Monday, 30 July 2001: 4:00 PM

The theoretical foundation for models of moist convection

The equations describing the dynamics and thermodynamics of cloudy air are derived assuming a continuum hypothesis. The diffusion of heat and momentum in the air and between the air and the hydrometeors is included. 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. The momentum equation includes terms describing the exchange of momentum during phase changes. For example, the condensation of water vapor to form falling drops requires the ascent of the moist air in the absence of forces. The thermodynamic energy equation includes the exchange of kinetic and internal energy during phase changes. Only the exchange of internal energy is significant for typical atmospheric conditions. The thermal energy equation also includes the dissipation of kinetic energy associated with the drag forces between the air and the hydrometeors. This dissipation is significant for large hydrometeors with mass mixing ratios of order 10-2. We also derive from the basic set of equations expressions for the conservation of virtual potential temperature and virtual potential vorticity. It is shown that an exact prognostic equation for the total entropy is not possible for multi-temperature flows.
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 time scales of heat and momentum are small compared to the dynamic time scales 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, an ice-liquid-water 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 time scales 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.

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