or
where is the specific entropy (defined here to be that associated with unit mass of dry air) (Brunt, 1939). The total entropy as represented by the RHS of eq.(9) is conserved following the adiabatic, reversible ascent/descent of a saturated air parcel provided that droplet freezing or precipitation does not occur. This then, defines the adiabatic reference process when the condensate is liquid water. It is possible to include the ice phase in the above entropy definition though this will not be pursued here (Dutton, 1986).
In deep clouds, precipitation ensures that cloud liquid water mixing ratios are far less than the level implied by the adiabatic reference process. The amount of entropy lost from a precipitating air parcel therefore depends on the microphysical processes which determine . A useful idealization is to assume that precipitation instantly removes liquid water so that r_l is always zero. The decrease in the moist entropy of an ascending air parcel must be equal to that lost in precipitation. Consider the entropy produced in precipitation as saturated air moves
from temperature and pressure to . The increment of mass of liquid water produced per kilogram of dry air is given by:
and so the entropy of the precipitation falling out of the parcel is in this infinitesimal change of parcel state.
The total entropy lost from the parcel during its saturated ascent or descent is therefore equal to:
or
Therefore, the sum of ) (see eq.(9)) and the above net entropy loss by precipitation must be conserved i.e.
or
where is the constant total entropy. This is known as the pseudo-adiabatic reference process (Saunders, 1957) and is usually assumed to be more accurate than the adiabatic reference process