J. Plasma Fusion Res. SERIES, Vol.5 (2002) 500-504

Equilibrium Galculation of Tohoku University Heliac
HARADA Koumei, KITAJIMA Sumio, HASHIZUME Hidetoshi and SUZUKI Yasuhiro'Tohoku University, Sendai 980-8579, JapanI Graduate School of Energy Science, Kyoto University(Received: 11 December 2001 / Accepted: 2 December 2002)

Abstract
Numerical analysis of equilibrium states in the heliac configuration is performed by using VMEC
code. In the heliac configuration, it is possible to produce flux surfaces with the magnetic well undervacuum condition. Mercier instability is stabilized in a wide region of the plasma column in the deep
well configuration. On the other hand, the local stability is limited only around a magnetic axis in the hill
configuration. In the Mercier criterion, the well term has favorable effect on the local stability, while thegeodesic curvature has negative effect.

Keywords:
heliac, plasma, VMEC, equilibrium, stability, beta

1. Introduction

transform and well depth are evaluated under the finite

In the heliac configuration [1,2], many kinds ofmagnetic configurations can be obtained easily bychanging in the ratio of coil currents. The heliacbeta and vacuum conditions using VMEC code, andthen the local stability is also investigated based on theMercier criterion [7]. This study is the first step forsuitable design of the high B DJHe plasma confinement

configuration can produce flux surfaces with a moderate
shear and a magnetic well under the vacuum condition.
There is possibility to achieve high beta plasma
confinement in the heliac configuration, which is reallyreactor by the heliac configuration.

2. Parameters for Magnetic Gonfigurationrequired for future fusion reactors. For the high p
plasma confinement, it is important to know how theIn the TU-Heliac, the parameters characterizing themagnetic field configurations are a ratio of center
conductor coil current to toroidal coil current (1. I RHo)and a ratio of vertical coil current to center conductor
coil current (8" I I"). The 1", R and F16 are center
conductor coil current, major radius and toroidal
magnetic field, respectively. RF16 is a quantity
proportional to toroidal coil current, and B" is a verticalmagnetic field. By changing I" / RHo we can control amagnetic surface configuration which is equivalent tocontrolling the rotational transform and shear. In thesame way, B, I I. affects radial shift of the surfaces

magnetic surface quantities, such as rotational transformand well depth, change with increase in p. One of theanalysis code of the MHD equilibrium solution isVMEC code [3,4] which assumes the nested flux
surfaces.

The magnetic surfaces of Tohoku University Heliac[5] (TU-Heliac) are produced by three sets of simple
circular coils, thirty two toroidal field coils, a center
conductor coil and two vertical field coils. which meansno needs of complicated helical coils. The difficulty inconstructing the magnetic coils in the heliac can besolved by realizing the demountable magnetic coilproposed in the ref. [6]. In this paper, both rotational

relating to magnetic well.
Figures 1(a) and l(b) show the magnetic well depth

Corre sponding author's e -mail : koumei.harada@ qse.tohoku.ac.jp

500@2002 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Harada K. et al., Equillbium Calculation of Tohoku University Heliacincreases with increase in B. On the other hand, therotational transform does not change when B is less than
3.5 Vo as shown in Fig. 2(a). The magnetic shear
increases monotonically with increase in minor radius,
while it changes to negative value around the magnetic
axis when F = 3.5 Vo. Figure 2(c) shows the stableregion evaluated from the Mercier criterion, where theand rotational transform under the vacuum condition,respectively. In these calculations, the value of I" I RHs
is selected to be 0.40. Since the period of magnetic field

is 4 in the T'U-Heliac, this configuration contains
rational surface of nlm = 8/5 (n: toroidal mode number,
m: poroidal mode number) which is located in the coreplasma region. As shown in Fig. l(a), the well depth
increases with decrease in Bu | 1". On the other hand, B" I

local stabilities are evaluated on the each magnetic flux
surfaces. The stable region expands with increase in B,
and reaches maximum when Bs 2 lVol, and then it=shrinks with increase in B. Figure 2(d) shows profiles ofterms for Mercier criterion. Stable state is achieved

1" does not aflect the rotational transform profiles, asshown in Fig. l(b).

3. Results
The equilibriums of the TU-Heliac are calculated
by using VMIIC code. Two typical configurations arechosen from conditions in Fig. l. In the vacuum
condition, one has the deep well (I"/Rfls = 0.40 and B, I
I" = 2.0), and the other has the high hill (1" / RIIo = 0.40
and B, /1" = 5.0).
Figure 2 shows finite B effect on the deep wellconfigration. Remarkable change of the well depth canbe obtained by increasing the B value. The well depth

1.8
1.7

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1.6
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po (%124ffi.".;ltc 0.5
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II er 2.5
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1.65
1.6

1.55
1.5

+\-&m-(d)-xi':-'

00.02
d0.04

0.06

(nr)

Fig. 2 The finite p effect on the well configuration. (a) therotational transform, (b) the well depth, (c)(b)0sn 1 108/:/+a'1.75+

1.5 108@0.02 0.04 0.06 0.08
a (m)Mercier stable region (mesh region) and (d) indexes for Mercier criterion. LCFS means Last
Closed Flux Surface. Local stability is achieved

when DM(ideal) (= DM(shear) + DM(well) +DM(J.B) + DM(GC)) > 0. DM(shear), DM(well),Fig. 1 The radial profile of {a} the magnetic well depthand (b) the rotational transform under vacuum
condition. Horizontal .axis represents mean
plasma radius, and B, / l" is chosen as a parameterin both figures.

DM(J.B) and DM(GC) are related to magnetic

shear, well depth, parallel current with magnetic
field and geodesic curvature, respectively.
501

Harada K. et al., Eqruilibium Calculation of Tohoku University Heliac

when DM (ideal) > 0, where DM (ideal) = DM (shear) +DM (well) + DM (J.B) + DM (GC). DM (shear),from the view point of the Mercier criterion in the heliac

DM(well), DM(J.B) and DM(GC) are related tomagnetic shear, well depth, parallel current withmagnetic field and geodesic curvature, respectively.

Figure 3 shows results in the high hill configuration. The magnetic well is produced in the finite B

configuration.

configuration even though the magnetic hill has been
generated under the vacuum condition. Local stability
exists only around the magnetic axis as shown in Fig.
3(c). Items of Mercier criterion in Fig. 3(d) show thesame tendency as those in the well configuration shown

Figure 2(d) indicates that the magnetic well is effective
for stability, but the geodesic curvature has negative
effect on the stability. Because absolute value of DM(shear) is smaller compared with DM (well) and DM(GC), the magnetic shear does not affect the stabilityin Fig. 2(d). The magnetic well contributes to the

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p., (%\\offis*$-dSN'!-

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I43z2105 107
0-5 107

IT LCFSI LCFS

01.5 1081108o 5107wqi9o==.-+ r"-'0t-5 107-1 108(d)0a(nr)

Fig. 3 The finite p effect on the hill configuration. (a) therotational transform, (b) the well depth, {c)

00.02
a(d)

0.04
(rn)

0.06

Fig. 4 The finite B effect on the well configuration
without 8/5 rational surface. (a) the rotational

Mercier stable region (mesh region) and (d) indexes for Mercier criterion. LCFS means Last
Closed Flux Surface, Local stability is achieved

transform, {b) the well depth, (c) Mercier stable region (mesh region) and (d) indexes for Mercier
criterion. LCFS means Last Closed Flux Surface.

when DM(ideal) (= DM(shear) + DM(well) +DM(J.B) + DM{GC)) > 0. DM(shear), DM(well),Local stability is achieved when DM(ideal) (=DM(shear) + DM(well) + DM(J.B) + DM(GC)) > 0.

DM(J.B) and DM(GC) are related to magnetic
shear, well depth, parallel current with magnetic
field and geodesic curvature, respectively.
Solid line region in figure (c) is stable region
shown in Fig. 2 (c).

DM(shear), DM(well), DM(J.B) and DM(GC) are re-lated to magnetic shear, well depth, parallel current with magnetic field and geodesic curvature,respectively.

Solid line region in figure (c) is stable region
shown in Fig.2 (c).
502

Harada K. et al., Equilibium Calculation of Tohoku University Heliac

stability, on the other hand the geodesic curvature does

where the magnetic configuration has the well or hillunder the vacuum condition, respectively. The rotational

not improve local stability, which is the same as the welltransform in Fig. 4 is similar to previously obtained
result in Fig. 2. In Fig. 4, converged solutions are notobtained in the case that B0 is lager than 4.0. Figure 5
shows results for the hill configuration under theconfiguration.

The equilibrium of the configuration which does
not contain the 8/5 rational surface is calculated. In

Figures 2 and 3, sufficient convergence of thecalculation is not obtained when the rotational transformis about 1.6, and thus the 8/5 rational surface might haveeffect on the convergence of the calculation. Results ofvacuum condition without the 8/5 rational surface. Thereis little stable region in this configuration. From Figs.
4(d) and 5(d), it is found that the magnetic well affects
the local stability and geodesic curvature enhances theinstability. This behavior is the same as that obtained
from Figs. 2 and3. As a result, equilibrium solutions are
not affected by the 8/5 rational surface in the VMEC

the stability calculation are shown in Figures 4 and 5,

1.9

1.85

E+calculations.

v

1.8

1.75

d1.65

The stability in a large-scale heliac device is#vacu0.0
aj 1.0+ 2.0
3.0
av! 4.0
5.0
5.5

A+ + +1I,

1.7
1.6

1.55

evaluated, where the size is 10 times larger than the TU-Heliac one. Major radius and mean plasma radius of thedevice are 4.8 m and 0.5 m, respectively. Here, the16\oparameters characterizing magnetic configuration are
selected to be 1" I RHs= 0.40 and Bu I I.- 2.0, whichcorrespond to the magnetic configuration with themagnetic well under vacuum condition. As shown inFig. 6, a stable region exists in the region of a = 0.2 0.4 at Bo = 1.5 Vo, which is similar to the TU-Heliac

12U4o\case. On the other hand, the stability is improved around7s

the magnetic axis, because there is a stable region forft= 5 - 6 l%ol near the axis. Scaling up of the confinement

region, therefore, contributes to the high B stability

43

around the axis. However, this different result of simply

215 1070-5 107-1 1oE

3500.02
a0.04(m)v40.06

q.3Fig. 5 The finite B effect on the hill configuration without
8/5 rational surface. (a) the rotational transform,(b) the well depth, (c) Mercier stable region (meshregion) and (d) indexes for Mercier criterion. LCFS
means Last Closed Flux Surface. Local stability isL@,rcrs
0achieved when DM(ideal) (= DM{shear} +DM(well) + DM(J.B) + DM{GC)) > 0. DM(shear},DM(well). DM(J.B) and DM(GC) are related tomagnetic shear, well depth, parallel current withmagnetic field and geodesic curvature, respec-0.2

0.4

0.6

A. (m)

Fig. 6 Mercier stable region (mesh region) in the wellconfiguration of Heliac. The size of Heliac is assumed to be ten times larger than TU-Heliac.
Solid line region is stable region shown in Fig. 2(c) expanded to 10 times radial size.

tively.

Solid line region in figure (c) is stable region
shown in Fig. 3 (c).
503

Harada K. et al., F,quilibrium Calculation of Tohoku University Heliac

References

scale-up device from TU-Heliac is not acceptable fromthe viewpoint of ideal MHD model. The reason can bethat initial conditions of each calculation, which are
obtained after repeated trial and error, are different.

t1l A.H. Boozer el al., Plasma Physics and Controlled
Nuclear Fusion Research (Proc. 9th Int. Conf. Bal'timore, 1982 XIAEA, Vienna, 1983) 3,129 (1983).
t2) S. Yoshikawa, Nucl. Fusion 23,667 (1983).
t3l S.P. Hirshman et aI.,Phys. Fluids 26, 3553 (1983).

4. Summary
The MHD equilibrium and Mercier stability[4] S.P. Hirshman et al.,Comp. Phys. Comm. 43, 143rss (1986).

calculations of the Tohoku University Heliac have been
performed by using the VMEC code. There is a widestable region in the configuration that has the magnetic
well under the vacuum condition when Bs isnearly 2Vo.
The well term in the Mercier criterion has a favorable
effect on the local stability, and the geodesic curvatureterm enhances instability. There is no effect of rationalsurface on the equilibrium solution and local stability

l5l S. Kitajima et aI., Jpn. J. Appl. Phys. 30, 2606
(1991).

t6l S. Ito, H. Hashizume and S. Kitajima, Intemational
Journal of Applied Electromagnetic and Mechanics
13, 14 (2001) (in printing) 'Study on mechanicaljointing of high temperature superconductors'.
[7] Mercier C., Nucl. Fusion 1,47 (1960 ).

with the VMEC code.

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