A simple solution is at hand if one accepts a basic state of constant static stability. In this case, multiplying eq.(41) by gives:
or
Using this in eq.(43) leads to:
and after conversion to flux form and integration over the domain yields the global conservation law:
The quadratic form of the potential energy in this definition is related to the available potential energy defined by Lorenz (1955). For many idealized studies in mesoscale and synoptic scale dynamics this form of energy definition is very useful. In general however, the constancy of the basic state stability could be a problem (e.g. when troposphere and stratosphere have to be represented in a model).
Another form of the energy equation can be obtained by defining a temperature variable such that:
is the temperature a parcel would have if its pressure is adjusted isentropically to the reference pressure at the parcel elevation.