Wikipedia

Draft:Origin of rogue waves

Although commonly described as a tsunami, The Great Wave off Kanagawa, by artist Hokusai, likely depicts a rogue wave off of the coast of Japan.

The exact origins deriving the mechanics of rogue waves has been a matter of active research and ongoing scientific debate.[1] Amongst scientific consensus, the development of rogue waves is likely influenced by several collective environmental factors, including wind, wave oscillations, currents, and possibly gale forces.[a] The universal cause explaining the origins of rogue waves exists in numerous hypotheses, the most prominent explanations include Diffractive focusing, nonlinear effects (modulational instability), and wind wave interactions. In human knowledge, rogue waves originated in myth, existing through anecdotal evidence given by early eyewitness accounts.[3][b] The irregular damage inflicted upon ships later suggested that large surface anomalies have long occurred; the application of modern technology and oceanographic studies confirmed the existence of unpredictable freak waves in later decades, and generated extensive research amongst the scientific community into several possible causes. The ambiguity surrounding rogue waves is deeply rooted in the unpredictability of wave propagation and the chaotic dynamics of wind waves attributing to their apparent randomness within evolving sea states.[5][6]

Rogue waves do not appear to have a single distinct cause.[7][c] The nature of a "freak" rogue wave are generally agreed to occur variably and without warning, yet can be observed to have highest predictability where a strong ocean current runs counter to the prevailing direction of the traveling waves. Further observations shows high risk areas where currents or winds cause large swells to travel at different speeds, creating a convergence of waves in which a significantly larger wave is created. A majority of research into the causes of rogue waves are focused on the interactions of linear superposition and wave breaking to explain the universal mechanics of rogue wave systems.[9][10][11][d] More recent studies have accounted for the interactions with localized features such as nonuniform topography, wave-current interactions, Antarctic Circumpolar Current, or crossing sea states at high crossing angles as additional theories behind the mechanisms of rogue waves.[14][11][e] Studies of nonlinear waves suggest that modulational instability can create an unpredictable sea state where the transfer of energy between waves can generate a much larger wave.

Scientific studies of rogue waves can be traced to the recording of an abnormally large wave off of the Gorm Field in the central North Sea. The measurement of the Draupner wave off the Draupner platform was the first rogue wave to be detected by a measuring instrument. Early scientific research of unusual waves began in the 19th century with the discovery of wave of translation by John Scott Russell in 1834, in which the modern study of solitons was formed. The use of statistical models beginning in the 19th century helped to predict wave height while the general knowledge was that wave heights were grouped around a central value equal to the average of the largest third.

Scientific works on "Freak Waves" began with Professor Laurence Draper in 20th century where he documented the efforts of the National Institute of Oceanography in the early 1960s to record wave height. The first scientific study to comprehensively prove that freak waves exist was published in 1997 and began an overall censuses amongst scientific authors that rogue waves exist with the caveat that wave models could not replicate rogue waves.[15] The 21st century saw the extensive discovery of rogue wave mechanics, with the successful replication of a wave with similar characteristics to the Draupner wave in 2019.[16]

Nature

edit
 
To simulate rogue waves, a Nonlinear Schrödinger's Equation (NLSE) is used, various simplifying assumptions allow different models to be built.

Rogue waves are defined as waves which are greater than twice the size of surrounding waves and contain limited predictable qualities.[17][18][19] The geometric definition of a rogue wave is given as a wave whose crest-to-trough height   exceeds a threshold relative to the significant wave height  . The significant wave height is defined as four times the SD of the sea surface elevation.[20][f] Rogue waves are ubiquitous in nature and do not appear to be effected in an ocean environment by the patterns of prevailing winds or general wave direction. A rogue wave is a natural ocean phenomenon that is not caused by land movement, only lasts briefly, occurs in a limited location, and most often happens far offshore. Besides water waves, they have been recently reported in liquid Helium, in nonlinear optics, and microwave cavities.[21] A rogue wave's spontaneous formation is key feature in its lack of scientific observation. [22] Unlike tsunamis caused by earthquakes, rogue waves are appear to be unpredictable and localized in space and time.[23][24][25] As the Schrödinger equation, governing the wave function of a quantum-mechanical systems, can be applied to waves in the ocean, the solutions can describe “rogue waves” with virtually infinite amplitude.[26][27][28] The journal Physics Letters A, on theoretical and experimental frontier physics, describes this function by stating:

They can appear from smooth initial conditions that are only slightly perturbed in a special way, and are given by our exact solutions. Thus, a slight perturbation on the ocean surface can dramatically increase the amplitude of the singular wave event that appears as a result.

— N. Akhmediev, A. Ankiewicz, M. Taki, Waves that appear from nowhere and disappear without a trace (Physics Letters A), Volume 373, Issue 6

Furthermore, the dynamics of surface gravity waves is described by a set of nonlinear partial differential equations known as the Euler equations for water waves.[29] In order to describe the mechanics of rogue waves, it has become standard to use nonlinear methods.[30] Therefore, Nonlinear Schrödinger (NLS) equations are used a simplified model of the original equations of motion. These nonlinear effects can also be the cause of Peregrine solitons, also known as breathers, which are waves characterized by an isolated high peak that first grows before dissipating. The features of Peregrine solitons, which contain congruent properties with rogue waves, are the basis of hypotheses linking their origins. The Technical University of Berlin successfully produced breather solutions of the NLS and applied the functions of nonlinear equations to examine the effects of rogue waves on offshore structures and ships, and attributed the effects of nonlinear interactions to the origins of rogue waves (See Research methods).[31][32][33]

Rogue waves occur most frequently off the southeast coast of South Africa, where the curvature of the Agulhas current generates steeper swells with shorter wavelengths.[34] Rogue waves have been observed to appear most in areas with strong winds and currents, such as the Roaring Forties and Furious Fifties, and in the North Atlantic and North Pacific Oceans.[35][36] Dr. Bengt Fornberg, who studied the phenomenon with the professor, Marius Gerber, of the University of Stellenbosch, South Africa, theorized a possible link between the nature of rogue wave origins and the formation of Eddy currents:

It’s the interaction of wave swell with the current. Eddies are often generated along the edges of currents, but they can survive for long times and are able to drift across oceans, forming very extensive eddy fields. These eddy fields in fact contain far more kinetic energy than the currents do. Within, and in the immediate vicinity of currents, rogue waves tend to be somewhat predictable—and they are confined to relatively small areas. On the other hand, energy focusing due to the chaotic, irregular and widely distributed eddies is somewhat less likely, and is essentially unpredictable, as these can occur almost everywhere.[37]

— Dr. Bengt Fornberg, National Geographic Education

The dynamics of wind wave behavior can been directly correlated with the nature of rogue waves. When waves form as wind energy is transferred to the ocean's surface, and as conditions generate stronger winds, wave patterns become more organized and begin traveling in one direction.[38] The resulting significant wave height, denotes the characteristic height of the random waves in a sea state, while the occurrence of a rogue wave forms as the mean of the largest third and independent of the mean wave height. The effect of a wave train occurs when waves organize themselves by wavelength, forming a series of waves that pass in a regular pattern.[39] The wave trains will travel several mile across ocean basins before encountering other wave trains as they move.

In statistics, the origin of rogue waves is defined as an unexpected wave that is α times larger than a set of one-sided (preceding) waves or two-sided (preceding and following) waves.[40] Moreover, the definition of unexpectedness, as it relates to the nature of rogue waves, refers to the time interval of apparent calm before or during which a wave is much taller than the neighboring waves. The American Meteorological Society (AMS) describes the origin of rogue waves through the principles of mathematical and statistical techniques, probability forecasts/models/distribution, and various stochastic models. Additionally, research proceedings held L.E. Borgman of the University of Wyoming, presented a general model for the probability distribution function for wave heights in storms with time-varying intensities.[41] Here, it is determined that the assumption that a time interval   during which a stationary sequence of Nw =  /Tm consecutive waves occurs on average, is convenient for the theoretical development of a probabilistic model. Furthermore, Borgman argues, “It seems reasonable to assume that a wave height is at most interdependent with the first several wave heights occurring before and after it and essentially independent with waves further back into the past or forward into the future.” The models presented by the AMS on the conditions of rogue wave events include the applications of the Andrea and Wave Crest Sensor Intercomparison Study (WACSIS), in which the following observation are presented while studying the occurrence of two rogue waves: "Both waves appeared without warning and their crests were nearly 2 times larger than the surrounding O(10) wave crests and thus unexpected."[42][43]

Derivatives and hypotheses

edit

While there is no scientific consensus on the universal cause of rogue waves, numerous natural variables exist that most likely influence their development.[44][45][46] This includes water depth, tidal forces, wind blowing across the water, physical objects such as islands that reflect waves, and interaction with other waves and ocean currents.[47][48][49] The research paper Physica D: Nonlinear Phenomena on Intricate dynamics of rogue waves governed by the Sasa–Satsuma equation describes as "large amplitude waves, localized in both space and time, thus making these events unexpected".[50][51] Hypothesized mechanisms for rogue waves include:

Nonlinear effects

edit

Effects of nonlinear dynamics of solitary waves and wave modulations have been theorized as a possible force behind the mechanics of rogue waves.[52] Sets of field data taken using various tools exhibit that the main generation mechanism for rogue waves is the constructive interference of elementary waves enhanced by second-order bound nonlinearities, possibly explaining that rogue waves are likely to be rare occurrences of weakly nonlinear random seas. In order to study the effects of nonlinearities, the theories proposing the distribution of linear waves are generalized to second-order waves, utilizing quasi-deterministic results on the expected shape of large waves. With the observation of the Draupner wave in 1995, two competing hypotheses of nonlinear focusing, featuring third-order quasi-resonant wave-wave interactions, and purely dispersive focusing of second-order non-resonant or bound harmonic waves, have been proposed to explain the physical mechanisms and occurrence of such waves.[53][54]

The models analyzing these effects include the efficacy of Gram–Charlier models in describing the effects of third-order nonlinearities on the distributions of wave heights. The application of Modulational Instability has been associated with third-order quasi-resonant interactions of rogue waves and described by the Nonlinear Schrödinger (NLS) equation as possible mechanisms.[55][56][57] These effects cause the statistics of weakly nonlinear gravity waves to significantly differ from the Gaussian structure of linear seas.[58][59] The process of modulational instability includes the late-stage occurrence of breathers that lead to the development large waves, in which energy is therefore ‘trapped’ as in a long wave-guide.[60][61]

Originating in the 1990s as a possible solution to the NLS equation for the mechanisms of rogue wave formation, whether second-order or third-order nonlinearities are a dominant factor in the origins of freak waves is a subject of considerable debate.[62][63] Recent theoretical studies show that third-order quasi-resonant interactions are insignificant to the formation of large waves in realistic oceanic seas. As typical oceanic wind seas contain short-crested, or multidirectional wave field features, nonlinear focusing due to modulational effects is diminished since energy can spread directionally.[64] Therefore, the process of modulation instabilities may have an insignificant effect in the development of wave patterns, especially in finite water depth where they are further reduced.[65][66][67]

Diffractive focusing

edit

Diffractive focusing refers to the focusing of light through the division and mutual interference of a propagating electromagnetic wave.[68] In the context of ocean waves, the effects of diffractive focusing on smaller wind and current-driven waves by underwater and coastal topology is often attributed to the development of rogue waves. During this process, coast shape or seabed shape directs several small waves to converge, therefore combining their crests to create a larger wave.[69][70]

Constructive interference

edit

According to this hypothesis, as these swells travel at different speeds and directions and pass through one another, their crests, troughs, and lengths coincide and reinforce each other.[71] This process can form abnormally large waves, and may last for several minutes before subsiding as swells travel in the same direction.

Wave train interaction

edit

Amongst another theory describing the origin of rogue waves includes the focusing of wave energy due to the interaction of opposing wave patterns. As an opposing water current is formed by the development of storm surges, the interaction between the opposite current and the normal wave direction results in a shortening of the wave frequency. This can result in the dynamic convergence of waves, subsequently forming a single larger wave. This effect can reportedly be seen around areas the Gulf Stream and where Agulhas current is countered by the westerlies.[72][73]

Further explanations

edit

Antarctica's choppy seas and wild winds can cause large waves to 'self amplify,' resulting in rogue wave frequency scientists had theorised for years, but could not yet verify in the ocean. Our observations now show that unique sea conditions with rogue waves arise during the 'young' stage of waves - when they are most responsive to wind. This suggests wind parameters are the missing. We show young waves display signs of self-amplifying and more likelihood of becoming rogue because of the wind. We recorded waves twice as high as their neighbours once every six hours.

Physical Review Letters, a study conducted by an international research team led by Professor Alessandro Toffoli of the University of Melbourne, reported direct observations of surface waves along with concurrent measurements of wind speed in the Southern Ocean.[74] While in the austral winter aboard the South African icebreaker S.A. Agulhas II in 2017, measurements of water surface elevation across a range of wave conditions spanning from early stages of wave growth to full development, helped support observations that rogue waves arise from strong wind forces and unpredictable wave shapes. The following studies proposed that the wind creates a chaotic pattern in which waves of different sizes and directions coexist. Furthermore, as wind increases wave length and speed, swells grow disproportionately at the expense of their neighbors in a self-reinforcing mechanism.[75]

Historical origins

edit
See also: List of rogue waves
 
The image of a rogue wave sighted off the western North Atlantic Ocean, was observed by the research vessel, Cape Henlopen, as part of the Atlantic Remote Sensing Land/Ocean Experiment (ARSLOE). The crest of the rogue propagates from the right to left.

For several centuries, reports of abnormally large waves was once the subject of folklore.[76]

Research methods

edit

See also

edit

Notes

edit
  1. ^ The nature behind what causes rogue waves remains unresolved.[2]
  2. ^ Before science could prove the existence of rogue waves, they were regarded as folklore.[4]
  3. ^ The nature behind what causes rogue waves remains unresolved.[8]
  4. ^ Häfner notes the linear superstition analysis on the origins of rogue waves.[12][13]
  5. ^ Modern studies present localized data for the causes of rogue waves.
  6. ^ Häfner uses a rogue wave criterion with a threshold of 2.0:.

References

edit
  1. ^ Onorato, M.; Residori, S.; Bortolozzo, U.; Montina, A.; Arecchi, F. T. (2013-07-10). "Rogue waves and their generating mechanisms in different physical contexts". Physics Reports. 528 (2): 47–89. Bibcode:2013PhR...528...47O. doi:10.1016/j.physrep.2013.03.001. ISSN 0370-1573.
  2. ^ US Department of Commerce, National Oceanic and Atmospheric Administration. "What is a rogue wave?". oceanservice.noaa.gov. Retrieved 2024-09-19.
  3. ^ US Department of Commerce, National Oceanic and Atmospheric Administration. "What is a rogue wave?". oceanservice.noaa.gov. Retrieved 2024-09-19.
  4. ^ US Department of Commerce, National Oceanic and Atmospheric Administration. "What is a rogue wave?". oceanservice.noaa.gov. Retrieved 2024-09-19.
  5. ^ "Wave Dynamics | Fluid Mechanis Lab". fluids.umn.edu. Retrieved 2024-09-19.
  6. ^ "Behaviour of waves". Science Learning Hub. Retrieved 2024-09-19.
  7. ^ NOAA 2024, p. 23.
  8. ^ US Department of Commerce, National Oceanic and Atmospheric Administration. "What is a rogue wave?". oceanservice.noaa.gov. Retrieved 2024-09-19.
  9. ^ Tayfun, M. Aziz; Fedele, Francesco (2007-08-01). "Wave-height distributions and nonlinear effects". Ocean Engineering. 34 (11): 1631–1649. Bibcode:2007OcEng..34.1631T. doi:10.1016/j.oceaneng.2006.11.006. ISSN 0029-8018.
  10. ^ Gemmrich, J.; Garrett, C. (2011-05-18). "Dynamical and statistical explanations of observed occurrence rates of rogue waves". Natural Hazards and Earth System Sciences. 11 (5): 1437–1446. Bibcode:2011NHESS..11.1437G. doi:10.5194/nhess-11-1437-2011. ISSN 1561-8633.
  11. ^ a b Häfner 2023, p. 1.
  12. ^ Tayfun, M. Aziz; Fedele, Francesco (2007-08-01). "Wave-height distributions and nonlinear effects". Ocean Engineering. 34 (11): 1631–1649. Bibcode:2007OcEng..34.1631T. doi:10.1016/j.oceaneng.2006.11.006. ISSN 0029-8018.
  13. ^ Gemmrich, J.; Garrett, C. (2011-05-18). "Dynamical and statistical explanations of observed occurrence rates of rogue waves". Natural Hazards and Earth System Sciences. 11 (5): 1437–1446. Bibcode:2011NHESS..11.1437G. doi:10.5194/nhess-11-1437-2011. ISSN 1561-8633.
  14. ^ Häfner, Dion; Gemmrich, Johannes; Jochum, Markus (2023). "Machine-guided discovery of a real-world rogue wave model". Proceedings of the National Academy of Sciences of the United States of America. 120 (48): e2306275120. arXiv:2311.12579. Bibcode:2023PNAS..12006275H. doi:10.1073/pnas.2306275120. ISSN 0027-8424. PMC 10691345. PMID 37983488.
  15. ^ J. Skourup, Hansen, Andreasen, J. , N.-E. O., K. K. (August 1, 1997). "Non-Gaussian Extreme Waves in the Central North Sea". asmedigitalcollection.asme.org. Retrieved 2024-09-19.{{cite web}}: CS1 maint: multiple names: authors list (link)
  16. ^ McAllister, M. L.; Draycott, S.; Adcock, T. a. A.; Taylor, P. H.; Bremer, T. S. van den (February 2019). "Laboratory recreation of the Draupner wave and the role of breaking in crossing seas". Journal of Fluid Mechanics. 860: 767–786. Bibcode:2019JFM...860..767M. doi:10.1017/jfm.2018.886. ISSN 0022-1120.
  17. ^ "Monsters of the deep". The Economist. ISSN 0013-0613. Retrieved 2024-09-20.
  18. ^ "Weird Science: Rogue Waves | manoa.hawaii.edu/ExploringOurFluidEarth". manoa.hawaii.edu. Retrieved 2024-09-19.
  19. ^ "Rogue Waves". education.nationalgeographic.org. Retrieved 2024-09-19.
  20. ^ Häfner 2023, p. 23.
  21. ^ Kharif, Pelinovsky, Slunyaev, Christian, Efim, Alexey (December 11, 2008). Rogue Waves in the Ocean (in 639-2). Springer Berlin Heidelberg (published 2008). pp. 2–29. ISBN 9783540884194.{{cite book}}: CS1 maint: date and year (link) CS1 maint: multiple names: authors list (link) CS1 maint: unrecognized language (link)
  22. ^ "Enormous 'rogue waves' can appear out of nowhere. Math is revealing their secrets". Premium. 2024-09-20. Retrieved 2024-09-20.
  23. ^ #author.fullName}. "Huge rogue waves rise from nowhere to sink ships. Can we predict them?". New Scientist. Retrieved 2024-09-23. {{cite web}}: |last= has generic name (help)
  24. ^ Service, National Weather. "National Weather Service - Tsunami Hazards". www.tsunami.gov. Retrieved 2024-09-23.
  25. ^ Trillo, Stefano; Chabchoub, Amin (2019-12-18). "A Unifying Framework for Describing Rogue Waves". Physics. 12: 146. doi:10.1103/PhysRevX.9.041057.
  26. ^ Gemmrich, Johannes; Cicon, Leah (2022-02-02). "Generation mechanism and prediction of an observed extreme rogue wave". Scientific Reports. 12 (1): 1718. doi:10.1038/s41598-022-05671-4. ISSN 2045-2322. PMC 8811055. PMID 35110586.
  27. ^ Bludov, Yu. V.; Konotop, V. V.; Akhmediev, N. (2009-09-15). "Matter rogue waves". Physical Review A. 80 (3): 033610. doi:10.1103/PhysRevA.80.033610.
  28. ^ Akhmediev, N.; Ankiewicz, A.; Taki, M. (2009-02-02). "Waves that appear from nowhere and disappear without a trace". Physics Letters A. 373 (6): 675–678. doi:10.1016/j.physleta.2008.12.036. ISSN 0375-9601.
  29. ^ Islam, M.N.; Hafez, M.G. (March 2024). "Propagation of dust acoustic shocks and oscillatory waves in a coupled complex plasma described by nonlinear evolution equations". Partial Differential Equations in Applied Mathematics. 9: 100628. doi:10.1016/j.padiff.2024.100628. ISSN 2666-8181.
  30. ^ Dysthe, Kristian B.; Trulsen, Karsten (1999-01-01). "Note on Breather Type Solutions of the NLS as Models for Freak-Waves". Physica Scripta. 1999 (T82): 48. doi:10.1238/Physica.Topical.082a00048. ISSN 1402-4896.
  31. ^ Clauss, Gunther F. (2002-06-01). "Dramas of the sea: episodic waves and their impact on offshore structures". Applied Ocean Research. 24 (3): 147–161. doi:10.1016/S0141-1187(02)00026-3. ISSN 0141-1187 – via NewScientist.
  32. ^ Kibler, B.; Fatome, J.; Finot, C.; Millot, G.; Dias, F.; Genty, G.; Akhmediev, N.; Dudley, J. M. (October 2010). "The Peregrine soliton in nonlinear fibre optics". Nature Physics. 6 (10): 790–795. doi:10.1038/nphys1740. ISSN 1745-2481.
  33. ^ Onorato, Miguel; Proment, Davide; Clauss, Günther; Klein, Marco (6 February 2013). "Rogue Waves: From Nonlinear Schrödinger Breather Solutions to Sea-Keeping Test". PLOS ONE. 8 (2): e54629. doi:10.1371/journal.pone.0054629. ISSN 1932-6203. PMC 3566097. PMID 23405086.  This article incorporates text from this source, which is available under the CC BY 4.0 license.
  34. ^ Toffoli, Alessandro (2024-04-16). "Africa: Rogue Waves in the Ocean Are Much More Common Than Anyone Suspected, Says New Study". The Conversation Africa. Retrieved 2024-09-19.
  35. ^ "What conditions led to the Draupner freak wave?". ECMWF. 2016-10-24. Retrieved 2024-09-20.
  36. ^ HAL (2023-05-02). "What are rogue waves and how frequent are they?". Global Solo Challenge. Retrieved 2024-09-20.
  37. ^ "Rogue Waves". education.nationalgeographic.org. Retrieved 21 September 2024.
  38. ^ US Department of Commerce, National Oceanic and Atmospheric Administration. "Why does the ocean have waves?". oceanservice.noaa.gov. Retrieved 2024-09-20.
  39. ^ Onorato, Miguel; Cavaleri, Luigi; Randoux, Stephane; Suret, Pierre; Ruiz, Maria Isabel; de Alfonso, Marta; Benetazzo, Alvise (2021-12-08). "Observation of a giant nonlinear wave-packet on the surface of the ocean". Scientific Reports. 11 (1): 23606. doi:10.1038/s41598-021-02875-y. ISSN 2045-2322. PMC 8654958. PMID 34880276.
  40. ^ Fedele, Francesco (2016-05-01). "Are Rogue Waves Really Unexpected?". Journal of Physical Oceanography. 46 (5): 1495–1508. doi:10.1175/JPO-D-15-0137.1. ISSN 0022-3670.
  41. ^ Borgman, L. E. (1970-01-29). "Maximum Wave Height Probabilities for a Random Number of Random Intensity Storms". Coastal Engineering Proceedings (12): 4. doi:10.9753/icce.v12.4. ISSN 2156-1028.
  42. ^ Magnusson, Anne (2003-01-01). "Extreme Wave Statistics from Time-Series Data". {{cite journal}}: Cite journal requires |journal= (help)
  43. ^ Forristall, George Z.; Barstow, Stephen F.; Krogstad, Harald E.; Prevosto, Marc; Taylor, Paul H.; Tromans, Peter S. (2004). "Wave Crest Sensor Intercomparison Study: An Overview of WACSIS". Journal of Offshore Mechanics and Arctic Engineering. 126 (1): 26–34. doi:10.1115/1.1641388.
  44. ^ "Are rogue waves predictable?". ScienceDaily. Retrieved 2024-09-23.
  45. ^ Newkey-Burden, Chas; published, The Week UK (2024-05-23). "What are rogue waves and what causes them?". theweek. Retrieved 2024-09-21.
  46. ^ Olagnon, Michel (June 15, 2017). Rogue Waves: Anatomy of a Monster (in 639-2). Bloomsbury Publishing (published 2017). p. 19. ISBN 9781472944429.{{cite book}}: CS1 maint: unrecognized language (link)
  47. ^ Didenkulova, Ekaterina (2020-04-15). "Catalogue of rogue waves occurred in the World Ocean from 2011 to 2018 reported by mass media sources". Ocean & Coastal Management. 188: 105076. doi:10.1016/j.ocecoaman.2019.105076. ISSN 0964-5691.
  48. ^ "How Rogue Waves Work". HowStuffWorks. 1970-01-01. Retrieved 2024-09-20.
  49. ^ Fedele, Francesco; Brennan, Joseph; Ponce de León, Sonia; Dudley, John; Dias, Frédéric (2016-06-21). "Real world ocean rogue waves explained without the modulational instability". Scientific Reports. 6 (1): 27715. Bibcode:2016NatSR...627715F. doi:10.1038/srep27715. ISSN 2045-2322. PMC 4914928. PMID 27323897.
  50. ^ Mu, Gui; Qin, Zhenyun; Grimshaw, Roger; Akhmediev, Nail (2020-01-15). "Intricate dynamics of rogue waves governed by the Sasa–Satsuma equation". Physica D: Nonlinear Phenomena. 402: 132252. doi:10.1016/j.physd.2019.132252. ISSN 0167-2789.
  51. ^ Trillo, Stefano; Chabchoub, Amin (2019-12-18). "A Unifying Framework for Describing Rogue Waves". Physics. 12: 146. doi:10.1103/PhysRevX.9.041057.
  52. ^ Slunyaev, A. V.; Kokorina, A. V.; Pelinovsky, E. N. (2023-12-01). "Nonlinear waves, modulations and rogue waves in the modular Korteweg–de Vries equation". Communications in Nonlinear Science and Numerical Simulation. 127: 107527. arXiv:2304.07104. doi:10.1016/j.cnsns.2023.107527. ISSN 1007-5704.
  53. ^ Bitner-Gregersen, E. M.; Fernandez, L.; Lefèvre, J. M.; Monbaliu, J.; Toffoli, A. (2014-06-05). "The North Sea Andrea storm and numerical simulations". Natural Hazards and Earth System Sciences. 14 (6): 1407–1415. doi:10.5194/nhess-14-1407-2014. ISSN 1561-8633.
  54. ^ Dysthe, Kristian; Krogstad, Harald E.; Müller, Peter (2008-01-01). "Oceanic Rogue Waves". Annual Review of Fluid Mechanics. 40 (1): 287–310. doi:10.1146/annurev.fluid.40.111406.102203. ISSN 0066-4189.
  55. ^ Crawford, Donald R.; Lake, Bruce M.; Saffman, Philip G.; Yuen, Henry C. (April 1981). "Stability of weakly nonlinear deep-water waves in two and three dimensions". Journal of Fluid Mechanics. 105: 177–191. doi:10.1017/S0022112081003169. ISSN 1469-7645.
  56. ^ "The effects of randomness on the stability of two-dimensional surface wavetrains". Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 363 (1715): 525–546. 1978-11-27. doi:10.1098/rspa.1978.0181. ISSN 0080-4630.
  57. ^ Fedele, Francesco; Brennan, Joseph; León, Sonia Ponce de; Dudley, John; Dias, Frédéric (2016). "Real world ocean rogue waves explained without the modulational instability". Scientific Reports. 6: 27715. doi:10.1038/srep27715. PMC 4914928. PMID 27323897.
  58. ^ Peregrine, D. H. (July 1983). "Water waves, nonlinear Schrödinger equations and their solutions". The ANZIAM Journal. 25 (1): 16–43. doi:10.1017/S0334270000003891. ISSN 1839-4078.
  59. ^ Janssen, Peter A. E. M. (2003-04-01). "Nonlinear Four-Wave Interactions and Freak Waves". Journal of Physical Oceanography. 33 (4): 863–884. doi:10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2. ISSN 0022-3670.
  60. ^ Bristol, University of. "2010: Peregrine 'soliton' | News and features | University of Bristol". www.bristol.ac.uk. Retrieved 2024-09-26.
  61. ^ Ankiewicz, Adrian; Devine, N.; Akhmediev, Nail (2009-10-19). "Are rogue waves robust against perturbations?". Physics Letters A. 373 (43): 3997–4000. doi:10.1016/j.physleta.2009.08.053. ISSN 0375-9601.
  62. ^ Tayfun, M. Aziz (2008-12-01). "Distributions of Envelope and Phase in Wind Waves". Journal of Physical Oceanography. 38 (12): 2784–2800. doi:10.1175/2008JPO4008.1. ISSN 0022-3670.
  63. ^ Fedele, Francesco (2008-08-15). "Rogue waves in oceanic turbulence". Physica D: Nonlinear Phenomena. Euler Equations: 250 Years On. 237 (14): 2127–2131. doi:10.1016/j.physd.2008.01.022. ISSN 0167-2789.
  64. ^ Onorato, M.; Cavaleri, L.; Fouques, S.; Gramstad, O.; Janssen, P. a. E. M.; Monbaliu, J.; Osborne, A. R.; Pakozdi, C.; Serio, M.; Stansberg, C. T.; Toffoli, A.; Trulsen, K. (May 2009). "Statistical properties of mechanically generated surface gravity waves: a laboratory experiment in a three-dimensional wave basin". Journal of Fluid Mechanics. 627: 235–257. doi:10.1017/S002211200900603X. hdl:2318/45746. ISSN 1469-7645.
  65. ^ "BBC - Science & Nature - Horizon - Freak Wave". www.bbc.co.uk. Retrieved 2024-09-26.
  66. ^ "Math explains water disasters - ScienceAlert". 2016-04-24. Archived from the original on 2016-04-24. Retrieved 2024-09-26.
  67. ^ Toffoli, A.; Benoit, M.; Onorato, M.; Bitner-Gregersen, E. M. (2009-02-24). "The effect of third-order nonlinearity on statistical properties of random directional waves in finite depth". Nonlinear Processes in Geophysics. 16 (1): 131–139. doi:10.5194/npg-16-131-2009. ISSN 1023-5809.
  68. ^ "Diffractive Optics - an overview | ScienceDirect Topics". www.sciencedirect.com. Retrieved 2024-09-23.
  69. ^ "Diffraction and Current Focusing". ffden-2.phys.uaf.edu. Retrieved 2024-09-23.
  70. ^ "Ocean Prediction Center--Rogue Waves". 2010-05-28. Archived from the original on 2010-05-28. Retrieved 2024-09-23.
  71. ^ "Physics Tutorial: Interference of Waves". www.physicsclassroom.com. Retrieved 2024-09-23.
  72. ^ Korde, Umesh A.; Ertekin, R. Cengiz (2014-02-01). "On wave energy focusing and conversion in open water". Renewable Energy. 62: 84–99. doi:10.1016/j.renene.2013.06.033. ISSN 0960-1481.
  73. ^ "A short history of wave energy". Retrieved 2024-09-26.
  74. ^ Toffoli, A.; Alberello, A.; Clarke, H.; Nelli, F.; Benetazzo, A.; Bergamasco, F.; Ntamba, B. Ntamba; Vichi, M.; Onorato, M. (2024-04-12). "Observations of Rogue Seas in the Southern Ocean". Physical Review Letters. 132 (15): 154101. doi:10.1103/PhysRevLett.132.154101.
  75. ^ Schirber, Michael (2017-04-07). "Making Rogue Waves with Wind and Water". Physics. 10: 37. doi:10.1103/PhysRevLett.118.144503.
  76. ^ "Weird Science: Rogue Waves | manoa.hawaii.edu/ExploringOurFluidEarth". manoa.hawaii.edu. Retrieved 2024-09-27.

Further reading

edit

Books

edit

Other

edit
edit
Retrieved from "https://en.wikipedia.org/w/index.php?title=Draft:Origin_of_rogue_waves&oldid=1248121111"