J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)199–202
000–000

The Effect of Reduced Pfirsch-Schlüter Current
on the Ideal MHD Stability and Alpha Particle Confinementin 2-period Compact Configurations
COOPER W. Anthony, ISAEV Maxim Yu.1 and OKAMURA Shoichi2
Centre de Recherches en Physique des Plasmas, EPFL, Lausanne, Switzerland
1Nuclear Fusion Institute, RRC “Kurchatov Institute”, Moscow, Russia
2
National Institute for Fusion Science, Toki, 509-5292, Japan(Received: 9 December 2003 / Accepted: 1 April 2004)

Abstract
The CHS-qa 2b32 [1] configuration has been considered as a starting point for the further numerical optimisation
towards small Pfirsch-Schlüter current. The configurations obtained have about 4 times smaller PS current and arestable with respect to Mercier modes for β values around 0.085. However, the α-particle confinement and the MagnetoHydro-Dynamic (MHD) stability with respect to the ballooning modes appear to be more difficult issues to improvein the configurations obtained. In particular, the maximum value of β, stable with respect to the ballooning modes,
has been obtained on the level of 0.02. Detailed calculations of the configurations computed with the 3D fixed boundary
ideal equilibrium code VMEC2000 [2], with the 3D ideal MHD stability code TERPSICHORE [3], which reconstruct
the equilibrium state in Boozer magnetic coordinates and with the MCT [4] code, which follows collisionless monoenergetic α -particles drift orbits, are presented.
Keywords:
stellarators, quasi-axisymmetry, 3D codes, CHS-qa

1. Introduction

beginning of the period) elongation E0 = 5.08, while therotational transform is designed to be limited between 1/3 and2/5. The Shafranov shift is reduced by about 30% comparedwith the 2w39 configuration, the drop of the central rotationaltransform is also suppressed and the β limit is above 0.05
with respect to the Mercier criterion [9]. Recently a newcandidate 2b32m3 has been found with a larger elongation
E0 = 5.33, small negative shear, larger aspect ratio A = 3.41
and even further reduced Shafranov shift. This case issomewhat more stable than the 2b32 candidate.
In this paper, a further optimisation of the 2b32m3 casehas been performed with respect to small Pfirsch-Schlüter(PS) current. Such optimisation should generally give a small
Shafranov shift and a high equilibrium β limit, theconfiguration becomes non-sensitive to the plasma pressure.
During this optimisation, we can impose different weightfactors for other desired characteristics of the magnetic
configurations such as quasi-axisymmetry, aspect ratio,
Mercier criterion, maximal elongation (for the coil design
restrictions), etc.
The paper is organised as follows. Section 2 presents thePfirsch-Schlüter current minimisation issues. Section 3 showsthe equilibrium, Mercier and ballooning stability as well as

Quasi-axisymmetric helical configurations have axiallysymmetric magnetic field in the Boozer magnetic coordinate
system [5], therefore tokamak-like neoclassical transport
properties can be expected. Because helical systems need noinductive current, a quasi-axisymmetric stellarator has
essentially no problem with respect to steady-state operation.
Such type of quasi-symmetry, quasi-axisymmetry [6],constitutes the basis of new stellarator projects, NCSX [7]and CHS-qa [8].
CHS-qa is a 2-period quasi-axisymmetric compact
concept with a low aspect ratios of A = 3.2 – 3.9 which iscurrently under design and optimisation at the National
Institute for Fusion Science (NIFS, Japan). During last few
years, several candidates of magnetic configuration for CHSqa with the different optimised features have been proposed.
The first engineering design was made for the 2w39
configuration with an aspect ratio A = 3.9 and a very weakly
increasing rotational transform profile approaching 0.4 at theplasma edge. The β value of this configuration is limited
below 0.02 where a large Shafranov shift accompanied witha drop of central rotational transform appears.
In accordance with VMEC2000 calculations, the next
candidate 2b32 with A = 3.36 has a larger initial (at the©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: wilfred.cooper@epfl.ch1199

Cooper W.A. et al., The Effect of Reduced Pfirsch-Schlüter Current on the Ideal MHD Stability and Alpha Particle Confinement

α-particle confinement properties of the configuration with
reduced PS current and is followed by a summary.

fdsdθ dζ )/dV of the PS components is performed using
different weight functions W. In the JMC code this weight isW JMC = g / | ∇S | 2, where g is Boozer jacobian. In theVMEC code this weight is WVMEC = g . We have performed
also the TERPSICHORE calculations of currents ratio 〈 j||2 〉/〈 j⊥2〉 with both WJMC and WVMEC weights.
The nice coincidence of the current ratio calculations
obtained from the JMC code and from the TERPSICHORE
codes with the WJMC weights and the PS ratio calculations
obtained from VMEC and from the TERPSICHORE codewith the WVMEC weights has been achieved for the test W-7X
configuration.

2. Pfirsch-Schlüter current minimisation
In this section, we present several details connected withPS current minimisation and its calculations with the VMEC,
JMC and TERPSICHORE codes. By definition, the PfirschSchlüter current j|| is that component of the plasma secondary
current j connected with the plasma pressure that is parallelto the magnetic field B:⎧( B×∇s )⎫j = j|| + j ⊥ = p′(s ) ⎨α B +⎬B2 ⎭⎩3. Optimisation

where the perpendicular component is j⊥, the plasma pressureis p(s), s is a flux surface label proportional to the toroidal
flux, s = 1 corresponds to the plasma edge and α is a periodic
function. A convenient nondimensional measure of thePfirsch-Schlüter current in different configurations is givenby the ratio of the average square of the PS current component
along B over the corresponding average of the perpendicularcomponent 〈 j||2 〉/〈 j⊥2 〉. This ratio can be used as a targetfunction to reduce the Pfirsch-Schlüter current and to obtaina large limit for the equilibrium β -value. Here andsubsequently, β indicates the 〈β 〉 value.
One example of PS minimisation, a 6-period quasihelically symmetric configuration with a high β ≈ 0.5
approaching the equilibrium limit, has been previously
obtained with 〈 j||2 〉/〈 j⊥2 〉 ≈ 0.1 [10]. The current ratio in [10]was calculated with the JMC code. It was found by using theparaxial approximation [11] that large elongation and large
rotational transform can significantly reduce Pfirsch-Schlüter
currents in quasi-symmetrical configurations. This conclusion
is also valid for quasi-axisymmetric configurations like CHSqa. However, the smaller number of periods (2 instead of 5or 6) and the additional restrictions from the coil designpreclude achieving the same level of PS currents as in Ref.
[10].
In our numerical optimisation with the help of the NAG
Fortran library E04UCF, based on a sequential quadratic
programming method, we use the 3D fixed boundary ideal
equilibrium VMEC code [2]. Subsequent improvements of theVMEC code, versions VMEC2000-6.80 [12] andVMEC2002-7.10, have extended the range of applicability forthe computation of plasma equilibria to low aspect ratio
systems with net toroidal current such as NCSX and QOS[13].
As a test of a system with small Pfirsch-Schlüter currents
and of the formulation used to calculate these currents, wehave made the comparison of the PS calculations obtainedfor the Wendelstein-7X (W7X, Greifswald, FRG)
configuration with the VMEC, JMC and TERPSICHORE
codes. The JMC code [14] and the TERPSICHORE code [3]calculate the magnetic field strength B in so called magnetic
or Boozer coordinates taking into account a diverging parallel
current density at rational values of the rotational transform(ι). The averaging over the plasma layer dV, 〈 f 〉 = ∫ (W ×The optimised configuration called chs67 is obtainedusing the current ratio target, constant aspect ratio, weakquasisymmetry condition (the axisymmetric mode is 2 times
larger than any non-symmetric mode, but only at the edge)
and edge ι value near 0.55. Such value of ι(1) is above thedangerous rational value 1/2 which has a possibly large
resonant effect. The pressure profile used in the calculations
is proportional to the function 14 – 22s + 8s2. During theoptimisation it was possible to decrease the PS current almost
4 times with respect to the initial level (see Fig. 1). The
strongly elongated (E0 = 10.4) configuration chs67 has a verysmall PS current, min〈 j||2 〉/〈 j⊥2 〉 = 0.36, which leads to a very
high limit for the equilibrium β-value above 0.20 for 2-period
configurations. The region s < 0.1 should not be taken into
account because of the errors of VMEC near the magneticaxis.
The Boozer Fourier spectrum of chs67 (Fig. 2) can becalled weakly quasi-axisymmetric only near the edge, here
the magnetic field strength B = Σ Bmn cos(mθ – nζ ), θ and ζ
are poloidal and toroidal Boozer angles. There the firstsubscript ‘m’ identifies poloidal index, the second ‘n’ - thetoroidal index per period. Near the axis, the dominantcomponent is a mirror-type mode (0,2), while another mirrortype component (0,1) is not very large for all radial points.

4CHS2B32
CHS2B32M3
CHS67
W7X

3.5

2.5

2
⊥

<j >/<j >32||21.5
1
0.5
00Fig. 1

2200

0.2

0.4

0.6
0.8
1
SThe current squared ratio versus flux surface label
obtained with the VMEC code for chs2b32, chs2b32m3,chs67 and W7X configurations with β = 0.05.

Cooper W.A. et al., The Effect of Reduced Pfirsch-Schlüter Current on the Ideal MHD Stability and Alpha Particle Confinement

One can say the constraint of small PS current significantly
destroy the quasi-axisymmetry.
Nonsymmetrical modes lead to the increasing of the α particle losses during the initial 10–4 s period of confinement,
see Fig. 3. The collisionless confinement of α-particles in
chs67, β = 0.085 is calculated with the guiding-centre orbit
code MCT [4] with launching surfaces Ψ0 = 0.0625 (lowercurve) and Ψ0 = 0.240 (upper curve). Normalisation: plasma
volume 103 m3, magnetic field 5 T. Two thousands particles
randomly distributed in the pitch angle variable and aroundthe birth surface are followed; the dashed lines indicate thepercentage of the reflected particles. Almost all trapped
particles are lost during a time of 0.01 s. However, this is thecase also for the starting configuration chs2b32m3, which canbe attributed to the high sensitivity of α particle confinement
in quasi-axisymmetrical systems to small nonsymmetricalcomponents.

100
2 : start surface Ψ0=0.24

Lost [%]

Fig. 3

MERCIER CRITERION

0.1

BMN

0.05
(0,1)
0(1,1)

0.4

0.6

0.810 3t[s]

10 2

10 1

0

10β=0.005
β=0.017β=0.035
β=0.051
β=0.085

STABLE

10864
20(1,0)0.2

10 412(0,2)

02 trapped
2 1 trapped
1Collisionless confinement in the chs67 configuration,
MCT code.

14(0,0)0.15

4005
100.15

0.1

60

20

0.2

0.05

1 : start surface Ψ0=0.06

80

UNSTABLE
21S

0

0.2

0.4

0.6

0.8

1SFig. 2

Main components of the magnetic field strenght
spectrum in the chs67 configuration, β = 0.085,
obtained with the JMC code. Normalisation: B0,0 (0) =1.00.

Fig. 4

Fig. 5

Plasma surface cross-sections at the beginning, at one quarter and at the middle of the period in a chs67 configuration, β =0.093 (top row); equilibrium profiles of the rotational transform, magnetic well and plasma pressure (bottom row).

3201

The Mercier criterion obtained with the TERPSICHORE
code for the chs67 configurations, β = 0.005, β = 0.017,
β = 0.035, β = 0.051, and β = 0.085.

Cooper W.A. et al., The Effect of Reduced Pfirsch-Schlüter Current on the Ideal MHD Stability and Alpha Particle Confinement

Acknowledgements

The TERPSICHORE calculations of the Mercier
criterion for chs67 configuration are shown in Fig. 4. Thiscase is stable with respect to the Mercier modes almost for
all plasma surfaces up to the β = 0.085, which is a little bit
higher than the β = 0.051 limit with respect to these modesin the chs2b32m3 configuration. However, the stability limit
for chs67 case with respect to the ballooning modes is β =0.017 and is lower than the ballooning stability limit of thechs2b32 configuration (β = 0.02). Here we take into accountonly the symmetrical ballooning modes.
Figure 5 presents the main equilibrium profiles of therotational transform, magnetic well, pressure and view ofmagnetic surface cross-sections in the chs67 configuration at
the beginning, at one quarter and at the middle of a period
for a value of β = 0.093.

The authors are thankful to Drs. S. Hirshman and R.
Zille for their help and providing us with the VMEC and JMC
codes. The work was supported by INTAS Grant 99-00592,
by Russian-German Agreements WTZ RUS 01-581, by theRussian Foundation of Basic Research, Grant N 00-02-17105,
by the Russian Federal Programme on Support of the Leading
Scientific School, Grant N 00-15-96526 and by the Fonds
National Suisse pour la Recherche Scientifique. A working
visit to the National Institute for Fusion Science (NIFS, Japan)of M. Isaev was initiated and supported by the Director ofNIFS Prof. M. Fujiwara and Prof. C. Namba (Grant-in-aid
from the Ministry of Education, Science, Culture, Sports andTechnology, Japan). The computational results wereperformed on the NEC-SX5 platforms at the Centro Svizzero
di Calcolo Scientifico in Manno, Switzerland and at theRechenzentrum Garching, Max-Planck-Institut fürPlasmaphysik, Germany.

4. Summary
The CHS-qa compact helical quasi-axisymmetric concept(NIFS, Japan) is currently in the process of the furtheroptimisation and configuration design so it is important toconsider all possible limits and extreme configurations toincrease the field of future selection for the device candidates.
Pfirsch-Schlüter current minimisation has been performed inseveral CHS-qa configurations. Such optimisation
significantly improved the equilibrium β-limit and the ideal
MHD stability properties with respect to Mercier modes.
However, the strong stability towards ballooning modes
becomes a more difficult issue to achieve. In theconfigurations obtained, an equilibrium β-limit is above 0.20,
the stability limits with respect to the Mercier and ballooning
modes are at the β ≈ 0.085 and β ≈ 0.02 levels, respectively.
The optimised configurations with small PS current havestrong elongation of the plasma boundary cross-section. Largeelongation can impose difficulties for the modular coil design,
which is currently underway for the configurations obtained.
We also almost did not impose quasi-axisymmetry constraints,
so the configurations obtained have a large mirror-type fieldcomponent (0,2). It is not clear yet how strong this component
will influence the plasma confinement. These problems
together with other possible restrictions, for example, due tothe bootstrap current, magnetic islands or free-boundary
equilibria issues, will be considered in future optimisations.

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