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Stability criterion for symmetric MHD equilibria by minimizing the potential energy

Published online by Cambridge University Press:  13 March 2009

Johann W. Edenstrasser
Johann W. Edenstrasser
Affiliation:
Institute for Theoretical Physics, University of Innsbruck, Austria
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Abstract

The potential energy of an ideal static MHD plasma is minimized using the invariants of motion as variational constraints and assuming a general symmetry (dependence on two space variables only). For simplicity only the plasma-on- the-wall case is considered. The first variation yields a generalized Shafranov equation, the second the desired stability criterion. It is found that equilibria with a longitudinal current increasing monotonicaily towards the boundary are always stable with respect to symmetric modes. For equilibria with an outwardly decreasing current a sufficient criterion (for symmetric modes) is derived, which only requires the solution of a linear eigenvalue problem. The theory is applied to the straight circular cylinder and to the axisymmetric torus.

Type
Research Article
Information
Journal of Plasma Physics , Volume 20 , Issue 3 , December 1978 , pp. 503 - 520
Copyright
Copyright © Cambridge University Press 1978

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References

REFERENCES

Bernstein, I. B., Frieman, E. A., Kruskal, M. D. & Kulsrud, R. M. 1957 Proc. Roy. Soc. A 244, 17.Google Scholar
Edenstrasser, J. W. 1978 Ph.D. thesis, University of Innsbruck.Google Scholar
Grad, H. & Rubin, H. 1958 Proceedings of the Second International Conference on the Peaceful Uses of Atomic Energy, Geneva, vol. 31, p. 190.Google Scholar
Herrnegger, F. & Nührenberg, 1975 Nucl. Fusion, 15, 1028.CrossRefGoogle Scholar
Keller, J. B. & Antman, S. 1969 Bifurcation Theory and Nonlinear Eigenvalue Problems. Benjamin.Google Scholar
Klötzler, R. 1970 Mehrdiniensionale Variationsrechnung, pp. 119, 130. Birkhäuser Verlag.CrossRefGoogle Scholar
Krüger, J. 1976 a J. Plasma Phys. 15, 15.CrossRefGoogle Scholar
Krüger, J. 1976 b J. Plasma Phys. 15, 31.CrossRefGoogle Scholar
Kruskal, M. D. & Kulsrud, R. M. 1958 Phys. Fluids, 1, 265.CrossRefGoogle Scholar
Schlüter, A. & Lüst, R. 1955 Z. Astrophysik, 38, 190.Google Scholar
Schmidt, G. 1966 Physics of High Temperature Plasmas, p. 90. Academic.Google Scholar
Shafranov, V. D. 1957 Soviet Phys. JETP, 33, 710.Google Scholar
Solov'ev, L. S. 1975 Reviews of Plasma Physics, vol. 6 (ed. Leontovich, M. A.). Consultants Bureau.Google Scholar
Taylor, J. B. 1975 Relaxation of Toroidal Discharges, Third Topical Conference on High-β Plasmas, Culham, 09 1975.Google Scholar
Woltjer, L. 1958 a Proc. Nat. Acad. Sci. 44, 833.CrossRefGoogle Scholar
Woltjer, L. 1958 b Proc. Nat. Acad. Sci. 44, 489.CrossRefGoogle Scholar