Proceedings of ITC/ISHW2007

Numerical Study of Resistive Magnetohydrodynamic ModeStructures in Typical Plasmas of the Large Helical Device
Yutaka MATSUMOTO1) , Kiyomasa WATANABE2) and Luis GARCIA3)
1)

Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan
2)
National Institute for Fusion Science, Toki 509-5292, Japan
3)
Universidad Carlos III, Madrid, Spain(Received 10 January 2006 / Accepted 24 January 2006)

We study the resistive instabilities typically observed in the Large Helical Device (LHD) in order to systematically investigate the eﬀects of the resistive magnetohydrodynamics (MHD) instabilities on the plasmaconfinement. The resistive instabilities with the low toroidal mode number are investigated by using the threedimensional linear resistive MHD code FAR3D. It is confirmed that both the growth rate and the mode widthobey the theoretical formula of the gravitational interchange mode independent of the toroidal mode number.
The range of the magnetic Reynolds number, in which the growth rate and/or the mode width obey the theoreticalformula, is clarified. It is found that the growth rate is proportional to the mode width when beta is constant.
Keywords: resistive MHD, LHD, FAR3D
DOI: 10.1585/pfr.1.001

1Introduction

It is important to understand the magnetohydrodynamic(MHD) equilibrium and stability properties for the realization of the nuclear fusion reactor. In a helical device,
such as the Large Helical Device (LHD)[1] in Japan, thecomplex three-dimensional (3D) MHD study is needed forthe successful operation. The MHD instability modes typically observed in the LHD experiments are summarized asfollows. Some modes, of which the dependence propertyon the magnetic Reynolds number is similar to the resistive gravitational interchange mode[2], are obtained in theperipheral region of the high beta plasma with the strong
magnetic shear. Others are frequently monitored in thecore region of the high aspect ratio plasma with the relatively weak magnetic shear.
It is the purpose of this study to systematically investigate the eﬀects of the resistive MHD instability on theplasma confinement in the LHD. Consequently, the typical MHD instability modes are numerically analyzed byusing the 3D resistive MHD code FAR3D[3]. We especially focus on the mode width, i.e., the resistive layer
thickness and systematically investigate the relationships
between mode width (W), beta (hβi), growth rate (γ) andmagnetic Reynolds number (S ).
In this paper, the resistive MHD instability modes in
the high beta plasma with the strong magnetic shear are described. The outline of the FAR3D code and the numericalconditions are given in Section 2. The results of the calculations are summarized in Section 3. Section 4 is devoted
to a brief summary.
author’s e-mail: shopon@qe.eng.hokudai.ac.jp

2Numerical Model

As mentioned in the preceding section, the FAR3D code
is a 3D resistive MHD code[3]. In this code, a set ofresistive MHD equations is time-advanced in the Boozer
coordinates[4]. And the perturbed quantities are expanded
in Fourier series in the generalized poloidal and toroidal
angles by using the poloidal mode number (n) and thetoroidal mode number (m). FAR3D appears in diﬀerent
forms depending on the approximation and the treatment
for the MHD equations. We adopt one of the some versions of FAR3D, in which the reduced equations[3]:∂ψeq= ∇p Φ + ηBζ Jζ ,∂t∂U∂t=++(1)−v · ∇U{()√√
β0 1 ∂ g ∂p ∂ g 1 ∂p2−S∂ρ ρ ∂θ2 2 ρ ∂θ ∂ρζ()1 ∂
ζ 1 ∂ψ 1 ∂Jeq ∂ψ ζρJeq, (2)∇p J −ρ ∂ρρ ∂θ ρ ∂θ ∂ρ anddpeq ∂Φ∂p= −v · ∇p +,∂tdρ ∂θ(3)

are linearly calculated, where the velocity and the magnetic√
field are described as v = g∇ζ ×∇Φ and B = R0 ∇ζ ×∇Ψ,respectively.
The used MHD equilibria, i.e., the high beta plasma
with the strong magnetic shear, is generated by using the3D equilibrium code VMEC[5]. The equilibrium is described by using 1,000 radial grid points, and 20 helicalProceedings of ITC/ISHW2007

1.0

1.2

0.8
3.05 %
2.56 %
1.99 %
1.51 %
0.87 %

0.8

0.6

P

ι / 2π

1.0

0.4

0.6

0.2

0.4
0.0

0.2

0.4

0.6

0.0
1.0

0.8

ρ
Fig. 1 Radial profiles of ι/2π and P in the equilibrium using inthis study.

(n , 0) and 3 toroidal (n = 0) Fourier modes. Figure
1 shows the rotational transform (ι/2π) and the pressure(P) of the used equilibrium, in which the volume-averaged
beta (hβi) are 0.87, 1.51, 1.99 2.56, and 3.05 %, respectively.
In this paper, the resistive instabilities with the low
toroidal mode numbers (n = 1, 2, 3) are described. It isseen from Fig. 1 that there are the rational surfaces ofm/n = 1/1, 2/1, 2/2, 3/2, 4/2, 3/3, 4/3, 5/3, 6/3.
As mentioned in Section 1, we pay attention to themode width, i.e., the resistive layer thickness. The modewidth (W) is defined as the region that the growing speedof the instability to the radial direction (Vρ ) is larger than
70 % of the peak value. It is noted that ∂ξ/∂t = Vρ , whereξ is the perturbation to the radial direction. A example of
the radial profiles of Vρ , i.e., the mode structure, is shownin Fig.2.

3Results

3.1

n = 1 mode

In the cases of hβi ≤ 1.51 %, both m/n = 1/1 and 2/1
modes are unstable and their peak values of Vρ s are almost
the same. When the calculation without the m/n = 2/1
mode is done to determine the primary unstable mode, γremains the same as the results of the calculations including the m/n = 2/1 mode. Both m/n = 1/1 and 2/1 modes
also appear in the cases of hβi ≥ 1.99 %. In these cases,
the peak value of Vρ of m/n = 1/1 mode is larger than thatof m/n = 2/1 mode and γ remains the same as the resultsof the calculation without m/n = 2/1 mode. Therefore, them/n = 1/1 mode is the primary unstable mode. There exist
the rational surfaces of m/n = 1/1 around ρ ' 0.89.
The relationships between S and γ are shown in Fig. 3,and the relationships between S and W are shown in Fig. 4.
It can be seen from Figs. 3 and 4 that both γ and W are proportional to S −1/3 . This dependency is similar to the resis10-1

10-2

10-3

γ

1.2

1.4

10

0.87 %
1.51 %
1.99 %
2.56 %
3.05 %-44S

567-1/3

8log10SFig. 3 Relationships between S and γ in the case of m/n = 1/1mode.

1.0

10-1

10-2

10-3

10-4

0.8W0.6Vρ4S = 10
5S = 106S = 107S = 108S = 10

0.4

0.2

0.0
0.0

0.2

0.4

0.6

0.8

1.0

ρ
Fig. 2 Radial profiles of Vρ of m/n = 1/1 modes in the case ofhβi = 1.99 %.

0.87 %
1.51 %
1.99 %
2.56 %
3.05 %

4S

567-1/3

8log10S
Fig. 4 Relationships between S and W in the case of m/n = 1/1mode.

Proceedings of ITC/ISHW2007
0.075
10010-110-210-310-4γ

0.050

0.87 %
1.51 %
1.99 %
2.56 %
3.05 %

0.0000.00

γ0.025

0.02

0.04

0.06

0.08

0.10

WFig. 5 Relationships between W and γ in the case of m/n = 1/1mode.

In the case of n = 2, only the m/n = 3/2 mode is observed
as the unstable mode. The rational surfaces of m/n = 3/2
exist around ρ ' 0.63.
The relationships between S and γ are shown in Fig. 6,
and the relationships between S and W are shown in Fig. 7.
Both γ and W are proportional to S −1/3 in the case of hβi =0.87 %. When hβi ≥ 1.51 % and log10 S > 5, however, γand W are almost independent of S . Both γ and W of them/n = 3/2 mode hardly increase with hβi in the cases ofhβi ≥ 1.99 % γ of the m/n = 3/2 mode as well as γ ofthe m/n = 1/1 mode is proportional to W. The slopes also
become steep with the increase of hβi (Fig. 8).
Compared these results of the m/n = 3/2 mode withthose of the m/n = 1/1 mode, γ of the m/n = 3/2 mode islarger than that of the m/n = 1/1 mode. On the other hand,
W of the m/n = 3/2 mode is slightly smaller than that in
the m/n = 1/1 mode. The range of S , in which γ and W∝ S −1/3 , in n = 2 case is narrower than that in n = 1 case.
In the cases of hβi ≥ 2.56 %, the dependency of γ and W
on hβi in n = 2 case is smaller than that in n = 1.

3.3n = 3 mode

In the case of hβi = 0.87 %, only the m/n = 5/3 mode
is unstable. The m/n = 4/3 unstable mode, however, appears in the calculations without the m/n = 5/3 mode. In
this case, γ is larger than that in the calculation including the m/n = 5/3 mode. Although both m/n = 4/3
and m/n = 5/3 modes are unstable in the hβi = 1.51 %567-1/3

8Fig. 6 Relationships between S and γ in the case of m/n = 3/2mode.

W

10010-1

10-210-3

10-4

0.87 %
1.51 %
1.99 %
2.56 %
3.05 %

4S

567-1/3

8log10SFig. 7 Relationships between S and W in the case of m/n = 3/2mode.

0.14
0.12
0.100.08

γn = 2 mode

4Slog10S

tive gravitational interchange mode[2]. Such dependencyof γ and W on S , however, becomes small when hβi = 3.05% and log10 S > 6.
Figure 5 shows the relationships between W and γ. In
each hβi cases, γ is proportional to W. The slopes become
steep with hβi increasing.

3.2

0.87 %
1.51 %
1.99 %
2.56 %
3.05 %

0.87 %
1.51 %
1.99 %
2.56 %
3.05 %

0.06
0.04
0.02
0.00
0.00

0.02

0.04

0.06

0.08

0.10

W
Fig. 8 Relationships between W and γ in the case of m/n = 3/2mode.

Proceedings of ITC/ISHW2007

100.160.12

γcase, Vρ of the m/n = 5/3 mode is larger than that of them/n = 4/3 mode. In the calculations without m/n = 5/3mode, however, the m/n = 4/3 mode becomes unstable
and its γ is larger than that in the calculation including them/n = 5/3 mode. When hβi ≥ 1.99 %, both m/n = 4/3
and m/n = 5/3 modes are unstable. However, Vρ of them/n = 4/3 mode is larger than that of the m/n = 5/3mode. These results indicate that m/n = 4/3 mode is theprimary unstable mode. There are the rational surfaces ofm/n = 4/3 around ρ ' 0.72.

0.87 %
1.51 %
1.99 %
2.56 %
3.05 %

0.08

0.04

0.00
0.000

00.025

0.050

0.075

W
10-1

10-2

10-3γ

Fig. 11 Relationships between W and γ in the case of m/n = 4/3mode.

100.87 %1.51 %
1.99 %
2.56 %
3.05 %

S-1/3-4456

78log10S
Fig. 9 Relationships between S and γ in the case of m/n = 4/3mode.

case of hβi ≥ 2.56 %. As shown in Fig. 11, γ of them/n = 4/3 mode is also proportional to W and the slopes
become steep with the increase of hβi.
Compared these results of n = 1, n = 2 and n = 3modes, γ of n = 3 mode is the largest in three n cases. On
the contrary, W of n = 3 mode is the smallest. The rangeof S , in which γ and W ∝ S −1/3 , in n = 3 case is narrower
than that in n = 2 case.

4-1

10-2

10-310-4W

10

0.87 %
1.51 %
1.99 %
2.56 %
3.05 %

45678log10SFig. 10 Relationships between S and W in the case of m/n = 4/3mode.

The relationships between S and γ are shown in Fig. 9,
and the relationships between S and W are shown in Fig.
10. γ is proportional to S −1/3 when hβi ≤ 1.51 %. On theother hand, W is proportional to S −1/3 only if hβi = 0.87%. In high beta and log10 S > 6 cases, the dependencyof γ and W on S become small. Such changes of the dependency of W occur in lower beta cases than those of γ.
The dependencies of γ and W of the m/n = 4/3 modeon hβi is as small as that of the m/n = 3/2 mode in theSummary

In this paper, the low n resistive instability modes in thetypical LHD high beta plasma with the strong magnetic
shear are analyzed by using the FAR3D code.
It is found that the primary unstable modes are m/n =1/1, 3/2, 4/3. The rational surfaces of these modes exist in the plasma periphery (ρ > 0.6). It is confirmed thatboth γ and W obey the theoretical formula of the gravitational interchange mode(∝ S −1/3 ) independent of n. Theirdependencies on S become small when hβi and/or S arehigh. The range of S , in which γ and/or W are proportional to S −1/3 , becomes narrow as n decreasing. Whenhβi = const., γ is proportional to W. The slope becomessteep with hβi increasing.
In near future, the resistive instabilities of n > 4 will
be studied. In addition, the resistive instabilities in the different equilibrium of the LHD will also be investigated.
This work was supported by the collaborative research
program of NIFS (NIFS06KLHH302).
[1] A. Iiyoshi et al., Nucl. Fusion 39, 1245 (1999).
[2] H. P. Furth, J. Killeen, M. N. Rosenbluth, Phys. Fluids 6, 459(1963) .
[3] L. Garcia, Proc. 25th EPS Conf. on Controlled Fusion andPlasma Physics (Prague, 1998), VOL. 22A, PartII, p. 1757.
[4] A. H. Boozer, Phys. Fluids 23, 904 (1980).
[5] S. P. Hirshman, Phys. Fluids 26, 3553 (1983).