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A model of water wave ‘horse-shoe’ patterns

Published online by Cambridge University Press:  26 April 2006

Victor I. Shrira ,
Sergei I. Badulin  and
Christian Kharif
Victor I. Shrira
Affiliation:
P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences, 23 Krasikov str., Moscow 117218, Russia
Sergei I. Badulin
Affiliation:
P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences, 23 Krasikov str., Moscow 117218, Russia
Christian Kharif
Affiliation:
Institut de Recherche sur les Phenomenes Hors Equilibre, Laboratoire Interactions Ocean Atmosphere, 163 Avenue de Luminy — Case 903, 13288 Marseille Cedex 9, France
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Abstract

The work suggests a simple qualitative model of the wind wave ‘horse-shoe’ patterns often seen on the sea surface. The model is aimed at explaining the persistent character of the patterns and their specific asymmetric shape. It is based on the idea that the dominant physical processes are quintet resonant interactions, input due to wind and dissipation, which balance each other. These processes are described at the lowest order in nonlinearity. The consideration is confined to the most essential modes: the central (basic) harmonic and two symmetric oblique satellites, the most rapidly growing ones due to the class II instability. The chosen harmonics are phase locked, i.e. all the waves have equal phase velocities in the direction of the basic wave. This fact along with the symmetry of the satellites ensures the quasi-stationary character of the resulting patterns.

Mathematically the model is a set of three coupled ordinary differential equations for the wave amplitudes. It is derived starting with the integro-differential formulation of water wave equations (Zakharov's equation) modified by taking into account small (of order of quartic nonlinearity) non-conservative effects. In the derivation the symmetry properties of the unperturbed Hamiltonian 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 cannot obtain solutions describing persistent asymmetric patterns, but the presence of small non-conservative effects changes drastically the system dynamics at large times. The main new feature is attractive equilibria, which are essentially distinct from the conservative ones. For the existence of the attractors a balance between nonlinearity and non-conservative effects is necessary. A wide class of initial configurations evolves to the attractors of the system, providing a likely scenario for the emergence of the long-lived three-dimensional wind wave patterns. The resulting structures reproduce all the main features of the experimentally observed horse-shoe patterns. In particular, the model provides the characteristic ‘crescent’ shape of the wave fronts oriented forward and the front-back asymmetry of the wave profiles.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 318 , 10 July 1996 , pp. 375 - 405
Copyright
© 1996 Cambridge University Press

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