Decay of capillary wave turbulence

We report on the observation of freely decaying capillary wave turbulence on the surface of a fluid. The capillary wave turbulence spectrum decay is found to be self-similar in time with the same power law exponent as the one found in the stationary regime, in agreement with weak turbulence predictions. The amplitude of all Fourier modes are found to decrease exponentially with time at the same damping rate. The longest wavelengths involved in the system are shown to be damped by a viscous surface boundary layer. These long waves play the role of an energy source during the decay that sustains nonlinear interactions to keep capillary waves in a wave turbulent state.

Figure 1

Decay of the wave amplitude, $\eta \left(t\right)$, as a function of time (solid line). Random forcing is working for $t<0$ and is stopped at $t=0$. Mercury. Dashed line corresponds to an exponential fit $\eta \left(t\right)=|{\eta }_0|e^{-t/\tau }$ with $\tau =15.5$ s and $|{\eta }_0|=3$ mm. (Inset) $\square$, Temporal decay of the averaged standard deviation of wave amplitude, $\langle {\sigma }_{\eta }\rangle$; $--$, exponential fit $\langle {\sigma }_{\eta }\left(t\right)\rangle =\langle {\sigma }_{\eta }\left(0\right)\rangle e^{-t/\tau }$, with $\tau =15.5$ s and $\langle {\sigma }_{\eta }\left(0\right)\rangle =1$ mm.

Figure 2

Averaged spectrogram $\langle S_{\eta }(f,t)\rangle$ of the wave amplitude during the decay as a function of frequency and time. Color bar is a logarithmic scale of the spectrum amplitude. Pixelization in time corresponds to the short time interval $\delta t=3.5$ s. $f_R^a$ is an eigenvalue of the vessel (see text) remnant of the forcing frequencies (0.1–6 Hz).

Figure 3

Averaged power spectrum $\langle S_{\eta }(f,t^{*})\rangle$ at different times $t^{*}$ of the decay. From top to bottom, $t^{*}=2$, 5.5, 9, 16, 26.5, 37, 47.5, 58, and 68.5 s. Dashed lines are power laws fits $\sim$$f^{-5.3}$ for gravity ($6<f<25$ Hz, top curve only) and $\sim$$f^{-2.9}$ for capillary waves ($25<f<100$ Hz). Black arrows indicate the cutoff frequencies $f_c$ of the capillary wave cascade. Red (light gray) dashed arrows indicate vessel eigenvalues: $f_R^a$ and $f_R^s$ remnant of the forcing frequencies (0.1–6 Hz), and their harmonics $2f_R^a$ and $4f_R^a$ (see text). (Inset) The rescaled wave height power spectra $\langle S_{\eta }(f,t^{*})\rangle /{\langle {\sigma }_{\eta }\left(t^{*}\right)\rangle }^2$.

Figure 4

Exponent $\alpha$ of the frequency power law spectrum of capillary wave turbulence as a function of time. One has $\alpha =2.9\pm 0.2$ during most of the decay ($t<3\tau =45$ s). The dashed line is the theoretical prediction for stationary capillary wave turbulence [1].

Figure 5

Cutoff frequency as a function of time. First set of experiments: $\delta t=1.95$ s, $\square$; 3.5 s, $\diamond$. Second set: $\delta t=1.95$ s, $\circ$. The solid line is an exponential fit $f_c\sim e^{-at/\tau }$, with $a=0.38$ a fitting parameter and $\tau =15.5$ s the damping time for mercury. (Inset) Same for water. First set, $\square$; second set, $\diamond$; and third set, $\circ$; $\delta t=1.5$ s. The solid line is an exponential fit $f_c\sim e^{-at/\tau }$, with $a=0.38$ a fitting parameter and $\tau =5$ s the damping time for water.

Figure 6

Decaying of the total energy ${\langle {\sigma }_{\eta }\left(t\right)\rangle }^2$ ($\square$) and of the Fourier modes $\langle S_{\eta }(f^{*},t)\rangle$ as a function of time. Each solid line corresponds to a mode at frequency $f^{*}$: both top $f_R^s=3.2\text{Hz}$ and $f_R^a=3.8$ Hz; from the third from the top to the bottom $f^{*}=13$, $23,...,93$ Hz, with 10-Hz steps. Dashed lines are exponential fits $\sim$$e^{-2t/\tau }$ for $\tau /3<t<3\tau$ (slow damping) and $\sim$$e^{-5t/\tau }$ for $t<\tau /3$ (fast damping). Vertical dashed lines correspond to $\tau /3$ and $3\tau$.

Figure 7

Decay rate, $1/\left(\tau 2\pi f_R^a\right)$, of the wave mode at $f_R^a=3.2$ Hz as a function of viscosity $\nu$ for various fluids ($\square$). Theoretical decay rate by viscous dissipation: $\circ$, surface boundary layer, ${\delta }_S$, from Eq. (2); $+$, bottom surface layer, ${\delta }_B$, from Eq. (3); $\diamond$, wall surface layer, ${\delta }_W$, from Eq. (4); and $\times$, viscous surface ${\delta }_{\nu }$ from Eq. (1). The dashed line is the total theoretical dissipation ${\delta }_T={\delta }_S+{\delta }_B+{\delta }_W$.

Figure 8

Stationary regime. Power spectrum $S_{\eta }\left(f\right)$ of wave height for different mean injected powers $\overline{I}$. From bottom to top, $\overline{I}=$ 5, 7, 10, 13, 16, 19, 26 mW. Dashed lines indicate power law fits of $\sim$$f^{-2.8}$. (Inset) The best rescaled spectrum $S_{\eta }/{\overline{I}}^{1\pm 0.2}$ (mercury).

Figure 9

Stationary regime. Cutoff frequency $f_c$ of the capillary wave spectrum as a function of the mean injected power $\overline{I}$. The solid line is a power law fit $f_c\sim {\overline{I}}^{0.2\pm 0.05}$ (mercury).