J. Plasma Fusion Res. SERIES, Vol.1 (1998) 75-79

MHD Physics on LHD
NAKAJIMA Noriyoshi
National Institute for Fusion Science, Toki 509_5292, Japan(Received: 30 September L997/Accepted:26 January 199g)

Abstract
The Large Helical Device (LHD) has a flexible coil system making various vacuum configurations.
The characteristics of such vacuum configurations are basically expressed in terms of toroidally bended
corresponding straight helical configurations. The MHD physics of finite LIHD plasma is understoodPfrom the competitive or synergetic relation between helicity and toroidicity. The former comes from thevacuum configurations and the properties still remain in the plasma periphery even for high equilibria.
BThe latter is brought about by the essentially axisymmetric large Shafranov shift, which is strongly influenced by the toroidal current conditions and pressure profiles, and such effects become significant in
the plasma core region as B increases. Throught the local and global properties of the finite p MHD
equilibria in the planar axis LHD vacuum configuration, the ideal MHD stability of interchange and ballooning modes is addressed.

Keywords:LHD, MHD equilibria and stability, interchange and ballooning modes

1. Introduction

the properties of the vacuum magnetic field still remain

The Large Helical Device (LHD)[l] has a flexible
coil system, i.e., one pair of two helical coils, three pairsof two poloidal coils, and ten coils making an (m/n) :(/l'1 tairly large magnetic island that acts as a Local
Island Divertor (LID)[2]. Each helical coil has such independent current feeders that the coil current can befed in a balanced or unbalanced manner. Also, eachhelical coil has three multiple layers with independent
current feeders, which is used to change the minor
radius of the current center of helical coils. The poloidal coil system can control the vacuum magnetic axis
shift, the shape of the poloidal cross sections, and soon. Such the flexible coil system creates four types oftypical vacuum configurations making the relatively low
rotational transform t near the magnetic axis, whichleads to a large Shafranov shift or Pfirsch-Schliter current even for fairly low B values. In contrast with it,
both the Pfirsch-Schliiter current and the bootstrap cur-in whole plasma region for fairly high B values. Thus,
such the flexible coil system and the large Shafranov
shift of LHD give us an absolutely contrastive concept
to W7-X as plasma confinement device. In this paper,
we will concentrate our attention on one of the key
issues of plasma performances, namely, the ideal MHD

stability of the MHD equilibria in the L/M : 2/L0
planar axis vacuum configurations without LID throughthe understanding of the vacuum configurations and theMHD equilibria.
In Section 2, the characteristics of the L/M: 2/ I0
planar axis vacuum configurations are described. The
characteristics of the finite B MHD equilibria in suchvacuum configurations are given in Sec. 3 including effects of pressure profile and net toroidal current. Sec-tion 4 is devoted to the characteristics of the ideal
MHD stability in such MHD equilibria, where theproperties of interchange and ballooning modes arerent are significantly reduced in W7-X[3] and hence

mainly examined. The summary is given in Sec. 5.
@1998 by The Japan Society of Plasma
Science and Nuclear Fusion Research' Correspond.ing author's e-mail: nakajima@ nifs.ac.jp75Nakajima N., MHD Physics on LHD

averaged magnetic hill, leading to the situtation favor-

2. Characteristacs of Vacuum Gonfigurations

able for interchange modes. In contrast with it, thelarge deformation of the local pitch of the magnetic
field lines leads to the disappearance of the local magnetic shear in the region with stellarator-like magnetic'fhe L/M: 2/10 planar axis vacuum configura-tions are examined. The characteristics of such vacuum
configurations are basically expressed in terms of toroidally bended L/M : 2/10 straight helical configurations, so that such vacuum configurations have the clear
separatrix magnetic field lines (4- M/L: 5), the stellarator-like magnetic shear, and the magnetic hill nearshear, which is not favorable for ballooning modes[5].

By using the stellarator expansion under the high-B ordering (B- O (t )), the Shafranov shift A is expressedby:the periphery.

By the control of the poloidal coil system, theA,/_\ :A\',d^(r): a4{ - o(\'- drmagnetic axis shift and the shape of flux surfaces arechanged keeping the distribution of the field strength
almost unchanged. Since the toroidal magnetic field isinversely proportional to the major radius, the vacuum
magnetic flux surfaces shift to the outside of the torus.
Thus, the inward shift of the vacuum magnetic axis canrestore straight helical-like properties, and the outward

4P''D/?\P(4:T(1)

From Eq.(1), we can consider effects of the pressure
profile[6] and the net toroidal current[6,7] on MHD
equilibria.
Peaked pressure profiles make a large Shafranov
shift, leading to the deep averaged magnetic well in thecore region and to the situation that maximum pressure
gradient easily exists in the magnetic well region. It follows, however, that broad pressure profiles create theopposite situation to that brought by peaked pressure
profiles. Note that the beam pressure profile of NBI
and deposition profiles by NBI, ECRH, and ICRF may
affect the pressure profile. Parameters of vacuum configurations or MHD equilibria creating the large Shafranov shift from the sense of Eq. (1) have a tendency tobring about broad pressure profiles reducing the Sha-shift enhances toroidal effects. By the inward shift ofthe vacuum magnetic axis, the equilibrium beta limit increases, the magnetic hill becomes high, and the orbitsof deeply trapped particles in helical ripples become

better. The outward shift makes the opposite properties.

3. Characteristics of MHD Equilibria
The characteristics of finite p MHD equilibria inthe L/M: 2/10 planar axis vacuum configurations arefranov shift from the aspect of particle orbits, and parameters making the Shafranov shift small have an opposite tendency. The experimental control of the pressure profile may become important.
The net toroidal currents such as bootstrap current, Ohkawa current, and ECRH-driven current affect
the Shafranov shift through the rotational transform t.
The bootstrap current in the t/ v collisionality regime,

examined by using VMEC code[4] under the fixedboundary condition. Since the LHD plasmas have arelatively high magnetic shear, the nested flux surfaces
are assumed to exist by neglecting the small magnetic
islands if any.

At first the currentless MHD equilibria are considered. The relatively low rotational transform t near
the magnetic axis in such planar axis vacuum configurations allows a large and essentially axisymmetric Shafranov shift or Pfirsch-Schliiter current even for fairly low
B values, which leads to a large deformation of both theflux surfaces and the local pitch of the magnetic fieldlines in the core region, almost leaving the distributionof the magnetic field strength as it is in the vacuum
configuration[5]. Thus, the essential characteristics of
MHD equilibria are described by nearly axisymmetricthe Ohkawa current due to co-injected NBI, andECRH-driven current increasing t reduce the Shafranov shift, whcih leads to the shallow magnetic well in
the core region and weak stellarator{ike magnetic shear
in the periphery. As a result of it, MHD stability isdeteriorated. The bootstrap current in the plateau collisionality regime, the Ohkawa current due to counterinjected NBI, and ECRH-driven current decreasing tenhance the Shafranov shift and are favorable for thelarge Shafranov shift (toroidicity dominant) in the coreregion and properties of the straight helical system (helicity dominant) near the periphery, resulting in thecompetitive or synergetic states between toroidicity andhelicity. Such the large Shafranov shift can create theaveraged magnetic well in the plasma core region withlow magnetic shear and the resultingly enhanced stellarater-likg magnetic shear in the periphery with theMHD stability. Note that the bootstrap current isstrongly affected by both vacuum configurations andMHD equilibria themselves and that it is reduced by
the positive radial electric field when electrons and ionsexist in the l/ v and plateau collisionality regimes, respectively[8,9].

76

Nakajima N., MHD Physics on LHD

4. Characteristics of MHD Stability
The ideal MHD stability is examined under thefixed boundary condition. Current-driven modes in
LHD are stable when the net toroidal current is lessthan about 300 kA (the net toroidal current due tothe equilibrium plimit in the sense that in the region
near the Mercier marginal stability with 4<0.2 thelow-mode-number interchange modes are stable andthat the high-mode-number interchange modes highly
localize on their mode rational surfaces with the quitebootstrap current is about 200 kA). Thus, examinations
are concentrated on pressure driven-modes, namely, in_terchange modes and ballooning modes. The amplitudes of the interchange modes do not change so muchalong the resonant magnetic field lines, so that the stabilization effects due to the averaged magnetic well andthe magnetic shear are important for their stability. In
contrast with it, the amplitudes of the ballooning modeschange along the magnetic field lines, so that the local

small growth rates[7].

As is well know, the ballooning modes are more
stringent than interchange modes. Moreover, it is foundout by using high-mode-number ballooning mode analysis that ballooning modes inherent to three dimensional (3-D) equilibria exist[6]. In the 3-D equilibria,
the eigenvalue of the high-mode-number ballooning
mode equation l" is a function with respect to q.t, 0r, anda, where tp and a are the labels of the flux surfaces andthe magnetic field lines in the covering space, respectively, and Q is the radial wave number comins fromWKB approxiamtion:

magnetic well or hill and the local magnetic shear
become important for their stability, the fact of whichindicates that the ballooning modes can easily feel thethree dimensional structure of the MHD equilibria.
From the previous discussion about the vacuum
configurations and the MHD equilibria, it is found thatpeaked pressure profiles and the currentless or ner roroidal current decreasing t and the vacuum configuration with outward axis shift are favorable for interchange modes. Figure 1 shows the interchange stability
diagram with equilibrium B-limit for currentless equilibria with the relatively peaked pressure profile: p :P(1 - ,')'l7l.Q indicates the Mercier criterion andthe five level surfaces of Q are drown with /Dr: 0.1,.
The horizontal axis denotes the vacuum axis shift /rr.
The standard LHD configuration corresponds to zlu :-15 cm. The standard LHD configuration is slightly
Mercier unstable, however, it is practically stable upto

7: 1(8, 0r, a)(2)Note that ,1 becomes independent of a in axisymmetrictokamaks. Therefore, properties of /. inherent to 3-D

equilibria appear throuth their a-dependence, whichcomes from the local magnetic curvature because thelocal magnetic curvature consists of two componenm,
namely, one part due to toroidicity and the other partdue to helicity. In the case of LHD, the locally unfavor-able curvatures due to toroidicity and helicity aresuperposed in the outside of the torus on the horizontally elongated poloidal cross sections, however, on thevertically elongated poloidal cross sections the locallyunfavorable curvature due to toroidicity is cancelled by
the local favorable curvature due to helicity leading tolocally favorable curvature even in the outside of thetorus. These properties of local magnetic curvature andthe disappearance of the local magnetic shear due tothe large Shafranov shift make high-mode-number ballooning modes unstable in the region with the stellara-

tor{ike global magnetic shear. Figure 2 shows bothschematic level surfaces of the unstable eigenvalue ,1, ofx

high-mode-number ballooning modes in (1l, 0u, a)space (in the left column) and schematic positions
where finite-mode-number ballooning modes would
occur (in the right column) for tokamaks (in the toprow), for strongly Mercier unstable 3-D equilibria (inthe middle row), and for slightly Mercier unstable orcompletely Mercier stable 3-D equilibria (in the bottom.!2xo6oorow), respectively. In the axisymmetric tokamaks, theAv (cm)level surfaces of unstable ,1 of high-mode-number ballooning modes are topologically cylindrical because L isindependent of a. In the strongly Mercier unstable 3-DFig. 1 Mercier stability diagram in the LHD configuration.

Dotted area corresponds to unstable region.

Dashed lines in this region are contours oI el7l.

equilibria, such tokamak-like cylindrical level surfaces
77

Nakajima N., MHD PhYsics on LHD

I /'----\IHtll||l{-t:}--- \'l sI l'--- ,)I \. ->)

d'\+-,- \"lolo\\.a----=.
dr\Jvl\Fig.3 Radial structure of the finite-mode-number
looning mode.

left to right, n: -56, -66, -76, and -86' From theseresults, we can expect that the finite-mode-number
modes leading to high-mode-number modes with thespheroidal level surfaces of the eigenvalues have quite
so large and many poloidal and toroidal mode numbersthat they localize in the toroidal direction as well as inFig.2 Schematic pictures of level surfaces of unstableeigenvalues (in the left column) and positions
where ballooning modes occur (in the right column)
for three types of equilibria.

and spheroidal level surfaces coexist, and only spheroidal level surfaces exist in the slightly Mercier unstableor completely Mercier stable 3-D equilibria' The highmode-number ballooning modes with the spheroidal

the poloidal direction.

5. Summary
The Large Helical Device (LHD) has a flexible coilsystem making various vacuum configurations. In thecase of the planar axis vacuum configurations, the inward shift of the magnetic axis creates more straightlevel surfaces are inherent to 3-D equilibria. As isshown in the right column of. Fig. 2, finite-mode-number modes leading to high-mode-number modes withcylindrical level surfaces of. )u are considered to occur
outside of the torus without localization in the toroidal
direction. In contrast with them, finite-mode-number
modes leading to high-mode-number modes with spheroidal level surfaces of,t are considered to localize inthe locally unfavorable region of the outside of thetorus. To confirm the properties of such ballooning
modes expected from the high-mode-number analysis,
the global mode analysis with finite-mode-number isbeing performed by using CAS3D stability code[l0]'where a strongly Mercier unstable equilibrium is used.
When the Fourier modes of the perturbation are se-helical-like properties, and the outward shift enhances

toroidal effects. The essential characteristics of MHD
equilibria of such planar axis vacuum configurations aredescribed by nearly axisymmetric large Shafranov shift(toroidicity dominant) in the core region and propertiesof the straight helical system (helicity dominant) near

the periphery. The net toroidal current and pressure
profiles mainly determined by the experimental condi
tions strongly influence on the Shafranov shift. Since
the Mercier stability of LHD is established by both themagnetic well in the core region and stellarator{ikemagnetic shear near the periphery, the negative or zero
toroidal currents and peaked pressure profiles are favorable for the Mercier stability. High-mode-number
ballooning modes are still destabilized by the disappearance of the local magnetic shear in the region with stellarator-like maguetic shear, even if the Mercier crite-

lected as centered in low toroidal mode numbers(n<< M : 10), only interchange modes appear- For
medium toroidal mode numbers (n-M1, takamak-likeballooning modes appear with only one dominant toroidal mode number. As the toroidal mode numbers areincreased more and more (n) i4), ballooning modes
with both poloidal and toroidal mode couplings appear.
One example is shown in Fig. 3 where four group exist
with the different toroidal mode numbers, namely, fromrion is satisfied. Those modes inherent to helical systems are expected to highly localize in the toroidal di-rection as well as the poloidal direction and to78

Nakajima N., MHD Physics on LHD

become harmless by the kinetic effects such as theFinite Larmor Radius (FLR).To clarify FLR effects onsuch ballooning modes, the dominant poloidal and toroidal mode numbers are being examined from global

mode analysis.
The analysis of pressure-driven resistive modes isanother important issue of MHD physics. Since themagnetic hill always exist in the plasma periphery, thepeaked pressure profiles are better for those modes aswell as ideal MHD modes. Kinetic and dissipative effects on MHD modes will be continuously examined inorder to clari! the MHD physics of LHD.

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