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August  2014, 34(8): 3061-3093. doi: 10.3934/dcds.2014.34.3061

On the nature of large and rogue waves

Mikhail Kovalyov 1,

1. 

Department of Mathematics, Sungkyunkwan University, Suwon, Gyeonggi-do, South Korea

Received  October 2012 Revised  October 2013 Published  January 2014

In this paper we discuss a model of large and rogue waves in non-necessarily shallow water. We assume that the relevant portion of the flow is restricted to a near-surface layer, assumption which enables us to use the Kadomtsev-Petviashvili equation. The shape and behavior of several types of waves predicted by some singular solutions of the Kadomtsev-Petviashvili equation is compared to the physical waves observed in the ocean.
Keywords: large wave, Kadomtsev-Petviashvili equation., Rogue waves, freak waves.
Mathematics Subject Classification: 76B15, 76B25, 35Q35, 35Q5.
Citation: Mikhail Kovalyov. On the nature of large and rogue waves. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3061-3093. doi: 10.3934/dcds.2014.34.3061
References:
[1]

S. Ackerman and M. Coslovich, 2011 Video, in Riptide, November/December, 184, p. 49. Also posted on December 15, 2011, 13:49 at http://www.riptidemag.com.au/. Google Scholar

[2]

A. Grawin, 2004, , ().   Google Scholar

[3]

G. Biondini, K. Maruno, M. Oikawa and H. Tsuji, Soliton interactions of the Kadomtsev-Petviashvili equation and generation of large-amplitude waves, Stud. Appl. Math., 122 (2009), 377-394. doi: 10.1111/j.1467-9590.2009.00439.x.  Google Scholar

[4]

H. Chanson, Gallery of photographs, (2010), Also availble at http://staff.civil.uq.edu.au/h.chanson/photo.html#Tidal_bores. Google Scholar

[5]

H. Chanson, D. Reungoat, B. Simon and P. Lubin, High-frequency turbulence and suspended sediment concentration measurements in the Garonne River tidal bore, in Estuarine, Coastal and Shelf Science, 95 2011, 298-306. Academic Press, Also available at http://espace.library.uq.edu.au/view/UQ:261649. Google Scholar

[6]

A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynamics Research, 40 (2008), 175-211. doi: 10.1016/j.fluiddyn.2007.06.004.  Google Scholar

[7]

A. Constantin, The trajectories of particles in Stokes waves, Inventiones mathematicae, 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[8]

A. Constantin and J. Escher, Particle trajectories in shallow water waves, Bulletin of the American Mathematical Society, 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[9]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Communications in Pure and Applied Mathematics, 63 (2010), 533-557. doi: 10.1002/cpa.20299.  Google Scholar

[10]

B. V. Divinsky, B. V. Levin, L. I. Lopatikhin, E. N. Pelinovsky and A. V. Slyungaev, A freak wave in the Black sea: Observations and simulation, Doklady, Earth Science, 395 (2004), 438-443. Google Scholar

[11]

W. Dudley and M. Lee, Tsunami, 2nd edition, page 97, University of Hawaii Press, 1998. Google Scholar

[12]

M. D. Earle, Extreme wave conditions during Hurricane Camille, Journal of Geophysical Research, 80 (1975), 377-379. Google Scholar

[13]

K. Freeze, Monster waves threaten rescue helicopters,, in Proceedings of US Naval Institute, (): 246.   Google Scholar

[14]

D. M. Graham, NOAA vessel swamped by rogue wave, 284 (2000). The quote may also be found at http://hal.archives-ouvertes.fr/docs/00/00/03/52/PDF/Rogue_wave_V1.pdf, bottom of page 4. Google Scholar

[15]

Y. S. Gutshabash and I. V. Lavrenov, Swell transformation in the Cape Agulhas current, Izvestiya, Atmospheric and Oceanic Physics, 22 (1986), 494-497. Google Scholar

[16]

D. E. Irvine and D. G. Tilley, Ocean wave directional spectra and wave-current interaction in the Agulhas from the shuttle imaging radar-B synthetic aperture radar, Journal of Geophysical Research, 93 (1988), 15389-15401. Google Scholar

[17]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Univ. Press, Cambridge, 1988. doi: 10.1017/CBO9780511624056.  Google Scholar

[18]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl, 15 (1970), 539-541. Google Scholar

[19]

M. Kovalyov and I. Bica, Some properties of slowly decaying oscillatory solutions of KP, Chaos, Solitons and Fractals, 25 (2005), 1979-1989. doi: 10.1016/j.chaos.2004.11.054.  Google Scholar

[20]

K. Mallory, Abnormal waves on the south-east of South Africa, Inst. Hydrog. Rev., 51 (1974), 89-129. Google Scholar

[21]

M. Maxted, Photo, in Riptide, 176 2010, 88-89. Also posted on June 15, 2010, 13:43 at http://www.riptidemag.com.au/. Google Scholar

[22]

N. Mori, P. C. Liun and T. Yasuda, Analysis of freak wave measurements in the sea of Japan, Ocean Engineering, 29 (2002), 1399-1414. Google Scholar

[23]

D. Mouaze, H. Chanson and B. Simon, Field Measurements in the tidal bore of the Selune River in the Bay of Mont Saint Michel (September 2010), in Hydraulic Model Report No. CH81/10, 2010, pp. 246-247. School of Civil Engineering, The University of Queensland, Brisbane, Australia. Google Scholar

[24]

P. Peterson, T. Soomere, J. Engelbreight and E. van Groesen, Soliton interaction as a possible model for extreme waves in shallow water, Nonlin. Proc. Geophys, 10 (2003), 503-510. Google Scholar

[25]

C. Rip, Tip 2 Tip - Seven Ghosts,, 2011, ().   Google Scholar

[26]

Riptide, Riptide Internet journal,, 2010, ().   Google Scholar

[27]

J. H. Simpson, N. R. Fisher and P. Wiles, Reynolds stress and TKE production in an estuary with a tidal bore, Estuarine, Coastal and Shelf Science, 60 (2004), 619-627. Google Scholar

[28]

Videos, http://www.youtube.com/watch?v=AC2UXYK65Tc&feature=related,, 2010, ().  http://www.youtube.com/watch?v=AC2UXYK65Tc&feature=related,+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[29]

E. Wolanski, D. Williams, S. Spagnol and H. Chanson, Undular tidal bore dynamics in the daly estuary, Estuarine, Coastal and Shelf Science, 60 (2004), 629-636. doi: 10.1016/j.ecss.2004.03.001.  Google Scholar

show all references

References:
[1]

S. Ackerman and M. Coslovich, 2011 Video, in Riptide, November/December, 184, p. 49. Also posted on December 15, 2011, 13:49 at http://www.riptidemag.com.au/. Google Scholar

[2]

A. Grawin, 2004, , ().   Google Scholar

[3]

G. Biondini, K. Maruno, M. Oikawa and H. Tsuji, Soliton interactions of the Kadomtsev-Petviashvili equation and generation of large-amplitude waves, Stud. Appl. Math., 122 (2009), 377-394. doi: 10.1111/j.1467-9590.2009.00439.x.  Google Scholar

[4]

H. Chanson, Gallery of photographs, (2010), Also availble at http://staff.civil.uq.edu.au/h.chanson/photo.html#Tidal_bores. Google Scholar

[5]

H. Chanson, D. Reungoat, B. Simon and P. Lubin, High-frequency turbulence and suspended sediment concentration measurements in the Garonne River tidal bore, in Estuarine, Coastal and Shelf Science, 95 2011, 298-306. Academic Press, Also available at http://espace.library.uq.edu.au/view/UQ:261649. Google Scholar

[6]

A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynamics Research, 40 (2008), 175-211. doi: 10.1016/j.fluiddyn.2007.06.004.  Google Scholar

[7]

A. Constantin, The trajectories of particles in Stokes waves, Inventiones mathematicae, 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[8]

A. Constantin and J. Escher, Particle trajectories in shallow water waves, Bulletin of the American Mathematical Society, 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[9]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Communications in Pure and Applied Mathematics, 63 (2010), 533-557. doi: 10.1002/cpa.20299.  Google Scholar

[10]

B. V. Divinsky, B. V. Levin, L. I. Lopatikhin, E. N. Pelinovsky and A. V. Slyungaev, A freak wave in the Black sea: Observations and simulation, Doklady, Earth Science, 395 (2004), 438-443. Google Scholar

[11]

W. Dudley and M. Lee, Tsunami, 2nd edition, page 97, University of Hawaii Press, 1998. Google Scholar

[12]

M. D. Earle, Extreme wave conditions during Hurricane Camille, Journal of Geophysical Research, 80 (1975), 377-379. Google Scholar

[13]

K. Freeze, Monster waves threaten rescue helicopters,, in Proceedings of US Naval Institute, (): 246.   Google Scholar

[14]

D. M. Graham, NOAA vessel swamped by rogue wave, 284 (2000). The quote may also be found at http://hal.archives-ouvertes.fr/docs/00/00/03/52/PDF/Rogue_wave_V1.pdf, bottom of page 4. Google Scholar

[15]

Y. S. Gutshabash and I. V. Lavrenov, Swell transformation in the Cape Agulhas current, Izvestiya, Atmospheric and Oceanic Physics, 22 (1986), 494-497. Google Scholar

[16]

D. E. Irvine and D. G. Tilley, Ocean wave directional spectra and wave-current interaction in the Agulhas from the shuttle imaging radar-B synthetic aperture radar, Journal of Geophysical Research, 93 (1988), 15389-15401. Google Scholar

[17]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Univ. Press, Cambridge, 1988. doi: 10.1017/CBO9780511624056.  Google Scholar

[18]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl, 15 (1970), 539-541. Google Scholar

[19]

M. Kovalyov and I. Bica, Some properties of slowly decaying oscillatory solutions of KP, Chaos, Solitons and Fractals, 25 (2005), 1979-1989. doi: 10.1016/j.chaos.2004.11.054.  Google Scholar

[20]

K. Mallory, Abnormal waves on the south-east of South Africa, Inst. Hydrog. Rev., 51 (1974), 89-129. Google Scholar

[21]

M. Maxted, Photo, in Riptide, 176 2010, 88-89. Also posted on June 15, 2010, 13:43 at http://www.riptidemag.com.au/. Google Scholar

[22]

N. Mori, P. C. Liun and T. Yasuda, Analysis of freak wave measurements in the sea of Japan, Ocean Engineering, 29 (2002), 1399-1414. Google Scholar

[23]

D. Mouaze, H. Chanson and B. Simon, Field Measurements in the tidal bore of the Selune River in the Bay of Mont Saint Michel (September 2010), in Hydraulic Model Report No. CH81/10, 2010, pp. 246-247. School of Civil Engineering, The University of Queensland, Brisbane, Australia. Google Scholar

[24]

P. Peterson, T. Soomere, J. Engelbreight and E. van Groesen, Soliton interaction as a possible model for extreme waves in shallow water, Nonlin. Proc. Geophys, 10 (2003), 503-510. Google Scholar

[25]

C. Rip, Tip 2 Tip - Seven Ghosts,, 2011, ().   Google Scholar

[26]

Riptide, Riptide Internet journal,, 2010, ().   Google Scholar

[27]

J. H. Simpson, N. R. Fisher and P. Wiles, Reynolds stress and TKE production in an estuary with a tidal bore, Estuarine, Coastal and Shelf Science, 60 (2004), 619-627. Google Scholar

[28]

Videos, http://www.youtube.com/watch?v=AC2UXYK65Tc&feature=related,, 2010, ().  http://www.youtube.com/watch?v=AC2UXYK65Tc&feature=related,+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[29]

E. Wolanski, D. Williams, S. Spagnol and H. Chanson, Undular tidal bore dynamics in the daly estuary, Estuarine, Coastal and Shelf Science, 60 (2004), 629-636. doi: 10.1016/j.ecss.2004.03.001.  Google Scholar

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