Biases in Thorpe-Scale Estimates of Turbulence Dissipation. Part II: Energetics Arguments and Turbulence Simulations

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

Search for other papers by Alberto Scotti in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

This paper uses the energetics framework developed by Scotti and White to provide a critical assessment of the widely used Thorpe-scale method, which is used to estimate dissipation and mixing rates in stratified turbulent flows from density measurements along vertical profiles. This study shows that the relevant displacement scale in general is not the rms value of the Thorpe displacement. Rather, the displacement field must be Reynolds decomposed to separate the mean from the turbulent component, and it is the turbulent component that ought to be used to diagnose mixing and dissipation. In general, the energetics of mixing in an overall stably stratified flow involves potentially complex exchanges among the available potential energy and kinetic energy associated with the mean and turbulent components of the flow. The author considers two limiting cases: shear-driven mixing, where mixing comes at the expense of the mean kinetic energy of the flow, and convective-driven mixing, which taps the available potential energy of the mean flow to drive mixing. In shear-driven flows, the rms of the Thorpe displacement, known as the Thorpe scale is shown to be equivalent to the turbulent component of the displacement. In this case, the Thorpe scale approximates the Ozmidov scale, or, which is the same, the Thorpe scale is the appropriate scale to diagnose mixing and dissipation. However, when mixing is driven by the available potential energy of the mean flow (convective-driven mixing), this study shows that the Thorpe scale is (much) larger than the Ozmidov scale. Using the rms of the Thorpe displacement overestimates dissipation and mixing, since the amount of turbulent available potential energy (measured by the turbulent displacement) is only a fraction of the total available potential energy (measured by the Thorpe scale). Corrective measures are discussed that can be used to diagnose mixing from knowledge of the Thorpe displacement. In a companion paper, Mater et al. analyze field data and show that the Thorpe scale can indeed be much larger than the Ozmidov scale.

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

Abstract

This paper uses the energetics framework developed by Scotti and White to provide a critical assessment of the widely used Thorpe-scale method, which is used to estimate dissipation and mixing rates in stratified turbulent flows from density measurements along vertical profiles. This study shows that the relevant displacement scale in general is not the rms value of the Thorpe displacement. Rather, the displacement field must be Reynolds decomposed to separate the mean from the turbulent component, and it is the turbulent component that ought to be used to diagnose mixing and dissipation. In general, the energetics of mixing in an overall stably stratified flow involves potentially complex exchanges among the available potential energy and kinetic energy associated with the mean and turbulent components of the flow. The author considers two limiting cases: shear-driven mixing, where mixing comes at the expense of the mean kinetic energy of the flow, and convective-driven mixing, which taps the available potential energy of the mean flow to drive mixing. In shear-driven flows, the rms of the Thorpe displacement, known as the Thorpe scale is shown to be equivalent to the turbulent component of the displacement. In this case, the Thorpe scale approximates the Ozmidov scale, or, which is the same, the Thorpe scale is the appropriate scale to diagnose mixing and dissipation. However, when mixing is driven by the available potential energy of the mean flow (convective-driven mixing), this study shows that the Thorpe scale is (much) larger than the Ozmidov scale. Using the rms of the Thorpe displacement overestimates dissipation and mixing, since the amount of turbulent available potential energy (measured by the turbulent displacement) is only a fraction of the total available potential energy (measured by the Thorpe scale). Corrective measures are discussed that can be used to diagnose mixing from knowledge of the Thorpe displacement. In a companion paper, Mater et al. analyze field data and show that the Thorpe scale can indeed be much larger than the Ozmidov scale.

Corresponding author address: Alberto Scotti, CB 3300, Dept. of Marine Sciences, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3300. E-mail: ascotti@unc.edu
Save
  • Alford, M. H., M. C. Gregg, and M. A. Merrifield, 2006: Structure, propagation, and mixing of energetic baroclinic tides in Mamala Bay, Oahu, Hawaii. J. Phys. Oceanogr., 36, 9971018, doi:10.1175/JPO2877.1.

    • Search Google Scholar
    • Export Citation
  • Alford, M. H., and Coauthors, 2011: Energy flux and dissipation in Luzon Strait: Two tales of two ridges. J. Phys. Oceanogr., 41, 22112222, doi:10.1175/JPO-D-11-073.1.

    • Search Google Scholar
    • Export Citation
  • Chalamalla, V. K., and S. Sarkar, 2015: Mixing, dissipation rate, and their overturn-based estimates in a near-bottom turbulent flow driven by internal tides. J. Phys. Oceanogr., 45, 19691987, doi:10.1175/JPO-D-14-0057.1.

    • Search Google Scholar
    • Export Citation
  • Chalamalla, V. K., B. Gayen, A. Scotti, and S. Sarkar, 2013: Turbulence during the reflection of internal gravity waves at critical and near-critical slopes. J. Fluid Mech., 729, 4768, doi:10.1017/jfm.2013.240.

    • Search Google Scholar
    • Export Citation
  • Chandrasekhar, S., 1981: Hydrodynamics and Hydromagnetic Stability. Dover, 652 pp.

  • Corrsin, S., 1958: Local isotropy in turbulent shear flow. NACA Research Memo. RM 58B11, 15 pp.

  • De Silva, I. P. D., J. Imberger, and G. N. Ivey, 1997: Localized mixing due to a breaking internal wave ray at a sloping bed. J. Fluid Mech., 350, 127, doi:10.1017/S0022112097006939.

    • Search Google Scholar
    • Export Citation
  • Dillon, T. M., 1982: Vertical overturns: A comparison of Thorpe and Ozmidov length scales. J. Geophys. Res., 87, 96019613, doi:10.1029/JC087iC12p09601.

    • Search Google Scholar
    • Export Citation
  • Dillon, T. M., 1984: The energetics of overturning structures: Implication for the theory of fossil turbulence. J. Phys. Oceanogr., 14, 541549, doi:10.1175/1520-0485(1984)014<0541:TEOOSI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Dillon, T. M., and M. M. Park, 1987: The available potential energy of overturns as an indicator of mixing in the seasonal thermocline. J. Geophys. Res., 92, 53455353, doi:10.1029/JC092iC05p05345.

    • Search Google Scholar
    • Export Citation
  • Ferron, B., H. Mercier, K. Speer, A. Gargett, and K. Polzin, 1998: Mixing in the Romanche Fracture Zone. J. Phys. Oceanogr., 28, 19291945, doi:10.1175/1520-0485(1998)028<1929:MITRFZ>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Gargett, A. E., and T. Garner, 2008: Determining Thorpe scales from ship-lowered CTD density profiles. J. Atmos. Oceanic Technol., 25, 16571670, doi:10.1175/2008JTECHO541.1.

    • Search Google Scholar
    • Export Citation
  • Ivey, G. N., K. B. Winters, and J. R. Koseff, 2008: Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech., 40, 169184, doi:10.1146/annurev.fluid.39.050905.110314.

    • 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, 17561775, doi:10.1175/2009JPO4085.1.

    • Search Google Scholar
    • Export Citation
  • Klymak, J. M., and S. M. Legg, 2010: A simple mixing scheme for models that resolve breaking internal waves. Ocean Modell., 33, 224234, doi:10.1016/j.ocemod.2010.02.005.

    • Search Google Scholar
    • Export Citation
  • Lawrie, A. G., and S. B. Dalziel, 2011: Rayleigh–Taylor mixing in an otherwise stable stratification. J. Fluid Mech., 688, 507527, doi:10.1017/jfm.2011.398.

    • Search Google Scholar
    • Export Citation
  • Lumpkin, R., and K. Speer, 2007: Global ocean meridional overturning. J. Phys. Oceanogr., 37, 25502562, doi:10.1175/JPO3130.1.

  • Mann, K., and J. Lazier, 2006: Dynamics of Marine Ecosystems: Biological-Physical Interactions in the Oceans. Blackwell, 496 pp.

  • Mater, B. D., and S. K. Venayagamoorthy, 2014: 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
  • Mater, B. D., S. K. Venayagamoorthy, L. S. Laurent, and J. N. Moum, 2015: Biases in Thorpe scale estimates of turbulence dissipation. Part I: Assessments from large-scale overturns in oceanographic data. J. Phys. Oceanogr., 45, 24972521, doi:10.1175/JPO-D-14-0128.1.

    • Search Google Scholar
    • Export Citation
  • Moum, J. N., 1996: Energy-containing scales of turbulence in the ocean thermocline. J. Geophys. Res., 101, 14 09514 109, doi:10.1029/96JC00507.

    • Search Google Scholar
    • Export Citation
  • Munk, W., 1966: Abyssal recipes. Deep-Sea Res. Oceanogr. Abstr., 13, 707730, doi:10.1016/0011-7471(66)90602-4.

  • Munk, W., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res., 45, 1977–2010, doi:10.1016/S0967-0637(98)00070-3.

    • Search Google Scholar
    • Export Citation
  • Nash, J. D., M. H. Alford, E. Kunze, K. Martini, and S. Kelly, 2007: Hotspots of deep ocean mixing on the Oregon continental slope. Geophys. Res. Lett., 34, L01605, doi:10.1029/2006GL028170.

    • Search Google Scholar
    • Export Citation
  • Osborn, T., 1980: Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr., 10, 8389, doi:10.1175/1520-0485(1980)010<0083:EOTLRO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Ozmidov, R. V., 1965: On the turbulent exchange in a stably stratified ocean. Izv. Akad. Sci. USSR Atmos. Oceanic Phys., 1, 861871.

  • Scotti, A., 2008: A numerical study of gravity currents propagating on a free-slip boundary. Theor. Comput. Fluid Dyn., 22, 383402, doi:10.1007/s00162-008-0081-6.

    • Search Google Scholar
    • Export Citation
  • Scotti, A., 2011: Inviscid critical and near-critical reflection of internal waves in the time domain. J. Fluid Mech., 674, 464488, doi:10.1017/S0022112011000097.

    • 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, 114135, doi:10.1017/jfm.2013.643.

    • Search Google Scholar
    • Export Citation
  • Smyth, W., J. Moum, and D. Caldwell, 2001: The efficiency of mixing in turbulent patches: Inferences from direct simulations and microstructure observations. J. Phys. Oceanogr., 31, 19691992, doi:10.1175/1520-0485(2001)031<1969:TEOMIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Tailleux, R. G. J., 2013: Available potential energy and exergy in stratified fluids. Annu. Rev. Fluid Mech., 45, 3558, doi:10.1146/annurev-fluid-011212-140620.

    • Search Google Scholar
    • Export Citation
  • Tennekes, H., and J. L. Lumley, 1972: A First Course in Turbulence. MIT Press, 300 pp.

  • Thorpe, S. A., 1977: Turbulence and mixing in a Scottish loch. Philos. Trans. Roy. Soc. London, A286, 125181, doi:10.1098/rsta.1977.0112.

    • Search Google Scholar
    • Export Citation
  • Thorpe, S. A., 1987: On the reflection of a train of finite-amplitude internal waves from a uniform slope. J. Fluid Mech., 178, 279302, doi:10.1017/S0022112087001228.

    • Search Google Scholar
    • Export Citation
  • Thorpe, S. A., 2005: The Turbulent Ocean. Cambridge University Press, 439 pp.

  • Toggweiler, J., and B. Samuels, 1998: On the ocean’s large-scale circulation near the limit of no vertical mixing. J. Phys. Oceanogr., 28, 18321852, doi:10.1175/1520-0485(1998)028<1832:OTOSLS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Tseng, Y.-H., and J. Ferziger, 2001: Mixing and available potential energy in stratified flows. Phys. Fluids, 13, 12811293, doi:10.1063/1.1358307.

    • Search Google Scholar
    • Export Citation
  • Waterhouse, A. F., and Coauthors, 2014: Global patterns of diapycnal mixing from measurements of the turbulent dissipation rate. J. Phys. Oceanogr., 44, 18541872, doi:10.1175/JPO-D-13-0104.1.

    • Search Google Scholar
    • Export Citation
  • Wesson, J. C., and M. C. Gregg, 1994: Mixing at the Camarinal Sill in the Strait of Gibraltar. J. Geophys. Res., 99, 98479878, doi:10.1029/94JC00256.

    • Search Google Scholar
    • Export Citation
  • Wilson, R., H. Luce, F. Dalaudier, and J. Lefrre, 2010: Turbulence patch identification in potential density or temperature profiles. J. Atmos. Oceanic Technol., 27, 977993, doi:10.1175/2010JTECHA1357.1.

    • 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, 115128, doi:10.1017/S002211209500125X.

    • Search Google Scholar
    • Export Citation
  • Wolfe, C. L., and P. Cessi, 2009: Overturning circulation in an eddy-resolving model: The effect of the pole-to-pole temperature gradient. J. Phys. Oceanogr., 39, 125142, doi:10.1175/2008JPO3991.1.

    • Search Google Scholar
    • Export Citation
  • Wolfe, C. L., and P. Cessi, 2010: What sets the strength of the middepth stratification and overturning circulation in eddying ocean models? J. Phys. Oceanogr., 40, 15201538, doi:10.1175/2010JPO4393.1.

    • 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, 281314, doi:10.1146/annurev.fluid.36.050802.122121.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 476 176 26
PDF Downloads 403 123 7