ON GENERAL TRANSFORMATIONS AND VARIATIONAL-PRINCIPLES FOR THE MAGNETOHYDRODYNAMICS OF IDEAL FLUIDS .3. STABILITY-CRITERIA FOR AXISYMMETRICAL FLOWS

The general theory developed in Part I of the present series is here a pplied to axisymmetric solutions of the equations governing the magnet ohydrodynamics of ideal incompressible fluids. We first show a helpful analogy between axisymmetric MHD flows and hows 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, namel y the vector potential of the magnetic field and the scalar field asso ciated with the 'modified vorticity field', the additional frozen-in f ield introduced in Part I. Using this Casimir, sufficient conditions f or linear stability to axisymmetric perturbations are obtained by stan dard Arnold techniques. We exploit Arnold's method to obtain sufficien t conditions for nonlinear (Lyapunov) stability of the MHD flows consi dered. The appropriate norm is a sum of the magnetic and kinetic energ ies and the mean square vector potential of the magnetic field.