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Nonlinear interactions of gravity-capillary waves: Lagrangian theory and effects on the spectrum

Published online by Cambridge University Press:  21 April 2006

Klaartje Van Gastel
Klaartje Van Gastel
Affiliation:
Royal Netherlands Meteorological Institute, PO Box 201, 3730 AE De Bilt, The Netherlands Present address: State University of Utrecht. Mathematical Institute, P.O. Box 80.010, 3508 TA Utrecht, The Netherlands.
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Abstract

A weakly nonlinear inviscid theory describing the interactions within a continuous spectrum of gravity-capillary waves is developed. The theory is based on the principle of least action and uses a Lagrangian in wavenumber-time space. Advantages of this approach compared to the method of Valenzuela & Laing (1972) are much simplified mathematics and final equations and validity on a longer timescale. It is shown that much of the development of the spectrum under the influence of nonlinear terms can be understood without actually having to integrate the equations. To this end multiwave space, a new concept comparable with phase space, is introduced. Using multiwave space the magnitude of the nonlinear transfer is estimated and it is shown how the energy goes through the spectrum. Also it is predicted that at fixed wavenumbers, the smallest being 520 m−1, finite peaks will arise in the spectrum. This is confirmed by numerical integrations. From the integrations it is also deduced that nonlinear interactions are at least as important to the development of the spectrum as wind growth. Finally it is shown numerically that the near-Gaussian statistics of the sea surface are unaffected by nonlinear interactions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 182 , September 1987 , pp. 499 - 523
Copyright
© 1987 Cambridge University Press

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