For the simplest QTCM, we use a version of the Betts-Miller (1986)
moist convective adjustment scheme, with convection restoring toward convective
profiles in both T and q. The convective profile
for moisture need not be purely a subsaturation parameterization, but
is assumed to have a dominant vertical dependence 
.
After some rearrangement this implies
where we use
,
with 
the
time scale of convection in strongly convecting regions and
a smoothed Heaviside function depending on
, a quantity related to CAPE projected on retained structures.
The quantity 
is a projection of moisture onto
basis function 
.
The thermodynamic equation and moisture equations are then
where the dry stability 
and the gross moisture stratification 
are the contributions of s and q to
in (11). The sum of these equations yields the moist static energy
equations (10) with the q terms simplified to
.