Sensitivity of rapidly rotating Rayleigh-Bénard convection to Ekman pumping

Abstract

The dependence of the heat transfer, as measured by the nondimensional Nusselt number N u, on Ekman pumping for rapidly rotating Rayleigh-Bénard convection in an infinite plane layer is examined for fluids with Prandtl number P r = 1. A joint effort utilizing simulations from the Composite Non-hydrostatic Quasi-Geostrophic model (CNH-QGM) and direct numerical simulations (DNS) of the incompressible fluid equations has mapped a wide range of the Rayleigh number Ra -Ekman number E parameter space within the geostrophic regime of rotating convection. Corroboration of the N u-Ra relation at E = 10 −7 from both methods along with higher E covered by DNS and lower E by the asymptotic model allows for this extensive range of the heat transfer results. For stress-free boundaries, the relation N u − 1 ∝ (RaE 4/3 ) α has the dissipation-free scaling of α = 3/2 for all E ≤ 10 −7 . This is directly related to a geostrophic turbulent interior that throttles the heat transport supplied to the thermal boundary layers. For no-slip boundaries, the existence of ageostrophic viscous boundary layers and their associated Ekman pumping yields a more complex 2D surface in N u(E, Ra) parameter space. For E < 10 −7 results suggest that the surface can be expressed as N u − 1 ∝ (1 + P (E))(RaE 4/3 ) 3/2 indicating the dissipation-free scaling law is enhanced by Ekman pumping by the multiplicative prefactor (1 + P (E)) where P (E) ≈ 5.97E 1/8 . It follows for E < 10 −7 that the geostrophic turbulent interior remains the flux bottleneck in rapidly rotating Rayleigh-Bénard convection. For E ∼ 10 −7 , where DNS and asymptotic simulations agree quantitatively, it is found that the effects of Ekman pumping are sufficiently strong to influence the heat transport with diminished exponent α ≈ 1.2 and N u − 1 ∝ (RaE 4/3 ) 1.2 . arXiv:1704.04696v1 [physics.flu-dyn]

Scaling Laws in Rayleigh‐Bénard Convection

“…This suggests the existence of a more complex 2‐D surface in Nu ( E , Ra ) parameter space (Kunnen et al, ; Stellmach et al, ). For E <10 −7 and Pr =1 fluids, Plumley et al () suggested that the surface can be expressed as indicating the dissipation‐free scaling law is enhanced by Ekman pumping by the multiplicative prefactor (1+ P ( E )) where P ( E )≈5.97 E 1/8 was fit empirically with their numerical simulations, as shown in Figure b. This indicates a singular transition in the sense that its dependence on supercriticality has an inverse relationship with E ; that is, it is increasingly delayed and moves to infinity as E →0.…”

“…Thus, scaling laws have been observed even at relatively low supercriticality. The asymptotic analysis and results showing convergence to scaling laws (Plumley et al, ) suggest that it is appropriate to benchmark theoretically deduced scalings with the numerical experimental results in the less supercritical, but accessible regime.…”

“…The models have shown high levels of quantitative agreement in comparison with DNS and experimental results in the overlaping region of E ∼10 −7 (Stellmach et al, ; Plumley et al, ). They have also been useful in mapping the distinct flow morphologies and heat transfer within the low E parameter space (Julien et al, ; Plumley et al, ; Rubio et al, ). Notably, a dissipation‐free scaling exponent of α =3/2 is observed by both asymptotic models and DNS for turbulent flow in the presence of stress‐free boundaries.…”

“…The fit incorporates a prefactor of (1+ P ( E )) where P ( E ) is a function of the Ekman number that accounts for the functional dependence of the scaling on E , as shown in (a). This surface is reprinted from Plumley et al (). The use of rather than R a improves the clarity of the plot.…”

“…The desired functional form of the scalings varies by application, but they typically relate an input parameter to an output value. Examples include the relationship between thermal forcing and heat transfer efficiency for nonrotating convection studies (Howard, ; Kraichnan, ), velocity and rotational strengths (Cheng et al, ; Ecke & Niemela, ; King & Aurnou, ; Plumley et al, ; Rubio et al, ), or the typical velocity and magnetic field strength for a given heat flux in dynamo studies (Christensen, ; Davidson, ). Another alternative is to uncover dominant force balances that may be utilized for model reduction or parameterizations appropriate for geophysical and astrophysical settings (Aubert, ; Julien et al, ).…”