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

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.