  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 $\delta W =-P {\rm d}\upsilon$,
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 $\delta
W_{ba}$, the corresponding approximation to the "heat" $\delta Q_{ba}$ 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.
