Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)508–511
000–000
J. Plasma Fusion

The Effect of Coil Misalignment
on Particle Transport in Quasi-Axisymmetric Systems
SHIMIZU Akihiro, OKAMURA Shoichi, ISOBE Mitsutaka, SUZUKI Chihiro,
NISHIMURA Shin, AKIYAMA Tsuyoshi, MATSUOKA Keisuke and NEMOV Victor V.1
National Institute for Fusion Science, Toki 509-5292, Japan
1Institute of Plasma Physics, National Science Center, 310108, Kharkov, Ukraine(Received: 11 December 2003 / Accepted: 4 March 2004)

Abstract
The effect of the misalignment of modular coils on neoclassical transport in CHS-qa is discussed in this paper. Inorder to calculate the effective helical ripple that characterizes the neoclassical transport of the stellarator configuration, NEO code is used which gives the effective helical ripple numerically. A displacement is put artificially on themodular coils of the CHS-qa, and the effect of this on the profile of the effective helical ripple is evaluated quantitatively. One objective of this study is to obtain useful information to determine the acceptable error level for the assembly of modular coils from the viewpoint of engineering. The calculation results of the NEO code show that if the levelof displacement is smaller than a few centim eters, it has only a very small effect on the neoclassical transport. The a
mount of change in the effective helical ripple is small. Therefore the displacement of the modular coils caused by theelectromagnetic force at a 1.5 T operation of CHS-qa does not lead to significant problems in terms of neoclassicaltransport.
Keywords:
quasi-axisymmetric stellarator, modular coil, effective helical ripple, error magnetic fieldcode uses the solution for the distribution function of thebounce averaged kinetic equation, the calculation time isshorter than for a Monte Carlo calculation. In this paper wewill show the dependence of the error in the position of thecoil on neoclassical transport and give useful information for
determining the acceptable level for the error in coil position.

1. Introduction
In recent years, we have continued our design work ofan advanced stellarator device, CHS-qa, that has a quasiaxisymmetric configuration [1,2]. By using a numerical optimization method, a modular coil system was designed for it,
which was reported on in our previous paper [3]. In the subsequent paper, we reported on the property of neoclassicaltransport for the configuration produced by this modular coilsystem [4].
In this paper, we will discuss the effect of the error in
the position of the coil on neoclassical transport. Its investigation is important for determining the acceptable level oferror for the modular coils, since it influences the total cost of
the device. We are also interested in the sensitivity of theerror in the position of the coil in terms of the transport,
because in the machine operation the modular coil is subject
to a large electromagnetic force, which causes changes in its
shape and position.
In order to calculate the effect of the error in the positionof the coil on neoclassical transport, NEO code [5] is used.
The advantage of this code is that, 1) the code can include theeffects of particles trapped in any magnetic field ripples.
Therefore, it can calculate quantitatively the effect of thecomplicated geometry of the magnetic field. 2) Since the2. Modular coil system
The reference configuration of CHS-qa, “2b32” version[6], is used in this paper. The major radius is 1.5 m, thetoroidal periodic number is 2, and the aspect ratio is 3.2. The
maximum magnetic field is 1.5 T.
The modular coil system optimized for this configuration is used. The total number of modular coils is 20, and forthe stellarator symmetry the coils consist of only five types ofshapes. All coils are located between two surfaces. One is asurface that is 26 cm away from the plasma boundary, andthe other is 45 cm away from the plasma boundary. The averaged minor radius of the coil is about 70 cm. The minimum
radius of curvature is 25 cm, and the minimum distance
between adjacent coils is 22 cm. The top view of the modular
coil system is shown in Fig. 1a). We will refer to the coil by
the number written in this figure. We use the vacuum mag©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: akihiro@nifs.ac.jp
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Shimizu A. et al., The Effect of Coil Misalignment on Particle Transport in Quasi-Axisymmetric Systems

netic field configuration produced by this modular coil system and ignore the finite beta effect of the plasma.
In order to evaluate the effect of the error in the coilposition, the modular coil is moved artificially. The displacements put on the modular coil system are as follows: In CaseA, COIL 2, COIL 4, COIL 8, COIL 12, COIL 14, C OIL 18
are moved in the positive direction along the z axis, andCOIL 3, COI L 7, COIL 9, COIL 13, COIL 17 and COIL 19
are moved in the negative direction along the z axis, as
shown in Fig. 1b). The amount of coil displacement is givenby ∆z. The coils are moved by the same distance, only thedirection is different. For the sake of simplicity of the calculations, the displacement is assumed to have stellarator symmetry. In Case B, COIL 2, COIL 4, COIL 7, COIL 9, COIL
12, COIL 14, COIL 17 and COIL 19 are moved in the +R
direction, while COIL 3, COIL 8, COIL 13, COIL 18 aremoved in the -R direction, as shown in Fig. 1c). The amountof coil displacement is given by ∆R in this case.
Firstly, we will present Poincare plots of the magnetic
surfaces. It is interesting to note the extent to which the magnetic surfaces are broken by the displacement of the coils. InFig. 2, the poincare plots for the displacement of ∆z = 6 cmin Case A and ∆R = 6 cm in Case B are compared with thoseof the standard configuration. In this figure, two poloidal
cross sections for each of the cases are shown. One is at thetoroidal angle of 0 degrees and the other is at 90 degrees. Thechange in the volume of the confinement region is little, andthe magnetic surfaces are insensitive to the displacement ofthe coils. A good magnetic surface is maintained, even for arelatively large displacement in ∆z = 0.1 m or ∆R = 0.1 m. In
the poincare plot for ∆z = 6 cm in Case B, islands appear near
the center of the plasma, since the rotational transform has
the rational number 0.4 at this position. However, the size ofa)b)yCOIL 6COIL 7COIL 5

3. Calculation result
For the standard stellarator configuration for which themagnetic field is characterized only by the toroidal ripple εt
and the helical ripple εh, the neoclassical diffusion coefficientin the 1/ν regime is proportional to ε3/2
h . If the configuration
has multi-helicity, the diffusion coefficient is no longer proportional to ε3/2
h . In this case, by replacing εh with εeff , the diffusion coefficient of the neoclassical transport can be characterized. The NEO code calculates numerically this effective
helical ripple, εeff , by using the solution of the bounce averaged kinetic equation for the distribution function of thetrapped particle. As described in paper [5], εeff can be calculated asπR20⎛⎛ Ls ds ⎛ ⎛ Ls ds
3/2
εeff=lim|∇ψ |L→∞⎝⎝⎝⎝s8 2BB
002jmax
absBmax /B0Ĥj,db'
abs /B
Bmix
Iˆj0
j=1

∫∑∫

1b'Ĥj =Iˆj =

∫∫

Sjmax

Sjmin

∫

-2Sjmax

Sjmin

dsBdsBB ⎛ B01⎛√b' − B ⎝4 B − b' ⎝ |∇ψ |k ,G0

√1 − B b' .
B

(1)0Here, R0 is the major radius, B0 is the reference magnetic
field, ψ is the magnetic surface label, kG ≡ (h × (h · ∇) · ∇ψ /|∇ψ | is the geodesic curvature of a magnetic field line, andabs
absand Bmax
are the minimum andh is a unit vector B/B. B minmaximum values of B within the interval from 0 to L s,
respectively. Ls is a maximum value of path length in theintegral along a magnetic field. s is the magnetic field length,a)Case AR

COIL 4COIL 8the island is small in this case.

COIL 3COIL 9COIL 2COIL 10COIL 1z
COIL 11COIL 20

COIL 12x c)

Case Bb)

COIL 19

COIL 13COIL 18

COIL 14COIL 15

COIL 17
COIL 16

Fig. 1 a) Top view of a modular coil system for CHS-qa. The
plasma boundary of “2b32” configuration is also
shown. The axis of the Cartesian coordinate systemand the R-axis of the cylindrical coordinate system
shown in this figure are used in this paper. b) The displacement of the coils in Case A. c) The displacementof the coils in Case B.

Fig. 2 a) Poincare plots at the toroidal angle of 0°, where the cross section is vertically elongated. Left: for thestandard configuration. Center: for ∆z = 6 cm in Case
A. Right: for ∆R = 6 cm in Case B. b) Poincare plots atthe toroidal angle of 90° where the cross section ishorizontally elongated. The solid line represents theoutermost magnetic surface of the ”2b32” configuration.

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Shimizu A. et al., The Effect of Coil Misalignment on Particle Transport in Quasi-Axisymmetric Systems

and sjmin and sjmax are the turning points of the trapped particle
j. b' is an integration variable, which corresponds tov2/(J⊥ B0), v is velocity and J⊥ is the perpendicular adiabaticinvariant.
The calculation result of the NEO code for Case A isshown in Fig. 3a). ∆z is changed from 0 to 10 cm. If the displacement is smaller than 2 cm, its effect is not seen in theprofile of the effective helical ripple. The local increase
appears at the averaged minor radius of 0.1 cm for ∆z = 6 cm
and at 0.2 cm for ∆z = 10 cm. This is for the effect of theisland. By the way, the ε3/2
eff of CHS, which is of the sameorder as the standard stellarator, is about 1.0 × 10-4 at theplasma center, and 0.2 at the edge. All of the effective helicalripples in Case A are smaller than for CHS and good characteristics are maintained. Thus, the property of the neoclassical transport of CHS-qa is insensitive to the vertical displacement of the modular coils.
The calculation result for Case B is shown in Fig. 3b).
The dependence of the displacement of coils on the transport
is similar to that in Case A. Therefore, the property of transport is also insensitive to the horizontal displacement of thecoils. The difference from Case A is that the local increase instandard∆z=2cm∆z=4cm-2

εeff3/2

104. Conclusion
The effect of the displacement of the modular coils on
neoclassical transport was investigated using NEO code. Ifthe displacement is smaller than 2 cm, its effect on the neoclassical transport is very small. By the way, we alreadyknew that the displacement of coils by the electromagnetic
forces in the machine operation of 1.5 T is below about a few
millimeters. Therefore, from the point of view of neoclassicaltransport, this displacement can be neglected. We can conclude that it is sufficient that the accuracy of the coil assembling should be below a few centimeters.

a)∆z=6cm∆z=10cm

Rotational transforma)

the effective helical ripple does not appear in Case B. This isbecause the profile of the rotational transform is different. InFig. 4a) and 4b), the profiles of the rotational transform forCase A and B are shown. The change in the rotational transform in Case A is larger than in Case B. In Case A, for ∆z = 6and 10 cm, the rational number 0.4 is crossed and n/m = 2/5
islands appear. This island structure causes the increase in theeffective helical ripple, as seen in Fig. 3a). On the other hand,in Case B, the change in the rotational transform is small.
Therefore, the rational number 0.4 is not crossed.

10-3

10-40.1
0.2
0.3
0.4
averaged minor radius (m)

0.45
0.4
0.35

standard∆R=2cm∆R=4cm-2

1000.5b)∆R=6cm∆R=10cm

Rotational transform

0εeff3/2∆z=6cm∆z=10cm

0.3

10-5b)

standard∆z=2cm∆z=4cm

0.5

10-3

10-4

10-5

0.1
0.2
0.3
0.4
Averaged minor radius (m)standard∆R=2cm∆R=4cm

0.50.5∆R=6cm∆R=10cm

0.45
0.4
0.35
0.3

00.1
0.2
0.3
0.4
Averaged minor radius (m)00.5

Fig. 3 a) The profiles of the effective helical ripple when thedisplacement of Case A is considered. At the averaged minor radius of 0.1 m for ∆z = 6 cm cm and 0.2m for ∆z = 10 cm, the increase in the effective ripple
can be seen. This is because of the effect of an island,
n/m = 2/5. b) The profiles of the effective helical ripple
when the displacement in Case B for a coil system isconsidered.

0.1
0.2
0.3
0.4
Averaged minor radius (m)0.5

Fig. 4 a) The profiles of the rotational transform when thedisplacement of Case A for the coil system is considered. b) The profiles of the rotational transform when
the displacement of Case B is considered.

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Shimizu A. et al., The Effect of Coil Misalignment on Particle Transport in Quasi-Axisymmetric Systems

However, there are issues remaining that should be considered. (1) In the calculation of the NEO code shown in thispaper, toroidal periodicity and stellarator symmetry areassumed for the magnetic configuration, namely for thegeometry of the displacement of the modular coils. In themore general case, this assumption may be inappropriate. Inorder to calculate the effective helical ripple for this situation,a modification of the NEO code is needed. This will be carried out in a future study. (2) The profile of the rotationaltransform of CHS-qa is low shear. Therefore, low mode large
islands may appear in the confinement region through achange in the rotational transform caused by the error in thecoil position. This may lead to a significant deterioration forconfinement. In addition, the appearance of a low mode
rational number of the rotational transform in the confinement region causes MHD (Magnetro Hydro Dynamics) instability, which may also adversely affect the confinement
through such as disruption.

Further detailed calculations on these two remaining
issues are required to determine the acceptable error level
more safely.

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