Three-wave and four-wave interactions in gravity wave turbulence

Weak-turbulence theory is a statistical framework to describe a large ensemble of nonlinearly interacting waves. The archetypal example of such system is the ocean surface that is made of interacting surface gravity waves. Here we describe a laboratory experiment dedicated to probe the statistical properties of turbulent gravity waves. We set up an isotropic state of interacting gravity waves in the Coriolis facility (13-m-diam circular wave tank) by exciting waves at 1 Hz by wedge wave makers. We implement a stereoscopic technique to obtain a measurement of the surface elevation that is resolved in both space and time. Fourier analysis shows that the laboratory spectra are systematically steeper than the theoretical predictions and the field observations in the Black Sea by Leckler et al. [F. Leckler et al., J. Phys. Oceanogr. 45, 2484 (2015)]. We identify a strong impact of surface dissipation on the scaling of the Fourier spectrum at the scales that are accessible in the experiments. We use bicoherence and tricoherence statistical tools in frequency and/or wave-vector space to identify the active nonlinear coupling. These analyses are also performed on the field data by Leckler et al. for comparison with the laboratory data. Three-wave coupling is characterized by and shown to involve mostly quasiresonances of waves with second- or higher-order harmonics. Four-wave coupling is not observed in the laboratory but is evidenced in the field data. We discuss temporal scale separation to explain our observations.

Figure 1

Picture of the experimental setup. Waves are generated by vertical oscillations of two wedge wave makers of size $1\times 1\times 1{\mathrm{m}}^3$ (visible in the inset). The deformation of the interface is measured in space and time using three cameras positioned above the surface. The common field of view of the cameras covers a surface of $2\times 1.5{\mathrm{m}}^2$. We seed the surface with polystyrene buoyant particles of 0.7 mm diameter. We perform cross correlation between the different views in order to reconstruct the interface deformation $\eta (x,y,t)$ as well as the velocity field $\mathbf{u}=(u,v,w)$ [24]. A set of four capacitive probes are also used to obtain single-point measurements.

Figure 2

Setup used to clean up the free surface. The upper water layer flows in a 1-m cylinder located near the wall of the circular tank and connected to an intermediate tank located below the main one. The water is then pumped through an active carbon filter before being reinjected at the surface at a location diametrically opposite to the skimmer.

Figure 3

(a) Snapshot of the 3D reconstruction of the free-surface elevation $\eta (x,y,t)$. (b) Temporal evolution of $\eta$ at one given location in the image. (c) Probability density function of $\eta$ normalized by the standard deviation of the wave height ${\sigma }_{\eta }=1.6$ cm. The black dashed line is the normal distribution. The red dashed line is the Tayfun distribution, which incorporates nonlinear second-order effects (see the text).

Figure 4

Two pictures of the free surface taken prior to the filtration (left) and after filtering for a few hours. The intensity of the forcing is equivalent for both case. The filtration changes significantly the surface and capillary waves become present.

Figure 5

(a) Temporal power spectrum $E^{\eta }\left(\omega \right)$ for two experiments with a similar wave steepness. The red curve is obtained without filtration of the free surface while the blue is obtained after a few hours of filtration. The exponent of the spectrum changes from about $-8$ initially to $-5$ for the cleanest water. (b) Distribution of the measured spectral exponent $\alpha$ as a function of the typical steepness of the waves ${\epsilon }_p$ (changed by tuning the magnitude of the forcing). Black triangles are previous measurements reported by Deike et al. [12]. The blue and red squares are, respectively, our measurements with and without filtration. The single purple point is an in situ measurement of gravity waves in the Black Sea [30].

Figure 6

(a) Space-time Fourier power spectrum of the vertical velocity of the waves $E^w(k,\omega )$ for a typical experiment where the mean wave steepness is $\epsilon =0.11$. The spectrum $E^w(\mathbf{k},\omega )$ has been integrated over directions of $\mathbf{k}$. (b) Four cross sections of the complete spectrum $E^w(\mathbf{k},\omega )$ at given frequencies (values indicated on the image). In both cases, the black solid line is the deep water linear dispersion relation $\omega =\sqrt{gk}$ and the dashed line is its second harmonic. Energy is color coded in logarithmic scale.

Figure 7

(a) Snapshot of ${\mathbf{u}}_{\mathbf{h}}·\mathbf{\nabla }\eta$, the nonlinear component of the vertical velocity field $w=\frac{\partial \eta }{\partial t}+{\mathbf{u}}_{\mathbf{h}}·\mathbf{\nabla }\eta$. (b) Coherence of the different components of the velocity relation (4) as a function of $k$. Assuming isotropy, the coherence is averaged over eight directions of $\mathbf{k}$ regularly spread over a circle. The black curve is the coherence $C^{\dot{\eta },w}$ between the vertical geometric velocity $\dot{\eta }$ and $w$. The blue curve is the coherence $C^{\dot{\eta },-{\mathbf{u}}_{\mathbf{h}}·\mathbf{\nabla }\eta }$ between $\dot{\eta }$ and $-{\mathbf{u}}_{\mathbf{h}}·\mathbf{\nabla }\eta$. The red curve is the coherence $C^{\dot{\eta },w-{\mathbf{u}}_{\mathbf{h}}}$ between $\dot{\eta }$ and $w-{\mathbf{u}}_{\mathbf{h}}·\mathbf{\nabla }\eta$ [see (5) for definitions].

Figure 8

(a) The 3D theoretical exact geometrical solution for three-wave resonant interactions. For simplicity, only the positive frequencies are represented. A first linear dispersion relation ${\omega }_1\left({\mathbf{k}}_1\right)$ is shown in light pink. We pick a value of the pair $({\mathbf{k}}_2,{\omega }_2)$ shown with the blue arrow. We then plot a second linear dispersion relation in black starting from the point $({\mathbf{k}}_2,{\omega }_2)$. This black surface is thus ${\omega }_2+{\omega }_3({\mathbf{k}}_3=\mathbf{k}-{\mathbf{k}}_2)$. Any intersection of both surfaces corresponds to resonance ${\omega }_1={\omega }_2+{\omega }_3$. As it is well known, we observe no intersection between the pink and black curves, as no three-wave resonances are geometrically allowed for waves following the linear dispersion relation $\omega =\sqrt{gk}$. The red surface is the second harmonic ${\omega }_1=\sqrt{2g{k}_1}$. The intersection of the red and the black surface indicates the presence of resonant solutions when the wave labeled 1 is a second harmonic of the linear dispersion relation and waves 2 and 3 are linear waves. These are usually known as bound harmonics where ${\mathbf{k}}_2+{\mathbf{k}}_3=2{\mathbf{k}}_1$. (b) Cross section of the 3D representation for $k_y/2\pi =0{\mathrm{m}}^{-1}$. The third harmonic $\omega =\sqrt{3gk}$ is added as the red dashed line and the second harmonic ${\omega }_3=\sqrt{2g{k}_3}$ is added as the black dashed line starting from the point $(k_2,{\omega }_2)$. Exact solutions are indicated by the green circles. Note that in the vicinity of some solutions (large values of $k_x$ on the right-hand side of the figure), the two dispersion relations remain close for a large range of frequencies, suggesting the possibility of quasiresonance (see the text). (c) Cross section of (a) for a given frequency $\omega /2\pi =3.56\text{Hz}$. Exact solutions are plotted with green circles.

Figure 9

(a) Normalized third-order correlation $C_3^{\omega }({\omega }_1,{\omega }_2,{\omega }_3)$ for a given wave ${\omega }_2/2\pi =2$ Hz. A high concentration of correlations is present along the resonant line ${\omega }_1={\omega }_2+{\omega }_3$, suggesting the presence of three-wave interactions. (b) Bicoherence $B_3^{\omega }({\omega }_2,{\omega }_3)$. A red spot is visible around ${\omega }_2={\omega }_3=2\pi \times 2.5$ Hz.

Figure 10

The 3D bicoherence $B_3^{k,\omega }({\mathbf{k}}_2,{\mathbf{k}}_3,{\omega }_2,{\omega }_3)$ for a given wave $({\mathbf{k}}_2,{\omega }_2)$ (with $|k_2|/2\pi =4{\mathrm{m}}^{-1})$. In order to allow a direct comparison, the axes are the same as those of Fig. 8 and show ${\mathbf{k}}_1={\mathbf{k}}_2+{\mathbf{k}}_3$ and ${\omega }_1={\omega }_2+{\omega }_3$. The $x$ axis is given by the direction of ${\mathbf{k}}_2$. The black arrows show the wave $({\mathbf{k}}_2,{\omega }_2)$. (a) Cross section in the plane $(k_x,k_y=0,\omega )$. The red and black curves are similar to the ones shown in Fig. 8. Exact resonances with harmonics are at the intersection of a black curve and a red one. A high correlation intensity is observed in these regions, suggesting the occurrence of interaction between linear waves and harmonics. (b)–(d) Cross sections in the planes corresponding to given values of (b) $\omega /2\pi =4$, (c) $\omega /2\pi =5$, and (d) $\omega /2\pi =6$ Hz. An angular dispersion of the interactions of about ${90}^{\circ }$ is observed.

Figure 11

(a) Power spectrum $E^{\eta }(k,\omega )$ of surface gravity waves measured by Leckler et al. [30] in the Black Sea. The inset shows the cross section of the complete spectrum at a given frequency of 1 Hz. The solid line is the linear dispersion relation and the dashed line is the second harmonic. (b) Cross section along ${\mathbf{k}}_2,{\omega }_2$ $\left(k_y/2\pi =0m^{-}1\right)$ of the 3D bicoherence $B_3^{k,\omega }({\mathbf{k}}_2,{\mathbf{k}}_3,{\omega }_2,{\omega }_3)$ for a given wave $({\mathbf{k}}_2,{\omega }_2)$ computed from the same data set. The red and black curves are similar to the ones shown in Fig. 8. Exact solutions are in the region where the black curve intersects the red one. A high correlation intensity is observed at this region, suggesting the occurrence of quasiresonant interaction between linear waves and harmonic.

Figure 12

(a) Normalized fourth-order correlations $C_4^{\omega }({\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4)$ for two given waves ${\omega }_1/2\pi =2.2$ Hz and ${\omega }_3/2\pi =3.2$ Hz in the experiment. A very weak value of correlations is present on the resonant line ${\omega }_1+{\omega }_2={\omega }_3+{\omega }_4$ (dashed line) at the highest frequencies $({\omega }_2/2\pi >5$ Hz). At low frequencies where linear waves exist, there is no evidence of four-wave interactions at the level of convergence, which is about ${10}^{-2.5}$. (b) Tricoherence $B_4^{\omega }({\omega }_1,{\omega }_3,{\omega }_4)$. Like in (a), only a weak correlation is observed at the highest frequencies. The light blue on the bottom left corner of the figure $({\omega }_3/2\pi <2.2$ Hz) is not statically converged. The two red lines correspond to the trivial solution ${\omega }_3={\omega }_1$ and ${\omega }_4={\omega }_2$ (or ${\omega }_4={\omega }_1$ and ${\omega }_3={\omega }_2)$.

Figure 13

(a) Normalized fourth-order correlation $C_4^{\omega }({\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4)$ computed for the in situ data set. Two arbitrary frequencies are given as ${\omega }_1/2\pi =2.2$ Hz and ${\omega }_3/2\pi =3.2$ Hz. A significant level of statistically converged correlation is present along the resonant line ${\omega }_1+{\omega }_2={\omega }_3+{\omega }_4$ (dashed line). (b) Tricoherence $T_4^{\omega }({\omega }_1,{\omega }_3,{\omega }_4)$ for ${\omega }_1/2\pi =1$ Hz. The two dark red lines correspond to the trivial solution ${\omega }_3={\omega }_1$ and ${\omega }_4={\omega }_2$ (or ${\omega }_4={\omega }_1$ and ${\omega }_3={\omega }_2)$. A significant level of correlation can be seen out of these trivial lines.

Figure 14

Time evolution of the power spectrum $E^{\eta }(\omega ,t)$ for four frequencies 1, 2, 3, and 4 Hz. An exponential decrease is observed (dashed lines) and leads to a typical time scale $\tau$ following $e^{-t/\tau }$. We assume that the dissipative time scale $T^d$ is $T^d\approx \tau$.