Different Regimes for Water Wave Turbulence

We present an experimental study on gravity capillary wave turbulence in water. By using space-time resolved Fourier transform profilometry, the behavior of the wave energy density $|{\eta }_{\mathbf{k},\omega }{|}^2$ in the 3D $(\mathbf{k},\omega )$ space is inspected for various forcing frequency bandwidths and forcing amplitudes. Depending on the bandwidth, the gravity spectral slope is found to be either forcing dependent, as classically observed in laboratory experiments, or forcing independent. In the latter case, the wave spectrum is consistent with the Zakharov-Filonenko cascade predicted within wave turbulence theory.

Figure 1

Typical instantaneous velocity fields $\dot{\eta }(\mathbf{r},t)$ (color scale is in $\mathrm{m}/\mathrm{s}$). Top: Experiments I ($A=22\mathrm{mm}$), the variance $\sigma \equiv \sqrt{⟨\dot{\eta }(\mathbf{r}{)}^2⟩}=0.16\mathrm{m}/\mathrm{s}$, skewness $S\equiv ⟨\dot{\eta }(\mathbf{r}{)}^3⟩/{\sigma }^3=0.15$ and kurtosis $K\equiv ⟨\dot{\eta }(\mathbf{r}{)}^4⟩/{\sigma }^4=3.54$. Bottom: Experiments II ($A=28\mathrm{mm}$), $\sigma =0.012\mathrm{m}/\mathrm{s}$, $S=0.023$, $K=2.95$. The insets show the probability density functions of the normalized wave velocity $\dot{\eta }(\mathbf{r})/\sigma$: circles for the presented field, solid line when averaged over six fields at different times, and dotted line the Gaussian fit with mean zero and unit variance.

Figure 2

Experiments I: Typical spectrum $|{\eta }_{k,\omega }{|}^2$ at low forcing ($A=3\mathrm{mm}$). Inset (i) shows a zoom of $|{\eta }_{k,\omega }{|}^2$ in the gravity wave regime at high forcing ($A=22\mathrm{mm}$). The solid line shows the linear dispersion relation $\omega (k)$ and the dotted lines the dispersion relation of the slave modes $N=2,3,4$. Inset (ii) illustrates the isotropy in the $(k_x,k_y)$ space by plotting $|{\eta }_{\mathbf{k},\omega }{|}^2$ for $\omega =138{\mathrm{s}}^{-1}$ ($A=3\mathrm{mm}$).

Figure 3

Spectra $|{\eta }_{\omega }{|}^2$ in experiments I for $A=1.5$, 3, 7.5, 15, 22, and 30 mm. Vertical dotted lines at $30{\mathrm{s}}^{-1}$ and at ${\omega }_c=85{\mathrm{s}}^{-1}$ delimitate the inertial range of the gravity regime. Top inset: Corresponding spectra $|{\dot{\eta }}_k{|}^2$. Bottom inset: Spectral exponent as a function of $A$; open circles in the gravity regime and plain circles in the capillary regime (multiple symbols correspond to multiple fits in a moving window within the inertial range).

Figure 4

Experiments II: Typical spectrum $|{\eta }_{k,\omega }{|}^2$ at forcing amplitude $A=20\mathrm{mm}$. Inset (i) shows the collected dispersion relations for $A=1$, 6, 12, 20, and 28 mm. The dashed line shows the linear dispersion relation and the dotted line the first nonlinear mode. Inset (ii) illustrates the isotropy in the $(k_x,k_y)$ space by plotting $|{\eta }_{\mathbf{k},\omega }{|}^2$ for $\omega =27.64{\mathrm{s}}^{-1}$.

Figure 5

Experiments II: Spectra $|{\eta }_{\omega }{|}^2$. Same representation as in Fig. 3 (only the gravity regime is accessible in these experiments).

Figure 6

Scaling laws of $|{\eta }_k{|}^2$ in the inertial range as a function of the forcing amplitudes ${\omega }_m^2{A}^2\propto P$. Black symbols come from $|{\dot{\eta }}_k{|}^2$ and gray (red) symbols are deduced from $|{\dot{\eta }}_{\omega }{|}^{2}d\omega /dk$ with $d\omega /dk$ determined from the linear dispersion relation. In experiments I, results are shown for both the gravity (circles) and capillary (diamond) regimes, and in experiments II, for the gravity regime. Solid and dotted lines have slope 1 and $1/3$. The inset shows $|{\eta }_k{|}_{\max }^2/{\omega }_m^2$ as a function of $A$ (solid line has a slope 2), suggesting that the injected power is proportional to $A^2$ in our experiments.