J.
6 (2004)
J. Plasma
Plasma Fusion
FusionRes.
Res.SERIES,SERIES,Vol.
Vol.
6 (2004)45–50
000–000

Growth Rates and Structures of MHD Modesin Stellarator/Heliotron
NAKAJIMA Noriyoshi, NÜHRENBERG Carolin1 and NÜHRENBERG Jürgen1
National Institute for Fusion Science, Toki, 509-5292, Japan
Max-Planck-Institut für Plasmaphysik, D-17491 Greifswald, Germany

1(Received: 21 January 2004 / Accepted: 11 March 2004)

Abstract
Recently, high-b plasmas with ·b Ò ~ 3 % have been established in two different types of helical devices, namely,
Large Helical Device (LHD) and Wendelstein 7 -AS (W7-AS). These devices are designed with complementary physics
concepts. The purpose of the present work is to theoretically understand the properties of ideal Magneto-HydroDynamics (MHD) instabilities in those high-b plasmas and to try to explain the experimental results on a common
theoretical basis, namely, the linearized ideal MHD global mode analyses using compressible perturbations. Physicallycorrect growth rates and mode structures make the direct comparison between theory and experiment possible.
Keywords:
ideal MHD, growth rate, stellarator, heliotron

1. Introduction

condition, so that physical growth rates and eigenfunctions
can be correctly evaluated.
Recently, high- b plasmas with ·b Ò ~ 3 % have been
established in two different types of helical devices, namely,
LHD [1] and W7-AS [2]. These devices are designed with socomplementary physics concepts as to make two maindifferent lines in researches of the three dimensional
configurations. In the case of LHD, high b plasmas are
established in the inward-shifted configurations with thevacuum magnetic axis Rax of 3.6 m, where pressure-driven
modes are theoretically more unstable compared with thosein the standard configuration with Rax = 3.75 m or in theoutward-shifted configurations with Rax ~ 3.9 m, when the
MHD equilibria with same pressure and current conditions
are used by only changing the plasma boundary shape. At
present, the typical high-b MHD equilibria in the inwardshifted configurations are not rigorously fixed from theexperimental point of view, thus, theoretically chosen MHD
equilibria are used in this work.
The second line of stellarators is represented by W7-AS
and by W7-X. In W7-AS, the maximum b was obtained with
B = 0.9 T, ·b Ò ~ 3.2 % with a flat top time of ~ 0.35 s. Above·b Ò ~ 2.4 % the Mirnov diagnostic shows the discharge to bequiescent. In the computational reconstruction of such a discharge (#51755), the rotational transform was found to vary
substantially. In keeping with the experimental measurements,
the low-node-number global ideal MHD modes (dominantly
n = 1) have computationally been found unstable up to ·b Ò ~In order to investigate the ideal MHD stability in stellarator/heliotron, linearized ideal MHD stability analyses
have been performed byassuming that the perturbations orÆdisplacement vectors x in the energy principle form areincompressible, since the stability criterion of pressuredriven modes with resonant rational surfaces, determined
only by the sign of the potential energy, does not change
whether the perturbations are incompressible or not.
Moreover, such incompressible analyses have been performed under the fixed boundary condition, namely the condition
that the normal displacement x s is not allowed on theplasma-vacuum boundary. These incompressible analyses
under the fixed boundary condition are not so time
consuming that they are suitable to the fast analyses of thereconstructed MHD equilibria from the experimental data or
analyses based on the theoretically prepared MHD equilibrium data base. In those incompressible stability analyses,
however, the growth rates do not have correct physicalmeaning and the profiles of eigenfunctions are not exact,
because the component of the displacement vector parallelto the equilibrium magnetic field x || is not included, or more
physically, the branch of the slow magnetosonic is notincluded. Therefore, in order to compare between theoreticaland experimental results, to be done firstly from the theoretical point of view is to perform the MHD stability analyses
avoiding a possible discrepancy between theoretical and experimental results. For such a purpose, here, ideal compressible MHD analyses are performed under the free boundary©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: nakajima@nifs.ac.jp
145Nakajima N. et al., Growth Rates and Structures of MHD Modes in Stellarator/Heliotron(dotted) lines, with the location of the dominant modes shownby open rectangles (free boundary) and open circles (fixedboundary), and the location of the dominant low-m rationalsurfaces, namely i = 0.4, 0.5, 0.6, 0.75 and 1.0. Since the
MHD equilibria are Mercier unstable, the toroidal mode
coupling inherent to helical systems is so weak [5,6] that thetoroidal mode number n can be used as a good quantum
number. Therefore, to distinguish the unstable eigenvalues by
the radial node number is also possible for both interchange
modes consisting of multiple Fourier modes and ballooning
modes. It is understood that 1) the unstable region moves
from plasma core to plasma periphery as b increases, 2) thelower the toroidal mode number n is, the wider the radialwidth is, and also this tendency becomes clear in low-b
plasmas, 3) the free boundary motions become important for
low-n modes with a global structure under the fixed boundary
condition, and/or for high-b plasmas, and 4) the unstableradial region does not depend on the toroidal mode number n
so much, leading to the simultaneous excitation of many
linearly independent modes around same radial location. For
more detail, see Fig. 3. The movement of the unstable region
is correlated with the magnetic well formation by theShafranov shift. When ·b Ò increases more, the Mercier stableregion appears in the plasma core region. This tendency is

2.4 %. Computationally obtained, maximum physical, growthrate is 14 kHz. Because interchange stability prevails in W7X, the present work clarifies the role of local ballooning
stability b limit.

2. Properties of pressure-driven modes ininward-shifted LHD configurations
To understand the overall properties of pressure-driven
modes in inward-shifted LHD configurations, three MHD
equilibria with ·b Ò = 1 %, 2 % and 3 % are created underboth fixed boundary and currentless conditions by using
VMEC code [3], where 300 radial meshes, and 7 poloidal
and 12 toroidal Fourier modes are used, respectively. The
used plasma boundary is so small that the plasma volume isaround 25 m3 and the rotational transform at the plasmaboundary is around 1.36. The used pressure profile is P (s) =P (0)(1 – s)(1 – s9), where s is the normalized toroidal flux,
which is related to the normalized minor radius r as r = ÷ s .
The rotational transform i and Mercier index DI for ·b Ò =1 %, 2 % and 3 % are shown in Fig. 1. The i almost
monotonically increases in any cases, and all equilibria are
Mercier-unstable except for plasma periphery. The MHD
equilibria are calculated under the fixed boundary condition,
so that the stabilizing effects by the free boundary motion of
the plasma boundary are reduced, leading to the increase of
D I as b increases. Note that it is considered that thetheoretically chosen MHD equilibria, used here, are moreunstable than experimentally observed ones.
The ideal MHD stability analyses for compressible perturbations are performed by the cas3d3 code [4], where 300
radial meshes, 20 poloidal and 20 toroidal Fourier modes in
the Boozer coordinates are used for the mapping ofequilibrium. As a perturbation,
60 Fourier modes are used toÆexpress each component in x^, the component perpendicular
to the equilibrium magnetic field, and 490 Fourier modes areused for x ||. Note that quite large Fourier space is needed tocorrectly express the incompressible properties of the perturbations in the compressible calculations [4]. The radialwidth of the most unstable mode for each toroidal mode
number n is shown in Fig. 2, where the radial width is roughly
defined as the region that the amplitude of the eigenfucntion
is larger than 70 % of the peak value. The radial width forthe free (fixed) boundary condition is indicated by the solidFig. 2

Fig. 1

Radial profiles of i and DI in inward-shifted LHD
plasmas. The solid, short dotted, and long dotted lines
correspond to ·b Ò = 1 %, 2 %, and 3 % respectively.

2
46

Radial width of unstable modes vs toroidal mode
number n in inward-shifted LHD with ·b Ò = 1 %, 2 %,and 3 %. The solid (dotted) lines correspond to free(fixed) boundary condition, and Open rectangles (circles) indicate the location of the dominant Fouriermodes for free (fixed) boundary condition. The locationof the dominant low-m rational surfaces is also shownby vertical lines with the value of i.

Nakajima N. et al., Growth Rates and Structures of MHD Modes in Stellarator/Heliotron

creases, 2) up to around 3 %, the growth rates for a specified
toroidal mode number do not change so much even if b -valuechanges, and for typical high-b LHD operation parameters,
the order of the growth rates is around 40 msec (g tA0 = 0.1
corresponds to around 40 msec), 3) the growth rates do notdepend on the toroidal mode number so much, and so taking
account of the the radial positions of the unstable modes inFig. 2, many modes may be simultaneously excited aroundthe same radial location, and 4) for high-b equilibria, freeboundary analyses are inevitable. As an example, the normal
displacement x s of n = 3 mode for ·b Ò = 3 % is shown in
Fig. 4 for both free and fixed boundary conditions. It is
understood that when the free boundary motion of theperturbation is allowed on the plasma-vacuum boundary, theFourier modes near the plasma periphery are excited, leading
to a more global profile. Another interesting point is that theexperimentally observed modes (m, n) = (2, 3) [7], whose
resonant surface is outside of the plasma, is excited near theplasma periphery.
The radial distributions of the potential energy WP and
kinetic energy WK are shown in Fig. 5 and Fig. 6, respectively,
for the ballooning mode under free boundary condition in Fig.
4. The potential energy is divided into the shear Alfvén term
noted by 1, the fast magnetosonic term by 2 (quite small),the slow magnetosonic term by 3 (magnified 5 times), thepressure-driven term by 4, the current-driven term by 5. WP
itself is indicated by 6. The kinetic energy is divided into two
x || (Wk (x ||)) indicated by 1, and the otherparts: oneÆis due toÆis due to x^ (WK (x^)) by 2. WK itself is indicated by 3. The
kinetic energy WK of the unstable modes mainly comes fromthe component of the displacement vector parallel to theÆequilibrium magnetic field x ||, namely Wk (x ||) >~ WK (x^). Thisis due to the fact that the incompressibility condition is almost
satisfied except for the mode rational surfaces as shown in
Fig. 5, so that a strong x || is created around the mode rationalsurfaces as shown in Fig. 6. This tendency becomes stronger
for interchange modes with an almost uniform amplitude of
the perturbed pressure along the equilibrium magnetic field
line, compared with ballooning modes where the amplitudeof the perturbed pressure is non-uniform along the equilibriumÆmagnetic field line. In the present analyses,
Wk (x ||)/WK (x^) ~Æ
4 for interchange modes, and Wk (x ||)/WK (x^) >~ 1 for ballooning modes. As b increases or modes become more unstable,

already reflected on the movement of the unstable region. In
some cases, the dominant modes changes from fixedboundary condition to free boundary condition, say, n = 2
and 4. Note that in these equilibria, the rational surface withi = 0.5 always exists inside of the plasma, leading to theglobal modes with (m, n) = (2, 1) even for low-b plasma, andthat the dominant poloidal mode numbers are fairly large forn > 5.
Figure 3 shows the growth rate normalized by the Alfvén
transit time on the magnetic axis: g tA0 and the type of modes.
For low-b case (·b Ò = 1 %), all unstable modes are interchange modes with only single dominant Fourier mode.
Except for (m, n) = (2, 1), all modes are localized modes as
is also understood from Fig. 2. As b increases, such interchange modes with single dominant Fourier mode change into
localized interchange modes consisting of multiple Fouriercomponents except for (m, n) = (2, 1) and (m, n) = (4, 2) for,say, ·b Ò = 2 %. And finally, almost all modes except for lown modes change into ballooning modes. There are several
important features: 1) there is a case that low-n global interchange modes under the fixed boundary condition change into
ballooning modes under the free boundary condition as b in-

Fig. 3

Normalized growth rates and type of modes vs toroidal
mode number n in inward-shifted LHD with ·b Ò = 1 %, 2%, and 3 %. The left and right columns correspond to
fixed and free boundary condition, respectively. In theupper two figures, the open circles, rectangles, triangles denote interchange modes, and the solid
triangles indicate ballooning modes. In the lower two
figures, s–I and m–I means interchange modes
consisting of single dominant Fourier mode andmultiple Fourier modes, respectively, and B means aballooning mode. Attached global (local) indicates a
global (local) mode.

Fig. 4

3
47

Radial profile of x s for n = 3 under fixed (left) and free(right) boundary condition in LHD with ·b Ò = 3 %. The
attached numbers indicate the poloidal mode numbers.

Nakajima N. et al., Growth Rates and Structures of MHD Modes in Stellarator/Heliotron

Fig. 7

Fig. 5Flux-surface cross-sections for the W7-AS discharge#51755 at ·b Ò = 3 % (NEMEC by S.P. Hirshman, calculation: courtesy J. Geiger).

cross-sections at ·b Ò ª 3.1 % are shown in Fig. 7. The rotational transform i changes significantly with the plasma-b :from monotonically increasing towards the plasma edge andi < 1/2 at b(0) ª 0 to monotonically decreasing with i = 1/2
inside the plasma for ·b Ò > 1 %. Also, i steepens with increasing b . In the global stability analyses (CAS3D, 101
radial grid points, 64 perturbation harmonics, free-boundary,finite adiabatic index 5/3, non-uniform equilibrium massdensity profile) the m = 2, n = 1 perturbation harmonics
mostly dominate (see Fig. 8 for normal displacements
harmonics at ·b Ò ª 0.84 %). A b -scan of the growth rates isgiven in Fig. 9; here, the electron number density was at ne (0)ª 2.4 ¥ 1020 m–3 at ·b Ò ª 3.1 % (comm. by A. Weller). The
growth-rate peak at ·b Ò ª 2 % is due to the occurrence of i =5/11 just outside the plasma boundary. The global mode
analysis finds stability beyond ·b Ò ª 2.3 %, which is in
keeping with the measurements that also see very little MHD
activity beyond t = 0.24 s or ·b Ò ª 2.3 % [2].

Radial profile of potential energy corresponding to Fig.
4. Slow magnetosonic term denoted by 3 is magnified
5 times.

4. Standard high-mirror W7-X

Fig. 6 Radial profile of kinetic energy corresponding to Fig. 4.

Global ballooning modes have been studied in the W7X standard high-mirror case (see Ref. [8] for the plasmaboundary and Fig. 10 for the flux-surface cross-sections at·b Ò ª 5 %) in order to understand the meaning of the theoretical local ballooning limit, which is ·b Ò ª 5 %. For the high—mirror cases, |Bmax – Bmin|/(2B0) ª 0.1 is the mirror field on
the magnetic axis. With b = 0.08 ¥ T [kV]n[1020]/B02[T], anumber density n (0) = 3 ¥ 1020 m–3, a temperature of T(0) =3 kV, and the magnetic field B0(0) = 2.5 T correspond to b (0)= 12 % or ·b Ò = 5 % for the profile used. In W7-X equilibria
the rotational transform typically changes only very little withincreasing b , for the standard case 5/6 < i < 5/5. All theequilibria studied here are locally Mercier stable. In Fig. 11,
the normal-displacement harmonics are shown for an unstable
perturbation at ·b Ò = 6.3 %. Stellarator-type coupling isimportant [(m = 14, n = 12) and (m = 20, n = 17)]. With its
many side-bands and the maximum-amplitude regions on theoutside of the torus, the perturbation is manifestly ballooningtype (101 radial points, 280 mode harmonics). The b -scan of
the growth-rates (compare Fig. 12) shows that all global
stability limits are above ·b Ò = 5 %, the higher poloidal-nodenumber perturbations (m = 55) being closest. With low-nodenumber global perturbations (m = 14) a global stability limitof ·b Ò = 6 % is found, which is above the local ballooning
limit.

Æthe contribution by x || becomes same as that by x^. Note that,in usual incompressible treatments, x || component is eliminated in both linear and nonlinear calculations.
In the incompressible perturbations, the counter parts of
the unstable modes are shear Alfvén modes, since the slow
magnetosonic waves are excluded. In contrast with it, thecounter parts of the unstable modes become slow magnetosonic waves for the compressible perturbations, since theshear Alfvén continuum spectrum moves up from themarginal point [4]. These properties in the stable side of theideal MHD spectrum will influence on the point spectra like
Toroidicity-induced Alfvén Eigenmodes, and Helicity-induced
Alfvén Eigenmodes in LHD configuration.

3. High-b W7-AS discharge
A series of equilibria representing the W7-AS discharge#51755 has been studied for its ideal MHD properties using
the CAS3D [8] global stability code. All equilibrium properties referred to below originate from the computational
reconstruction (NEMEC code by S.P. Hirshman, calculation:
courtesy J. Geiger) and match the experimental measurements
according to the quality of the reconstruction. In the discharge
W7-AS #51755 ·b Ò ª 3.1 % was achieved with b (0) ª 7 %for a flat top time of > 0.1 s [2]. The VMEC flux-surface
448

Nakajima N. et al., Growth Rates and Structures of MHD Modes in Stellarator/Heliotron

Fig. 8Fig. 11 CAS3D normal-displacement harmonics of an unstablelow-m perturbation in the standard high-mirror W7-X at·b Ò = 6.3 %.

CAS3D normal-displacement harmonics of an unstableessentially m = 2 perturbation versus s in W7-AS#51755 at ·b Ò = 0.84 %.

Fig. 12 Plasma-b scan of MHD growth rates, g , for the W7-X
standard high-mirror case.
Fig. 9Plasma-b scan of MHD growth rates, g , in W7-AS
discharge #51755. The inset shows perturbed pressure
contours (j = 4 , essentially m = 2, n = 1, ·b Ò = 0.84 %,compare Fig. 7).

MHD stability theory, as it is, is useful or not. If there still
were discrepancies between theory and experiment forreconstructed MHD equilibria, then we have to consider the
non-ideal effects like two-fluids effects, nonlinear effects andso on. Moreover, we have to answer why ideal MHD theory
is not applicable to inward-shifted LHD configurations,because there are fairly good consistency between theory andexperimental results in W7-AS as shown here and tokamaks.
Related to the discrepancy, if any, the treatment of thestochastic region surrounding the plasma in the nested flux
surface region is also important issue. In the case of
tokamaks, there is no such a stochastic region basically, andin W7-AS the connection length in the stochastic region isso short compared with LHD that such a stochastic region istreated as vacuum region. In the case of LHD configurations,however, the connection length in the stochastic region is solong compared with parallel mean free path of particles that
there is a possibility that a force free ideal or resistive plasma
exists there, which leads to stabilizing effects compared withvacuum. In order to clarify above problems, comparative
studies among LHD, W7-AS, W7-X and tokamaks are quite
useful, which leads to deeper understanding of the ideal
MHD theory in torus systems.

Fig. 10 Flux-surface cross-sections for the W7-X standard highmirror case at ·b Ò = 5 % [2].

5. Discussions
It is thought from results on inward-shifted LHD configurations that used fixed boundary MHD equilibria areconsiderably more unstable than experimentally observed
plasmas. Therefore, more detail analyses of ideal MHD
instabilities in inward-shifted LHD configurations should be
performed by using the reconstructed MHD equilibria basedon the experimental data, in order to clarify whether ideal
549

Nakajima N. et al., Growth Rates and Structures of MHD Modes in Stellarator/Heliotron

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[7] S. Sakakibara et al., Plasma Phys. Control. Fusion 44,
A217 (2002).
[8] C. Nührenberg, Phys. Plasmas 3, 2401 (1996).

[1] O. Motojima et al., Nucl. Fusion 43, 1674 (2003).
[2] A. Weller et al., Proc. 13th ISW Canberra at http://www.
rsphysse.anu.edu.au/admin/stellarator/[3] S.P. Hirshman, Phys. Fluids 26, 3553 (1983).
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