Journal of Fluid Mechanics. 714:476-488.
An exact expression E-a for available potential energy density in Boussinesq fluid flows (Roullet & Klein, J. Fluid Mech., vol. 624, 2009, pp. 45-55; Holliday & McIntyre, J. Fluid Mech., vol. 107, 1981, pp. 221-225) is shown explicitly to integrate to the available potential energy E-a of Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115-128). E-a is a positive definite function of position and time consisting of two terms. The first, which is simply the indefinitely signed integrand in the Winters et al. definition of E-a, quantifies the expenditure or release of potential energy in the relocation of individual fluid parcels to their equilibrium height. When integrated over all parcels, this term yields the total available potential energy E-a. The second term describes the energetic consequences of the compensatory displacements necessary under the Boussinesq approximation to conserve vertical volume flux with each parcel relocation. On a pointwise basis, this term adds to the first in such a way that a positive definite contribution to E-a is guaranteed. Globally, however, the second term vanishes when integrated over all fluid parcels and therefore contributes nothing to E-a. In effect, it filters the components of the first term that cancel upon integration, isolating the positive definite residuals. E-a can be used to construct spatial maps of local contributions to E-a for direct numerical simulations of density stratified flows. Because E-a integrates to E-a, these maps are explicitly connected to known, exact, temporal evolution equations for kinetic, available and background potential energies.