Plane-symmetric capillary turbulence: Five-wave interactions

The theory of isotropic capillary turbulence was developed in the late 1960s by Zakharov and Filonenko. To date, the analytical solution of the kinetic equation describing the stationary transfer of energy to small scales due to three-wave resonant interactions, called the Zakharov–Filonenko spectrum, has been confirmed with high accuracy. However, in the case of strong anisotropy in wave propagation, where all waves are collinear, the situation changes significantly. In such a degenerate geometry, the conditions of resonant interaction cease to be fulfilled not only for three waves, but also for four interacting waves. In this work, we perform fully nonlinear simulations of plane-symmetric capillary turbulence. We demonstrate that the system of interacting waves evolves into a quasi-stationary state with a direct energy cascade, despite the absence of low-order resonances. The calculated spectra of surface elevations are accurately described by analytical estimates derived dimensionally under the assumption of the dominant influence of five-wave resonant interactions. A detailed study of the statistical characteristics of the weakly turbulent state does not reveal the influence of any coherent or strongly nonlinear structures. The performed high-order correlation analysis indicates a variety of non-trivial five-wave resonances. We show that the process of wave decay into two pairs of counter-propagating waves is responsible for the local energy transfer to small scales. Overall, the calculation results are in good agreement with both the weak turbulence theory and recent experiments made by Ricard and Falcon.