An official website of the United States government
null
(Ed.)
In this study, we investigate and develop scaling laws as a function of external non-dimensional control parameters for heat and momentum transport for non-rotating, 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 exist 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 non-rotating limit (NRL) and the slowly rotating limit (SRL). 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 (RRL), 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 non-dimensional transport parameters --- local Reynolds $$Re_\ell = U \ell /\nu$$; local P\'eclet $$Pe_\ell = U \ell /\kappa$$; and Nusselt number $$Nu = U \vartheta/(\kappa \Delta T/H)$$ --- through the selection of physically relevant estimates for length $$\ell$$, velocity $$U$$ and temperature scales $$\vartheta$$ in each regime. Emergent from the scaling analyses is a unified continuum based on a single external control parameter, the convective Rossby number\JMA{,} $$\RoC = \sqrt{g \alpha \Delta T / 4 \Omega^2 H}$$, that strikingly appears in each regime by consideration of the local, convection-scale Rossby number $$\Rol=U/(2\Omega \ell)$$. Thus we show that $$\RoC$$ scales with the local Rossby number $$\Rol$$ in both the slowly rotating and the rapidly rotating regimes, explaining the ubiquity of $$\RoC$$ in rotating convection studies. We show in non-, slowly, and rapidly rotating systems that the convective heat transport, parameterized via $$Pe_\ell$$, 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 $$Ro_H \sim Ro_{CIA} \sim \RoC^2$$, demonstrating that these momentum transport scalings are identical in the limit of rapidly rotating turbulent heat transfer.
more »
« less
