Prolateness of the solar tachocline inferred from latitudinal force balance in a magnetohydrodynamic shallow-water model

Motivated by recent helioseismic observations concerning solar tachocline shape and thickness and by the theoretical development of MHD shallow-water equations for the tachocline, we compute the prolateness of the tachocline using an MHD shallow-water model, in which the shape and thickness are determined from the latitudinal force balance equation. We show that a strong toroidal magnetic field stored at or below the overshoot part of the tachocline leads to a pileup of fluid at high latitude, owing to the poleward magnetic curvature stress which has to be balanced by an equatorward latitudinal hydrostatic pressure gradient. For toroidal fields of solar amplitude (similar to 100 kG), results for differentially rotating and uniformly rotating tachoclines are almost the same. In contrast, the unmagnetized differentially rotating tachocline would always be weakly oblate. We propose that a strong toroidal held in the overshoot part of the tachocline should tend to suppress the overshooting, thereby increasing the magnetic storage capacity of the layer since the stratification there should become more subadiabatic. We illustrate the effect of this process on the shape and thickness of the layer by assuming its effective gravity is a function of field strength. If toroidal fields are concentrated in relatively narrow bands which migrate toward the equator with the advance of the sunspot cycle, then they should be accompanied by a thickness front advancing at the same rate. Applying our model to the prolateness estimate of Charbonneau et al. yields toroidal fields of 60-150 kG in the overshoot layer, consistent with other considerations. Their prolateness in the radiative part of the tachocline would require similar to 600 kG fields to be present.