A MODEL OF WATER-WAVE HORSE-SHOE PATTERNS

The work suggests a simple qualitative model of the wind wave 'horse-s hoe' patterns often seen on the sea surface. The model is aimed at exp laining the persistent character of the patterns and their specific as ymmetric shape. It is based on the idea that the dominant physical pro cesses are quintet resonant interactions, input due to wind and dissip ation, which balance each other. These processes are described at the lowest order in nonlinearity. The consideration is confined to the mos t essential modes: the central (basic) harmonic and two symmetric obli que satellites, the most rapidly growing ones due to the class II inst ability. The chosen harmonics are phase locked, i.e. all the waves hav e equal phase velocities in the direction of the basic wave, This fact along with the symmetry of the satellites ensures the quasi-stationar y character of the resulting patterns. Mathematically the model is a s et of three coupled ordinary differential equations for the wave ampli tudes. It is derived starting with the integro-differential formulatio n of water wave equations (Zakharov's equation) modified by taking int o account small (of order of quartic nonlinearity) non-conservative ef fects. In the derivation the symmetry properties of the unperturbed Ha miltonian system were used by taking special canonical transformations , which allow one exactly to reduce the Zakharov equation to the model . The study of system dynamics is focused on its qualitative aspects. It is shown that if the non-conservative effects are neglected one can not obtain solutions describing persistent asymmetric patterns, but th e presence of small non-conservative effects changes drastically the s ystem dynamics at large times. The main new feature is attractive equi libria, which are essentially distinct from the conservative ones. For the existence of the attractors a balance between nonlinearity and no n-conservative effects is necessary. A wide class of initial configura tions evolves to the attractors of the system, providing a likely scen ario for the emergence of the long-lived three-dimensional wind wave p atterns. The resulting structures reproduce all the main features of t he experimentally observed horse-shoe patterns. In particular, the mod el provides the characteristic 'crescent' shape of the wave fronts ori ented forward and the front-back asymmetry of the wave profiles.