Available potential energy density for a multicomponent Boussinesq fluid with arbitrary nonlinear equation of state
Tailleux, R. (2013) Available potential energy density for a multicomponent Boussinesq fluid with arbitrary nonlinear equation of state. Journal of Fluid Mechanics, 735. pp. 499-518. ISSN 0022-1120
To link to this article DOI: 10.1017/jfm.2013.509
In this paper, the concept of available potential energy (APE) density is extended to a multicomponent Boussinesq ﬂuid with a nonlinear equation of state. As shown by previous studies, the APE density is naturally interpreted as the work against buoyancy forces that a parcel needs to perform to move from a notional reference position at which its buoyancy vanishes to its actual position; because buoyancy can be deﬁned relative to an arbitrary reference state, so can APE density. The concept of APE density is therefore best viewed as deﬁning a class of locally deﬁned energy quantities, each tied to a diﬀerent reference state, rather than as a single energy variable. An important result, for which a new proof is given, is that the volume integrated APE density always exceeds Lorenz’s globally deﬁned APE, except when the reference state coincides with Lorenz’s adiabatically re-arranged reference state of minimum potential energy. A parcel reference position is systematically deﬁned as a level of neutral buoyancy (LNB): depending on the nature of the ﬂuid and on how the reference state is deﬁned, a parcel may have one, none, or multiple LNB within the ﬂuid. Multiple LNB are only possible for a multicomponent ﬂuid whose density depends on pressure. When no LNB exists within the ﬂuid, a parcel reference position is assigned at the minimum or maximum geopotential height. The class of APE densities thus deﬁned admits local and global balance equations, which all exhibit a conversion with kinetic energy, a production term by boundary buoyancy ﬂuxes, and a dissipation term by internal diﬀusive eﬀects. Diﬀerent reference states alter the partition between APE production and dissipation, but neither aﬀect the net conversion between kinetic energy and APE, nor the diﬀerence between APE production and dissipation. We argue that the possibility of constructing APE-like budgets based on reference states other than Lorenz’s reference state is more important than has been previously assumed, and we illustrate the feasibility of doing so in the context of an idealised and realistic oceanic example, using as reference states one with constant density and another one deﬁned as the horizontal mean density ﬁeld; in the latter case, the resulting APE density is found to be a reasonable approximation of the APE density constructed from Lorenz’s reference state, while being computationally cheaper.