Internal-Wave-Driven Mixing: Global Geography and Budgets

Eric Kunze NorthWest Research Associates, Redmond, Washington

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Abstract

Internal-wave-driven dissipation rates ε and diapycnal diffusivities K are inferred globally using a finescale parameterization based on vertical strain applied to ~30 000 hydrographic casts. Global dissipations are 2.0 ± 0.6 TW, consistent with internal wave power sources of 2.1 ± 0.7 TW from tides and wind. Vertically integrated dissipation rates vary by three to four orders of magnitude with elevated values over abrupt topography in the western Indian and Pacific as well as midocean slow spreading ridges, consistent with internal tide sources. But dependence on bottom forcing is much weaker than linear wave generation theory, pointing to horizontal dispersion by internal waves and relatively little local dissipation when forcing is strong. Stratified turbulent bottom boundary layer thickness variability is not consistent with OGCM parameterizations of tidal mixing. Average diffusivities K = (0.3–0.4) × 10−4 m2 s−1 depend only weakly on depth, indicating that ε = KN2/γ scales as N2 such that the bulk of the dissipation is in the pycnocline and less than 0.08-TW dissipation below 2000-m depth. Average diffusivities K approach 10−4 m2 s−1 in the bottom 500 meters above bottom (mab) in height above bottom coordinates with a 2000-m e-folding scale. Average dissipation rates ε are 10−9 W kg−1 within 500 mab then diminish to background deep values of 0.15 × 10−9 W kg−1 by 1000 mab. No incontrovertible support is found for high dissipation rates in Antarctic Circumpolar Currents or parametric subharmonic instability being a significant pathway to elevated dissipation rates for semidiurnal or diurnal internal tides equatorward of 28° and 14° latitudes, respectively, although elevated K is found about 30° latitude in the North and South Pacific.

Denotes content that is immediately available upon publication as open access.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Eric Kunze, kunze@nwra.com

Abstract

Internal-wave-driven dissipation rates ε and diapycnal diffusivities K are inferred globally using a finescale parameterization based on vertical strain applied to ~30 000 hydrographic casts. Global dissipations are 2.0 ± 0.6 TW, consistent with internal wave power sources of 2.1 ± 0.7 TW from tides and wind. Vertically integrated dissipation rates vary by three to four orders of magnitude with elevated values over abrupt topography in the western Indian and Pacific as well as midocean slow spreading ridges, consistent with internal tide sources. But dependence on bottom forcing is much weaker than linear wave generation theory, pointing to horizontal dispersion by internal waves and relatively little local dissipation when forcing is strong. Stratified turbulent bottom boundary layer thickness variability is not consistent with OGCM parameterizations of tidal mixing. Average diffusivities K = (0.3–0.4) × 10−4 m2 s−1 depend only weakly on depth, indicating that ε = KN2/γ scales as N2 such that the bulk of the dissipation is in the pycnocline and less than 0.08-TW dissipation below 2000-m depth. Average diffusivities K approach 10−4 m2 s−1 in the bottom 500 meters above bottom (mab) in height above bottom coordinates with a 2000-m e-folding scale. Average dissipation rates ε are 10−9 W kg−1 within 500 mab then diminish to background deep values of 0.15 × 10−9 W kg−1 by 1000 mab. No incontrovertible support is found for high dissipation rates in Antarctic Circumpolar Currents or parametric subharmonic instability being a significant pathway to elevated dissipation rates for semidiurnal or diurnal internal tides equatorward of 28° and 14° latitudes, respectively, although elevated K is found about 30° latitude in the North and South Pacific.

Denotes content that is immediately available upon publication as open access.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Eric Kunze, kunze@nwra.com

1. Introduction

Quantifying and understanding ocean mixing remains one of the most challenging problems in physical oceanography because of its spatial and temporal heterogeneity. Much of the turbulent mixing is concentrated in localized hot spots so that average mixing can only be accurately assessed from large amounts of data with well-distributed geographical coverage (Kunze et al. 2006; Whalen et al. 2012; Waterhouse et al. 2014). A wide range of features on time scales of months to millennia, from precipitation in the western equatorial Pacific (Jochum 2009) to the strength of the deep meridional overturning circulation, equatorial upwelling, and the Southern Hemisphere westerlies (Friedrich et al. 2011; Melet et al. 2016), are sensitive to how diapycnal mixing is parameterized in global OGCMs, linking diapycnal mixing not just to the ocean circulation but also biogeochemical cycles, weather, and long-term climate. Jochum (2009) found that applying the latitude dependence to mixing (Gregg et al. 2003) improved the skill of OGCMs in reproducing equatorial SST and precipitation, the spiciness of Labrador Seawater, and the Gulf Stream path.

In the bulk of the stratified ocean interior, internal wave breaking is the dominant source of turbulent mixing (Munk and Wunsch 1998). The primary sources of deep-ocean internal waves are tide/topography generation of internal tides at ~1.0 TW (Egbert and Ray 2001; Nycander 2005); wind-forced, near-inertial waves at 0.2–1.1 TW (Alford 2001; Plueddemann and Farrar 2006; Furuichi et al. 2008; Rimac et al. 2013); and subinertial flow/topography generation of internal lee waves at 0.2–0.7 TW (Scott et al. 2011; Nikurashin and Ferrari 2011; Wright et al. 2014). Thus, total internal wave power input is 2.1 ± 0.7 TW with most of the uncertainty in (i) near-inertial wave production by winds, associated with the temporal resolution of global wind products at high latitudes and mixed-layer depth assumptions, and (ii) lee-wave dissipation based on microstructure measurements being up to an order of magnitude below predictions (Waterman et al. 2014). This generated internal wave energy is redistributed vertically and horizontally by propagation of low-mode internal waves (Ray and Mitchum 1997; Alford 2001; Zhao et al. 2016) and from large to small scales by wave–wave and wave–mean flow interactions. Away from direct forcing by wind and currents at boundaries, turbulence production is controlled by the rate at which energy cascades from large to small vertical scales. Internal wave–wave interaction theory (McComas and Müller 1981; Henyey et al. 1986; Henyey 1991) has provided a parameterization for turbulence production that can be expressed in terms of finescale internal wave shear Vz and/or strain ξz (Gregg 1989; Gregg and Kunze 1991; Polzin et al. 1995; Gregg et al. 2003). Comparison with direct microstructure measurements validates the shear-and-strain finescale parameterizations to within factors of 2–3 (Polzin et al. 1995; Polzin et al. 2014; Whalen et al. 2015), although care is needed in its implementation to avoid overestimation, particularly in low-N environments where sensor noise becomes problematic and where N exhibits large changes with depth.

The finescale parameterization has recently been reviewed by Polzin et al. (2014). In contrast to direct microstructure measurements, its inferences represent turbulent dissipation and mixing with built-in averaging over internal wave time and space scales. It has been used to (i) infer that elevated dye mixing in Santa Monica Basin (Ledwell and Watson 1991) was being driven by an energetic internal wave field on the slopes (Gregg and Kunze 1991); (ii) predict elevated turbulent mixing above seamount summits (Kunze et al. 1992) before microstructure confirmation (Lueck and Mudge 1997; Toole et al. 1997; Kunze and Toole 1997); (iii) infer weak mixing over smooth bottom topography and elevated mixing over rough topography (D’Asaro and Morison 1992; Wijesekera et al. 1993; Kunze and Sanford 1996; Mauritzen et al. 2002; Walter et al. 2005; Kunze et al. 2006; Stöber et al. 2008; MacKinnon et al. 2008), consistent with deep direct microstructure measurements (Toole et al. 1994; Polzin et al. 1997); (iv) identify sites where Antarctic Circumpolar Currents interact with bottom topography to generate elevated turbulence northeast of the Kerguelan Plateau (Polzin and Firing 1997; Kunze et al. 2006) and in Drake Passage (Naveira Garabato et al. 2004; Wu et al. 2011; Damerell et al. 2012), subsequently verified with direct microstructure measurements (St. Laurent et al. 2012; Sheen et al. 2013; Waterman et al. 2013); (v) contribute to the argument that parametric subharmonic instability (PSI) may transfer energy from low-mode internal tides to high-wavenumber near-inertial shear of half the frequency (Hibiya et al. 2006) and thence to turbulence immediately equatorward of 28° (MacKinnon and Winters 2005; Carter and Gregg 2006; Simmons 2008; MacKinnon et al. 2013a,b; Sun and Pinkel 2013); (vi) determine there is little seasonal variability in upper-ocean mixing except under fall–winter storm tracks (30°–40°) (Whalen et al. 2015); and (vii) assess the role of turbulent diapycnal mixing in the meridional overturning circulation and large-scale property budgets in the Indian Ocean (Huussen et al. 2012).

In this paper, a global assessment of deep-ocean, internal-wave-driven turbulent dissipation rates ε and diapycnal diffusivities K will be inferred by applying a parameterization based on finescale internal-wave strain ξz (Gregg and Kunze 1991; Wijesekera et al. 1993; Polzin et al. 1995; Gregg et al. 2003) to ~30 000 CTD profiles. Data are absent from the Arctic, Weddell, and Ross Seas and are limited poleward of 60°S in the Southern Ocean but otherwise are well-distributed with latitude and longitude in all the major ocean basins. Strain-based inference of internal-wave-driven turbulence dissipation rates ε and diffusivities K has seen widespread use (Mauritzen et al. 2002; Sloyan 2005; Lauderdale et al. 2008; Wu et al. 2011; Whalen et al. 2012; 2015; Damerell et al. 2012). The analysis here expands on previous shear-and-strain estimates based on ~3500 full-depth LADCP/CTD profiles (Kunze et al. 2006) with an order of magnitude more data to provide much more comprehensive global geographical coverage, expanding the scope to the South Atlantic and improving sampling in the North Atlantic, western Pacific, and Southern Oceans. The data and methods are detailed in section 2, global maps and sections are discussed in section 3, averages and budgets are in section 4, zonally averaged structure is in section 5, an evaluation of OGCM tidal mixing parameterizations is in section 6, and finally the conclusions and discussion are found in section 7. In a companion paper (Kunze 2017, manuscript submitted to J. Phys. Oceanogr.), the inferred diapycnal diffusivities K and dissipation rates ε are used to compute the interior internal-wave-driven diabatic meridional overturning circulation and compare it with diapycnal transports driven by near-bottom buoyancy-flux divergence.

2. Data and methods

CTD profiles from WOCE/CLIVAR hydrographic sections (https://cchdo.ucsd.edu/) were employed in this analysis. These include about 73 sections from the Indian, 422 from the Pacific [although over 200 of these are from the Hawaii Ocean Timeseries (HOT) site], and 146 from the Atlantic with roughly 6700, 13 000, and 10 500 usable casts, respectively; casts shorter than 300 m or with resolution coarser than 2 m are excluded. No data are included from the Arctic, Weddell, or Ross Seas and are sparse south of 60°S in the Southern Ocean. Roughly 10% of the retained profiles span less than 25% of the water column. The T, S, and p files were downloaded and converted to a uniform format. Density variables σp, σθ, σ3, and γn and stratification N2 were then computed.

Internal-wave-driven turbulent dissipation rates ε and diapycnal diffusivities K are inferred from a finescale parameterization based on internal wave–wave interaction theory (McComas and Müller 1981; Henyey et al. 1986) that was first tested using 10-m vertical shear (Gregg 1989), then modified to estimate variances spectrally in vertical wavenumber space and incorporate internal wave strain as both (i) an independent means of inferring internal wave spectral levels (Gregg and Kunze 1991; Wijesekera et al. 1993) and (ii) to account for deviations of the internal wave aspect ratio or frequency content from the Garrett–Munk (GM) model (Polzin et al. 1995). The strain-based form of the parameterization used here is
e1
(e.g., Kunze et al. 2006), where K0 = 0.05 × 10−4 m2 s−1 for a mixing efficiency γ = 0.2, 〈ξz2〉 is the strain variance with strain estimated as ξz = (N2N2fit)/N2fit following Polzin et al. (1995), and N2fit represents a quadratic fit to half-overlapping, 256-m profile segments; alternative fitting procedures were attempted including to log(N2) rather than N2 and different functional forms, but these were found to be biased compared to the simpler fitting scheme. The normalizing GM model strain variance GMξz2〉 is computed over the same wavenumber band as the observed strain. The GM75 model vertical wavenumber kz spectrum for strain is given by
e2
(Cairns and Williams 1976; Gregg and Kunze 1991), where the canonical nondimensional spectral energy level E0 = 6.3 × 10−5, stratification length scale b = 1300 m, peak mode number j* = 3, and corresponding vertical wavenumber kz* = (πj*/b)(N/N0). Integrated to 10-m vertical wavelengths, the GM75 strain variance in (2) is 0.24. The dependence on shear/strain variance ratio Rω = 〈Vz2〉/(〈N2〉〈ξz2〉) is
e3
(Fig. 1a), which provides a crude measure of the wave field’s aspect ratio (or frequency) content for internal waves dominated by lower frequencies (Henyey 1991; Kunze et al. 1990; Polzin et al. 1995). The dependence on the ratio of buoyancy to Coriolis frequencies N/f is
e4
where f30 = f(30°) = 7.3 × 10−5 rad s−1 and N0 = 5.2 × 10−3 rad s−1. The latitude dependence [(4)] vanishes at the equator because f goes to zero more rapidly than Arccosh(N/f) goes to infinity; the appendix argues that internal waves at the equator will be equatorially trapped meridional modes with minimum frequencies set by off-equatorial Coriolis frequencies at turning latitudes, but it is not known how this will impact the cascade. The dissipation rate ε = KN2〉/γ (Osborn 1980), where mixing efficiency γ = 0.2 is assumed (Oakey 1982; Itsweire et al. 1986; Moum 1996; St. Laurent and Schmitt 1999) following standard practice in the ocean microstructure observational community.
Fig. 1.
Fig. 1.

(a) Correction h(Rω) [(3)] for the strain-only turbulence parameterization [(1)] as a function of shear/strain variance ratio Rω. (b) Probability distribution function of Rω based on the LADCP/CTD analysis of Kunze et al. (2006) with vertical lines denoting the GM Rω = 3 as well as the measured mean (10.6) and mode (6) in the data. Almost identical distributions are found when the data are split into nine buoyancy frequency N and nine latitude bins, indicating that this distribution is a robust feature of the ocean internal wave field. Means (modes) decrease for larger strain variances that will contribute more to mixing, dropping to 8 (4) for 〈ξz2〉/GM exceeding 1.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

For the GM model, the shear/strain variance ratio GMRω = 3. But the ocean’s average Rω appears to be higher (Fig. 1b), signifying that the ocean is more inertial than the GM model on average. For measured Rω data (Kunze et al. 2006), the distribution has mean 10.6 and mode 6.1, independent of buoyancy frequency N and latitude; 80% of the data had Rω below 16. The distribution’s mean and mode diminish for the larger strain variances that will dominate internal-wave-driven mixing (not shown). For shear/strain ratios greater than 3, h(Rω) is an increasing function of Rω; Rω = 7 will be used here, which produces dissipation rates ε and diffusivities K a factor of 3 larger than for Rω = 3 and a factor of 3 smaller than for Rω = 10 (Fig. 1a), so that factor of 3 uncertainties are expected; Kunze et al. (2006) reported maximum factor of 2 differences between shear-and-strain and strain-only diffusivities for Rω = 7.

Finescale parameterization (1) only accounts for weakly nonlinear internal-wave-driven turbulence. It will fail in environments where a weakly nonlinear wavenumber cascade is not expected either because of (i) lack of bandwidth such as on continental shelves (MacKinnon and Gregg 2003; Carter et al. 2005), (ii) short-circuiting of the cascade because of near-critical bottom reflection (Carter and Gregg 2002; Nash et al. 2004; Kunze et al. 2012), or (iii) direct boundary forcing of turbulence (e.g., Klymak et al. 2008, 2010). It does not account for mixing due to hydraulically controlled flow (Ferron et al. 1998) or breaking solitary waves (MacKinnon and Gregg 2003). While these regions occupy a small fraction of the ocean, they may be important. For example, density overturns of O(100) m [e.g., in Luzon Strait (Alford et al. 2011), Samoan Passage (Alford et al. 2013), and Romanche Fracture Zone (Ferron et al. 1998)] imply local diffusivities 104–105 times the background and so they need only occupy 0.01%–0.1% of the ocean to produce basin-averaged diffusivities of 10−4 m2 s−1. The shear-and-strain parameterization overestimates turbulent dissipation rates on the flanks of the Florida Strait (Winkel et al. 2002) and overlying regions where lee-wave generation is expected (Waterman et al. 2014).

Strain variance for (1) is estimated spectrally from strain ξz = (N2N2fit)/N2fit for half-overlapping, 256-m profile segments starting at the bottom up to the depth of the highest N2 in the upper 150 m of the water column (to exclude the surface mixed layer). This yields roughly 500 000 usable estimates. Strain variances are computed by integrating the strain spectra S[ξz](kz) from the lowest resolved vertical wavenumber (λz = 256 m) to the wavenumber where variance exceeds a threshold value of 0.05, which, for a GM-level spectrum, corresponds to λz = 50 m, in part to avoid contamination by ship heave near 10-m wavelengths (Polzin et al. 2014). By stopping the integration short of the rolloff wavenumber, it is assumed that the strain spectrum is flat like the GM model; redder (more negative spectral slope) spectra, as typically found here, will result in overestimation of the strain variance by at most a factor of 1.3, while bluer (more positive spectral slope) spectra will result in underestimation. The 256-m profile segment length is a compromise between resolution in depth and strain variance (diffusivity). Strain variance is computed below the rolloff wavenumber kc, which behaves as (0.2π m)EGM/E (Fritts 1984; Gargett 1990; D’Asaro and Lien 2000). With diffusivities K(EGM) ~ 10−5 m2 s−1 and K ~ E2, 256-m segment lengths can resolve strain variance ratios 〈ξz2/GMξz2〉 less than 10 and diffusivities K less than O(10−3) m2 s−1.

Strain is assumed to be dominated by finescale internal waves. While contamination by finescale geostrophic motions (Pinkel 2014), thermohaline staircases (Gregg 1989), and interleaving cannot be ruled out on dynamical grounds, in earlier high-resolution profiler and CTD analyses, the only contamination signals that stood out were sharp pycnoclines at low latitudes (Polzin et al. 1995; Mauritzen et al. 2002; Kunze et al. 2006), with contamination by water-mass intrusions and geostrophic motions appearing to be confined to wavelengths less than 10 m (Polzin et al. 2003) and greater than 200 m (Kunze et al. 2006). Therefore, the contamination is largely filtered out here by the chosen 50–256-m band of integration. Pinkel (2014) reports subinertial strain confined to near the base of the mixed layer. Double-diffusive interleaving may contribute where there is strong water-mass variability on isopycnals (S. Merrifield 2016, personal communication). Double diffusion tends to produce thermohaline staircases in only a few known locales of weak internal-wave-driven mixing (Gregg and Sanford 1987; Kunze 2003), which are of little interest here, or at lateral water-mass boundaries. Thermohaline staircases escape the spectral filter at 1200–1800-m depth beneath the Mediterranean salt tongue between 30° and 40°N in the eastern North Atlantic and are also expected east of Barbados, in the Tyrrhenian Sea, and under the Red Sea outflow in the Arabian Sea (Schmitt 2003).

To avoid contamination by sharp pycnoclines, the shallowest two segments (corresponding to the upper 380 m of the water column) are omitted from analysis because they often, and unpredictably, exhibit unrealistically high strain variances. This problem has previously been recognized and dealt with in a similar manner by Mauritzen et al. (2002), Kunze et al. (2006), and Whalen et al. (2012). Whalen et al. (2015) compared Argo float strain-based diffusivities K to average microstructure K profiles at six sites and found that 81% agreed to within a factor of 2 and 96% agreed to within a factor of 3 below 250-m depth. A more conservative 380-m depth was chosen here because of contamination by very deep mixed layers at high latitudes. Qualitative evidence that this is sufficient can be seen in Figs. 4, 6, and 9 (shown below), where the shallowest plotted diffusivities and strain variances just below 380 m are similar to those at greater depths. Profile segments were also excluded if their average buoyancy frequency 〈N〉 fell below the noise threshold 3 × 10−4 rad s−1, as these are dominated by digitization noise (Whalen et al. 2015). With the expectation that N > 2f is a minimal frequency bandwidth to allow internal wave–wave interactions, segments with 〈N〉 less than 2f were also excluded; these largely overlap with the 〈N〉 noise threshold, representing 10% of the data, 30% within 380 meters above bottom (mab), and 17% within 1000 mab in abyssal basins. Very low stratification is found (i) throughout much of the water column at high latitudes in the Southern Ocean, particularly the eastern Atlantic sector; (ii) at middepth around southern Greenland; and (iii) in abyssal basins in the Caribbean, Angola Basin, and North Pacific. These will have small diapycnal buoyancy fluxes 〈wb′〉 = −KN2〉, where w′ and b′ are turbulent vertical velocity and buoyancy fluctuations, because their stratification is weak, so their omission has little impact. While internal waves can exist for N < f such that topographic generation of internal waves is still possible in very weakly stratified bottom boundary layers, these will be unable to propagate into regions where N exceeds f and so will be confined near the bottom. How this weak stratification might impact internal tide and lee-wave generation, or near-bottom turbulence, has not been investigated to the author’s knowledge.

3. Geography

Maps of depth-integrated dissipation rate ∫ε = ρ0εdz1 from 380-m depth to the bottom (Fig. 2) show elevated values associated with abrupt topography and slow spreading ridges, particularly (i) in the western Indian over the Southwest Indian Ridge (30°–40°S, 40°–60°E), the Seychelles and Mascarene Ridge east of Madagascar (2°–20°S, 50°–60°E), in the western Arabian Sea over the Owen Fracture Zone (10°N, 60°E) and Carlsberg Ridge, and in the wake of the Kerguelan Plateau (50°–60°S, 70°–80°E; Polzin and Firing 1997); (ii) in the western Pacific over abrupt ridge/trench topography associated with subduction; (iii) in the central Pacific associated with island archipelagos such as the Hawaiian Island chain (20°–30°N, 150°–180°W) and Tuamotu Archipelago (20°–30°S, 130°–160°W); and (iv) in the Atlantic over continental slopes such as in the Bay of Biscay (45°–50°N, 0°–10°W), the Flemish Cap (40°–50°N, 35°–40°W), and Mid-Atlantic Ridge. Low values are found over smooth abyssal basins such as (i) the central Arabian Sea (10°–15°N, 55°E) and Bay of Bengal (10°–20°N, 90°E) and south Indian Basin south-southeast of Sri Lanka (0°–20°S, 80°E) in the Indian Ocean, (ii) in the subpolar North Pacific and eastern Pacific Ocean, (iii) in the Angola Basin in the eastern South Atlantic (0°–20°S, 0°–20°E), and (iv) in the Southern Ocean south of ~58°S, where N is low throughout the water column. Overall, high vertically integrated dissipation rates are consistent with predicted sites of high internal tide generation (e.g., Egbert and Ray 2001; Simmons et al. 2004; Nycander 2005). Low values in the Southern Ocean do not support lee waves being a major dissipative conduit for the Antarctic Circumpolar Currents (Nikurashin and Ferrari 2011; Scott et al. 2011), consistent with microstructure measurements at Kerguelen Plateau and Drake Passage finding dissipation rates up to an order of magnitude below lee-wave generation predictions (Waterman et al. 2014). The WOCE/CLIVAR hydrography is not well suited for examining the upper ocean’s response to storm forcing because of its temporal intermittency. Argo profiling floats provide better temporal sampling of the seasonal cycle of upper-ocean mixing, which appears to be confined to 30°–40° latitude (Whalen et al. 2012).

Fig. 2.
Fig. 2.

Maps of the vertically integrated dissipation rate ∫ε = ∫ρε dz in mW m−2 for the full water column excluding the upper 380 m. Low values are found in the Southern Ocean, the Indian’s Bay of Bengal, and South Atlantic’s Angola Basin. High values are associated with abrupt topography in the western Indian, western and central Pacific, and over midocean ridges. Because of the N2 scaling of ε, dissipation rates integrated over the pycnocline (<2000 m) show similar patterns and magnitudes.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

Binning vertically integrated dissipation rates ∫ε by longitude shows that elevated values in the western Indian and western to central Pacific are related to topography (Fig. 3), not tidal flows, which is more uniform. This interpretation augments that of Hibiya et al. (1999), who predicted western intensification of turbulence in the North Pacific because of the hot spot of near-inertial wave generation near 40°N and west of the date line (Alford 2001). On average, most of the dissipation occurs in the pycnocline rather than near the bottom (Fig. 3a), in contrast to OCGM tidal mixing parameterizations (Jayne and St. Laurent 2001; Decloedt and Luther 2010).

Fig. 3.
Fig. 3.

Vertically integrated dissipation rates ∫ε (a) reveal that most of the dissipation occurs in the pycnocline (red). Peaks in the western Indian and western Pacific (left two gray bars) appear to be most correlated with 30 km × 30 km topographic roughness h2 (b), whereas rms tidal velocities (c) are more uniform. Elevated dissipation rates and topography over 285°–300°E (60°–75°W) are associated with both Drake Passage and the Caribbean. These ∫ε exclude the upper 380 m.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

Repeat sections of inferred diapycnal diffusivity K illustrate that elevated variability related to topography is reproducible. While elevated diffusivities K above weak rough topography are sometimes confined to within 500 mab over stronger and more extensive topography, it often extends throughout the entire water column (Fig. 4), consistent with Fig. 3a and in contrast with the fixed decay scale of 500 m implemented in OGCM tidal mixing parameterizations (Simmons et al. 2004; Saenko and Merryfield 2005; Jayne 2009; Friedrich et al. 2011) based on microstructure measurements on the Mid-Atlantic Ridge bounding the eastern side of the Brazil Basin (St. Laurent et al. 2001). This is more clearly illustrated in the joint probability density function of diffusivity above 2000-m depth Kpycno versus below 2000-m depth Kdeep (Fig. 5), which reveals a correlation between diffusivities in these two depth ranges with Kpycno ~ Kdeep/2. Since ε = KN2/γ and N2 exhibit more variability than K, most of the dissipation will be in the high stratification of the pycnocline. Figure 4 also illustrates some of the variety of bottom geometries that can contribute to elevated strain variance.

Fig. 4.
Fig. 4.

Repeat sections of inferred turbulent diapycnal diffusivity K show the influence of rough topography including (a05) crossing of the Mid-Atlantic Ridge near 25°N, (p14) crossing of the Aleutian Island Ridge, (p06) crossing of the Colville Ridge and Louisville Seamount Chain near 33°S in the subtropical South Pacific, (i05) crossing the ridges in the western Indian near 34°S, and (p06) crossing between eastern Australia and north of New Zealand near 30°S. While elevated diffusivities K over rough topography are sometimes confined near the bottom, they often extend through the entire water column, which will produce particularly strong dissipation rates in the pycnocline where N is high.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

Fig. 5.
Fig. 5.

Probability distribution function of diapycnal diffusivity K in the pycnocline (<2000-m depth) and in the deep (>2000-m depth), illustrating that elevated pycnocline diffusivities are correlated with deeper diffusivities but weaker by a factor of 2 (lower thin dotted line).

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

Equatorial crossings consistently show elevated strain variance within ±2° of the equator (Fig. 6). There is little corresponding signal in diapycnal diffusivity K because a reduced cascade rate as f → 0 in (4) allows more variance to accumulate at lower wavenumbers for the same dissipation rate (Gregg et al. 2003). This contrasts with the stripes of elevated diffusivity K or integrated dissipation rate ∫ε reported flanking the equator by Whalen et al. (2012); the absence of this signal here appears to be due to the hydrography casts missing frequent but intermittent mixing bursts associated with negative La Niña and neutral ENSO conditions (C. B. Whalen 2016, personal communication).

Fig. 6.
Fig. 6.

Repeat sections illustrating equatorial crossings including (i04) in the Indian Ocean along 80°E and (p18) south of Baja California in the eastern Pacific. Strain variance is elevated near the equator because the cascade proceeds more slowly, as represented by the j(N/f) term in (1). The N/f scaling [(4)] compensates for the excess strain near the equator for more uniform diffusivities K that are forced to zero at the equator as f → 0. In the Pacific section, there is also elevated strain variance associated with rough topography at 5°S and 10°N.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

4. Averages and budgets

The global-integrated dissipation rate , computed as times the ocean area, where < > is the average of all the profiles, is 1.5 ± 0.4 TW (4.3 ± 1.0 mW m−2 per unit area) below 380-m depth. Averages per unit area are largest in the North Pacific (5.3 mW m−2) and smallest in the South Atlantic (2.3 mW m−2). The above value is missing the contribution above 380-m depth, which is potentially significant in light of the N2 scaling of ε, so this fraction is now estimated. Assuming a conservative, that is, low average pycnocline diffusivity K = 0.1 × 10−4 m2 s−1 (Gregg 1989; Waterhouse et al. 2014) and ε = KN2/γ above 380 m implies an additional 0.5 ± 0.2 TW in the upper 380 m, likely an underestimate because shipboard sampling is biased toward fair weather so will miss some of the wind-forced contribution in the 30°–40° latitude band (Whalen et al. 2015). The total inferred dissipation of 2.0 ± 0.6 TW is then consistent with the sum of internal wave power inputs of 1.0–1.2 TW from the tide (Egbert and Ray 2001; Nycander 2005), 0.2–1.1 TW from wind (Alford 2001; Plueddemann and Farrar 2006; Furuichi et al. 2008; Rimac et al. 2013), and 0.2–0.7 TW from lee-wave generation (Scott et al. 2011; Nikurashin and Ferrari 2011; Melet et al. 2014; Wright et al. 2014) and thus is sufficient to close the internal wave energy budget within the present large uncertainties for both sources and sinks of internal waves.

Average profiles are similar in all ocean basins so only global averages are shown (Fig. 7). Average GM-normalized strain variances 〈ξz2〉/GM ~ 2, almost independent of depth z, but increase to ~3 at the bottom in height above bottom coordinates h. Dissipation rates ε exhibit the most variability with respect to depth z and buoyancy B ≈ −(n/ρ0), where γn is neutral density, while GM-normalized strain variance and diapycnal diffusivity K vary the most with respect to height above bottom h. Dissipation rates ε decrease with depth z. They have a minimum of 0.2 × 10−9 W kg−1 between 1000 and 3000 mab and increase to a maximum of 10−9 W kg−1 within 500-mab of the bottom. But much of this increase is contributed by the pycnocline. Averaging only waters below 2000-m depth, the average dissipation rate is 0.3 × 10−9 W kg−1 at the bottom and 0.1 × 10−9 W kg−1 above 1000 mab in height above bottom coordinates.

Fig. 7.
Fig. 7.

Global average profiles of, from left to right, the number of data points n, buoyancy frequency N, GM-normalized strain variance 〈ξz2〉/GM, diapycnal diffusivity K, and dissipation rate ε as functions of (top) depth z, (middle) height above bottom h, and (bottom) buoyancy with neutral density γn indicated along the rightmost axis. Values are not plotted for n < 300 and are plotted gray for n < 3000. Dotted curves in height above bottom coordinates exclude data shallower than 2000 m. Normalized strain variance and diapycnal diffusivity K are nearly independent of depth (top row) at 1.5–2 and (0.3–0.4) × 10−4 m2 s−1, respectively, but K decreases from 10−4 m2 s−1 at the bottom to 0.15 × 10−4 m2 s−1 above 5000 mab in height above bottom coordinates h (middle row) with a 2000-m e-folding scale, and the dissipation rate ε is elevated below 1000 mab because of both increasing K as h → 0 and elevated N below 1000 mab; much of the elevated N and ε BBL is contributed by shallow water less than 2000 m deep. Strain-inferred diffusivities exhibit less variability with buoyancy B (bottom row) than the Lumpkin and Speer (2003) inverse estimates though agreeing within their uncertainties except over B = −0.267 to −0.269 m s−2 (γn = 28.0–28.2).

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

Average diffusivities K are (0.3–0.4) × 10−4 m2 s−1 with little dependence on depth z but increase toward 10−4 m2 s−1 at the bottom in height above bottom coordinates h with an e-folding scale of ~2000 m. This difference arises because the bottom is not always at the same depth. Diffusivities K increase from 0.3 × 10−4 m2 s−1 for buoyancy B > −0.27 m s−2 to K = 0.7 × 10−4 m2 s−1 at lower buoyancy (higher density), showing weaker dependence than the Lumpkin and Speer (2003) inverse estimates, though lying within the latter’s uncertainties except in −0.267 > B > −0.269 m s−2 (γn = 28.0–28.2). While average diffusivity profiles here are lower than the 10−4 m2 s−1 reported below 1000-m depth from 17 microstructure sites in Waterhouse et al. (2014), comparison of finescale parameterization [(1)] inferences in the vicinity of these 17 sites were consistent with microstructure averages. Most of the sites considered in Waterhouse et al. have predicted sources larger than depth-integrated dissipation rates, implying that the sites were mostly located in net internal wave sources rather than net sinks. This suggests that (i) the strain parameterization is reasonable on average and (ii) microstructure sampling has been biased toward regions of stronger forcing, which is consistent with Waterhouse et al. reporting that internal wave sources exceeded sinks at most microstructure sites, which have undersampled regions of low tidal power (their Fig. 5b).

The strain-based average diffusivities K are a factor of 2–3 higher than shear-and-strain-based values in Kunze et al. (2006) above 3000-m depth but comparable below 3000-m depth and comparable to their strain-based estimates. They are higher for heights above bottom h below 1000 mab; Kunze et al. reported average dissipation rates ε increasing monotonically with height above bottom.

Cumulative dissipation rates, substituting for the upper 380 m as above, are concentrated in the upper pycnocline (Fig. 8), with 80% of the dissipation below 380-m depth contributed above 1000 m. Only 0.08 TW dissipates below 2000-m depth, suggesting very little local mixing in the abyss. Roughly 20% (30%) of the dissipation is found below h < 500 mab (1000 mab), that is, 0.4 TW (0.6 TW). Again, these differences reflect that the bottom is not always at the same depth.

Fig. 8.
Fig. 8.

Global cumulative dissipation rates ∫ε as a function of (top) depth z and (bottom) height above bottom h illustrate that 50% (80%) of the dissipation occurs above 500-m (700 m) depth. Dissipation rates are not accumulated above 400-m depth. Only 20% of the dissipation is found below h = 500 mab.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

5. Average zonal and meridional structure

Zonally averaged depth–latitude sections reveal small differences between the three major oceans (Fig. 9). All the oceans show a pool of elevated dissipation rates ε shallower than 2000-m depth for latitudes equatorward of 50°–60° (Fig. 9c), coinciding with the higher stratification in the main pycnocline (Fig. 9b). This is not reflected in the diffusivity K, which is vertically more uniform (Fig. 9d), consistent with average dissipation rates ε scaling as N2 (Gregg and Sanford 1988). Weaker diffusivities straddling the equator reflect the N/f scaling in (4) but may be biased low (Whalen et al. 2012; Thurnherr et al. 2015) because the rich equatorial wave field is outside the scope of the internal gravity wave–wave interaction theory behind the finescale parameterization. Indian and Atlantic diffusivities K exceed 10−4 m2 s−1 at all depths for latitudes poleward of 40°–50°, associated with weaker high-latitude stratification, while the Pacific diffusivities are more uniformly weak at subpolar latitudes and its stratification is stronger. An alternative explanation is that this latitude band is associated with internal lee-wave generation by Antarctic Circumpolar Currents interacting with topography, where the finescale parameterization overestimates turbulence dissipation rates ε by as much as an order of magnitude (Waterman et al. 2014). In the Indian and Atlantic, diffusivities seem to weaken slightly south of 60°S, while they become more elevated north of 60°N in the Atlantic.

Fig. 9.
Fig. 9.

Depth–latitude sections of (a) number of data points n, (b) average buoyancy frequency N, (c) dissipation rate ε, and (d) diapycnal diffusivity K for the (left) Indian, (center) Pacific, and (right) Atlantic. Dissipation rates ε (c) are elevated in the pycnocline (latitudes < 50°–60°) mirroring the stratification N in (b). Diffusivities K in (d) are O(0.1 × 10−4) m2 s−1 in much of the oceans but are elevated in the Indian and Atlantic sectors of the Southern Ocean (latitudes below 40°S) and in the northern North Atlantic (latitudes above 40°N) and spottily near the bottom. There are also hints of a band of elevated K just equatorward of the semidiurnal PSI critical latitude of 28° (Alford et al. 2007; MacKinnon et al. 2013a,b) in the North and South Pacific but not the Atlantic or Indian. Black contours are density surfaces.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

Zonally averaged vertically integrated dissipation rates ∫ε (Fig. 10a) appear to correlate with topographic roughness h2 (Fig. 10b), while tidal flows are more uniform (Fig. 10c). In the Southern Hemisphere, a peak in ∫ε poleward of 30°S is not consistent with the predictions of parametric subharmonic instability enhancing the cascade of low-mode internal tide energy equatorward of 28° and 14° (MacKinnon and Winters 2005; Hibiya et al. 2006; Alford et al. 2007; MacKinnon et al. 2013a,b). A signature of elevated ∫ε equatorward of 30°N in the Northern Hemisphere may be tied to either parametric subharmonic instability or elevated topographic roughness over 10°–22°N (Fig. 10b) latitude in both the Pacific and Atlantic.

Fig. 10.
Fig. 10.

Zonal averages of (a) vertically integrated dissipation rates ∫ε, (b) topographic roughness h2, and (c) rms tidal currents U as a function of latitude. Integrated dissipation rates are for the water column below 380-m depth (black) in the pycnocline between 380- and 2000-m depth (red and below 2000-m depth (blue).

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

Meridionally averaged depth–longitude sections show widespread elevated diffusivities poleward of 50° in the Atlantic and Indian (Fig. 11) and a slight tendency toward higher values in the upper ocean near western boundaries. In the 50°–70°S latitude bin, the longitude dependence resembles that of wind forcing (Kilbourne 2015) with elevated values of O(10−4) m2 s−1 in the Indian and Atlantic sectors of the Southern Ocean but weak mixing O(10−5) m2 s−1 in the eastern Pacific sector. However, this pattern is also seen in higher stratification N2 in the eastern Pacific sector compared to the Indian and Atlantic (Fig. 11), and, as already mentioned, this is the latitude band where the finescale parameterization overestimates turbulent dissipation rates in Antarctic Circumpolar Currents (Waterman et al. 2014).

Fig. 11.
Fig. 11.

Depth–longitude sections of turbulent diapycnal diffusivities K and buoyancy frequencies N binned by latitude. Diffusivities are O(10−5) m2 s−1 in most of the ocean but become higher in the North Atlantic (60°W–0°) in the upper ocean of the western boundaries and in parts of the Southern Ocean (30°E –180°, 70°–10°W), where the stratification is weak. Elevated diffusivities in the Southern Ocean correspond to the longitude bands where N is weak and there is elevated inertial wind forcing (Kilbourne 2015) but are also at latitudes of the Antarctic Circumpolar Current where the finescale parameterization overestimates turbulence (Waterman et al. 2014). Elevated diffusivities in the shallowest layer at high latitudes are likely biased high by sharp pynoclines so are not to be trusted.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

6. Tidal mixing parameterizations

The last decade has seen the development (Jayne and St. Laurent 2001; Polzin 2004; Decloedt and Luther 2010) and implementation (St. Laurent et al. 2002; Simmons et al. 2004; Saenko and Merryfield 2005; Jayne 2009; Friedrich et al. 2011; Melet et al. 2013, 2014) of subgrid-scale parameterizations for local tidally driven mixing in OGCMs. In general, the dissipation rate ε can be expressed as
e5
(Jayne and St. Laurent 2001), where E(x, y) is the laterally variable bottom forcing, F(z) is the vertical structure, and q is the fraction lost to turbulent dissipation locally in the overlying water column. OGCM implementations of (5) have assumed constant q = 0.3 and a constant decay scale of 500 m in F(z). As already shown (Figs. 45), turbulent bottom boundary layer thicknesses are extremely variable, often extending through the entire water column. This supports a more dynamically variable turbulent bottom boundary layer thickness, such as Polzin (2004) as implemented in Melet et al. (2013), or Olbers and Eden (2013). Most of the dissipation occurs in the pycnocline (Figs. 3, 7, 8, 9, 10). This suggests that bottom-generated internal tides freely propagate up through the water column, largely dissipating in the upper ocean where higher stratification amplifies the nonlinear cascade.
Likewise, it is known that q is much lower over steep isolated topography (Althaus et al. 2003; Klymak et al. 2006) than over the abyssal hills’ topography characterizing slow midocean spreading ridges (St. Laurent and Garrett 2002). Here, we compare vertically integrated dissipation rates ∫ε with topographic forcing predictions to reiterate that q is not constant but appears to decrease with increasing forcing. In Fig. 12, the vertically integrated dissipation rates ∫ε (Fig. 2) are binned with (i) linear internal tide power input (Bell 1975)
e6
(ii) linear internal lee-wave generation theory (Bell 1975)
e7
and (iii) topographic roughness (height variance) h2, where N is the bottom buoyancy frequency; U is the rms barotropic tidal velocity from TPXO.3 (Egbert and Ray 2001); h2 is the topographic roughness (height variance) on length scales less than that of the mode-one internal tide, which here is taken to be the variance in 30 km × 30 km domains in the Smith and Sandwell (1997) global bathymetric database (10 km × 10 km domains yielded similar dependences); and k is a characteristic horizontal wavenumber that is taken as a free parameter to best match rms global surface tide elevation (Simmons et al. 2004). These linear theories are applicable for weak topography (topographic height h much less than the water depth H and topographic slope s much less than the wave ray slope k/m), which is valid at the semidiurnal frequency for 97% of the bottom based on Smith and Sandwell (1997) bottom slopes (though only 75% above 1500-m water depth). At 1.0–1.2 TW (Egbert and Ray 2001; Nycander 2005), internal tide generation [(6)] is thought to dominate (Waterhouse et al. 2014) over the less certain wind-generated inertial waves power input of 0.2–1.1 TW (Alford 2001; Plueddemann and Farrar 2006; Furuichi et al. 2008; Rimac et al. 2013) and internal lee-wave generation [(7)] of 0.2–0.7 TW (Scott et al. 2011; Nikurashin and Ferrari 2011; Wright et al. 2014; Waterman et al. 2014), but neither these other sources nor remote tidal dissipation are separable in our estimates.
Fig. 12.
Fig. 12.

Full water column vertically integrated dissipation rates ∫ε binned by (left) bottom roughness variance <h2> over 30 km × 30 km, (center) linear internal lee-wave generation [(7)], and (right) linear internal tide generation [(6)]. The upper row displays the joint probability distributions and the lower row the means and standard errors. All integrated dissipation rates ∫ε exclude the upper 380 m where strain estimates of ε may be contaminated by sharp changes in background stratification. Dotted curves in the lower row correspond to the probability distributions of the lower-axis variable. Thick diagonal lines correspond to a linear dependence on the horizontal axes (e.g., h2 in the left set of panels); thin lines indicate the corresponding square root and quartic root of the same. Levels are not meaningful.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

Observed dependence for ∫ε on topographic forcing (Fig. 12) is much weaker than one to one and similar to the h1/2 dependence reported by Kunze et al. (2006). Since saturation has been discounted above, we interpret this as signifying that bottom-forced internal waves are not all locally dissipated (q < 1), and horizontal radiation redistributes a significant fraction of the forcing before dissipation and mixing; that is, most internal-wave-driven mixing is remote from sources. This is consistent with Waterhouse et al. (2014), who reported that most of the 17 microstructure measurement sites they considered had excess forcing compared to dissipation. The weak dependence of ∫ε on forcing in Fig. 12 points to q decreasing with increasing forcing, but there is order-of-magnitude scatter, suggesting unresolved physics.

While Figs. 4 and 12 point to problems with choosing constant local dissipation fractions q and decay scales as in existing tidal mixing parameterizations, it would be premature to suggest better scalings. Order of magnitude scatter is evident in the raw scatterplots (Fig. 12) that may be related to topographic details lost in the coarse Smith and Sandwell topographic roughness h2 and rms tidal flows U used here. Some of the largest internal tide sources are associated with abrupt topography (Fig. 2) such as the Luzon Strait, Hawaiian Ridge, Tuamotu Archipelago, Aleutian Island chain, and so on (Ray and Mitchum 1997; Lee et al. 2006; Simmons et al. 2004; Zhao et al. 2016), for which the weak-topography approximation does not apply. No effort was made to isolate topographic scales relevant to internal tide and lee-wave generation or to include fortnightly tidal modulation or subinertial flows. We have no way of robustly isolating tidal, lee-wave, and wind-forced dissipation sources. Nor do we need internal wave sources and sinks to be correlated in space/time because internal waves propagate (Ray and Mitchum 1997; Alford 2001, Zhao et al. 2016), carrying energy away to dissipate elsewhere and at other times. For example, the canonical q = 0.3 implies 70% of the energy is not dissipated locally. We cannot distinguish locally and remotely forced dissipations, both of which Oka and Niwa (2013) find are necessary to explain the Pacific thermohaline circulation.

Future research is planned to try to better tease apart conditions needed for the turbulent bottom boundary layer to extend through the whole water column and how q depends on topographic forcing, but, because of the limitations in the data, process-oriented numerical modeling may be the best way to explore this parameter space.

7. Conclusions

A finescale parameterization for internal-wave-driven turbulent dissipation rates ε and diapycnal diffusivities K was applied to ~30 000 CTD casts from all the major oceans (Fig. 2), though excluding the Arctic, Weddell, and Ross Seas. The global integrated dissipation of 2.0 ± 0.6 TW is consistent with the 2.1 ± 0.7 TW tide, wind, and lee-wave sources for internal gravity waves, so there may be no need to invoke missing or “dark” turbulent mixing on continental slopes and canyons (Kunze et al. 2006; Waterhouse et al. 2014), though we caution that uncertainties are large enough that, for example, there need be little or no contribution from internal lee waves (Waterman et al. 2014). Of this dissipation, 80%–90% occurs above 1000-m depth and less than 0.08 TW below 2000 m (Fig. 8), compared to the 0.3 TW required to maintain deep stratification in a vertical advective–diffusive balance that ignores horizontal advection (Munk 1966; Munk and Wunsch 1998; Wunsch and Ferrari 2004). As a caveat, because the bulk of the dissipation occurs in the upper pycnocline, deep (>2000-m depth) and abyssal (>4000-m depth) mixing are poorly constrained by the global bulk budget. Nevertheless, it can be concluded that most mixing is remote from deep generation sites. The 256-m half-overlapping spectral bins limit vertical resolution and so may not resolve thin stratified turbulent bottom boundary layers. Vertically integrated dissipation rates ∫ε vary by three to four orders of magnitude (Fig. 2) with elevated values in the western Indian and Pacific Oceans associated with abrupt topography (Fig. 3), consistent with internal tide generation site predictions (Egbert and Ray 2001; Simmons et al. 2004; Nycander 2005). These do not scale with the predictions of linear theories for internal tide or lee-wave generation (Bell 1975; Fig. 12), suggesting that the locally dissipated fraction q decreases with increasing forcing, and significant horizontal redistribution of wave energy occurs before dissipation. Because there is little local dissipation/mixing and considerable redistribution by internal wave propagation, prediction of where and when turbulent dissipation will occur is not straightforward.

Spatial patterns are repeatable (Figs. 4, 6) and show variable turbulent bottom boundary layer thicknesses that often extend throughout the whole water column over rough topography (Figs. 45) in contrast to the fixed 500-m decay scale assumed in OGCM tidal mixing parameterizations (Simmons et al. 2004; Saenko and Merryfield 2005; Jayne 2009; Friedrich et al. 2011). This supports use of a more dynamically motivated parameterization such as Polzin (2004) as implemented in Melet et al. (2013), or Olbers and Eden (2013). Further testing is needed to better determine how local dissipative fraction q and decay scale depend on topography, tidal flows, and other environmental properties.

The global-averaged turbulent diapycnal diffusivity K is almost independent of depth z at (0.3–0.4) × 10−4 m2 s−1 but increases from 10−5 m2 s−1 at 6000 mab to 10−4 m2 s−1 at the bottom (h = 0) in height above bottom h coordinates with an average e-folding scale of ~2000 m (Fig. 7), though this decay scale is not constant (Fig. 4). The difference between the z and h coordinate systems arises because the bottom (h = 0) is not always at the same depth z. Diffusivities vary by two orders of magnitude but cannot be estimated reliably for values exceeding 10 × 10−4 m2 s−1 because of limitations in the methodology. On average, the dissipation rate ε decreases with depth z and density (Fig. 7), though it may display a weak increase with neutral densities greater than 28.2. It is elevated to 10−9 W kg−1 in the bottom 500 mab in height above bottom coordinates but exhibits little variability over this bottom layer, the largest gradient being between 700 and 1200 mab, and it is 0.15 × 10−9 W kg−1 between 1000- and 3000-mab before increasing slowly as h increases. This contrasts with the commonly assumed exponential decay over 500 mab above rough topography in OGCMs based on microstructure measurements in the Brazil Basin (St. Laurent et al. 2001). This may reflect a difference between global and regional behavior or that the finescale parameterization is underestimating near-bottom directly forced turbulence. Consistent with the former interpretation, an average profile that excludes the upper 2000 m (pycnocline) produces a turbulent bottom boundary layer that more closely resembles the canonical decay scale of ~500 mab (dotted curves with h in Fig. 7). The average dissipation rate is 0.2 × 10−9 W kg−1 for buoyancy B < −0.268 (γn > 28.2), increasing to values greater than 2 × 10−9 W kg−1 for B > −0.262 (γn < 27.0). Finescale inferences agree with average direct microstructure measurements at the 17 sites highlighted in Waterhouse et al. (2014).

Zonal averages in all three oceans (Fig. 9) are similar, with weak diffusivities along the equator despite elevated strain variance (Fig. 6) because the N/f dependence in (4) moderates the elevated strain; this contrasts with the off-equatorial stripes of elevated dissipation rate reported in the pycnocline by Whalen et al. (2012) based on more extensive Argo profiling float sampling. Higher diffusivities are found at subpolar latitudes in the Indian and Atlantic but not Pacific, reflecting both their stratification and wind-forcing patterns. However, the finescale parameterization is also known to overestimate turbulence in the Antarctic Circumpolar Current at these latitudes (Waterman et al. 2014). The overall uniformity of K is consistent with ε ~ KN2, and most features in the zonally averaged ε can be related to variability in the stratification rather than diffusivity. No compelling support for tidal parametric subharmonic instability (PSI) enhancing turbulence production equatorward of 14° and 28° latitudes was found (Figs. 910) since K is elevated near 30° in the Pacific but not Atlantic or Indian. Likewise, integrated dissipation rates south of 40°S are less than 0.03 TW (Figs. 2, 10), which does not support 0.1–0.3 TW lee-wave dissipation of Antarctic Circumpolar Currents (Nikurashin and Ferrari 2011; Scott et al. 2011; Melet et al. 2014; Wright et al. 2014) but is consistent with microstructure measurements finding dissipations as much as an order of magnitude below theoretical predictions (Waterman et al. 2014). Again, the finescale parameterization overestimates turbulent dissipation rates in Antarctic Circumpolar Currents (Waterman et al. 2014).

The proxy dataset for global ocean mixing assembled here has shown reasonable skill in reproducing direct microstructure and preconceptions in a broad brush view. While well suited for studying semisteady turbulent sources such as internal tides, the dataset may not be suitable for studying wind-forced internal-wave-driven mixing because of its limited temporal sampling; the Argo profiling float dataset has proven more appropriate for exploring these dependencies (Whalen et al. 2012) and finds little seasonal variability outside storm-forced latitudes 30°–40° (Whalen et al. 2015). The finescale parameterization has also been found to overestimate turbulence where internal lee waves are expected to be the source (Waterman et al. 2014). How well this dataset can reproduce the details of internal-wave-driven mixing despite these known limitations awaits further analysis and comparison, specifically exploration of the decay scale and dissipative fraction q of the stratified turbulent bottom boundary layer associated with internal tide generation in the context of predictions based on internal wave–wave interaction theory (Polzin 2004; Olbers and Eden 2013).

Acknowledgments

For Walter Munk, who started it all, on his 100th birthday. The author acknowledges the efforts of the hundreds of scientists and technicians who collected, processed, and quality controlled the hydrographic data used in this study. Barry Ma and Fiona Lo assisted with data extraction. Discussions with Cimarron Wortham on fitting procedures were valuable. Comments from two anonymous reviewers led to improvements of the manuscript. This research was supported by NSF Grant OCE-1153692. (Internal-wave-driven inferred turbulence dataset is available at ftp.nwra.com/outgoing/kunze/iwturb.)

APPENDIX

Equatorial Internal Waves

Internal gravity waves on the equator are trapped inside a meridional waveguide by β = ∂f/∂y such that they feel off-equatorial rotation f = βy by virtue of having finite meridional scales. Therefore, the equatorial internal wave dispersion relation might be written as
ea1
where ℓ is proportional to an effective meridional wavenumber, k is the zonal wavenumber, and m is the vertical wavenumber. Equation (A1) has a minimum frequency at k = 0 and ℓ, satisfying
ea2
with corresponding frequency
ea3
(Fig. A1). If this scaling is correct, it may be more appropriate to use ωmn [(A3)] instead of f in the N/f scaling [(4)] near the equator to allow finite turbulence production, though Fig. A2 illustrates that wavelengths O(10) m are meridionally confined within ~0.4° of the equator and so will have very low ωmn ~ 10−6 rad s−1 (~2-month periods). In contrast, low vertical modes with O(1000) m vertical wavelengths are confined within ~2° latitude with ωmn ~ 5 × 10−6 rad s−1 (~2-week periods).
Fig. A1.
Fig. A1.

(left) Minimum frequency ωmn (A3) and corresponding latitude (upper axis) where ωmn = f as functions of vertical wavenumber m (vertical coordinate). (right) Meridional length scale L = 1/ℓmn [(A2)] corresponding to the minimum frequency ωmn [(A3)] as a function of vertical wavenumber m. Low vertical wavenumbers are influenced by latitudes ~3°, while finescale waves are much more closely confined within 20 km in their off-equatorial influence.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

Fig. A2.
Fig. A2.

Latitude dependence fArccosh(N/f)/f30Arccosh(N0/f30) [(4)] in the finescale parameterization. The thick solid curve corresponds to the geophysical Coriolis frequency; the thick dotted curve replaces f with ωmn = 10−6 rad s−1 for a vertical wavelength ~O(10) m; the thin dotted curve replaces f with ωmn = 6 × 10−6 rad s−1 corresponding to a vertical wavelength ~O(1000) m.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0141.1

REFERENCES

  • Alford, M. H., 2001: Internal swell generation: The spatial distribution of energy flux from wind to mixed-layer near-inertial motions. J. Phys. Oceanogr., 31, 23592368, doi:10.1175/1520-0485(2001)031<2359:ISGTSD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Alford, M. H., J. A. MacKinnon, Z.-X. Zhao, R. Pinkel, J. Klymak, and T. Peacock, 2007: Internal waves across the Pacific. Geophys. Res. Lett., 34, L24601, doi:10.1029/2007GL031566.

    • Crossref
    • 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.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Alford, M. H., J. B. Girton, G. Voet, G. S. Carter, J. B. Mickett, and J. M. Klymak, 2013: Turbulent mixing and hydraulic control of abyssal water in the Samoan Passage. Geophys. Res. Lett., 40, 46684674, doi:10.1002/grl.50684.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Althaus, A. M., E. Kunze, and T. B. Sanford, 2003: Internal tide radiation from Mendocino Escarpment. J. Phys. Oceanogr., 33, 15101527, doi:10.1175/1520-0485(2003)033<1510:ITRFME>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bell, T. H., 1975: Topographically-generated internal waves in the open ocean. J. Geophys. Res., 80, 320327, doi:10.1029/JC080i003p00320.

  • Cairns, J. L., and G. O. Williams, 1976: Internal-wave observations from a midwater float, 2. J. Geophys. Res., 81, 19431950, doi:10.1029/JC081i012p01943.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Carter, G. S., and M. C. Gregg, 2002: Intense variable mixing near the head of Monterey Submarine Canyon. J. Phys. Oceanogr., 32, 31453165, doi:10.1175/1520-0485(2002)032<3145:IVMNTH>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Carter, G. S., and M. C. Gregg, 2006: Persistent near-diurnal internal waves observed above a site of M2 barotropic-to-baroclinic conversion. J. Phys. Oceanogr., 36, 11361147, doi:10.1175/JPO2884.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Carter, G. S., M. C. Gregg, and R.-C. Lien, 2005: Internal waves, solitary-like waves and mixing on the Monterey Bay shelf. Cont. Shelf Res., 25, 14991520, doi:10.1016/j.csr.2005.04.011.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Damerell, G. M., K. J. Heywood, D. P. Stevens, and A. C. Naveira Garabato, 2012: Temporal variability of diapycnal mixing in Shag Rocks Passage. J. Phys. Oceanogr., 42, 370385, doi:10.1175/2011JPO4573.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • D’Asaro, E. A., and J. H. Morison, 1992: Internal waves and mixing in the Arctic Ocean. Deep-Sea Res., 39A, S459S484, doi:10.1016/S0198-0149(06)80016-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • D’Asaro, E. A., and R.-C. Lien, 2000: The wave–turbulence transition for stratified flows. J. Phys. Oceanogr., 30, 16691678, doi:10.1175/1520-0485(2000)030<1669:TWTTFS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Decloedt, T., and D. S. Luther, 2010: On a simple empirical parameterization of topography-catalyzed diapycnal mixing in the abyssal ocean. J. Phys. Oceanogr., 40, 487508, doi:10.1175/2009JPO4275.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Egbert, G. D., and R. Ray, 2001: Estimates of M2 tidal energy dissipation from TOPEX/Poseidon altimeter data. J. Geophys. Res., 106, 22 47522 502, doi:10.1029/2000JC000699.

    • Crossref
    • 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.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Friedrich, T., A. Timmermann, T. Decloedt, D. S. Luther, and A. Mouchet, 2011: The effect of topography-enhanced diapynal mixing on ocean and atmospheric circulation and marine biogeochemistry. Ocean Modell., 39, 262274, doi:10.1016/j.ocemod.2011.04.012.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fritts, D. C., 1984: Gravity wave saturation in the middle atmosphere: A review of theory and observations. Rev. Geophys., 22, 275308, doi:10.1029/RG022i003p00275.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Furuichi, N., T. Hibiya, and Y. Niwa, 2008: Model-predicted distribution of wind-induced internal wave energy in the world’s oceans. J. Geophys. Res., 113, C09034, doi:10.1029/2008JC004768.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gargett, A. E., 1990: Do we really know how to scale the turbulent kinetic energy dissipation rate ε due to breaking of oceanic internal waves? J. Geophys. Res., 95, 15 97115 974, doi:10.1029/JC095iC09p15971.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gregg, M. C., 1989: Scaling turbulent dissipation in the thermocline. J. Geophys. Res., 94, 96869698, doi:10.1029/JC094iC07p09686.

  • Gregg, M. C., and T. B. Sanford, 1987: Shear and turbulence in thermohaline staircases. Deep-Sea Res., 34, 16891696, doi:10.1016/0198-0149(87)90017-3.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gregg, M. C., and T. B. Sanford, 1988: The dependence of turbulent dissipation on stratification in a diffusively stable thermocline. J. Geophys. Res., 93, 12 38112 392, doi:10.1029/JC093iC10p12381.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gregg, M. C., and E. Kunze, 1991: Internal wave shear and strain in Santa Monica Basin. J. Geophys. Res., 96, 16 70916 719, doi:10.1029/91JC01385.

  • Gregg, M. C., T. B. Sanford, and D. P. Winkel, 2003: Reduced mixing from the breaking of internal waves in equatorial ocean waters. Nature, 422, 513515, doi:10.1038/nature01507.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Henyey, F. S., 1991: Scaling of internal wave predictions for ε. Dynamics of Internal Gravity Waves in the Ocean: Proc. ‘Aha Huliko’a Hawaiian Winter Workshop, Honolulu, HI, University of Hawai‘i at Mānoa, 233–236.

  • Henyey, F. S., J. Wright, and S. M. Flatte, 1986: Energy and action flow through the internal wave field: An eikonal approach. J. Geophys. Res., 91, 84878495, doi:10.1029/JC091iC07p08487.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hibiya, T., M. Nagasawa, and Y. Niwa, 1999: Model-predicted distribution of internal wave energy for diapycnal mixing processes in the deep waters of the North Pacific. Dynamics of Oceanic Internal Gravity Waves II: Proc.‘Aha Huliko’a Hawaiian Winter Workshop, Honolulu, HI, University of Hawai‘i at Mānoa, 205–213.

  • Hibiya, T., M. Nagasawa, and Y. Niwa, 2006: Global mapping of diapycnal diffusivity in the deep ocean based on the results of expendable current profiler (XCP) surveys. Geophys. Res. Lett., 33, L03611, doi:10.1029/2005GL025218.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Huussen, T., A. C. Naveira Garabato, H. L. Bryden, and E. L. McDogangh, 2012: Is the Indian Ocean MOC driven by breaking internal waves? J. Geophys. Res., 117, C08024, doi:10.1029/2012JC008236.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Itsweire, E. C., K. N. Helland, and C. W. van Atta, 1986: The evolution of grid-generated turbulence in a stably-stratified fluid. J. Fluid Mech., 162, 299338, doi:10.1017/S0022112086002069.

    • Crossref
    • 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.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jayne, S. R., and L. C. St. Laurent, 2001: Parameterizing tidal dissipation over rough topography. Geophys. Res. Lett., 28, 811814, doi:10.1029/2000GL012044.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jochum, M., 2009: Impact of latitudinal variations in vertical diffusivity on climate simulations. J. Geophys. Res., 114, C01010, doi:10.1029/2008JC005030.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kilbourne, B. F., 2015: On the topic of ocean variability near the Coriolis frequency: Generation mechanisms, observations and implications for interior mixing. Ph.D. thesis, University of Washington, 124 pp.

  • Klymak, J. M., and Coauthors, 2006: An estimate of tidal energy lost to turbulence at the Hawaiian Ridge. J. Phys. Oceanogr., 36, 11481164, doi:10.1175/JPO2885.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Klymak, J. M., R. Pinkel, and L. Rainville, 2008: Direct breaking of the internal tide near topography: Kaena Ridge, Hawaii. J. Phys. Oceanogr., 38, 380399, doi:10.1175/2007JPO3728.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Klymak, J. M., S. Legg, and R. Pinkel, 2010: A simple parameterization of turbulent tidal mixing near supercritical topography. J. Phys. Oceanogr., 40, 20592074, doi:10.1175/2010JPO4396.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kunze, E., 2003: A review of oceanic salt-fingering theory. Prog. Oceanogr., 56, 399417, doi:10.1016/S0079-6611(03)00027-2.

  • Kunze, E., and T. B. Sanford, 1996: Abyssal mixing: Where it isn’t. J. Phys. Oceanogr., 26, 22862296, doi:10.1175/1520-0485(1996)026<2286:AMWIIN>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kunze, E., and J. M. Toole, 1997: Tidally driven vorticity, diurnal shear, and turbulence atop Fieberling Seamount. J. Phys. Oceanogr., 27, 26632693, doi:10.1175/1520-0485(1997)027<2663:TDVDSA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kunze, E., A. J. Williams III, and M. G. Briscoe, 1990: Observations of shear and vertical stability from a neutrally buoyant float. J. Geophys. Res., 95, 18 12718 142, doi:10.1029/JC095iC10p18127.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kunze, E., M. A. Kennelly, and T. B. Sanford, 1992: The depth dependence of shear finestructure off Point Arena and near Pioneer Seamount. J. Phys. Oceanogr., 22, 2941, doi:10.1175/1520-0485-22.1.29.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kunze, E., E. Firing, J. M. Hummon, T. K. Chereskin, and A. M. Thurnherr, 2006: Global abyssal mixing inferred from lowered ADCP shear and CTD strain profiles. J. Phys. Oceanogr., 36, 15531576, doi:10.1175/JPO2926.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kunze, E., C. MacKay, E. E. McPhee-Shaw, K. Morrice, J. B. Girton, and S. R. Terker, 2012: Turbulent mixing and exchange with interior waters on sloping boundaries. J. Phys. Oceanogr., 42, 910927, doi:10.1175/JPO-D-11-075.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lauderdale, J. M., S. Bacon, A. C. Naveira Garabato, and N. P. Holliday, 2008: Intensified turbulent mixing in the boundary current system of southern Greenland. Geophys. Res. Lett., 35, L04611, doi:10.1029/2007GL032785.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ledwell, J. R., and A. J. Watson, 1991: The Santa Monica Basin tracer experiment: A study of diapycnal and isopycnal mixing. J. Geophys. Res., 96, 86958718, doi:10.1029/91JC00102.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lee, C. M., E. Kunze, T. B. Sanford, J. D. Nash, M. A. Merrifield, and P. E. Holloway, 2006: Internal tides and turbulence along the 3000-m isobath of the Hawaiian Ridge. J. Phys. Oceanogr., 36, 11651183, doi:10.1175/JPO2886.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lueck, R. G., and T. D. Mudge, 1997: Topographically induced mixing around a shallow seamount. Science, 276, 18311833, doi:10.1126/science.276.5320.1831.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lumpkin, R., and K. Speer, 2003: Large-scale vertical and horizontal circulation in the North Atlantic Ocean. J. Phys. Oceanogr., 33, 19021920, doi:10.1175/1520-0485(2003)033<1902:LVAHCI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • MacKinnon, J. A., and