Experimental Confirmation of Stokes Drift Measurement by High-Frequency Radars

Abïgaëlle Dussol aInstitut des sciences de la mer de Rimouski, Université du Québec à Rimouski, Rimouski, Quebec, Canada

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Cédric Chavanne aInstitut des sciences de la mer de Rimouski, Université du Québec à Rimouski, Rimouski, Quebec, Canada

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Sandy Gregorio aInstitut des sciences de la mer de Rimouski, Université du Québec à Rimouski, Rimouski, Quebec, Canada

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Dany Dumont aInstitut des sciences de la mer de Rimouski, Université du Québec à Rimouski, Rimouski, Quebec, Canada

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Abstract

High-frequency radars (HFR) remotely measure ocean surface currents based on the Doppler shift of electromagnetic waves backscattered by surface gravity waves with one-half of the electromagnetic wavelength, called Bragg waves. Their phase velocity is affected by their interactions with the mean Eulerian currents and with all of the other waves present at the sea surface. Therefore, HFRs should measure a quantity related to the Stokes drift in addition to mean Eulerian currents. However, different expressions have been proposed for this quantity: the filtered surface Stokes drift, one-half of the surface Stokes drift, and the weighted depth-averaged Stokes drift. We evaluate these quantities using directional wave spectra measured by bottom-mounted acoustic wave and current (AWAC) profilers in the lower Saint Lawrence Estuary, Quebec, Canada, deployed in an area covered by four HFRs: two Wellen radars (WERA) and two coastal ocean dynamics applications radars (CODAR). Since HFRs measure the weighted depth-averaged Eulerian currents, we extrapolate the Eulerian currents measured by the AWACs to the sea surface assuming linear Ekman dynamics to perform the weighted depth averaging. During summer 2013, when winds are weak, correlations between the AWAC and HFR currents are stronger (0.93) than during winter 2016/17 (0.42–0.62), when winds are high. After adding the different wave-induced quantities to the Eulerian currents measured by the AWACs, however, correlations during winter 2016/17 significantly increase. Among the different expressions tested, the highest correlations (0.80–0.96) are obtained using one-half of the surface Stokes drift, suggesting that HFRs measure the latter in addition to mean Eulerian currents.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Abïgaëlle Dussol, abigaelle.dussol@gmail.com

Abstract

High-frequency radars (HFR) remotely measure ocean surface currents based on the Doppler shift of electromagnetic waves backscattered by surface gravity waves with one-half of the electromagnetic wavelength, called Bragg waves. Their phase velocity is affected by their interactions with the mean Eulerian currents and with all of the other waves present at the sea surface. Therefore, HFRs should measure a quantity related to the Stokes drift in addition to mean Eulerian currents. However, different expressions have been proposed for this quantity: the filtered surface Stokes drift, one-half of the surface Stokes drift, and the weighted depth-averaged Stokes drift. We evaluate these quantities using directional wave spectra measured by bottom-mounted acoustic wave and current (AWAC) profilers in the lower Saint Lawrence Estuary, Quebec, Canada, deployed in an area covered by four HFRs: two Wellen radars (WERA) and two coastal ocean dynamics applications radars (CODAR). Since HFRs measure the weighted depth-averaged Eulerian currents, we extrapolate the Eulerian currents measured by the AWACs to the sea surface assuming linear Ekman dynamics to perform the weighted depth averaging. During summer 2013, when winds are weak, correlations between the AWAC and HFR currents are stronger (0.93) than during winter 2016/17 (0.42–0.62), when winds are high. After adding the different wave-induced quantities to the Eulerian currents measured by the AWACs, however, correlations during winter 2016/17 significantly increase. Among the different expressions tested, the highest correlations (0.80–0.96) are obtained using one-half of the surface Stokes drift, suggesting that HFRs measure the latter in addition to mean Eulerian currents.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Abïgaëlle Dussol, abigaelle.dussol@gmail.com

1. Introduction

High-frequency (HF) surface-wave radars remotely sense ocean surface currents by using radio waves backscattered from ocean surface gravity waves (Crombie 1955; Barrick 1972; Stewart and Joy 1974). HF radars map currents from the coast to more than 100 km offshore, with a horizontal resolution of a few kilometers or less, and a temporal resolution of one hour or less. The high temporal and spatial resolution and the wide spatial coverage make HF radars valuable tools for aiding search and rescue operations (Breivik et al. 2013) and oil spill mitigation (Abascal et al. 2009), and for monitoring coastal erosion (Irvine 2015) and pollutant transport and dispersion (Washburn et al. 2005).

HF radars measure ocean surface currents by transmitting radio waves in the HF band, which covers frequencies between 3 and 30 MHz, with wavelengths ranging from 10 to 100 m. The recorded signal is dominated by the first-order backscattering caused by ocean surface gravity waves, called Bragg waves, having a wavelength of one-half of the radio wavelength and propagating exactly toward or away from the radar (Crombie 1955; see Fig. 1). These waves have wavelengths ranging from 5 to 50 m, with periods ranging from 2 to 6 s, and are locally generated by the wind.

Fig. 1.
Fig. 1.

A schematic showing how HF radars measure ocean surface currents by sending radio waves and recording the signal backscattered by ocean surface gravity waves. The recorded signal is dominated by radio waves backscattered from ocean surface waves with one-half of the radio wavelength, called Bragg waves. These waves propagate at a known linear phase velocity c0, causing a Doppler shift in the frequency of the backscattered signal. However, wave–current interactions induced by the presence of a mean Eulerian current uE and wave–wave interactions related to the Stokes drift uS modify their phase velocity (ΔcE and ΔcS, respectively), causing a measurable additional Doppler shift.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

The received radio waves experience a Doppler shift caused by any movement of the sea surface in the direction of the radar. In the absence of mean Eulerian currents and waves other than infinitesimally small-amplitude Bragg waves, the Doppler shift is proportional to the linear phase velocity of the Bragg waves, given by (in the deep-water approximation)
c0=±g/kB,
where g is the gravitational acceleration and kB is the Bragg wavenumber magnitude. For the HF radio band, c0 ranges from 3 to 9 m s−1.
When the sea surface moves with mean Eulerian currents, the linear phase velocity of the Bragg waves differs from c0 by a quantity ΔcE given by (Stewart and Joy 1974):
ΔcE=2kB0UE(z)exp(2kBz)dz,
where UE(z) is the mean Eulerian current component in the direction of the radar, also called radial current. By measuring ΔcE, HF radars perform a weighted depth averaging of the ocean currents, with weights exponentially decaying from the sea surface downward. This theoretical result has been confirmed experimentally in many studies (e.g., Stewart and Joy 1974; Ha 1979; Teague 1986).

In the presence of other waves with different wavenumbers, the phase velocity of the Bragg waves with finite amplitude differs from c0 + ΔcE by an additional quantity ΔcS due to wave–wave interactions, which is related to the Stokes drift (Stokes 1847; Longuet-Higgins and Phillips 1962; Huang and Tung 1976; Weber and Barrick 1977; Laws 2001; see Fig. 1). Since both mean Eulerian currents and surface gravity waves can affect the phase velocity of the Bragg waves, HF radars should therefore measure a quantity related to the Stokes drift in addition to mean Eulerian currents.

However, the literature is inconsistent, both theoretically and experimentally, on whether the Stokes drift, or part of it, is measured by HF radars (Chavanne 2018). In some studies, HF radar currents are interpreted as Eulerian currents (i.e., not including the Stokes drift; e.g., Röhrs et al. 2015; Sentchev et al. 2017). Other studies (e.g., Stewart and Joy 1974; Laws 2001) have suggested that HF radars measure the Stokes drift in the same way as mean Eulerian currents, replacing UE(z) in Eq. (2) with the Stokes drift (Kenyon 1969)
US(z)=20ω(k)kS(k)e2kzdk,
where k is the wavenumber magnitude, ω(k) is the wave angular frequency associated with k, and S(k) is the unidirectional wave spectrum in the direction of the radar. However, it is not justified to use a mean Lagrangian current in Eq. (2). Others have proposed that HF radars measure instead one-half of the surface Stokes drift (Huang and Tung 1976) or a quantity numerically close to, but different from, the surface Stokes drift, namely, the filtered surface Stokes drift (Longuet-Higgins and Phillips 1962; Weber and Barrick 1977; Ardhuin et al. 2009).

Our published preliminary experimental results (Dussol et al. 2019) suggested that HF radars measure the filtered surface Stokes drift in addition to mean Eulerian currents. That study was based on comparisons between currents measured by AWACs and HF radars only at the radars’ bearings toward the AWACs. Unfortunately, upon further analysis, we realized that this conclusion was based on flawed radar current data, resulting from a problem when loading the radar data into the MATLAB software, which caused some azimuths to be duplicated. Here, we have corrected this error and also now compare the AWACs currents with the HF radars currents at all bearings.

In the present work, we revisit whether HF radars measure a quantity related to the Stokes drift in addition to mean Eulerian currents, and what this quantity is, based on calibrated and quality-controlled data. The paper is organized as follows. The study area and experimental data are documented in section 2. Our methods are presented in section 3. Results are given in section 4, followed by a discussion and conclusions in section 5.

2. Study area and experimental data

a. Study area

The area of this study is located in the lower Saint Lawrence Estuary (LSLE), Quebec, Canada (Fig. 2). The LSLE has a width ranging from 20 km upstream to 50 km downstream and extends from the mouth of the Saguenay River northeast to Pointe-des-Monts where it opens to the Gulf of Saint Lawrence (El-Sabh 1988). The tide is predominantly semidiurnal, and its amplitude increases upstream (Drapeau 1992).

Fig. 2.
Fig. 2.

Map of the estuary and Gulf of Saint Lawrence. The black-outlined rectangle and upper-right inset delimit the study area in the lower Saint Lawrence Estuary. The instrument locations are indicated, and a typical HF radar coverage (current vectors) is shown by the gray arrows. Red dots show scatterplots of the eastward velocity u and northward velocity υ at each moored AWAC.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

In general, waves observed in the LSLE are produced locally by winds blowing predominantly from the southwest. The surface winds are steered along the longitudinal axis of the estuary by the mountains located on both shores and are strongest in winter (Koutitonsky and Bugden 1991). Even though surface waves are often fetch limited, significant wave heights can frequently reach 1 m and sometimes 3 m or more (Didier et al. 2017).

During winter 2016/17 (from October 2016 to April 2017), the study area was dominated by moderate 5–15 m s−1 winds, from a wide range of directions (Fig. 3a). During winter storms, winds can reach over 20 m s−1. These weather conditions are favorable to the formation of waves. During summer 2013 (from May to October), the study area was dominated by weak winds (3–8 m s−1), with dominant southwesterly and northeasterly directions (Fig. 3b).

Fig. 3.
Fig. 3.

Wind rose for (a) winter 2016/17 (from October 2016 to April 2017) at the Bic weather station (Fig. 2, green square) and (b) summer 2013 (from May 2013 to October 2013) at the PMZA-Riki buoy (Fig. 2, magenta circle). For each direction, the cumulative frequency is indicated, with wind speeds increasing from the center outward.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

b. Data

1) HF radar currents

Four HF radars have been deployed and operated in the LSLE, measuring hourly surface currents since November 2012. In this study, we use radar measurements from October 2016 to April 2017 (winter 2016/17) and from May 2013 to October 2013 (summer 2013). A typical HF radar coverage is illustrated by the gray arrows in Fig. 2.

Two coastal ocean dynamics applications radars (CODAR) were deployed on the south shore of the LSLE at Pointe-au-Père (PAP; 48.5171°N, 68.4706°W) and at Sainte-Flavie (STF; 48.6095°N, 68.2347°W) (Fig. 2, red triangles). Two Wellen radars (WERA) were deployed at Pointe-aux-Outardes (PAO; 49.0426°N, 68.4580°W) and Pointe-à-Boisvert (PAB; 48.5733°N, 69.1344°W) on the north shore (Fig. 2, orange triangles). The characteristics of each HF radar are summarized in Table 1.

Table 1

Characteristics of the HF radars.

Table 1

CODARs and WERAs use different techniques for estimating radial currents: direction finding and beam forming, respectively. Radial currents for CODARs were computed using the processing toolbox developed by B. Emery (Emery 2018), based on the multiple signal classification (MUSIC) algorithm (Schmidt 1986), which provides current estimates in 1° directional bins. For summer 2013, the CODAR currents at STF have been processed using antenna patterns measured on 25 July 2013 (Fig. 4). For winter 2016/17, the CODAR currents at PAP have been processed using antenna patterns measured on 27 October 2016 (Fig. 4).

Fig. 4.
Fig. 4.

Antenna patterns measured on 27 Oct 2016 at PAP and on 25 Jul 2013 at STF. Beamforming patterns for PAO and PAB in the directions of the AWACs are indicated.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

Radial currents for WERAs were computed using the software developed by K.-W. Gurgel (Gurgel et al. 1999), based on the beamforming technique with an array of 12 receiving antennas. For each WERA site, beamforming patterns were calculated with an ultraspherical window and using positions of the antennas measured with an accuracy of ∼1 cm with a differential global positioning system (GPS). Phase differences due to receiving cables were calibrated on 22 October 2012 at PAB and were used to process the currents for the two periods. Phase differences were calibrated in subsequent years, but currents processed using these new values did not improve our results. Phase differences due to receiving cables were calibrated on 23 October 2012 at PAO and were used to process the currents for the summer 2013. Antennas were knocked down in December 2014 by a winter storm and were repositioned a few hundred meters away in autumn 2015. Phase differences between antennas were calibrated with a shipborne transmitter on 12 July 2016 and were used to process the currents for the winter 2016/17. Radial currents are provided on a polar grid with 1° azimuthal spacing, although this does not correspond to the azimuthal resolution of beamforming since the width of the beams is much larger (Table 1; Fig. 4).

Each radar station transmits a chirped continuous wave with a 100-kHz bandwidth, which gives a radial resolution of 1.5 km. To reduce the noise of the measurements, we performed 1-h moving averaging, and we applied most of the quality control tests listed in the QARTOD Quality Manual for HF radars (Bushnell and Worthington 2016). One of these tests is based on a minimum signal-to-noise ratio (SNR). For the WERAs, we chose a threshold of 8 dB for the SNR of Doppler lines (Cosoli et al. 2018). For the CODARs, we chose a threshold of 5 dB for the SNR of the monopole receive element (Kirincich et al. 2012).

2) AWAC data

Three 1000-kHz Nortek acoustic waves and currents profilers (AWAC) with acoustic surface tracking (AST) were moored, respectively, in front of Forestville (48.6590°N, 69.0440°W), in Bic Channel (48.3792°N, 68.8105°W), and in front of Saint-Ulric (48.7210°N, 67.9841°W) within the HF radars’ range (Fig. 2, blue squares). The instruments in front of Forestville and in Bic Channel collected data from 14 October 2016 to 23 April 2017, and the AWAC moored in front of Saint-Ulric collected data from 30 May to 30 October 2013. The depth of the bottom-mounted AWACs varied between 10 and 16 m for Forestville, 8 and 15 m for the Bic Channel, and 15 and 20 m for Saint-Ulric, depending on the tide. Current velocity data are provided every 20 min as 1-min averages, with a vertical resolution of 1 m. The shallowest profiler measurements were 2 m below the mean ocean surface. For consistency with HF radars data, AWAC currents were hourly averaged.

Directional wave spectra are needed to obtain the nonlinear correction ΔcS to the phase velocity of Bragg waves. The maximum entropy method has been applied to the three wave orbital velocity components and the surface elevation measured by the AWACs to estimate the two-dimensional wave spectra (Lygre and Krogstad 1986). Wave data are provided every hour as 17-min averages. An example of directional wave spectrum is presented in Fig. 5. This example features northeasterly locally wind-generated waves at frequencies between 0.2 and 0.45 Hz, encompassing the HF radars Bragg frequencies (Table 1).

Fig. 5.
Fig. 5.

Example of directional wave spectrum obtained from measurements by the AWAC near Forestville (Fig. 2, blue square). The direction of origin of the waves is given by the angular position of the peak. The frequency in hertz is indicated by the distance to the center of the diagram.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

The wave spectra are measured in the frequency domain, but they need to be expressed in the wavenumber domain to compute the nonlinear corrections to the phase velocity of Bragg waves. We tested both using the full dispersion relation requiring bottom depth and the deep-water dispersion relation. The latter yielded slightly better results (an increase in correlation coefficients of approximately 7%), and it was used thereafter.

3. Methods

a. Wave-induced contributions

1) Theoretical expressions

Different expressions have been proposed in the literature for the contribution of wave–wave interactions to HF radar current measurements.

(i) Surface Stokes drift
We compute the surface Stokes drift, which is given by Eq. (3) at z = 0 (Kenyon 1969):
US(0)=20ω(k)kS(k)dk.
(ii) The filtered surface Stokes drift
Weber and Barrick (1977) give a general but complicated expression for the nonlinear correction to the phase velocity of Bragg waves, which we call ΔcL62 [see Eqs. (23), (25), (29), and (30) in Weber and Barrick 1977]. Assuming waves are propagating in parallel directions, Barrick and Weber (1977) showed that this expression reduces to that obtained by Longuet-Higgins and Phillips (1962), which is
ΔcL62ΔcL62*=±20kBω(k)kS(k)dk ± 2kBkBω(k)S(k)dk.
The positive sign is used for waves propagating in the same direction as the Bragg waves, and the negative sign is used for waves propagating in the opposite direction.

Broche et al. (1983) showed that even for a full directional wave spectrum, with waves propagating in different directions, Eq. (5) remains a good approximation to the full expression given by Weber and Barrick (1977). Ardhuin et al. (2009) called it the filtered surface Stokes drift, because the first term on the right-hand side of Eq. (5) has the same form as the surface Stokes drift [Eq. (4)] except that the integration is carried over only wavenumbers smaller than the Bragg wavenumber. Note however that there is a contribution from waves shorter than the Bragg waves [the second term on the right-hand side of Eq. (5)]. Note also that ΔcL62* involves the Bragg wavenumber and therefore depends on the radar frequency.

To get an idea of the typical magnitude of ΔcL62*, consider the JONSWAP wave spectrum (Hasselmann et al. 1973)
S(ω)=αg2ω5exp[54(ωpω)4]γr,
where α, γ, ωp (the peak frequency), r, and σ are given by
α=0.076(U102Fg)0.22,γ=3.3,ωp=22(g2U102F)1/3,r=exp[(ωωp)22σ2ωp2],andσ={0.07ifω<ωp0.09ifω>ωp,
where F is the fetch and U10 is the wind speed at 10 m above sea level.

The JONSWAP spectrum is a reasonable choice since waves in the LSLE are mainly fetch limited. For example, waves cannot be entirely developed for daily winds exceeding 8 m s−1, the wind speed for which waves are limited by the longest fetch of about 300 km (WMO 1998). For this fetch, ΔcL62* is shown in Fig. 6a (green line) as a function of wind speed. It varies from 0.02 to 0.37 m s−1 for winds ranging from 1 to 25 m s−1, respectively, and is typically 0.04 m s−1 weaker than the surface Stokes drift (red line).

Fig. 6.
Fig. 6.

Surface Stokes drift (red), filtered surface Stokes drift (green), weighted depth-averaged Stokes drift (blue), and one-half of surface Stokes drift (black), as a function of wind speed for (a) an HF radar operating at 12.5 MHz and a JONSWAP unidirectional wave spectrum with a fetch of 300 km and (b) the CODAR at PAP, operating at 12.5 MHz, and directional wave spectra obtained from the AWAC at Forestville. In (b), we used the wind speed data at the meteorological station near Bic (green square in Fig. 2).

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

(iii) One-half of the surface Stokes drift
Huang and Tung (1976) obtained an expression for the nonlinear correction to the phase velocity of Bragg waves that differs from that of Weber and Barrick (1977), which we call ΔcH76 [see Eq. (17) of Huang and Tung 1976]. For waves propagating in the same direction, the expression simplifies to
ΔcH76ΔcH76*=0ω(k)kS(k)dk=12US(0),
where US(0) is the surface Stokes drift. Contrary to the filtered surface Stokes drift, ΔcH76* is independent of the radar frequency; ΔcH76* is shown in Fig. 6a (black line) for the JONSWAP spectrum as a function of wind speed. It reaches 0.21 m s−1 for winds of 25 m s−1.
(iv) The weighted depth-averaged Stokes drift
Laws (2001) derived an expression [his Eq. (2.21)] where he assumed that the Stokes drift is measured by the HF radars in the same way as mean Eulerian currents, which we call ΔcL01. In the case of a unidirectional wave spectrum, he obtained
ΔcL01ΔcL01*=±2kB0ω(k)kk+kBS(k)dk,
where the positive sign is used when the waves are in the same direction as the Bragg waves, and the negative sign is used when the waves are in the opposite direction. As for the filtered surface Stokes drift, ΔcL01* involves the Bragg wavenumber and therefore depends on the radar frequency; ΔcL01* is shown in Fig. 6a (blue line) for the JONSWAP spectrum. It is typically 0.08 m s−1 weaker than the surface Stokes drift (red line).

2) Estimation of the wave-induced contributions from the observed wave spectra

The observed wave spectra are only resolved for frequencies f = ω/(2π) smaller than 0.5 Hz (or wavenumbers k smaller than 1 rad m−1), which is only slightly larger than the Bragg wave frequencies (Fig. 7; Table 1). To carry out the integrals in wavenumber for the different wave-induced contributions, it is necessary to extrapolate the observed wave spectra to higher wavenumbers. A common approach, based on the Phillips (1958) spectrum, is to extend the observed spectral tails with a slope of k−4, implying f−5 (e.g., Röhrs et al. 2015). However, many studies show that observed spectral tails are closer to k−3.5, implying f−4 (Kitaigorodskii 1983; Phillips 1985; Prevosto et al. 1996).

Fig. 7.
Fig. 7.

An observed wave spectrum (black) at Forestville with k−3.5 spectral tail from the energy peak (red dashed) or from the highest measured frequency (blue dashed). The Bragg wavenumber for the HFR at PAO is indicated in magenta.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

To check whether the k−3.5 spectral tail could be used to extend our observed spectra to higher frequencies, hourly measured spectra were normalized by the maximum energy density, and frequencies were normalized by the corresponding peak frequency. These normalized spectra were averaged during the whole observational periods (from May to October 2013 for Saint-Ulric and from October 2016 to April 2017 for Forestville and Bic). As an example, the resulting average normalized spectrum at Forestville is shown in Fig. 8 (black line). Our data suggest a k−3.5 power decay (Fig. 8, red line) and do not support the original theory of a k−4 high-frequency tail (Fig. 8, green line).

Fig. 8.
Fig. 8.

The average normalized spectrum at Forestville (black), with a k−3.5 (red) and k−4 (green) spectral tail. The standard deviation of the measured normalized spectra is shown in light gray.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

We tested two options to extend the observed spectra to high frequencies: by adding the k−3.5 tail either to the peak wavenumber (Fig. 7, dashed red line) or to the highest observed wavenumber (Fig. 7, dashed blue line). While the former is less sensitive to measurement noise, the latter yielded the best results in the subsequent analysis and was used thereafter. The different possible wave-related contributions to the HF radar current measurements are computed by integrating the theoretical expressions to a maximum wavenumber of 100 m−1. Note that with a k−3.5 high-frequency extension, the surface Stokes drift does not converge as k tends to infinity, which requires imposing a maximum wavenumber. We also tested using a k−4 high-frequency extension, for which the surface Stokes drift converges. Our results remained similar, with only slightly lower correlation values.

The different possible wave-related contributions to the HF radar current measurements as a function of wind speed are shown in Fig. 6b for the directional wave spectra obtained from the AWAC at Forestville. The magnitude and wind dependence of the different contributions are similar to those obtained for the JONSWAP spectrum with a fetch of 300 km (Fig. 6a). These contributions are strongly correlated to each other but differ in magnitude, with the filtered surface Stokes drift (ΔcL62) being the strongest and one-half of the surface Stokes drift (ΔcH76) being the weakest. Differences between ΔcL62 and the two other contributions range from 0.01 to 0.15 m s−1 for winds up to 25 m s−1.

b. Estimation of wind-driven shear

The effective measurement depths of mean Eulerian currents for our HF radars typically range from 0.4 to 1.0 m (Stewart and Joy 1974; Chavanne 2018). To compare the currents measured by the HF radars and the AWACs, one should perform the same weighted depth averaging on the current profiles from the AWACs that HF radars perform on mean Eulerian currents [Eq. (2)]. However, the AWAC currents only extend to 2-m depth because of sidelobe contamination from the sea surface. Therefore, extrapolating the currents of the AWACs to the surface is necessary to mimic HF radar measurements.

Tamtare et al. (2021) have proposed a method to extrapolate currents measured at a given depth to the sea surface with the assumption that the upper-ocean currents satisfy the linear time-varying Ekman dynamics:
U(υ,z)=H(υ,z)T(υ),
where U, H, and T are the Fourier transforms of the current u(t, z), the impulse response function h(t′, z), where t′ is the time lag, and the wind stress τ(tt′).
Using Eq. (10), currents at a depth zi can be inferred from currents observed at a depth zi+1:
U(υ,zi)=H(υ,zi)H(υ,zi+1)U(υ,zi+1).
Since the transfer function H only accounts for wind-driven dynamics, geostrophic and tidal currents must be first removed from the observed currents (Tamtare et al. 2021).
Here, we use a slightly different approach and choose to extrapolate the current shear
ΔU(υ,zi,zi+1)U(υ,zi)U(υ,zi+1)=[H(υ,zi)H(υ,zi+1)]τ(υ),
which filters out geostrophic and tidal currents since their vertical shear near the surface is usually negligible relative to the wind-driven shear. The current shear is extrapolated using the ratio of shear at different depths:
Ru(υ,zi+1)ΔU(υ,zi,zi+1)ΔU(υ,zi+1,zi+2).
Using Eq. (12), this ratio can be estimated using the transfer function H:
RH(υ,zi+1)=H(υ,zi)H(υ,zi+1)H(υ,zi+1)H(υ,zi+2).
Next, assuming Ru(υ, zi+1) = RH(υ, zi+1), we obtain
U(υ,zi)=U(υ,zi+1)+RH(υ,zi+1)[U(υ,zi+1)U(υ,zi+2)].
Last, the velocity u(t, zi) is obtained by taking the inverse Fourier transform of U(υ, zi).
The transfer function H is obtained assuming time-dependent linear Ekman dynamics for a horizontally homogeneous ocean (Elipot and Gille 2009). The specific form of H depends on the boundary conditions and the parameterization of the vertical turbulent stress. Using the classical K-profile parameterization, Elipot and Gille (2009) consider nine models with three different vertical profiles for the vertical viscosity K, and three different bottom boundary conditions (their Fig. 1). We rejected three models with infinite velocity at the surface. Of the six remaining models, three required a finite boundary layer depth h, which we set to the mooring depth of each AWAC (which were bottom mounted), whereas the other models assume an infinite ocean. Three models assumed a constant vertical viscosity K = K0, and the other models assumed a vertical viscosity linearly increasing with depth, K = K0 + K1z. To determine the boundary condition and vertical viscosity that best allow an Ekman-type model to represent the observed AWAC data, the ratios of observed shears Ru are compared with the six theoretical ratios RH. We seek the optimal parameters (i.e., K0 and K1) that minimize the cost function
L=υ,zi|Ru(υ,zi)RH(υ,zi)|w(υ,zi),
where ǁ designates the absolute value and w(υ, zi) is a weighting function, taken to be the squared coherence between ΔU(υ, zi−1, zi) and ΔU(υ, zi, zi+1).
To identify the optimal Ekman-model configuration, we assess which of the models has the smallest L. For all tested models, values of L range from 1.24 to 4.82 in arbitrary units. The smallest value of L is obtained for the model assuming a finite boundary layer and a constant vertical viscosity K0 = 0.016 m2 s−1, for which the transfer function H is (Elipot and Gille 2009)
H(υ,z)=eiπ/4ρ(2πυ+f)K0sinh[(1+i)(hz)/δ]cosh[(1+i)h/δ],
where ρ is the density of seawater, f is the Coriolis parameter, and δ=2K0/(2πυ+f). The theoretical ratio RH obtained with this transfer function compares favorably to the observed ratio Ru (Fig. 9), contrary to the other models (not shown).
Fig. 9.
Fig. 9.

Vertical profiles of the (left) modulus and (right) phase of (a),(b) the theoretical ratio RH and (c),(d) the observed ratio Ru of the current shear.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

Using Eq. (15), we extrapolate the currents measured by the AWACs to the surface at 1-m depth increments (Figs. 10a,b), and obtain improved correlations between the radial currents measured by the HF radars and the AWAC currents extrapolated at the sea surface (Fig. 10c), confirming that the radars measure the currents near the surface. Note that the correlations between the radar and the observed AWAC currents decrease above 5-m depth at Saint-Ulric, so we extrapolated the currents between 5 m and the surface for this AWAC.

Fig. 10.
Fig. 10.

Vertical profiles of (a) amplitude and (b) direction of observed mean AWAC currents (solid lines). (c) Correlation coefficient between the currents measured by the HF radars and the AWAC currents at different depths. In each panel, the green, blue, and black lines have been obtained for the Saint-Ulrich, Forestville, and Bic AWACs data, respectively. The dashed lines show the extrapolated currents.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

c. Statistical analysis of data

To compare the currents from HF radars with currents from in situ instruments, it is better to project the latter into the directions of the radars and compare these projected currents with the radial currents measured by the radars, UHFR, rather than combining radial currents from two or more radars into 2D vector currents, to avoid the associated geometric amplification of errors (e.g., Chavanne et al. 2010). The 2D vector currents measured by each AWAC, including their extrapolation to the sea surface, UAWAC(z), are therefore projected in the direction from each HFR toward the AWAC, and vertically integrated with exponentially decaying weights [as in Eq. (2)] to obtain
UAWAC-e=2kBH0[UAWAC(z)i]exp(2kBz)dz,
where H is the deepest measurement depth of the AWAC and i is the unit vector pointing from the HFR toward the AWAC.

The range of the HF radar measurements is determined by the frequency shift between the frequency-modulated continuous-wave (FMCW) transmitted signal, called a chirp, and the received signal. It is a robust estimation since it does not require any calibration and it is not affected by environmental parameters. Therefore, the range of HF radar currents closest to the range of the AWAC, with a maximum separation distance of 1 km, is selected for the comparison. In contrast, the determination of the azimuth of the HF radar measurements depends on the accuracy of the calibration of the receive antenna elements and on environmental parameters, which can lead to significant azimuth, or direction-of-arrival (DOA), errors (Emery et al. 2004; Emery and Washburn 2019). To detect and quantify such errors, the radial currents from each azimuth available at the selected range are compared with UAWAC-e. In the absence of DOA errors, one would expect the best comparison to occur at the bearing toward the AWAC.

To examine the comparison of radar and AWAC currents, two metrics are usually used: (i) the root-mean-square (RMS) difference of the radar and AWAC currents and (ii) the Pearson’s correlation coefficient. This correlation coefficient is an estimator based on the mean. It is an efficient estimator for normally distributed variables. However, in the presence of strong outliers, the performance of this correlation coefficient is badly affected (Shevlyakov 1997; Kim and Fessler 2004). A new robust correlation coefficient has been proposed in the literature and has been called median-product correlation coefficient. This robust correlation coefficient is an estimator based on the median (med) instead of the mean (Shevlyakov et al. 2012):
r(x,y)=med2|a|med2|b|med2|a|+med2|b|,
where a = [x − med(x)] + [y − med(y)] and b = [x − med(x)] − [y − med(y)].
Despite the quality control applied to the HF radars data some strong outliers remain. Thus, to quantify the comparison between the HF radar and AWAC currents, we used two robust metrics: the median-product correlation coefficient r between UHFR and UAWAC-e, and their “normalized robust difference,” defined as
D=med(|UHFRUAWAC-e|)U97.5U02.5,
where U is the concatenation of UHFR and UAWAC-e and U97.5 and U02.5 are the quantiles corresponding to 97.5% and 2.5%, respectively. Thus, U97.5U02.5 is the range of values containing 95% of the observations. Confidence intervals on r are obtained using the bootstrap method with 1500 bootstrap samples of the residuals from the original linear regression (Emery and Thomson 2001).
The HF radar azimuth yielding the best comparison with the AWAC currents is the azimuth θ* at which D is minimized. The corresponding correlation coefficient is r* = r(θ*). We define bearing offsets Δθ as in Emery et al. (2004):
Δθ=θAWACθ*,
where θAWAC is the azimuth toward the AWAC.

To test whether HF radar currents include a wave-induced contribution, we also compute r and D between UHFR and UAWAC-e + Δci, where Δci with i = (L62, H76, L01) are the different possible contributions from wave–wave interactions. The Δci are computed from the wave spectra observed by the AWACs using both the full expressions given in the original papers mentioned in section 3a(1) and the approximations given in Eqs. (5), (8), and (9). For completeness, we also compute r and D between UHFR and UAWAC-e + US(0).

The comparisons between the WERA at PAB and the AWAC at Forestville, and between the WERA at PAO and the AWAC at Saint-Ulric, have both been discarded because of the location of the AWACs on the edges of the domains covered by the HF radars, where the beams formed with the 12-element receive antennas have a very broad main lobe and relatively strong sidelobes, which may contaminate the radial currents toward the AWACs with currents from other directions (Fig. 4).

4. Results

Columns 3–5 of Table 2 summarize the statistical comparisons between UHFR and UAWAC-e for all HF radar–mooring pairs. For the radar azimuths toward the AWACs, values of r are in the range 0.17–0.91 (column 3 of Table 2). For example, radial currents measured by the WERA at PAB and the AWAC in the Bic channel are compared in Fig. 11b. Another example is given in Fig. 12b for the radial currents measured by the CODAR at PAP and the AWAC at Forestville. Column 4 of Table 2 shows Δθ for each of the HF radar–mooring pairs; Δθ ranges from −12° to −1°. After correcting for bearing offsets, correlations r* are in the range 0.42–0.93 (column 5 of Table 2).

Fig. 11.
Fig. 11.

Scatterplots of (a) the surface Stokes drift, US(0), vs the Bic AWAC surface current, UAWAC(0), and (b) the HF radar radial currents for the WERA at PAB vs the Eulerian radial currents for the Bic AWAC (UAWAC-e) for the bearing to the AWAC. The different possible contributions of the Stokes drift are added to the AWAC currents (UAWAC-e): (c) the surface Stokes drift [US(0)], (d) one-half of the surface Stokes drift (ΔcH76), (e) the filtered surface Stokes drift (ΔcL62), and (f) the weighted depth-average Stokes drift (ΔcL01). The correlation coefficients r and the x = y line are indicated in each panel. The colors indicate the number of observations in velocity bins.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

Fig. 12.
Fig. 12.

As in Fig. 11, but for the radial currents of the CODAR at PAP vs the AWAC at Forestville.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

Table 2

Correlation coefficients (with 95% confidence intervals) between HFR radial currents and AWAC currents (UAWAC-e) to which the different possible contributions of the Stokes drift were added. In the right section, UHFR vs UAWAC-e + Δci [where i = (H76, L01, L62) for one-half of the surface Stokes drift, the weighted depth-averaged Stokes drift and the filtered surface Stokes drift, respectively], the second line indicates the correlation coefficients that have been calculated with the approximations to the nonlinear corrections. The best correlations for each radar station are highlighted in boldface type. The bearing offsets are given for each radar–moored current meter pair.

Table 2

During winter 2016/17, when adding the possible contributions of the Stokes drift to the Eulerian radial currents measured by the AWACs, correlations with the HF radar currents increase significantly for all possible contributions (columns 6–13 of Table 2; Figs. 11 and 12). These experimental results support the theoretical arguments implying that HF radars measure a quantity related to the Stokes drift in addition to mean Eulerian currents. Furthermore, the bearing offsets Δθi with i = [US(0), H76, L01, L62] decrease in magnitude for all radar–mooring pairs (Table 2; Figs. 13 and 14), except for the WERA at PAO and the AWAC at Forestville.

Fig. 13.
Fig. 13.

Comparison between the radial currents measured by the WERA at PAB and the AWAC in Bic Channel, UAWAC-e, as a function of the radar azimuth. (a) Number of data points, (b) correlation r (left y axis; solid line) and normalized robust differences D (right y axis; dashed line). Additional panels show the comparisons between the radar currents and the AWAC currents, UAWAC-e, to which the different possible contributions of the Stokes drift were added: (c) US(0), (d) ΔcL62, (e) ΔcL01, and (f) ΔcH76. The correlation r* at the minimum of D, the bearing of the AWAC (brown dashed line), the bearing of the minimum of D (solid color line), and the bearing offsets Δθ are indicated in each panel.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

Fig. 14.
Fig. 14.

As in Fig. 13, but for the CODAR at PAP vs the AWAC at Forestville.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

For winter 2016/17, the strongest correlations (in boldface type in Table 2) are obtained when adding one-half of the surface Stokes drift to the Eulerian currents measured by the AWACs. In contrast, the weakest correlations are obtained when adding the surface Stokes drift to the AWAC currents. When adding the filtered surface Stokes drift to the AWAC currents, correlations obtained are significantly different from the correlations obtained when adding one-half of the surface Stokes drift, according to the 95% confidence intervals on the correlation values. When adding the weighted depth-averaged Stokes drift to the AWAC currents, the correlations are weaker than, albeit not always significantly different from (within the 95% level) the correlations obtained when adding one-half of the surface Stokes drift.

The second lines of each mooring of the columns 6–13 in Table 2 are the correlations between HF radars and the AWACs to which the approximations to the different contributions of the Stokes drift have been added. Correlation coefficients are slightly reduced, but not significantly different (within the 95% level) from those obtained without the approximations. This result shows that the approximate expressions of the wave-induced contributions assuming unidirectional wave spectra are reasonable approximations to the full expressions for directional wave spectra.

To further test whether one-half of the surface Stokes drift is the true quantity that HF radars measure due to wave–wave interactions, we computed the correlations obtained between the radar and AWAC currents, to which a quantity proportional to the surface Stokes drift, αUS(0), was added, for the CODAR at PAP and the AWAC at Forestville (Fig. 15). The maximum correlation is obtained for α = αH76 = 0.53 (magenta line), which is exactly the value of the slope of the regression line between ΔcH76 and US(0) and is very close to the value αH76*=0.50 (black line), corresponding to the approximation ΔcH76*. In contrast, the other contributions proposed in the literature, αL01 ≈ 0.71 for the weighted depth-averaged Stokes drift (blue line), αL62 ≈ 0.82 for the filtered surface Stokes drift (green line) and α = 1 for the surface Stokes drift (red line), have all weaker correlations. This result further confirms that one-half of the surface Stokes drift is what HF radars measure in addition to mean Eulerian currents.

Fig. 15.
Fig. 15.

Correlations r (left y axis; black line) and normalized robust differences D (right y axis; black dashed line) between the radial currents measured by the CODAR at PAP and the Forestville AWAC currents, UAWAC-e, to which a quantity proportional to the surface Stokes drift, αUS(0), was added. Vertical lines are the correlations obtained when adding αH76US(0), αL01US(0), αL62US(0), and US(0).

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

Similarly, for the other HF radar–mooring pairs, the strongest correlations are always obtained when adding one-half of the surface Stokes drift to the AWAC currents (Table 2). It is encouraging that the results obtained are similar for the three AWACs in different places in the study area, and three HF radars, which look in different directions and have different properties (type of radar and operating frequency).

However, for the summer 2013, the addition of one-half of the surface Stokes drift, or any other possible contribution of the Stokes drift to the Eulerian radial currents measured by the AWACs, does not increase the correlations with the HF radars currents (Table 2). For the winter 2016/17, one-half of the surface Stokes drift is typically on the order of 0.10 m s−1 (Fig. 16b) and can reach sometimes over 0.25 m s−1. For the summer 2013, when the winds are weak (Fig. 17a), one-half of the surface Stokes drift is typically on the order of 0.05 m s−1 or less and is about an order of magnitude weaker than the mean Eulerian currents (Fig. 17b).

Fig. 16.
Fig. 16.

From 1 to 6 Jan 2017, time series of (a) wind speed (left y axis; blue line) and wind direction (right y axis; orange line) at the Bic weather station, (b) Forestville AWAC currents UAWAC-e (magenta) and one-half of the surface Stokes drift (black), and (c) HF radar radial currents for the CODAR at PAP (green) and the Forestville AWAC currents (UAWAC-e) plus one-half of the surface Stokes drift (black and magenta).

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

Fig. 17.
Fig. 17.

From 9 to 16 Jun 2013, time series of (a) wind speed (left y axis; blue line) and wind direction (right y axis; orange line) from buoy PMZA-RIKI; (b),(c) as in Fig. 16, but for the radial currents from the CODAR at PAP and the AWAC at Saint-Ulric.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

5. Discussion

The correlation coefficients we obtained between the HF radar and AWAC currents are all statistically different from zero at the 95% confidence level. In summer, when winds are weak, the correlation coefficient between the CODAR at PAP and the AWAC at Saint-Ulric is 0.93, while in winter, when winds are strong, the correlation coefficients between the various radar and AWAC currents are only moderate (r < 0.63), despite the fact that the wind-induced shear has been taken into account by extrapolating the AWAC currents to the sea surface assuming Ekman dynamics. However, when adding one-half of the surface Stokes drift to the Eulerian AWAC currents, correlation coefficients significantly increase in winter (r ≥ 0.80). Interestingly, the increase in correlation coefficients is larger for the AWAC at Forestville than for the AWAC at the Bic (Table 2). This could be due to the different locations of the AWACs relative to the coastline (Fig. 4) and the dominant southerly wind directions in winter (Fig. 3a), resulting in a larger fetch and stronger waves at Forestville than at the Bic.

Another interesting result is that the offset Δθ, between the AWAC azimuth and the optimal azimuth θ*, generally decreases when adding one-half of the surface Stokes drift to the AWAC currents (Table 2). A graphical explanation is proposed in Fig. 18 for an idealized scenario with uniform alongshore Eulerian currents and uniform cross-shore winds.

Fig. 18.
Fig. 18.

(a) A schematic showing the azimuth offset Δθ, between the AWAC azimuth θAWAC and the optimal azimuth θ* that minimizes the differences between the radial currents measured by the HF radar UHFR and the radial Eulerian currents measured by the AWAC UAWAC. (b) Since the HF radar measures the Eulerian currents UE plus one-half of the surface Stokes drift (1/2)US(0), Δθ = 0 when adding the projection of (1/2)US(0) toward the radar to the AWAC currents UAWAC.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0025.1

The experimental results presented here strongly suggest that HF radars measure one-half of the surface Stokes drift rather than the weighted depth-averaged Stokes drift, confirming the theoretical expectation that HF radars should not measure the Stokes drift (a Lagrangian current) in the same way as they measure the mean Eulerian currents, due to the differences between wave–current and wave–wave interactions (Teague 1986; Chavanne 2018).

Our experimental results do not support the theoretical results of Longuet-Higgins and Phillips (1962) or Weber and Barrick (1977), but rather support the theoretical results of Huang and Tung (1976), who noted that Longuet-Higgins and Phillips (1962) “considered only the case of discrete [wave] components under resonant interaction conditions.” As noted by Chavanne (2018), the theoretical results of Weber and Barrick (1977), implying that HF radars should measure the filtered surface Stokes drift, has a singularity at the Bragg wavenumber (see Fig. 1, dashed black line, in Chavanne 2018) contrary to the result of Huang and Tung (1976). Barrick and Weber (1977) did not provide any physical explanation as to why this should be the case. However, the result that HF radars measure one-half of the surface Stokes drift can be physically interpreted, since the latter is a quasi-Eulerian quantity that would be measured by a current meter at a fixed horizontal position but allowed to follow the free surface moving vertically up and down with the passage of the waves (Phillips 1960). This is consistent with the view that HF radars are not expected to measure the full surface Stokes drift (a purely Lagrangian quantity) since they monitor fixed horizontal areas.

Note, however, that adding any of the 4 different contributions of the Stokes drift we have tested significantly improves correlations between HF radar and AWAC currents, in agreement with previously published experimental results suggesting that HF radars measure the filtered surface Stokes drift (e.g., Chevallier et al. 2014, see their Fig. 7). However, we obtained the best improvement using one-half of the surface Stokes drift, which previously published works had not considered.

Recent experimental results (Röhrs et al. 2015) suggested that HF radars do not measure the surface Stokes drift nor the filtered surface Stokes drift. Röhrs et al. (2015) argued that “the view that HF radar currents should be Eulerian is motivated by the fact that the radar retrieves its signal from fixed regions in space, hence not following particle motions.” This argument can be used to partly explain differences between surface drifter trajectories and the trajectories computed from HF radar measurements, even in weak wave conditions when the Stokes drift is negligible (Rypina et al. 2014). However, the fact that HF radars measure currents by using the departure of the Bragg waves phase velocity from its linear expression, and the fact that the Bragg waves phase velocity is modified by all the other gravity waves present at the sea surface, imply that HF radars should measure, in addition to mean Eulerian currents, a quantity related to the Stokes drift.

Since one-half of the surface Stokes drift is smaller than both the surface Stokes drift and the filtered surface Stokes drift, Röhrs et al. (2015) overcorrected the ADCP currents and drifting buoy velocities before comparing them with the HF radar measurements. Therefore, the results published by Röhrs et al. (2015) are not inconsistent with our result that HF radars measure one-half of the surface Stokes drift, which they did not investigate.

Our method to estimate the wind-driven shear is based on linear Ekman dynamics, and does not account for all wind-induced processes (other than the Stokes drift) that could generate vertical shear near the sea surface, such as nonlinear Ekman dynamics and wave breaking. It could therefore be argued that a study site with waves that are not generated by the local winds, such as open ocean swell, would be better suited for a study like ours than a fetch-limited area without swell. However, our study site has the advantage to feature conditions with very high winds in winter, and thus the Stokes drift constitutes a sufficiently large signal to resolve.

Our result that HF radars measure one-half of the surface Stokes drift has some practical implications. One is that the Stokes drift contribution to the HF radar current measurements should be independent from the operating frequency [Eq. (8); see also Fig. 3 of Chavanne 2018]. Therefore, one could easily remove this contribution using a multifrequency HF radar and subtracting the currents measured at one frequency from those measured at a different frequency, yielding a purely Eulerian vertical shear measurement since the Stokes drift contributions will exactly cancel out (the same bandwidth, and therefore range resolution, should be used for all frequencies). Another practical implication is that one-half of the surface Stokes drift should be added to the HF radar measurements of surface currents to account for the full surface Stokes drift in applications for which the surface drift is of interest, such as oil spill tracking. Conversely, one-half of the surface Stokes drift should be subtracted from the HF radar measurements of surface currents before assimilating such measurements into purely Eulerian ocean numerical models. In both cases, we need a way to reliably estimate the surface Stokes drift. Given the regional variability of the wave field in our study area, a single AWAC measuring the wave spectrum is not sufficient to estimate the surface Stokes drift over the entire area covered by the HF radars. Indeed, the amplitude of the complex correlation coefficient between the surface Stokes drifts obtained from the Bic and Forestville AWACs is not significantly different from 0 (|r| = 0.08). Given that HF radars can also measure part of the wave spectrum, perhaps the surface Stokes drift could be estimated directly from the HF radar measurements, but this is beyond the scope of this paper.

Acknowledgments.

We acknowledge the Fonds de recherche du Québec–Nature et technologies (FRQNT), the Marine Environmental, Observation, Prediction and Response Network (MEOPAR), Canada Economic Development for Quebec Regions, and Québec-Océan for their financial support. AWAC data were collected by Urs Neumeier with funding from the Transport Department of Quebec as part of the Green Fund actions and the 2013–20 Climate Change Action Plan of the Government of Quebec. We thank Jean Clary for the quality control of the radar data. We thank Louis-Philippe Nadeau for suggesting the analysis shown in Fig. 15. We thank the reviewers for their careful reviews and helpful suggestions on how to improve the paper.

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    • Export Citation
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    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Phillips, O., 1960: The mean horizontal momentum and surface velocity of finite-amplitude random gravity waves. J. Geophys. Res., 65, 34733476, https://doi.org/10.1029/JZ065i010p03473.

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    • Search Google Scholar
    • Export Citation
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    • Export Citation
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    • Export Citation
  • Röhrs, J., A. Sperrevik, K. Christensen, G. Broström, and Ø. Breivik, 2015: Comparison of HF radar measurements with Eulerian and Lagrangian surface currents. Ocean Dyn., 65, 679690, https://doi.org/10.1007/s10236-015-0828-8.

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    • Search Google Scholar
    • Export Citation
  • Rypina, I. I., A. Kirincich, R. Limeburner, and I. A. Udovydchenkov, 2014: Eulerian and Lagrangian correspondence of high-frequency radar and surface drifter data: Effects of radar resolution and flow components. J. Atmos. Oceanic Technol., 31, 945966, https://doi.org/10.1175/JTECH-D-13-00146.1.

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    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sentchev, A., P. Forget, and P. Fraunié, 2017: Surface current dynamics under sea breeze conditions observed by simultaneous HF radar, ADCP and drifter measurements. Ocean Dyn., 67, 499512, https://doi.org/10.1007/s10236-017-1035-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Shevlyakov, G., 1997: On robust estimation of a correlation coefficient. J. Math. Sci., 83, 434438, https://doi.org/10.1007/BF02400929.

  • Shevlyakov, G., P. Smirnov, V. I. Shin, and K. Kim, 2012: Asymptotically minimax bias estimation of the correlation coefficient for bivariate independent component distributions. J. Multivar. Anal., 111, 5965, https://doi.org/10.1016/j.jmva.2012.04.020.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stewart, R., and J. Joy, 1974: HF radio measurements of surface currents. Deep-Sea Res., 21, 3949, https://doi.org/10.1016/0011-7471(74)90066-7.

    • Search Google Scholar
    • Export Citation
  • Stokes, G., 1847: On the theory of oscillatory waves. Trans. Cambridge Philos. Soc., 8, 441473.

  • Tamtare, T., D. Dumont, and C. Chavanne