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  • View in gallery

    Factors multiplying the wave spectrum in the integrals of the expressions for the phase shift of a Bragg wave as a result of wave–wave interactions, as a function of the normalized wavenumber ; m−1 is the Bragg wavenumber for an HF radar operating at 13.5 MHz.

  • View in gallery

    Surface Stokes drift (red), weighted depth-averaged Stokes drift (blue), filtered surface Stokes drift (green), half of surface Stokes drift (thick black), and half of surface Bragg Stokes drift (thin black), as a function of wind speed, for an HF radar operating at 13.5 MHz and a saturated Phillips wave spectrum with . The Stokes drifts at the respective effective measurement depths are shown (dashed lines).

  • View in gallery

    HF radar effective measurement depths as a function of radar frequency, for mean Eulerian currents with exponential (red) and logarithmic (magenta) vertical profiles, and for the weighted depth-averaged Stokes drift (blue), the filtered surface Stokes drift (green), and half of the surface Stokes drift (black; for different wind speeds indicated above the lines). The different Stokes drift contributions are for a saturated Phillips wave spectrum with . The radar frequencies allocated by the International Telecommunication Union are indicated (vertical dashed lines; ITU 2014).

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Do High-Frequency Radars Measure the Wave-Induced Stokes Drift?

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  • 1 Institut des Sciences de la Mer de Rimouski, Université du Québec à Rimouski, Rimouski, Quebec, Canada
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ABSTRACT

High-frequency (HF) radars remotely measure ocean near-surface currents based on the Doppler shift of electromagnetic waves backscattered by surface gravity waves with half the electromagnetic wavelength, called Bragg waves. Since their phase velocity is affected not only by wave–current interactions with vertically sheared mean Eulerian currents but also by wave–wave interactions with all the other waves present at the sea surface, HF radars should measure a quantity related to the Stokes drift in addition to mean Eulerian currents. However, the literature is inconsistent—both theoretically and experimentally—on the specific expression and even on the existence of the Stokes drift contribution to the HF radar measurements. Three different expressions that have been proposed in the literature are reviewed and discussed in light of the relevant published experimental results: 1) the weighted depth-averaged Stokes drift, 2) the filtered surface Stokes drift, and 3) half of the surface Stokes drift. Effective measurement depths for these three expressions are derived for the Phillips wave spectrum. Recent experimental results tend to discard the second expression but are not inconsistent with the first and third expressions. The latter is physically appealing, since it 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. A definitive answer will require further experimental investigations.

Denotes content that is immediately available upon publication as open access.

© 2018 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: Cédric Chavanne, cedric.chavanne@ensta.org

ABSTRACT

High-frequency (HF) radars remotely measure ocean near-surface currents based on the Doppler shift of electromagnetic waves backscattered by surface gravity waves with half the electromagnetic wavelength, called Bragg waves. Since their phase velocity is affected not only by wave–current interactions with vertically sheared mean Eulerian currents but also by wave–wave interactions with all the other waves present at the sea surface, HF radars should measure a quantity related to the Stokes drift in addition to mean Eulerian currents. However, the literature is inconsistent—both theoretically and experimentally—on the specific expression and even on the existence of the Stokes drift contribution to the HF radar measurements. Three different expressions that have been proposed in the literature are reviewed and discussed in light of the relevant published experimental results: 1) the weighted depth-averaged Stokes drift, 2) the filtered surface Stokes drift, and 3) half of the surface Stokes drift. Effective measurement depths for these three expressions are derived for the Phillips wave spectrum. Recent experimental results tend to discard the second expression but are not inconsistent with the first and third expressions. The latter is physically appealing, since it 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. A definitive answer will require further experimental investigations.

Denotes content that is immediately available upon publication as open access.

© 2018 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: Cédric Chavanne, cedric.chavanne@ensta.org

1. Introduction

High-frequency (HF) radars remotely measure ocean surface currents and gravity waves by sending electromagnetic (EM) waves in the HF radio band (3–30 MHz) and recording the EM waves backscattered by ocean surface gravity waves (Crombie 1955). The recorded signal is dominated by EM waves backscattered from ocean surface waves with half the EM wavelength, called Bragg waves, propagating exactly toward or away from the radar. For the HF radio band, Bragg waves have wavelengths ranging from 5 to 50 m, corresponding to periods ranging from 1.8 to 5.7 s, that is, locally wind-generated waves. In the absence of mean Eulerian currents and surface gravity waves other than the Bragg waves, and for infinitesimally small Bragg waves, the backscattered EM waves are Doppler shifted by the linear phase velocity of the Bragg waves, which is given in the deep-water approximation by
e1
where g is the gravitational acceleration and is the Bragg wavenumber. The approaching and receding Bragg waves therefore cause two peaks in the recorded energy spectrum at frequencies symmetrically shifted around the frequency of the transmitted signal by an amount , where is the EM wavelength.

In the presence of mean Eulerian currents, other surface gravity waves, and finite-amplitude Bragg waves, the phase velocity of the latter differs from by an amount , causing an additional frequency shift of the two peaks, which is not symmetric around the frequency of the transmitted signal. This additional frequency shift is what HF radars measure. Since both mean Eulerian currents and surface gravity waves can affect the phase velocity of the Bragg waves (Stewart and Joy 1974; Stokes 1847; Longuet-Higgins and Phillips 1962; Huang and Tung 1976), the question arises as to whether HF radars measure a quantity related to the wave-induced Stokes drift in addition to mean Eulerian currents. This question has important implications for assimilating HF radar currents into numerical ocean models (Lewis et al. 1998; Breivik and Sætra 2001) and for practical applications, such as the use of HF radars in search and rescue operations (Breivik et al. 2013, and references therein) and oil spill mitigation (Abascal et al. 2009).

However, the literature is inconsistent on this question, both theoretically and experimentally. For example, Barrick (1986, p. 291) investigated the effect of wave–wave interactions on the dispersion relation used to interpret HF radar observations and concluded that “the third-order correction to the dispersion relation can be ignored in HF radar current measurements, for the “error” in its neglect that gets added is actually the Stokes drift . . . HF radar current measurements therefore include Stokes drift.” In contrast, Röhrs et al. (2015) have recently argued that HF radars measure only mean Eulerian currents, based on comparisons of HF radar measurements with moored current meters and near-surface drifters. The purpose of this paper is to review the different theoretical considerations that have been put forward in the literature (section 2) and the relevant experimental results that support or contradict these theoretical considerations (section 3). The experimental results are discussed in light of the theoretical considerations in section 4.

2. Theoretical considerations

a. caused by mean Eulerian currents

Stewart and Joy (1974) and Ha (1979) showed that the phase velocity of a Bragg wave propagating on a mean Eulerian horizontal current , assumed to vary only with depth, is modified from by an amount
e2
where is the mean current component in the direction of Bragg wave propagation; is assumed to be small relative to the Bragg wave phase velocity, that is, , which is generally the case at high frequencies ( ranges from 2.8 to 8.8 m for frequencies ranging from 30 to 3 MHz). Therefore, HF radars perform a weighted depth averaging of mean Eulerian currents, with weights exponentially decreasing from the surface as fast as the Stokes drift associated with the Bragg wave (Stokes 1847),
e3
where is the Bragg wave amplitude.

b. caused by the Bragg wave nonlinear self-interaction

Even in the absence of mean Eulerian current, a finite-amplitude Bragg wave has a phase velocity (Stokes 1847),
e4
which differs from . For small-amplitude waves (), the departure from is half the surface Stokes drift,
e5
For example, for a radar operating at 13.5 MHz, a typical phase velocity modification by the self-interaction effect would be m (see section 2d).

Stewart and Joy (1974) remarked that if one replaces the mean Eulerian current in Eq. (2) with the Stokes drift given in Eq. (3), which is a mean Lagrangian current, one obtains the same phase velocity shift as given in Eq. (5). Stewart and Joy (1974, 1040–1041) concluded that “the ocean wave cannot distinguish between currents produced by wave-wave interactions and wind-driven currents”; in other words, HF radars should measure the Stokes drift in the same way as they measure mean Eulerian currents, such as wind-driven currents. Although it is not justified to use a mean Lagrangian current in Eq. (2), let us assume their conclusion to be true and generalize it in the next section to the Stokes drift associated with the entire surface wave spectrum.

c. caused by nonlinear wave–wave interactions: Option 1

For a directional wave spectrum , the Stokes drift is given in the deep-water approximation by (Kenyon 1969)
e6
where ω is the frequency and is the wavenumber magnitude. To simplify the expression given above, let us assume that all the waves are propagating in the same direction, that is, the direction of the Bragg wave. Then, the Stokes drift becomes
e7
where is the unidirectional wave spectrum.
Replacing in Eq. (2) with the Stokes drift given in Eq. (7), one obtains the weighted depth-averaged Stokes drift (Laws 2001)
e8
Laws (2001) noted that for a monochromatic wave spectrum containing only the Bragg wave, , where δ is the Dirac delta function, , and the result of Stokes (1847) is recovered [Eq. (5)]. This is illustrated in Fig. 1, where the factor multiplying the wave spectrum in the integral of Eq. (8) is shown as a function of the normalized wavenumber (blue line). For , it is indeed equal to the factor obtained by Stokes (1847) (red star).
Fig. 1.
Fig. 1.

Factors multiplying the wave spectrum in the integrals of the expressions for the phase shift of a Bragg wave as a result of wave–wave interactions, as a function of the normalized wavenumber ; m−1 is the Bragg wavenumber for an HF radar operating at 13.5 MHz.

Citation: Journal of Atmospheric and Oceanic Technology 35, 5; 10.1175/JTECH-D-17-0099.1

To get an idea of the typical magnitude of , consider the Phillips wave spectrum (Phillips 1958)
e9
where and is the peak wavenumber with being the wind speed. For this spectrum, is shown in Fig. 2 (blue line) as a function of wind speed. It varies from 0.01 to 0.3 m s−1 for winds ranging from 4 to 25 m s−1, respectively, and is typically 0.09 m s−1 weaker than the surface Stokes drift (red line).
Fig. 2.
Fig. 2.

Surface Stokes drift (red), weighted depth-averaged Stokes drift (blue), filtered surface Stokes drift (green), half of surface Stokes drift (thick black), and half of surface Bragg Stokes drift (thin black), as a function of wind speed, for an HF radar operating at 13.5 MHz and a saturated Phillips wave spectrum with . The Stokes drifts at the respective effective measurement depths are shown (dashed lines).

Citation: Journal of Atmospheric and Oceanic Technology 35, 5; 10.1175/JTECH-D-17-0099.1

d. caused by nonlinear wave–wave interactions: Option 2

Although the approach taken in the previous section has been adopted by some authors (e.g., Teague 1986; Essen 1993; Laws 2001), it is not justified to use a mean Lagrangian current in Eq. (2), as acknowledged by Teague (1986). A direct approach must be taken by considering the nonlinear interactions between different waves. Longuet-Higgins and Phillips (1962) considered the case of two small-amplitude waves moving in the same direction. The increase in phase velocity of the wave with amplitude , wavenumber , and phase velocity , resulting from the interaction with the wave , is given by
e10
For , the phase velocity modification of wave 2 equals the surface Stokes drift of wave 1, not half the surface Stokes drift as one would obtain by using , the Stokes drift of wave 1, in Eq. (2). For , the phase velocity modification of wave 2 equals times the surface Stokes drift of wave 1, again in general different from half the surface Stokes drift of wave 1. For , the result of Longuet-Higgins and Phillips (1962) gives , which is twice the result of Stokes (1847) [Eq. (5)]. Longuet-Higgins and Phillips (1962, p. 336) remarked that “this is a curious result,” but they did not provide a physical explanation for the difference between their result and that predicted by Stokes (1847). If correct, their result shows that the interaction between two different waves is not a simple advection of one wave by the other wave’s Stokes drift.
Longuet-Higgins and Phillips (1962) then directly extended their result for the nonlinear interaction between two different waves to the case of a continuous spectrum of waves all traveling in the same direction, to obtain (integrating over wavenumber instead of frequency)
e11
is shown in Fig. 2 (green line) for the Phillips spectrum. As for the surface Stokes drift, it varies linearly with wind speed and is about 0.04 m s−1 weaker than the surface Stokes drift. The factor multiplying the wave spectrum in Eq. (11) is shown in Fig. 1 (green line). Its value is twice that predicted by Stokes (1847) for (red star). The factors predicted by Longuet-Higgins and Phillips (1962) and Laws (2001) converge at the same values for and , but they differ significantly over a wide range of wavenumbers (cf. the blue and green lines in Fig. 1), resulting in different phase velocity modifications (cf. the blue and green lines in Fig. 2).
Weber and Barrick (1977) extended the analysis of Stokes (1847) to the general case of a two-dimensional directional wave spectrum, and Barrick and Weber (1977) showed that this generalized result reduces in the case of waves propagating in the same direction to
e12
recovering the results of both Stokes (1847) [Eq. (5)] and Longuet-Higgins and Phillips (1962) [Eq. (10)]. Although their generalized result unifies these two particular results, it is peculiar that the phase velocity shift of the Bragg wave as a result of its interaction with another wave propagating in the same direction has a singularity at the Bragg wavenumber (see Fig. 1, dashed black line). Barrick and Weber (1977) did not provide any physical explanation as to why this should be the case. Finally, neglecting the self-interaction effect [first term on the right-hand side of Eq. (12)], they arrived at the same expression as that obtained by Longuet-Higgins and Phillips (1962) [Eq. (11)]. The magnitude of the self-interaction effect for the Phillips spectrum and a radar operating at 13.5 MHz is m s−1 (Fig. 2, thin black line). It is negligible relative to the mutual interactions effect (green line) for strong winds but not for weak winds. Nevertheless, Eq. (11) was also used by Broche et al. (1983) and Ardhuin et al. (2009). Because the first term on the right-hand side of Eq. (11) has the same form as the surface Stokes drift [taking z = 0 in Eq. (7)], except that the integration is carried over only wavelengths longer than the Bragg wavelength, Ardhuin et al. (2009) called it the filtered surface Stokes drift, although there is also a contribution from waves shorter than the Bragg wave [second term on the right-hand side of Eq. (11)].

e. caused by nonlinear wave–wave interactions: Option 3

Huang and Tung (1976) noted that Longuet-Higgins and Phillips (1962) considered only the case of discrete spectral components under resonant interaction conditions. Solving the nonlinear equations for a continuous wave spectrum, Huang and Tung (1976) obtained a different result from that of Longuet-Higgins and Phillips (1962), namely, half of the surface Stokes drift,
e13
is shown in Fig. 2 (thick black line) for the Phillips spectrum. It varies linearly with wind speed and reaches 0.2 m s−1 for winds of 25 m s−1.

The factor multiplying the wave spectrum in Eq. (13) is shown in Fig. 1 (thick black line). Its value agrees with that predicted by Stokes (1847) for (red star), contrary to the result of Longuet-Higgins and Phillips (1962). Furthermore, Huang and Tung (1976) propose a physical interpretation of their result. Phillips (1960) computed the mean horizontal fluid velocity at the free surface at a fixed horizontal position in the presence of a random sea and found it is precisely half the surface Stokes drift. The former velocity is a quasi-Eulerian quantity in the sense that it is measured at a fixed horizontal position but at the free surface that is moving vertically up and down, while the Stokes drift is a Lagrangian quantity. Therefore, Huang and Tung (1976) remarked that the change in phase velocity they obtained is precisely a Doppler shift caused by the local (quasi Eulerian) mean velocity field.

f. Effective measurement depths

It was shown in the previous sections that at least three different expressions have been proposed in the literature for the phase shift of a Bragg wave caused by its nonlinear interactions with other waves traveling in the same direction: [Eq. (8)], [Eq. (11)], and [Eq. (13)]. To test these theoretical predictions with observations, it is necessary to compare HF radar measurements with in situ measurements of Eulerian and Lagrangian currents by moored current meters or profilers and drifters, respectively. Because of the different measurement characteristics of the instruments, it is useful to express the currents measured by HF radars as if they were measured at a specific depth, called the effective depth. For Eulerian currents, the effective depth will depend on their vertical profile [see Eq. (2)]. For the Stokes drift, the effective depth will depend on the wave spectrum [see Eq. (7)].

First, let us review the effective depths obtained for different Eulerian current profiles. For an exponential profile (e.g., Ekman spiral, neglecting rotation of the currents with depth),
e14
where is the surface current and m is the inverse of the e-folding depth; the effective depth is (Stewart and Joy 1974)
e15
provided , that is, the current profile does not decay with depth too fast. Stewart and Joy (1974) also noted that the same effective depth is obtained for a linear current profile.
For a logarithmic profile (e.g., wind drift),
e16
where is the friction velocity, κ is the von Kármán constant, and is the surface roughness length; the effective depth is (Ha 1979)
e17
where is the Euler constant.
To determine the effective measurement depth for the Stokes drift, one needs to consider a particular wave spectrum. Breivik et al. (2016) showed that the Phillips spectrum provides a good approximation to the Stokes drift profile for a variety of other parametric spectra as well as for numerically modeled and observed spectra, especially close to the surface. Since the simple analytical form of the Phillips spectrum [Eq. (9)] allows one to obtain analytical expressions for the effective depths, let us therefore consider the Phillips spectrum in Eq. (7), which gives the Phillips Stokes drift (Breivik et al. 2016)
e18
where erfc is the complementary error function.
Now, using the Phillips spectrum in Eq. (8),
e19
The effective depth at which is
e20
provided . The term is shown in Fig. 2 (dashed blue line) and is close to (thick blue line) except for low wind speeds at which is not much smaller than .
Next, using the Phillips spectrum in Eq. (11),
e21
assuming , otherwise there would be no Bragg wave under the Phillips spectrum and hence no HF radar first-order signal. The effective depth at which is
e22
provided . The term is shown in Fig. 2 (dashed green line) and is close to (thick green line) except for low wind speeds at which is not much smaller than .
Finally, using the Phillips spectrum in Eq. (13),
e23
The effective depth at which is
e24
Term is shown in Fig. 2 (dashed black line) and is indistinguishable from (thick black line).

The different effective measurement depths are listed in Table 1 and are shown in Fig. 3 as a function of radar frequency. The effective depth for the weighted depth-averaged Stokes drift [Eq. (20), blue line in Fig. 3] is shallower than the effective depth for exponentially decaying mean Eulerian currents [Eq. (15), red line in Fig. 3] but deeper than the effective depth for logarithmically decaying mean Eulerian currents [Eq. (17), magenta line in Fig. 3]. In contrast, the effective depth for the filtered surface Stokes drift [Eq. (22), green line in Fig. 3] is much shallower, ranging from 0.06 m at 25 MHz to 0.37 m at 4.5 MHz. The effective depth for half of the surface Stokes drift [Eq. (24), black lines in Fig. 3] depends on the wind speed but not on the radar frequency.

Table 1.

Effective depths for HF radar measurements, considering a saturated Phillips wave spectrum with for the Stokes drift contributions. The Bragg wavelength is represented by , is the Euler constant, is the wind speed, and g is the gravitational acceleration.

Table 1.
Fig. 3.
Fig. 3.

HF radar effective measurement depths as a function of radar frequency, for mean Eulerian currents with exponential (red) and logarithmic (magenta) vertical profiles, and for the weighted depth-averaged Stokes drift (blue), the filtered surface Stokes drift (green), and half of the surface Stokes drift (black; for different wind speeds indicated above the lines). The different Stokes drift contributions are for a saturated Phillips wave spectrum with . The radar frequencies allocated by the International Telecommunication Union are indicated (vertical dashed lines; ITU 2014).

Citation: Journal of Atmospheric and Oceanic Technology 35, 5; 10.1175/JTECH-D-17-0099.1

3. Experimental results

While the way HF radars measure mean Eulerian currents has been well established theoretically (Stewart and Joy 1974; Ha 1979) and confirmed experimentally (e.g., Stewart and Joy 1974; Ha 1979; Teague 1986; Teague et al. 2001; Sentchev et al. 2017), the way HF radars measure the Stokes drift has not been established consistently in the literature from a theoretical point of view, as shown in section 2. Some relevant experimental results are reviewed here in light of the different theoretical considerations.

The first published comparisons between currents measured by HF radars and near-surface drifting buoys were reported by Stewart and Joy (1974). They obtained good agreement (within 2 cm s−1) for buoys with drogues placed at the HF radar effective measurement depth for linearly or exponentially decaying mean Eulerian currents [Eq. (15)], confirming that HF radars perform a weighted depth averaging of the currents [Eq. (2)]. Winds were weak during their experiment ( m s−1) and therefore the Stokes drift was probably negligible ( m s−1 according to Fig. 2). One of the earlier comparisons between currents measured by HF radars and moored current meters was reported by Schott et al. (1985). They found some discrepancies for the zonal current component between their Coastal Ocean Dynamics Applications Radar (CODAR) HF radar and their PEGASUS current profiler, and mentioned that the “Stokes drift of longer surface waves . . . could superimpose a westward near-surface current on the waves seen by CODAR which would be decayed at the depth of the first PEGASUS measurement” (Schott et al. (1985, p. 9016).

Teague’s (1986) experimental results were perhaps the first to quantitatively suggest that the Stokes drift is part of HF radar measurements. Using a multifrequency ship-mounted HF radar, he showed that the vertical current shear measured as the difference between currents from radar measurements at different frequencies was higher than expected from either wind-induced or wave-induced shears but comparable to their sum. A decade later, Fernandez et al. (1996) reported measurements of current shear in the top meter of the ocean using a shore-based dual-frequency HF radar. They observed that the current vector rotated 13°–23° in the clockwise direction between effective measurement depths of 30 and 50 cm, a much higher value than expected from Ekman currents for typical Ekman depths. The current shear between these depths was aligned with the prevailing wind direction, again consistent with either wind-induced or wave-induced shear, or both. Paduan and Rosenfeld (1996) observed that the largest differences between radar-derived and ADCP currents occurred during the strongest winds. They noted that “the radar-derived currents include a Stokes drift component that is not present in the ADCP currents” (p. 20 677) and that the Stokes drift could therefore account for some of the observed differences between radar-derived and ADCP currents. Graber et al. (1997) went on to estimate the expected contributions from different geophysical processes, including the Stokes drift, to the observed root-mean-square (rms) differences between radar-derived currents and near-surface current meter measurements. They found that the Stokes drift could explain between 3% and 14% of the observed rms differences, a value that is lower than those explained by Ekman or baroclinic current shears but nonetheless is not negligible.

Laws (2001) carried out a thorough investigation of the relationship between radar-derived currents and the Stokes drift estimated from wave spectra measurements by a buoy moored in the radars coverage area. He assumed that the Stokes drift is measured by the HF radars in the same way as mean Eulerian currents are (see section 2c). He found significant correlation between radar-derived currents and the assumed Stokes drift contribution only for the radar operating frequency of 6.8 MHz, with a regression slope of 9, indicating that the radar-derived currents that correlate with the assumed Stokes drift contribution are 9 times as large as the latter in magnitude. This may be because the Stokes drift is itself correlated with the local wind and therefore with the wind-induced drift currents also measured by HF radars. This highlights the difficulty in separating the Stokes and wind-induced drift currents in HF radar measurements. Ullman et al. (2006) found no dependence on wind speed for the difference between CODAR-measured and drifter velocities when the CODAR and drifter effective depths are similar (0.5 and 0.65 m, respectively). In contrast, they found some dependence when the effective depths differed from each other (2.4 and 0.65 m, respectively), which they could mostly account for by considering the vertical shear resulting from the steady-state Ekman spiral and the Stokes drift. These results suggest that HF radars measure the Stokes drift at a similar effective depth as for mean Eulerian currents.

Mao and Heron (2008) examined the response of surface currents measured by HF radars to winds in both short-fetch and long-fetch conditions. They showed that the ratio of surface current speed to wind speed is larger under the long-fetch condition, while the angle between the surface current vector and wind vector is larger under the short-fetch condition. They interpreted these results by decomposing the surface current into a surface Ekman current, which was proportional to the square of the wind velocity and rotated by 45° from the wind direction, and a surface Stokes drift, which was proportional to the wind velocity and aligned with the wind direction. They found that the Stokes drift dominates the surface currents under the long-fetch condition when the sea state is more developed, while the Stokes drift and Ekman current are almost equally important under the short-fetch condition. Their results provide observational evidence that HF radars measure the Stokes drift and highlight the importance of the fetch conditions. Ardhuin et al. (2009) also examined the response of surface currents measured by HF radars to winds. They estimated the surface Stokes drift from a realistic numerical wave model and found that it generally increases quadratically with the wind speed, contrary to the linear relationship obtained by Mao and Heron (2008), who assumed that the Stokes drift was dominated by the wave component at the spectral peak. Ardhuin et al. (2009) found that the coherence between radar-derived currents and wind is reduced when the filtered surface Stokes drift [Eq. (11)] is subtracted from the radars measurements, providing more observational evidence that HF radars measure at least part of the surface Stokes drift. Abascal et al. (2009) used HF radar data to predict the trajectories of surface drifters. They implicitly assumed that HF radars do not measure the Stokes drift, since they independently added the latter (estimated from an operational wave forecasting system) in their trajectory prediction model. However, they found that the effect of Stokes drift was not significant. This result suggests that either the Stokes drift is already included in the HF radar measurements or that it was negligible under the prevailing wind conditions, which were unfortunately not reported.

Finally, Röhrs et al. (2015) compared measurements from HF radars, moored current meters, and near-surface drifters. They estimated the Stokes drift from numerically modeled wave spectra adjusted to observed wave spectra. They found that the correlation between HF radar and ADCP currents was decreased when adding either the surface Stokes drift or the filtered surface Stokes drift to the Eulerian ADCP currents. Conversely, they also found that the correlation between HF radar and near-surface drifter currents was increased when subtracting the Stokes drift at the drifter depth from the Lagrangian drifter currents. These results suggest that the Stokes drift is not included in the HF radar measurements, contrary to the earlier experimental results reviewed above. However, Röhrs et al. (2015) remarked that the uncertainties in their HF radar current estimates are larger than the filtered surface Stokes drift, so their results should be interpreted with caution.

4. Discussion

As noted in the introduction, the literature is inconsistent on the question of whether HF radars measure the surface Stokes drift, or a related quantity, both theoretically (section 2) and experimentally (section 3). Among the different theoretical considerations proposed in the literature (sections 2ce), is there one that would be consistent with the majority of the experimental results reviewed in section 3?

Let us start with the results of Röhrs et al. (2015), which seem to contradict all the earlier experimental results reviewed in section 3. Röhrs et al. tested whether HF radars measure the surface Stokes drift or the filtered surface Stokes drift and found experimental suggestion that they do not, although measurement uncertainties prevent reaching a definitive conclusion. They did not test whether HF radars measure the weighted depth-averaged Stokes drift as for mean Eulerian currents [Eq. (8)] or half of the surface Stokes drift [Eq. (13)]. Since these two quantities are smaller than both the surface Stokes drift and its filtered version (except for low wind conditions for half of the surface Stokes drift; see Fig. 2), it could be that they overcorrected the ADCP and drifter measurements before comparing them with the HF radar currents. Their results actually hint that this may be the case: they obtained no correlation between the surface Stokes drift and the difference between currents from HF radar and from drifters drifting at 1-m depth (Röhrs et al. 2015, their Fig. 7b). If HF radars measured only the Eulerian currents, then the difference between HF radar and drifter currents should be correlated with the Stokes drift at 1-m depth, which is not negligible during the drifters’ deployment period (see Röhrs et al. 2015, Fig. 3a). Instead, if HF radars measured the Stokes drift in the same way as they measure mean Eulerian currents, then the effective depth for the Stokes drift measurement would be (assuming a Phillips wave spectrum) m. This is close to the drifters’ depth, so the Stokes drift measured by the HF radars would be similar to the Stokes drift felt by the drifters, which is consistent with the absence of correlation between the Stokes drift and the difference between HF radar and drifter currents. Alternatively, if HF radars measured half of the surface Stokes drift, then the Stokes drift measured by the HF radars would also be similar to the Stokes drift felt by the drifters, since the Stokes drift at 1-m depth was about half of the surface Stokes drift during most of their experiment (see Röhrs et al. 2015, Fig. 3a). Therefore, the results of Röhrs et al. (2015) are not inconsistent with the possibilities that HF radars measure either the weighted depth-averaged Stokes drift as for mean Eulerian currents or half of the surface Stokes drift.

The results reported by Ullman et al. (2006) suggest that HF radars measure the Stokes drift at a similar effective depth as for mean Eulerian currents. If HF radars measured half of the surface Stokes drift, then the effective depth for the Stokes drift contribution would depend quadratically on the wind speed [Eq. (24)], and the difference between CODAR-measured and drifter velocities would always be correlated with the wind whatever the drifter depth, contrary to the observations of Ullman et al. (2006). However, from a theoretical point of view, it is not justified to use a mean Lagrangian current (the Stokes drift) in Eq. (2). Therefore, one should not expect HF radars to measure the Stokes drift in the same way as they measure mean Eulerian currents. The consequence of the result of Huang and Tung (1976) that HF radars should measure half of the surface Stokes drift is physically appealing, since it 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). An interesting result is that the Stokes drift contribution to the radar measurement would not depend on the radar frequency. Therefore, differences between HF radar measurements at different frequencies would not be correlated with the Stokes drift. Laws (2001) indeed found that differences between HF radar measurements at different frequencies were not significantly (to the 95% significance level) correlated with the Stokes drift, while they were significantly correlated with the wind.

In conclusion, a definitive answer to the question of whether HF radars measure the surface Stokes drift, or a related quantity, will require further experimental investigations. Perhaps the simplest approach would be to compare large amounts of current measurements from HF radars with the different Stokes drift contributions discussed in section 2, which could be computed from observed or numerically predicted wave spectra. The proposed Stokes drift contributions differ from each other by typically 5 cm s−1 or more, especially for strong winds (Fig. 2), which ought to be measurable using large amounts of data. Until this question is resolved, uncertainties of this magnitude will remain about the Stokes drift contribution to HF radar measurements. Knowing exactly what HF radars measure is important, especially for assimilating HF radar currents into numerical ocean models and for computing Lagrangian trajectories of surface drifting objects or particles.

Acknowledgments

The motivation for this paper stemmed from preparing a course on remote sensing of the ocean surface by HF radars, for the winter school course on marine environmental prediction, funded by the Marine Environmental Observation, Prediction and Response Network (MEOPAR; 1-02-01-022.3) of the Network of Centres of Excellence, that took place in Rimouski, Canada, in March 2017. I thank both reviewers for their careful reviews and helpful suggestions on how to improve the paper.

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