On the diffusive instability of some simple steady magnetohydrodynamic flows

Abstract

The stability characteristics of some simple steady magnetohydrodynamic flows within an axisymmetric container of arbitrary electrical conductivity are investigated. Attention is focused upon rapidly rotating fluids in which the unperturbed velocity and magnetic field are axially symmetric and purely zonal. Detailed solutions are obtained for the particularly simple basic state representing a rigidly rotating homogeneous fluid with a uniform axial current. The theory of dynamic (dissipationless) instabilities is reviewed and its shortcomings are elucidated. A stability criterion is derived for an inviscid fluid of small electrical conductivity within a perfectly conducting axisymmetric container and it is shown that a certain class of inertial modes is unstable for any non-zero magnetic field strength. When the effects of container conductivity are included it is found that a class of slow modes with westward phase speed may be unstable. These modes are shown to be unstable within a cylinder but appear to be stable within a sphere. The influence of density gradients within a spherical container is investigated and it is found that for a certain class of exceptional slow modes with westward phase speed, a bottom-heavy density gradient is destabilizing. This surprising behaviour is explained in terms of a new branch of the stability curve developed by Eltayeb & Kumar (1977).