J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)227–229
000–000

HINT Computation of LHD Equilibriumwith Zero Rotational Transform Surface
KANNO Ryutaro, TOI Kazuo, WATANABE Kiyomasa, HAYASHI Takaya, MIURA Hideaki,
NAKAJIMA Noriyoshi and OKAMOTO Masao
National Institute for Fusion Science, Toki, 509-5292, Japan(Received: 9 December 2003 / Accepted: 20 February 2004)

Abstract
A Large Helical Device equilibrium having a zero rotational transform surface is studied by using the threedimensional MHD equilibrium code, HINT. We find existence of the equilibrium but with formation of the two or
three n = 0 islands composing a homoclinic-type structure near the center, where n is a toroidal mode number. The
LHD equilibrium maintains the structure, when the equilibrium beta increases.
Keywords:
MHD equilibrium, zero rotational transform surface, Large Helical Device

1. Introduction

is shown as follows. The first step (A-step) is a relaxation
process of pressure along magnetic field lines under acondition of a fixed magnetic field B. To expedite the pressure
relaxation, we make an average of pressure, p–, along a field
line, and replace a value of pressure at each grid point with–:
the average p

In the Large Helical Device (LHD) [1], an MHD
equilibrium with both deep magnetic well and high magnetic
shear in the plasma core region attracts much attention froma point of view of improved MHD stability and plasma
confinement. In the LHD, such an equilibrium can be realized
by a large Ohkawa current [2-4] induced by counter neutralbeam injection. In a plasma with a net subtractive toroidal
current of about –100 kA/T, the rotational transform isexpected to be below zero around the magnetic axis.
A helical equilibrium with a zero rotational transformsurface was studied in Heliotron E experiment [5]. When arotational transform at the center was below zero, strong
MHD activities were observed and were guessed to beexplained by an m/n = 1/0 resistive tearing mode, where m isa poloidal mode number and n is a toroidal mode number.
The result seemed to be understood by a numerical analysis
employing a low beta resistive MHD model for a straightheliotron-like configuration [6]. The numerical study of ref.
[6] also showed that when the resonant surface existed near
the axis, the m/n = 1/0 tearing instability was weak and themagnetic island width saturated. This result suggests thepossibility of existence of the equilibrium having a zerorotational transform surface. From these previous studies[5,6], at first we should start to investigate whether or not an
LHD equilibrium having a zero rotational transform surface
can be allowed.

p (u1, u 2, u 3 ) ⇒ p (u1, u 2, u 3 ) =∫ drp B ,∫ dr B(1)wherer is a length along a field line starting from a grid
point (u1,u2,u3). Toroidal periods of tracing a field line are
typically 20–60 in calculations of this article. In the second
step (B-step), calculation of a net toroidal current density jnet ,e.g. Ohkawa current, is carried out under conditions of fixedpressure and magnetic field, if a net toroidal current exists inthe equilibrium. The third step (C-step) is a relaxation processof magnetic field under a fixed pressure profile:∂v= −∇p + j × B,∂t(2)

∂B= −∇ × E = ∇ × [v × B − η ( j − jnet )] ,∂t(3)ρwhere j =(1/µ0) ∇ × B is the total current density. To calculate
an equilibrium with a net toroidal current, we modify the
Faraday equation, as shown in equation (3), which is givenby the following idea. When an equilibrium with a net
toroidal current has flux surfaces, Ohm’s law can be modified
as E + v × B = η ( j – B〈 j ·B〉net / 〈B2〉), where 〈 …〉 means theflux surface average. Since the existence of nested flux

2. HINT computation
Numerical analysis of the equilibrium is carried out by
using the HINT code [7-10]. The standard scheme of HINT

©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: kanno@nifs.ac.jp1227

Kanno R. et al., HINT Computation of LHD Equilibrium with Zero Rotational Transform Surface

surfaces is not assumed in the HINT code, a toroidal flux isgiven by using contour lines of pressure. In an island and/orthe ergodic region, the flux is estimated by an interpolation.
Then the Faraday equation can be extended ass), where B0 is the magnetic field strength at the magneticaxis, R0 is the major radius of the axis, p0 is pressure at theaxis, and s is the normalized toroidal flux. As shown in Fig.
1, we find that an LHD equilibrium having a zero rotationaltransform surface is possible to exist. Here the equilibrium
beta value β is 0.56 %, and the total net toroidal current It is–100 kA/T. We assume that a net toroidal current density
modeling the Ohkawa current is given as j ∝ –(p/p0)2. Profilesof rotational transforms and pressure are plotted in Fig. 2. In
the field line structure of Fig. 1, we see two islands. Thecentral island has a negative ι/2π, and the other island with
the n = 0 mode, located around the central one, has a zerorotational transform around an O-point of the central island,as shown in Fig. 2. When the equilibrium beta increases to1.7 %, the inner region sufficiently away from the separatrixof the later island is split into two parts, and a doublet-type n= 0 island having only one X-point located at Z = 0 is formed,see Fig. 3. Note that the LHD equilibrium maintains ahomoclinic-type structure [9,11] composed by the islands near
the center, when β increases; see Figs. 1, 3 and 4. We canconsider that this field line structure is general for both helicaland tokamak plasmas, because the toroidal mode number ofthe islands is zero. The magnetic axis and the central
rotational transform are plotted with an absolute value of the〈 j ⋅ B〉 net ⎞∂B⎛
= −∇ × E = ∇ × v × B − η ⎜ j − B⎟ . (4)⎝⎠∂t〈B2〉
If there exist flux surfaces in a steady state, we have 〈E· B〉 =0 = η(〈 j · B〉 – 〈 j · B〉net). Thus 〈 j · B〉 becomes 〈 j · B〉net in asteady state, and we obtain an equilibrium with a net toroidal
current. Note that in the HINT computation, a numericalequilibrium on the way to a steady state is not meaningful ina sense of the MHD physics, and the scheme is justified when
the numerical equilibrium relaxes sufficiently into a steadystate.
A relaxation process computed by the HINT code starts
from the vacuum configuration with B0 = 1.5 T and R0 = 3.75m, and the initial pressure profile given as p = p0(1 – s 4)(1 –Fig. 1

Poincaré plots of field lines at the horizontallyelongated poloidal cross section in the LHD equilibriumwith β ≈0.56 %. The total net toroidal current It is –100kA/T.

2.5

1.0Fig. 3

Poincaré plots of field lines at the horizontallyelongated poloidal cross section in the LHD equilibriumwith β ≈ 1.7 %. The total net toroidal current It is –100kA/T.

Fig. 4

Poincaré plots of field lines at the horizontallyelongated poloidal cross section in the LHD equilibriumwith β ≈ 3.8 %. The total net toroidal current It is –100kA/T.

2.0

1.0

0.5

p/p0

ι/2π

1.5

0.5

0.0

-0.5
2.5

3.0

3.5

4.0

4.5

0.0
5.0

R (m)

Fig. 2

Profiles of rotational transform ι /2π (solid circle) andpressure (open circle) along Z = 0 for β ≈ 0.56 % and It =–100 kA/T. Open triangles represent ι /2π in the vacuum
field.

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4.25

0.50

4.00

0.25

3.75

0.00

3.50

-0.25

3.25

-0.500Fig. 5

comparison to the tokamak plasma with the current hole.
Recently, an idea of an axisymmetric tri-magnetic-islands(ATMI) equilibrium has been proposed in ref. [12], where theATMI equilibrium is one of the candidates for a tokamak
equilibrium having a zero rotational transform surface. The
ATMI equilibrium has three islands with n = 0 near the center,and they compose the heteroclinic-type structure [9,11]. As
contrasted with ref. [12], in this article we have suggested
that the homoclinic-type structure is also allowed. Note thatthe structure shown in this article is general in both helicaland tokamak plasmas because of the islands having thetoroidal mode number n = 0.
A numerical study of the nonlinear stability in atokamak-like equilibrium having a negative central current[13] suggests that the equilibria obtained here require to be
examined on an MHD stability of n = 0 modes. Then thenonlinear stability of the equilibria are studied now by using
the nonlinear simulation code, which is a natural extensionof the HINT code [14], and the results will be reported innear future.

ι/2πR (m)Kanno R. et al., HINT Computation of LHD Equilibrium with Zero Rotational Transform Surface

50|It| (kA/T)

100

The Shafranov shift of the axis for 1) open circle: β ≈0.56 %, 2) open square: β ≈ 1.7 %, and 3) open triangle:
β ≈ 3.8 %. A dotted line represents the positions of theaxis in the vacuum. The central rotational transforms
for 1) cross (×): β ≈ 0.56 %, 2) asterisk (*): β ≈ 1.7 %, and3) plus (+): β ≈ 3.8 % are also shown.

ReferencesFig. 6[1] O. Motojima, N. Ohyabu, A. Komori et al., Proc. 19th
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Poincaré plots of field lines at the horizontallyelongated poloidal cross section in the LHD equilibriumwith β ≈ 1.7 %. The total net toroidal current It is –(100/1.5) kA/T.

total net toroidal current |It | in Fig. 5. As |It | increases, theaxis shift to the outside of the torus increases and therotational transform around the center decreases to zero.
When the central rotational transform crosses zero, however,the Shafranov shift of the axis reduces. This is because of thetopological change of the field line structure around thecenter; for example, see Figs. 3 and 6.

3. Conclusions
By using the HINT code, we have found that an LHD
equilibrium having a zero rotational transform surface ispossible to exist. And we have seen that the LHD equilibrium
maintains the homoclinic-type structure composed by theislands near the center, when β increases. Let us consider

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