Nonlinear evolution of global magnetoshear instabilities in a three-dimensional thin-shell model of the solar tachocline

We investigate global instabilities of toroidal fields and differential rotation in the solar tachocline using a three-dimensional thin-shell model. We initiate our nonlinear numerical simulations by superposing random, high-wavenumber perturbations on an equilibrium state and we then allow the system to evolve freely as the instabilities develop, grow exponentially, and saturate. For broad toroidal field profiles the dominant mode is the clamshell instability previously identified in two-dimensional and shallow-water investigations in which loops of field in the northern and southern hemispheres tilt and reconnect, eventually becoming perpendicular to the equatorial plane. If the initial toroidal field is instead confined to thin bands of alternating polarity in the northern and southern hemispheres, the evolution is more complex. At early times, a previously unidentified instability occurs near the edges of each band that is characterized by a high longitudinal wavenumber m. These edge instabilities are later superseded by an m = 1 tipping instability whereby field lines tilt in latitude as in the broad-field case. The tipping instability saturates by forming a jet of fluid within each band, which provides gyroscopic stabilization. Both the clamshell and the tipping instabilities are quasi - two-dimensional and proceed most efficiently near the impenetrable boundaries where the vertical velocity vanishes. Strong stable stratification enhances this tendency by decoupling horizontal layers. These simulations demonstrate the robustness of global m = 1 instabilities in differentially rotating spherical shells threaded by toroidal magnetic fields and as such have important implications for tachocline dynamics, including dynamo processes and tachocline confinement.