## Abstract

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 etÂ al.Â (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 etÂ al.Â 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.

Original language | English |
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Pages (from-to) | 476-488 |

Number of pages | 13 |

Journal | Journal of Fluid Mechanics |

Volume | 714 |

DOIs | |

State | Published - 10 Jan 2013 |

Externally published | Yes |

## Keywords

- baroclinic flows
- computational methods
- stratified flows