Compressible turbulent convection at very high Rayleigh numbers

Planetary and stellar convection, which are compressible and turbulent, remain not well understood. We report numerical results on the scaling of the Nusselt number (Nu) and Reynolds number (Re) for extreme turbulence. Using the computationally efficient MacCormack-TVD finite difference method, we simulate compressible turbulent convection in a two-dimensional Cartesian box up to Ra = $10^{16}$, the highest Ra achieved so far, and in a three-dimensional box up to Ra = $10^{13}$. We show that Nu $\propto \mathrm{Ra}^{0.3}$ (classical scaling) that differs strongly from the ultimate-regime scaling, which is Nu $\propto \mathrm{Ra}^{1/2}$. The bulk temperature drops adiabatically along the vertical even for high Ra, which is in contrast to the constant bulk temperature in Rayleigh-B\'{e}nard convection (RBC). Unlike RBC, the density decreases with height. In addition, the vertical pressure-gradient ($-dp/dz$) nearly matches the buoyancy term ($\rho g$). However, the difference, $-dp/dz-\rho g$, is equal to the non-linear term that leads to the Reynolds number $ \mathrm{Re} \propto \mathrm{Ra}^{1/2}$. We also find that the boundary layers become turbulent at high Ra. In 2D, both velocity ($u^+$) and temperature ($T^+$) profiles show logarithmic regions, while in 3D only $T^+$ does.