Baryy Saltzman's 1962 paper
First get a copy of
Saltzman, Barry, Finite Amplitude Free Convection as an Initial Value Problem -- I, Journal for the Atmospheric Sciences, 19, (1962), pp. 329-341.
The main goal of the project is to understand Saltzman's derivation of a set of ordinary differential equations from the governing equations of atmospheric fluid dynamics (the four equations at the bottom of page 336).