Phenomenological model of weakly damped Faraday waves and the associated mean flow

Abstract

A phenomenological model of parametric surface waves (Faraday waves) is introduced in the limit of small viscous dissipation that accounts for the coupling between surface motion and slowly varying streaming and large-scale flows (mean flow). The primary bifurcation of the model is to a set of standing waves (stripes, given the functional form of the model nonlinearities chosen here). Our results for the secondary instabilities of the primary wave show that the mean flow leads to a weak destabilization of the base state against Eckhaus and transverse amplitude modulation instabilities, and introduces a longitudinal oscillatory instability which is absent without the coupling. We compare our results with recent one-dimensional amplitude equations for this system systematically derived from the governing hydrodynamic equations.