Scaling behaviour of rotating convection in a spherical shell with different Prandtl numbers

Abstract

Rayleigh–Bénard convection in a rotating spherical shell provides a simplified model for convective dynamics of planetary and stellar interiors. Over the past decades, the problem has been studied extensively via numerical simulations, but most previous simulations set the Prandtl number $Pr$ to unity. In this study we build more than 200 numerical models of rotating convection in a spherical shell over a wide range of $Pr$ ( $10^{-2}\le Pr \le 10^2$ ). By increasing the Rayleigh number $Ra$ , we characterise four different flow regimes, starting from the linear onset to multiple modes, then transitioning to the geostrophic turbulence and eventually approaching the weakly rotating regime. In the multiple modes regime, we show evidence of triadic resonances in numerical models with different $Pr$ , which may provide a generic mechanism for the transition from laminar to turbulence in rotating convection. We analyse scaling behaviours of the heat transfer and convective flow speeds in numerical simulations, paying particular attention to the $Pr$ dependence. We find that the so-called diffusion-free scaling for the heat transfer cannot reconcile all numerical models with different $Pr$ in the geostrophic turbulence regime. However, the characteristic flow speeds at different $Pr$ roughly follow a unified scaling that can be described by visco-Archimedean–Coriolis force balances, though the scaling tends to approach the Coriolis-inertial-Archimedean force balance at low $Pr$ . We also show that transition behaviours from rotating to non-rotating convection depend on $Pr$ . The transition criteria based on heat transfer and flow morphology would be rather different when $Pr>1$ , but the two criteria are consistent for cases with $Pr\le 1$ . Both scaling behaviours and transition behaviours suggest that the heat transfer is controlled by the boundary layers while the convective flow speeds are mainly determined by the force balance in the bulk for cases with $Pr>1$ , which is in line with recent experimental results with moderate to high $Pr$ . For cases with $Pr \le 1$ , both the heat transfer and convective velocities are approaching the inviscid dynamics in the bulk. We also briefly analysed the magnitude and scaling of zonal flows at different $Pr$ , showing that the zonal flow amplitude rapidly increases as $Pr$ decreases.