Weakly nonlinear and stochastic properties of ocean wave fields. Application to an extreme wave event

There has been much interest in freak or rogue waves in recent years, especially after the Draupner “New Year Wave” that occurred in the central North Sea on January 1st 1995. From the beginning there have been two main research directions, deterministic and statistical. The deterministic approach has concentrated on focusing mechanisms and modulational instabilities, these are explained in Chap. 3 and some examples are also given in Chap. 4. A problem with many of these deterministic theories is that they require initial conditions that are just as unlikely as the freak wave itself, or they require idealized instabilities such as Benjamin-Feir instability to act over unrealistically long distances or times. For this reason the deterministic theories alone are not very useful for understanding how exceptional the freak waves are. On the other hand, a purely statistical approach based on data analysis is difficult due to the unusual character of these waves. Recently a third research direction has proved promising, stochastic analysis based on Monte-Carlo simulations with phase-resolving deterministic models. This approach accounts for all possible mechanisms for generating freak waves, within a sea state that is hopefully as realistic as possible. This chapter presents several different modified nonlinear Schrödinger (MNLS) equations as candidates for simplified phase-resolving models, followed by an introduction to some essential elements of stochastic analysis. The material is aimed at readers with some background in nonlinear wave modeling, but little background in stochastic modeling. Despite their simplicity, the MNLS equations capture remarkably non-trivial physics of the sea surface such as the establishment of a quasi-stationary spectrum with ω

−4

power law for the high-frequency tail, and nonlinear probability distributions for extreme waves. In the end we will suggest how often one should expect a “New Year Wave” within the sea state in which it occurred.