J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)349–352
000–000

Low m/n Mode Behavior of MHD Plasma in LHD
MIURA Hideaki and HAYASHI Takaya
National Institute for Fusion Science, Toki 509-5292, Japan(Received: 9 December 2003 / Accepted: 22 January 2004)

Abstract
Behaviors of low poloidal (m) and toroidal (n) Fourier modes in the Large Helical Device (LHD) are investigatedby means of direct numerical simulations (DNS) of fully three-dimensional, nonlinear and compressible
magnetohydrodynamics (MHD) equations. Starting from an ideal equilibrium with the position of vacuum magneticaxis Rax = 3.6 m and β0 = 4% finite pressure, a m/n = 2/1 mode grows in the DNS. Fluid motions on poloidal sections
are governed by the two pairs of anti-parallel vortex pairs associated with the m/n = 2/1 modes. The vortex pairs
transport plasma pressure from the core to edge region and bring about large pressure deformations. It is also shownthat the toroidal part in the kinetic energy and the enstrophy are comparable to the poloidal parts of them. The
numerical results demonstrate importance of investigating three-dimensional behaviors of MHD plasmas in LHD.
Keywords:
magnetohydrodynamic, direct numerical simulation, m/n = 2/1 mode, vortex pair

1. Introduction

fluid motions.

Understanding MHD behaviors is one of the mostimportant subjects to understand physics of plasma
confinment in LHD. MHD fluids in LHD are exposed tovarious physical mechanism such as the steep pressure
gradient and the curvature effects of the magnetic field lines.
These mechanism have been expected to cause MHD
instabilities when so-called inward-shifted magnetic fieldconfiguration is adopted. However, recent experiments with
Rax = 3.6 m (inward-shifted) vacuum magnetic axis revealed
that the plasma was confined very well in spite of theMercier-unstable nature of the magnetic field [1]. It isreported in the article that, even though an m/n = 2/1 MHD
activity appeared when the averaged beta 〈β 〉 stayed for 1%< 〈β 〉 < 2.2%, it did not destroy the confinment. It suggeststhat there should be some stabilization mechanism which
suppress instabilities.
In order to investigate MHD behaviors in full threedimensional (3D) geometry of LHD, direct numericalsimulations (DNS) of MHD equations is one of the best
approaches. In this article, we conduct DNS of full 3D,
compressible, dissipative and nonlinear MHD equations
which start from an equillibrium with Rax = 3.6 m and reporton low-n behaviors of the MHD plasmas. Though the fulltorus evolution of the MHD plasma with the same initial
equilibrium is partially reported in the previous work [2], only
a few aspects of the numerical results are described anddetailed fluid motions are not presented there. In this article,
the simulation results are reported with special attention to2. Direct numerical simulations
The MHD equations are described in the helical-toroidal
coordinate system. Detailed information about our numericalcode is reported in Ref. 3. The number of grid points is 97 ×
97 on poloidal sections and 640 in the toroidal direction. The
MHD fluid is assumed to obey to the equation of an ideal
gas with the ratio of specific heats γ = 1.4. The nondimensional parameters of the MHD equations are the heat
conductivity κ = 1 × 10-6, the resistivity µ = 1 × 10-5 and theviscosity ν = 2 × 10-3.
The initial condition is provided by making use of theHINT code [4]. The HINT computation gives a 3D MHDequilibrium in a helical system. In our previous work, anMHD equilibrium in the LHD system with Rax = 3.6 m and
β0 = 4% was obtained [5]. Starting from the equilibrium, theplasma is dominated by the resistive ballooning instability if
the system is under the stellarator (half-pitch) symmetry. Herewe investigate behaviors of the MHD plasma starting fromthe same equilibrium but without the half-pitch symmetry.
In Fig. 1, the time evolution of the kinetic energy 〈k〉 =〈ρ |v|2/2〉 is shown, where v = (v1, v2, v3) is the velocity vector
and the brackets 〈·〉 represent the volume average. The kinetic
energy begins to grow rapidly at t ~– 200 τA. At t ~– 200 τA, weobserve ballooning-like fluctuations on horizontally-elongatedpoloidal sections. In Fig. 2, the pressure profile on ahorizontally-elongated poloidal section is shown at t = 180,©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: miura@toki.theory.nifs.ac.jp1349

Miura H. et al., Low m/n Mode Behavior of MHD Plasma in LHD

280, 350, 500, 600 and 700 τA. The pressure is higher for
darker shades. The two thick solid ellipses represent magnetic
surfaces with the rotational transform ι /2π = 0.5 and 0.66 at
the initial time, respectively. Thin lines represent streamlines
depicted with the poloidal components of the velocity vector
vpol = (v 1, v 2). As time evolves from the initial state tot = 180, 280 and 350 τA, the two anti-parallel vortex pairs
associated with the m/n = 2/1 mode are strongly excited.
Because of the mutual advection of the anti-parallel vortex
pairs, the high-pressure regions are divided into two groups,advected and stretched toward the edge region. Consequently,
two mushroom-like structures are formed on poloidalsections. The formation of the mushroom structures leads togeneration of the secondary vortex structures, as are observed
at t = 500 τA. After t = 500 τA, the vortices become weaker
and the mushroom structures become ambiguous. The plasma
finaly forms a broad pressure profile at the core and relaxes
to inactive state t = 700 τA.
Now we should pay attentions to the point thatstreamlines in Fig. 2 have sinks or sources. Thoughstreamlines do not represent strength of streams directly, thedense convergent streamlines suggest that there are strong
sinks or sources there. The sinks or sources of the twodimensional stremlines come either from three-dimensionality
or from compressibility of the velocity field. In Fig. 3(a), themean kinetic energy 〈k〉 = 〈ρ | v|2/2〉 is compared to its twodimensional contribution 〈kpol〉 = 〈ρ | vpol |2/2〉. We find that〈kpol〉 (dotted line) is as large as one-half of 〈k〉 (solid line) at
most. It implies that the toroidal component of the velocity

1e-06

1e-07

1e-08

1e-09

1e-10
0

100

200

300

400

500

600

700

800t

Fig. 1 Time evolution of the kinetic energy.

Fig. 2 Pressure and streamlines on a poloidal section at a few time snapshots.

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Miura H. et al., Low m/n Mode Behavior of MHD Plasma in LHD

Fig. 4Fig. 3(a) Time evolutions of mean values of the threedimensional velocity magnitude 〈k 〉 (solid line) and thetwo-dimensional magnitude 〈kpol 〉 (dashed line). (b)Time evolutions of the enstrophy 〈Q 〉 (solid line), its
toroidal component 〈Qtol 〉 (dashed line) and the meansquared dilatation 〈(∇· v)2〉.

(a) Isosurface of the pressure. (b) Isosurfaces of thetoroidal vorticity ω tol . Dark and light shades represent
positive and negative signs of ω tol , respectively.

(typically n = 10) modes, although m/n = 2/1 is the mostdominant mode. The three-dimensional structures of thetoroidal vorticity ω tol in Fig. 4(b) are more complicated thanthose of p. We find that dark- and light-shaded isosurfaces(for positive and negative ω 3, respectively) come to the innerside of the torus alternately ten times. It means that n = 10modes have finite amplitudes. It is quite reasonable toconsider that the growth of multiple modes is one of themechanism which brings about (nonlinear) saturation of thekinetic energy growth in Fig. 1.

v 3 occupies a significant part of the kinetic energy. InFig. 3(b), the mean enstrophy 〈Q〉 = 〈| ω |2/2〉 is compared tothe toroidal component of the vorticity 〈Qtol〉 = 〈ω 3ω 3/2〉,
where ω = ∇ × v is the vorticity and ω 3 (ω 3) is covariant(contravariant) component (calculated by v1 and v2) of thevorticity vector. We find that 〈Qtol〉 occupies 2/3 of 〈Q〉 atmost. These two comparisons indicate that the fluid motions
are fully three-dimensional and that the toroidal motions of
the plasma is as important as those on poloidal sections. InFig. 3(b), the mean squared dilatation 〈(∇ · v)2〉 is comparedto 〈Q〉 and 〈Q tol〉. It is clear that 〈(∇ · v)2〉 (dashed line) is
quite smaller than the other two quantities. It indicates that
direct contributions of compressibility to the fluid motions
are very small, though there can be some indirectcompressibility effectsto fluid motions. (Note that, in neutral
fluid turbulence, small compressibility can cause significant
changes in vortex structures [5].)Three-dimensional natures of fluid motions are moreclearly seen in isosurfaces of the pressure p in Fig. 4(a) andω 3 in Fig. 4(b). A clear m/n = 2/1 structure is observed inFig. 4(a). However, the isosurfaces are strongly distorted near
the horizontally-elongated cross-sections. It suggests that thestructure is not very simple but consists of some other

3. Summary
We have conducted fully toroidal DNS of MHD plasma
in LHD and observed growth of m/n = 2/1 modes. The m/n =2/1 anti-parallel vortex pairs transport plasma pressure fromthe core to edge region, and are relaxed to an inactive state.
Though toroidal motions have not been investigated very
much, our numerical results suggest that toroidal motions are
as important as poloidal motions (vortex structures) forpressure transports in MHD. Detailed analysis on growth,saturation and relaxation of m/n = 2/1 modes and pressuretransport associated with the m/n = 2/1 mode will be reported
elsewhere.

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[2] H. Harafuji et al., J. Comp. Phys. 81 169, (1989).
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Miura H. et al., Low m/n Mode Behavior of MHD Plasma in LHD[5] H. Miura, in Proc. IUTAM Symposium on Reynolds
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[4] H. Miura et al., J. Plasma Fusion Res. SERIES 4 476,
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