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Resonant structures within incompressible ideal MHD

Published online by Cambridge University Press:  13 March 2009

C. Zorzan  and
P. S. Cally
C. Zorzan
Affiliation:
Department of Mathematics, Royal Melbourne Institute of Technology, Melbourne, Victoria 3000, Australia
P. S. Cally
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia
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Abstract

The resonant characteristics of an incompressible ideal MHD fluid are highly structured. To help expose this structure, an equivalent electrical analogue of the MHD system is developed. The model, in the form of a transmission line, makes it possible to identify a number of new and important concepts, one of which is the effective impedance. This in turn enables entire regions of MHD fluid to be replaced with equivalent impedances. When fully exploited, the model also provides a more consistent interpretation of the spectrum of ideal MHD. The discrete Alfvén modes are found to be highly degenerate, while the transition to a discontinuous profile is accompanied by a redistribution of an uncountably infinite number of ‘poles’ from the continuous spectrum and onto the Alfvén modes. In addition, the electrical analogue shows that within a continuously structured fluid the characteristic behaviour is not necessarily dominated by the ‘surface mode’ alone. This view is also supported by the results of a numerical simulation of the linear MHD equations. Depending on the initial conditions, the collective behaviour can have any frequency within the range spanned by the transition zone. The energy itself is monitored using a new pair of energy and flux expressions derived from a variational (Lagrangian) description of the MHD system. Again the electrical model is used to provide a physical interpretation of the individual terms within these expressions. In particular, it allows a partition of the total energy into separate kinetic- and potential-energy terms.

Type
Research Article
Information
Journal of Plasma Physics , Volume 47 , Issue 2 , April 1992 , pp. 321 - 347
Copyright
Copyright © Cambridge University Press 1992

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