Magnetohydrodynamic instabilities in stellar radiative regions. I. Linear study of shear-driven instabilities

This paper is the first in a series investigating magnetohydrodynamic instabilities that may contribute to angular-momentum transport and magnetic-field evolution in stellar radiative zones. We focus on shear-driven instabilities, specifically the Goldreich-Schubert-Fricke (GSF) instability and the magnetorotational instability (MRI), which are expected to play key roles in the internal dynamics of radiative regions. We carried out a local linear stability analysis using a numerical approach that extends beyond classical limiting cases and includes stabilizing effects such as stratification and magnetic tension, allowing the exploration of realistic flow regimes. These results were validated through a global mode analysis in a Taylor-Couette configuration. We recovered the standard MRI and azimuthal MRI stability criteria and quantified the effects of stratification, magnetic tension, and diffusion on their growth. In strongly sheared regimes, we derived a new criterion for the magnetised GSF (MGSF) instability and clarified the transition from SMRI to MGSF as stratification and magnetic effects narrow the unstable domain. We also provided approximate growth-time formulae that identify the dominant instability under given stellar conditions and can be implemented in 1D stellar evolution codes. Global Taylor-Couette calculations validate the local WKB analysis. Applied to subgiants and young red giants, our results show that shear-driven instabilities can grow rapidly for magnetic fields below 100 kG. Strong axial fields (100 kG) confined to the hydrogen-burning shell suppress instabilities unless the shear is sufficiently distant. These results support incorporating our criteria and growth estimates into stellar evolution models to assess the efficiency of shear-driven transport.