On the calculation of the available potential energy of an isolated perturbation in a density-stratified fluid

Two methods for calculating the available potential energy (APE) of an isolated feature in a density-stratified fluid, such as an internal solitary wave or an eddy, are compared. The first formulation calculates the APE by integrating the perturbation potential energy density Ew. The second uses an available potential energy density Ea. Both formulations are based on the reference density obtained by adiabatically rearranging the density field to a state of minimum energy. It is shown, under more general conditions than used previously, that (i) the integrals of Ew and Ea over a finite domain are identical; and (ii) that for an isolated feature in an unbounded domain, the far-field density $\bar\rho(z)$ can be used as the reference density if Ea is used to find the APE. This is not the case when Ew is used, hence use of the available potential energy density formulation is simpler in this situation.