Space-time-resolved capillary wave turbulence

We report experiments on the full space- and time-resolved statistics of capillary wave turbulence at the air-water interface. The three-dimensional shape of the free interface is measured as a function of time by using the optical method of diffusing light photography associated with a fast camera. Linear and nonlinear dispersion relations are extracted from the spatiotemporal power spectrum of wave amplitude. When wave turbulence regime is reached, we observe power-law spectra both in frequency and in wave number whose exponents are found to agree with the predictions of capillary wave turbulence theory. Finally, the temporal dynamics of the spatial energy spectrum highlight the occurrence of stochastic bursts transferring wave energy through the spatial scales.

Figure 1

(a) Experimental setup (see text). (b) Snapshot of the wave field $h(x,y)$. Random forcing between 4 and $6$Hz. ${\sigma }_h=3.4$ mm and ${\sigma }_s=0.30$ (see text). Color scale is in millimeters. Wave maker is parallel to the $y$ axis and located at $x=-12$ mm. (c) Snapshot of the spatial gradient of wave elevation $\left|\right|\nabla h(x,y)\left|\right|$. Same parameters.

Figure 2

Spatiotemporal spectrum of wave elevation $S_h(\omega ,k)$ in sinusoidal forcing with an excitation frequency of 40 Hz. Amplitude ${\sigma }_h=0.22$mm and ${\sigma }_s=0.16$. ($+$) Experimental dispersion relation extracted from the maxima of $S_h(\omega ,k)$. Red (dark gray) dashed line: linear theoretical dispersion relation. Color scale corresponds to ${\log }_{10}\left[S_h(\omega ,k)\right]$. Camera frame rate: 200 Hz.

Figure 3

(a) Spatiotemporal spectrum of wave elevation $S_h(\omega ,k)$ for two different random forcing amplitudes: (a) ${\sigma }_h=2.7$mm and ${\sigma }_s=0.26$. (b) ${\sigma }_h=3.6$mm and ${\sigma }_s=0.34$. ($+$) Experimental dispersion relation extracted from the maxima of $S_h(\omega ,k)$. Red (dark gray) dashed line: linear theoretical dispersion relation. White dot-dashed line: theoretical dispersion relation with a nonlinear shift (see text). Color scales correspond to ${\log }_{10}\left[S_h(\omega ,k)\right]$. Camera frame rates: (a) 1 kHz and (b) 200 Hz.

Figure 4

Spectrum $S_h(\omega ,{{\lambda }_x}^{-1},{{\lambda }_y}^{-1})$ for fixed $f=10$, 20, 40, and 80 Hz (${\text{log}}_{10}$ color scale) showing anisotropy of wave field at high forcing amplitude (${\sigma }_h=3.6$mm and ${\sigma }_s=0.34$). Dashed black circle: Linear dispersion relation of Eq. (2). Wave amplitudes are larger in the direction of forcing ($x$ axis), revealing anisotropy in all frequency ranges.

Figure 5

(a) Temporal power spectra $S_h\left(\omega \right)$ for different forcing amplitudes. From bottom to top: ${\sigma }_h=$1.3, 2.7, 2.1, 3.4, and 3.6 mm, and ${\sigma }_s=0.15$, 0.26, 0.19, 0.3, and 0.34. Solid black line is the capillary prediction $f^{-17/6}$. The second measurement is recorded with a 1-kHz frame rate; the other ones are recorded at 200 Hz. (b) Inset: $S_h\left(\omega \right)$ exponents vs ${\sigma }_h$ (fits from $20\le f\le 100$Hz). Dashed line shows the theoretical value $-17/6$. (c) Spatial power spectra $S_h\left(k\right)$ for the same measurements. Solid black line is the capillary prediction $k^{-15/4}$. (d) Inset: $S_h\left(k\right)$ exponents vs ${\sigma }_h$ (fits from $0.094\le {\lambda }^{-1}\le 0.30$ mm${}^{-1}$). Dashed line shows the theoretical value $-15/4$. ${\lambda }_d$ is a characteristic scale of dissipation. ${\lambda }_c$ is the gravity-capillary crossover and ${\lambda }_{☆}^{-1}=0.094$ mm${}^{-1}$ (see Fig 6). (e) Spatial spectrum of the energy $E\left(k\right)$ ($•$). Contributions of gravitational $E_{\mathrm{grav}}\sim S_h\left(k\right)$ and capillary $E_{\mathrm{cap}}\sim k^2{S}_h\left(k\right)$ energies are well separated. The solid black line is the prediction for the capillary energy spectrum $\sim$$k^{-7/4}$. ${\sigma }_h=3.6$ mm and ${\sigma }_s=0.34$.

Figure 6

(a) Evolution of the wave energy $E(k,t)$ in the ($t$,${\lambda }^{-1}$) space. Color scale in ${\log }_{10}$ scale. (b) Probability density function of wave energy $E(k_{☆},t)/\langle E(k_{☆},t)\rangle$ at a mode ${\lambda }_{☆}^{-1}=0.094$ mm${}^{-1}$: ($*$) moderate forcing (${\sigma }_h=2.0$mm, ${\sigma }_s=0.19$), and ($\circ$) high forcing (${\sigma }_h=3.6$mm, ${\sigma }_s=0.34$). Dashed line is the exponential distribution: $\exp (-x)$.