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 (
), where
is the geopotential at the lowest pressure level 
.
is surface elevation, 
is surface pressure, 
is
atmospheric near-surface density, and 
has been neglected.
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, 
.
