This is the potential temperature attained by a cloudy air parcel if the liquid water contained within it is evaporated in a reversible adiabatic process. The moist entropy (given by eq.(9) of the thermodynamics section) is conserved in such a process and can be written as:
where is the total water mixing ratio with . If the cloudy parcel follows a reversible adiabatic process to temperature at which point the liquid water is assumed to vanish and the potential temperature equals then:
Ignoring the temperature dependence of L_v and equating the above two expressions then gives:
Now if with then
to order and eq.(53) may be written as:
where
Taking , , , and , then and the error in resulting from the assumption that is . However if , the error rises to implying that this simplifying assumption is only valid for shallow clouds.