Sporadic wind wave horse-shoe patterns

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 condition s. The paper suggests a qualitative model of these structures aimed at expl aining their sporadic nature, physical mechanisms of their selection and th eir specific asymmetric form. First, the phenomenon of sporadic horse-shoe patterns is studied numericall y 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 wide-spectrum primordial noise and subjected to small non-conservative effects typically follows the simple evolution scen ario: most of the time the system can be considered as consisting of a basi c wave and a single pair of oblique satellites, although the choice of this pair tends to be different at different instants. Despite the effective lo w-dimensionality of the multimodal system dynamics at relatively short time spans, the role of small satellites is important: in particular, they enla rge the maxima of the developed satellites. The presence of Benjamin-Feir s atellites appears to be of no qualitative importance at the timescales unde r consideration. The selection mechanism has been linked to the quartic resonant interaction s among the oblique satellites lying in the domain of five-wave (McLean's c lass II) instability of the basic wave: the satellites tend to push each ot her out of the resonance zone due to the frequency shifts caused by the qua rtic interactions. Since the instability domain is narrow (of order of cube of the basic wave steepness), eventually in a generic situation only a sin gle pair survives and attains considerable amplitude. The specific front as ymmetry is found to result from the interplay of quartic and quintet intera ctions and non-conservative effects: the growing and grown satellites have a specific value of phase with respect to the basic wave that corresponds t o downwind orientation of the convex sides of wave fronts. As soon as the p hase relation is violated, the satellite's amplitude quickly decreases down to the noise level.