Predicting rainfall with governing equations
The Barotropic vorticity equation is insufficient to model rain. Part of this is due to the word 'Barotropic,' which indicates a lack of thermal gradient. The other reason is that while vorticity is a useful quantity, it lacks explicit information on the atmospheric state.

The primitive equations are a system of partial differential equations that are used by most numerical weather prediction models to model various quantities. Rainfall rate is a parameterized variable, a diagnosed formula (or formulas) based on atmospheric state, based on research. Sometimes the results of the research can give wildly different answers given the same data, which is a weakness of parameterizations. The idea is to model the Reynolds-averaged primitive equations and intermittently invoke the parameterizations (including turbulence, radiation, cloud cover, and so on).

Here is a good book that discusses parameterization schemes in general.