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] and [], where the dry stability and the gross moisture stratification are the contributions of s and q to in (). The sum of these equations yields the moist static energy equations () with the q terms simplified to .