Flux-conservative thermodynamic equations in a mass-weighted framework

A set of equations, ready for discretization, is presented for the purely thermodynamic part of atmospheric energetics along a vertical column. Considerations of kinetic energy budgets and detailed turbulence laws are left for further study. The equations are derived in a total mass-based framework, both for the vertical coordinate system and for the conservation laws. This results in the use of the full barycentric velocity as the vector of advection. Under these conditions, the equations are derived from first principles on the basis of an a priori defined set of simplifying hypotheses. The originality of the resulting set of equations is twofold. First, even in the presence of a full prognostic treatment of cloud and precipitation processes, there exists a flux-conservative form for all relevant budgets, including that of the thermodynamic equation. Secondly, the form of the state law that is obtained for the multiphase system allows the flux-conservative form to be kept when going from the hydrostatic primitive equations system to the fully compressible system and projecting then the heat source/sink on both temperature and pressure tendencies.