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Nonlinear wave interaction for broad-banded, open seas - deterministic and stochastic theory

Nonlinear interaction, along with wind input and dissipation, is one of the key drivers of wave evolution at sea, and is included in every modern wave-forecast model. The mechanism behind the nonlinear interaction terms in such models is based on the kinetic equation for wave spectra first derived by Hasselmann. This does not allow, for example, for statistically inhomogeneous wave fields, and is restricted to very long time-scales.

Beginning with the incompressible Euler equations, one may find simpler deterministic equations for the third-order problem, such as the Zakharov equation or the nonlinear Schrödinger equation. Using these, I will sketch the derivation of a discretized equation for the fast evolution of random, inhomogeneous surface wave fields, along lines first laid out by Crawford, Saffman, and Yuen. This allows for a more general treatment of the stability and long-time behaviour of broad-banded sea states, and progress beyond the stability criteria proposed by Alber.