ON GENERAL TRANSFORMATIONS AND VARIATIONAL-PRINCIPLES FOR THE MAGNETOHYDRODYNAMICS OF IDEAL FLUIDS .2. STABILITY-CRITERIA FOR 2-DIMENSIONALFLOWS

The techniques developed in Part 1 of the present series are here appl ied to two-dimensional solutions of the equations governing the magnet ohydrodynamics of ideal incompressible fluids. We first demonstrate an isomorphism between such flows and the flow of a stratified fluid sub jected to a field of force that we describe as 'pseudogravitational'. We then construct a general Casimir as an integral of an arbitrary fun ction of two conserved fields, namely the vector potential of the magn etic field, and the analogous potential of the 'modified vorticity fie ld', the additional frozen held introduced in Part 1. Using this Casim ir, a linear stability criterion is obtained by standard techniques. I n closed integral 4, the (Arnold) techniques of nonlinear stability ar e developed, and bounds are placed on the second variation of the sum of the energy and the Casimir of the problem. This leads to criteria f or nonlinear (Lyapunov) stability of the MHD flows considered. The app ropriate norm is a sum of the magnetic and kinetic energies and the me an-square vector potential of the magnetic held.