## 1. Introduction

Flow coupling across the air–water interface in oceanic regions takes place within boundary layers where various properties adjust, over a relatively small fraction of the depth of the atmosphere and ocean, between their values in the interior of each fluid. The atmospheric and oceanic surface layers are the sublayers of these boundary layers located nearest to the air–water interface, occupying about 10% of their depth, which have a decisive importance in mediating the turbulent fluxes of momentum, heat, and gases between the atmosphere and the ocean (Csanady 2004), and where these fluxes are approximately constant. Hereafter, “surface layer” will always be used with this meaning, although the term is often adopted in an oceanographic context to denote the whole oceanic boundary layer.

Whereas the atmospheric surface layer over land has a no-slip bottom boundary condition applied at the ground, the atmospheric and oceanic surface layers in ocean regions are characterized by continuity of velocity and stress at the mobile air–water interface that separates them. This, on the one hand, leads to the generation of a wind-induced current in the oceanic surface layer, and on the other hand allows the generation of surface waves at the air–water interface. Both of these aspects considerably complicate the physics of these surface layers, especially the oceanic one, as is widely recognized (Thorpe 2005) and will be further discussed here.

Nevertheless, the oceanic surface layer is still largely understood and modeled based on the transposition to the ocean of theories developed for the atmospheric surface layer over land, where the effects of surface waves are not represented (Kraus and Businger 1994). Deficiencies in this approach become apparent when one realizes that key parameters in surface layer theory, such as the friction velocity *z*_{0}, are deemed to take values in the ocean that seem to be inconsistent with the values of the shear stress and the geometric properties of the air–water interface, respectively.

Standard surface layer theory is based on Monin–Obukhov scaling, which in the limit of neutral stratification reduces to a theory for the logarithmic mean wind profile. In the ocean, or in underwater flows measured in the laboratory, such a theory has been applied, with varying degrees of success, to model the mean current induced by the wind. However, it has often been detected that the value of *z*_{0} obtained by extrapolating the logarithmic current profile up to the surface is often much larger than would be expected based on the size of the surface corrugations deforming the air–water interface, and exceeds by several orders of magnitude the air-side value of *z*_{0} (Csanady 1984; Burchard 2001; Soloviev and Lukas 2003; Sullivan et al. 2004; Kudryavtsev et al. 2008).

There is some awareness that the first aspect is due to the fact that a fraction of the surface stress is carried by surface waves, and therefore does not support as much shear as if the waves were absent. On the other hand, the increased values of *z*_{0} have been attributed to the effect of surface waves as roughness elements seen from below, or to wave breaking, but the exact mechanism by which this enhancement arises remains rather mysterious. The huge disparity between the estimated values of *z*_{0} as seen from the air side or from the water side of the air–water interface is especially puzzling, since the amplitude of the corrugations is the same. Even if the flow on both sides of the air–water interface could be assumed to be aerodynamically smooth, the differences in the value of

Craig and Banner (1994) and Craig (1996) developed a model of the oceanic surface layer that produces profiles of the mean current and of the associated dissipation rate of turbulent kinetic energy (TKE), which showed some success in predicting both quantities, and was subsequently adapted and used by a number of researchers (e.g., Drennan et al. 1996; Terray et al. 1999; Gemmrich and Farmer 1999; Burchard 2001; Rascle et al. 2006; Feddersen et al. 2007; Rascle and Ardhuin 2009; Gerbi et al. 2009; Kukulka and Harcourt 2017). That model is based on an approximate balance between the turbulent fluxes of TKE and dissipation and produces a substantial surface dissipation enhancement, which is consistent with the observations of Gargett (1989), Agrawal et al. (1992), Terray et al. (1996), and Drennan et al. (1996). However, it requires adjusting *z*_{0} for each dataset, yielding values of this quantity on the order of the height or wavelength of the surface waves, which is much larger than estimated for an aerodynamically smooth boundary, or from the Charnock relation. Both Craig and Banner (1994) themselves and, more recently, Grant and Belcher (2009) recognized that this need to adjust *z*_{0} in order to fit measurements is a weakness of the model.

More recently, Kudryavtsev et al. (2008) developed a rather elaborate model based on the momentum and TKE budgets and assuming a balance between turbulence production by wave breaking and dissipation. This model avoids the strong dependence on *z*_{0} displayed by the model of Craig and Banner (1994), but contains many ad hoc assumptions and approximations (e.g., the parameterization of the TKE production by wave breaking, or the mixing length definition), and nevertheless is so complicated that the corresponding equations can only be solved numerically. Although it predicts satisfactorily the qualitative behavior of the mean current profiles measured in the laboratory experiments of Cheung and Street (1988) and the aforementioned surface dissipation enhancement, it produces dissipation profiles that look somewhat artificial and seem to underestimate most datasets at small depths (see their Fig. 7). Although this model succeeds in predicting the enhanced values of the apparent *z*_{0} in the experiments of Cheung and Street (1988), it does not explain the reduced values of

In this study a very simple model is developed, based on the partition of the shear stress in the surface layer between shear-related and wave-related parts, that reconciles all these results, explaining in particular the discrepancies between expected and observed values of *z*_{0} in the oceanic surface layer, purely due to the effect of nonbreaking waves (unlike Kudryavtsev et al. 2008). The model draws heavily on that developed by Teixeira (2012), which is inspired by rapid distortion theory (RDT) calculations, and is essentially analytical, being much simpler than the one proposed by Kudryavtsev et al. (2008), but is able to produce more accurate results. It has the advantage of being formulated as a variant of Monin–Obukhov scaling, where instead of the Obukhov stability parameter, the key dimensionless parameters account for the effects of surface waves. These parameters are the well-known turbulent Langmuir number La_{t} and (as in Monin–Obukhov theory) a dimensionless depth, here normalized by the wavenumber of the dominant surface waves. An extended version of this model was shown by Teixeira (2012) to give good predictions of the dissipation rate by comparison with field data from various sources (Terray et al. 1996; Drennan et al. 1996; Burchard 2001; Feddersen et al. 2007; Jones and Monismith 2008; Gerbi et al. 2009). The model is tested here by comparison with the data of Cheung and Street (1988), showing good agreement, despite the fact that [unlike the model of Kudryavtsev et al. (2008)] it uses a monochromatic wave approximation and neglects the viscous boundary layer.

This paper is organized as follows: section 2 presents the proposed model, including its version for a vertically uniform shear stress and its extension for a shear stress that decreases linearly with depth. Section 3 contains the results, starting with tests to the model as a function of its input parameters, and proceeding with its comparison with laboratory data. Finally, in section 4, the main conclusions of this study are summarized.

## 2. Theoretical model

It will be assumed that the rotation of Earth and stratification of the water in the oceanic surface layer can be neglected. The first assumption is generally acceptable in the surface layer, where the flow is by definition dominated by turbulent fluxes (and throughout the whole oceanic boundary layer in equatorial regions, where the Coriolis parameter is zero). The second assumption is acceptable if some other dynamical process (in the present case, the effect of surface waves) is stronger than that of buoyancy. The effect of breaking surface waves will also be neglected. This is a working hypothesis, which is not as justifiable as the previous two, but was shown to be a plausible approximation given the level of agreement achieved between the model of Teixeira (2012) and dissipation data (for further details concerning its motivation, see that paper).

The water-side friction velocity *z*_{0} will be specified according to their most fundamental definitions: as the square root of the surface value of the kinematic shear stress and as the depth at which the current velocity relative to its surface value is zero (without assuming a displacement height), respectively, rather than based on the slope and intercept of the current profiles (which would be misleading in the present context).

The point of departure for the model is that turbulence in the surface layer is dominated by the transfer of kinetic energy from the mean wind-driven current and the Stokes drift of surface waves to the turbulence, via the shear production and Stokes drift production terms in the TKE budget (see, e.g., McWilliams et al. 1997), which are assumed to be balanced locally by the dissipation rate, as in Teixeira (2011b, 2012). This balance, although of questionable accuracy, has been motivated in Teixeira (2012) by the TKE budgets presented in the large-eddy simulation (LES) studies of Polton and Belcher (2007), Grant and Belcher (2009), and Kukulka et al. (2010) (which did not account for the effects of wave breaking). More recent supporting evidence for this balance is provided by Van Roekel et al. (2012) and Kukulka and Harcourt (2017).

*x*direction)where

*ρ*is the density. It will be assumed hereafter that

### a. Scaling of the oceanic surface layer

*a*

_{w}, wavenumber

*k*

_{w}, and angular frequency

*σ*

_{w}at a depth

*z*is given by (Phillips 1977)and, to a first approximation, in the surface layer the mean current shear satisfieswhere

*κ*is the von Kármán constant. To evaluate the relative importance of the Stokes drift strain rate and mean shear rate of the current, the ratio of (2) and (3) may be taken at a representative depth where the flow is affected by surface waves, say

*R*attains its maximum (cf. Teixeira and Belcher 2010; Teixeira 2011a).

Consider first the magnitude of *R* in the atmosphere. Although one does not often think about Stokes drift in the atmosphere, its magnitude is similar to that in the ocean, since the wave orbital motions (usually immersed in a tangle of turbulent eddies) are likewise of similar magnitude. Parameter *k*_{w} ≈ 0.06–6.3 m^{−1}, and using the linear dispersion relation of deep-water gravity waves, *c*_{w} ≈ 1.25–12.5 m s^{−1}, with the limits swapped relative to those of *k*_{w}. Taking a typical value of the friction velocity in the atmosphere,

For the oceanic surface layer, although the same estimates for the wave characteristics may be used, it must be noted that, to a first approximation, the shear stress *τ* is continuous across the air–water interface in steady flow, and since by definition *R* substantially higher than 1. In addition, it is quite possible that *R*. This means that in the ocean it is unacceptable to ignore the effect of the Stokes drift of surface waves, and this difference is what gives oceanic turbulence its distinctive character, as shown by McWilliams et al. (1997) using LES and Teixeira and Belcher (2002, 2010) and Teixeira (2011a) using RDT.

### b. Shear stress partition

_{t}, (6) may be alternatively expressed aswhere it has been noted that in the surface layer the shear stress

*γ*is an adjustable (positive) coefficient. The calibration of this coefficient may be exploited to account for extraneous effects, such as the possibility that the waves are nonmonochromatic, and the fact that the current profile is not perfectly logarithmic. Assuming that

*γ*is constant with depth is likely to be less accurate outside the surface layer, because the above assumptions about the behavior of

Note that (8) has the properties of approaching the usual definition of the total shear stress as either *γ* possibly not being treated as a constant.

_{t}is low. This is in agreement with LES results by, for example, McWilliams et al. (1997), Li et al. (2005), and Grant and Belcher (2009) showing that shear in the current profile decreases markedly for a constant wind stress

*τ*as La

_{t}decreases (see section 3). One advantage of (9) is that it allows the definition of friction velocities due to shear and due to the wave that are additive, yielding the sum

### c. A model for the current profile

*ε*= 2 from (8), but will hereafter be kept as an adjustable parameter for maximum generality. As for

*γ*, the adjustment of

*ε*may be exploited to account for various extraneous effects, such as the presence of nonmonochromatic waves. The connection with this latter aspect is even closer, since

*ε*controls the vertical penetration of wave effects, which may depend not only on the dominant wavelength, but also on the wave energy distribution by scale.

Note that *z* is, arguably, the simplest possible that benefits from these properties. The partition of the shear stress into shear-induced and wave-induced components, conjugated with the use of a first-order turbulence closure in (10), parallels the approach, used in a numerical modeling context, of Harcourt (2013). However, the partition itself was originally suggested by Teixeira (2011a) based on the shear stress budget in (5) and used in the present form by Teixeira (2012).

*z*. This yieldsIf velocities are normalized by

*k*

_{w}, (14) may be rewritten

*ν*is the kinematic viscosity of water. Using these definitions, (15) can be expressed asThe advantage of expressing the lower limit of integration in this form is that for aerodynamically smooth flow,

*z*

_{0}.

*κ*= 0.4.

*z*

_{0}is the height at which

*z*

_{0w}is much larger than the true

*z*

_{0}when the effect of waves is important, because of the break point (or more precisely, transition region) existing in the current profile at |

*z*| ≈ 1/(

*ε*

*k*

_{w}). Variable

*z*

_{0w}can be obtained by integrating (11) between

*z*

_{0}and ∞ and then (3) back to

*z*

_{0w}. This yieldsEquations (9), (17), and (21) form the basis of the calculations presented in this paper.

It is worth noting that the formulation of the shear stress on which these equations are based, (10), is strictly local, neglecting any transport effects, whereby

#### Model for a linearly decreasing shear stress

*δ*is the depth where

Note that, according to (10) and (22), for *U* varies indefinitely). Defining arbitrarily

As a caution, it should be emphasized that the assumption of a nonconstant shear stress, expressed by (22), may not be strictly consistent with statistically steady and horizontally homogeneous flow (implicit in surface layer theory), requiring either a time evolution of the mean current or a mean horizontal pressure gradient, but hopefully this assumption is still acceptable for the present purposes. A model with a linearly decreasing shear stress, such as the one just presented, might be thought of as a very simple representation of the whole oceanic boundary layer (of depth *δ*) instead of just the surface layer. However, its applicability to real cases is limited by neglect of the effect of Earth’s rotation, the choices made to approximate (7) as (8), and the Monin–Obukhov approach inherent to (11) and (12). These are confined to the surface layer, and would require modification in order to extend the model.

## 3. Results

It is instructive first of all to explore the model behavior for a few representative cases, because this illustrates in the “cleanest” possible way the range of behavior of the model and its impact on the perceived values of the water-side values of *z*_{0}. More detailed comparisons with laboratory experiments follow. In all of these cases, *γ* and *ε* will be treated as adjustable parameters.

### a. Generic behavior of the model

Figure 1 shows profiles of _{t} = 0.5, 1, 2, assuming that *γ* = 1 and *ε* = 1, for simplicity. Note that these values of *γ* and *ε* are of the same order of magnitude as those adopted by Teixeira (2012). The results are not qualitatively very sensitive to

The value La_{t} = 2 intends to represent shear-dominated turbulence, La_{t} = 0.5 refers to Langmuir (i.e., wave dominated) turbulence, and La_{t} = 1 to turbulence with a transitional character. As can be seen in Fig. 1, the current profiles (denoted by the solid curves) have a lower portion with invariant slope for larger depths. This slope, when expressed in terms of _{t} = 2, _{t} = 1, and _{t} = 0.5. On the other hand, if the lower portion of the current profile is prolonged toward the surface (dotted line asymptotes), one obtains an “effective” value of the roughness length, expressed by (21), which would be obtained by ignoring the upper portion of the current profile. For La_{t} = 2, _{t} = 1, _{t} = 0.5, *z*_{0w} may become various orders of magnitude larger than *z*_{0} as La_{t} decreases (see further discussion below).

Note that, according to the present model, if measurements are taken at a range of depths well below the transition region located around *z*_{0} will be strongly overestimated as *z*_{0w}. Conversely, if measurements are taken at a range of depths above this transition region (if that is feasible), *z*_{0} will be correctly diagnosed from the current profile, but *z*_{0} and to an underestimation of

Circumstantial evidence that this is so is provided by the reported need to change (more specifically decrease) the value of the von Kármán constant to achieve an adequate collapse of measured current profiles in wall coordinates (Howe et al. 1982; Cheung and Street 1988; Craig and Banner 1994; Siddiqui and Loewen 2007), unless the friction velocity used to define

Although both a decrease of the friction velocity and an increase of the roughness length, as diagnosed from current profiles, might be expected as a result of vertical mixing of momentum due to wave breaking, the remarkable property of the model proposed here is that this phenomenon arises simply due to the partition of the shear stress imposed by nonbreaking waves, something that can be traced back to the production terms of the shear stress budget in (5), and is thus much easier to pinpoint physically. It is, of course, possible, and even likely, that both processes act in concert when wave breaking does occur, but it is striking that the present mechanism does not require wave breaking.

Figure 2 shows the variation of _{t} for different values of the calibrating constant *γ*, from (9). Unsurprisingly, this ratio takes values that range from ≈1 for large La_{t} to ≪1 for small La_{t}. Clearly, what matters for a correct representation of the variation in between is the value of *γ*, with large values corresponding to strong wave effects and small values to weaker wave effects. This partition of the friction velocity, or between the corresponding shear-induced and wave-induced stresses, is not an often measured or calculated quantity, but Fig. 5 of Bourassa (2000) presents an example with some relevance, even if a quantitative comparison is not easy. If an increase in wind speed is equated with a decrease of La_{t} (an idea that is suggested by the comparisons of the next subsection), and the ratio of the aqueous shear stress to the total atmospheric stress is equated with _{t} to the wind speed at the highest wind speeds, which is corroborated by the comparisons presented in the next subsection. Both results are compatible with the established idea that in well-developed seas in the real ocean, La_{t} becomes largely independent of the wind speed.

Figure 3 presents the variation of *k*_{w}*z*_{0w} and *z*_{0w}/*z*_{0} as a function of La_{t} from (21) for *γ* = 1 and *ε* = 1 (as assumed in Fig. 1) and different values of *k*_{w}*z*_{0}. As expected, *k*_{w}*z*_{0w} approaches *k*_{w}*z*_{0} for large values of La_{t} but tends to a value independent of *k*_{w}*z*_{0} at small La_{t}. What this means is that at low La_{t}, *z*_{0w} scales with *z*_{0}, that is, *z*_{0w} is proportional to the wavelength of the dominant waves, not to any property of small-scale capillary waves (neglected in the model), or to the amplitude of the dominant waves *a*_{w}. This behavior is confirmed by the ratio *z*_{0w}/*z*_{0}, which only approaches 1 for large values of La_{t}, whereas it tends to be very high for small La_{t}. As is consistent with the behavior of *k*_{w}*z*_{0w}, *z*_{0w}/*z*_{0} at low La_{t} is inversely proportional to *k*_{w}*z*_{0}. Since in real situations *k*_{w}*z*_{0} may easily be as small as 10^{−5}, the amplification of the apparent roughness length can be very pronounced. A qualitative comparison with Fig. 3 of Bourassa (2000) is pertinent. Although the dependence of *z*_{0} (which should probably be taken as *z*_{0w} in the present notation) with *z*_{0} [see (26) below] and (21) via the definition of La_{t}, the important point to retain from Fig. 3 of Bourassa (2000) is the enormous amplification of *z*_{0}. Bourassa (2000) notes that *z*_{0} is about 10^{5} larger than expected from Charnock’s relation (and therefore much higher than the values estimated for the true *z*_{0} in the next subsection).

### b. Comparison with Cheung and Street (1988)

Finding adequate datasets to test the present model is challenging, because usually the quantities required as input to the model are not measured. First of all, measuring current profiles in the field with the required accuracy is extremely difficult, hence the most relevant studies typically involve laboratory experiments. Even in those cases, almost invariably not all relevant wave quantities are measured (Bourassa 2000; Siddiqui and Loewen 2007; Longo et al. 2012), and often the shear stress is not measured directly, but rather estimated from the current profiles (Bourassa 2000; Siddiqui and Loewen 2007), which makes comparisons more difficult [the erratic behavior of the current speeds measured by Siddiqui and Loewen (2007) as a function of the wind speed is another reason to exclude their data]. A notable exception is the laboratory experiments of Cheung and Street (1988) of the current beneath surface waves generated by the wind. The relevant quantities are presented in their Table 1. As Kudryavtsev et al. (2008) do for the comparison presented in their Fig. 10, only wind-generated waves are considered here and the case among these waves with the lowest wind speed (where the wave amplitude is so small as to be barely measurable) is ignored.

The experiments with mechanical waves are excluded from this comparison because the assumption of the model that

For a reasonable range of input parameters, the present model predicted almost no difference between the current profiles beneath wind waves for the two lowest wind speeds in Table 1 of Cheung and Street (1988). This justifies (following Kudryavtsev et al. 2008) ignoring the profile for the lowest wind speed, 1.5 m s^{−1}, which has a roughness length smaller than that expected for an aerodynamically smooth flow, and might be affected by some inaccuracy.

#### 1) Unbounded model

The first comparison to be made uses an uncalibrated version of the “unbounded” model described in section 2c. The values of *a*_{w}*k*_{w}*c*_{w} is taken as *a*_{w} here], the angular frequency *σ*_{w} is equated to 2*πf*_{D}, where *f*_{D} is the frequency (in cycles) of the dominant waves, and the corresponding wavenumber is *z*_{0}. As a first approximation the definition valid for aerodynamically smooth flow is adopted: *ν* = 10^{−6} m^{2} s^{−1}. Figure 4 shows a comparison of the model with the data presented in Fig. 1 of Cheung and Street (1988) (excluding the upward-pointing triangles for the reasons explained above), assuming *ε* = 2 and *γ* = 2, as in Teixeira (2012) (Fig. 4a) and using the adjusted values *ε* = 0.5 and *γ* = 0.5 (Fig. 4b).

Parameters of the datasets from Cheung and Street (1988) used here and derived parameters: wind speed, depth of the boundary layer *δ*, wavelength of the dominant waves *λ*_{w}, depth of penetration of the wave stress 1/(*εk*_{w}), surface Stokes drift velocity *U*_{S}(*z* = 0), and turbulent Langmuir number La_{t}. Parameters 1/(*εk*_{w}), *U*_{S}(*z* = 0), and La_{t} were estimated from the dominant wave parameters using a monochromatic wave approximation (see text). Parameter 1/(*εk*_{w}) is estimated for the cases displayed in Figs. 4b and 5a, where *ε* = 0.5 (the lowest value of *ε* considered). For other cases, *ε* must be changed accordingly.

It can be seen in Fig. 4a that the behavior of the measured currents is reasonably well reproduced qualitatively, with a decrease of the overall normalized current speed as the wind speed increases. In terms of the input parameters of the model, this is due to a decrease of the turbulent Langmuir number La_{t} as the wind speed increases for the lowest wind speeds, but mostly due to an increase in penetration of the wave motion at the highest wind speeds, for which La_{t} actually changes very little (see Table 1). Noteworthy disagreements are that the range of variation of the current speed in the model is much too wide compared with the data, in particular, the current speed in wall coordinates is overestimated for the lowest wind speed and quite underestimated for the highest wind speeds. Additionally, although two logarithmic portions of the current profile exist in the model at the highest wind speeds (lowest values of La_{t}), these portions do not coincide with the data that show a reduced slope (e.g., stars and open circles). Finally, the detailed variation with the wind speed is not reproduced. While most of the variation occurs at the lowest wind speeds in the model and weakens roughly monotonically as the wind speed increases, the rate of variation seems to increase again at the highest wind speeds in the data.

When Fig. 4a is compared with Fig. 10 of Kudryavtsev et al. (2008), it may be noticed that the agreement with the data is somewhat less satisfactory. Although the performance of the model of Kudryavtsev et al. (2008) is itself far from perfect, its consideration of the effect of the viscous boundary layer for the current profile with the lowest wind speed substantially improves the agreement at small depths compared with the present model. Additionally, the model of Kudryavtsev et al. (2008) does not underestimate the current as much at the highest wind speeds. Curiously, it has some deficiencies similar to those of the present model, namely, it overestimates the sensitivity of the normalized current to the wind speed at intermediate values of that parameter and, on the contrary, has a too weak dependence for the highest values. On the other hand, the model of Kudryavtsev et al. (2008) is unable to capture the apparent reduction of

Clearly, the comparison presented in Fig. 4a indicates an overestimation of parameter *γ* in the present model. One might wonder why this happens, given that this calibration seemed to work for predictions of the dissipation rate by Teixeira (2012), and also in his preliminary calibration procedure using current profiles from the LES of Li et al. (2005). Possible reasons are speculative, but might have to do with inadvertently accounting for the effect of wave breaking in the first case and adopting a value of *γ* suitable for monochromatic waves in the second, both conditions which are not applicable here. It seems fortuitous that both of these distinct differences should lead to a similar value of *γ*.

To improve agreement with the data of Cheung and Street (1988), *γ* and *ε* may be readjusted. Figure 4b shows a comparison similar to that of Fig. 4a, but where *γ* = 0.5 and *ε* = 0.5 are assumed, presumably to account for both the absence of wave breaking in the experiments of Cheung and Street (1988) and the fact that the waves are nonmonochromatic. The adjusted values of these parameters improve the agreement, particularly for the dataset with the highest wind speed (making it almost perfect by construction), but this turns out not to be sufficient. The variation of the normalized current speed for intermediate wind speeds is still affected by the problems pointed out above.

*c*

_{1}and

*c*

_{2}are coefficients, and the second term is of a form analogous to the Charnock relation but using the friction velocity in the water. In what follows,

*γ*,

*ε*,

*c*

_{1}, and

*c*

_{2}are adjusted to produce the best possible agreement with the data of Cheung and Street (1988). The values found for the unbounded model are

*γ*= 0.25,

*ε*= 0.5,

*c*

_{1}= 0.2, and

*c*

_{2}= 0.9.

Figure 5a shows a comparison of the model with the data of Cheung and Street (1988) using these adjusted parameters. The agreement is much better than in Fig. 4, in particular for the rate of variation of the normalized current profiles at intermediate wind speeds (this is not surprising, being a result of the calibration procedure). Agreement is less close for the lowest wind speed considered at small depths, due to the absence of a viscous boundary layer in the model, but this is a minor limitation. The transition of the datasets from a slope corresponding to ^{−1}. However, at these intermediate wind speeds, the current at the smallest depths covered by the data is somewhat underestimated by the model (the shear suggested by the data at those depths is weaker than expected). Additionally while the current is slightly underestimated for a wind speed of 4.7 m s^{−1}, it is on the contrary slightly overestimated for a wind speed of 3.2 m s^{−1}. It is perhaps risky to attach too much relevance to these discrepancies in detail, given the limited precision of the measurements (which are, nevertheless, among the most precise that could be found).

The value of *γ* was already discussed above. The value of *ε* adopted for this comparison would correspond to the Stokes drift of a monochromatic wave with a wavelength 4 times larger than the wavelength of the dominant waves, obtained from the data. The significance of this mismatch for nonmonochromatic waves (such as the ones under consideration) is not obvious, but indicates a larger depth of penetration of the wave-induced stress than would be expected. The Stokes drift gradient of a wave spectrum is known to be characterized by a larger penetration depth than a monochromatic wave with the same dominant wavelength (Fig. 18 of Li and Garrett 1993), and this may perhaps account for a similar effect on the wave-induced stress.

Concerning parameters estimated for (26), *c*_{1} = 0.2 is substantially larger than the value of 0.11 most commonly accepted for aerodynamically smooth flow. It is worth noting that, in Fig. 10 of Kudryavtsev et al. (2008), the thin line (corresponding to aerodynamically smooth flow) assumes *c*_{2}, the Charnock relation, when expressed in terms of the friction velocity in the airflow, usually has a coefficient of 0.015. Taking into account continuity of the shear stress at the air–water interface, when that relation is expressed in terms of the friction velocity in the water, the coefficient should become 833 × 0.015 = 12.5. This is clearly much larger than *c*_{2} = 0.9 used here, but it should be noted that the Charnock relation, as usually formulated, is valid in the open ocean and for a fully developed wave field, which are very distinct conditions from those produced in the experiments of Cheung and Street (1988). Additionally, continuity of the shear stress at the air–water interface (used in the above calculation) assumes equilibrium, which is not warranted in these experiments either. Nevertheless, a reassuring aspect is that, on dimensional grounds, the quantities on which (26) depends are still likely to be the most relevant.

It might be argued that the agreement between model and measurements in Fig. 5a was artificially improved by allowing *z*_{0} to vary according to (26). To test this, Fig. 5b shows a similar comparison, but where wave effects are ignored altogether, and only the dependence of *z*_{0} on *c*_{1} and *c*_{2}). It is clear that this dependence, by itself, is unable to produce a satisfactory agreement with the measurements, particularly at the highest wind speeds, and naturally does not represent the decrease of the apparent value of *z*_{0} required to match the data. Relatedly, (26) contributes significantly to the weakening of the current speed at the highest wind speeds, which is important to improve agreement with the data relative to Fig. 4.

#### 2) Finite-depth model

Figure 6 shows a similar comparison to Fig. 5, but using the finite-depth model developed at the end of section 2c. Because of the log-linear form of the current profile, the current solutions are no longer composed of straight line segments when using a logarithmic depth scale but tend to have a reduction in shear at the depths near where the shear stress becomes zero (and the current speed stabilizes), marked by the vertical lines in Fig. 6. Below those levels the shear obviously becomes zero, as is denoted by the horizontal lines in Fig. 6. However, some modified form of the current slope transition at depth *γ* = 0.25, *ε* = 1, *c*_{1} = 0.2, and *c*_{2} = 0.9. The agreement between the model and measurements is roughly as satisfactory as in Fig. 5, with essentially the same deficiencies in the midrange of wind speeds. At the largest depths considered (near to *ε* to be employed in Fig. 6.

A noteworthy property of this finite-depth model is that it enables an estimation of the magnitude of the surface current speed

## 4. Concluding remarks

This study presents a simple model for the wind-driven current existing in the oceanic boundary layer in the presence of surface waves generated by the wind. The model sheds light on two puzzling aspects that have been noted repeatedly about these currents, for which a logarithmic profile model, with the friction velocity *z*_{0} as basic parameters, has often been adopted. First, if the current speed is scaled using the total friction velocity, measured independently, for example, using the surface wind stress, the friction velocity diagnosed from shear in the current profile is smaller than expected, being only a fraction of the total friction velocity. Second, the roughness length diagnosed from the same fitting procedure is much larger than expected, by various orders of magnitude, being inconsistent with the roughness length that would be estimated either for an aerodynamically smooth flow, or aerodynamically rough flow affected by waves. The corresponding Charnock parameter appears to be enormously amplified (Bourassa 2000).

Both of these features are explained here as resulting from a partition of the total turbulent shear stress into a shear-induced component and a wave-induced component, which result from the local mechanical production of this stress by the mean shear in the current profile, and by the Lagrangian strain rate associated Stokes drift of the waves, respectively, when the effect of nonbreaking waves is included in the equations of motion via the Craik–Leibovich vortex force. In this framework, the wave-associated part of the shear stress is not a property of the wave itself, as assumed by some authors, but is a stress created on the turbulence (which coexists with the shear-induced stress) by Stokes drift straining of turbulent vorticity into the streamwise direction (the assumed direction of both the mean current and the Stokes drift; Teixeira 2011a). This is independent from any vertical mixing associated with preexisting turbulence, or turbulence injected into the water by wave breaking.

It is likely that this mechanism associated with nonbreaking waves acts in concert with other mechanisms related to wave breaking, and with the transport of turbulence by itself in general, but the fact that it can account for the two phenomena mentioned above, and that its dependence on the turbulent Langmuir number appears to be confirmed by measurements, support its relevance.

The model predicts that the part of the turbulent shear stress induced by shear in the surface layer becomes a progressively smaller fraction of the total stress near the surface and down to a depth of the order the wavelength of the dominant surface waves as La_{t} decreases. This leads to the perceived reduction of the friction velocity. The model also predicts that the roughness length inferred if the uppermost portion of the current profile is disregarded is amplified by various orders of magnitude as La_{t} decreases, and scales with _{t}. The profile of the wind-induced current becomes flatter (i.e., less different from its surface value) as La_{t} decreases.

If the parameters in the model are adjusted appropriately, departing from their values assumed in Teixeira (2012) (presumably to account for the facts that there is no substantial wave breaking in the experiments and the waves are not monochromatic), good agreement is found with the laboratory measurements of Cheung and Street (1988), which appear to be the only dataset that is precise and comprehensive enough for this purpose. Other more recent datasets (Siddiqui and Loewen 2007; Longo et al. 2012) either seem unreliable, or do not provide complete enough information about the characteristics of the wave field or of the total shear stress. In the experiments of Cheung and Street (1988), the current profile becomes flatter as the wind speed increases. Using the present model, this is interpreted as being primarily due to a decrease in La_{t} at the lowest wind speeds, and due to an increasingly deeper penetration of the wave stress, conjugated with a higher real roughness length, at the highest wind speeds.

As in the present model, a recent study of Sinha et al. (2015) uses insights from Teixeira (2012) to develop a turbulence closure that includes wave effects. However, the dataset they use to test their model, from the LES of Tejada-Martinez et al. (2013), refers to shallow water flow, and is thus strongly affected by the bottom boundary layer. Sinha et al. (2015) primarily focus on an analysis of the current profile in wall coordinates within the bottom boundary layer, but the full-depth current profiles shown by them (e.g., their Figs. 19 and 21) suggest a relatively modest agreement between their model in the top boundary layer adjacent to the air–water interface, despite the fact that they include a term in the shear stress definition that is nonlocal, accounting for turbulent transport of TKE (which is not considered here).

To bring the model presented here closer to real oceanic conditions, and thus increase its usefulness, it is probably not only necessary to account for nonlocal mixing (which is important in some datasets), but also for the effect of Earth’s rotation, as wind-driven currents are known to be typically misaligned with the surface stress and rotate with depth, in accordance with Ekman layer theory. However, within the surface layer where the shear stress is the primary mechanism shaping the current, shear at least is necessarily aligned with the wind stress, and thus the model presented here may still be directly applicable to the streamwise component of the current.

Defining precisely the range of applicability of the present model is complicated (when compared to the atmosphere) by the presence of surface waves, as their influence may in some cases be confined to the oceanic surface layer (as happens here), and in others extend below it. To a first approximation, the surface layer might be defined as the layer in which there is little fractional change in the vertical of both the shear stress and the current direction.

The results reported here are presented in dimensionless form, which should facilitate their transposition to real oceanic conditions, enabling the development of physically based parameterizations for the turbulent momentum flux in the wave-affected boundary layer for ocean circulation models.

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