On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. Part III. Stability criteria for axisymmetric flows

Abstract

The general theory developed in Part I of the present series is here applied to axisymmetric solutions of the equations governing the magnetohydrodynamics of ideal incompressible fluids. We first show a helpful analogy between axisymmetric MHD flows and flows of a stratified fluid in the Boussinesq approximation. We then construct a general Casimir as an integral of an arbitrary function of two conserved fields, namely the vector potential of the magnetic field and the scalar field associated with the ‘modified vorticity field’, the additional frozen-in field introduced in Part I. Using this Casimir, sufficient conditions for linear stability to axisymmetric perturbations are obtained by standard Arnold techniques. We exploit Arnold's method to obtain sufficient conditions for nonlinear (Lyapunov) stability of the MHD flows considered. The appropriate norm is a sum of the magnetic and kinetic energies and the mean square vector potential of the magnetic field.