Sensitivity of rapidly rotating Rayleigh-Bénard convection to Ekman pumping

Plumley M., Julien K., Marti P., Stellmach S.

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 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 extensive range of the heat transfer results. For stress-free boundaries, the relation Nu-1 (RaE4/3)α has the dissipation-free scaling of α=3/2 for all E≤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 Nu(E,Ra) parameter space. For E<10-7 results suggest that the surface can be expressed as Nu-1 [1+P(E)](RaE4/3)3/2 indicating the dissipation-free scaling law is enhanced by Ekman pumping by the multiplicative prefactor [1+P(E)] where P(E)≈5.97E1/8. It follows for E<10-7 that the geostrophic turbulent interior remains the flux bottleneck in rapidly rotating Rayleigh-Bénard convection. For E∼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 α≈1.2 and Nu-1 (RaE4/3)1.2.