coupling of
the 77;-mode to the resistive interchange mode [14]. This coupling acts toenhance instability in all parameter regimes. Indeed, solution of the eigenmode equation for the coupled system reveals a smooth spectrum of roots
from the pure RPGDT limit (non-adiabatic electron) to the pure r^-mode(adiabatic electron) limit. At higher temperatures (i.e. k?,Vj-f/uv > 1)
the J7,-mode character dominates. In particular, fy-modes are broader andmore robust than semi-collisional resistive interchange modes. Results fromrenormalized turbulence theory [15] indicate that the resulting ion thermal
diffusivity is given by
^(^)2

(6)

where c(Re) ~ £n(l •+ j^) and n is the average toroidal mode number. It
follows that for ZT-40 parameters, the predicted energy confinement time is
TE ~ .5msec. Balancing with Ohmic heating (for Te ~ T-) yields the scaling
relations T ~ Ip/NllA and 0d ~ N3/4/Ip. Hence, fo may be expected todegrade with current (for N ~ Ip) if ?j,-modes are excited.

672

AN et al.

The implications of the various studies of RFP thermal transport can besummarized as follows. In the core, magnetic fluctuations driven by m =1 tearing modes dominate. Since BjB remains constant as Ip increases,
central confinement is always poor. Nearer the edge, RPGDT and 77,-modes
control the local transport. Results indicate that both mechanisms resultin the degradation of /?,» as Ip increases. Thus, the implications appear
quite pessimistic at first glance. However, it is important to recall thatsteep density gradients tend to suppress ift-modes {ni < ?7i,Crit) and to quench
RPGDT losses via diamagnetic effects (Eq. (5)). Hence, improving particle
confinement and steepening the density profile may be expected to resultin improved energy confinement! Thus, pellet injection is suggested asa potentially viable mechanism for improving RFP performance at highcurrents.
ACKNOWLEDGMENT
This research was supported by the Office of Fusion Energy, U.S.D.O.EV
under Contract No. DE-FG05-80ET-53088 with The University of Texas
and Contract No. DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc.
REFERENCES[1] WATT, R.G., NEBEL, R.A., Phys. Fluids 26 (1983) 1168.
[2] DIAMOND, P.H., et al., Phys. Fluids 27 (1984) 1449.
[3] AN, Z.G., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984 (Proc. 10th
Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 231.
[4] STRAUSS, H.R., Phys. Fluids 27 (1984) 2580.
[5] HUTCHINSON, I.H., et al., Nucl. Fusion 24 (1984) 59.
[6] LEE, G.S., PhD Thesis, The University of Texas at Austin, 1986.
[7] MOFFATT, H.K., Magnetic Field Generation in Electrically Conducting Fluids, Cambridge
Univ. Press (1978).
[8] DRAKE, J.F., et al., Phys. Fluid 26 (1983) 2509.
[9] SCOTT, B.D., et al., Phys. Rev. Lett. 54 (1985) 1027.
[10] CARRERAS, B.A., et al., Rep. No. 247, Institute for Fusion Studies, Texas Univ. (1986); tobe published in Phys. Fluids.
[11] CARRERAS, B.A., et al., Paper IAEA-CN-47/D-V-2, these Proceedings, Vol. 2.
[12] DIAMOND, P.H., et al., Phys. Fluids 28 (1985) 1116.
[13] COPPI, B., et al., Phys. Fluids 10 (1967) 582.
[14] AN, Z.G., et al., in Compact Toroids (Proc. 7th Int. Conf. Santa Fe, NM, 1985), Los Alamos
National Lab., NM.
[15] LEE, G.S., DIAMOND, P.H., Phys. Fluids 29 (1986).

1AEA-CN-47/D-V-9

THEORETICAL AND EXPERIMENTAL STUDIESOF FIELD REVERSED CONFIGURATIONS*
D.C. BARNES
Science Applications International Corporation,
Austin, Texas, United States of America
W.T. ARMSTRONG, E.J. CARAMANA, R.E. CHRIEN,
W.N. HUGRASS, H.R. LEWIS, R.K. LINFORD, K.M. LING,
K.F. McKENNA, D.J. REJ, J.L. SCHWARZMEIER,
R.E. SIEMON, M.M. TUSZEWSKI, K.A. WERLEY
Los Alamos National Laboratory,
Los Alamos, New Mexico, United States of America
D.S. HARNED
Courant Institute, New York University,
New York, N.Y., United States of America
S.) OKADA
Osaka University,
Osaka, Japan
C.E. SEYLER
Cornell University,
Ithaca, N.Y., United States of America
D.E. SHUMAKER
National Magnetic Fusion Computer Center,
Livermore, California, United States of America
H. TUCZEK
Essen University,
Essen, Federal Republic of Germany
G. VLASES, R.D. BROOKS, Z.A. PIETRZYK,
K.D. HAHN, D. LOTZ, R. RAMEN, J. WIGHT
University of Washington,
Seattle, Washington, United States of America
Abstract
THEORETICAL AND EXPERIMENTAL STUDIES OF FIELD REVERSED CONFIGURATIONS.
Important new Field Reversed Configuration (FRC) results are presented. A kinetic ion modelis used to calculate the growth rate of the internal tilting mode, including exact ion orbits in a selfconsistent, axisymmetric FRC. Previous calculations, which used the MHD mode as a trial function,* Work supported by the United States Department of Energy.

673

674

BARNES et al.

are extended to determine the kinetic eigenfunction consistently. A regularity condition must be imposedon the solution at the internal field null (0-point). This condition is derived for an appropriate choiceof the independent variable. The new results on kinetic stabilization in present experiments show kinetic
modifications of the tilt mode and reductions in growth rate for large s comparable to those found inprevious studies. — Transport calculations using both 1W-D and 1V2-D models show that experimentallyobserved transport exceeds classical expectations by roughly one order of magnitude. — The FRX-C/T
formation region has been enlarged in diameter by 50%, and quasi-steady cusp coils have been installed
to compare tearing and non-tearing formation. FRCs with significantly larger poloidal flux ( < 8 mWb)
and s ( < 4) have been formed. However, their flux confinement was degraded compared with earlier
FRX-C results. — Initial results from the Coaxial Slow Source (CSS) are reported and compared withnumerical modeling. A long 'annular F R C is formed at lower voltages and is maintained for 30 to
60 /is. Flux builds up to levels of 7 to 10 mWb with a flux confinement time of about 15 us.

KINETIC STABILITY CALCULATIONS
A long—standing question concerning FRC observations has been
the lack of experimental observation of the internal tilt mode,even though MHD calculations predict a virulent instability. In
the Los Alamos FRX-C experiment, for example, FRCs show noevidence of the internal tilt mode for the transport-determined
lifetime of up to 300 /xs, which is over an order of magnitudelarger than the MHD growth time. Profile effects [1], nonlinear
effects [2] and finite Larmor radius [3,4] theory do not explain
the stability. Recent Vlasov-fluid model computations [5] showthat large orbit ions have a strong stabilizing effect on thetilt mode. In the present computation, a dispersion functional(quadratic form) [6] is constructed, which includes exact ion
orbits in a self-consistent 2-D FRC equilibrium. Efficient useof data generated by a Monte Carlo sampling of particle orbits
makes the calculation feasible numerically. For a representative
FRC equilibrium, the kinetic value of the growth rate has been
calculated vs. s (Fig. 1), where s measures the number of local
ion Larmor radii from the field null to the separatrix. Valuesof s in present experiments for cases that optimize confinement
are in the range of 1 to 2. In initial calculations, which used
the MHD eigenmode as a trial function in the kinetic dispersion
functional, the growth rate for s ^ 2.0 is less than the MHD
value by more than a factor of 10. When s increases to about 7,
the growth rate increases to about one half of the MHD value.
In the stabilized regime (s ^ 2) strong kinetic effects may
cause the kinetic mode to differ significantly from the MHD mode.
The effect of this variation on the growth rate is investigatedby extending the previous calculations so that the kinetic
eigenmodeis determined
self-consistently
along with the
eigenfrequency.
As described previously [5], the field-line
displacement £ is represented in terms of a single function
iz(i')- where V
is the magnetic flux and {zthe axial
displacement. An approximation of (z was first determined by

675

IAEA-CN-47/D-V-9' 15.7.

3.

2.

1.5

0.75

0.00

FIG. 1. Variation of normalized kinetic growth rate with system size parameter s. As s — 0,mode becomes kinetically stabilized. As s — oo, kinetic growth rate approaches MHD growth rate.
Points obtained numerically are indicated. Lower scale indicates s~' as the abscissa, the upper s(increasing to left).

expanding ( in terms of a finite basis set of functions of i>,then requiring the variation of the dispersion functional withrespect to the conjugate expansion coefficients to vanish.
However, this procedure led to very inaccurate results. This
inaccuracy appears to have been caused by admixture of acomponent of the singular solution of the integro-differential
normal mode equation near the o-point, which is a singular pointof the magnetic flux coordinates. It has proven to be effective
to consider the regularity problem in terms of an independent
variable p instead of VThe variable p is a radius-likecoordinate defined by p = -jip — V o . where V o >s the value of if/ at
the o-point. The requirement that the magnetic field and current
density be nonsingular at the o-point implies that iz be twice
differentiable with respect to V- In terms of p, this conditionmeans that the first and third derivatives of (2 with respect top vanish at p = 0.
Results have been obtained by imposing this regularity
condition and the condition that (z vanish at the separatrix onto
the basis of B-splines over a uniform p mesh. That is, linear
combinations of the B-splines which satisfy these boundary
conditions are first formed; then (z is represented as a linear
combination of these resulting basis elements, the coefficientsof which are determined by the variational procedure. Figure 2
shows a comparison of the kinetic mode with the previously
determined MHD mode for an MHD-like case (s = 15). For thisapproximation, the range of p from the o-point to the separatrix
was divided into 9 uniform intervals. Even for this large s
case, there is a modification of the mode near the o-point wherekinetic effects dominate. The mode also develops an imaginary
part (phase is chosen to make fz real at the o-point), so that
there is a twisting of the tilt as position varies across theBARNES et al.

676
-1.0

0.50-

-0.S0

-to
FIG. 2. Comparison ofMHD eigenmode (solid curve) with real (dashed curve) and imaginary (dottedcurve) parts of kinetic mode for s = 15. Normalization is such that area under magnitude of kinetic
mode is same as MHD mode and max of MHD mode is unity.

flux
surfaces.
The
growthrate
determined
with
this
self-consistent mode is 0.76 of the MUD value, compared to 0.75
obtained using the MHD mode as a trial function. Calculations of
the self-consistent mode in the kinetically stabilized regime are
now being completed.

FRC TRANSPORT CALCULATIONSTransport simulations of FRCs were performed using both1-1/2-D I 7 ! and 1-1/4-D [8,9] transport codes. The 1-1/2-D code
alternates between an 2-D equilibrium and a 1—D transport
calculation, and includes Braginskii and lower-hybrid drift (LHD)transport coefficients. The 1-1/4-D code refers to a 1-D FRC
model with some 2-D effects. This latter code describes radialtransport in both the closed and open field line regions, while
incorporating
sucheffects
asaxial
contractionoftheseparatrix, flux surface averaging and axial streaming andconduction along open field lines. By multiplying the classicalcollision frequencies by appropriate factors, the confinement
times for the FRX-C experiment can be duplicated to within 10% by
both transport codes. The 1-1/2-D code predicts that for the 20
mtorr case the e-i collision frequency must be increased by afactor of about 4.5 at the o-point while only a factor of one at
the separatrix. For the 5 mtorr case the resisitivity anomaly
factor at the o-point is 11.
Ion thermal conduction isapproximately classical, while the electron thermal conductivity
is strongly anomalous. For example, in the 5 mtorr case, theelectron thermal conductivity is 22 times classical. For both

IAEA-CN-47/D-V-9

677

the 5 and 20 mtorr cases only half of the LHD anomalous
resistivity is used. Comparison of the 1-1/4-D code with the
1-1/2-D code has been done for the case of thermal conduction
being the only transport mechanism, and the agreement is good.
Simulations of the FRX-C experiment with the 1-1/4-D code predictfor 20 mtorr: K^K^,
ffe=6.5*Kcl,' T?=5.2*?JCJ ; and for 5 mtorr:<j=f c j. * e =36.*K C J, V-9**)ci- In the 20 mtorr case, electron-ion
equilibration dominates and holds Tj=T . In the 5 mtorr case,ions on the open field lines are col 1isionless thereby permitting
Tj to remain high for the discharge duration.
It is unclear
whether the difference between the 5 and 20 mtorr cases is due
to some transport scaling or whether the plasma is exhibiting
completely different transport physics regimes.

STUDIES OF TEARING AND NON-TEARING FORMATION IN LSM
A. Introduction
The FRX-C/T device[10] has been modified by enlarging thediameter of the FRC formation region by 50%.
This larger
formation region (termed LSM for Large Source Modification) will
be more suitable for future translation and adiabatic compression
experiments; it has to date been used (without translation) to
study the processes governing FRC formation.
An important
experimental objective is to achieve the maximum possible trapped
poloidal flux <p in the larger source (corresponding to maximums) in order to obtain optimum confinement and to look for thetilt instability. Improved FRC formation in small (0.2 •» 0.3 m)
diameter devices has been reported when field line tearing andreconnection is avoidedf11,12].
Installation of quasi-steadycusp coils on LSM has permitted comparisons of tearing andnon-tearing formation in a larger device which are reported here.
It was found that larger-radius FRCs (up to 0.21 m ) , withincreased values of the inferred </> ($ 8 mWb) and s (^4), could
be formed in LSM. However, their flux confinement, in bothtearing and non—tearing operation, was degraded compared withthat obtained in earlier FRX-C experiments[13,14].
B. Description of Experiment
The measurements were obtained with the formation geometry
and magnetic characteristics listed in Table I. The plasma wascreated in an initial 10 mtorr static fill of deuterium gas(except for 2 mtorr results in section E . ) . The preionization
method consisted of an initial seed ionization produced by a 10
MHz generator coupled to antennas at each end of the coil,followed by full ionization produced by a 175 kHz ringing i5-pinch
discharge.
The capacitor bank energies and voltages were
unchanged from earlier FRX-C operation.
Thus the azimuthal
electric field at the quartz tube during field-reversal was
decreased from 37 kV/m in FRX-C to 28 kV/m in LSM.
Anothersignificant change was the decrease in tf-pinch coil elongation

678

BARNES et al.

TABLE I. CHARACTERISTICS OF LSM
Formation Geometry•tf-Pinch Coil Length (overall)
Coil Central Diameter (1.4 m long)
Passive Mirror Diam. (0.3 m each end)
Magnetic Mirror Ratio (on axis)
Quartz Chamber I.D.
Magnetic Fields
Main Field Swing (vacuum)
Main Field Rise Time ( T / 4 )Main Field Decay Time (L/R)
tf-Preionization Field Amplitude
Bias Field (maximum)
Cusp Field (maximum on axis)
Cusp Field Rise Time ( T / 4 )2.01 m0.76 m0.66 m1.20
0.60 m
0.6 T
6.7 f.AS
500 /AS
0.06 T
0.13 T
0.30 T
4 ms(length between mirror peaks to central diameter ratio) from 3.6
to 2.1, which tended to reduce the FRC elongation and increase
the influence of the mirror fields on the FRC equilibrium. The
passive mirror ratio of 1.20 was sufficient to prevent FRCs fromdrifting axially out of the coil following formation (a mirror
ratio of 1.11 was found to be inadequate).
C. Tearing versus Non-Tearing Formation
After performing initial tests of tearing formation, cusp
coils were installed at each end of the iJ-pinch coil to permit
direct comparison of tearing and non-tearing formation. The twoformationtechniques
areillustrated
by
the
2-D
MHD
simulations[l5] shown in Fig. 3. The simulations indicated thatcusp field strength B c u^ 0.05 + 0.10 T was needed to avoid
tearing and ejection of part of the plasma at each end. Thistransition was qualitatively confirmed by a large reduction invisible light observed at the end during non-tearing formation.
Cusp fields were found to enhance flux trapping.
Thisenhancement was particularly noticeable at weak bias field B^(B^ < 0.0B T) for which the "zero-crossing" flux, <pzc = to(Bw=0),
was increased by 1.3 •* 2.5 times that obtained with B c u= 0.
The use of cusp coils was observed to improve the symmetry(about the axial midplane) of FRC formation. One way to quantify
the improved symmetry was to observe the maximum axial speed of
the FRCs separatrix volume.
The corresponding axial kinetic
energy decreased monotonical ly from 28 J to 8 J as B c
wasincreased from 0 to 0.2 •• 0.3 T. The observed improvement information symmetry was consistent with the transition fromtearing to non-tearing formation as predicted by the simulations.
The axial contraction tended to be stronger for non-tearing
formation. The strength of the axial contraction was estimatedusing the minimum elongation cm\n defined as the ratio of the

IAEA-CN-47/D-V-9

679

FIG. 3. Configuration of LSMformation region showing simulations of (a) tearing and (b) non-tearing
formation. In tearing formation, closed field lines tear and reconnect under passive mirrors after main
field has risen. In non-tearing formation, cusp coil fields drive preformed FRC separatrix inwards inaxial direction shortly after field reversal, thereby preventing tearing.

(FWHM) length of the FRC measured using an axial array ofinterferometers to the maximum diameter inferred from theseparatrix radius profile.
The e m i n values were 1.2 -» 1.8
without tearing and 8.0 •> 3 5 with tearing (for B^ = 0.07 -• 0.09
T).
The axial contraction also took place earlier without
tearing.
For the formation conditions which optimized confinement,
larger <p values could be retained in the equilibrium FRC with
non-tearing formation. An example of the parameters of one of
the best FRCs formed using non-tearing formation is shown in
Fig. 4. The largest average <p values (5.8+0.6 mWb, where theerror estimates denote standard deviations) were obtained forBcusp = °' 2 •*0 3 T a n d ^b = 0 0 8 * °' 0 9 T ("cusp on" condition).
When B c u= 0 ("cusp off" condition), <p was lower (2.3+0.5 mWb
for B b = 0.065 •+ 0.075 T ) . However, larger <p with tearing
formation(3.5±0.6 mWb) could be obtained
undersimilarconditions before the cusp coils were installed ("no cusp"
condition). Thus it appeared that the currents induced in thecusp hardware were detrimental to the tearing formation process.
Corresponding to the larger <p , larger values of s (up to 4)
were also attained with non-tearing formation. In the "cusp on"
condition, s values of 2.7±0.4 were observed at 30 /is after
field-reversal. In the "cusp off" condition, the s values were1.9±0.4. Both <p and s were inferred from external measurements
assuming a typical diffuse profile[l4] that matched FRX-C data.
Further work is needed to determine the most appropriate profile
for LSM conditions.

680

BARNES et al.

b
m

'E

0.4
0

4.0

—

I

i

•

i

i

r

)

i

i

|

—

2.0
200.0
100.0
SO

100

150

200

TIME (/>s)

FIG. 4. Time evolution of external magnetic field, separatrix radius, average density, and average temperature ((Tt + TJ/2) for one of the best FRCs formed without tearing. Equilibrium (t = 30 us) ischaracterized by xs = 0.50, <l>p = 6.7mWb, and s = 3.0. Confinement times (t = 30 — 100 us):
T+ = 130 )is, TN = 250 us, and TE = 85 us.

D. FRC Confinement
The confinement times of flux (T ), particles ( T N ) . andenergy ( T E ) were determined over an interval from the start of
the axial equilibrium (t = 30 jus) to the appearance of the n=2
mode (t = 70+10 /*s). When no n=2 oscillations occurred, the time
the separatrix radius began to decay more rapidly was chosen to
end the interval.
The flux confinement of FRCs produced in LSM was, on average,
degradedcomparedwiththat
observed
in earlier
FRX-C
experiments. Similar average T values were observed for "cuspon" ( T = 58±40 /xs) and "cusp off" ( T = 50+22 /us) conditions.
Somewhat better average flux confinement was observed during
tearing formation in the "no cusp" condition ( T = 110±45
The maximum T was about 200 /is. The absence of T e measurements
prevented detailed comparisons with classical flux confinement
predictions. However, the resistivity anomaly with respect tothe classical resistivity of a 100 eV plasma (assuming a rigid
rotor FRC profile) was in the range 15 -> 60.
This wassignificantly
larger than the resistivity
anomaly(3 -* 7)
measured in FRX-C[13,14].
The
particleconfinementin LSM improved
withr( T N = 1 . 2 T ). For those FRCs with greater than average T
values, r»j was, on average, a factor of two smaller than thepredictions of a transport theory based on lower hybrid drift
turbulence[16]. The global energy confinement times[17] scaledwith T N (Tg « T N / 2 ) i n a manner consistent with an energy balance
dominated By particle convection.
E. Low Fill Pressure Operation
A data run at 2 mtorr was conducted in which hotter(T + Tj sj 0.8 keV) and less dense (n « 5x1020 nT 3 ) FRCs were
(75±35) its, could only beformed.
The best confinement,<P

IAEA-CN-47/D-V-9

681

achieved with non—tearing formation (B C U S p « 0.25 T) and weakbias (Bjj = 0.04 T ) . Tearing formation at 2 mtorr resulted inpoor confinement with r < 35 fis for any B^ • For non-tearing
formation strong axial contractions were observed and theequilibrium values of <p (= 2.6+0.3 mWb) and s (= 1.0) were
smaller than those inferred at 10 mtorr because of the lower B^
and higher Tj .
F. Discussion
LSM experiments (and formation simulations) have shown
several advantages in the use of auxiliary quasi—steady cusp
coils for FRC formation.
First, cusp fields in sufficientstrength permit FRC formation without field line tearing andreconnection driven by mirror fields.
This eliminates thenecessity for those mirror fields during formation.
Second,non-tearing formation is more symmetric, resulting in less axial
speed. Third, FRCs with larger <p and s can be formed in the
non-tearing mode. Fourth, FRC formation was possible at low (2mtorr)
fillpressure with non-tearing
formation.
These
advantages were attained in LSM without the complexity of rapidly
pulsed auxiliary coils and precise timing necessary for"balloon"[ll] or "programmed"[12] formation modes.
Flux confinement was degraded in LSM compared with earlier
tearing formation results in FRX—C, independent of the formation
mode used. The degraded confinement is not presently understood.
Some possible explanations are poor formation, low T e , FRC
elongations that are too small (either transiently or in
equilibrium), interaction with mirror fields, and instability to
the tilt mode. The possibility of low T e will be tested byThomson scattering measurements. Elongation and mirror effects
can be ameliorated by a change in coil geometry (e. g. a 0.70 n
diam. coil without passive mirrors) or by translation experiments
using the FRX—C/T translation region. We have attempted to
identify tilt instability by searching for m=l, n=l magnetic
perturbations (B^) outside the quartz vacuum chamber.
Noevidence for the tilt mode has been detected by this method (butit is not clear whether the probe sensitivities (10""' T) are
adequate to detect the largely internal mode predicted by linear
theory).
It is also worth noting that formation at low bias
field or fill pressure resulted in FRCs with the same range of s
as in FRX-C and yet continued to show lower T values. This
indicates that formation or equilibrium problems are more likely
than tilt instability to be responsible for the degraded flux
confinement.
LOW VOLTAGE GENERATION OF ANNULAR FRC'S IN THE CSS DEVICE
The conventional modified i5-pinch method of forming FRCs, as
exemplified by the FRX-C device, requires flux insertion on
Alfven time scales, with concomitant high-voltage technology. We
report here on early results with a system called the Coaxial

682

BARNES et al.

PlasmaOuter coil(b)
Outer T 1 / a = 6 0 0 jus

Time

,8

32

FIG. 5. (a) Schematic of CSS; (b) CSS timing.

0.4

-0.2

-0.4
-20

0

20

40

60

Time (jus)FIG. 6. Bz at midplane adjacent to inner (below) and outer (above) coils.

683

IAEA-CN-47/D-V-9> 14 mtorr• 22 mtorr
a 37 mtorr

-10

0

10

20

30

40

50

60

FIG. 7. Flux within separatrix measured and predicted on the basis of resistivities shown.

Slow Source (CSS) [18], which is designed to form "annular FRCs"(Fig. 5a) at low voltage, with flux build-up on diffusive timescales. In an optimized system, the flux inside the inner coilgoes to zero as the inner coil current reaches its maximum(3rd-l/4 cycle, Fig. 5b) so that no flux links both coils and theplasmoid can, in principle, be translated into a simple
cylindrical chamber, and coalesce into a true compact toroid.
The CSS coils are 95 cm long and have radii of 7 and 21 cm. The
loop voltage is typically 1.25-2 kV, with stored energies up to
25 kJ and an effective system rise time of 30 /xs. Deuterium fillpressures used range from 10 to 50 mtorr. Design of the device
and experiment interpolation have been aided by the use of
calculations based on a 2—D resistive MHD code coupled to a
vacuum field solver [19]., The principal results of the initial
testing phase of the program are as follows:• The fieldreversedconfiguration,
characterized byopositely directed B fields at the inner and outer walls, isestablished quickly after start of the outer coil current,
for all fill pressures examined. It lasts from 30 to 60 ^is,depending on coil voltage ratio and fill pressure (Fig. 6 ) .
• The flux within the separatrix increases for approximately
25 /us, to values of 7-10 mWb, depending on charging voltage
and fairly independent of fill pressure. Figure 7 shows acomparison of experiment and simulation.
• The plasma radial position can be controlled by choice of
coil voltage ratio.
The length remains long under mostcases, with X—points near the coil ends, similar tosimulation plots of Fig. 8.

684

BARNES et al.

10

14

18

Radius (cm)FIG. 8. Calculated flux contours.

4 8 MS

FIG. 9. Holographically determined electron density.

• Density
profilesaremeasuredholographically,asillustrated in Fig. 9. Pressure balance total temperatures
range from 50 eV at P Q = 50 mtorr to about 200 eV at
P Q = 10 mtorr.
• The flux lifetime is on the order of 15 us, nearly
and thus T e + Tj. This suggests theindependent of P o>o>presenceofananomalous,
drift-parameter
dependent,
effective resistivity as is seen in convenentional FRCs,
although low T e values resulting from radiation barriers may
play a role as wel1.
In conclusion, slow flux buildup to FKX-C like levels has
been achieved on long time scales, and at about l/50th of the
FRX-C loop voltage. Efforts are now concentrating on improving
plasma lifetime.

IAEA-CN-47/D-V-9

685

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]

SCHWARZMEIER, J.L., BARNES, D.C., HEWETT, D.W., SEYLER, C.E.,
SHESTAKOV, A.I., SPENCER, R.L., Phys. Fluids 26 (1983) 1295.
SCHNACK, D.D., BARNES, D.C., AYDEMIR, A.H., ANDERSON, D.V., submitted toPhys. Rev. Lett.
SEYLER, C.E., BARNES, D.C., Phys. Fluids 25 (1981) 1989.
SCHWARZMEIER, J.L., SEYLER, C.E., Phys. Fluids 27 (1984) 2151.
BARNES, D.C., SCHWARZMEIER, J.L., LEWIS, H.R., SEYLER, C.E., Phys. Fluids 29(1986) 2616.
LEWIS, H.R., BARNES, D.C., SCHWARZMEIER, J.L., SEYLER, C.E., Phys. Fluids 28(1985) 2546.
SHUMAKER, D.E., Rep. UCRL-94433, Lawrence Livermore National Lab.; submitted to
J. Comput. Phys.
CARAMANA, E.J., Phys. Fluids 28 (1985) 3557.
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REJ, D.J., et al., Phys. Fluids 29 (1986) 852.
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SLOUGH, J.T., et al., Nucl. Fusion 24 (1984) 1537.
TUSZEWSKI, M., et al., Phys. Fluids 25 (1982) 1696.
SIEMON, R.E., et al., Fusion Technol. 9 (1986) 13.
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TUSZEWSKI, M., LINFORD, R.K., Phys. Fluids 25 (1982) 765.
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BARNES, D.C., SCHNACK, D.D., Bull. Am. Phys. Soc. 30 (1985) 1514.

IAEA-CN-47/D-V-10

EXPERIMENT ON
RELATIVISTIC ELECTRON BEAM
RING CORE SPHERATOR
K. NARIHARA, T. TOMITA, S. KUBO, M. HASEGAWA1,
M. MATSUI, T. TSUZUKI, A. MOHRI
Institute of Plasma Physics,
Nagoya University,
Nagoya, Japan
Abstract
EXPERIMENT ON RELATIVISTIC ELECTRON BEAM RING CORE SPHERATOR.
Intense relativistic electron beam (REB) rings with a peak current of 30 kA and a
lifetime of 40 ms are realized in SPAC-VII by using a novel REB injector improved in
impurity reduction and REB trapping efficiency. A very quiescent phase at a fluctuationlevel |B p /B p | as low as 10"s continues for more than 15 ms at a ring current of 20 kA.
There, the activity of velocity anisotropy instabilities becomes very weak, owing to thestrong decrease of the acceleration field induced by the current decay. The RF produced
plasma is demonstrated to be confined around the ring with a particle confinement timeof about 1 ms.

1.

INTRODUCTION

It has been shown [ 1 ] that a magnetic configuration such as spherator(Astron-Spherator [2] or REB Ring Core Spherator [1]) can be formed byinjecting a pulsed, intense relativistic electron beam (REB) into the toroidal
device SPAC-VII. This configuration has marked features that distinguish itfrom ordinary tokamak. No sawtooth oscillations occur although the q-value
varies from less than 1 at the centre to 3 to 4 in the outer confinement region.
The current profile peaks more sharply than that of a tokamak but no MHDlike instabilities are present. Owing to the strong vertical field necessary forring equilibrium, a toroidal magnetic configuration with a deep magnetic well,together with high shear, is formed. Therefore, high 0 plasma may, in principle,
be confined. For example, the stability criteria for both Mercier's condition
and ballooning modes lead to a (3-Iimit above 20%, excluding kink modes whichhave not been observed in the experiment.
The experiments on SPAC-VII were carried out in an attempt to solve thefollowing key problems in order to improve the chances for an REB ring core
Mitsubishi Electric Corporation, Tokyo 100, Japan.

687

688

NARIHARA et al.

resistive shelni•«

ithodeo c o r n p r e s s i o n"Icathodei,
REB-plasma
/toil

\YS
f
so

toroidal coilL'^OOkA),

FIG. 1. Schematic drawing of ring formation section of toroidal device SPAC-VII.

spherator to function as a fusion plasma container: (i) effective formation of along lived, high current REB ring, (ii) conditions for achieving a stable REB ringand (iii) clarifying the characteristics of a confined plasma.

2.

FORMATION OF INTENSE LONG LIVED REB RING

The ring formation section is drawn schematically in Fig. 1. To generate an
REB (normally, of 900 keV, 30 kA, 700 ns), a high voltage pulse was sent on a
7 m long transmission line from a Marx generator to an REB injector installedinside the formation section. The toroidal field produced by a current in a central
conductor was 3 kG at a major radius of R = 30 cm; the vertical magnetic field
was pre-programmed to control the orbital motion of the injected REB andthe equilibrium position of the formed ring.
To achieve high current REB rings, the trapping efficiency of the injected
REB was substantially improved by developing a new injector system. A cathodeof conical shape was placed near the side wall at R = 58 cm and connected toa metallic rod attached to the end of the transmission line (Fig. 1). Both thecathode and the rod were immersed in a gun produced plasma before thepulse was imposed. The rod worked as a magnetically insulated transmission
line (MITL) with the plasma operating as the outer conductor. This is anextension of the plasma anode method [3]. Owing to the small cross-section of
the rod MITL and the action of the electromagnetic fields around the injector,
a bombardment of the injected REB by the injector after circulating aroundthe torus was prevented to such an extent that the loss in injected REB as wellas the resulting release of metallic impurities were substantially reduced.
A factor which limits the attainable ring current is the deceleration of
injected electrons by an electric field induced by the rising ring current [4].
Injection of an REB with higher energy is, therefore, favourable in order to689

IAEA-CN-47/D-V-10

30.0-

0.8

1.0
Vk (MV)

1.2

FIG. 2. Ring current versus cathode voltage.

maintain sufficient kinetic momentum of the beam electrons in the ring. In
Fig. 2, the ring current I R is plotted versus the acceleration voltage V k for
different conditions. The upper envelope of I R increases with Vk up toV£ ~ 1MV, but saturates in the range Vk V^, where electric breakdown isobserved around the injector. The voltage V^ was increased with the new
injector so that REB injection became possible at higher beam energy without
encountering a fatal impurity problem.
When the new injector was used, the intensity of the bremsstrahlung X-rays
decreased by more than one order of magnitude compared with the case of theold injectors. To find out the origin of residual impurities, different materials(C, Ti, Ta, Mo) were attached on the wall and/or used for the cathode, andimpurity lines were diagnosed spectroscopically. The residual impurities weremainly released from the wall and not from the cathode. These concentrations
were evaluated in the plasma by comparing X-ray intensities before and afterargon gas puffing. The relation X <* 2 n z Z 2 was used, where X is the intensity
and n z the density of particles with atomic number Z. Thus, it was found thatn z Z 2 /n e < 20. This means that further reduction of the impurities isnecessary in order to attain better plasma confinement and a longer lifetime of
the REB ring.

3.

STABILITY OF REB RING

Frequently, REB rings are accompanied by two kinds of instabilities(Fig. 3 (a)): one is characterized by stepwise current drops and the other one
by a train of short pulse increments in line emission, X-rays and diamagnetic
signals [5]. When the former instability occurred, the ring position moved
inward so as to retain equilibrium. This instability is, as yet, beyond control
but seems to be related to the strength of the toroidal field. An externallyapplied m = 1 dominant error field B er triggered the instability when the ratio
B e r /Bj reached a threshold value.

690

NARIHARA et al.

(a)

I I 1 I I I I I I I I I I

20.0

10.0
time (ms)

(b)

0=1
3 5

+^-vy—j

-5

a

J—i—

l.iu.J ,.

l
f I
10.0

20.0

30.0

40.0

time (ms)

(ms)

o
iao
r(cm)

45.0

FIG. 3. (a) Traces exhibiting typical instabilities observed in REB ring. Ring currently
and Up radiation, fb) Behaviour of REB ring with REB injected into helium plasma.
Ring current, IR; fluctuation ofpoloidal magnetic field, B; He II radiation; and electron line
density, nJ.. (c) Time evolution of minor radius profile of X-ray emissivity.

691

IAEA-CN-47/D-V-10
i—r~ -1—1—1—1—I—1—I—I—I

-1

I

I 1

•

i

r
—'—^—

1.0
0
„ 10
3

A.7cm

1

o 1.0-

0.7

3.3

h i
0
1.0

A

AAWVWV

H A

6.7

§

A A M A A ]\J\l\S\A A. A .

< o
£0.5
i

i

5.0

tiffiili JM
10.0

15.0

20.0

time(ms)

FIG. 4. Time evolution of electron line densities along different chords during RF power
application. Distance between ring centre and each chord is indicated. For reference,
RF antenna current, 7A, and ring current, /R, are shown.

The latter instability, arising with pulsive signals, is in many respects
similar to tail anisotropy instabilities observed in tokamak plasmas withsuprathermal or runaway electrons. However, the plasma, in this experiment,
is in the cope > w c e regime, in contrast to the tokamak case, co p e < o>ce: The repetition
rate of the pulsive signals became higher for faster current decay. This fact suggeststhat the beam electrons were accelerated by the electric field induced by the decayof the self-magnetic ring flux.
When the REB was injected into a helium plasma with a lower neutral
particle concentration, the lifetime of the ring was extended to 40 ms or more;
it decayed exponentially with a time constant of 50 ms (Fig. 3(b)). It wasnoted that a very quiescent phase continued for 15 ms, while the fluctuationlevel of the poloidal field, |B p /B p |, was as low as 10~ s . The kinetic instabilities
caused by the velocity anisotropy to be noticed in Fig. 3 (a) disappeared, in thiscase, because of the weak acceleration of the tail electrons (if present). The
profile of the X-ray emissivity, which reflected the profile of the REB electrons,
maintained its peaked shape during such a long decay phase (Fig. 3(c)).
4.

PLASMA CONFINEMENT

The electron density profile measured with a five channel HCN laser interferometer system [6] shows that the plasma electrons produced in the REB ringregion diffused towards the wall. To examine the particle confinement properties
in greater detail, a pulsed RF (100 kW, 3.7 MHz, 0.2 ms pulse width with 25%duty) was applied to a one-third turn antenna set in the toroidal direction.
Figure 4 shows the line integrated electron densities ne£ at five different chords

692

NARIHARA et al.

during the RF pulse train. The decay rate of ne£ at the centre was lower thanthat at the periphery, which indicates that the central region had better particle
confinement than the periphery. The global electron confinement times
estimated from these data are 1 ms for a ring current of I R = 20 kA and 0.5 ms
for IR = 12 kA. From the spectroscopic data, the electron temperature wasestimated to be lower than 10 eV. For the present plasma parameters(n e = 5 X 1012 cm" 3 , T e ~ 10 eV, B T = 3 kG, R/a = 3), the classical diffusion
time is about 10 ms, which is about ten times longer than the observed value.

REFERENCES
[1]

[2]

[3]

[4]
[5]
[6]

MOHRI, A., NARIHARA, K., TOMITA, Y., HASEGAWA, M., KUBO, S., TSUZUKI, T.,
KOBATA, T., FLEISCHMANN, H.H., in Plasma Physics and Controlled Nuclear Fusion
Research 1984 (Proc. 10th Int. Conf. London, 1984), Vol. 3, IAEA, Vienna (1985) 395.
YOSHIKAWA, S., CHRISTOFILOS, N.C., in Plasma Physics and Controlled Nuclear
Fusion Research 1971 (Proc. 4th Int. Conf. Madison, 1971), Vol. 2, IAEA, Vienna(1971)375.
MOHRI, A., NARIHARA, K., TSUZUKI, T., KUBOTA, Y., TOMITA, Y., IKUTA, K.,
MASUZAKI, M., in Plasma Physics and Controlled Nuclear Fusion Research 1978
(Proc. 7th Int. Conf. Innsbruck, 1979), Vol. 3, IAEA, Vienna (1979) 311.
NARIHARA, K., Jpn. J. Appl. Phys. 20 (1981) L683.
TOMITA, Y., NARIHARA, K., KUBO, S., TSUZUKI, T., MATSUI, M., MOHRI, A.,
in High-Power Particle Beams (Proc. 6th Int. Conf. Kobe, 1986) (1986) 399.
KUBO, S., NARIHARA, K., TOMITA, Y., TSUZUKI, T., MOHRI, A., in Controlled
Fusion and Plasma Heating (Proc. 13the Europ. Conf. Schliersee, 1986), Vol. 1,
European Physical Society (1986) 364.

IAEA-CN-47/D-V-ll

THREE-DIMENSIONAL MHD STUDIES OF THE
MAGNETIC ENERGY RELAXATION PROCESSES
T. SATO, K. KUSANO
Institute for Fusion Theory,
Hiroshima University,
Higashisenda-machi, Naka-ku, Hiroshima
R. HORIUCHI
Science Project Corporation,
Kuchita-minami, Asakita-ku, HiroshimaJapan
Abstract
THREE-DIMENSIONAL MHD STUDIES OF THE MAGNETIC ENERGY RELAXATIONPROCESSES.
A unified theory of MHD relaxation is presented. Driven reconnection is found to be the only
fundamental process governing the relaxation. The MHD relaxation is categorized into two types, i.e.
'global' relaxation, where the global magnetic topology is changed, and 'local' relaxation, where thetopology remains the same. Global relaxation is associated with 'total' reconnection, which is reconnection of antiparallel fields. It is proved that local magnetic helicity vanishes at the reconnection point of
total reconnection; thus, total helicity is conserved. Local relaxation, on the other hand, is associatedwith 'partial' reconnection, which is reconnection of obliquely intersecting fields. Helicity is, therefore,
not zero at the reconnection point of partial reconnection; thus, total helicity is not conserved.

1.

INTRODUCTION

The frozen-in principle is the basis of magnetic confinement in closed systems,
e.g. stellarators, tokamaks, spheromaks, and FRCs. An ideal plasma can forever be
confined within a closed magnetic surface. This is true, regardless of whether theconfiguration is ideally stable or unstable since the magnetic topology remains
unchanged even locally against an ideal instability.
There are two ways a confined plasma can escape from a closed magnetic surface. One is slippage between the plasma and the magnetic surface or percolation of
the plasma through the magnetic surface. This occurs without any change of the magnetic topology, neither globally nor locally. The other way is break of the magnetic
surface. The first process may simply be called a 'diffusion' and the other one a'reconnection' process. In both processes, 'resistivity' is the key parameter, whether
it is classical or anomalous.
In this paper we survey the roles of the reconnection process in a magnetohydrodynamic (MHD) plasma and propose a unified view of MHD confinement.
693

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SATO et al.

2.

TOKAMAK-TYPE RELAXATION AND RFP-TYPE RELAXATION

We first assume that the diffusion process defined above is a sufficiently slow
process on our time-scale of interest, i.e. on the MHD time-scale. Hence, the percolative escape of the plasma through a magnetic surface is negligible. In otherwords, unless a magnetic surface is broken by reconnection, the plasma confined bymagnetic surface remains confined forever. Secondly, we assume that the effect ofthe diamagnetic current is negligible compared to that of the force-free current, i.e.
the pressure effect is negligible.
On these assumptions, let us consider a force-free equilibrium confined by aclosed conducting wall. If the equilibrium concerned is ideally stable, it can be a goodcandidate for magnetic confinement geometry. The so-called minimum energy state
under a realistic constraint, notably, the constancy of the total magnetic helicity[1-4], is one of the ideally stable states. In other words, if the energy level of theinitial equilibrium is higher than that of the minimum energy state, the equilibrium
is ideally unstable, subject to a current driven (kink) instability [2-4].
According to the above definition of the minimum energy state, i.e. that thestate is ideally stable, the tokamak configuration with an on-axis q-value larger than
one and the RFP configuration can both be minimum energy states. The tokamak
configuration with qajljS < 1 and the RFP configuration in the setting-up phase are,however, both ideally kink-unstable. Therefore, they are not the minimum energy
state. Thus, they tend to relax towards the corresponding minimum energy states. We
notice an essential difference between the two relaxations: the initial and final
tokamak configurations are 'topologically' identical, although some minor 'geometrical' difference may exist. In contrast, the initial and final RFP configurations are'topologically' different.

3. PARTIAL AND TOTAL RECONNECTION
Let us here consider the physical reason for and the physical significance of thedifference between the two relaxations. Of course, magnetic reconnection takes placein both relaxations. Therefore, some significant difference between the two reconnection processes must exist. As can be seen in Fig. 1, the initial and final topologiesof the reduced field B* are different. Here, the reduced part B* is defined by
B* = B - B H , where B is the true field and BH is the resonant (n = 1) helical field.
When we express B = Bp0 + Bt<£ and BH = qBp0 + Bt<£, where 0 is the poloidal
and 4> the toroidal unit vector, we have B* = (1 -q)B p 0. To see the topologies of
the true fields we must, therefore, add the helical part BH to B*. Then, it turns outthat the final topology is generally the same as the initial one, in the usual tokamak
operation.
The above argument indicates that reconnection involved in the tokamak relaxation is 'partial' reconnection. 'Partial' means that only a fraction of the total magnetic

IAEA-CN-47/D-V-ll
q=1

resonant

Initial695

surface

Final

FIG. 1. Topologies of reduced field component of tokamak field, B* = B-BH, where B is tokamak
field and BH is resonant helical field of q = 1, before and after reconnection is triggered by
m = 1/n = 1 kink mode.

field takes part in the reconnection. More specifically, when a magnetic component
exists perpendicular to the reconnection plane (BH in the present example), i.e.
when oblique field lines reconnect with each other, we speak of a 'partial' reconnection. Incidentally, it is also interesting to note that the m= l/n= 1 kink instability relevant to the tokamak relaxation is a resonant (non-global) instability. When
antiparallel fields reconnect we shall, on the other hand, call this phenomenon a'total' reconnection.
In the RFP relaxation, the global topology changes during the relaxation
process. Since the m = 1 kink instability relevant to the relaxation in this case is a
global (non-resonant) instability, the entire plasma is deformed by the global flow
induced by the global instability, and, consequently, non-linear driven reconnection
takes place at certain places whereby the magnetic topology is globally changed[2-4]. This strongly suggests that the relevant reconnection process must be the'total' rather than the 'partial' process.
Let us prove that the Taylor-type relaxation (which is accompanied by a global
topological change) is caused by 'total' reconnection. As was mentioned previously,'total' reconnection should not have a magnetic field component perpendicular to thereconnection plane. The previous simulation of RFP self-reversal strongly supports
this hypothesis. Here we reproduce one example of Ref. [4], in Fig. 2. This figure
shows that the reconnection point is more or less coincident with a point on thedemarcation circle between the positive and negative toroidal fields, i.e. on the nulltoroidal field circle. Another evidence is that reconnection takes place at the nullpoints of local magnetic helicity as is shown in Fig. 3 reproduced from Ref. [3]. Let
us discuss this point analytically in the next section.
4.

HELICITY CONSERVATION IN TOTAL RECONNECTION

Without loss of generality, we can define the reconnection plane to be the x-yplane, in rectangular co-ordinates (x,y,z). Then, at the reconnection point, the mag-

SATO et al.

696
(a)

t= 0

FIG. 2. Contours of helical flux (a) and negative toroidal field (b) at different times for single helicity
simulation in RFP setting-up phase.

netic field B has only the z-component Bz, i.e. B = Bzz. Thus, the local magnetic
helicity Kx at the reconnection point is given by
K, = A B = AZBZ

(1)Using the simulation result that the local helicity is zero at the (non-linearly formed)
reconnection point, we may conclude that
B, = 0(2)

697

IAEA-CN-47/D-V-llt / t . = 24

t/tA =0'rrrr:::::;;/ i! I'" * * •'11 * • * I i I i •1:::::::::.'•

t/tA=8

t/tA

=40

»•
•
t/1. = 1 6

t/t.

=60

F/G. 3. Contour maps of magnetic helicity density andpoloidal magnetic field lines of force in poloidal
plane at six different times.

Thus, we can reach two important conclusions: non-linear driven reconnection as
observed in Refs [2-4] is 'total' reconnection, and 'total' reconnection occurs at the'zero' helicity point. Conversely speaking, since Taylor relaxation assumes 'total'
reconnection resulting in a true topology change, the magnetic helicity does not decay
at the reconnection point; hence, the total helicity is conserved.

5. GLOBAL AND LOCAL RELAXATION
We have seen above that there are two classes of MHD relaxation: one isassociated with 'partial' reconnection and no global topology change occurs in themagnetic field. Tokamak relaxation generally corresponds to this class. The otherclass is associated with 'total' reconnection, and the magnetic topology is changed
globally. The RFP (Taylor-type) relaxation corresponds to this class. We here propose to call the former class 'local relaxation' and the latter class 'global relaxation'.

698

SATO et al.

6.

NECESSITY OF DRIVEN RECONNECTION IN MHD RELAXATION

In what follows we shall discuss the time-scale of the MHD relaxation since thisis the essential point in this relaxation process. The MHD relaxation should be
defined as the process which evolves on the MHD time-scale, tA0 (i.e. the Alfve'ntime-scale). Thus, the energy decay process associated with the resistive (diffusion)
time-scale, tR0, should be excluded from MHD relaxation. The key issue to beresolved is, therefore, whether there exists a fundamental process leading to magnetic
energy dissipation on the MHD time-scale, not on the resistive time-scale. Since theonly mechanism causing magnetic energy dissipation is resistive diffusion (Ohmic
heating), it appears at a glance that the answer is no. We may suppose here that thetearing mode instability is a candidate. Certainly, the tearing mode instability accelerates dissipation and reduces the time-scale substantially. But still the time-scale (theinverse of the growth rate) is largely dependent on the resistivity (proportional toS3/5, S being the magnetic Reynolds number). Therefore, when resistivity is
infinitesimally small, the tearing time-scale is infinitely large compared to the MHD
time-scale. Thus, the tearing mode can be excluded from the MHD relaxation definedat present.
Reconnection is unavoidable in the relaxation. Therefore, the above question
can be rephrased as whether there exists fast reconnection that evolves on the Alfve'ntime-scale, or not. There is one reconnection meeting this condition: 'driven' reconnection [5, 6]. Here, we reproduce some results that represent the general featuresof 'driven' reconnection. Figure 4(a) shows the temporal evolution of the electricfields at the reconnection point (reconnection rate) for different resistivities with
fixed driving force (driving flow speed); Fig. 4(b) shows this evolution for different
driving forces with fixed resistivity. As can be clearly seen in these figures, the risetime and the saturation value are independent of the resistivity (Fig. 4(a)) but largely
dependent on the driving force (Fig. 4(b)). We further note that the time-scale is of
the order of the MHD time-scale. The only dependence of the driven reconnection
process on the resistivity is the onset time of reconnection (Fig. 4(a)). These features
are observed consistently in quite different simulation processes [4, 7].
These features of driven reconnection can be physically explained as follows:
Driven reconnection essentially requires 'compressibility' for its onset since a converging flow is necessary [8]. The essential point of driven reconnection is that the
local characteristic spatial scale (scale of the current profile) is narrowed at the flow
converging point (or line), i.e. current peaking takes place there. Let the magnitudeof the converging flow, the initial spatial characteristic scale and the narrowed spatial
scale be designated by V, L and d, respectively. Then,
TR = M 2 /r? = TR0(d/L)2(3)

where TR0 = /iOL2/i7,17 is the resistivity and /n0 is the vacuum permeability. It can beroughly said that driven reconnection is triggered when the local diffusion speed,
VR, reaches the converging (driving) flow speed

699

IAEA-CN-47/D-V-ll

at neutral point

5 0.15 h A o = 0.2
ic =a = 0.05/' / / a = 0.02

0.10

(a)

///a = 0.005

E 0.05111

10
15
TIME (normalized)

10

15
20
25
TIME (normalized)

20

25

35

FIG. 4. (a) Development of neutral point electric fields for three different resistivity coefficients a.
(b) Development of neutral point electric fields for four different input fluxes A^

V = VR a d/TR

(4)

From (3) and (4), we obtaind/L = (V A0 /V)S' 1(5)

where VA0 = L/TA0 and S = TR0/TAO. Since the reconnection rate A is represented
by v/V A0 (VA0 is the Alfven speed defined by the initial values), it is, as a matterof fact, independent of resistivity. Specifically, when the induction electric field
associated with the plasma motion penetrates into the reconnection (X) point, driven
reconnection starts, and the magnitude of the electric field is independent of theresistivity. Resistivity only takes part in the determination of the time (penetrationtime) during which the induction electric field penetrates through the diffusion
layer d. Since, usually, V/VA0 ~ O(1(T2) and S > 104, we have d/L > 10' 2 .
Therefore, the penetration time, d/VR = d/V, is, at most, of the order of the Alfven
transit time TA0- This indicates that the delay time of the reconnection onset is given

700

SATO et al.

by TR, which is proportional to resistivity, but is limited by the Alfve'n transit time,
TA0. Thus, it is concluded that the driven reconnection process is almost independent of resistivity (within the uncertainty of the onset time, of the order of TA0) andthat the evolution time is described by (VA0/V)TA0.

7.

CONCLUSIONS

The fundamental physical process governing MHD relaxation is 'driven' (nonlinear) reconnection, which is essentially different from spontaneous tearing reconnection. Because of the participation of driven reconnection, a non-minimum energy
state can relax towards the minimum energy state, stepwise with the falling time of
the MHD time-scale.
The relaxation can be categorized into two types global and local relaxaton. In
global relaxation, 'total' reconnection participates, in which case the global magnetic
topology is changed. At reconnection points of total reconnection, where the physicalquantity can dissipate selectively, local magnetic helicity vanishes; hence, total
manetic helicity is nearly conserved during the process, while the magnetic energy
dissipates. RFP relaxation is a typical example for global relaxation.
In the local relaxation, on the other hand, 'partial' reconnection does take part.
Partial reconnection contributes to a partial redistribution of the current distribution,
but does not change the global magnetic topology. Since a field component exists,
perpendicular to the reconnection plane at the reconnection points, local magnetic
helicity does not vanish. Therefore, in local relaxation, magnetic helicity is not conserved. Tokamak relaxation (including low q tokamak relaxation) is a typicalexample for local relaxation.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]

TAYLOR, J.B., Phys. Rev. Lett. 33 (1974) 1139.
HORIUCHI, R., SATO, T., Phys. Fluids 29' (1986) 1161.
HORIUCHI, R., SATO, T., Phys. Fluids 29 (1986) 1304.
KUSANO, K., SATO, T., Nucl. Fusion 26 (1986) 1051.
SATO, T., HAYASHI, T., Phys. Fluids 22 (1979) 1189.
SATO, T., in Plasma Physics (Proc. Int. Conf. 1986), Commission of the European Communities, Brussels (1986) 963.
[7] MIN, K., OKUDA, H., SATO, T., J. Geophys. Res. 90 (1985) 4035.
[8] SATO, T., KUSANO, K., Phys. Rev. Lett. 54 (1985) 808.

IAEA-CN-47/D-V-12

MODELLING AND TRANSPORT SIMULATONSOF OSCILLATING FIELD CURRENT DRIVE(F-0 PUMPING) IN A REVERSED FIELD PINCH*
R.A. NEBEL, K.A. WERLEY
Los Alamos National Laboratory,
Los Alamos, New Mexico
R.A. SCARDOVELLI, G.H. MILEY
Fusion Studies Laboratory,
University of Illinois,
Urbana, Illinois
D.S. HARNED, H.R. STRAUSSCourant Institute of Mathematical Sciences,
New York University,
New York, New YorkD.D. SCHNACK, Z. MIKIC
Science Applications, Inc.,
La Jolla, California
United States of America
Abstract
MODELLING AND TRANSPORT SIMULATIONS OF OSCILLATING FIELD CURRENT DRIVE(F-0 PUMPING) IN A REVERSED FIELD PINCH.
Oscillating field current drive has been proposed earlier as a means of maintaining quasi-steadystate currents in a reversed field pinch. Previous analyses relied on the concepts of helicity injection
and relaxation to a force-free state. The paper presents an investigation of the associated MHD processes
which uses combinations of a one-dimensional transport model, a linear resistive MHD stability model,
Kadomtsev's nonlinear global reconnection model, a one-dimensional model including hyper-resistive
terms derived from the nonlinear interaction of tearing modes, and a three-dimensional nonlinear MHD
model.

INTRODUCTION.
Oscillating field current drive (OFCD) has been
proposed [1,2] as a means of maintaining quasi-steady-state
currents in an RFP. Previous analyses relied on the conceptsof helicity injection and relaxation to a force-free (Taylor13]) state. We investigate the associated MHD processes using
combinations of a 1-D transport model, a linear resistive MHD
stability model, Kadomtsev's nonlinear global reconnection*Work performed under the auspices of the United States Department of Energy.

701

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NEBEL et al.

model, a 1-D model including hyper-resistive terms derived fromthe nonlinear interaction of tearing modes, and a 3-D nonlinear
MHD model.
1.

H = 1 MODES FOR OFCD

In this section we derive a relation between the mean
toroidal current density and the V x B term on axis, whichindicates the instabilities needed for an efficient current
drive. The torus is approximated as a periodic cylinder oflength 2rtR and radius a.
Integration of Faraday's law on 0 = 9' = constant surface
from the magnetic axis at r = 0 to the conducting wall at r = aleads to)Ez(a,G',z',t) - E 2 (0,9', Z ',t)ld Z ' (1)where <f> (9',t) is the poloidal flux through the consideredsurface and E is the toroidal electric field.
OFCD requires the time-averaged toroidal electric field at
the wall to vanish and the time-averaged poloidal flux to be aconstant. Consequently, taking the time average of Eq.
(1)yields
lim

±|

r

T

• ^" Jt

dt'l

E_(0,9',z',t') dz' = 0

(2)

Jo

We now consider the z-component of the resistive MHD Ohm's law
on axis, which reads[Ez + (V x B ) z ] r = 0 = [*, J z ] r = Q(3)

with the resistivity Y\ a time-independent scalar function.
Expanding each term in Eq.
(3) in Fourier components,
substituting them into Eq. (2) and integrating over a toroidal
period in the z-direction yields the following expression for
the time-averaged (m = 0, n = 0) component:
,t+x
¥1

L. L.
m=0n=-»

'

'

r=0

(A)
Regularity conditions at r = 0 constrain the radial andpoloidal components of V and B such that only the m = 1 modes

IAEA-CN-47/D-V-12

703

TABLE I. RELATION BETWEEN POLOIDALFLUX AND INTERACTION REGION
Increase inpoloidal flux (%)

Interaction
region (%)

0.5

36.0

1.0

42.6

2.0

48.0

5.9

65.0

12.74

73.7

16.1

81.2

do not vanish as r—> 0. Since OFCD requires that the first
integral in(4) be greater than 0, a necessary criterion for
OFCD is that the second integral in Eq. (4) also be greaterthan 0. Note that all of the terms under the second integral
in Eq. (4) vanish identically, due to regularity constraints,
except for those components with m = 1. Consequently, m = 1
modes are required for OFCD within the resistive MHD model in a
finite cylinder.
A class of m = 1 tearing modes have been identified thatsatisfy the necessary criterion of Eq. (4). The linear andnonlinear behavior of those modes was described in detail in apaper by Caramana, Nebel and Schnack[4].
These modes are
driven unstable by off-axis peaks in the X profile (X = Jy/B)
and nonlinearly result in an increase in the poloidal flux.
Single helicity nonlinear simulations[4] have shown that the
final globallyreconnectedstate is well described by
Kadomtsev's reconnection model [5JIn order to determine if these
instabilities cansignificantly increase the poloidal flux in the plasma,we have
performed a stability and reconnection study with various
off-axis peaked X profiles. X was parameterized as a power
series with X vanishing at the wall. Parameters were chosen sothat F and 9 are close to ZT-40M values. Table I shows that asignificant increase in the poloidal flux (1%—10%) can beachieved with an interaction region of less than 70% of theplasma radius.

2.

SIMULATIONS OF THE EXPERIMENTAL DATA

Simulations of OFCD discharges have been
performed,
utilizing
three different
models: 1-D transport, 1-D
hyper-resistive, and 3-D MHD. A standard 150 kA, 200 eV OFCD

704

NEBEL et al.

FIG. 1. <B^> and 1$ versus time for the transport simulation (dots) compared with the experimental
data (solid curves).

discharge from ZT-40M was simulated. Details of the experimentare described in Refs [6, ?]•The purpose of the 1-D transport simulations is todetermine if the m = 1 modes required for OFCD are present in
the experimental data. Since complete experimental informationon temperature and impurity profiles is not yet available, wehave assumed these profiles and calculated the resistivity fromthem. Our model contains no "dynamo" term; thus agreement withexperimental data indicates that the dynamo is not active.
Similarly,
disagreement between simulation and experimentstrongly suggests that a dynamo mechanism is present [8J. The
applied boundary conditions are the experimentally measured
toroidal magnetic field and electric field at the wall. The
dynamical evolution of the 1-D model of the plasma thendetermines the toroidal current I* and the toroidal flux 4>+

IAEA-CN-47/D-V-12

705

FIG. 2. \(r) at (a) 1 = 7.80 ms, (b) t = 7.925 ms, and (c) t - 8.05 ms, from 1-D transport.

which can be compared with the experimental values. We also
monitor the linear stability of the tearing modes as the
profiles evolve.
Our best transport simulation of the base case OFCD
discharge is shown in Fig. 1. The only phase we were able tosimulate was the flux decay phase due to the presence of the
RFP "dynamo" during the other phase. A reasonable match toexperimental data required flat temperature profiles over 95%
of the plasma radius, with the temperature falling to 10 eV at
the wall.
X-profiles at various times in the simulation are shown in
Fig. 2. Note that there are two off-axis peaks: the innermost
is a result of plasma compression driven by V^, while theoutermost is a skin current driven by VQ.
Stability analysisof these profiles indicates that they are weakly unstable tohigh k (n ~ 20-25) m = 1 tearing modes. This is a result of
the peaks in X residing in a region where high k modes areresonant.
Helical flux values for these high k modes are larger on
the axis than at the wall; hence the instability cannot
reconnect to the axis as is necessary for the generation ofpoloidal flux (and current drive). Reconnections to the wall,
however, would tend to flatten the X-profile outward and may
account for our difficulty in exactly matching the experimental
valuesof the toroidal
flux
and toroidal
current.
Consequently, we do not see any indication of current-drive
related modes in the data. This is apparently because thepeaks in X are too far out in radius for the discharge
simulated.
Theories have been developed recently in which theevolution of the RFP is described by a modified OHM's law.
(See, for instance, Refs [9 and 10].) From fluctuating fields(due to the nonlinear interaction of tearing modes) one obtains
a hyper-resistive term that modifies Ohm's law for the mean

NEBEL et al.

706

F

D

0.6 -0.4 -O.I«

03 0.4 O.« 0.0 10

12 1.4l.« T.I

FIG. 3. (a) Experimental F-6 diagram.
(b) Base case (super-FOT) hyper-resistive simulation F-6 diagram.
(c) Sub-FOT hyper-resistive F-6 diagram.

IAEA-CN-47/D-V-12

707

fields.
The hyper-resistive term is of the form BV • (DVX),
where the hyper-resistivity D is given by (B^Vr) /(k • B) .
Here Br^ represents the radial magnetic perturbation ofmode k.
Close to the rational surfaces the denominators are modified byresistivity and inertia. The hyper-resistive terms in Ohm's
law have the property that they dissipate energy
while
conserving
total magnetic
helicity.
The equations aredescribed in detail in the Refs [9 and 11]. The model is well
suited for modeling OFCD because it provides a method of
performing inexpensive one-dimensional computations that may
approximate the complicated three-dimensional interactions that
produce relaxation.
The form of the hyper-resistivity profile is determined bymatching
a steady driven discharge to the experimentallyobserved values of F and 9.
For the case of the ZT-40H
discharge
described
above, we
find
that
when
the
hyper-resistivity is such that D=f|a , the plasma relaxes toa state close to that measured in ZT-40H in which F = -0.13 and
9 = 1.47. The experimentally measured F-9 diagram is given inFig.
3a. The F-9 diagram from the hyper-resistive model isgiven in Fig. 3b. Note that this limit cycle is similar tothat of the experiment and earlier 0-D simulations [2J. The
form of the hyper-resistivity profile is determined by matchinga steady driven discharge to the experimentally observed valuesof F and 9. For these parameters, as expected, the oscillating
fields do not produce a large effect. The plasma relaxation
time due to hyper-resistivity is near the oscillation period.
This result indicates that the plasma in ZT-40M was not fully
relaxed during the oscillations. In order to see the effect ofa given oscillation period on the plasma response, we havevaried the frequency while holding the remaining
plasmaparameters fixed. There appears to be a frequency operationthreshold (FOT) below which the plasma
undergoeslarge
oscillations. The FOT corresponds roughly to the point where
the oscillation period is comparable to the hyper-resistive
relaxation time. An F-9 diagram for a case in which theoscillation frequency has been reduced by a factor of three, sothat the oscillation period exceeds the relaxation time, isshown in Fig. 3c. A similar effect to that shown in Fig. 3coccursif the hyper-resistivity is increased beyond thecomputed value for the experiment.
The one-dimensional
hyper-resistive
model
has
the
advantage of computational efficiency, but it cannot model thefull dynamics of the three-dimensional nonlinear evolution of
the plasma. Three-dimensional nonlinear computations for the
RFP are difficult because features such as compressibility,
which are important for an accurate model, produce time step
constraints that are too severe to allow simulations on theresistive
time
scales. A new computational method forthree-dimensional MHD [12,13] has been developedusingsemi-implicit
techniques!14]
that makesuch
long-time
simulations practical. The semi-implicit method has been used

708

NEBEL et al.

.to

(a)

NX

.05
.00

.05
.10
.15
.10
.25
.10

.15
»

w>

n

o

N

»

o

1.0

.9
.0
-.5

\

t.o
t.9

t.o
1.9
JO

9.9
t.o
(.5
5.0

FIG. 4. (a) T = 50 rA (super-FOT).
(b) T = 700 TA (sub-FOT).

to eliminate both the fast congressional and the shear Alfve'ntime-step constraints.
Using the semi-implicit method, we have performed 3-D MHD
simulations of OFCD for many periods of the oscillating applied
voltage (typically significant fractions of the resistive
diffusiontimeatS = 10 ) .
The
code is resistive,
compressible, and uses the pseudospectral approximation for thepoloidal and axial (toroidal) directions [15]. We employ 100
radial mesh points, and retain 3 poloidal (m = 0,1,2) and 11
axial (-5 < n < 5) modes. The resistivity profile is the same

IAEA-CN-47/D-V-12709

as that used in the hyper-resistive calculations described
previously.
The results of these 3-D simulations are shown in Figs
4a and 4b, where we display the variations in F and 8 for
driving periods of 50 and 700 Alfve*n times, respectively. Notethat large oscillations occur unless the frequency exceeds the
FOT. The qualitative agreement of these relatively coarse 3-D
results with those obtained with the 1-D hyper-resistive model
indicatesthat
the experimentally significant relaxation
processes occur at long wavelengths. 3-D simulations withfiner resolution are in progress, and will be reported infuture publications.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]

BEV1R, M.K., GRAY, J.W., in Reversed Field Pinch Theory (Proc. Workshop, 1980)
Paper IHA-3 (1980).
SCHOENBERG, K.F., GRBBLE, R.F., BAKER, D.A., J. Appl. Phys. 56 (1984) 2519;
Rep. LA-9161-MS, Los Alamos National Lab., NM (1984).
TAYLOR, J.B., Phys. Rev. Lett. 35 (1974) 1139.
CARAMANA, E.J., NEBEL, R.A., SCHNACK, D.D., Phys. Fluids 26 (1983) 1305.
KADOMTSEV, B.B., Fiz. Plazmy 1 (1975) 710; Sov. J. Plasma Phys. 1 (1975) 389.
SCHOENBERG, K.F., et al., Phys. Fluids 27 (1984) 548.
SCHOENBERG, K.F., et al., Paper IAEA-CN-47/D-H-3, these Proceedings, Vol. 2.
WERLEY, K.A., NEBEL, R.A., WURDEN, G.A., Phys. Fluids 28 (1985) 1450.
STRAUSS, H.R., Phys. Fluids 28 (1986) 2786.
BOOZER, A.H., J. Plasma Phys. 35 (1986) 133.
STRAUSS, H.R., HARNED, D.S., to be published in Phys. Fluids.
HARNED, D.S., KERNER, W., 3. Comput. Phys. 60 (1985) 62.
HARNED, D.S., SCHNACK, D.D., J. Comput. Phys. 65 (1986) 57.
ROBERT, A.J., in Proc. WMO/IUGG Symp. on Numerical Weather Prediction, Tokyo (1969).
SCHNACK, D.D., BARNES, D.C., MIKIC, Z., HARNED, D.S., CARAMANA, E.J., to be
published in J. Comput. Phys.

IAEA-CN-47/D-V-13

RELAXATION PHENOMENAIN REVERSED FIELD PINCHES AND TOKAMAKS
A. BHATTACHARJEE, C.K. CHU, Y.-C. KWOK
Department of Applied Physics and Nuclear Engineering,
Columbia University,
New York, New YorkE. HAMEIRI, H.R. STRAUSSCourant Institute of Mathematical Sciences,
New York University,
New York, New YorkA.H. BOOZER
Princeton Plasma Physics Laboratory,
Princeton University,
Princeton, New Jersey
United States of America

Abstract
RELAXATION PHENOMENA IN REVERSED FIELD PINCHES AND TOKAMAKS.
A self-consistent theory is given for the field * * • <v"| X B,> generated by fluctuations (v*, andB, are the velocity and magnetic fields, respectively) within the framework of resistive MHD, including the effects of plasma compressibility. Rigorous functional forms of Tdue to tearing and resistive
interchange modes are given. It is shown that, in reversed field pinches, Tproduces relaxed states withreversed toroidal field in the vicinity of the 'Taylor state' T = XQB*, but with finite pressure gradients.
In tokamaks, the theory is applied to calculate the turbulent resistivity governing current relaxation andto investigate MHD current drive ('F - 8 pumping') and steady state operation using bootstrap currents.
Numerical results on finite beta plasmas, based on reduced MHD equations, are presented.

INTRODUCTION
The purpose of this paper is to give a self-consistenttheory for t = <v"2xB"i>, conventionally known as the "dynamo"
field in astrophysics, due to resistive modes. The theory isapplied to describe relaxation phenomena in reversed-field
pinches (RFP's) and tokamaks.

1.

MEAN FIELD e IN INCOMPRESSIBLE PLASMAS

We consider an incompressible plasma inside a perfectly
conducting cylindrical shell (aligned with the z-axis) to

711

712

BHATTACHARJEE et al.

which the velocity and magnetic fields are tangential. Allmean quantities depend only on the radius r, and the mean magnetic field lies on surfaces r = constant. Superimposed on
the mean field, denoted by subscript o, are fluctuations ofzero mean, denoted by subscript 1. We will use the following
assumptions: (1) the mean state, containing only small flows
induced by diffusion, evolves on a resistive time-scale andvaries spatially on the scale of the minor radius a of thetorus, i.e., a|v"o| = 0 ( n ) , a 2 9/3t = 0 ( n ) , a|v| = 0 ( 1 ) , where
n is the resistivity of the plasma; (2) the energy of thefluctuations, be it magnetic,.kinetic or thermal, is smaller
than the mean magnetic energy, i.e., B*i2, pv"i2, pi << B o 2 ; and(3) the fluctuations vary on a smaller space-scale and a fastertime-scale than the nusan quantities, i.e., a 2 3/3t >> 0(n),a|V| >> 0(1). Since J\ - VxBj, the fluctuating current may belarge; we allow a|j]J/|B o | = 0 ( 1 ) . Averaging Ohm's law, we get
3B o /3t - Vx(nJ o - e") = 0, t = <v 1 xB 1 >(1)

The average is either over an ensemble, or is a space-time
average over the scales of fluctuations. We note that v o does
not enter Eq. (1) for an incompressible plasma
It can be shown that e obeys the following Theorems:
Theorem 1:/dt e".3o = ~ fdtn<J\2> , to 0(n)Theorem 2:
fdi t.%o = 0, to O h )where the integrals are taken over the total volume of theplasma. Theorem 1 may be proved by averaging the energy equation
9/9t (B2/2 + pv2/2) + V.(pv2v/2 + pv + (nJ - vxB)xB)+ nJ2= 0 (2)(p is the density and p, the plasma pressure), and keeping
terms to 0(n) . We get
3/3t(Bo2/2) = -nJ o 2 - n <Jx2> + V.Cj
where Cj represents an energy flux.
3/8t(Bo2/2)= t.Jo - rJ o 2 + V.C 2(3)
Now, from Eq. (2), we get(4)

The surface integral /(Ci~C2).ds represents the energy flux due
to fluctuations which vanishes on a perfectly conducting wall.
From (3) and (4), Theorem 1 follows. For a proof of Theorem 2,
the reader is referred to Ref. 1. Also, both Theorems 1 and 2
can be shown to carry over to the case of the compressible plasma, with v o and ~e redefined 2 .
2.

-»- 1,3
HEURISTIC DERIVATION OF G,,

From Theorem 2, we may write e,, = | B O | ~ 2 ( V . ^ ) B O , where C, isa vector function representing_^the flux of helicity due to thefluctuations, and satisfying § C n d s = ° o n t h e plasma boundary.

IAEA-CN-47/D-V-13

713

For a nearly :|orce-free plasma, Theorem 1 now implies /£.VXdT>0,
where X = J O .B O /B O 2 . We envision the turbulence as generated bya bath of local resistive modes, where it is known that eachmode behaves non-ideally only in the vicinity of some resonant
magnetic surface. Therefore, /C.VXdx>0 has to be satisfied
pointwise, that is, £.VX.>p . We expect that c, - K 2 VX, where K 2
is a positive function; thust,, = B O / | B O | 2 V.(K 2 VX)(5)

The functional form (7) encompasses the calculations of e,, inRefs.4 and 5, which are based, respectively, on a kinetic and aquasi-linear fluid model. In the next section, we strengthen
the heuristic argument by explicitly deriving e,, from the dynamical equations of tearing modes.

3.

CALCULATION OF e,, FOR TEARING MODES

We envision the fluctuations to be due to tearing modes,
each resonant on some magnetic surface, and ignore the nonlinear interaction between them. The mean fields now represent
equilibrium fieldte, and the fluctuations are the linearized
perturbations. For a particular mode with the Fourier dependence exp (qt + ik.r), k = kz + m9/r^ the dominant contribution
to e comes from the resonance layer k.B o = 0 . Outside thelayer the mode behaves ideally, with zero contribution to ~£ .
It is seen 1 ' 2 that to leading ordere.Bo = V. [ A 8 ? 2 <(r.B1)(r.v1)>r]Irxk.B o(6)

J

Using the notation of Ref. 6, we may rewrite Eq. (6) as e.B o =- V. (nr~2Seo2y'Q<H'oE>r) , where p = rBZo/Beo» primes denote r derivatives and Q, 4"o^and E are the normalized growth rate, theradial components of Bj and the displacement vector, respectively. The average is now interpreted as an integral over thedimensionless inner layer variable. Since Vo = constant, only
the even part of ¥ contributes. We finally obtain 1 ' 2

E.B O = V. (anSVX'r)(7)

where a = Q " 1 / 2 ^ f (n•), and f (u') is some positive function,a has the dimensions of length squared and depends on the
global equilibrium profile through the mode amplitude ^o and Q,
both of which depend on A1 . In a model calculation, such as
the one presented later in the paper, a is taken to be a Q a 2 ,where a o is a positive constant.
We have shown thus, from the linear theory of tearing modes,that the functional form for etl is given by Eq. (5) , and an
explicit impression has been obtained for K 2 , which we propose
to call "eddy"2 (Of "hyper"7) resistivity. The calculation

714

BHATTACHARJEE et al.

given above has been extended using the direct interaction
approximation 8 , and estimates show that eddy resistivity canbe more.important than the enhancement of the classical resistivity due to tearing mode turbulence.
The tearing mode model described in this section may also
be used to calculate ej_. The calculation, however, is much
more tedious and seems unprofitable for linear theory. It is
more rewarding to derive an exact expression for e for nonlinear resistive interchange modes. We summarize our principal findings in the next section? for details, the reader isreferred to Refs. 1 and 2.

4.

CALCULATION OF Ej_ FOR RESISTIVE INTERCHANGE MODES

Recently, the nonlinear evolution of resistive interchange
modes and the consequences for energy confinement in RFP's,
have been studied 9 ' °. These studies, however, suffer fromusing an ad hoc pressure equation 9 ' 1 0 and inconsistent approximations in the treatment of perturbed fields 1 0 . We have,therefore, developed an independent treatment of the problem ,and shown that previous scaling laws derived by similarity
methods 1 0 are incorrect. The correct dynamical equations canbe solved exactly in a low-pressure, weak-s^hear limit to obtain an analytic expression for e. Since £„ = 0 to leading
order for these modes, the only surviving component of e isperpendicular to B, and given by
r x B

(8)

where B o is a positive, dimensionless constant, y is the specific heat and we have dropped the subscript o on mean quantities.
5.

STEADY-STATE SOLUTIONS IN RFP's

We consider an RFP plasma in a bath of resistive modes, withtearing modes contributing primarily to £„ and resistive interchange modes, to Ej_. The relaxed state is given by nJ - e =Ez , where the constant E is the applied toroidal voltag«» perunit^length, and t,, is given by Eq. (5), with K 2 = a o a 2 n B 2 ,and e_|_ by Eq. (8). Thus, we solve the equation

where the pressure p, for given B, is determined from theequilibrium condition (p + B 2 / 2 ) ' + Bg 2 /r = 0, with p(a) = 0.

715

IAEA-CN-47/D-V-13
1.4
1.2
1.0
0.8

p

0.6
0.4
0.2

0

0.2

0.4

0.6

0.8

1.0

-0.2

4.0 ir

FIG. I. Pressure and Bz profiles for a0 = 0 and 0O = 50 (dotted lines) and a0 = 20 and |3 0 = 50
(solid lines). In both cases, E = 12. (Bz is normalized to the volume averaged Bz (= BJ, p to Sf,
r to a and E to l

We impose the boundary condition X 1 = 0 at r = a, in conformity
with Theorem 2.
If c<o = 0, no solutions with toroidal field reversal are
possible. This is because the 0-component of Eq. (9) is of theform B z ' + fB z = 0, f >_0, which implies that B z decays exponentially from the center, but is never negative. Also, large aQ
and B o imply that X' = 0{,ao~1),^ ~ ^ ( B Q " 1 ^ , and correspondasymptotically to the Taylor state J = X Q B, with constant X o .
We note that, in reality, the plasma pressure will be determinedby additional loss mechanisms, most importantly, by anomalous
electron heat conduction, which has to be modelled separately.
While the dominant effect of 3 O causes B z to decay exponentiallyfrom the center, a finite value of a o is necessary to explain
reversal of B. 1.2
This is seen in Fig. 1, which is computedfor typical ZT-40 parameters.

716
6.

BHATTACHARJEE et al.
STEADY-STATE SOLUTIONS IN TOKAMAKS

Eq. (9) also yields relaxed states in tokamaks. Whereas in
ZT-40, to the extent it is known , the electron temperature
profile T is almost flat, in TFTR, under a wide variety ofoperating conditions, T Q (r) = T e (0) exp . (-2Ar2/3), where A =q(a) + 0.5 11 . This gives n(r) = n(0) exp (Ar2) . It can beshown'-2 then that for small o o (which is consistent with a low
level of magnetic fluctuations in tokamaks), the effect of e,,leads to an enhancement of the classical resistivity by approximately a factor Of 2 for TFTR, which is not of great importance, given the large uncertainties in the measurement ofZ e ff. What is important is that the theory 12 explains why
this enhancement of resistivity does not necessarily lead to
RFP-like profiles.

7.

HELICITY CONTENT AND TOKAMAK APPLICATIONS OF HELICITY 13

The definition of magnetic helicity K Q = /dTA.B (B = VxA)
for an integration volume bounded by a magnetic surface, canbe extended to the case when the bounding surface has an arbitrary shape and time dependence, for which the proper definition i s 1 4
KQ = /dTA.B - iA.dxiA.dx(10)provided the gauge is chosen correctly, where the line integrals are in the poloidal (p) and toroidal (t) directions. It
can be shown 13 that the time-derivation of Kg is given bydl^/dt = 2V41 - 2/dT E.B - 2/ (<H - 3r/9t.A)B.dl(11)

where V is the average loop voltage, ¥ , the toroidal flux enclosed by the bounding^:egion^ E, the electric field, and $,the scalar potential (E = - 8A/3t - V$).
We investigate the possibility of MHD current drive ("F - 9
pumping"). From (11) we get
dK o /dt = 2(V - Rjjl) V(12)

Where Ro, = (/dTE.B)/I4/ is the resistivity of the plasma in Ohms
and I, the plasma current in amperes. Assuming that I = l o exp(-t/x), with x constant, and oscillating the loop voltage andthe toroidal flux so that V r Vj cos wt and V = Vo + V\ cos tot,
one can show from (12) that the conditions K Q > 0 and ^i-if,-, canbe satisfied if cox £. 2.326. We remark that a critical assumption in the analysis is the rigidity of the current profile,which may be difficult to obtain experimentally.
The second application of helicity is to the possible existence of an intrinsically steady-state tokamak due to the

IAEA-CN-47/D-V-13

717

interaction of the bootstrap current and tearing modes. It canbe shown
from the neoclassical equations that a strong bootstrap effect distorts the current distribution into a hollow
profile and makes the q-profile non-monotonic, which are known
to be unstable to tearing modes. These modes, however, may bebeneficial since they create additional poloidal flux. In fact,
it can be shown 13 that if r = b is the edge of the tearing unstable region, the condition for poloidal flux maintenancethrough the bootstrap effect is 2<r)> = n(b) , where <n> is thevolume average of the resistivity inside r = b .

8. NUMERICAL SIMULATION OF FINITE BETA RFP PLASMAS 15
Reduced mean-field equations have been derived for resistive
MHD with classical collisional transport. The evolution of themean profiles is treated without any essential approximation
except neglect of inertia. The fluctuations satisfy linear
equations, except that parallel gradients are calculated withthe total magnetic field. The main benefits of the approach
are that it is tractable analytically, and that it requires
less computation than the full MHD equations.
Numerical computations in 1-D show a thermal instability due
to the temperature dependence of the transport coefficients.
In 3-D, the transport due to m=l tearing modes gives a stationary state which agrees qualitatively with experiment in several
respects. The F-6 diagram is qualitatively consistent withexperiment. The value of (3,g « 0.5, which is not unreasonably
larger than in the OHTE experiment. Finally, | B I | / | B | " S " 1 / 3 ,where S is the magnetic Reynolds number, for relaxation dominated by a few m=l modes, which is consistent with experimental data on HBTX - 1A.

ACKNOWLEDGMENTS
This research is supported by U.S.D.O.E. Grant Nos.
DE-FG02-86ER53222, DE-FG02-86ER53223 and DE-AC02-76-CH -3073.

REFERENCES
[1]
[2]
[3]
[4]
[5]

BHATTACHARJEE, A., HAMEIRI, E., Phys. Rev. Lett. 57 (1986) 206.
HAMEIRI, E., BHATTACHARJEE, A., submitted to Phys. Fluids.
BOOZER, A.H., to be published in J. Plasma Phys.
JACOBSON, A.R., MOSES, R.W., Jr., Phys. Rev., A 29 (1984) 3335.
STRAUSS, H.R., Phys. Fluids 28 (1985) 2786.

718

[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]

BHATTACHARJEE et al.

COPPI, B., GREENE, J.M., JOHNSON, J.L., Nucl. Fusion 6 (1966) 101.
STRAUSS, H.R., to be published in Phys. Fluids.
KRAICHNAN, R.H., J. Fluid Mech. 5 (1959) 497.
AN, Z.G., DIAMOND, P.H., CHIEUH, T.H., in US-Japan Workshop on Compact Toroid
Research, Princeton University, Princeton, NJ (1984).
CONNOR, J.W., TAYLOR, J.B., Phys. Fluids 27 (1984) 2676.
MURAKAMI, M., et al., to be published in Nucl. Fusion.
BHATTACHARJEE, A., HAMEIRI, E., submitted to Phys. Fluids.
BOOZER, A.H., to be published.
BEVIR, M., GRAY, J., in Reversed Field Pinch Theory (Proc. Workshop, 1980), Vol. 3,
Los Alamos National Lab., NM (1980).
STRAUSS, H.R., Phys. Fluids 29 (1986) 3008.

IAEA-CN-47/D-V-14

PHYSICS CONSIDERATIONSON A FUSION PLASMAIN A FIELD REVERSED CONFIGURATION
H. MOMOTA, M. OKAMOTO, Y. NOMURA
Institute of Plasma Physics,
Nagoya University, Nagoya
K. YOSHIKAWA, M. OHNISHI
Institute of Atomic Energy,
Kyoto University, Uji, Kyoto
A. ISHIDA
Department of Physics,
Niigata University, Niigata
K. SATO
Department of Electricity,
Himeji Institute of Technology, Himeji
S. OHI
Plasma Physics Laboratory,
Faculty of Technology,
Osaka University, Suita, OsakaJapan
H.L. BERK, T. TAJIMA
Institute of Fusion Studies,
University of Texas at Austin,
Austin, Texas,
United States of America
Abstract
PHYSICS CONSIDERATIONS ON A FUSION PLASMA IN A FIELD REVERSEDCONFIGURATION.
FRC/DD-3He fusion has advantages such as a smaller neutron fraction and a higher
overall plant efficiency. Physics issues inherent in FRC/DD-3He fusion reactors are discussed,
particularly, achievement of ignited plasma, current sustainment by fusion produced protons,
and stabilization of tilt modes by means of plasma rotation.

719

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1.

MOMOTA et al.
INTRODUCTION

In thermonuclear fusion based on deuterium and tritium fuel, most of thefusion energy is released as 14 MeV neutrons. The engineering requirements
for a commercial reactor are, accordingly, very hard to achieve. Advanced fuels
have been studied so as to mitigate these engineering problems and, possibly,to provide a reasonable reactor system with higher plant efficiency.
In advanced fuels such as Cat. D [1], DD-3He [2], and D-3He fuel cycles, notritium breeding is necessary, and a reduction of the 14 MeV neutrons is possible. Thispermits a large degree of freedom in the selection of structural reactor materials.
The DD- 3 He fuel cycle is very similar to Cat. D. The fusion plasma is fed with
deuterium, which yields tritium and 3He together with other fusion products
through D-D reactions. A certain amount of tritium and 3He is burned in situ,
and the remainder diffuses out of the plasma. The extracted tritium is stored
in a bank where tritium is converted to 3 He through the beta decay process.
3
He produced in this bank is refuelled into the plasma, together with theexctracted 3He so as to promote D-3He reactions.
The confinement parameter, the radiation parameter [3], and the 14 MeV
neutron fraction for self-ignited DD-3He fusion plasmas are shown in Fig. 1 asfunctions of temperature T. The corresponding curves for D-T and Cat. D
plasmas are also drawn, for the convenience of comparison. Both the required
confinement parameter and the temperature for the DD- 3 He fusion are higher
than the corresponding values for D-T fusion. On the other hand, the radiation
parameter is very large, and the 14 MeV neutron fraction is very small in a highbeta plasma. The overall plant efficiency, thus, can be expected to be as high as
70%, with a high efficiency direct energy converter of the multistage Venetianblind type (DEC) [2],Very high values of plasma beta, plasma temperature, and accessibility toa high power DEC are required in order to utilize the favourable characteristicsof the DD-3He fuel cycle. Hence, one finds that an FRC is the most promising
candidate for a DD- 3 He reactor. The representative parameters of an FRC
plasma and its favourable features are shown in Table I.
As far as physics and engineering are concerned, there are many problems
to be solved for DD-3He FRC fusion. Some physics key issues are as follows:
heating the plasma up to 100 keV, realizing steady burn, and achieving stabilityof the burning FRC plasma.
In Section 2, we discuss plasma evolution from D-T to DD- 3 He without
any external heating. Section 3 deals with the steady maintenance of FRC
with applied seed current and fuelling. The plasma current sustained by fusion
products in DD- 3 He ignited plasmas and resulting plasma rotation are studiedin Section 4. In Section 5, the effects of plasma rotation on tilt mode stability
are analysed. Summary and discussions are presented in the last section.

721

1AEA-CN-47/D-V-14

ioa

1 1 1

I " " I

IgnitionDD"He--Cat.D

fa)
1022 r,0=0.3

2

.2

10"

10w

10
100
Temperature T (keV)

1000

a.

z

I
10

100

Temperature T (keV)

1000

10

100

Temperature T (keV)

FIG. 1. (a) Confinement parameter nr versus temperature 77 (b) radiation parameter \prversus temperature T; (c) 14MeVneutron fraction normalized by fusion power versus
temperature T.

2.

TRANSITION OF PLASMA OPERATION FROM D-T TO DD- 3 He

The operating temperature of a DD- 3 He reactor is required to be as highas 100 keV, from the viewpoint of overall plant efficiency. Although plasma
heating up to such a high temperature seems to be formidable, efficient useof alpha particles produced by the D-T reactions may make this difficult task
easier. Then, we study the transition of plasma operation from D-T to DD-3He
only controlling the externally applied magnetic field and the fuelling rate,
without any external heating.

1000

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MOMOTA et al.

TABLE I. REPRESENTATIVE PARAMETERS AND FAVOURABLEFEATURES OF DD-3He FUEL
D-T

Cat. DDD- 3 He

2.2 X 1 0 2 0

4 X 10 21

10 21

Temperature (keV)

14

40

100

Obtainable plant efficiency (%)

40

51

71

14 MeV neutrons in 1 GW plant (GW)

1.8

0.65

0.18

Required confinement (s-m3)

The basic equations used for this study are: scaling of the average beta, <0),in long, thin equilibrium approximation; the relation between the trappedmagnetic flux, <pi; and the magnetic field at the wall, B w , derived from a high flux,
sharp boundary model; the energy balance equation in a non-adiabatic situation;and the particle balance equations for the ions of species j , i.e.

(1)

(2)

£ = -IV+Q(3)(4)

where x s denotes the ratio of the separatrix radius rs, to the wall radius, r w .
The rest of the notations is conventional. The fuelling rates are given by
So = W>/TP+S[D)(5)

ST = g(nT/rp+sP)

(6)

Sm

(7)

= (l-g)n.T/Tp+Ti3ffe/Tp

where SD » ST andS3He are the fuelling rates of D, T and 3He respectively,
and g stands for the tritium refuelling parameter. For the D-T fuel cycle,
g s 1 and for the DD-3He fuel cycle, g = 0.

723

IAEA-CN-47/D-V-14

DD-'HeIgnition

Goal _10"c•o

I i i nl

10100

temperature (keV)

FIG. 2. Trajectory of transition from D-T to DD-3He in nr-T diagram.

To increase the plasma temperature in a controlled manner, the following
feedback control of the external magnetic field is applied to the plasma [4]:=-G(T-T0)/T0(8)

where r$ and G are the time constant of the change in the magnetic field and thegain of the feedback control. TQ denotes an 'objective' temperature, which is,for example in this study, changed stepwise from 15 keV to 100 keV, by 1 keV
every second.
When the tritium refuelling parameter g is changed from g = 1 to g = 0,
simultaneously with magnetic field control, the plasma temperature faithfully
follows the objective temperature, and the density of each ion species approaches
the value obtained in the DD-3He fuel cycle. The trajectory of the transitionin the nr-T diagram is shown in Fig. 2. The trapped flux is supplied intermittently during the transition so as to keep the plasma length constant. It isdemonstrated that transition from D-T to DD- 3 He can be attained mainly byfusion product heating. Then, in a DD- 3 He fusion plasma, there may not be
any essential difficulty once a D-T self-ignited plasma has been attained.

3.

STEADY STATE FRC

A method is considered for sustaining the diamagnetic plasma current byexternally supplying a seed current at a field null point. The steady state solution
is obtained by solving the fluid equations, including equilibrium and transport
on the following assumptions: 1) injected beam ions such that the seed
(Ohkawa current) does not impart any significant momentum to the plasma;

724

MOMOTA et al.

2) ion and electron temperatures are spatially uniform; 3) the FRC plasma iselongated in the axial (z-) direction; 4) a particle source, S p , is introduced,
compensating for the particle diffusion loss. The equations are then reduced toa radial, one-dimensional problem. The stationary equation in cylindricalco-ordinates (r,0,z) are given bydr~~(rpvr)=Sp-PT<
r civ

LJ

Mh = ~^r:ur(9)

The magnetic flux function is defined by

where

From Eq. (9), we obtain

f2)

(10)

Here, we have decomposed the resistivity 17 into

where rj^Cy) is the spatially varying part of the resistivity and TJQ is constant.
The other new variables are defined by(ID
where a, S, and J* are 'virtual sink', normalized particle source, and seedcurrent density, respectively. By using parity properties, Eq. (10) can be
decomposed into even and odd parts to produce

IAEA-CN-47/D-V-14

725

02)

03)

Equation (12) determines \j/, and subsequently a is calculated from Eq. (13).
Integration of Eq. (12) yields

with

and
S(y) = (l/j?fi(y)) 1'1/2
The boundary condition for Eq. (14) is/(1)= A /1-/3 S ,Vd)=O and V(0)=0

where jis is the plasma beta value at the separatrix

S(y) is associated with the particle source S p and is proportional to S o :(15)Here, Tg and r ^ are the magnetic diffusion and particle confinement times,
respectively. The numerical calculation shows that the seed current needed
to keep the specified steady configuration depends strongly on the value of SQIt should be emphasized that the energy balance equation imposes a constraint
on the value of S o .

726
4.

MOMOTA et al.
ROLE OF FUSION PRODUCTS

A large amount of fusion protons is generated in a DD-3He burning
plasma, which may stabilize FRC plasmas 15]. In this section, we demonstrate
another important role of the energetic fusion protons: they can also contributea large fraction of the current needed to support the configuration.
Protons are likely to be lost immediately when they intersect the separatrixof the FRC. Thus, some of the energetic protons will be lost directly while others
will be trapped. The direct loss allows a directed flow of the fusion proton
component that is trapped, and this flow gives rise to an electric current if acertain shortening mechanism exists, preventing radial electric fields fromgrowing. The distribution function of trapped protons in an axisymmetric FRC
equilibrium can be described as a function of the speed v and canonical angularmomentump=rv+U(U=e<p/N)
The kinetic equation for the distribution function f(u,p) of protons in a steadystate is found to be

(16)

where

a=r$, <Q> = [Jrv>\p-V\

Q(r,z)drdz/f

drdz

Jrv>\p-l>\is the function of p and v,
SPp)(r,z)
is the source density of the fusion protons per unit time, p(r,z) is the electrondrag rate, and vc is the speed at which the drags due to the background electrons
and the pitch angle scattering on the plasma ions are equal.
The trapping condition is described by Heaviside's function h ( p - au).
The solution for f(r,z) is easily obtained through the path integral, and theresulting current density carried by the protons is given by the formula:

727

IAEA-CN-47/D-V-14

FIG. 3. Current density carried by energetic protons in DD-3He fusion plasma.

= (q/4r2)

PO(P.I').VO/I CPo(P»w)-ai;o] dp(17)where pg = pg(p,u) is the birth angular momentum of a proton born withspeed v0, that reaches the point (r,u) along the characteristics.
The expression for this current density has been evaluated numericallyby a reasonable choice ofin an extremely elongated Hill's vortex model. The resulting proton current
density produced in a DD- 3 He fusion plasma is shown in Fig. 3, where themaximum value is approximately 1.2 of the edge plasma current density.
Consistent results have been obtained with a Monte Carlo code.
We find the possibility of obtaining a large fraction of the current needed
to support the configuration from fusion produced protons. At the same time,the drag due to plasma particles is found to give rise to a radial electric field
and, consequently, to cause an azimuthal rotation of the plasma. This will be
studied in the next section.

728

MOMOTA et al.
10*

101

a
10'

10

FIG. 4. Internal tilt stability as a function of scale factor a and elongation b/a. Plotted areexperimental data: 1 and l' forPIACE [7], 2 for NUCTE [8], 3 for FRX-B [9], 4 and 4' for
FRX-C [P], and 5 for FRX-C-LSM [10].

5.

TILT STABILITY

In this section, we investigate the tilt stability of FRC plasmas. Previous
theoretical studies [6] have predicted that the internal tilt mode in elongated
FRCs is violently unstable while long lived elongated FRCs have been observedin a number of experiments [7-9]. This discrepancy can be explained by taking
into account the rotation of the plasma. This rotation may be more significant
in a self-ignited DD- 3 He plasma in an FRC, as was mentioned in the previous
section.
We have adopted the two-fluid model in which the ions are described asa rigidly rotating fluid with finite temperature and the electrons as a cold fluidat rest. Adopting Hill's vortex-like solution as an equilibrium and prescribing
the displacements for the internal tilt mode,•O=(\-rz/a2-zz/b2)x-](rz/b2+diz/b2-zr/a2)eie
we have a cubic dispersion relation that includes an elongation factor b/a and ascale factor a = (a/6) 2 as parameters. Here, b is the half-length of the separatrix,
X the shaping parameter, and 6 the collisionless skin depth of the ion fluid.
The results are shown in Fig. 4, where we have reasonably chosen X = 2.
The two solid lines are the stability boundaries. On the left (right) hand side ofthe dashed line, the usual MHD potential W mhd is positive (negative). Note that

IAEA-CN-47/D-V-14

729

W mhd depends on the parameter A and is positive (negative) for b/a < (>) 1
when X = 1. On the right (left) hand side of the dot-dashed line the 'ion fluid
itself, by which we mean the rotating ion fluid including no interaction withthe electrons, is stable (unstable). The newly found stable region on the righthand side, which is given by b/a > 0.93 y/a for a. > 20, is sustained by therotation of the ion fluid while the left hand side, conventionally stable regionis sustained by the usual MHD potential energy. Since the ratio of rotational
to thermal energy of the ion fluid at the field null point is nearly equal to 8/a,this ratio is small enough to yield a new stable region. To compare the above
results with experiments, several experimental data are plotted in Fig. 4. The
present theory can hardly explain the long stable periods observed in FRX-C(4 and 4' of Fig. 4) because the growth periods of the internal tilt instability
for the parameters relevant to these are much shorter than the corresponding
observed stable periods [9]. It seems, however, to explain well the otherexperimental results shown in the figure.

6.

SUMMARY AND DISCUSSIONSIn developing a DD- 3 He fusion reactor of FRC configuration, some physics
issues inherent in the FRC plasma have been studied. Once a D-T ignited plasma
has been obtained, a self-ignited plasma of 100 keV is found to be subsequently
achieved by controlling the fuelling of the components and the magnetic field.
The steady state operation of the FRC plasma is examined, showing that thecurrent is sustained by an applied seed current and is also carried by fusion
produced protons when the DD- 3 He plasma is ignited. The effects of the plasma
rotation on the internal tilt mode stability are also studied in connection withplasma configuration of a spheroid shape. It is found that the mode can besuppressed solely by plasma rotation when both elongation and scale factors
are chosen suitably.
Further physics issues should be studied such as kinetic and rotational
effects on the plasma stability, ambient plasma behaviour, ambipolar potentialbuildup, plasma equilibrium consistent with transport including fusion products,
and achievement of D-T ignited FRC plasma.

REFERENCES[1] McNALLY, J.R,, Jr., in Engineering Problems of Fusion Research (Proc. 6th Symp.
San Diego, CA, 1985), IEEE Publ., Piscatway (1976) 1012.
[2] MOMOTA, H., et al., to be published in Fusion Technol.
[3] MILEY, G.H., in Fusion Energy Conversion, American Nuclear Society, LaGrange Park,IL, (1976) Ch. 2.
[4] OKAMOTO, M., OHNISHI, M., J. Nucl. Sci. Tech. 20 (1983) 366.

730

MOMOTA et a).

[5] FINN, J.M., SUDAN, R.N., Nucl. Fusion 22(1982) 1443.
[6] SHESTAKOV, A.I., et al., in Compact Toruses and Energetic Particle Injection (Proc.
Japan-US Joint Symp. Princeton, NJ, 1979), Plasma Physics Lab., Princeton Univ., NJ,(1979)126.
CLEMENTE, R.A., MILOVICH, J.L., Phys. Lett., A 85 (1981) 148.
SCHWARZMEIER, J.L., et al., Phys. Fluids 26 (1983) 1295.
CLEMENTE, R.A., MILOVICH, J.L., Phys. Fluids 26 (1983) 1874.
BARNES, D.C., et al., Phys. Fluids 29 (1985) 2616.
[7] OHI, S., et al., Phys. Rev. Lett. 51 (1983) 1042.
[8] TANJO, M., et al., in Plasma Physics and Controlled Nuclear Fusion Research 1984
(Proc. 10th Int. Conf. London, 1984), Vol. 2, IAEA, Vienna (1985) 523.
[9] SIEMON, R.E., et al., Fusion Technol. 9 (1986) 13.
[10] CHRIEN, R.E., Bull. Am. Phys. Soc. 30 (1985) 1456.

IAEA-CN-47/D-V-15

PRELIMINARY EXPERIMENTS
IN A COMPACT TOROIDFORMED BY FOUR C-GUNS
S. SINMAN
Plasma Engineering Laboratory,
Electrical and Electronics Engineering Department,
Middle East Technical University
A. SINMAN
Nuclear Fusion Laboratory,
Nuclear Research and Training Centre
Ankara, Turkey

Abstract
PRELIMINARY EXPERIMENTS IN A COMPACT TOROID FORMED BY FOUR C-GUNS.
Experiments have been performed in a compact toroid (CT) formation system withfour C-guns located in an octagonal floating flux conserver. The C-gun is similar to theelectromagnetically driven shock tube. As the conventional T-tube, the C-gun has a pairof vertical electrodes and backstrap components, but the vertical electrodes are located inside
the flux conserver. Two different CT production schemes have been realized with the C-gun,depending upon the operational mode. In the under-damping mode (UDM) a stable toroid
is obtained which begins at the onset of cyclotron wave damping and lasts exponentially for
2.8 ms. The critical damping mode (CDM) also begins at the onset of the ion cyclotron
instability; however, in this case the instability is not damped but is transformed to a drift
wave instability of 26 kHz, in the same way as in the current drive method. In the CDM it
has been possible to generate a toroid, originating from the modulated current channel, fora maximum duration of 20 ms. The duration is strongly dependent upon the backgroundhelium gas pressure, which is in the range of 0.05-0.07 torr. The results obtained with
Spitzer's resistivity formula and the measurements with the plasma resistivity probe showedthat the electron temperature is in the range of 50-75 eV in the CDM. However, a maximum
electron temperature of about 90 eV has been measured with a paramagnetic loop of an areaof 24 cm2 . In the UDM, these values are lower. For instance, depending upon the background helium gas pressure in the range of 0.1-0.2 torr, the electron temperature ranges
between 20 and 35 eV. According to the Langmuir probe results in the UDM and CDM
schemes, the electron densities are 5 X 1015 cm"3 and 1014-1015 cm"3, respectively. The
loop current of the C-gun is about 100-150 kA and the helical toroidal current is about 6 kA.

1.

INTRODUCTION

In view of the recent rapid development of toroidal confinement systems
towards a breakeven plasma state, the question arises whether the final goal of
a commercially feasible nuclear fusion reactor can indeed be reached with the

731

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SINMAN and SINMAN

present tokamak approach. Research on compact toroids (CT) [1, 2] is complementary to tokamak research and will improve the understanding of plasma
physics in toroidal devices, with CTs permitting an overall optimization of thetoroidal experimental arrangements.
In 1985, we presented a conceptual system design for the realization ofa CT, using four magnetized T-tubes, and discussed the results to be expected [3].
Further, we constructed a small-scale device with a single magnetized T-tube andpresented preliminary experimental results [4], Subsequently, modifying theconventional T-tube, we developed a C-gun and showed that a CT may be
produced with four C-guns inserted into an octagonal floating flux conserver.
In the latest study [5] the findings were evaluated in the light of recent
knowledge.
In the present study, the formation of a CT with four C-guns is investigatedin more detail. The modes of operation of the C-guns are determined and theeffects of these modes on CT generation investigated by experiments. The CT
production mechanisms of the proposed alternative schemes are interpreted.

2.

SYSTEM

2.1. General
The experimental arrangement has a 40 L octagonal floating flux conserverwhich is also the vacuum chamber. The back-pressure in the flux conserver is
5 X 10~6 torr. Four C-guns, 90° apart, are arranged around the flux conserver.
The system consists of a capacitor bank of 2 kJ for each C-gun, spark-gap
switches controlled by a self-generated UV ring, and other diagnostic equipment.
2.2. The C-gun and its operational modes
The C-gun is a modification of the conventional T-tube. Since the plasma
is in electromagnetically driven shock tubes, the plasma column is pushed out
by the backstrap. The vertical electrodes of the C-gun are located inside theflux conserver; suitable insulation is used. In this structure there are no horizontal
or vertical legs as in a T-tube. The backstrap is directly attached to the external
wall of the flux conserver; this has been made possible by a special insulation.
The discharge circuit consists of the capacitor bank, the controlled sparkgap switch, the backstrap and the vertical electrodes. Under the influence ofthe magnetic field generated by the backstrap, the current belt between thevertical electrodes is pushed out towards the centre of the flux conserver.
Since this configuration has the shape of the letter C, this injector is called a
C-gun. The C-gun has two modes of operation - the under-damping modeand the critical damping mode.

IAEA-CN-47/D-V-15

733

2.2.1. Under-damping mode (UDM)
When the background gas pressure is selected to be in the range of
0.07-0.2 torr, the discharge is in the UDM and passes only through the verticalelectrodes. Characteristically, this discharge consists of the well known exponentially damping sinusoidal waves (voltage or current).
Under the experimental conditions, the period of this discharge is about
10-12 ms and the logarithmic decrement is 0.58. The working voltage rangeof the C-gun is 10-15 kV and the maximum current of the operating loop
is 150 kA.
2.2.2. Critical damping mode (CDM)
When the distance between the vertical electrodes is not changed according
to the background gas pressure in the critical range of 0.04-0.07 torr, thedischarge through the vertical electrodes of the C-gun occurs with the aid of theinner walls of the flux conserver. This result is very natural, since the distance
between the vertical electrodes is larger than the distance from these electrodes
to the wall. In this condition, the currents passing through the inner wall andthe backstrap are in opposite directions. Because of the mutual inductance
between the wall and the backstrap, the serial leakage inductance of the capacitor
bank circuit becomes smaller. Thus the discharge ends as can be expected in
the CDM.
2.3. Diagnostics used
Generally, in this study, diagnostic techniques as used in high beta systems
have been applied. These techniques may be divided into two main groups,
namely internal and external diagnostics.
For the internal techniques, toroidal and poloidal magnetic loops, movablemagnetic probes at an angle of 135° in the poloidal plane with respect to the
C-gun, Langmuir probes, resistivity probes and paramagnetic loops have beenused.
For the external techniques, charge exchange cells, Rogowski coils,
external magnetic loops, voltage dividers, fast integrators and visible lightdetectors with filters have been used.
In addition, analog-to-digital data processing systems have been applied.

3.

FORMATION OF THE COMPACT TOROID

3.1. UDM scheme
Under the influence of the self-generated toroidal magnetic field of the
C-gun of about 10 kG, warm electrons (5-15 eV) are produced in the shock

734

SINMAN and SINMAN

II, 1 1 |ii (til Mill A Ili.fliHMil

I l l i

WlllllliinrA'A

•

I I I

•

I I I

pi

Mil

1 I'l i
FIG. 1. Ion cyclotron wave damping in the UDM scheme.
Time-scale: 200 ns/div., vertical scale: relative units.

heated paraboloid plasma volume, drawing a helical orbit around this toroidal
magnetic field and reconnecting a CT.
According to measurements carried out by the magnetic probe, with anarea of 31.4 X 10" 3 cm2 and an angular position of 135° on the poloidal plane,a total mean magnetic flux density of about 650 G is found. The corresponding
helical toroidal current is approximately 6 kA. In this mode an exponentiallydecaying stable CT with 2.5-3.0 ms lifetime is obtained.
3.2. CDM scheme
By means of a passive conducting rod, located near the centre of the flux
conserver and parallel to the vertical wall of the C-gun, the upper and lower
walls of the flux conserver are short circuited; also, the current passing throughthe wall (Section 2.2.2) is transformed to a closed-loop current, the discharge
on the capacitor bank circuit is suddenly cut off and a potential difference
between the electrodes of the C-gun of up to 70 kV occurs. This voltage decays
exponentially in 1.5 MS to 6-8 kV, which is the minimum threshold voltageof the capacitors; according to the time constant of the external discharge
circuit (250 ms), this value continues for the lifetime of the CT. In other words,
the inductive coupling between the toroidal current of the flux conserver andthe capacitor bank is completely exhausted.
In this mode, microinstabilities are effective and the toroidal current
channel is modulated by a drift wave of 26 kHz with a lifetime of up to 20 ms,
depending upon the background gas pressure, which is in the range of
0.04-0.06 torr.

735

IAEA-CN-47/D-V-15

1,II ft =
III
III' n 1]111rf¥
1p i 1
- L.

ii ii

I

• ''•TIII

I III

III 1

1 II 1

III 1

1 1i

FIG. 2. Typical magnetic probe data in the UDM scheme. Variations of the toroidal
magnetic field generated by the C-gun for 60 JUT, after ion cyclotron wave damping.
Successively, a part of the toroidal magnetic field variation of the stable CT decays
exponentially in 2.8 ms. Time-scale: 10 jisjdiv., vertical scale: relative units.

4.

EXPERIMENTAL OBSERVATIONSPreliminary experiments have been carried out with a single C-gun. The
modes of operation of the C-gun in the UDM and CDM schemes depend upon
the background helium gas pressure. In the pressure range of 0.05-0.07 torr
the C-gun is operated in the CDM scheme and in the range of 0.07-0.2 torr the
C-gun is operated in the UDM scheme. These modes of operation lead to the
production of two separate CTs.
According to the data from the magnetic probes, the CT generated by the
UDM scheme shows generally MHD behaviour, whereas in the CDM scheme
drift wave instabilities become effective during CT formation.
4.1. MH D behaviour in the U DM
In the UDM scheme, the first operational half-period of the C-gun ofbetween 5 and 6 ms changes in dependence of the gas pressure.
The oscillograms taken by the magnetic probes, moving through the toroidal
and poloidal planes at an angular distance of 135° from the C-gun, show thatat the end of 2-3 ms a damping ion cyclotron wave occurs (Fig. 1).
From time dependent evaluations it is seen that at the end of the firsthalf-period a wave form is obtained which has both poloidal and toroidal
magnetic field components and damping in 40-60 /Lts, with an exponential
zero average (Fig. 2). The derivation of the exponential portion of this magnetic
probe signal lasted 600 jus.

736

SINMAN and SINMAN_£.
FIG. 3. Oscillogram taken from the charge exchange cell in the CDM scheme.
Time-scale: 0.1 msjdiv., vertical scale: relative units.

At 7 cm distance from the wall of the flux conserver, the toroidal andpoloidal magnetic flux densities are almost the same, namely about 600-700 G.
The decay times of the fields change between 2.2 and 2.8 ms; they are calculatedwith the expression t B = B/(dB/dt).
4.2. Drift wave instability in the CDM
In this scheme, shortly after the occurrence of the undamped ion cyclotron
wave instability (10-15 MHz) of a duration of 5 ms, drift wave instabilities
commence.
The data taken from the charge exchange cell (Fig. 3) illustrate that thisinstability has a low frequency (26 kHz). In the analysis of the oscillograms
obtained from magnetic probes or from toroidal and poloidal magnetic loops
with an effective area of 3 cm2 and located at a distance of 2 cm from the wallof the flux conserver, 135° from the C-gun on the poloidal plane, it is foundthat this instability depends on the background gas pressure and that the closed
loop configuration continues for 5-20 ms.
At the end of this instability, a toroidal current channel modulated
completely with a frequency of 26 kHz (wave electron ring) comes into existence.
This mechanism is similar to low frequency self-generated current drive.
The signals from the integrated paramagnetic loop and the Langmuir probe
recorded by the sampling technique are shown in Fig. 4. In these experiments
the paramagnetic loop and the Langmuir probe were at an angular positionof 135° in the poloidal plane with respect to the C-gun.

IAEA-CN-47/D-V-15

Illk. i A

l

737

l

1

\

/

h •1

ll

-

N

FIG. 4. Signals from the integrated paramagnetic loop and the Langmuir probe versus timein the CDM scheme. Time-scale: 5 ms/div., vertical scale: relative units.

5.

CONCLUDING REMARKS

A new electromagnetically driven shock tube — the C-gun — has beendeveloped. Without the use of crowbarring techniques the operational regimeof this gun can be transformed from the UDM scheme to the CDM scheme, only
by changing the background gas pressure.
By means of the self-magnetic field of the C-gun it has been possible togenerate ion cyclotron waves in the two operational modes — the UDM schemeand the CDM scheme. In the UDM scheme, which occurs in the magnetic beachmodel, the ion cyclotron wave is damped, causing thermalization. In the CDM
scheme, on the other hand, the ion cyclotron wave is transformed to the drift
wave instability, with a current channel modulated by 26 kHz.
These processes have led to the two different ways of CT formation in the
floating octagonal flux conserver under background gas pressure. The CT with
MHD behaviour was stable and had a lifetime of 2.8 ms. The CT formed by atoroidal current loop modulated by the drift wave kept its properties for about
20 ms, although instabilities existed.
The electron temperature range for the UDM scheme was 20-35 eV andthe range for the CDM scheme was 50-75 eV. The electron density changed
between 1014 cm" 3 and 5 X 1015 cm" 3 .

ACKNOWLEDGEMENT
This work was performed under a co-operative agreement between the
Turkish Atomic Energy Authority and the IAEA, Contract No. 3823/R1/RB.

738

SINMAN and SINMAN
REFERENCES

[1]
[2]
[3]
[4]
[5]

KATSURAI, M., YAMADA, M., Nucl. Fusion 22 (1982) 1407.
YAMADA, M., in Advances in Compact Torus Research (Proc. IAEA Tech. Comm. Mtg
Sydney, 1985), IAEA-TECDOC-399, IAEA, Vienna (1986) 175.
SINMAN, S., SINMAN, A., in Controlled Fusion and Plasma Physics (Proc. 12th Europ.
Conf. Budapest, 1985), Vol. 9F, Part I, European Physical Society (1985) 642.
SINMAN, A., SINMAN, S., in Controlled Fusion and Plasma Heating (Proc. 13th Europ.
Conf. Schliersee, 1986), Vol. IOC, Part I, European Physical Society (1986) 377.
SINMAN, S., SINMAN, A., ibid., p.381.

IAEA-CN-47/D-V-22

LINEAR, HIGH BETA, £= 1 AND HELIAC
STELLARATOR EXPERIMENTS
F.L. RIBE, M.E. KOEPKE, E.R. HEDIN, C M . GREENFIELD
University of Washington,
Seattle, Washington,
United States of America
Abstract
LINEAR, HIGH BETA, I = 1 AND HELIAC STELLARATOR EXPERIMENTS.
High beta, linear stellarator plasma is produced in a low compression, 3 m theta pinch whose
compression coils (22 cm inner diameter) are modified to have a helical offset of 2 cm and an axial helical period of 40 cm. This I = 1 stellarator configuration is easily transformed to a heliac configuration
by installing a current-carrying conductor along the cylindrical axis. — For the I = 1 stellarator, internal probes were used to measure the spatial structure and temporal behaviour of the magnetic field,
which agree well with the predictions of ideal MHD equilibrium theory. The higher temperature, lower
density, diffuse profile quasi-equilibria are observed to be perturbed by a stable, m = l , k <= 0, oscillatory mode. Measured properties of the mode agree with predictions based on a particle-fluid model
which includes finite ion Larmor radius (FLR) effects. Cooler, higher density, sharp boundary plasma
can also be produced and is observed to be unstable to an exponentially growing (0.5 us e-folding time)m = 1, k = 0 mode. This agrees with ideal MHD sharp boundary theory. The higher temperature,
diffuse profiles provide FLR stabilization in the presence of a nearby conducting wall. At lower temperatures, the profile has a sharp boundary with 0 = 1 , and the mode becomes MHD unstable. — For
the heliac, similar measurements were made for the purpose of experimentally determining flux contours and other plasma properties during formation and quasi-equilibrium. As expected from theory,
there is no indication of MHD mode activity.

1.

Apparatus and Measurements

We investigate the high-beta properties of the heliac and £ = 1
stellarator in the limit of infinite aspect ratio. For the present work,
the High Beta 0 Machine1 (HBQM) is modified to permit plasma production
about a helical axis.
The 90-kV HBOM high-voltage theta-pinch discharge is produced in a 3meter segmented aluminum compression coil whose inner surface is helicalwith wall radius rw = 11 cm and closely resembles an i = 1 stellarator
flux surface with no interior separatrix (Fig. 1). The discharge inside ahelical quartz discharge tube produces fully-ionized high-beta deuterium
plasma with n *« 1015 cm"3, Te+Ti » 50-200 eV, <6> = 0.1-0.4, with particle
end-loss times of the order of 30 usec. The longitudinal compression
739

740

RIBE et al.
segmented helicalcompression coil.crowbar( gap

helical discharge
tube

front
collector platemagnetic probe

oooo
ooo

ooo
.interferometerbeam

Ocollecter
plate

O

O /

Q

O

O

IIIIII

interferometer
beam

power crowbar transformer

flux loop

FIG. 1. Top view of High Beta Q Machine showing nine start spark gaps, nine passive crowbar gaps,
helically displaced compression coil segments, helical discharge tube, and nine power crowbar transformer modules.

field BOR^4.5 kG has a rise time of 450 nsec and is power-crowbarred togive a nearly constant field for ~30 gsec.
Diagnostic equipment (F1g. 1) Includes internal field magnetic probesfor Bp» Be, and B2, axial fractional-fringe laser interferometry, anddiamagnetic loop-magnetic probe combinations to measure average plasmabeta. Helical symmetry 1s invoked to associate various (r, z, 8) magnetic
probe positions with corresponding points (r, 9+hz) on a common plane perpendicular to the axis.
2.

The i " 1 Stellarator Experiments

The purpose of these experiments is to measure the Internal magnetic
fields, plasma current, and plasma beta as functions of r and u = hz+8 at
various values of average beta J , density n, and temperature T and tocompare them with theoretical predictions of equilibrium and stability.
Figure 2 shows traces of Bp, Be and B2 at u = 90° where there should be no
Rp component. (The oscillation before the main discharge at 14.5 ysec 1s
the preionization B field.) At a f i l l i n g pressure pp = 7 mT, Br shows an

741

IAEA-CN-47/D-V-22
r = 3.5cm, u = 90°
0.4

Bz

(a)

0.3
PF=0mtorr .

0.2
0.1

Br

0
2-o.i
- 0.4
3

Bz

. (b)

I 0.3
: 0.2

/-"

F^= 7 mtorr

5- 0.1;oV

3-o ; ;.

ft,

0.3 . (c)
0.2
F

0.1

9?-15 mtorr

0
-0.1

10

20
30
time (/is)

ft

40

SO

FIG. 2. Internal magnetic field waveforms qfBr(t), Be(t), and Bz(t), in t = I stellarator: (a) vacuum,(b) 7 mtorr, and (c) 15 mtorr discharges.

o s c i l l a t i o n [ F i g . 2 ( b ) ] , while at pF = 15 mT, colder, denser plasma is
produced and the B r perturbation rises quickly to saturation while the Bz
f i e l d is almost e n t i r e l y excluded from the plasma i n t e r i o r .

Measurements

show that the longitudinal wavenumber k z » h in both cases.
In Fig. 2{b) and F i g . 2(c) the internal Bz f i e l d i s much smaller thanthe external f i e l d , represented approximately by the vacuum case of
Fig. 2(a), because of the large plasma diamagnetism (6 m a x > 0.8).

The

loss of 8 in approximately 10 psec is due to electron thermal conduction
to the ends.

Figure 3(a) shows the quasiequilibrium Bz fields taken at

8 usec after the main discharge, at the zero of the o s c i l l a t i o n ofFig. 2(b).

Figure 3(b) shows the i - 1 component of B z , obtained bytaking the component of Bz 1n F i g . 3(a) that is antisymmetric about x = 0.
The theoretical curves are from the ideal MHD theory of Freidberg. 2

Also,

we observe an antisymmetric component of Bg, comparable in magnitude tothe (symmetric) equilibrium component, that can be accounted for by theaxial plasma current produced by flux (or iota) conservation during theRIBE et al.

742

Exp't Theory<j3>'25% •0% *1.0.8-(b)

5 .6* .4*S 2 | oS .4

I*

-

.8
-10 -8 -6 - 4
U=3TT/2

-2

0
X(cm)

2

4

6

8

10

U = TT/2

FIG. 3. Measured and analytically predicted axial magnetic field profiles for plasma aand vacuum discharges: (a) total, flz • B^-B^,(b) antisymmetric component, Bu.

i n i t i a l formation of the helical plasma (magnetic axis radius = 2.6 cm at•ff = 25*).
An analysis of the stable oscillation shows that i t corresponds to a(k «• 0, m = 1) rotation of the plasma about the cylindrical axis. The
radial structure of the node is that of incompressible flow with no radial
node of the stream function x (r) between the cylindrical axis and plasmaedge. The stream function is given by ?=-?z*n> where L is the displacement vector. The perturbation field i s T p = Vx(?x1e), where"Be is theequilibrium f i e l d . Observations of the mode agree well with predictions
from MHO theory as corrected by Schmidt and MiUer 3 for f i n i t e Larmor
radius (FLR) effects according to the method of Pearlstein andFreidberg.4 The prediction for the experimental case of F1g. 2(b) is

IAEA-CN-47/D-V-22

2

743

4
6
radius (cm)FIG. 4. Sharp boundary plasma profiles: (a) axial magnetic field measurements at various radii withbest fit sharp boundary profile; (b) beta profile calculated from magnetic field profile of (a).

uj = 0, and uR = 600 sec"1. The stabilizing effect is that of a diffuse
plasma in proximity to the conducting compression-coil wall, where FLR
effects act upon the internal structure of the mode.
The unstable mode for pp = 15 mT exhibits exponential growth with
uj = 0.5 usec. The plasma column excludes virtually all of the axial
f i e l d , characteristic of s^-1.0, until t =< 22 usec in Fig. 2(c), when theplasma is cooled by wall contact. Figure 4 shows that during the instab i l i t y at t = 2 psec the plasma has a very sharp boundary with B«* 1. For
6 = 0.98, sharp-boundary theory predicts a growth rate of 0.54 usec.

3.

Heiiac Experiments
In order to produce a heliac configuration '** a "hard core" current
conductor, consisting of a tensioned, 1-cm 0.0., aluminum rod Inside a
1.4-cm 0.0. quartz tube, is installed along the axis of the i = 1 system
of Fig. 1. A crowbarred I z current with a rise time of 8 usec is driven
by a 7.2 yF capacitor bank and initiated 21 usec before the main capacitor
bank discharge. As in the t = 1 case,Br, 8 e , and B2 fields are measured
simultaneously at four r-values and at various e-and z-values for

RIBE et al.

744

Bo= 2.9 kG

B0=2.4kG

FIG. 5. Magnetic flux contours derivedfrom probe measurements. Flux values associated with contours
have constant A ^ spacing.

successive discharges.

Helical symmetry is invoked to map a l l of theprobe positions to a single z = constant plane.

The experimental valuesof the helical flux function f = Jdr(B 9 +hrB 2 ) are calculated at various( r , u) values.

After a bicubic spline interpolator produces a regularly

spaced grid from the original i r r e g u l a r l y spaced g r i d , contours ofconstant ty are then plotted on t h i s r-u plane.
Figure 5 shows experimental flux contours for vacuum and pp = 7 mT
f i l l i n g pressure for I z = 38 kA, a case which corresponds to B2s« 2.7 kG
and (Big^ax = 320 G.

In the vacuum case, the bean-shaped flux contours

around the helical axis at r = 5 cm do not change as functions of time.
In the high-beta plasma case, the wider spacing of the flux contours at
earlier times is evident, disappearing at later times as the plasma beta
decays by thermal conduction to the ends.

An axial measurementof thedeuterium ion temperature by Doppler broadening of the CV l i n e from 10%

IAEA-CN-47/D-V-22745m e t h a n e in t h e filling g a s gives T^ = 9 0 ± 20 eV, averaged along a
paraxial line at r «* 2 c m . T h e r e is no indication of M H D instability o ro s c i l l a t i o n o f a m a g n i t u d e s i m i l a r t o t h o s e o f t h e 1=1

stellarator.

ACKNOWLEDGEMENTS
We are indebted t o G.C. Harper f o r technical support of these experiments.

The work was performed under the auspices of the U.S. Departmentof Energy.

REFERENCES
[1]
[2]
[3]
[4]
[5]

[6]
[7J

KNOX, S.O., MEUTH, H., RIBE, F.L., SEVILLANO, E., Phys. Fluids 25 (1982) 262.
FREIDBERG, J.P., Rev. Mod. Phys. 54 (1982) 845.
SCHMIDT, M.J., MILLER, G., Phys. Fluids 24 (1981) 2306.
PEARLSTEIN, L.D., FREIDBERG, J.P., Phys. Fluids 20 (1977) 2128.
FURTH, H.P., KILLEEN, J., ROSENBLUTH, M.N., COPPI, B., in Plasma Physics and Controlled Nuclear Fusion Research (Proc. 2nd Int. Conf. Culham, 1965), Vol. 1, IAEA, Vienna(1966) 103.
SOLOVEV, L.S., SHAFRANOV, V.D., in Reviews of Plasma Physics, Vol. 2, Consultants
Bureau, New York (1966) 195.
CEKIC, M., private communication, 1986.

CHAIRMEN OF SESSIONS
Volume 2
Session E-I

D.B. NELSON

USA

Session E-II

R.R. PARKER

USA

Session C-I

D.D. RYUTOV

USSR

Session D-I

L.M. KOVRIZHNYKH

USSR

Session D-II

T. UCHIDA

Japan

Session D-III

G. GRIEGER

Germany, Fed. Rep

Session D-IV

F.L. RIBE

USA

SECRETARIAT OF THE CONFERENCE
Scientific
Secretaries:

M. LEISER
A.A. SHURYGIN

Division of Research andLaboratories, IAEA

Administrative
Secretary:

Edith PILLER

Division of External
Relations, IAEA

Editors:

J.W. Weil
Maria DEMIR

Division of Publications,
IAEA

Records Officer:

J. RICHARDSONDivision of Languages,
IAEA

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