Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T15:24:44.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Available potential energy density for a multicomponent Boussinesq fluid with arbitrary nonlinear equation of state

Published online by Cambridge University Press:  25 October 2013

Rémi Tailleux
Rémi Tailleux*
Affiliation:
Department of Meteorology, University of Reading, Earley Gate, PO Box 243, Reading RG6 6BB, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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 to 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 affects 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 idealized 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.

JFM classification

Mathematical Foundations: General fluid mechanics Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 735 , 25 November 2013 , pp. 499 - 518
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andrews, D. G. 1981 A note on potential energy density in a stratified compressible fluid. J. Fluid Mech. 107, 227–236.Google Scholar
Antonov, J. I., Seidov, D., Boyer, T. P., Locarnini, R. A., Mishonov, A. V., Garcia, H. E., Baranova, O. K., Zweng, M. M. & Johnson, D. R. (ed. Levitus, S.). World Ocean Atlas 2009, Volume 2: Salinity NOAA Atlas NESDIS 69, 2010 US Government Printing Office.Google Scholar
Bannon, P. 2003 Hamiltonian description of idealized binary geophysical fluids. J. Atmos. Sci. 60, 2809–2819.2.0.CO;2>CrossRefGoogle Scholar
Codoban, S. & Shepherd, T. G. 2003 Energetics of a symmetric circulation including momentum constraints. J. Atmos. Sci. 60, 2019–2028.Google Scholar
Gregory, J. M. & Tailleux, R. 2010 Energetic analysis of simulated time-dependent changes in the Atlantic meriidonal overturning circulation in response to increasing ${\mathrm{CO} }_{2} $ concentration. Clim. Dyn. 37, 893–914.CrossRefGoogle Scholar
Holliday, D. & McIntyre, M. E. 1981 On potential energy density in an incompressible stratified fluid. J. Fluid Mech. 107, 221–225.Google Scholar
Huang, R. X. 2005 Available potential energy in the world’s oceans. J. Mar. Res. 63, 141–158.CrossRefGoogle Scholar
Hughes, G. O, Hogg, A. & Griffiths, R. W. 2009 Available potential energy and irreversible mixing in the meridional overturning circulation. J. Phys. Oceanogr. 39, 3130–3146.CrossRefGoogle Scholar
IOC, SCOR & IAPSO, 2010 The international thermodynamic equation of seawater – 2010: calculations and use of thermodynamic properties. Intergovernmental Oceanographic Commission, Manuals and Guides No. 56.Google Scholar
Jackett, D. R., McDougall, T. J., Feistel, R., Wright, D. G. & Griffies, S. M. 2006 Algorithms for density, potential temperature, conservative temperature, and freezing temperature of seawater. J. Atmos. Oceanic Technol. 23, 1709–1728.Google Scholar
Kucharski, F. 1997 On the concept of exergy and available potential energy. Q. J. R. Meteorol. Soc. 123, 2141–2156.Google Scholar
Lamb, K. G. 2007 Energy and pseudoenergy flux in the internal wave field generated by tidal flow over topography. Cont. Shelf Res. 27, 1208–1232.Google Scholar
Lamb, K. G. 2008 On the calculation of the available potential energy of an isolated perturbation in a density stratified fluid. J. Fluid Mech. 597, 415–427.Google Scholar
Locarnini, R. A., Mishonov, A. V., Antonov, J. I., Boyer, T. P., Garcia, H. E., Baranova, O. K., Zweng, M. M. & Johnson, D. R. 2010 World Ocean Atlas 2009, Volume 1: Temperature (ed. Levitus, S.), NOAA Atlas NESDIS 68, US Government Printing Office.Google Scholar
Lorenz, E. N. 1955 Available potential energy and the maintenance of the general circulation. Tellus 7, 157–167.Google Scholar
Margules, M. 1903 Über die Energie der Stürme. Jahrb. Zentralanst. Meteorol. Wien 40, 1–26.Google Scholar
McDougall, T. J. 2003 Potential enthalpy: a conservative oceanic variable for evaluating heat content and heat fluxes. J. Phys. Oceanogr. 33, 945–963.2.0.CO;2>CrossRefGoogle Scholar
Molemaker, M. J. & McWilliams, J. C. 2010 Local balance and cross-scale flux of available potential energy. J. Fluid Mech. 645, 295–314.CrossRefGoogle Scholar
Nycander, J. 2010 Horizontal convection with a nonlinear equation of state: generalization of a theorem of Paparella and Young. Tellus 62A, 134–137.Google Scholar
O’Gorman, P. A. 2010 Understanding the varied response of the extratropical storm tracks to climate change. Proc. Natl Acad. Sci. 107, 19176–19180.Google Scholar
Oort, A. H., Anderson, L. A. & Peixoto, J. P. 1994 Estimates of the energy cycle of the oceans. J. Geophys. Res. 99 (C4), 7665–7688.Google Scholar
Pauluis, O. 2007 Sources and sinks of available potential energy in a moist atmosphere. J. Atmos. Sci. 64, 2627–2641.Google Scholar
Peixoto, J. P. & Oort, A. H. 1992 Physics of Climate. American Institute of Physics.CrossRefGoogle Scholar
Roquet, F. 2013 Dynamical potential energy: a new approach to ocean energetics. J. Phys. Oceanogr. 43, 457–476.Google Scholar
Roullet, G., Capet, X. & Maze, G. 2013 Oceanic interior mesoscale turbulence revealed by ARGO floats. Geophys. Res. Lett. (submitted).Google Scholar
Roullet, G. & Klein, P. 2009 Available potential energy diagnosis in a direct numerical simulation of rotating stratified turbulence. J. Fluid Mech. 624, 45–55.CrossRefGoogle Scholar
Scotti, A., Beardsley, R. & Rutman, B. 2006 On the interpretation of energy and energy fluxes of nonlinear internal waves: an example from Massachussetts Bay. J. Fluid Mech. 561, 103–112.Google Scholar
Scotti, A. & White, B. 2011 Is horizontal convection really ‘non-turbulent’? Geophys. Res. Lett. 38, doi:10.1029/2011GL049701.Google Scholar
Shepherd, T. G. 1993 A unified theory of available potential energy. Atmos.-Ocean 31, 1–26.CrossRefGoogle Scholar
von Storch, J.-S., Eden, C., Fast, I., Haak, H., Hernandez-Deckers, D., Maier-Reimer, E., Marotzke, J. & Stammer, D. 2012 An estimate of the Lorenz energy cycle for the world ocean based on the STORM/NCEP simulation. J. Phys. Oceanogr. 42, 2185–2205.Google Scholar
Tailleux, R. 2009 On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models, and the ocean heat engine controversy. J. Fluid Mech. 638, 339–382.Google Scholar
Tailleux, R. 2012 Thermodynamics/dynamics coupling in weakly compressible turbulent stratified fluids. ISRN Thermodyn. 2012:609701.Google Scholar
Tailleux, R. 2013 Available potential energy and exergy in stratified fluids. Annu. Rev. Fluid Mech. 45, 35–58.Google Scholar
Tailleux, R. & Grandpeix, J. Y. 2004 On the seemingly incompatible parcel and globally integrated views of the energetics of triggered atmospheric deep convection over land. Q. J. R. Meteorol. Soc. 130, 3223–3243.Google Scholar
Tailleux, R. & Rouleau, L. 2010 The effect of mechanical stirring on horizontal convection. Tellus A 62, 138–153.Google Scholar
Vallis, G. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.Google Scholar
Winters, K. B. & Barkan, R. 2013 Available potential energy density for Boussinesq fluid flow. J. Fluid Mech. 714, 476–488.Google Scholar
Winters, K. B., Lombard, P. N., Riley, J. J. & d’Asaro, 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115–228.Google Scholar
Young, W. R. 2010 Dynamic enthalpy, conservative temperature, and the seawater Boussinesq approximation. J. Phys. Oceanogr. 40, 394–400.CrossRefGoogle Scholar