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Rogue Waves and Their Generating Mechanisms in Different Physical Contexts

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Rogue Waves and Their Generating Mechanisms in Different Physical Contexts Physics Reports 528 (2013) 47–89 Contents lists available at SciVerse ScienceDirect Physics Reports journal homepage: www.elsevier.com/locate/physrep Rogue waves and their generating mechanisms in different physical contexts M. Onorato a,b, S. Residori c,∗, U. Bortolozzo c, A. Montina d, F.T. Arecchi e,f a Dipartimento di Fisica Generale, Università degli Studi di Torino, Via Pietro Giuria 1, 10125 Torino, Italy b INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy c INLN, Université de Nice Sophia-Antipolis, CNRS, 1361 route des Lucioles, 06560 Valbonne, France d Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5 Canada e Dipartimento di Fisica, Università di Firenze, Italy f CNR-INO, largo E. Fermi 6, 50125 Firenze, Italy article info a b s t r a c t Article history: Rogue waves is the name given by oceanographers to isolated large amplitude waves, Accepted 19 December 2012 that occur more frequently than expected for normal, Gaussian distributed, statistical Available online 14 March 2013 events. Rogue waves are ubiquitous in nature and appear in a variety of different contexts. editor: G.I. Stegeman Besides water waves, they have been recently reported in liquid Helium, in nonlinear optics, microwave cavities, etc. The first part of the review is dedicated to rogue waves in the oceans and to their laboratory counterpart with experiments performed in water basins. Most of the work and interpretation of the experimental results will be based on the nonlinear Schrödinger equation, an universal model, that rules the dynamics of weakly nonlinear, narrow band surface gravity waves. Then, we present examples of rogue waves occurring in different physical contexts and we discuss the related anomalous statistics of the wave amplitude, which deviates from the Gaussian behavior that were expected for random waves. The third part of the review is dedicated to optical rogue waves, with examples taken from the supercontinuum generation in photonic crystal fibers, laser fiber systems and two-dimensional spatiotemporal systems. In particular, the extreme waves observed in a two-dimensional spatially extended optical cavity allow us to introduce a description based on two essential conditions for the generation of rogue waves: nonlinear coupling and nonlocal coupling. The first requirement is needed in order to introduce an elementary size, such as that of the solitons or breathers, whereas the second requirement implies inhomogeneity, a mechanism needed to produce the events of mutual collisions and mutual amplification between the elementary solitons or wavepackets. The concepts of ``granularity'' and ``inhomogeneity'' as joint generators of optical rogue waves are introduced on the basis of a linear experiment. By extending these concepts to other systems, rogue waves can be classified as phenomena occurring in the presence of many uncorrelated ``grains'' of activity inhomogeneously distributed in large spatial domains, the ``grains'' being of linear or nonlinear origin, as in the case of wavepackets or solitons. ' 2013 Elsevier B.V. All rights reserved. Contents 1. Introduction............................................................................................................................................................................................. 48 2. Introduction to the statistical properties of ocean waves.................................................................................................................... 49 ∗ Corresponding author. Tel.: +33 0492967317. E-mail address: [email protected] (S. Residori). 0370-1573/$ – see front matter ' 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physrep.2013.03.001 48 M. Onorato et al. / Physics Reports 528 (2013) 47–89 2.1. The wave spectrum..................................................................................................................................................................... 49 2.2. Rayleigh distribution for the wave envelope: linear theory.................................................................................................... 50 3. Experimental evidence of rogue waves in the ocean ........................................................................................................................... 52 4. Nonlinearity and rogue waves in one horizontal dimension............................................................................................................... 53 4.1. The nonlinear Schrödinger equation for water waves in infinite water depth ...................................................................... 54 4.1.1. The NLS in nondimensional form: the Benjamin–Feir-Index ................................................................................... 54 4.2. The modulational instability...................................................................................................................................................... 55 4.3. Breather solutions....................................................................................................................................................................... 55 4.3.1. Relation between the breather solutions and standard solitons.............................................................................. 57 4.3.2. Higher order breathers................................................................................................................................................ 57 4.3.3. Experimental/numerical verification of the breather solutions in wave tanks....................................................... 57 4.4. Finite depth effects ..................................................................................................................................................................... 59 4.5. The fully nonlinear approach in one horizontal dimension..................................................................................................... 60 4.6. Rogue waves in random sea states ............................................................................................................................................ 61 4.7. Some experimental results in wave tank laboratories for 1-dimensional propagation ........................................................ 61 5. Kurtosis and skewness of the surface elevation.................................................................................................................................... 63 5.1. The contribution to the kurtosis from free modes ................................................................................................................... 63 5.2. The contribution to deviations from Gaussian statistics by bound modes............................................................................. 64 6. Rogue waves in two horizontal dimensions.......................................................................................................................................... 64 7. The role of inhomogeneity: the Alber–Saffman approach ................................................................................................................... 66 8. Rogue waves in crossing seas................................................................................................................................................................. 67 9. Rogue waves and currents...................................................................................................................................................................... 68 10. Rogue waves in other physical systems ................................................................................................................................................ 68 10.1. Wave turbulence and rogue waves in superfluid Helium........................................................................................................ 69 10.2. Microwave experiments: two-dimensional rogue waves in the linear regime ..................................................................... 70 10.3. Extreme events in parametrically driven capillary waves....................................................................................................... 70 10.4. Rogue waves in plasmas............................................................................................................................................................. 71 11. Rogue waves in optics............................................................................................................................................................................. 72 12. Optical rogue waves in fiber supercontinuum generation................................................................................................................... 73 12.1. The 1D generalized nonlinear Schrödinger equation............................................................................................................... 74 12.1.1. Numerical simulations ................................................................................................................................................ 75 12.1.2. Statistical properties.................................................................................................................................................... 75 12.2. Optical rogue waves in continuous pumped supercontinuum................................................................................................ 75 12.3. Optical rogue waves
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