Nonlinear stability in anisotropic magnetohydrodynamics

Formal stability, i.e., positive energy linear stability, and nonlinear stability are defined. The generic way in which a system with infinite degrees of freedom may be nonlinearly unstable in spite of formal stability is illustrated. The energy-Casimir method of proof of nonlinear stability is applied to relaxed states of anisotropic magnetohydrodynamics (MHD). Such states are consistent with ergodic magnetic field lines and are formally stable under mild restrictions. Specifically, there is no free energy to drive interchange modes. Nonlinear stability is guaranteed under somewhat more restrictive conditions than is formal stability.