1. Introduction
The dissipation flux coefficient Γ is related to mixing efficiency (i.e., the ratio of the background potential energy gain to the available mechanical energy loss due to irreversible mixing) or the flux Richardson number Rf as Γ = Rf/(1 − Rf). Conventionally, Rf is defined as the ratio of the buoyancy flux term to the shear production term in the TKE equation. As noted by Venayagamoorthy and Koseff (2016), however, Rf in this definition represents the mixing efficiency only for the case of stationary homogeneous shear-driven turbulence. Actually, for the case of convective-driven turbulence (Scotti 2015), the buoyancy flux represents not a sink of TKE to the background potential energy, but a source of TKE from the available potential energy, so that the conventional definition of Rf is useless for a general representation of the mixing efficiency. To more rigorously account for irreversible energy conversions due to turbulent mixing, several authors (Peltier and Caulfield 2003; Venayagamoorthy and Koseff 2016) redefined Rf as the ratio of the turbulent available potential energy (TAPE) dissipation (i.e., the background potential energy production) to the total turbulent energy dissipation. In this framework, Rf directly represents the efficiency of irreversible mixing, and Γ is given by the TAPE/TKE dissipation ratio.
Direct numerical simulations (DNSs) of stratified turbulent flows have shown that values of Γ are highly variable, depending on different triggering mechanisms and evolution stages of turbulent mixing. Smyth et al. (2001) showed that simulated Γ varies by more than an order of magnitude over the time elapsed from the onset of Kelvin–Helmholtz (KH) instability: the shear-driven mixing is very efficient such that Γ reaches O(1) during the initial growth of KH billows, but Γ decreases to O(0.1) after the collapse of the billows into more complicated flows. They also showed that parameter ROT, defined as the ratio of the Ozmidov scale (Ozmidov 1965) to the Thorpe scale (Thorpe 1977), monotonically increases with time from O(0.1) to O(1), so that a negative relationship between Γ and ROT is obtained. Such time-dependent behavior, however, is not found for the case of convective-driven mixing. Scotti (2015) showed that efficient mixing with Γ ~ O(1) and ROT ~ O(0.1) found during the young stage of shear-driven mixing continues over the entire time from the onset of convective instability.
In addition to ROT, the buoyancy Reynolds number Reb, sometimes called the Gibson number, has been thought to be another key parameter that controls Γ. Shih et al. (2005) analyzed DNS data obtained during the mature stage of shear-driven mixing to show that Γ decreases with Reb as 
The variability of Γ thus demonstrated by a series of DNS studies has a significant impact on OGCM results (de Lavergne et al. 2016; Mashayek et al. 2017); its applicability to the real ocean, therefore, should be comprehensively assessed. However, there is little observational evidence supporting the variability of Γ against ROT and Reb. In particular, the previous estimates of Γ were made mostly in the strongly stratified upper ocean, but rarely in the weakly stratified deep ocean where more obvious variations of Γ might be observed.
This study therefore examines variations of Γ using deep microstructure profiles collected in various regions of the North Pacific and Southern Oceans. Details of the data and methods are described in section 2. Observed spatial variations of Γ and relationships of Γ to ROT and Reb are presented in sections 3a and 3b, respectively. Furthermore, a theoretical scaling of Γ consistent with the observed results is explored in section 3c. These results have important implications for underlying mechanisms of turbulent mixing, the applicability of the widely used Thorpe-scale-based parameterization of ε, and closure of the global overturning circulation, which are discussed in section 4. Finally, conclusions are presented in section 5.
2. Data and methods
Data collection sites in the (a) western and (b) central North Pacific Ocean and (c) Southern Ocean superposed on bathymetric contours with 1000-m intervals. The color of each dot denotes the vertical distance of the maximum measurement depth from the bottom.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Data collection sites in the (a) western and (b) central North Pacific Ocean and (c) Southern Ocean superposed on bathymetric contours with 1000-m intervals. The color of each dot denotes the vertical distance of the maximum measurement depth from the bottom.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Data collection sites in the (a) western and (b) central North Pacific Ocean and (c) Southern Ocean superposed on bathymetric contours with 1000-m intervals. The color of each dot denotes the vertical distance of the maximum measurement depth from the bottom.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
a. Data collection sites
Out of the total 55 VMP-5500 casts, 23 casts were made in the Izu–Ogasawara Ridge (Fig. 1a) during several cruises of the training vessel (T/V) Oshoro-Maru of Hokkaido University in November 2008 and December 2011 and the T/V Shinyo-Maru of Tokyo University of Marine Science and Technology in October 2012, October 2013, December 2014, and December 2016. Most of the casts reached down to within 200 m above the bottom. The Izu–Ogasawara Ridge is one of the prominent generation sites of semidiurnal internal tides (e.g., Niwa and Hibiya 2014) and crosses the critical latitude of 28.8°N for parametric subharmonic instability (PSI; McComas and Bretherton 1977) of the semidiurnal internal tides from north to south, so that PSI-induced strong turbulent mixing is expected (e.g., Hibiya et al. 2002). Actually, the microstructure data show higher dissipation rates in the Izu–Ogasawara Ridge (Fig. 2).
Histogram of the TKE dissipation rate ε in each observed region.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Histogram of the TKE dissipation rate ε in each observed region.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Histogram of the TKE dissipation rate ε in each observed region.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
In the Aleutian Ridge (Fig. 1b), 11 casts were made during cruises of the T/V Oshoro-Maru in July 2007, June 2008, and June 2009. More than half of the casts reached down to within 200 m above the bottom. Although a significant amount of semidiurnal internal tidal energy is generated also in the Aleutian Ridge (e.g., Niwa and Hibiya 2014), most of it is thought to be unavailable for the local mixing because the Aleutian Ridge lies far north of the critical latitude for PSI of the semidiurnal internal tides (e.g., Hibiya et al. 2002). The microstructure data actually show lower dissipation rates in the Aleutian Ridge than in the Izu–Ogasawara Ridge (Fig. 2).
In the Australian–Antarctic Basin (Fig. 1c), seven casts were made during a cruise of the T/V Umitaka-Maru of Tokyo University of Marine Science and Technology in January 2015. Most of the casts reached down to within 200 m above the bottom. There is growing evidence that turbulent mixing is greatly enhanced in strong frontal regions of the Antarctic Circumpolar Current (St. Laurent et al. 2012; Waterman et al. 2013; Sheen et al. 2013) and over the Antarctic continental slope (Mead Silvester et al. 2014; Fer et al. 2016), while being weak in most other regions of the Southern Ocean (Ledwell et al. 2011). Because the casts were made in non- or weak frontal regions of the Australian–Antarctic basin characterized by a smooth abyssal plain, much lower dissipation rates were observed (Fig. 2). Note that several density-compensated intrusions were observed, particularly in a transition zone between Lower Circumpolar Deep Water and Antarctic Bottom Water, where the method using (3) cannot be applied (section 2d).
In the Kerama Gap (Fig. 1a), seven casts were made during a cruise of the T/V Kagoshima-Maru of Kagoshima University in June 2013 (Nishina et al. 2016). The Kerama Gap is the deepest channel connecting the East China Sea to the northwestern North Pacific Ocean, where intermediate water is thought to be modified due to strong mixing over sills (Nakamura et al. 2013). Unfortunately, the VMP-5500 was tethered throughout the cruise because of trouble with a ballast release system, so the maximum depth of the casts was limited to 900 m, leaving near-bottom mixing not fully observed.
In addition to the major datasets described above, we used small datasets collected over seamounts and in bays to include the most diverse oceanic environments possible in the following analysis. During the cruises in July 2007 and June 2008, four casts were made over the Emperor Seamounts (Fig. 1b), where bottom-intensified mixing as observed over Fieberling Seamount (Kunze and Toole 1997; Toole et al. 1997) was expected. During the cruises in October 2012 and October 2013, three casts were made in the deep troughs of Suruga Bay and Sagami Bay (Fig. 1a), where large-amplitude internal tides had been observed (Ohwaki et al. 1991; Matsuyama et al. 1993; Kitade and Matsuyama 1997).
b. Microstructure data processing
Samples of the vertical wavenumber shear spectrum Ψ∂u/∂z(k) and the temperature gradient spectrum Ψ∂T/∂z(k), together with the corresponding theoretical spectra for the case of (a) stronger or (b) weaker dissipation. The Kolmogorov wavenumber kK and the Batchelor wavenumber kB are denoted by the red and blue triangles, respectively. The upper integration limits in (4) and (5) are denoted by the red and blue dashed lines, respectively. The threshold noise level for Ψ∂T/∂z(k) is denoted by the black line.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Samples of the vertical wavenumber shear spectrum Ψ∂u/∂z(k) and the temperature gradient spectrum Ψ∂T/∂z(k), together with the corresponding theoretical spectra for the case of (a) stronger or (b) weaker dissipation. The Kolmogorov wavenumber kK and the Batchelor wavenumber kB are denoted by the red and blue triangles, respectively. The upper integration limits in (4) and (5) are denoted by the red and blue dashed lines, respectively. The threshold noise level for Ψ∂T/∂z(k) is denoted by the black line.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Samples of the vertical wavenumber shear spectrum Ψ∂u/∂z(k) and the temperature gradient spectrum Ψ∂T/∂z(k), together with the corresponding theoretical spectra for the case of (a) stronger or (b) weaker dissipation. The Kolmogorov wavenumber kK and the Batchelor wavenumber kB are denoted by the red and blue triangles, respectively. The upper integration limits in (4) and (5) are denoted by the red and blue dashed lines, respectively. The threshold noise level for Ψ∂T/∂z(k) is denoted by the black line.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
c. Hydrographic data processing
Spurious salinity spikes arising from dissimilar response characteristics of the SeaBird conductivity–temperature (CT) sensors (Horne and Toole 1980) and thermal inertia of the conductivity cell (Lueck and Picklo 1990) were substantially reduced by applying several filters to CT signals. These included a response-matching filter estimated from the cross-spectrum of CT (Anderson 1993) and a thermal-lag correction filter (Morison et al. 1994). A median filter was also applied to CT with bin widths ranging from 2 to 20 dbar that were determined by the amount and size of residual salinity spikes. In return, however, genuine overturns would also be smoothed out by this SeaBird data processing. To overcome this problem, we combined the high-resolution FP07 data with the smoothed SeaBird data to estimate 
d. Turbulent patch identification
Following Mater et al. (2015), we identified each turbulent patch using the cumulative Thorpe displacements ΣδT: ΣδT remains nonzero within a patch but becomes zero at its boundaries (Fig. 4c). If the width of a turbulent patch thus identified was less than 5 dbar, several adjacent patches were merged into one composite patch whose width became larger than 5 dbar (Fig. 4f), so as to inevitably include bins for the calculations of ε and χT (section 2b). Here, δT was calculated as the difference in depth of each fluid parcel between the unsorted and sorted profiles of Θ (Figs. 4b,e). Since Θ was used as a surrogate for potential density, a near-surface temperature minimum layer at high latitudes must be excluded from this reordering process. We therefore excluded the uppermost 500-m layer in the Aleutian Ridge and the uppermost 800-m layer in the Australian–Antarctic basin, but the uppermost 50-m layer otherwise. We also excluded turbulent patches with 
Samples of the (left) potential temperature Θ, (center) Thorpe displacement δT, and (right) cumulative Thorpe displacements ΣδT for the case that the turbulent patch enclosed by the horizontal lines is identified (top) without or (bottom) with the merging process.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Samples of the (left) potential temperature Θ, (center) Thorpe displacement δT, and (right) cumulative Thorpe displacements ΣδT for the case that the turbulent patch enclosed by the horizontal lines is identified (top) without or (bottom) with the merging process.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Samples of the (left) potential temperature Θ, (center) Thorpe displacement δT, and (right) cumulative Thorpe displacements ΣδT for the case that the turbulent patch enclosed by the horizontal lines is identified (top) without or (bottom) with the merging process.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
e. Patch-averaged stratification
There are two types of 
3. Results
a. Observed spatial variations of Γ
We first examine the validity of the constant Γ in the stratified ocean interior that has been widely assumed. Figure 5 shows histograms of Γ classified in terms of regions and depth ranges. Note that turbulent patches with Reb ≤ 40 were not counted in these histograms, which might be affected by differential diffusion (Jackson and Rehmann 2014) and anisotropic turbulence (Yamazaki and Osborn 1990). This exclusion caused the decrease of available patches, particularly in the Australian–Antarctic basin where extremely low dissipation rates were observed (Fig. 2), resulting in the distribution of Γ far from reliable (Fig. 5c). We therefore begin with the histograms of Γ in the Izu–Ogasawara Ridge (thick lines in Fig. 5a), the most reliable ones because of their larger number of patch elements with higher dissipation rates (Fig. 2). It is apparent that observed Γ varies widely in a lognormal fashion as observed by the previous studies (Moum 1996; Ruddick et al. 1997). Of special notice is that the peak of the histogram significantly shifts to a larger value as the depth deepens: the peak is found near the conventional value of 0.2 in the uppermost 500 m, largely consistent with the previous observations as reviewed by Gregg et al. (2018), but beyond 1 in a depth range of 2500–3000 m. Such a tendency can be found also in the other regions (Figs. 5b–f), although the number of elements in each histogram may not be sufficient.
Histograms of Γ classified in terms of regions and depth ranges. Turbulent patches with Reb ≤ 40 are not counted in these histograms. Note that thin lines in (a) show additional histograms of Γ in the Izu–Ogasawara Ridge using only turbulent patches with 〈χT〉 larger than 5.8 × 10−11 °C2 s−1, the median of the χT distribution in this region.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Histograms of Γ classified in terms of regions and depth ranges. Turbulent patches with Reb ≤ 40 are not counted in these histograms. Note that thin lines in (a) show additional histograms of Γ in the Izu–Ogasawara Ridge using only turbulent patches with 〈χT〉 larger than 5.8 × 10−11 °C2 s−1, the median of the χT distribution in this region.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Histograms of Γ classified in terms of regions and depth ranges. Turbulent patches with Reb ≤ 40 are not counted in these histograms. Note that thin lines in (a) show additional histograms of Γ in the Izu–Ogasawara Ridge using only turbulent patches with 〈χT〉 larger than 5.8 × 10−11 °C2 s−1, the median of the χT distribution in this region.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
However, there is a concern that the observed larger values of Γ in the deeper ocean might be contaminated by noise: Γ [(3)] might be overestimated if low values of χT were not fully resolved. To address this concern, we show additional histograms of Γ in the Izu–Ogasawara Ridge (thin lines in Fig. 5a) using only turbulent patches with 〈χT〉 larger than the median of the χT distribution in this region (specifically, 5.8 × 10−11 °C2 s−1). A similar peak shift with depth can also be found from these histograms. On this basis, we believe that the observed spatial variations of Γ are actually reflecting the nature of oceanic turbulence. This result suggests the highly variable nature of Γ and, more importantly, larger values of Kρ as well as Γ than previously thought, particularly in the deep ocean, whose possible impacts on closure of the global overturning circulation are discussed in section 4c.
b. Observed relationships of Γ to
We next address which parameter controls the variations of Γ thus observed (section 3a). Here, we focus on two observed parameters, Reb and ROT, although there might be some other parameters controlling Γ, such as the Richardson number (Mashayek et al. 2013) and the Reynolds number (Mashayek and Peltier 2013). Figures 6 and 7 show observed variations of Γ against Reb and ROT. It appears that there are no definite relationships between Γ and Reb regardless of the observed regions (see scatterplots in Fig. 6 and histograms of Γ in Fig. 7). In particular, the scatterplots for ROT > 1 (blue dots in Fig. 6) do not support the negative relationship 
Observed variations of Γ against Reb classified in terms of regions and ROT ranges.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Observed variations of Γ against Reb classified in terms of regions and ROT ranges.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Observed variations of Γ against Reb classified in terms of regions and ROT ranges.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Observed variations of Γ against ROT classified in terms of regions and Reb ranges.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Observed variations of Γ against ROT classified in terms of regions and Reb ranges.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Observed variations of Γ against ROT classified in terms of regions and Reb ranges.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
In contrast, significant variations of Γ against ROT can be confirmed regardless of the Reb ranges and the observed regions (see scatterplots in Fig. 7 and histograms of Γ in Fig. 6): Γ ~ O(1) for ROT ~ O(0.1), whereas Γ ~ O(0.1) for ROT ~ O(1). Such negative relationship was also obtained by Smyth et al. (2001) through the analysis of upper-ocean datasets as well as DNS results. Note that we used the deep-ocean datasets to confirm the negative relationship between Γ and ROT for the range of ROT lower than that covered by Smyth et al. (2001). Combining this observed result and the previous DNS results gives an insight into underlying mechanisms of deep-ocean mixing, which is discussed in section 4a. In addition, the observed significant variations of ROT (Fig. 7) suggest the limitation of the validity of the Thorpe-scale-based parameterization of ε that has been widely applied to hydrographic datasets, which is discussed in section 4b.
c. Theoretical scaling of Γ in terms of
This simple theoretical scaling is consistent with the observed negative relationship between Γ and ROT (Fig. 8). Since the formulation of Γ under the high Reynolds number limit has thus been supported, we argue that Γ in the ocean interior should not be scaled in terms of the viscosity-related parameter Reb, particularly in the energetic regime with Reb > 100. This argument is consistent with the observations (section 3b) but inconsistent with the DNS studies (Shih et al. 2005; Bouffard and Boegman 2013). Such inconsistency may be explained from the fact that DNS still cannot simulate sufficiently high Reynolds number turbulent flows. Garrett (2001, p. 6) also commented on this point: “Perhaps an apparent dependence of Γ on 
Observed relationship of Γ to ROT (black dots and bars) and the theoretical scaling 
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Observed relationship of Γ to ROT (black dots and bars) and the theoretical scaling
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Observed relationship of Γ to ROT (black dots and bars) and the theoretical scaling
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
We note that our theoretical and observational results exhibit the ROT dependence of Γ slightly stronger than Smyth et al. (2001)’s DNS results. We suspect that their DNS might be performed with Reynolds numbers not high enough to quantitatively address Γ in the real ocean; actually, their simulated Γ depends not only on ROT but also on the Prandtl number, another viscosity-related parameter. The ROT dependence of Γ should be examined through DNS with much higher Reynolds numbers in the future.
4. Discussion
a. Underlying mechanisms of turbulent mixing in the ocean interior
Combining the results from the previous DNS (Smyth et al. 2001; Scotti 2015) and from our observations (sections 3a and 3b) gives an insight into possible underlying mechanisms of turbulent mixing in the ocean interior: moderate mixing with Γ ~ O(0.1) and ROT ~ O(1) observed in the upper ocean may reflect the mature stage of shear-driven mixing, whereas efficient mixing with Γ ~ O(1) and ROT ~ O(0.1) observed in the deep ocean may reflect convective-driven mixing and/or the young stage of shear-driven mixing. We expect that the shear-driven mixing would reach the mature stage more slowly in the deep ocean than in the upper ocean because the turbulent time scale is negatively related to the stratification in the high Reynolds number limit (Baumert and Peters 2004). In the deep ocean, bottom-generated internal waves may then break, causing mixing before the previously induced mixing reaches the mature stage and resulting in the observed overall young and efficient mixing. Considering large amplitudes of bottom-generated internal waves, the convective-driven mixing seems to occur in the deep ocean. Nevertheless, the results from the observations (section 3c) are consistent with the theoretical scaling of Γ [(8)] that is not applicable to the convective-driven mixing, suggesting that the efficient mixing with Γ ~ O(1) and ROT ~ O(0.1) observed in the deep ocean may reflect the young stage of the shear-driven mixing. It should be noted that the moderate mixing with Γ ~ O(0.1) and ROT ~ O(1) is not universal throughout the upper ocean, as in the Luzon Strait where Mater et al. (2015) found small values of ROT associated with convective collapse of large-amplitude internal waves. Obviously, more observational, numerical, and theoretical studies are necessary to clarify actual mechanisms of turbulent mixing in the ocean interior.
b. Applicability of the Thorpe-scale-based parameterization of ε
Scatterplots of microstructure-based estimates of the TKE dissipation rate εmicro against its Thorpe-scale-based estimates εThorpe classified in terms of regions and depth ranges. Here, εThorpe is estimated from (11) with ROT = 1. Turbulent patches with Reb ≤ 40 are excluded from these plots.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Scatterplots of microstructure-based estimates of the TKE dissipation rate εmicro against its Thorpe-scale-based estimates εThorpe classified in terms of regions and depth ranges. Here, εThorpe is estimated from (11) with ROT = 1. Turbulent patches with Reb ≤ 40 are excluded from these plots.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Scatterplots of microstructure-based estimates of the TKE dissipation rate εmicro against its Thorpe-scale-based estimates εThorpe classified in terms of regions and depth ranges. Here, εThorpe is estimated from (11) with ROT = 1. Turbulent patches with Reb ≤ 40 are excluded from these plots.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Here, we cannot avoid mentioning inconsistency between this study and Nishina et al. (2016), both of which used the same microstructure profiles collected in the Kerama Gap (section 2a) to assess the Thorpe-scale-based parameterization of ε. Nishina et al. (2016) showed that the parameterization tends to overestimate ε by an order of magnitude (Fig. 5 of Nishina et al. 2016), whereas this study has shown its high performance in this region (Fig. 9d). We suspect that Nishina et al. (2016) failed to calculate 〈N〉 in (11) appropriately: 〈N〉 was calculated not from the bulk gradient (section 2e) but from the smoothed mean gradient (H. Nakamura and A. Nishina 2017, personal communication), so that turbulent patch properties might be lost considerably.
c. Closure of the global overturning circulation
5. Conclusions
It is still controversial whether or not the dissipation flux coefficient Γ in the Osborn’s eddy diffusivity model is constant throughout the stratified ocean interior. Motivated by lack of observational estimates of Γ, particularly under weakly stratified deep-ocean conditions, we have examined variations of Γ using deep microstructure profiles collected in various regions of the North Pacific and Southern Oceans. We have shown that Γ is not constant but varies significantly with the Ozmidov/Thorpe scale ratio ROT in a fashion similar to that obtained by the previous DNS study on the evolution of shear-driven mixing (Smyth et al. 2001). Of special notice is that efficient mixing events with Γ ~ O(1) and ROT ~ O(0.1) tend to be frequently observed in the deep ocean (i.e., weak stratification), whereas moderate mixing events with Γ ~ O(0.1) and ROT ~ O(1) tend to be observed in the upper ocean (i.e., strong stratification). Referring to the DNS and theoretical studies (Smyth et al. 2001; Baumert and Peters 2004), we have speculated that the vertical distributions of Γ and ROT thus observed may reflect the stratification dependence of the time required for the shear-driven mixing to reach the mature stage. The observed small values of ROT and large values of Γ in the deep ocean, respectively, imply overestimates of the TKE dissipation rate by the widely used Thorpe-scale-based parameterization and underestimates of Γ by the conventional fixed model, resulting in less biased Thorpe-scale-based estimates of eddy diffusivity. The observed vertical distribution of Γ implies that upwelling due to interior mixing may be weaker than previously thought. Given the potential importance of these implications, the universality of the observed distributions of Γ and ROT should be checked through many more observations in the near future.
Furthermore, using classical turbulent theories under the high Reynolds number limit, we have derived the simple scaling 
Admittedly, the above conclusions are based on indirect estimates of Γ; the TAPE dissipation rate is approximated in terms of the thermal variance dissipation rate,2 so that estimated Γ might be biased under the presence of differential diffusion or double diffusion. However, we believe that our observed results are not largely biased since we have discarded data favorable to differential diffusion (Jackson and Rehmann 2014) and double diffusion (St. Laurent and Schmitt 1999) using Reb and the density stability ratio. Another concerning issue is that observations cannot deal with a three-dimensionally sorted density field, from which the background stratification should be ideally calculated (Scotti and White 2014). Using DNS, Arthur et al. (2017) showed that the usual method to calculate the background stratification from a locally sorted vertical density profile yields biased estimates of Γ in an inhomogeneous turbulent flow near a boundary. Nevertheless, we believe that this does not seriously matter in the ocean interior, most of our observed oceanic environments, where locally homogeneous turbulence would dominate.
Acknowledgments
This study was supported by MEXT KAKENHI Grant JP15H05824 and JSPS KAKENHI Grant JP15H02131. The authors express their gratitude to two anonymous reviewers for their invaluable comments on the original manuscript. Data used for producing the results herein may be requested by contacting the authors.
APPENDIX
Validity of Patch-Averaged Stratification Estimates
As described in section 2e, we used 
Scatterplots of Γ against (a) Reb and (b) ROT for the case of 
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Scatterplots of Γ against (a) Reb and (b) ROT for the case of
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
Scatterplots of Γ against (a) Reb and (b) ROT for the case of
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0275.1
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Remember that ROT > 1 corresponds to the mature stage of shear-driven mixing according to Smyth et al. (2001), as introduced in section 1.
Remember that Γ is defined as the TAPE/TKE dissipation ratio.
