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  • View in gallery

    Positions of the 91 repeated CTD–LADCP stations (yellow dots) along the A25 hydrographic line. This hydrographic line was repeated every 2 years from 2002 to 2012 (see Table 1). Topography from Smith and Sandwell (1997). Some distance-from-Portugal markers (km) are plotted to facilitate the reading of the following figures.

  • View in gallery

    (a) Turbulent kinetic energy dissipation rate 〈ε2002–2012〉 (W kg−1) averaged from the six estimates of the repeated A25 hydrographic line. Black isolines depict the 2002–12 mean buoyancy frequency N [log10(N), s−1, interval 0.2 s−1]. The specific value log10(N) = −2.8 s−1 is plotted as a thick black line. (b) Ratio of the local maximum to minimum dissipation rate over the period 2002–12. Color scales are nonlinear. IbB, Iberian Plain; ABR, Azores–Biscay Rise; ETR, East Thulean Rise; ES, Eriador Seamount; RR, Reykjanes Ridge; IrB, Irminger basin (see Fig. 1).

  • View in gallery

    Dissipation rate estimates of the turbulent kinetic energy (color, nonlinear scale, W kg−1) along the repeated A25 hydrographic line for years 2002–12. White colors correspond to an absence of data or to the surface layer. Black isolines depict the buoyancy frequency N [log10(N), s−1, interval 0.2 s−1]. The specific value log10(N) = −2.8 s−1 is plotted as a thick black line for the corresponding year and by the white line for its 2002–12 average.

  • View in gallery

    Ratio of the 2002–12 mean clockwise variance to the 2002–12 mean counterclockwise variance.

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    (a) The 2002–12 mean low-frequency factor. (b) The 2002–12 mean high-frequency factor. (c) Ratio of the 2002–12 mean low- to high-frequency factors. A white area denotes a location where no estimate was computed (at the surface) or no physically meaningful estimate was found (at the bottom).

  • View in gallery

    Dissipation rates averaged (a) within 500 m of the bottom and (b) in the thermocline between 100 and 500 m depth. (c) Barotropic to baroclinic tidal energy conversion (FIT). (d) Surface eddy kinetic energy interpolated at the station location. Each line is color coded by year and the 2002–12 mean is in black. The gray line in (c) is the 2002–12 mean tidal conversion estimated with the single-beam bathymetry. Letters denote the location of nine events of large bottom dissipation rates discussed in the text.

  • View in gallery

    Transect average of vertical dissipation rate profiles whose bottommost values are larger than 5 × 10−10 W kg−1 (thick lines). The average was computed as a function of height above the bottom. The color depicts the transect year and the thin line is a fit of an exponential profile decaying with a 1000-m vertical scale. Depending on the year, 8–12 profiles were averaged.

  • View in gallery

    Geostrophic eddy velocity field (arrows) from sea surface height maps at the time of the largest 2002–12 bottom dissipation rate event B2 (yellow triangle) that occurred in 2008 with bathymetry (Smith and Sandwell 1997) colored in blue.

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Variability of the Turbulent Kinetic Energy Dissipation along the A25 Greenland–Portugal Transect Repeated from 2002 to 2012

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  • 1 Laboratoire de Physique des Océans, CNRS, UMR 6523 CNRS-IFREMER-IRD-UBO, IFREMER Centre de Brest, Plouzané, France
  • 2 Laboratoire de Physique des Océans, UBO, UMR 6523 CNRS-IFREMER-IRD-UBO, Brest, France
  • 3 Laboratoire de Physique des Océans, IFREMER, UMR 6523 CNRS-IFREMER-IRD-UBO, IFREMER Centre de Brest, Plouzané, France
  • 4 Instituto de Investigaciones Marinas de Vigo, Consejo Superior de Investigaciones Científicas, Vigo, Spain
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Abstract

The variability of the turbulent kinetic energy dissipation due to internal waves is quantified using a finescale parameterization applied to the A25 Greenland–Portugal transect repeated every two years from 2002 to 2012. The internal wave velocity shear and strain are estimated for each cruise at 91 stations from full depth vertical profiles of density and velocity. The 2002–12 averaged dissipation rate 〈ε 2002–2012〉 in the upper ocean lays in the range 1–10 × 10−10 W kg−1. At depth, 〈ε 2002–2012〉 is smaller than 1 × 10−10 W kg−1 except over rough topography found at the continental slopes, the Reykjanes Ridge, and in a region delimited by the Azores–Biscay Rise and Eriador Seamount. There, the vertical energy flux of internal waves is preferentially oriented toward the surface and 〈ε 2002–2012〉 is in the range 1–20 × 10−10 W kg−1. The interannual variability in the dissipation rates is remarkably small over the whole transect. A few strong dissipation rate events exceeding the uncertainty of the finescale parameterization occur at depth between the Azores–Biscay Rise and Eriador Seamount. This region is also marked by mesoscale eddying flows resulting in enhanced surface energy level and enhanced bottom velocities. Estimates of the vertical energy fluxes into the internal tide and into topographic internal waves suggest that the latter are responsible for the strong dissipation events. At Eriador Seamount, both topographic internal waves and the internal tide contribute with the same order of magnitude to the dissipation rate while around the Reykjanes Ridge the internal tide provides the bulk of the dissipation rate.

Corresponding author address: Bruno Ferron, Laboratoire de Physique des Océans, UMR 6523 CNRS-IFREMER-IRD-UBO, IFREMER Centre de Brest, CS 10070, 29280 Plouzané, France. E-mail: bruno.ferron@ifremer.fr

Abstract

The variability of the turbulent kinetic energy dissipation due to internal waves is quantified using a finescale parameterization applied to the A25 Greenland–Portugal transect repeated every two years from 2002 to 2012. The internal wave velocity shear and strain are estimated for each cruise at 91 stations from full depth vertical profiles of density and velocity. The 2002–12 averaged dissipation rate 〈ε 2002–2012〉 in the upper ocean lays in the range 1–10 × 10−10 W kg−1. At depth, 〈ε 2002–2012〉 is smaller than 1 × 10−10 W kg−1 except over rough topography found at the continental slopes, the Reykjanes Ridge, and in a region delimited by the Azores–Biscay Rise and Eriador Seamount. There, the vertical energy flux of internal waves is preferentially oriented toward the surface and 〈ε 2002–2012〉 is in the range 1–20 × 10−10 W kg−1. The interannual variability in the dissipation rates is remarkably small over the whole transect. A few strong dissipation rate events exceeding the uncertainty of the finescale parameterization occur at depth between the Azores–Biscay Rise and Eriador Seamount. This region is also marked by mesoscale eddying flows resulting in enhanced surface energy level and enhanced bottom velocities. Estimates of the vertical energy fluxes into the internal tide and into topographic internal waves suggest that the latter are responsible for the strong dissipation events. At Eriador Seamount, both topographic internal waves and the internal tide contribute with the same order of magnitude to the dissipation rate while around the Reykjanes Ridge the internal tide provides the bulk of the dissipation rate.

Corresponding author address: Bruno Ferron, Laboratoire de Physique des Océans, UMR 6523 CNRS-IFREMER-IRD-UBO, IFREMER Centre de Brest, CS 10070, 29280 Plouzané, France. E-mail: bruno.ferron@ifremer.fr

1. Introduction

The diapycnal mixing in the interior of the open ocean is generated by a variety of mechanisms. Among the major sources of mechanical energy for the mixing are internal waves forced by tides, winds, or mean flows interacting with topography (Ferrari and Wunsch 2009). All these mechanical energy sources are strongly dependent on time.

Finescale parameterizations of the diapycnal mixing due to the internal waves are nowadays routinely used. They are frequently applied to datasets obtained from a unique realization of a transect, which gives us a snapshot of the mixing intensity at the time of the observations. While this approach is certainly important to map the geographical heterogeneity in the distribution of the diapycnal mixing and points toward mechanisms, one can question how representative such snapshots are because generating mechanisms leading to mixing processes are intermittent. Some recent studies used finescale and microscale observations from profilers, gliders, and towed vehicles to quantify the mean mixing and its temporal variability (Moum et al. 2009; Fer et al. 2014; Palmer et al. 2015; Johnston et al. 2011). Most of these studies focused on the upper ocean (0–500 m). Much less is known about the large-scale variability of the deep ocean mixing from an observation point of view.

In the present paper, we address this question using hydrographic observations from the A25 Greenland–Portugal transect that was repeated six times from 2002 to 2012. We examine the variability of the turbulent kinetic energy dissipation due to internal waves in light of some possible generating mechanisms. Section 2 briefly describes the available dataset and some common diagnostics based on velocity and density profiles. Section 3 focuses on the dissipation rate variability, on some characteristics of the internal wave field, and explores the contribution due to the internal tide and topographic internal waves. Results are summarized and discussed in section 4.

2. Data and methods

a. Datasets

1) Hydrography

In this study we used full-depth CTD and lowered-ADCP (LADCP) vertical profiles of the A25 Greenland–Portugal transect, that was repeated every two years from 2002 to 2012 (Fig. 1). All stations were collected around the same period of the year (late spring to early summer) and over a time span of 3–4 weeks depending on the year (Table 1). For each transect, we selected the 91 stations that were sampled almost systematically. The sensor configuration of the rosette evolved over the years (Table 1). Nevertheless, all LADCP data were processed with the velocity-inversion method initially developed by Visbeck (2002) that provided observations of vertical shear of horizontal currents with a 16-m vertical resolution.

Fig. 1.
Fig. 1.

Positions of the 91 repeated CTD–LADCP stations (yellow dots) along the A25 hydrographic line. This hydrographic line was repeated every 2 years from 2002 to 2012 (see Table 1). Topography from Smith and Sandwell (1997). Some distance-from-Portugal markers (km) are plotted to facilitate the reading of the following figures.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0186.1

Table 1.

Cruise names, years, and periods of each data acquisition and CTD and LADCP settings. The sampling period corresponds to the time taken to carry out the 91 repeated CTD stations of the A25 line. Manufacturer acronyms are: RD Instruments (RDI), Laboratoire de Physique des Océans (LPO), and Sea-Bird Electronics (SBE).

Table 1.

2) Topography

Two topographic datasets were used. The first was the 1-min gridded Global Topography version 18.1 (Smith and Sandwell 1997), which resolves scales larger than 10–20 km. It was used to estimate the vertical energy flux into the internal tide. The second was the single beam sounding measured along the transect. It has a horizontal resolution of about 150 m (depends on ship velocity) and thus correctly resolves topographic scales larger than about 750 m. It was used to estimate the vertical energy flux into topographic internal waves.

3) Sea surface height

To calculate the surface geostrophic eddy kinetic energy, we used the daily global absolute dynamic topography on a ¼° × ¼° horizontal resolution grid. Those fields are produced in delayed time by SSALTO/Duacs and distributed by AVISO.

b. Diagnosed quantities

1) Dissipation rates from vertical shear and isopycnal strain variances

Estimates of the turbulent kinetic energy dissipation rate associated with the vertical shear of horizontal velocities and with the isopycnal strain generated by the internal wave field were computed using Gregg et al.’s (2003) finescale parameterization:
e1
where ε0 = 7 × 10−10 W kg−1 and N/N0 and are the buoyancy frequency and the shear variance, respectively, scaled by their respective values from the Garrett and Munk (GM) model (Garrett and Munk 1975; Cairns and Williams 1976). The frequency correction function h reads:
eq1
where is the shear (Vz)-to-strain (ζz) variance ratio (Polzin et al. 1995). The function j reads
eq2
where f is the inertial frequency and f30 its value at a latitude of 30°.

During year 2008, vertical microstructure profiles (VMPs) were collected along with CTD–LADCP profiles at 30 stations. A comparison between dissipation rates εG03 estimated from the fine structure and εVMP observed from the microstructure showed that 90% of εG03 estimates are within a factor of 4 of εVMP values (Ferron et al. 2014). In this paper, we followed exactly the same processing for calculating εG03 as that detailed in Ferron et al. (2014), including the use of 640-m long segments with 75% overlap. We obtained profiles of 160-m vertical mean values of εG03 at each CTD–LADCP station and for each transect.

To get some clues about turbulent mechanisms that could possibly explain some of the observed time variability in the dissipation rates, several diagnostics described in the following subsections were computed.

2) Rotary velocity variances

We used ratios of clockwise (CW) to counterclockwise (CCW) velocity variances to estimate the dominant direction of the vertical energy flux of the internal wave field (Gonella 1972; Leaman 1976). The velocity vector of a near-inertial wave rotates CW (CCW) with depth when the phase propagates upward and the energy downward (upward) in the Northern Hemisphere. A ratio of CW to CCW velocity variances above unity is then interpreted as a dominant downward energy flux. This decomposition and interpretation becomes useless for high-frequency waves as they are more linearly polarized than low-frequency waves. Such a decomposition based on velocity spectra emphasizes lower wavenumbers as compared with the shear spectra used to estimate the dissipation rate [Eq. (1)]. Thus, although useful to give us a clue on the internal wave generation, the use of rotary velocity variances does not necessarily describe the waves causing the kinetic energy dissipation. The CW and CCW variances were calculated by integrating the rotary spectra on the same vertical segments and same wavenumber band as those used to estimate shear variances for εG03.

3) Spectral distortion from the GM model

For estimating εG03, we used the observed shear and strain variances scaled by their GM counterpart (, ). To get some indications on the frequency distortion of observed spectra as compared with the GM spectra, we followed Hibiya et al.’s (2012) approach that decomposes the observed shear and strain variances as follows:
eq3
eq4
where α (β) is the observed departure from the GM model in the high-frequency (2fN) [low-frequency (f–2f)] band. While α and β are measures of the observed spectral levels as compared with the GM model, the ratio β/α is a measure of the spectral distortion. Variables α and β were calculated on the same vertical segments and same wavenumber band as those used to estimate shear variances for εG03. The shear and strain variances, which are calculated from a limited number of wavenumbers, follow a distribution, assuming that the number of degrees of freedom n is equal to twice the number of wavenumbers used in the variance. Their statistical uncertainties follow from this distribution leading to standard deviations in α and β of about one.

4) Surface eddy kinetic energy

Geostrophic velocities were computed from the daily absolute dynamic topography from which the 2002–12 mean topography was removed. The eddy kinetic energy (EKE) of the surface geostrophic velocities reads
eq5
where ug and υg are the zonal and meridional geostrophic velocities. The EKE at the day of a station was interpolated at the station position.

5) Barotropic to baroclinic tidal flux

The barotropic tide to baroclinic internal tide energy flux was estimated using a scale relation derived for subcritical topography (Jayne and St. Laurent 2001) as
e2
where ρ is the density of seawater, Nb is the near bottom buoyancy frequency, u is the barotropic tidal speed, and (κ, h) are the characteristic wavenumber and amplitude of the topography. The topography is “subcritical” if the topographic slopes are less than the slope of the internal wave generated by the tides. In the formulation Eq. (2), κ is an adjustable parameter (Jayne and St. Laurent 2001; Jayne 2009). As Jayne and St. Laurent (2001) stressed, Eq. (2) is not a precise specification of the internal tide energy flux but results from a scaling approach. To avoid an excessive smearing in the estimated flux (Green and Nycander 2013), h was calculated as h(x, y) = L(x, y)Smax(x, y), where Smax is the maximum slope over a distance L bounded by a maximum of 20 km around a given position (x, y). The effect of rotation is taken into account by multiplying Eq. (2) by (1 − f2/ω2)1/2, where ω is the tidal frequency (Llewellyn Smith and Young 2003; Nycander 2005). Thus, Eq. (2) was replaced by
e3
The constant C = 1/(4π) in Eq. (3), equal to that of Nycander (2005), gives a reasonable consistency between FIT and the related dissipation rate along the A25 line (see section 3). For the latitudinal range of the A25 line, tidal waves at diurnal frequencies are bottom trapped (ω < f). The energy flux was thus restricted to the contributions of the semidiurnal frequencies (M2, S2, N2, K2). The daily amplitude of the barotropic velocity used in Eq. (3) was computed from the TOPEX/Poseidon Global Inverse Solution 7.2 (TPXO7.2) database (Egbert and Erofeeva 2002). Variable Nb was computed from the CTD observations as the 160-m mean buoyancy frequency [last segment of N used in Eq. (1)]. The topographic scale L and slope Smax were inferred from the global topography (Smith and Sandwell 1997). More complex parameterizations of the tidal flux exist (e.g., Nycander 2005; Pétrélis et al. 2006), but, as in Eq. (3), they all depend on the square of the velocity. Since we are interested in the relative variations of FIT over the years, which are mostly driven by variations in the barotropic tidal velocity, we then expect that the formulation Eq. (3) allows us to derive a reasonable estimate of the time variability in the tidal flux.

6) Mean flow to topographic internal wave energy conversion rate

Topographic internal waves (also called lee waves or mountain waves) are generated when a mean current Umean flows over some topographic roughness in a stratified fluid. The linear theory of internal wave generation in the presence of topography developed by Bell (1975) gives the energy conversion rate from the mean flow into topographic internal waves as
e4
where k = (k, l) is the horizontal wavevector, P(k, l) is the spectrum of a topographic patch, and ω = (Umeank) is the wave frequency. The wavenumber integration is done over the range of radiating waves for which |f| < ω < N and does not take into account topographic trapped waves. Aguilar and Sutherland (2006) showed that, when the vertical Froude number, defined as Frυ = Umean/Nh (where Umean is the flow speed in the direction of the wavevector k) is smaller than a critical value Frc (e.g., for large amplitude topography), the topography partially blocks the flow and reduces the conversion rate such that
e5
For sinusoidal and triangular hills, Aguilar and Sutherland (2006) found Frc = 0.75−1. This dependence was confirmed by numerical simulations with Frc = 0.7−1 (Nikurashin and Ferrari 2010). In this study we chose Frc = 0.75−1.

To evaluate FTW using (4) and (5), several ingredients are needed. Since we do not have time measurements to quantify the mean flow, the bottom LADCP velocity averaged within 500 m of the bottom is assumed to be representative of the mean flow. A constant buoyancy frequency of 5 × 10−4 s−1, representative of the bottom stratification, is taken from the CTD stations located to the southeast of the Reykjanes Ridge where Eqs. (4) and (5) are applied. Given the latitudinal range of the events and the bottom-averaged LADCP velocity, waves can radiate if topographic horizontal length scales are in the range 0.5–8 km. Such scales are usually not resolved by the global topographic map (Smith and Sandwell 1997). To characterize those horizontal scales k along with their vertical scales h at a given location, we used the single beam sounding along the transect that correctly resolves topographic scales larger than about 750 m. Finally, FTW was calculated using the observed scales k and h and assuming that the kilometer-scale hills have an axisymmetric sinusoidal shape, which fully determines P(k, l) in Eq. (4).

7) Conversion of vertical energy fluxes into dissipation rates

In the following, FIT and FTW are converted to dissipation rates to quantify their respective contributions to the observed dissipation rates. This is done assuming that an average q = 30% of the energy is dissipated locally and that the vertical decay scale of the dissipation rate is about 1000 m. While the latter can be diagnosed from our set of observations, we took a conservative value for q that was estimated for the internal tide dissipation in the Brazil basin (St. Laurent and Garrett 2002). The morphology of the topography in our region of interest shares some similarities with that of the Brazil basin, both being located on the flank of the Mid-Atlantic Ridge. One can, however, expect some local variability in the parameter q for the internal tide. To our knowledge, no such estimate of q exists for topographic internal waves. Following Nikurashin and Ferrari (2013), we then chose the same value for both types of internal waves.

3. Results

a. Mean dissipation rate and observed variability range

The regions having the smallest 2002–12 mean dissipation rate, 〈ε2002–2012〉, are located below 2000 m east of 21°W (distance from Portugal dPort < 1450 km) from the Iberian basin to the northwest of the Azores–Biscay Rise with values smaller than 1 × 10−10 W kg−1 (Fig. 2a). Such small values are also found in the weakly stratified center of the Irminger basin, but this time at all depths. Those two deep regions are characterized by smooth topography as in the Brazil basin for which weak dissipation rates are also found (Polzin et al. 1997). The 〈ε2002–2012〉 is enhanced at depth from 21° to 26°W (1450 km < dPort < 2200 km). This region starts to the southwest of the East Thulean Rise (ETR; see Fig. 2) to Eriador Seamount (ES; Fig. 2 and map in Fig. 1). Because of the proximity of the Mid-Atlantic Ridge, it is characterized by an increase in topographic roughness having typical horizontal length scales of 3–40 km and vertical length scales of several hundreds of meters. There 〈ε2002–2012〉 ranges over 1–20 × 10−10 W kg−1. Enhanced 〈ε2002–2012〉 are also found farther north, on the flanks of the Reykjanes Ridge and on the Greenland slope, with maximum values of 5–10 × 10−10 W kg−1. Those enhanced dissipation rates at depth vertically extend up to 500–1500 m above the bottom. In the upper ocean, 〈ε2002–2012〉 is in the range 1–10 × 10−10 W kg−1 for buoyancy frequencies N > 1.6 × 10−3 s−1 (Fig. 2a, thick line). Those values are typically found in the main thermocline (e.g., Gregg 1989).

Fig. 2.
Fig. 2.

(a) Turbulent kinetic energy dissipation rate 〈ε2002–2012〉 (W kg−1) averaged from the six estimates of the repeated A25 hydrographic line. Black isolines depict the 2002–12 mean buoyancy frequency N [log10(N), s−1, interval 0.2 s−1]. The specific value log10(N) = −2.8 s−1 is plotted as a thick black line. (b) Ratio of the local maximum to minimum dissipation rate over the period 2002–12. Color scales are nonlinear. IbB, Iberian Plain; ABR, Azores–Biscay Rise; ETR, East Thulean Rise; ES, Eriador Seamount; RR, Reykjanes Ridge; IrB, Irminger basin (see Fig. 1).

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0186.1

To quantify the amplitude of the dissipation rate variability along the A25 line over the 2002–12 period, the ratio of the maximum to the minimum dissipation rate was calculated for each 160-m vertical segment (Fig. 2b). Over most of the section this ratio is below 5. Having in mind that the dissipation rate as estimated by Eq. (1) typically under- or overestimates the observed dissipation rate from microstructure by a factor of 4 (Ferron et al. 2014), the amplitude of the variability is remarkably small along this section for the late spring/early summer period. In the thermocline, the ratio is slightly larger, ranging from 5 to 15 between 17° and 26°W (1200 km < dPort < 2200 km), as well as in the Irminger Sea above 1500 m from 31° to 39°W (2800 km < dPort < 3200 km). The largest variability (ratio from 5 to 100) is found at depth in regions where <ε2002–2012> is small (dPort < 1450 km and in the Irminger Sea) or where it is enhanced (1450 km < dPort < 2200 km). There is also more variability on the northwestern flank of the Reykjanes Ridge than on the southeastern one.

The year-by-year distribution of εG03 confirms that the isoline N = 1.6 × 10−3 s−1 is a good predictor of the value εG03 = 1 × 10−10 W kg−1 that delimits the usually small dissipation rates in the deep ocean from the moderate ones in the thermocline (Fig. 3, thick black lines). The variability of εG03 in the Iberian basin (Fig. 2b) is mostly due to one occurrence of enhanced values observed in 2010 and to a lesser extent in 2008 (Fig. 3). At depth in the region (1450 km < dPort < 2200 km), εG03 is often enhanced locally around seamounts, with maximum values of 20–100 × 10−10 W kg−1 reached in 2008 west of the ETR (22°W, dPort = 1640 km) and of 1–50 × 10−10 W kg−1 reached in 2010 near ES. In the main thermocline, the largest εG03 (>5 × 10−10 W kg−1) are almost systematically located from the Azores–Biscay Rise to Eriador Seamount and above the Reykjanes Ridge. In the Irminger basin, the small dissipation rates of the upper water column tend to spread to the southeast with time, consistently with the migration of the isoline N = 1.6 × 10−3 s−1 in that same direction (cf. the thick black line with thin white lines in Fig. 3). This is consistent with a strengthening of the winter deep convection observed since 2008 in the Irminger basin (Piron et al. 2016).

Fig. 3.
Fig. 3.

Dissipation rate estimates of the turbulent kinetic energy (color, nonlinear scale, W kg−1) along the repeated A25 hydrographic line for years 2002–12. White colors correspond to an absence of data or to the surface layer. Black isolines depict the buoyancy frequency N [log10(N), s−1, interval 0.2 s−1]. The specific value log10(N) = −2.8 s−1 is plotted as a thick black line for the corresponding year and by the white line for its 2002–12 average.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0186.1

b. Internal wave energy propagation and spectral distortion from the GM model

The 2002–12 ratio of mean clockwise to mean counterclockwise variances shows a dominance of counterclockwise variance at depth with two exceptions: the Iceland basin to the northwest of Eriador Seamount and the center of the Irminger basin (Fig. 4). This suggests that the energy in the internal wave field preferentially propagates upward above the seafloor, which is compatible with an energy source at the bottom. Above the Reykjanes Ridge, a dominant upward energy propagation is also found from the bottom up to the near surface. Those features observed with the mean 2002–12 variances are representative of what happens on a year-by-year basis.

Fig. 4.
Fig. 4.

Ratio of the 2002–12 mean clockwise variance to the 2002–12 mean counterclockwise variance.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0186.1

The distributions of the 2002–12 mean high (α) and mean low (β) frequency factors show that they remain below a factor of three on most of the section (Figs. 5a,b). This reveals that the observed level of internal wave field energy is close to the GM model level. This is particularly true for the upper ocean. This is consistent with the observation that the isolines N = 10−2.8 s−1 and ε = 1 × 10−10 W kg−1 follow the same path (Figs. 2a, 3). Indeed, for a wave field close to the GM model, Eq. (1) mainly depends on N/N0. In the vicinity of the bathymetry, departures from the GM model are stronger: α and β both reach the range 5–7 with a few peak values around 10–12 for α and around 8–10 for β.

Fig. 5.
Fig. 5.

(a) The 2002–12 mean low-frequency factor. (b) The 2002–12 mean high-frequency factor. (c) Ratio of the 2002–12 mean low- to high-frequency factors. A white area denotes a location where no estimate was computed (at the surface) or no physically meaningful estimate was found (at the bottom).

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0186.1

Note that the low-frequency band (f–2f) includes the semidiurnal frequency. It is then not surprising that, close to the bathymetry, our observed energy level in the low-frequency band is higher than that of the GM model (i.e., β > 1) since the latter does not take the spectral peak of the internal tide into account. Hibiya et al. (2012) also reported such large values of α and β over the rough topography of the North Pacific. The spectral distortion as measured by the ratio β/α is moderate (⅓ < β/α < 3), even in the region where we observed stronger energy levels than the GM level (Fig. 5c). It is, however, worth noting that we systematically observe low values of β/α over the Reykjanes Ridge. This bias toward higher frequency possibly suggests that high-frequency waves are generated by interactions between the tide and the topography of the Reykjanes Ridge (Gerkema et al. 2006; Gerkema 2006; Pairaud et al. 2010). Low values of β/α are also found in the Iberian basin at and below the depth of the Mediterranean outflow water, between 1000 and 1800 m. For an environment located below the Mediterranean outflow water, at a depth of 1400–1500 m in the Canary basin, van Haren and Gostiaux (2009) reported that the temperature spectra follow a ω−1 slope in the frequency domain for the internal wave band instead of the ω−2 slope of the GM model. Compared with the GM model, this amounts to having a larger contribution of the high-frequency band than of the low-frequency band. Indeed, for an internal wave field with a ω−1 slope, the ratio β/α can be readily expressed as
e6
Applying this relation to the Mediterranean water region delimited by dPort < 1000 km (i.e., maximum latitudinal extent slightly to the northwest of the Azores–Biscay Rise) and to depths in the range 1000–1800 m, with an averaged N = 2.1 × 10−3 s−1 and f = f(41.5°), we get a typical β/α ratio of 3.8−1 for an internal wave field following a ω−1 slope. The observed β/α averaged over the same region is estimated at 3.1−1 (Fig. 5c). Thus, there is some consistency between our estimate and the ω−1 slope that van Haren and Gostiaux (2009) observed accurately with temperature sensors. They attributed the change in slope to the presence of density layering and intermittency. Our CTD data also show a layered structure at and below the depth of the Mediterranean outflow.

c. Relation between bottom dissipation rates, internal tide and topographic internal wave generations, and surface EKE

To gain more insight about some possible mechanisms at the origin of the observed variability, dissipation rates εG03 were averaged within 500 m of the bottom (εbottom) and within the upper thermocline between 100 m and 500 m depth (εthermocline). It clearly appears that the time variability of εbottom (εthermocline) lies within a factor of 4–5 (2–3) around its 2002–12 mean value (Figs. 6a,b). The interannual dispersion of εbottom is usually remarkably weak in the Iberian, Iceland, and Irminger basins and stronger in the region from the ETR to ES, the Reykjanes Ridge, and the Greenland slope. The dispersion of εthermocline follows the same pattern: it is enhanced between the ABR to the Reykjanes Ridge but reduced in the Iberian, Iceland, and Irminger basins.

Fig. 6.
Fig. 6.

Dissipation rates averaged (a) within 500 m of the bottom and (b) in the thermocline between 100 and 500 m depth. (c) Barotropic to baroclinic tidal energy conversion (FIT). (d) Surface eddy kinetic energy interpolated at the station location. Each line is color coded by year and the 2002–12 mean is in black. The gray line in (c) is the 2002–12 mean tidal conversion estimated with the single-beam bathymetry. Letters denote the location of nine events of large bottom dissipation rates discussed in the text.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0186.1

The barotropic to baroclinic tidal energy conversion FIT as diagnosed from Eq. (3) shows an energy flux that varies because of the fortnight variability of the barotropic tide (Fig. 6c). Note that a flux FIT = 1 mW m−2 is equivalent to a dissipation rate ε = 2.3 × 10−10 W kg−1, providing that about 30% of the wave energy is dissipated locally (St. Laurent and Garrett 2002) and that ε decreases over a vertical length scale of about 1000 m. Such a vertical length scale is observed on average for the bottom-enhanced ε profiles of our dataset, although some departures exist (Fig. 7). The constant C in Eq. (3) has a reasonable value since FIT, converted to a dissipation rate, is consistent with εbottom over the Reykjanes Ridge (dPort = 2800 km) where a strong internal tide was observed (van Haren 2007). While some expected consistency is found in the location of enhanced εbottom and enhanced FIT in terms of 2002–12 mean quantities (e.g., around ES, over the Reykjanes Ridge, and along the continental slopes), there is obviously a lack of consistency on a year-by-year basis. The interannual change in tides may sign its influence on εbottom at two locations: the Iberian basin, for which FIT and εbottom both have maximum values for year 2010 (Figs. 6a,c), and the summit of the Reykjanes Ridge at dPort = 2800 km, for which the events with the three largest dissipation rates have also the largest FIT. Elsewhere, this lack of consistency is either due to εbottom being related to more complex forcing mechanisms or to the signal-to-noise ratio being close to one. Indeed, the time variability of εbottom lies within a factor of 4–5, which is similar to the dispersion between fine- and microstructure estimates of εG03 (Ferron et al. 2014).

Fig. 7.
Fig. 7.

Transect average of vertical dissipation rate profiles whose bottommost values are larger than 5 × 10−10 W kg−1 (thick lines). The average was computed as a function of height above the bottom. The color depicts the transect year and the thin line is a fit of an exponential profile decaying with a 1000-m vertical scale. Depending on the year, 8–12 profiles were averaged.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0186.1

The surface eddy kinetic energy averaged over 2002–12 is enhanced by a factor of 3–7 between the Azores–Biscay Rise to the ETR (1000 km < dPort < 1900 km) compared with an averaged observed value of 50 cm2 s−2 outside of this region (Fig. 6d, black line). To the north of the Azores–Biscay Rise, the surface EKE presents a large dispersion with peak values reaching 4 times their mean value and 4–10 times their median value. The most energetic region close to the ETR (1500 km < dPort < 1900 km), with a 2002–12 mean EKE larger than 200 cm2 s−2, has also a mean εthermocline among the largest of the transect (εthermocline > 5 × 10−10 W kg−1). On the Iceland basin side of the Reykjanes Ridge, the 2002–12 mean EKE is weak (about 40 cm2 s−2 between 2500 km < dPort < 2800 km), while on the Irminger basin side it is enhanced and reaches 100 cm2 s−2 with a strong interannual variability. On a year-by-year basis, no obvious relationship is found between the level of EKE and that of εthermocline or εbottom except for some energetic events.

Nine events having the largest departure in εbottom from its median value calculated over the six transects were selected in order to study their characteristics. They are depicted by letters A to E in the region from the Azores–Biscay Rise to Eriador Seamount, and by I1 to I3 in the Iberian basin (Fig. 6, Table 2). All those events present an εbottom that departs by at least a factor of 6 from the station median value (Table 2, column 7). The interpolated EKE at the location of the events A to E is moderate (close to or larger than 200 cm2 s−2) to moderately strong (>800 cm2 s−2), except for event D (Table 2, column 4). For the events I1 to I3 located to the southeast, the EKE is less energetic and ranges from 50 to 140 cm2 s−2. At the exception of event D, the EKE is larger than its median value by at least a factor of 2 (Table 2, column 8). The bottom velocity averaged within 500 m of the bottom frequently shows elevated values at the location of the events. From the Azores–Biscay Rise to Eriador Seamount, it ranges from 8 to 16 cm s−1 (Table 2, column 5). In the Iberian basin it reaches 4 to 11 cm s−1.

Table 2.

Bottom-intensified dissipation rate events for which εbottom/εmedian > 5. Events A to E are located between the Azores–Biscay Rise and the Reykjanes Ridge. The three less turbulent I events are located in the Iberian basin. Surface EKE interpolated at the location and time of the events and estimated from AVISO gridded sea surface height. Magnitude of the LADCP velocities and dissipation rates averaged within 500 m of the bottom (〈ULADCPbottom and εbottom, respectively); ratio of εbottom (EKE) to its median value is calculated from the six transects (Figs. 6a and 6d, respectively).

Table 2.

The largest observed peak in εbottom of the 2002–12 period reaches 40 × 10−10 W kg−1. It occurred during the year 2008 to the southeast of the ETR at dPort = 1640 km (Table 2, event B2). This peak in εbottom is associated with a large peak in surface EKE that reaches 830 cm2 s−2 (Fig. 6d, Table 2). It is the second-largest surface EKE value observed on the transect for the 2002–12 period. Geostrophic eddy velocities show that the hydrographic station associated with the event is located on the northeastern edge of a cyclonic eddy (Fig. 8). At the same location, the second-largest peak in εbottom (11 × 10−10 W kg−1) is observed in 2006, for which we also observe the second-largest peak in EKE (320 cm2 s−2). The four other EKE values at dPort = 1640 km lie in the range 40–100 cm2 s−2 (Fig. 6d). Contrastingly, a tidal flux of 0.7 mW m−2 is diagnosed for 2008 and a smaller one (0.3 mW m−2) for the year 2006 at dPort = 1640 km. The largest tidal flux for this area occurs in 2012 and reaches 1.3 mW m−2. Thus, the change in εbottom at dPort = 1640 km is more consistent with the change in EKE level than with that of the tidal flux.

Fig. 8.
Fig. 8.

Geostrophic eddy velocity field (arrows) from sea surface height maps at the time of the largest 2002–12 bottom dissipation rate event B2 (yellow triangle) that occurred in 2008 with bathymetry (Smith and Sandwell 1997) colored in blue.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0186.1

To evaluate the order of magnitude of the energy flux from the mean flow into topographic internal waves using Eqs. (4) and (5), the characteristic height and length scale of appropriate hills (i.e., such that |f| < ω < N) surrounding the location of each events are given in Table 3. For most of the events, the vertical Froude number is large (i.e., Frυ−1 < 0.75) so that the energy conversion Eq. (4) derived from linear theory can be used. For events A, B1, and I2, the conversion rate is reduced according to Eq. (5) (values in parentheses in Table 3). The ratio ω/f varies in the range 1.1–3.4 while ω/N is in the range 0.25–0.7. The dissipation rate associated with the energy flux from the mean flow into topographic internal waves FTW is compared with that associated with the internal tide FIT and to the observed 500-m bottom-averaged dissipation rate (last three columns of Table 3). Two comments are worth noting. First, the values of εTW are consistent with those of εbottom within a factor of 2–3, except for events B1 and I2 for which εTWεbottom. For these two events, the blocking effect by the topography decreases the conversion rate by 60% for event B1 and by 96% for event I2. Second, except for event I3 in the Iberian basin and event E along Eriador Seamount, εIT is much smaller by an order of magnitude than εbottom. The dataset thus suggests that the strong events of bottom dissipation rates observed to the southeast of the Reykjanes Ridge are rather sustained by the interaction of mean currents with the topography than by the internal tide. In this region, eddies drift slowly with time scales much larger than an inertial period. Moreover, Mercier and Colin de Verdière (1985) showed that these eddies have as much energy in the first baroclinic mode as in the barotropic mode. When they interact with the topography, these eddies provide a source for the mean currents at depth, as shown by the consistent increase in bottom dissipation rates and surface EKE for most of the studied events.

Table 3.

Parameters used to estimate the energy conversion rate FTW from the mean flow Umean into topographic internal waves of intrinsic frequency ω. Characteristic wavelength λ and peak-to-peak height h of topographic hills surrounding the location of the event as measured from single-beam soundings. The vertical (Frυ) and horizontal (Frh) Froude numbers used a constant buoyancy frequency N = 5 × 10−4 s−1. Variable f is the Coriolis frequency at the event location. Dissipation rates εTW, εbottom, and εIT were estimated from FTW [Eqs. (4) or (5), values in parentheses, when Frυ−1 > 0.75], from finescale parameterization [Eq. (1)], and from FIT [Eq. (2)], respectively.

Table 3.

4. Summary and discussion

We have examined the time variability in the dissipation rate along the A25 Greenland–Portugal transect repeated every 2 years. Applying a finescale parameterization that uses CTD and LADCP profiles, we showed that the dissipation rate is usually remarkably stable in time, at least not larger than the uncertainty of the method. However, this is not the case for some stations located to the southeast of the Iceland basin where the time variability of bottom dissipation rates is largely increased. We used estimates of the energy flux into the internal tide and topographic internal waves in order to discriminate between various processes leading to a possible time variability in the turbulent mixing. It was shown that if the tides are a major source for the mixing around the Reykjanes Ridge, topographic internal waves generated by mean currents seem to be at the origin of strong dissipative events to the southeast of the Iceland basin. In the following, we present and discuss uncertainties and limitations associated with the approach.

The finescale parameterization provides dissipation rate estimates that were shown to be, at 90% over the whole transect, within a factor of 3–4 larger or smaller than microstructure estimates (Ferron et al. 2014). The geographical distribution of this dispersion sometimes exhibits a random character but also local biases. Local biases may not be persistent in time if the underlying dynamics leading to small-scale turbulence and kinetic energy dissipation evolves in time. This “natural” dispersion of the finescale method is relatively low if one considers that dissipation rates typically evolve over six orders of magnitude (10−11 to 10−6 W kg−1) in the open ocean. One can argue that it is relatively large for the temporal variability that we found on this transect. That is why we have only discussed a few events of enhanced dissipation rate above the bottom that stand out of the natural dispersion of the finescale method. Those events depart from the median by at least a factor of 6. Most of those events are located in a region between the south of the East Thulean Rise and Eriador Seamount. In this region, the finescale parameterization tends to systematically underestimate the microstructure dissipation rate by a factor of 2, for moderate dissipation rates (1–5 × 10−10 W kg−1), to 5, for enhanced dissipation rates (10–50 × 10−10 W kg−1; Ferron et al. 2014). The finescale parameterization produces a low-biased estimate but with a smaller dispersion. Therefore, the events of enhanced dissipation rates that were identified can be reasonably related to physical mechanisms rather than to uncertainties in the finescale parameterization.

The energy flux from the barotropic tide into the internal tide that we used has several limitations. Indeed, Eq. (3) does not take into account the orientation of the tidal ellipse relative to the topographic gradient (e.g., Nycander 2005). We also did not apply the correction for finite depth that tends to suppress wave generation when the horizontal topographic scale L is longer than the horizontal wavelength Lc associated with the first vertical mode (Llewellyn Smith and Young 2002). Suppression occurs when L > Lc = ND/(ω2f2)1/2, where D is the ocean depth and Lc varies from 80 km at the southern part of the transect to 40 km around the summit of the Reykjanes Ridge. The effect of this correction for the semidiurnal tide will be small here since Lc is always larger than L along the transect.

On the other hand, the unresolved topography of the global map (Smith and Sandwell 1997), that is, scales shorter than about 20 km, as well as bottom-trapped waves associated with diurnal tidal frequencies could each add a contribution to Eq. (3). To assess the importance of the former, the 2002–12 mean tidal flux was computed with the single-beam sounding topography that resolves horizontal scales larger than about 750 m only in the direction of the transect. It clearly appears that the two estimates of the 2002–12 mean tidal fluxes are consistent, with those estimated from the global topography being usually slightly larger than those from the single-beam sounding (Fig. 6c, black and gray curves, respectively). Only one hill to the southeast of Eriador Seamount exhibits a single-beam-based tidal flux significantly larger (by a factor of 8) than that of the global lower-resolution topography (Fig. 6c; dPort = 1850 km). The single-beam sounding shows that this hill is slightly supercritical to semidiurnal tidal waves, that is, H/L < ω/N. Thus, the associated 6 mW m−2 tidal flux is an upper bound of what the hill can produce. Elsewhere on the section, the topography is subcritical. We then do not expect a significant contribution from unresolved topographic scale to the tidal flux for this transect. Falahat and Nycander (2015) quantified the energy flux from the barotropic tide to bottom-trapped tidal waves. They showed that this energy conversion is usually weak compared to the energy input into freely propagating waves. Regionally, the contribution can, however, be significant, as mentioned for the diurnal internal tide around the Kuril Islands (see also Niwa and Hibiya 2011). In our region of interest, since the kinetic energy in the diurnal components of the barotropic tide represents only 10% of the semidiurnal component kinetic energy, we do not expect that diurnal trapped waves contribute significantly to our estimated energy flux into the internal tide. Semidiurnal waves can also be bottom trapped whenever the maximum critical trapping frequency ωT = N sinα is larger than the semidiurnal frequency, where α is the angle of the bottom slope to the horizontal (Rhines 1970). This would increase the ratio of internal wave energy dissipated locally to that radiated away from the generation region. However, apart from over the slopes of the continental shelf, ωT is always smaller than the semidiurnal frequency along the transect.

The estimate of the energy flux into topographic internal waves is based on kilometer-scale topographic features present in our single-beam sounding record. This record is only one dimensional while a two-dimensional high-resolution map would be necessary. We have assumed that the observed relevant scale of the hills located in a reasonable distance (maximum of a few kilometers) from the station were typical of the area. Other topographic small-scale hills that were not sampled by the single-beam survey certainly exist and can change the estimated flux. Furthermore, without this two-dimensional high-resolution topography, we also had to arbitrarily choose a shape for the hills. We chose an axisymmetric sinusoidal shape. For instance, an assumed axisymmetric sinusoidal hill would underestimate by a factor of 3 the real flux if the hill was actually a sinusoidal ridge.

Despite all these limitations, we see that the selected events, all located in a region of rough topography to the southeast of the Iceland basin, share some common points: 1) a horizontal length scale of the relevant topography that frequently lies between 2 and 4 km for a vertical length scale of 100 to 200 m, 2) a bottom-intensified current whose magnitude is O(10) cm s−1 associated with the subtropical–subpolar intergyre of moderate surface intensified eddy kinetic energy that varies in the range 200–800 cm2 s−2, and 3) an estimated energy flux into topographic internal waves that is most of the time an order of magnitude larger than the estimated semidiurnal tidal flux. Unfortunately, we do not have time measurements of bottom currents to characterize their frequency modulation. Given the geographical and dynamical context of the region, it seems reasonable to assume that the energy of the bottom-enhanced currents is fed by the subsurface mesoscale as in other areas (Gille et al. 2000; Liang and Thurnherr 2012; St Laurent et al. 2012). A high-resolution model clearly showed that the link between surface-intensified eddying geostrophic flows, bottom-enhanced velocities, topographic internal gravity waves, and energy dissipation is of primary importance for the energetic of the Antarctic Circumpolar Current (Nikurashin et al. 2013). Observations of turbulent mixing also support this scenario in the Southern Ocean (Heywood et al. 2002; Naveira Garabato et al. 2004; Sloyan 2005; Sheen et al. 2013).

The region of the Northeastern Atlantic located between the Iceland basin and the Azores–Biscay Rise exhibits a smaller and less constant eddy kinetic energy level than that of the Southern Ocean or that of the Gulf Stream. But even in this less energetic part of the ocean, this set of observations suggests the existence of the same link between surface-intensified eddying geostrophic flows, bottom-enhanced velocities, and kinetic energy dissipation. The surface eddy kinetic energy level of the region varies in time over three orders of magnitude. According to altimeter data, it is on average larger by 25% during the months of January–June than it is between August and December for the 2002–12 period. One may then question the representativeness of our estimates derived from summer instantaneous profiles. To get an idea of the annual mean value of the bottom dissipation rate due to topographic internal waves along the transect, we calculated the number of days per year for which the surface eddy kinetic energy level is in the range 200–600 cm2 s−2 and larger than 600 cm2 s−2 for the period 2002–12. Based on our observations, if we assign a 500-m bottom-averaged dissipation rate of 0 W kg−1 to an EKE smaller than 200 cm2 s−2, of 10 × 10−10 W kg−1 to an EKE in the range 200–600 cm2 s−2, and of 40 × 10−10 W kg−1 to an EKE larger than 600 cm2 s−2, one can very crudely estimate the annual mean dissipation rate due to topographic internal waves and its range of variation (Table 4). This annual mean expected dissipation rate is usually weak in the Iberian basin and increases while moving north of the Azores–Biscay Rise. An annual mean value of 3–7 × 10−10 W kg−1 is expected at the location of events A, (B1/B2), C, and D where the EKE level is the largest of the transect. This annual mean value increases by a factor 2–3 for years having the largest EKE levels. This simple calculation favors the idea that the contribution to the annual mean dissipation rate of the generation of topographic internal waves will dominate that of the internal tide at the location of the events A to D (columns 4–5, Table 4). For all the other event locations, the dissipation rate associated with the internal tide is expected to be larger than that of topographic internal waves, although the latter has the same order of magnitude for the most energetic years. We can also expect to get larger energy fluxes into topographic internal waves and dissipation rates westward of our transect since the EKE level is larger there while the topography remains rough.

Table 4.

Annual mean and interannual range of the number of days per year for which the local surface EKE is in the range 200–600 cm2 s−2 or larger than 600 cm2 s−2 at each event location and for the period 2002–12. Annual mean and interannual range of the expected bottom dissipation rates within 500 m of the bottom are due to topographic internal waves εTW, assuming that an EKE of 200–600 cm2 s−2 produces a dissipation rate of 10 × 10−10 W kg−1 and that an EKE larger than 600 cm2 s−2 produces a dissipation rate of 40 × 10−10 W kg−1. Annual mean dissipation rate within 500 m of the bottom due to internal tides εIT.

Table 4.

It is not clear whether the global estimates of the energy flux into topographic internal waves calculated from the bottom velocities of a global model at ⅛° horizontal resolution (Nikurashin and Ferrari 2011) are able to capture the flux that we diagnose in our study. This model probably underestimates the bottom velocities associated with mesoscale–bathymetry interactions since, at this resolution, the smaller (but poorly) resolved scales are about 20 km at 50°N. This scale is similar to the large-scale roughness of the region. We can anticipate that new configurations with at least 1/50° resolution will give more accurate estimates of enhanced bottom velocities to indirectly estimate the energy flux into topographic internal waves, but it would still not resolve the generation of topographic internal waves that would require a topography and a model with at least 1/200° (500 m) of resolution.

Acknowledgments

We thank all crew members of the R/V Thalassa, R/V Maria S. Merian, and R/V Sarmiento de Gamboa. We also wish to thank all technicians and engineers who contributed to the CTD/LADCP data acquisition and processing (A. Billant, P. Branellec, J.-P. Gouillou, M. Hamon, C. Kermabon, C. Lagadec, P. Le Bot, S. Leizour, O. Ménage, O. Péden, and the Unidad de Tecnología Marina). The OVIDE project was supported by IFREMER, CNRS/INSU and the National Program LEFE. B. Ferron, H. Mercier, and T. Huck were supported by the French National Center for Scientific Research (CNRS), P. Lherminier and V. Thierry by the French Institute for Marine Science (Ifremer), and F. Kokoszka by the European University of Brittany. Aida F. Ríos was funded by the Spanish Ministry of Economy and Competitiveness (BES-2014-070449) through the BOCATS project (CTM2013-41048-P). The CATARINA cruise was supported the Spanish Ministry of Sciences and Innovation cofunded by the Fondo Europeo de Desarrollo Regional 2007-2012 (FEDER) through the CATARINA project (CTM2010-17141). We thank the anonymous reviewers for their useful comments and Aurélien Ponte for some helpful discussions on internal wave dynamics.

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