Optimal heat transport in rotating Rayleigh-B\'{e}nard convection at large Rayleigh numbers

Abstract

The heat transport in rotating Rayleigh-B\'enard convection (RBC) can be significantly enhanced for moderate rotation, i.e., for an intermediate range of Rossby numbers $Ro$, compared to the non-rotating case. At Rayleigh numbers $Ra\lesssim5\cdot10^8$, the largest heat transport enhancement (HTE) is achieved when the thicknesses of kinetic and thermal boundary layer are equal. However, experimental and numerical observations show that, at larger $Ra$ ($\gtrsim5\cdot10^8$), the maximal HTE starts to deviate from the expected optimal boundary layer ratio and its amplitude decreases drastically. We present data from direct numerical simulations of rotating RBC in a periodic domain in the range of $10^7\leq Ra\leq10^{10}$ and $0\leq Ro^{-1}\leq40$ for Prandtl number $Pr=4.38$ and $6.4$ to identify the reason for the transition to this large $Ra$ regime of HTE. Our analysis reveals that the transition occurs once the bulk flow at the optimal boundary layer ratio changes to geostrophic turbulence for large $Ra$. In that flow state, the vertically coherent vortices, which support HTE by Ekman pumping at smaller $Ra$, dissolve into vertically decorrelated structures in the bulk, such that the enhancing effect of Ekman pumping and the influence of the boundary layer ratio become small. Additionally, more heat leaks out of the Ekman vortices as the fraction of thermal dissipation in the bulk increases. We find that the rotation induced shearing at the plates helps to increase the thermal dissipation in the bulk, and thus acts as a limiting factor for HTE at large $Ra$: beyond a certain ratio of wall shear stress to vortex strength, the heat transport decreases irrespectively of the boundary layer ratio. This $Pr$ dependent threshold, which roughly corresponds to a bulk accounting for $\approx1/3$ of the total thermal dissipation, sets the maximal HTE and the optimal rotation rate at large $Ra$.