## 1. Introduction

Turbulent mixing plays a key role in the global water mass budget of the ocean and contributes to the maintenance of the stratification. Since turbulent mixing is a key component of the global overturning circulation (Munk and Wunsch 1998) and because of its strong spatial heterogeneity and temporal intermittence, continuous efforts to measure its distribution and magnitude and to understand the underlying mechanisms need to be pursued. Turbulent mixing is usually quantified locally from microstructure data obtained from specific instruments with *O*(1 cm) resolution (e.g., Oakey 1982) and from the finestructure data obtained with the widely used conductivity–temperature–depth (CTD) and lowered acoustic Doppler current profilers (LADCP). The finestructure is either used to quantify Thorpe scales (Thorpe 1977; Dillon 1982) or to diagnose the intensity of the internal wave field (Gregg 1989). At larger scales, turbulent mixing can also be quantified from tracer release experiments and from tracer budgets (e.g., Ledwell et al. 1993, 2000; Hogg et al. 1982). During the last decade, an increasing number of studies used the finescale properties of the internal wave field to quantify turbulent mixing due to the internal wave environment (e.g., Hibiya and Nagasawa 2004; Naveira Garabato et al. 2004a,b; Walter et al. 2005; Kunze et al. 2006; Lauderdale et al. 2008; Stöber et al. 2008; MacKinnon et al. 2008; Daae et al. 2009; Tian et al. 2009). Other studies have compared and validated the local approaches based on microstructure, which contains mixing from different mechanisms, and finescale parameterizations (Gregg 1989; Polzin et al. 1995; Gregg et al. 2003; Hibiya et al. 2012; Frants et al. 2013; Sheen et al. 2013; Waterman et al. 2013) based on the internal wave field, which is presumed to be a major contributor of ocean interior mixing.

*ε*

_{0}= 7 × 10

^{−10}W kg

^{−1}, and

*N*/

*N*

_{0}and

*h*leading to a new parameterization, defined aswhereand

*V*

_{z})-to-strain (

*ζ*

_{z}) variance ratio. Gregg et al. (2003) added a term

*j*dependent on latitude to improve the consistency with the dissipation rates found in the equatorial region that are weaker than at midlatitudes. Gregg et al.’s (2003) parameterization readswherewhere

*f*is the inertial frequency and

*ε*

_{G89},

*ε*

_{P95}, and

*ε*

_{G03}, and compare them with dissipation rates measured with a microstructure profiler along the OVIDE transect that spans both a large latitudinal band (40°–59°N; Fig. 1) and a large range of buoyancy frequencies from 1.6 × 10

^{−4}s

^{−1}to 3.2 × 10

^{−3}s

^{−1}.

## 2. Dataset

The OVIDE project consisted of repeating the A25 Greenland–Portugal hydrographic line every 2 years from 2002 to 2010 (Mercier et al. 2014). Each survey was composed of nearly 100 CTD–LADCP profiles (Fig. 1, yellow and red dots). Measurements lasted 3 weeks usually in June/July. During 2008, an autonomous vertical microstructure profiler (VMP; www.rocklandscientific.com) able to dive to 6000 m was deployed at 30 CTD locations (Fig. 1, red dots). The geographical distribution of the VMP deployments was not as uniform as that of the CTD. In particular, no VMP stations were done on either the southeastern flank of the Reykjanes Ridge because of heavy sea conditions or on the Greenland continental slope because of a technical problem with the instrument. The VMP was deployed 20–30 min before the CTD cast began and about half a nautical mile away from the CTD station location to avoid any collision. The VMP was retrieved after the end of the CTD upcast. The VMP was programmed before deployment to stop its profile 70 m above the bottom, while the CTD–LADCP profiles were stopped 10–15 m above the bottom.

### a. Microstructure data

The VMP is a 3-m-long instrument that weights 150 kg. Microstructure sensors are located at the bottom of the instrument, so that only data from the downcast are useful. On the OVIDE cruise, the typical downcast fall rate of the instrument was 0.7 m s^{−1}, a velocity high enough to apply Taylor’s frozen turbulence hypothesis, which allows a conversion from time to space using a smoothed instrument fall speed profile.

The VMP carried two Rockland Scientific fast thermistors (FP-07), one Sea-bird Electronics (SBE) microconductivity sensor (SBE-7), and two Rockland Scientific velocity shear probes (SPM-38). Microstructure channels were sampled at 512 Hz. A low-pass analog filter with a cutoff frequency of 165 Hz was applied on all microstructure channels to avoid aliasing.

Data from the two shear probes were used to calculate dissipation rates. The first 10 m of the dissipation profiles were discarded, since data were frequently contaminated by ship wake. In this study, fast thermistors and microconductivity data were only used to visually check the turbulence signal given by the shear probes. Shear data were first edited to remove spikes. Assuming isotropy, the dissipation rate of turbulent kinetic energy per unit mass reads *ε* = 7.5*ν*〈*u*_{z}^{2}〉, where *ν* is the viscosity of seawater and *u*_{z} is the vertical shear of the horizontal velocity *u*. The velocity shear variance 〈*u*_{z}^{2}〉 was estimated by a spectral analysis of the velocity shear signals. The velocity shear spectra were calculated using half-overlapping 512-point segments. Each segment was linearly detrended and windowed with a Hanning taper function prior to FFT calculation. Spectral coherence between the velocity shear data and the three accelerometer data was used to remove vibrational contamination (Levine and Lueck 1999; Goodman et al. 2006). Because of their finite size, velocity shear probes tend to average the smallest turbulent eddies, thereby underestimating shear variance at high wavenumbers (Oakey 1982). Velocity shear spectra were corrected for velocity shear probe averaging using a single-pole filter for the wavenumber response of the probe with a cutoff wavenumber of 48 cycles per meter (cpm; Macoun and Lueck 2004). To obtain velocity shear variance, shear spectra were integrated up to a cutoff wavenumber *k*_{c} found iteratively. The procedure followed that described by Moum et al. (1995). For the first iteration, *k*_{c} was set to 15 cpm and gave a first dissipation rate estimate from which the Kolmogorov wavenumber *k*_{c} was increased to *k*_{s} or, to avoid noise contamination at high wavenumbers, to a maximum of 50 cpm. The Kolmogorov wavenumber was then updated. The iterative process was stopped when the relative difference between *k*_{c} and *k*_{s} was lower than 0.001%, or when *k*_{c} was equal to 50 cpm or after 10 iterations. At the end of the iteration process, if *k*_{c} < *k*_{s}, then velocity shear variance loss was accounted for assuming Nasmyth (1970) turbulence spectrum form. The background noise on the dissipation rate profile was set to 5 × 10^{−11} W kg^{−1}. It was estimated from quiescent profiles of the deep ocean located in the Iberian Abyssal Plain. Following Gregg (1999) criteria, we observed that for dissipation rates lower than 5 × 10^{−11} W kg^{−1}, spectra of microstructure velocity shear tend to depart from Nasmyth spectra. Examples of microstructure velocity shear spectra calculated at two different depths are provided as an illustration for a profile located at 42°35′N, 15°27′W (Fig. 2). While the spectrum followed the Nasmyth spectral shape at a depth of 4200 m and for a dissipation rate of 7.2 × 10^{−11} W kg^{−1} (Fig. 2, gray curves), it clearly departed from the Nasmyth spectral shape at a depth of 4270 m for a dissipation rate of 2.6 × 10^{−11} W kg^{−1} (Fig. 2, black curves). Note that for those two examples, power spectral density (PSD) rise was due to the presence of noise for wavenumbers larger than 13 cpm. This does not influence the dissipation rate estimate though, since Kolmogorov wavenumbers (viscous cutoff wavenumber) that set the upper bound of the spectral integration in the dissipation rate calculation are lower than 13 cpm (Fig. 2, vertical black and gray lines).

### b. Finestructure data

On the OVIDE 2008 cruise, one upward-looking 300-kHz and one downward-looking 150-kHz RD Instruments workhorse LADCPs mounted on the CTD carousel provided observations of vertical shear of horizontal currents with 16-m vertical resolution. LADCP data were processed with the velocity-inversion method initially developed by Visbeck (2002).

To compute the velocity shear variance, each velocity profile was split into 640-m segments with 75% overlap. Each segment was linearly detrended prior to FFT calculation. Buoyancy-normalized shear variance was calculated as the integral, from the lowest wavenumber *k*_{min} = 2*π*/640 rad m^{−1} up to a maximum wavenumber *k*_{max} determined for each segment, of the velocity spectra *S*(*u*) multiplied by a transfer function *T*(*k*)(see discussion below): *k*_{max} is the minimum of *k*_{noise} and *k*_{c}, *k*_{noise} is the wavenumber above which noise dominates the signal (see discussion below), and *k*_{c} is the cutoff wavenumber for which the integrated velocity shear variance equals 0.66 *k*_{min} to *k*_{max}. Finding *k*_{c} for which *π*/10 m^{−1}) takes into account the fact that the cutoff wavenumber decreases with increasing energy level in the shear (Polzin et al. 1995; Gregg et al. 2003).

Polzin et al. (2002) showed that spectral corrections *T*(*k*) need to be applied to compensate for loss of velocity variance at a wavelength shorter than about 200 m because of the various steps of the acquisition and processing methods of the LADCP data. While Polzin et al. (2002) processed their LADCP data using the shear method (Firing and Gordon 1990; Fischer and Visbeck 1993), Thurnherr (2012) updated the spectral corrections for the velocity-inversion method and showed that it naturally produces less damping at short wavelengths. With the velocity-inversion method, the shear spectra need to be corrected at least for range averaging and depth binning. A supplementary term arises when data are prebinned into superensembles before the inversion, as done for our dataset. When instrument tilting is important, an additional term also applies. Considering the weak averaged tilt (mean tilt angle of 1.8°) due to our carousel configuration, no tilt correction was applied in this study. As Polzin et al. (2002) suggested, beam separation effects were ignored. Spectral correction for our data processing then takes the form of the transfer function *k* is the vertical wavenumber and Δ*z* = 16 m represents the ADCP transmit pulse length, range gate length, superensemble preaveraging interval, and bin length of the output grid. An illustration of an ensemble-averaged raw (uncorrected) velocity shear spectrum scaled by the buoyancy frequency and its corrected version is given for an ensemble of profiles that were taken away from steep bathymetry, so that we expected a relatively flat spectral shape close to that of the GM model (Fig. 3). The ensemble was made of 589 spectra in the depth range 96–2656 m, from longitude 10° to 25°W and from latitude 40° to 52°N. In this example, the raw spectrum falls off by 60% from wavenumbers 0.04 rad m^{−1} (wavelengths of 160 m) to wavenumbers 0.1 rad m^{−1} (wavelength of 60 m). Such a damping is typical of the velocity-inversion method (see Thurnherr 2012, his Fig. 1). After correction, the spectral shape is relatively flat up to wavenumbers close to 0.1 rad m^{−1}, above which noise dominates.

Ensemble-averaged shear spectra clearly show a depth-dependent contamination by noise at high wavenumbers for which the rms noise *S*_{rms} strongly dominates the signal (Fig. 4a). To estimate the increase in the single-ping error *S*_{err} depth, a fit to the high wavenumber part of each spectrum is done, which provides the rms noise *S*_{rms}. The number of averaged pings *N*_{pings} into each superensemble varies as a function of depth, since the carousel was stopped more frequently to close Niskin bottles when getting to shallower depth during the upcast (Table 1). The single-ping error [*S*_{err} = *S*_{rms}(*N*_{pings})^{1/2}] increased with depth from 0.06 m s^{−1} above 1400 m to 0.11 m s^{−1} deeper than 3900 m (Table 1). This is consistent with the single-ping error profile provided by the LADCP software for each station. This single-ping error increase with depth was likely due to a decrease in the number of scatterers with depth. To be consistent with the variable observed noise levels, a depth-dependent wavenumber *k*_{noise} was chosen in the variance computation (Table 1).

Number of pings and single-ping error by depth range. Given the number of pings, the single-ping error was estimated as a fit to the corresponding shear spectrum (Fig. 4).

A SBE 911 CTD was mounted on the carousel. CTD data were processed to provide 1-m vertical resolution density profiles consistent with World Ocean Circulation Experiment (WOCE) standards. Strain defined as the vertical derivative of the isopycnal displacement *ζ* was estimated on the same vertical segments as the velocity shear. For a given downcast CTD profile, *ζ* were calculated from density fluctuations as *ρ*_{HP} is the density profile filtered by a high-pass Butterworth filter with a cutoff wavelength of 200 m, *ρ*_{0} is a reference density, and *g* is the standard gravity. The filter cutoff frequency was chosen to avoid contamination by background stratification (Kunze et al. 2006). The use of a local filter does not make any assumption on the shape of the density profile on the segment. Fitting a polynomial (e.g., quadratic fit for Kunze et al. 2006; linear fit for Lauderdale et al. 2008) to remove the low-frequency part from the density profile can produce in the thermocline increased high-frequency density fluctuations if the degree of the polynomial is not consistent with the shape of the density profile. Strain variances were computed as the integral of the isopycnal displacement spectra *S*(*ζ*) from ^{−1} and ^{−1} and a cutoff wavenumber for which the integrated strain variance equals 0.22 (i.e., the variance of the GM strain spectra integrated up to its cutoff wavenumber of 2π/10 rad m^{−1}). The GM strain spectra were also integrated from

Since wavenumber ranges for shear and strain variances are not equal, the shear-to-strain ratio is calculated as *k*_{min;} *k*_{max}], both strain terms of the denominator are evaluated on the range [*k*^{ζ}_{min}; *k*^{ζ}_{max}], and

## 3. Results

### a. Dissipation rates from microstructure

The distribution of dissipation rates was first inferred from microstructure velocity shear data (Fig. 5a) and averaged on the same grid as the finestructure estimates (160-m-long bins; *ε*_{bin} = ∑*ε*_{i}/*N*_{bin}, where *ε*_{bin} is the linear average of the *N*_{bin} individual microstructure dissipation rate estimates *ε*_{i} present in the bin). The weakest dissipation rate is found below 2000 m in the Iberian Abyssal Plain, to the north of the Azores–Biscay Rise and in the center of the Irminger basin with values mostly below 0.5 × 10^{−10} W kg^{−1}. Above 2000 m, dissipation rates remain weak on the transect, usually around 1–5 × 10^{−10} W kg^{−1} with a few peaks in the range 10–50 × 10^{−10} W kg^{−1}. The most intense values are found above the northwestern flank of the Reykjanes Ridge with values in the range of 5–100 × 10^{−10} W kg^{−1} and at 48°N, 21°W (blue arrow in Fig. 1) from 3400 m down to the seafloor at 4400 m with dissipation rates ranging from 10 to 100 × 10^{−10} W kg^{−1}.

To quantify the associated vertical diffusivity *K*, an equilibrium between production of turbulent kinetic energy by vertical velocity shear, increase in potential energy (mixing) and dissipation rate is often assumed leading to *K* = Γ*εN*^{−2} (Osborn 1980) with a constant mixing efficiency Γ = 0.2. Shih et al. (2005) and more recently Bouffard and Boegman (2013) pointed to a dependency of the diffusivity upon the turbulent intensity parameter Re_{b} = *ε*/*(νN*^{2}). Depending on the value of Re_{b}, four regimes for the diffusivity were identified: molecular (Re_{b} < 1.7), buoyancy-controlled (1.7 < Re_{b} < 8.5), transitional (8.5 < Re_{b} < 400) and energetic (Re_{b} > 400). The boundaries between the various regimes are given for a Prandtl number of 7 and are supported by field data (Bouffard and Boegman 2013). In the transitional regime, the Osborn relation with Γ = 0.2 can be applied whereas in the energetic regime *K* = 4*ν*(Re_{b})^{1/2} is more appropriate. In the buoyancy-controlled (energetic) regime the use of the Osborn relation with Γ = 0.2 underestimates (overestimates) the diffusivity. In this study, Re_{b} was derived from microstructure data (Fig. 6) and used to calculate the diffusivity. Most of the section belongs to the transitional regime for which the Osborn relation is valid. The energetic regime is found at a depth mainly at 21°W and on the northwestern flank of the Reykjanes Ridge.

The vertical diffusivities were usually below 0.5 × 10^{−4} m^{2} s^{−1} on the transect, with values well below 0.11 × 10^{−4} m^{2} s^{−1} in the upper 2000 m of the Iberian Abyssal Plain and at depth in the Irminger Basin (Fig. 5b). The two major regions with enhanced diffusivity concerned the bottom of the station at 21°W and 48°N where *K* reaches 5 × 10^{−4} m^{2} s^{−1} to 50 × 10^{−4} m^{2} s^{−1} and the flank of the Reykjanes Ridge with moderate values ranging from 1 to 5 × 10^{−4} m^{2} s^{−1}. Note that for those two regions, using the Osborn relation with Γ = 0.2 as done in most of previous studies would lead to a diffusivity larger by a factor of 3–4 along the Reykjanes Ridge and by a factor 4–8 at 21°W and 48°N.

### b. Comparison between finestructure and microstructure estimates of the dissipation rate

Using the procedure described in section 2b, finescale estimates *ε*_{G89}, *ε*_{P95}, and *ε*_{G03} are calculated using vertical profiles of velocity and density. The comparison of those estimates with the microstructure dissipation rate *ε*_{vmp} allows us to quantify how the different terms entering the parameterizations improve the consistency with *ε*_{vmp} (Fig. 7). Since the dissipation rate varies on several orders of magnitude and in order to compare equally the under- and overestimation of *ε*_{vmp}, we defined the ratio *R*_{ε} = log_{10}(*ε*_{finestructure}/*ε*_{vmp}), whose mean and standard deviation calculated for each of the tested parameterization is a measure of the bias between the parameterization and *ε*_{vmp} and of the dispersion of the estimates, respectively (Table 2).

Averaged (〈 *R*_{ε}〉) and standard deviation (*σ*_{Rε}) of the magnitude of the ratio between the finestructure and the microstructure estimates of *ε* with *R*_{ε} = log_{10}(*ε*_{finestructure}/*ε*_{microstructure}). Columns *F*(*m*) gives the cumulative distribution *P*(1/*m* < *ε*_{finestructure}/*ε*_{microstructure} < *m*) with *P* denoting the probability that *ε*_{finestructure}/*ε*_{microstructure} lies in the interval (1/*m*, *m*). Dissipation *ε**_{G89} (*ε**_{ζ}) is the same as *ε*_{G89} (*ε*_{ζ}) but restricted to the region where 〈*ω*〉 > 1.1*f*. *ε*_{G03}^{cons} uses conservative fixed wavenumber bands to estimate shear and strain variances (see text for details).

The simplest parameterization, *ε*_{G89}, only agrees on average within an order of magnitude with *ε*_{vmp} (Fig. 7a) and clearly overestimates *ε*_{vmp}. The mean *R*_{ε} of 0.17 (Table 2) corresponds to an averaged overestimation by 50% (10^{0.17} = 1.5). Weakly stratified regions are mostly responsible for this overestimation (Fig. 7a, dark blue dots). The exclusion of these regions where the buoyancy frequency *N*^{2} falls below 10^{−7} s^{−2} improves the results and leads to an overestimation of 30%. The standard deviation of the ratio is high (0.52) when all segments are considered and slightly decreases to 0.44 when weakly stratified regions are omitted. Taking the frequency modulation term *h*(*R*_{ω}) into account as done in the parameterization *ε*_{P95} significantly improves the consistency of the parameterization with *ε*_{vmp} (Fig. 7b). The term *h*(*R*_{ω}) is lower than unity over most of the section (Figs. 8a,b). This strongly reduces the positive bias observed with *ε*_{G89}. The lowest values of *h*(*R*_{ω}) are found in weakly stratified regions where *h*(*R*_{ω}) is close to 0.2. Most of the values larger than one are found in the upper 2000 m east of 21°W (distance to Portugal < 1500 km). The mean of *R*_{ε} falls close to zero (−0.02) and its standard deviation decreases to a value of 0.39 with all segments taken into account. The added term *j*(*f*/*N*) in the parameterization *ε*_{G03} does not have a large impact on our dataset. Gregg et al. (2003) showed its strong impact at low latitudes. But it has also a significant influence in the case of a weak stratification. Indeed, this term decreases the estimated dissipation rate by a factor of 2–3 in the weakly stratified regions (Fig. 8c) of the Iberian Abyssal Plains. Elsewhere on the section, the term *j*(*f*/*N*) tends to decrease the dissipation rates below 1500 m and to increase it above, with a maximum increase of 20%–30% in the stratified region of the North Atlantic Drift between 48°N, 21°W and the Reykjanes Ridge. Overall, the contribution of *j*(*f*/*N*) slightly reduces the scatter in *ε*_{G03} versus *ε*_{vmp} (Fig. 7c). The mean of *R*_{ε} is −0.05, slightly larger than for *ε*_{P95}, and its standard deviation is slightly improved with a value of 0.37 when all segments are taken into account (Table 2). Consistently, the cumulative distribution *F*(*m*) = *P*(1/*m* < *ε*_{finestructure}/*ε*_{vmp} < *m*), with *P* being the probability that *ε*_{finestructure}/*ε*_{vmp} lies in the interval [1/*m*, *m*], gives better results for *ε*_{P95} than for *ε*_{G89} (Table 2). The cumulative distributions of *ε*_{P95}/*ε*_{vmp} and *ε*_{G03}/*ε*_{vmp} are similar. The fact that the standard deviation is nonetheless better for *ε*_{G03} than for *ε*_{P95} is associated with the fewer extreme points in the *ε*_{G03} parameterization. The probability density function (PDF) of the ratio *ε*_{G03}/*ε*_{vmp} is symmetric about unity (Fig. 9). For 3% of the segments, the ratio *ε*_{G03}/*ε*_{vmp} presents extreme values larger than 7 or weaker than *ε*_{P95}/*ε*_{vmp}) .

The regional distribution of the ratio *ε*_{G03}/*ε*_{vmp} does not provide a clear picture (Fig. 10). Some specific features stand out: *ε*_{G03} tends to overestimate *ε*_{vmp} in the upper 1000 m of the stratified waters south of the Reykjanes Ridge and to underestimate *ε*_{vmp} in the upper 1500 m of the weakly stratified waters of the Irminger basin; in the deepest part of the transect, *ε*_{G03} overestimates *ε*_{vmp} in the poorly stratified waters of the Iberian Abyssal Plain (*z* > 4500 m) and in the deepest part of the Irminger Basin, while it underestimates *ε*_{vmp} from 21° to 26°W and on the northwestern flank of the Reykjanes Ridge.

To test the sensitivity of *ε*_{G03} to the spectral ranges chosen to calculate the shear and strain variances, another estimate *ε*_{G03}^{cons} is presented that also uses Gregg et al. (2003) finescale parameterization with “conservative” choices taken from Kunze et al. (2006): segments are 320 m long, strain (shear) variances are calculated using a [30–150] m ([150–320] m) wavenumber range. The new estimate *ε*_{G03}^{cons} exhibits a larger dispersion than *ε*_{G03}, especially for observed dissipation rates *ε*_{vmp} < 1 × 10^{−10} W kg^{−1}and buoyancy frequencies *N* < 1.3 × 10^{−3} s^{−1} (Fig. 7d; Table 2). A closer inspection of the results shows that most of the increase in the dispersion comes from the shear variance estimate being restricted to a shorter wavenumber range. Indeed, only two spectral components are used in the range [150–320] m to calculate the shear variance for *ε*_{G03}^{cons}, which produces a larger uncertainty than for *ε*_{G03}. The moderate reduction in the range of wavenumbers for the strain variance only slightly increases the dispersion but not significantly (*ε*_{G03}^{cons} dispersion *σ*_{Rε} increased by 0.02 compared with *ε*_{G03}).

### c. Dissipation rates and diffusivity from finestructure on the full transect

To get a complete view of the distribution of the dissipation rate on the OVIDE transect, the parameterization *ε*_{G03} is applied to all the CTD–LADCP casts and the diffusivity is derived following Bouffard and Boegman (2013; Fig. 11). For buoyancy frequencies weaker than 1.4 s^{−1} (potential temperature *θ* < 4.5°C), the dissipation rate usually reaches weak values, *ε*_{G03} < 10^{−10} W kg^{−1}. High values of the dissipation rate are found above the bottom at some locations as the microstructure data suggested. The southeastern flank of the Reykjanes Ridge, for which no microstructure data were available because of a heavy sea state, shows the same moderate intensification as that found on the northwestern flank. A moderate intensification is also visible along the Greenland slope.

Since the diffusivity scales as *ε*_{G03}*N*^{−2}, the diffusivity distribution is somehow opposite to that of the dissipation. The largest values are found at a depth where the stratification is weak. The most intense values are found in a 1000-m-thick layer above the seafloor from the Iberian Plain up to 21°W (from 1 to 10 × 10^{−4} m^{2} s^{−1}), around the Reykjanes Ridge (1–5 × 10^{−4} m^{2} s^{−1}) and above 1200 m on the Greenland slope (1–5 × 10^{−4} m^{2} s^{−1}). Elsewhere on the transect, the diffusivity falls below 5 × 10^{−5} m^{2} s^{−1}.

In regions of weakly stratified waters or when only CTD data are available, Kunze et al. (2006) suggested the use of a strain-based parameterization *ε*_{ζ} as presented in section 2. The application of such a parameterization with a constant shear-to-strain ratio representative of the section-averaged *R*_{ω} gives contrasted results. From 10° to 23°W, the estimated *ε*_{ζ} below a depth of 2500–3000 m strongly underestimates *ε*_{vmp} by a factor of 3–1000 (Fig. 12). The factor 1000 is reached for the profile at 21°W, where the microstructure showed enhanced levels above the bottom. For comparison, *ε*_{G03} underestimated *ε*_{vmp} by a moderate factor of 3–5 at the bottom of this profile. Parameterization *ε*_{ζ} completely misses the intensification of the dissipation rate at 21°W (cf. Figs. 5a, 12). Note that the region between 10° and 23°W is weakly stratified at depth, which hinders good estimates of the strain from one instantaneous vertical density profile. Moreover, the strain-based parameterization is expected to underestimate dissipation rates there, since this region is biased toward low frequencies (i.e., high values of *R*_{ω}; Fig. 8a), which means weak signatures on the strain. In contrast, *ε*_{ζ} overestimates *ε*_{vmp} by a factor of 2–10 above 1500–2000 m and south of the Reykjanes Ridge (Fig. 12b), where *R*_{ω} is relatively low (region biased toward higher frequencies). Again, *ε*_{G03} provides a better estimate despite a similar bias in this region. North of the Reykjanes Ridge *ε*_{ζ} and *ε*_{G03} are similar. The averaged ratio log_{10}(*ε*_{ζ}/*ε*_{vmp}) is −0.12 with a standard deviation of 0.48 (Table 2). Those statistics are close to that obtained with *ε*_{G89} that similarly assumes a constant shear-to-strain ratio [*h*(*R*_{ω}) = 1 <=> *R*_{ω} = *R*_{ω}^{GM} = 3] together with *j*(*f*/*N*) = 1. However, *ε*_{G89} produces reasonable estimates where *ε*_{vmp} was enhanced at depth. From this comparison, we conclude that the use of *ε*_{ζ} based on a single density profile with a constant *R*_{ω} has to be used with care and may miss important features and produce regionally strong biases even when a regional value of *R*_{ω} is chosen.

To gain more insight into the frequency content of the internal wave field, an estimate of the wave-field-averaged frequency 〈*ω*〉 is diagnosed from *R*_{ω} [see Polzin et al. 1995, Eq. (15)]. For the whole transect, 〈*ω*〉 is close to the inertial frequency (between *f* and 2*f*) apart from a small region located between 14° and 16°N and at 1200–2200-m depth for which 〈*ω*〉 is nearly 3 times as large as *f* (Fig. 13a). Hibiya et al. (2012) used another approach to estimate the relative contribution of *α*, the “high” (2*f*-*N*) frequency band, and of *β*, the “low” (*f*-2*f*) frequency band, to the observed shear and strain variances. The ratio *β*/*α* measures the spectral distortion from the GM spectra. Applied to our dataset, we found that the high-frequency band is close to the GM level (*α* < 2) apart from the southeastern flank of the Reykjanes Ridge (6 < *α* < 10) and in a small region centered at 15°N and 1700 dbar (4 < *α* < 7). The contribution of the low-frequency band is usually close to the GM level (*β* < 2) above 2000 dbar. The largest values (*β* > 6) are found at depth below 3500 dbar to the south of 26°N and on the northwestern flank of the Reykjanes Ridge. Elsewhere, the low-frequency band is only moderately higher than the GM prescription (2 < *β* < 5). Interestingly, the ratio *β*/*α* shows a good consistency with 〈*ω*〉 (Fig. 13b). The high-frequency band tends to dominate (*β*/*α* < ⅓) in a few patches above 2000 m located south of 22°N. At depth, the ratio *β*/*α* is also smaller than ⅓ on the southern flank of the Reykjanes Ridge but not on its northern flank. Elsewhere, the low-frequency band mostly dominates (*β*/*α* > 4). Note that 〈*ω*〉 and *β*/*α* are indirect estimates of the averaged frequency content of the wave field subject to uncertainties. Several criteria based on either *α*, *β*, *β*/*α*, or 〈*ω*〉 have been tested to define a validity threshold above which *ε*_{ζ} remains consistent with *ε*_{vmp}, thus avoiding the large discrepancy found at depth in the weakly stratified waters. The only acceptable criterion found is to consider the use of *ε*_{ζ} only when 〈*ω*〉 > 1.1*f*, with *f* being the Coriolis parameter (Fig. 13a, white line). Using this criterion, the statistics of the ratio *ε*_{ζ}/*ε*_{vmp} are not significantly different from those of *ε*_{G03}/*ε*_{vmp} (Table 2, line *ε**_{ζ}). This confirms that the sole isopycnal strain data diagnosed from the CTD cannot be used to quantify dissipation rates in a near-inertial wave field with 〈*ω*〉 < 1.1*f* and in the presence of a weakly stratified environment. No criterion was found that improves the statistics of the ratio *ε*_{G89}/*ε*_{vmp} to the level of *ε*_{G03}/*ε*_{vmp}, although restricting the calculation to either 〈*ω*〉 > 1.1*f* or *β* < 10 improves by 4% the cumulative distribution while still showing a mean positive bias (+25%)(Table 2, line *ε**_{G89}).

## 4. Summary and discussion

For our dataset, the largest overestimation of *ε*_{vmp} by the finescale estimate appears in the Iberian Abyssal Plain below a depth of 4400 m (Fig. 10). In this region of weak stratification (*N* ≈ 3*f*), both the terms *h*(*R*_{ω}) and *j*(*f*/*N*) contribute to improve the consistency of *ε*_{G03} with *ε*_{vmp} (see the evolution of the dark blue dots from Fig. 7a to Fig. 7c). However, some of the segments remain strongly inconsistent (by a factor of 10). From an analysis of the shear-to-strain ratio *R*_{ω}, Kunze et al. (2006) argued that weakly stratified waters are associated with velocity profiles that are potentially strongly contaminated by noise, since the number of scatterers there is often low. They reported very high values of *R*_{ω} ranging from 30 to more than 1000 for *N* < 4.5 × 10^{−4} s^{−1} (see their . Below 4000 m in the Iberian Plain, R_{ω} also exhibits values ranging from 30 to 300 for a buoyancy frequency *N* < 3.2 × 10^{−4} s^{−1} (Fig. 8a). The noise associated with those deep profiles (*z* > 3900 m) is more intense by a factor of 2 compared to depths *z* < 2700 m and is clearly visible for wavenumbers larger than 0.07 rad m^{−1} (Table 1; Fig. 4). At low wavenumbers (*k* < 0.05 rad m^{−1}), however, the spectral shape of the buoyancy-normalized shear is flat as it is at shallower depths (Fig. 4). This means that if noise exists on velocity profiles for *k* < 0.05 rad m^{−1}, then it coincidentally has the same *k*^{−2} dependence before changing to white noise for *k* > 0.07 rad m^{−1}. Although one cannot exclude that noise is responsible for the overestimation of the dissipation rate in those regions, one could also argue that the scaling of observed shear by the traditional GM shear may not be appropriate for weakly stratified waters. Indeed, Gerkema and Shrira (2005) and Gerkema and Exarchou (2008) showed that nontraditional effects (effects of the Coriolis terms proportional to the cosine of latitude) can significantly change internal wave properties in weakly stratified waters.

Shallower than 3500 m, the averaged transect value of *R*_{ω} is 5.2 (standard deviation of 3), nearly twice the GM value of 3. Thus, there is on average more energy in near-inertial frequencies in our dataset than there is in the GM model (there is no peak of energy at tidal frequencies in the GM model). Kunze et al. (2006) found an averaged *R*_{ω} of 7 for various sections distributed over the Atlantic, Pacific and Indian Oceans. The ratio of the low (*f*-2*f*) to high (2*f*-*N*) wavenumber band contribution to the observed shear and strain variances confirms that at depth spectra are biased toward low frequencies compared to GM spectra. This is somewhat expected, since GM spectra does not take the tidal forcing into account. Interestingly the Reykjanes Ridge shows a significant cross-ridge contrast, the northwestern flank being biased toward low frequencies, while the southeastern flank is biased toward high frequencies. Over the Emperor seamount chain, Hibiya et al. (2012) also found some regions where *β* and *α* dominate at depth (e.g., their profiles H3 and H1, respectively). They attributed the bias toward high frequencies to tidal interactions with topographic features, which also possibly occur on the southeastern flank of the Reykjanes Ridge. Additional temporal measurements from moorings are needed to clearly quantify whether the contrast as diagnosed from *β*/*α* and 〈*ω*〉 is really significant. One can, however, note from altimetric data that there is also a cross-ridge asymmetry in the mesoscale eddy kinetic energy distribution with much higher levels to the northwest than to the southeast of the ridge. Nikurashin and Ferrari (2010) report enhanced near-inertial motions due to lee-wave generation by geostrophic currents, a process that is possibly at play to the north of the ridge.

On most parts of the section, the finescale parameterization *ε*_{G03} is consistent with the microstructure dissipation rate *ε*_{vmp}. This suggests that most of the turbulent activity recorded in this dataset is due to the internal wave field. Over 75% of the transect, the agreement is within a factor of 2.5 and almost 95% within a factor 4.5. Although we did not average the segments into bins as done in Polzin et al. (1995), which would reduce the scatter, those statistics are consistent with previous published studies (e.g., Polzin et al. 1995, 2002; Gregg et al. 2003). The use of the frequency-based correction term *h(R*_{ω}) improves the consistency as Polzin et al. (1995) found. The latitude/buoyancy frequency term slightly improves the parameterization, especially for weakly stratified waters. On average, the parameterization *ε*_{G03} tends to underestimate the microstructure dissipation rate by 11%. Most of the transect takes place away from internal wave generation sites (away from the surface and away from rough topography). The environment is then favorable for the comparison with finescale parameterizations based on the GM model, which is representative of the internal wave field away from boundaries.

Apart from a few locations near the topography, diffusivity remains weak. It is lower than 1 × 10^{−5} m^{2} s^{−1} over 44% of the transect and lower than 5 × 10^{−5} m^{2} s^{−1} over 90% of the transect. The picture emerging from this dataset is consistent with the finescale estimate of the dissipation rate along the A16N WOCE hydrographic line in the eastern North Atlantic for the same latitude range (Kunze et al. 2006, their Fig. 11). Stöber et al. (2008) reported frequent occurrences of enhanced diffusivities larger than 1 × 10^{−4} m^{2} s^{−1} on the Greenland slope off Cape Farewell (Eirik Ridge) for profiles located over isobaths ranging from 2700 up to 1100 m (their Fig. 4e). The part of our transect above the Greenland slope is located 100 km eastward and only exhibits such enhanced diffusivities for slopes shallower than 1150 m despite similar velocity amplitudes of the boundary current (maxima of 0.2–0.3 m s^{−1}), which extends down to the isobath 2500 m. Consistently, Lauderdale et al. (2008) reported bottom-intensified diffusivities frequently beyond 10 × 10^{−4} m^{2} s^{−1} around the Eirik Ridge and a factor of 5–10 less for a transect located on the Greenland slope near ours. Note that Lauderdale et al. (2008) used a strain-based parameterization with a constant “conservative” shear-to-strain ratio of 7.6 [*h*_{ζ} (*R*_{ω}) = 3 in Eq. (4)]. At 48°N and from 22° to 14°W, Walter et al. (2005) applied the shear variance [Eq. (3)] to a set of LADCP and CTD profiles to calculate dissipation and diffusivity. They found enhanced diffusivities that are systematically in the range of 10–100 × 10^{−4} m^{2} s^{−1} near the bottom and that they gradually decreased to 1 × 10^{−4} m^{2} s^{−1} at a distance of more than 1000 m above the seafloor. Such a strong intensification widely distributed in the eastern basin was partly attributed to the roughness of the bathymetry due to the proximity of the Mid-Atlantic Ridge. They also pointed out that from 40° to 50° of latitude, subinertial wave energy can be trapped (Gerkema and Shrira 2005) and lead to such intense diffusivities. In the region of our transect located in the same eastern basin, we indeed observe an intensification of turbulent diffusivities but 1) it is one to two orders of magnitude less important near the bottom (at most 1–5 × 10^{−4} m^{2} s^{−1}; Fig. 11a) than Walter et al. (2005) found, and 2) it is not associated with an intensification of the dissipation rate (Fig. 11b) but results from the weak values of the buoyancy frequency *N* used to calculated diffusivities from the Osborn (1980) relation. East of 21°W, our microstructure data do not show an intensification of the dissipation rate at depth either. Consistent with our study, Kunze et al. (2006) only find a few spots of elevated diffusivity that hardly reach 1 × 10^{−4} m^{2} s^{−1i} in the same deep basin. We believe that part of the discrepancy observed in this region at depth comes from the fact that Walter et al. (2005) did not use the scaling term function of the buoyancy frequency {cosh^{−1}(*N*/*f*)/[cosh^{−1}(*N*_{0}/*f*_{30})]} in *j*(*f*/*N*) [Eq. (3)]. Our dataset showed that for weakly stratified waters, this term decreased the finescale estimate of the dissipation rates and diffusivity, and improved the consistency with microstructure observations of the dissipation rates. Thus, our microstructure data and our finestructure shear/strain dataset do not support the idea of a basinwide intensification of the turbulent diffusivities at depth because of wave trapping between 40° and 50°N. The canonical globally averaged value of order 10^{−4} m^{2} s^{−1} needed to maintain the abyssal stratification (Munk 1966) is only reached on the flank of the Reykjanes Ridge and in the region around 21°W. Enhanced dissipation rates and diffusivity above the Reykjanes Ridge are potentially sustained by enhanced levels of semidiurnal tidal energy observed above the Reykjanes Ridge (van Haren 2007) and by breaking of lee waves generated by bottom geostrophic flows above this rough topography (Nikurashin and Ferrari 2011).

The microstructure profiler was funded by the French Agence Nationale de la Recherche (ANR) through Grant ANR-JC05_50690 and by the French Institute for Marine Science (IFREMER). B. Ferron and H. Mercier were supported by the French National Center for Scientific Research (CNRS), F. Kokoszka by a doctoral grant from the Ministère de la Rechercher et de l'Enseignement Supérieur, and P. Lherminier by IFREMER.

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