Variational principles and stability criteria for two-dimensional, gravitating, ideal magnetohydrodynamic configurations

摘要

In previous papers we derived sufficient conditions for linear stability of spherically symmetric and spherical, axisyinmetric, gravitating configurations in ideal magnetohydrodynamics by using the general theory of Arnold and Vladimirov at al. The same general theory is now used to deduce sufficient conditions for linear stability of two-dimensional gravitating configurations with perpendicular magnetic field. Again a helpful analogy between two-dimensional magnetohydrodynamic flows subjected to a self-gravitating force field (or a pseudo-gravitating one) and flows of stratified fluids iu the Vladimirov-Boussinesq approximation is obtained. A "modified vorticity field" is considered which turns out to be an additional frozen-in field like the vector potential of the magnetic field. This allows to construct a general Casimir functional. Prom the latter a linear stability criterion is obtained.