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

Abstract

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.

Oscillatory thermal–inertial flows in liquid metal rotating convection

“…in agreement with Guervilly et al (2019), Maffei et al (2020) and Aurnou et al (2020). In contrast to figure 5(d), the u z,max measurements are not well collapsed by the system-scale Rossby number, Ro z,max = u z,max /(2ΩH), which estimates the ratio of flow inertia and system-scale Coriolis force and whose values lie in the range 3 × 10 −3 Ro z,max 3 × 10 −1 .…”

“…In this regime, the thermal wind and free-fall velocities should be comparable (Aurnou et al. 2020), which explains why these low rotating convection velocities are approaching the non-rotating free-fall limit.…”

“…With this normalization, all the data are clustered in the vicinity of unity (0.7 < u z,max /u TW < 1.8) (e.g. Aurnou et al 2020), showing that the thermal wind scaling holds well. Comparing figures 5(a) and 5(c) shows that our thermal-inertial flows are simultaneously in thermal wind balance and in inertial balance.…”

“…Our UDV and DNS results demonstrate that the thermal–inertial oscillatory convection velocities simultaneously attain rotationally dominated thermal wind values, , and near free-fall values, , for which thermal buoyancy is transferred completely into fluid inertia (cf. Aurnou, Horn & Julien 2020). Multi-modal bulk oscillations dominate the velocity field over the whole range of supercriticalities investigated.…”

Rotation suppresses giant-scale solar convection

“…Our analysis shows that interior flows likely exist in a quasigeostrophic (QG) state, with a joint Coriolis–inertial–Archimedean (CIA) balance (e.g., refs. 7 – 10 ) remaining after accounting for leading-order geostrophy. * Rotation strongly influences solar convection as a result.…”