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

Space missions such as Kepler have brought new constraints, along with new questions, on stellar evolution. A key issue is the unexpected spin-down of low-mass stellar cores, pointing to an efficient angular momentum transport mechanism in which magnetic fields are likely to play a role. This has renewed interest in the origin and impact of magnetic fields in stars. This paper is the first in a series investigating magnetohydrodynamic instabilities that might contribute to angular momentum transport and magnetic-field evolution in stellar radiative zones. Here, we focus on shear-driven instabilities and, specifically, the Goldreich-Schubert-Fricke (GSF) instability and the magnetorotational instability (MRI), which could play key roles in the internal dynamics of stellar radiative regions. We performed a detailed local linear stability analysis using a numerical approach that extends beyond classical limiting cases and incorporates stabilizing effects such as stratification and magnetic tension, enabling the exploration of more realistic flow regimes. These local results were then validated through a global mode analysis in a Taylor-Couette configuration. Together, these methods allow us to identify unstable regions, quantify growth rates, and assess the astrophysical relevance of the instabilities. Finally, we applied our results to evolutionary models of subgiant and young red giant stars constrained by recent observations. We recovered the known standard MRI (SMRI) and azimuthal MRI stability criteria and quantified how stratification, magnetic tension, and diffusion affect their growth. In strongly sheared regimes, we derived a new criterion for the magnetised GSF (MGSF) instability and clarified how magnetic and stratification effects narrow the unstable domain, illuminating the transition from SMRI to MGSF. We also provided approximate growth time formulae that identify which instability (SMRI or MGSF) dominates under given stellar conditions and can be directly implemented in 1D stellar evolution codes to model angular momentum transport more realistically. Global Taylor-Couette calculations validate the local Wentzel-Kramers-Brillouin analysis, confirming that it is able to reliably predicts unstable regions and mode behaviour. In its application to subgiants and young red giants, our results show that shear-driven instabilities can grow rapidly for magnetic fields below 100, The analytical criteria indicate where SMRI or MGSF modes should occur depending on the shear amplitude and location. Conversely, strong axial fields $(∼ 100, )$ confined to the hydrogen-burning shell suppress instabilities unless the shear lies sufficiently far from the shell. These findings support incorporating our instability criteria and growth estimates into stellar evolution models to assess the efficiency of shear-driven transport.