An energetically and thermodynamically consistent Boussinesq model

TL;DR

This work addresses the ambiguous energetics of the Boussinesq approximation by constructing an energetically and thermodynamically consistent Boussinesq model derived from the two-component compressible Navier–Stokes equations using the static energy framework $\Sigma(\eta,S,p,\Phi)$. It introduces an exact thermodynamically soundproof (TS) model that preserves the full compressible dynamics while enforcing a pressure-independent linear equation of state, and pairs it with a Boussinesq limit that retains the same thermodynamic structure. The authors explicitly formulate diffusion closures for heat and salt that ensure non-negative entropy production, decompose potential energy into internal and gravitational components, and clarify the role of gravitational potential energy and diabatic divergence in stratified turbulence. The resulting framework reconciles classical Boussinesq practice with compressible energetics, providing robust tools for analyzing stratified mixing, mixing efficiency, and energy budgets, while offering a benchmark through the TS model for assessing soundproof approximations. Overall, the paper delivers a transparent, traceable pathway from compressible energetics to energetically consistent Boussinesq formulations and highlights the importance of subtle thermodynamic terms in governing irreversible processes and energy partitioning.

Abstract

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 \textit{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 and potential temperature explicitly, and incorporates diffusive fluxes that homogenise the correct thermodynamic potentials, ensuring consistent non-negative entropy production. The result clarifies the status of gravitational potential energy, resolves ambiguities surrounding salinity--entropy coupling, and retains the small terms carriers of key information about thermodynamics. Alongside this, we introduce an exact thermodynamically soundproof (TS) model whose weak diabatic divergence highlights the role of compressibility effects in stratified turbulence. Together, these models provide a transparent framework that reconciles classical approximations with compressible energetics, offering better-defined pathways for analysing stratified mixing, mixing efficiency, and the energy budgets of geophysical flows.