Confinement effects on gravity-capillary wave turbulence

The statistical properties of a large number of weakly nonlinear waves can be described in the framework of the weak turbulence theory. The theory is based on the hypothesis of an asymptotically large system. In experiments, the systems have a finite size and the predictions of the theory may not apply because of the presence of discrete modes rather than a continuum of free waves. Our study focuses on the case of waves at the surface of water at scales close to the gravity-capillarity crossover (of order 1 cm). Wave turbulence has peculiar properties in this regime because one-dimensional resonant interactions can occur, as shown by Aubourg and Mordant [Phys. Rev. Lett. 114, 144501 (2015)]. Here we investigate the influence of the confinement on the properties of wave turbulence by reducing gradually the size of our wave tank along one of its axes, the size in the other direction being unchanged. We use space-time resolved profilometry to reconstruct the deformed surface of water. We observe an original regime of coexistence of weak wave turbulence along the length of the vessel and discrete turbulence in the confined direction.

Figure 1

Sketch of the experimental setup. Waves are generated by oscillating horizontally the container. The width of the wave tank can be tuned by a mobile wall so that to modify the horizontal confinement. The wave height is measured by using a Fourier transform profilometry technique. See text for details.

Figure 2

(a) Frequency spectra of the vertical velocity field $E^v\left(\omega \right)$ for all confinements. The spectra have been shifted vertically for clarity in the main figure. The nonshifted spectra are shown in the insert. (b) Frequency spectra for the $l_4=7$ cm case. The three curves correspond to the full spectrum (black), the spectrum of the purely longitudinal waves (blue), and the purely transverse waves (red). The vertical dashed lines correspond to the frequency of the transverse modes (Table 2).

Figure 3

Space-time Fourier spectrum of the vertical velocity field of the longitudinal waves $E^v(k_x,k_y=0,\omega )$ [panels (a) and (b)] and transverse waves $E^v(k_x=0,k_y,\omega )$ [panels (c) and (d)]. In panels (a) and (c), the width of the tank is equal to $l_1=32.5$ cm. In panels (b) and (d), the width of the tank is equal to $l_4=7$ cm. In all cases, the solid black line is the theoretical deep water linear dispersion relation for gravity-capillary waves in pure water (1). The energy magnitude is displayed in a logarithmic scale.

Figure 4

Frequency spectra of the longitudinal (a) and transverse (b) waves for all confinements. The width of the vessel is indicated in the legend. The spectra have been shifted vertically for a better visualization, with confinement increasing from the top to the bottom. The inserts show the unshifted curves. The dashed line is a $f^{-5}$ as an eye guide.

Figure 5

(a) Cut of the spectrum for the full wave field for the maximum confinement at a frequency which lies in between two successive transverse eigenmodes of the vessel $E^v(k_x,k_y,\frac{\omega }{2\pi }=15.9\mathrm{Hz})$. (b) Cut of the spectrum for the full wave field for the maximum confinement at a frequency which corresponds to a transverse eigenmode of the vessel $E^v(k_x,k_y,\frac{\omega }{2\pi }=20.9\mathrm{Hz})$. In both panels (a) and (b), the solid black circle is the theoretical deep water linear dispersion relation for gravity-capillary waves in pure water. The energy magnitude is in a logarithmic scale.

Figure 6

Frequency spectra of the transverse waves in the case $l_3$ with nonlinearity (average slope) $\epsilon$ increasing from the bottom to the top.

Figure 7

Domain of existence of resonant waves (2) in the frequency domain for gravity-capillary waves. The solid black line is the border of this domain and corresponds to interactions between three copropagating waves. The gray pattern area above the solid black line represents solutions where the directions of the three waves are no longer the same. In the white area below the solid black line, no resonant solutions exist.

Figure 8

Bicoherence (6) of the velocity field for (a) the experiment with the weakest confinement $l_1=32.5$ cm and (b) for the experiment with the strongest confinement $l_4=7$ cm. The black line corresponds to the 1D three-wave resonant solutions.

Figure 9

Bicoherence of the longitudinal waves (a) and transverse waves (b) for the experiment with the most important confinement corresponding to a width of the vessel equal to 7 cm. The black line corresponds to the 1D solutions. The frequencies associated with the discrete transverse modes appear as black rectangle in panel (b).

Figure 10

Bicoherence of the wave field for experiments carried out in a high-aspect-ratio channel of size 1.5 m $\times$ 10 cm with a rms slope of $3\%$. The black line corresponds to the 1D solutions. (a) Full wave field, (b) longitudinal waves, and (c) transverse waves. The frequencies associated with the discrete transverse modes appear as black rectangle in panel (c).