  In models of oceanic and atmospheric flows, thermodynamic functions and
conservative variables may be defined up to a certain degree of arbitrariness,
in the sense that, for a given set of observable quantities such as pressure
and temperature, the predictions of the model are insensitive to, e.g. some
reference enthalpies, entropies or pressures.Since the compressible
Navier-Stokes-Fourier 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,
even conceptually, from it must be invariant at least to the same extent. Upon
examination, it is found that the dependence of usual conservative variables to
a reference pressure propagates to their fluxes and gradients, and to
down-gradient closures based on them.Conversely, closure relationships between
adequately defined "reduced" gradients and fluxes of enthalpy and entropy, are
guaranteed to lead to invariant models, even with multiple turbulent
diffusivities and cross-diffusivities. More work is required to address the
invariance of more sophisticated closures, especially shallow and deep
convective closures.
