Skip to main content Accessibility help

On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models and the ocean heat engine controversy



In this paper, the available potential energy (APE) framework of Winters et al. (J. Fluid Mech., vol. 289, 1995, p. 115) is extended to the fully compressible Navier–Stokes equations, with the aims of clarifying (i) the nature of the energy conversions taking place in turbulent thermally stratified fluids; and (ii) the role of surface buoyancy fluxes in the Munk & Wunsch (Deep-Sea Res., vol. 45, 1998, p. 1977) constraint on the mechanical energy sources of stirring required to maintain diapycnal mixing in the oceans. The new framework reveals that the observed turbulent rate of increase in the background gravitational potential energy GPEr, commonly thought to occur at the expense of the diffusively dissipated APE, actually occurs at the expense of internal energy, as in the laminar case. The APE dissipated by molecular diffusion, on the other hand, is found to be converted into internal energy (IE), similar to the viscously dissipated kinetic energy KE. Turbulent stirring, therefore, does not introduce a new APE/GPEr mechanical-to-mechanical energy conversion, but simply enhances the existing IE/GPEr conversion rate, in addition to enhancing the viscous dissipation and the entropy production rates. This, in turn, implies that molecular diffusion contributes to the dissipation of the available mechanical energy ME = APE + KE, along with viscous dissipation. This result has important implications for the interpretation of the concepts of mixing efficiency γmixing and flux Richardson number Rf, for which new physically based definitions are proposed and contrasted with previous definitions.

The new framework allows for a more rigorous and general re-derivation from the first principles of Munk & Wunsch (1998, hereafter MW98)'s constraint, also valid for a non-Boussinesq ocean: where G(KE) is the work rate done by the mechanical forcing, Wr, forcing is the rate of loss of GPEr due to high-latitude cooling and ξ is a nonlinearity parameter such that ξ = 1 for a linear equation of state (as considered by MW98), but ξ < 1 otherwise. The most important result is that G(APE), the work rate done by the surface buoyancy fluxes, must be numerically as large as Wr, forcing and, therefore, as important as the mechanical forcing in stirring and driving the oceans. As a consequence, the overall mixing efficiency of the oceans is likely to be larger than the value γmixing = 0.2 presently used, thereby possibly eliminating the apparent shortfall in mechanical stirring energy that results from using γmixing = 0.2 in the above formula.


Corresponding author

Email address for correspondence:


Hide All
Balmforth, N. J. & Young, W. R. 2003 Diffusion-limited scalar cascades. J. Fluid Mech. 482, 91100.
Bannon, P. R. 2004 Lagrangian available energetics and parcel instabilities. J. Atmos. Sci. 61, 17541767.
Barry, M. E., Ivey, G. N., Winters, K. B. & Imberger, J. 2001 Measurements of diapycnal diffusivities in stratified fluids. J. Fluid Mech. 442, 267291.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bejan, A. 1997 Advanced Engineering Thermodynamics. John Wiley.
Bouruet-Aubertot, P., Sommeria, J. & Staquet, C. 1996 Stratified turbulence produced by internal wave breaking: two-dimensional numerical experiments. Dyn. Atm. Oceans 23, 357369.
Caulfield, C. P. & Peltier, W. R. 2000 The anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid Mech. 413, 147.
Colin de Verdière, A. 1993 On the oceanic thermohaline circulation. In Modelling Oceanic Climate Interactions (ed. Willebrand, J. & Anderson, D. L. T.), vol. 2, pp. 151183. Nato-Asi Series I. Springer.
Dalziel, S. B., Patterson, M. D., Caulfield, C. P. & Coomaraswamy, I. A. 2008 Mixing efficiency in high-aspect-ratio Rayleigh–Taylor experiments. Phys. Fluids 20, 065106.
de Groot, S. R. & Mazur, P. 1962 Non-Equilibrium Thermodynamics. North Holland.
de Szoeke, R. A. & Samelson, R. M. 2002 The duality between the Boussinesq and non-Boussinesq hydrostatic equations of motion. J. Phys. Oceanogr. 32, 21942203.
Eckart, C. 1948 An analysis of the stirring and mixing processes in incompressible fluids. J. Mar. Res. 7, 265275.
Feistel, R. 2003 A new extended Gibbs thermodynamic potential of seawater. Prog. Oceanogr. 58, 43114.
Fofonoff, N. P. 1962 Physical properties of seawater In The Sea (ed. Hill, M. N.), vol. 1, pp. 330. Wiley-Interscience.
Fofonoff, N. P. 1998 Nonlinear limits to ocean thermal structure. J. Mar. Res. 56, 793811.
Fofonoff, N. P. 2001 Thermal stability of the world ocean thermoclines. J. Phys. Oceanogr. 31, 21692177.
Gibbs, W. 1878 On the equilibrium of heterogeneous substances. Trans. Conn. Acad. 3, 343524.
Gnanadesikan, A., Slater, R. D., Swathi, P. S. & Vallis, G. K. 2005 The energetics of the ocean heat transport. J. Climate 18, 26042616.
Gregg, M. C. 1987 Diapycnal mixing in the thermocline: a review. J. Geophys. Res. 92 (C5), 52495286.
Griffies, S. M. 2004 Fundamentals of Ocean Climate Models. Princeton University Press.
Haltiner, G. J. & Williams, R. T. 1980 Numerical Prediction and Dynamic Meteorology. John Wiley.
Holliday, D. & McIntyre, M. E. 1981 On potential energy density in an incompressible stratified fluid. J. Fluid Mech. 107, 221225.
Holloway, G. 1986 Considerations on the theory of temperature spectra in stably stratified turbulence. J. Phys. Oceanogr. 16, 21792183.
Huang, R. X. 1998 Mixing and available potential energy in a Boussinesq ocean. J. Phys. Oceanogr. 28, 669678.
Huang, R. X. 2004 Ocean, energy flows in. Encycl. Energy 4, 497509.
Huang, R. X. 2005 Available potential energy in the world's oceans. J. Mar. Res. 63, 141158.
Hughes, G. O. & Griffiths, R. W. 2006 A simple convective model of the global overturning circulation, including effects of entrainment into sinking regions. Ocean Modelling 12, 4679.
Hughes, G. O. & Griffiths, R. W. 2008 Horizontal convection. Annu. Rev. Fluid Mech. 40, 185208.
Hughes, G. O., Hogg, A. McC. & Griffiths, R. W. 2009 Available potential energy and irreversible mixing in the meridional overturning circulation. J. Phys. Oceanogr. (in press).
Jeffreys, H. 1925 On fluid motions produced by differences of temperature and humidity. Q. J. Roy. Met. Soc. 51, 347356.
Kondepudi, D. & Prigogine, I. 1998 Modern Thermodynamics. From Heat Engines to Dissipative structures. John Wiley & Sons.
Kuhlbrodt, T. 2008 On Sandström's inferences from his tank experiments: a hundred years later. Tellus 60, 819836.
Kuhlbrodt, T., Griesel, A., Levermann, A., Hofmann, M. & Rahmstorf, S. 2007 On the driving processes of the Atlantic meridional overturning circulation. Rev. Geophys. 45, doi:10.1029/2004RG00166.
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon Press.
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207242.
Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13, 323.
Linden, P. F. & Redondo, J. M. 1991 Molecular mixing in Rayleigh–Taylor instability. Part 1: global mixing. Phys. Fluids A 3, 12691277.
Lorenz, E. N. 1955 Available potential energy and the maintenance of the general circulation. Tellus 7, 157167.
Margules, M. 1903 Über die Energie der Stürme. Jahrb. Zentralanst. Meteorol. Wien 40, 126.
Marquet, P. 1991 On the concept of exergy and available enthalpy: application to atmospheric energetics. Q. J. Roy. Met. Soc. 117, 449475.
McDougall, T. J., Church, J. A. & Jackett, D. R. 2003 Does the nonlinearity of the equation of state impose an upper bound on the buoyancy frequency? J. Mar. Res. 61, 745764.
McEwan, A. D. 1983 a The kinematics of stratified mixing through internal wavebreaking. J. Fluid Mech. 128 4757.
McEwan, A. D. 1983 b Internal mixing in stratified fluids. J. Fluid Mech. 128, 5980.
Molemaker, M. J. & McWilliams, J. C. 2009 A complete energy budget using a local available potential energy density. J. Fluid Mech. (in press).
Mullarney, J. C., Griffiths, R. W. & Hughes, G. O. 2004 Convection driven by differential heating at a horizontal boundary. J. Fluid Mech. 516, 181209.
Munk, W. & Wunsch, C. 1998 Abyssal recipes II: energetics of tidal and wind mixing. Deep-Sea Res. 45, 19772010.
Nakamura, N. 1996 Two-dimensional mixing, edge formation, and permeability diagnosed in area coordinates. J. Atmos. Sci. 53, 15241537.
Nycander, J., Nilsson, J., Döös, & Broström, G. 2007 Thermodynamic analysis of the ocean circulation. J. Phys. Oceanogr. 37, 20382052.
Oakey, N. S. 1982 Determination of the rate of dissipation of turbulent energy from simultaneous temperature and velocity shear microstructure measurements. J. Phys. Oceanogr. 22, 256271.
Oort, A. H., Anderson, L. A. & Peixoto, J. P. 1994 Estimates of the energy cycle of the oceans. J. Geophys. Res. 99 (C4), 76657688.
Osborn, T. R. & Cox, C. S. 1972 Oceanic fine structure. Geophys. Astr. Fluid Dyn. 3, 321345.
Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 8389.
Ottinger, H. C. 2005 Beyond equilibrium thermodynamics. Wiley Intersciences.
Paparella, F. & Young, W. R. 2002 Horizontal convection is non turbulent. J. Fluid Mech. 466, 205214.
Pauluis, O. 2007 Sources and sinks of available potential energy in a moist atmosphere. J. Atmos. Sci. 64, 26272641.
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135167.
Roullet, G. & Klein, P. 2009 Available potential energy diagnosis in a direct numerical simulation of rotating stratified turbulence. J. Fluid Mech. 624, 4555.
Sandström, J. W. 1908 Dynamische versuche mit meerwasser. Ann. Hydrodynam. Marine Meteorol. 36, 623.
Siggers, J. H., Kerswell, R. R. & Balmforth, N. J. 2004 Bounds on horizontal convection. J. Fluid Mech. 517, 5570.
Staquet, C. 2000 Mixing in a stably stratified shear layer: two- and three-dimensional numerical experiments. Fluid Dyn. Res. 27, 367404.
Tailleux, R. 2009 Understanding mixing efficiency in the oceans: Do the nonlinearities of the equation of state matter? Ocean Sciences 5, 271283.
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. Roy. Met. Soc. 130, 32233243.
Tailleux, R. & Rouleau, L. 2009 The effect of mechanical stirring on horizontal convection. Tellus A (in press).
Thorpe, S. A. 2005 The Turbulent Ocean. Cambridge University Press.
Tseng, Y.-H. & Ferziger, J. 2001 Mixing and available potential energy in stratified flows. Phys. Fluids 13, 12811293.
Wang, W. & Huang, R. X. 2005 An experimental study on thermal circulation driven by horizontal differential heating. J. Fluid Mech. 540, 4973.
Winters, K. B. & d'Asaro, E. A. 1996 Diascalar flux and the rate of fluid mixing. J. Fluid Mech. 317, 179193.
Winters, K. B., Lombard, P. N.Riley, J. J. & d'Asaro, E. A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115228.
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281314.
Zeldovich, Y. B. 1937 The asymptotic law of heat transfer at small velocities in the finite domain problem. Zh. Eskp. Teoret. Fiz. 7, 12, 14661468. [Reprinted in English in the Selected Works of Yakov Borisovich Zeldovich, vol. 1: Chemical Physics and Hydrodynamics (ed. J. P. Ostriker, G. I. Barrenblatt & R. A. Sunyaev). Princeton University Press, 1992].
Zilitinkevich, S. S., Elperin, T., Kleeorin, N., Rogachevskii, I., Esau, I., Mauritsen, T. & Miles, M. W. 2008 Turbulence energetics in stably stratified flows: strong and weak mixing regimes. Q. J. Roy. Met. Soc. 134, 793799.
MathJax is a JavaScript display engine for mathematics. For more information see

Related content

Powered by UNSILO

On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models and the ocean heat engine controversy



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.