J. Plasma Fusion Res. SERIES, Yol.5 (2002) 57-62

MHD Stability of 3D Plasma confinement Gonfigurations
with Finite Plasma Current
CIOOPER W. Anthony, ISAEV Maxim Yu.t, OKAMURA Shoichi2 and yAMAZAKI Kozo2

Centre de Recherches en Physique des Plasmas, Association Euratom-Suisse, Ecole
Polytechnique Fdddrale de Inusanne, CHl015 Inusanne, Switzerland

I Russian Research Center "Kurchatov Institute", 123182 Moscow, Russia
2

National Institute for Fusion Science, 322-6, Toki 509-5292, JapanlReceived: 11 December 2001 / Accepted: I May 2002)

Abstract
The ideal magnetohydrodynamic stability properties of reactor-sized sphellamak andquasiaxisymmetric stellarator configurations with finite toroidal plasma current are investigated. In
sphellamak systems, a peaked toroidal current profile yields maximum-B conditions in the central regionof the plasma as a consequence of the paramagnetic enhancement of the magnetic field induced by thecurrent. The resulting poloidally closed contours of the magnetic field strength prove instrumental in thefavourable confinement predictions for a-particles. Under these conditions, the sphellamak is stable or
weakly unstable to local Mercier and ballooning modes as well as low order kink instabilities for volume
average B* - 8Vo. Unfortunately, the bootstrap current calculated in the collisionless regime provides
only a small fraction of the total plasma current that is required. In a two field-period quasiaxisymmetric
reactor device, the bootstrap current in the collisionless limit can enhance the edge rotational transform
when reactor-relevant values of F* < 5Eo are approached. Global external kinks and internal resonantmode slructures associated with the rational surface t = ll2 can be destabilised. This effect is examinedfor various pressure profiles indicating the possibility of stabilisation via a judicious profile choice ormodest counter current drive.

Keywords:
bootstrap current, QAS, sphellamak, ideal MHD stability, global extemal kink, p*, collisionless l/yregime rotational transform

1. lntroduction

which helical coils wound on a spheroidal structure

Three dimensional (3D) magnetic confinement

combined with a set of vertical field coils to compensate
for the current in the connecting arc segments that linkconfigurations such as the sphellamak [1] andquasiaxisymmetric stellarator [2,3,4,5] (QAS) can offer

two helical windings provide a small initial translation
ofthe field lines. A strong toroidal current is required toprovide the confining fields in this type of device which
also paramagnetically enhances the toroidal magnetic
field. Because the equilibrium magnetic field stnrcture is
3D, the dynamo effect is not required for the generation

attractive potential for a reactor system such as stable
operation with respect to global ideal magnetohydrodynamic (MHD) modes at B* values in excess of
5Vo, adeqtate confinement of a-particles, compactness
and acceptable neoclassical transport properties.

The sphellamak system [1] is a coreless device in@20O2 by The Japan Society of Plasma

Corresponding author's e-mail: wilfred.cooper@ epfl.ch

Science and Nuclear Fusion Research57

Cooper W.A. et al., MHD Stability of 3D Plasma Confinement Configurations with Finite Plasma Current

current profile produces a maximum-B configuration in
the central region of the plasma as shown in Fig. 1 for acase obtained with a standard pressure profile p(s) =p(0)t0.9(1 - s)2 + 0.1(1 - s)l at B* - 8Vo. This system isunstable to a low order extemal kink dominatedby a mln = lll component because the rotational transform near
the edge of the plasma approaches unity. The spectrumof the dominant components of the radial displacementof the toroidal magnetic field component. This system
displays features of spherical tokamaks, stellarators,
spheromaks and reversed field pinches. The main
advantage with respect to spherical tokamaks is theabsence of a central conductor and the concomitantshielding requirements against large neutron fluxes
expected in a reactor environment. We specificallyinvestigate the the sensitivity of the maximum-B
magnetic field structure generated by a peaked toroidal
plasma current and the bootstrap current (BC) [6,7] tovarious pressure profiles at B* - 87o as well as theirdependence with respect to the current in the helicalcoils (for a toroidal current fixed at 20 MA) and as avector are shown in Fig. 2a. At B* - 8Vo, thecollisionless BC contributes a relatively modest lUVo to
the total toroidal current. Therefore, the bulk of thecurrent must be inductively driven. This constitutes themain shortcoming of the sphellamak concept. The

profile of the imposed toroidal current and those of the
BC resulting from a pressure profilep(s) =p(0)(1 - s2)rwith ft = 3,4 and 5 are displayed in Fig. 2b. For thiswork, we have considered a single ion and electron

function of B*.
Quasiaxisymmetric stellarator (QAS) reactors are
3D systems in which the magnetic field structure in
Boozer coordinates [8] is dominantly axisymmetric likein tokamaks. This property causes the neoclassicaltransport and the a-particle confinement to behave in asimilar way as in a tokamak. For reactor relevant valuesof P*, the QAS system can develop a substantial

plasma of equal temperature and normalised density
N(s) = (1 - 0.999s2). The BC increases from 1.58 MA to1.68 MA as we increase the pressure profile peakedness

factor ft from 3 to 5. For the standard p(s) profile, theBC increases linearly from 1.63 MA to 1.97 MA as B*
varies from 7 .4Vo to 9Vo. The BC varies from 1 .7 MA to1.88 MA as the current in the helical coils 1gg changes
from 54 MA to 66 MA. In this range, the global kinkeigenvalue reaches a minimum at IHs - 60 MA. The
variation of the rotational transform is minimal withrespect to changes in 1g6 orp(s) peakedness. However,
the magnitude of the external kink unstable eigenvalue
decreases as the p(s) peakedness parameter varies from
3 to 5. The Mercier criterion is weakly unstable in the
inner half of the plasma volume. For the standard p(s)profile, the ballooning modes are unstable in the inner
20Vo of the plasma volume and also near the edge of theplasma around s - 0.9. For the pressure profiles p(s) =p(0)(l - s2;t, the vanishing of the pressure gradient near
the axis stabilises the ballooning modes in that region.
For k > 3, the unstable region near s - 0.9 disappears,
but for k > 5 a new region of weak instability emerges
near s - 0.3. The confinement of a-particles born near
1/4 of the plasma volume in the region of poloidallybootstrap current that can significantly alter therotational transform. We examine the global ideal MHD
stability properties with respect to internal and externalkink modes using the TERPSICHORE code l9l in a 2field period QAS at F* - 5Eo with a self consistentbootstrap current evaluated in the collisionless l/v
regime and determine the sensitivity to various pressure

profiles. In this work, we employ the definition B* =zpo"[iZf,)t<82>, where V is the plasma volume, p is
the pressure, B is the magnetic field strength and <A> =I aVe, A""uuse this constitutes a more relevant
parameter with respect to fusion power production. B*
represents the root mean square value of B.

2. The Sphellamak Reactor System
The sphellamak system has a large toroidal plasma
current that can cause it to become susceptible to loworder kink instabilities. We have identified a
configuration which is stable to localised ballooning and
Mercier modes and global n = I kink modes at ff - 1Vo,
but as it has a plasma volume V of less than 500 m3, wehave not pursued it further as a reactor option. We haveconcentrated on a configuration with V - 1000 m3 toclosed mod-82 contours shows a negligible loss after 0.1

s. For those particles born near half volume(corresponding to a normalised radius of - 0.7), the loss
pattem approaches 4OVo corresponding to the bulk ofthetrapped population as illustrated in Fig. 2c. This fractionof a-particles experiences the full 3D structure of themagnetic field nearer the edge.

perform sensitivity studies of the mod-82 structure andthe collisionless BC to variations in the width of thepressure profile, the magnitude of the helical coil current
and B*. The toroidal plasma current is prescribed as
2n J'(s) = Ztc J'(0)[3(l -.r)5 + (l - s5)2]/4 such that the

total toroidal current 2n J(l) = Ip = 2O MA. This type of58

Cooper W.A. et al., MHD Stability of 3D Plasma Confinement Configurations with Finite Plasma Current
k-F

didbdonh SrHhmt

SPHELLAMAK

#i
0.1

o.2

0.3

0.4

GLOBALST€ILIN-........:r"r"'-.aJ';;;;;;i;i,;;;;"','',,,':::::':""
0.5

0.6

0.7

TOROIDAL CURRENT PROFILES IN SPHEUMK

Mod-gr diiibdid

in Srdbmt

llttlrr.''J

,.w

S
I
c
F

z

:l

F
F

z

p

Mod-82 diftdon

in S.hdhndl

blue: q40 €s6

ro'

10'

ro'

,,#*,

10

"

ro

'

roo

Fig. 2 a) (top) The radial displacement vector spectrum

Fig. 1 The mod-82 structure in a sphellamak reactor
system with V- 1000 m3, peaked toroidal current/p = 20 MA and B* - 8%" at the beginning of a field

profiles, b) (middle) the prescribed toroidal current and collisionless bootstrap current profiles
for p(s) = p(0)(1 - s2lk, k= 3 -+ 5 and c) (bottom)
the loss pattern in time of a-particles born at halfvolume for two p(s) profiles in a sphellamak withpeaked toroidal current and F - 8%.

period (top), at 114 ot the period (middle) and at
midperiod (bottom). This structure remains resilient to variations of the pressure profile, the heli-cal coil current and 0.
59

Cooper W.A. et al., MHD Stability of 3D Plasma Confinement Configurations with Finite Plasma Current

3. 2-Period OAS Reactor
We investigate in this section the global ideal
MHD stability properties of a 2-field period QAS
reactor in which the rotational transform results from the

combined action of the 3D plasma shaping and the selfconsistent bootstrap current (BC) computed in thecollisionless llv rcgime. [6,7] In particular, we examine
the sensitivity of the BC and the global kink properties
with respect to various types of pressure profiles atoreactor relevant values of F* - sEo. The computation of
the VMEC equilibria [10] and the BC is undertakenp(s)=pol1-s2-0.2(1-s11y0.8

iteratively until a converged BC and corresponding{

profile are achieved typically in 6 + 12 iterations. The
pressure profiles are chosen to have vanishing gradientsnear the edge of the plasma to prevent discontinuities in
the BC at the plasma-vacuum interface. The BC profiles
for various pressure profiles are presented in Fig. 3a atp(s)=poF- s- 0.10- s1o)Y0.9

B* - 5Vo. The toroidal component of the BC (multipliedby the permeability of free space ;ra) as a function of B*
for a nearly parabolic pressure profilep(s) =p(0)[1 - s 0.1(l - s1)1/0.9 is shown in Fig.3b. The structure ofzEo

GdA", where 68" is the radial component of theperturbed magnetic field, dominated by a mln = 2/l

6
F

6
o

contribution near the edge of the plasma is displayed at

ff - Sqo in Fig. 3c. To further understand the sensitivityof the ideal MHD instability properties with respect tothe BC, we multiply the toroidal component of the BC
that is used as input for the VMEC code by an artificial
enhancement/suppression factor Z. For a case with a0nearly parabolic pressure profile, the global kinkeigenvalues as a function of Z at B* - 4.97o, therotational transform profiles for different values of Z
and the spectrum of the radial component of the1g 6ff dlddnhNrys

displacement vector (that shows a dominating mln = 2/lterm) are presented in the left hand column of Fig. 4. It
can be seen that a modest 207o suppression of thebootstrap current can stabilise the extemal kink mode. A
corresponding set of graphs associated with a slightly
more peaked pressure profile p(s) = p(0)[1 - s - 0.2(1 r5)l/0.8 at P* - 5.2Vo appears in the right hand columnof Fig. 4. For this profile, the BC must be decreased by
one half to guarantee stability. Enhancement of the BC

Fig.3 a) (top) The bootstrap current profiles in thecollisionless 1/v regime at p* - 5Y. tor variouspressure profiles. b) (middle) The bootstrap current as a function of B* for a nearly parabolic p(s)= p(0)11 - s - 0.1(1 - slo)l/0.9 pressure profile. c)(bottom) The f968" distribution near the edge ofthe plasma in a 2-period quasiaxisymmetricstellarator at B* - 5% that shows a mln = 211
dominated unstable external kink obtained with adestabilises more localised mln = 413, 615, 7 /5, 917 , etc
mode components. This also happens when more peakedpressure profiles such as p(s) = p(0)(l - s.)2 areconsidered.

4. Conclusions
We have examined the ideal MHD stability
properties of two 3D magnetic confinement reactor

p(s) = p(0)[1 - s-0.211 - s5]l/0.8 pressure profile.
60

Cooper W.A. et al., MHD Stability of 3D Plasma Confinement Configurations with Finite Plasma Current
GLOBAL IDilL MHD STABILIfr IN 2- PERIODQAS

GLOaAL IDEAL MHD STABILITY IN 2- PERIoD QAs

g' = 4.s%

9' = s.z"h
PoJ = 10.9p(s)=potl- s 0.1 (1- slo)Y0.9

p(s)=po[l- s- 0.2(1-

=

z

r

z

F

F

z

z
E

5

F

o

c

B = 527"
poJ = 1 0.54p(s)=pol(1- s)- 0.2(1- s1Y0.8

r= - 0.0365

Fig.4 The global kink eigenvalue as a function of the toroidal bootstrap current enhancemenVsuppression factor f(top row). the rotational transform profiles for different values of f (middle row) and the spectrum profiles of theradial component of the displacement vector dominated by n = 1 external kink modes for f =1 (bottom row) in aquasiaxisymmetric stellarator reactor with a nearly parabolic pressure profile p(s) = p(0)[1 - s- 0.1(1 - s10)]/0.9(left hand column) and for a somewhat more peaked pressure profile p(s) = p(0)t1 - s- 0.2(1 - s5)l/0.8 (righthand column).

6lCooper W.A. et al., MHD Stability of 3D Plasma Confinement Configurations with Finite Plasma Current

could reasonably yield fully stable conditions at reactor

systems with finite plasma current which constitute themost compact 3D devices proposed so far. A detailed
comparative study between the two configurations is not
attempted here because they display very different

relevant values of B*. Alternatively, some modest
counter current drive in the range of I MA could not
only realise external kink stability but could also assist
in profile control. Enhancement of the BC or more
peaked pressure profiles cause the destabilisation ofinternal/external kinks of higher order. We could,

characteristics. A peaked toroidal current in thesphellamak device guarantees a maximum-B con-figuration near the central region of the plasma.

nevertheless, hope and anticipate that the plasma will
naturally adopt a pressure profile that does not violate
kink and ballooning stability conditions under real

Energetic a-particles bom within the domain of closed
mod-B contours remain extremely well confined, while
the trapped fraction born closer to the edge of theplasma experiences the full 3D structure of the magnetic
fields and consequently drift out within a slowing downtime. We have analysed the sensitivity of the magnetic
field structure and the bootstrap current in a sphellamak

experimental operation.

Acknowledgements
This research was partially sponsored by the Fonds
National Suisse de la Recherche Scientifique, Euratom
and INTAS. We thank Dr. S.P. Hirshman for the use ofconfiguration that is mildly unstable to global external

ideal MHD modes dominated by the mln = lll
component because the rotational transform under theconditions tested approaches the critical value of unity
at the edge of the plasma. The bootstrap current in thecollisionless l/v regime contributes only about lo%b of
the total toroidal plasma current in the device at B'( 87o. This bootstrap current increases with pressure

the VMEC code. The numerical calculations wereprimarily performed on a NEC/SX5 platform at theCentro Svizzero di Calcolo Scientifico, Manno,

profile peakedness, helical coil current and ff, while the[]variations of these parameters. Though the Mercier
criterion is weakly unstable in the central region of theplasma, the ballooning modes are stable for p(s) =p(0)(1 - s2)a and only marginally unstable for otherpressure profiles with small gradient on axis. Although
it may be difficult to control the current and pressurepan (1998).
[2] J. Niihrenberg et al., Proc. Joint Varenna-LausanneInt. Workshop on Theory of Fusion Plasmas,
Editrice Compositori, Bologna (1994).
[3] P.R. Garabedian, Phys. Plasmas 3,2483 (1996).

Switzerland.

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mod-B structure is essentially unaffected by the

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profiles in an actual experiment, the global external kinkproperties may force a peaked toroidal current and theballooning properties may adapt the pressure profile tothat which we have found to be nearly optimal.
We have furthermore analysed the effect of a self

t5l G.H. Neilson et al., Phys. Plasmas 7, l9ll (2000).
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t7l K.Y. Watanabe et al.,Ntcl. Fusion 35, 335 (1995).
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34 (1990).
[10] S.P. Hirshman et al., Comput. Physics Commun.
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bootstrap current increases the rotational transformabove the critical value of t = ll2 at B* < 5Vo which candestabilise a mln = 2/l external kink mode. We haveinvestigated the sensitivity of the external kink modes inthis reactor system to the BC by altering the pressure
profile and by artificially enhancing or decreasing the
BC by multiplying its profile by a fixed factor Z. We
have found that a nearly parabolic pressure profile thatsuppresses the BC by 207o can stabilise global external
kinks. For a slightly more peaked pressure profile, thesuppression factor required increases to 507o. Thissuggests that a more judicious choice of pressure profile
62