Invariance(s) of (un)averaged models of geophysical motion

Weather and climate modelling systems result, at least conceptually, from a sequence of approximations and coarse-graining operations, starting from a "perfect" model such as the compressible Navier-Stokes equations. In general, the resulting models consist of a reversible set of fluid equations complemented by so-called closure relationships. This system should possess invariance properties inherited from its "perfect" ancestor. I will consider two quite different kinds of invariance : with respect to a change of space-time coordinates, and with respect to a change of certain thermodynamic constants.
I will show that dynamical-geometrical approximations such as traditional shallow-atmosphere and beta-plane are concisely expressed in a space-time covariant formulation involving a geopotential-Coriolis-metric covariant tensor. This formulation helps drawing a line between "dynamics", which is the same for geophysical fluid dynamics and standard hydrodynamics, and "gravity", which differs between them, especially for geophysical models featuring a curved space or lacking an inertial frame.
Regarding thermodynamics, a very practical question is whether a given model, starting from an initial condition characterized by observable variables such as temperature and pressure, leads to the same prediction independently from the value given to those model constants that are arbitrary, such as the reference pressure used to define potential temperature or reference enthalpies and entropies appearing in thermodynamic potentials. I will show that, in general, flux-gradient relationships involving (variants of) potential temperature violate this invariance, and propose a systematic construction of invariant flux-gradient relationships.