On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. Part 2. Stability criteria for two-dimensional flows

The techniques developed in Part 1 of the present series are here applied to two-dimensional solutions of the equations governing the magnetohydrodynamics of ideal incompressible fluids. We first demonstrate an isomorphism between such flows and the flow of a stratified fluid subjected to a field of force that we describe as 'pseudogravitational'. 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 analogous potential of the 'modified vorticity field', the additional frozen field introduced in Part 1. Using this Casimir, a linear stability criterion is obtained by standard techniques. In §4, the (Arnold) techniques of nonlinear stability are 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 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.