23P076-P
Effect of Resistivity on Mode Structure of Interchange Instability
交換型不安定性のモード構造に対する抵抗率の影響
Ryosuke Ueda, Yutaka Matsumoto, Masafumi Itagaki, Shun-ichi Oikawa,Kiyomasa Watanabe and Masahiko Sato
上田亮介, 松本 裕，板垣正文，及川俊一，渡邊清政，佐藤雅彦
Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan
National Institute for Fusion Science, Oroshi-cho 322-6, Toki, 509-5292, Japan
The interchange instability mode of the straight heliotron plasma is analyzed by use of the reduced
MHD equations transformed to eigenvalue equation. The differences of the instabilities between thefirst eigenmode and the second eigenmode are investigated. It is found that the instabilities due tofirst eigenmode are larger than second eigenmode. It is also found that the differences of the mode
structure appear substantially when the magnetic Reynolds number or β become large.

1. Introduction
It is one of the important subjects for a realization of the fusion reactor to investigate the Magnetohydrodynamics (MHD) instabilities. The interchange mode which is one of the pressuredriven instabilities is considered to play an important role in the plasma confinement property.
In LHD, the interchange instability modes areobserved in various magnetic Reynolds numbers.
In this study, the interchange mode instabilitiesof the straight heliotron plasma with various
magnetic Reynolds numbers and various beta
values are analyzed by eigenvalue analysis. Especially the difference of the effect on mode structure due to the first and second eigenmodes isstudied.
2. Numerical Method
In this study, the interchange mode with the various magnetic Reynolds number S (104 ≦ S ≦ 1010)
are analyzed. For such analyses, the following normalized reduced MHD equations [1] are useful.
These equations with the cylindrical coordinates (r,θ, z) are written as
(1)
(2)
(3)
where(4)(5)
Here, φ is the stream function, p is the pressure. Thepoloidal flux is ψ = A + ψh, where A and ψh are dueto the plasma current and the helical coils, respectively. In Eq. (1), the ∇Ω term denotes the contribution of the averaged magnetic curvature. The η isresistivity expressed by η=S-1.
We assume thatare written as
and linearize
Eqs. (1)-(3). The number m and n indicates thepoloidal mode number and toroidal mode number.
Then the following equations are obtained:(6)(7)
where k∥ = mι-n, k⊥ = m/r and ι is the rotationaltransform. is expressed by(8)).
N is the pitch number and is the pole number ofheliotron device [3].
We solve the above equations in 0 ≦ r ≦ 1. The
boundary conditions are set as followings:(9)
The equilibrium profile of the pressure and thepoloidal flux are given by
(10)
.
(11)
-3
Here, pa=10 , ιa=0.461, ιb=1.1.
In this way, the growth rate γ and the mode structure of instabilities can be obtained. We focus on
the instabilities with (m, n) = (1, 1) mode.
3. Saturation levelWe are unable to obtain the saturation level ofmode is larger than the second eigenmode. Moreover, when β > 3% and S > 108, the growth rate isdifferent substantially.

Fig. 1 Flattened pressure profile
p=p0+p1, around the rational surface.

Fig. 3 Growth rate as a function of the magnetic
Reynolds number in the case of first eigenmode(left) and in the case of second eigenmode (right).

the instability by only use of the linear eigenvalue
analysis. We assumed pressure perturbation p1 sothat the pressure profile (p=p0+p1) around ι=1 rational surface is flattened. Under such an assumption, the saturation level can be defined. .
4. Differences between the first and second eigenmodes.
Equations (6) and (7) have multiple eigenvalues.
We evaluate the differences between the first andsecond eigenmodes. Specifically we investigate themode structure, the growth rate and the mode widthwith various magnetic Reynolds numbers.
4.1 Mode Structures of instabilities
Figure 2 shows the mode structure of ψ and φ (S= 104 and β = 2%).The left figure of Fig. 2 is themode structure of first eigenmode and the right isthat of the second eigenmode. It can be seen thefirst eigenmode structure of φ is even around therational surface. On the other hand, the second eigenmode structure of φ is odd. The peak value andthe mode width of φ of the first eigenmode are larger than those of the second eigenmode. It is notedthat the mode width is evaluated by FWHM (Fullwidth at half maximum).

4.3 FWHM of displacement
We evaluate the FWHM of the displacement
as a function of the magnetic Reynolds
number. Here, the displacement is written as(12)Here, p1 is the perturbation of pressure. Figure 4 shows the FWHM ofas a functionof S. It can be seen that the FWHM of first
eigenmode is larger than those of the second
eigenmode. The dependence of the FWHM on
S is different in the high β cases. Moreover,
we can find that the dependence of FWHM
on S is close to theory () in low β cases.

Fig. 4 FWHM of displacement ξr in the case of
first eigenmode (left) and second eigenmode(right).

6. SummaryFig. 2 Mode structure of ψ (solid) and φ (dashed).
Left figure is the case of first eigenvalue andright is the case of second eigenvalue.

4.2 Growth rate
Figure 3 shows the growth rate of the first eigenmode (left) and the second eigenmode (right). It
can be seen that the growth rate of the first eigen-

We investigate the interchange instabilitiesof the heliotron plasma. The differences of theinstabilities between the first eigenmode andthe second eigenmode are shown. As a result ofanalyses, we found the instabilities due to first
eigenmode are larger than second eigenmode.
Additionally, when the magnetic Reynolds
number or β become large, the differences ofthe mode structure appear substantially. For
this reason, it may be expected that the second
eigenmode also affects on the interchange instabilities.

References[1] H. R. Strauss, Plasma Phys., 22, 733 (1980).
[2] K. Miyamoto, Nucl. Fusion 18, 243 (1978).