The asymptotic equivalence of fixed heat flux and fixed temperature thermal boundary conditions for rapidly rotating convection

Calkins, Michael A.; Hale, Kevin; Julien, Keith; Nieves, David; Driggs, Derek; Marti, Philippe

doi:10.1017/jfm.2015.606

Cited by 21 publications

(20 citation statements)

References 33 publications

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“…The agreement between the independent scalings Eqs. (20) and (21) shows that Re ∼ Re H and Pe ∼ Pe H , consistent with our assumption that ∼ H in NRL and SRL. Lastly, multiplying by Ek, the momentum transport scalings Eqs.…”

Section: The Nonrotating and Slowly Rotating Limitssupporting

confidence: 85%

“…(1b), we use the abbreviated notation (u • ∇θ ) = u • ∇θ − ∇ • (uθ ). Convective motions in this system are driven by an unstable, system-scale temperature gradient ∂ g T = O( T /H ) measured in the direction of gravityê g , where T is the temperature drop across the fluid layer of system depth H. Here T is sustained either via fixed temperature boundaries or via an applied heat flux Q [21]. Depending on the setup,ê g can be oriented in the axial directionê z [22], the cylindrically radial directionê s [23], or the spherically radial directionê r [16].…”

Section: Governing Equations and Parametersmentioning

confidence: 99%

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Connections between nonrotating, slowly rotating, and rapidly rotating turbulent convection transport scalings

Aurnou

Horn

Julien

2020

Phys. Rev. Research

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In this paper, we investigate and develop scaling laws as a function of external nondimensional control parameters for heat and momentum transport for nonrotating, slowly rotating, and rapidly rotating turbulent convection systems, with the end goal of forging connections and bridging the various gaps between these regimes. Two perspectives are considered, one where turbulent convection is viewed from the standpoint of an applied temperature drop across the domain and the other with a viewpoint in terms of an applied heat flux. While a straightforward transformation exists between the two perspectives, indicating equivalence, it is found the former provides a clear set of connections that bridge between the three regimes. Our generic convection scalings, based upon an inertial-Archimedean balance, produce the classic diffusion-free scalings for the nonrotating limit and the slowly rotating limit. This is characterized by a free-falling fluid parcel on the global scale possessing a thermal anomaly on par with the temperature drop across the domain. In the rapidly rotating limit, the generic convection scalings are based on a Coriolis-inertial-Archimedean (CIA) balance, along with a local fluctuating-mean advective temperature balance. This produces a scenario in which anisotropic fluid parcels attain a thermal wind velocity and where the thermal anomalies are greatly attenuated compared to the total temperature drop. We find that turbulent scalings may be deduced simply by consideration of the generic nondimensional transport parameters-local Reynolds Re = U /ν; local Péclet Pe = U /κ; and Nusselt number Nu = U ϑ/(κ T /H)-through the selection of physically relevant estimates for length , velocity U , and temperature scales ϑ in each regime. Emergent from the scaling analyses is a unified continuum based on a single external control parameter, the convective Rossby number, Ro C = gα T /4 2 H , that strikingly appears in each regime by consideration of the local, convection-scale Rossby number Ro = U/(2). Thus we show that Ro C scales with the local Rossby number Ro in both the slowly rotating and the rapidly rotating regimes, explaining the ubiquity of Ro C in rotating convection studies. We show in non-, slowly, and rapidly rotating systems that the convective heat transport, parametrized via Pe , scales with the total heat transport parameterized via the Nusselt number Nu. Within the rapidly rotating limit, momentum transport arguments generate a scaling for the system-scale Rossby number, Ro H , that, recast in terms of the total heat flux through the system, is shown to be synonymous with the classical flux-based CIA scaling, Ro CIA. These, in turn, are then shown to asymptote to Ro H ∼ Ro CIA ∼ Ro 2 C , demonstrating that these momentum transport scalings are identical in the limit of rapidly rotating turbulent heat transfer.

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Section: The Nonrotating and Slowly Rotating Limitssupporting

confidence: 85%

Section: Governing Equations and Parametersmentioning

confidence: 99%

Connections between nonrotating, slowly rotating, and rapidly rotating turbulent convection transport scalings

Aurnou

Horn

Julien

2020

Phys. Rev. Research

Self Cite

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“…However, it should be noted that the specific form of the thermal boundary conditions is unimportant in the limit of rapid rotation (Calkins et al. 2015), and the present model can be generalised to no-slip mechanical boundary conditions (Julien et al. 2016).…”

Section: Methodsmentioning

confidence: 96%

On the inverse cascade and flow speed scaling behaviour in rapidly rotating Rayleigh–Bénard convection

et al. 2021

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“…Choosing fixed-temperature or fixed-flux boundary conditions results in different formulations of the Rayleigh and Nusselt numbers; these differences are outlined in §2. However, after appropriate translation between the fixed-flux and fixed-temperature formulations the Nusselt-Rayleigh scaling is the same in both cases (Otero et al 2002;Ahlers et al 2009;Johnston & Doering 2009;Calkins et al 2015;Goluskin 2015).…”

Section: Heterogeneous Boundary Conditions For the Corementioning

confidence: 98%

Heat transfer in rapidly rotating convection with heterogeneous thermal boundary conditions

Mound

Davies

2017

J. Fluid Mech.

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Convection in the metallic cores of terrestrial planets is likely to be subjected to lateral variations in heat flux through the outer boundary imposed by creeping flow in the overlying silicate mantles. Boundary anomalies can significantly influence global diagnostics of core convection when the Rayleigh number, Ra, is weakly supercritical; however, little is known about the strongly supercritical regime appropriate for planets. We perform numerical simulations of rapidly rotating convection in a spherical shell geometry and impose two patterns of boundary heat flow heterogeneity: a hemispherical Y 1 1 spherical harmonic pattern; and one derived from seismic tomography of Earth's lower mantle. We consider Ekman numbers 10 −4 E 10 −6 , flux-based Rayleigh numbers up to ∼ 800 times critical, and Prandtl number unity. The amplitude of the lateral variation in heat flux is characterised by q * L = 0, 2.3, 5.0, the peak-to-peak amplitude of the outer boundary heat flux divided by its mean. We find that the Nusselt number, Nu, can be increased by up to ∼ 25% relative to the equivalent homogeneous case due to boundary-induced correlations between the radial velocity and temperature anomalies near the top of the shell. The Nu enhancement tends to become greater as the amplitude and length scale of the boundary heterogeneity are increased and as the system becomes more supercritical. This Ra dependence can steepen the Nu ∝ Ra γ scaling in the rotationally dominated regime, with γ for our most extreme case approximately 20% greater than the equivalent homogeneous scaling. Therefore, it may be important to consider boundary heterogeneity when extrapolating numerical results to planetary conditions. Key words: Authors should not enter keywords on the manuscript, as these must be chosen by the author during the online submission process and will then be added during the typesetting process (see http://journals.cambridge.org/data/relatedlink/jfmkeywords.pdf for the full list)

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The asymptotic equivalence of fixed heat flux and fixed temperature thermal boundary conditions for rapidly rotating convection

Cited by 21 publications

References 33 publications

Connections between nonrotating, slowly rotating, and rapidly rotating turbulent convection transport scalings

Connections between nonrotating, slowly rotating, and rapidly rotating turbulent convection transport scalings

On the inverse cascade and flow speed scaling behaviour in rapidly rotating Rayleigh–Bénard convection

Heat transfer in rapidly rotating convection with heterogeneous thermal boundary conditions

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