Thermodynamically consistent versions of approximations used in modelling moist air
Some existing approaches to modelling the thermodynamics of moist air make approximations that break thermodynamic consistency, such that the resulting thermodynamics does not obey the first and second laws or has other inconsistencies. Recently, an approach to avoid such inconsistency has been suggested: the use of thermodynamic potentials in terms of their natural variables, from which all thermodynamic quantities and relationships (equations of state) are derived. In this article, we develop this approach for unapproximated moist-air thermodynamics and two widely used approximations: the constant-κ approximation and the dry heat capacities approximation. The (consistent) constant-κ approximation is particularly attractive because it leads to, with the appropriate choice of thermodynamic variable, adiabatic dynamics that depend only on total mass and are independent of the breakdown between water forms. Additionally, a wide variety of material from different sources in the literature on thermodynamics in atmospheric modelling is brought together. It is hoped that this article provides a comprehensive reference for the use of thermodynamic potentials in atmospheric modelling, especially for the three systems considered here.