## 1. Introduction

Away from surface and bottom boundary layers the magnitude and geography of diapycnal mixing in the ocean interior is largely set by the dynamics of breaking internal gravity waves. Over the last two decades it has become clear that wave breaking, and the resultant turbulent mixing, is strongly inhomogeneous in both space and time. Limited microstructure measurements show typical diapycnal diffusivities in the main thermocline of order 10^{−5} m^{2} s^{−1} (Gregg 1987), but recent estimates inferred from finescale strain show spatial variations over two orders of magnitude (Whalen et al. 2012). The patterns are controlled in part by the geography of internal wave generation, propagation, interaction, and dissipation. In turn, the patchiness of diapycnal mixing has significant consequences for both regional and global flow patterns (Jochum and Potemra 2008; Harrison and Hallberg 2008; Jayne 2009). Current climate models include little if any of these patterns; improvement requires both a better map of where mixing is happening and a better dynamical understanding of why, so that mixing may be incorporated in models of both present and future climate (Wunsch and Ferrari 2004).

Much recent work has focused on striking patterns of elevated mixing near the generation sites of internal tides (Polzin et al. 1997) or wind-driven near-inertial waves (Alford and Gregg 2001). Internal tides are generated where the barotropic tide flows over rough topography, generating internal waves on a variety of scales. Waves with smaller vertical scales tend to break nearby, producing a global pattern of mixing hotspots that resembles a map of internal tide generation (St. Laurent et al. 2002; Polzin 2004). However, at some locations, particularly near tall, steep topography like the Hawaiian Ridge, most generated internal tide energy escapes to radiate up to thousands of km away; where and how these waves break is as yet unknown (St. Laurent and Garrett 2002; Klymak et al. 2006; Zhao et al. 2010). As a community we are challenged to map “farfield” patterns of energy loss from the propagating component of the tide and associated patterns of mixing across ocean basins. Candidates for wave dissipation include scattering off deep sea topography (Müller and Xu 1992; Johnston et al. 2003) and reflection and wave breaking at the continental shelf break (Nash et al. 2004, 2007; Martini et al. 2011; Kelly et al. 2012). Alternately, a propagating internal tide can steadily lose energy through nonlinear interactions with other waves or mesoscale features (Gregg 1989; Polzin 2008). Differentiating between these candidates is an essential part of developing proper understanding and parameterizations of mixing. Part of the difficulty in differentiating the role of each mechanism is the relative sparseness of mixing observations (Gregg 1998), the most accurate of which are expensive and require specialized microstructure programs. Recent techniques that allow rough estimates of the turbulent mixing rate from finescale measurements of shear and strain have considerably broadened the available data, though great care must be taken in interpreting the results (section 3b).

The Internal Waves Across the Pacific (IWAP) experiment was designed to investigate the long-range fate of a propagating low-mode internal tide through a combination of moored arrays, spatial surveys, and intensive time series (Fig. 1). One of the leading hypotheses was that the tide would lose significant energy to small-scale mixing through parametric subharmonic instability (PSI) near a critical latitude of 29°N (Hibiya and Nagasawa 2004; MacKinnon and Winters 2005; Young et al. 2008). Preliminary analysis by Alford et al. (2007, hereafter A07) show suggestive evidence of PSI, but no catastrophic loss of tidal energy. In a companion paper (MacKinnon et al. 2013) we do a more thorough analysis using bispectra and show phase-locking characteristic of PSI, but only observe modest energy transfer rates. In fact, analysis of altimetry shows a mode-one internal tide propagating relatively unscathed for thousands of kilometers across the Pacific (Zhao et al. 2010; Dushaw et al. 2011).

Resultant mixing rates are thus not expected to be catastrophic. However, even a modest elevation of diffusivity due to PSI may be an important pattern to include in climate models (Jochum 2009). Alford et al. (2007) present a simple estimate of diffusivity from finescale shear measured by a shipboard Doppler sonar showing elevation near and equatorward of the critical latitude. Here we present a more thorough analysis of IWAP data, estimating mixing rates from several types of instruments spanning 12 degrees of latitude. We find shear from the shipboard sonar to be consistently elevated near and equatorward of 29°N across four nearly synoptic ship transects. Resultant depth-averaged diffusivities show a factor of 2–4 elevation, which is consistent within error between all instruments and methods presented here. Below we begin with a description of observational tools and methods before moving on to discussion of the results.

## 2. Instruments and experimental design

The following observations were made over a 60-day period encompassing two cruises aboard R/V *Revelle* during spring 2006.

### a. Moored profiler array

Six moorings were deployed at locations shown in Fig. 1 (labeled MP1–MP6) for 50-day time series. A McLane Moored Profiler (MMP) on each mooring crawled from 85–1400 m every 1.5 h (each way) measuring temperature, salinity, and horizontal velocity (Doherty et al. 1999). MP4 stopped profiling after 23 days owing to a broken drive axle, and MP5 did not sample the upper 400 m after about 5 days owing to heavy ballasting. Each profiler was equipped with an Acoustic Current Meter and CTD from Falmouth Scientific. To reduce the effects of salinity spiking, the cross- spectrum between conductivity and temperature is estimated, using data from a region of near-uniform salinity. The magnitude and phase of conductivity are then altered to match those of temperature. The matching technique is described in more detail in Anderson (1993). To remove residual sensor noise, temperature and conductivity data were smoothed to 3 m. Velocity data were smoothed to 0.1 cpm. The filtered density has an approximate noise level of 6 × 10^{−4} kg m^{−3}. Figure 2 shows an example of the zig-zag pattern of MMP measurements in depth and time.

### b. Spatial velocity surveys

Two northward and two southward 1400-km transits of the line were conducted from 25° to 37°N, as well as an eastward transit along 28.8°N (Fig. 1, white). Velocity and shear were inferred from the Doppler shift measured using *Revelle*’s Hydrographic Sonar System (HDSS). Using the 50-kHz sonar system we calculate velocity in 8-m vertical bins (16-m pulse length) between approximately 50- and 950-m depth. To remove a noise floor as seen in spectral analysis, data are filtered to 0.02 cpm.

### c. High-resolution time series

Intensive time series were conducted with a Fast CTD (FCTD) platform developed by R. Pinkel. Four-day time series were made in three locations (TS1–3 in Fig. 1, numbered in order of occupation). TS1 was located about a kilometer from MP3. The profiler measures temperature and conductivity down to 1000 m every 10 min with a SBE49 CTD. FCTD data are also corrected for sensor mismatch, and low-pass filtered to 0.5 m. The filtered density has an approximate noise level of 3 × 10^{−4} kg m^{−3}. The shipboard HDSS sonar system was also operational during these time series, measuring both components of horizontal velocity down to 1000 m (Fig. 2, lower panel).

## 3. Methods of estimating mixing

*ε*is often related approximately to the diapycnal eddy diffusivity (hereafter simply the diffusivity) through

*N*is the buoyancy frequency. Though laboratory studies suggest Γ may vary (Shih et al. 2005; Ivey et al. 2008), particularly with very low stratification, we follow the vast majority of mixing literature and take Γ ≈ 0.2 (Osborn 1980; Oakey 1982).

Microstructure instruments estimate *ε* by measuring well into the inertial subrange, but are rare and somewhat costly. Several methods exist for estimating the dissipation rate by capturing the downscale energy transfer at an earlier point in the process, either at the outer scales of turbulent overturns (section 3a) or at the small-scale end of the internal-wave continuum (section 3b). In a steady state this rate of downscale energy transfer is assumed equal to the dissipation rate, and hence can be used to estimate diffusivity. As a general rule, the further one steps back in this energy cascade (the larger the scales used to estimate the dissipation rate), the larger the number of assumptions made and the greater the uncertainty of the diffusivity estimate.

### a. Overturns

The observational analog is known as the Thorpe scale (*L _{T}*), defined as the root-mean-squared displacement a parcel has moved between a measured density profile with a density inversion (overturn) and the sorted version of the same profile. The Thorpe scale has been shown to be highly correlated with the Ozmidov scale, so CTD measurements of density inversions can be used to estimate

*ε*through Eq. (2) (Thorpe 1977; Dillon 1982; Ferron et al. 1998). Following Dillon (1982), we use

*L*= 0.8

_{O}*L*. The results generally compare well with microstructure estimates (e.g., Seim and Gregg 1994; Ferron et al. 1998; Klymak et al. 2008). Care must be taken though to avoid interpreting noise in density measurements as genuine overturns (Galbraith and Kelley 1996). Uncertainty and bias in this measurement are discussed further in the appendix.

_{T}### b. Finescale parameterization

Stepping to larger scales, the rate of downscale energy transfer through the internal wave field can be estimated by combining finescale measurements (order 10–100 m) with theoretical models of energy transfer through wave–wave interaction (Gregg 1989; Polzin et al. 1995; Gregg et al. 2003). Many formulations are based on the canonical Garrett–Munk (GM) internal wave spectra of shear and strain, both of which are nearly flat at lower wavenumbers, then drop off with a −1 slope beyond a cutoff wavenumber *k _{c}* (Gregg and Kunze 1991). Physically, motions at scales much larger than the cutoff (smaller wavenumbers) are interpreted as weakly nonlinear internal waves, while motions at smaller scales become more strongly nonlinear, eventually leading to wave breaking (D’Asaro and Lien 2000). For the empirically derived GM spectrum, the spectral slope transition occurs at a wavelength of 2

*π*/

*k*= 10 m. For other observations, the cutoff appears to move toward lower wavenumbers with higher spectral energy levels (Gargett 1990). Polzin et al. (1995) suggest a family of spectra in which the cutoff occurs at a point in which cumulatively integrated stratification normalized shear variance (essentially an inverse Richardson number) is 0.66, the value the GM spectra achieves at 10 m. They show that even when observed spectra have wavenumber slopes or frequency content somewhat different than the GM spectrum, the finescale method still gives estimates that are within a factor of 2 of microstructure values.

_{c}*k*) is calculated by requiring a set value of observed shear variance—we use

_{c}*R*is the shear/strain ratio,

_{w}*k*. For linear internal waves

_{c}*R*provides a measure of the average frequency content of a wavefield. For a GM spectrum

_{w}*R*= 3. The

_{w}*R*term in

_{w}*L*(

*R*,

_{w}*θ*) term in Eq. (3) includes the theoretical dependence on downscale energy transfer rate on both average wavefield frequency content (through

*R*) and latitude (Polzin et al. 1995; Gregg et al. 2003).

_{w}## 4. Results

### a. Finescale shear and strain

Since application of the finescale parameterization is dependent on assumptions about shear and strain spectra, we start with a closer look at the observed shear and strain. Full time series of shear from each mooring are presented in A. Pickering et al. (2012, unpublished manuscript) and will not be repeated here. Four shear sections from the HDSS sonar are shown in Fig. 3. Here shear has been normalized by stratification, with average stratification profiles from each mooring interpolated to the latitudes in between. Results are similar if stratification from climatology is used instead. The normalization accounts for expected scaling of shear spectra and is the first step toward applying the finescale parameterization (Gregg 1989; Polzin et al. 1995).

As reported by Alford et al. (2007), there is a tendency for enhanced stratification-normalized shear variance equatorward of about 30°N visible in all four sections. A. Pickering et al. (2012, unpublished manuscript) interpret the coherent sloping features visible in Fig. 3 as near-inertial internal waves and discuss their phenomenology in greater detail. The same latitudinal pattern is visible in depth-averaged stratification-normalized shear variance (Fig. 4, upper left). There is a cloud of elevated shear south of 31°N, with a few spikes at higher latitude.

However, the latitudinal pattern is not as clear in the mooring data (Fig. 4, blue stars). Time series at each mooring show significant variability in time (Fig. 4), with typical ranges comparable to the latitude variation from HDSS shear (the MP5 plot only includes data deeper than 500 m, which is significantly less variable). HDSS shear estimates (green dots in Fig. 4) were not always a good measure of the time-averaged value at each mooring location. The discrepancy is particularly large at MP6, where moored shear variance was elevated for several days early in the record following passage of a strong storm.

Shear spectra were computed in half-overlapping 300-m windows using both MMP and HDSS data, with the results normalized by stratification (Kunze et al. 2006). Strain spectra were computed from density profiles in the same vertical windows for both FCTD and MMP density. Here strain is taken as

Comparing average strain spectra across all depths and latitudes, observed strain is GM-like (Fig. 5) in both shape and level. The strain spectral level is slightly above GM at lower latitudes, and declines to at or below GM at MP6. Where FCTD and MMP measurements coincide, strain spectral estimates are close (red versus orange in Fig. 5, upper right). In contrast, shear spectra at all latitudes deviate significantly from the flat GM shape at low wavenumbers, often with a peak at 100–300-m wavelengths. Related analysis suggests these peaks are primarily from coherent near-inertial motions, attributable to a combination of PSI and wind-generated near-inertial waves (MacKinnon et al. 2013; A. Pickering et al. 2012, unpublished manuscript). Reflecting these differences, the shear/strain ratio (*R _{w}*) is always greater than the GM value of 3, with a tendency to increase toward higher latitudes as strain variance declines (Fig. 5).

Average profiles of diffusivity estimated using the finescale parameterization applied to all instruments are shown in Fig. 6. Diffusivity estimates from available data are calculated using two methods. First, the full finescale method of Eq. (3) is used, incorporating both shear and strain spectra from MMP data (Fig. 6, blue). For the HDSS data, shear spectra are integrated to a maximum wavenumber of 0.02 cpm because of instrument resolution limitations and an average *R _{w}* from MMP data is interpolated between mooring latitudes (Fig. 6, green). Second, for comparison with other published work, diffusivity is also calculated using a strain-only version of Eq. (3) for both MMP and FCTD data. In this case

*k*, at which cumulative strain variance is 0.22 (Kunze et al. 2006), and an assumed shear/strain ratio of

_{c}*R*= 3 (Fig. 6, red and orange). Uncertainty in finescale parameterization estimates is discussed in the appendix.

_{w}Reflecting the temporal variability of shear variance (Fig. 4), finescale parameterization estimates of diffusivity varied considerably in time at each mooring (Fig. 7). Here time series of diffusivity are calculated by averaging shear and strain spectra over one day (appendix), then applying Eq. (3). At MP1, MP3, and MP6 the diffusivity slowly varies by an order of magnitude over the 50-day record. The depth-averaged diffusivities from all instruments and methods are shown in the upper left panel of Fig. 7.

Shear from the single east–west HDSS section shows considerable lateral variability along 28.8°N (Fig. 1). Shear variance shows a clear enhancement at all depths between 197° and 198°E, just to the east of MP3 (Fig. 8). Shear spectra from this section are normalized using the average stratification profile from MP3. The depth- and longitude-averaged shear spectrum between 197° and 198°E is enhanced at all wavenumbers compared to that at MP3, particularly in the 200–300-m wavelength range (Fig. 8, right). The depth-averaged shear variance between 197° and 198°E is three times that at MP3. Diffusivity is estimated from finescale shear along this passage using an assumed *R _{w}* = 3. Estimated depth-averaged diffusivity rises to 1 × 10

^{−3}m

^{2}s

^{−1}between 197 and 198, a factor of 30 higher than the depth average at MP3.

### b. Thorpe scales

Several previous studies have calculated Thorpe scales from temperature measurements, which tend to have lower noise levels (and hence require less vertical smoothing), but the presence of substantial numbers of density-compensated intrusions throughout this dataset required the use of density instead. The accuracy of the estimate is limited by instrument noise in the density measurement and the size of resolvable overturns. Here density noise has been limited by smoothing data in depth, so vertical resolution is the primary constraint (Johnson and Garrett 2004). For the FCTD, the Thorpe displacement (*L*_{th}) was computed by subtracting observed from sorted density profiles, where sorting is performed over the entire profile. The overturn or patch size (*L*_{ot}) was taken as the region over which Σ*L*_{th} = 0, and the Thorpe scale (*L _{T}*) the rms of the Thorpe displacement over each patch. Here it is important to use a buoyancy frequency in Eq. (2) that is computed from the sorted density profile within each overturn, not an average or smoothed value (Alford and Pinkel 2000). Overturns were only allowed if they met a minimum overturn patch size criteria (2.5 m), satisfied the run length criteria of Galbraith and Kelley (1996), and had density that deviated from the sorted profile by at least twice the instrument noise (section 2). Statistical analysis in the appendix suggests FCTD resolution is adequate to resolve most of the dissipation, within a general factor of 2 uncertainty. However, the MMP could resolve only a small fraction of the dissipation-containing overturns, hence dissipation estimates are presented using FCTD data only).

## 5. Discussion

### a. Method comparison

Depth profiles of diffusivity from all methods at all locations are shown in Fig. 6. Given that many of these profiles represent averages of different time periods, the agreement is reasonably good. Diffusivity estimates from HDSS shear are somewhat biased in this environment owing to the combination of red shear spectral shapes and limited vertical resolution of the instrument, but average profiles are still within a factor of 2 from those inferred from finescale mooring shear and strain (appendix). Finescale estimates from shear and strain (variable *R _{w}*) and strain only (fixed

*R*= 3) are in good agreement at lower latitudes where

_{w}*R*is close to the GM value of 3, but diverge at higher latitudes where

_{w}*R*rises. As a reminder, the strain-only estimates are presented here mostly as illustration of the relative importance of the

_{w}*R*correction term, for comparison with other work; we expect the full shear–strain calculations to be the more accurate estimate. Thorpe-scale estimates of diffusivity (black) generally agree with finescale estimates, within uncertainty and the tendency for unresolved overturns to bias our estimates low (appendix). Note here we use the conservative

_{w}*L*= 0.8

_{O}*L*in (2), whereas some other research has suggested a proportionality constant of 1.0 (Ferron et al. 1998). If we had used

_{T}*L*=

_{O}*L*instead, the Thorpe-scale diffusivity estimates at TS1, 2, and 3 (black stars in Fig. 7, upper left) would be closer to the estimates from other instruments.

_{T}### b. Patterns

Finescale diffusivity estimates are essentially another way of looking at patterns in finescale shear or strain variance, multiplied by the additional analytical functions in Eq. (3) that accounts for the effects of latitude and shear/strain ratio variability on expected wave–wave interaction rates. Depth-averaged diffusivities from all methods are shown as a function of latitude in Fig. 7 (upper left). The depth-average diffusivity estimated from HDSS shear shows a factor of ~4 elevation south of the critical latitude. About half of this increase can be explained by the meridional pattern of shear variance seen in Figs. 3 and 4, while the other half is due to the meridional variation of shear/strain ratio through the *R _{w}* term in Eq. (3).

Diffusivity estimates from the moorings show a muted version of the same pattern. Data from MP1–MP5 show a factor of ~2 change with latitude. The MP6 average diffusivity value is higher, largely reflecting the elevated shear early in the record following a strong storm; if only diffusivity after the storm were used for the average (onward of yearday 130 in Fig. 7, bottom right), the average MP6 diffusivity would be closer to that at MP5, approximately half the value at lower latitudes. It is interesting to note that the strain-only finescale diffusivity parameterization (Fig. 7, red stars), while biased low, also show a clear factor of 2 decrease for the two northern moorings, with values consistent with those reported by Whalen et al. (2012).

The single east–west section of shear from the HDSS sonar showed considerable longitudinal variability as well, with a factor of 30 elevation near 197.5°E. One possible interpretation is that PSI-related mixing might be significantly higher in concentrated “beams” of tidal energy. Such beams are best interpreted as interference patterns that can shift with changes in the mesoscale features that control tidal ray paths (Rainville et al. 2010). The altimetric flux patterns visible in Fig. 1 represent only the coherent portion of low-mode tidal flux, essentially the part that remains after shifting interference patterns are averaged over years (Zhao et al. 2010). Regional tidal numerical simulations that include different mesoscale states show more focused tidal beams that shift back and forth through the longitudinal range of enhanced shear in Fig. 8 (E. Zaron 2006, personal communication). The elevated finescale shear near 197.5°E could also be related to passing wind-generated near-inertial wave packets or concentrations of internal wave energy owing to interaction with the mesoscale (Buhler and McIntyre 2005; Polzin 2008), further discussion of which is beyond the scope of this work.

## 6. Conclusions

Here we present estimates of diapycnal diffusivity across 12 degrees of latitude primarily based on finescale parameterization techniques applied to shear and strain from several different instruments. The order of magnitude temporal variability of diffusivity at each mooring is a major result of this study and suggests caution be used in drawing significant conclusions from published individual profiles of diffusivity using this technique. Four north-south sections show persistently elevated shear variance near and equatorward of 29°N, consistent with an interpretation of PSI acting at this latitude (Alford et al. 2007; MacKinnon et al. 2013). As an upper bound, this suggests a diffusivity increase of up to a factor of 4. Diffusivity estimates from finescale shear and strain at six moorings suggest a more moderate factor of ~2 enhancement near and equatorward of 29°N. Given factors of 2–3 uncertainty associated with the methods used here, such patterns are on the edge of statistical significance. Diffusivity at the northernmost mooring (MP6, 37°N) may be comparatively elevated on top of this pattern because of the proximity to the storm track. A. Pickering et al. (2012, unpublished manuscript) report that estimated input of power into near-inertial waves is a factor of 2–4 higher at MP6 than at MP1–4. A single zonal shear section provides suggestive evidence of higher shear and inferred diffusivity in a concentrated region to the east, at the critical latitude. However, given the time variability of shear at each mooring, it would be premature to assign large dynamical significance to this observation.

Overall, the mixing rates presented here are quite modest, often close to GM levels, supporting the conclusion that mixing in the midocean thermocline is typically weak. Though even a moderate elevation associated with PSI at some latitudes may be an important mixing pattern to include in global models (Jochum 2009), these rates do not imply a significant drain of energy from the internal tide. Though in some parts of the world the order-one balance appears to be local dissipation of generated internal tides (Polzin 2009), our results are hence consistent with a growing conclusion that most of the low-mode tidal energy propagating in the central North Pacific must be dissipated elsewhere, perhaps at the boundaries.

## Acknowledgments

This work was sponsored by NSF OCE 04-25283. We thank the tireless captain and crew of the R/V *Revelle*, for making our two months at sea productive and enjoyable. Eric Boget and Andrew Cookson played an essential role in the design, construction, deployment, and recovery of the moorings. Tom Peacock, Paula Echeverri, and Kim Martini provided valuable assistance at sea.

## APPENDIX

### Uncertainty Estimates

#### a. Uncertainty of overturn estimates of dissipation

The Thorpe scale calculation (section 4) was performed initially with both the FCTD and MMP. A natural question with the method is whether the resolution limitation leads to a significant underestimate of dissipation from potentially important small overturns. Because of instrument resolution, overturns were only allowed if they met a minimum overturn patch size criteria of 2.5 m for FCTD and 9 m for MMP (Galbraith and Kelley 1996). The distributions of Thorpe displacements (*L _{T}*) and overturn patch sizes for the FCTD both look close to lognormal (Fig. A1, top panels, red), consistent with most previous observations (Moum et al. 1992; Wijesekera et al. 1993; Alford and Pinkel 2000). The lack of resolution at the smallest scales produces a dropoff in the Thorpe displacement pdf around a meter. Comparison with a Gaussian fit (black curve) suggests the FCTD is capturing about 80% of the Thorpe displacement variance. For a more conservative estimate, using a model for the spectrum of

*L*, Dillon and Park (1987) argue that 90% of

_{T}*L*variance is resolved when the vertical resolution is

_{T}*L*/5. For the FCTD resolution of 0.5 m, their criteria suggest we are only accurately capturing events with

_{T}*L*> 2.5, which essentially suggests that the Gaussian fit curve in Fig. A1 (upper left) is an underestimate.

_{T}Fortunately, the resolved overturns contribute a significant portion of the total dissipation. Specifically, the relative importance of each overturn size class can be seen by adding the cumulative contribution of each overturn class to the total dissipation rate, integrating from large (resolved) to small scales (Fig. A1, lower panels). As with Alford and Pinkel (2000), the relative contribution is approximately linear with decreasing patch size/displacement for overturns less than 5–6 m. This linearity allows an estimate of the unresolved dissipation by extrapolating the line leftwards toward small scales. The resulting intercept is near 1.2, suggesting the FCTD is only missing around 20% of the total dissipation. Though admittedly adhoc, this calculation reassures that our estimates are not off by an order of magnitude. If we instead use the rubric of Dillon and Park (1987), the “well-resolved” events with *L _{T}* > 2.5 contribute over half the observed dissipation, suggesting the estimates presented here are valid within a factor of 2.

The resolution limitation becomes more serious at higher stratification, as the overturning scale is expected to decrease, meaning the same resolution instrument will resolve a smaller proportion of overturns. The degree of underestimation is difficult to assess quantitatively, as the size of overturns, the resultant dissipation rate, and the frequency of overturning events are all likely to change in not well understood ways with changing average stratification (Alford and Pinkel 2000; Klymak et al. 2008). As a very rough guideline, we note that if the dissipation rate scales with stratification, *ε* ∝ *N*^{2}, then the Thorpe scale should scale as *N* for the data shown in Fig. A1 is 2.5 cph. For double the average value (5 cph), the peak of the Gaussian distribution may shift about *N* is above 5 cph, which is shallower than 250 m at TS3 and shallower than 150 m at TS1 and TS2.

For the MMP, the resolution limitation is more serious. Though the FCTD is only sampling for a fraction of the MMP time series, we can roughly estimate the statistical limitations of the MMP data by comparison with the FCTD data. For example, comparison of the patch size PDFs suggests the MMPs are only resolving 15% of the overturns (Fig. A1, upper panels, blue), and less than a third of the total dissipation. It is worth noting that the severity of this constraint is environment dependent. For example, Alford (2010) gets much better statistics from the a similar instrument due not only to a quieter CTD, but also to lower stratification and a generally higher-energy environment, both of which lead to significantly larger overturns. For this dataset the MMP with FSI CTD appears inadequate for using the Thorpe method; overturn estimates of dissipation are hence calculated using the FCTD only.

In addition to the possibility that dissipation estimates are biased low owing to instrument resolution, there are also significant uncertainties associated with natural spread in the data. A bootstrap analysis suggests 95% confidence levels of a factor of 2 for average dissipation values, consistent with other published uncertainty estimates. A horizontal bar showing a factor of 2 uncertainty is shown in Fig. 6.

#### b. Uncertainty of finescale estimates of dissipation

It is difficult to assess uncertainty associated with finescale estimates of mixing rates, and in fact most published profiles do not show error bars. Expected uncertainty in the second- and fourth-order moments of shear is discussed by Gregg et al. (1993) in terms of number of independent samples. Following the run length test they propose, we find integrated shear spectral variance in each depth bin to be uncorrelated at the 95% level for samples 1 day apart. When the ship is underway the HDSS shear estimates are uncorrelated on shorter time scales, but when calculating diffusivities we average all spectra over one day before applying the finescale parameterization. This gives us 50 independent samples in each depth bin for MMP data. Gregg et al. (1993) suggest associated confidence limits of about a factor of 2 for this number of independent samples.

However, equally if not more important are the large sources of uncertainty and bias associated with many dynamical assumptions made in the derivation and implementation of the finescale method (Eq. 3). For example, HDSS and MP estimates of diffusivity can diverge even when evaluated over the same time period (Fig. A2, left). The explanation revolves partially around the nonwhite nature of the shear spectra (Fig. 5). Owing to instrument resolution limitations, the HDSS shear spectra are integrated out to a maximum wavenumber of 0.02 cpm, not the “true” *k _{c}* from Eq. (3). When spectra are red, this systematically biases diffusivity estimates high. Comparing all depth bins using only the four days at MP3 when we have concurrent HDSS and MP shear estimates, the high bias of the HDSS estimate is consistent but not of a uniform magnitude (Fig. A2, left). In part that is because the spectral shape varies with depth (Fig. A2, right). However the situation is complicated by the differing time resolution of the two instruments. In particular, the zig-zag time–depth sampling of the MMP (Fig. 2) means that it observes a subset of the shear observed by the HDSS, so spectra from the two instruments sometimes diverge even in the wavenumber range they are both able to resolve. The time series were located about a kilometer apart, so were sampling different water masses. Given the shallow slopes and coherent nature of the most energetic shear features visible in Fig. 3, we do not expect this to be a major effect, on average. This example serves to illustrate the sensitive nature of the finescale parameterization. Overall, Polzin et al. (1996) suggest that even in the best of circumstances the method is uncertain with a factor of ~2–4. Bootstrap analysis of average diffusivity from finecsale estimates in each depth bin return a value much lower than this. A factor of 3 uncertainty in all finescale estimates is hence indicated in Fig. 6 (Naveira Garabato et al. 2004) as a rough guide.

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