1. Introduction
Sea surface waves generated by wind are random and thus have to be described in statistical terms. Modern theoretical analysis of wave statistics was initiated by Longuet-Higgins (1952) who showed that if the temporal variation of the surface elevation can be described by a narrow-banded Gaussian frequency spectrum, the wave heights follow the Rayleigh distribution. One of the first field studies of sea waves by Kinsman (1965) showed that the distribution of surface elevation, although skewed owing to the wave nonlinearity, have shape that is close to Gaussian, while the waves’ heights have exceedance probabilities that approximately follow Rayleigh distribution. It was also demonstrated that the probability distribution of the highest waves may deviate notably from the predictions based on the Rayleigh distribution. The knowledge of probability of steepest waves for a given wave state is of great practical importance for marine traffic and offshore engineering. Forristall (1978) reported that Rayleigh distribution overestimates the probability of high waves owing to their nonlinearity, while Tayfun (1980) attributes this discrepancy to wave breaking. For more recent statistical studies of wave height distribution in nonlinear wave field see, for example, Forristall (2005), Tayfun and Fedele (2007), and references therein.
Extensive experimental studies of statistics of mechanically excited unidirectional random water waves carried out in large wave tanks by Socquet-Juglard et al. (2005), Onorato et al. (2005, 2006), Mori et al. (2007), Shemer and Sergeeva (2009), and Shemer et al. (2010a,b), demonstrated significant deviations from the Gaussian behavior that were shown to be dependent on wave nonlinearity and spectral width. In these studies the evolution along the tank of the statistical parameters characterizing the wave field was investigated. The effect of the directional spreading was in the focus of large wave basin experiments by Onorato et al. (2009). In all these studies no wave energy was introduced to the wave field once the waves were generated by the wavemaker. Since the effects of dissipation were of relatively minor importance, the waves in those experiments constituted an essentially conservative system. Spatial evolution of the wave field statistics thus resulted nearly solely from the nonlinear interactions among waves.
Wind waves in general, and in a laboratory tank in particular, represent a qualitatively different wave system. Not only energy and momentum are transferred from wind to waves everywhere in the wave field, the dissipation, mainly because of breaking, in general cannot be neglected. The energy input and the energy sink are distributed nonuniformly among the wave frequencies and may be more prominent at different spectral ranges. Thus, the evolution of the statistical parameters in this case is very different from the conservative case; it is mostly dependent on wind input and dissipation.
It is generally accepted that the high-frequency tail of the wave spectrum is better approximated by a ω−4 shape than by the Phillips ω−5 power law; the coefficient of proportionality is independent of the wave age.
In the present study the spectral shapes, the wave height probability distributions, as well as numerous additional statistical parameters, are investigated under carefully controlled conditions in a laboratory wind wave flume. The experiments are carried out at a number of wind velocities and at numerous fetches along the test section.
2. Experimental facility and procedure
The experiment was conducted in a laboratory wind wave flume that consists of a closed-loop wind tunnel over a 5-m-long test section with the cross section 0.4 m by 0.5 m. The wind tunnel is equipped with large settling chambers at the inlet and at the exit of the test section. Side walls and the bottom of the test section are made of clear glass to enable flow visualization of the wave field from all directions. The test section is covered by transparent removable Perspex plates with a partially sealed slot along the center line to facilitate the sensors positioning. Water depth in the test section was kept at about 0.2 m, satisfying deep water conditions for wind wave lengths observed in this study; a flexible flap connects the bottom of the converging nozzle to the test section slightly above the mean water level height to ensure smooth airflow. A computer-controlled blower enables maximum wind speed in the test section that may exceed 15 m s−1. A heat exchanger connected to an external water chiller and controlled by a temperature controller was installed in the system and helped to eliminate excessive water evaporation because of air heating in the closed system. The chiller was set to keep the air temperature independent of wind velocity and constant at 22°C.
Capacitance-type wave gauge made of 0.3 mm tantalum wires was used for measuring instantaneous surface elevation, while a 1 mm ID Pitot tube connected to a sensitive pressure transducer (MAMAC Systems, INC 2 PR274) with a resolution of 2.5 · 10−5 Pa determined for the local mean air velocity. The sensors were supported by a carriage that was manually positioned at a desired fetch along the test section. The wave gauge was mounted on a computer-controlled vertical stage to enable its static calibration. At any given fetch and wind velocity, the calibration of the wave gauge was performed by a computer and covered the expected range of surface elevation variation. The Pitot tube was mounted on a separate accurate computer-controlled vertical stage that enabled its positioning at any prescribed height above the water surface. Temperature in the test section was monitored using the PT-100 resistance thermometer.
A high level of automation requiring minimum human intervention in conducting the experiment was achieved by the means of a LabView program that enables controlling the wind speed in the tunnel, calibration of the wave gauge before and after each measurement session, and data acquisition. For more detailed description of the experimental facility, the available instrumentation and the calibration and data acquisition procedures employed see Liberzon and Shemer (2011).
At each fetch and airflow rate characterized by the maximum wind velocity in the test section, Umax, the instantaneous variation of the surface elevation was continuously recorded for 90 min at the sampling rate 120 Hz, thus enabling to accumulate a very large ensemble of experimental data. Measurements were carried out at 8 fetches (x = 100, 140, 180, 220, 260, 300, 340, and 380 cm), and at 9 airflow rates in the test section. Values of Umax and of the corresponding friction velocities, u*, are summarized in Table 1. Methods used to determine the friction velocities from measurements of wind velocity profiles are described in Zavadsky and Shemer (2012).
Experimental conditions.
3. Results
The frequency spectra of the surface elevation variation with time η(t) were calculated by dividing each record into windows that contained 8192 points (duration of each window about 70 s), with 50% overlap, resulting in 94 data segments for each fetch and airflow rate. Variation with fetch of the surface elevation power spectra averaged over all windows is presented in Figs. 1a,b for two representative airflow rates.
Variation of surface elevation spectra along the test section: (a) Umax = 5.5 m s−1 and (b) Umax = 12.3 m s−1.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
Growth of the total wave energy along the channel is observed for all wind velocities. At each wind speed, the peak frequency decreases with fetch; comparison of Figs. 1a,b demonstrates the shift of the peak toward lower frequencies with increasing wind speed, so that for every given fetch the wavelength increases with increasing wind speed. The results of Fig. 1 are in agreement with Liberzon and Shemer (2011).
Since fdom is an integral quantity and thus less subject to fluctuations in the experimentally estimated wave power spectra; it represents the most robust characteristic frequency at each fetch and airflow rate. Note that in (9) the integration is carried out over the free waves domain only; in the present study this domain for each spectrum ωmin < ω < ωmax is taken within ±60% of the peak frequency.
More often, however, the frequency of the spectral peak fp is chosen as the characteristic of the wave field rather than fdom since it constitutes a more intuitively straightforward physical and visual parameter. To mitigate the inevitable scatter in the experimentally determined power spectra, see Fig. 1, the value of fp is defined here as the frequency of the peak of the parabolic fit performed for ±20 data points (about ±0.3 Hz) of the measured spectra around the corresponding maximum values.
The peak frequencies measured at all fetches and wind velocities in the present experiments are summarized in Fig. 2. As expected, the values of fp decrease at each fetch with wind velocity; for any given wind flow rate, the peak frequency decreases with fetch.
Variation of the peak frequency fp along the tank.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
The comparison of the values of fdom and fp is presented in Fig. 3. Although the difference between these two parameters is small, the dominant frequency is consistently larger than fp owing to higher weight of higher frequencies in the computation of the spectral moments m1, see (9). The linear fit that has a slope of 0.96 does not contain data points corresponding to two lowest wind flow rates; for those wind velocities the waves in the test section are characterized by small amplitudes and wide spectra without well-defined peaks. The peak frequency at these conditions cannot be determined with sufficient accuracy and the results exhibit considerable scatter.
Comparison of fdom and fp.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
Note that the characteristic wave amplitude can be presented in (11) either by the rms value of the surface elevation 
Figures 4 and 5 present the measured dependence of the dimensionless peak frequency 
Dimensionless peak frequency 
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
Dimensionless wave amplitude 
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
As noticed above, not all data points fit the power law pattern. Notable deviations from relations (13) and (14) were also observed by Mitsuyasu (1968) and Mitsuyasu and Honda (1974). The points in Figs. 4 and 5 that do not fit the general trend and marked by filled symbols generally correspond to the lowest wind velocities in the present experiments (and thus higher dimensionless fetches 
Variation of the phase velocities of dominant waves along the test section.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
Wave age vs mean wind velocity.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
This result differs somewhat from the generally accepted 
The significant wave height Hs (mm).
The characteristic steepness of random waves ɛ may be defined as the product of the representative wave amplitude calculated as an rms value of the surface elevation, and the peak wavenumber kp = k(fp); 
The characteristic wave steepness: (a) as a function of fetch; and (b) as a function of wave age.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
Water surface state at the most severe experimental conditions (fetch around 340 cm, Umax = 11.2 m s−1).
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
When plotted against the wave age, Fig. 8b, the characteristic steepness values decrease following a linear trend for the range of parameters investigated. Note the variation of the slope of the pattern for each wind flow rate in the test section, from a nearly vertical at low wind velocities, corresponding to considerable change in steepness coupled with a nearly independent of fetch wave age at these conditions, to close to horizontal for stronger winds, where the steepness remains practically constant while the wave age increases along the test section.
The surface elevation power spectra are now examined in greater detail in Fig. 10. In Fig. 10a the spectra obtained at a constant airflow rate are plotted in a logarithmic scale as a function of the dimensional frequency, whereas in Fig. 10b the power spectra for a wide range of wind velocities are normalized by their peak values and presented as a function of f/fp. The line corresponding to the power law with n = 4 is also plotted in Fig. 10.
The power spectrum of the surface elevation for (a) different fetches at Umax = 6.6 m s−1 and (b) various wind velocities at x = 260 cm.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
The spectra in Fig. 10 are characterized by clearly visible peaks at the second and the third harmonics of the peak frequency. At low wind velocities even weaker peaks corresponding fourth harmonic can be identified. Those peaks signify the bound waves contribution to the nonlinear wave field. At frequencies exceeding the third or the fourth harmonics of the peak frequencies, the bound waves’ contributions cannot be distinguished; those frequencies correspond to the so-called quasi-saturated high-frequency tail. The normalized power spectra of the surface elevation for a range of wind flow rates in Fig. 10b seem to collapse on a single curve, as long as the frequency is below about 3fp. The tail behavior is only in a qualitative agreement with the n = 4 slope; notable spread of the tail slopes is clearly visible in Fig. 10b.
Variation of the spectral width ν along the tank.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
The values of ν characterize the spectral shape at lower frequencies around fp. A closer look at the power law behavior of the high-frequency spectral tail is presented in Fig. 12 that is plotted in normalized coordinates as in Fig. 10.
High-frequency part of the surface elevation power spectrum for Umax = 8.9 m s−1 and various fetches.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
The power n defined by the slope of the linear fit in Fig. 12 increases with fetch from n = 3.1 for x = 100 cm to n = 3.74 for the longest fetch, x = 380 cm. The solid line representing the “−4 power law” is also plotted for comparison. The fit range was chosen as from 3 < f/fp < 6, as the spectra in this part are free of noticeable higher harmonic peaks which can affect the slope.
The values of C and n of the high-frequency tail part of the spectrum retrieved from the fit (17) are summarized in Fig. 13. Only measurements performed at airflow rates corresponding to Umax ≥ 6.6 m s−1 were considered. The values of n plotted as a function of wave-age in Fig. 13a seem to follow a common trend growing monotonically with wave age from n = 3 to about n = 4 for more mature waves at larger fetches and relatively low wind velocities, thus attaining the typical value for the fully developed sea (Toba 1973; Badulin et al. 2007). The coefficients C in (17) presented in Fig. 13b are scattered around the value of 0.1, the scatter of data seems to decrease with wave age.
(a) The power n as a function of wave-age and (b) coefficient of proportionality as a function of wave-age.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
The skewness coefficient λ3.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
The kurtosis coefficient λ4.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
The kurtosis coefficient λ4 plotted in Fig. 15 is below the Gaussian value of 3 for all wind velocities and at all fetches, except for the low wind velocity at moderate fetches where it is slightly above that value. For wind velocities exceeding about 7 m s−1, the kurtosis coefficient seems to be nearly constant, with λ4 being in the range 2.4–2.6 and essentially independent of both the fetch and the wind velocity. These values of λ4 indicate at a relative deficit of large amplitude waves in the distribution. The values of kurtosis in Fig. 15 are in qualitative agreement with Huang and Long (1980), as well as with the experimental results of Shemer et al. (2010a,b) obtained for mechanically generated unidirectional random waves at comparable values of the spectral width ν. They disagree, however, with the results of wind wave simulations by Annenkov and Shrira (2009) based on the Zakharov equation who reported on λ4 > 3.
Probability distribution for wave heights: (a) Umax = 3.3 m s−1 and (b) Umax = 12.3 m s−1. Solid curves with markers denote the TF3 distributions at corresponding fetches.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
Longuet-Higgins (1963) also has shown that nonlinearities can cause the deviation of surface elevation distributions F(H) from the Gaussian statistics and approximated F(H) by a Gram-Charlier expansion. Huang and Long (1980) carried out a comparison of the experimentally measured wind waves’ height distributions with the Longuet-Higgins (1963) model and obtained a reasonable agreement between their results and the theory, although some problems in application of the Gram-Charlier approximation called for additional studies. Later, Tayfun and Fedele (2007) suggested coupling of the Gram-Charlier expansion with the Tayfun (1980, 2006) model for narrow-banded sea.
In Fig. 16 the exceedance distributions obtained in the present experiments are compared with the Rayleigh distribution as well as with the 3rd order Tayfun and Fedele (2007) model (TF3). For all experimental conditions, notable deviations from the Rayleigh distribution are indeed clearly visible in this Figure. For the Rayleigh distribution, the significant wave height Hs ≈ 4
4. Discussion and conclusions
A random quasi-steady wave field evolving in a small-scale wind wave flume was studied. Compared to field studies, laboratory experiments offer an obvious considerable advantage of accurate control of flow conditions, repeatability, and possibility of detailed and extended measurements under steady conditions. To assess the relevance of experiments in a small flume, the extent of similarity between waves excited in such laboratory facility and wind-generated waves in the sea was examined. For that purpose extended and statistically significant temporal records of instantaneous water surface fluctuations under variety of steady wind forcing conditions and at numerous fetches along the flume were accumulated. Statistical and spectral parameters of the wave field evolving along the test section were compared with the available theoretical and experimental data accumulated during both laboratory and field studies. Spectral integral moments were used to calculate the characteristic wave amplitudes and frequencies, as well as the spectral widths. The appropriate dimensionless parameters governing wind water waves’ excitation and evolution were considered to enable comparison between the results obtained investigating wave fields in laboratory and in the sea, as those differ notably by their spatial and temporal scales. Relations between such parameters in the present experimental facility were obtained and compared with available data derived on the basis of laboratory and field studies. Finally, a statistical characterization of wave heights distribution was evaluated by calculating skewness and kurtosis coefficients, as well as the wave height exceedance distributions.
Wave energy growth with fetch was documented for a range of wind speeds. The friction velocity u* was used as a characteristic wind velocity being a more natural characteristic in the laboratory facility than the U10 velocity often used in the field measurements. Relation between the dimensionless characteristic surface elevation amplitude and the dimensionless peak frequency depicted in Fig. 4 corresponds to power law with the power value of −1.87, not very different from the empirical value of −1.5 suggested by Toba (1972) on the basis of field data analysis. The wave amplitude growth with the dimensionless fetch observed in the present experiments and summarized in Fig. 5 is also in good agreement with the available laboratory results by Mitsuyasu and Honda (1974), Mitsuyasu (1968), as well as the field observations (Kahma 1981; Hasselmann et al. 1980). The agreement becomes less impressive for lower wind velocities, Umax < 7 m s−1, where the wave field is characterized by wide-band waves of very small amplitudes, so the results are less reliable and exhibit considerable scatter.
The power spectra were obtained by averaging a large number of statistically independent estimates, taking advantage of steady conditions in the experimental facility. The averaged spectra for each fetch and wind velocity covered up to 5 decades. Two spectral domains, the high-frequency tail, and the vicinity of the spectral peak, were treated separately. To estimate the spectral tail behavior care was taken to consider frequencies exceeding about 3.5fp. At those frequencies the direct effect of bound waves on the spectral shape is not discernible anymore, see Figs. 10 and 12; it thus can be assumed that the slope represents mainly the contribution of free waves. This distinction allows carrying out comparison of the present results with theoretical estimates based on considering free waves only—see, for example, Badulin et al. (2007). The spectral tails dependence on frequency was found to follow the power law (17) with n about 3 at lower wave ages and then increasing with c/u*, but generally remaining somewhat below the values of n = 4 suggested by Toba (1973) and Badulin et al. (2007). The exponents n attain the value of n = 4 in the present study only at the largest wave ages. It should be stressed, however, that those wave ages correspond to the very small wind waves at shorter fetches and lower wind velocities; as stressed above, the results at these conditions are less accurate. Moreover, the waves at the frequency range considered are strongly affected by surface tension, while Badulin et al. (2007) consider pure gravity waves. The values of the coefficient C in the power relation (17) presented in Fig. 13b are also close to C = 0.1 suggested by Toba. It thus can be concluded that the behavior of the high frequency spectral tail in the present experiments does not differ significantly from that obtained in the previous theoretical and field studies.
The spectral shape (23) was therefore fitted to the data plotted in Fig. 17. The coefficients γ, σR, and σL were obtained from this fit for each fetch using the corresponding dimensionless spectral widths ν averaged over wind velocities, see Fig. 11. The fitted shapes are in a good agreement with the data at both sides of the peak value 
Averaged over all fetches exceeding 100 cm normalized power spectra as a function of 
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
A very good agreement was obtained for the longer waves at the left side of the spectrum; the fit for the higher frequency part of the spectrum deviated slightly from the experimental data starting from about 
The peak power of the JONSWAP spectrum is determined by the coefficient β, see (22). Values of β calculated from the measured spectra using significant wave heights HS presented in Table 2 are plotted in Fig. 18 as a function of wave age. Calculation for the value of γ obtained in the present study according to Goda (2000) yields β ≈ 0.21. Figure 18 demonstrates that for all experimental conditions except of high values of 
The peak energy coefficient β in the JONSWAP spectrum as a function of c/u*.
Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-0103.1
Next, the waves shape and the surface elevation probability distributions were also compared with the corresponding values characterizing sea waves. Steepness of the dominant waves component, ɛ, grew with the increase in Umax, while its dependence on fetch quickly diminishes (Fig. 8). At all experimental conditions values of ɛ did not exceed the apparent saturation value of about 0.2, constrained by wave energy dissipation that at the length scales characteristic for the present experiments is mainly due to microbreaking and the appearance of parasitic ripples, see Caulliez et al. (2008). For stronger winds the wave steepness remained virtually constant and independent of fetch and wave age, see Fig. 8a. Although the dissipation mechanism in the present experiments may differ from that in the open sea, the limiting value of the wave steepness was found to be similar to the one observed in field measurements.
Deviations of the surface elevation variation from that corresponding to Gaussian were examined using higher distribution moments of skewness λ3 and kurtosis λ4, see (20). The wave height exceedance distribution was compared with the Rayleigh distribution. For all experimental conditions values of λ3 were positive and in general agreement with the theoretical estimates of Tayfun (2006) and the experimental results of Shemer and Sergeeva (2009) for a narrow-banded random wave field. Values of kurtosis coefficient λ4 were below 3, indicating relative deficit in large amplitudes in the ensemble, in agreement with the findings of Shemer et al. (2010a) for unidirectional mechanically generated random waves studied in a very large experimental wave tank. The low kurtosis values suggested that the deviation from Gaussian distribution of wave heights probability can be expected. Indeed deviation from Rayleigh distribution for all experimental conditions was obtained for all experimental conditions (Fig. 16). While the probability of relatively small waves was higher than that corresponding to the Rayleigh distribution, the probability of very high waves was vanishingly small. The experimental results showed good qualitative agreement with the TF3 model of broad banded wind waves heights distribution. Quantitative agreement with the TF3 model improves for higher wind speeds and at longer fetches and thus longer waves dominating the wave field; these conditions correspond to narrower surface elevation spectra (Fig. 10). Results obtained for waves driven by wind-induced stresses are again in agreement with Shemer et al. (2010a) who demonstrated that for surface stress-free wavemaker-generated waves with relatively broad spectra, the probability of the so-called freak waves is virtually zero.
To conclude, extensive statistical characteristics of wind-generated waves were accumulated in the present experiments carried out in a small experimental facility. The results are presented using appropriate dimensionless parameters, thus enabling detailed comparison with the available data obtained either in larger experimental installations, mostly in the absence of wind, or during field measurements. In particular, a dimensionless spectral form for the energy containing domain around the peak wave frequency was suggested allowing direct quantitative comparison of numerous spectra obtained at various conditions. Strong similarity was obtained between the present results and those obtained in the open sea. The presented study thus demonstrates that in spite of certain limitations imposed by size, experimental research on wind generated water waves in relatively small experimental facilities may yield valuable results, applicable at considerably larger scales. Experiments in small- and midsize laboratory facilities, beyond being relatively inexpensive, have significant benefits as they allow detailed investigation of diverse processes associated with the generation of water waves by wind under repeatable conditions in a controlled environment and with temporal and spatial resolutions that cannot be achieved during field measurements.
Acknowledgments
The authors gratefully acknowledge support of this study by the Israel Science Foundation under Grant 153/11. We also acknowledge the help offered by Dr. A. Sergeeva.
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