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 .