Observation of the Nonlinear Dispersion Relation and Spatial Statistics of Wave Turbulence on the Surface of a Fluid

We report experiments on gravity-capillary wave turbulence on the surface of a fluid. The wave amplitudes are measured simultaneously in time and space by using an optical method. The full space-time power spectrum shows that the wave energy is localized on several branches in the wave-vector-frequency space. The number of branches depends on the power injected within the waves. The measurement of the nonlinear dispersion relation is found to be well described by a law suggesting that the energy transfer mechanisms involved in wave turbulence are restricted not only to purely resonant interaction between nonlinear waves. The power-law scaling of the spatial spectrum and the probability distribution of the wave amplitudes at a given wave number are also measured and compared to the theoretical predictions.

Figure 1

Space-time spectrum $E(k,f)$ of the vertical velocity of surface waves: moderate (a) and strong (b) injected powers [$P^{1/2}=5.2$ (a) and 24.7 (b) in arbitrary units]. Forcing: 1–4 Hz. Colors are log scaled. Solid white lines are ${\Omega }_N(k)$ with $N=1$, 2, and 3 (see text). The slope of the dotted line corresponds to a constant phase velocity $\omega (k)/k$.

Figure 2

Nonlinear dispersion relation $k(f)$ computed from the lines of maximum energy of $E(k,f)$ for different forcings [$P^{1/2}=1$ ($+$), 5.2, (○), 10.5 ($\times$), and 24.7 ($*$)]. The solid thick lines around each branch correspond to the branch width averaged for all forcings. The solid lines are ${\Omega }_N(k)$ with $N=1$, 2, and 3 (see the text—the same as in Fig. 1). Inset: Snapshot of the wave amplitudes at strong forcing ($P^{1/2}=24.7$).

Figure 3

Space-time spectrum $E(k_x,f)$ of velocity at $P^{1/2}=10.5$ located at $k_y=0$. The same forcing bandwidth as in Fig. 1. Inset: Space spectrum $E(k_x,k_y)$ located at $f=5$ (a) and 9 Hz (b). Dashed lines: Forcing directions. Log scaled colors are different for each plot.

Figure 4

Spatial power spectrum $E(k)$ of the velocity for $P^{1/2}=1$, 5.2, 10.5, and 24.7 (from bottom to top). Dashed lines have slopes $-4.6$, $-4.5$, and $-4.2$ (from bottom to top). Dot-dashed line: ${\lambda }_{\star }^{-1}=25.2{\mathrm{m}}^{-1}$ (see Fig. 5).

Figure 5

PDF of the Fourier wave amplitude, $|{\widehat{\eta }}_{k_{\star }}{|}^2/{\sigma }_{|{\widehat{\eta }}_{k_{\star }}{|}^2}$, at $k_{\star }\equiv 2\pi /{\lambda }_{\star }$ with ${\lambda }_{\star }^{-1}=25.2\pm 0.6{\mathrm{m}}^{-1}$ (see Fig. 4) for two forcings $P^{1/2}=5.2$ (○) and 10.5 ($*$). $⟨|{\widehat{\eta }}_{k_{\star }}{|}^2⟩=1.1$ (○) and 1.3 ($*$). Solid lines have slopes $-0.46$ and $-0.58$. Curves have been shifted vertically for clarity. Inset: $|{\widehat{\eta }}_{k_{\star }}{|}^2$ vs time. ${\lambda }_{\star }^{-1}=25.2\pm 0.6{\mathrm{m}}^{-1}$. $P^{1/2}=24.7$.