  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, heat flux,
and kinetic and thermal dissipation rates are all diffusion-free, meaning that
they are independent of the viscosity and thermal diffusivity. Our direct
numerical simulations (DNS) of rotating Rayleigh--B\'enard 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 $Ra=3\times10^{13}$ and the Ekman number
down to $Ek=5\times10^{-9}$) indeed demonstrate these diffusion-free scaling
relations, in particular, that the dimensionless convective heat transport
scales with the supercriticality parameter $\widetilde{Ra}\equiv Ra\,Ek^{4/3}$
as $Nu-1\propto \widetilde{Ra}^{3/2}$, where $Nu$ is the Nusselt number. We
further derive and verify in the DNS that with the decreasing $\widetilde{Ra}$
the geostrophic turbulence regime undergoes a transition into another
geostrophic regime where the convective heat transport scales as $Nu-1\propto
\widetilde{Ra}^{3}$.
