a. Initial information

Surface drift currents play a significant role in marine shipping, the spread of surface waste, and marine industry safety. Thus, studying and understanding this phenomenon, which was comprehensively described in the recent review by Bremer and Breivik (2017), is of physical and practical importance. From the physical perspective, the drift currents are interesting due to their observability, availability for measurements, and possibility of theoretical description (e.g., Stokes 1847; Longuet-Higgins 1953; Craik 1982; Bremer and Breivik 2017, and references therein). For this reason, a large number of surface drift current measurements have been obtained over long periods, started by Shemdin (1972), Wu (1975, 1983), Churchill and Csanady (1983), and Bye (1988), and continued at present (e.g., Longo 2012; Longo et al. 2012; Zavadsky and Shemer 2017). The earlier work was well described by Tsahalis (1979), and an extended list of recent work was reported by Bremer and Breivik (2017). In addition, extensive theoretical work has addressed numerous aspects of the dynamics of currents induced by waves in water, namely, the generation of current vorticity due to viscosity, particle trajectories in water, turbulence induced by waves, the oscillation of boundary layers, the occurrence of return currents in a tank, the origin of Langmuir circulations, and so on. Most of these topics have been well described in certain papers (e.g., Longuet-Higgins 1953; Craik 1982; Teixeira and Belcher 2002; Babanin and Haus 2009; Teixeira 2018), surveys (e.g., Sullivan and McWilliams 2010; Bremer and Breivik 2017), and monographs (e.g., Phillips 1977; Janssen 2004; Bühler 2014). However, in this paper, we consider measurements of time-averaged surface currents induced by waves in water. The focus of the study is on the experimental aspects of the surface current problem rather than particle trajectories near the surface, turbulence generation by waves, or the depth profile of induced currents.

Despite the evident simplicity of surface drift current measurements, accurate measurements of a wavy water surface can have many sources of error. Field measurements (e.g., Churchill and Csanady 1983; Babanin 1988; Malinovsky et al. 2007; Kudryavtsev et al. 2008) are influenced by the nonstationarity of the wind field, uncontrolled background currents U_b, the stratification of air and water in situ, numerous technical difficulties in field measurements, and other factors. In laboratory (tank) measurements (Toba 1973; Wu 1975; Tsahalis 1979; Longo 2012; Longo et al. 2012; Paprota et al. 2016; Zavadsky and Shemer 2017), the sources of errors include the influence of the upper and lateral boundaries of the tank, presence of return currents, short wind and wave fetches, and difficulty of placing equipment in a tank. Some specific problems related to surface current measurements will be discussed in this paper.

All these limitations influence the wind and current profiles and establish dependencies of the surface drift on the wind and wave parameters of the air–water interface. Besides, small tank dimensions limit the range of wave state parameters, thereby decreasing the ability to identify the abovementioned dependencies. However, tank measurements have the evident advantage that all parameters in the air–water system can be completely controlled during an experiment. Therefore, the most exact and reliable results for surface drift currents have been obtained in laboratory measurements (Wu 1975, 1983; Longo 2012; Longo et al. 2012; Zavadsky and Shemer 2017). Nevertheless, many features of the drift current phenomenon are still not completely understood. In this paper, some of these issues will be addressed based on our own tank measurements.

b. Specification of the tasks

For completeness, consider two principally different cases: (i) mechanical waves and (ii) wind waves in water.

Regarding mechanical waves, one may expect that the surface drift current U_d should be of the same order as the Stokes mass-transport current U_St if the ideal fluid assumptions are applicable (Stokes 1847). In the potential approximation and the case of a monochromatic nonlinear gravity wave with amplitude a, frequency ω, and wavenumber k = ω²/g (g is the gravity acceleration), the theory predicts the ratio (Stokes 1847; Phillips 1977)

where the first equality corresponds to the scaling of U_St by the phase velocity, ω/k, and the second equality corresponds to the scaling of U_St by the orbital velocity, ωa. The factor ka is the steepness of the wave (typically ~0.1), which determines the scale of the Stokes drift with respect to the mentioned velocities. For the wave spectrum S(ω), Eq. (1) for U_St can be generalized. In the case of quasi-monochromatic (narrowband) waves, Eq. (1) can be replaced by the following formula:

where ω_p and k_p are the spectral peak frequency and the peak wavenumber, respectively. In the case of mechanical waves with wideband spectra, such as Pierson–Moscowitz (PM) or Joint North Sea Wave Observation Project (JONSWAP; JW) spectra (Komen et al. 1994), the Stokes drift can be estimated by the following formula (Churchill and Csanady 1983):

Here, [ω_min, ω_max] is the available frequency band of the wave spectrum S(ω), and z is the vertical coordinate directed positively upward. Equation (3) allows the Stokes drift for a given spectrum S(ω) to be estimated at both the surface and at any depth z; besides, it shows that the Stokes drift decays exponentially with depth z. Note that ratios (2) and (3) are also used for the estimation of U_St(0) in the presence of wind waves (Wu 1983; Toba 1973; Clarke and Gorder 2018).

Unfortunately, direct measurements of real surface drift currents U_d, induced by mechanical waves, are rather limited (according to Bremer and Breivik 2017). For example, Craik (1982) described the depth profile formation of currents induced by different kinds of waves in viscous and contaminated water. Herterich and Hasselmann (1982) provided results for the diffusion of tracers by surface waves. Monismith et al. (2007) presented results for return current formation in a water body of finite depth. Paprota et al. (2016) and Breivik et al. (2014) measured Stokes drift profiles. In all these works, the authors noted a discrepancy between the measured currents U_St(z) and the theoretical results (e.g., Longuet-Higgins 1953; Craik 1982; Constantin 2006). However, none of the researchers did quantify the measured surface drift current U_d in comparison with the theoretical value U_St(0). This relation needs further specification, and this issue will be addressed in this paper based on tank measurements.

In the case of wind waves, the total surface drift (in the absence of background currents) includes two terms (the case of one-dimensional tank measurements is assumed):

where U_dw is the wind-induced part of the surface drift current U_d. To estimate U_dw, the Stokes drift needs to be excluded from U_d in Eq. (4); to this aim, the values of U_St(0) are often estimated using Eq. (2). Based on such estimates, previous studies (Wu 1975, 1983; Churchill and Csanady 1983; Longo 2012; Longo et al. 2012; Zavadsky and Shemer 2017) showed that the Stokes drift at the surface U_St(0) has a typical empirical value of approximately 10%–15% of the wind-induced drift U_dw. This indicates that U_St should be taken into account in measurements and practical tasks, especially in the presence of intensive long waves that have high phase velocities (e.g., swell). Moreover, in the field measurements by Clarke and Gorder (2018), the Stokes drift was considered as the total surface drift current, with no attention given to the wind-induced part U_dw.

In practice, the wind-induced drift currents U_dw have values on the order of the friction velocity in air $u_{*a}$. With no swell, such currents can be described by the simple ratio (Wu 1975; Longo et al. 2012; Zavadsky and Shemer 2017)

where coefficient c_d varies in the range from 0.3 in the laboratory measurements (Zavadsky and Shemer 2017) to 0.53 (Wu 1975), and from 0.2 (Babanin 1988) to 1.5 (Tsahalis 1979) in the field measurements. In all cases, no authors considered the dependence of U_dw on specific wave parameters. In the present paper, we address this relation in detail.

Until recently, it was unclear why the surface drift current U_dw has values on the order of the friction velocity in air $u_{*a}$ rather than the friction velocity in water, $u_{*w}\approx 0.03u_{*a}$, as should follow from the condition of momentum-flux continuity at the interface. Second, it was unknown why the empirical ratio $U_d/u_{*a}$ does not manifest any dependence on wave parameters. Despite of numerous theoretical work in this field (e.g., Huang 1979; Kitaigorodskii et al. 1983; Lumley and Terray 1983; Anis and Moum 1995; Mellor 2001; Rascle and Ardhuin 2009; Chalikov and Rainchik 2011; Babanin and Chalikov 2012; Benilov 2012; Teixeira 2018, and references therein), the answer to these questions was found only in the recent paper by Polnikov (2019).

In Polnikov (2019), a simple semi-phenomenological model was constructed based on the theoretical concept of the three-layer structure of the air–water interface (Polnikov 2011). Processing the numerical simulation data obtained by Chalikov and Rainchik (2011) and Polnikov (2011) showed that the profile of the total drift current, U_d(z), corresponding to Eq. (4), is linearly related to depth z in the so-called “wave zone,” where the air and the water alternatively exist. This finding was empirically confirmed by the measurements of Longo (2012) and Longo et al. (2012) (see details in Polnikov 2019). This means that the wave zone plays the role of a friction layer between the air and the water layers.

Polnikov (2019) proposed that in the wave zone the following ratio is valid

where ${\tau }_{tw}/{\rho }_w\simeq 0.8u_{*a}^2$ is the vertical wind-induced momentum flux in the water, normalized by the water density ρ_w and related to drift current production; U_dw(z) is the wind-induced drift; K_t = f_t(.)U_dh is the effective eddy viscosity in the water part of the wave zone, where h is the wave height and f_t(.) is a dimensionless fitting function that depends on surface wave parameters [the symbol (.) indicates the possible dependence of function f on the parameters of the interface]. As the values τ_tw and ∂U_dw(z)/∂z are constant in the wave zone, Eq. (6) yields

where C_p is a constant of the order of air-to-water density ratio ρ_a/ρ_w. Thus, the empirical ratio (5) describing the relation between U_dw and $u_{*a}$ is theoretically justified. If the values of f_t(.) vary within the range of 0.01–0.03, ratio (7) quantitatively coincides with ratio (5) [note that in the above consideration, the smallness of U_St(0) with respect to U_dw is taken into account].

One may expect that the wind-induced drift velocity U_dw is dependent on the following dimensionless parameters of the wave state: the steepness σ = ak_p, the wave age $A=g/\left({\omega }_p{u}_{*}\right)$, and the breaking intensity Br. It was suggested in Polnikov (2019) that this dependence could be “hidden” in the fitting function f_t(.). However, the function f_t(.) cannot be theoretically derived because it has a turbulent (statistical) nature. Therefore, the proposed assumption must be empirically verified.

Based on our own direct drift current measurements obtained with surface floats in a tank, the above dependencies were successfully revealed. In parallel, we quantified the ratio between the measured surface drift currents U_d, induced by mechanical waves in water, and the expected Stokes drift U_St(0), provided by Eq. (3). Obtaining a solution for the first task is the main objective of our experiment, while the second task is an auxiliary consideration following from the limitations of the studies listed above. On this way, we encountered some methodological difficulties related to measuring the surface drift currents with floats, and these issues are worthy of discussion.

The layout of the paper is as follows. Section 2 describes the equipment and measurement methods used. The results and analysis for mechanical waves and wind waves are presented in sections 3 and 4, respectively. Section 5 contains a general discussion, and the conclusions are given in section 6.

a. Results for regular waves

The results are summarized in Table 1. First, we should note a reasonable (up to 30%) difference between the estimations of the surface Stokes drift U_St found with Eqs. (2) and (3) (hereafter, the notation of the surface level, z = 0, is omitted for simplicity). An example of the relation between these estimates of U_St for regular waves is shown in Fig. 3a, and an example of the spectrum for narrowband waves, used to get the above estimations, is shown in Fig. 2a. Importance of using Eq. (3) is evident. Notably, in the integral of Eq. (3), we restricted the upper boundary at f_max = 15 Hz due to the accuracy restrictions of the WGs with a 1-mm wave height. The difference between the mentioned estimates suggests that Eq. (3) is the more accurate equation. Hereafter, only such estimations of U_St are presented.

Table 1.

Parameters of waves and surface currents for regular waves. Note that the two values for the Stokes drift, separated by a slash, correspond to the estimations with Eqs. (2) and (3), respectively; “n-dim” indicates nondimensional.

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Relation between the Stokes drift estimations with Eq. (2), Ust2, and with Eq. (3), Ust3. (a) Regular mechanical waves (Table 1): the solid trend line (y₁ = 1.30x₁) corresponds to f_p = 1.5 Hz; the dashed trend line (y₂ = 1.27x₂) corresponds to f_p = 1.0 Hz; the dotted trend line (y₃ = 1.18x₃) corresponds to f_p = 0.7 Hz. (b) Irregular waves with PM spectra (Table 2): the solid trend line (y₁ = 1.98x₁) corresponds to f_p = 1.5 Hz; the dashed trend line (y₂ = 2.73x₂) corresponds to f_p = 1.0 Hz.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0009.1

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Relation between the Stokes drift estimations with Eq. (2), Ust2, and with Eq. (3), Ust3. (a) Regular mechanical waves (Table 1): the solid trend line (y₁ = 1.30x₁) corresponds to f_p = 1.5 Hz; the dashed trend line (y₂ = 1.27x₂) corresponds to f_p = 1.0 Hz; the dotted trend line (y₃ = 1.18x₃) corresponds to f_p = 0.7 Hz. (b) Irregular waves with PM spectra (Table 2): the solid trend line (y₁ = 1.98x₁) corresponds to f_p = 1.5 Hz; the dashed trend line (y₂ = 2.73x₂) corresponds to f_p = 1.0 Hz.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0009.1

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Relation between the Stokes drift estimations with Eq. (2), Ust2, and with Eq. (3), Ust3. (a) Regular mechanical waves (Table 1): the solid trend line (y₁ = 1.30x₁) corresponds to f_p = 1.5 Hz; the dashed trend line (y₂ = 1.27x₂) corresponds to f_p = 1.0 Hz; the dotted trend line (y₃ = 1.18x₃) corresponds to f_p = 0.7 Hz. (b) Irregular waves with PM spectra (Table 2): the solid trend line (y₁ = 1.98x₁) corresponds to f_p = 1.5 Hz; the dashed trend line (y₂ = 2.73x₂) corresponds to f_p = 1.0 Hz.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0009.1

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The second peculiarity of the measurements is that the ratio U_d/U_St varies in the range of 0.5–0.8 (the second-to-last column in Table 1), reflecting a permanent underestimation of the expected surface Stokes drift. Specifically, there is no evident dependence of this ratio on the intensity of wave breaking (presented in the rightmost column in Table 1). Following the review by Bremer and Breivik (2017), the estimates of U_d/U_St for the tank measurements are pioneering, despite numerous empirical reports of the abovementioned discrepancies between measurements and theoretical values. A more detailed analysis will be made in section 3c.

b. Results for irregular waves

These results are summarized in Table 2. In this case, the peculiarities are similar to those in the previous case. The role of spectral shape is well shown in Figs. 2a,b and 3a,b. From these figures, one can see that for a wideband spectrum, the estimates of U_st based on Eqs. (2) and (3) can be more than 2 times different.

Table 2.

Parameters of waves and surface currents for irregular waves; “n-dim” indicates nondimensional.

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The additional difference in the surface drift for irregular waves is a slightly greater ratio U_d/U_St varying from 0.6 to 0.9. In this case, the integral in Eq. (3) should be greater due to the consideration of a wider spectral band than in the previous case (Fig. 2b); therefore, the only explanation for the higher values of U_d/U_St is accounting for the small steepness Ϭ of irregular waves (see Table 2). Theoretically, it is reasonable to assume that for smaller wave steepness values, the observations of U_d should be closer to the theoretical values (Stokes 1847).

c. Combined analysis of mechanical waves

The presented results are novel, despite numerous empirical reports on the discrepancies between measurements and theoretical values of Stokes drift [e.g., Craik (1982), Breivik et al. (2014), Paprota et al. (2016), and so on, to the review by Bremer and Breivik (2017)]. Notably, for the first time, we obtained empirically quantified ratios U_d/U_St for regular and irregular mechanical waves in a tank. These ratios are interesting from three points of view: (i) understanding the reasons for the empirical underestimation of the surface drift currents, (ii) methodical points of measurements, and (iii) identifying possible applications of these results.

Underestimating U_d is not new; nevertheless, this issue requires separate consideration. The following reasons for the empirical underestimation of surface drift currents in experiments are well known: (i) the viscosity of the water (which disrupts the potentiality of wave motion), (ii) return currents, (iii) water surface contamination, and (iv) the sailing of floats, including dynamic-pressure braking (McNally and White 1985). The first three reasons were discussed in previous studies (e.g., Longuet-Higgins 1953; Craik 1982; Herterich and Hasselmann 1982; see also Breivik et al. 2014; Paprota et al. 2016). We cannot add more to this discussion, noting that the first two reasons are inevitable and cannot be regulated; however, the third influence can be minimized by cleaning the water surface after each change in frequency, thus the third item should be excluded from the list of reasons discussed. Therefore, we do not dwell on the first three reasons, but the phenomenon of the float sailing is worthy of discussion.

Besides the return current influence (Monismith et al. 2007), which cannot be quantified in our case, in our mind, the sailing of floats on the water surface is the main reason for the abovementioned discrepancy between the experimental and theoretical values of the Stokes drift. To validate this assumption, we tried to regulate the effect of sailing by choosing different sizes of floats. Indeed, when we used plastic balls with a diameter of 5 mm as the floats, the drift current values were fixed at values in 20%–30% smaller than those for floats that were 3 mm in diameter. Thereby, the assumption stated above is justified.

The nature of the sailing process is relatively evident. In the case of mechanical waves and spherical floats, the sailing has two negative constituents, namely, the air and the water components. The air component is associated with the air-induced resistance to float propulsion due to the viscosity and negative dynamic pressure of the ambient motionless air. On the other hand, the water sailing is influenced by the same manner due to the rapid decrease in the total drift current with depth, described by profile U_d(z). For mechanical waves, this decrease follows from Eq. (3). In the case of wind waves, the rapid decrease in the wind-induced current U_dw(z) has been empirically shown by Longo et al. (2012) and described in Polnikov (2011, 2019).

These considerations account for mismatch between theoretical and experimental values, and one may go on to methodical points of the surface-drift current measurements.

From a methodological perspective, a number of previous experiments (e.g., Wu 1975, 1983; Zavadsky and Shemer 2017), in which the volumetric surface floats were used, have suffered from the abovementioned problems, resulting in the underestimation of surface drift in these experiments. Therefore, it is important to make certain methodological changes when obtaining future measurements. To avoid the sailing of floats, it is reasonable to use floats that are fully flat. For example, floats could be pieces of thin flexible tissue (similar to gauze) with a net density less than that of water. Such experiments have been performed in the past (see references in Tsahalis 1979), and they should be planned in the future to further improvement the results presented above.

Another alternative approach involves using PIV methods (e.g., Longo et al. 2012; Paprota et al. 2016) or the laser-warming method (thermal-marking velocimetry) (Savelyev et al. 2012) to measure surface drift currents. However, due to technical reasons related to the optical equipment location, such methods are mainly appropriate for small tanks. But in such cases, the tank boundaries could result in additional sources of errors. Another problem associated with using the optical methods to obtain current measurements in tanks is the restriction to equipment location and movement along a tank. Ideally, technical progress will help overcome the mentioned methodological problems.

In practical applications, the obtained results can be used as the first guess for expected drift current estimates in the engineering of harbors and other marine infrastructure, taking in mind the influence by swell. On other hand, from the physical point of view, the presented results are useful for accurately estimating the observed wind-induced drift currents U_dw, based on Eq. (4). It is this idea (rather than simple estimating ratio U_d/U_St) was stimulating our measurements of surface drift currents U_d for mechanical waves, the results of which will be actively used below in the data analysis for wind waves.

a. Analysis of the accuracy of the results

First, let us estimate the accuracy of the results starting with the wind parameter $u_{*}$. The friction velocity $u_{*}$ is usually obtained from Pitot tube measurements based on the traditional assumption of a logarithmic wind profile (Wu 1975; Longo 2012; Zavadsky and Shemer 2017). In our case, only three upper tubes of the PT set provide relatively accurate estimations of $u_{*}$. The mean error of these estimations, obtained using Microsoft Excel, is 5%–7%. It is important to note that the values of $u_{*}$ (as well as U_d) reasonably increase with the fetch (in 1.5–2 times), what is typical for the tank measurements (Toba 1973; Longo 2012; Longo et al. 2012). This fact should be taken into account to obtain accurate values of $U_{dw}/u_{*}$ based the data obtained in different parts of a long tank.

The wave parameters ω_p = 2πf_p, Ϭ, and A were estimated with very high accuracy (errors less than 1%) from the wave spectra calculated within the MATLAB software (see section 2). For the completeness of the wind-wave spectra description, their spatial evolution is presented in Fig. 4. Due to a wide spectral band for wind waves, using Eq. (3) is quite important for estimation of theoretical value for the Stokes drift. In our case, these values have errors of approximately 5%–7%, what is defined mainly by the value of the upper limit of the integral in Eq. (3). It was chosen to be 15 Hz due to the technical restrictions of the WGs used. As shown in Fig. 4, the wave spectra are radically changing along the tank (this feature was ignored for mechanical waves), therefore the theoretical estimates of U_St are also varying, which plays a crucial role in the correct estimating the wind-induced drift currents U_dw based on using Eq. (3).

Wind-wave spectra at points P1, P2, and P4 for the run with W = 8 m s⁻¹. The straight line with digits shows the frequency-decay law of spectra in the tail range (the error is ~ 3%).

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0009.1

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Wind-wave spectra at points P1, P2, and P4 for the run with W = 8 m s⁻¹. The straight line with digits shows the frequency-decay law of spectra in the tail range (the error is ~ 3%).

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0009.1

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Wind-wave spectra at points P1, P2, and P4 for the run with W = 8 m s⁻¹. The straight line with digits shows the frequency-decay law of spectra in the tail range (the error is ~ 3%).

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0009.1

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The visually obtained float movement timing resulted in the error on the order of 10% for the measured total mean drift U_d. In turn, the intensity of wave breaking was estimated (by two persons in parallel) with the same error. These values influence the accuracy of the empirical results for the wind-induced drift currents U_dw.

b. Analysis of the main results

In the case of wind waves, we estimated the wind-induced drift currents as the difference between the measured value U_d and the theoretically estimated value U_St by assuming that the positive floats sailing in the air (due to the wind) is compensated for by the negative floats sailing in the water. This assumption was verified by the measurements with plastic balls of different diameters (3 and 5 mm) used as the floats. This means that the values of U_dw presented in Table 3 are the smallest possible values, taking in mind the results for mechanical waves discussed above in section 3a.

Based on Table 3, the determined empirical ratio $U_{dw}/u_{*}$ varies in the wide range of 0.65–1.2, which corresponds to the range presented in the literature (see section 1b). This peculiarity of our results will be separately discussed below in section 5, but here, the main attention is paid on identifying the proper parameterization for the dependence of ratio $U_{dw}/u_{*}$ on the parameters of the system under consideration. The theoretical basis for such parameterization is the following.

According to the theoretical model of the wind-induced drift currents proposed in a previous study (Polnikov 2019), one may expect $U_{dw}/u_{*}$ to be dependent on the following wave parameters: steepness σ = ak_p, wave age $A=g/\left({\omega }_p{u}_{*}\right)$, and breaking intensity (Br). After several attempts (see the assumption for the general approximation below), we obtained the following relation:

In a graphic form, the relation between the empirical and parametrical values of $U_{dw}/u_{*}$ is shown in Fig. 5. From Table 3 and Fig. 5, it is seen that parameterization (8) is valid with a mean error of approximately 10%, which is very good for experimental data.

Correspondence between empirical and parametrical values of ratio $U_{dw}/u_{*}$ estimated with Eq. (8) for wind waves. The trend line is described by the equation y = 1.03x with a mean accuracy of 10%.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0009.1

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Correspondence between empirical and parametrical values of ratio $U_{dw}/u_{*}$ estimated with Eq. (8) for wind waves. The trend line is described by the equation y = 1.03x with a mean accuracy of 10%.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0009.1

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Correspondence between empirical and parametrical values of ratio $U_{dw}/u_{*}$ estimated with Eq. (8) for wind waves. The trend line is described by the equation y = 1.03x with a mean accuracy of 10%.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0009.1

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Equation (8) provides the dependence relation between the wind-induced surface drift currents and the wind and wave parameters, which is the main aim of this paper. This dependence is established for the first time during a long history of empirical investigations (section 1a). Below, it will be shown that ratio (8) corresponds relatively well to the results published earlier for the tank measurements (Wu 1975; Longo 2012; Longo et al. 2012; Zavadsky and Shemer 2017). In the future, parameterization (8) could be checked for the field data based on accurate measurements of wind and wave parameters.