A Multilayer Magnetohydrodynamic Shallow-water Model of The Solar Tachocline: Equilibrium Shape and Thickness

Abstract We build a multilayer magnetohydrodynamic shallow-water model to study the thickness and shape of the solar tachocline. This allows us to include characteristics of both the overshoot and the radiative parts of the tachocline. The equations derived include equilibrium in latitude among Coriolis, pressure gradient, and magnetic curvature stresses for each layer, and magnetohydrostatic equilibrium in the radial direction. In each layer, the total mass is conserved; mass is redistributed for different amplitudes and latitude positions of toroidal bands, thus producing variations in tachocline shape and thickness with solar cycle phases. While we solve here for equilibrium of two layers, the equations can be readily generalized for additional layers. In pure hydrodynamic tachocline with no differential rotation, thickness and shape are independent of latitude. With differential rotation and/or magnetic fields, the tachocline is, in general, oblate in equatorial regions but prolate in polar latitudes. A local bump occurs at the poleward side of tachocline toroidal band. Hence, depending on latitude-location and amplitude of magnetic band as function of solar cycle, the local bump drifts equatorward trailing the magnetic field. Oblateness and prolateness are much larger in the overshoot than in the radiative layer, due to its lower effective gravity. Our results can provide guidance for interpreting helioseismic estimates of variations in tachocline shape and thickness in latitude, including upper limits to banded toroidal field amplitudes.