Thermodynamically consistent versions of approximations used in modelling moist air

TL;DR

This paper develops thermodynamically consistent formulations for moist air modeling using thermodynamic potentials, addressing common approximations and unifying various approaches for improved atmospheric simulations.

Contribution

It introduces a thermodynamic potential framework for moist air, ensuring consistency in common approximations like constant fect and dry heat capacities, and unifies existing literature.

Findings

Consistent formulation for moist air thermodynamics using potentials.

Adiabatic dynamics depend only on total mass with the constant fect approximation.

Provides a comprehensive reference for thermodynamic modeling in atmospheric science.

Abstract

Some existing approaches to modeling the thermodynamics of moist air make approximations that break $\textit{thermodynamic consistency}$, such that the resulting thermodynamics do not obey the 1st and 2nd laws or have other inconsistencies. Recently, an approach to avoid such inconsistency has been suggested: the use of $\textit{thermodynamic potentials}$ in terms of their $\textit{natural variables}$, from which all thermodynamic quantities and relationships are derived. In this paper, we develop this approach for $\textit{unapproximated}$ moist air thermodynamics and two widely used approximations: the constant $\kappa$ approximation and the dry heat capacities approximation. The consistent constant $\kappa$ approximation is particularly attractive because it leads to, when using virtual potential temperature $\theta_v$ as the thermodynamic variable, adiabatic dynamics that depend only on total mass, 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 paper provides a comprehensive reference for the use of thermodynamic potentials in atmospheric modelling, especially for the three systems considered here.