Weakly dissipative Faraday waves in 2D large aspect ratio annuli

Weakly dissipative parametrically excited (by vertical vibration) surface gravity-capillary waves in a two-dimensional, horizontally periodic container are considered. A set of equations is derived for the coupled evolution of the left- and right-traveling surface waves and the associated mean flow, in the case when the container depth is small compared to its length but large compared to the wavelength of the excited waves. The stability of the spatially uniform standing waves (SWs) is first analyzed and then the large time spatio-temporal behavior of the system beyond threshold is numerically studied. The viscous mean flow is found to drastically affect the dynamics of the system and the resulting surface wave patterns.