Stability analysis of tachocline latitudinal differential rotation and coexisting toroidal band using a shallow-water model

Recently global, quasi-two-dimensional instabilities of tachocline latitudinal differential rotation have been studied using a so-called shallow-water model. While purely hydrodynamic shallow-water type disturbances were found to destabilize only the overshoot tachocline, the MHD analysis showed that in the presence of a broad toroidal field, both the radiative and overshoot parts of the tachocline can be unstable. We explore here instability in the shallow-water solar tachocline with concentrated toroidal bands placed at a wide range of latitudes, emulating different phases of the solar cycle. In equilibrium, the poleward magnetic curvature stress of the band is balanced either by an equatorward hydrostatic pressure gradient or by the Coriolis force from a prograde jet inside the band. We find that toroidal bands placed almost at all latitudes make the system unstable to shallow-water disturbances. For bands without prograde jets, the instability persists well above 100 kG peak field, while a jet stabilizes the band at a field of ∼40 kG. The jet imparts gyroscopic inertia to the toroidal band inhibiting it from unstably " tipping " its axis away from rotation axis. Like previously studied HD and MHD shallow-water instabilities in the tachocline, unstable shallow-water modes found here produce kinetic helicity and hence a tachocline α-effect; these narrow kinetic helicity profiles should generate narrowly confined poloidal fields, which will help formation of the narrow toroidal field. Toroidal bands poleward of 15° latitude suppress midlatitude hydrodynamic α-effects. However, even strong toroidal bands equatorward of 15° allow this hydrodynamic α-effect. Such bands should occur during the late declining phase of a solar cycle and, thus, could help the onset of a new cycle by switching on the mid latitude α-effect.