epl draft Experimental study of the inverse cascade in gravity wave tur- bulence

We perform experiments to study the inverse cascade regime of gravity wave turbulence on the surface of a fluid. Surface waves are forced at an intermediate scale corresponding to the gravity-capillary wavelength. In response to this forcing, waves at larger scales are observed. The spectrum of their amplitudes exhibits a frequency-power law at high enough forcing. Both observations are ascribed to the upscale wave action transfers of gravity wave turbulence. The spectrum exponent is close to the value predicted by the weak turbulence theory. The spectrum amplitude is found to scale linearly with the mean injected power. We measure also the distributions of the injected power fluctuations in the presence of upscale (inverse) transfers or in the presence of a downscale (direct) cascade in gravity wave turbulence. Introduction. – Waves on ocean surface is the most common example of wave turbulence. Wave turbulence concerns the dynamical and statistical study of a field of dispersive waves in nonlinear interaction. It is a rather universal phenomenon since it occurs in various physical contexts including geophysics, plasma physics, non linear optics or solid state physics [1]. One of the most important result of wave turbulence theory is the existence of out-of-equilibrium stationary solutions for the wave spectrum that follow Kolmogorov-like cascades of flux of conserved quantities [1]. This cascade type behavior is similar to two-dimensional turbulence ones where both a direct cascade of enstrophy (rms vorticity) and an inverse cascade of energy occur [2, 3]. In wave turbulence, the cascades are governed by the nonlinear interaction process between waves: for a 4-wave process (as for gravity surface waves) the energy and the wave action are conserved [1,4], whereas for a 3-wave process (as for capillary waves) only the energy is conserved [1]. Frequency-power law solutions for gravity wave spectrum then exist corresponding to either a constant flux of energy from large to small scale (direct cascade) or a constant flux of wave action from small to large scale (inverse cascade) [5]. The direct gravity cascade has been observed in open seas [6], in wellcontrolled laboratory experiment [7, 8], and in numerical (a)Corresponding author: [email protected] simulations [9]. Existence of an inverse cascade in gravity wave turbulence has been confirmed recently using numerical simulations [10, 11]. These simulations notably show the formation of both inverse and direct gravity cascades without interacting each other and of the “condensation” of waves at large scale as an analogous of Bose-Einstein condensation in condensed matter physics [11]. Experiments related to the observation of an inverse cascade in wave turbulence are scarce. It concerns either non linear optics [12] or heat wave turbulence in a superfluid [13]. To our knowledge, no observation of an inverse cascade in gravity wave turbulence has been reported so far. In this letter, we report the observation of upscale transfers in gravity wave turbulence. The energy injected into the waves at an intermediate scale (corresponding to the gravity-capillary length) generates waves of larger and larger scales due to the nonlinear wave interactions. A stationary state is reached and presents properties close to the inverse cascade of wave action predicted by wave turbulence theory [1]. We show that the large scale cut-off of this upscale transfer regime is related to the horizontal finite size of the set-up. We characterized this regime by measuring the scaling of the wave amplitude spectrum with the frequency scale and with the power injected into the waves. The probability density of the injected power fluctuations is also measured in the presence or in the absence of the upscale transfer regime of gravity wave turp-1 ha l-0 06 24 71 9, v er si on 1 19 S ep 2 01 1 Author manuscript, published in "EPL 96 (2011) 34004" DOI : 10.1209/0295-5075/96/34004