Editorial Type:
Article
Article Type:
Research Article

The Mixing Efficiency of Stratified Turbulent Boundary Layers

Alberto Scotti Department of Marine Sciences, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina

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Brian White Department of Marine Sciences, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina

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Print Publication:
01 Oct 2016
DOI:
https://doi.org/10.1175/JPO-D-16-0095.1
Page(s):
3181–3191
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Abstract

The mixing efficiency observed in stratified turbulent boundary layers is considered within the framework of the Monin–Obukhov similarity theory. It is shown that the efficiency within the layer increases with distance from the boundary. Near the boundary, the efficiency is proportional to the distance from the boundary scaled with the Monin–Obukhov length. Far from the boundary, the efficiency relaxes to a value that depends on the overall thickness of the layer relative to the Monin–Obukhov layer. This value approaches 1/6 when the thickness is larger than 1/2 of the Monin–Obukhov length. The same analysis shows that the buoyancy Reynolds number cannot be used to unequivocally predict the efficiency. The −1/2 scaling between the efficiency and buoyancy Reynolds number that has been observed in field measurements and experiments is shown to depend on an extra dimensional scale and thus is not universal.

Corresponding author address: Alberto Scotti, Dept. of Marine Sciences, University of North Carolina, 3117H Venable Hall, Chapel Hill, NC 27599-3300. E-mail: ascotti@unc.edu

Abstract

The mixing efficiency observed in stratified turbulent boundary layers is considered within the framework of the Monin–Obukhov similarity theory. It is shown that the efficiency within the layer increases with distance from the boundary. Near the boundary, the efficiency is proportional to the distance from the boundary scaled with the Monin–Obukhov length. Far from the boundary, the efficiency relaxes to a value that depends on the overall thickness of the layer relative to the Monin–Obukhov layer. This value approaches 1/6 when the thickness is larger than 1/2 of the Monin–Obukhov length. The same analysis shows that the buoyancy Reynolds number cannot be used to unequivocally predict the efficiency. The −1/2 scaling between the efficiency and buoyancy Reynolds number that has been observed in field measurements and experiments is shown to depend on an extra dimensional scale and thus is not universal.

Corresponding author address: Alberto Scotti, Dept. of Marine Sciences, University of North Carolina, 3117H Venable Hall, Chapel Hill, NC 27599-3300. E-mail: ascotti@unc.edu
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  • Barenblatt, G. I., 1996: Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press, 386 pp.

  • Barry, M. E., 2002: Mixing in stratified turbulence. Ph.D. thesis, University of Western Australia.

  • Barry, M. E., G. N. Ivey, K. B. Winters, and J. Imberger, 2001: Measurements of diapycnal diffusivities in stratified fluids. J. Fluid Mech., 442, 267–291, doi:10.1017/S0022112001005080.

    • Search Google Scholar
    • Export Citation

  • Baumert, H., and H. Peters, 2004: Turbulence closure, steady state, and collapse into waves. J. Phys. Oceanogr., 34, 505–512, doi:10.1175/1520-0485(2004)034<0505:TCSSAC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Bluteau, C. E., N. L. Jones, and G. N. Ivey, 2013: Turbulent mixing efficiency at an energetic ocean site. J. Geophys. Res. Oceans, 118, 4662–4672, doi:10.1002/jgrc.20292.

    • Search Google Scholar
    • Export Citation

  • Burchard, H., O. Petersen, and T. P. Rippeth, 1998: Comparing the performance of the Mellor-Yamada and the κ-ε two-equation turbulence models. J. Geophys. Res., 103, 10 543–10 554, doi:10.1029/98JC00261.

    • Search Google Scholar
    • Export Citation

  • Cantwell, B. J., 2002: Introduction to Symmetry Analysis. Cambridge University Press, 612 pp.

  • Canuto, V., A. Howard, Y. Cheng, and M. Dubovikov, 2001: Ocean turbulence. Part I: One-point closure model—Momentum and heat vertical diffusivities. J. Phys. Oceanogr., 31, 1413–1426, doi:10.1175/1520-0485(2001)031<1413:OTPIOP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • de Lavergne, C., G. Madec, J. Le Sommer, A. G. Nurser, and A. C. Naveira Garabato, 2016: The impact of a variable mixing efficiency on the abyssal overturning. J. Phys. Oceanogr., 46, 663–681, doi:10.1175/JPO-D-14-0259.1.

    • Search Google Scholar
    • Export Citation

  • Foken, T., 2008: Micrometeorology. Springer, 306 pp.

  • Frisch, U., 1995: Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press, 296 pp.

  • Gargett, A. E., 1988: The scaling of turbulence in the presence of stable stratification. J. Geophys. Res., 93, 5021–5036, doi:10.1029/JC093iC05p05021.

    • Search Google Scholar
    • Export Citation

  • Gibson, C. H., 1980: Fossil temperature, salinity, and vorticity turbulence in the ocean. Marine Turbulence, J. C. J. Nihoul, Elsevier Oceanography Series, Vol. 28, 221–257, doi:10.1016/S0422-9894(08)71223-6.

  • Grachev, A. A., E. L Andreas, C. W. Fairall, P. S. Guest, and P. O. G. Persson, 2015: Similarity theory based on the Dougherty–Ozmidov length scale. Quart. J. Roy. Meteor. Soc., 141, 1845–1856, doi:10.1002/qj.2488.

    • Search Google Scholar
    • Export Citation

  • Incropera, F. P., and D. P. DeWitt, 1990: Fundamentals of Heat and Mass Transfer. Wiley, 919 pp.

  • Ivey, G. N., K. B. Winters, and J. R. Koseff, 2008: Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech., 40, 169–184, doi:10.1146/annurev.fluid.39.050905.110314.

    • Search Google Scholar
    • Export Citation

  • Jackson, P. R., and C. R. Rehmann, 2014: Experiments on differential scalar mixing in turbulence in a sheared, stratified flow. J. Phys. Oceanogr., 44, 2661–2680, doi:10.1175/JPO-D-14-0027.1.

    • Search Google Scholar
    • Export Citation

  • Jayne, S. R., 2009: The impact of abyssal mixing parameterizations in an ocean general circulation model. J. Phys. Oceanogr., 39, 1756–1775, doi:10.1175/2009JPO4085.1.

    • Search Google Scholar
    • Export Citation

  • Linden, P., 1979: Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn., 13, 3–23, doi:10.1080/03091927908243758.

  • Lozovatsky, I., and H. Fernando, 2013: Mixing efficiency in natural flows. Philos. Trans. Roy. Soc., A371, 20120213, doi:10.1098/rsta.2012.0213.

    • Search Google Scholar
    • Export Citation

  • Mater, B. D., and S. K. Venayagamoorthy, 2014a: The quest for an unambiguous parameterization of mixing efficiency in stably stratified geophysical flows. Geophys. Res. Lett., 41, 4646–4653, doi:10.1002/2014GL060571.

    • Search Google Scholar
    • Export Citation

  • Mater, B. D., and S. K. Venayagamoorthy, 2014b: A unifying framework for parameterizing stably stratified shear-flow turbulence. Phys. Fluids, 26, 036601, doi:10.1063/1.4868142.

    • Search Google Scholar
    • Export Citation

  • Mater, B. D., S. M. Schaad, and S. K. Venayagamoorthy, 2013: Relevance of the Thorpe length scale in stably stratified turbulence. Phys. Fluids, 25, 076604, doi:10.1063/1.4813809.

    • Search Google Scholar
    • Export Citation

  • Melet, A., R. Hallberg, S. Legg, and K. Polzin, 2013: Sensitivity of the ocean state to the vertical distribution of internal-tide-driven mixing. J. Phys. Oceanogr., 43, 602–615, doi:10.1175/JPO-D-12-055.1.

    • Search Google Scholar
    • Export Citation

  • Olbers, D., and C. Eden, 2013: A global model for the diapycnal diffusivity induced by internal gravity waves. J. Phys. Oceanogr., 43, 1759–1779, doi:10.1175/JPO-D-12-0207.1.

    • Search Google Scholar
    • Export Citation

  • Osborn, T., 1980: Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr., 10, 83–89, doi:10.1175/1520-0485(1980)010<0083:EOTLRO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Pardyjak, E., P. Monti, and H. Fernando, 2002: Flux Richardson number measurements in stable atmospheric shear flows. J. Fluid Mech., 459, 307–316, doi:10.1017/S0022112002008406.

    • Search Google Scholar
    • Export Citation

  • Polzin, K. L., 2009: An abyssal recipe. Ocean Modell., 30, 298–309, doi:10.1016/j.ocemod.2009.07.006.

  • Rudin, W., 1987: Real and Complex Analysis. McGraw-Hill, 416 pp.

  • Scotti, A., 2015: Biases in Thorpe-scale estimates of turbulence dissipation. Part II: Energetics arguments and turbulence simulations. J. Phys. Oceanogr., 45, 2522–2543, doi:10.1175/JPO-D-14-0092.1.

    • Search Google Scholar
    • Export Citation

  • Scotti, A., and B. White, 2014: Diagnosing mixing in stratified turbulent flows with a locally defined available potential energy. J. Fluid Mech., 740, 114–135, doi:10.1017/jfm.2013.643.

    • Search Google Scholar
    • Export Citation

  • Shih, L. H., J. R. Koseff, G. N. Ivey, and J. H. Ferziger, 2005: Parameterization of turbulent fluxes and scales using homogeneous sheared stably stratified turbulence simulations. J. Fluid Mech., 525, 193–214, doi:10.1017/S0022112004002587.

    • Search Google Scholar
    • Export Citation

  • St. Laurent, L., H. Simmons, and S. Jayne, 2002: Estimating tidally driven mixing in the deep ocean. Geophys. Res. Lett., 29, 2106, doi:10.1029/2002GL015633.

    • Search Google Scholar
    • Export Citation

  • Stretch, D. D., J. W. Rottman, S. K. Venayagamoorthy, K. K. Nomura, and C. R. Rehmann, 2010: Mixing efficiency in decaying stably stratified turbulence. Dyn. Atmos. Oceans, 49, 25–36, doi:10.1016/j.dynatmoce.2008.11.002.

    • Search Google Scholar
    • Export Citation

  • Tailleux, R. G. J., 2013: Available potential energy and exergy in stratified fluids. Annu. Rev. Fluid Mech., 45, 35–58, doi:10.1146/annurev-fluid-011212-140620.

    • Search Google Scholar
    • Export Citation

  • Taylor, J. R., and S. Sarkar, 2008: Stratification effects in a bottom Ekman layer. J. Phys. Oceanogr., 38, 2535–2555, doi:10.1175/2008JPO3942.1.

    • Search Google Scholar
    • Export Citation

  • Walter, R. K., M. E. Squibb, C. B. Woodson, J. R. Koseff, and S. G. Monismith, 2014: Stratified turbulence in the nearshore coastal ocean: Dynamics and evolution in the presence of internal bores. J. Geophys. Res. Oceans, 119, 8709–8730, doi:10.1002/2014JC010396.

    • Search Google Scholar
    • Export Citation

  • Wells, M., C. Cenedese, and C. P. Caulfield, 2010: The relationship between flux coefficient and entrainment ratio in density currents. J. Phys. Oceanogr., 40, 2713–2727, doi:10.1175/2010JPO4225.1.

    • Search Google Scholar
    • Export Citation

  • Winters, K. B., and E. A. D’Asaro, 1996: Diascalar flux and the rate of fluid mixing. J. Fluid Mech., 317, 179–193, doi:10.1017/S0022112096000717.

    • Search Google Scholar
    • Export Citation

  • Winters, K. B., P. N. Lombard, J. J. Riley, and E. A. D’Asaro, 1995: Available potential energy and mixing in density stratified fluids. J. Fluid Mech., 289, 115–128, doi:10.1017/S002211209500125X.

    • Search Google Scholar
    • Export Citation

  • Wunsch, C., and R. Ferrari, 2004: Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech., 36, 281–314, doi:10.1146/annurev.fluid.36.050802.122121.

    • Search Google Scholar
    • Export Citation
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