MHD spectrum of a Z-pinch in an external magnetic multipole field

Abstract

MHD stability properties of previously calculated equilibria for Extrap type of configurations, generalized to an arbitrary number (N) of external conductors, are investigated. A perturbation technique with a small parameter epsilon measuring the non-circularity of the plasma, is used. An important issue concerns the degeneracy of modes corresponding to various poloidal model numbers m, and some general results, due solely to the angular periodicity of the configuration, are derived. In particular, it is shown that for the mode m=1 the degeneracy prevails to all orders in epsilon if N>2 or, equivalently, if the equilibrium has no quadrupole component. This turns out to be extremely important for the possibility of stabilizing long kink modes, and probably explains why only 'square-shaped' equilibria have been found to be stable in earlier numerical computations. In the four-conductor Extrap, the mode which is most strongly affected by the external field is m=2, for which the degeneracy is removed already in first order. Leading order frequency shifts for the modes m=1 and m=2 are explicitly calculated for surface- and constant-current pinches.