Two-dimensional Rayleigh-B\'enard convection without boundaries

We study the effects of Prandtl number $Pr$ and Rayleigh number $Ra$ in two-dimensional Rayleigh-B\'enard 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 $10^{-3} \leq Pr \leq 10^2$, the Nusselt number tends to follow the `ultimate' scaling $Nu \propto Pr^{1/2} Ra^{1/2}$, and the viscous dissipation scales as $\epsilon_\nu \propto Pr^{1/2} Ra^{-1/4}$. The latter scaling is based on the observation that enstrophy $\langle \omega^2 \rangle \propto Pr^0 Ra^{1/4}$. 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.