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

TL;DR

This study demonstrates that highly turbulent compressible convection exhibits classical 1/3 Nusselt number scaling at extremely high Rayleigh numbers, supported by extensive numerical simulations in 2D and 3D.

Contribution

The paper provides the first high-Rayleigh-number simulations showing classical Nusselt number scaling in compressible turbulent convection.

Findings

Nusselt number scales as Ra^{0.32} in 2D

Nusselt number scales as Ra^{0.31} in 3D

Achieved Ra up to 10^{16} in 2D simulations

Abstract

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 ($\mathrm{Nu}$) and Reynolds number ($\mathrm{Re}$) for extreme convection. Using computationally-efficient MacCormack-TVD finite difference method, we simulate compressible turbulent convection in a two-dimensional Cartesian box up to $\mathrm{Ra} = 10^{16}$, the highest $\mathrm{Ra}$ achieved so far, and in a three-dimensional box up to $\mathrm{Ra} = 10^{11}$. We show adiabatic temperature drop in the bulk flow, leading to the Reynolds number scaling $\mathrm{Ra}^{1/2}$. More significantly, we show classical $1/3$ Nusselt number scaling: $\mathrm{Nu} \propto \mathrm{Ra}^{0.32}$ in 2D, and $\mathrm{Nu} \propto \mathrm{Ra}^{0.31}$ in 3D up to the highest $\mathrm{Ra}$.