An energetically and thermodynamically consistent Boussinesq model

The Boussinesq approximation is a cornerstone of geophysical fluid dynamics, yet its thermodynamic and energetic underpinnings have remained ambiguous. In standard formulations, the links with the fully compressible Navier–Stokes equations are obscured, internal energy is only implicit, and the representation of diffusion and irreversibility remains ad hoc. Here we derive a new Boussinesq model in a fully traceable way from the two-component compressible Navier–Stokes equations, ensuring exact energy conservation and consistent thermodynamics. Assuming a linear equation of state, our model treats density as a proxy for specific volume, distinguishes in situ temperature and potential temperature explicitly, and incorporates diffusive fluxes that homogenise the correct thermodynamic potentials, ensuring consistent non-negative entropy production. The resulting formulation makes gravitational and internal energy explicit, resolves the salinity – entropy coupling required for non-negative entropy production, and yields an exactly closed total energy budget. We also introduce a thermodynamically soundproof model that retains the compressible dynamical structure for the same simplified equation of state and therefore provides a reference for interpreting the Boussinesq approximation. The purpose of these models is primarily to improve structure rather than accuracy: they establish a transparent and thermodynamically admissible framework for energy budget analysis and for future studies of stratified mixing.