J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)519–522
000–000

Effects of Current Profile on Global Ideal MHD Stability
in a Compact Quasi-Axisymmetric Stellarator
SUZUKI Chihiro, OKAMURA Shoichi, ISOBE Mitsutaka, NISHIMURA Shin,
SHIMIZU Akihiro, AKIYAMA Tsuyoshi, MATSUOKA Keisuke, NAKAJIMA Noriyoshi,
COOPER Wilfred A.1 and NÜHRENBERG Carolin2
National Institute for Fusion Science, Toki 509-5292, Japan
Centre de Recherches en Physique des Plasmas, Association Euratom-Suisse,
Ecole Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland
2Max-Planck-Institut für Plasmaphysik, D-17491, Greifswald, Germany

1(Received: 9 December 2003 / Accepted: 26 February 2004)

Abstract
The global ideal magnetohydrodynamic (MHD) stability for a proposed compact quasi-axisymmetric stellarator
CHS-qa has been investigated taking the effect of bootstrap current into account. Assuming experimentally achievable
density and temperature profiles, the stability properties of global low-n modes have been studied by using threedimensional numerical codes based on fixed boundary MHD equilibria including self-consistent bootstrap current for
the CHS-qa reference configuration. Consequently it has been shown that values of edge rotational transform play a
crucial role in triggering external kink instability. Concerning a lot of other possibilities in experimental practice tochange the total parallel current, we have also studied equilibria with increased or decreased parallel current, but fixedprofile. The onset of external kink modes depends on rotational transform or current profile, and we found a stableequilibrium in spite of the edge rotational transform above 0.5. The results imply the possibility of stabilizing externalkink modes through current and/or pressure profile control in high beta equilibria.
Keywords:
quasi-axisymmetric stellarator, CHS-qa, bootstrap current, ideal MHD stability, external kink mode,
rotational transform, current profile

1. Introduction

dimensional configurations.
Since basic information of CHS-qa has already been
published [1], only a brief explanation is given here. Average
major and minor radii of CHS-qa are 1.5 and 0.4 m,
respectively. The maximum magnetic field strength generated
by 20 modular coils is 1.5 T. The current reference
configuration of CHS-qa is called “2b32”, which means thenumber of field periods is Np = 2, aspect ratio is 3.2, and it
has been optimized to stabilize local ballooning modes on asingle flux surface. We discuss this reference configuration
only throughout this article.

It is intrinsically unavoidable that relatively large amountof bootstrap current (BSC) flows so as to increase rotationaltransform in a quasi-axisymmetric (QA) stellarator according
to its tokamak-like magnetic field structure. Therefore the roleof the BSC on ideal magnetohydrodynamic (MHD) stability
properties in a QA configuration is more important than thosein other stellarators. Accordingly ideal MHD stability analysis
including plasma current is essential for exploring beta limit
and avoiding serious instabilities in the physics design processof a QA device.
In this study we discuss the effects of current profile onideal MHD stability properties in a QA stellarator CHS-qa
proposed as a candidate for the next step compact stellarator
in Japan. In order to assess ideal MHD instabilities driven by
the BSC, three-dimensional global low-n mode analyses basedon calculations of eigenmode structure are necessary inaddition to local mode analyses. Recent progress insupercomputer performance makes it possible to calculate
immediately ideal MHD modes in such complicated three

2. Numerical procedure
The entire procedure to solve finite beta equilibria
including self-consistent BSC has already been described
elsewhere [2]. We have employed the SPBSC code which
executes calculations of fixed boundary MHD equilibrium andanalytical BSC formula [3] alternately until convergence. The
assumed density and temperature profiles in this study are©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: csuzuki@nifs.ac.jp1519

Suzuki C. et al., Effects of Current Profile on Global Ideal MHD Stability in a Compact Quasi-Axisymmetric Stellaratorn(s) = n0 (1 – 0.8s + 1.3s 2 – 1.5s 3) and T(s) = T 0 (1 – s),respectively, where s is the normalized toroidal flux and thesubscript “0” means central value. These profiles are chosen
to be compatible with experimentally achievable ones inexisting devices.
Since it is difficult to get good convergence of BSC in
the high beta equilibria investigated in this study, only
axisymmetric components of the magnetic field are includedin the BSC calculation, while all components are used in theequilibrium calculation. This implies that the calculated BSC
should correspond to the upper limit in that equilibrium.
The global low-n ideal MHD stability is calculated
mainly with the TERPSICHORE code [4]. Since the fluid
compression term in the energy integral is ignored forsimplicity, the calculated eigenvalue does not strictly agreewith the physical growth rate. However, the magnitude of theeigenvalue is still valid for a rough estimation, and themarginal point for stability is correct. We have just started
calculation with the CAS3D3 code [5] including the fluid
compression term for obtaining physical growth rates, which
will be studied extensively in the future.
As is well known, global ideal MHD mode structures in
non-axisymmetric configurations constitute “mode families”
due to mutual mode coupling. In CHS-qa (Np = 2), eventoroidal mode numbers make up the N = 0 family, while odd
numbers the N = 1 family. In this paper, we display the resultsof the N = 1 family only, because the N = 0 family has asimilar tendency to the N = 1 family and it is meaningless todisplay the results of both families. The mode selection table
for the N = 1 family is shown in Fig. 1, where 81 perturbation
Fourier modes included in the calculation are indicated byblack dots. We consider all modes resonant with rotationaltransform from 0.3 to 0.8 and coupling with the principal
equilibrium helical component (m,n) = (1,2) to covertransform profiles of all equilibria examined in this study. The
toroidal perturbation mode numbers up to 7 are considered toFig. 1

meet the scope of this study (low-n mode instability) and tosave the calculation time.

3. Effect of bootstrap current
In this study, three representative cases of plasma
parameter sets are investigated as shown in Table 1. The
central electron density and temperature, volume average beta,
and total calculated BSC per toroidal magnetic field of 1 T
are listed in Table 1. The ion temperature is assumed to be
three quarters of the electron temperature. The profiles of selfconsistent BSC and the resulting rotational transform areillustrated in Fig. 2. Though the volume average beta in cases
A and B is identical (〈β 〉 = 3 %), the plasma density in A is

Table 1 Plasma parameters of three representative cases forstability calculations. The ion temperature is assumed
to be three quarters of the electron temperature. Theaverage beta and the BSC are calculated per toroidal
magnetic field of 1 T.

Perturbation mode selection table for N = 1 family.
Modes between the two solid lines are resonant withrotational transform from 0.3 to 0.8. In addition, themode coupling with the equilibrium component (m,n) =(1,2) is considered.

2520

ne (1020 m-3)

Te (keV)〈β 〉 (%)

IBSC (kA)

A

1.0

1.04

3.0

103

B

0.2

5.2

3.0

168

C

1.5

1.04

4.6

122

Fig. 2

The radial profiles of (a) bootstrap current density and(b) rotational transform for the three cases listed inTable 1. The rotational transform in vacuum is also
displayed in (b) for reference.

Suzuki C. et al., Effects of Current Profile on Global Ideal MHD Stability in a Compact Quasi-Axisymmetric Stellarator

Fig. 3

Fourier components of the most unstable perturbation
normal to the flux surfaces in the case B.

Fig. 4 The most unstable eigenvalues as functions of total
current for three representative cases.

five times higher than that in B. This means that thecollisional regimes are different, which results in differencesin the magnitude of BSC and rotational transform as shownin Fig. 2. Note that the edge rotational transform exceeds 0.5
for B, while not for A. The plasma density in the case C
corresponds to 1.5 times the case A, which results in higher
average beta (〈β 〉 = 4.6 %). The current profile is slightly
shifted inward in comparison with A, and the edge rotationaltransform is kept below 0.5.
Under these conditions, the global ideal MHD stability
is analyzed in the manner described in Sec. 2. As a result,instability appears only in the case B. The eigenmodestructure normal to flux surfaces for the most unstable
perturbation for this case is shown in Fig. 3. The amplitudeof the dominant perturbation mode number (m,n) = (2,1)
increases toward the edge, which clearly indicates thecharacteristics of an external kink instability caused by therational surface of 0.5. This indicates that values of edge
rotational transform play a crucial role in triggering externalkink instability.
The calculation of this instability has also been
performed with the CAS3D code including the fluid
compression term (CAS3D3), and similar results have beenobtained with slightly distorted eigenfunction. The resulting
physical growth rate is ω = 1.0×105 s-1.

perturbation modes for the N = 1 family as functions of thetotal parallel current. Positive and negative eigenvalues
correspond to stability and instability, respectively. The mode
number of the dominant Fourier component in the unstableeigenfunction is (m,n) = (2,1) for 150–225 kA, and (5,3) for
250 kA. The edge rotational transform, which is closely
associated with external kink instability, traverses 0.5 between
125 and 150 kA and 0.6 between 225 and 250 kA for all thecases studied. This implies that the onset of destabilization
above 150 kA in the cases A and B clearly corresponds tothe crossing of the edge rotational transform beyond 0.5 and0.6. However, case B is apparently more unstable incomparison with A. Furthermore, if the total current isincreased above 200 kA, the eigenvalue for A tends todecrease with total current, while it continues to increase for
B. Since the pressure profile and average beta are completely
identical for these two cases, this difference can be attributed
to the difference in rotational transform profile closely
associated with the BSC profile. One possible reason for thisphenomenon is the fact that the edge shear is stronger andthe current density near the edge is lower in A compared withthose in B as shown in Fig. 2 (b).
The result of the case C is more interesting. In this casethe global mode is kept stable up to 250 kA total current
despite the edge rotational transform being raised above 0.5.
Apparently it seems to be reasonable that it could be
attributed to the difference in current profile. However, if theaverage beta is artificially decreased to the same level (〈β 〉 =3.0 %) as the other two cases while keeping the current profile
fixed, the (m,n) = (2,1) external kink mode is destabilized(not shown in the figure). Therefore, other reasons should beconsidered to explain this phenomenon. Since fixed boundary
calculations at high beta above 4 % generate equilibrium
solutions with strongly distorted flux surfaces, more realistic
calculations based on free boundary VMEC equilibria arenecessary to investigate this implication. Nevertheless, thisstudy demonstrated the importance of rotational transformprofile on global MHD stability in the QA configuration andthe possibility of stabilization of the external kink mode bycontrolling the parallel current through pressure profile

4. Dependence on total parallel current
Since non-axisymmetric Fourier components of themagnetic field are excluded in the BSC calculation shown inSection 3, the magnitude of BSC is actually reduced to someextent from the values listed in Table 1. On the other hand, itwould be possible to further elevate the rotational transformby external current drive techniques such as neutral beam
injection, ohmic current, etc. Therefore it is important toinvestigate the dependence of the stability properties on themagnitude of the total parallel current. For this purpose, wehave artificially changed the magnitude of the parallel current
from 25 to 250 kA (per 1 T) while keeping its profile shownin Fig. 2 (a).
Figure 4 shows eigenvalues of the most unstable3521

Suzuki C. et al., Effects of Current Profile on Global Ideal MHD Stability in a Compact Quasi-Axisymmetric Stellarator

term to the overall energy integrals should be investigatedwith the CAS3D3 or modified TERPSICHORE codes [6].
Additionally, more realistic calculations based on freeboundary MHD equilibria are required to estimate more
precisely the onset of current-driven instabilities.

control or various current drive methods.

5. Conclusion
We have investigated the global ideal MHD stability
properties of a proposed compact QA stellarator CHS-qa
reference configuration considering a self-consistent BSC.
With a reasonably achievable plasma pressure, external kinkmodes resonant with the (m,n) = (2,1) rational surface aredestabilized in the low density case at an average beta of
3.0 % because the edge rotational transform crosses 0.5. The
dependence of the eigenvalue on the total parallel current hasrevealed that the global mode stability properties are stronglyaffected by the rotational transform or the parallel current
profile, and it may be completely stabilized under appropriateconditions. The results imply the possibility of stabilizationof external kink modes by controlling the current and/or thepressure profiles. In order to clarify the destabilizing
mechanism in more detail, contributions of each destabilizing

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[5] C. Nührenberg, Phys. Plasmas 6, 137 (1999).
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Driving Mechanismsin 3D Heliotron and QuasiAxisymmetric Systems”, J. Plasma Fusion Res. SERIES
6, P1-03 (2004).

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