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Turbulent fluxes of entropy and internal energy in temperature stratified flows

Part of: Macroscopic Randomness

Published online by Cambridge University Press:  03 August 2015

I. Rogachevskii  and
N. Kleeorin
I. Rogachevskii*
Affiliation:
Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden
N. Kleeorin
Affiliation:
Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden
*
†Email address for correspondence: gary@bgu.ac.il
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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 velocity–entropy correlation, $\overline{\boldsymbol{u}s}$, in the limits of small and large Péclet 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.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 5 , October 2015 , 395810504
Copyright
© Cambridge University Press 2015 

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