Predicting rainfall with governing equations Which model is a good starting point for predicting precipitation (rain) for 24 hours in advance? I looked at Kalnay's Atmospheric modeling, data assimilation and predictability, and Holton's An Introduction to Dynamic Meteorology as well as various other sources. These are great references but I could not figure out which part I could use to compute what I needed. It seems there are governing equations which are solved numerically, the output is used for a prediction. I am knowledgeable on numerically solving ODEs, PDEs, and am familiar with Matlab, Python, but there aren't any references that I could find providing clear formula -> data -> code -> output.

The most talked about model with clear solution is the Barotropic Vorticity Equation, which, I believe, models wind and pressure. This model was proposed by Charney, Fjortoft and von Neumann during the 50s, and solved with a computer, the results are recreated by Peter Lynch on 2008 (Matlab code available). Could this model be used for rainfall prediction? 

If one has to dig in the basic equations, which part of the governing equations should I focus on? Any pointers, references with running code is most welcome.
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.