Scaling regimes in rapidly rotating thermal convection at extreme Rayleigh numbers

The geostrophic turbulence in rapidly rotating thermal convection exhibits characteristics shared by many highly turbulent geophysical and astrophysical flows. In this regime, the convective length and velocity scales and heat flux are all diffusion-free, i.e. independent of the viscosity and thermal diffusivity. Our direct numerical simulations (DNS) of rotating Rayleigh–Bénard convection in domains with no-slip top and bottom and periodic lateral boundary conditions for a fluid with the Prandtl number $Pr=1$ and extreme buoyancy and rotation parameters (the Rayleigh number up to Ra=3× 10¹³ and the Ekman number down to Ek=5× 10⁻⁹ ) indeed demonstrate all these diffusion-free scaling relations, in particular, that the dimensionless convective heat transport scales with the supercriticality parameter Ra≡ Ra\, Ek4/3 as Nu-1∝ Ra3/2 , where $Nu$ is the Nusselt number. We further derive and verify in the DNS that with the decreasing Ra , the geostrophic turbulence regime undergoes a transition into another geostrophic regime, the convective heat transport in this regime is characterized by a very steep Ra -dependence, Nu-1∝ Ra³ .