The onset of low Prandtl number thermal convection in thin spherical shells

Abstract

This study considers the onset of stress-free Boussinesq thermal convection in rotating spherical shells with aspect ratio $\hat{I}\cdot =r_i/r_o=0.9$ ($r_i$ and $r_o$ being the inner and outer radius), Prandtl numbers $\mathrm{Pr}\in [10^{-4},10^{-1}]$, and Taylor numbers $\mathrm{Ta}\in [10^4,10^{12}]$. We are particularly interested in the form of the convective cell pattern that develops, and in its time scales, since this may have observational consequences. For a fixed $\mathrm{Ta}<10^9$ and by decreasing $\mathrm{Pr}$ from 0.1 to $10^{-4}$ a transition between spiralling columnar (SC) and equatorially-attached (EA) modes, and a transition between EA and equatorially antisymmetric or symmetric polar (AP/SP) weakly multicellular modes are found. The latter modes are preferred at very low $\mathrm{Pr}$. Surprisingly, for $\mathrm{Ta}>3\times 10^9$ the unicellular polar modes become also preferred at moderate $\mathrm{Pr}\sim 10^{-2}$ because two new transition curves between EA and AP/SP and between AP/SP and SC modes are born at a triple-point bifurcation. The dependence on $\mathrm{Pr}$ and $\mathrm{Ta}$ of the transitions is studied to estimate the type of modes, and their critical parameters, preferred at different stellar regimes.

Accepted for publication in Physical Review Fluids. Contains 17 pages, 8 figures and 3 tables. Added brief erratum correcting values used for estimates of neutron star ocean viscosity