Impact of dissipation on the energy spectrum of experimental turbulence of gravity surface waves

We discuss the impact of dissipation on the development of the energy spectrum in wave turbulence of gravity surface waves with emphasis on the effect of surface contamination. We performed experiments in the Coriolis facility, which is a 13-m-diam wave tank. We took care of cleaning surface contamination as well as possible, considering that the surface of water exceeds $100{\mathrm{m}}^2$. We observe that for the cleanest condition the frequency energy spectrum shows a power-law decay extending up to the gravity capillary crossover (14 Hz) with a spectral exponent that is increasing with the forcing strength and decaying with surface contamination. Although slightly higher than reported previously in the literature, the exponent for the cleanest water remains significantly below the prediction from the weak turbulence theory. By discussing length and time scales, we show that weak turbulence cannot be expected at frequencies above 3 Hz. We observe with a stereoscopic reconstruction technique that the increase with the forcing strength of energy spectrum beyond 3 Hz is mostly due to the formation and strengthening of bound waves.

Figure 1

Additional dissipation factor $y\left(f\right)$ as a function of frequency using Przadka et al. parameters in Eq. (2) for a commercial paint pigment [16]. A maximum is observed at about 4 Hz with a peak value close to 15.

Figure 2

Measured spectral exponent $\alpha$ of the temporal spectrum as a function of the typical steepness of the waves $\epsilon$ (changed by tuning the magnitude of the forcing and dependent as well on the surface contamination). Dark triangles are previous measurements reported by Deike et al. [9]. The blue and red squares are measurements by Aubourg [28, 29] with and without filtration, respectively. The single purple point is an in situ measurement of gravity waves in the Black Sea [21]. Our present data for the cleanest case are the two cyan circles.

Figure 3

(a) Global schematics of the setup in the Coriolis facility (seen from above). The tank is 13 m in diameter and the water is 0.9 m deep. The positions of the two wave makers are shown (black ovals) and those of the ten capacitive probes are shown as red dots. The field of view of the stereoscopic reconstruction (see Sec. 6) is the green rectangle at the center. (b) Schematics of a wave maker. It is a wedge wave maker (horizontal size $2\times 1{\mathrm{m}}^2$ at the top) set into vertical oscillation by an eccentric cam. The off-center distance is 20 mm in the two experiments reported here. The rotation frequency is changed randomly in a given interval $0.585\pm 0.15$ Hz and $0.78\pm 0.15$ Hz for the two experiments.

Figure 4

Spectra after several days of almost continuous filtration for the strong case (top) and the weak case (bottom). The spectra have been averaged over several capacitive probes. The dashed lines correspond to decays of $1/f^{5.2}$ (top) and $1/f^6$ (bottom). The gravity-capillarity crossover occurs at 14 Hz, which corresponds to the observed change of slope of the spectrum. The signal reaches the noise level at about 100 Hz.

Figure 5

Evolution of the wave spectrum versus time for weakest forcing. Each curve corresponds to an average over the ten probes and a time average over 1 h of continuous recording (seven successive records from red to blue). (a) Wave elevation spectrum $E^{\eta }\left(f\right)$. (b) Compensated spectrum $f^6{E}^{\eta }\left(f\right)$ in the low-frequency range. The dashed line is proportional to $f^{-0.5}$.

Figure 6

Temporal evolution of the wave spectrum for weak forcing relative to the initial spectrum (same colors as in Fig. 5). This representation highlights the deficit of energy in the spectrum as the surface becomes contaminated. The solid black curve represents the extra dissipation factor $1/y\left(f\right)$ for oleic acid (taken from Alpers and Hühnerfuss [4]). For this data set the forcing occurs at a central frequency of 0.585 Hz.

Figure 7

Variability of spectra measured in similar conditions over time for (a) weak and (b) strong forcing. In (a) we show 28 spectra (averaged over the 10 probes), each corresponding to a 1-h-long record. In (b) we show 19 similar spectra. The spectra have been compensated by $f^6$ and $f^5$, respectively. The dashed line shows an $f^{-1/2}$ decay. The variability is lower at stronger forcing.

Figure 8

(a) Ratio $\mathrm{\Delta }/k$ which provides the dissipation length scale in terms of wavelengths. Blue represents the case of a perfectly clean surface, red the case of a contaminated surface with oleic acid, and black $y\left(f\right)$ for oleic acid. (b) Time scale ratio of the dissipation scale $T_d$ over the period of the wave $T$. The color code is the same as in (a).

Figure 9

Variation of spectra when floating particles are present or not. Blue (red) represents the weak case without (with) particles and black (green) represents strong forcing without (with) particles. The inset shows that the spectra with particles have been normalized by the spectra without particles.

Figure 10

(a) The $k\text{−}\omega$ spectrum $E^{\eta }(k,\omega )$ for $\epsilon =0.11$ (weak case). The spectrum has been normalized by its maximum value and is displayed in ${log}_{10}$ scale. The red line is the linear dispersion relation (6). Dotted lines correspond to bound waves made from successive harmonics of the forcing peak and waves on the dispersion relation. (b) Frequency spectrum constructed by adding successively the spectral contributions $E_n^{\eta }$ of the bound waves following ${\omega }^{(\pm n)}={\omega }_{\mathrm{LDR}}(k\mp n{k}_0)\pm n{\omega }_0$ with $n>0$ (see the text). Here $E^{\eta }\left(\omega \right)$ is the full spectrum, $E_{\mathrm{LDR}}^{\eta }\left(\omega \right)$ is the contribution of the linear dispersion relation, and $E_n^{\eta }\left(\omega \right)$ is the contribution of the $n$ th bound waves as described in the text. The red dashed line is the spectrum obtained from the local probes. The divergence of the black and dashed red curves at about 7 Hz on the black line is due to the noise level of the stereoscopic measurement. (c) Similar construction for the wave-number spectrum $E^{\eta }\left(k\right)$.