J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)214–217
000–000

MHD Stability of Low-n Ideal External Modein Large-Helical-Device Plasma
NARUSHIMA Yoshiro, WATANABE Kiyomasa, SAKAKIBARA Satoru, NAKAJIMA Noriyoshi,
NISHIMURA Kiyohiko, YAMADA Hiroshi, YAMAZAKI Kozo,
LHD experimental group and COOPER Wilfred A.1
National Institute for Fusion Science, Toki, 509-5292, Japan
1
Centre de Recherches en Physique des Plasmas,
Association Euratom / Confederartion Suisse, EPFL, 1015 Lausanne, Switzerland(Received: 10 December 2003 / Accepted: 25 April 2004)

Abstract
The characteristics of low-n ideal external modes in the Large-Helical-Device are studied. The low-n analysis iscarried out with the TERPSICHORE code [W.A. Cooper, Plasma Phys. Control. Fusion 34, 1011 (1992)]. Under
fixed boundary conditions, the low-n instability region is limited within the Mercier unstable region and the external
mode does not appear. On the other hand, with free boundary conditions, the low-n unstable region spreads and theexternal modes become unstable. The effects of a conducting wall similar in shape to that of the plasma show thatstability to external modes of higher growth rate can be obtained even with the wall further away from the plasma.
Keywords:
ideal MHD, LHD, external mode, conducting wall, TERPSICHORE

1. Introduction

in Sec. 3, and Section 4 will summarize these discussions.

Recent magnetohydrodynamics (MHD) studies in theLarge Helical Device (LHD) experiment show that MHD
activity clearly appears around the peripheral region of highβ plasmas [1]. The LHD plasma is generated and stably
maintained even in Mercier unstable regions. This shows thatthe high-n mode does not affect the LHD plasma (Here, n is
the toroidal mode number). The magnetic measurement
clearly indicates that the low-n mode fluctuation originatedfrom a pressure gradient driven instability. In general the lown interchange mode structures spread across to the wholeplasma, while the high-n mode structures are narrow, whichmeans that the low-n mode has much more spatial effect on
plasma. Therefore, we focus on the low-n mode in this study.
Although serious MHD instabilities leading to thetermination of the discharge have not been observed, it isworthwhile to investigate the characteristics of the MHD
activities in the LHD plasma theoretically and experimentally.
In this study, we focus on external modes (particularly m/n =1/1 mode) which occurs near the peripheral region in LHD.
A low-n ideal MHD analysis is performed for the n = 1 modefamily in the LHD plasma using the three-dimensional
stability code TERPSICHORE [2]. We will show theequilibrium and the method of analysis in the following
section. The characteristics of the unstable region and themode structures affected by the conducting wall are examined

2. Stability analysis and equilibrium
2.1 Equilibrium
The equilibria of LHD plasma are calculated with VMEC
[3] with free boundary conditions. The pressure profile p(ρ) isassumed to be p(ρ) = p0(1 – ρ2)0.5(1 – ρ8) here, p0 and ρ are
the plasma pressure at the magnetic axis and normalized
averaged minor radius, respectively. We examine the effect ofthe profile of the rotational transform, because the external
mode is sensitive to the rotational transform at the edge ι(1)/2π [4]. In the equilibrium calculation the various ι(1)/2π are
obtained by changing the position of the plasma surface andβ [5].

2.2 Stability
Because we mainly examine the external mode(particularly m/n = 1/1 mode) in this study, the low-n ideal
MHD analysis is performed for the n = 1 mode family in theLHD plasma using TERPSICHORE, which is able tocalculate the stability under fixed and free boundary
conditions. In the free boundary condition, various wallshapes can be prescribed. To examine the wall effect against
the instabilities, the analysis of the stability is carried out withvarious walls that conform to the shape of the plasma. The©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: narusima@LHD.nifs.ac.jp1214Narushima Y. et al., MHD Stability of Low-n Ideal External Mode in Large-Helical-Device Plasma

Fig. 1 Shapes of the conducting wall and plasma.

Fig. 3

Profile of the rotational transform ι /2π, Mercier
parameter D I and Fourier amplitude under the fixedboundary condition. Solid, dashed and dotted lines
mean the m/n = 2/1 and 1/1 mode respectively.

3. MHD Unstable regionThe VMEC equilibrium points are plotted in the planeof beta vs. rotational transform at the plasma edge ι (1)/2π inFig. 2. The boundary conditions of the fixed (awall = 1.0) andfree (awall = 4.0) are shown in Figs. 2(a) and 2(b), respectively.
The open circles indicate growth rates γ > 10–5, which isdefined as unstable for the n = 1 mode. The solid and dashedcontour lines indicate the Mercier parameter DI = 0 and DI =0.2 at ι (1)/2π = 0.5, respectively.

3.1 Fixed boundaryFig. 2

As shown in Fig. 2(a), the n = 1 modes are destabilized
within the Mercier unstable region with DI (at ι /2π = 0.5) >0.2 regardless of the value of ι (1)/2π . A typical mode structurewith the rotational transform and DI profiles are shown in Fig.
3 for a plasma that has the parameters (β, ι (1)/2π ) = (1.36%,1.1). The m/n = 2/1 mode shows the peak at the resonantsurface of ι /2π = 0.5 within the Mercier unstable region,
which displays the characteristic of an internal mode.
Under the fixed boundary condition, the m/n = 1/1 mode
has a smaller Fourier amplitude than the 2/1 mode.

Unstable region under (a) Fixed and (b) Free boundary
conditions. Closed and open circle means the stableand unstable respectively. The solid and dashedcontour lines indicate the Mercier parameter D I = 0 and
D I = 0.2 at ι /2π = 0.5, respectively.

parameter of the wall shape awall is defined as the ratio of theaverage radius of the wall to the average plasma radius. The
range of awall adopted in this study is 1.0 < awall < 4.0 (Fig.
1). Here, awall = 1.0 is equivalent to the fixed boundarycondition.

3.2 Free boundary
Under the free boundary condition, the low-n unstablearea expands to encompass the part of the Mercier stable
domain with ι (1)/2π < 1.0 which is mostly stable under fixed2215

Narushima Y. et al., MHD Stability of Low-n Ideal External Mode in Large-Helical-Device Plasma

Fig. 5

Low-n unstable region modified by the conductingwall. The open symbols mean unstable. The circle,square, triangle, and star indicate the growth rates of γ> 10–5, γ > 10–3, γ > 2 × 10-3, and γ > 3 × 10–3, respectively.
The dotted, double dot-dashed, dot-dashed, dashed,and solid lines indicate the boundary between thestable and unstable region at each wall position of awall= 1.5, 1.3, 1.2, 1.1, and 1.0 (fixed boundary),
respectively.

In case of ι (1)/2π < 1.0, the expanded unstable region
exists even in the Mercier stable region. The mode structurein this region clearly shows the characteristics of an external
mode (Fig. 4(b)) in which the m/n = 1/1 mode remarkably
appears. According to the toroidal mode coupling effect, them/n = 2/1 component is also destabilized as a nonresonantly
coupled mode when the resonance surface of ι (1)/2π = 0.5
does not exist.

3.3 Effect of the position of the conductingwall

Fig. 4The unstable region and mode structures depend on theposition of the conducting wall. The change of the unstableregion is shown in Fig. 5. The open symbols mean unstable.
The circle, square, triangle, and star indicate the growth rateof γ > 10 –5, γ > 10 –3 , γ > 2 × 10 –3, and γ > 3 × 10 –3 ,respectively. The dotted, double dot-dashed, dot-dashed,dashed, and solid lines indicate the boundary between thestable and unstable region at each wall position of awall = 1.5,1.3, 1.2, 1.1, and 1.0 (fixed boundary), respectively.
The conducting walls change the unstable region when
ι (1)/2π < 1.0. On the other hand, the boundary around thelow-β region is not modified by the wall. The plasmas withhigher growth rate of γ > 10–3 are stabilized with a wall
further away from the plasma. For example, almost all thepoints with γ > 3 × 10–3 (Star) plasmas are stabilized by thewall with awall = 1.3. Fig. 5 shows that the conducting wall
position required to stabilize the external mode depends onthe growth rate. The physical mechanism of this interesting
phenomenon is the future work. The difference of thestabilizing behavior is shown in Fig. 6. The free boundary

Profile of the rotational transform ι /2π, Mercier
parameter D I and Fourier amplitude under the freeboundary condition. (a) ι (1) /2π > 1.0, (b) ι (1) /2π < 1.0.

boundary conditions (Fig. 2(b)). In the region of ι (1)/2π > 1.0,
the low-n unstable area remains in the Mercier unstableregion. Fig. 4(a) shows a typical mode structure in this area,
which is similar to an internal mode under the fixed boundary
condition.
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Narushima Y. et al., MHD Stability of Low-n Ideal External Mode in Large-Helical-Device Plasma

Fig. 6

Relationship between the awall and γ. In lower growth rate (a) γ =1.1 × 10–3, mode structures do not significantly vary with awall .
On the other hand, in higher growth rate (b) γ = 3.3 × 10–3, the amplitude of the m/n = 1/1 mode component decreases incomparison with that of m/n = 2/1 mode component.

condition (awall = 4.0), the plasmas with (β, ι (1)/2π) = (1.9%,0.96) and (2.2%, 0.81) have growth rates of γ = 1.1 × 10–3,and γ = 3.3 × 10–3 respectively. In the former case, the growthrate hardly decreases with the wall and reduces to zero atawall < 1.14 as shown in Fig. 6(a). The mode structures (m/n= 2/1 and 1/1) are also shown in Figs. 6(a)(1)–(3). Thesestructures do not significantly vary with awall. On the otherhand, the latter case (Fig. 6(b)) shows that a wall with larger
effective radius remarkably reduces the growth rate and theplasma is stabilized at awall < 1.4. The wall modifies the mode
structures (Figs. 6(b)(1)–(3)). The amplitude of the m/n = 1/1
mode component decreases in comparison with that of m/n =2/1 mode component.

roughly constant at lower β and γ . The external mode isstabilized by the wall effect. However, the internal mode (m/n = 2/1) is not stabilized.

Acknowledgement
This study was supported by a Grant-in-Aid for Young
Scientists(B)(No.15760627) from the Ministry of Education,Culture, Sports, Science and Technology of Japan.

References[1] S. Sakakibara, H. Yamada, K.Y. Watanabe, Y. Narushima
et al., Plasma Phys. Control. Fusion 44, A217 (2002).
[2] W.A. Cooper, Plasma Phys. Control. Fusion 34, 1011
(1992).
[3] S.P. Hirshman, W.I. Van Rije and P. Merkel, Comput.
Phys. Commun. 43, 143 (1986).
[4] Y. Narushima, S. Sakakibara, K.Y. Watanabe, W.A.
Cooper et al., J. Plasma Fusion Res. SERIES 5, 514(2002).
[5] Y. Narushima, K.Y. Watanabe, S. Sakakibara, N.
Nakajima, W.A. Cooper et al., Proceedings of the 29thEPS Conference on Plasma Physics and Controlled
Fusion Montreux, 17–21 June 2002 ECA 26B, P-1.080(2002) (http://elise.epfl.ch/pdf/P1_080.pdf).

4. Summary
We have mainly examined the stability of external modesin the LHD plasma. The external mode occurs in equilibria
without the resonant surface of ι /2π = 1.0 under the freeboundary condition, which has m/n = 1/1 component. Theinternal mode (m/n = 2/1) remains in the Mercier unstableregion under the fixed boundary condition. The conducting
wall suppresses the external mode. In free boundary
conditions, the growth rate and the mode structure changewith wall positions at higher β and γ value, while they remain

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