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 (Chandrasekhar 1961). 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 (Niiler & Bisshopp 1965;Heard & Veronis 1971). By contrast, in the nonlinear regime recent laboratory experiments and supporting numerical simulations are now providing evidence that the efficiency of heat transport remains strongly influenced by Ekman pumping in the rapidly rotating limit (Stellmach et al. 2014;Cheng et al. 2015). 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 (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 a heat transport that diverges to infinity. 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 this behavior 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.

Rotating convective turbulence in Earth and planetary cores

“…It should be possible to extrapolate this result to low E geophysical settings, so long as the horizontal length scale of boundary variations greatly exceeds ' ' E 1=3 H. Thus, we hypothesize that Ekman pumping effects may prove to be important in terrestrial planetary cores Julien et al, 2015), while having little relevance in gas planets and stars (e.g., Julien et al, 2012a; Barker et al, 2014). ).…”

“…However, the reduced equations have been shown to be well-defined at all latitudes Calkins et al, 2013). Julien et al (2015) have recently constructed an extended reduced framework that includes the effects of Ekman pumping. Furthermore, Calkins et al (2015b) have developed a set of multi-scale reduced MHD equations that are capable of simulating fully nonlinear dynamo action via quasi-geostrophic convection.…”

Scaling Laws in Rayleigh‐Bénard Convection

“…Figure shows the difference in heat transfer due to boundary condition as indicated by filled (no slip) and open (stress free) symbols. Stellmach et al (); Julien et al () have demonstrated that this effect occurs as a transition within an O ( E 1/3 H ) thermal‐wind layer sandwiched between adjacent an O ( E 1/2 H ) Ekman layers and the fluid interior. The interval of the transition is found to be finite in that the heat transport ultimately resettles to a pretransition exponent but with a significantly enhanced prefactor (Plumley et al, ; demonstrated in Figure b), which illustrates a transition between regimes where Ekman pumping effects are negligible and dominant (Julien et al, ; Plumley et al, ).…”

Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers

“…In Calkins et al [10] it was shown that scale separation can also be assured when Rm = O(1) (i.e. P m = O(1)) for the case of rapidly rotating, anisotropic motions, though for this case α can no longer be written in a form as simple as that given by equation (32). Moreover, the first-order-smoothing approximation, characterized by the absence of terms involving products of u ′ and b ′ in the fluctuating induction equation (e.g.…”

“…To determine the influence of k on dynamo behavior two types of solutions were investigated: (1) the horizontal wavenumber was fixed at the critical (linear) value k = k c for all Ra; and (2) the wavenumber was calculated to maximize the heat transfer as Ra increased [e.g. see 26,32].…”