Interplay between magnetic fields and differential rotation in a stably stratified stellar radiative zone

The present study aims at studying the flow and field produced by a stellar radiative zone which is initially made to rotate differentially in the presence of a large-scale poloidal magnetic field threading the whole domain. We focus both on the axisymmetric configurations produced by the initial winding-up of the magnetic field lines and on the possible instabilities of those configurations. We aim in particular at assessing the role of the stratification at stabilising the system. We perform 2D and 3D global Boussinesq numerical simulations started from an initial radial or cylindrical differential rotation and a large-scale poloidal magnetic field. Under the conditions of a large rotation frequency compared to the Alfv\'en frequency, a magnetic configuration strongly dominated by its toroidal component is built. The parameters of the simulations are chosen to respect the ordering of time scales of a typical stellar radiative zone. In this framework, the axisymmetric evolution is studied by varying the relative effects of the thermal diffusion, the Br\"unt-V\"ais\"al\"a frequency, the rotation and the initial poloidal field strength. We find that the axisymmetric state only depends on $t_{es}/t_{Ap}$, the ratio between the Eddington-Sweet circulation time scale and the Alfv\'en time scale. A scale analysis of the Boussinesq equations allows us to recover this result. In the cylindrical case, a magneto-rotational instability develops when the thermal diffusivity is sufficiently high to enable the favored wavenumbers to be insensitive to the effects of the stable stratification. In the radial case, the magneto-rotational instability is driven by the latitudinal shear created by the back-reaction of the Lorentz force on the flow. Increasing the level of stratification then leaves the growth rate of the instability mainly unaffected while its horizontal length scale grows.