A nonlinear model for rotationally constrained convection with Ekman pumping

Abstract

It is a well established result of linear theory that the influence of differing mechanical boundary conditions, i.e., stress-free or no-slip, on the primary instability in rotating convection becomes asymptotically small in the limit of rapid rotation. This is accounted for by the diminishing impact of the viscous stresses exerted within Ekman boundary layers and the associated vertical momentum transport by Ekman pumping. By contrast, in the nonlinear regime recent experiments and supporting simulations are now providing evidence that the efficiency of heat transport remains strongly influenced by Ekman pumping in the rapidly rotating limit. In this paper, a reduced model is developed for the case of 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 predominately 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 for spatially resolving 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 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 a heat transport that diverges to infinity. This layer arrests this blowup resulting in finite heat transport at a significantly enhanced value.

38 pages, 14 figures

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• Oscillatory thermal-inertial flows in liquid metal rotating convection
• Experimental observation of the geostrophic turbulence regime of rapidly rotating convection
• Heat transport scaling and transition in geostrophic rotating convection with varying aspect ratio
• The sensitivity of rapidly rotating Rayleigh--Bénard convection to Ekman pumping
• The Radius and Entropy of a Magnetized, Rotating Fully-convective Star: Analysis With Depth-dependent Mixing Length Theories
• Diffusion-free scaling in rotating spherical Rayleigh-Bénard convection
• Bridging the Rossby number gap in rapidly rotating thermal convection
• Effects of kinematic and magnetic boundary conditions on the dynamics of convection-driven plane layer dynamos
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• Internally heated convection with rotation: bounds on heat transport
• Asymptotic approximations for convection onset with Ekman pumping at low wavenumbers
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• Bounds on solutions of the rotating, stratified, incompressible, non-hydrostatic, three-dimensional Boussinesq equations
• Transition to inverse cascade in turbulent rotating convection in absence of the large-scale vortex