UC Berkeley

In the weakly inviscid regime parametrically driven surface gravity-capillary waves generate oscillatory viscous boundary layers along the container walls and the free surface. Through nonlinear rectification these generate Reynolds stresses which drive a streaming flow in the nominally inviscid bulk; this flow in turn advects the waves responsible for the boundary layers. The resulting system is described by amplitude equations coupled to a Navier-Stokes-like equation for the bulk streaming flow, with boundary conditions obtained by matching to the boundary layers, and represents a novel type of pattern-forming system. The coupling is responsible for new types of secondary instabilities of standing waves leading to chaotic dynamics, and in appropriate regimes can lead to the presence of relaxation oscillations. These are present because in the nearly inviscid regime the streaming flow decays much more slowly than the waves, and resemble a class of oscillations discovered by Simonelli and Gollub [J. Fluid Mech. 199 (1989), 349] in a domain with an almost square cross-section.