Analysis of instability of latitudinal differential rotation and toroidal field in the solar tachocline using a magnetohydrodynamic shallow-water model. I. Instability for broad toroidal field profiles

We examine the global MHD instability of solar tachocline latitudinal differential rotation and the coexisting broad toroidal magnetic field, using a "shallow-water" model that captures the simplest effects of subadiabatic stratification. We assume a single fluid shell that has a fixed bottom but variable thickness. This model is the MHD generalization of a hydrodynamic model that we have previously applied to the tachocline, although the solution method is somewhat different. Stratification in the model is characterized by an "effective gravity" G (G = 0 for adiabatic stratification). The radiative (lower) part of the tachocline thus has high G (∼10²) and the overshoot part, low G (less than 1). We obtain growth rates, phase velocities, and spatial structures of unstable modes for a wide range of toroidal field strengths and effective gravities, as well as differential rotations that are consistent with helioseismic observations. We recover known two-dimensional MHD stability results in the limit of large G and hydrodynamic instability results in the limit of vanishing toroidal field. For strong magnetic fields, only longitudinal wavenumber m = 1 is unstable, but for weak fields m = 2 is also. For peak toroidal fields of 20 kG and above, the growth rates and disturbance structures are essentially independent of the effective gravity, until it becomes so small that the fluid shell shrinks to zero in low latitudes, whereupon the instability is cut off. In contrast, the instability evolves radically at low G when toroidal field is increased from zero. In both overshoot and radiative parts of the tachocline, unstable modes grow fastest for toroidal fields of the order of 10² kG. The structure of the unstable disturbances is always governed by the latitude location of singular or critical points at which the Doppler-shifted phase velocity of the disturbance equals the local (angular) Alfvén speed. All unstable disturbances possess kinetic helicity, narrowly concentrated in the neighborhood of the same critical points. Just as shown by Dikpati & Gilman for the hydrodynamic case, such disturbances could provide an "a-effect" for the solar dynamo. But unlike the hydrodynamic case, this en-effect would be a function of the toroidal field itself.