J. Plasma Fusion Res. SERIES, Vol.4 (2001) 75-81

Gontrol of Radial Electric Field and Low-Frequency
Fluctuations in Open-Ended Systems
INUTAKE Masaaki*
Tohoku University, Sendai 980-8579, Japan(Received: 5 December 2000 / Accepted,:27 October 2001)

Abstract
Both drift mode and flute mode instabilities are found to depend sensitively on radial electric fieldstrength in the GAMMAl0 tandem mirror, when the electric field is controlled by use of end-platebiasing. In order to clarify effects ofboth radial electric field and its shear on low-frequency fluctuations,systematic experiments and precise measurements have been done in a small linear device, Qr-U ofTohoku University. Drift-mode fluctuations depend on both radial electric field and its shear. The driftmode is destabilized in the region of weakly negative electric field. In the strong E. region, the mode issuppressed, irrespective of its sign. This behavior agrees qualitatively with that observed in the ECH
mode of GAMMAI0. The drift-mode in the Q7-U is clearly stabilized by the increase in the net ion drift
shear which is defined as the sum of .E x I drift shear and ion diamagnetic drift shear. Flute-mode
observed in the Q7-U is identified as Kelvin-Helmholtz instability driven by strong E x B drift shear.

Keywords:
open system, tandem mirror, end plate biasing, electric field shear, flow shear stabilization, drift wave,

Kelvin-Helmholtz instabilitv

1. Introduction

ment in a fusion-oriented device, electric field or electric

Effect of radial electric field and its shear on lowfrequency fluctuations has been investigated since theso-called H/L mode-transition of improving radial

field shear. This may be caused by the difficulty ofmeasuring precisely radial electric field profile andconfinement was found in both open and closed systems

in large-scale fusion-oriented devices. Furthermore,

t1-31. Biased-electrode experiments have been

when qualitatively comparing with experiment, it often
is necessary to account for the radial character of theelectric field in cylindrical geometry. However, a kinetic
theory on the transverse electric field shear effects on adrift mode is available only for slab geometry. In a slab
geometry, the transverse electric field shear increases
with the increase of the electric field shear with a fixed

identifying instabilities concerned with radial transport

performed in linear devices [4-7] and also in toroidal
devices [8,9] in order to control radial potential profiles

and fluctuations. Radial potential control in the
GAMMA10 tandem mirror [0] is very effective tosuppress low-frequency fluctuations [1,12] and toachieve a stable plasma with hot ion temperature of up
to 10 kev [13]. It is found in GAMMAI0 that both fluremode and drift mode instabilities depend sensitively onradial electric field strength.
It is not yet clarified which is the most effective forsuppressing fluctuations and improving radial confine*shear length. In experiments, this situation is notnecessarily true. Therefore, basic experiments have been

done under various combinations of the electric field
and its shear in order to discriminate both effects of theCorresponding author's e-mail: inutake @ ecei.tohoku. ac.jp75@2001 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchInutake M., Control of Radial Electric Field and Low-Frequency Fluctuations in Open-Ended Systems

rotation and the rotation shear.
In order to investigate effects of both radial electric

I Magnetfield and its shear on low-frequency fluctuations, basic
experiments have been carried out in a small linear

Field Linedevice, Q1U of Tohoku University, in which systematic

control and precise measurements of radial potential

I Plasma

profile can be done t14,151.
Numerous theoretical works have been done fortransverse electric-field shear driven Kelvin-Helmholtz

Potentiali(KH) modes [16-19] in both slab and cylindrical

End-Plategeometry, to explain the laboratory experiments in the
Q-machine [5,20] and the observation in space plasmas.
The effects of transverse electric-field shear on drift
modes are also theoretically studied l2l-231 in slab
geometry. In a tandem mirror with a parabolic potential
profile, a rotation-driven drift and flute mode isFig. 1 Schematic of GAMMAl0 tandem mirror.

predicted to be destabilized but has a stable window

non-uniformity is expected to be small at least in thearound the rotation frequency shifted by the electron andlong central cell where the fluctuations and the electric
potential are mostly measured and analyzed.
There are two representative modes of operation in

ion diamagnetic frequency, respectively [24].To explain
shear-driven instabilities in space plasmas and related

GAMMA 10. One is the ECH (Electron Cyclotron

laboratory experiments 125,261, inhomogeneous energydensity-driven (IEDD) instability mechanism [27] areproposed and compared with KH instabilities in slab andcylindrical geometry [28]. The IEDD mechanism is also

Heating) start-up mode and the other is the RF (RadioFrequency) start-up mode. The ECH mode has a centralcell plasma with lower density of less than 1.0 x l0r8
m-3 on axis and a higher central-cell potential of around
I kV with electron temperature higher than 100 eV. The

studied for ion-cyclotron, ion-acoustic and drift modes

in slab geomety [291.
In this paper, results of radial electric field E,
control in the GAMMAI0 and the Q7-U devices are

density profile is rather flat near axis and steep near theedge. On the other hand, the RF mode has a peaked

central-cell density profile with a relatively higher

presented and compared with each other.

density of 3.0 x 1gt8 ttt-3 and a higher ion temperature ofaround 5 keV.

2. Radial Electric Field Gontrol in the
GAMMA10 Tandem Mirror
2.1 GAMMA10 Tandem Mirror
GAMMAIO consists of an axi-symmetric central
cell for confining a main plasma, anchor cells with

2.2 Control of E, in the ECH ModeIn order to control radial electric field of thecentral-cell plasma, a bias voltage is applied only on #l
and #2 segments of the end plates. Potential profiles aremeasured by a beam probe [31] in the barrier region forminimum-B field for suppressing MHD instabilities, andaxisymmetric end mirrors for forming both plug/thermal
barrier [30], as shown schematically in Fig.l. At eachend is provided a segmented end-plate, which consistsof five concentric annular rings, electrically insulatedwith each other. The innermost ring is called #1 and theoutermost #5. Each segment of the end plates isconventionally floated electrically from the vacuum
vessel through I MA resistor and can be biased

various end-plate biasing. Density fluctuations aremeasured by a microwave Fraunhofer diffraction
method. Density fluctuations, which is concluded to be
due to electron drift waves are reduced fromlOVo to lVoby the increase in radial electric field, irrespective of its
sign [11]2.3 Control of E, in the RF ModeA stability boundary for a flute interchange mode
based on the average mimimum B criterion has been
experimentally confirmed in the RF mode [10], since
ion temperature is enough high for diamagnetic loop
signals to be measured accurately. Typical ion

arbitrarily at a voltage to control radial electric field E
in the confining region. The radial electric field may be
non-uniform in the axial direction due to the nonuniform magnetic configuration in the tandem mirror.
However, as the plug/thermal barrier potential is not
actively created in the end-plate biasing experiments, the76

Inutake M., Control of Radial Electric Field and Low-Frequency Fluctuations in Open-Ended Systems

J
(d

E
c
o
o
(L

Mo ECH

t=72(ms)

1.2

Bias

o.4

q

-:::, -0.5
-1 .5 - l

-o.4

Fts

4

i5

0

0.8

oL

00.5

I

Ft

\
''

E
.!2

E=^

.---An;-^

0

i-i

ECH

@c-

0.8

.g
=
c

0.4

o
o

1

0

o-

- 80

oE#1-2 (kv)

E

3 40

Fig. 2 Central-cell on-axis potential Oc(O) and floated-

limiter potential @. versus end-plate potential @r.
Radial potential profile is controlled by biasing a
voltage on #1 and #2 annular segments of end-

0

50

60

70

80

Time (ms)90100plate.

(b) vo= -1.o(kv)*1 1 2465

Bias

temperature perpendicular and parallel to the magnetic

I

t-]

ECH

field line is 5 keV and 400 eV, respectively. The
stability boundary is determined by a critical beta ratio

Frtl|ot.Here 861 and Bo, are central-cell beta andanchor-cell beta perpendicular to the field line,

E
.9;

I

PB

4

J

0.8

9=

F8
i5

respectively. Above the critical ratio, violent density
fluctuations appear and the plasma cannot be sustained.
It is found that plasma stability depends nor only on

6
E
c

o
o

the critical beta ratio but also on radial electric field E,.

0

0.4
0

o-

In Fig.2, on-axis central-cell potential @g(0) and thelimiter potential <\- are plotted against the end-plate

30
-d 20

potential <D6 biased on #1 and #2 segments. Only the on--P 10

axis plasma potential is available, because the gold

0

neutral beam probe in the central cell cannot be radiallyscanned. Positive and negative radial potential

so 60

11'" t'30

eo

1oo

Fig. 3 Effects of end-plate biasing and plug-ECH

differences 4@6 = @c(0) - @66 in Fig.2 correspond to

injection on diamagnetism, on-axis potential @"(0)and egde fluctuations on probe current /,s in thecentral cell. (a) biasing voltage Vt=+O.i kV, (b) yb

hill-type and well-type potential profile, respectively.
Here, the edge plasma potential Q6 is estimated from= -1.0 kV.

the floating potential of the central-cell limiter @1
together with the edge electron temperature T" of 25 ey

byusingqG=q+3Ledge fluctuations.

When the plug-ECH power is injected for the plug

For a given value of the perpendicular beta ratio

potential formation, the central-cell diamagnetic signal

FctlFot, the central-cell plasma is sustained for the full
duration for a certain range of the radial potential

often decreases and edge-fluctuations on Langmuirprobe current suddenly increase as seen in Fig.3(a).
Edge fluctuations are not necessarily correlated to the
ECH power injection. As shown in Fig.3(b), the edge
activities are suppressed by end-plate biasing evendifference 4@6. These results are plotted by open circles

in Fig.4 . It is clearly seen that there is a criticalboundary on the perpendicular beta ratio Frrl\o_,indicated by a broken line, above which the plasma
cannot be sustained. The critical beta ratio has its peaknear A@c = 0, that is, zero radial electric field andduring the ECH power injection. This shows clearly thatthe radial electric field is responsible for low-frequency
77

Inutake M., Control of Radial Electric Field and Low-Frequency Fluctuations in Open-Ended Systems
A<D"= OO 131r1"/e+O,-)

101

4

{

a

-?

100

Na

10-1

3

co-

(u

I

0
-

0.8

-o.4

0

-:Ne

10- 2

0.8
Aoc(kv)

0-

-0.8

CDn

a:
o'a oo
1.8
a\

aK8

r'i

t

a

!a

o

0.4

o

a

I

-0.4

0

0.4

0.8

A@c (kv)

Fig.4 Perpendicular beta ratio B"'lBot versus radial

Fig. 5 Dependence of fluctuation level on radial potential
difference A<D" = 6"191 - @ro for shots with beta
ratio between 1 and 2 indicated by solid circles inpotential difference AO. = 6"191 - (D.o. Broken line
is an experimentally-determined envelope above

which the plasma dumps. Solid circles

correspond to the data which are FFT-analyzed to
identify modes of fluctuations shown in Fig.5.

Fig.4.

A@c < 0. as seen in Fig.5.

In GAMMAI0, effect of E x B drift shear cannot
be discussed due to the lack of the precise radialpotential profile up to the plasma edge in the central

decreases with an increase in E, irrespective of its sign.

This means that there is a stability window with respect
to A@g.

Fluctuations in the edge plasma are measured byuse of an azimuthal array of electrostatic probes set on
the central-cell limiter. Fluctuations are FFT-analyzed inorder to clarify dependence of the fluctuation level on
the radial electric field by selecting shots with the beta
ratio between I and 2, which are indicated by solid
circles in Fig.4. The results are shown in Fig.5. The
fluctuation levels is kept low in the weak E, region andincrease drastically when E, increases, leading to theplasma-dump. Comparing Fig.4 with Fig.5, it is clearlycell. It should be noted that the potential profile attainedin the RF mode of GAMMA1O by keeping all of theend-plate segments floating is actually very effective toseen that the larger the beta ratio 861/Bs is, thenanower the stability window is.
In order to identify modes of instabilities, phase

Device3.1 O--U Linear Device
The experiment is performed in the Q7-U linear
device of Tohoku University [14], whic is shown inkeep the radial electric field weak and to suppress lowfrequency fluctuations in the central cell, because in thiscase most of the radial potential drop appears on theend-plate segments instead of appearing in the centralcell.

3. Control of E, and lts Shear in the Or'U

velocity vot is experimentally obtained from the phase
spectral analyses of the fluctuations. Two modes ofFig.6. An argon plasma with electron density re = l016
m-3, electron temperature T, = 7 eY and ion temperature
4 = 1 eV is produced by electron cyclotron resonance(ECR) heating. Magnetic field strength is almost
uniform and 2.3 kG in the experimental region except
for in the ECR plasma source region. Profiles of

instabilities are identified; one is drift mode withazimuthal mode number m = l, propagating in theelectron diamagnetic direction and the other is m = 1
flute mode rotating with E x B drift velocity. The flute
mode and drift modes are discriminated by open andsolid circles, respectively, in Fig.5. It is concluded froma limited number of available data the flute modes are
dominated in strong positive E., while the electron drift
modes are in strong negative E. It should be noted thatelectron density, electron temperature and spacepotential are measured by Langmuir probes. Radial
profile of density in the experimental region is rather

flat near axis and has a steep gradient near the edge.
Flute and drift mode instabilities are identified by FFTanalyzing the observed fluctuations for various E andthe stability window on 4@6 seems to be symmetricwith respect to A@6 = 0 but different modes offluctuations are dominated in the region of A@6 > 0 andits shear. Flute-mode and drift-mode fluctuations are78

Inutake M., Control of Radial Electric Field and Low-Frequency Fluctuations in Open-Ended Systems
Horn-Type

Sogmented

Gla!3 Plrte

Anlennr+6I

t|3lal Endplate

3.5

Pump

tm
Ar Ga3

q-Uilachl|!
L-450cmD- 20cm

3-

I

N

r

-:a

2.5 -br-Probc^uJ-E=I -->
d)

.^7

uJ

o)

x

Source Erporlmental
Reglon R€lonou.c (/)

Fig. 6 Schematic of 01U device.

-4-3-2-101234
Radial Electric Field (V/cm)

50

O(Uo

a

Fig. 8 Contour-plot of flute-mode fluctuation intensities

for radial electric field and rotation-frequency

3+oshear.

(E

.E

oto
o.
20
a,
o
6

of fluctuations on two probes which are located ataxially-different positions is observed within anexperimental error. Then it is concluded that these

CL
(,10

fluctuations are due to flute instabilities which propagate

azimuthally with the E x B rotation frequency. Ion
0
-8

-6

4

diamagnetic drift frequency is very low in the present
plasma due to the low ion temperature. The fluctuation-2

localized mainly in the radial region with steep density

intensities are contour-plotted on the plane of E, andshear of rotation frequency in Fig.S. The fluctuation
intensities increase drastically with the increase in E x B
rotation shear, while those are suppressed by theincrease in E, regardless of its sign. The slope of a line
drawn from the origin is propotional to the inverse ofcharacteristic shear length I. The line on the abscissameans the rigid rotation without any rotation-frequency

gradient.

shear.

The end-plate has ten concentric rings in order toprecisely control both radial electric field and its shear

in a wide range [5]. The radial electric field isconfirmed to be axially uniform in the experimental

The observed instabilitis are driven even at a low
rotation frequency as far as the rotation shear is enoughstrong. The observed frequencies are near the rotaion
frequecies which are much lower than those predicted

region. Radial electric field E, and its shear are obtainedby the IDDEI [27] and the transverse current-driven

Radial Position (cm)Fig. 7 Typical profiles of radial potential in OrU. Bothelectric field and its shear are varied. Fitted curves
are 6th-order polynomials.

by fitting 6th-order polynomial functions to theinstabilities [32]. The azimuthal mode numbers larger
than m - 2 may be suppressed by the finite Larmor
radius effect and The m = | mode by the watl effect of
the cylindrical metal limiter, which is located between
the plasma source region and the experimental region.
As shown in Fig.8, the instability grows in the region
where the electric field rotation shear is larger than themeasured plasma potential profiles, as shown in Fig.7.

Low-frequency fluctuations are mostly localized in the
shadowed region in this figure.

3.2 Ettect of E and lts Shear on Flute Mode
Rotation frequencies of one mode are almost
constant at any radial position, propagating azimuthally
in the E x I direction with azimuthal mode number rr =2, The frequency agrees well with the E x B drrft

diamagnetic drift shear and where the shear length I, isfrequency calculated by using globatly-determinedcomparable to and larger than the ion Larmor radius pci.
From these experimental results, this mode in the er-U
device is concluded to be a low-frequency transverse

electric field. In the axial direction, no phase difference

KH instability [5,20].
79

Inutake M., Control of Radial Electric Field and Low-Frequency Fluctuations in Open-Ended Systems

IEB-t;14

3.3 Effect of 4 and lits Shear on Drift-wave
Mmode

Y

4\

N

^_r

Frequencies of the other mode of the fluctuations

J.C tZ

are found to agree with electron diamagnetic drift-Efrequency (at/2n = 5 kHz - 13 kHz) which isulcalculated by using the density profile. From varioust_)experiments, these fluctuations are concluded to be due(I)to electron drift-wave instabilities with m = 2.

uJ

;
10)

Fluctuation intensities due to the drift-mode depend on
both radial electric field and its shear, as shown in Fig.9.
The drift-mode is destabilized in the weakly negative E
region, where the E x B rotation frequency is around aquarter of the electron diamagnetic drift frequency. In
the strong E. region, the mode is suppressed, regardlessof its sign.
When the E x I rotation shear increases with E.
being fixed, the drift-mode is destabilized once in ao.s 0)(g-4-3-2-101234

0o)aRadial Electric Field (V/cm)
Fig. 9 Contour-plot of drift-mode fluctuation intensities

for radial electric field and rotation-frequency
shea r.

drift modes are consistent with the prediction by

weak shear region, attains the maximum amplitudes andthen stabilized in the stronger shear region. The E x B
rotation shear at the maximum amplitudes is nearly
equal to the estimated ion diamagnetic drift shear in

Chaudhry et al. 1237, though the theory treats a slab
model with E x B shear as well as E.
In the RF mode of GAMMA 10, flute modes arestrongly excited in the strong positive E, region, while
drift waves are strongly excited in the strong negative E
region. The experimentally-observed stability window
for the flute mode and drift mode seems to be located at
A@c < 0 and A@6 > 0, respectively, as is expected from
Fig.5. These behaviors are consistent with a rotationdriven flute [16] and drift l24l modes, respectively.
These E dependences in the RF mode are quite opposite
to those in the ECH mode of GAMMA l0 and in Qr-U,
where no modes are excited in the strong E. region.
regardless of its sign. On the other hand, in the strong E
x B drift shear region in Q7-U, Kelvin-Helmholtz modeis excited. These different behaviors in GAMMA10 and
Q7-U may be due to the fact that the magnetic field line
in Q7-U is straight and has almost no curvature, whilethat in GAMMA 10 has a bad curvature in the central
cell. The effect ofE x B drift shear cannot be discussed
in GAMMA 10 due to the lack of the precise radialpotential profile up to the plasma edge in the central

magnitude with opposite sign and therefore the drift
shear is effectively cancelled by each other. Thissuggests the net drift shear defined by the sum of E x B

drift shear and ion diamagnetic drift shear is responsible
for suppressing the drift mode.
In the paper [29], the drift wave growth is generally
lowered by transverse velocity shear due to thedissipative effects of the IEDD mechanism, while thismode is destabilized by the reactive effects of the IEDD
mechanism at a sufficiently large flow shear and therotation frequency approaching to ion cyclotron
frequency. The destabilization and stabilization behavior
observed in the Qr-U seems apparently to be different
from this theoretical prediction, although more carefulconsideration of this mechanism is necessary by taking
account ofthe differences ofthe geometry as well as thedensity and electric field profiles.

4. Comparison between GAMMA10 and OtU Results

cell.

In the ECH mode of GAMMAI0, the drift mode is

5. Conclusions

destabilized in the weak E region, as shown in Fig.Z,
with the maximum fluctuation level near E = 0 andstabilized in the stronger E region, regardless of its sign
[ 1]. This behavior agrees qualitatively with that of thedrift mode observed in the Qr-U [5], though the
maximum fluctuations in Q7-U are located at a weakly
negative E.. In the strong E. region, the mode issuppressed, regardless of its sign. These behaviors ofIn the ECH mode of GAMMA 10, the drift-mode
fluctuation amplitude has its peak near the weak E.
region and suppressed in the strong .EI region, regardlessits sign. In the RF mode of GAMMA 10, on thecontrary, the flute and drift modes are destabilized in thestrong positive and negative E. region, respectively.
These are qualitatively consistent with theoretically80

Inutake M., Control of Radial Electric Field and Low-Frequency Fluctuations in Open-Ended Sysrems

predicted rotation-driven flute and drift modes.

tl0l M. Inutake et al., Open Plasma Confinment System
for Fusion, Novosibirsk, 1993, World Scientific,

The flute-mode observed in Q7-U is identified as a

low frequency Kelvin-Helmholtz instability driven bystrong E x B flow shear. Fluctuation intensities due to a

Singapore,5l (1994).

tl ll A. Mase et a/., Phys. Rev. Leu. 64,2281(1990).
|21T. Tokuzawa et al., Jpn. J. Appl. Phys. 33, 807

drift mode depend on both radial electric field and its
shear. The drift-mode is destabilized in the region of(1994\.

weakly negative E. In the strong 4 region, the mode issuppressed irrespective of its sign. The drift-mode is

tl3l Y. Kiwamoto et al., Phys. of Plasmas 3, 578

clearly stabilized by the increase in the net ion drift
shear which is defined as the sum of .E x B drift drift
shear and ion diamagnetic drift shear.

t14l M. Yoshinuma et al., Phys. Letters A 255, 301(1996).

(199e).

t15l M. Yoshinuma et al., llth Int. Toki Conference,
Toki-city, PI-69, 2000.

Acknowledgements

tl6l M.N. Rosenbluth et al., Phys. Fluids 8, 1300

The author deeply acknowledges the GAMMAIO

fl965).

group of University of Tsukuba and Drs. M.

[7] F.W. Perkins et al.,Phys. Fluids 27, 2067 (1984).

Yoshinuma, T. Kaneko, A. Ando, K. Hattori and Profs.
R. Hatakeyama and N. Sato for their collaboration. This

tl8l G. Ganguli et al.,Phys. Fluids 31,823 (1988).
[19] H. Hojo et a/., J. Phys. Soc. Jpn 64, 4073 (1995).
[20] G.I. Kent et a/., Phys. Fluids 12, 2140 (1969).
[2llH. Sanuki et al.,Phys. Fluids 27,2500 (1984).

work was supported partly by a Grant-in-Aid for
Scientific Research from the Ministry of Education,
Science and Culture of Japan and partly by the LHD122)H.Hojo et a/., J. Phys. Soc. Jpn. 57,7|t (1988).
[23] M.B. Chaudhry et al., J. Phys. Soc. Jpn. 57, 3043

Joint Planning Research Program at National Institutefor Fusion Science.

(1e88).

[241W. Horton et al.,Phys. Fluids 27, 2061 (1984).
[25] M.E. Koepke et al., Phys. Rev. Letr. 72, 3355

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81