Storage and Equilibrium of Toroidal Magnetic Fields in the Solar Tachocline: A Comparison between MHD Shallow-Water and Full MHD Approaches

Recently Dikpati & Gilman have shown, using a shallow-water model of the solar tachocline that allows the top surface to deform, that a tachocline with the observed broad differential rotation and a strong toroidal field is prolate. A strong toroidal field ring requires extra mass on its poleward side to provide a hydrostatic latitudinal pressure gradient to balance the poleward curvature stress. In a parallel study using a different approach, Rempel, Schüssler, & T?th have shown that such a latitudinal pressure gradient is found in a strongly subadiabatic stratification, whereas a weakly subadiabatic stratification leads to a Acomplementary equilibrium state of the overshoot tachocline in which the magnetic curvature stress is balanced by a prograde rotational jet inside the toroidal ring. We show that the shallow-water model with height deformation is a first-order approach to the equilibrium state found by Rempel, Schüssler, & T?th for a strongly subadiabatic stratification. We also show that the shallow-water model can be generalized to allow for the equilibrium state found for a weakly subadiabatic stratification by suppressing the shell deformation associated with the toroidal field and allowing the differential rotation to be modified.