Advancements in cutting-edge observational technologies, combined with refined helioseismic techniques, have revealed that the surface differential rotation (which has been observed for hundreds of years) permeates throughout the convection zone of the Sun, below which the radiative zone rotates approximately as a solid body. The thin layer bounding these two regions is known as the solar tachocline and is expected to play a vital role in many of the most interesting and least understood solar mechanisms.
This thesis studies the local instabilities of differentially rotating, stably stratified, magnetised stellar interiors, which are relevant for the lower regions of the tachocline among many other astrophysical contexts. Our primary focus is on the ability of these instabilities to transport angular momentum in the regions where they operate. We use a local box model to analyse and simulate a small patch of the stellar radiative zone within the tachocline.
Initially, we systematically explore the hydrodynamic regime, derive the dispersion relation for axisymmetric modes, and identify key stability criteria. Our findings reveal two dominant instabilities: an adiabatic centrifugal instability and the diffusive Goldreich-Schubert-Fricke (GSF) instability. Nonlinear simulations with generalised differential rotation profiles show that these instabilities lead to the formation of 'zonal jets', whose orientation and angular momentum transport properties depend heavily on the local rotation profile.
Transitioning to the magnetohydrodynamic system, we introduce a poloidal magnetic field and observe its interaction with the centrifugal and GSF instabilities, as well as the new axisymmetric instabilities enabled by its presence, in particular the magnetorotational instability. We use linear theory in combination with an analysis of the energetic properties of the system to derive stability criteria and better understand the instabilities in the magnetised system. Nonlinear simulations provide insights into the dynamo-generating and angular momentum-transporting capabilities of such flows in the presence of a magnetic field.
