Observation of Gravity-Capillary Wave Turbulence

We report the observation of the crossover between gravity and capillary wave turbulence on the surface of mercury. The probability density functions of the turbulent wave height are found to be asymmetric and thus non-Gaussian. The surface wave height displays power-law spectra in both regimes. In the capillary region, the exponent is in fair agreement with weak turbulence theory. In the gravity region, it depends on the forcing parameters. This can be related to the finite size of the container. In addition, the scaling of those spectra with the mean energy flux is found in disagreement with weak turbulence theory for both regimes.

Figure 1

Schematic view of the experimental setup showing a typical time recording of the surface wave height, $\eta (t)$, at a given location during 50 s. $⟨\eta ⟩\simeq 0$.

Figure 2

Probability density functions of the waveheight, $\eta$, for the maximum excitation amplitude ($U_{\mathrm{rms}}=0.9\mathrm{V}$) and for 6 different values of the fluid depth, from $h=18$, 35, 55, 80, 110, to 140 mm (see the arrow). The frequency band is $0\le f\le 6\mathrm{Hz}$. Inset: Same PDFs displayed using the reduced variable $\eta /\sqrt{⟨{\eta }^2⟩}$. Gaussian fit with zero mean and unit standard deviation (dashed line).

Figure 3

Power spectra of the surface waveheight for two different driving voltages $U_{\mathrm{rms}}=0.2$ and 0.9 V (from bottom to top). The frequency band is $0\le f\le 6\mathrm{Hz}$. Dashed lines have slopes $-4.3$ and $-3.2$. Inset: The frequency band is $0\le f\le 4\mathrm{Hz}$, and $U_{\mathrm{rms}}=0.9\mathrm{V}$. Dashed lines had slopes of $-6.1$ and $-2.8$.

Figure 4

Slopes of surface-height spectra for gravity (solid symbols) and capillary (open symbols) waves for different forcing bandwidths and intensities: (○) 0 to 4 Hz, (▽) 0 to 5 Hz, and (□) 0 to 6 Hz. Power-law exponents of gravity wave spectrum (dash-dotted line) and capillary waves spectrum (dashed line) as predicted by WT theory [Eq. (1)]. Inset: Crossover frequency between gravity and capillary regimes as a function of the forcing intensity and bandwidth.

Figure 5

Spectra of the surface wave amplitude divided by the variance ${\sigma }_V^2$ of the velocity of the wave maker for different forcing amplitudes, ${\sigma }_V=2.1$, 2.6, 3.5, and $4.1\mathrm{cm}/\mathrm{s}$. The frequency band is $0\le f\le 4\mathrm{Hz}$. The dashed line has slope $-5.5$ whereas the solid line has slope $-17/6$. The mean injected power is displayed as a function of ${\sigma }_V^2$ in the inset. The best fit gives a slope $11.5\mathrm{W}/(\mathrm{m}/{\mathrm{s}}^2)$.