Equations of motion and conservation laws in a theory of stably   stratified turbulence
  The letter considers non-isothermal fluid flows and revises simplifications of basic hydrodynamic equations for such flows arriving eventually to a generalization of the Oberbeck-Boussinesq approximation valid for arbitrary equation of state including both non-ideal gases as well as liquids. The proposed approach is based on a suggested general definition of potential temperature. Special attention is put on the energy conservation principle, and it is shown that the proposed approximation exactly preserves the total mechanical energy by approximate equations of motion. The principal importance for any turbulent boundary layer model to respect the conservation laws is emphasized explicitly. 
Victor S. L'vov and Oleksii Rudenko