The dependence of the heat transfer, as measured by the nondimensional Nusselt number $\text{Nu}$, on Ekman pumping for rapidly rotating Rayleigh-Bénard convection in an infinite plane layer is examined for fluids with Prandtl number $\text{Pr}=1$. A joint effort utilizing simulations from the composite non-hydrostatic quasi-geostrophic model and direct numerical simulations (DNS) of the incompressible fluid equations has mapped a wide range of the Rayleigh number $\text{Ra}$-Ekman number $E$ parameter space within the geostrophic regime of rotating convection. Corroboration of the $\text{Nu}\text{−}\text{Ra}$ relation at $E={10}^{-7}$ from both methods along with higher $E$ covered by DNS and lower $E$ by the asymptotic model allows for this extensive range of the heat transfer results. For stress-free boundaries, the relation $\text{Nu}-1\propto {\left(\text{Ra}{E}^{4/3}\right)}^{\alpha }$ has the dissipation-free scaling of $\alpha =3/2$ for all $E\le {10}^{-7}$. This is directly related to a geostrophic turbulent interior that throttles the heat transport supplied to the thermal boundary layers. For no-slip boundaries, the existence of ageostrophic viscous boundary layers and their associated Ekman pumping yields a more complex two-dimensional surface in $\text{Nu}(E,\text{Ra})$ parameter space. For $E<{10}^{-7}$ results suggest that the surface can be expressed as $\text{Nu}-1\propto [1+P\left(E\right)]{\left(\text{Ra}{E}^{4/3}\right)}^{3/2}$ indicating the dissipation-free scaling law is enhanced by Ekman pumping by the multiplicative prefactor $[1+P(E\left)\right]$ where $P\left(E\right)\approx 5.97E^{1/8}$. It follows for $E<{10}^{-7}$ that the geostrophic turbulent interior remains the flux bottleneck in rapidly rotating Rayleigh-Bénard convection. For $E\sim {10}^{-7}$, where DNS and asymptotic simulations agree quantitatively, it is found that the effects of Ekman pumping are sufficiently strong to influence the heat transport with diminished exponent $\alpha \approx 1.2$ and $\text{Nu}-1\propto {\left(\text{Ra}{E}^{4/3}\right)}^{1.2}$.

• Received 14 April 2017

DOI:https://doi.org/10.1103/PhysRevFluids.2.094801