Available potential energy density for Boussinesq fluid flow

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.