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
.

Wed Aug 26 16:58:31 PDT 1998