Shear instabilities in shallow-water magnetohydrodynamics

TL;DR

This paper analyzes the linear instability of shear flows in shallow-water magnetohydrodynamics, extending classical instability results and exploring how magnetic fields and stratification influence stability through analytical and numerical methods.

Contribution

It extends classical instability theorems to shallow-water MHD, analyzes specific profiles, and elucidates the stabilizing effects of magnetic fields and stratification.

Findings

Magnetic field and stratification generally stabilize shear flows.

Weak instabilities can occur at high Froude numbers.

Counter-propagating Rossby waves explain primary instability mechanisms.

Abstract

Within the framework of shallow-water magnetohydrodynamics, we investigate the linear instability of horizontal shear flows, influenced by an aligned magnetic field and stratification. Various classical instability results, such as H{\o}iland's growth rate bound and Howard's semi-circle theorem, are extended to this shallow-water system for quite general profiles. Two specific piecewise-constant velocity profiles, the vortex sheet and the rectangular jet, are studied analytically and asymptotically; it is found that the magnetic field and stratification (as measured by the Froude number) are generally both stabilising, but weak instabilities can be found at arbitrarily large Froude number. Numerical solutions are computed for corresponding smooth velocity profiles, the hyperbolic-tangent shear layer and the Bickley jet, for a uniform background field. A generalisation of the long-wave asymptotic analysis of Drazin & Howard (1962) is employed in order to understand the instability characteristics for both profiles. For the shear layer, the mechanism underlying the primary instability is interpreted in terms of counter-propagating Rossby waves, thereby allowing an explication of the stabilising effects of the magnetic field and stratification.