The sensitivity of rapidly rotating Rayleigh--B\'enard convection to Ekman pumping

Abstract

The dependence of the heat transfer, as measured by the nondimensional Nusselt number $Nu$, on Ekman pumping for rapidly rotating Rayleigh-B\'enard convection in an infinite plane layer is examined for fluids with Prandtl number $Pr = 1$. A joint effort utilizing simulations from the Composite Non-hydrostatic Quasi-Geostrophic model (CNH-QGM) and direct numerical simulations (DNS) of the incompressible fluid equations has mapped a wide range of the Rayleigh number $Ra$ - Ekman number $E$ parameter space within the geostrophic regime of rotating convection. Corroboration of the $Nu$-$Ra$ 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 range of the heat transfer results. For stress-free boundaries, the relation $Nu-1 \propto (Ra E^{4/3} )^{\alpha} $ has the dissipation-free scaling of $\alpha = 3/2$ for all $E \leq 10^{-7}$. 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 2D surface in $Nu(E,Ra)$ parameter space. For $E<10^{-7}$ results suggest that the surface can be expressed as $Nu-1 \propto (1+ P(E)) (Ra E^{4/3} )^{3/2}$ indicating the dissipation-free scaling law is enhanced by Ekman pumping by the multiplicative prefactor $(1+ P(E))$ where $P(E) \approx 5.97 E^{1/8}$. It follows for $E<10^{-7}$ that the geostrophic turbulent interior remains the flux bottleneck in rapidly rotating Rayleigh-B\'enard convection. For $E\sim10^{-7}$, 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 $Nu-1 \propto (Ra E^{4/3} )^{1.2}$.