Proceedings of ITC/ISHW2007

Search for very-high-β MHD stable quasi-isodynamic
configurations
V.R. Bovshuk1) , W.A. Cooper3) , M.I. Mikhailov1) , J. Nührenberg2) , and V.D. Shafranov1)
1)

NFI, Kurchatov Institute, Russia
IPP, EURATOM Association, Germany
3)
CRPP, EURATOM Association, EPFL, Switzerland
2)(Received October 12, 2007/ Accepted ?)Quasi-isodynamic configurations offer the possibility of very good energetic particle confinement. They
seem to offer the possibility of achieving very high plasma β, too.
Keywords: Plasma and Fusion Research, Stellarators, Quasi-isodynamicity, MHD stability, Very high plasma β1Introduction

Quasi-isodynamic [1] (qi) configurations have been previously found by computational optimization with high stability β limit, good neoclassical confinement properties andexcellent fast particle collisionless confinement for configurations with poloidally closed contours of the strength ofthe magnetic field B [2,3]. It was shown analytically [3]that the secondary parallel current density in qi configurations remains contained within each plasma field period,
namely, between the cross-sections with maximal magnetic field strength B. In the qi configurations consideredearlier, the divergence of the current density perpendicular
to the magnetic field lines changes sign only once along
the magnetic field within one field period. From this it follows that the parallel current density cannot change signalong the magnetic field within one period. Thus, becauseof the vanishing net parallel current, the parallel current
density exhibits a dipole component which impairs MHD
stability at very high β in the qi situation considered here
for configurations with shallow magnetic well in the associated vacuum magnetic field. The search for possible
ways to diminish this current density in quasi-isodynamic
configurations was the subject of [4].
For this search a two-staged approach had beentaken. In a model investigation it was clarified that quasiisodynamicity is compatible with vanishing dipole current
density in a more elaborate structure of the topography ofB exploiting the possibility of detailed toroidal design of
B in 3d configurations; then a configurational investigation
established a geometry realizing the essential features ofthis model and is seen in Fig. 1, Fig. 3 (left) and Fig. 4(left).
In this work it is investigated whether MHD-stableequilibria of this type of configuration exist.

author’s e-mail: mikhail@nfi.kiae.ru

2Current result at hβi ≈ 0.2

The configuration of Figs. 1 and 4 (left) had been obtainedat zero β. It exhibits a significant magnetic hill so that oneof the essential ingredients of its optimization towards highβ has been the transition to a vacuum field magnetic well
as a prerequisit for MHD stability. As seen from Fig. 1,the starting point of the optimization is characterized by aroughly hexagonal plasma shape, i.e. by as little curvature
as compatible with the straight sections encompassing themaxima of B. Since a magnetic well necessitates plasma
curvature and higher order poloidal shaping (triangularity,indentation, ...), it is plausible that the local plasma column curvature had to increase. This is seen in Fig. 2, but
most clearly in Fig. 9 (while not prominent in Fig. 10).
The subsequent achievement of high β is accompanied by
two characteristic features. As in the initial condition, theFourier components (in magnetic coordinates) of B and the√volume element g corresponding to axisymmetric curvature are very small, simulating an aspect ratio of several
hundred, see Fig. 5. Also, since the strong poloidal as wellas toroidal shaping, see Fig. 4 (right), drives higher order√
Fourier components of g, the optimization needed to exploit (and strictly observe) a window in rotational transform value, here chosen to be 67 < ι < 66 . While introducing curvature is necessary for stability, see Fig. 6, ballooning instability will limit it. Preliminary evaluation of ballooning instability considering the local ballooning equation for symmetric ballooning at the four symmetry pointsof the field period and flux surface label s ≈ 0.3 shows theabscence of strong-ballooning instability, see Fig. 7. This
analysis needs to be completed following the procedureused in [3], in particular because of the gap in marginal β
found there between local and non-local ballooning analysis. Some features observed in earlier stable configurations
are prominent in the configuration obtained here, too: thetriangular shape of the flux-surface cross-sections at theProceedings of ITC/ISHW2007

Fig. 1 Boundary magnetic surface showing the magnetic topography of [4].

6655
poloidal Boozer angle

poloidal Boozer angle

Fig. 2 Boundary magnetic surface showing the magnetic topography of the stable configuration.

43232

1

40

3−1

2−2−31

10000−412
345
toroidal Boozer angle, one period

612345
toroidal Boozer angle, one period

6Fig. 3 Left: the structure of the parallel current density ( j|| /B) of the configuration of Fig. 1 showing j||1,0 = 0; Right: the structure of
the parallel current density ( j|| /B) of the configuration of Fig. 2.
June 11, 2007

October 2, 2007

Fig. 4 Cross sections of magnetic surfaces of the configurations in Figs. 1 and 2 along half a period beginning with the minimum ofB and ending at the maximum of B.

Proceedings of ITC/ISHW2007−3

10x 10solution of ballooning modes equation

toroidal terms in B and Jacobian

4BJacobian3210−100.2

0.40.60.8√Fig. 5 Fourier coefficients of B and g in magnetic coordinatescorresponding to toroidal curvature.

6420.5130

1Mercier modes
Resistive modes

1l

1.5

250.5

20

10.5
0

1

540

25
−0.504530
0−0.5

10−1−150

350.5

15−0.5

55

340

120

1082Fig. 7 Solutions of the ballooning equation along fieldlines covering two periods and passing through the four symmetry
points on the flux surface at s = 0.3.

12Mercier and resistive modes stability

8001sθ0=0.5, ζ0=0.5
θ0=0.0, ζ0=0.0
θ0=0.5, ζ0=0.0
θ0=0.0, ζ0=0.5

35

20
−1−10

1

15

−1−190

10130

64570

161120

65

4

0.5

0.5
80

60
0110

0.5

00

100

−0.5

70 −0.5

90

55
−0.5

250
−1−10

1

45

−1
−10

1

60

−1
−180
0170

0
00.2

0.4

0.6

0.8

1sFig. 6 Mercier and resistive-interchange stability.

minimum of B and indentation in the range of strongest
curvature. It remains to be investigated whether the higherorder poloidal and toroidal shaping found, e.g. the quadrangularity at the maximum of B and in Fig. 4 (right), isreally necessary to achieve MHD stability.
The neoclassical physics properties have not in detailbeen part of this high-β optimization; they should be benign in view of the contours of the second adiabatic invariant, see Fig. 8 (right), but their detailed investigation
and, eventually, optimization remains to be done in order
to complete this case study of a very-high-β configuration.

3SummaryIn the context of quasi-isodynamic stellarators withpoloidally closed contours of the magnetic field strength
it is investigated whether very-high-β MHD-stable equilibria exist. With a previously introduced new structure ofFig. 8 Structure of J in the configuration obtained for different
values of B, Bref , at which trapped particles are reflected;
1 - near the minimum of B, 6 - near the maximum of B.

a period as a starting point equilibria are found which are
Mercier, resistive-interchange and strong balooning (symmetric) stable at hβi ≈ 0.2. Further work will concern further MHD-stability analysis and the details of the neoclassical physics properties of this type of configuration.

4Acknowledgment

Part of the computations of this work has been performedon the NIFS LHD Numerical Analysis Computer SX-8.
[1] S. Gori, W. Lotz, and J. Nührenberg, Proc. Joint VarennaLausanne Int. Workshop on Theory of Fusion Plasmas(Bologna: Editrice Compositori, 1996) p. 335.
[2] M.I. Mikhailov et al, Nucl. Fusion 42, L23 (2002).
[3] A.A. Subbotin et al, Nucl. Fusion 46, 921 (2006).
[4] V.R. Bovshuk et al, 34th EPS Conf. on Plasma Phys. andControl. Fusion, P4.103.

Proceedings of ITC/ISHW2007

Fig. 9 Top view of the configuration shown in Fig. 2.

Fig. 10 Equatorial view of the configuration shown in Fig. 2.