J. Plasma Fusion Res. SERIES, Vol'1 (1998)

62-65

Beta Studies in Ouasi-Symmetric Gonfigurations
Jiirgen* and zILLE Regine
GORI Silvio, NUHRENBERG carolin, NUHRENSERG
G reifswald'
M ax- Planck- I nstitut fi)r P lasmaphysik' Teilinstitut
I PP - EII RATOM Ass" D - I 7 509 Greifswald'

Germany(Received: 30 September L997/Accepted:22 October

1997)

Abstractandquasi-helically symmetric (qH), quasi-axisymmetric (qA)
Quasi-symmetrical configurations configurato their MHD stability properties' QH
quasi-isodynamic (qI) - are investigated with respect*0'o6' if no net toroibe MHD stable at < g}toappeartions with aspect ratio {a(10) arefound whichperiods, aspect ratio - 4 and t-tr ate found by combined
dal current is present. QA tokamaks with twosymmetryandstabilityoptimizationwhichappeartobestablear<p>=0.03foraratioofexternallyqI and magnetic-

As for qI stellarators simultaneous
to current-created rotational transform of about 2.
good collisionless particle confinement and awell optimization at vanishing and finite B shows that very
are compatible'-ugn"ti" well in the vacuum field configuration

Keywords:
quasi-symmetries, quasi-helical symmetry, quasi-axisymmetry,
local and global MHD-stability

quasi-isodynamicity' configuration study'

types of modes can readily be investigated' Ballooning
modes also aPpear to be one important obstacle to1. Introduction
Stellarators may exhibit large neo-classical transport in the long-mean-free-path regime and poor aparticle confinement so that these issues generally

high-beta qA configurations if the externally createdrotational transform is strong. As for qI configurations
with aa more elementary problem is their compatibility
papervacuum magnetic well. In the remainder of thisd"r"*" critical attention in configurational studies'quasi-helically
Quasi-symmetrical configurations -case studies of these problems are presented'(qH) symmetric [1], quasi-axisymmetric (qA) [2,3] andquasi-isodynamic (q! [ ] - alleviate these problems' soih"t ,h"i, MHD behavior is an important remaining
conarea of study. As far as high-beta equilibria areisthecerned probably the most challenging situation
qIandhigh-beta qA tokamak, while for qH symmetric
WZ-X type) configurations high-beta equilibria

2. Ballooning Modes in a Ouasi-Helically Symmetric Configuration
The qH symmetric stellarator was the first example
of a true three-dimensional configuration which defeats
the traditional stellarator plagues of large neo-classicaltransport and poor a-particle confinement' At moder-1orpos" less problems. In the area of MHD stability several
qH
specific problems have been identified' High-beta
intransform
rotationalwithsymmetric configurations
foundbeen
haveplasmaboundarycreasing towards thewhich are stable to Mercier and resistive interchange
modes but unstable to local as well as non-local baltwo
looning modes, so that the relationship of the latter

ate aspect ratio lA- 0 (10)l these configurations

aregenuinely three-dimensional in plasma shape, i'e' notclose to a helically symmetric shape of their cross-sec-tions. They are capable of very high equilibrium-B
values since they do not exhibit a Shafranov shift'

Theytoalso show quite large stability-B values with respect@1998 by The Japan Society of Plasma' Corresponding author's e-mail: kgtc@

ipp- garching'mpg'de62

Science and Nuclear Fusion Research

Gori S. er a/., Beta Studies in euasi_symmetriccouplings. Non-local ballooning modes occurat < p>values above 0.06, while local ballooning instability(obtained from the ballooning equation along
a field
line [5]) prevails for <B> values above 0.05. ThelDry@

Fig. 1

Configurations

case<f> o0.1 flux-surface cross-sections at the be_ginning, a quarter, and half of a field_periodof an
4=6 qH configuration with aspect ratio A=11 andvacuum magnetic well depth -0.02. For<f> -0.1 the rotational transform varies between

considered here exhibits a rotational transform profile
increasing towards the plasma edge. A discrepancy
be_tween non-local and local ballooning was observedin

W7-X type cases, too, as opposed to the standard toka_
mak situation characterized by a rotational transformdecreasing towards the plasma edget.=6/5 and 12/9.

[6].

3. Ballooning Modes in a Ouasi-AxisymmetricTokamak

5 0.10

QA configurations have been studied with tokamak [2] as well as stellarator emphasis[3,7,g,9]. Here,a case intermediate to these extremes with a rotationalEo(,

ggo o.05

transform (without net current) of!

6or.<2/5 with
No:2 and aspect ratio A- 4 is studied. Tiis configura-oELtion also served to investigate the problem of control

I o.ooofthe quasi-axisymmetry for free-boundary equilibria
withtoroidal current by external coil currents (see Ref.
10][).
The configuration shown in Fig. 3 was obtained bysimultaneous optimization of quasi-axisymmetryandresistive-interchange stability at vanishing toroidal

0.

flux lobel sFig.2 Result from the cas3o stability code: An unstablenon-local ballooning-type mode (poloidal node_cur_rent and fixed <B>:0.02. Non{ocal ballooning sta_bility investigations have been started for cases with to_roidal current, here simply by fixing the rotationaltransform profile in the equilibrium calculations. For

number M-2Ol detected in the equilibrium shownin Fig. 1. A more detailed description is given in themain text. Note that this equilibrium is itable withrespect to the Mercier criterion. Computation par_ameters: radial grid size ff"= 192, - 1OO perturba_
tion Fourier components, CAS3D2MN[12j. ln a se_quence of equilibria with lower <d> the marginal
point given by such medium-node_number penur_

two cases with slightly increasing and slightly decreasing
profiles (0.51 < r(0.56, 0.56> r>0.51) marginalsta_
bility is found for < B> *0.026 and <p> *0.022, r"_bations is obtained for ( p> - 9.65.

spectively, compare Fig. 4.

These results are preliminary, because simulta_
neous consideration of a ballooning optimization[11]
still needs to be done.

Mercier and resistive interchange modes, see Fig. 1 fora case with six field periods, i.e. No- 6, so that theirballooning stability properties are interesting. Results
concerning this question are shown in Fig. 2, in which
the Fourier harmonics of the scalar component of the4. Ouasi-lsodynamic Configurations with a
Magnetic Well

ideal MHD displacement vector normal to flux surfaces
are plotted versus the normalized flux label (which iss:0 on the magnetic axis and s-L on the plasmaboundary). For this specific perturbation the value ofthe mode resonant rotational transform is tol.25:
25/20. Out of 100 perturbation Fourier harmonics L5
are above the plotting threshold. The dominant har_monic (curve 1 in Fig. 2) has poloidal Fourier index
m:2O and toroidal Fourier index n:-25. Curves 2
and 8 represent tokamak-like couplings (same toroidal
index as principal harmonic, i.e. n:-25, m:19 and,

Excellent orbit confinement in a 3_D structure ofB(* 0, fi of a vacuum field configuration was demon_strated recently [4]. Here, combined optimization ofm:21), curves 6, 7 (n:-19, m-I6, 15) and curves
L2, 13 (n:-31, m:24, 25) ttre dominant helicalFig. 3 Same as Fig. l, but for a qa tokamak withpect ratio Ao 4 at <63ly'^ = 2. as_f > = 0.0g. Here, 0.51 < ,"< O.Se.

Gori S. er a/., Beta Studies in Quasi-symmetric Configurations

olLo

5 0'0oql stel-Fig. 5 Same as Fig. 1, but for the vacuum field of alarator witn rur= s, aspect ratio = 10' The magnetic
well is =0.01, 1""'" and L.ds"=1, and l.in=0'93'"

flux lobcl sFig.4 Result from the cns3o stability code: Flux-surface
averaged energy terms for an unstable perturbation in the qA case of Fig. 3. The various contributions are driven by the field line bending (label 1)'

local shear and parallel current density (2), the fieldcompression (3), curvature terms (4). and sum up toO'W \sl. Computation parameters: poloidal nodenumber -60, radial grid size N"=192, -80 perturbation Fourier components, CAS3D2MN' Extrapolation of < B> = 0.03 and 0.027 results leads to a marginal value of 0'026; local ballooning is found to beunstable tor <B> =0.017' The corresponding casethe collisionless a-particle confinement (simultaneousat <P>:0 and 0.05) and the magnetic well property(necessary for MHD stability) was used to find a configuration with very good confinement of core a-particles (for details see Fig. 6) and a vacuum field magnetic well of -0.01. The results of Fig' 6 suggest thatthe qI condition cannot be satisfied at low aspect ratio

within this class of configurations' The optimizations (in6 became= 200 variables) leading to the results of Fig'

with outwardly decreasing rotational transform

only possible using B and its partial derivatives on aleads to a marginal value of < P> = 0'022'

3-D grid (radially 90, poloidally 140, half a field period

Lost [%lLost [7o]

to-s

to-1/

to-3

lo-4 to-3

rc-z to-lt ["1to-2

10- It ["1Lost [%l,^-5lu

10-s lo-1 l0-3 rc-z t 19_t["]rn-4rurn-3 rc-z
rvru

10-lt^ttl

Normalizations used:8=5 T' average minor plasma
Fig,6 Collisionless a-particle confinement, ql and w7-X stellarators'
are the histories of losses of samples of particles Shown
MeV.
3.6
radius a=1.5 m, kinetic energy of d-particles
and the pitch angle - up topoloidal and toroidal variables
started at various surfaces with random distributions in theeach symbol indicating the loss of a particle at the timeway,
cumulative
ina
approximately the slowing-down-time
o: 0'08' n" O'24' +: 0'4' A : 0'56' Left column for the qlf
lux:of its loss. Symbols indicating start surfaces in normalizedg,bottom frames for < B> = 0'05'
case of Fig. 5, right column to-r wz-x. Top frames tor B =64

Gori S. er a/., Beta Studies in Quasisymmetric Configurations

toroidally 80 exploiting stellarator symmetry) andachieving =0.02 TFlops on 256 processors of theCRAY T3E of the Rechenzentrum Garching. Evaluation of the local ballooning criterion for such qI configurations indicates stability for < B> = 0.05, while theReferences
J.
Ntihrenberg
andR. Zille, phys. Lett. A 1^}g,[1]113 (1e88).

[2] J. Niihrenberg, W. Lotz and S. Gori, Theory ofFusion Plasmas Varenna 1994 (Ed. Comp., Bo-global-mode stability limit at < d> = 0.058 as given by
the cAS3D stability code [12] cas is slightly less stringent. These results are preliminary, since simultaneouslogna, 1994), p.3.

[3] P. R. Garabedian, Phys. Plasmas 4,1617 (L997).
[4] S. Gori, W. I-otz, and J. Ntihrenberg, Theory ofoptimization of the rotational transform profile still

Fusion Plasmas Varenna 1996 (Ed. Comp., Bologna, 1997), p.335.
[5] D. Correa-Restrepo and Z. Naturforsch. 33a, 789(1e78).

needs to be done.

5. Gonclusion
QH configurations should be investigated with respect to the transition to very large B values as NocA
increase and to the influence of a selfconsistent boot-[6] C. Schwab, European Conference Abstracts (EpS,Petit-Lancy, 1993), Vol. ITC,partl, p.417.
[7] A.H. Reiman et al., Plasma Physics Rep. 23, 472

strap current on these results, QA configurationsshould be optimized - in addition - against ballooning.
The results for qI configurations show that the qI condition can probably not be satisfied at low aspect ratio(Lee7).

[8] K. Matsuoka et al., Plasma Physics Rep. 23, 542(ree7).

[9] N. Nakajima et al., Plasma physics Rep. 23, 460

in vacuum field configurations in the configurational(tee7).

neighborhood of W7-X.

[10] M. Drevlak, in these Proceedings, p.a26 e998).
[11] J. Niihrenberg et al., Int. Sherwood Th. Conf.

1989,lD4.

[I2lC. Schwab, Phys. Fluids B 5, 3195 (1993).

65