Examining the physical principles behind the motion of moist air: Which expressions are sound?

TL;DR

This paper critically examines two conflicting formulations of moist air motion equations, clarifies misconceptions about reactive motion, and emphasizes the importance of consistent momentum conservation in atmospheric modeling.

Contribution

It clarifies the physical principles behind moist air motion equations, refutes the reactive motion concept, and highlights the need for consistent momentum treatment in models.

Findings

Reactive motion concept is based on a misunderstanding of momentum conservation.

The mixed equation used in some models is physically inconsistent.

Total momentum consideration clarifies the correct formulation.

Abstract

The physical equations determining the motion of moist atmospheric air in the presence of condensation remain controversial. Two distinct formulations have been proposed, published and cited. The equation of Bannon [2002, J. Atmos. Sci. 59: 1967--1982] includes a term for a "reactive motion" that arises when water vapor condenses and droplets begin to fall; according to this term the remaining gas moves upwards so as to conserve momentum. In the equation of Ooyama[2001, J. Atmos. Sci. 58: 2073--2102] the reactive motion term is absent. Both equations contain a term for condensate loading, but in the formulation of Ooyama [2001] there are two additional terms. In some modern nonhydrostatic models of moist atmospheric circulation, however, formulations have been mixed. Here we examine the contrasting equations for the motion of moist air. We discuss inconsistencies in the application of Newton's second and third laws to an air and condensate mixture. We show that the concept of reactive motion in this context is based on a misunderstanding of the conservation of momentum in the presence of a gravitational field: such a reactive motion does not exist. We show that the "mixed" equation used in some models is not physically consistent either. We examine why consideration of total momentum, that is air and condensate combined, has been misleading in the search for valid equations of motion in the presence of phase transitions.