On the thermodynamic invariance of fine‐grain and coarse‐grain fluid models

Abstract Thermodynamics is an essential ingredient of fine‐grain and coarse‐grain models of oceanic and atmospheric flows. Thermodynamic functions and conservative variables, for a given set of observable quantities such as pressure, temperature, and composition, may be defined up to a certain degree of arbitrariness, in the sense that the predictions of the fine‐grain model are insensitive to, for example, some reference enthalpies, entropies, or pressures. Since the compressible Navier–Stokes model, regarded as a “mother” fine‐grain model, is invariant with respect to arbitrary changes in reference enthalpies and entropies, restricted only by phase change, any coarse‐grain model obtained from it, even conceptually, must be invariant at least to the same extent. A framework is devised that enables the systematic examination of this invariance. It is found that the dependence of usual conservative variables on a reference pressure propagates to their fluxes and gradients, and to downgradient closures based on them. To resolve this issue, it is shown how to construct invariant “reduced” gradients and fluxes of enthalpy and entropy. Closure relationships between such “reduced” fluxes and gradients are then guaranteed to be invariant. More work is required to address the invariance of more sophisticated closures, especially shallow and deep convective closures.