Shear instabilities in shallow-water magnetohydrodynamics

The interaction of horizontal shear flows and magnetic fields in stably stratified layers is central to many problems in astrophysical fluid dynamics. Motions in such stratified systems, such as the solar tachocline, may be studied within the shallow-water approximation, valid when the horizontal length scales associated with the motion are long compared to the vertical scales. Shallow-water systems have the advantage that it captures the fundamental dynamics resulting from stratification, but there is no explicit dependence on the vertical co-ordinate, and is thus mathematically simpler than the continuously stratified, three-dimensional fluid equations. Here, we study the shear instability problem within the framework of shallow-water magnetohydrodynamics. A standard linear analysis is first carried out, where we derive theorems satisfied by general basic states (growth rate bounds, semi-circle theorems, stability criteria, parity results), investigate the instabilities associated with idealised, piecewise-constant profiles (the vortex sheet and rectangular jet), and investigate the instabilities associated with two prototypical smooth profiles (hyperbolictangent shear-layer and Bickley jet); these are studied via analytical, numerical and asymptotic methods. The nonlinear development of the instabilities associated with the smooth profiles is then investigated numerically, focussing first on the changes to the nonlinear evolution arising from MHD effects, before investigating the differences arising from shallow-water effects. We finally investigate the interplay between MHD and shallow-water effects on the nonlinear evolution.