Theoretical Foundations for Models of Moist Convection

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Abstract
The equations describing the dynamics and thermodynamics of cloudy air are derived using the theories of multicomponent fluids and multiphase flows. The formulation is completely general and allows the hydrometeors to have temperatures and velocities that differ from those of the dry air and water vapor. The equations conserve mass, momentum, and total thermodynamic energy. They form a complete set once terms describing the radiative processes and the microphysical processes of condensation, sublimation, and freezing are provided. An equation for the total entropy documents the entropy sources for multitemperature flows that include the exchange of mass, momentum, and energy between the hydrometeors and the moist air. It is shown, for example, that the evaporation of raindrops in unsaturated air need not produce an increase in entropy when the drops are cooler than the air. An expression for the potential vorticity in terms of the density of the moist air and the virtual potential temperature is shown to be the correct extension of Ertel's potential vorticity to moist flows. This virtual potential vorticity, along with the density field of the hydrometeors, can be inverted to obtain the other flow variables for a balanced flow. In their most general form the equations include prognostic equations for the hydrometeors' temperature and velocity. Diagnostic equations for these fields are shown to be valid provided the diffusive timescales of heat and momentum are small compared to the dynamic timescales of interest. As a consequence of this approximation, the forces and heating acting on the hydrometeors are added to those acting on the moist air. Then the momentum equation for the moist air contains a drag force proportional to the weight of the hydrometeors, a hydrometeor loading. Similarly, the thermal energy equation for the moist air contains the heating of the hydrometeors. This additional heating of the moist air implies a diabatic loading for which the heating of the hydrometeors is realized by the moist air. The validity of the diagnostic equations fails for large raindrops, hail, and graupel. In these cases the thermal diffusive timescales of the hydrometeors can be several minutes, and prognostic rather than diagnostic equations for their temperatures must be solved. However, their diagnostic momentum equations remain valid. Anelastic and Boussinesq versions of the equations are also described.

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