Identifying four-wave-resonant interactions in a surface gravity wave turbulence experiment

The nonlinear dynamics of waves at the sea surface is believed to be ruled by the weak-turbulence framework. In order to investigate the nonlinear coupling among gravity surface waves, we developed an experiment in the Coriolis facility, which is a 13-m-diam circular tank. An isotropic and statistically stationary wave turbulence of average steepness of 10% is maintained by two wedge wave makers. The space- and time-resolved wave elevation is measured using a stereoscopic technique. Wave-wave interactions are analyzed through third- and fourth-order correlations. We investigate specifically the role of bound waves generated by nonresonant three-wave coupling. Specifically, we implement a space-time filter to separate the dynamics of free waves (i.e., following the dispersion relation) from the bound waves. We observe that the free-wave dynamics causes weakly resonant four-wave correlations. A weak level of correlation is actually the basis of the weak-turbulence theory. Thus our observations support the use of the weak turbulence to model gravity wave turbulence as is currently being done in the operational models of wave forecasting. Although in the theory bound waves are not supposed to contribute to the energy cascade, our observation raises the question of the impact of bound waves on dissipation and thus on energy transfers as well.

Figure 1

(a) Global schematics of the setup in the Coriolis facility. A 13-m-diam circular tank is filled with $0.9\mathrm{m}$ of water. Surface gravity waves are generated by two wedge wave makers. Surface elevation is recorded by ten capacitive wave gauges (red points) and by a stereoscopic reconstruction on a $2\times 2{\mathrm{m}}^2$ area at the center of the tank (green square). (b) Perspective view of a wedge wave maker. It is oscillating vertically at a randomly modulated frequency ${\omega }_f/2\pi =0.59\pm 0.15\mathrm{Hz}$ with an amplitude of 2 cm.

Figure 2

(a), (c), and (d) Instantaneous surface elevation $\eta$ and (b), (d), and (f) norm of its spatial gradient $\left|\mathbf{\nabla }\eta \right|$ for (a) and (b) raw data $\eta$, (c) and (d) filtered data ${\eta }_0$ (free waves), and (e) and (f) ${\eta }_{-1}+{\eta }_0+{\eta }_1$ (see the text for definitions). One sees that fine-scale details are associated with bound waves.

Figure 3

Probability density function of (a) the norm of the surface elevation $\eta$, (b) local slope $\left|\mathbf{\nabla }\eta \right|$, (c) the time derivative of surface elevation ${\partial }_t\eta$, and (d) the squared local slope ${\left|\mathbf{\nabla }\eta \right|}^2$. Raw data $\eta$, filtered data $|\mathbf{\nabla }{\eta }_0|$ and $\left|\mathbf{\nabla }\right({\eta }_{-1}+{\eta }_0+{\eta }_1\left)\right|$, and data from probes are represented by black, red, green, and black dashed lines, respectively. Dotted lines represent a Gaussian distribution in (a) and (c) and a Rayleigh distribution in (d). Departure from Gaussianity can clearly be attributed to the contribution of bound waves.

Figure 4

Frequency power spectrum of the surface elevation from stereoscopic measurements and capacitive probes. The black curve is the spectrum of the full signal measured by the stereoscopic PIV. The other solid curves correspond to filtered data as shown in the legend. The red dashed line is the spectrum measured by the capacitive probes. The black dashed straight line is a guide to the eye for the ${\omega }^{-4}$ theoretical behavior. The spectra measured by capacitive probes and by the stereoscopic contributions are in agreement up to about 7 Hz. At higher frequencies, the stereoscopic spectrum deviates from that of the local probes due to measurement noise. The difference between the spectrum of ${\eta }_0$ (free waves) and that of the full signal corresponds to the energy of the bound waves.

Figure 5

(a) Normalized angular average of the spatiotemporal spectrum of the surface elevation. The red line corresponds to the linear dispersion relation of gravity waves $k_{\text{LDR}}\left(\omega \right)$ and dotted lines correspond to $k_n\left(\omega \right)$, with $n\in [-2,-1,1,2]$ [Eq. (12)]. (b) Spatiotemporal spectrum of the surface elevation for $\omega /2\pi =2.3\mathrm{Hz}$. Solid and dashed lines represent the expected wave number for free and bound waves (with $n=\pm 1$), respectively. In both figures most of the energy lies on the linear dispersion relation, but a significant part of the energy is spread over a large area with a special concentration on the specific bound waves $k_n\left(\omega \right)$.

Figure 6

Normalized third-order correlation of the temporal Fourier transform of the wave elevation $C_3({\omega }_1,{\omega }_2,{\omega }_3)$ for ${\omega }_1/2\pi =2.13\mathrm{Hz}$ (a) averaged over all data and (b) conditionally averaged over data for which $|\partial \eta /\partial t|\le 3.5v_{\text{rms}}$ ($99.8\%$ of total data). Dotted lines indicate the resonant line ${\omega }_3={\omega }_1-{\omega }_2$, which shows that spurious events in the stereoscopic reconstruction are responsible for the high background level in (a).

Figure 7

Shown at the top left is an example of the bicoherence of the temporal Fourier transform of the wave elevation $B({\omega }_2,{\omega }_3)$ defined by Eqs. (27) and (28). This correlation intrinsically has the symmetries $B({\omega }_2,{\omega }_3)=B({\omega }_3,{\omega }_2)=B(-{\omega }_2,-{\omega }_3)$. For the sake of simplicity, the nonredundant part of the correlation is represented here, i.e., ${\omega }_3>{\omega }_2$ and ${\omega }_1={\omega }_2+{\omega }_3>0$. Dashed lines indicate ${\omega }_1,{\omega }_2=\pm n{\omega }_0$, with $n=1,2$. Several components can be seen in this picture: discrete lines highlighted by the dashed lines and a diffuse continuum. All these contributions are due to bound waves as free waves have no three-wave resonance possible.

Figure 8

Cut of the bicoherence $\mathcal{B}$ for ${\omega }_2/2\pi =3\mathrm{Hz}$ and ${\mathbf{k}}_2=k_{\text{LDR}}\left({\omega }_2\right){\mathbf{e}}_x$ in the planes defined by (a) $k_{1,y}=k_{3,y}=0$, (b) ${\omega }_3={\omega }_1-{\omega }_2=-0.7\times 2\pi \mathrm{rad}{\mathrm{s}}^{-1}$, and (c) ${\omega }_3=1.4\times 2\pi \mathrm{rad}{\mathrm{s}}^{-1}$. The red line corresponds to the linear dispersion relation ${\mathbf{k}}_1={\mathbf{k}}_{\text{LDR}}\left({\omega }_1\right)$ shifted by ${\omega }_2$ in frequency and ${\mathbf{k}}_2$ in wave vector, i.e., ${\mathbf{k}}_1={\mathbf{k}}_2+{\mathbf{k}}_{\text{LDR}}({\omega }_1-{\omega }_2)$. This equation corresponds to a surface in the 3D space $({\mathbf{k}}_1,{\omega }_1)$ whose cuts in the various panels are shown by the red lines. The green solid lines are cuts of the linear dispersion relation. The green dashed lines are cuts of the bound-wave dispersion relations ${\mathbf{k}}_n\left(\omega \right)$ for $n=-3,...,3$. Wave resonances lie at the intersection of red and green lines. In (b) and (c) the red arrows represent ${\mathbf{k}}_2$. High levels of bicoherence are clearly shown near the intersections of the free-wave dispersion relation (red) and the bound waves (green), highlighting the contribution of the latter. No such intersection (three-wave resonances) exists for free waves only, except for trivial solutions ${\mathbf{k}}_1={\mathbf{k}}_2$ and ${\omega }_1={\omega }_2$. Thus the observed nonzero bicoherence is clearly attributed to wave resonances involving at least one bound wave.

Figure 9

Shown at the top left is the bicoherence of the filtered temporal Fourier transform of the wave elevation $B_{n_{1}00}({\omega }_2,{\omega }_3)$ with (a) $n_1=0$, (b) $n_1=1$, (c) $n_1=2$, and (d) $n_1=-1$. This correlation intrinsically has symmetries (see the text). For the sake of simplicity, the minimalistic part of the correlation is represented here. Dashed lines indicate ${\omega }_1,{\omega }_2=\pm n{\omega }_0$, with $n=1,2$. No correlation is seen in $B_{000}$ (except at zero frequency, which is not relevant), as expected from the fact that no three-free-wave resonance exists. In contrast, high correlation levels can be observed in $B_{100}, B_{200}$, and $B_{-100}$ around the vertical dashed line that corresponds to the relevant bound wave.

Figure 10

Normalized fourth-order correlation of the temporal Fourier transform of the wave elevation $C_4({\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4)$ for ${\omega }_2/2\pi =2.13\mathrm{Hz}$ and ${\omega }_3/2\pi =1.07\mathrm{Hz}$. The black dashed line is the resonant line ${\omega }_4={\omega }_2-{\omega }_3+{\omega }_1$. A line of correlation emerges on the resonant line, as expected for a stationary signal. The red dot is a special case of ${\omega }_1={\omega }_3$ and ${\omega }_2={\omega }_4$ for which one expects trivially a correlation level close to 1.

Figure 11

Explanation of symmetries of the tricoherence. (a) Example of a tricoherence estimation of the temporal Fourier transform of the wave elevation $T({\omega }_2,{\omega }_3,{\omega }_4)$ for ${\omega }_2/2\pi =3.2\mathrm{Hz}$. (b) Sketch of the various zones in the $({\omega }_3,{\omega }_4)$ plane that are equivalent due symmetries under permutations of indices in the definition of the correlation. Regions A (yellow) are equivalent and correspond to $2\leftrightarrow 2$ interactions of waves (bound or free). Similarly, regions B are equivalent and correspond to $3\leftrightarrow 1$ wave interactions. Region C also corresponds to $3\leftrightarrow 1$ interactions. The correlation picture is also equivalent by symmetry around the main diagonal. Thus the area surrounded by the thick black dashed line is nonredundant and only this subregion will be shown in the following figures.

Figure 12

Tricoherence of the temporal Fourier transform of the wave elevation $T({\omega }_2,{\omega }_3,{\omega }_4)$ for (a) ${\omega }_2/2\pi =1.6$ Hz, (b) ${\omega }_2/2\pi =2.13$ Hz, (c) ${\omega }_2/2\pi =3.2$ Hz, and (d) ${\omega }_2/2\pi =5.2\mathrm{Hz}$. This correlation intrinsically have symmetries. For the sake of simplicity, the minimalistic part of the correlation is represented here (see Fig. 11). Black lines indicate ${\omega }_1,{\omega }_3=0$. Distinct regions of significant correlation levels are visible. First, vertical and horizontal lines of correlation level equal to 1 (in red and reddish colors) due to trivial cases (corresponding to the red dot in Fig. 10), second, more faint lines in yellow and yellowish colors, and third a continuous variation in the background (see the text for descriptions).

Figure 13

Tricoherence of the filtered temporal Fourier transform of the wave elevation $T_{0000}({\omega }_2,{\omega }_3,{\omega }_4)$ for (a) ${\omega }_2/2/\pi =1.6$ Hz, (b) ${\omega }_2/2/\pi =2.13$ Hz, (c) ${\omega }_2/2/\pi =3.2$ Hz, and (d) ${\omega }_2/2/\pi =5.2\mathrm{Hz}$, which corresponds to four-free-wave correlations. This correlation intrinsically has symmetries. For the sake of simplicity, the minimalistic part of the correlation is represented here. Black lines indicate ${\omega }_1,{\omega }_3=0$. The levels and the map of the correlation are very different than for the unfiltered signal of Fig. 12. In particular, the correlation level is much weaker. See the text for more details.

Figure 14

Tricoherence of the filtered temporal Fourier transform of the wave elevation (a) $T_{1000}$, (b) $T_{1010}$, (c) $T_{-1010}$, and (d) $T_{2020}$ for ${\omega }_2/2\pi =3.2\mathrm{Hz}$. This correlation intrinsically has symmetries. For the sake of simplicity, the minimalistic part of the correlations are represented here. We notice that among the represented correlations, by definition of the tricoherence, the horizontal lines defined by ${\omega }_4={\omega }_2$ for $T_{n0n0}$ are the only ones to be trivially equal to 1. Black lines indicate ${\omega }_1,{\omega }_3=0$. Dotted lines indicate ${\omega }_1,{\omega }_3,{\omega }_4=\pm {\omega }_0$. (e)–(h) Same as (a)–(d), but here colored dots are the exact 1D solutions of the resonance (45) (see Appendix pp1 for details). One can see that high levels of correlation (in red) are always associated with regions of 1D resonances with bound waves.

Figure 15

Bicoherence computed from field data measured in the Black Sea [18] and reproduced from Ref. [24]. For this data set the peak of the spectrum occurs at $\omega /2\pi \approx 0.35\mathrm{Hz}$. Thus the secondary lines are most likely contributions of bound waves as in the experiment. These bound waves have actually been observed in the $(\mathbf{k},\omega )$ spectrum in [18].

Figure 16

Graphical representation of solutions for quasiresonant equation (A1) in the collinear wave-vector case, i.e., ${\theta }_i={\theta }_2(mod\pi )$ with $i\in (1,3,4)$, for (a) ${\omega }_2/2\pi =3.2\mathrm{Hz}$ and ${\omega }_3=2\mathrm{Hz}$ and (b) ${\omega }_2/2\pi =3.2\mathrm{Hz}$ and ${\omega }_3=7\mathrm{Hz}$. Solutions are localized at the crossing between the green and red curves and are highlighted by colored dots. (c) Graphical representation of 2D solutions for quasiresonant equations for $({\omega }_2,{\omega }_3,{\omega }_4)/2\pi =(3.2,4,2.5)\mathrm{Hz}$ and $({\theta }_2,{\theta }_3)=(0,\pi /6)$. Green and red curves are defined by $k_{\text{LDR}}({\omega }_3+{\omega }_4-{\omega }_2)\left(\begin{array}{c}cos\left({\theta }_1\right)\\ sin\left({\theta }_1\right)\end{array}\right)+k_{\text{LDR}}\left({\omega }_2\right)\left(\begin{array}{c}cos\left({\theta }_2\right)\\ sin\left({\theta }_2\right)\end{array}\right)$ and $k_{\text{LDR}}\left({\omega }_3\right)\left(\begin{array}{c}cos\left({\theta }_3\right)\\ sin\left({\theta }_3\right)\end{array}\right)+k_{\text{LDR}}\left({\omega }_4\right)\left(\begin{array}{c}cos\left({\theta }_4\right)\\ sin\left({\theta }_4\right)\end{array}\right)$, respectively. The 2D solutions are localized at the crossing between green and red circles.

Figure 17

Tricoherence of the temporal Fourier transform of the wave elevation $T({\omega }_2,{\omega }_3,{\omega }_4)$ for (a) ${\omega }_2/2\pi =1.6$ Hz, (b) ${\omega }_2/2\pi =2.13$ Hz, (c) ${\omega }_2/2\pi =3.2$ Hz, and (d) ${\omega }_2/2\pi =5.2\mathrm{Hz}$. This correlation intrinsically has symmetries. For the sake of simplicity, the minimalistic part of the correlation is represented here. Black lines indicate ${\omega }_1,{\omega }_3=0$. Dotted lines indicate ${\omega }_1,{\omega }_3,{\omega }_4=\pm {\omega }_0$. (e)–(h) Same as (a)–(d), but here colored dots are the 1D solutions of the resonant equations (A2) with the same colors as in Fig. 16.

Figure 18

On the bottom is the fraction of angle ${\theta }_3$ for which solutions of quasiresonant equations (A1) exist with ${\omega }_2/2\pi =3.2$ and ${\theta }_2=0$.

Figure 19

Tricoherence of the filtered temporal Fourier transform of the wave elevation $T_{0000}({\omega }_2,{\omega }_3,{\omega }_4)$ for (a) ${\omega }_2/2\pi =1.6$ Hz, (b) ${\omega }_2/2\pi =2.13$ Hz, (c) ${\omega }_2/2\pi =3.2$ Hz, and (d) ${\omega }_2/2\pi =5.2\mathrm{Hz}$. This correlation intrinsically has symmetries. For the sake of simplicity, the minimalistic part of the correlation is represented here. Black lines indicate ${\omega }_1,{\omega }_3=0$. Dotted lines indicate ${\omega }_1,{\omega }_3,{\omega }_4=\pm {\omega }_0$. (e)–(h) Same as (a)–(d), but here colored dots are the 1D solutions of the resonant equations (A2).