The Formation of Parasitic Capillary Ripples on Gravity–Capillary Waves and the Underlying Vortical Structures

Abstract

The evolution of moderately short, steep two-dimensional gravity?capillary waves, from the onset of the parasitic capillary ripples to a fully developed quasi-steady stage, is studied numerically using a spectrally accurate model. The study focuses on understanding the precise mechanism of capillary generation, and on characterizing surface roughness and the underlying vortical structure associated with parasitic capillary waves. It is found that initiation of the first capillary ripple is triggered by the fore?aft asymmetry of the otherwise symmetric carrier wave, which then forms a localized pressure disturbance on the forward face near the crest, and subsequently develops an oscillatory train of capillary waves. Systematic numerical experiments reveal that there exists a minimum crest curvature of the carrier gravity?capillary wave for the formation of parasitic capillary ripples, and such a threshold curvature (≈0.25 cm?1) is almost independent of the carrier wavelength. The characteristics of the parasitic capillary wave train and the induced underlying vortical structures exhibit a strong dependence on the carrier wavelength. For a steep gravity?capillary wave with a shorter wavelength (e.g., 5 cm), the parasitic capillary wave train is distributed over the entire carrier wave surface at the stage when capillary ripples are fully developed. Immediately underneath the capillary wave train, weak vortices are observed to confine within a thin layer beneath the ripple crests whereas strong vortical layers with opposite orientation of vorticity are shed from the ripple troughs. These strong vortical layers are then convected upstream and accumulate within the carrier wave crest, forming a strong ?capillary roller? as postulated by Longuet-Higgins. In contrast, as the wavelength of the gravity?capillary wave increases (e.g., 10 cm), parasitic capillary ripples appear as being trapped in the forward slope of the carrier wave. The strength of the vortical layer shed underneath the parasitic capillaries weakens, and its thickness and extent reduces. The vortices accumulating within the crest of the carrier wave, therefore, are not as pronounced as those observed in the shorter gravity?capillary waves.