Nonlinear Four-wave Interactions and Freak Waves

Wave forecasting is about forecasting the mean sea state, as reflected by the ocean wave spectrum, and for quite some time it was thought that it was not possible to make statements about extreme events. Recently, however, it has been shown how to relate fluctuations around the mean sea state to the wave spectrum. Therefore, when the wave spectrum is known, the probability distribution function (pdf) of the sea surface elevation can be determined. The tails of the pdf give vital information on the occurrence of extreme events such as freak waves.

It is important to try to understand why extreme waves do occur. Nowadays it is accepted that at least three mechanisms are responsible for the formation of extreme waves. The first one is just linear superposition of waves, in this case the surface elevation probability distribution is Gaussian. The second mechanism is the interaction of waves with non-uniform currents, linear theory can explain the formation of extreme waves using ray theory. The third mechanism is regarded by the present author as the most promising mechanism as it may provide an explanation for the formation of freak waves on the open ocean: the generation of extreme events as a result of the modulational instability, a four wave quasi-resonant interaction process. This process will result in deviations from the normal distribution of waves, in particular in the case of long-crested waves which implies almost one-dimensional propagation.

Starting from the Hamiltonian description of surface gravity waves it is shown that the short-term dynamics of ocean waves is governed by the Zakharov equation. I have used its one-dimensional version to study the statistical properties of the generation of extreme events using a Monte Carlo simulations. Indeed, deviations from the Normal distribution are shown to be related to the mean sea state. Good agreement with an approximate statistical theory is found, which at the same time describes the evolution in time of the wave spectrum owing to quasi-resonant four wave interactions.

In order to better understand the formation of extreme events we study the properties of the narrow-band version of the Zakharov equation, which turns out to be the well-known Nonlinear Schr\"{o}dinger (NLS) equation. For one-dimensional propagation, the NLS equation may be solved by means of the Inverse Scattering approach and for large times an initial disturbance evolves towards a train of envelope solitary waves, explaining the formation of extreme events. In fact, if the ocean would be truly one-dimensional, shipping would be a hazardous enterprise. In the case of two-dimensional propagation however, envelope solitons are unstable to transverse perturbations and therefore in that event the formation of freak waves is less frequent.

Peter Janssen, European Center for Medium Range Weather Forecasts, United Kingdom


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