In this document is described how one can combine the heat equations including convection with the Navier-Stokes equations:
However, as far as I can see, for our purposes we also need to include some mechanism by which the energy is lost to space. Otherwise we need to invent some fictitious "Top of the Atmosphere" which is precisely what I want to avoid.
I think this could be accomplished by including an energy sink S at all altitudes which ought to be a function of the density, the temperature and not to forget the total mass above that altitude. In other words, the more mass aloft the less heat is lost directly into space.
So there you have it. Now we only need someone who can solve all these equations..