Turbulent fluxes of entropy and internal energy in temperature stratified flows

TL;DR

This paper derives exact equations for mean entropy and internal energy fluxes in low-Mach-number temperature stratified turbulence, highlighting differences from the well-known turbulent convective flux of internal energy, with implications for astrophysical and geophysical flows.

Contribution

It provides the first exact derivation of turbulent flux equations for entropy and internal energy in low-Mach-number stratified turbulence, clarifying their differences and limitations.

Findings

Turbulent flux of entropy is given by mean density times mean velocity-entropy correlation.

Turbulent flux of internal energy differs from the entropy flux and cannot replace it in entropy equations.

Derived equations for velocity-entropy correlation in different Peclet number regimes.

Abstract

We derive equations for the mean entropy and the mean internal energy in the low-Mach-number temperature stratified turbulence (i.e., for turbulent convection or stably stratified turbulence), and show that turbulent flux of entropy is given by ${\bf F}_s=\overline{\rho} \, \overline{{\bf u} s}$, where $\overline{\rho}$ is the mean fluid density, $s$ are fluctuations of entropy and overbars denote averaging over an ensemble of turbulent velocity field, ${\bf u}$. We demonstrate that the turbulent flux of entropy is different from the turbulent convective flux, ${\bf F}_c=\overline{T} \, \overline{\rho} \, \overline{{\bf u} s}$, of the fluid internal energy, where $\overline{T}$ is the mean fluid temperature. This turbulent convective flux is well-known in the astrophysical and geophysical literature, and it cannot be used as a turbulent flux in the equation for the mean entropy. This result is exact for low-Mach-number temperature stratified turbulence and is independent of the model used. We also derive equations for the velocity-entropy correlation, $\overline{{\bf u} s}$, in the limits of small and large Peclet numbers, using the quasi-linear approach and the spectral tau approximation, respectively. This study is important in view of different applications to the astrophysical and geophysical temperature stratified turbulence.