Stability and Instability Criteria for Magnetohydrodynamics

摘要

The motion of an electrically conducting ideal fluid is governed by the equationsof magnetohydrodynamics (MHD) which reduce to the Euler equations in the absence of a magnetic field. Arnold's treatment for the stability of the Euler equations has been extended to the augmented system for ideal MHD by Holm et al (1985) and Friedlander and Vishik (1990). In contrast with the purely hydrodynamic problem it is possible for MHD to obtain 3-dimensional examples in which the second variation of the energy is in fact definite. We note, however, that it is still a difficult task to check the remaining conditions associated with nonlinear stability concerning convexity and regularity. In this paper we examine restrictions on the class of examples of MHD equilibria for which the second variation of the energy is definite. We follow the approach of Sadun and Vishik (1993) and consider the Arnold functional corresponding to localized high frequency disturbances.