Two-dimensional Rayleigh-Bénard convection without boundaries

Abstract

We study the effects of Prandtl number $Pr$ and Rayleigh number $Ra$ in two-dimensional Rayleigh-Bénard convection without boundaries, i.e. with periodic boundary conditions. In the limits of $Pr\to 0$ and $\infty$, we find that the dynamics are dominated by vertically oriented elevator modes that grow without bound, even at high Rayleigh numbers and with large scale dissipation. For finite Prandtl number in the range $10^{-3}\le Pr\le 10^2$, the Nusselt number tends to follow the `ultimate' scaling $Nu\propto P{r}^{1/2}R{a}^{1/2}$, and the viscous dissipation scales as ${\epsilon }_{\nu }\propto P{r}^{1/2}R{a}^{-1/4}$. The latter scaling is based on the observation that enstrophy $⟨{\omega }^2⟩\propto P{r}^{0}R{a}^{1/4}$. The inverse cascade of kinetic energy forms the power-law spectrum ${\hat{E}}_u(k)\propto k^{-2.3}$, while the direct cascade of potential energy forms the power-law spectrum ${\hat{E}}_{\theta }(k)\propto k^{-1.2}$, with the exponents and the turbulent convective dynamics in the inertial range found to be independent of Prandtl number. Finally, the kinetic and potential energy fluxes are not constant in the inertial range, invalidating one of the assumptions underlying Bolgiano-Obukhov phenomenology.

16 pages, 9 figures