Local stability of differential rotation in magnetised radiation zones and the solar tachocline

We study local magnetohydrodynamical (MHD) instabilities of differential rotation in magnetised, stably-stratified regions of stars and planets using a Cartesian Boussinesq model. We consider arbitrary latitudes and general shears (with gravity direction misaligned from this by an angle φ), to model radial (φ = 0), latitudinal (φ = ±90°), and mixed differential rotations, and study both non-diffusive (including magnetorotational, MRI, and Solberg-Høiland instabilities) and diffusive instabilities (including Goldreich-Schubert-Fricke, GSF, and MRI with diffusion). These instabilities could drive turbulent transport and mixing in radiative regions, including the solar tachocline and the cores of red giant stars, but their dynamics are incompletely understood. We revisit linear axisymmetric instabilities with and without diffusion and analyse their properties in the presence of magnetic fields, including deriving stability criteria and computing growth rates, wavevectors and energetics, both analytically and numerically. We present a more comprehensive analysis of axisymmetric local instabilities than prior work, exploring arbitrary differential rotations and diffusive processes. The presence of a magnetic field leads to stability criteria depending upon angular velocity rather than angular momentum gradients. We find MRI operates for much weaker differential rotations than the hydrodynamic GSF instability, and that it typically prefers much larger lengthscales, while the GSF instability is impeded by realistic strength magnetic fields. We anticipate MRI to be more important for turbulent transport in the solar tachocline than the GSF instability when φ > 0 in the northern (and vice versa in the southern) hemisphere, though the latter could operate just below the convection zone when MRI is absent for φ < 0.