J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)241–244
000–000

The Effect of the Aspect Ratio on the External Kink-Ballooning
Instability in High-β Tokamaks
AIBA Nobuyuki, TOKUDA Shinji1, ISHIZAWA Tomoko2 and OKAMOTO Masao3
1

The graduate school for Advanced studies, Toki 509-5292, Japan
Naka Fusion Research Establishment, Japan Atomic Energy Research Institute, Ibaraki 311-0193, Japan
2
Research Organization for Information Science and Technology, Ibaraki 319-11, Japan
3
National Institute for Fusion Science, Toki 509-5292, Japan(Received: 9 December 2003 / Accepted: 8 March 2004)

Abstract
A theory for the stability analysis of ideal external modes is developed with the property of the Newcomb
equation. This theory is also useful for stabilizing resistive wall modes by feedback controls. As an application of thistheory, the effect of the aspect ratio on the ideal magnetohydrodynamic (MHD) stability limits of tokamak plasmas
are examined numerically. The toroidal mode number n = 1 external modes are stabilized by making the aspect ratio
smaller.
Keywords:
newcomb equation, ideal MHD stability, external mode, resistive wall mode, tokamak

1. Introduction

δK + δWp + δWV = 0

Ideal kink-ballooning instability limits the performanceof tokamaks, and resistive wall modes (RWMs) [1] should bestabilized or be controlled in order to realize stationary highperformance tokamaks. RWMs are ideal kink-ballooning
modes that are stable when the plasma is surrounded by aperfect conducting wall, but become unstable due to theresistivity of a wall.
We developed a theory where the change of a potential
energy δW p for external modes in a toroidal system isexpressed in terms of the boundary values by using theproperty of the Newcomb equation [2]. We apply this theory
to analyze the stability of external kink-ballooning modes inhigh- β tokamaks. Emphasis is put on the aspect ratio
dependence of a kink-ballooning stability, which attracts
attention recently in the design research on high-performance
tokamaks.
We first describe the analytical method with the propertyof the Newcomb equation briefly. Next, we show a benchmark
test of MARG2D code [3] comparing with ERATOJ code [4].
The effect of the aspect ratio on the stability of ideal external
modes is investigated in Sec. 4.

(1)where δK, δWp and δWV are the change of a kinetic energy, aplasma potential energy and a vacuum energy, respectively.
The Newcomb equation is the inertia free linear ideal
magnetohydrodynamic (MHD) equation. In the case that theplasma displacement ξ satisfies the Newcomb equation, whichmeans that an equilibrium is marginally stable, δW p isexpressed by the boundary values of an equilibrium. On thebasis of this theory, δWp + δWV can be expressed by thestability matrix [5], A, asδWp [ξ , ξ ] + δWV [ξ , ξ ] = 〈 x | M p | x 〉 + 〈 x | MV | x 〉= 〈x | A | x〉x = ( x − Lf ,..., x Lf )ξ=∑x m Y m (r )(2)mwhere Lf is the maximum poloidal mode number to analyze
the stability, Y m(r) are the basis functions satisfying the
Newcomb equation, and Mp and MV are the matrices forexpressing δWp and δWV by the displacement at a plasma
surface x. MARG2D code constructs Y m(r) by solving the
Newcomb equation, and obtains the matrix A. We analyze thestability of ideal external modes by investigating the sign ofthe minimum eigenvalue of A. This theory is explained
minutely in Ref. [3].

2. External mode theory with theproperty of the Newcomb equation
We analyze the stability of ideal external modes
according to the quadratic form;©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: aiba@tcsc.nifs.ac.jp1241

Aiba N. et al., The Effect of the Aspect Ratio on the External Kink-Ballooning Instability in High-β Tokamaks

This analysis with the stability matrix has advantages forstabilizing resistive wall modes (RWMs) by feedback controls[5]. One reason is that we can analyze even when external
modes are stable. The other reason is that this analysis isexecuted in short time, because M p does not change
approximately when an equilibrium is stable against idealinternal modes.

The shape parameters of equilibria are fixed as the elongation
κ = 1.8, the triangulity δ = 0.45. The profiles of dp/dψ arealso fixed. The toroidal magnetic field at the magnetic axis,
Bt , and the poloidal field current, Ip , are changed as

3. Benchmark test of MARG2D code
We first compare the result obtained with the stability
matrix, calculated by MARG2D code, with that of ERATOJ
code as a benchmark test. The equilibrium has a circularcross-section and the poloidal beta βp is very small, as shownin Fig. 1. The plasma pressure and the safety factor areexpressed as p and q, and s is defined as s = ⎯√ψ , where ψ isthe poloidal flux function normalized to unity at a plasma
surface. In this equilibrium, Only ideal external kink modes
are unstable.
Figure 2 shows the dependence of the stability of n = 1
external kink modes on the safety factor at a plasma surface
qa , where n is the toroidal mode number. The minimum
eigenvalues obtained with MARG2D and ERATOJ are
expressed as µ0 and λ0–ERT , respectively. An equilibrium with
arbitrary value of q a is obtained by σ -scaling [3]. The
marginally stable point obtained with the stability matrix wellagree with that calculated with ERATOJ.
We also compare the eigenfunctions near the marginallystable point, q = 2.9. Both eigenfunctions obtained withMARG2D and ERATOJ, shown in Fig. 3, have the profiles
peaking near a plasma surface. It is a well-known feature ofideal external kink modes in low-β tokamaks [6,7].
From these comparisons, the analysis with the stability
matrix is effective for examining the stability of ideal externalmodes.

Fig. 1Equilibrium for the benchmark test. (a) Cross-section.
(b) Profiles of the pressure and the safety factor. s isdefined as √⎯ψ.

Fig. 2

Dependence of the stability of ideal external kinkmodes on qa. µ0 and λ0–ERT is the minimum eigenvalue
obtained with MARG2D and ERATOJ, respectively.

4. Effect of the aspect ratio on thestability of external modes
As an application of the analysis with the stabilitymatrix, we analyze the effect of the aspect ratio on thestability of n = 1 external modes with the stability matrix.

Fig. 3 Eigenfunctions of ideal external kink modes at qa = 2.9. (a) MARG2D code. (b) ERATOJ code.

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Aiba N. et al., The Effect of the Aspect Ratio on the External Kink-Ballooning Instability in High-β Tokamaks

Fig. 4Equilibrium for analyzing the effect of aspect ratio. Parameters are A = 3.26 and βN = 6.0. (a) Cross-section. (b) Profiles of thepressure and the safety factor.

Fig. 5

Dependence of the βN limit against n = 1 ideal externalkink modes on the aspect ratio. No perfect conducting
wall surrounds the equilibrium.

Bt = Bt 0 ×

Rc,
Rc 0(3)

I p0 ×

Rc 0,
Rc(4)

IpFig. 6

Dependence of the βN limit against n = 1 ideal externalkink modes on the position of a perfect conducting
wall.

Figure 6 shows the dependence of the βN limit against n= 1 ideal external modes on the position of a perfectconducting wall. The position of a conducting wall Rext isnormalized with a. When βN < 7.5, a small aspect ratio
equilibrium is more stable than a large aspect ratio one as in
a no wall case.
The eigenfunctions, when βN = 6.0 and the wall positionis slightly more far from a plasma surface than the marginallystable position, are shown in Fig. 7. In A = 3.26 case, thegrowth rate calculated with ERATOJ is γ 2 = –1.6 × 1.0-5; thatis normalized with the toroidal Alfvén transit time at themagnetic axis. These eigenfunctions have a global mode
structures unlike in a low-β case shown in Fig. 3.
The m = 3 Fourier components obtained with MARG2Din each aspect ratio cases are pointed near the outer q = 3.0
rational surface. However, since the widths of these points
are nearly 0.001 measured in s, a general structure of thewhere Rc is the major radius and the subscript 0 means thevalues when the aspect ratio A = 3.26, as Bt0 = 3.36 and Rc0= 2.93. Ip0 of the normalized beta βN = 5.0 equilibrium is set
to 4.0. Though the q profile varies when A and βN arechanged, q value at s = 0.95 is fixed as q95 = 3.5 by adjusting
Ip . The profiles of the A = 3.26 equilibrium are shown inFig. 4.
We first investigate the dependence of the βN limit on
the aspect ratio against n = 1 ideal external modes without aperfect conducting wall. As shown in Fig. 5, a low aspectratio equilibrium is more stable against n = 1 external modes.
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Aiba N. et al., The Effect of the Aspect Ratio on the External Kink-Ballooning Instability in High-β Tokamaks

Fig. 7

Eigenfunctions when βN = 6.0. (a) A = 2.44 obtained with MARG2D code. (b) A = 3.26 with MARG2D. (c) A = 4.0 with MARG2D.
(d) A = 3.26 obtained with ERATOJ code.

eigenfunction is almost the same as that obtained withERATOJ.

is made small, and the eigenfunction of the most unstable
external mode in high- β tokamaks has a global mode
structure. MARG2D is effective for analyzing the stability ofideal external modes, and for the design research on highperformance tokamaks.

5. Summary
We developed a theory that the change of a potential
energy of a plasma is expressed in terms of the boundary
values when the plasma displacement satisfies the Newcomb
equation. On the basis of this theory, we are able to analyze
the stability of ideal external modes numerically with thestability matrix. This analysis has advantages for stabilizing
resistive wall modes by feedback controls, because thisanalysis can be executed even when external modes are stable;
and since the matrix expressing the change of a plasma
potential energy does not change approximately in the casethat ideal internal modes are stable, we can analyze thestability of external modes in short time.
We executed a benchmark test of the analysis with thestability matrix, computed with MARG2D code, bycomparing that with ERATOJ code, and confirmed that theresult obtained with MARG2D well agree with that examinedby ERATOJ.
As an application of MARG2D, we analyzed the effectof the aspect ratio on the stability of external modes. From
this result, external modes are stabilized when the aspect ratio

Acknowledgements
The authors express their gratitude to Dr. H. Ninomiya,Dr. Y. Kishimoto for supporting our research. The authors also
thank Dr. Y. Kamada and Dr. G. Kurita for helpful discussions
and suggestions.

References[1] D. Pfirsch and H. Tasso, Nucl. Fusion 11, 259 (1971).
[2] W.A. Newcomb, Ann. Phys. 10, 232 (1960).
[3] S. Tokuda and N. Aiba, “Theory of the Newcomb
equation and applications to MHD stability analysis of atokamak”, PI-18, this conference.
[4] R. Gruber et al., Comput. Phys. Commun. 21, 323(1981).
[5] A.H. Boozer, Phys. Plasmas 5, 3350 (1998).
[6] V.D. Shafranov, Sov. Phys.-Tech. 15, 175 (1970).
[7] J.A. Wesson, Nucl. Fusion 18, 87 (1978).

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