Scaling in two-dimensional Rayleigh-B\'enard convection

An equation for the evolution of mean kinetic energy, $E$, in a 2-D or 3-D Rayleigh-B\'enard system with domain height $L$ is derived. Assuming classical Nusselt number scaling, $Nu \sim Ra^{1/3}$, and that mean enstrophy, in the absence of a downscale energy cascade, scales as $\sim E/L^2$, we find that the Reynolds number scales as $Re \sim Pr^{-1}Ra^{2/3}$ in the 2-D system, where $Ra$ is the Rayleigh number and $Pr$ the Prandtl number, which is a much stronger scaling than in the 3-D system. Using the evolution equation and the Reynolds number scaling, it is shown that $\tilde{\tau} > c Pr^{-1/2}Ra^{1/2}$, where $\tilde{\tau}$ is the non-dimensional convergence time scale and $c$ is a non-dimensional constant. For the 3-D system, we make the estimate $\tilde{\tau} \gtrsim Ra^{1/6}$ for $Pr = 1$. It is estimated that the total computational cost of reaching the high $Ra$ limit in a simulation is comparable between 2-D and 3-D. The results of the analysis are compared to DNS data and it is concluded that the theory of the `ultimate state' is not valid in 2-D. Despite the big difference between the 2-D and 3-D systems in the scaling of $Re$ and $\tilde{\tau}$, the Nusselt number scaling is similar. This observation supports the hypothesis of Malkus (1954) that the heat transfer is not regulated by the dynamics in the interior of the convection cell, but by the dynamics in the boundary layers.