Abstract
Two freak waves were observed a day apart in October 2009 at a 5000-m deep moored station in the northwest Pacific Ocean. As the typhoon passed by, the wave system transitioned within a day from a narrow and unimodal spectrum to the broad and bi-modal spectrum. The occurrence probability of a freak wave is known to increase due to a modulational instability; however, whether the modulational instability survives under a realistic directional sea state has not been conclusively determined yet. In this study, a phase-resolving wave model was used to obtain ensembles of realizations based on observed and simulated directional spectra. Unlike previous studies that focused only on the probability of freak wave occurrence, this study focuses on wave shape. It reveals that the front-to-rear asymmetry and crescent shape deformation of the crest are more pronounced for narrower spectrum and longer-lifetime freak waves; this distortion of wave shape and extended lifetime are both characteristics of nonlinear wave groups. This study also shows that the distribution of the lifetime of a freak wave depends on the sea state and that the number of nonlinear wave groups increases for a narrower spectrum. We therefore conjecture that both the four-wave quasi-resonance and dispersive focusing are responsible for freak wave generation, but their relative significance depends on the spectral broadness. Investigating the total kurtosis or occurrence probability alone is insufficient to unravel the underlying mechanisms of individual freak-wave generation.
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References
Adcock TAA, Taylor PH (2016) Fast and local non-linear evolution of steep wave-groups on deep water: a comparison of approximate models to fully non-linear simulations. Phys Fluids 28:016601. https://doi.org/10.1063/1.4938144
Ardhuin F, Rogers WE, Babanin AV, Filipot JF, Magne R, Roland A et al (2010) Semiempirical dissipation source functions for ocean waves. Part I: definition, calibration, and validation. J Phys Oceanogr 40:1917–1941. https://doi.org/10.1175/2010JPO4324.1
Bitner-Gregersen EM, Gramstad O (2018) Impact of sampling variability on sea surface characteristics of nonlinear waves. In: ASME 2018 37th International Conference on Ocean, Offshore and Arctic Engineering. Madrid, Spain
Bitner-Gregersen EM, Fernandez L, Lefèvre J-M, Monbaliu J, Toffoli A (2014) The North Sea Andrea storm and numerical simulations. Nat Hazards Earth Syst Sci 14:1407–1415. https://doi.org/10.5194/nhess-14-1407-2014
Boccotti P (1983) Some new results on statistical properties of wind waves. Appl Ocean Res 5:134–140. https://doi.org/10.1016/0141-1187(83)90067-6
Clamond D, Francius M, Grue J, Kharif C (2006) Long time interaction of envelope solitons and freak wave formations. Eur J Mech B/Fluids 25:536–553. https://doi.org/10.1016/j.euromechflu.2006.02.007
Dommermuth DG (2000) The initialization of nonlinear waves using an adjustment scheme. Wave Motion 32:307–317. https://doi.org/10.1016/S0165-2125(00)00047-0
Dommermuth DG, Yue DKP (1987) A high-order spectral method for the study of nonlinear gravity waves. J Fluid Mech 184:267–288
Dysthe KB (1979) Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc R Soc A Math Phys Eng Sci 369:105–114. https://doi.org/10.1098/rspa.1979.0154
ECMWF (2006) IFS documentation CY36r4. Part VII: ECMWF wave model. ECMWF model documentation, technical report
Fedele F (2015) On the kurtosis of deep-water gravity waves. J Fluid Mech 782:25–36. https://doi.org/10.1017/jfm.2015.538
Fedele F, Tayfun MA (2009) On nonlinear wave groups and crest statistics. J Fluid Mech 620:221–239. https://doi.org/10.1017/S0022112008004424
Fedele F, Brennan J, Ponce de León S, Dudley J, Dias F (2016) Real world ocean rogue waves explained without the modulational instability. Sci Rep 6:1–11. https://doi.org/10.1038/srep27715
Flanagan JD, Dias F, Terray E, et al (2016) Extreme water waves off the west coast of Ireland: analysis of ADCP measurements. In: The 26th International Ocean and Polar Engineering Conference, 26 June–2 July, Rhodes, Greece. International Society of Offshore and Polar Engineers, p 589
Fonseca N, Guedes Soares C, Pascoal R (2006) Structural loads induced in a containership by abnormal wave conditions. J Mar Sci Technol 11:245–259. https://doi.org/10.1007/s00773-006-0222-9
Fujimoto W, Waseda T (2016) The relationship between the shape of freak waves and nonlinear wave interactions. In: ASME 2016 35th International Conference on Ocean, Offshore and Arctic Engineering. ASME, Busan, South Korea, p V003T02A003
Gibson R, Swan C (2007) The evolution of large ocean waves: the role of local and rapid spectral changes. Proc R Soc A Math Phys Eng Sci 463:21–48. https://doi.org/10.1098/rspa.2006.1729
Gramstad O, Trulsen K (2007) Influence of crest and group length on the occurrence of freak waves. J Fluid Mech 582:463–472
Guedes Soares C, Cherneva Z, Antao EM, Antão E (2003) Characteristics of abnormal waves in North Sea storm sea states. Appl Ocean Res 25:337–344. https://doi.org/10.1016/j.apor.2004.02.005
Guedes Soares C, Cherneva Z, Antao EM (2004) Steepness and asymmetry of the largest waves in storm sea states. Ocean Eng 31:1147–1167. https://doi.org/10.1016/j.oceaneng.2003.10.014
Guedes Soares C, Pascoal R, Antão EM, Voogt AJ, Buchner B (2007) An approach to calculate the probability of wave impact on an FPSO bow. J Offshore Mech Arct Eng 129:73. https://doi.org/10.1115/1.2426983
Guedes Soares C, Fonseca N, Pascoal R (2008) Abnormal wave-induced load effects in ship structures. J Ship Res 52:30–44
Hashimoto N (1997) Analysis of the directional wave spectrum from field data. In: Advances in Coastal and Ocean Engineering. Vol. 3. World Scientific, pp 103–143
Hasselmann S, Hasselmann K (1985) Computations and parameterizations of the nonlinear energy transfer in a gravity-wave spectrum. Part I: a new method for efficient computations of the exact nonlinear transfer. Integral. J Phys Oceanogr 15:1369–1377
Haver S (2004) A possible freak wave event measured at the Draupner jacket January 1 1995. Rogue waves, Proc. Rogue Waves 20–22 October. Brest: IFREMER
Houtani H, Tanizawa K, Waseda T, Sawada H (2016) An experimental investigation on the influence of the temporal variation of freak wave geometry on the elastic response of a container ship. In: Proceedings of 3rd International Conference on Violent Flows. Osaka, pp 9–11
Janssen PAEM (2003) Nonlinear four-wave interactions and freak waves. J Phys Oceanogr 33:863–884. https://doi.org/10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2
Janssen PAEM (2004) The interaction of ocean waves and wind. Cambridge University Press
Janssen PAEM (2009) On some consequences of the canonical transformation in the Hamiltonian theory of water waves. J Fluid Mech 637:1. https://doi.org/10.1017/S0022112009008131
Janssen PAEM, Onorato M (2007) The intermediate water depth limit of the Zakharov equation and consequences for wave prediction. J Phys Oceanogr 37:2389–2400. https://doi.org/10.1175/JPO3128.1
Jensen JJ (1996) Second-order wave kinematics conditional on a given wave crest. Appl Ocean Res 18:119–128. https://doi.org/10.1016/0141-1187(96)00008-9
Jensen JJ (2005) Conditional second-order short-crested water waves applied to extreme wave episodes. J Fluid Mech 545:29. https://doi.org/10.1017/S0022112005006841
Kitamoto A (NII) (2016) Digital typhoon: typhoon images and information. http://agora.ex.nii.ac.jp/digital-typhoon/index.html.en. Accessed 31 Aug 2016
Krasitskii VP (1994) On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J Fluid Mech 272:1–20. https://doi.org/10.1017/S0022112094004350
Lindgren G (1970) Some properties of a normal process near a local maximum. Ann Math Stat 41:1870–1883
Lo EY, Mei CC (1984) A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation. J Fluid Mech 150:395. https://doi.org/10.1017/S0022112085000180
Lo EY, Mei CC (1987) Slow evolution of nonlinear deep water waves in two horizontal directions: a numerical study. Wave Motion 9:245–259. https://doi.org/10.1016/0165-2125(87)90014-X
Magnusson AK, Donelan MA (2013) The Andrea wave characteristics of a measured North Sea rogue wave. J Offshore Mech Arct Eng 135:31108. https://doi.org/10.1115/1.4023800
Miyazawa Y, Zhang R, Guo X, Tamura H, Ambe D, Lee JS, Okuno A, Yoshinari H, Setou T, Komatsu K (2009) Water mass variability in the western North Pacific detected in a 15-year eddy resolving ocean reanalysis. J Oceanogr 65:737–756. https://doi.org/10.1007/s10872-009-0063-3
Mori N, Janssen PAEM (2006) On kurtosis and occurrence probability of freak waves. J Phys Oceanogr 36:1471–1483. https://doi.org/10.1175/JPO2922.1
Mori N, Yasuda T (2002) Effects of high-order nonlinear interactions on unidirectional wave trains. Ocean Eng 29:1233–1245. https://doi.org/10.1016/S0029-8018(01)00074-9
Myrhaug D, Kjeldsen SP (1986) Steepness and asymmetry of extreme waves and the highest waves in deep water. Ocean Eng 13:549–568. https://doi.org/10.1016/0029-8018(86)90039-9
Onorato M, Osborne AR, Serio M (2002) Extreme wave events in directional, random oceanic sea states. Phys Fluids 14:L25–L28. https://doi.org/10.1063/1.1453466
Onorato M, Osborne AR, Serio M (2007) On the relation between two numerical methods for the computation of random surface gravity waves. Eur J Mech B/Fluids 26:43–48. https://doi.org/10.1016/j.euromechflu.2006.05.001
Osborne AR, Ponce de León S (2017) Properties of rogue waves and the shape of the ocean wave power spectrum. In: ASME 2017 36th International Conference on Ocean, Offshore and Arctic Engineering. ASME, Trondheim, Norway, p V03AT02A013
Ponce de León S, Osborne AR, Guedes Soares C (2018) On the importance of the exact nonlinear interactions in the spectral characterization of rogue waves. In: ASME 2018 37th International Conference on Ocean, Offshore and Arctic Engineering. Madrid, Spain
Saha S, Moorthi S, Pan H-L, Wu X, Wang J, Nadiga S et al (2010) The NCEP climate forecast system reanalysis. Bull Am Meteorol Soc 91:1015–1057. https://doi.org/10.1175/2010BAMS3001.1
Socquet-Juglard H, Dysthe KB, Trulsen K, Krogstad HE, Liu J (2005) Probability distributions of surface gravity waves during spectral changes. J Fluid Mech 542:195. https://doi.org/10.1017/S0022112005006312
Tanaka M (2001) A method of studying nonlinear random field of surface gravity waves by direct numerical simulation. Fluid Dyn Res 28:41–60. https://doi.org/10.1016/S0169-5983(00)00011-3
Tanaka M, Yokoyama N (2004) Effects of discretization of the spectrum in water-wave turbulence. Fluid Dyn Res 34:199–216. https://doi.org/10.1016/J.FLUIDDYN.2003.12.001
Tian Z, Perlin M, Choi W (2010) Energy dissipation in two-dimensional unsteady plunging breakers and an eddy viscosity model. J Fluid Mech 655:217–257. https://doi.org/10.1017/S0022112010000832
Tobisch E (2014) What can go wrong when applying wave turbulence theory. arXiv
Toffoli A, Gramstad O, Trulsen K et al (2010) Evolution of weakly nonlinear random directional waves: laboratory experiments and numerical simulations. J Fluid Mech 664:313–336. https://doi.org/10.1017/S002211201000385X
Tolman HL (2014) User manual and system documentation of WAVEWATCH III version 4.18. NOAA/NWS/NCEP/MMAB Tech Note. doi: https://doi.org/10.3390/ijerph2006030011
Tolman HL, Chalikov D (1996) Source terms in a third-generation wind wave model. J Phys Oceanogr 26:2497–2518. https://doi.org/10.1175/1520-0485(1996)026<2497:STIATG>2.0.CO;2
Trulsen K, Nieto Borge JC, Gramstad O, Aouf L, Lefèvre JM (2015) Crossing sea state and rogue wave probability during the Prestige accident. J Geophys Res Ocean 120:7113–7136. https://doi.org/10.1002/2015JC011161
Waseda T, Kinoshita T, Tamura H (2009) Evolution of a random directional wave and freak wave occurrence. J Phys Oceanogr 39:621–639. https://doi.org/10.1175/2008JPO4031.1
Waseda T, Shinchi M, Nishida T, Tamura H, Miyazawa Y, Kawai Y, … & Taniguchi K (2011) GPS-based wave observation using a moored oceanographic buoy in the deep ocean. In: Proc. Twenty-first Int. Offshore Polar Eng. Conf.
Waseda T, Tamura H, Kinoshita T (2012) Freakish sea index and sea states during ship accidents. J Mar Sci Technol 17:305–314. https://doi.org/10.1007/s00773-012-0171-4
Waseda T, Sinchi M, Kiyomatsu K, Nishida T, Takahashi S, Asaumi S, … & Miyazawa Y (2014) Deep water observations of extreme waves with moored and free GPS buoys. Ocean Dyn 1269–1280. doi: https://doi.org/10.1007/s10236-014-0751-4
Waseda T, Webb A, Kiyomatsu K et al (2016) Marine energy resource assessment at reconnaissance to feasibility study stages; wave power, ocean and tidal current power, and ocean temperature power. J Japan Soc Nav Archit Ocean Eng 23:189–198
Webb A, Waseda T, Fujimoto W, et al (2016) A high-resolution, wave and current resource assessment of Japan: the Web GIS dataset. In: Proceedings of Asian Wave and Tidal Energy Conference
West BJ, Brueckner KA, Janda RS, Milder DM, Milton RL (1987) A new numerical method for surface hydrodynamics. J Geophys Res 92:11803–11824
Xiao W, Liu Y, Wu G, Yue DKP (2013) Rogue wave occurrence and dynamics by direct simulations of nonlinear wave-field evolution. J Fluid Mech 720:357–392. https://doi.org/10.1017/jfm.2013.37
Young IR, Hasselmann S, Hasselmann K (1987) Computations of the response of a wave spectrum to a sudden change in wind direction. J Phys Oceanogr 17:1317–1338
Yuen HC, Lake BM (1982) Nonlinear dynamics of deep-water gravity waves. In: Advances in applied mechanics. Elsevier
Zakharov VE (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. J Appl Mech Tech Phys 9:190–194
Acknowledgements
Alessandro Toffoli (University of Melbourne) and Miguel Onorato (University of Torino) provided the original HOSM model code that was modified and used. Peter Janssen (ECMWF) and Miguel Onorato gave us valuable comments with regard to the canonical transformation of the Zakharov equation and the relationship between the HOSM and the Zakharov equation. We are also grateful to the reviewers for their suggestions.
Funding
W.F. acknowledges the support from the Fundamental Research Developing Association for Shipbuilding and Offshore (REDAS) in Japan. This research was funded by the Japan Society for the Promotion of Science (JSPS) and Grants-in-Aid for Scientific Research (KAKENHI).
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Responsible Editor: Jose-Henrique Alves
This article is part of the Topical Collection on the 15th International Workshop on Wave Hindcasting and Forecasting in Liverpool, UK, September 10–15, 2017
Appendix: relationship between HOSM and Zakharov equation
Appendix: relationship between HOSM and Zakharov equation
Onorato et al. (2007) illustrated that the HOSM with M = 2 and M = 3 is equivalent to the second and third-order Hamiltonian dynamical equations, respectively, for \( \widehat{\eta}\left(\boldsymbol{k}\right) \) and \( \widehat{\varPhi}\left(\boldsymbol{k}\right) \). Likewise, as illustrated on page 16 of Krasitskii (1994), the nonlinear order of the Zakharov equation A(k) coincides with that of the Hamiltonian dynamical equations. The second-order kernels of the Zakharov equation, \( {U}_{0,1,2}^{(3)} \) and \( {U}_{0,1,2}^{(1)} \) (corresponding to \( {V}_{1,2,3}^{\left(+\right)} \) and \( {V}_{1,2,3}^{\left(-\right)} \) in this paper), are expressed by the second-order kernel, \( {E}_{0,1,2}^{(3)} \), of the Hamiltonian dynamical equation. Similarly, the third-order kernel, \( {V}_{0,1,2,3}^{(2)} \) (corresponding to \( {W}_{0,1,2,3}^{(2)} \)), is defined from the third-order kernel, \( {E}_{0,1,2,3}^{(4)} \), of the Hamiltonian dynamical equation. Therefore, the nonlinear order of the HSOM is consistent with that of the Zakharov equation Eq. (13) from Sect. 4.1.
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Fujimoto, W., Waseda, T. & Webb, A. Impact of the four-wave quasi-resonance on freak wave shapes in the ocean. Ocean Dynamics 69, 101–121 (2019). https://doi.org/10.1007/s10236-018-1234-9
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DOI: https://doi.org/10.1007/s10236-018-1234-9