Theoretical considerations on the mass and energy consistent treatment of precipitation in cloudy atmospheres

The purpose of this paper is to present a rigorous derivation of a model equation system for the mass and energy open precipitating atmosphere. The derivation is based on the treatment of the atmosphere in terms of three open thermodynamic subsystems consisting of cloudy air, rain water and snow, respectively. Each of the three subsystems moves with a different velocity as caused by the different fall velocities of rain and snow in comparison to the cloudy air.A basic assumption of the model consists in the neglect of the inertia terms in the momentum equation of the falling hydrometeors. It will be shown that this simplification yields additional source terms in the budget equations of the total momentum and the total kinetic energy of the system. However, the prognostic equations for the atmospheric wind field and the corresponding kinetic energy equation remain free of these source terms. The model equations strictly conserve the total mass and energy of the system. It is concluded that the combination of the barycentric velocity of the cloudy air and the total density of the system is the best choice to describe the time evolution of the atmospheric wind field.