  Rayleigh--B\'enard (RB) convection, the flow in a fluid layer heated from
below and cooled from above, is used to analyze the transition to the
geostrophic regime of thermal convection. In the geostrophic regime, which is
of direct relevance to most geo- and astrophysical flows, the system is
strongly rotated while maintaining a sufficiently large thermal driving to
generate turbulence. We directly simulate the Navier--Stokes equations for two
values of the thermal forcing, i.e. $Ra=10^{10}$ and $Ra=5\cdot10^{10}$, a
constant Prandtl number~$Pr=1$, and vary the Ekman number in the range
$Ek=1.3\cdot10^{-7}$ to $Ek=2\cdot10^{-6}$ which satisfies both requirements of
super-criticality and strong rotation. We focus on the differences between the
application of no-slip vs. stress-free boundary conditions on the horizontal
plates. The transition is found at roughly the same parameter values for both
boundary conditions, i.e. at~$Ek\approx 9\times 10^{-7}$ for~$Ra=1\times
10^{10}$ and at~$Ek\approx 3\times 10^{-7}$ for~$Ra=5\times 10^{10}$. However,
the transition is gradual and it does not exactly coincide in~$Ek$ for
different flow indicators. In particular, we report the characteristics of the
transitions in the heat transfer scaling laws, the boundary-layer thicknesses,
the bulk/boundary-layer distribution of dissipations and the mean temperature
gradient in the bulk. The flow phenomenology in the geostrophic regime evolves
differently for no-slip and stress-free plates. For stress-free conditions the
formation of a large-scale barotropic vortex with associated inverse energy
cascade is apparent. For no-slip plates, a turbulent state without large-scale
coherent structures is found; the absence of large-scale structure formation is
reflected in the energy transfer in the sense that the inverse cascade, present
for stress-free boundary conditions, vanishes.

Fluid Dynamics