Dissipation and Mixing in Circulation forced by Differential Surface Heating in a Rotating Basin

We consider the influence of rotation on circulation forced by a surface buoyancy gradient in a rectangular basin. Direct numerical simulations are reported for a rotating $f$-plane ocean with an applied surface temperature differential, Prandtl number $Pr=5$ and two values of the Rayleigh number which, in the absence of rotation, relate to the viscous ($Ra=7.4\times10^8$) and inertial ($Ra=7.4\times10^{11}$) regimes. The effects of rotation depend primarily on the parameter $Q$, the square ratio of the relative thicknesses of the thermal boundary layer to the Ekman layer. The strongly rotating regime is governed by geostrophic balance within the thermal boundary layer and causes this layer to become thicker, reducing the Nusselt number as $Nu\sim Q^{-1/3}$. At very strong rotation, the smaller $Ra$ gives a conduction dominated regime with strong interior stratification and $Nu\sim 1$. The dynamics at larger $Ra$ are rich, with small-scale turbulent convection in the thermal boundary layer, similar to the non-rotating case. Eddies extend through the depth of the interior outside the boundary layer over the heated region and can advect horizontally to the cooled region, thus eddy flux contributes significantly to the heat transport. The viscous dissipation remains unchanged by rotation to relatively large $Q$, and we attribute this to the larger Rossby numbers of small scale convection in the boundary layer. In contrast, the rate of mixing decreases throughout the geostrophic regime, scaling with $Nu$. Thus the global mixing efficiency $\eta$ decreases with increasing rotation; however it remains consistent with the theoretical prediction $\eta=1-(HNu/L)^{-1}$ (where $H$ is height and $L$ is length of the domain) for $HNu/L\gg 2$. The efficiency also approaches unity in the conduction regime at extremely rapid rotation, and hence there is a trough in minimum $\eta$ near the onset of rotational effects on turbulent dissipation scales in the boundary layer. With or without rotation, $\eta$ approaches unity at large $Nu$, and therefore at large $Ra$.