• 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
  • View in gallery

    Schematic diagram portraying the relationship between the reservoirs of mean and turbulent (left) KE and (right) APE (adapted from SW).

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    Primary energy pathways in a shear-driven mixing event. The mean flow provides mean KE, and the shear production term transfers energy to the turbulent KE. From the latter, a fraction equal to Γ/(1 + Γ) is transferred by the buoyancy flux to turbulent APE for mixing, while the dissipation term removes the rest.

  • View in gallery

    Primary energy pathways in a convective-driven mixing event. The mean flow provides mean APE. From this reservoir, energy flows to the turbulent KE reservoir via the turbulent buoyancy flux term (which has thus opposite sign relative to the shear-driven case), the turbulent APE reservoir via the turbulent diapycnal flux, and the mean KE via the mean buoyancy flux. Unlike the first two, the last term is reversible.

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    (left) Reduced-gravity distribution at t = 0 in the high aspect ratio case. Note how the colors are saturated to highlight the overturned region. (right) Reduced-gravity profile along a vertical transect across the overturned region (solid line) and the isochorically restratified profile (dashed line). The displacement LT is obtained by measuring the distance between the solid and dashed line along the vertical.

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    Representative profiles from a stratified Couette flow. (a) Vertical profiles of isochorically restratified reduced gravity (, circles), randomly chosen vertical profile of reduced gravity b (solid line), and reference state calculated from the randomly chosen b profile (dashed–dotted line). (b) Vertical profile of the absolute value of the Thorpe displacement calculated from the random reduced-gravity profile shown in (a) (solid line) and rms value of lT (dashed line). (c) The full turbulent APE (solid line), quadratic limit [Eq. (11)] (open circles), and turbulent KE (dashed line). In all cases, Ri = 0.03 and Re =110 000.

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    Comparing displacements calculated from 1D vertical casts and using the fully three-dimensional b field. (left) PDF of |δT|, calculated using 1 (squares), 10 (circles), 100 (crosses), and 1000 (stars) randomly chosen vertical casts. For reference, the PDF of the displacements |lT| calculated from the entire dataset is shown as a solid line. (right) PDF of Thorpe scale LT obtained from 1000 individual casts. Each cast yields a value for LT. Values are normalized with the mean value of lT. Data from stably stratified Couette flow at Ri = 0.03 and Re =110 000.

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    (top) Ratio of turbulent APE to turbulent KE as a function of the ratio Γ ≡ ϵp/ϵk in steady, shear-driven flows; (bottom) normalized dissipation time scale Γ−1/2 as a function of Γ.

  • View in gallery

    PDF of turbulent KE dissipation in steady shear-driven flows normalized with the estimate based on the Thorpe scale and dissipation time [Eq. (18)]: Re = 57 000 (solid line), Re = 72 000 (dashed line), Re = 112 000 (dashed–dotted line), Re = 220 000 (stars), and Re = 110 000 (squares). On average, Eq. (18) overestimates the mean dissipation in the core layer by less than a factor 2 across all experiments.

  • View in gallery

    (a) Evolution of mean APE (solid line), mean KE (dashed line), and BPE (dashed–dotted line) as a function of time in overturning flows. Energies are normalized so that the APE at t = 0 is one. Time is normalized with Ntan(A). (b) Mixing and dissipation. BPE evolution [normalized as in (a)] during the turbulent phase: high aspect ratio (solid line) and isotropic overturn (circles); cumulative turbulent KE dissipation: high aspect ratio (dashed line) and isotropic overturn (squares). Time normalized with N. For the isotropic case, time is shifted by 10 buoyancy periods.

  • View in gallery

    (a) Evolution of turbulent APE (solid line) and KE (dashed line) in a high aspect ratio overturning flow. The energies are normalized with the amount of mean APE at t = 0, time with N; (b) dissipative time scale for turbulent APE (solid line) and KE (dashed line) during the same period.

  • View in gallery

    (left) Mean and (right) turbulent Thorpe scale at three times during the turbulent episode in a high aspect ratio overturn. The overturned region is originally centered around the origin. As time evolves, it spreads radially outward, and turbulence develops primarily along the front.

  • View in gallery

    Time evolution of energy, dissipation, and turbulent scales within the turbulent core of a high aspect ratio overturn. (a) Mean APE (solid line) and turbulent APE (dashed line) with the corresponding estimates based on the mean (crosses) and turbulent (circles) component of the displacement. All energies are normalized with the initial amount of total APE. The turbulent APE is further multiplied by 10. (b) Dissipation (solid line) and estimates of dissipation based on the theory presented in this paper [Eq. (18)] with α = 5 and Γ = 1 (circles) and the Thorpe scale following the standard recipe [Eq. (4)]. All dissipations are normalized with the peak of the actual dissipation. (c) Ratio of turbulent displacement to Ozmidov scale (solid line) and Thorpe scale to Ozmidov scale (dashed–dotted line).

  • View in gallery

    PDFs of dissipation within the turbulent core of a high aspect ratio overturn at different times: when the turbulent KE attains its maximum (Nt = 23, solid line) and at two later times [Nt = 25 (dashed–dotted line) and Nt = 27 (dashed line)]. In all cases, the dissipation is normalized with the estimate based on the turbulent Thorpe scale and the buoyancy frequency [Eq. (18)], with α = 5 and Γ = 1, and the PDF is calculated conditional to the local turbulent APE being larger than a cutoff value, to exclude the largely laminar regions outside the turbulent patch.

  • View in gallery

    Turbulent APE averaged along a vertical transect near the middle of a high aspect ratio overturns as a function of time. The solid line shows the actual averaged over the full ensemble. The stars show obtained using a top-hat filter to extract an approximation to the turbulent Thorpe scale along five randomly chosen transects.

  • View in gallery

    Evolution of an overturning patch generated during the reflection of a semidiurnal internal wave beam (ωb = 1.45 × 10−4 s−1) off a sloping bottom near the equator (f = 0). The red arrows indicate the directions of the incoming and reflected beam. The thick black line indicates the local plumb line. The background stratification N = 8 × 10−4 s−1 is representative of a weakly stratified near-bottom region. The incoming beamwidth is ~200 m, about one wavelength in the across-beam direction. Time increases along the direction of the arrow to the left, each panel showing snapshots spaced 2.5 h. The thin black lines show isopycnals, whereas the color indicates the strength of the local Richardson gradient number Rig. Yellow if Rig > ¼, white if Rig < 0, and with gradation of blue for intermediate values. The letter A in the top panel indicates the patch whose APE and KE is calculated as it evolves in time. Note how, with the exception of very thin layers mostly at the bottom side of the patch, Rig is negative within the patch. The three panels have the same aspect ratio.

  • View in gallery

    The ratio of APE to total energy within the overturned patch indicated with the latter A in Fig. A1. The thick line shows the e-folding scale of Rayleigh–Taylor instabilities for the specific case considered here.

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

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  • 1 Department of Marine Sciences, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina
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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

1. Introduction

Small-scale mixing in the stratified ocean interior plays an important role in regulating the meridional overturning circulation (MOC) (see, e.g., Munk 1966; Munk and Wunsch 1998; Wunsch and Ferrari 2004), though recent evidence suggests that wind-driven upwelling in the Southern Ocean may be just as important as interior mixing in this respect (Toggweiler and Samuels 1998; Wolfe and Cessi 2009, 2010; Lumpkin and Speer 2007). Regardless, many geochemical and biological processes are impacted locally by mixing (Mann and Lazier 2006).

From a theoretical point of view, the problem has been traditionally framed in terms of the rate ϵk at which turbulent kinetic energy (TKE) is lost to friction; the background stratification, characterized by the value of the Brunt–Väisälä (BV) frequency N; and the mixing efficiency, which is intended to quantify the fraction of the overall energy lost to diabatic processes, which goes into irreversibly raising the center of mass of the system. From these quantities, length and time scales characteristic of the turbulence are derived. Vast theoretical, laboratory, and observational efforts have been devoted to provide robust estimates of these quantities in terms of the properties of the large-scale flow that drives the turbulence [see, e.g., Ivey et al. (2008), for a review], though in general a simple parameterization may not be possible (Mater et al. 2013; Mater and Venayagamoorthy 2014). In this note we focus on the use of the Thorpe scale LT to diagnose rates of energy dissipation and mixing in flows that are overall stably stratified.1 The method holds the promise of estimating dissipation from relatively inexpensive vertical measurements of density obtained from CTD casts, free-falling profilers, or even moored sensors and is being widely used (see, e.g., Waterhouse et al. 2014). It has even been applied to atmospheric datasets (Wilson et al. 2010). The method relies on the assumption that LT is a suitable proxy for the Ozmidov scale LO ≡ (ϵk/N3)1/2, which is supported by open-ocean measurements (Dillon 1982; Wesson and Gregg 1994; Moum 1996; Ferron et al. 1998).

In a companion paper, Mater et al. (2015, hereinafter MVSM) compare estimates of dissipation from microstructure profilers with estimates based on the Thorpe scale in field data and find that the latter can overestimate the former, that is, LTLO, when the vertical size of the overturns is large. Increasing the resolution of global circulation models makes them more sensitive to how mixing is parameterized (Jayne 2009), and it is therefore important to critically reassess the theoretical derivation of the standard recipe linking LT to dissipation in order to establish its range of applicability. This is accomplished in this paper by combining theoretical arguments based on the turbulent decomposition of kinetic energy (KE) and available potential energy (APE) of Scotti and White (2014, hereinafter SW) with highly resolved direct numerical simulations (DNS) of stratified mixing. SW’s approach allows the separation of APE into turbulent and mean components and highlights the differences between flows in which the energy for mixing comes from the kinetic energy associated with the mean flow (shear-driven mixing) and flows in which the energy for mixing comes primarily from the available potential energy of the mean field (convective-driven mixing).

While a more precise definition will be given further below, we emphasize that the distinction is based on where the energy for mixing is initially contained in the large-scale field. For example, the classical scenario of mixing driven by the onset of Kelvin–Helmholtz instabilities is an example of shear-driven mixing according to our classification because even though the instabilities may build up APE in coherent structures (the billows), which is eventually transferred to turbulent KE and turbulent APE, the billows are not part of the large-scale flow but rather an instability of the latter. Conversely, in the appendix we provide an example from the reflection of internal wave beams off a sloping bottom where the large-scale flow develops, as part of its own dynamics (i.e., not because of instabilities), regions of elevated APE and little KE. Instabilities will then use a fraction of the APE to drive mixing. Clearly shear- and convective-driven mixing as defined here are idealizations. Any real mixing event in the ocean will combine elements of the two. However, it is important to recognize that they involve different energetics pathways. In particular, the standard recipe that gives turbulent dissipation and mixing in terms of LT is appropriate for shear-driven turbulence. For convective-driven mixing, the use of the standard recipe overestimates the amount of dissipation. The crux of the matter is that the Thorpe scale measures the total amount of APE (mean plus turbulent), whereas for estimating dissipation and mixing it is the amount of APE associated with the turbulent eddies that matters. This is why in flows where the mean field has little or no APE (shear-driven turbulence) the standard recipe works. We show that if the length scale used is suitably redefined to isolate the contribution of the actively turbulent eddies, good agreement can be obtained even in the case of convective-driven mixing, though how to practically extract the component due to the mixing eddies may not be so easy.

MVSM assume that the mixing events in their datasets are shear-driven events and discuss the bias in LT/LO as a result of sampling “young” events. Our analysis highlights a second possibility for biases in LT/LO, related to the nature of the mixing event.

The rest of the paper is organized as follows: Section 2 reviews briefly SW’s theory and presents the theoretical analysis; the main hypotheses to be tested are discussed in section 3. Section 4 describes the details of the numerical simulations and contains a rationale for the choice of flows, followed by section 5, which uses the datasets to verify the assumptions at the heart of the analysis presented in section 2. Finally, in section 6 we summarize the results and discuss possible applications to real ocean data. An appendix discusses a scenario where convective-driven mixing, as defined in this paper, can be expected to occur in the ocean.

2. Theory

a. Dimensional analysis

Dimensional analysis of turbulent mixing is based on the assumption that threedimensional quantities control to leading order the process, namely, the background stratification, as measured by the Brunt–Väisälä frequency N; the background shear S; and the rate of dissipation of turbulent kinetic energy ϵk. Two length scales can be derived from these quantities, the Ozmidov scale
e1
which is usually interpreted as the vertical size of the largest eddies that can overturn (Ozmidov 1965), and the Corrsin scale
e2
which bounds from below the size of eddies that are deformed by shear (Corrsin 1958). The ratio LC/LO = Ri3/4, where Ri ≡ N2/S2 is the gradient Richardson number, indicates that in shear-driven turbulent mixing flows, the Ozmidov scale is the largest of the two. If a suitable proxy for LO can found, and N is known, ϵk can be obtained from Eq. (1).

b. Oceanographic practice

Over the last four decades, beginning with the work of Thorpe (1977), the Thorpe-scale method to estimate LO from CTD casts has gained considerable popularity. Briefly stated, the method starts from a vertical density profile obtained from a CTD cast, a free-falling instrument, or from a set of moored instruments. Under the assumption that turbulent eddies displace water parcels from their undisturbed position, the first step is to reconstruct the undisturbed profile. This is achieved through an algorithm that takes the observed profile ρ(z), which in general will have regions where /dz > 0, and returns a profile such that (i) and (ii) the mass of the particles along the transect whose density lies between two arbitrary values ρ1 and ρ2 is the same in the measured profile and in the undisturbed profile. Practically, is obtained by resorting in descending order the observed densities after a suitable binning. Since is monotonic, we can invert it to obtain , the undisturbed elevation of a parcel of density ρ. Let z be the in situ elevation of a parcel of density ρ. The Thorpe displacement is defined as
e3
Finally, the Thorpe scale LT is defined as the rms value of δT over the profile, though Thorpe points out that the rms should be calculated over many profiles (see Thorpe 2005, his footnote 7). In the latter case, is calculated for each cast to provide an ensemble of Thorpe displacement profiles (more about this below). Field measurements have established that LO and LT are proportional in shear-dominated mixing (Dillon 1982, 1984; Dillon and Park 1987; Wesson and Gregg 1994; Ferron et al. 1998), and LT has by now become the de facto estimate for the Ozmidov scale.
The Thorpe scale is attractive as a diagnostic tool since it can be calculated from vertical density profiles alone. Thus, in the field a simple vertical profile of temperature and salinity (or just temperature, if the TS relationship is known) is needed to estimate ϵk,2 since substituting LT for LO in Eq. (1) above yields
e4
The quantity C is a dimensionless constant assumed O(1), though a priori should depend on LC/LO, and N is the Brunt–Väisälä frequency calculated from the resorted profile. Once ϵk is known, the diffusivity is recovered via the well-known relation
e5
where Γ is the ratio of turbulent buoyancy flux to ϵk, generally assumed to be constant and equal to 0.2 (Osborn 1980). Even though Thorpe (2005, p. 176, note 6) cautions against using it in situations other than purely shear-driven mixing, Eq. (4) has become the standard way to estimate ϵk from CTD casts regardless of the mechanism that sustains turbulence (Alford et al. 2006; Waterhouse et al. 2014). It has also been recently proposed as a way to parameterize turbulent mixing in numerical simulations of large-scale overturns where the resolution employed is not enough to capture the turbulent scales responsible for the actual mixing (Klymak and Legg 2010). MVSM provide strong field evidence that LT/LO can be (much) greater than one in flows characterized by large overturns, in which case Eq. (4) overestimates dissipation and mixing. It is therefore essential to determine the range of applicability of Eq. (4). Unfortunately, a derivation based solely on dimensional analysis is not enough. Equation (1) provides a length scale, for which we have a name, the Ozmidov scale, but dimensional analysis does not tell us how to calculate it without a priori knowledge of ϵk, an estimate of which of course is what we are seeking. In the next section, we provide an alternative derivation based on energetics. Its point of departure can be traced to the original work of Thorpe (1977), as expanded by Dillon (1984) and Dillon and Park (1987).

c. An alternative approach based on energetics

Dillon’s seminal analysis relied on a number of assumptions, which were in part due to the limitations of the observational datasets available and in part due to the still early stage of development of the APE concept, which, since then, has been developed into a powerful tool to diagnose stratified flows [for a recent review, see Tailleux (2013), and references therein]. Particularly relevant to the problem at hand is the work recently published by SW, in which the authors show that just as the kinetic energy of a turbulent flow can be divided into a mean and a turbulent component, so can the APE. Such a split is possible because we can define the APE of a single parcel of fluid (though still dependent on a reference state) and then apply the machinery of Reynolds decomposition. Thus, we have now at our disposal a much more precise diagnostic tool. Further, direct numerical simulation, in its early infancy in the 1980s, provides datasets where we can test hypothesis without relying on ad hoc assumptions.

1) Reynolds decomposition of kinetic and potential energy: Shear- and convective-driven mixing

The basic elements of the analysis contained in SW are presented here. For details, the reader is referred to the original publication. To fix the notation, let f be a generic field. We denote its Reynolds decomposition , where the overline represents the averaged component and the prime represents the turbulent fluctuation. In turbulence theory, the standard averaging operator is the ensemble operator, though in practice we will consider a suitable proxy, such as averaging in directions along which the flow is homogenous. Further, let Sij be the strain rate tensor. In analogy, we define ΘI ≡ ∂ib, with bg(ρρ0)/ρ0, where ρ0 is a reference density. Just as for the globally defined APE, we need to introduce a reference state, analogous to used in the definition of the Thorpe displacement. This is achieved by means of an isochoric rearrangement of the field b(x, t) to yield a field , such that (Winters et al. 1995; Tseng and Ferziger 2001). Note that the reference state is in principle a function of time, since mixing will alter the density distribution of the fluid over time. The connection between and (aside from the obvious rescaling) will be discussed later. We are now in a position to define the APE of a parcel of fluid at location x = (x, y, z) and time t as
e6
Note how the integrand mirrors the definition of the Thorpe displacement given earlier, since is the height of a fluid parcel of reduced gravity b in the isochorically resorted density field. Also, from now on we will drop the parametric dependence on t for clarity. Like any other field variable, the APE can be Reynolds decomposed, and appropriate transport equations can be written for its mean and turbulent components, just as transport equations can be written for the mean and turbulent component of the KE [see, e.g., Tennekes and Lumley 1972, their Eq. (3.2.1)]. Figure 1 provides a schematic representation of how the total energy within a given volume (which can be infinitesimal, given the locality of the theory) is apportioned between the mean and turbulent KE and APE reservoirs. Each quantity is by definition positive and is connected to the other reservoirs by conversion terms. The values and are the shear production term and dissipation term, respectively, unmodified from the case of unstratified turbulence. The values (turbulent diapycnal flux) and (turbulent APE dissipation rate or mixing rate) are their respective counterpart for APE, κ being the diffusivity of the stratifying agent and . Finally, the turbulent buoyancy flux measures the exchange between turbulent KE and mean APE and the mean buoyancy flux , with , quantifies the exchange between mean KE and mean APE. Fluxes (diabatic and/or material), not shown in the diagram, complete the balance. They are not important for the foregoing discussion.
Fig. 1.
Fig. 1.

Schematic diagram portraying the relationship between the reservoirs of mean and turbulent (left) KE and (right) APE (adapted from SW).

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

The turbulent KE and APE reservoirs are directly connected to mixing, whence it follows that quantification of mixing should depend on the amount of energy contained in these reservoirs and a suitably defined residence time. Energy is injected by the mean field into turbulence, where it is dissipated either as heat via ϵk or goes into raising the center of mass of the isochorically resorted system via ϵp.

In principle, the evolution of a flow involves nontrivial exchanges among all reservoirs. Two idealized limits can be considered: One is to assume that on a scale large with respect to turbulent scales, the energy of the mean flow is present as KE. In this limiting case, we further assume that the dynamics of the large-scale flow (i.e., discounting the nonlinear evolution of the instabilities that lead to turbulence) do not transfer energy to APE. Under such conditions, the mean field possesses mean KE that is transferred via ϕk into turbulence. Figure 2 shows the resulting simplified flow of energy. There will be turbulent APE generated as a result of the instabilities, but generally the energy flow is out of mean KE into turbulence. Under the standard assumptions of quasi equilibrium between production and dissipation (Osborn 1980), is also equal to the mixing term ϵp, and the efficiency
e7
is related to the flux Richardson number
e8
via the well-known relation
e9
We call this the shear-driven mixing case. A well-studied example is the case of shear leading to turbulence via the Kelvin–Helmholtz route to instability (Smyth et al. 2001).
Fig. 2.
Fig. 2.

Primary energy pathways in a shear-driven mixing event. The mean flow provides mean KE, and the shear production term transfers energy to the turbulent KE. From the latter, a fraction equal to Γ/(1 + Γ) is transferred by the buoyancy flux to turbulent APE for mixing, while the dissipation term removes the rest.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

At the opposite end of the spectrum, we consider flows whose large-scale dynamics creates statically unstable localized regions with little or no shear. That is, the energy of the large-scale flow is contained in the mean APE reservoir. Within the unstable region, turbulence derives its energy from the mean APE reservoir, and the energetics in this case is shown in Fig. 3. Note that in this case, the turbulent buoyancy flux transfers energy from the mean APE to turbulent KE, and thus it is negative. Also, since we assume Rif ≪ −1, is a measure of turbulent KE dissipation. We still allow the mean APE to exchange energy with the mean KE field. This for two reasons: First, the large-scale dynamics that leads to the development of these regions may involve exchange between the mean APE and mean KE; and nonlinearities may drain energy from the mean APE adiabatically, for example, via the generation of higher-frequency internal waves that are still considered part of the mean field in the sense that their evolution is assumed to be identical for all the members of the turbulent ensemble. We call this case convective-driven mixing. In the appendix, we show that internal tide beams reflecting a sloping bottom under appropriate circumstances can generate regions that approximate the conditions of convective-driven mixing.

Fig. 3.
Fig. 3.

Primary energy pathways in a convective-driven mixing event. The mean flow provides mean APE. From this reservoir, energy flows to the turbulent KE reservoir via the turbulent buoyancy flux term (which has thus opposite sign relative to the shear-driven case), the turbulent APE reservoir via the turbulent diapycnal flux, and the mean KE via the mean buoyancy flux. Unlike the first two, the last term is reversible.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

According to these definitions, shear- and convective-driven mixing represent opposite ends of the spectrum of conditions under which mixing occurs in the ocean. While any real mixing event will likely encompass both shear and convective features, it provides a useful classification. In particular, we will show that Eq. (4) provides a valid estimate of dissipation for shear-driven mixing [i.e., LT/LO = O(1)] but likely overestimates mixing for convective-driven mixing (LT/LO ≫ 1).

2) A displacement-based estimate of mean and turbulent APE

From its definition, with a simple change of variables we can expand the total APE as a power series of lT:
e10
so that to lowest order
e11
Simple algebra [see Eq. (2.29)–(2.31) in SW] shows that
e12
so that replacing with in Eq. (11) results in an error . More specifically, the approximations involved in this analysis break down if the range of variation of b in the volume under consideration straddles a region of rapid change in the derivative of the reference profile. But in this case, it is no more legitimate to treat lT as a small quantity. Indeed, the precise formulation of the small parameter is
e13
and the APE averaged over the parcels in the volume under consideration is
e14
(from now on we drop the and the explicit reference to in N2 for simplicity). In passing, if the resorted field is linear, then Eq. (11) is exact. Indeed, μ measures the departure of the resorted profile from linearity over the range of vertical motions associated with the flow. Further, we can Reynolds decompose the field lT(z, b) to obtain the mean , and turbulent displacements, in terms of which
e15
We leave it to the reader to verify that to the same degree of approximation, and defined above coincide with the mean and turbulent APE defined in SW.

3) From displacements to dissipation

Next, we define the time scales of turbulent KE and APE dissipation as and , where indicate turbulent KE and APE and ϵp is the turbulent APE dissipation rate (Fig. 1). We assume that
e16
where at this point α is an as yet unspecified nondimensional quantity.3 The equality of the dissipation times is based on the assumption that the loss of KE and APE is due to the action of the same eddies. Note that this does not imply that the dissipation rates are the same. However, it does imply that the ratio of turbulent APE dissipation rate to turbulent KE dissipation rate equals the ratio of turbulent APE to turbulent KE
e17
Combining Eqs. (15) with (17), we obtain
e18
Finally, the turbulent diffusivity becomes
e19
which does not depend explicitly on Γ, though α may still depend on it.

3. From theory to practice

In the preceding section we have derived an estimate for the dissipation within a turbulent stratified flow [Eq. (18)] that rests on two assumptions:

  • Hypothesis A1: The term μ is small or equivalently Eq. (11) is a good approximation for the APE.
  • Hypothesis A2: The dissipative time scales for turbulent KE and turbulent APE are equal.
The first goal of the paper will be to test how accurate these assumptions are as a function of the large-scale conditions that can sustain mixing in an overall stably stratified fluid.

The second goal is to investigate when the formula for the dissipation in terms of the Thorpe scale Eq. (4) provides a reasonable estimate of the actual dissipation. MVSM frame this last problem in terms of the ratio LT/LO, which should be close to unity if Eq. (4) describes dissipation accurately. Here, in light of the theory developed in the preceding section, assuming for the time being that Eq. (18) is correct, we see that Eqs. (4) and (18) are equal if the following are met:

  • Hypothesis B1: We identify the constant C with (2αΓ)−1.
  • Hypothesis B2: Calculating the reference state by resorting the density profile given by each vertical cast as is oceanographic practice is statistically equivalent to using the isochorically restratified profile as defined in the APE theory.
  • Hypothesis B3: The Thorpe scale LT coincides with the turbulent displacement .
The dependence of 2αΓ on how turbulence is sustained will be addressed with the help of the numerical experiments described in the next sections. To be valid, estimates of dissipation based on the Thorpe scale require 2αΓ to be at most a weak function of the parameters of the flow.

The second condition is necessary to ensure that the BV frequencies used in Eqs. (4) and (18) agree. In the numerical calculations, we have access to the full b field, and the domain contains all of the relevant scales. Therefore, the isochoric resorting provides the “true” background state (Winters et al. 1995). The situation when dealing with field measurements is different. For obvious practical reasons multiple vertical casts cannot be guaranteed to sample the same statistical ensemble since large-scale processes (e.g., internal tides) can change the background stratification between one cast and the next. Thus, using multiple casts to determine a single reference state would alias processes (e.g., displacements due to internal tides) that do not have anything to do with the turbulence responsible for the small-scale inversions, resulting in a biased estimate of the Thorpe scale. Rather, each cast must be seen as traversing a separate realization out of an ensemble of random processes, each characterized by a potentially different background state that can be approximated by resorting the vertical density profile measured during the cast (Dillon 1984). This hypothesis can be examined with the aid of the numerical datasets comparing the PDF obtained from the isochoric restratification with the PDF obtained from n vertical casts, as a function of n.

The second condition is also a necessary condition for the third condition to apply but not sufficient. To appreciate this, assume for the time being that B2 applies, whence it follows that statistically δT and lT are equivalent (assuming that a suitably large ensemble for δT exists), and as such
e20
with the equality holding if and only if . This occurs in the special case in which the isochorically restratified density field is very close to the averaged density field, and all of the APE is contained in the turbulent component. Note that in this case there is no mean APE. The energy to drive mixing comes entirely from the mean KE reservoir via shear production, what we have called shear-driven mixing. More generally, within the SW framework, in the limit of small μ and assuming statistical equivalence between δT and lT, the Thorpe scale measures the total APE in the region of interest. However, what we need is a measure of the fraction of APE of the eddies directly engaged in the turbulent process, that is the turbulent APE. As the rest of the paper will show, in mixing events that are closer to being convective than shear driven, , and Eq. (4) overestimates dissipation.

4. Numerical simulations

We solve the Navier–Stokes equations in the Boussinesq approximation (DNS), assuming a linear dependency between the reduced gravity bg(ρρ0)/ρ0, where g is the local gravity, ρ is the density, and ρ0 is the reference density, and the concentration of the stratifying agent
e21
e22
e23
Here, u is the velocity, and P is the pressure. The nondimensional form of the equations depends on five parameters. Three are dynamical, namely, the Reynolds number Re, which measures the strength of diabatic processes; the bulk Richardson number Ri, which couples buoyancy to momentum; and the ratio of viscosity to diffusivity Pr. In addition, the geometry of the domain and possibly the initial conditions introduce two additional aspect ratios.

The equations are solved on a staggered grid, whereby the fluxes are computed at the corresponding cell edges, and scalar quantities (pressure and reduced gravity) are calculated at cell centers. The equations are solved in time using a projection method. Convergence is checked ensuring that the spectra of velocity and density increments capture the peak and rolloff. Details can found in Scotti (2008).

a. Choice of flows

When considering numerical simulations, a choice out of necessity must be made regarding what scales can be realistically simulated as opposed to what scales need to be idealized or modeled. In this paper, we focus on the scales that experience turbulent motions, whose dynamics we want to capture realistically. To achieve a meaningful scale separation between the Ozmidov and the Kolmogorov scales, the number of grid points becomes quite elevated, reaching close to one billion nodes in one of the simulations (see Table 1). Yet, despite such a large number of grid points, we are still forced to consider very idealized driving conditions, detailed below, which are usually nowhere to be found in the real ocean. As a justification, we offer the following argument.

Table 1.

DNS parameters. The simulations are characterized by three nondimensional dynamical parameters: Pr is the ratio of viscosity to diffusivity of the stratifying agent; a bulk Richardson number Ri that measures how strongly buoyancy is coupled to the vertical momentum; and a Reynolds number Re that controls the strength of the viscous term. In the Couette flows they are based on the velocity and buoyancy difference across the channel and the channel height giving Re ≡ ΔUH/ν and Ri ≡ ΔbHU2. In the convective case, we use N−1 as a time scale, and the thickness of the overturned layer δo as a length scale. Then , while Ri = 1. In the isotropic case, the horizontal extension of the overturn is equal to its vertical extent, while in the anisotropic case it is 10 times larger. In shear-driven flows, the turbulent Prandtl number approaches unity in the mixing layer where the focus of the analysis resides. Setting Pr = 1 makes the thin viscous (for momentum) and diffusional (for buoyancy) layers near the upper and lower walls equal, which economizes numerical resources. In the convective case, we do not have to deal with boundary layers, and we can use a more realistic value of Pr. The aspect ratio, defined only for the convective cases, is the ratio of horizontal to vertical size of the overturned region.

Table 1.

Any parameterization of turbulent processes, of which Eq. (18) is one, relies on the assumption that, in flows that share similar, though not necessarily identical, driving mechanisms, a degree of universality exists on the scales experiencing turbulence. It is of course possible to reject such an assumption, but since in the real ocean no two mixing events are ever identical, then no parameterization would be possible. Conversely, if we are willing to concede that the parameterization of turbulent processes is possible, the relevant question shifts from “does flow A exist in the ocean?” to “does flow A capture the essential features of a class of turbulent flows that exists in the ocean ?” The latter is the question we considered when choosing what flows to simulate.

b. Shear-driven mixing

As an example of highly idealized shear-driven mixing, we consider the flow in a channel driven by a velocity differential applied at the lower and upper boundary (stratified Couette flow). Periodic conditions are imposed on the horizontal directions, and Dirichlet conditions are imposed on velocity and reduced gravity at the upper and lower boundary. In this configuration Ri = ΔBHU2 and Re = HΔU/ν, where ΔB and ΔU are the reduced-gravity and velocity differences imposed at the boundaries, H is the height of the channel and ν is the viscosity. This type of flow must be considered as the simplest case of shear-driven mixing that can be maintained in steady state. The purpose is to test the theoretical prediction of the previous section, for which we need a flow that is turbulent, stratified, and in which a reasonable separation exists between the Ozmidov scale and the Kolmogorov scale on one hand and the Ozmidov scale and the boundaries on the other, while at the same time μ is small. Stratified Couette flows with the parameters considered here fit these requirements. We could have considered the case of a time-evolving mixing layer in which Kelvin–Helmholtz instabilities drive turbulence (Smyth et al. 2001). However, a Couette flow has several advantages: The flow is maintained at steady state by the injection of momentum, buoyancy, and energy at the boundaries, so we do not have to worry with external time scales imposed by the large-scale evolution of the flow. Moreover, unlike a time-evolving mixing layer, in stratified Couette flows we can separately vary the ratio L0/ηk, where ηk is the Kolmogorov scale and the ratio Γ, which is roughly proportional to Ri (Scotti and White 2014, manuscript submitted to J. Fluid Mech.), which allows us to test the scaling ideas over a wider range of parameters. Finally, since the flow is homogeneous along the horizontal directions and in time, averages along the x and y directions and time t are used as a proxy for ensemble averages, resulting in very robust statistical estimates. Table 1 indicates the parameters of these simulations. The buoyancy Reynolds number Reb ≡ (LO/ηk)4/3 of the simulations are representative of values encountered in the field (Smyth et al. 2001).

c. Convective-driven mixing

As an example of convective-driven mixing, we consider a uniformly stratified fluid in which at t = 0 an isolated patch of unstably stratified fluid is released. The patch is obtained by exchanging parcels of fluids across the midplane near the center of the domain. Practically, this means that vertical resorting yields very close estimates to the actual reference state. The fluid is enclosed by no-slip boundaries at the top and bottom and is otherwise periodic in the remaining directions. A small-amplitude sinusoidal perturbation with a wavelength equal to the spanwise length of the domain is introduced along the y direction to ensure that the flow becomes three-dimensional. Figure 4 shows the initial condition in reduced gravity, the reduced-gravity profile at t = 0 along a vertical transect across the middle of the patch, and the reference reduced-gravity profile obtained by an isochoric restratification, which, as mentioned earlier, coincides with the vertically resorted profile by construction. In an unbounded domain, energy can radiate away as internal waves. This is prevented in our simulations by the solid boundaries at the top and bottom. Rather, we observe the development of a pattern of standing waves.

Fig. 4.
Fig. 4.

(left) Reduced-gravity distribution at t = 0 in the high aspect ratio case. Note how the colors are saturated to highlight the overturned region. (right) Reduced-gravity profile along a vertical transect across the overturned region (solid line) and the isochorically restratified profile (dashed line). The displacement LT is obtained by measuring the distance between the solid and dashed line along the vertical.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

In these flows, we use N and the vertical scale h of the overturn to nondimensionalize the problem, given Ri = 1 and Re = h2N/ν = 4000. This value is not far from what can be expected in overturned regions generated by internal tide beams reflecting off a sloping bottom in the weakly stratified ocean interior (see the appendix). MVSM interpret as the ratio of the Thorpe scale to the diffusive time scale over a buoyancy period ( in MVSM) and find that even in the most energetic dataset conducted during the Internal Waves in Straits Experiment (IWISE) experiment, the Reynolds number in a significant fraction of the observation does not exceed 10 000.

The aspect ratio of the overturned patch Lh/h measures the horizontal extent of the patch. Here, we consider two cases: a high aspect ratio patch (Lh/h = 10), which we take as being representative of “oceanic” conditions, and an isotropic patch (Lh/h = 1).

This flow represents a highly idealized case of convective-driven mixing. We choose it because it represents a flow in which the energy for mixing is provided initially by mean APE. As the flow evolves, its energetics involve in a nontrivial way the four energy reservoirs in the form of a large-scale, largely adiabatic exchange between mean APE and mean KE due to the standing waves and a turbulent episode involving the transfer of energy to turbulent APE and turbulent KE. Note that in a more realistic setting, we would expect that mean APE and mean KE would instead be associated with radiating internal waves. In the appendix we provide an example of oceanic significance (reflection of internal tide beams of a sloping topography) that shows how thin, near-horizontal regions can develop where the stratification is locally unstable with little shear present. Moreover, we show that under the weakly stratified conditions often encountered below the main thermocline, the time scale of convective instabilities is long enough for the large-scale flow to grow the overturned region to finite size before we expect turbulence and mixing to substantially alter the flow.

Since the patch is localized in the xz plane, and evolving in time, averages along the y direction are used as a proxy for the ensemble average.

5. Results

a. Shear-driven flows

1) Thorpe displacement versus

We discuss here the flow with the highest value of Reb, but very similar results apply for smaller values of Reb as well. Figure 5a shows the reduced-gravity profile along a randomly chosen vertical transect and, for comparison, both the isochorically resorted profile and the reference profile obtained by resorting the vertical profile shown. Notable is the fact that in the profile chosen tends to track the reduced-gravity profile closely in regions where the latter is monotonically decreasing (e.g., in the regions 0.1 < z < 0.2 and near z = 0.6), resulting in low Thorpe displacement in these regions (Fig. 5b). We have calculated δT from 1000 randomly chosen vertical casts, using the profile in each cast to calculate a reference state, according to the oceanographic practice outline earlier. If we compare the PDF of δT versus the PDF of lT, we note that the PDF of δT reasonably follows the PDF of lT even when only a relatively few casts are included, though the agreement obviously improves with the number of casts because of the departure from Gaussianity along the tails of the distribution. However, there is a persistent bias favoring displacements smaller than 0.05 [Fig. 6(left)]. If we now calculate for each cast LT, its PDF peaks at about 80% of the rms value of lT [Fig. 6 (right)]. Considering the asymmetric shape of the PDF of LT, the expected value of LT is close to 90% of the value calculated from the actual reference state, obtained from an isochoric restratification. While not perfect, the agreement is certainly very good, and therefore we can conclude that the statistics derived using are comparable to the statistics obtained with .

Fig. 5.
Fig. 5.

Representative profiles from a stratified Couette flow. (a) Vertical profiles of isochorically restratified reduced gravity (, circles), randomly chosen vertical profile of reduced gravity b (solid line), and reference state calculated from the randomly chosen b profile (dashed–dotted line). (b) Vertical profile of the absolute value of the Thorpe displacement calculated from the random reduced-gravity profile shown in (a) (solid line) and rms value of lT (dashed line). (c) The full turbulent APE (solid line), quadratic limit [Eq. (11)] (open circles), and turbulent KE (dashed line). In all cases, Ri = 0.03 and Re =110 000.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

Fig. 6.
Fig. 6.

Comparing displacements calculated from 1D vertical casts and using the fully three-dimensional b field. (left) PDF of |δT|, calculated using 1 (squares), 10 (circles), 100 (crosses), and 1000 (stars) randomly chosen vertical casts. For reference, the PDF of the displacements |lT| calculated from the entire dataset is shown as a solid line. (right) PDF of Thorpe scale LT obtained from 1000 individual casts. Each cast yields a value for LT. Values are normalized with the mean value of lT. Data from stably stratified Couette flow at Ri = 0.03 and Re =110 000.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

2) Dissipation time scale and turbulent KE to turbulent APE ratios

As expected from our analysis, away from the viscous layers near the boundaries where N2 varies rapidly and μ is not small, the mean turbulent APE profile (Fig. 5c) is very well approximated using , which under the present circumstances coincides with the rms value of lT. This validates hypothesis A1. Because the flow is at steady state, . Considering next the relative amount of energy in each reservoir, we find that does indeed match Γ [Fig. 7 (top)], as expected under the hypothesis that the dissipative time τ is the same for KE and APE (A2). Further, Γ−1/2 appears to be constant to first order [Fig. 7 (bottom)], and we find that α ≃ 4.5Γ1/2. It is oceanographic practice to assume a value close to 1/5 for Γ; hence, (2αΓ)−1 = 1.2, slightly larger than the range of values reported in the literature (0.6–0.9) (Dillon 1982; Ferron et al. 1998), though since α tends to increase somewhat with the buoyancy Reynolds number, the discrepancy is not significant.

Fig. 7.
Fig. 7.

(top) Ratio of turbulent APE to turbulent KE as a function of the ratio Γ ≡ ϵp/ϵk in steady, shear-driven flows; (bottom) normalized dissipation time scale Γ−1/2 as a function of Γ.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

In the core of the flow, the dissipation obeys a lognormal distribution. When normalized with the value given by Eq. (18), the PDFs collapse nicely. The peak of the distribution appears to be slightly less than what is predicted by Eq. (18), but overall the dissipation is well represented (Fig. 8).

Fig. 8.
Fig. 8.

PDF of turbulent KE dissipation in steady shear-driven flows normalized with the estimate based on the Thorpe scale and dissipation time [Eq. (18)]: Re = 57 000 (solid line), Re = 72 000 (dashed line), Re = 112 000 (dashed–dotted line), Re = 220 000 (stars), and Re = 110 000 (squares). On average, Eq. (18) overestimates the mean dissipation in the core layer by less than a factor 2 across all experiments.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

Note that as the background stratification weakens, the rms value of lT increases, but it never exceeds more than 15% of the channel depth. Thus, we can confidently assume that the physical boundaries do not artificially constrain the size of the largest mixing eddies.

Overall, in shear-driven mixing flows the conditions under which the quadratic approximation to APE holds and turbulent KE and turbulent APE have the same dissipative time scale are satisfied (hypothesis A1–A2). Likewise, (2αΓ)−1 is found to be close to unity (hypothesis B1) and individually resorted profiles are statistically close to the actual reference state (hypothesis B2). Hypothesis B3 is satisfied by definition in shear-driven mixing.

b. Convective-driven mixing

In the setup chosen, the initial buoyancy distribution is obtained from a reference state by exchanging particles across the horizontal midplane. Thus, by construction the reference state obtained via a isochoric restratification coincides with the state obtained by resorting a vertical profile, no matter where the latter is chosen. Hence, hypothesis B2 is trivially verified. The subsequent development of turbulence may invalidate locally the symmetry along the midplane, but we find that the differences are small. Thus, as in the shear-driven case, we can assume that for statistical purposes and are equivalent. It also means that the Thorpe scale satisfies
e24

The energetics of the adjustment process of a localized region where the density field has been overturned involves a nontrivial interaction between the four reservoirs (Fig. 3). The mean APE and KE exchange energy as standing waves, characterized by a time scale that is O[tan(A) N−1], where A is the aspect ratio of the overturned patch (Fig. 9a). There is also a single turbulent episode, which occurs near the beginning of the simulation (tN = 20; see Fig. 10) where part of the energy initially contained in the mean APE is transferred to the turbulent APE and turbulent KE. The time scale of the turbulent episode is O(N−1). The turbulent episode shows up clearly when we consider the evolution of the background potential energy (BPE), defined as the difference between the potential energy bz and the APE defined in Eq. (6). Before turbulence sets in, and after the turbulence disappears, the BPE is constant [once the trivial diffusive component given by is subtracted], increasing only during the turbulent phase (Fig. 9a). The ratio of the BPE gain ΔEBPE to the initial amount of energy contained in the system (all in the APE reservoir at t = 0) measures the overall mixing efficiency of the process, that is, how much of the available energy is lost to irreversibly raising the center of mass of the system. For both aspect ratios, we find an efficiency slightly larger than 10%. Note that while in shear-driven mixing this definition of mixing efficiency coincides with the flux Richardson number, it does not in the present case. In addition to losing energy to turbulent APE and from there to mixing, mean APE loses energy to turbulent KE and dissipation at a rate equal or slightly larger in the high aspect ratio and somewhat smaller (but comparable) in the isotropic case (Fig. 9b). Therefore, roughly three-quarters of the APE initially contained in the overturned region remain available at the end of the turbulent episode. The substantial equality between energy lost to mixing and energy lost to turbulent kinetic energy dissipation is also found in Rayleigh–Benard convection (SW) and experiments of mixing induced by Rayleigh–Taylor (RT) instabilities in an otherwise stratified medium (Lawrie and Dalziel 2011, their Fig. 8); hence, Γ ≃ 1. It can be explained by noticing that the rate at which mean APE is fed into turbulent KE is (a positive value indicates energy flowing to turbulent KE), whereas the rate at which mean APE flows to turbulent APE is , where N2 is the BV frequency of the resorted field (positive by definition) (Fig. 1). Within the overturn, the gradient of the mean reduced gravity is directed upward and the magnitude ; hence, the two fluxes are similar.

Fig. 9.
Fig. 9.

(a) Evolution of mean APE (solid line), mean KE (dashed line), and BPE (dashed–dotted line) as a function of time in overturning flows. Energies are normalized so that the APE at t = 0 is one. Time is normalized with Ntan(A). (b) Mixing and dissipation. BPE evolution [normalized as in (a)] during the turbulent phase: high aspect ratio (solid line) and isotropic overturn (circles); cumulative turbulent KE dissipation: high aspect ratio (dashed line) and isotropic overturn (squares). Time normalized with N. For the isotropic case, time is shifted by 10 buoyancy periods.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

Fig. 10.
Fig. 10.

(a) Evolution of turbulent APE (solid line) and KE (dashed line) in a high aspect ratio overturning flow. The energies are normalized with the amount of mean APE at t = 0, time with N; (b) dissipative time scale for turbulent APE (solid line) and KE (dashed line) during the same period.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

c. Time scales

As before, we define the dissipative time scale during the turbulent episode as the ratio of turbulent APE or turbulent KE to the corresponding dissipation rates. Figure 10b shows the evolution of the respective ratios as a function of time. It remains remarkably constant for the kinetic energy, whereas when calculated with the APE it is initially larger but within the same order of magnitude and eventually settles to a constant value during the decaying phase. Regardless, the two values are reasonably close so that we can consider a single value equal to 0.8 buoyancy periods as representative. This value is also consistent with the overall duration of the turbulent event, about 1.6 buoyancy periods.4 It is also largely independent of the initial aspect ratio of the overturn, unlike the standing wave period, which depends on the aspect ratio. This is consistent with the notion that the dissipative time scale is set by the physics of the local instabilities.

As is the case for steady-state shear-driven mixing, the energy partition ratio mirrors the ratio of the corresponding fluxes, in this case close to unity. This can be seen clearly in the time evolution of turbulent KE and APE (Fig. 10a).

Snapshots of and are shown in Fig. 11 at different times during the turbulent episode. The largest values of are found at the leading edge of the flattening overturned region as it spreads outward, and they are substantially smaller than . Consistent with the theory developed earlier, estimates of mean and turbulent APE based on the mean and turbulent components of lT scale as the mean and turbulent APE calculated from the DNS (Fig. 12a). Likewise, the dissipation during the turbulent episode estimated with Eq. (18), assuming Γ = 1 (i.e., equipartition of the energy lost to dissipation and mixing) and α = 5, agrees very well at all times, whereas estimating the dissipation using the standard formula based on the Thorpe scale [Eq. (4)] consistently overestimates the actual dissipation (Fig. 12b). Thus, in this flows hypotheses A1 and A2 are satisfied, but neither B1, since (2αΓ) ≃ 10−1, nor B3.

Fig. 11.
Fig. 11.

(left) Mean and (right) turbulent Thorpe scale at three times during the turbulent episode in a high aspect ratio overturn. The overturned region is originally centered around the origin. As time evolves, it spreads radially outward, and turbulence develops primarily along the front.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

Fig. 12.
Fig. 12.

Time evolution of energy, dissipation, and turbulent scales within the turbulent core of a high aspect ratio overturn. (a) Mean APE (solid line) and turbulent APE (dashed line) with the corresponding estimates based on the mean (crosses) and turbulent (circles) component of the displacement. All energies are normalized with the initial amount of total APE. The turbulent APE is further multiplied by 10. (b) Dissipation (solid line) and estimates of dissipation based on the theory presented in this paper [Eq. (18)] with α = 5 and Γ = 1 (circles) and the Thorpe scale following the standard recipe [Eq. (4)]. All dissipations are normalized with the peak of the actual dissipation. (c) Ratio of turbulent displacement to Ozmidov scale (solid line) and Thorpe scale to Ozmidov scale (dashed–dotted line).

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

Given the disparity in magnitude between the mean and turbulent displacements, the need to base the estimation of dissipation on the turbulent component of lT becomes apparent. Using Eq. (18) with α = 5 and Γ = 1 to normalize the PDF of turbulent KE dissipation, we obtain a good collapse of the PDFs (Fig. 13), with the peak being reasonably close to unity.

Fig. 13.
Fig. 13.

PDFs of dissipation within the turbulent core of a high aspect ratio overturn at different times: when the turbulent KE attains its maximum (Nt = 23, solid line) and at two later times [Nt = 25 (dashed–dotted line) and Nt = 27 (dashed line)]. In all cases, the dissipation is normalized with the estimate based on the turbulent Thorpe scale and the buoyancy frequency [Eq. (18)], with α = 5 and Γ = 1, and the PDF is calculated conditional to the local turbulent APE being larger than a cutoff value, to exclude the largely laminar regions outside the turbulent patch.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

Likewise, the ratio of the Thorpe to Ozmidov scales in these flows is always larger and at times much larger than one. However, we must bear in mind that this ratio cannot be interpreted as an “age” of the event, as is the case for the shear-driven instability. In the latter case, the instability initially transfers energy to APE, but until secondary instabilities energize the dissipative scales, dissipation is small. Thus, LT/LO is large at the beginning and decreases with time as turbulent dissipation increases, while the APE frontloaded at the beginning is quickly lost. Here, LT is a measure of the total APE and not of the APE that is part of the turbulent processes. Thus, the ratio LT/LO, while always greater than one, does not decrease monotonically during the event. It decreases initially, mostly because as turbulence develops LO increases, but as the turbulent event comes to an end, LT/LO increases (Fig. 12c). Thus, the mixing context matters when interpreting the ratio LT/LO.

6. Discussion

In this paper, we have revisited the use of the Thorpe scale to diagnose mixing and turbulent dissipation in turbulent-stratified flows. Specifically, we have used an energetics framework to investigate under what conditions the widely used formula that relates the Thorpe scale LT to dissipation ϵk and stratification N,
e25
is expected to be valid.

Traditionally, the justification for using the Thorpe scale to estimate dissipation lies in the assumption that it approximates the Ozmidov scale. In this note, we show that an energetics argument can be used to derive an estimate of turbulent KE dissipation and mixing using a suitably defined length scale and the background value of the BV frequency. The theory assumes the knowledge of the true reference state in order to calculate how far a parcel is displaced from its rest position lT. In oceanographic practice, lT is approximated by the Thorpe displacement δT, in which the rest state is calculated by resorting the vertical density cast along a profile. We show that as long as sufficient, many casts are included, statistically speaking δTlT.

The alternative approach is based on the following assumptions:

  1. The displacement are small relative to the vertical scale of variation of N2.
  2. The dissipation time scale for kinetic energy and available potential energy are the same.
  3. The dissipation time scales as N−1.
  4. The partition ratio of turbulent APE dissipation to turbulent KE dissipation is known and, because of assumption 2, equals the ratio of turbulent APE to turbulent KE.
Under these conditions, we derive an expression for the turbulent KE dissipation that superficially resembles the one traditionally employed (after all, we are still subject to the same dimensional constraints), the substantive difference being how the length scale is calculated.

While in the standard formula, the length scale is usually defined as the rms value of the Thorpe displacement δT (i.e., the Thorpe scale); in our approach, we rely on a Reynolds decomposition of the displacement lT. For a shear-driven flow, the Thorpe scale coincides with the rms value of lT since the mean value of lT, that is, the component due to the mean flow, is zero. Thus, LT/LO = O(1), which validates the standard oceanographic practice.

However, for mixing driven by a large-scale buildup of APE resulting in statically unstable regions (convective-driven mixing), the mean component of lT can be much larger than the turbulent component. Since lT and δT are statistically equivalent, the Thorpe scale is dominated by the mean component of lT, and following the standard oceanographic practice overestimates the amount of mixing, that is, LTLO. It would amount to assuming that the entire amount of APE in the statically unstable region is dissipated locally within a few buoyancy periods. However, the large-scale dynamics must still retain a degree of control on the mean APE, part of which may be transferred back to the mean KE. An incorrect estimation of dissipation and mixing can skew important parameters, such as the conversion rate from barotropic to baroclinic energy at topographic features (Alford et al. 2011).

Diagnostic versus prognostic use of the Thorpe scale

The need to extract the component due to turbulence from profiles of Thorpe displacement poses different problems for a diagnostic use (i.e., estimate dissipation and mixing from field data) as opposed to a prognostic use (e.g., as a parameterization of mixing). In the former case, what is necessary is to accumulate a sufficiently large number of data points in order to be able to statistically separate the mean from the turbulent component. Ideally, as we have shown, multiple casts should be used to create the statistical ensemble. However, in practice this may not be feasible, since the large-scale flow may evolve on a time scale comparable to the time it takes to obtain a single profile so that two consecutive casts may yield realizations from distinct ensembles. A possible solution would be to use a low-pass filter along the vertical direction to separate the mean from the turbulent component. We tested this idea on the high aspect ratio convective mixing dataset. At any given time, a single random vertical profile is chosen within the turbulent patch, and the Thorpe displacement δT is calculated in the usual way. To define the approximated mean Thorpe scale, δT is convolved with a top-hat filter (z) of width equal to δ0/5, where δ0 is the initial thickness of the overturned region:
e26
The factor 5 was chosen based on a visual inspection of the data to separate the turbulent fluctuations from the large-scale overturn. From field data, it would be necessary to determine if a scale separation exists between the large vertical displacement associated with the mean field and the small-scale displacements due to turbulence and select the width of the filter accordingly. We can define an approximated turbulent APE as
e27
When integrated along the transect, the approximated turbulent APE compares favorably with the actual turbulent APE once the turbulent patch develops (Fig. 14), especially considering the typical scatter involved in field data.
Fig. 14.
Fig. 14.

Turbulent APE averaged along a vertical transect near the middle of a high aspect ratio overturns as a function of time. The solid line shows the actual averaged over the full ensemble. The stars show obtained using a top-hat filter to extract an approximation to the turbulent Thorpe scale along five randomly chosen transects.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

The situation is different if the Thorpe scale is used prognostically to parameterize unresolved mixing in regions that are statically unstable (Klymak and Legg 2010). In this case, by definition we do not have access to even a single realization of the ensemble, but only to the mean value of the Thorpe scale. The present work suggests that using LT to estimate dissipation via Eq. (4) may overestimate the dissipation and consequently provides at best an upper limit to the turbulent diffusivity but offers no robust guidance on how to provide a more accurate parameterization, since the fraction of APE actually involved in mixing, which is 10% in the present simulations, may be different in other flows. Likewise, the time scale for mixing, measured in buoyancy periods, may be dependent on other factors as well.

In summary, SW’s energetics approach indicates that mixing events inhabit a continuum bracketed by two idealized conditions: shear- and convective-driven mixing. This framework can be used to justify the use of a displacement scale to estimate dissipation and mixing, provided that the displacement scale used reflects only the contribution of turbulent fluctuations. For shear-driven flows at typical values of the flux Richardson number, our approach in practice coincides with the oceanographic practice because in such flows the Thorpe scale is a good measure of the displacements due to turbulent eddies and thus can be used to approximate the turbulent APE.

For flows driven by an excess of mean APE (convective-driven mixing), using the standard oceanographic approach leads to potentially overestimating the dissipation. As a diagnostic tool, we have shown that a suitable displacement scale can be obtained from the Thorpe displacement δT even with relatively few data points, using a vertical spatial filter as a proxy for the ensemble average. A final remark is in order. The two flows considered here represent limiting cases of what in realistic situations is a continuous spectrum. As such, a vertical filter applied to the Thorpe displacements should always be used to separate the turbulent contribution. However, determination of dissipation and eddy diffusivity depends also on two nondimensional coefficients Cϵ ≡ (2αΓ)−1 and Cmix ≡ (2α)−1. The coefficient Cmix varies between 0.1 for pure convective-driven flows and 0.25 for pure shear-driven flows, a rather modest spread. The coefficient Cϵ on the other hand varies from 0.1 to 1.2, a more substantial variation. Therefore, if the goal is to diagnose the turbulent diffusivity, the crucial quantity is , which can be determined solely from CTD profiles. For dissipation, information about background shear should also be collected to determine where in the convective-to-shear continuum the flow resides.

While this paper is based solely on theoretical considerations backed by numerical calculations, comparison of estimates based on Eq. (4) with actual measurements from microstructure profilers in field data presented in a companion paper (MVSM), as well as turbulence-resolving simulations of overturns driven by the interaction of the internal tide with the bottom (Chalamalla and Sarkar 2015), reach similar conclusions, namely, that using Eq. (4) can overestimate ϵk by an order of magnitude when the vertical size of the overturned region is large.

Acknowledgments

The author thanks B. D. Mater, L. St. Laurent, S. K. Venayagamoorthy, and J. Moum for engaging discussions on the subject and the three anonymous reviewers for their constructive criticism. This work was supported by ONR under Grant N00014-09-1-0288.

APPENDIX

Large Overturns during the Reflection of Internal Wave Beams

The purpose of this appendix is to show that reflecting internal wave beams can generate overturns in which most of the energy of the mean field is contained as mean APE over a significant period. The intent is not to discuss the physics of these events but to show a “realistic” example of convective-driven mixing.

We begin by recalling a result derived by Thorpe (1987): As the amplitude of a freely propagating internal wave approaches a finite critical value Ac, the maximum5 slope of the isopycnals locally approaches infinity (hence, the local BV frequency N2 → 0), while at the same time the shear S approaches zero in such a way that the gradient Richardson number has a discontinuity at the critical amplitude
ea1
where A is the amplitude of the wave. From a kinematic point of view, locally the flow is approximated by a solid body rotation (since S = 0) that tilts isopycnals past their stability point. From an instability point of view, locally the flow becomes statically unstable without going, during the runup, through a stage where shear instabilities can grow.

The reflection of an internal wave beam off a slope provides a striking example of this phenomena, namely, the occurrence of zero-shear statically unstable regions that originate away from boundaries. They have been observed in laboratory experiments (De Silva et al. 1997) and DNS (Chalamalla et al. 2013) and possibly over continental shelves (Nash et al. 2007). Scotti (2011) obtained an analytical solution for the reflection of beams in the time domain that shows how the development of such regions occurs even for relatively weak incoming beams. This analytical solution forms the basis for the analysis presented here. It applies to an inviscid fluid, and outside the near-wall boundary layer it was found to be in excellent agreement with fully nonlinear simulations (Chalamalla et al. 2013). We consider a sloping bottom near the equator (hence, rotation is negligible) making an angle γ = 7° with the horizontal, upon which a semidiurnal beam (ωb = 1.45 × 10−4 s−1) impinges and reflects. The ambient stratification is N = 8 × 10−4 s−1, and the along-beam velocity is ≃1 cm s−1. Let R ≡ (sinα + sinγ)/sinγ, where α is the angle the beam axis makes with the horizontal and signed so that α < 0 means the group velocity of the incoming beam is propagating downward (Fig. A1). For the present case, R = −½, so that the reflected beam propagates upslope. The incoming beam is described by a streamfunction whose maximum amplitude ψmax obeys ψmaxk2/N = 0.01, where N is the background BV frequency and k is the dominant wavenumber of the beam. The width of the beam is about one wavelength.

Fig. A1.
Fig. A1.

Evolution of an overturning patch generated during the reflection of a semidiurnal internal wave beam (ωb = 1.45 × 10−4 s−1) off a sloping bottom near the equator (f = 0). The red arrows indicate the directions of the incoming and reflected beam. The thick black line indicates the local plumb line. The background stratification N = 8 × 10−4 s−1 is representative of a weakly stratified near-bottom region. The incoming beamwidth is ~200 m, about one wavelength in the across-beam direction. Time increases along the direction of the arrow to the left, each panel showing snapshots spaced 2.5 h. The thin black lines show isopycnals, whereas the color indicates the strength of the local Richardson gradient number Rig. Yellow if Rig > ¼, white if Rig < 0, and with gradation of blue for intermediate values. The letter A in the top panel indicates the patch whose APE and KE is calculated as it evolves in time. Note how, with the exception of very thin layers mostly at the bottom side of the patch, Rig is negative within the patch. The three panels have the same aspect ratio.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

Figure A1 shows the region along the bottom centered on the incoming beam axis at three different times. The overturned region of interest (marked with A) is the one that forms off wall. Initially small, it grows over time as a flattened region (i.e., with a high aspect ratio), slowly moving upslope and getting closer to the wall, eventually merging with the statically unstable region that is attached to the wall. Note how with the possible exception of a very thin layer at the edge of the overturned region, the gradient Richardson number is everywhere greater than ¼ except within the overturned region itself, where it is negative.

To study the energetics within the statically unstable patch, we integrate the APE and the KE within such a region (KE is referenced to a frame of reference that moves with the mean velocity of the patch). For a newly born patch, over 95% of the energy is contained as APE, decreasing over the next several hours to 80% (Fig. A2). This shows how a large-scale flow can create and maintain regions characterized by a high ratio of APE to KE over a significant period of time that are not due to instabilities [which are not allowed in the Scotti (2011) solution]. Of course, higher-order nonlinearities not accounted in the analytic solution will clearly lead to a more complicated evolution, but in general we expect that energy for mixing will flow out of the (mean) APE reservoir where the mean (i.e., large scale) flow is injecting it.

Fig. A2.
Fig. A2.

The ratio of APE to total energy within the overturned patch indicated with the latter A in Fig. A1. The thick line shows the e-folding scale of Rayleigh–Taylor instabilities for the specific case considered here.

Citation: Journal of Physical Oceanography 45, 10; 10.1175/JPO-D-14-0092.1

If we entertain the hypothesis that the primary pathway to instability is via RT instabilities, we can further consider how the time scale of the instabilities depends on the thickness of the patch Hp. Referring the reader to Chandrasekhar (1981, p. 447, Table XLVI) for details, the essential point is that the time scale of the most unstable RT modes is TRT ~ (ν/g2)1/3, where g′ is the reduced gravity across the unstable layer and ν is the viscosity that, translated in terms of the properties of the patch, gives
ea2
where Np is the value of the BV frequency within the resorted patch and
ea3
The coefficient 50 that appears in the formula is appropriate for RT instabilities in the low Atwood number regime. For the particular case considered here, where Re = 7200, TRT ≃ 2 h, small enough to be able to extract APE from the patch before the external dynamics undoes the overturning, yet slow enough so that sufficient APE can accumulate during the initial evolution of the patch. Interestingly, if we apply the scaling Eq. (A2) to our simulations, for which Re = 4000, we obtain TRTNp ≃ 3, not too far from the dissipative time scale that was calculated from the simulation.

REFERENCES

  • 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
1

By “overall” we mean that locally the stratification can be statically unstable, even on scales much larger than the turbulent scales but that such statically unstable regions are isolated and immersed in an otherwise stably stratified environment.

2

Note, however, that in regions where the salinity plays an important role in determining the density, there are significant technical hurdles that need to be addressed (see, e.g., Gargett and Garner 2008).

3

The idea that N provides a time scale for the problem is already present in Dillon (1984). Here, we introduce the further hypothesis that the time scale is the same for turbulent KE and APE dissipation.

4

One buoyancy period is equal to 2πN−1.

5

The maximum is evaluated over a wave cycle.

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