The sensitivity of rapidly rotating Rayleigh--Bénard 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-Bénard 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 $E\le 10^{-7}$. 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 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 $P(E)\approx 5.97E^{1/8}$. 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 $Nu-1\propto (Ra{E}^{4/3}{)}^{1.2}$.

9 pages, 14 figures