General stability analysis of force-free magnetic fields. Part 1. General theory

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Abstract
We formulate, in the framework of MHD, a simple eigenvalue problem, capable of treating the stability of force-free magnetic fields curl B = αB in different geometries. We prove that a force-free field surrounded by a rigid wall is stable, if the eigenvalue α corresponds to the lowest value ofcompatible with the geometry considered. We extend this result to the case where α is a function of position; and we recliscuss it from the viewpoint of the first-order variation. We give various theorems and criteria for stability, for continuous as well as for admissible discontinuous or infinite displacements. A general upper limit for the growth rate is, where andand are respectively the maximum values ofand υAthe Alfvén velocity inthe plasma volume.

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