Rigorous bounds on the heat transport of rotating convection with Ekman pumping

Abstract

We establish rigorous upper bounds on the time-averaged heat transport for a model of rotating Rayleigh-Benard convection between no-slip boundaries at infinite Prandtl number and with Ekman pumping. The analysis is based on the asymptotically reduced equations derived for rotationally constrained dynamics with no-slip boundaries, and hence includes a lower order correction that accounts for the Ekman layer and corresponding Ekman pumping into the bulk. Using the auxiliary functional method we find that, to leading order, the temporally averaged heat transport is bounded above as a function of the Rayleigh and Ekman numbers Ra and Ek according to $Nu\le 0.3704R{a}^{2}E{k}^2$. Dependent on the relative values of the thermal forcing represented by $Ra$ and the effects of rotation represented by $Ek$, this bound is both an improvement on earlier rigorous upper bounds, and provides a partial explanation of recent numerical and experimental results that were consistent yet surprising relative to the previously derived upper bound of $Nu\lesssim R{a}^3{k}^4$.

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