Thermodynamics/dynamics coupling and thermodynamic consistency of Boussinesq and anelastic binary fluids with an arbitrary nonlinear equation of state

This paper shows that the energetics of Boussinesq and anelastic fluids possesses a term that can be identified as the approximation $\delta W_{ba}$ to the compressible work of expansion/contraction $\delta W=-P\mathrm{d}\upsilon$, where $P$ is the pressure and $\upsilon$ is the specific volume. It follows that Boussinesq and anelastic fluids admit explicit compressible effects and conversions between internal energy and mechanical energy, under the form of apparent changes in gravitational potential energy resulting from changes in density by diabatic and adiabatic effects. From the knowledge of $\delta W_{ba}$,thecorrespondingapproximationtothe\mathrm{"}heat\mathrm{"}$\delta Q_{ba}$ can be constructed in a consistent way by requiring that the Maxwell relationships be satisfied, ultimately leading to the construction of a well defined approximation to the internal energy and ultimately of the full range of known thermodynamic potentials. These properties make it possible to endow common forms of the Boussinesq and anelastic approximations with fully consistent energetics and thermodynamics, even when diabatic effects and an arbitrary nonlinear equation of state for a binary fluid are retained, without loss of accuracy. In that case, it can be shown that the sum of kinetic energy and enthalpy is a conservative quantity, which plays the role of the total energy in the Boussinesq and anelastic approximations for both diabatic and adiabatic motions. This implies that gravitational potential energy can be regarded as the difference between enthalpy and internal energy, and hence as a pure thermodynamic property of the fluid. The results have implications for our understanding of turbulent mixing in stratified fluids, as well as for correcting the energetics of current numerical ocean general circulation models, which are discussed.