Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T22:19:41.588Z Has data issue: false hasContentIssue false

On the meaning of mixing efficiency for buoyancy-driven mixing in stratified turbulent flows

Published online by Cambridge University Press:  17 September 2015

Megan S. Davies Wykes*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Graham O. Hughes
Affiliation:
Research School of Earth Sciences, The Australian National University, Canberra, ACT 0200, Australia
Stuart B. Dalziel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
Email address for correspondence: [email protected]

Abstract

The concept of a mixing efficiency is widely used to relate the amount of irreversible diabatic mixing in a stratified flow to the amount of energy available to support mixing. This common measure of mixing in a flow is based on the change in the background potential energy, which is the minimum gravitational potential energy of the fluid that can be achieved by an adiabatic rearrangement of the instantaneous density field. However, this paper highlights examples of mixing that is primarily ‘buoyancy-driven’ (i.e. energy is released to the flow predominantly from a source of available potential energy) to demonstrate that the mixing efficiency depends not only on the specific characteristics of the turbulence in the region of the flow that is mixing, but also on the density profile in regions remote from where mixing physically occurs. We show that this behaviour is due to the irreversible and direct conversion of available potential energy into background potential energy in those remote regions (a mechanism not previously described). This process (here termed ‘relabelling’) occurs without requiring either a local flow or local mixing, or any other process that affects the internal energy of that fluid. Relabelling is caused by initially available potential energy, associated with identifiable parcels of fluid, becoming dynamically inaccessible to the flow due to mixing elsewhere. These results have wider relevance to characterising mixing in stratified turbulent flows, including those involving an external supply of kinetic energy.

Type
Papers
Copyright
© Crown Copyright. Published by Cambridge University Press 2015 

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

Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 121.CrossRefGoogle Scholar
Bluteau, C. E., Jones, N. L. & Ivey, G. N. 2013 Turbulent mixing efficiency at an energetic ocean site. J. Geophys. Res. 118 (9), 46624672.CrossRefGoogle Scholar
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 (6), 065106.CrossRefGoogle Scholar
Davies Wykes, M. S. & Dalziel, S. B. 2014 Efficient mixing in stratified flows: experimental study of a Rayleigh–Taylor unstable interface within an otherwise stable stratification. J. Fluid Mech. 756, 10271057.CrossRefGoogle Scholar
Dunckley, J. F., Koseff, J. R., Steinbuck, J. V., Monismith, S. G. & Genin, A. 2012 Comparison of mixing efficiency and vertical diffusivity models from temperature microstructure. J. Geophys. Res. 117 (C10), 112.CrossRefGoogle Scholar
Fernando, H. J. S. 1991 Turbulent mixing in stratified flows. Annu. Rev. Fluid Mech. 368 (1), 455493.CrossRefGoogle Scholar
Gayen, B., Griffiths, R. W., Hughes, G. O. & Saenz, J. A. 2013 Energetics of horizontal convection. J. Fluid Mech. 716, R10.CrossRefGoogle Scholar
Gayen, B., Griffiths, R. W. & Hughes, G. O. 2014 Stability transitions and turbulence in horizontal convection. J. Fluid Mech. 751, 698724.CrossRefGoogle Scholar
Hult, E. L., Troy, C. D. & Koseff, J. R. 2011a The mixing efficiency of interfacial waves breaking at a ridge: 1. Overall mixing efficiency. J. Geophys. Res. 116 (C02), 110.Google Scholar
Hult, E. L., Troy, C. D. & Koseff, J. R. 2011b The mixing efficiency of interfacial waves breaking at a ridge: 2. Local mixing processes. J. Geophys. Res. 116 (C02), 113.Google Scholar
Ivey, G. N. & Imberger, J. 1991 On the nature of turbulence in a stratified fluid. Part I. The energetics of mixing. J. Phys. Oceanogr. 21 (5), 650658.2.0.CO;2>CrossRefGoogle Scholar
Jacobs, J. W. & Dalziel, S. B. 2005 Rayleigh–Taylor instability in complex stratifications. J. Fluid Mech. 542, 251279.CrossRefGoogle Scholar
Jones, S. W. 1991 The enhancement of mixing by chaotic advection. Phys. Fluids A 3 (5), 10811086.CrossRefGoogle Scholar
Lawrie, A. G. W. & Dalziel, S. B. 2011a Rayleigh–Taylor mixing in an otherwise stable stratification. J. Fluid Mech. 688, 507527.CrossRefGoogle Scholar
Lawrie, A. G. W. & Dalziel, S. B. 2011b Turbulent diffusion in tall tubes. Part II. Confinement by stratification. Phys. Fluids 23 (8), 085110.CrossRefGoogle Scholar
Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13 (1), 323.CrossRefGoogle Scholar
Lorenz, E. N. 1955 Available potential energy and the maintenance of the general circulation. Tellus 7 (2), 157167.CrossRefGoogle Scholar
Lozovatsky, I. D. & Fernando, H. J. S. 2013 Mixing efficiency in natural flows. Phil. Trans. R. Soc. Lond. A 371 (1982), 20120213.Google ScholarPubMed
Mashayek, A., Caulfield, C. P. & Peltier, W. R. 2013 Time-dependent, non-monotonic mixing in stratified turbulent shear flows: implications for oceanographic estimates of buoyancy flux. J. Fluid Mech. 736, 570593.Google Scholar
Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10 (1), 8389.2.0.CO;2>CrossRefGoogle Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35 (1), 135167.CrossRefGoogle Scholar
Prastowo, T., Griffiths, R. W., Hughes, G. O. & Hogg, A. Mc C. 2009 Effects of topography on the cumulative mixing efficiency in exchange flows. J. Geophys. Res. 114 (C8), 112.CrossRefGoogle Scholar
Rayleigh, Lord 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14 (1), 8.Google Scholar
Scotti, A. & White, B. 2011 Is horizontal convection really ‘non-turbulent?’. Geophys. Res. Lett. 38 (21), 15.CrossRefGoogle 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, 339382.CrossRefGoogle Scholar
Tailleux, R. 2013 Available potential energy and exergy in stratified fluids. Annu. Rev. Fluid Mech. 45 (1), 3558.CrossRefGoogle Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201 (1065), 192196.Google Scholar
Tseng, Y. & Ferziger, J. H. 2001 Mixing and available potential energy in stratified flows. Phys. Fluids 13 (5), 12811293.CrossRefGoogle Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar
Winters, K. B. & D’Asaro, E. A. 1996 Diascalar flux and the rate of fluid mixing. J. Fluid Mech. 317, 179193.CrossRefGoogle Scholar
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, 115128.CrossRefGoogle Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36 (1), 281314.CrossRefGoogle Scholar