Furthermore, is typically of order and so the exponential in eq.(54) may be expanded to give:
which is the form commonly used in numerical modelling (e.g. Deardorff, 1976). Since , and are conserved for reversible, adiabatic changes in non-precipitating cloudy air then, from eq.(52), is also conserved. Deardorff emphasises the usefulness of as independent variables in shallow cloud modelling. Important advantages are the reduction to for unsaturated air and the computational economy which results from not having to store as a separate dependent variable.
Another approach to representing moist thermodynamics is through a simplified form of the energy equation and at the same time we shall incorporate the effect of virtual temperature. Consider the modified virtual temperature defined by analogy with eq.(46) i.e.
where, ignoring the small.difference between and ,
Now taking the logarithm of eq.(55) and differentiating following the motion gives:
and on using the hydrostatic equation for the basic state pressure one has:
Defining , multiplying the above by and rearranging then leads to: