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Applied mathematics, Probability
This paper presents a generalized model for simulating wavefields associated with the sea surface. This includes the case when `freak waves' may occur through an effect compounded in the nonlinear (cubic) Schrodinger equation. After providing brief introductions to linear sea wave models, `freak waves' and the linear and nonlinear Schrodinger equations, we present a unified model that provides for a piecewise continuous transition from a linear to a nonlinear state. This is based on introducing a fractional time derivative to develop a fractional nonlinear partial differential equation with a stochastic source function. In order to explore the characteristics of this equation, we consider a separation of variables approach in order to derive governing equations for the spatial and temporal behaviour.
Models for the source function (which, in physical terms, describes the conversion of wind energy into wave energy) are also considered on a separable basis. With regard to the temporal characteristics, we provide a new model that is based on assuming Levy processes for the time-dependent wind velocity informed by experimental data. We consider a spatial frequency model that is based on a generalization of Berman and Ornstein-Uhlenbeck processes. This provides a statistically self-affine source function which has a synergy with the Pierson-Moskowitz model for the spectral form of fully developed wind driven seas based on `similarity theory'.
Having presented the source function models, solutions to the governing nonlinear wave equations are explored using a Green's function transformation under a low frequency bandwidth condition. Iterative methods of solution are then considered in threedimensions and then in two-dimensions. Example results are presented based on considering a first order solution that is equivalent to the application of the Born approximation for the linear Schrodinger equation. The simulations provide evidence for the formation of freak waves being related to the fact that the wind force (as a function of time) is non-Gaussian distributed. Consequently, freak waves are more common than would be expected using Gaussian statistics.
Blackledge, J.: A Generalized Nonlinear Model for the Evolution of Low Frequency Freak Waves. International Journal of Applied Mathematics, Vol. 41, Issue 1, pp.33-55. 2010.