to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Recent observations show that the probability of encountering an extremely large rogue wave in the open ocean is much larger than expected from ordinary wave-amplitude statistics. Although considerable effort has been directed towards understanding the physics behind these mysterious and potentially destructive events, the complete picture remains uncertain. Furthermore, rogue waves have not yet been observed in other physical systems. Here, we introduce the concept of optical rogue waves, a counterpart of the infamous rare water waves. Using a new real-time detection technique, we study a system that exposes extremely steep, large waves as rare outcomes from an almost identically prepared initial population of waves. Specifically, we report the observation of rogue waves in an optical system, based on a microstructured optical fibre, near the threshold of soliton-fission supercontinuum generation--a noise-sensitive nonlinear process in which extremely broadband radiation is generated from a narrowband input. We model the generation of these rogue waves using the generalized nonlinear Schrodinger equation and demonstrate that they arise infrequently from initially smooth pulses owing to power transfer seeded by a small noise perturbation.

Optical rogue waves

Waves that appear from nowhere and disappear without a trace

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrodinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.

/pdf/physical-mechanisms-of-the-rogue-wave-phenomenon-iao5kemdkh.pdf

Physical Mechanisms of the Rogue Wave Phenomenon

http://personalpages.to.infn.it/~onorato/publications_files/2013_Onorato_etal_PhysRep.pdf

Rogue waves and their generating mechanisms in different physical contexts

Preface.- Freak Waves: Peculiarities of Numerical Simulations.- Rogue Waves in High-Order Nonlinear Schrodinger Models.- Non-Gaussian Properties of Shallow Water Waves in Crossing Seas.- Modeling of Rogue Wave Shapes in Shallow Water.- Runup of Long Irregular Waves on Plane Beach.- Symbolic Computation for Nonlinear Wave Resonances.- Searching for Factors that Limit Observed Extreme Maximum Wave Height Distributions in the North Sea.- Extremes and Decadal Variations of the Northern Baltic Sea Wave Conditions.- Extreme Waves Generated by Cyclones in Guadeloupe.- An Analytical Model of Large Amplitude Internal Solitary Waves.

Extreme ocean waves

The influence of wind on extreme wave events in deep water is investigated experimentally and numerically A series of experiments conducted in the Large Air–Sea Interactions Facility (LASIF-Marseille, France) shows that wind blowing over a short wave group due to the dispersive focusing of a longer frequency-modulated wavetrain (chirped wave packet) may increase the time duration of the extreme wave event by delaying the defocusing stage A detailed analysis of the experimental results suggests that extreme wave events may be sustained longer by the air flow separation occurring on the leeward side of the steep crests Furthermore it is found that the frequency downshifting observed during the formation of the extreme wave event is more important when the wind velocity is larger These experiments have pointed out that the transfer of momentum and energy is strongly increased during extreme wave eventsTwo series of numerical simulations have been performed using a pressure distribution over the steep crests given by the Jeffreys sheltering theory The first series corresponding to the dispersive focusing confirms the experimental results The second series which corresponds to extreme wave events due to modulational instability, shows that wind sustains steep waves which then evolve into breaking waves Furthermore, it was shown numerically that during extreme wave events the wind-driven current could play a significant role in their persistence

/pdf/influence-of-wind-on-extreme-wave-events-experimental-and-5gaf81cmfc.pdf

Influence of wind on extreme wave events : experimental and numerical approaches

This paper concerns long time interaction of envelope solitary gravity waves propagating at the surface of a two-dimensional deep fluid in potential flow. Fully nonlinear numerical simulations show how an initially long wave group slowly splits into a number of solitary wave groups. In the example presented, three large wave events are formed during the evolution. They occur during a time scale that is beyond the time range of validity of simplified equations like the nonlinear Schrodinger (NLS) equation or modifications of this equation. A Fourier analysis shows that these large wave events are caused by significant transfer to side-band modes of the carrier waves. Temporary downshiftings of the dominant wavenumber of the spectrum coincide with the formation large wave events. The wave slope at maximal amplifications is about three times higher than the initial wave slope. The results show how interacting solitary wave groups that emerge from a long wave packet can produce freak wave events. Our reference numerical simulation are performed with the fully nonlinear model of Clamond and Grue [D. Clamond, J. Grue, A fast method for fully nonlinear water wave computations, J. Fluid Mech. 447 (2001) 337–355]. The results of this model are compared with that of two weakly nonlinear models, the NLS equation and its higher-order extension derived by Trulsen et al. [K. Trulsen, I. Kliakhandler, K.B. Dysthe, M.G. Velarde, On weakly nonlinear modulation of waves on deep water, Phys. Fluids 12 (10) (2000) 2432–2437]. They are also compared with the results obtained with a high-order spectral method (HOSM) based on the formulation of West et al. [B.J. West, K.A. Brueckner, R.S. Janda, A method of studying nonlinear random field of surface gravity waves by direct numerical simulation, J. Geophys. Res. 92 (C11) (1987) 11 803–11 824]. An important issue concerning the representation and the treatment of the vertical velocity in the HOSM formulation is highlighted here for the study of long-time evolutions.

Long time interaction of envelope solitons and freak wave formations

The freak wave formation due to the dispersive focusing mechanism is investigated experimentally without wind and in presence of wind. An asymmetric behaviour between the focusing and defocusing stages is found when the wind is blowing over the mechanically generated gravity wave group. This feature corresponds physically to the sustain of the freak wave mechanism on longer periods of time. Furthermore, a weak amplification of the freak wave and a shift in the downstream direction of the point where the waves merge are observed. The experimental results suggest that the Jeffreys' sheltering mechanism could play a key role in the coherence of the group of the freak wave. Hence, the Jeffreys' sheltering theory is introduced in a fully nonlinear model. The results of the numerical simulations confirm that the duration of the freak wave event increases with the wind velocity.

Christian Kharif

Papers

Physical Mechanisms of the Rogue Wave Phenomenon

Extreme ocean waves

Influence of wind on extreme wave events : experimental and numerical approaches

Long time interaction of envelope solitons and freak wave formations

Freak waves under the action of wind: experiments and simulations