### abstract

- We derive equations for the mean entropy and the mean internal energy in 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 $\boldsymbol{F}_{s}=\overline{{\it\rho}}\,\overline{\boldsymbol{u}s}$ , where $\overline{{\it\rho}}$ is the mean fluid density, $s$ is fluctuation of entropy and overbars denote averaging over an ensemble of turbulent velocity fields, $\boldsymbol{u}$ . We demonstrate that the turbulent flux of entropy is different from the turbulent convective flux, $\boldsymbol{F}_{c}=\overline{T}\,\overline{{\it\rho}}\,\overline{\boldsymbol{u}s}$ , 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 velocityâ€“entropy correlation, $\overline{\boldsymbol{u}s}$ , in the limits of small and large Peclet numbers, using the quasi-linear approach and the spectral ${\it\tau}$ approximation, respectively. This study is important in view of different applications to astrophysical and geophysical temperature stratified turbulence.