Search

A one-dimensional model of magnetic relaxation in a pressureless low-resistivity plasma is considered. The initial two-component magnetic field $\boldsymbol{b}(\boldsymbol{x},t)$ is strongly helical, with non-uniform helicity density. The magnetic pressure gradient drives a velocity field that is dissipated by viscosity. Relaxation occurs in two phases. The first is a rapid initial phase in which the magnetic energy drops sharply and the magnetic pressure becomes approximately uniform; the helicity density is redistributed during this phase but remains non-uniform, and although the total helicity remains relatively constant, a Taylor state is not established. The second phase is one of slow diffusion, in which the velocity is weak, though still driven by persistent weak non-uniformity of magnetic pressure; during this phase, magnetic energy and helicity decay slowly and at constant ratio through the combined effects of pressure equalisation and finite resistivity. The density field, initially uniform, develops rapidly (in association with the magnetic field) during the initial phase, and continues to evolve, developing sharp maxima, throughout the diffusive stage. Finally it is proved that, if the resistivity is zero, the spatial mean $\langle (\boldsymbol{b}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\times \boldsymbol{b})/b^{2}\rangle$ is an invariant of the governing one-dimensional induction equation.

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.

We construct numerical examples of a ‘modon’ (counter-rotating vortices) in an Euler flow by exploiting the analogy between steady Euler flows and magnetostatic equilibria in a perfectly conducting fluid. A numerical modon solution can be found by determining its corresponding magnetostatic equilibrium, which we refer to as a ‘magnetic moclon’. Such an equilibrium is obtained numerically by a relaxation procedure that conserves the topology of the relaxing field. Our numerical results show how the shape of a magnetic modon depends on its ‘signature’ (magnetic flux profile), and that these magnetic modons are unexpectedly unstable to non-symmetric perturbations. Diffusion can change the topology of the field through a reconnection process and separate the two magnetic eddies. We further show that the analogous Euler flow (or modon) behaves similar to a perturbed Hill's vortex.