A Modern Introduction to the Mathematical Theory of Water Waves. By R. S. Johnson. Cambridge University Press, 1997. xiv+445 pp. Hardback ISBN 0 521 59172 4 £55.00; paperback 0 521 59832 X £19.95.

There is much speculation that the largest and steepest waves may need to be modelled with different physics to the majority of the waves on the open ocean. This review examines the various physical mechanisms which may play an important role in the dynamics of extreme waves. We examine the evidence for these mechanisms in numerical and physical wavetanks, and look at the evidence that such mechanisms might also exist in the real ocean.

http://iopscience.iop.org/article/10.1088/0034-4885/77/10/105901/pdf

The physics of anomalous (‘rogue’) ocean waves

A Modern Introduction to the Mathematical Theory of Water Waves: Bibliography

We consider the problem of short-term prediction of rare, extreme water waves in irregular unidirectional fields, a critical topic for ocean structures and naval operations. One possible mechanism for the occurrence of such rare, unusually intense waves is nonlinear wave focusing. Recent results have demonstrated that random localizations of energy, induced by the linear dispersive mixing of different harmonics, can grow significantly due to modulation instability. Here we show how the interplay between (i) modulation instability properties of localized wave groups and (ii) statistical properties of wave groups that follow a given spectrum defines a critical length scale associated with the formation of extreme events. The energy that is locally concentrated over this length scale acts as the ‘trigger’ of nonlinear focusing for wave groups and the formation of subsequent rare events. We use this property to develop inexpensive, short-term predictors of large water waves, circumventing the need for solving the governing equations. Specifically, we show that by merely tracking the energy of the wave field over the critical length scale allows for the robust, inexpensive prediction of the location of intense waves with a prediction window of 25 wave periods. We demonstrate our results in numerical experiments of unidirectional water wave fields described by the modified nonlinear Schrodinger equation. The presented approach introduces a new paradigm for understanding and predicting intermittent and localized events in dynamical systems characterized by uncertainty and potentially strong nonlinear mechanisms.

Reduced-order precursors of rare events in unidirectional nonlinear water waves

Extreme events in fluid flows, waves, or structures interacting with them are critical for a wide range of areas, including reliability and design in engineering, as well as modeling risk of natura...

Statistics of Extreme Events in Fluid Flows and Waves

The evolution of steep waves in the open ocean is nonlinear. In narrow-banded but directionally spread seas, this nonlinearity does not produce significant extra elevation but does lead to a large change in the shape of the wave group, causing, relative to linear evolution, contraction in the mean wave direction and lateral expansion. We use the nonlinear Schrodinger equation (NLSE) to derive an approximate analytical relationship for these changes in group shape. This shows excellent agreement with the numerical results both for the NLSE and for the full water wave equations. We also consider the application of scaling laws from the NLSE in terms of wave steepness and bandwidth to solutions of the full water wave equations. We investigate these numerically. While some aspects of water wave evolution do not scale, the major changes that a wave group undergoes as it evolves scale very well.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2012.0029

Richard H. Gibbs

Papers

The nonlinear evolution and approximate scaling of directionally spread wave groups on deep water