Metastability of stratified magnetohydrostatic equilibria and their relaxation

Motivated by explosive releases of energy in fusion, space and astrophysical plasmas, we consider the nonlinear stability of stratified magnetohydrodynamic equilibria against two-dimensional interchanges of straight magnetic-flux tubes. We demonstrate that, even within this restricted class of dynamics, the linear stability of an equilibrium does not guarantee its nonlinear stability: equilibria can be metastable. We show that the minimum-energy state accessible to a metastable equilibrium under non-diffusive two-dimensional dynamics can be found by solving a combinatorial optimisation problem. These minimum-energy states are, to good approximation, the final states reached by our simulations of destabilised metastable equilibria for which turbulent mixing is suppressed by viscosity. To predict the result of fully turbulent relaxation, we construct a statistical mechanical theory based on the maximisation of Boltzmann's mixing entropy. This theory is analogous to the Lynden-Bell statistical mechanics of collisionless stellar systems and plasma, and to the Robert-Sommeria-Miller theory of two-dimensional vortex turbulence. Our theory reproduces well the results of our numerical simulations for sufficiently large perturbations to the metastable equilibrium.