Variational principles and stability criteria for spherical axisymmetric, gravitating, ideally magnetohydrodynamic configurations

摘要

Following a previous work, Nuovo Cimento B, 120 (2005) 1313, wherein we have obtained sufficient conditions for linear stability of spherically symmetric, gravitating, ideally magnetohydrodynamic configurations, by using the general theory of Arnold (Dokl. Akad. Nauk SSSR, 162 (1965) 975; Izv. Vyssh. Uchebn. Zaved. Mat., 54 (1966) 3) and Vladimirov et al. (J. Plasma Phys., 57 (1997) 89), here we obtain sufficient conditions for linear stability of spherical axisymmetric, gravitating, ideally magnetohydrodynamic configurations. We show a helpful analogy between spherical axisymmetric MHD flows subjected to a self-gravitating force field (or a pseudo-gravitating one) and flows of stratified fluids in a Boussinesqlike approximation. We construct a general Casimir functional as an integral of an arbitrary function of two conserved fields, namely the vector potential of the magnetic field and the scalar field associated with the “modified vorticity field", an additional frozen-in field. Using this Casimir functional, a linear stability criterion is obtained. The application to magnetized gas clouds is considered as a preliminary approximation.