Abstract
The problem of unidirectional shoaling of a water-wave field with a narrow energy spectrum is treated by using a new Alber equation. The stability of the linear stationary solution to small non-stationary disturbances is analysed; and numerical solutions for its subsequent long-distance evolution are presented. The results quantify the physics which causes the gradual decay in the probability of freak-wave occurrence, when moving from deep to shallow coastal waters.
Highlights
Beginning with a Schrödinger equation in infinite or constant depth as their starting point, Longuet-Higgins (1976) and Alber (1978) obtained two rather different stochastic evolution equations.Longuet-Higgins assumed that the wave field is a homogeneous and nearly Gaussian random process
Alber’s findings are the stochastic counterpart of the well-known deterministic Benjamin–Feir instability, which can be described with the cubic Schrödinger equation
The deterministic evolution of a nonlinear, shoaling, ocean wave field with a narrow spectral band is governed by a cubic Schrödinger equation, see (Iusim & Stiassnie 1985):
Summary
Beginning with a Schrödinger equation in infinite or constant depth as their starting point, Longuet-Higgins (1976) and Alber (1978) obtained two rather different stochastic evolution equations. Longuet-Higgins assumed that the wave field is a homogeneous and nearly Gaussian random process His result is the narrow-band limit of the Hasselmann kinetic equation. On the other hand, enabled the random process to be inhomogeneous but required Gaussianity He used his equation to study the instability of a homogeneous wave field to inhomogeneous disturbances. Stiassnie, Regev & Agnon (2008) solved the Alber equation in infinite depth numerically and showed that a stochastic parallel to the Fermi–Pasta–Ulam recurrence exists. This stochastic recurrence enabled them to study the probability of waves that are higher than.
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