Observation of depth-induced properties in wave turbulence on the surface of a fluid

We report the observation of changes in the wave turbulence properties of gravitycapillary surface waves due to a finite depth effect. When the fluid depth is decreased, a hump is observed on the wave spectrum in the capillary regime at a scale that depends on the depth. The possible origin of this hump is discussed. In the gravity regime, the wave spectrum still shows a power law but with an exponent that strongly depends on the depth. A change in the scaling of the gravity spectrum with the mean injected power is also reported. Finally, the probability density function of the wave amplitude rescaled by its rms value is found to be independent of the fluid depth and to be well described by a Tayfun distribution. Introduction. – Wave turbulence is ubiquitous in nature. It ranges from surface or internal waves in oceanography, Alfvén waves in solar winds, Rossby waves in geophysics, elastic or spin waves in solids, waves in optics, and quantum waves in Bose condensates (for reviews see [1,2]). Wave turbulence theory (a statistical theory describing an ensemble of weakly nonlinear interacting waves) predicts analytical solutions of the kinetic equations of weak turbulence at the equilibrium or in a stationary out-of equilibrium regime in various systems involving wave dynamics [3]. Surprisingly, well-controlled laboratory experiments on wave turbulence were scarce [4] up to last years where new observations was reported such as intermittency [5], fluctuations of the energy flux [6], finite size effect of the system [7, 8] and the full space-time power spectrum of wave amplitudes [9]. Several questions still are open, notably about the validity domain of the theory in experiments (horizontal finite size effects, role of strongly nonlinear coherent structures), and the possible existence of weak turbulence solutions for nondispersive (ie, for 2D acoustic waves) or weakly dispersive systems [10, 11]. Indeed, the lack of dispersion could lead to cumulative nonlinear effects leading to shock wave formation. It is thus of primary interest to know the evolution of the wave energy spectrum when the dispersion relation of a system is (a)Corresponding author: [email protected] changed from a dispersive to a nondispersive regime. A simple way to experimentally do this is to change the depth of a fluid on which surface waves propagate. Indeed, gravity waves are known to become nondispersive in a shallow water limit. In this limit, weak turbulence predicts power spectra of wave amplitude much less steep than in the deep regime for both gravity [12, 13] and capillary [14] wave turbulence. The prediction for the gravity regime has been tested in few laboratory experiments using a large-scale wave flume with a sloping bottom [15,16]. At a more applied level, in situ observations exist in oceanography when ocean waves propagate from deep water into shallow coastal areas [16]. Notably, when ocean surface waves approach the shore, bottom friction and depth-induced wave breaking are no more negligible, and near-resonant triad interactions could play a dominant role instead of the 4-wave interaction process [17,18]. In this letter, we study gravity-capillary wave turbulence on the surface of a fluid within a constant depth tank. When the depth is decreased from a deep to a thin fluid layer, a hump is observed on the power spectrum in the capillary regime at a scale that depends on the depth. In the gravity regime, the wave spectrum still shows a power-law but with an exponent that depends on the depth. The scaling of the power spectrum with the mean injected power is measured for different fluid depths as well as the probability distribution of the wave ampli-