J. Plasma Fusion Res. SERIES, Vol. 5 (2002) 355-359

Compatibility of Different Elements of Gonfinement
Gharacteristics in Ouasi-axisymmetric Stellarators
OKAMURA ShOiChi, NISHIMURA ShiN, ISOBE MitSUtAKA, SUZUKI ChihirO,MURAKAMI Sadayoshi, YOKOYAMA Masayuki, SHIMIZU Akihirorand MATSUOKA Keisuke
National Institute for Fusion Science, Toki 509-5292, Japan
Nagoya University, Nagoya 464-8603, Japan(Received: ll December 2001 / Accepted: 3 August 2002)

Abstract

In order to understand the relation between several confinement characteristics of quasiaxisymmetric stellarators, optimization runs were made with a special weight on a selected characteristicsof the configuration without considering other aspects. A limit of highly axisymmetric configuration wasobtainecl with helical ripple of 0.12 7o at the plasma boundary sacrificing the magnetic well and rotationaltransform. Another design with reduced helical ripple with high rotational transform gave improved
transport in highly collisionless regime but it also lost magnetic well. On the other hand the configuration
with recluced Shafranov shift and improved MHD stability was designed with slightly smaller helicalripples. The method to obtain simultaneously high quasi-symmetry and good MHD property has not beenfound.

Keywords:
quasi-axisymmetric stellarator, helical ripple, Shafranov shift, optimization, MHD stability, collisionlesstranspolt

1. lntroduction
In the magnetic configuration design for thetoroidal confinement system, it is generally very
difficult to satisfy all important conditions (particle

characteristics is alreadv established at least in theorbits, magnetic shear, MHD stability, etc.) necessary
for the good confinement characteristics. Usually we try
to find a good compromising point for those conditions
following to the objectives of the design work or theexperimental plan. One of typical examples is found in
the magnetic configuration of heliotron/torsatron device
where good particle orbits and the linear ideal MHD
stability are contradicting with each other depending on
the parameter of magnetic axis position. In this case, thedecision of delermining a compromising point can bemade straightforwardly since the relation between thosefrequently called advanced stellarators. ThisC o r re sp ondin g autho r' s e - mai I : o kamura @ nifs. actheoretical work.

A quasi-axisymmetric stellarator concept [1,2] is
one of new generations of helical systems which areconfiguration is actually of a stellarator because it has
the rotational transform (iota=llq) given by the external

colls. Iota value typically ranges from 0.2 to 0.5
depending on the individual design. Most importantcharacteristic of this concept is that the drift orbits oftrapped particles are almost equivalent to those intokamaks (negligibly small number of helically trappedparticles). Because the high beta stability is a basic
requisite for the new advanced stellarators, configuration@2002 by The Japan Society of Plasma
Science and Nuclear Fusion Researchjp355

Okamura S. et al., Compatibility of Different Elements of Confinement Characteristics in Quasi-axisymmetric Stellarators

designs have been made for a good combination of thefor the equilibrium, the Boozer spectrum of magnetic

axisymmetry and MHD stability for high beta

field was calculated. By summing up amplitudes of nonaxisymmetric components of the spectrum over all
magnetic surfaces, the level of axisymmetry isevaluated. The 41 variables are gradually modified toequilibrium [3,4]. From the aspect of designing a goodexperimental device, present designs might be sufficientfor obtaining a good plasma confinement with verysmall helical ripples because the important issue in theconfinement would be anomalous transport.
However, when we consider the prospect to thethe direction to decrease this value. The aspect ratio wasfixed. In the quasi-axisymmetric stellarator, therotational transform is created by the non-axisymmetry

reactor for quasi-axisymmetric stellarators, theconfinement of high energy particles in the highlyin the geometric shape of the torus. Since thegeometrically more symmetric configuration naturally
has better axisymmetry in the Boozer spectrum, theoptimization with Boozer spectrum gradually push thecollisionless regime must be discussed. One solution
was found [5] with additional mirror ripples which iseven less axisymmetric than the standard designs ofquasi-axisymmetric stellarators. In this paper, we tried

to find a solution by reducing the helical ripple
components as much as possible. We intentionallyabandon to consider the MHD stability and high level ofrotational transform in this work, in order to find an
extreme limit of quasi-axisymmetry with smaller but
acceptable level of external rotational transport. Suchprocess is also instructive to find which confinement
characteristics are cooperative or contradictory to thequasi-axisymmetry. Other examples of configuration
design are also given with the consideration of high€f 0.00s!

E-EO(l)E -0.00s

Eolevel of rotational transform or the MHD stability.
Based on these results, the discussions about thecompatibility of confinement characteristics of quasi--0.01-0.015

axisymmetric stellarators are given.

o.z

0.4

0.6

0.8

rho

2. Configuration with Very Small HelicalFig. 1 Boozer spectrum of the magnetic field ripples for
Sym-1 configuration. Thin lines are axisymmetric

Ripplescomponents.

The characteristic of quasi-axisymmetry is clearlyshown when the magnetic field structure is represented
in the Boozer coordinates. Since the particle orbits canbe described with only the spectra of magnetic fieldstrength in this coordinates [6], the axisymmetry in theBoozer spectrum guarantees the tokamak-like trapped
particle motion. The configuration design was made tofind out how much axisymmetry in Boozer spectrum
can be obtained if we improve the symmetry without

considering the MHD stability tentatively. We started
from one of quasi-axisymmetric configurations which
has toroidal periods N=2 and an averaged aspect ratio
3.9. The rotational transform at the magnetic axis is 0.23
slightly increasing to 0.28 at the plasma boundary. Thisconfiguration was designed with care of both quasi-51015?Oaxisymmetry and the magnetic well.
The improvement of configuration was made with
4l variables of Fourier coefficients for the boundaryshape. Based on the fixed boundary VMEC calculation

Toroidal tums
Fig. 2 Variation of magnetic field strength along fieldlines of Sym-l configuration for three minor ra-dius.
356

Okamura S. et al., Compatibility of Different Elements of Confinement Characteristics in Quasi-axisymmetric Stellarators

solution to the axisymmetric geometric shape of thetorus which has lower value of the rotational transform.

of this configuration with a standard configuration(2b32) for CHS-qa device design [3], the crescentshapes are similar for the vertically elongated cross

We stopped this work when the rotational transformdecreased to 0.2 at the boundary.

sections while the shape of the horizontally elongatedcross section is more circular and the direction of thetriangularity is opposite to each other (a tip of triangle
faces toward the torus center in Fig. 3). The magnetic

Figure 1 shows the Boozer spectrum of the finalsolution (Sym-1). The amplitudes of the spectra are therelative values to the toroidal field. The largest nonaxisymmetric component is the mirror term (m, n)=(0,l) where m and n are the poloidal and toroidal mode
numbers per period respectively. At the half minor
radius, the relative amplitudes of non-axisymmetriccomponents are less than 0.013 Vo of the toroidal field.
Figure 2 shorvs the variation of the magnetic fieldstrength along the field line for three minor radii. All
three lines show toroidicity with negligible levels ofwell is almost vanishing which is a direct consequenceof the direction of the triangularity. The Shafranov shiftis very large and consequently the equilibrium beta ismuch lower than a standard configuration of CHS-qa.
This solution lost good MHD property in exchange forthe good symmetry.

3. Configuration with Small Helical Ripples
and High Rotational Transform

non-axisymmetric (helical) ripples.
Three poloidal cross sections of magnetic surfaces
are shown in ttig. 3. If we compare the boundary shape

1.0

Flux Surfaces
Phi = 0o

Flux Surfaces
Phi = 45"

MN

0.0

0.5

1.0

Even though the solution Sym-l has a very goodquasi-axisymmetry, the calculation of alpha particle

01.5

Flux Surfaces
Phi = 900

@2.0

1.5

RR

Fig. 3 Three cross sections of magnetic surfaces for Sym-1 configuration

Ffux Surfaces
Phi =Flux Surfaces
Phi = 450

1.0

0.0

-0.5

-1.0

1.0

DN
1.5

2.0

1.0

1.5

2.0

Flux Surfaces
Phi = 900

1.0

r.5

2.0

Fig. 4 Three cross sections of magnetic surfaces for Sym-2 configuration,
357

Okamura S. et al., Compatibility of Different Elements of Confinement Characteristics in Quasi-axisymmetric Stellaratorsof the toroidal magnetic field which is smaller by afactor of4 than those in2b32 configuration.

confinement did not show very promising results for its
vacuum configuration. Small rotational transform causes
relatively large direct losses of alpha particles due to thebigger Larmor radius. In order to solve this problem, weThe Monte Carlo calculation was made for this new

configuration to evaluate the neoclassical diffusion
coefficient. Figure 5 shows the transport coefficients fornext started from the CHS-qa standard configuration
2b32. lt has a toroidal periods N=2, the rotational

1 keV electrons for 2b32 and Sym-2 configurations withadditional parameters of 1.5 m major radius and 1.5 T
magnetic field. The diffusion coefficient is evaluated byfollowing the motions of electrons that are initiallydistributed at the magnetic surface of half minor radiusand diffuse by collisions. The diffusion coefficients forthe conventional helical device CHS are also shown fortransform of 0.4 at the boundary and the averaged aspectratio 3.2. The configuration improvement was madetowards the better quasi-axisymmetry with a conditionof keeping the edge rotational transform at 0.4 but again
without considering the MHD stability. The magnetic
surfaces of the final solution of this optimization work

the comparison. Significant improvement in theconfinement for a very low collisionality regime is(Sym-2) are shown in Fig. 4. The non-axisymmetric
Boozer spectrum components were decreased below 17oshown for the Sym-2 configuration.

However, the magnetic well is lost again for thisconfiguration. In Fig. 4, the similar change in the shapeof cross section is recognized to the case of Fig. 3, thatis, the direction of the triangularity in the horizontallyelongated cross section is opposite to 2b32
configuration. By considering these two examples, it isnoticed that the shape of the horizontally elongated1oou,cross section determines the depth of the magnetic wellt 0.1

although the crescent shape of the vertically elongatedocross section must also help to create the magnetic well.
0.01

0.001

4. Gonfiguration lmproved for the MHD
Stability
Finally we try to improve the configuration with
1000

10110smore emphasis on the MHD property. The optimization
was made, started from 2b32 configuration, for the106

collisionality (st)reduction of the Shafranov shift and the Mercier

Fig. 5 Diffusion coefficients tor 2b32 and Sym-2 configurations. Those values for two configurations inCHS are also shown.

F1ux SurfacesPhi =stability together with the condition of the better quasiaxisymmetry. The improvement was successful for theFl-ux Surfaces
Phi = 45o

Flux surfaces
Phi = 90o

0.5

N

0.0

-0.5
-1 n

0.5

1-0

DN
1-.0

t-.5
R

2.O

2.0

1-0RFig. 6 Three cross sections of magnetic surfaces for improved MHD configuration.

358

Okamura S. et al., Compatibility of Different Elements of Confinement Characteristics in Quasi-axisymmetric Stellarators

MHD properties without affecting the axisymmetry.

simultaneously, the reduction of helical ripples and theAbout 20 7o reduction of the Shafranov shift is obtained
for the new configuration. The Mercier stability is also

control of the direction of triangularity in thehorizontally elongated cross section (facing toward theoutside of torus). The remaining problems are 1) to try

improved. However the improvement in the quasiaxisymmetry was very small even by making furtheroptimization, as far as these conditions of MHD

to obtain a favorable direction of the triangularity
starting from the good axisymmetric solution and 2) toproperty were applied. Figure 6 shows the cross sections

try to improve the MHD property of good axisymmetryof magnetic surfaces for the improved configuration. It

with the effect of the plasma current.

is very distinct that the triangularity of the horizontally

References

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(Proc. Workshop Varenna, 1994), Editrice
5. Conclusion

Compositori, Bologna, p.3 Q999.

By comparing shapes of cross sections of Figures[2] P. Garabedian, Phys. Plasmas 3,2483 (1996).
[3] S. Okamuta et al., Nucl. Fusion 41, 1865 (2001).
t4l G.H. Neilson et al.,Phys. Plasmas 7,l9ll (2000).
[5] S. Gori et al.,Plasma Phys. Control. Fusion 43, 137

3,4 and 6, it rs clear that the good MHD property isdetermined by the shape of the horizontally elongatedcross section. Especially important is the triangularity ofthese cross sections. From the experiences of several
examples of the optimization procedures we have made(2001).

[6] A.H. Boozer, Phys. Fluids 23,9O4 (1980).

so far. it was difficult to obtain two results359