Heat Transport in Low-Rossby-Number Rayleigh-Bénard Convection

Abstract

We demonstrate, via simulations of asymptotically reduced equations describing rotationally constrained Rayleigh-Bénard convection, that the efficiency of turbulent motion in the fluid bulk limits overall heat transport and determines the scaling of the nondimensional Nusselt number Nu with the Rayleigh number Ra, the Ekman number E, and the Prandtl number σ. For E << 1 inviscid scaling theory predicts and simulations confirm the large Ra scaling law Nu-1 ≈ C(1)σ(-1/2)Ra(3/2)E(2), where C(1) is a constant, estimated as C(1) ≈ 0.04 ± 0.0025. In contrast, the corresponding result for nonrotating convection, Nu-1 ≈ C(2)Ra(α), is determined by the efficiency of the thermal boundary layers (laminar: 0.28 ≤ α ≤ 0.31, turbulent: α ~ 0.38). The 3/2 scaling law breaks down at Rayleigh numbers at which the thermal boundary layer loses rotational constraint, i.e., when the local Rossby number ≈ 1. The breakdown takes place while the bulk Rossby number is still small and results in a gradual transition to the nonrotating scaling law. For low Ekman numbers the location of this transition is independent of the mechanical boundary conditions.

Laboratory-numerical models of rapidly rotating convection in planetary cores

“…Data points below Nu = 1.3 are not considered in these fits, as they correspond to a shallower Nu-Ra near onset (e.g. Julien et al 2012b). For the steep scaling regime, we find that β monotonically increases with decreasing E, with a roughly linear trend between log (E) and β.…”

“…The resulting transition scaling, Ra T ∼ E −7/4 , does not strongly collapse our present, lower E heat transfer data. However, the essential concept posited in King et al (2009)-that boundary layer processes underly the heat transfer transition-are not refuted (Niemela & Sreenivasan 2006;Cébron et al 2010;Julien et al 2012b). Fig.…”

“…Although we have carried out a limited number of numerical simulations in this low E, steep scaling regime, the best-fitting trend is statistically distinct from a cubic law. Julien et al (2012b) argue that the β = 3 law is among a family of plausible solutions for rapidly rotating convection, found to contain marginally unstable thermal boundary layers. However, our β = 3.6 result implies that the marginal rotating boundary layer mechanism Figure 5.…”

Turbulent convection in liquid metal with and without rotation

“…King et al argue that the heat transfer by rotating RBC with Pr ≥ 1 is determined by the marginal stability of the boundary layer, which gives a heat transfer scaling law Nu = ðRa=Ra c Þ 3 (37). Julien et al analyze an asymptotically reduced system of equations for rapidly rotating RBC, finding the emergence of an "ultimate" regime where Nu ∝ ðRa=Ra c Þ 3=2 (38). In this ultimate regime, the primary bottleneck for heat transfer is the rotationally constrained interior turbulence, not thermal boundary layers.…”

“…When Ra > Ra t , the boundary layer is instead stabilized predominantly by rotationally independent diffusive effects, leading to convective heat transfer that is not influenced by rotation ( [1] (38).…”