Classical 1/3 Nusselt number scaling in highly turbulent compressible convection

Planetary and stellar convection, which are compressible and turbulent, remain poorly understood. In this paper, we report numerical results on the scaling of Nusselt number ($\mathrm{N}\mathrm{u}$) and Reynolds number ($\mathrm{R}\mathrm{e}$) for extreme convection. Using computationally-efficient MacCormack-TVD finite difference method, we simulate compressible turbulent convection in a two-dimensional Cartesian box up to $\mathrm{R}\mathrm{a}={10}^{16}$, the highest $\mathrm{R}\mathrm{a}$ achieved so far, and in a three-dimensional box up to $\mathrm{R}\mathrm{a}={10}^{11}$. We show adiabatic temperature drop in the bulk flow, leading to the Reynolds number scaling ${\mathrm{R}\mathrm{a}}^{1\mathrm{/}2}$. More significantly, we show classical $1\mathrm{/}3$ Nusselt number scaling: $\mathrm{N}\mathrm{u}\propto {\mathrm{R}\mathrm{a}}^{0.32}$ in 2D, and $\mathrm{N}\mathrm{u}\propto {\mathrm{R}\mathrm{a}}^{0.31}$ in 3D up to the highest $\mathrm{R}\mathrm{a}$.