Transition to geostrophic convection: the role of boundary conditions

The so-called geostrophic regime of rapidly rotating Rayleigh--B\'enard convection is dominated by rotation with strong enough thermal forcing to attain a turbulent flow. It is the appropriate regime for the description of the large-scale geophysical and astrophysical convective flows. Only very recently, numerical simulations and experiments have become able to enter into this regime with distinctly different scalings than the traditional rotation-affected regime, with many open questions remaining. We explore the transition to the geostrophic regime using direct numerical simulations of the Navier--Stokes and heat equations by varying the rotation rate (Ekman number~$Ek$) at two constant values of the thermal forcing (Rayleigh number~$Ra=1\times 10^{10}$ and~$5\times 10^{10}$) and constant Prandtl number~$Pr=1$. We focus on the differences between the application of no-slip or stress-free boundary conditions on the horizontal plates. We find the transition as changes in heat transfer, boundary-layer thickness, bulk/boundary-layer distribution of dissipation and bulk mean temperature gradient. The transition is gradual: many statistics reveal a change in scaling, but not sharp and not at exactly matching~$Ek$.