Do High-Frequency Radars Measure the Wave-Induced Stokes Drift?
1. Introduction
In the presence of mean Eulerian currents, other surface gravity waves, and finite-amplitude Bragg waves, the phase velocity of the latter differs from 
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. 
b. 
Stewart and Joy (1974) remarked that if one replaces the mean Eulerian current 
c. 
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 
Citation: Journal of Atmospheric and Oceanic Technology 35, 5; 10.1175/JTECH-D-17-0099.1
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
Citation: Journal of Atmospheric and Oceanic Technology 35, 5; 10.1175/JTECH-D-17-0099.1
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
Citation: Journal of Atmospheric and Oceanic Technology 35, 5; 10.1175/JTECH-D-17-0099.1
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 
Citation: Journal of Atmospheric and Oceanic Technology 35, 5; 10.1175/JTECH-D-17-0099.1
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
Citation: Journal of Atmospheric and Oceanic Technology 35, 5; 10.1175/JTECH-D-17-0099.1
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
Citation: Journal of Atmospheric and Oceanic Technology 35, 5; 10.1175/JTECH-D-17-0099.1
d. 
e. 
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 
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: 
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.
Effective depths for HF radar measurements, considering a saturated Phillips wave spectrum with 
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 
Citation: Journal of Atmospheric and Oceanic Technology 35, 5; 10.1175/JTECH-D-17-0099.1
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
Citation: Journal of Atmospheric and Oceanic Technology 35, 5; 10.1175/JTECH-D-17-0099.1
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
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 (
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 2c–e), 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) 
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.
REFERENCES
Abascal, A. J., S. Castanedo, R. Medina, I. J. Losada, and E. Alvarez-Fanjul, 2009: Application of HF radar currents to oil spill modelling. Mar. Pollut. Bull., 58, 238–248, https://doi.org/10.1016/j.marpolbul.2008.09.020.
Ardhuin, F., L. Marié, N. Rascle, P. Forget, and A. Roland, 2009: Observation and estimation of Lagrangian, Stokes, and Eulerian currents induced by wind and waves at the sea surface. J. Phys. Oceanogr., 39, 2820–2838, https://doi.org/10.1175/2009JPO4169.1.
Barrick, D. E., 1986: The role of the gravity-wave dispersion relation in HF radar measurements of the sea surface. IEEE J. Oceanic Eng., 11, 286–292, https://doi.org/10.1109/JOE.1986.1145156.
Barrick, D. E., and B. L. Weber, 1977: On the nonlinear theory for gravity waves on the ocean’s surface. Part II: Interpretation and applications. J. Phys. Oceanogr., 7, 11–21, https://doi.org/10.1175/1520-0485(1977)007<0011:OTNTFG>2.0.CO;2.
Breivik, Ø., and Ø. Sætra, 2001: Real time assimilation of HF radar currents into a coastal ocean model. J. Mar. Syst., 28, 161–182, https://doi.org/10.1016/S0924-7963(01)00002-1.
Breivik, Ø., A. A. Allen, C. Maisondieu, and M. Olagnon, 2013: Advances in search and rescue at sea. Ocean Dyn., 63, 83–88, https://doi.org/10.1007/s10236-012-0581-1.
Breivik, Ø., J.-R. Bidlot, and P. A. Janssen, 2016: A Stokes drift approximation based on the Phillips spectrum. Ocean Modell., 100, 49–56, https://doi.org/10.1016/j.ocemod.2016.01.005.
Broche, P., J. C. Demaistre, and P. Forget, 1983: Mesure par radar décamétrique cohérent des courants superficiels engendrés par le vent. Oceanol. Acta, 6, 43–53.
Crombie, D. D., 1955: Doppler spectrum of sea echo at 13.56 Mc./s. Nature, 175, 681–682, https://doi.org/10.1038/175681a0.
Essen, H.-H., 1993: Ekman portion of surface currents, as measured by radar in different areas. Dtsch. Hydrogr. Z., 45, 57–85, https://doi.org/10.1007/BF02226315.
Fernandez, D. M., J. F. Vesecky, and C. C. Teague, 1996: Measurements of upper ocean surface current shear with high-frequency radar. J. Geophys. Res., 101, 28 615–28 625, https://doi.org/10.1029/96JC03108.
Graber, H. C., B. K. Haus, R. D. Chapman, and L. K. Shay, 1997: HF radar comparisons with moored estimates of current speed and direction: Expected differences and implications. J. Geophys. Res., 102, 18 749–18 766, https://doi.org/10.1029/97JC01190.
Ha, E.-C., 1979: Remote sensing of ocean surface current and current shear by HF backscatter radar. Ph.D. thesis, Stanford University, 134 pp.
Huang, N. E., and C.-C. Tung, 1976: The dispersion relation for a nonlinear random gravity wave field. J. Fluid Mech., 75, 337–345, https://doi.org/10.1017/S0022112076000256.
ITU, 2014: Guidelines for the use of spectrum by oceanographic radars in the frequency range 3 to 50 MHz. International Telecommunication Union Tech. Rep. ITU-R M.2321-0, 36 pp.
Kenyon, K. E., 1969: Stokes drift for random gravity waves. J. Geophys. Res., 74, 6991–6994, https://doi.org/10.1029/JC074i028p06991.
Laws, K., 2001: Measurements of near surface ocean currents using HF radar. Ph.D. thesis, University of California, Santa Cruz, 171 pp.
Lewis, J. K., I. Shulman, and A. F. Blumberg, 1998: Assimilation of Doppler radar current data into numerical ocean models. Cont. Shelf Res., 18, 541–559, https://doi.org/10.1016/S0278-4343(98)00006-5.
Longuet-Higgins, M. S., and O. M. Phillips, 1962: Phase velocity effects in tertiary wave interactions. J. Fluid Mech., 12, 333–336, https://doi.org/10.1017/S0022112062000245.
Mao, Y., and M. L. Heron, 2008: The influence of fetch on the response of surface currents to wind studied by HF ocean surface radar. J. Phys. Oceanogr., 38, 1107–1121, https://doi.org/10.1175/2007JPO3709.1.
Paduan, J. D., and L. K. Rosenfeld, 1996: Remotely sensed surface currents in Monterey Bay from shore-based HF radar (Coastal Ocean Dynamics Application Radar). J. Geophys. Res., 101, 20 669–20 686, https://doi.org/10.1029/96JC01663.
Phillips, O. M., 1958: The equilibrium range in the spectrum of wind-generated waves. J. Fluid Mech., 4, 426–434, https://doi.org/10.1017/S0022112058000550.
Phillips, O. M., 1960: The mean horizontal momentum and surface velocity of finite-amplitude random gravity waves. J. Geophys. Res., 65, 3473–3476, https://doi.org/10.1029/JZ065i010p03473.
Röhrs, J., A. K. Sperrevik, K. H. Christensen, G. Broström, and Ø. Breivik, 2015: Comparison of HF radar measurements with Eulerian and Lagrangian surface currents. Ocean Dyn., 65, 679–690, https://doi.org/10.1007/s10236-015-0828-8.
Schott, F., A. S. Frisch, K. Leaman, G. Samuels, and I. P. Fotino, 1985: High-frequency Doppler radar measurements of the Florida Current in summer 1983. J. Geophys. Res., 90, 9006–9016, https://doi.org/10.1029/JC090iC05p09006.
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, 499–512, https://doi.org/10.1007/s10236-017-1035-6.
Stewart, R. H., and J. W. Joy, 1974: HF radio measurements of surface currents. Deep-Sea Res. Oceanogr. Abstr., 21, 1039–1049, https://doi.org/10.1016/0011-7471(74)90066-7.
Stokes, G. G., 1847: On the theory of oscillatory waves. Trans. Cambridge Philos. Soc., 8, 441–473.
Teague, C. C., 1986: Multifrequency HF radar observations of currents and current shears. IEEE J. Oceanic Eng., 11, 258–269, https://doi.org/10.1109/JOE.1986.1145178.
Teague, C. C., J. F. Vesecky, and Z. R. Hallock, 2001: A comparison of multifrequency HF radar and ADCP measurements of near-surface currents during COPE-3. IEEE J. Oceanic Eng., 26, 399–405, https://doi.org/10.1109/48.946513.
Ullman, D. S., J. O’Donnell, J. Kohut, T. Fake, and A. Allen, 2006: Trajectory prediction using HF radar surface currents: Monte Carlo simulations of prediction uncertainties. J. Geophys. Res., 111, C12005, https://doi.org/10.1029/2006JC003715.
Weber, B. L., and D. E. Barrick, 1977: On the nonlinear theory for gravity waves on the ocean’s surface. Part I: Derivations. J. Phys. Oceanogr., 7, 3–10, https://doi.org/10.1175/1520-0485(1977)007<0003:OTNTFG>2.0.CO;2.
