Thermodynamic Consistency of the Anelastic Approximation for a Moist Atmosphere

The primary goal of this paper is to validate the use of the anelastic approximation for fluids with a complex equation of state such as moist air or seawater. The anelastic approximation is based on a leading-order expansion of the equations of motion for a compressible fluid in terms of density. Its application to atmospheric flows has been based on a dry framework that treats phase transitions as an external energy source. However, cloudy air is more accurately described as a two-phase fluid in which condensed water and water vapor are in thermodynamic equilibrium. Thermodynamic equilibrium reduces to three the number of state variables necessary to describe the thermodynamic state of moist air, and leads to a discontinuity in the partial derivatives of the equation of state at the saturation point. A version of the anelastic approximation for a moist atmosphere is derived here by considering the atmospheric density as a small perturbation from a moist-adiabatic reference profile, and using moist entropy and total water content as prognostic variables, with buoyancy determined from the complete nonlinear equation of state. The key finding of this paper is that this implementation of the anelastic approximation conserves energy. The total energy is equal to the sum of the kinetic energy and the thermodynamic energy. The latter is found to be equal to the sum of the enthalpy and geopotential energy of the parcel. Furthermore, the state relationships between this thermodynamic energy, entropy, and other state variables are the same as those for moist air after replacing the total pressure with the reference state pressure. This guarantees that, as long as the pressure perturbation remains small, the thermodynamic behavior of a fluid under the anelastic approximation is fully consistent with both the first and second laws of thermodynamics. Two implications of this finding are also discussed. First, it is shown that the first and second laws of thermodynamics constrain the vertically integrated buoyancy flux. This is equivalent to deriving the total work performed in a compressible atmosphere from its entropy and energy budgets. Second, it is argued that an anelastic model can be built with temperature or enthalpy as a prognostic variable instead of entropy. The rate of change for this new state variable can be obtained from energy conservation, so that such a model explicitly obeys the first law of thermodynamics. The entropy in this model is equal to the entropy of the parcel evaluated at the reference pressure, and its evolution obeys the second law of thermodynamics.