3.1. Basin and wave generation

The experiments were performed in a large rectangular wave basin, $15~\text{m}\times 10~\text{m}$ , at the Ecole Centrale de Nantes, France. The basin is filled with water to a uniform depth fixed at 1.90 m. Surface waves are generated by a 10 m wide, rectangular flap wavemaker. The latter is located at one basin width as shown in figure 1. This flap is moved by hydraulic cylinders, driven in-phase and controlled by a computer. The wavemaker has a frequency cut-off at 2 Hz due to mechanical parts. A linear variable displacement transducer (LVDT) is fixed on top of the wavemaker to infer its temporal displacement especially to allow feedback control of the wavemaker position with respect to a prescribed shape spectrum.

Figure 1.

Sketch of the wave basin. The location of the wavemaker is shown. The boundary opposite the wavemaker is either a beach or a removable wall. The locations of the array of probes are also visible: capacitive probes $C_{1}$ and $C_{2}$ , and resistive ones $R_{1}$ and $R_{2}$ .

The wavemaker generates irregular waves which are randomly distributed in amplitude and in frequency within a certain bandwidth. The wavemaker is driven either by a bandpass-filtered random noise (FRN) within a bandwidth ${\rm\Delta}f$ around a frequency $f_{m}$ or by a unidirectional JONSWAP spectrum (JON – see figure 2). The latter has been used in oceanography to model the wave energy in the frequency domain, and is based on a parametrization of the wave spectrum measurements in the North Sea (see Komen et al. Reference Komen, Cavaleri, Donelan, Hasselmann, Hasselmann and Janssen1994). In both cases, the forcing parameters are controlled by the frequency bandwidth ${\rm\Delta}f$ of the spectrum around its maximal value of frequency $f_{m}$ , and by the wavemaker amplitude. Instead of the latter, we will use in the following the value of the root-mean-square (r.m.s.) wave amplitude ${\it\sigma}_{{\it\eta}}\equiv \sqrt{\overline{{\it\eta}^{2}(t)}}$ at the gauge locations (temporal averages are denoted by an overline). Typically, $f_{m}\approx 1$  Hz (corresponding to a wavelength ${\it\lambda}_{m}\approx 1.5$  m), $0.3~\text{Hz}\leqslant {\rm\Delta}f\leqslant 1.3~\text{Hz}$ , and $0.5~\text{cm}<{\it\sigma}_{{\it\eta}}<7~\text{cm}$ . Subsequently, frequency bandwidth will be considered in two typical ranges: narrow-band for ${\rm\Delta}f<0.5~\text{Hz}$ , and broad-band for ${\rm\Delta}f\geqslant 0.5~\text{Hz}$ . The forcing parameters are summarized in table 2.

Figure 2.

Power spectrum density of wavemaker displacement. FRN (solid) and JON (dashed) forcings for a broad frequency bandwidth ${\rm\Delta}f=f_{max}-f_{min}$ . See table 2 for corresponding forcing parameters.

Table 2.

Forcing parameters to generate a prescribed spectrum of wavemaker displacement with a spectral shape (JON or FRN), a frequency bandwidth ${\rm\Delta}f=f_{max}-f_{min}$ , and a maximum spectrum amplitude at frequency $f_{m}$ (see figure 2). The corresponding wavelengths are ${\it\lambda}_{m}\approx 1.5~\text{m}$ (broad) and 1.2 m (narrow), respectively.

The typical power spectrum density of wavemaker displacement is shown in figure 2 for a broad-band forcing. It is computed from the displacement sensor fixed on top of the wavemaker. No significant change is observed in the spectrum shape when changing the forcing type (JON or FRN), whereas the frequency bandwidth and the amplitude of the spectrum peak are well controlled by ${\rm\Delta}f$ and ${\it\sigma}_{{\it\eta}}$ , respectively. Thus, the forcing type (JON or FRN) will not be distinguished in the discussion.

3.2. Boundary conditions

Two boundary conditions were tested, as illustrated in figure 1. First, the wave basin is equipped with an absorbing sloping beach at the opposite end of the basin to the wavemaker, in order to strongly reduce wave reflections. The beach is a porous beach made of stones with a weak slope of the order of $1/3$ for the first 3.2 m, the last 3.5 m being almost flat. This enables wave absorption by wave breaking and porosity. The amplitude of reflections is estimated to be less than 10 % after 5 min of irregular wave generation of peak period of 1 s (Bonnefoy Reference Bonnefoy2005). Thus, waves propagate up to the beach, with almost no reflections going back ( ${<}10\,\%$ ). This boundary condition will be subsequently referred to as the absorbing boundary condition. The second configuration consists of a wooden wall vertically fixed in the wave basin in front of the beach (see figure 1). This case, called the reflecting boundary condition, corresponds to a closed basin, a situation already tested in previous laboratory experiments on gravity wave turbulence of various basin sizes (Denissenko et al. Reference Denissenko, Lukaschuk and Nazarenko2007; Falcon et al. Reference Falcon, Laroche and Fauve2007b ; Herbert et al. Reference Herbert, Mordant and Falcon2010; Nazarenko et al. Reference Nazarenko, Lukaschuk, McLelland and Denissenko2010; Cobelli et al. Reference Cobelli, Przadka, Petitjeans, Lagubeau, Pagneux and Maurel2011; Bedard, Lukaschuk & Nazarenko Reference Bedard, Lukaschuk, Nazarenko, Shrira and Nazarenko2013a ; Bedard et al. Reference Bedard, Nazarenko and Lukaschuk2013b ). We will show in the following that the boundary conditions play an important role on the dynamics of the wave field.

3.3. Wave gauges

We use an array of four wave gauges (two capacitive and two resistive) to measure the wave amplitude, ${\it\eta}(t)$ , as a function of time with a sampling frequency of 500 Hz during typically $T=10$ or 19 min. Resistive gauges are 80 cm in height. Their vertical resolution is approximately 0.1 mm, and their frequency resolution is close to 10 Hz (Bonnefoy Reference Bonnefoy2005). The capacitive gauges are 60 cm in height and are homemade (Falcon et al. Reference Falcon, Laroche and Fauve2007b ). Their vertical resolution is approximately 0.1 mm and the frequency resolution up to 200 Hz. The location of the probe array in the basin is shown in figure 1. They are located in the middle of the basin, 7.5 m from the wavemaker, corresponding to a distance of $5{\it\lambda}_{m}$ for the smallest value of $f_{m}$ used. The measurement can thus be considered ‘far’ from the wavemaker with respect to previous experiments (see table 2) in which the basin size was of the order of ${\it\lambda}_{m}$ (Falcon et al. Reference Falcon, Laroche and Fauve2007b ; Herbert et al. Reference Herbert, Mordant and Falcon2010; Cobelli et al. Reference Cobelli, Przadka, Petitjeans, Lagubeau, Pagneux and Maurel2011). Indeed, if the forcing scale ${\it\lambda}_{m}$ is not in the inertial range of the cascade (see § 5.1), one should use a scale ${\it\xi}$ at the beginning of the cascade, rather than ${\it\lambda}_{m}$ (e.g. ${\it\xi}\simeq 17~\text{cm}$ corresponding to a cascade beginning at 3 Hz – see § 5.1). The gauges are then located at a distance from the wavemaker corresponding to 42 spatial scales ${\it\xi}$ ( $L/{\it\xi}\simeq 84$ ). We have also verified that the wave spectrum measured in the vicinity of the wavemaker is different from that measured in the far field in the centre of the basin. All the results obtained here are found to be independent of the gauge type in the working range of the gauges, and of the spectral shapes prescribed to the wavemaker. Moreover, they do not depend significantly on the gauge location within the basin except when the gauges are too close to the boundaries (wavemaker, beach or walls). Typically, ${\it\sigma}_{{\it\eta}}$ varies less than 5 % for different gauge locations, keeping all the other parameters fixed.

Figure 3.

Amplitude of the wave spectrum amplitude $S_{{\it\eta}}(f)$ measured at $f=3~\text{Hz}$ as a function of mean wave steepness $s$ . The dashed line has slope 1. Forcing parameters: broad bandwidth ( $f_{m}=1~\text{Hz}$ , ${\rm\Delta}f=0.6~\text{Hz}$ ) for FRN (▫) or JON (○) forcings; narrow bandwidth ( $f_{m}=1~\text{Hz}$ , ${\rm\Delta}f=0.3~\text{Hz}$ ) for FRN ( $+$ ) or JON ( $\times$ ) forcings. Black: beach. Red (light-grey): wall.

3.4. Wave amplitude parameter

Several parameters have been used in the literature to quantify irregular wave amplitudes. A natural choice is the r.m.s. wave amplitude, ${\it\sigma}_{{\it\eta}}$ , a value directly related to the area under the wave spectrum, $S_{{\it\eta}}(f)$ . In oceanography, the significant wave height $H_{s}$ or $H_{1/3}$ was traditionally defined as the mean wave height (trough to crest) of the highest third of the waves. Now, it is usually defined as $H_{s}=4{\it\sigma}_{{\it\eta}}$ , and with this choice $H_{s}$ and ${\it\sigma}_{{\it\eta}}$ are equivalent. The mean injected power by the wavemaker within the system has also been used previously. However, an unknown amount of energy is injected into the bulk and not into the waves (Deike et al. Reference Deike, Berhanu and Falcon2014a ). The mean wave steepness (or wave slope) $s$ is useful to quantify the degree of nonlinearity of the wave field. It is usually defined as $s\equiv {\it\sigma}_{{\it\eta}}k_{m}$ , with $k_{m}$ the wavenumber corresponding to the maximum amplitude of the spectrum. In all our experiments, $k_{m}$ is roughly constant, and is located in the forcing range $k_{m}=2{\rm\pi}/{\it\lambda}_{m}\approx 4.2~\text{m}^{-1}$ . The range of the nonlinearity parameter is $0.02<s<0.25$ . The value of the spectrum amplitude at the beginning of the cascade, but outside the forcing frequency range, is a more relevant parameter (Nazarenko et al. Reference Nazarenko, Lukaschuk, McLelland and Denissenko2010). Since there is no trivial relation between the input energy by the wavemaker, and the energy flux cascading through the wave scales, the spectrum amplitude at the beginning of the cascade is actually a relevant parameter to quantify the magnitude of the cascade of gravity wave turbulence. For instance, for a forcing frequency bandwidth close to 1 Hz, the amplitude of the wave spectrum measured at 3 Hz corresponds roughly to the beginning of the cascade of gravity wave turbulence, and is also well separated from the first harmonic of the forcing. In the following, we will choose the spectrum amplitude at 3 Hz, $S_{{\it\eta}}(3~\text{Hz})$ , as the parameter that characterizes the forcing amplitude. As shown in figure 3, we found that this parameter increases monotonically with the mean wave steepness but not with a simple scaling (nonlinearly at small $s$ , then linearly at high $s$ ). However, this relationship does not depend on the basin boundary conditions.

4.1. Direct observation of the wave field

Irregular waves (of random frequency and amplitude) are generated by the wavemaker as explained in § 3. The typical temporal evolution of wave amplitude ${\it\eta}(t)$ and the corresponding picture of the wave field in a stationary regime are shown in figure 4 for two different boundary conditions: beach (figure 4 a,b) or wall (figure 4 c,d). Direct observation of the wave field shows that its spatial structure depends strongly on the absorbing or reflecting boundary condition. In the absorbing case (beach), a quasi-one-dimensional field of nonlinear waves propagates from the wavemaker before being damped by the beach (see figure 4 b). In the reflecting case (wall), such coherent structures are not visible. Instead, a multidirectional wave field is observed (see figure 4 d) due to nonlinear interactions between waves and multiple reflections occurring from the basin walls. Note that the direction of forcing is one-dimensional in both cases, and the wave steepnesses are of the same order.

The temporal evolution of the wave amplitude ${\it\eta}(t)$ is shown in figure 4(a,c). Both signals are erratic, showing rare large wave events, as well as higher frequency components than the forcing ones. Note that sharp crest events seem more probable in the beach case (as emphasized by the (red) circles in figure 4 a), occurring only rarely with a wall (figure 4 c). The displayed sample for each boundary condition is representative of the whole time series. Moreover, for the highest forcing amplitudes, we occasionally observe the presence of breaking events during the propagation similar to those studied in laboratory flumes (Melville, Veron & White Reference Melville, Veron and White2002; Perlin, Choi & Tian Reference Perlin, Choi and Tian2013). At sufficiently high forcing amplitude, and for both boundary conditions, we find that the probability distribution function of wave amplitude is well described by a Tayfun distribution (the first quadratic nonlinear correction to the Gaussian) (Tayfun Reference Tayfun1980; Socquet-Juglard et al. Reference Socquet-Juglard, Dysthe, Trulsen, Krogstad and Liu2005) as already observed in laboratory experiments (Onorato et al. Reference Onorato, Osborne, Serio, Cavaleri, Brandini and Stansberg2004, Reference Onorato, Cavaleri, Fouques, Gramstad, Janssen, Monbaliu, Osborne, Pakozdi, Serio and Stansberg2009; Falcon et al. Reference Falcon, Laroche and Fauve2007b ; Falcon & Laroche Reference Falcon and Laroche2011) or in oceanography (Ochi Reference Ochi1998; Forristall Reference Forristall2000).

4.2. Spatial correlation of the wave field

We compute the spatial correlation between the wave gauges to quantify the basic spatial properties of the wave field. The correlation between the wave gauges $i$ and $j$ reads $C_{ij}({\it\tau})=\lim _{T\rightarrow \infty }(1/T)\int _{-T}^{T}{\it\eta}_{i}(t){\it\eta}_{j}(t+{\it\tau})\,\text{d}t\,/\sqrt{C_{ii}C_{jj}}$ , where $C_{ii}=\lim _{T\rightarrow \infty }(1/T)\int _{-T}^{T}{\it\eta}_{i}(t){\it\eta}_{i}(t+{\it\tau})\,\text{d}t$ is the autocorrelation function. The correlation function is thus normalized between  $-1$ and  $+1$ . The maximum over the time ${\it\tau}$ of the correlation, $C_{ij}^{m}$ , gives information on the wave field mixing and propagation properties between the gauges: $C_{ij}^{m}=1$ occurs when signals from gauges $i$ and $j$ are totally correlated, while $C_{ij}^{m}=0$ corresponds to two signals completely uncorrelated. Note that the correlation between two gauges depends on the linear dispersion of a wave packet, the propagation direction of the waves, as well as decorrelation induced by nonlinear interactions.

Figure 5.

Maximum of correlation amplitude, $C_{1,2}^{m}$ , between the wave gauges 1 and 2 as a function of r.m.s. wave amplitude, ${\it\sigma}_{{\it\eta}}$ , for two different boundary conditions: absorbing (beach – black) and reflecting (wall – light grey (red)). Symbols correspond to the same forcing parameters as in figure 3.

Figure 5 shows the maximum of correlation amplitude, $C_{1,2}^{m}$ , between two probes facing the wavemaker (wave gauges $C_{1}$ and $C_{2}$ ). These probes are located at the same distance from the wavemaker and are 2 m apart (see figure 1). The maximum correlation is reached for ${\it\tau}\simeq 0$ . We found that $C_{1,2}^{m}$ depends strongly on the basin boundary conditions. For the absorbing condition (beach), the wave amplitudes are highly correlated whatever the forcing ( $C_{1,2}^{m}$ close to 1), while for the reflecting boundary condition (wall), the correlation is low ( $C_{1,2}^{m}<0.4$ ). A two-point correlation close to 1 means that the same wave train is observed at the two probes at the same time. This confirms quantitatively the fact that, in the case of the beach, the wave field remains almost one-dimensional during the propagation. For the wall case, the correlation is much lower due to the multiple reflections occurring on the basin walls enhancing nonlinear wave interactions. The resulting wave field is thus more complex than observed in figure 4(d). The two-point correlation thus confirms direct observation of the wave field pictures. Note that similar results are found for the correlation between two probes aligned with the forcing direction. These spatial properties obtained from temporal measurements (even if spatio-temporal ones should be ideally obtained) are mainly related to the forcing properties and to the boundary conditions. We have to keep in mind these simple spatial properties when discussing the wave spectrum in § 5.

5.1. Wave spectrum

Figure 6 shows the wave amplitude spectra, $S_{{\it\eta}}(f)$ , for increasing forcing amplitudes for reflecting (figure 6 a) or absorbing (figure 6 b) boundary conditions. Surprisingly, both conditions lead to the same qualitative shape of the spectra as the forcing is increased. For small forcing amplitude, peaks related to the forcing and its harmonics are visible in the low-frequency part of the spectrum and no power law is observed. At sufficiently high forcing, those peaks are smoothed out and a power law, $S_{{\it\eta}}(f)\sim f^{{\it\alpha}}$ , can be fitted. This corresponds to the cascade of gravity wave turbulence over a one decade frequency range from roughly 1.5 Hz (the higher forcing frequency) up to the gravity–capillary crossover frequency $f_{gc}\equiv \sqrt{2g/l_{c}}/(2{\rm\pi})\simeq 14~\text{Hz}$ with $l_{c}\equiv \sqrt{{\it\gamma}/({\it\rho}g)}$ the capillary length, $g=9.81~\text{m}~\text{s}^{-2}$ the acceleration due to gravity, ${\it\gamma}=70~\text{mN}~\text{m}^{-1}$ the surface tension, and ${\it\rho}=1000~\text{kg}~\text{m}^{-3}$ the water density (Falcon et al. Reference Falcon, Laroche and Fauve2007b ). When the forcing is further increased, the slope of the power-law spectrum becomes less steep, corresponding to an increase in the exponent ${\it\alpha}$ . Finally, for the highest forcings, the slope seems to saturate to a constant value (see dashed line), although the peak amplitude of the forcing frequencies still increases. For both boundary conditions, this value is close to $-4$ , the exponent predicted by gravity wave turbulence theory (Zakharov & Filonenko Reference Zakharov and Filonenko1967a ). This may be coincidental, since the effects of dissipation and nonlinear coherent structures are strongly involved experimentally but are not taken into account in weak turbulence theory (see § 5.2). Note that the role of dissipation has recently been studied theoretically (Zakharov et al. Reference Zakharov, Korotkevich, Pushkarev and Resio2007).

Figure 6.

Power spectra of wave amplitude, $S_{{\it\eta}}(f)$ , for reflecting (a) or absorbing (b) boundary conditions. The forcing amplitude increases from bottom to top. (a) Weak turbulence predictions for gravity regime $S_{{\it\eta}}\sim f^{-4}$ (dashed line) and capillary regime $S_{{\it\eta}}\sim f^{-17/6}$ (dot-dashed line). (b) Best fit ${\sim}f^{-4.4}$ in the gravity regime (dashed line) and $f^{-17/6}$ in the capillary regime (dot-dashed line). Vertical grey lines indicate the forcing frequency range. The same forcing parameters were used for both panels (a) and (b). FRN forcing with broad bandwidth ( $f_{m}=1~\text{Hz}$ , ${\rm\Delta}f=0.6~\text{Hz}$ and $0.6~\text{cm}\leqslant {\it\sigma}_{{\it\eta}}\leqslant 3.7~\text{cm}$ ).

Figure 7.

Frequency power-law exponent ${\it\alpha}$ of the gravity wave spectrum $S_{{\it\eta}}\sim f^{{\it\alpha}}$ as a function of the value of the spectrum amplitude at 3 Hz, $S_{{\it\eta}}$ (3 Hz). Boundary conditions: wall (light grey (red)) and beach (black). The dashed line corresponds to the prediction of weak turbulence theory ${\it\alpha}=-4$ . Fitted frequency range: 3–10 Hz. Error bar on ${\it\alpha}$ is $\pm 0.2$ . Symbols correspond to the same forcing parameters as in figure 3 and $0.5~\text{cm}<{\it\sigma}_{{\it\eta}}<5~\text{cm}$ . The $\bullet$ symbols correspond to the Hull experiments in figure 6 (left) of Nazarenko et al. (Reference Nazarenko, Lukaschuk, McLelland and Denissenko2010).

Let us now look at the high-frequency part of the spectrum, corresponding to the capillary range, where the spectrum shape changes. At sufficiently high forcing, a second power law is indeed observed over a one-decade frequency range ( $f_{gc}<f<100~\text{Hz}$ ). The slope is much less steep than that for the gravity range and is close to the capillary wave turbulence prediction $f^{-17/6}$ (see dot-dashed lines) (Zakharov & Filonenko Reference Zakharov and Filonenko1967b ). Note that the observation of both direct cascades of gravity and capillary wave turbulence was practically unattainable in previous large basin facilities. It is possible here due to both the high sensitivity and low noise level of the capacitive probes, the latter being reached for $f\gtrsim 200~\text{Hz}$ . These results are found to be independent of the forcing parameters (spectral shape and frequency bandwidth).

As discussed in § 4, the propagation of a quasi-one-dimensional field of nonlinear waves is observed in the presence of a beach, whereas the presence of a wall leads to numerous propagation directions and consequently a multidirectional wave field. Although the different boundary conditions yield pronounced differences in wave field structure, there is surprisingly no significant difference in the corresponding wave spectra.

The frequency power law of the gravity wave spectrum, $S_{{\it\eta}}(f)\sim f^{{\it\alpha}}$ , is found to depend on the forcing amplitude. Figure 7 shows ${\it\alpha}$ as a function of the forcing strength for both the absorbing and reflecting boundary conditions. We choose to plot it as a function of $S_{{\it\eta}}$ (3 Hz), the value of the spectrum amplitude at 3 Hz (a forcing strength parameter more relevant than the mean wave steepness $s$ , or the r.m.s. wave amplitude ${\it\sigma}_{{\it\eta}}$ , as explained in § 3.4). The exponent ${\it\alpha}$ is found to increase with the forcing strength for both boundary conditions. In a closed basin, ${\it\alpha}$ seems to saturate at high forcing near $-4$ within the data scattering. In the presence of a beach, the highest value reached by ${\it\alpha}$ is also $-4$ but occurs at a smaller $S_{{\it\eta}}$ (3 Hz). For both boundary conditions, ${\it\alpha}$ is thus found to be close to $-4$ at sufficiently high forcing. The maximum value of $S_{{\it\eta}}$ (3 Hz) reached in the presence of a beach is less than that obtained with a wall for the same forcing parameters. This arises from the fact that the dissipated power is stronger in the presence of a beach than within a closed basin. As stated earlier, our results are independent of the spectral shapes prescribed to the wavemaker. Finally, when comparing our results performed in the closed basin with those reported in the Hull experiments (Nazarenko et al. Reference Nazarenko, Lukaschuk, McLelland and Denissenko2010) (see $\bullet$ symbols in figure 7), a good overall agreement is found, although a smaller value of $S_{{\it\eta}}$ (3 Hz) is needed in our case to reach the same value of  ${\it\alpha}$ .

7.1. Experimental protocol

We use the same protocol as in previous studies on freely decaying wave turbulence on thin elastic plates (Miquel & Mordant Reference Miquel and Mordant2011b ; Deike, Bacri & Falcon Reference Deike, Bacri and Falcon2013) or on the surface of a fluid (Deike, Berhanu & Falcon Reference Deike, Berhanu and Falcon2012; Bedard et al. Reference Bedard, Lukaschuk, Nazarenko, Shrira and Nazarenko2013a ,Reference Bedard, Nazarenko and Lukaschuk b ). A typical experiment is as follows. First, surface waves are generated during 7 min, a sufficiently long time to reach a stationary wave turbulence state. The forcing is then stopped at $t=0$ , and the temporal decay of the wave amplitude ${\it\eta}(t)$ is recorded with a 500 Hz sampling frequency by means of two capacitive probes ( $C_{1}$ and $C_{2}$ ) over a period of 15 min, a sufficiently long time to observe the wave damping up to a still state. The experiment is then repeated 20 times to improve statistics, and the results are averaged. Accuracy on the wavemaker stopping time is within 2 s. The results are found to be independent of the locations of the probes on the 4 m probe rack. The results reported in this section do not depend qualitatively on the initial forcing conditions used in table 2.

Figure 10.

Decay of wave amplitude ${\it\eta}(t)$ as a function of time for the reflecting boundary condition. Forcing is working for $t<0$ and is stopped at $t=0$ . Initial forcing conditions: FRN forcing with a narrow bandwidth ( $f_{m}=1.15~\text{Hz}$ , ${\rm\Delta}f=0.3~\text{Hz}$ , ${\it\sigma}_{{\it\eta}}=4.6~\text{cm}$ ).

7.2. Temporal decay of the wave amplitude

The temporal decay of the wave amplitude ${\it\eta}(t)$ is shown in figure 10 for a reflecting boundary condition. Time $t=0$ corresponds to the moment the wavemaker stops. The decay lasts roughly 900 s, including the very slow relaxation of the transverse modes of the tank. Wave energy is dissipated by viscous mechanisms (in bulk, on the free interface and on the tank sides), and transferred to other scales by nonlinear interactions. For the absorbing boundary condition and for the same initial forcing conditions, the decay is much faster ( ${\sim}50~\text{s}$ , roughly corresponding to the propagation time of the last generated wave train) since the beach absorbs most of the wave energy. Thus, we will only report below results on the decay within the closed basin.

7.3. Temporal decay of the spectrum

To analyse the different steps of the decay of ${\it\eta}(t)$ , the time–frequency wave amplitude spectrum $S_{{\it\eta}}(f,t)$ is computed by means of a spectrogram analysis (MATLAB function), for each experiment over short temporal windows $[t,t+{\it\delta}t]$ with ${\it\delta}t=8~\text{s}$ , and $0\leqslant t\leqslant 800~\text{s}$ . Then $S_{{\it\eta}}(f,t)$ is averaged, first over the two probe signals, and then over 20 different realizations, leading to the averaged spectrum $\langle S_{{\it\eta}}(f,t)\rangle$ , with $\langle \,\cdot \,\rangle$ denoting ensemble average. Figure 11 shows $\langle S_{{\it\eta}}(f,t^{\ast })\rangle$ as a function of the frequency at different decay times $t^{\ast }$ . At the beginning of the decay (top curve), the spectrum displays a frequency power law ${\sim}f^{{\it\alpha}}$ in the gravity frequency range ( $1~\text{Hz}\leqslant f\leqslant 10~\text{Hz}$ ) with an exponent ${\it\alpha}$ close to its value in the stationary regime. When $t^{\ast }$ increases, the power-law spectrum becomes progressively steeper, with ${\it\alpha}$ decreasing over time as shown in the inset of figure 11. No self-similar decay is thus observed in the gravity regime. On the contrary, in the capillary frequency range, when measurements are not too noisy, it can be observed that the shape of the power-law spectrum does not depend significantly on the decay time. This last result is compatible with the self-similar decay of capillary wave turbulence observed previously in a small container (Deike et al. Reference Deike, Berhanu and Falcon2012).

Figure 11.

Wave spectrum $\langle S_{{\it\eta}}(f,t^{\ast })\rangle$ at different times $t^{\ast }$ of the decay. From top to bottom: $t^{\ast }=25$ , 81, 161, 241, 401 and 641 s. The dashed line is a power-law fit ${\sim}f^{{\it\alpha}}$ with ${\it\alpha}=-4.7$ in gravity frequency range ( $1~\text{Hz}\leqslant f\leqslant 10~\text{Hz}$ ). The dot-dashed line corresponds to the stationary capillary wave turbulence prediction of $f^{-17/6}$ . The three top curves have been shifted vertically for clarity by a factor 10, 5 and 2, respectively. Inset: gravity exponent ${\it\alpha}$ as a function of time. Closed basin. Same initial forcing conditions as in figure 10.

Figure 12.

Frequency power-law exponent, ${\it\alpha}$ , of the gravity wave spectrum $S_{{\it\eta}}\sim f^{{\it\alpha}}$ as a function of the value of the spectrum amplitude at 3 Hz, $S_{{\it\eta}}$ (3 Hz). Decaying regime (black) or stationary regime (light grey (red), same data as in figure 7). Reflecting boundary condition. The dashed line corresponds to the prediction of weak turbulence theory ${\it\alpha}=-4$ . FRN forcing with a broad (▫) or narrow ( $+$ ) bandwidth.

7.4. Frequency power-law exponent of the gravity spectrum during the decay

The frequency power-law exponent ${\it\alpha}$ of the spectrum is estimated within the gravity frequency range, at each instant of the decay (see inset of figure 11) and is displayed in figure 12 (black symbols) as a function of $S_{{\it\eta}}$ (3 Hz), the value of the spectrum amplitude at 3 Hz. Just after the forcing is stopped (corresponding to the highest value of $S_{{\it\eta}}$ (3 Hz)), the exponent of the power spectrum is similar to that observed in the stationary regime, close to $-4$ . During the decay, the spectrum decreases in amplitude (smaller values of $S_{{\it\eta}}$ (3 Hz)) and is steeper (see figure 11). This leads to the exponent ${\it\alpha}$ strongly depending on $S_{{\it\eta}}$ (3 Hz) as shown in figure 12. The values of ${\it\alpha}$ in the decaying regime are then compared in figure 12 with those obtained in the stationary regime (light grey (red) symbols) of § 5, both being performed in the closed basin. We observe that ${\it\alpha}$ increases with the spectrum amplitude in both the stationary and decaying regimes with the same trend. As a first approximation, this means that decaying wave turbulence can be seen, at each time point of the decay, as wave turbulence in a stationary regime but with the corresponding decreasing wave energy. This feature has also been observed for capillary wave turbulence decay (Kolmakov et al. Reference Kolmakov, Levchenko, Brazhnikov, Mezhov-Deglin, Silchenko and McClintock2004; Deike et al. Reference Deike, Berhanu and Falcon2012). Note that the data for the decaying regime in figure 12 are more scattered than in the stationary regime, as the non-stationary spectra involve less statistics and thus a lower signal-to-noise ratio.

Figure 13.

Temporal evolution of the wave spectrum $\langle S_{{\it\eta}}(f^{\ast },t)\rangle$ for Fourier components $f^{\ast }=1.2$ , 4.4, 8.3, 18.1 and 28.8 Hz (from top to bottom). Dashed (red) lines: $t^{-1/2}$ law predicted for four-wave interactions (gravity). Dot-dashed (blue) lines: $t^{-1}$ law predicted for three-wave interactions (capillary). Solid (magenta) lines: exponential decay $\text{e}^{-t/{\it\tau}_{d}(f^{\ast })}$ as expected for a viscous damping, with ${\it\tau}_{d}(f^{\ast })$ the damping time. Closed basin.

7.5. Temporal decay of the spectrum and energy Fourier modes

Figure 13 shows the temporal evolution of $\langle S_{{\it\eta}}(f^{\ast },t)\rangle$ for different Fourier components $f^{\ast }$ . In the first stage of the decay ( $t<200~\text{s}$ ), the Fourier modes in the gravity frequency range are observed to decrease over time as $t^{-1/2}$ (see dashed lines) as predicted for a nonlinear wave decay involving four-wave interactions (Bedard et al. Reference Bedard, Lukaschuk, Nazarenko, Shrira and Nazarenko2013a ). This confirms with more accuracy the experimental $t^{-1/2}$ scaling found by Bedard et al. (Reference Bedard, Nazarenko and Lukaschuk2013b ). In the capillary frequency range, each Fourier component of the spectrum decreases over time as $t^{-1}$ as expected for three-wave interactions (Falkovich, Shapiro & Shtilman Reference Falkovich, Shapiro and Shtilman1995) (see dot-dashed lines). This first stage of the decay is thus related to nonlinear mechanisms. For longer decay times ( $t>200~\text{s}$ ), the Fourier modes decay roughly exponentially with time as $\text{e}^{-t/{\it\tau}_{d}(f^{\ast })}$ , as expected for a linear viscous dissipation. Viscous dissipation could arise from surface boundary layers on the bottom and side walls as well as on the free surface due mostly to surface contamination (Lamb Reference Lamb1932; van Dorn Reference van Dorn1966; Miles Reference Miles1967). The viscous damping time ${\it\tau}_{d}(f^{\ast })$ is fitted empirically and is found to decrease with the Fourier mode frequency from 300 to 100 s typically. This second stage of the decay is thus driven by the viscous decay of the waves.

Now, let $E_{f}(t)$ be the wave energy of the Fourier mode at frequency $f$ at time $t$ . At $t=0$ , the forcing is stopped and the decaying wave energy can be modelled by

(7.1) $$\begin{eqnarray}\frac{\text{d}E_{f}(t)}{\text{d}t}=-a_{1}E_{f}(t)-a_{2}E_{f}^{2}(t)-a_{3}E_{f}^{3}(t),\end{eqnarray}$$

with $a_{1}$ , $a_{2}$ and $a_{3}$ taking positive values depending on the frequency $f$ . The first term on the right-hand side corresponds to a usual viscous linear dissipation, the second and third terms modelling nonlinear dissipation from three-wave and four-wave nonlinear interactions, respectively. These nonlinear dissipations result from the difference at a fixed frequency between the in-flux from low frequencies and the out-flux towards high frequencies. We solve this equation by considering only one non-zero dissipation coefficient $a_{1}$ , $a_{2}$ or $a_{3}$ in order to compare the analytical solutions and the experiment results with a unique fitting parameter. The linear case leads to $\text{d}E_{f}(t)/\text{d}t=-a_{1}E_{f}(t)$ , and thus the wave energy of the Fourier mode decays exponentially in time as

(7.2) $$\begin{eqnarray}E_{f}(t)=E_{f}(0)\exp [-t/{\it\tau}_{d}],\end{eqnarray}$$

with $1/{\it\tau}_{d}=a_{1}$ the linear dissipative time scale, and $E_{f}(0)$ the energy when the forcing is stopped. For a quadratic nonlinearity (three-wave interaction such as for capillary waves), $\text{d}E_{f}(t)/\text{d}t=-a_{2}{E_{f}(t)}^{2}$ , and thus

(7.3) $$\begin{eqnarray}E_{f}(t)=E_{f}(0)[1+t/{\it\tau}_{nl}^{c}]^{-1},\end{eqnarray}$$

with $1/{\it\tau}_{nl}^{c}=a_{2}E_{f}(0)$ the nonlinear decay time of capillary waves. For $t\gg {\it\tau}_{nl}^{c}$ , this becomes $E_{f}(t)\sim t^{-1}$ . Finally, for a cubic nonlinearity (four-wave interaction such as for gravity waves), one obtains

(7.4) $$\begin{eqnarray}E_{f}(t)=E_{f}(0)[1+2t/{\it\tau}_{nl}^{g}]^{-1/2},\end{eqnarray}$$

with $1/{\it\tau}_{nl}^{g}=a_{3}{E_{f}(0)}^{2}$ the nonlinear decay time of gravity waves. For $t\gg {\it\tau}_{nl}^{g}$ , it follows that $E_{f}(t)\sim t^{-1/2}$ . Note that ${\it\tau}_{d}$ , ${\it\tau}_{nl}^{c}$ and ${\it\tau}_{nl}^{g}$ depend on the scale $f$ .

The temporal decay of the wave energy $E_{f^{\ast }}(t)$ at frequency $f^{\ast }$ is related to the power spectrum of wave height at the same component, $S_{{\it\eta}}(f^{\ast },t)$ , by

(7.5) $$\begin{eqnarray}E_{f^{\ast }}(t)=gS_{{\it\eta}}(f^{\ast },t)+\frac{{\it\gamma}}{{\it\rho}}k^{2}(f^{\ast })S_{{\it\eta}}(f^{\ast },t),\end{eqnarray}$$

with $k(f)$ given by the dispersion relation of linear gravity–capillary waves.

The temporal decay of the wave energy $E_{f^{\ast }}(t)$ at each frequency $f^{\ast }$ is thus inferred experimentally from that of the wave spectra, $S_{{\it\eta}}(f^{\ast },t)$ (see figure 13) by using (7.5). For a fixed $f^{\ast }$ in the gravity range ( $0.5~\text{Hz}\leqslant f^{\ast }\leqslant 10~\text{Hz}$ ), (7.4) is found to be a good fit for $E_{f^{\ast }}(t)$ over short time periods ( $0\leqslant t\leqslant 100~\text{s}$ ), leading to an experimental estimate of ${\it\tau}_{nl}^{g}(f^{\ast })$ , $E_{f}(0)$ being given by the value of ${\it\sigma}_{{\it\eta}}$ in the stationary regime $(t\leqslant 0)$ . Similarly, for a fixed $f^{\ast }$ in the capillary regime ( $10~\text{Hz}<f^{\ast }\leqslant 50~\text{Hz}$ ), (7.3) is a good fit for $E_{f^{\ast }}(t)$ for small $t$ , leading to an estimate of ${\it\tau}_{nl}^{c}(f^{\ast })$ . For long times ( $t>200~\text{s}$ ), $E_{f^{\ast }}(t)$ is found to decay exponentially in both regimes as in (7.2), thus leading to an estimate ${\it\tau}_{d}(f^{\ast })$ . Finally, reiterating these fits for various $f^{\ast }$ gives the frequency dependence of time scales ${\it\tau}_{nl}^{g}$ , ${\it\tau}_{nl}^{c}$ and ${\it\tau}_{d}$ .

7.6. Time-scale separations

Let us now consider the typical time scales involved in our experiment. Weak turbulence theory assumes a time-scale separation ${\it\tau}_{l}(f)\ll {\it\tau}_{nl}(f)\ll {\it\tau}_{d}(f)$ , between the linear propagation time, ${\it\tau}_{l}$ , the nonlinear interaction time, ${\it\tau}_{nl}$ , and the dissipation time, ${\it\tau}_{d}$ . To our knowledge, such a time-scale separation has been tested experimentally in only two different wave turbulence systems (Miquel & Mordant Reference Miquel and Mordant2011a ; Deike et al. Reference Deike, Bacri and Falcon2013), but has never been investigated experimentally for gravity wave turbulence. The linear propagation time is ${\it\tau}_{l}=1/f$ , whereas ${\it\tau}_{d}(f)$ and ${\it\tau}_{nl}(f)$ are inferred from freely decaying experiments using the results of § 7.5. These time scales are displayed in figure 14. The dissipative (viscous) linear time scale ${\it\tau}_{d}(f)$ is found to be of the order of 100 s and varies smoothly by a factor of 3 within the gravity and capillary frequency ranges. For comparison, a theoretical viscous decay time assuming dissipation due to a viscous surface boundary layer with an inextensible film (Lamb Reference Lamb1932; van Dorn Reference van Dorn1966; Miles Reference Miles1967; Deike et al. Reference Deike, Berhanu and Falcon2012) reads

(7.6) $$\begin{eqnarray}{\it\tau}_{d}^{theo}=\frac{2\sqrt{2}}{k({\it\omega})\sqrt{{\it\omega}{\it\nu}}},\end{eqnarray}$$

with ${\it\nu}$ the kinematic viscosity of water, and $k({\it\omega})$ given by the gravity–capillary dispersion relation. This dissipation comes from the presence of surfactants and/or contaminants at the interface, which leads to an inextensible surface where fluid tangential velocity should be cancelled at the interface. This type of dissipation is known to strongly affect the stability of large-scale gravity waves in the ocean (Henderson & Segur Reference Henderson and Segur2013). For all frequencies, ${\it\tau}_{d}$ is found to be much larger than ${\it\tau}_{d}^{theo}$ except at the forcing frequencies ${\sim}1~\text{Hz}$ where the two curves intersect. This observation and the fact that ${\it\tau}_{d}(f)$ varies smoothly compared to ${\it\tau}_{d}^{theo}(f)$ mean that the decay of a largest scale mode (near the forcing scale) transfers energy continuously in time towards smaller scales. Thus, the decay of all Fourier modes is driven by the viscous decay of a large-scale mode as has been observed in small container experiments (Deike et al. Reference Deike, Berhanu and Falcon2012). Consequently, the estimated nonlinear decay time scales ${\it\tau}_{nl}^{g}(f)$ and ${\it\tau}_{nl}^{c}(f)$ include a contribution due to the cumulative energy transfer from this large-scale mode, in addition to that from nonlinear wave interactions. Indeed, ${\it\tau}_{nl}^{g}(f)$ is found to roughly decrease as $f^{-3/2}$ in the gravity inertial range, whereas the four-wave nonlinear interaction time scale reads dimensionally ${\it\tau}_{4w}^{g}\equiv c{\it\epsilon}^{-2/3}g^{2}f^{-3}$ (Connaughton et al. Reference Connaughton, Nazarenko and Newell2003; Newell & Rumpf Reference Newell and Rumpf2011) with ${\it\epsilon}$ the mean energy flux as estimated in § 7.7, and $c$ a non-dimensional constant. Constant $c$ is then adjusted to have ${\it\tau}_{4w}^{g}={\it\tau}_{nl}^{g}\simeq 40~\text{s}$ at the forcing frequency $f=1~\text{Hz}$ . This leads to ${\it\tau}_{nl}^{g}(f)\gg {\it\tau}_{4w}^{g}(f)$ for $f>1~\text{Hz}$ , as displayed in figure 14. More interestingly, we observe that the scale separation ${\it\tau}_{l}(f)\ll {\it\tau}_{4w}^{g}(f)\ll {\it\tau}_{d}(f)$ is satisfied but in a quite narrow frequency band ( $1~\text{Hz}<f<6~\text{Hz}$ ) despite the use of a large basin. Note that a similar analysis can be experimentally performed for the capillary regime (see Deike et al. (Reference Deike, Fuster, Berhanu and Falcon2014b ) for direct numerical simulations).

Figure 14.

Typical time scales as a function of the frequency, $f$ . Dissipative linear time scale ${\it\tau}_{d}$ ( $\diamondsuit$ ). Nonlinear decay times of gravity ${\it\tau}_{nl}^{g}$ (○) and capillary ${\it\tau}_{nl}^{c}$ (●) regimes. Times ${\it\tau}_{d}$ , ${\it\tau}_{nl}^{g}$ and ${\it\tau}_{nl}^{c}$ are inferred from fits of $E(f,t)$ (data of figure 13 using (7.5)) using (7.2), (7.4) and (7.3), respectively. Solid (green) line: linear time scale ${\it\tau}_{l}=f^{-1}$ . Dot-dashed (red) line: theoretical nonlinear interaction time scale of gravity ${\it\tau}_{4w}^{g}\equiv c{\it\epsilon}^{-2/3}g^{2}f^{-3}$ (see text). Thin dotted (red) line: best fit ${\sim}f^{-3/2}$ . Dashed (black) line: theoretical dissipation time scale ${\it\tau}_{d}^{theo}$ from (7.6) with ${\it\nu}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$ . Same initial forcing conditions as in figure 10. Closed basin.

Thus, non-stationary experiments make it possible to estimate for the first time the dissipative and nonlinear time scales in gravity wave turbulence at all scales of the cascade by extrapolating their values from that of the forcing scale. We show that an important part of this nonlinear time comes from the cumulative energy transfer from a large-scale mode, and thus appears as an upper limit of the four-wave nonlinear interaction time scale of weak turbulence. This large-scale mode thus plays a crucial role in gravity wave turbulence in large basins.

7.7. Estimations of the mean energy flux and Kolmogorov constant

The mean energy flux cascading from large scales to small scales is a key quantity in hydrodynamics turbulence (Pope Reference Pope2006). In wave turbulence, one way to estimate the mean energy flux ${\it\epsilon}$ is to measure the wave energy decay rate after switching off the wavemaker (Denissenko et al. Reference Denissenko, Lukaschuk and Nazarenko2007; Nazarenko et al. Reference Nazarenko, Lukaschuk, McLelland and Denissenko2010). This method gives a good estimate of the mean energy flux, provided large-scale dissipation is negligible (otherwise the large-scale waves lose most of their energy through large-scale dissipation rather than by transferring energy to smaller scales). Here, the estimate of ${\it\epsilon}$ is obtained just at the beginning of the energy decay, thus avoiding this bias. Assuming no forcing and dissipation, the power budget then reads $\text{d}E(t)/\text{d}t=-{\it\epsilon}$ , where $E(t)$ is the wave energy per unit surface and fluid density at time $t$ , and ${\it\epsilon}$ the mean energy flux per unit surface and density. The energy of linear gravity waves (neglecting capillary waves) averaged over a small time lag reads $E(t)=g{\it\sigma}_{{\it\eta}}^{2}(t)$ , where $g$ is the acceleration due to gravity. Combining the two expressions then leads to an estimation of the mean energy flux in the stationary regime ( $t\leqslant 0$ ) as

(7.7) $$\begin{eqnarray}{\it\epsilon}=-g\left.\frac{\text{d}{\it\sigma}_{{\it\eta}}^{2}(t)}{\text{d}t}\right|_{t=0}.\end{eqnarray}$$

Figure 15 shows the temporal evolution of $E(t)$ after switching off the wavemaker at $t=0$ . The tangent to the curve at $t=0$ then gives ${\it\epsilon}=100\pm 30~\text{cm}^{3}~\text{s}^{-3}$ . Note that this value is much smaller than the critical flux $({\it\gamma}g/{\it\rho})^{3/4}\approx 2200~\text{cm}^{3}~\text{s}^{-3}$ corresponding to the breakdown of weak turbulence at the transition between the gravity and capillary regimes (Newell & Zakharov Reference Newell and Zakharov1992). The values of ${\it\epsilon}$ estimated in our experiments are such that ${\it\epsilon}<({\it\gamma}g/{\it\rho})^{3/4}$ . Our estimate of ${\it\epsilon}$ from the decay of the wave energy is found to increase as expected when the initial wave amplitude increases.

Figure 15.

Temporal evolution of the gravity wave energy per unit surface and density. The wavemaker is stopped at $t=0$ . Dashed line: tangent at $t=0$ of slope ${\it\epsilon}=1\times 10^{-4}~\text{m}^{3}~\text{s}^{-3}$ – see (7.7). Each value of ${\it\sigma}_{{\it\eta}}$ is averaged over 20 s and averaged over 19 runs to have a good statistical convergence. Same initial forcing conditions as in figure 10. Closed basin. Inset: wave power spectrum in the stationary regime for the same forcing conditions. Dashed line: theoretical weak turbulence spectrum $C{\it\epsilon}^{1/3}g{\it\omega}^{-4}$ for gravity waves with $C=1.8$ and ${\it\epsilon}=1\times 10^{-4}~\text{m}^{3}~\text{s}^{-3}$ . Dot-dashed line: $f^{-17/6}$ power-law fit. Closed basin.

It is now possible, knowing the value of ${\it\epsilon}$ , to evaluate experimentally the Kolmogorov constant $C$ of (1.1) from the gravity wave spectrum obtained in the stationary regime at sufficiently high forcing. The inset of figure 15 shows such a spectrum displaying good agreement with the ${\it\omega}^{-4}$ power-law scaling expected in the gravity wave turbulence regime and the ${\it\omega}^{-17/6}$ scaling expected in the capillary regime. Using the ${\it\omega}^{-4}$ fitting parameter, the value of ${\it\epsilon}$ obtained above and the expression for the non-dimensional Kolmogorov–Zakharov constant (Zakharov & Filonenko Reference Zakharov and Filonenko1967a )

(7.8) $$\begin{eqnarray}C=\frac{S_{{\it\eta}}({\it\omega}){\it\omega}^{4}}{{\it\epsilon}^{1/3}g},\end{eqnarray}$$

( $S_{{\it\eta}}({\it\omega})$ has dimension $L^{2}T$ and ${\it\epsilon}$ has dimension $L^{3}/T^{3}$ ), one finds a value of the constant $C=1.8\pm 0.2$ , of the same order of magnitude as a theoretical value of 2.75 estimated by Zakharov (Reference Zakharov2010). Note that Badulin et al. (Reference Badulin, Pushkarev, Resio and Zakharov2005) found a numerical constant value of 0.5. Our study therefore reports the first experimental estimation of the Kolmogorov–Zakharov constant for gravity wave turbulence, the latter being compatible with a recently obtained theoretical value.