Rogue waves and analogies in optics and oceanography

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Abstract

Over a decade ago, an analogy was drawn between the generation of large ocean waves and the propagation of light fields in optical fibres. This analogy drove numerous experimental studies in both systems, which we review here. In optics, we focus on results arising from the use of real-time measurement techniques, whereas in oceanography we consider insights obtained from analysis of real-world ocean wave data and controlled experiments in wave tanks. This Review of the work in hydrodynamics includes results that support both nonlinear and linear interpretations of rogue wave formation in the ocean, and in optics, we also provide an overview of the emerging area of research applying the measurement techniques developed for the study of rogue waves to dissipative soliton systems. We discuss the insights gained from the analogy between the two systems and its limitations in modelling real-world ocean wave scenarios that include physical effects that go beyond a one-dimensional propagation model.

Key points

  • An analogy between wave propagation on the ocean and in optical fibres has provided new insights into the physical mechanisms and dynamical features that underpin the occurrence of rogue waves.

  • Real-time measurement techniques studying instabilities in fibre optics have highlighted the emergence of localized breather structures associated with nonlinear focusing, a scenario confirmed in wave-tank experiments.

  • The experimental techniques developed for rogue wave measurement in optics have also yielded improved understanding of transient dynamics and dissipative soliton structures in lasers.

  • Advanced analysis and hindcasting of real-world ocean wave data have revealed the central role of directionality and the superposition of random wave trains in the formation of ocean rogue waves.

  • The emergence of oceanic rogue waves in the general case is likely to arise from both linear and nonlinear mechanisms to different degrees depending on the prevalent wind and sea state conditions.

  • Machine learning could play a key role in future efforts to forecast and predict ocean rogue waves and to identify new areas of physical analogy and overlap between optics and hydrodynamics.

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Fig. 1: Localization properties of nonlinear focusing dynamics in the nonlinear Schrödinger equation for three different cases.
Fig. 2: Timeline illustrating the parallel developments in fibre optics (top) and hydrodynamics (bottom).
Fig. 3: Optical rogue wave measurements.
Fig. 4: Characterization of transient instabilities in lasers.
Fig. 5: Ocean and wave-tank measurements of rogue waves.

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Acknowledgements

J.M.D. acknowledges support from the French Investissements d’Avenir programme, project ISITE-BFC (contract ANR-15-IDEX-0003). G.G. acknowledges support from the Academy of Finland (grants 298463 and 318082). A.M. acknowledges support from the Fonds Européen de Développement Economique Régional (project HEAFISY), the Labex CEMPI (ANR-11-LABX-0007) and Equipex FLUX (ANR-11-EQPX-0017) and the French Investissements d’Avenir programme. F.D. acknowledges support from Science Foundation Ireland (SFI) under the research project ‘Understanding extreme nearshore wave events through studies of coastal boulder transport’ (14/US/E3111). Earlier but critical financial support to J.M.D. and F.D. was provided by the European Research Council (ERC-2011-AdG 290562-MULTIWAVE). The authors' understanding of the physics and applications of rogue waves in many different physical systems has benefited from collaboration and discussion with numerous colleagues and friends whom the authors thank. The authors also thank C. Billet for assistance in figure preparation.

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Glossary

Rogue waves

Large-amplitude waves satisfying the common definition that their height from trough to crest exceeds twice the significant wave height. In optics, the definition is the same, but expressed in terms of optical intensity.

Supercontinuum

A broadband optical spectrum typically spanning from the visible to the infrared that is generated by a number of different nonlinear spectral broadening processes.

Envelope

A slowly varying function that modulates the amplitude of optical or water carrier waves.

Deep water

In hydrodynamics, deep water waves are those that propagate in water of depth d that is much greater than half their wavelength λ; that is, \(d\gg \lambda /2\).

Soliton fission

The break-up of a higher-order background-free soliton into constituent fundamental soliton components due to perturbations going beyond the strict nonlinear Schrödinger equation description.

Carrier oscillations

Individual cycles of a propagating wave underneath a group or pulse envelope.

Modulation instability

Exponential growth of a weak perturbation on a continuous-wave excitation in any nonlinear system.

Akhmediev breather

A soliton on a finite background solution to the nonlinear Schrödinger equation that describes a single cycle of growth and decay along the propagation direction ξ with periodic behaviour along the time axis τ.

Continuous wave

Also known as plane wave, this is a wave of constant amplitude or intensity.

Solitons

Coherent structures in a nonlinear dispersive system that display either stationary or recurrent behaviour with propagation, including stationary background-free sech-solitons and solitons on finite background that are also known as breathers.

Stokes wave

The surface elevation of water waves including ‘bound’ harmonic components, which to second order is written η(t, z) = a cos(θ) + 1/2ka2 cos(2θ) where θ = kz − ωt and k is the wavenumber given here for deep water.

Significant wave height

Mean wave height from trough to crest of the upper third of all events in a recorded time series of surface elevation. In optics, the equivalent quantity is ‘significant intensity’, the mean intensity (from zero) of the upper third of all events in a recorded intensity time series.

Dispersive Fourier transform

Also known as time stretch and used for real-time spectroscopy, this technique temporally stretches an ultrashort pulse through linear dispersion such that its temporal intensity assumes the form of its spectrum.

Dissipative soliton

Stable localized structure that is localized as a result of balance between nonlinearity, dispersion and energy exchange (gain or loss) with an environment.

Peregrine soliton

A limiting case of the Akhmediev breather and Kuznetsov–Ma soliton solutions that is doubly localized along the propagation direction ξ and τ.

Kuznetsov–Ma soliton

A soliton on a finite background solution to the NLSE describing periodic oscillation along the propagation direction ξ with localization along τ.

Long-tailed distribution

A characteristic of statistical distributions in which the tails decrease very slowly and contain a subpopulation of extreme events.

Mode-locked lasers

Lasers typically emitting picosecond-duration or femtosecond-duration pulses as a result of either active or passive phase synchronization of the longitudinal modes of the laser cavity.

Coherence

Phase stability of carrier oscillations of a single frequency wave, or the stability of the phase difference between the carrier oscillations of two waves.

Long-crested and short-crested waves

The crest of a wave is equivalent to its transverse extent, with ocean waves classified as long-crested or short-crested respectively depending on whether they predominantly propagate in one direction or consist of a superposition of waves propagating in different directions.

Space–time extreme

The maximum wave surface height observed over a given area during a time interval, and not just at a given point.

Hindcasting

Also known as backtesting, an approach used to test a mathematical model by predicting wave elevation properties based on archival inputs such as directional wave energy spectra at an earlier time and comparing this with known results.

Directional wave energy spectrum

The distribution of wave energy in frequency and direction, often used to provide initial conditions for multi-dimensional linear and nonlinear wave modelling.

Crossing seas

A sea state with two independent wave systems travelling at oblique angles.

Steepness

For water waves only, the steepness is given by ka = 2πa/λ where k is the wavenumber, λ the wavelength and a the amplitude.

Shallow water

In hydrodynamics, shallow water waves are those that propagate in water of depth d that is much less than half their wavelength λ, that is, \(d\ll \lambda /2\). The intermediate depth regime lies between that of shallow and deep water.

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Dudley, J.M., Genty, G., Mussot, A. et al. Rogue waves and analogies in optics and oceanography. Nat Rev Phys 1, 675–689 (2019) doi:10.1038/s42254-019-0100-0

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