A nonlinear model for rotationally constrained convection with Ekman pumping

A reduced model is developed for low-Rossby-number convection in a plane layer geometry with no-slip upper and lower boundaries held at fixed temperatures. A complete description of the dynamics requires the existence of three distinct regions within the fluid layer: a geostrophically balanced interior where fluid motions are predominantly aligned with the axis of rotation, Ekman boundary layers immediately adjacent to the bounding plates, and thermal wind layers driven by Ekman pumping in between. The reduced model uses a classical Ekman pumping parameterization to alleviate the need to resolve the Ekman boundary layers. Results are presented for both linear stability theory and a special class of nonlinear solutions described by a single horizontal spatial wavenumber. It is shown that Ekman pumping (which correlates positively with interior convection) allows for significant enhancement in the heat transport relative to that observed in simulations with stress-free boundaries. Without the intermediate thermal wind layer, the nonlinear feedback from Ekman pumping would be able to generate heat transport that diverges to infinity at finite Rayleigh number. This layer arrests this blowup, resulting in finite heat transport at a significantly enhanced value. With increasing buoyancy forcing, the heat transport transitions to a more efficient regime, a transition that is always achieved within the regime of asymptotic validity of the theory, suggesting that this behaviour may be prevalent in geophysical and astrophysical settings. As the rotation rate increases, the slope of the heat transport curve below this transition steepens, a result that is in agreement with observations from laboratory experiments and direct numerical simulations.