3.1 Water surface response to activation of the blower

The variation of the surface drift velocity during the initial stage of the wind–wave field evolution, with the wind starting to blow over an initially smooth water surface, is presented first. The drift velocity excited by shear stress at the air–water interface due to wind constitutes the boundary condition for determination of the wind velocity profile and is thus essential for the wind–wave momentum exchange (Caulliez et al. Reference Caulliez, Makin and Kudryavtsev2008; Liberzon & Shemer Reference Liberzon and Shemer2011). The rate of adjustment of the current at the water surface to the varying wind velocity was measured in a separate series of experiments. At each fetch, measurements were performed while the wind velocity in the test section grows from zero to one of the prescribed steady-state values, $U=6.5$ , 8.5 and $10.5~\text{m}~\text{s}^{-1}$ . Following Liberzon & Shemer (Reference Liberzon and Shemer2011), particle tracking velocimetry (PTV) technique was used for this purpose. Black 6 mm diameter paper disks were spread over the still water surface; a video camera operating at 60 frames per second with resolution of 1280 pixel $\times$ 720 pixel located above the water surface recorded the position of the floaters which move with water surface following activation of the blower. The camera imaged water surface area of approximately 15 cm in the wind direction and approximately 10 cm across the test section. The initiation of video recording was synchronized with the instant of the blower activation. For each fetch and wind velocity, from 5 to 7 independent experimental runs were performed, with approximately 200 floaters tracked in each run.

The PTV results are shown in figure 4. It can be seen that the time needed for the drift velocity to attain a quasi-steady value does not vary significantly with fetch and with wind velocity, and is approximately 4–5 s. In all cases, in the course of the initial 4 s or so, the wind velocity increases nearly linearly with time and attains the value of approximately $6{-}7~\text{m}~\text{s}^{-1}$ (see figure 2). During this time interval, the surface drift velocity is apparently independent of fetch, attaining a value of approximately $13~\text{cm}~\text{s}^{-1}$ , i.e. approximately 2 % of the wind velocity. At longer times that correspond to higher wind velocities, the surface drift velocity ceases to grow monotonically. Note that at those wind velocities, in particular at larger fetches, the PTV measurements become less accurate, with the orbital motion of the emerged wind–wave system becoming a significant factor in the movement of the floaters; moreover, some floaters get submerged resulting in spurious recorded velocities. Nevertheless, it is apparent that once the wind velocity in the test section exceeds approximately $6~\text{m}~\text{s}^{-1}$ , the surface drift velocity attains quasi-steady values that are scattered in the range of $15{-}25~\text{cm}~\text{s}^{-1}$ corresponding to steady forcing, with an obvious trend to faster drift velocities at higher values of $U$ . Fluctuations in surface drift velocity are discussed in greater detail in § 3.5.

Figure 4.

Variation of the PTV-measured surface drift velocity with the increasing wind velocity $U$ . Different curves correspond to different duration $t_{0}$ of the wind acceleration stage.

The temporal variation of the surface drift velocity can be estimated theoretically assuming that during the initial stages, the water flow induced by wind shear is laminar. As can be seen in figure 2, the temporal variation of $U$ can be approximated as $U~(\text{m}~\text{s}^{-1})\approx 2.1\cdot t^{\prime }~(\text{s})$ , where in order to reflect the delay of the air flow in the test section relative to the blower driving signal due to the system inertia, the effective time $t^{\prime }$ is shifted relative to the elapsed time $t$ by $t_{sh}=1$  s, so that $t^{\prime }=t-t_{sh}$ . It is assumed that the behaviour of the boundary layer in the air above the water is quasi-steady, and thus the interfacial shear stress $\unicode[STIX]{x1D70F}_{0}$ at any instant is adjusted to the instantaneous mean wind velocity $U$ . Under this assumption, the interfacial shear stress $\unicode[STIX]{x1D70F}_{0}=\unicode[STIX]{x1D70C}_{air}u_{\ast }^{2}$ , where $\unicode[STIX]{x1D70C}_{air}$ is air density; the friction velocity $u_{\ast }$ corresponds to the wind velocity $U$ under steady forcing, see § 2. Thus, the interfacial shear is a quadratic function of the shifted elapsed time $t^{\prime }$ during wind acceleration stage, $0<t^{\prime }<t_{0}$ , and retains a constant value $\unicode[STIX]{x1D70F}_{0}=\unicode[STIX]{x1D70F}_{0}(t^{\prime }=t_{0})$ at $t^{\prime }>t_{0}$ . For spatially constant interfacial shear stress $\unicode[STIX]{x1D70F}_{0}(t^{\prime })$ , the temporal variation of the induced current in water at depth $z$ , $u_{w}(z,t)$ , is described by one-dimensional diffusion equation (see, e.g. Veron & Melville Reference Veron and Melville2001). The solution of this problem for arbitrary time dependent $\unicode[STIX]{x1D70F}_{0}$ is given by Carslaw & Jaeger (Reference Carslaw and Jaeger1959) in the form of Duhamel’s integral. The surface drift velocity $U_{d}=u_{w}(z=0,t)$ is obtained from this solution as

In (3.1), $\unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D70C}_{w}$ are respectively kinematic viscosity and density of water. The solution (3.1a ) for the surface drift velocity $U_{d}$ during the wind acceleration stage is identical to the expression derived by Veron & Melville (Reference Veron and Melville2001) for interfacial shear stress growing quadratically with time.

It thus can be concluded that during the initial stage of wind acceleration, the water flow induced by shear stress is laminar, whereas the boundary layer in the air indeed adjusts fast to the instantaneous wind velocity, so that the interfacial shear stress induced by the turbulent air flow above the water surface at each instant can be determined from the value of $U$ at that instant. The solution (3.1) that predicts continuous increase in time of the surface velocity ceases to agree with the experimental results when the surface drift velocity increases beyond approximately $15{-}20~\text{cm}~\text{s}^{-1}$ and the mean velocity does not grow anymore. These results are consistent with measurements of drift velocity $U_{d}$ in field experiments where it was estimated as approximately 3 % of the wind velocity (Kudryavtsev et al. Reference Kudryavtsev, Shrira, Dulov and Malinovsky2008). The termination of the mean drift velocity growth in the laboratory experiments may be attributed to different reasons, one of them being the finite length of the experimental facility that leads to stagnation at the far end of the test section and accompanied by surface set-up.

It is instructive to examine representative time variation records of the simultaneously acquired surface elevation $\unicode[STIX]{x1D702}$ and of the two components of surface slope, $\unicode[STIX]{x2202}\unicode[STIX]{x1D702}/\unicode[STIX]{x2202}x$ and $\unicode[STIX]{x2202}\unicode[STIX]{x1D702}/\unicode[STIX]{x2202}y$ , plotted in figure 5. The records were made at $x=220$  cm and wind velocity $U=8.5~\text{m}~\text{s}^{-1}$ . The values of the slope components are multiplied by a factor of 10 and shifted vertically for convenience. Visible fluctuations of the surface elevation, as well as of the surface slope components, appear in figure 5 only after approximately 3.5 s elapse since the activation of the blower. Comparison of wind velocity $U(t)$ shown in figure 2 with the variation of the water surface characteristics in figure 5 demonstrates that it can be assumed that detectable development of the wave field starts only after the constant wind velocity in the test section has been attained. For target wind velocities not exceeding approximately $U=8.5~\text{m}~\text{s}^{-1}$ , the wind forcing in the present study can thus effectively be presented by a step function. The quasi steady-state level of the fluctuations of the surface elevation $\unicode[STIX]{x1D702}$ in figure 5 is only attained at $t>9$  s, more than 5 s after the appearance of the initial visible disturbances at the water surface. Contrary to that, the quasi-steady level of fluctuations of $\unicode[STIX]{x2202}\unicode[STIX]{x1D702}/\unicode[STIX]{x2202}x$ is attained already at $t\approx 4$  s, only approximately 1 s after the inception of disturbances. The fluctuations of $\unicode[STIX]{x2202}\unicode[STIX]{x1D702}/\unicode[STIX]{x2202}y$ attain their quasi steady level at comparable times.

While the quasi-steady levels of the range of fluctuations in all records plotted in figure 5 are attained relatively fast, the characteristic frequencies of those oscillations appear to vary over substantially longer times. The decrease in the characteristic frequency of $\unicode[STIX]{x2202}\unicode[STIX]{x1D702}/\unicode[STIX]{x2202}t$ is clearly visible in this figure; the records of $\unicode[STIX]{x2202}\unicode[STIX]{x1D702}/\unicode[STIX]{x2202}x$ and $\unicode[STIX]{x2202}\unicode[STIX]{x1D702}/\unicode[STIX]{x2202}y$ seem to have somewhat higher frequencies.

Figure 5.

Single realization records of the surface elevation $\unicode[STIX]{x1D702}(t)$ and the instantaneous surface slope components, $\unicode[STIX]{x1D702}_{x}(t)$ and $\unicode[STIX]{x1D702}_{y}(t)$ acquired at $x=220$  cm and $U=8.5~\text{m}~\text{s}^{-1}$ .

Prior to emergence of wind waves, appearance of longitudinal streaks can be identified at the water surface along the whole tank. A snapshot of these streaks taken at the fetch of $x=220$  cm at approximately 3 s after the initiation of the blower is given in figure 6. Fast video imaging indicates that the longitudinal streaks remain visible for less than 1 s. The existence of those streaks was noticed before (see e.g. Caulliez, Ricci & Dupont Reference Caulliez, Ricci and Dupont1998). Melville, Shear & Veron (Reference Melville, Shear and Veron1998) and Veron & Melville (Reference Veron and Melville2001) studied the appearance of the streaks within the general framework of Langmuir circulation (LC) excited in water by a slowly accelerating wind. Phillips (Reference Phillips2005) demonstrated theoretically that the preferred LC cross-wind streak spacing agrees well with their observations. Recently, similar streaks on the surface of a viscous liquid were studied in greater detail by Paquier, Moisy & Rabaud (Reference Paquier, Moisy and Rabaud2015).

Figure 6.

Snapshot of the lengthwise streaks at the pre-growing stage at $x=220$  cm (image size approximately 30 by 10 cm $^{2}$ ).

Figure 7.

(a) The ensemble-averaged wavelet-derived dominant frequency at $x=120$  cm,  $U=6.5~\text{m}~\text{s}^{-1}$ and $x=340$  cm,  $U=10.5~\text{m}~\text{s}^{-1}$ during the initial stage of wind wave emergence based on the records of different parameters. (b) Comparison of the dominant frequency during the quasi-steady stage retrieved by two methods.

3.2 Characterization of the evolving wind–wave field by ensemble-averaged parameters

Synchronous temporal records of the surface variation provided by different sensors in each realization enable estimates of the instantaneous characteristic frequency for each signal independently. The variation of the dominant wave frequency with the time elapsed from the blower activation was performed separately using wavelet analysis of the surface elevation, $\unicode[STIX]{x1D702}(t)$ , and of the two components of the surface slope, $\unicode[STIX]{x1D702}_{x}(t)$ and $\unicode[STIX]{x1D702}_{y}(t)$ . The representative results for two extreme cases corresponding to the shortest fetch and lowest wind velocity, and the longest fetch and highest wind velocity, are presented in figure 7(a) for the initial stages of the wind–wave evolution. There are considerable similarities between all curves in this figure. At both locations and both wind velocities, reliable results can be obtained starting from approximately 3.5–4 s after the blower activation. The response of the surface slope sensor precedes somewhat the response of the wave gauge. When waves in the records of $\unicode[STIX]{x1D702}(t)$ , $\unicode[STIX]{x1D702}_{x}(t)$ and $\unicode[STIX]{x1D702}_{y}(t)$ first become detectable, their dominant frequency is in the range of 16–20 Hz, with no pronounced dependence on fetch or wind velocity. Veron & Melville (Reference Veron and Melville2001) reported on similar frequencies of the first detectable waves in a notably larger experimental facility. During the initial few seconds of the evolution process, the dominant frequencies of the surface slope components are higher to some extent than those obtained from the surface elevation records. Note that for the fetch $x=340$  cm and $U=10.5~\text{m}~\text{s}^{-1}$ , water surface disturbances with frequency of approximately 14 Hz were identified in the $\unicode[STIX]{x1D702}_{y}(t)$ record at $3.1~\text{s}<t<3.4~\text{s}$ . This period is within the interval when the longitudinal streaks shown in figure 6 were observed. Those streaks have very small amplitudes and their existence can only occasionally be detected. Due to their orientation, the longitudinal streaks are usually seen in $\unicode[STIX]{x1D702}_{y}(t)$ records.

One can see in figure 7(a) that within 8–9 s following the activation of the blower, the variation of the dominant frequency with time practically depends neither on fetch nor on the wind velocity. The differences between the curves during this time interval can be attributed at least partially to inevitable inaccuracies associated with the weakness of the signals produced by those very young wind waves. The low intensities in the wavelet map corresponding to dominant frequencies for $t<9$  s as seen in figure 3(b) lead to a considerable scatter of results. Nevertheless, it is important to stress that different sensors yield a consistent pattern of frequency variation, which decreases from the initial maximum around $f=18$  Hz at $t\approx 4$  s to approximately $f=7$  Hz at $t\approx 8{-}9$  s. This duration-limited stage then terminates, and the dominant frequencies at different fetches start to diverge. The frequencies eventually attain their corresponding fetch- and wind velocity-dependent steady-state values, however, the differences between the frequencies obtained at every fetch and wind velocity from the two slope components and from the surface elevation signal nearly vanish. To examine the accuracy of the wavelet-derived dominant frequencies, Fourier analysis was also applied for the quasi-steady stage of the wave evolution. The fast Fourier transform (FFT) was used to compute the surface elevation spectra during the 60 s long segments recorded at $60~\text{s}<t<120~\text{s}$ . The resulting power spectra were averaged over all realizations to determine the frequency of the spectral peak. The peak frequencies for all fetches and wind velocities are compared in figure 7(b) with the dominant frequencies obtained applying the wavelet procedure on the same records. The wavelet-derived dominant frequencies in this figure represent the mean values over the whole duration of the segment. The discrepancy between the frequencies obtained by the two methods does not exceed a few per cent.

Figure 8.

Temporal variation of the wave parameters, columns of the figures denote fetch increasing from left to right, rows denote wind velocities increasing from up to down: lines with circle markers denote $\sqrt{\langle \unicode[STIX]{x1D702}^{2}\rangle }$ , squares – instantaneous dominant frequency of $\unicode[STIX]{x1D702}$ , $f_{dom}$ , triangles – $\sqrt{\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle }$ , pentagrams – dominant wavelength, $\unicode[STIX]{x1D706}_{dom}$ .

For a given dominant frequency, the dominant wavelength at each stage of the evolution can be determined provided the dispersion relation is known. For the range of wave frequencies in figure 7(a), the Doppler shift due to the presence of the induced shear current cannot be neglected. Since the surface drift current follows faithfully the instantaneous wind velocity with no dependence on fetch, and attains steady values prior to development of notable waves in the tank, figure 4, application of the empirical dispersion relation suggested for wind–wave field under steady forcing in the presence of induced shear current is justified. By measuring independently wave frequency and phase velocity, Liberzon & Shemer (Reference Liberzon and Shemer2011) suggested empirical dispersion relation in the following form:

where $k_{gc}$ is the wavenumber that corresponds to the measured radian frequency $\unicode[STIX]{x1D714}=2\unicode[STIX]{x03C0}f$ according to the gravity–capillary dispersion relation for deep water

$\unicode[STIX]{x1D70E}$ being water surface tension coefficient and $\unicode[STIX]{x1D70C}$ water density. Zavadsky et al. (Reference Zavadsky, Benetazzo and Shemer2017) obtained the dispersion relation in the form given by (3.2) measuring independently frequency and wavenumber of wind waves by wave and laser slope gauges. The coefficients $a=0.006$  m and $b=-2.2\times 10^{-5}~\text{m}^{2}$ obtained in their study are close to those presented by Liberzon & Shemer (Reference Liberzon and Shemer2011). The dominant wavelength at each instant is thus $\unicode[STIX]{x1D706}=2\unicode[STIX]{x03C0}/k$ , where $k$ is the wavenumber calculated from (3.2) for the instantaneous value of the dominant frequency.

The temporal variation of main ensemble-averaged wave parameters, denoted by angle brackets, starting from the instant of the blower activation up to approaching the steady state, is plotted in figure 8 for three fetches and five wind velocities. In each panel, the instantaneous r.m.s. values of the surface elevation that characterize wave amplitude, $\sqrt{\langle \unicode[STIX]{x1D702}^{2}\rangle }$ , and of the surface slope $\sqrt{\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle }$ that can serve as indication of characteristic wave steepness, are plotted together with the dominant frequency and the corresponding wavelength. Note that the behaviour of $\langle \unicode[STIX]{x1D702}_{y}^{2}\rangle ^{1/2}$ is similar to that of $\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle ^{1/2}$ ; the corresponding curves are not plotted in order not to overload figure 8. The dominant frequency $f_{dom}$ is obtained using the records of the surface elevation $\unicode[STIX]{x1D702}$ ; the corresponding dominant wavelength $\unicode[STIX]{x1D706}_{dom}$ calculated using the empirical dispersion relation (3.2).

The high temporal resolution of the ensemble-averaged results that corresponds to the sampling frequency of 300 Hz makes it possible to distinguish between different stages in the wave field evolution. At all fetches and wind velocities, during the initial 4 s or so, the disturbances appearing at the water surface are too small to be characterized quantitatively. Following this wind acceleration stage, wave field is characterized by a rapid growth of very short waves with the r.m.s. values of surface elevation not exceeding approximately 1 mm over the whole length of the test section. The salient feature of this process is a nearly impulsive increase in the r.m.s. values of the surface slope $\sqrt{\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle }$ that represent the wave steepness; they rapidly attain quasi-steady values. The characteristic slopes are practically independent of fetch at a given wind velocity, and increase somewhat with $U$ , varying from approximately $\sqrt{\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle }=0.17$ for $U=6.5~\text{m}~\text{s}^{-1}$ to $\sqrt{\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle }=0.22$ for higher wind velocities. A delay of about 0.8 s in appearance of those ripples exists at fetches exceeding 120 cm at the lowest wind velocity ( $U=6.5~\text{m}~\text{s}^{-1}$ ); at $U>6.5~\text{m}~\text{s}^{-1}$ no such delay can be identified. The characteristic frequency of the shortest identifiable ripples exceeds 15 Hz at all fetches and wind velocities; the values of $f_{dom}$ then decrease.

At longer fetches and for stronger winds presented in figure 8, the dominant frequency decreases to approximately 10 Hz by the time when quasi-steady steepness is attained. While surface slopes attain their steady-state value quite fast, the process of evolution of other wind–wave parameters plotted in figure 8 is notably slower. As can be seen by comparing panels in each row of that figure, the increase in the values of $\sqrt{\langle \unicode[STIX]{x1D702}^{2}\rangle }$ is initially nearly identical for all fetches at the given wind velocity. Similarly, the decrease in the dominant frequency at those initial stages is essentially common for all panels. At $x=120$  cm, both the r.m.s. values of $\unicode[STIX]{x1D702}$ and the dominant frequency $f_{dom}$ attain their quasi-steady values at approximately 7–8 s after activation of the blower for all wind velocities. At longer fetches, the values of $\sqrt{\langle \unicode[STIX]{x1D702}^{2}\rangle }$ and of $f_{dom}$ continue to evolve, attaining quasi-steady values at approximately 12 s for the fetch $x=220$  cm, and at approximately 16–17 s for $x=340$  cm.

To verify this LSG-measured almost immediate growth of the ensemble-averaged surface slope $\sqrt{\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle }$ , an independent slope evaluation can be carried out using the wave-gauge records as $\sqrt{\langle \unicode[STIX]{x1D702}_{x}(t)^{2}\rangle }=k(t)\sqrt{\langle \unicode[STIX]{x1D702}(t)^{2}\rangle }$ . Figure 9 shows the comparison between the LSG-measured variation of the surface slope $\sqrt{\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle }$ and the wave-gauge-derived temporal variation of the slope reconstructed from $\sqrt{\langle \unicode[STIX]{x1D702}^{2}\rangle }$ and the wavenumber $k(t)=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}(t)$ as plotted in figure 8. Results presented for $U=10.5~\text{m}~\text{s}^{-1}$ and $x=120$  cm demonstrate consistent behaviour. Thus the joint variation of wave amplitude and wavelength following the initial ‘jump’ in $\sqrt{\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle }$ keeps the wave slope nearly constant during the whole growth stage.

Figure 9.

Comparison of temporal variation of the surface slope obtained by two methods at $U=10.5~\text{m}~\text{s}^{-1}$ , $x=120$  cm.

3.3 Scenario for the initial wind–wave system growth under impulsive forcing

Careful comparison of the variation of $\sqrt{\langle \unicode[STIX]{x1D702}^{2}\rangle }$ and $f_{dom}$ in each row of figure 8 (for an identical wind velocity $U$ ) reveals that at longer fetches, at the instant when the dominant frequencies $f_{dom}$ attain the value corresponding to the steady state $f_{dom}$ at one of the shorter fetches, the instantaneous ensemble-averaged values of $\sqrt{\langle \unicode[STIX]{x1D702}^{2}\rangle }$ are also identical to the steady r.m.s. values of the surface elevation at that shorter fetch. This observation leads to a conjecture that in the process of temporal growth of wind waves, once a harmonic with the frequency $\tilde{f}$ attains its equilibrium amplitude at a certain fetch $\tilde{x}(\tilde{f})$ , the characteristic amplitude of this harmonic remains constant from there on and does not vary significantly at longer fetches, $x>\tilde{x}(\tilde{f})$ . Thus, only low frequency harmonics with $f<\tilde{f}$ continue to contribute to wave growth at those fetches.

This conjecture suggests the following scenario describing the wind–wave field evolution under impulsive wind forcing. With activation of the blower over water at rest, a spatially homogeneous wave field is initially excited. This wave field is dominated by short ripples, but in fact contains also longer wave harmonics of vanishing amplitudes. Under the action of wind, different harmonics grow at their distinct growth rates, as studied by Liberzon & Shemer (Reference Liberzon and Shemer2011), until they attain the equilibrium amplitude and steepness for the given steady wind forcing. In the course of growing, each harmonic propagates with its group velocity $c_{g}$ . Longer gravity waves obviously propagate faster and for a given characteristic equilibrium steepness, attain greater wave heights; they need, however, more time and longer fetches to reach their equilibrium state. The growth stage terminates at each fetch once all waves with $f\leqslant f_{dom}$ have had sufficient time to propagate to that fetch. The overall duration of the growth stage at each fetch and wind velocity, $t_{tot}(x)$ , can thus be estimated as $t_{tot}=x/c_{g}(f_{dom}(x))$ , where $c_{g}=\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}k$ can be estimated using the empirical dispersion relation (3.2). The resulting group velocity is shown in figure 10 as a function of the wavenumber $k$ . The values of group velocity corresponding to the dispersion relation of gravity–capillary waves (3.3), $c_{g,gc}$ , are also plotted in this figure for comparison. The two curves collapse only for approximately $k<10~\text{m}^{-1}$ ( $\unicode[STIX]{x1D706}>0.6$  m). For the range of wavelengths studied in the present experiments, the Doppler shift due to mean shear current causes a notable increase in the effective group velocity and thus cannot be neglected.

Figure 10.

Wavenumber dependence of group velocity $c_{g,gc}$ in the absence of mean current compared with that corresponding to the empirical dispersion relation in presence of induced shear current, $c_{g}$ .

Figure 11.

Comparison of the growth duration obtained by two methods.

The total duration of the growth stage was therefore calculated using the empirical values of group velocity for each dominant wavenumber. The estimated values of the dominant frequency $f_{dom}$ , the wavenumber, $k_{dom}$ , corresponding to $f_{dom}$ according to (3.2) and of the corresponding group velocity, $c_{g}(f_{dom})$ are given in table 2. To extend the range of experimental parameters, the table contains results of an additional series of measurements performed at $x=340$  cm for a lower wind velocity, $U=5.3~\text{m}~\text{s}^{-1}$ .

The validity of the suggested scenario can be assessed estimating the actual duration of the growth stage from the experimental data presented in figure 8. The initial reference point may be taken at the instant when blower attains its maximum capacity, i.e. at 3.5–5.5 s after the activation of the blower, depending on the ultimate wind velocity, see figure 2. The duration of the growth stage, $t_{tot}$ , estimated from figure 8, is compared in figure 11 with the values calculated using the empirical group velocity. As expected, the data points indeed lie in close proximity to the $45^{\circ }$ line.

Figure 12.

Identification of stages of the temporal development of the wave field at a fixed fetch $x=340$  cm and wind velocity $U=7.5~\text{m}~\text{s}^{-1}$ .

Table 2.

Parameters of growing wave field.

3.4 Comparison with the results by Kawai (Reference Kawai1979) and the theory by Phillips (Reference Phillips1957)

The data on the temporal variation of the wind–wave field initially at rest under the action of nearly impulsive turbulent air flow, accumulated in the present study, allow quantitative comparison with some available experimental and theoretical results.

A closer look at the temporal variation of $\langle \unicode[STIX]{x1D702}^{2}\rangle ^{1/2}$ , $f_{dom}$ , and the dominant wavelength, $\unicode[STIX]{x1D706}_{dom}$ , at longer fetches reveals a number of notable changes in the slopes of the curves. For each fetch and wind velocity, these clearly detectable slope variations occur essentially simultaneously for all those parameters and suggest division of the temporal evolution of the wave field into distinct stages. To define the evolution stages, consider the variation with the elapsed time of the characteristic wave amplitude $\langle \unicode[STIX]{x1D702}^{2}\rangle ^{1/2}$ plotted in figure 12 for representative conditions (fetch $x=340$  cm and wind velocity $U=7.5~\text{m}~\text{s}^{-1}$ ).

The first detectable wavelets appear at the elapsed time $t_{1}$ , by this instant the wind velocity in this case has already attained its target value, see figure 2. Those initial wavelets grow fast, but after a short time, the growth rate abruptly decreases at $t=t_{2}$ . Note that at this instant the waves are quite small with characteristic amplitude less than 0.5 mm. The relatively slow wave growth continues for $t_{2}<t<t_{3}$ ; at $t_{3}$ the rate of wave amplitude growth increases again, until quasi-steady equilibrium state is attained at  $t_{4}$ .

Table 3.

Transition times as defined in figure 12.

The transition times between the evolution stages, $t_{1}{-}t_{3}$ , as well as the total durations of the wave field development from the instant of the blower activation to emergence of equilibrium state, $t_{4}$ , evaluated from figure 8 are summarized in table 3. Blanks denote cases when the corresponding instants cannot be clearly identified. As can be seen in table 3, the initial wavelets at all fetches and wind velocities appear at $t\approx 3.6{-}4.45$  s. At the shorter fetch, $x=120$  cm, the whole evolution process is short as apparent in the values of $t_{4}$ that do not exceed approximately 9 s; it is thus difficult to identify transition between the stages. The instant $t_{2}$ when the fast growth of wavelets slows down is approximately constant at $t_{2}\approx 5$  s for lower wind velocities ( $6.5~\text{m}~\text{s}^{-1}\leqslant U\leqslant 8.5~\text{m}~\text{s}^{-1}$ ) at both longer fetches. At higher wind velocities the value of $t_{2}$ increases somewhat and exceeds 6 s. The transition between the second and the third stage in the evolution can be identified in all cases except for lower wind velocities at the shortest fetch ( $x=120$  cm). The existence of distinct stages in the temporal growth of the wind–wave field identified in figure 12 suggests that the wave growth during each stage may be governed by a different mechanism. We therefore attempt to analyse results accumulated during each evolution stage separately invoking theoretical models that predict behaviour compatible with the experimental findings.

A closer look at the initial response of the water surface to impulsive wind forcing is presented in figure 13(a) for all fetches and the two lower wind velocities for which the target value of $U$ is attained at times $t\leqslant t_{1}$ . The plot, in semi-logarithmic coordinates, makes it obvious that the growth of the characteristic amplitude of the initial ripples with time is exponential. The energy growth of the initial wavelets thus can be approximated as

where $\unicode[STIX]{x1D6FD}$ is the growth parameter and $\langle \unicode[STIX]{x1D702}_{0}^{2}\rangle$ energy of the initial disturbance. The values of $\unicode[STIX]{x1D6FD}$ were evaluated for all fetches and wind velocities and summarized in table 4. The corresponding friction velocities $u_{\ast }$ are given in table 1.

For all cases presented in figure 13(a), the duration of the exponential growth is less than 0.5 s, so unambiguous determination of the growth parameter from the experimental data indeed requires high sampling rate of $f_{s}=300$  Hz used in this study.

Figure 13.

Zoom in on the initial response of the water surface: (a) an effectively impulsive forcing for lower wind velocities; (b) effect of the wind acceleration at high wind velocity.

Table 4.

Growth parameter $\unicode[STIX]{x1D6FD}$ .

The exponential growth of wavelets in figure 13(a) that was attributed to instability mechanism starts at $t_{1}$ given in table 3. The exponential growth stage in this figure thus lasts only for approximately 0.3 s. The growth rate parameters $\unicode[STIX]{x1D6FD}$ presented in table 4 are in general agreement with the computations by Kawai (Reference Kawai1979) and with his experimental growth rates. Extrapolation of his results to stronger wind forcing is needed for comparison since lower effective wind velocities and thus smaller values of the friction velocity $u_{\ast }$ were employed in his study. Note that for a constant wind velocity, the growth rate parameter $\unicode[STIX]{x1D6FD}$ decreases with fetch. For the case of high wind velocities, this stage is notably longer as shown in figure 13(b). For both wind velocities in this figure, exponential growth also starts at approximately $t_{1}=3.5$  s, as in figure 13(a). For those velocities acceleration of wind still continues during the growth of the initial wavelets. The growth rate is not constant; the slope of the curves initially resembles that at lower wind velocities as seen in figure 13(a), however, at $t\approx 4$  s the growth rate of wavelets decreases notably. The values of the growth coefficient in these cases were calculated from $\langle \unicode[STIX]{x1D702}^{2}\rangle$ within the time interval ranging from $t_{1}$ to approximately 4 s and are also given in table 4. The fetch dependence however is less prominent for these cases; the exponential growth terminates for all fetches and wind velocities in figure 13(b) at about $t_{2}=5.5$  s. The growth rates of the wavelets at $4~\text{s}<t<t_{2}$ are similar for the cases presented in figure 13(b) with $\unicode[STIX]{x1D6FD}$ ranging approximately from 1.7 s $^{-1}$ to 2.5 s $^{-1}$ .

The variation of the total slope, $(\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle +\langle \unicode[STIX]{x1D702}_{y}^{2}\rangle )^{1/2}$ during the initial stage of the temporal evolution of the wave field is compared in figure 14 with the temporal variation of $\langle \unicode[STIX]{x1D702}^{2}\rangle$ at all fetches at the extreme wind velocities considered in this study.

As noted above the wind velocity at the lower blower settings effectively attains its steady value prior to excitation of the first wavelets. Figure 14(a) demonstrates that at the lowest wind velocity ( $U=6.5~\text{m}~\text{s}^{-1}$ ), the quasi-steady value of the characteristic wave steepness is attained practically simultaneously at all fetches, with the termination of the exponential growth stage of the characteristic energy $\langle \unicode[STIX]{x1D702}^{2}\rangle$ of the initial wavelets at the elapsed time $t=t_{2}$ . Similarly to behaviour of $\langle \unicode[STIX]{x1D702}^{2}\rangle$ in figure 13(b), the growth of the characteristic wave steepness in the lower panel of figure 14 is also characterized by two different slopes, sharp decrease in the slope of $(\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle +\langle \unicode[STIX]{x1D702}_{y}^{2}\rangle )^{1/2}$ at each fetch occurs about $t=3.9{-}4.1$  s, depending on fetch, simultaneously with the corresponding change in the wave energy growth coefficient.

It thus can be concluded that the maximum quasi-steady characteristic slope is attained at the end of the stage corresponding to the exponential growth of initial wavelets, at $t=t_{2}$ ; the subsequent growth of the waves occurs while the slope remains nearly constant.

Figure 14.

Comparison of the temporal variation of $\langle \unicode[STIX]{x1D702}^{2}\rangle$ and $\left(\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle +\langle \unicode[STIX]{x1D702}_{y}^{2}\rangle \right)^{1/2}$ .

Appearance of the initial detectable ripples on the water surface was treated differently by Phillips (Reference Phillips1957); it was predicted that the resonance mechanism between the pressure fluctuations in turbulent air flow over the water causes the appearance of most prominent ripples with minimum phase velocity and wavelength of 1.7 cm (frequency $f=13.5$  Hz) that propagate at an angle relative to the wind direction. As stressed above, the initial detectable waves have frequencies in the range 14–18 Hz, notably above the Phillips prediction. It should be noted, however, that higher ripple frequencies observed in the present experiments may result from the Doppler shift due to the induced shear current that was not accounted for in the Phillips theory. Nevertheless, the results of figure 13 favour the Kawai viscous instability mechanism that predicts exponential growth of initial wavelets in time.

To describe wave evolution at later stages that follow the appearance of initial ripples, $t>t_{2}$ , the theory presented by Phillips (Reference Phillips1957) can be invoked. The wind–wave evolution at the early stage of the development corresponding to elapsed times $t_{2}<t<t_{3}$ , when the duration of the evolution is still very short, can be attributed to the initial stage of the water surface response, as defined by Phillips. Phillips did not offer a closed relation for the wind–wave growth at this early stage, however, his theory predicts that the mean squared values of the surface elevation of gravity–capillary waves that span different spatial scales initially grow linearly with time. The variation of $\langle \unicode[STIX]{x1D702}^{2}\rangle$ during this stage is plotted in figure 15 for all wind velocities and two fetches, $x=220$ and $x=340$  cm. The instant of the initiation of this stage, $t=t_{2}$ , serves as the reference in this plot. In spite of considerable scatter of the data, the results of figure 15 seem to comply in general with the linear in time wave energy growth prediction. As wind velocity $U$ increases, the growth rate increases as well, while the duration of this stage become shorter. The slopes of the fitted linear growth lines $K$ are summarized in table 5. The values for different fetches at each wind velocity appear to be very similar.

Figure 15.

The ‘initial growth stage’ (according to Phillips) of the temporal development of the wave field.

Table 5.

Linear growth parameter $K$ .

At the elapsed time $t=t_{3}$ , the rate of change of the characteristic wave energy $\langle \unicode[STIX]{x1D702}^{2}\rangle$ increases notably. The development of the wind–wave field at the elapsed times $t>t_{3}$ , until the quasi-steady equilibrium state is attained at $t=t_{4}$ , may be attributed to ‘the principal stage’ of the wave field development of Phillips (Reference Phillips1957). Note that at this stage, the waves at fetches $x=220$  cm and $x=340$  cm are already long enough with lengths exceeding 7 cm (see figure 8), and thus are practically unaffected by capillarity; they therefore can be considered as purely gravity waves. The following relation for temporal growth of gravity waves at this principal stage was suggested by Phillips:

Here, $\overline{p^{2}}$ is a mean square pressure fluctuations at the surface, $U_{c}$ is the convection velocity of surface pressure fluctuations that in the Phillips mechanism is related to the water-wave phase velocity, and $\unicode[STIX]{x1D70C}_{w}$ is water density. It should be stressed that direct measurements of turbulent pressure fluctuations were not possible more than half a century ago, as is the case nowadays as well. To enable comparison of his model predictions with field measurements, Phillips (Reference Phillips1957) therefore assumed that the characteristic pressure fluctuations are proportional to interfacial shear stress, $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70C}_{a}u_{\ast }^{2}$ , where $u_{\ast }$ is the friction velocity at air–water interface, so that $\overline{p^{2}}\sim u_{\ast }^{4}$ . Moreover, an assumption was made that pressure fluctuations are convected at the velocity $U_{c}\approx U$ , the wind velocity is related to the friction velocity as $U=18u_{\ast }$ , thus allowing to rewrite (3.5) as

The theory of Phillips therefore predicts that the mean square surface displacement of short gravity waves grows linearly with time, with the growth rate proportional to the third power of wind velocity. As can be seen in figure 10, the characteristic wave velocity is in fact nearly constant for the range of short gravity waves (10 cm ${<}\unicode[STIX]{x1D706}<$ 30 cm), with variations not exceeding approximately 10 %. It is substantially lower than the wind velocity and essentially independent of $U$ . It is thus reasonable to adopt approximation that the convection velocity $U_{c}$ is constant for the conditions prevailing in the present experiments. Retaining the relation between the pressure fluctuations and the wind velocity $U$ as adopted in (3.6) allows us to obtain from (3.5) the following relation for the ensemble-averaged squared surface elevation

where in the general framework of Phillips approach $C$ is supposed to be a constant dimensional coefficient.

The validity of relation (3.7) for description of variation of $\langle \unicode[STIX]{x1D702}^{2}\rangle$ with time for short gravity waves is now examined. The results on the growth of short gravity waves in figure 16 are presented here for two fetches ( $x=220$  cm and 340 cm) and 3 values of the wind velocity ( $U=8.5$ , 9.5 and $10.5~\text{m}~\text{s}^{-1}$ ). The instant of the initiation of the principal stage of wind–wave development, $t=t_{3}$ , see figure 12, is taken for each curve separately from table 3. This instant serves as the reference in figure 16. The wave energy $\langle \unicode[STIX]{x1D702}^{2}(t)\rangle$ variation is also presented relative to the initial values at the instant of transition, $t_{3}$ . The linearity of the dependence of $\langle \unicode[STIX]{x1D702}^{2}\rangle$ on time is clearly seen. The slopes of the curves are not identical; for $x=220$  cm they do not exhibit any detectable trend with variation of wind velocity; whereas for $x=340$  cm the slopes for various velocities are approximately constant and somewhat higher than those for $x=220$  cm. The slopes of all curves in figure 16, however, are of the same order of magnitude, varying in the range of $(1{-}3)\times 10^{-9}~\text{s}^{3}~\text{m}^{-2}$ .

Figure 16.

Variation of $\langle \unicode[STIX]{x1D702}^{2}\rangle /U^{4}$ of short gravity waves with time.

Careful examination of figure 8 prompted an attempt to assess a different normalization approach. In figure 17 the ensemble-averaged values of $\langle \unicode[STIX]{x1D702}^{2}\rangle$ for three wind velocities ( $U=8.5$ , 9.5 and $10.5~\text{m}~\text{s}^{-1}$ ) at fetches $x=220$  cm (in red) and $x=340$  cm (in black) were normalized by corresponding steady-state mean values. Note that no distinction is made in figure 17 between the different stages of gravity–capillary and short gravity waves development stages.

The results of figure 17 show that at each fetch, there is no prominent difference between curves corresponding to different wind velocities $U$ . It thus seems reasonable to average the results for all wind velocities taken at each fetch. The curves corresponding to the averaged values are also plotted in figure 17 by thick solid lines. The averaged curves allow delineating the stages of wind–wave growth: first, exponentially growing initial short wavelets appear simultaneously at both fetches; these wavelets dominate the wave field up to $t\approx 5$  s. This stage is followed by a relatively slow and close to linear growth of the energy of gravity–capillary waves, up to approximately $t_{3}=8$  s for $x=220$  cm and $t_{3}=13$  s for $x=340$  cm. Note that the growth rates at this stage are clearly fetch dependent. Next, more rapid and essentially linear growth of $\langle \unicode[STIX]{x1D702}^{2}\rangle$ is observed, with slope that seems to be only weakly dependent on fetch. Finally, the quasi-steady state is attained. The similarity between curves corresponding to different wind velocities $U$ for a fixed fetch indicate that the transition times between the development stages $t_{2}$ and $t_{3}$ as well as the total duration of the evolution, $t_{tot}$ , depend on fetch only and not on $U$ , see table 3. The weak dependence of the total duration of the wave growth on wind velocity $U$ can also be seen in figure 7 where the points corresponding to a constant fetch form clusters. The spread within each cluster does not exceed approximately 10 %.

Figure 17.

Temporal growth of normalized values of $\langle \unicode[STIX]{x1D702}^{2}\rangle$ for different wind velocities. Thick lines denote averaged over all wind velocities results. Red curves: $x=220$  cm; black curves: $x=340$  cm. Thin dashed lines denote three highest wind velocities applied in the experiment.

3.5 Three-dimensional structure of waves under impulsive wind forcing

Simultaneous measurements of the surface slope components $\unicode[STIX]{x1D702}_{x}$ and $\unicode[STIX]{x1D702}_{y}$ allow studying the temporal variation of the three-dimensional surface geometry during the initial wave growth stage. This can be done in terms of the probability density function (PDF) of the instantaneous azimuthal angle defined as $\unicode[STIX]{x1D719}=\tan ^{-1}(\unicode[STIX]{x1D702}_{y}/\unicode[STIX]{x1D702}_{x})$ that represents the direction of projection of the vector normal to the instantaneous surface onto the horizontal plane. The azimuthal angle $\unicode[STIX]{x1D719}=180^{\circ }$ corresponds to the wind direction. The PDF obtained for ensembles of instantaneous azimuthal angles accumulated at several instants after activation of the blower for two extreme conditions are presented in figure 18 for the shorter fetch and lower wind velocity (a) and for the longer fetch and higher wind velocity (b). The PDFs are calculated for 3000 instantaneous values of $\unicode[STIX]{x1D719}$ accumulated from the measured ensembles of two slope components sampled at the frequency of 300 Hz during the time interval of $\pm 0.05$  s around the specified instant for 100 realizations at each fetch and wind velocity.

For both cases considered in figure 18, the probability distribution of the slope inclination direction is symmetric around the two peaks at $\unicode[STIX]{x1D719}=0^{\circ }$ and $\unicode[STIX]{x1D719}=180^{\circ }$ . The difference in peak heights indicates upwind–downwind asymmetry of the wave shape; lower peaks correspond to wider directional spreading. It should be stressed that all inclination directions of the instantaneous surface have a non-negligible probability for both conditions and all instants plotted in figure 18.

Figure 18.

Probability density function of the instantaneous azimuthal angle $\unicode[STIX]{x1D719}$ at several instants during the initial wave growth stage.

In figure 18, the PDF of the azimuthal angle distribution varies strongly with time. The distributions corresponding to the initial recognizable ripples are strongly asymmetric, indicating that in a representative wave, the leeward part is notably longer than the upwind part. As the waves grow, the probability distributions become more symmetric and the peaks that correspond to the upwind part grow. Near perfect front–back symmetry is attained at $t=4.4$  s, (a), and at $t=4.2$  s, (b). Eventually, a steady state is attained that is characterized by a wider directional spreading and upwind–downwind asymmetry, with the downwind part of the wave typically shorter than its upwind part. The steady-state results are in agreement with those presented in Zavadsky et al. (Reference Zavadsky, Benetazzo and Shemer2017).

The variation of the PDF of the distribution of the azimuthal propagation angle is related to different rate of change of the characteristic slopes in the along-wind and cross-wind directions during the growth of the initial wavelets. Examination of the single realization record presented in figure 5 indicates that the growth of $\unicode[STIX]{x1D702}_{x}$ seems to precede that of that $\unicode[STIX]{x1D702}_{y}$ . A closer look at the variation of the characteristic slope variation amplitudes, $\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle ^{1/2}$ and $\langle \unicode[STIX]{x1D702}_{y}^{2}\rangle ^{1/2}$ , during the early stages of the evolution is presented in figure 19 for the conditions corresponding to those in figure 12. The wave slopes in the cross-wind direction appear with a pronounced delay as compared to $\unicode[STIX]{x1D702}_{x}$ , they grow notably slower and attain their quasi-steady-state value significantly later. The initial wavelets are therefore largely unidirectional.

To obtain an additional independent insight into the three-dimensional (3-D) structure of wind waves during the initial stages of excitation, it is instructive to revisit the PTV records of the temporal variation of the surface velocity. The ensemble-averaged mean drift velocity plotted in figure 4 has only one component in the wind direction $x$ ; the cross-wind mean velocity component, $V$ , is virtually absent. This information can be utilized to estimate from the available PTV records the values of velocity fluctuations in both $x$ and $y$ directions. For each tracer, the velocity fluctuations in the wind direction, $u$ , are obtained as the difference between the instantaneous and ensemble-averaged values at every elapsed time; the corresponding cross-wind components $v$ contain fluctuations only.

Figure 19.

Zoom in on the initial variation of $\langle \unicode[STIX]{x1D702}_{x}^{2}\rangle ^{1/2}$ and $\langle \unicode[STIX]{x1D702}_{y}^{2}\rangle ^{1/2}$ at a fixed fetch $x=340$  cm and wind velocity $U=7.5~\text{m}~\text{s}^{-1}$ .

The variation with the elapsed time of the PTV-derived r.m.s. values of ensemble-averaged velocity fluctuations in the wind direction $\sqrt{\langle u^{2}\rangle }$ and cross-wind direction $\sqrt{\langle v^{2}\rangle }$ are plotted in figure 20 for two wind velocities in the test section. The surface velocity fluctuations become notable after 1.5–2 s following the activation of the blower, already at the stage when surface waves are still undetectable. The fluctuations in the wind direction are stronger as compared to $\sqrt{\langle v^{2}\rangle }$ ; however, $\sqrt{\langle u^{2}\rangle }$ and $\sqrt{\langle v^{2}\rangle }$ remain of the same order at all times. Starting from about 4 s, i.e. approximately when the wind attains its prescribed velocity, the rate of growth of the fluctuations of both components the surface velocity increases somewhat, although the cross-wind velocity grows notably slower. Both components of the surface velocity fluctuations are higher in figure 20(b), corresponding to stronger wind.

Figure 20.

PTV-derived ensemble-averaged surface velocity fluctuations in along- and cross-wind directions. The characteristic r.m.s. values of orbital velocity defined as $\unicode[STIX]{x1D714}_{dom}\sqrt{\langle \unicode[STIX]{x1D702}^{2}\rangle }$ are plotted for comparison. (a) Wind velocity $U=6.5~\text{m}~\text{s}^{-1}$ , (b) $U=8.5~\text{m}~\text{s}^{-1}$ .

The existence of surface velocity fluctuations in both along-wind and cross-wind directions suggests that they may be related to the orbital motion induced by the 3-D wind waves. The essentially 3-D nature of short waves under steady wind forcing in our facility has been demonstrated in Zavadsky et al. (Reference Zavadsky, Benetazzo and Shemer2017) and Zavadsky & Shemer (Reference Zavadsky and Shemer2017). The characteristic scale of the velocity fluctuations due to the orbital motion of a 2-D wave can be defined as $v_{orb}=\unicode[STIX]{x1D714}_{dom}\sqrt{\langle \unicode[STIX]{x1D702}^{2}\rangle }$ , where the angular dominant frequency $\unicode[STIX]{x1D714}_{dom}=2\unicode[STIX]{x03C0}f_{dom}$ . Since this definition of $v_{orb}$ represents the absolute value of the velocity fluctuations at the surface, $v_{orb}$ can be seen as an upper bound of the $x$ and $y$ components of the characteristic velocity fluctuations due to 3-D orbital motion. The variation with time of $v_{orb}$ also plotted in figure 20 demonstrates that $v_{orb}$ indeed exceeds both $\sqrt{\langle u^{2}\rangle }$ and $\sqrt{\langle v^{2}\rangle }$ , but is of the same order. These findings indicate that the orbital motion in an essentially 3-D wind–wave field contributes significantly to the observed fluctuations of the surface drift velocity, in agreement with the results presented in figures 18 and 19.

3.6 Higher moments of the surface elevation

Temporal variation of ensemble-averaged time-dependent skewness $\unicode[STIX]{x1D706}_{3}$ and kurtosis $\unicode[STIX]{x1D706}_{4}$ defined as

is now studied. The variation of the vertical wave asymmetry as represented by the skewness coefficient $\unicode[STIX]{x1D706}_{3}$ is plotted in figure 21 for all fetches and the two extreme wind velocities.

Figure 21.

Temporal variation of the skewness coefficient $\unicode[STIX]{x1D706}_{3}$ .

The skewness $\unicode[STIX]{x1D706}_{3}$ of the initial ripples is negative and increases with time. The values of $\unicode[STIX]{x1D706}_{3}$ cross the horizontal axis at elapsed times approximately corresponding to the duration of the existence of spatially homogenous wind–wave field in the tank (cf. figure 8). The skewness coefficients then continue to increase in time; the quasi-steady state is attained at elapsed times $t\approx \unicode[STIX]{x1D70F}_{2}$ that denotes the total duration of the wave field development in figure 2. For lower wind velocity, figure 21(a), the steady-state values of $\unicode[STIX]{x1D706}_{3}$ increase with fetch, while for the highest wind velocity employed in the present study, the dependence of the skewness coefficient on the fetch is less pronounced, being close to $\unicode[STIX]{x1D706}_{3}=0.4$ for all fetches (b). The results on steady-state values of the skewness coefficient are in agreement with those reported in previous studies of wind waves in a laboratory tank (Huang & Long Reference Huang and Long1980; Hatori Reference Hatori1984; Zavadsky et al. Reference Zavadsky, Liberzon and Shemer2013; Zavadsky & Shemer Reference Zavadsky and Shemer2017).

The negative values of $\unicode[STIX]{x1D706}_{3}$ observed everywhere in the tank during the initial stage of wave growth deserve special attention. As already mentioned in § 3.2, waves at this early stage have frequencies of approximately 18 Hz and thus are strongly affected by surface tension. A typical example of the surface elevation variation in time recorded in the present experiments is plotted in figure 22(a). The probability density function of the surface elevation calculated for the segment duration of 0.7 s as shown in this figure and averaged over the whole set of realizations is given in figure 22(b). The waves are characterized by low flat crests and sharp large troughs, resulting in strongly asymmetric PDF and in negative skewness. The shape of the wave in figure 22 strongly resembles that obtained analytically by Crapper (Reference Crapper1957) for finite amplitude capillary deep-water waves, although the ripples observed during the initial growth stage belong to short gravity–capillary range. The similarity of the initial ripples with the shape of capillary finite amplitude waves was noticed by Huang & Long (Reference Huang and Long1980) who also obtained negative skewness for the short ripples under steady wind forcing.

Figure 22.

Initial ripples at the water surface: (a) segment of the individual surface elevation record normalized by the r.m.s. value; (b) PDF of the surface elevation averaged over all realizations for the whole interval plotted in (a).

The variation of the ensemble-averaged kurtosis coefficient $\unicode[STIX]{x1D706}_{4}$ for the conditions of figure 21 is plotted in figure 23. The values of $\unicode[STIX]{x1D706}_{4}$ at the initial stage of wave growth exceed $\unicode[STIX]{x1D706}_{4}=3$ that corresponds to the Gaussian probability distribution of the surface elevation, thus indicating that the probability of very high waves during this stage is relatively large. The extreme wave heights for those gravity–capillary waves, however, result not from very high crests, but rather from deep troughs, see figure 23(a). The deep trough at $t=4.71$  s in figure 23 indeed exceeds the r.m.s. value by a factor of 4.1. The highest waves during this stage can thus formally be seen as the so-called ‘rogue holes’. It should be kept in mind, though, that the absolute heights of these short ripples are limited to just few millimetres.

The ensemble-averaged values of kurtosis then decrease fast, and their steady-state values are notably below $\unicode[STIX]{x1D706}_{4}=3$ , in agreement with Hatori (Reference Hatori1984), Zavadsky et al. (Reference Zavadsky, Liberzon and Shemer2013) and Zavadsky & Shemer (Reference Zavadsky and Shemer2017). The steady-state kurtosis values are attained at elapsed time comparable to those of other dominant wave parameters. The consistently low values of $\unicode[STIX]{x1D706}_{4}$ corroborate the exceedance probability distributions presented for extremely large ensembles of waves in Zavadsky et al. (Reference Zavadsky, Liberzon and Shemer2013) and Zavadsky & Shemer (Reference Zavadsky and Shemer2017) that indicate very low probability of rogue waves in a young wind–wave field at different fetches under a range of steady wind-forcing velocities.

Figure 23.

Temporal variation of the kurtosis coefficient $\unicode[STIX]{x1D706}_{4}$ .