Shear Flow Instabilities in Quasigeostrophic Shallow-Water Magnetohydrodynamics

Motivated by astrophysical considerations, the rotating shallow-water equations, originally derived in the context of the terrestrial atmosphere, have recently been extended to include a magnetic component. These are quasi-2D equations which can improve our ability to simulate the dynamics of thin, stratified, rotating magnetised layers, such as the tachocline, in a simpler setting. In the rapidly rotating limit a set of quasi-geostrophic shallow-water (QG SW) equations can be obtained; the analogous QG equations in the hydrodynamic case have been used successfully to model large-scale flows in the terrestrial atmosphere and oceans.

The QG SWMHD equations have not previously been used to study shear instability. Approximating the solar differential rotation as a zonal shear flow, we investigate the effects of rotation, stratification, and a magnetic field on the linear and nonlinear development of shear flow instabilities in the QG SWMHD system by deriving general linear stability results, investigating the linear instability of particular velocity profiles, and performing fully nonlinear direct numerical simulations.

We will first derive necessary conditions for the presence of linearly unstable modes and bound the phase speed and growth rate of these instabilities. We then consider the linear instability of the vortex sheet profile which can represent the limiting case of a wide class of shear profiles, and for which analytic solutions can be derived. We then investigate a smooth profile, the hyperbolic tangent profile, which naturally extends the study of the vortex sheet, and show that a second unstable mode appears which may be particularly relevant when the magnetic field is strong. We conclude by performing fully nonlinear simulations of the QG SWMHD equations, which provide new insight into features of the long-term evolution of the QG SWMHD system including flux expulsion and vortex disruption.