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This is a rather peculiar question to ask because as far as I know these two have sort of independent definitions, i.e. stratification is not necessarily associated with fluid motion on the other hand compressibility is mostly related to high Mach number flows.

To be more specific I'm working on an LES diffusion scheme applied for momentum and thermodynamic energy equation in an atmospheric circulation model. The closure problem for the effects of unresolved scales is treated with so-called Dynamic Smagorinsky Model (DSM) and restricted only to horizontal motion:

\begin{aligned} {\partial_i (\rho u_iu_j)} &\to {\partial_i (\tilde{\rho}\ \overline{u_i}\ \overline{u_j})} \\ {\partial_i (\rho u_iT)} &\to {\partial_i (\tilde{\rho}\ \overline{u_i}\ \overline{T})}, \end{aligned}

where $i,j = 1,2$. And corresponding unresolved scale stress and flux are given as;

\begin{aligned} {\tau}_{ij} &= \overline{u_iu_j} - \overline{u_i}\ \overline{u_j} \\ {\theta}_i &= \overline{u_iT} - \overline{u_i}\ \overline{T} \end{aligned}

Since there is stratification and thermodynamic effects in atmosphere, it is sort of an a priori act to account for density variations (LES formulations for compressible flows handle this via Favre averaging, as shown above). However to my understanding variations in density for such a fluid problem are not caused by flow motion (Boussinesq approximation used in our simulation model completely ignores sound waves) thus for some reason I cannot wrap my head around the idea of formulating the turbulence closure with divergence as given above on a term including density. Some numerical results also indicate the very minor contribution of density variations in previous atmospheric simulations.

Briefly to my knowledge turbulence closure schemes are entirely based on movement of the flow and thus as long as motion itself does not introduce compressibility (as it is for atmosphere), why do we include divergence of density in equations of motion?

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