The thermodynamic equation and moisture equations, for temperature and
moisture projections 
and 
associated with basis functions 
and 
are:
For QTCM1, we use a version of the Betts-Miller (1986)
moist convective adjustment scheme, with convection restoring temperature
toward convective
profile 
. The moisture sink and convective heating terms are given by
where
,
with 
the
time scale of convection in strongly convecting regions and
a smoothed Heaviside function depending on
, given in square brackets, a quantity related to CAPE projected on
retained structures.
The convective profile of temperature is given as 
a reference profile
independent of space, plus departures 
following a moist adiabat from the
boundary layer with vertical profile of . An additional closure
assumption is required, which links to and after some
manipulation. Examples of this closure are discussed in [1]
where the dry stability 
and the gross moisture stratification 
are the contributions of s and q to
in (5.12). The sum of these equations yields the moist static energy
equations (5.11) with the q terms simplified to
.
