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Magnetohydrodynamic stability of plasmas with ideal and relaxed regions

Published online by Cambridge University Press:  01 October 2009

R. L. MILLS ,
M. J. HOLE  and
R. L. DEWAR
R. L. MILLS
Affiliation:
Department of Theoretical Physics and Plasma Research Laboratory, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 0200, Australia (ruth.mills@anu.edu.au)
M. J. HOLE
Affiliation:
Department of Theoretical Physics and Plasma Research Laboratory, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 0200, Australia (ruth.mills@anu.edu.au)
R. L. DEWAR
Affiliation:
Department of Theoretical Physics and Plasma Research Laboratory, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 0200, Australia (ruth.mills@anu.edu.au)
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Abstract

A unified energy principle approach is presented for analysing the magnetohydrodynamic (MHD) stability of plasmas consisting of multiple ideal and relaxed regions. The gauge a = ξ × B for the vector potential, a, of linearized perturbations is used, with the equilibrium magnetic field B obeying a Beltrami equation, ∇ × B = αB, in relaxed regions. In a region with such a force-free equilibrium Beltrami field we show that ξ obeys the same Euler–Lagrange equation whether ideal or relaxed MHD is used for perturbations, except in the neighbourhood of the magnetic surfaces where B · ∇ is singular. The difference at singular surfaces is analysed in cylindrical geometry: in ideal MHD only Newcomb's small solutions are allowed, whereas in relaxed MHD only the odd-parity large solution and even-parity small solution are allowed. A procedure for constructing global multi-region solutions in cylindrical geometry is presented. Focusing on the limit where the two interfaces approach each other arbitrarily closely, it is shown that the singular-limit problem encountered previously by Hole et al. in multi-region relaxed MHD is stabilized if the relaxed-MHD region between the coalescing interfaces is replaced by an ideal-MHD region. We then present a stable (k, pressure) phase-space plot, which allows us to determine the form a stable pressure and field profile must take in the region between the interfaces. From this knowledge, we conclude that there exists a class of single-interface plasmas that were found to be stable by Kaiser and Uecker, but are shown to be unstable when the interface is resolved.

Type
Papers
Information
Journal of Plasma Physics , Volume 75 , Issue 5 , October 2009 , pp. 637 - 659
Copyright
Copyright © Cambridge University Press 2009

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