Global MHD instabilities in a three-dimensional thin-shell model of solar tachocline

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 ≤20 kG is similar to the corresponding shallow water case, but for substantially higher (perhaps unrealistic) toroidal fields there is no low-subadiabaticity cutoff, and modes appear with much higher growth rate and increasingly negative phase velocities in longitude, analogous to those found earlier by Cally. All of these results are only modestly sensitive to the tachocline thickness chosen. For broad toroidal field profiles, the instability results are similar to the two-dimensional and shallow water cases for toroidal field peaks up to at least 80 kG, unless the shell has a thickness that is a substantial fraction of a pressure scale height. For thinner shells, only above about 94 kG do the high growth rate, low phase velocity modes appear, with structure similar to the "polar kink" instability Cally found. But even in this case, we find that the slower growing "clam-shell" instability eventually replaces the faster growing polar kink modes. We conclude that for conditions most likely to occur in the solar tachocline, such as peak toroidal fields limited by the dynamo to 20 kG or less, the two-dimensional and shallow water type unstable modes are likely to predominate even when the tachocline has finite thickness and the modes can have radial structure.