Wave turbulence in a two-layer fluid: coupling between free surface and interface waves

We experimentally study gravity-capillary wave turbulence on the interface between two immiscible fluids of close density with free upper surface. We locally measure the wave height at the interface between both fluids by means of a highly sensitive laser Doppler vibrometer. We show that the inertial range of the capillary wave turbulence regime is significantly extended when the upper fluid depth is increased: The crossover frequency between the gravity and capillary wave turbulence regimes is found to decrease whereas the dissipative cut-off frequency of the spectrum is found to increase. We explain these observations by the progressive decoupling between waves propagating at the interface and the ones at the free surface, using the full dispersion relation of gravity-capillary waves in a two-layer fluid of finite depths. The cut-off evolution is due to the disappearance of parasitic capillaries responsible for the main wave dissipation for a single fluid. Introduction. – Stratified fluids are ubiquitous in Nature such as in ocean or in atmosphere. The density stratification is usually due to a temperature or salinity gradient with the depth in oceans, or a temperature or humidity gradient with altitude in the atmosphere. The simplest stratified fluid consists in two superimposed homogeneous fluids, the fluid with higher density being below the fluid with lower density. In this situation, waves can propagate at the interface between the two fluid layers but also at the free surface of the top one. Under certain conditions, surface and interface waves interact together [1, 2]. An astonishing manifestation of this phenomenon is the dead-water effect first observed in 1904 on the sea surface [3], and recently reproduced in experiments [4, 5]. Indeed, ships evolving on a calm sea can slow or even stop sailing in a two-layer fluid due to the extra-drag generated by large interface waves. The coupling between the surface and interface waves in a two-layer fluid also generates narrow nested V-shaped wakes observed behind ships [6,7], as well as the damping of ocean surface waves over a layer of fluid mud [8]. Such a coupling is also involved in Faraday instability of floating droplets on a liquid bath [9, 10], or during the long-wave instabilities in thin two-layer liquid films (< 100 nm) in chemical physics [11]. In industrial applications like metal refining, such interactions can also have an influence on the ripples created during dewetting [12]. At last, the coupling between the surface and interface waves of large amplitudes occur in many physical and biological situations (involving or not elasticity), and lead to numerous challenging studies in applied mathematics such as the predictions of new solitary waves [13, 14]. When a set of stochastic waves, propagating on a free surface, have large enough amplitudes, interactions between nonlinear waves can generate a wave turbulence regime. These interactions transfer the wave energy from the large scales, where it is injected, to the small scales where it is dissipated. This generic phenomenon concerns various domains at different scales: Surface and internal waves in oceans, elastic waves on plates, spin waves in solids, magnetohydrodynamic waves in astrophysical plasma (for reviews, see [15–18]). Weak turbulence theory developed in the 60’s [19–21] leads to predictions on the wave turbulence regime in almost all domains of physics involving waves [16, 17]. The past decade has seen an important experimental effort to test the validity domain of weak turbulence theory on different wave systems (e.g. hydrodynamics, optics, hydro-elastic or elastic waves) [22]. In this paper, we study gravity-capillary wave turbu-