3.1. Effect of current on energy and steepness of young wind waves

Characteristic wave amplitudes are represented by the root mean square (rms) values of the instantaneous surface elevation and related to the wave energy $E$ by $\eta _{rms}(x)=\overline {\eta ^2}(x)^{1/2}= E(x)^{1/2}$. The values of $\eta _{rms}$ are presented in figure 2 as a function of water current velocity $U_w$ at a single fetch $x = 245$ cm, and for all wind velocities. In the presence of turbulent mean water flow, the water surface in the test section ceases to be smooth. As expected, the values of $\eta _{rms}$ with no airflow in the test section applied, also plotted in figure 2, do not depend notably on the flow direction but grow with increase in water velocity. Figure 2 demonstrates that the relative contribution of irregularities at the surface induced by the mean water current to the resulting wind-wave field is insignificant. At all wind conditions, increase in water velocity $U_w$ in the wind direction has only a minor effect on $\eta _{rms}$. Contrary to that, when the mean current is directed against the wind, the characteristic wave amplitudes increase significantly with $|U_w|$.

Figure 2. Variation of characteristics wave amplitude $\eta _{rms}$ as a function of water current velocity $U_w$ at $x = 245$ cm for different wind velocities $U_a$.

The variation with fetch of the characteristic wave amplitude $\eta _{rms}(x)$, measured using both the conventional wave gauge (open symbols) and the optical wave gauge (solid symbols), is plotted in figure 3 for all wind velocities $U_a$. The dimensionless fetch $\hat {x}=x(g/u_{\ast }^2)$ and the dimensionless characteristic wave amplitude $\hat {\eta }_{rms}(x)=(g /(u_{\ast }^2))\eta _{rms}$ introduced by Kitaigorodskii (Reference Kitaigorodskii1961) are now adopted. The values of dimensionless fetch $\hat {x}$ in the present experiments vary from 30 to 243. The solid lines in figure 3 correspond to the fit $\eta _{rms}(\hat {x})=\hat {\eta }_0(u_{\ast }^2/g)\hat {x}^n$, where $\hat {\eta }_0=\hat {\eta }_{rms}(\hat {x}=1)$, with a single value of the exponent ($n=0.5$) corresponding to the Mitsuyasu (Reference Mitsuyasu1970) law, which was adopted for all experimental conditions. The reference value of the dimensionless fetch $\hat {x}=1$ in the present experiments corresponds to a very short dimensional distance from the inlet of a few mm; the value of $\hat {\eta }_0$ can thus be seen as the effective dimensionless initial characteristic wave amplitude. For all wind velocities, water flow in the wind direction ($U_w>0$) has no notable effect on $\hat {\eta }_0$; the reference value $\hat {\eta }_0=0.013$ estimated in the present experiments agrees reasonably well with the results of Wilson (Reference Wilson1965) and Mitsuyasu (Reference Mitsuyasu1970), as well as with the previous measurements in our facility (Zavadsky, Liberzon & Shemer Reference Zavadsky, Liberzon and Shemer2013; Shemer Reference Shemer2019). The values of $\hat {\eta }_0$ also remain unaffected by water current in the wind direction, $U_w>0$, for all wind velocities. For the counter-wind current case, however, the wave energy at each fetch increases notably with $|U_w|$, and the initial effective dimensionless wave amplitude increases from $\hat {\eta }_0=0.017$ for $U_w=-0.05$ m s$^{-1}$ to $\hat {\eta }_0=0.023$ for the strongest counter-current, $U_w=-0.12$ m s$^{-1}$.

Figure 3. Effect of water current $U_w$ on variation with fetch $x$ of characteristic wave amplitude $\eta _{rms}$ for different values of the wind velocity $U_a$.

Time records of orthogonal slope components enable direct estimates of mean wave steepness defined as $\overline {ak}=(\overline {\eta _x^2}+\overline {\eta _y^2})^{1/2}$ that is a measure of wave field nonlinearity. The variation with fetch $x$ of the wave steepness for two wind velocities, $U_a = 6.83$ m s$^{-1}$ and $U_a = 9.35$ m s$^{-1}$, presented in figure 4, shows that its values remain almost constant along the test section for both wind velocities $U_a$. Somewhat higher values of steepness were measured as the value of $U_a$ increases. In the absence of current, these results are consistent with the steepness behaviour reported by Zavadsky et al. (Reference Zavadsky, Benetazzo and Shemer2017), Zavadsky & Shemer (Reference Zavadsky and Shemer2017a). Mean water current in wind direction practically does not affect $\overline {ak}$, while the counter-wind current causes a slight increase in the wave steepness at all fetches (with a possible exception of the shortest one).

Figure 4. Effect of mean current on variation of mean steepness $\overline {ak}$ along the fetch $x$ for two wind velocities $U_a$. Notation as in figure 2(a).

The effect of wind velocity $U_a$ in the presence of water current is studied further in figure 5. The variation with current velocity $U_w$ plotted in this figure at a single fetch $x = 245$ cm for all wind velocities $U_a$ clearly demonstrates that the mean steepness $\overline {ak}$ increases with the wind velocity; the change in the steepness with $U_a$ does not depend significantly on $U_w$. For all wind and water velocities in the present experiments, the values of mean steepness exceed approximately 0.15, indicating that the young wind-wave field is notably affected by nonlinearity; the waves become very steep and attain high values of $\overline {ak} \approx 0.25$ for adverse water current. The shape of all curves in this figure is quite similar to that of the corresponding curves showing the variation of the characteristic wave amplitude with current in figure 2.

Figure 5. Effect of mean current $U_w$ on variation of the mean steepness $\overline {ak}$ at $x = 245$ cm for all wind velocities $U_a$. Notation as in figure 2.

3.2. Variation of wave spectra with mean current

All spectra plotted in the sequel are based on 900 s long time records of $\eta (t)$ for a given fetch $x$, $U_a$ and $U_w$ that are divided into 10 s long segments with 50 $\%$ overlap. The resulting spectrum represents the average of 180 power spectra computed for each segment with frequency resolution ${\rm \Delta} f = 0.1$ Hz. The variation with fetch of the surface elevation power spectra $E_{\eta } (f)$ at multiple locations is presented in figure 6 for wind velocity $U_a = 6.83$ m s$^{-1}$ and water current velocities $U_w = 0$ m s$^{-1}$ and $U_w =\pm 0.12$ m s$^{-1}$. The downshifting of peak frequency and increase in overall wave energy with fetch are clearly visible in both plots of this figure. The presence of current significantly modifies the spectral shapes. The counter-wind current (figure 6a) results in the spectrum dominated by longer waves with a lower peak frequency and much higher energy, compared to the wind-only power spectra in figure 6(b). The current in wind direction, while not affecting notably the total wave energy (cf. figure 3), results in spectra that are flatter and wider, with waves characterized by higher peak frequency and smaller amplitudes, including at the spectral peak; see figure 6(c).

Figure 6. Wave spectra $E_{\eta }(f)$ at $U_a = 6.83$ m s$^{-1}$ and $U_w$ values (a) $-$0.12 m s$^{-1}$, (b) 0 m s$^{-1}$ and (c) 0.12 m s$^{-1}$.

The effect of the wind velocity $U_a$ on the wave energy spectra $E_{\eta }(f)$ in the presence of current is examined further in figure 7 at a single fetch $x = 296$ cm for two extreme wind velocities. As expected, for each water current $U_w$, an increase in the wind velocity causes a rise in the peak spectral amplitudes accompanied by a decrease in peak frequencies. Figures 7(a,b) emphasize the notable effect of the water current on the spectral shapes, with the co-wind current resulting in a more uniform wave energy distribution among the frequency harmonic, while the opposite effect is observed for the counter-wind current.

Figure 7. Power spectra of surface elevation $E_{\eta }(f)$ at $x = 296$ cm for (a) $U_a = 6.83$ m s$^{-1}$ and (b) $U_a=9.35$ m s$^{-1}$. Colour scheme as in figure 3.

Since the spectra plotted in figures 6 and 7 exhibit certain scatter, it is advantageous to use robust integral spectral moments to determine the dominant frequency $f_{dom}$ as the principal statistical parameter, rather than the peak frequency $f_p$. The $j$th spectral moment of the omnidirectional power spectrum of the surface elevation $E_\eta (f)$ is defined as

(3.1)\begin{equation} m_j = \int_{f_{min}}^{f_{max}} f^j E_{\eta}(f) \,{\rm d} f, \quad j = 0, 1, 2,\ldots .\end{equation}

The integration in (3.1) is carried out within the free wave domain around the peak frequency $f_p$, defined here by $f_{min}=0.5f_p$ and $f_{max}=1.5f_p$. The imposed restriction limits the analysis to free waves only and eliminates the excessive contribution of second-order bound waves to higher-order spectral moments. The frequency $f_p$ for each spectrum is defined by applying the parabolic fit in the vicinity of the spectral peak. Note that the zeroth moment $m_0$ also defines the total energy of free waves; the adopted integration limits in (3.1) result in characteristic wave amplitudes that are somewhat below the $\eta _{rms}$ values; however, the difference is less than a few per cent. The dominant frequency $f_{dom}$, defined as

(3.2)\begin{equation} f_{dom} = m_1/m_0,\end{equation}

is less prone to experimental error than $f_p$; however, the values of $f_{dom}$ and $f_p$ usually do not differ significantly. The variation of $f_{dom}$ with fetch is plotted in figure 8 for wind waves propagating over water with no mean current, as well as for the two extreme values of the water current velocity $U_w$. The plots correspond to each wind velocity $U_a$. The dominant frequency $f_{dom}$ decreases with fetch for all current conditions and wind velocities; for all operational conditions at any given fetch; $f_{dom}$ is higher for current in the wind direction, and lower for counter-wind water current, as compared to the corresponding $f_{dom}$ for $U_w = 0$. For all current conditions, the dominant frequency at all fetches decreases with an increase in current. The solid lines in figure 8 correspond to the power-law fit for the dependence of the dimensionless dominant frequency $\hat {f}_{dom}=f_{dom} (g/u_{\ast })$ on the dimensionless fetch $\hat {x}=x(g/u_{\ast }^2)$,

(3.3)\begin{equation} \hat{f}_{dom} = \hat{f}_{dom,0} (\hat{x})^{{-}1/3},\end{equation}

where the fitting coefficient is $\hat {f}_{dom,0}=\hat {f}_{dom}(\hat {x}=1)$. Note that the fitting exponent in the power law for all cases in figure 8 is close to $n = -0.33$, as suggested for deep-water wind waves, in agreement with previous studies cited in relation to figure 3. The coefficient $\hat {f}_{dom,0}$ is close to unity in the absence of current, again in agreement with Mitsuyasu (Reference Mitsuyasu1970), Zavadsky et al. (Reference Zavadsky, Liberzon and Shemer2013) and Shemer (Reference Shemer2019). All curves for $U_w>0$ in figure 8 are above those corresponding to the $U_w=0$ cases. Accordingly, the reference values of $\hat {f}_{dom,0}$ are somewhat higher. The largest value of $\hat {f}_{dom,0} = 1.3$ is attained at low wind velocity; the effect of co-wind current decreases with an increase in $U_a$, approaching $\hat {f}_{dom,0} \approx 1$ at higher wind velocities. The values of $\hat {f}_{dom,0}$ in case of counter-wind current are consistently below unity and seem to be independent of $U_a$; they vary from $\hat {f}_{dom,0} \approx 0.57$ for $U_w = -0.12$ m s$^{-1}$ to $\hat {f}_{dom,0} \approx 0.78$ for $U_w = -0.05$ m s$^{-1}$.

Figure 8. Variation of the dominant frequency with fetch $f_{dom}(x)$ for three values of the water current velocity $U_w$ and four values of the wind velocity $U_a$; colour scheme as in figure 3. Open symbols indicate wire gauge data; closed symbols indicate optical wave gauge data; solid lines indicate power-law fit.

3.3. Effect of current on statistical properties defined by higher spectral moments

Additional integral statistical parameters, such as dimensionless spectral width $\nu$, skewness $\lambda _3$ and kurtosis $\lambda _4$, are now examined. The spectral wave energy concentration around $f_{dom}$ is estimated using the second-order central moment $\tilde {m}_2=\int (f-f_{dom})^2\,\hat {E}(f) \, {\rm d} f$; rendered dimensionless by $f_{dom}^2 m_0$, it defines the spectral width $\nu$ (Massel Reference Massel1996):

(3.4)\begin{equation} \nu = \sqrt{\frac{m_0 m_2}{m_1^2} - 1}.\end{equation}

The values of $\nu$ plotted in figure 9 for two wind velocities allow quantitative refinement of the qualitative assessment of the mean current effect on $\nu$ made by examining the spectral shapes in figures 6 and 7. For both wind velocities and at any given fetch $x$, the spectral width $\nu$ for the co-wind current conditions (figure 9c) is higher than that corresponding to the values of $\nu$ with no mean current (figure 9b) as observed in figures 6 and 7. In figures 9(b,c), the spectral width $\nu$ decreases slightly with fetch. In the absence of current, such a decrease in $\nu$ was reported by Zavadsky et al. (Reference Zavadsky, Liberzon and Shemer2013). For counter-wind current, no clear pattern can be seen in the dependence $\nu (x)$ in figure 9(a); for lower wind velocity $U_a$, the spectral width $\nu$ in this plot does not vary significantly with fetch and is comparable to that in figure 9(b) with $U_w=0$ m s$^{-1}$, while for $U_a = 9.35$ m s$^{-1}$, the values of the dimensionless spectral width show a tendency to grow with fetch.

Figure 9. Variation of the spectral width $\nu$ with fetch for $U_w$ values (a) $-0.12$ m s$^{-1}$, (b) $0$ m s$^{-1}$ and (c) $0.12$ m s$^{-1}$, with $U_a = 6.83$ m s$^{-1}$ and 9.35 m s$^{-1}$.

Higher-order statistical moments such as skewness $\lambda _3$ and kurtosis $\lambda _4$, as well as asymmetry $A$, provide additional insight into the effect of water current on the wind waves’ shape. These statistical parameters are presented in figures 10 and 12 as functions of water velocity $U_w$ for two values of $U_a$, i.e. $6.83$ m s$^{-1}$ and $9.35$ m s$^{-1}$, at three fetches, $x = 100$, 200 and 300 cm. The dimensionless $n$th-order moments are defined as

(3.5)\begin{equation} \lambda_n = \frac{\overline{\eta^n}}{\overline{\eta^2}^{n/2}}.\end{equation}

Figure 10. Variation with current velocity $U_w$ of (a,c) skewness $\lambda _3$, and (b,d) kurtosis $\lambda _4$, at three fetches $x$ and wind velocities (a,b) $U_a = 6.83$ m s$^{-1}$ and (c,d) $U_a = 9.35$ m s$^{-1}$.

The third-order moment $\lambda _3$ (skewness) characterizes wave asymmetry with respect to the horizontal axis. The positive skewness of gravity waves with notable steepness represents wave crests that are larger than troughs, mainly due to the contribution of second-order bound waves. The values of $\lambda _3$ plotted in figures 10(a,c) are indeed positive for all values of $U_w$. In the absence of mean current, the skewness of wind waves measured in the present facility for a wide range of wind velocities $U_a$ and several fetches by Zavadsky & Shemer (Reference Zavadsky and Shemer2017a) and Shemer (Reference Shemer2019) remains positive and tends to increase with $U_a$ and with fetch $x$; this result is confirmed in figures 10(a,c). For counter-wind current, $U_w<0$, the waves become steeper as the current becomes stronger; the more pronounced nonlinearity contributes to increase in $\lambda _3$. When wind and current are aligned, $U_w>0$, $\lambda _3$ decreases slightly with increase in $U_w$.

The fourth-order moment, kurtosis $\lambda _4$, characterizes the wave height probability distribution; $\lambda _4=3$ corresponds to the normal (Gaussian) distribution. In the absence of current, the kurtosis does not vary notably with fetch and $U_a$, remaining close to $\lambda _4=2.5$, thus indicating that the wave height distribution is somewhat narrower than Gaussian. As seen in figures 10(b,d), this observation is still valid in the presence of current, although for $U_w>0$ the values of $\lambda _4$ tend to increase with $U_w$, tending to the Gaussian value $\lambda _4=3$. The effect of current on wave height distribution can also be examined by plotting the exceedance distribution function of wave height $h_w$ that for narrow-banded linear Gaussian waves has Rayleigh distribution

(3.6)\begin{equation} f(h_w) = \exp \left( -\frac{h_w^2}{8\overline{\eta^2}} \right) .\end{equation}

The height $h_w$ of each individual wave is defined as the difference between the consecutive crest and trough between two consecutive positive zero crossings; for more details, see Kumar et al. (Reference Kumar, Fershtman, Barnea and Shemer2021). The exceedance functions estimated at two extreme fetches $x = 100$ and 300 cm at $U_a = 6.83$ m s$^{-1}$ and water flowing in both directions are compared with the no-current case in figure 11 as a function of the normalized $h_w$. Note that for no-current and counter-wind current cases at fetch $x = 100$ cm, the characteristic wave height is still very small (see figure 3), the probability of higher waves with heights exceeding approximately $4\eta _{rms}$ is below that corresponding to the Rayleigh distribution. At higher fetch $x = 300$ cm, the probability of waves with $h_w>6\eta _{rms}$ falls below that of the Rayleigh curve. No extremely steep waves or rogue waves with heights exceeding $8\eta _{rms}$ were observed.

Figure 11. Exceedance distribution function $F(h_w)$ of wave height $h_w$ at fetch $x$ values (a) $100$ cm and (b) $300$ cm, at $U_a = 6.83$ m s$^{-1}$, for $U_w = 0$ and ${\pm }0.12$ m s$^{-1}$.

The symmetry of the wave shape relative to the vertical axis is often defined using the differences in location between successive crests and troughs Babanin et al. (Reference Babanin, Chalikov, Young and Savelyev2010). However, for a three-dimensional random wind-wave field, Elgar & Guza (Reference Elgar and Guza1985) and Elgar (Reference Elgar1987) offered a more robust estimate of vertical wave asymmetry $A$ based on the Hilbert transform $H(\eta )$ of the measured surface elevation $\eta (t)$:

(3.7)\begin{equation} A ={-}\frac{\overline{{\rm Im}(H(\eta))^3}}{\overline{\eta^2}^{3/2}}.\end{equation}

The wave asymmetry $A$ calculated according to (3.7) plotted in figures 12(a,b) is positive for all fetches and all current velocities, indicating that for young wind waves, the windward part of the wave is on average longer than the leeward part. In the absence of current, the asymmetry $A$ decreases somewhat with fetch, in agreement with observations by Shemer (Reference Shemer2019); both co- and counter-wind currents tend to decrease the asymmetry with an increase in absolute current velocity.

Figure 12. Variation with current velocity $U_w$ of asymmetry $A$ at three fetches $x$ and wind velocity $U_a$ values (a) $6.83$ m s$^{-1}$ and (b) $9.35$ m s$^{-1}$.

3.4. Dispersion relation in presence of current

The Doppler-shifted linear dispersion relation for gravity-capillary waves propagating over uniform current with velocity $U_w$ is

(3.8)\begin{equation} (\omega - k U_w)^2=\left( gk+\frac{\sigma}{\rho}\,k^3 \right) \tanh{kh},\end{equation}

where $\omega = 2{\rm \pi} f$ is the angular frequency, $\rho$ is the water density, $\sigma$ is the water surface tension, and $h$ is the water depth. Single-point synchronous measurement of surface elevation and surface slope components $\eta _x$ and $\eta _y$ allows direct determination of the relation between the wave frequency and the absolute value of its wave vector $k(f)=(k_x^2(f)+k_y^2(f))^{1/2}$. In the absence of mean current, those spectra have been presented in Zavadsky & Shemer (Reference Zavadsky and Shemer2017a); the present study examined the power spectra $E_{\eta _x}=k_x^2\,E_{\eta }(f)$ of $\eta _x(t)$, and $E_{\eta _y}=k_y^2\,E_{\eta }(f)$ of $\eta _y (t)$, and found that they are not affected qualitatively by the mean current and therefore not given here. The wavenumber $k(f)$ as a function of frequency $f$ is determined from the power spectra of the synchronous single-point measurements of $\eta (t)$, $\eta _x(t)$ and $\eta _y(t)$ as in Longuet-Higgins, Cartwright & Smith (Reference Longuet-Higgins, Cartwright and Smith1961):

(3.9)\begin{equation} k(f) = \sqrt{\frac{E_{\eta_x}(f)+E_{\eta_y}(f)}{E_{\eta}(f)}}.\end{equation}

In the present experiments, wavenumbers $k(f)$ estimated at each fetch $x$ at dominant frequency $f_{dom}(x)$ are plotted in figure 13; the error bars represent the standard deviation of $k(f)$ estimated at frequencies around $f_{dom}(x)$ that satisfy the condition on the magnitude squared coherence, $M_{\eta \eta _x}=|E_{\eta }^{\ast }E_{\eta _x}|^2/(E_\eta E_{\eta _x})>0.8$. At any given frequency $f$, due to wind-induced shear current, the experimentally determined wavenumbers $k$ for wind waves in the absence of current are consistently below those corresponding to the gravity-capillary dispersion relation, $k_{gc}$. This effective Doppler shift is accounted for in the empirical dispersion relation by Zavadsky & Shemer (Reference Zavadsky and Shemer2017a) in the form

(3.10)\begin{equation} \frac{k_{gc}(f)}{k} = 1+ak+bk^2,\end{equation}

where $a=0.006$ m and $b=-2.2\times 10^{-5}\ {\rm m}^{2}$ are the empirical fitting coefficients and agree well with the present experimental results.

Figure 13. Dominant wavenumber $k_{dom}$ as a function of frequency $f_{dom}$.

The ratio $k_{gc}/k$ suggested in the dispersion relation by Zavadsky & Shemer (Reference Zavadsky and Shemer2017a) is substituted into (3.9) to account for additional Doppler shift induced by uniform current $U_w$; the result is plotted in figure 13 by solid lines. For co-wind current, the dominant frequencies $f_{dom}$ are shifted to higher values, and for counter-wind current they are shifted to lower values, as compared to the no-current case. The empirical dispersion relation by Zavadsky & Shemer (Reference Zavadsky and Shemer2017a) seems to retain its validity also in the presence of uniform current.

3.5. Temporal coherence of wind waves propagating over mean current

To illustrate the essentially random nature of wind waves with and without mean current, it is instructive to look at the temporal variation of the ‘amplitude’ of instantaneous ‘frequencies’ using wavelet analysis (Kumar & Shemer Reference Kumar and Shemer2024). The 20 s long Morlet spectrograms starting at $t = 20$ s are plotted in figure 14 for a single fetch $x$, a single wind velocity $U_a$, and two extreme values of water current; the spectrogram in the absence of mean water current is also presented for comparison. The vertical section of the wavelet ‘spectrum’, or ‘map’, at any time $t$ provides an instantaneous ‘amplitude’ as a function of frequency $f$. The horizontal solid lines in all plots of figure 14 correspond to the dominant frequency $f_{dom}(x = 245\ {\rm cm})$ plotted in figure 8. The characteristic frequencies at which the highest amplitudes in each wavelet spectrogram are attained indeed agree well with their corresponding $f_{dom}$.

Figure 14. Wavelet spectrograms of surface elevation $S_\eta (f,t)$ as functions of frequency and time a $x = 250$ cm and $U_a = 6.83$ m s$^{-1}$, for three values of $U_w$.

At all current velocities, the spectrograms exhibit significant variability with time, thus confirming the essentially random character of wind waves, as demonstrated in figure 14. To quantify the effect of mean water current on the temporal wave coherence, the auto-correlation analysis of the temporal records of $\eta (t)$ is studied here. The dependence of the auto-correlation coefficient $R(\tau )$ on the time shift $\tau$ for a function with zero mean $F(t)$ is defined as (Bendat & Piersol Reference Bendat and Piersol1971)

(3.11)\begin{equation} R(\tau) = \frac{\overline{F(t)\,F(t+\tau)}}{\overline{F^2}}.\end{equation}

Statistically reliable estimates of $R$ are obtained by computing the values of $R(t)$ over each one of the 10 s long segment cuts of the whole record, and averaging the outcome over all segments. The resulting averaged auto-correlation coefficient is plotted in figure 15 at two extreme fetches, $x = 100$ and 300 cm, for two extreme current velocities. The lag $\tau$ is normalized by the local dominant frequency $f_{dom}$ (see figure 8). In each plot, the values of $R$ oscillate at the local dominant frequency and decay within a few dominant wave periods. The decay of the local maximum of the auto-coherence coefficient at each fetch $x$ is approximated by an exponent as ${\sim }{\rm e}^{-\alpha _d(\tau /T_d)}$, where $\alpha _d$ is the dimensionless decay rate. In the absence of mean current, the decay rate $\alpha _d$ in figure 15(b) decreases with fetch and thus with the wavelength, in agreement with Shemer & Singh (Reference Shemer and Singh2021) and Kumar et al. (Reference Kumar, Singh and Shemer2022). Waves propagating over water flowing against the wind retain their coherence longer compared to waves in the absence of current, while waves propagating over co-wind current lose their coherence within less than one dominant wave period; see figure 15(c).

Figure 15. Temporal coherence function $R(\tau )$ at $U_a = 6.83$ m s$^{-1}$ and $U_w$ values (a) $-0.12$ m s$^{-1}$, (b) 0 m s$^{-1}$ and (c) 0.12 m s$^{-1}$.

The dimensionless decay rates $\alpha _d$ for all operational conditions employed in the present experiments are summarized in figure 16 for all forcing conditions; they are presented as functions of dominant wavelength $\lambda _{d}$. For the reference wind-only cases, decay rate $\alpha _d$ decreases with increase in both wind velocity and wavelength, in agreement with Shemer & Singh (Reference Shemer and Singh2021). The dimensionless decay rates for waves in the absence of mean current and for counter-wind current show similar dependence on $\lambda _{d}$. Contrary to that, for the along-wind current, the values of $\alpha _d$ are significantly higher and widely spread.

Figure 16. Dimensionless decay rate $\alpha _d$ as a function of dominant wavelength $\lambda _d$ for all wind velocities.

3.6. Two-dimensional characterization of the wind-wave field in the presence of current

The two simultaneously measured slope components allow two-dimensional characterization of the instantaneous surface by projecting the vector normal to the surface on the horizontal plane. Instantaneous surface slope is defined as $\theta = \arctan (\eta _y/\eta _x)$. The probability distribution function (p.d.f.) of instantaneous surface slope at a single fetch and for all wind forcing conditions is presented in figure 17 for extreme values of current velocities and compared with $U_w = 0$ m s$^{-1}$.

Figure 17. The p.d.f.s of instantaneous surface slope $\theta$ at fetch $x = 335$ cm for $U_w$ values (a) $-0.12$ m s$^{-1}$, (b) $0$ m s$^{-1}$ and (c) $0.12$ m s$^{-1}$.

All p.d.f.s in figure 17 are characterized by two distinctive peaks at $\theta = 0$ and ${\rm \pi}$, indicating the prevailing direction of waves along the wind. For all wind and current forcing conditions, the probability of a forward-leaning slope ($\theta = 0$) is smaller than that of the windward slope at $\theta = {\rm \pi}$, showing front–back asymmetry. This asymmetry decreases with an increase in wind velocity, and nearly vanishes when stronger wind forcing is applied. For a given wind velocity, the angular asymmetry behaves similarly for co-wind current and no-current conditions, while in counter-wind current, the angular distribution approaches the symmetrical one. This is in agreement with the asymmetry plotted in figure 12. For a random wind-wave field with directional spreading, the probability of a wave vector normal to the surface that is perpendicular to the mean wind direction, $\theta =\pm {\rm \pi}/2$ is non-negligible for all wind and water current conditions. Though there is a well-defined wave asymmetry along the wind direction as documented in figures 17 and 12, there is a left–right symmetry near the vicinity of $\theta =\pm n {\rm \pi}/2$, thus effectively showing cross-wind symmetry of the p.d.f.