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

Rayleigh-Benard convection in a rotating spherical shell provides a simplified model for convective dynamics of planetary and stellar interiors. In this study, we build more than 200 numerical models of rotating convection in a spherical shell over a wide range of $\Pran$ ($10^{-2}\le\Pran\le10^2$). As increasing the Rayleigh number $Ra$, we characterise four different flow regimes, starting from the linear onset to multiple modes, then transiting 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 $\Pran$, 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 $\Pran$-dependence. We find that the so-called diffusion-free scaling for the heat transfer cannot reconcile all numerical models with different $\Pran$ in the geostrophic turbulence regime. However, the characteristic flow speeds at different $\Pran$ roughly follow a unified scaling that can be described by VAC force balances, though the scaling tends to approach the CIA force balance at low $\Pran$. 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 $\Pran>1$, which is in line with recent experimental results with moderate to high $\Pran$. For cases with $\Pran \le 1$, both the heat transfer and convective velocities are approaching the inviscid dynamics in the bulk.