Bounds on the stability of 3D magnetic equilibria in the solar corona

Abstract

A necessary and a sufficient condition are derived for the ideal magnetohydrodynamic stability of any 3D magnetohydrostatic equilibrium using the energy method and incorporating photospheric line-tying. The theory is demonstrated by application to a simple class of theoretical 3D equilibria. The main thrust of the method is the formulation of the stability conditions as two sets of ordinary differential equations together with appropriate boundary conditions which may be numerically integrated along tied field lines one at a time. In the case of the shearless fields with non-negligible plasma pressure treated here the conditions for stability arenecessary and sufficient. The method employs as a trial function a destabilizing ‘ballooning’ mode, of large wave number vector perpendicular to the equilibrium field lines. These modes may not be picked up in a solution of the full partial differential equations which arise from a direct treatment of the problem.