Transition to the ultimate regime in two-dimensional turbulent Rayleigh-B\'enard convection

The heat transfer in a RB system is determined by the Rayleigh number $Ra$ and the Prandtl number $Pr$. Various natural heat transfer phenomenon involve $Ra \ga 10^{20}$ and thus extrapolations to this high Ra number regime are required. Here we present results from DNS for two-dimensional RBC with $Pr = 1$ in an aspect ratio $\Gamma=D/L=0.23$, where $D$ and $L$ are the width and height of the box, respectively and achieve $Ra$ up to about $10^{13}$. For $Ra<1\times10^{10}$ the Nusselt number varies nearly as the $1/3$ power of $Ra$. However, for $Ra>1\times10^{10}$ we find a sharp transition towards a regime where the Nusselt number varies nearly as the $1/2$ power of $Ra$. A visualization of the simulation results reveals that the transition in the $Nu$ number scaling are caused by a break-up of the large scale structures that are observed at lower $Ra$ numbers.