Global MHD Instabilities in a Three‐dimensional Thin‐Shell Model of Solar Tachocline

Abstract

We generalize the linear analysis of the global instability of coexisting differential rotation and toroidal magnetic fields in the solar tachocline to include continuous radial stratification, thermodynamics, and finite tachocline thickness, as perturbed by three-dimensional disturbances of longitudinal wavenumbers m=1, 2. For radiative tachocline stratification, the instability for both banded and broad toroidal field profiles is similar to the two-dimensional and shallow water cases studied previously, even though the unstable modes have substantial vertical structure. For overshoot tachocline stratification, instability for banded toroidal fields with peaks