Articles | Volume 6, issue 1
31 Mar 1999
31 Mar 1999

Sporadic wind wave horse-shoe patterns

S. Yu. Annenkov and V. I. Shrira

Abstract. The work considers three-dimensional crescent-shaped patterns often seen on water surface in natural basins and observed in wave tank experiments. The most common of these 'horse-shoe-like' patterns appear to be sporadic, i.e., emerging and disappearing spontaneously even under steady wind conditions. The paper suggests a qualitative model of these structures aimed at explaining their sporadic nature, physical mechanisms of their selection and their specific asymmetric form. First, the phenomenon of sporadic horse-shoe patterns is studied numerically using the novel algorithm of water waves simulation recently developed by the authors (Annenkov and Shrira, 1999). The simulations show that a steep gravity wave embedded into widespectrum primordial noise and subjected to small nonconservative effects typically follows the simple evolution scenario: most of the time the system can be considered as consisting of a basic wave and a single pair of oblique satellites, although the choice of this pair tends to be different at different instants. Despite the effective low-dimensionality of the multimodal system dynamics at relatively sho ' rt time spans, the role of small satellites is important: in particular, they enlarge the maxima of the developed satellites. The presence of Benjamin-Feir satellites appears to be of no qualitative importance at the timescales under consideration. The selection mechanism has been linked to the quartic resonant interactions among the oblique satellites lying in the domain of five-wave (McLean's class II) instability of the basic wave: the satellites tend to push each other out of the resonance zone due to the frequency shifts caused by the quartic interactions. Since the instability domain is narrow (of order of cube of the basic wave steepness), eventually in a generic situation only a single pair survives and attains considerable amplitude. The specific front asymmetry is found to result from the interplay of quartic and quintet interactions and non-conservative effects: the growing and grown satellites have a specific value of phase with respect to the basic wave that corresponds to downwind orientation of the convex sides of wave fronts. As soon as the phase relation is violated, the satellite's amplitude quickly decreases down to the noise level.