J. Plasma Fusion Res. SERIES, Vol.5 (2002) 366-369

Design of Magnetic Coil System for GHS-qa
SHIMIZU AKihirO, OKAMURA ShOiChiI, ISOBE MiISUIAKAI, SUZUKI ChihirOl,
NISHIMURA Shinl and MATSUOKA Keisuker
Nagoya University, Nagoya 464-8603, Japanfor Fusion Science, Toki 509-5292, Japan

1 National Institute(Received: 31 January 2OO2

I Accepted: 18 June 2002)

Abstract
A modular coil system for a quasi-axisymmetric stellarator, CHS-qa, is designed. In the design
process of modular coils, the normal component of produced magnetic field is evaluated on the outermost

magnetic surface that is given from the configuration design. However, such an evaluation of the normalcomponent on the surface is not sufficient to obtain the designed quasi-axisymmetric configuration in agood precision. It is because the mirror ripple component produced by modular coils does not largely
affect the normal component of magnetic field on the surface. To improve the modular coil design, theevaluation of mirror ripple component at magnetic axis is included.

Keywords:
advanced stellarator, quasi-axisymmetry, CHS-qa, modular coil

1. Introduction

system to produce the magnetic field which realizes all

Recent development of computer technology
expanded the designing freedom of 3-D configuration ofphysical properties of advanced stellarator. Modular
coils are normally used for the coil system of advanced

toroidal confinement. Based on such a progress, manystellarators.

advanced stellarator configurations [1-6] were proposed.

The shape of outermost magnetic surface uniquely

Design objectives of these configurations are better
property in the neoclassical transport and improved
MHD stability for high beta plasma.
A quasi-axisymmetric configuration [4,5] is one ofdetermines all magnetic field configuration inside of thesurface. The modular coil system is designed to reducethese advanced stellarator concepts. It has theaxisymmetric field strength in Boozer coordinates [7]the normal component of produced magnetic field on
the outermost magnetic surface. When the normalcomponent on the surface is reduced to be zero, thedesigned quasi-axisymmetric magnetic configuration isand therefore has good neoclassical transport similar toproduced in an experimental device [0]. However, thetokamak. At NIFS, the quasi-axisymmetric device,
CHS-qa [8,9], with a major radius of 1.5 m and an

total number of modular coils is limited. Moreover.
from the engineering point of view, conditions are
imposed for the radius of curvature and the distance
between coils. Therefore, the perfect reduction of thenormal component cannot be achieved in the real

average aspect ratio of 3.2 was proposed and its design

work has been continued. An advanced stellarator
configuration is usually designed using the equilibrium
solving code based on the data of outermost magnetic
surface shape. When the magnetic configuration isdetermined, the next important step is the design of coilC o rre sp ondin g aut ho r' s e - mai I : aki hi ro @ nifs. ac

modular coil design. Due to such a residual normalcomponents on the surface caused by the technical@20O2 by The Japan Society of Plasmajp

Science and Nuclear Fusion Research366

Shimizu A. et al., Design of Magnetic Coil System for CHS-qa

constraint of the coils, the possibility comes out thatsome physical properties could not be reproducedcorrectly by minimizing the normal components on thesurface. In this paper, it is reported that adjustment of
the mirror component based on the evaluation of thetokamak. To reduce this level lower, number of coils
must be increased or the size of coils must be larger.

However, it causes other difficulties of coilmanufacturing: larger size and higher total cost.
Therefore we select the tolerant magnitude of normalcomponent to this level. Among several physicalnormal component on the outermost magnetic surface isproperties of magnetic field, it was found that avariation of mirror ripple component make very littlechange in the normal component of magnetic field on
the surface. The magnetic field strength can bedifficult and that the separate treatment of thiscomponent is required to design a good modular coilsystem.

2. Modular coil optimization including anevaluation of mirror ripple component

expressed by using Fourier series in Boozer coordinates
as fBf = I.,n B-,"(p) cos(m0-nN@), Here, 0 and Q are

poloidal and toroidal angle in the Boozer coordinates
respectively. N is the toroidal periodic number ofconfiguration and it is 2 for CHS-qa. The m and n are

The modular coil system of CHS-qa consists of 20coils. This number is determined by considering
magnitude of bumpy ripple produced by discrete coils.
By the stellarator symmetry, these coils have only 5
types of shapes. A shape of modular coil is expressedusing Fourier series on a toroidal surface. In thedesigning process of modular coils, values of Fourier
coefficients are varied, and finally a set of coefficientsthe poloidal and toroidal mode number respectively. The

mirror ripple component is expressed by Bo,r. Even if

modular coils are optimized so that the normalcomponent on the surface satisfies the tolerant level(l-2 Vo), the mirror ripple component at the magnetic
axis remains in a finite level. Figure 2(a) shows thevariation of magnetic field strength along the magnetic
axis for the first version of modular coil. Horizontal axis

which gives the minimum normal component ofmagnetic field on the outermost magnetic surface, isobtained as a rosult of optimization process. An exampleof modular coil system is shown in Fig. l. In this figure,five different types ofcoils are marked by coil I throughcoil 5. The coils having same type of shape are marked
same coil number.
Because of the technical reason described above.
the threshold level of the normal component of magnetic
field is set at I - 2 Vo of toroidal magnetic field strength,
This level is comparable to the magnitude of bumpy
ripple produced by discrete toroidal field coils in

1.04
1.02
1

0.98

0.96
1.04
1.02
1

Coil 2

Coil 2

0.98

Coil 1

Coil 1

0.96
Coil 12n

Coil 13naFig. 2 Variation of magnetic field along magnetic axis
produced by modular coils. (a) magnetic field proCoil 5

duced by modular coils which is optimized byonly the evaluation of normal component on theoutermost magnetic surface. (b) magnetic field
produced by modular coils with the evaluation ofmirror component.

Coit 5

Fig. 1 Top view of 20 modular coils for CHS-qa.
367

Shimizu A. et al., Design of Magnetic Coil System for CHS-qa

corresponds to the toroidal angle in Boozer coordinates.

The magnitude of mirror ripple is about 37o of thetoroidal magnetic field that is more than the acceptablelevel for the quasi-axisymmetry. To reduce this mirror
ripple at magnetic axis, the evaluation of this ripple isincluded in the coil optimization process. By thisinclusion, the mirror ripple at magnetic axis is greatlyreduced. In Fig. 2(b), the result is shown for theimproved version of modular coils. The other Fouriercomponents of magnetic field and physical properties
such as rotational transform, magnetic well depth areretained to be the same. The constraint imposed on coilstructure from the engineering point of view, such as thedistance between coils and the radius of modular coils
are also the same.
The profiles of normal component of magnetic fieldon the outermost magnetic surface are compared forthese two cases. In Fig. 3(a), lB.nlllBl on the surface isshown for the case that the mirror ripple exists. Here, z
is the normal unit vector of the surface. In Fig. 3(b), thecase that the mirror ripple is reduced is shown. Averagevalues of lB.nlllBl on the surface are l.45Vo for bothcases. A clear difference is not seen between thesefigures. The dominant contribution to lB.n l/lBl is the
bumpy ripple produced by discrete modular coils. The
difference of mirror ripple between two cases does notclearly appear in lB.nlllBl structures which is the reason
why the mirror ripple is not correctly determined by theminimization procedure for the normal component ofmagnetic field.
To investigate the relation between the averagevalue of normal component on the surface and themirror ripple at the magnetic axis, the mirror ripple isintroduced artificially by changing the current ratio oftwo groups of modular coils. The improved version ofmodular coil system is used in this study. Modular coils
are separated into two groups (group A and B). Group A
consists of coil I, coil 2 in Fig. 1. These coils are besidevertically elongated cross-section and the total numberof coils is 8 for full torus. Group B consists of the othercoils, namely coil 3, coil 4, coil 5, and the total number
is 12. Current of group A is reduced and the mirror
ripple component is artificially controlled. By changing
current as this, a mirror ripple, Be1, is added. TheFig.3 Normal component (lB.nlll4l of produced magnetic field on the outermost magnetic surface. (a)for modular coils that produces a mirror ripple on
the magnetic axis. (b) for modular coils that is designed to reduce a mirror ripple at the magneticaxis.

^2.5

Eo2-E t.S;1"r Large Bo, CoilSmall Bo, Coil

0.5
0-:-Artificial Gontrol

024681030, / Boo (%)
Fig. 4 Averaged normal component when mirror ripple
at magnetic axis is artificially controlled (symbolsof white circle). Mirror ripple is controlled by theaveraged value of normal component on the outermost

magnetic surface as a function of mirror ripple at themagnetic axis is shown in Fig. 4. Each open circle
corresponds to the case that current of group A isreduced to the level of 97.5 Vo,95.0 Vo,92.5 Va,90.0 Vo,85.0 Vo, 80.0 Vo (written here in order of Bs1lBss) ofratio of currents in two groups of modular coils.
The white rectangle symbol corresponds to improved modular coil system. The crossed rectangle symbol corresponds to modular coil systemthat produces a mirror ripple on the magneticaxis.
368

Shimizu A. et al., Design of Magnetic Coil System for CHS-qa

current of group B. Open circles show that the mirror
ripple has small effect on lB.nlllBl when the mirror
ripple is smaller than 3 7o. Even if the mirror ripple 861/
Be6 is decreased from 3 Vo tomirror ripple component, Bs1, is very small. Therefore,
the optimization of modular coils by only the evaluationof the normal component on the outermost magnetic
surface does not effectively suppress mirror ripples. In
order to reduce the mirror ripple on the magnetic axis,I Vo, the accompanying

change inlB.nlllBl is only 0.1 7o.

the evaluation of this ripple is included in theIn this Figure, the white rectangle corresponds toimproved modular coil system. The crossed rectangle
corresponds to the first version of coil system that
produces mirror ripple at the magnetic axis (Magnetic
field is same to one shown in Fig. 2(a)). Since thedependence of 861 on lB-nlllBl is very weak where lB.zll
lBl - 1.5 Vo, the Bo1 can not always be reduced below
2 Vo by only reducing lB.nlllBl. The amount of 2 Vo is
the level of largest component in other nonaxisymmetric components, and it is the maximum
allowable level for the Bs1 component. It is not effective
to reduce the rnirror component (namely, to reproduce
the designed quasi-axisymmetric configuration) by only
evaluation of the normal component, therefore a direct
evaluation of Bs1 is needed and included in our coil

optimization process of modular coils. By this inclusion,
improved modular coils are designed and the quasiaxisymmetric configuration is obtained with a goodprecision.

References[1] J. Niiehrenberg and R. Zille, Phys. Lett. A129,ll3(l988).

[2] J. Niiehrenberg et al., Transactions of Fusion Technology 27,71 (1995).
t3l M. Yokoyama et al., Plasma Fusion Research 70,
s42 (1994).

[4] J. Niiehrenberg, W. Lotz and S. Gori, Theory ofFusion Plasmas (International School of Plasma
Physics, SIF, Bologna, 1994),3.
[5] P. Garabedian, Phys. Plasmas 3,2483 (1996).
t6l D.A. Spong, S.P. Hirshman et a/., Nucl. Fusion 41,7rr (20or).
[7] A.H. Boozer, Phys. Fluids 23,904 (1980).
t8l K. Matsuoka et al., Plasma Phys. R.ep. 23, 542

optimization plocess.

3. Conclusion
A modular coil system is designed by minimizing
the normal component of produced magnetic field on
the outermost rnagnetic surface. However, the magnetic(1997\.

field produced by this modular coils has mirror ripple atthe magnetic axis which is not negligible for the quasi-

t9l S. Okamura et al., I. Plasma Fusion Res. SERIES1, 164 (1998).
[10] P. Merkel, Nucl. Fusion 27,867 (1987).

axisymmetry.

It is found that the modification of the shape ofoutermost magnetic surface caused by the change of369