All QTCM versions are derived from the nonlinear primitive equations
where temperature T and specific humidity q are in energy units, are the long-wave radiative fluxes depending on T, q and cloudiness over the column, and surface temperature . Shortwave fluxes S are positive downward. The operators , and include horizontal diffusion and horizontal advection terms. The vertical fluxes of sensible heat and moisture by nonconvective, diffusive transport, and , and stress , vanish at the top and have drag laws at the surface for , evaporation E, and . The hydrostatic equation has been used for baroclinic pressure gradients in the momentum equation (3), where is the geopotential at the lowest pressure level, .
An important subsidiary relation that holds for all convective parameterizations is the ``convective heating'' and ``moistening'' terms and cannot change the vertically integrated moist static energy budget, i.e. or
where , , absorbing into all fluxes for brevity, and denotes vertical averaging over the troposphere, from the surface, , to the tropopause, .