J. Pfasma Fusion Res. SERIES, Vol.5 (2002) 377-380

Finite B Equilibria of Heliotron J Plasmas
SUZUKI Yasuhiro, HAYASHI Takaya', NAKAMURA Yuji and KONDO Karsumi
Graduate School of Energy Science, Kyoto University, Gokasho, uji 611-0011, Japan
1

National Institute for Fusion Science, Toki 509-5292, Japan(Received: ll December 2001 / Accepted: l7 May 2002)

Abstract
Finite plasma pressure equilibria of Heliotron J is studied using a three-dimensional equilibrium
calculation code, HINT, which does not assume the existence of nested flux surfaces. The magneticisland lbrmation investigated for Heliotron J plasmas. In the standard configuration of Heliotron Jplasmas, the equilibria with magnetic islands can be appeared at p>. 1 Vo.When beta value on the axis isabout 2.7 Vo, the width of a magnetic island is half of the averaged minor radius. In this study, we also
show the approach to improve the convergence of the HINT calculation by using the equilibrium
obtained from the VMEC code as an initial guess. The result for applying the improved method to thelow beta equilibrium of Heliotron J plasmas is shown.

Keywords:
MHD, equilibrium, magnetic island, HINT code, VMEC code, Heliotron J

1. Introduction
In helical systems, nested flux surfaces can beinverse solver based on the energy principle using theflux coordinates [2]. Therefore, the equilibrium

generated by external coils without plasma current andincluding the magnetic island or the stochastic region

the analysis of the MHD equilibrium is performed

generated by the plasma pressure was not obtained. Inassuming net-current free and existence of nested flux
surfaces in many cases. However, the shape of theplasma boundary will be changed by the plasma
equilibrium current at the finite pressure. The magnetic
island generated at the finite plasma pressure will also
break the flux surface in non-axisymmetric system forsome cases. Therefore, the analysis including the changeof the flux surface at the finite pressure are required

this study, we investigate the finite pressure equilibria inthe standard configuration of Heliotron J using theHINT code [3]. At high B, nested flux surfaces is broken
by the magnetic island due to the finite pressure.

2. Computation Method of HINT GodeThe HINT code is a three-dimensional equilibrium
calculation code based on the time-dependent relaxation

without assuming existence of flux surfaces.
Heliotron J device is an Z = llM = 4 helical-axis
heliotron, which has large flexibility []. The main
parameters are R = t.2 m, (ao) - 0.1-0.2 m, B < 1.5 T
and t - 0.4-0.8. Heliotron J has a low shear and themagnetic configuration is very sensitive to the plasmapressure effect. In the previous study, Heliotron J
plasma is mainly studied using the VMEC code under
the fixed boundary constraint. The VMEC code is anmethod (the initial value problem), which uses nonorthogonal helical coordinates (ut, u', a3; (see ref. [3]).
The calculation is performed in the following steps.

The first step (A-process) is a relaxation process
along the magnetic field lines under the fixed magnetic
filed. This relaxation process calculates the averagedpressure along the field line which starts from a grid
point of the helical coordinates (ut, u2, u3),
@20O2 by The Japan Society of Plasma

Corre sponding author's e -mail : suzuki@ center.iae.kyoto-u. ac.jp

377

Science and Nuclear Fusion ResearchSuzuki Y. et aI., Finite B Equilibria of Heliotron J Plasmas[,#'=r #satisfied [5].

In this study, we also introduce the method toimprove the convergence of the HINT computing byusing the equilibrium obtained from the VMEC code as
an initial guess. For this purpose, we develop the IDH
code, which translates the equilibrium obtained by theVMEC from the flux coordinates (s, 0, () to the helicalcoordinates (u', u2, u3) inside and outside the VMEC

(1)

and uses it as updated pressure on a grid. The normal
I on all grid points. How-pressure relaxation calculates

ever, in the low shear configuration which has magnetic

islands in the plasma region for example, the field line
should be traced for very long distance to obtain the appropriate pressure on a grid. Since it is not efficient tocalculate I on all grid points, we use the new scheme ofA-process in ref. [4] to improve the computational time.

plasma boundary, respectively. For inside the boundary,This scheme does not trace the field lines on all grid
points, but calculates the field lines started from the lim-

u3 = -C fixed for simplicity. On the other hand, for

the IDH code makes the inverse mapping of theequilibrium from the flux coordinates to the helicalcoordinate using the method in ref. [6]. It calculates s
and 0to coincide with x(ul, u2)wdet the condition with

ited start points. The averaged pressure B on a grid point

outside the boundary, the field due to the plasma
equilibrium currents is calculated using the DIAGNO

is obtained from the pressure on nearly field lines

code [7] and translates to the helical coordinate also.

started from the sample points using the linear interpola-tion method.
The next step (B-process) is a relaxation of themagnetic field under the plasma pressure distribution isfixed. This process is obtained from the following time

3. Finite p Equilibria of Heliotron J Plasmas
In Fig. l, the flux surfaces of standard configuration of Heliotron J plasmas is shown in thevacuum. This configuration has a flat profile of theevolution equations,rotational transform (r - 0.55) and the magnetic well inthe vacuum as Fig. 2. The finite pressure equilibrium
obtained from the HINT code at P = 2.7 7o is shown in
Fig. 3. Here we usep(s) * (1 - s)2 as an initial pressure${n )--Yp+ixB,(2)$=vx(rxB -nU-j)),(3)

distribution, where s denotes normalized toroidal flux.
Large islands which correspond to the resonance of4l7
are generated in the plasma region and the width ofisland is half of the averaged radius. In this study, theresonance of 4/7 is appeared when B >- 0.8 Vo. The
resonance of , = 418 is not appeared for the standard

where pni = V x B andie is the coil current if it exists
in a computational domain. The final equilibrium state issolved by these process iteratively until the net-current
free condition or the prescribed net-current condition issatisfied. If there are coils in the computational region,
the Courant-Friedrichs-Levy conditon (CFL condition)is very severe because the Alfvdn velocity is very fast
due to the strong field near a coil. Therefore, the factor"fcas,, -l,"-1\Ifor B<8"(n"n)' for B> B.

(4)

is calculated on all grid points, where B" is specified to
the value between the plasma region and the region neara coil. The factor,/s, multiples right side of eq. (2) andthe Lorentz force due to the coil current is also excluded

from the second term of right side in eq. (2). According
to this scheme, the propagation velocity is uniform in
the computational region and the CFL condition is alsoFig. 1 Flux surfaces of the vacuum configuration. Rightside is the cross section at M0 = 0, left side is Mf

378

Suzuki Y. et al., Finite B Equilibria of Heliotron J Plasmas

0,6(a)

E3 o.sH o.r-- - -f-----f-----F--ttl____L____L____L____qoH

D-do'3

I'Io.z- - - -l- - - - -f- - - - -F - - - -ttl____L____L____L____ttl
ttl-- - -l------F----F-- -ltl

d3 o.r
h0.0

5.0
10.0
15.0
awerag€d radiuB

1.0

ao(b)

lt#t
____L____t 6___L____
ILFI
tAl____L__ t_L____L____
t^gt
t_6t
___L____L____tl
tl

0.5ItdoIr 10-2tl
tl____L____L__d_L__
ttpt

0.8!

20. 0

0.4

II

o.2

0.0

0.0

5.0

10.0
15.0
av€ragad radLua

20.0

r 10-2

Fig. 2 Profiles of the rotational transform and the magnetic well of standard configuration in Heliotron J
plasmas.

Fig. 4 Flux surfaces of the equilibrium for F = 1 "/" ob-tained from the conventional method (a) and newmethod (b). Right side is the cross section at MO 0, feft side is MQ = n.

::i'/ir!

obtained from the conventional method and our new
method are shown. In both equilibria, the results arevery similar, because both the width and the phase of%,$'',/al"itffit;

rVl^iislands are similar.The computational time is also

ilft'

improved about25 Vo.

^q4. Conclusions
We study the equilibria of Heliotron J plasmas
using the HINT code. It is found in the standard
configuration of Heliotron J plasmas, the magnetic
island of t = 417 resonance is generated by finite

Fig.3 Flux surfaces of the equilibrium for P = 2.7 y".
Right side is the ooss section at MQ - 0, left sideis MQ= 7s'

pressure effect when beta ratio is above 0.8 Vo - 0.85 7o.

We also improve the HINT computing scheme using theequilibrium obtained from the VMEC code as an initial
guess. For this purpose, we develop the IDH code, andwe can reduce the relaxation time steps with this new
method. In the present study, we study standard

configuration.
Finally, we show the equilibrium for B - 1 7o using

new method described in previous section. Thisequilibrium uses the result obtained from the VMEC at
F =l Vo as an initial guess. In Fig. 4, the equilibria

configuration only with specified pressure profile.
3't9

Suzuki Y. et al., Finite B Equilibria of Heliotron J Plasmas

Studying the effect of pressure profile or the vertical
filed to the equilibrium are future subjects.

I4l S. Lloyd et al., Proc. Conf. Computational Physics
2000, MonAM-2-4, Univ. Queensland, 2000.
t5l T. Hayashi et al., J. Plasma Fusion Res. 77, 594(2001).

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