24C10
MHD equilibrium including a static magnetic island for the reduced
MHD equations in straight heliotron plasmas
直線ヘリオトロン配位における静的磁気島を含んだ簡約化MHD方程式に対するMHD平衡
Kinya Saitoa, Katsuji Ichiguchia,b and Ryuichi Ishizakia,b齋藤欣也, 市口勝治，石崎龍一
aThe Graduate University for Advanced Studies
Oroshicho 322-6, Toki, Gifu 509-5292, Japanb
National Institute for Fusion Science,
Oroshicho 322-6, Toki, Gifu 509-5292, Japana
総合研究大学院大学 〒509-5292 岐阜県土岐市下石町322-6b
核融合科学研究所 〒509-5292 岐阜県土岐市下石町322-6

MHD equilibria including static magnetic islands for the reduced MHD equations instraight heliotron plasmas are studied. The equilibria are obtained by solving thesimultaneous equations of the constant pressure along each field line and of the forcebalance. There exist two kinds of solutions depending on the numerical method in solving theequation of the constant pressure along each field line. One is the pressure profile that islocally flat at not only the O-point but also the X-point, which is obtained by employing a
diffusion equation parallel to the field line. The other is the profile that is locally flat only at
the O-point, which is obtained by employing a field line tracing method.

1. Introduction
We study the interaction between static magnetic
islands generated by an external field and resistive
interchange modes by using the reduced MHD(magnetohydrodynamics) equations[1] in straightheliotron plasmas. In the previous work, a profile
corresponding to nested flux surfaces wasemployed for the equilibrium pressure. In this case,interchange modes grow despite the existing staticislands. The island width is changed by thenonlinear saturation of the interchange mode[2,3].
This pressure profile, however, does notsatisfy the equilibrium equations with themagnetic field including the islands. The
equilibrium pressure consistent with thetopology of the static islands should be
deformed to have local flat structure at theisland region. The local flat structure is
expected to affect the growth of the interchange
mode. For the study of the deformation effecton the stability, equilibria with the pressure
profile consistent with the static islands areneeded. In the present study, we calculate theequilibria by solving the simultaneous equationsof the constant pressure along each field line
and of the force balance.

2. Simultaneous equations for equilibrium
MHD equilibria including a static magnetic
island with the mode number of (m,n)=(1,1) arestudied by using the reduced MHD equations in astraight heliotron configuration. Here, m and n are
the polidal and toroidal mode numbers, respectively.
These equations are suitable for the analysis of suchlow mode number physics. The reduced MHD
equations are composed of Ohm’s law, vorticity
equation and the pressure equation for the ploidal
flux Ψ(r,θ), the stream function Φ(r,θ) and theplasma pressure P(r,θ). The normalized equationsin the cylindrical coordinates are given by(1)(2)
and(3)
Here,themagnetic field is expressed as. The equilibrium corresponding toEq.(1)-(3) with static islands has to satisfy thefollowing two equations. One is the force balance

(4)which is obtained byin Eq.(2). The
other is the constraint that the pressure is constantalong field line with arbitrary topology,(5)Equations (4) and (5) are the simultaneous
equations for Ψ(r,θ) and P(r,θ).

profile is locally flat even at the X-point.
The stability of the interchange modes for theseequilibria is studied in a future.
(a)-3[10 ]
1.5
Plasma Pressure

equation given by

3. Calculation methods
Equations (4) and (5) are separately solved as in
the case of the HINT code[4]. In the first step, P
satisfying Eq.(5) is solved with Ψ fixed. In thesecond step, Ψ satisfying Eq.(4) is solved with P
fixed. In the first step, we employ two methods.
One is the method utilizing the diffusion equation
parallel to the field line

1.0

0.5

0-1

Psym=0(1-r4)2
Equilibrium Pressure0r



1



1(b)

We follow the time evolution of P from the initial
profile. The solution is obtained when thesteady state is achieved. The other is the method
tracing the field line. In this method, P(r,θ=0) is set
to be. We follow the field line from θ=0 andset P(r,θ=0) to the pressure along the field line. Ineither case, the two steps are iterated until theconvergence of the island width is obtained.
4. Results
Figure1(a) shows the equilibrium pressure profilein the case with the diffusion equation. The initialpressure profile is, where
denotes the beta value at r=0. Here,= 0.16% isemployed. Figure1(b) shows the equilibriumpressure profile in the case with the tracing fieldline. In both case, O-point of the island is located at
θ=π and X-point is located at θ=0. Blue lines
indicate the positions of separatrix of the magneticisland. Dotted lines are the position of the rationalsurface. In Fig.1(a), the pressure profile is locallyflat at not only the O-point but also the X-point. InFig.(b), the pressure profile is locally flat only at theO-point.
5. Summary
MHD equilibria including static magnetic islands
for the reduced MHD equations are studied instraight heliotron plasmas. The equilibria with twokinds of pressure profile are found. One has local
flat structure in the region corresponding to theisland. In this case, the pressure gradient is kept at
the X-point. The other has local flat structure that isalmost uniform is the θ direction. In this case, the-3[10 ]
1.5

Plasma Pressure

(6)

1.0

0.5

0-1

Psym=1-r )Equilibrium Pressure
4 20r

Figure1: Equibrium pressure profiles in the cases withmethods of (a) diffusion equation and (b) tracing fieldline. Blue lines denote the position of the separatrix ofthe magnetic island. The dotted lines show the positionsof the rational surface.

Acknowledgments
This work is partly supported by budget
NIFS11KNXN222 of the National Institute for
Fusion Science and by a Grant-in-Aid for Scientific
Research (C) 22560822 from JSPS, Japan.
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17 (2010) 062504.
[4] Harafuji Kenji, Takaya Hayashi, and Tetsuya Sato, J.
Comput. Phys. 81, (1989) 169.