Fluxes through a uniform double-diffusive staircase at low Prandtl number
Double-diffusive convection in the diffusive regime is often observed to be in a long-lived layered state, well-known in the Arctic ocean and in volcanic lakes, and thought to exist in the interiors of giant planets. The density profile resembles a `staircase', with stacks of convective layers of uniform density separated by strongly stratified interfaces. The governing parameters are the Prandtl number $\mathrm{Pr}$, diffusivity ratio $\tau$, density ratio $R_\rho$, and Rayleigh number $\mathrm{Ra}$. In this work, we perform direct numerical simulations in triply periodic domains to systematically investigate the properties of statistically stationary double-diffusive staircases with $\tau \le \mathrm{Pr} \ll 1$. We demonstrate that they can exist for significantly larger density ratios than previously reported, and propose a new criterion for the existence of long-lived layered states. Notably, this criterion depends weakly on $\mathrm{Ra}$. In all simulations, we measure the total fluxes of density $F^{\mathrm{tot}}_\rho$, temperature $F^{\mathrm{tot}}_T$ and composition $F^{\mathrm{tot}}_C$, and the flux ratio $\gamma =F^{\mathrm{tot}}_C/F^{\mathrm{tot}}_T$. We confirm that $\gamma$ is approximately constant for moderate to large $R_\rho$ but also depends weakly on $\mathrm{Ra}$. We show that the Nusselt numbers for composition and temperature scale as $\mathrm{Ra}^{1/3}$ in this limit, which notably differs from the $(\mathrm{RaPr})^{1/3}$ scaling previously found at lower $R_\rho$. We propose a new model for these findings that is based on the rise and turbulent mixing of density anomalies that form at the edges of the interfaces. The model explains the existing low $\mathrm{Pr}$ data with reasonable accuracy.