A comparison of no-slip, stress-free and inviscid models of rapidly rotating fluid in a spherical shell

We investigate how the choice of either no-slip or stress-free boundary conditions affects numerical models of rapidly rotating flow in Earth's core by computing solutions of the weakly-viscous magnetostrophic equations within a spherical shell, driven by a prescribed body force. For non-axisymmetric solutions, we show that models with either choice of boundary condition have thin boundary layers of depth E^(1/2), where E is the Ekman number, and a free-stream flow that converges to the formally inviscid solution. At Earth-like values of viscosity, the boundary layer thickness is approximately 1m, for either choice of condition. In contrast, the axisymmetric flows depend crucially on the choice of boundary condition, in both their structure and magnitude (either E^(-1/2) or E^(-1)). These very large zonal flows arise from requiring viscosity to balance residual axisymmetric torques. We demonstrate that switching the mechanical boundary conditions can cause a distinct change of structure of the flow, including a sign-change close to the equator, even at asymptotically low viscosity. Thus implementation of stress-free boundary conditions, compared with no-slip conditions, may yield qualitatively different dynamics in weakly-viscous magnetostrophic models of Earth's core. We further show that convergence of the free-stream flow to its asymptotic structure requires E ≤10^(-5).