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, .

Wed Aug 26 16:58:31 PDT 1998