Experimental quasi-1D capillary-wave turbulence

Abstract

Wave turbulence in quasi-1D geometry is usually not investigated experimentally since low-order resonant wave interactions are theoretically prohibited. Here, we report on the first observation of unidirectional capillary-wave turbulence on the surface of a fluid in a canal. We also show that five-wave interactions are the lowest-order resonant process subsisting at small scales, and are thus probably the one generating such quasi-1D capillary wave turbulence. We show that the wave spectrum is compatible with the corresponding dimensional analysis prediction. The main assumptions of weak turbulence theory are also verified experimentally. Quasi-1D wave turbulence could be thus highlighted in other fields of wave turbulence. Introduction. – Wave turbulence is a phenomenon occurring within a large number of random nonlinear interacting waves [1–3]. These nonlinear interactions lead to an energy cascade from a large (forcing) scale down to a small (dissipative) scale, predicted by weak turbulence theory (WTT). These predictions have been applied in many different domains such as ocean surface waves, plasma waves, hydroelastic or elastic waves, internal waves, and optical waves [1–3]. This theory has then been assessed experimentally in various wave systems propagating in 2D or 3D [4–9], but rarely in 1D since generally no low-order resonant wave interactions are expected theoretically in this geometry [2]. To our knowledge, the unique experimental study concerns 1D nonlinear optics focusing only on inverse cascade towards large scales [10]. For capillary waves on the surface of a fluid, the WTT [11, 12] is rather well confirmed experimentally in 2D (see review [13]), whereas for a unidirectional propagation, the theory forbids low-order wave resonant interactions [2,14], and thus a wave turbulence regime. Nevertheless, a quasi-1D capillary wave turbulence regime has been recently reported numerically [15]. An experimental observation of such a regime would pave the way to other fields of wave turbulence due to easier calculations, and measurements in 1D geometry. In this letter we report the first observation of quasi-1D capillary-wave turbulence on the surface of a low-viscous fluid (mercury). With this specific fluid, a weak nonlinearity is sufficient to reach a wave turbulence regime without coherent structures. Using high-order correlations of wave elevations, we quantify the occurrence of three-, four(a)E-mail: [email protected] (corresponding author) , and five-wave interactions. Although quasi-resonant interactions are observed at low orders, five-wave resonant interactions are found to be the lowest-resonant order subsisting in the capillary range, and are thus probably the mechanism generating the observed 1D capillary-wave turbulence. This differs from the usual 2D capillary-wave turbulence involving three-wave resonant interactions [2, 11]. We also show that the wave spectra in frequency and in wavenumber are compatible with the corresponding dimensional analysis predictions. Moreover, the energy flux cascading towards small scales is roughly found to be constant as expected, and the main WTT assumptions are verified experimentally. Note that this 1D wave turbulence differs basically from 1D integrable turbulence (involving coherent structures such as solitons within stochastic waves) [16], recently observed [17]. Note also that an idealized 1D model of wave turbulence showed strong deviations from the WTT due to these coherent structures [14,18]. Moreover, our results should not be confused with the existence of unidirectional resonant interaction highlighted in 2D gravity-capillary wave turbulence, near the gravity-capillary crossover [19]. Theoretical backgrounds. – The dispersion relation of linear deep-water gravity-capillary surface waves reads ω = gk + (γ/ρ)k, with ω = 2πf the angular frequency, k the wave number, g the acceleration of gravity, γ the surface tension, and ρ the liquid density [20]. The theoretical crossover between the gravity and capillary regimes occurs for kgc = √ ρg/γ and fgc = (g ρ/γ)/( √ 2π) [21]. Both contributions coexist near fgc [13], and we denote fc the frequency from which gravity becomes negligible (see below). In the two limits