Axisymmetric Instabilities in Stellar and Planetary Magnetohydrodynamics

The large-scale circulations of many geophysical and astrophysical bodies are dominated by parallel flows. It is natural to enquire about the stability of these flows, and how they depend upon background rotation, shear and stratification. In the terrestrial atmosphere, inertial instabilities may result and are most likely to occur close to the equator. We therefore investigate whether corresponding instabilities may occur in stellar and planetary atmospheres, where large-scale magnetic fields may have significant dynamical effects; in particular, we focus on applications to the solar tachocline, and the upper atmospheres of Hot Jupiters.

We investigate the axisymmetric linear stability of various parallel flows with latitudinal shear in the presence of background rotation, stratification and magnetic field. Using the Boussinesq and magnetohydrostatic approximations, we derive general stability criteria and growth rate bounds, taking account of the magnetic field. We show that sufficiently strong magnetic field stabilises the system.

We show that a uniform shear flow on an f-plane allows magnetically modified inertial instabilities and purely magnetic instabilities to occur, the latter being analogous to magnetorotational instabilities in limiting cases. Allowing for non-zero kinematic viscosity, thermal diffusivity and magnetic diffusivity leads to several steady and oscillatory stability criteria. We categorise various double-diffusive instabilities, some of which occur at vertical wavenumbers that would otherwise be stable in the ideal regime.

We also consider the nonlinear evolution of a hyperbolic shear layer at mid-latitudes to understand the effect of a locally unstable region, the entire time evolution of the nonlinear instability, the redistribution of vorticity and the resulting change to the mean flow. The redistribution of vorticity is most substantial in the hydrodynamically stable regime.

Finally we consider the stability of a jet profile in an equatorial atmosphere, which admits localised solutions about a central latitude. For the ideal case, instabilities can occur at any vertical wavenumber given sufficiently weak magnetic field; the most unstable modes are found to occur for infinite vertical wavenumbers. In the presence of diffusion, instabilities are constrained to finite vertical wavenumbers and are often purely magnetic, occurring in the hydrodynamically stable regime.