A quantitative description of wind-wave momentum transfer in high wind conditions is necessary for accurate wave models, storm and hurricane forecasting, and models that require atmosphere–ocean coupling such as circulation and mixed layer models. In this work, a static pressure probe mounted on a vertical wave follower to investigate relatively strong winds (U10 up to 26.9 m s−1 and U10/Cp up to 16.6) above waves in laboratory conditions. The main goal of the paper is to quantify the effect of wave shape and airflow sheltering on the momentum transfer and wave growth. Primary results are formulated in terms of wind forcing and wave steepness ak, where a is wave amplitude and k is wave number. It is suggested that, within the studied range (ak up to 0.19), the airflow is best described by the nonseparated sheltering theory. Notably, a small amount of spray and breaking waves was present at the highest wind speeds; however, their effect on the momentum flux was not found to be significant within studied conditions.
Momentum transfer from a shear flow to a wavy boundary has been of great interest throughout the past century. Solution of this problem for light wind conditions has lead to a better understanding of air–sea interaction and its influence on ocean and atmosphere dynamics. To fully parameterize air–sea fluxes, the influence of the surface wave state must be taken into account. Therefore, it is important to resolve both wind-wave and wind-current momentum fluxes for various wind, wave, and current conditions.
As wind speed increases, the underlying physical processes at the air–sea interface change dramatically, moving from smooth, linear cases to turbulence-dominated rough airflow regimes. This study seeks to fill in the gap in understanding and parameterization of wind-wave momentum transfer in strongly forced wind-wave conditions.
Present theories are able to describe a regime corresponding to light wind blowing over sinusoidal waves of small steepness. A wave causes compression and decompression of airflow streamlines along its surface, and thus pressure differences appear between windward and leeward sides of the wave. The part of the wave-induced surface pressure pattern that is in phase with the slope of the surface acts to deposit wind momentum into the irrotational flow field of the wave. This process was described in analytical solutions by Miles (1957, 1959, 1960). He introduced a nondimensional wave growth function and suggested its dependence on wind forcing. Moreover, when waves of various frequencies are superimposed and represented as wave spectra, it is possible to parameterize wind-wave momentum transfer using a spectral wave growth function (i.e., Snyder et al. 1981), which is defined as
where ρa and ρw are air and water densities, E(ω) is the spectrum of surface elevation, and ω is the wave radian frequency.
Wave growth parameterization through one function that depends on wave age only is alluring, because it can be easily used by numerical spectral wave models to describe wave motion throughout the oceans. The theory, however, provided a solution only for light wind forcing. Also, according to Miles, the controlling parameter for wind-wave forcing is the curvature of the wind profile at the “matched layer” height. The matched layer is defined by the height at which the horizontal wind speed is equal to the wave phase speed. Such a parameter, however, was not found to be useful in practice. First, a matched layer does not exist in wind opposing wave cases; second, even in moderate wind forcing situations, the matched layer height is near or within the viscous sublayer, which has a linear wind speed profile. For these reasons, a lot of experimental effort was expended to reinforce the theoretical solution. Resulting empirical parameterizations use the dimensionless ratio U10/Cp as a more appropriate wind-wave forcing parameter, where U10 is wind speed at 10-m height and Cp is wave phase speed. A logarithmic wind speed profile assumption (Smith et al. 1992) is typically used to extrapolate wind speed to a desired height reference.
The most obvious approach to estimate momentum transfer experimentally was to use a static air pressure probe (i.e., Elliott 1972b; Nishiyama and Bedard 1991) while simultaneously measuring surface elevation (Elliott 1972a; Hsu et al. 1982). Momentum transfer from wind to waves is then given by the pressure–slope correlation,
where ∂η/∂x is the wave surface slope, p0 is the static air pressure at the surface, and the averaging 〈 〉 is done over one wavelength. However, it was soon realized that wave-induced static pressure fluctuations quickly decay with height. Because the lowest position of the static pressure probe must be higher than the highest wave crest, such measurements do not adequately sample static pressure at the water surface, especially above wave troughs.
As a solution to this problem, vertical arrays of probes were used to extrapolate the magnitude of wave-induced fluctuations to the surface (Snyder et al. 1981; Hristov et al. 2003; Donelan et al. 2005b, among others). These measurements indicated that wave-induced pressure fluctuations decay proportionally to e−αkz, where k is wavenumber, z is height, and α ≈ 1. However, the exact form of the pressure fluctuation decay near the surface, especially in strongly forced conditions, is still unknown. Therefore, one of the goals of the present study is to provide guidance for pressure extrapolation to the surface for future stationary probe measurements.
Another more direct experimental solution was to mount a static pressure probe on a frame that moves with surface elevation, thereby maintaining the probe at a small constant height from the surface. This method can potentially provide the most accurate measurements; however, it is technically challenging. In one of the first attempts (Dobson 1971), a surface buoyant platform was used to carry the pressure probe. In other attempts, coupled surface elevation sensors and vertical motor systems were used. Shemdin and Hsu (1967) reported successful following of a predetermined monochromatic wave in laboratory conditions. However, until recently, most of the attempts to use a wave follower for a random wave field faced technical difficulties, especially in the field. Among the most successful were Snyder et al. (1981) (Bight of Abaco, Bahamas, experiment), Harris and DeCicco (1993) (Chesapeake Light Tower), Donelan (1999) (laboratory experiment), Jacobs et al. (2002) (Meetpost Noordwijk research and monitoring platform), and Donelan et al. (2006) (Lake George experiment). The resulting empirical estimates of γ(U10/Cp) have some agreement for calm mature seas; however, the limited data available for strong wind forcing conditions are not sufficient to form a complete understanding of underlying processes.
Strong wind over young waves with intense wave breaking and spray generation are typical in the North Sea, in the Southern Ocean, and during storms and hurricanes. The shape of the ocean surface is such that it can no longer be described by a linear wave theory. Moreover, wave breakers and spray add completely new physical elements to the problem. Andreas and Emanuel (2001) suggested that reentrant sea spray is responsible for a large part of the total air–sea momentum flux in high winds. Kudryavtsev and Makin (2001) suggested that additional momentum flux due to airflow separation behind breaking waves reaches 50% of the total momentum flux in high winds.
It is clear that, as wind forcing and surface roughness increase, wind-wave momentum flux also increases at a certain rate. A more interesting question is, however, what happens to the nondimensional wave growth rate γ? It is generally agreed that the wind forcing and wave shape in some form (i.e., wave steepness, crest sharpness, or even wave breaking probability) are the parameters that control the deviation of γ from the values derived from linear theory. However, it is still debated whether the net effect of nonlinear corrections should be positive or negative. Analytical solutions by Van Duin (1996) and Belcher (1999) suggest a reduction of the wave growth rate with steepness and/or wind forcing. They assume that waves are steep enough to shelter their lee side but smooth enough (ak < 0.2) so that the airflow does not separate. Kudryavtsev and Makin (2001), on the other hand, describe airflow separation behind steep breaking waves and suggest a mechanism through which additional momentum is pumped into waves. Few experimental studies provide sufficient data to validate these theories. Banner (1990) observed a sharp increase in momentum transfer above a breaking wave; Donelan et al. (2006) concluded that wave steepness has positive effect on the wave growth rate; and Touboul et al. (2009) found that, over very steep waves, the wind-wave momentum flux predicted by Miles (1957) results in a better agreement with experiments if enhanced by a sheltering coefficient. However, a recent comprehensive review paper on the subject (Peirson and Garcia 2008) summarized all available data (their Fig. 7) and concluded that the growth rate γ first decreases with ak up to about 0.22 and then increases as steepness increases farther into the wave breaking region. In the present paper, we confirm the decrease of γ in the range of ak between 0.03 and 0.19.
An example of the airflow pattern in a nonseparated sheltering regime is shown in Fig. 1 (bottom). Such airflow structure with streamlines following the surface of the wave is predicted by linear theory. A finite-amplitude correction (Belcher 1999) to the linear theory suggests a modest reduction of the wave growth rate, which only becomes significant for very high wind forcing and wave steepness. The correction proposed by Van Duin (1996) is more significant; he argues that wave nonlinearity has a stabilizing effect on the wave growth and also provides an expression for the maximum wave amplitude. Both papers mention that above a certain steepness (ak ~ 0.2) an airflow separation bubble forms and the theory no longer applies. Kudryavtsev and Makin (2001), on the other hand, emphasize the role of the airflow separation (Fig. 1, top; see also Ryn et al. 2007; Reul et al. 2008) in the vertical flux of momentum. According to their model, the separation effect causes a sharp pressure drop behind a breaking wave; therefore, the momentum flux is growing faster than expected by linear theory.
From the empirical point of view, it is hard to distinguish separated and nonseparated sheltering, because within a given wave field wave steepness varies but ensemble averaging is needed to reconstruct the pattern of wave-induced airflow pressure fluctuations. The averaged pattern includes both separated and nonseparated cases, and the resulting parameterization for γ often incorporates both effects with unknown weights. Nonetheless, the sheltering effect (separated or not) can be quantified through a sheltering coefficient G,
providing an empirical solution for various practical applications such as numerical wave modeling. Coefficient G can be used in a variety of ways; in our work, its definition and purpose are discussed in detail in section 3d.
It is the main goal of the present paper to investigate the shape of function (3) by means of a laboratory experiment. The sheltering coefficient G is expected to mainly depend on wave shape and wind forcing; therefore, they are the parameters that were artificially controlled to vary over a wide range. Other wind and wave parameters such as wind speed and gustiness, spray volume, wave breaking probability, wave asymmetries, and nonlinear interactions are also expected to have an effect on G (Babanin and Makin 2008). Some of them are addressed in this study, but others are left for future investigations.
The experiment design and ranges of investigated parameters in this study were chosen such that the observed phenomenon would be as similar as possible to the wind-wave interaction in the ocean. Nonetheless, it is important to note that the wind-wave momentum fluxes in the field and in the laboratory can be different for a number of reasons, and strictly speaking they cannot be directly compared. The major sources of differences are the following: First, length scales, such as the wavelength and amplitude, are typically much smaller in the laboratory. Second, in the field, waves typically have a three-dimensional shape, whereas they are nearly two dimensional in the laboratory. Third, a wave produced by a mechanical wave maker in the laboratory is monochromatic and strictly periodic, whereas in the field the wind-wave spectrum is wide, leading to a large variety of possible wave shapes. According to the linearized theory, these differences should not influence the results, presented in nondimensional space. However, because our study covers nonlinear and strongly forced conditions, we are likely to exceed the limits of applicability of linear theory.
On the other hand, laboratory work has a number of advantages over experiments conducted in the filed. Unlike the real ocean, conditions in a wave tank are repeatable; the input parameters are controllable; and, finally, high winds pose less of a threat to equipment and researchers. Therefore, we see the purpose of our experiment in the extension of field measurements to higher winds, as well as in quantitative clarification of underlying physical processes observed in the field (e.g., the role of wave steepness in the wave growth).
2. Experiment setup
The experimental data presented in this work were acquired in the Air-Sea Interaction Saltwater Tank (ASIST) at the University of Miami Rosenstiel School of Marine and Atmospheric Science. ASIST is a 15 m × 1 m × 1 m wind-wave flume. It was filled with freshwater up to a 42-cm level (Fig. 2) and equipped with fully programmable wind, current, and surface gravity wave generators. The tank is capable of producing winds up to 30 m s−1 at the centerline, current up to 0.5 m s−1, and mechanically generated waves with up to 10-cm height. Fully transparent acrylic glass walls, bottom, and top of the tank allow using optical nonintrusive methods for flow measurements and visualization.
An “Elliott” type of static air pressure probe (Elliott 1972b) was used to study the wind-wave momentum transfer. Because wave-induced airflow pressure fluctuations quickly decay with height, it was important to measure pressure as close to the wave surface as possible. Therefore, the probe was continuously moved vertically and kept at a small distance (1–3 cm) from the wave surface.
To make wave following possible in strong wind conditions with occasional spray and wave breaking, a robust and fast response elevation gauge was developed. The new Digital Laser Elevation Gauge (DLEG) essentially is a vertical laser beam crossing the water surface. Fluorescein added to the water makes the laser beam highly visible; this creates a brightness contrast as the beam crosses the air–water interface. A digital line scan camera looks at the beam through the tank’s sidewall. The brightness threshold location on a line image signifies the water elevation. Line images, sampled at 250 Hz, were acquired through a firewire board and processed by a National Instruments Labview code in real time. The code had an advanced edge detection algorithm and thus provided a surface elevation signal clean from whitecap and spray-related spikes. The output signal had 4-ms temporal and 0.5-mm spatial resolution. The same software acquired data from other instruments and controlled the entire experiment. A detailed description of this system can be found in Savelyev (2009). The edge detection algorithm used to determine the water surface location is described in the appendix.
The heart of the wave follower is a linear servo motor with its programmable controller. To provide the optimal following trajectory, at every time step, in addition to a new water elevation coordinate, the current motor position was considered to make a new motion decision. This allowed smooth reattachment of water elevation and wave-follower trajectories, limited unnecessary vibrations, and disabled positive feedbacks. The key principle behind the smooth motion was that instead of position; only the velocity of the motor was controlled. Every time step, it was chosen as
where Xw is current surface elevation, Xm is current motor elevation, and Δt is sampling time step. This allowed the motor to be in constant motion and thus eliminated stop and go vibration, the so-called jerking problem, which is harmful for the motor and pressure sensors. The follower motion lagged about 30 ms behind surface elevation. To compensate for this lag, another elevation gauge was installed 4 cm upstream from the pressure probe location.
The static pressure Elliott probe is essentially a metal disk 2 cm in diameter with an inlet in the middle of each side. Both inlets are connected to a single pressure transducer. The shape of the probe is such that airflow, disturbed by the disk edge, restores by the time it reaches the center of the disk. This ensures that the measured pressure is static pressure, as long as the wind direction is within ±12° of the disk plane. More detailed information on the probe design and specifications can be found in Elliott (1972b). In ASIST, for the given experimental setup, the dominant wind speed components are in vertical and along-tank directions. Thus, the disk was mounted vertically along the tank direction on the bottom of a vertically moving shaft. In the event of flooding due to wave breaking or spray, a backflow was initiated through the probe.
The observed static pressure difference between the windward and leeward sides of a wind wave was in the 1–30-Pa range. At such small fluctuation magnitudes, a number of factors that can contaminate the pressure measurements need to be considered. Vertical accelerations generate additional pressure fluctuations in the air column between the pressure probe and the transducer,
where ρa is air density, af is wave-follower acceleration, and Δh is vertical distance between pressure probe and pressure transducer. This is compensated for in the analysis, because the acceleration of the probe may be deduced from the time history of its position. Also, to minimize these fluctuations, the transducer was mounted, with its membrane vertical, just 10 cm above the probe. As a result, the correction (5) was within 1% of the wave-induced airflow pressure fluctuations.
There are also errors associated with the transducer membrane’s finite mass, noise cancelling electronics, and the pressure wave propagation time between the probe and the pressure transducer. These effects were investigated using a controlled pressure chamber and are presented in the form of a phase–frequency response function (Fig. 3) and an amplitude–frequency response function (Fig. 4). These response functions were applied to all measured pressure signals in Fourier space. More information on the pressure probe calibration can be found in Donelan et al. (2005a), where similar procedures were followed.
Within each 30-min run, the wind speed was held constant and a monochromatic wave of a constant frequency and amplitude was generated mechanically. During the propagation from wave maker to the test location (8.7 m), mechanically generated waves were observed to keep constant wavelength; however, their amplitude increased significantly, especially under strong wind forcing. Therefore, all wind and wave parameters were measured at the exact location of the Elliott pressure probe (see Fig. 2) and the results represent growth rates only at that particular point.
The friction force, applied by the wind to the water surface, generated a near-surface current with the magnitude of 3–10 cm s−1, depending on the wind speed. The current reversed its direction at ~(10–15)-cm depth to form a compensating backflow. This effect introduces a Doppler shift to the wave dispersion relationship, effectively increasing the wavelength of a given wave frequency. However, the effect was neglected in this study, because the drift current was typically 20–50 times slower than the wave phase speed.
After data acquisition, correction, and calibration, the water surface elevation, pressure probe elevation, and static air pressure time series were obtained at 250-Hz sampling frequency. Table 1 provides a summary of all successful runs, as well as corresponding wave parameters, obtained from these measured time series.
The parameters were selected to extract momentum flux from the wind to a single wave component at various wind speeds and wave amplitudes. All experiments were repeated at three wave frequencies (0.75, 1, and 1.25 Hz) to test the scaling invariance. The minimum amplitude was chosen to be ~0.5 cm, because at lower values the pressure fluctuations were too small to be accurately sampled. The maximum amplitude (~4.5 cm) limitation was due to the pressure fluctuations exceeding the upper limit of the pressure transducers. The minimum wind speed was chosen to be U10 = 7 m s−1 in order to provide a sufficient pressure fluctuation signal to be above the noise floor of the transducer.
The largest effort was expended to increase the maximum wind speed limitation (U10 = 26.9 m s−1). Wave breaking and spray started to be visually noticeable around the wind speed U10 = 22 m s−1. At higher wind speeds, the amount of whitecaps and spray rapidly increased, eventually causing unacceptable degradation of the wave-follower performance, as well as repeated clogging of the pressure probe by water intrusion. For these reasons, most of our attempts at higher wind speeds did not produce satisfactory results, with the exception of two runs with U10 = 26.9 m s−1. The role of spray and wave breaking is thoroughly investigated in section 3d; however, because we only sampled at the inception of such conditions, our conclusions lack statistical confidence and call for additional experiments in higher winds.
3. Data analysis and results
a. Pressure profiles
Wind-wave momentum flux is given by (2), for which the static air pressure at the surface is required. Because pressure was measured at a small but finite height, it must be extrapolated to the surface. However, the exact form of the extrapolation function is unknown,
where z is the pressure probe’s instantaneous elevation above water surface, p is the measured pressure, p0 is the pressure at the surface, and k is the wavenumber. Potential theory, as well as previous experiments in low winds (e.g., Banner 1990; Hristov et al. 2003), suggests
where α is an empirically determined constant. In strong wind forcing, the vertical decay of the pressure fluctuation has not been previously observed.
For this purpose an averaged function p(z, x) was measured for each run, which was conducted under constant wind and mechanical wave forcing for 30 min. The vertical distance z is the elevation above the average surface at the wave phase x.
Using a numerical bandpass filter, the mechanically generated wave frequency was extracted from the surface elevation time series. For every data point, the real and imaginary parts of a Hilbert transform determined wave phase. Once the phase of the wave was known, phase-resolved averages of surface elevation reproduced the mean shape of the long wave. Pressure measurements, collected and averaged at multiple levels above each wave phase, enabled pressure extrapolation to the surface (Fig. 5). For the purpose of illustration of p(z, x), pressure measurements were averaged over areas λ/72 wide and 5 mm high, where λ is the wavelength of the long (mechanically generated) wave.
During each run, the pressure was sampled at a range of heights (above the wavy surface) starting from the closest nonwetting height [~(1–3 cm)] up to about 1/10 of the wavelength. Although the pressure probe can only sample one elevation z and phase x at a time, over the course of 30 min there was sufficiently dense sampling to provide bin-averaged mapping of the pressure distributions with elevation and phase. By averaging to obtain p(z, x), the turbulent component of the pressure fluctuations as well as the effects of random realizations of wind waves were removed.
The vertical pressure profile was found to vary with wave phase. It sometimes has a local minimum above the leeward side of the wave crest. For example, two cases with similar wind forcing but different wave steepness are shown in Fig. 5. Although airflow streamlines were not resolved in this experiment, in some cases a pressure minimum is visible above the lee side of a steeper wave, such as in Fig. 5 (bottom). Therefore, the bottom and top panels in Fig. 5 possibly illustrate averaged pressure patterns with and without the airflow separation, respectively. Further analysis (sections 3e and 4), however, shows that the airflow separation is weak in these conditions and plays a minor role in wind-wave momentum flux.
Analysis of all runs revealed that for each wave phase in the given height range the pressure profile is best described by a linear fit. The pressure measurements were within a small height above the surface (~1/10 of the wavelength); therefore, only the linear term of the Taylor series expansion of the pressure was preserved. This assumption allows estimation of the rate of the vertical exponential decay based on the available data,
where ∂p/∂z is the slope of the pressure profile near the surface and p0 is the pressure, linearly extrapolated to the surface. This provides an estimate for α at each phase Θ of the studied wave. Averaging α(Θ) over all available runs gives the phase dependent rate of vertical exponential decay of pressure fluctuations (Fig. 6).
The wide spread of 95% confidence intervals shows that for various wind-wave conditions the vertical decay of pressure near the surface is not necessarily exponential. Also, these results clearly suggest that the exponential decay coefficient α depends on phase. The minimum mean-square fit yields
with the corresponding mean surface elevation (shown by solid line in Fig. 6) having a fitted cosine wave [η = −cos(Θ)] shown by dashed line.
In agreement with a priori expectations, the observed value of α fluctuates around 1. However, these fluctuations were found to be wave phase dependent. More experimental work on this topic is needed to confirm this dependence and to investigate the effect of wind forcing severity on it. It is likely that the wave phase dependence exists because of sheltering, and, if so, α bears some dependence on wind forcing and/or wave shape.
b. Momentum flux from wind to waves
Once static air pressure at the surface and surface slope are obtained, momentum flux is given by (2). It was calculated for each run and is shown in Fig. 7 (top). As expected, momentum flux intensifies as wind speed increases, and the steepness of the imposed wave (shown by symbols) strongly influences the momentum input as well. The bulk air–sea momentum flux τ is often parameterized as a function of wind speed U10 and a drag coefficient Cd, , where U* is the friction velocity (e.g., Powell et al. 2003; Donelan et al. 2004). As momentum flux crosses the water surface, part of it transfers into the wave motion because of normal pressure force and the other part transfers directly into the near-surface current because of friction force. Therefore, the drag coefficient Cd can be represented by two components, Cd = Cdw + Cdf, where Cdw is responsible for the form drag and Cdf is responsible for the friction drag. This way, the wind-wave momentum flux measured in this study can be expressed as
A drag coefficient is expected to primarily depend on surface shape. We have found that within our dataset it can be well described by
Equations (10) and (11) are shown in Fig. 7 (bottom) by a solid line, which is strongly correlated (correlation coefficient 0.97) with our data. The relationship appears to hold throughout the entire range of tested parameters, U10 from 7 to 26.9 m s−1, a from 0.6 to 4.5 cm, and k from 2.76 to 6.35 m−1. This result leads to two conclusions: first, in these strongly forced conditions, the rate of momentum flux into the wave field does not depend on the actual size of waves but is controlled only by the surface slope. Second, unlike the bulk air–sea momentum flux drag coefficient, wind-wave drag coefficient does not directly depend on wind speed. However, it is likely that in realistic conditions ak tends to increase with wind speed, forming an indirect relationship between Cdw and U10. More detailed discussion on the applicability of this result to the realistic ocean environment is given in section 3f.
c. Wave growth rate as a function of wind forcing
The nondimensional wave growth function (1) is the ratio of rate of work done on a progressive surface wave by atmospheric pressure to the total energy of the wave (scaled by the ratio of water density to air density). In other words, it is a relative growth rate of energy. It may be calculated as
where 〈p(ω)η(ω)*〉 is the quadrature spectrum (Snyder et al. 1981). However, it is impossible to obtain the entire function based on one 30-min run because of the rapidly increasing error in the tail of the wave spectrum. Therefore, each run was used to calculate only one value, corresponding to the frequency of the mechanically generated wave. This simplifies Eq. (12) further,
where σ2 is the variance of the surface elevation of the mechanically generated wave at the given frequency. All variables on the right-hand side of Eq. (13) are known and the resulting wave growth function can be calculated by varying the frequency of interest. It is shown in Fig. 8 together with parameterizations obtained by Snyder et al. (1981), Hsiao and Shemdin (1983), Hasselmann and Bösenberg (1991), Donelan (1999), and Donelan et al. (2006).
According to the wave growth theory (Miles 1957), we expect γ to depend primarily on wind forcing. Here, wind forcing is represented by (Uλ/2/Cp − 1)2, where Uλ/2 is the wind speed at the height equal to half of the wavelength. A least squares fit to the data yields the dependence
d. Analysis of the wave growth rate sensitivity to secondary parameters
The close match between our results and previously observed growth rates (Fig. 9) is encouraging, but both the present data and the dataset compiled by Plant (1982) have significant scatter around their mean values. In a weakly nonlinear setting, the main cause for this scatter is anticipated to be the variation in wave steepness, which has been shown to influence the growth rate both experimentally (Peirson and Garcia 2008) and theoretically (e.g., Belcher 1999). As the wave field becomes strongly nonlinear, occurrences of breaking waves and spray are also expected to contribute to the observed scatter. All of these parameters, including the wind forcing, are not completely independent; therefore, further correlation analysis and empirical function construction must be approached with caution.
Let us assume that the final empirical wave growth function will take the form
where G(ζ) is an unknown function that depends on one or more of the wind-wave parameters. For further simplification and because of the lack of data, we further assume that G is a linear polynomial. Next, we examine if ζ can be one of the following parameters: wave amplitude a; wave frequency f; wave steepness ak; wave crest sharpness s, which is defined as the ratio between the crest elevation above the mean water level and the crest width at half the elevation; Uλ/2; Uλ/23; or the percentage of waves breaking Pbr. A wave was considered to be breaking if at any point its orbital velocity exceeded half of its phase velocity (Zhang 2008). The orbital velocity was calculated by means of a Hilbert transformation (Melville 1983). In the following analysis of the wave breaking probability, only runs with Pbr > 0 were considered.
The least mean-square fits of the linear functions G(ζ) are summarized in Table 2. The relative range value is the measure of the function’s G sensitivity to the current parameter ζ. It is defined as the difference in G between the largest and the smallest values of ζ, normalized by mean G and multiplied by 100%. For example, we see a strong dependence of G on ak (40%), f (35%), Uλ/23 (30%), Uλ/2 (28%), and Pbr (26%) within the investigated range of these variables. However, this does not necessarily mean that γ depends on all of them, because some of these parameters are not independent. To keep track of all correlations, the full correlation matrix is given in Table 3. For example, wave steepness is strongly correlated with crest sharpness and breaking probability. Less noticeable is the correlation between ak and f: lower-frequency waves tend to be less steep, and steeper waves tend to have higher amplitude. Note that, although some of these interdependencies are typical to ocean conditions, others are purely artificial and unique to this dataset. For example, the Uλ/2 correlation with f(−0.344) reflects not a fundamental wave growth law but merely our choices of wind speed and wave frequency. For these reasons, to avoid spurious correlations, function G(ζ) was investigated as a function of only one secondary parameter at a time. Also, in some cases below, function G(ζ) retains dimensions of the secondary parameter ζ, whereas the right-hand side of Eq. (15) is expected to be nondimensional. This was done simply to show a qualitative role of a given parameter, and such polynomials are not intended to be used as parameterizations, mainly because of large confidence intervals.
An a priori assumption for this study was that the wave growth function, in addition to its dependence on wind forcing also strongly depends on ak; therefore, these parameters were purposely varied during the experiment. Indeed, there was a strong sensitivity of G to ak, resulting in a 40% drop of the growth rate as ak varied from its minimum to maximum observed values. The linear regression slope and its 95% confidence interval was −2.52 ± 1.26. This dependence on wave steepness is in accord with previous experimental observations and theoretical predictions, as discussed in more detail in section 3e.
The wave growth also exhibited an unexpected sensitivity to the wave frequency, with G = −0.77f + 1.85. The dependence of G on f can be attributed to their indirect relationship through wave steepness (longer waves tend to be less steep). However, the correlation between ak and f is only moderate (0.237), prompting us to look further into the G(f) function behavior. In the present study, only three wave frequencies were studied (0.75, 1, and 1.25 Hz); therefore, no conclusive statements can be made on the matter. In future work, it will be useful to conduct a set of experiments with constant wind forcing and wave steepness but various wave frequencies to reveal the nature of G(f).
Based on experimental work (Banner 1990; Touboul et al. 2009) and the theoretical model of Kudryavtsev and Makin (2001), we know that a sharp wave crest, particularly the crest of a breaking wave, causes airflow separation, which in turn enhances the wind-wave momentum flux. A number of experimental (e.g., Reul et al. 2008) and numerical (e.g., Ryn et al. 2007) works have covered the topic of airflow separation, but its detailed description in terms of wind-wave parameters is still incomplete. Although it is an important research subject, the present study did not directly measure airflow velocity streamlines and as such could not resolve flow separation. Pressure patterns shown in Fig. 5 (top and bottom) may provide some information on the matter; for example, the local pressure minimum above the lee side of the wave in Fig. 5 (top) may suggest the existence of the separation bubble. However, such pressure patterns are not individual events but averages over hundreds or thousands of wave periods, which are likely to include both separated and nonseparated cases. Moreover, short wind waves that are not visible on these averaged images were likely to introduce an unknown variability to the location of the separation bubble. Therefore, these limitations do not allow us to establish a direct relationship between the airflow separation and the wind-wave momentum flux. Nonetheless, one might hypothesize that, in addition to wind forcing and wave steepness, the effect of the airflow separation on the wave growth is controlled by the wave crest sharpness s and/or wave breaking probability Pbr. However, in our experiments, the parameter s was not found to have any influence on the wave growth (Table 2). Function G was found to increase with breaking probability (by 26% within the studied range; Table 2). However, most of the already limited number of runs with Pbr > 0 has low breaking probability. Therefore, the 95% confidence interval of the linear regression slope is wide, 0.66 ± 0.73. Such statistical error does not allow making definite conclusions and calls for more data from future experiments.
Additionally, runs with breaking probability Pbr > 0 on average result in values of the wave growth rate 4% below the parameterization given by Eq. (14), whereas cases with Pbr = 0 are 22% higher. This suggests an opposite effect: that is, reduction of γ because of wave breaking. However, in the following section it will be shown that the wave growth rate is sensitive to wave steepness. It is likely that cases with Pbr = 0 produce higher growth rates, compared to Pbr > 0, mainly because of lower wave steepness. These two effects are hard to isolate within our dataset, because the correlation coefficient between ak and Pbr is 0.79 (Table 3).
A limited amount of spray was observed during some runs within this study. Therefore, it is interesting to investigate how spray presence affected the wind-wave momentum flux. On one hand, spray is expected to increase the bulk air–sea momentum flux: spray droplets are accelerated by the airflow before they reenter the water column. This slows down the wind speed near the surface (Pielke and Lee 1991), causing weaker wave-induced pressure fluctuations. Therefore, we would see a reduction in the wave growth function. On the other hand, based on Powell et al. (2003) field measurements, Makin (2005) suggested a reduction of the net air–sea momentum flux and an increase of the near-surface wind speed due to spray. However, an alternative model by Troitskaya and Rybushkina (2008) is able to account for the drag reduction with no regard to spray. In any case, Powell et al. (2003) observed this effect only at U10 exceeding 33 m s−1, which is well above the range of wind speeds in our study.
To investigate the role of spray, we have conducted an additional experiment and obtained a rough estimate of the spray concentration in the air. As a part of the digital laser elevation gauge technique, described in section 2, a digital line scan camera acquired images of spray droplets, crossing the path of the vertical laser beam (~2 mm in diameter). The number of spray droplets that crossed the beam in 1 s at the height range of 15–30 cm above the mean water level is shown as a function of wind speed in Fig. 10. It can be seen that, apart from occasional droplets, the spray amount is significantly smaller for U10 below 22 m s−1, which is also around the cutoff wind speed for most of our dataset (with the exception of two runs with U10 = 26.9 m s−1). This is not a coincidence, because most of our attempts to conduct wave following and pressure measurements in heavy spray conditions did not produce useful data.
Large asterisks in the bottom panel of Fig. 7 show two successful runs conducted in conditions where spray concentration is expected to be the highest (i.e., U10 = 26.9 m s−1). Both of them produced τw values lower than the fitted curve, thus suggesting the negative effect of spray on the wind-wave momentum flux (i.e., Pielke and Lee 1991). Clearly, more data at higher winds are needed to confirm and quantify this effect. Andreas (1998) expects nearly an order of magnitude increase in spray volume as wind speed increases from 22 to 32 m s−1. At that point, spray-related effects might dominate the wind-wave momentum flux. However, because of technical limitations of our experimental approach, described in section 2, we were unable to proceed to such high winds. For more detailed discussion regarding the effect of spray on the air–sea momentum flux in higher winds, the reader is referred to Andreas (2004).
e. Wave growth dependence on wave steepness
In accord with a priori expectations, the wave growth function was found to be particularly sensitive to the wave steepness among other secondary parameters. Next, we perform a more detailed analysis of this dependence and compare it to previous findings.
Previously we assumed a possibility of the separation of variables (15) within the function γ(Uλ/2/Cp, ak) = F(Uλ/2/Cp)G(ak), where F = 0.52[(Uλ/2/Cp) − 1]2. To test this assumption, the experiment was designed in a way that forms vertical clusters of points with nearly constant wind forcing but variable γ (Fig. 8). These clusters were created on purpose to provide cross sections of γ(ak) and thus map the entire surface γ(Uλ/2/Cp, ak) as the true function of two variables. Each of these clusters was analyzed separately; all of them are shown as with corresponding linear fits in Fig. 11. This figure gives a glimpse of the complex structure of γ(Uλ/2/Cp, ak), but, given the limited amount of data, we have to approximate each γ(ak) dependence as a linear fit. It is evident that a decline of γ with steepness takes place in almost all cases. Therefore, it is reasonable to generalize the data as having a linear decline of γ with increasing ak for any wind forcing. For simplicity, this relative decline is further assumed to be independent of wind forcing in the studied range. To generalize this dependence, each γ(ak) function was normalized by its average 〈γ〉 so that the linear fit through all available data gives an average γ(ak) linear fit. The resulting averaged dependence of the wave growth function on wave steepness is given by
where the 95% confidence interval of the linear regression slope is −1.9 ± 0.58. The present data with a corresponding linear fit (solid line) are shown in Fig. 12. The dashed–dotted line is a simplified representation of the dataset compiled by Peirson and Garcia (2008, their Fig. 7), and the dashed line is the nonlinear correction to the wave growth rate proposed by Belcher [1999, Eq. (4.9), β ≈ 20]. Both lines were normalized by their mean values 〈γ〉 averaged over the range of ak from 0.03 to 0.19 for the purpose of slope comparison ∂γ/∂(ak) with Eq. (16).
In contrast with this study, one of the conclusions of Donelan et al. (2006) suggests a proportionality between the wave growth rate and wave steepness (i.e., γ ∝ ak). Two proportionality coefficients are given for low and high wind forcing, but both of them are positive. We are unable to give a definitive explanation for this contradiction, in part because of the large statistical uncertainties of both results. The difference in γ(ak) might be in part attributed to the difference in the wave spectrum in the laboratory and in the field. A broad wind-wave spectrum results in a large range of possible wave steepnesses within a given wave field, whereas steepness stays nearly constant for each wave in the laboratory. Therefore, for example, if the mean steepness ak = 0.1, both in the laboratory and in the field, unlike the laboratory, waves in the field will frequently exceed the critical steepness and break. According to Banner (1990), this will enhance the wind-wave momentum flux. Laboratory waves, on the other hand, will not yet experience the full airflow separation at this steepness. Hence, the growth rate measured in the field might appear to be increasing with steepness. To test this hypothesis, it would be useful to compare momentum fluxes from wind to both monochromatic and wind-wave spectra in the future experiments.
f. Results applicability
The growth rate correction due to wave nonlinearity obtained in this study [Eq. (16)] is in close agreement with the theoretical prediction by Belcher (1999). This suggests that the theory of nonseparated sheltering is relevant to the airflow regime within the studied range of parameters. The empirical nonlinear correction for γ, compiled by Peirson and Garcia (2008), suggests a sharp decline while ak is below 0.09 and a nearly constant value while ak is between 0.09 and 0.23. As wave steepness approaches breaking threshold, a sharp increase is expected because of the airflow separation. Our data were collected in the range of ak between 0.03 and 0.19 and are in agreement with Peirson and Garcia (2008) within the range of 95% confidence intervals. Although there are indications of airflow separation in some cases (e.g., Fig. 5b), it appears that our study mostly covers the range of ak where nonseparated sheltering is the dominant mechanism for the wind-wave momentum transfer. Full airflow separation is expected to take a more dominant role and to enhance the wave growth for wave steepness above the range covered within this study.
In an ideal environment of pure wind waves, wave steepness can be related to the wind forcing using Toba’s law [Toba 1972, his Eq. (3.18) and Fig. 4],
In Fig. 13, Eq. (18) is shown as a solid line. The other three lines illustrate the wave growth function [Eq. (15)], with G given by Eq. (16). The three curves correspond to various values of ak, and the dots correspond to the data points listed in Table 1. The resulting correlation coefficient between the two-dimensional function γ(Uλ/2/Cp, ak) and the data points is 0.78.
Because the wave steepness and wind forcing are controllable in the laboratory, they can and often do violate Toba’s law [Eq. (17)], which is only applicable for pure wind waves. According to Eq. (17), for wind forcing Uλ/2/Cp > 6 the wave steepness is expected to be beyond the range covered in this study, ak > 0.2. Therefore, our results [i.e., Eqs. (15), (16), or (18)] are only applicable for pure wind-wave conditions for wind forcing Uλ/2/Cp within the range from 4 (i.e., the minimal wind forcing within this study) to 6.
Toba’s law predicts pure wind waves to be steeper as wind forcing increases. In addition, our results show that the growth rate γ decreases with steepness. Therefore, growth rate increases because of wind forcing but also decreases because of higher steepness caused by the wind forcing. In laboratory experiments, on the other hand, the steepness remains arbitrary, independent of wind forcing. This can possibly explain the tendency of laboratory-based parameterizations γ(Uλ/2/Cp) to yield faster growing values compared to field data (e.g., Fig. 8). Based on this conclusion, we suggest that, for practical purposes, laboratory-based parameterizations γ(Uλ/2/Cp) should only be used for the wave steepness they were measured at, or ideally they should include ak as one of the input parameters, γ(Uλ/2/Cp, ak) [e.g., Eqs. (15) and (16)]. On the other hand, field measurements, conducted in pure wind-wave conditions, can produce correct γ(Uλ/2/Cp), because Toba’s law is inherently satisfied. However, the application of such parameterizations is limited to pure wind waves.
For wind forcing Uλ/2/Cp above 6, the parameterization given by Eqs. (15) and (16) is not applicable for a pure wind sea, because the tested range of wave steepness is too low. This parameterization is expected to be useful in conditions that include strong winds blowing over swell or in situations where the wind sharply changes direction with respect to wave fronts, effectively reducing the wave steepness.
Similar narrow limitations apply to our wind-wave momentum flux results. For the lowest values of wind forcing U10/Cp tested in this study, Toba’s law is satisfied and can be combined with Eqs. (10) and (11), yielding Cdw = 0.0142U10/Cp, or
If Eq. (19) is extrapolated far outside of the tested region, toward the state of equilibrium (i.e., Cp/U10 = 1.37), momentum flux becomes simply (i.e., Cdw = 10−3), independent of any wind-wave conditions. In future studies, it will be interesting to observe if this is true for the state of equilibrium and, if not, what the limits are of the applicability of Eq. (19) in mature seas.
In this work, an experimental effort was undertaken to improve our understanding of the airflow pressure fluctuations near the air–sea interface. Pressure measurements were used to obtain wind-wave momentum flux and parameterize the wave growth rate in high winds.
The momentum flux τw from wind to a mechanically generated wave was found to be a function of wind speed, wavelength, and wave amplitude. It can be parameterized as
where Cdw = 0.146(ak)2 is the wind-wave drag coefficient (Fig. 7, bottom). The momentum flux measurements were further used to obtain the wave growth function (1) dependence on various wind-wave parameters. Primarily it was found to be sensitive to the wind forcing. The empirical dependence is given by
where G is a nondimensional function that takes the role of a correction coefficient due to waves nonlinearity, wave breaking, spray, etc. Some of the related parameters were investigated (Table 2), and the following simplified linear relationships were established.
First and foremost, G was found to decrease with the wave steepness, G = ak(−1.9 ± 0.58) + 1.2, where ±0.58 is the 95% confidence interval on the linear regression slope. The range of ak studied was 0.03 to 0.19. This result was found to be in agreement with previous observations (Peirson and Garcia 2008) and with the nonseparated sheltering theory of Belcher (1999). Although airflow streamlines were not resolved in this study and therefore airflow separation was not directly observed, we hypothesize that the nonseparated sheltering is the dominant mechanism controlling the wind-wave momentum transfer in the studied range of ak (0.03–0.19). Furthermore, we suggest that the separated sheltering mechanism, described by Kudryavtsev and Makin (2001), becomes dominant and enhances the wind-wave momentum flux for wave steepness above the range covered in this study.
Although the coefficient G potentially can be a function of many wind-wave parameters, the most robust dependence is on the wave steepness. Other parameters, considered in our analysis, were investigated over narrow ranges; therefore, the resulting parameterizations have only qualitative and preliminary meaning, calling for additional data to build statistical confidence.
According to the hypothesis of Andreas and Emanuel (2001), after spray particles detach from the water surface they are accelerated by the airflow. Therefore, as these particles reenter the water column, they transfer some of the wind momentum with them. This mechanism slows down the wind speed in close proximity to the water surface and hence is responsible for weaker pressure fluctuations and slower wave growth. Our rough estimate of spray concentration in the wind tunnel suggested that only two runs (with top wind speeds within our experiment) were conducted in heavy spray conditions. Although the results from these runs support the hypothesis, more data are needed to construct reliable quantitative parameterization.
Banner (1990) measured airflow pressure fluctuations above breaking waves and observed an enhancement of the wind-wave momentum flux. Our data suggests a similar dependence: that is, G(Pbr) = Pbr(0.66 ± 0.73) + 0.88. However, because of the limited amount of runs with the breaking probability Pbr > 0, large statistical uncertainly within the G(Pbr) function does not allow definite conclusions to be made. For wave fields with larger breaking probability and wave steepness, airflow streamlines are expected to switch to the full separation regime and sharply increase the value of the function γ(Uλ/2/Cp, ak). However, a more detailed experiment is needed to observe and quantify this phenomenon.
The authors gratefully acknowledge the support of the National Science Foundation (Grants OCE 0526318 and AGS 0933942) and the Office of Naval Research (Grant ONR N000140610288). We thank Mike Rebozo, Tom Snowden, and Hector Garcia for technical assistance during the experiments.
The Edge Detection Algorithm
One of the greatest technical challenges of the wave following experiment in rough conditions is the reliable and precise detection of the water surface location. An edge detection algorithm was developed specifically for the purpose of water surface measurements in rough wind-wave conditions. The algorithm was development for line scan image processing within the DLEG technique, as well as for areal images of a laser sheet crossing the water surface.
The idea behind the edge detection algorithm was to attempt to reconstruct the logic a human would use to find the water surface on an image such as Fig. A1. There are two main challenges that a rough surface poses for edge detection: first, spray particles are as bright as the surface and have a risk of being recognized as such; second, wave breaking foam, attached to the surface, misrepresents the actual water elevation. Both of these features are easily identified by a human eye but pose a significant challenge in automation development. To deal with the first problem, an image is defocused, where its resolution is decreased by a factor of 64 × 64. Because spray particles are small, their high brightness has little impact on mean brightness of 64 × 64 pixel areas. On the defocused image a raw estimate of the brightness edge can be easily found without the risk of spray contamination.
Although the coarse surface edge estimate effectively deals with spray, it would still have an error due to the foam. To filter that effect out, a critical surface wave curvature criterion was used. If the curvature of the coarse surface elevation signal ∂2η/∂x2 exceeded a critical value, such points were considered contaminated by the foam and were replaced by a function, smoothly filling the gap. Such function was found using small smoothing iterations, starting with unchanged function until the curvature criterion was satisfied.
The curvature threshold was optimized individually for each run, depending on wave conditions (i.e., expected maximum crest sharpness). In the image dimension, the typical critical curvature value was one vertical pixel per (one horizontal pixel × one horizontal pixel). Whether the replacement in form of the smooth function actually represents the true surface is an open question, because the air–water interface line does not exist within the foam. Nonetheless, the described method gives a good estimate on where the surface would have been in the absence of foam.
Once the coarse step of the edge detection is complete, the image resolution is increased by an arbitrary factor and the same edge detection principle is applied for the part of the image within close proximity to the rough elevation. The second step has the highest computation cost; therefore, such reduction in edge search area improves algorithm efficiency and reduces the chance of picking up an edge somewhere within air or water away from the surface. The third step increases the image resolution to the maximum and simply interpolates the surface elevation curve to provide elevation data for each pixel column of the original image.