Bounds on Heat Transport in Rapidly Rotating Rayleigh-Bénard Convection

Abstract

The heat transport in rotating Rayleigh-Bénard convection is considered in the limit of rapid rotation (small Ekman number $E$) and strong thermal forcing (large Rayleigh number $Ra$). The analysis proceeds from a set of asymptotically reduced equations appropriate for rotationally constrained dynamics; the conjectured range of validity for these equations is $Ra\lesssim E^{-8/5}$. A rigorous bound on heat transport of $Nu\le 20.56R{a}^3{E}^4$ is derived in the limit of infinite Prandtl number using the background method. We demonstrate that the exponent in this bound cannot be improved on using a piece-wise monotonic background temperature profile like the one used here. This is true for finite Prandtl numbers as well, i.e. $Nu\lesssim R{a}^3$ is the best upper bound for this particular setup of the background method. The feature that obstructs the availability of a better bound in this case is the appearance of small-scale thermal plumes emanating from (or entering) the thermal boundary layer.

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