Skip to main content Accessibility help
×
Home
Hostname: page-component-768ffcd9cc-96qlp Total loading time: 0.413 Render date: 2022-12-06T23:00:12.886Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Rapid spectral evolution of steep surface wave groups with directional spreading

Published online by Cambridge University Press:  25 November 2020

D. Barratt*
Affiliation:
Department of Engineering Science, University of Oxford, OxfordOX1 3PJ, UK
H. B. Bingham
Affiliation:
Department of Mechanical Engineering, Technical University of Denmark (DTU), 2800Lyngby, Denmark
P. H. Taylor
Affiliation:
Faculty of Engineering and Mathematical Sciences, University of Western Australia, Crawley, WA6009, Australia
T. S. van den Bremer
Affiliation:
Department of Engineering Science, University of Oxford, OxfordOX1 3PJ, UK
T. A. A. Adcock
Affiliation:
Department of Engineering Science, University of Oxford, OxfordOX1 3PJ, UK
*
Email address for correspondence: dylan.barratt@eng.ox.ac.uk

Abstract

We have investigated steep three-dimensional surface gravity wave groups formed by dispersive focusing using a fully nonlinear potential flow solver. We find that third-order resonant interactions result in rapid energy transfers to higher wavenumbers and reduced directional spreading during focusing, followed by spectral broadening during defocusing, forming steep wave groups with augmented kinematics and a prolonged lifespan. If the wave group is initially narrow-banded, quasi-degenerate interactions arise, characterised by energy transfers along the resonance angle, ${\pm }35.26^{\circ }$, of the Phillips ‘figure-of-eight’ loop. Spectral broadening due to the quasi-degenerate interactions facilitates non-degenerate interactions, characterised by oblique energy transfers at approximately ${\pm }55^{\circ }$ to the spectral peak. We consider the influence of steepness, finite depth, directional spreading and the high-wavenumber tail on spectral evolution. Steepness is found to augment both the quasi-degenerate and non-degenerate interactions similarly. However, a reduction in depth is found to weaken the quasi-degenerate interactions more severely than the non-degenerate interactions. We observe that increased directional spreading reduces spectral evolution, partially because wave groups with more spreading focus for a shorter duration due to linear dispersion. However, we also find that directional spreading reduces the peak rates of energy transfer. Inclusion of the high-wavenumber tail of the Joint North Sea Wave Project spectrum further reduces rates of energy transfer compared with a Gaussian wavenumber spectrum. Thus, directional spreading and the high-wavenumber tail may be integral to a form of spectral equilibrium that reduces rapid energy transfers during a steep wave event.

Type
JFM Papers
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Adcock, T. A. A. & Taylor, P. H. 2014 The physics of anomalous (‘rogue’) ocean waves. Rep. Prog. Phys. 77, 105901.CrossRefGoogle Scholar
Adcock, T. A. A. & Taylor, P. H. 2016 Fast and local non-linear evolution of steep wave-groups on deep water: a comparison of approximate models to fully-nonlinear simulations. Phys. Fluids 28, 016601.CrossRefGoogle Scholar
Adcock, T. A. A., Taylor, P. H. & Draper, S. 2015 Nonlinear dynamics of wave-groups in random seas: unexpected walls of water in the open ocean. Proc. R. Soc. Lond. A 471, 20150660.Google Scholar
Alber, I. E. 1978 The effects of randomness on the stability of two-dimensional surface wavetrains. Proc. R. Soc. Lond. A 363, 525546.Google Scholar
Alberello, A., Chabchoub, A., Monty, J. P., Nelli, F., Lee, J. H., Elsnab, J. & Toffoli, A. 2018 An experimental comparison of velocities underneath focussed breaking waves. Ocean Engng 155, 201210.CrossRefGoogle Scholar
Annenkov, S. Y. & Shrira, V. I. 2006 Role of non-resonant interactions in the evolution of nonlinear random water wave fields. J. Fluid Mech. 561, 181207.CrossRefGoogle Scholar
Annenkov, S. Y. & Shrira, V. I. 2018 Spectral evolution of weakly nonlinear random waves: kinetic description versus direct numerical simulations. J. Fluid Mech. 844, 766795.CrossRefGoogle Scholar
Banner, M. L. & Young, I. R. 1994 Model spectral dissipation in the evolution of wind waves. Part 1. Assessment of existing model performance. J. Phys. Oceanogr. 24 (7), 15501571.2.0.CO;2>CrossRefGoogle Scholar
Barratt, D., Bingham, H. B. & Adcock, T. A. A. 2020 Nonlinear evolution of a steep, focusing wave group in deep water simulated with OceanWave3D. J. Offshore Mech. Arctic Engng 142, 021201.CrossRefGoogle Scholar
Barthelemy, X., Banner, M. L., Peirson, W. L., Fedele, F., Allis, M. & Dias, F. 2018 On a unified breaking onset threshold for gravity waves in deep and intermediate depth water. J. Fluid Mech. 841, 463488.CrossRefGoogle Scholar
Benjamin, T. B. 1967 Instability of periodic wavetrains in nonlinear dispersive systems. Proc. R. Soc. Lond. A 299 (1456), 5976.Google Scholar
Benney, D. J. & Roskes, G. J. 1969 Wave instabilities. Stud. Appl. Maths 48 (4), 377385.CrossRefGoogle Scholar
Boccotti, P. 1983 Some new results on statistical properties of wind waves. Appl. Ocean Res. 5 (3), 134140.CrossRefGoogle Scholar
Boccotti, P. 2000 Wave Mechanics for Ocean Engineering. Elsevier.Google Scholar
Chawla, A. & Kirby, J. T. 2002 Monochromatic and random wave breaking at blocking points. J. Geophys. Res. 107 (C7), 4–1.Google Scholar
Crawford, D. R., Lake, B. M., Saffman, P. G. & Yuen, H. C. 1981 Stability of weakly nonlinear deep-water waves in two and three dimensions. J. Fluid Mech. 105, 177191.CrossRefGoogle Scholar
Dalzell, J. F. 1999 A note on finite depth second-order wave–wave interactions. 21 (3), 105111.Google Scholar
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338, 101110.Google Scholar
Dysthe, K., Krogstad, H. E. & Müller, P. 2008 Oceanic rogue waves. Annu. Rev. Fluid Mech. 40, 287310.CrossRefGoogle Scholar
Dysthe, K. B. 1979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369 (1736), 105114.Google Scholar
Dysthe, K. B., Trulsen, K., Krogstad, H. E. & Socquet-Juglard, H. 2003 Evolution of a narrow-band spectrum of random surface gravity waves. J. Fluid Mech. 478, 110.CrossRefGoogle Scholar
Engsig-Karup, A. P., Bingham, H. B. & Lindberg, O. 2009 An efficient flexible-order model for 3d nonlinear water waves. J. Comput. Phys. 228 (6), 21002118.CrossRefGoogle Scholar
Ewans, K. C. 1998 Observations of the directional spectrum of fetch-limited waves. J. Phys. Oceanogr. 28 (3), 495512.2.0.CO;2>CrossRefGoogle Scholar
Fadaeiazar, E., Alberello, A., Onorato, M., Leontini, J., Frascoli, F., Waseda, T. & Toffoli, A. 2018 Wave turbulence and intermittency in directional wave fields. Wave Motion 83, 94101.CrossRefGoogle Scholar
Fedele, F. 2014 On certain properties of the compact Zakharov equation. J. Fluid Mech. 748, 692711.CrossRefGoogle Scholar
Fedele, F., Brennan, J., De León, S. P., Dudley, J. & Dias, F. 2016 Real world ocean rogue waves explained without the modulational instability. Sci. Rep. 6, 27715.CrossRefGoogle ScholarPubMed
Fitzgerald, C. J., Taylor, P. H., Eatock Taylor, R. E., Grice, J. & Zang, J. 2014 Phase manipulation and the harmonic components of ringing forces on a surface-piercing column. Proc. R. Soc. Lond. A 470, 105901.Google Scholar
Francius, M. & Kharif, C. 2003 On the disappearance of the lowest-order instability for steep gravity waves in finite depth. Phys. Fluids 15 (8), 24452448.CrossRefGoogle Scholar
Fujimoto, W., Waseda, T. & Webb, A. 2019 Impact of the four-wave quasi-resonance on freak wave shapes in the ocean. Ocean Dyn. 69, 101121.CrossRefGoogle Scholar
Gibbs, R. H. & Taylor, P. H. 2005 Formation of walls of water in ‘fully’ nonlinear simulations. Appl. Ocean Res. 27, 142157.CrossRefGoogle Scholar
Gibson, R. S. & Swan, C. 2007 The evolution of large ocean waves: the role of local and rapid spectral changes. Proc. R. Soc. Lond. A 463, 2148.Google Scholar
Gramstad, O. & Trulsen, K. 2007 Influence of crest and group length on the occurrence of freak waves. J. Fluid Mech. 582, 463472.CrossRefGoogle Scholar
Hara, T. & Mei, C. C. 1991 Frequency downshift in narrowbanded surface waves under the influence of wind. J. Fluid Mech. 230, 429477.CrossRefGoogle Scholar
Hasselmann, K. 1962 On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory. J. Fluid Mech. 12, 481500.CrossRefGoogle Scholar
Hasselmann, K., Barnett, T. P., Bouws, E., Carlson, H., Cartwright, D. E., Enke, K., Ewing, J. A., Gienapp, H., Hasselmann, D. E. & Kruseman, P. et al. 1973 Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Dtsch. Hydrogr. Z. A8, 195.Google Scholar
Hwang, P. A., Wang, D. W., Walsh, E. J., Krabill, W. B. & Swift, R. N. 2000 Airborne measurements of the wavenumber spectra of ocean surface waves. Part 2. Directional distribution. J. Phys. Oceanogr. 30 (11), 27682787.2.0.CO;2>CrossRefGoogle Scholar
Janssen, P. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33, 863884.2.0.CO;2>CrossRefGoogle Scholar
Janssen, P. A. E. M. & Onorato, M. 2007 The intermediate water depth limit of the Zakharov equation and consequences for wave prediction. J. Phys. Oceanogr. 37 (10), 23892400.CrossRefGoogle Scholar
Kharif, C., Giovanangeli, J. P., Touboul, J., Grare, L. & Pelinovsky, E. 2008 a Influence of wind on extreme wave events: experimental and numerical approaches. J. Fluid Mech. 594, 209247.CrossRefGoogle Scholar
Kharif, C. & Pelinovsky, E. 2003 Physical mechanisms of the rogue wave phenomenon. Eur. J. Mech. B/Fluids 22, 603634.CrossRefGoogle Scholar
Kharif, C., Pelinovsky, E. & Slunyaev, A. 2008 b Rogue Waves in the Ocean. Springer Science and Business Media.Google Scholar
Komen, G. J., Cavaleri, L., Donelan, M., Hasselmann, K., Hasselmann, S. & Janssen, P. A. E. M. 1994 Dynamics and Modelling of Ocean Waves. Cambridge University Press.CrossRefGoogle Scholar
Lake, B. M., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. 1977 Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83 (1), 4974.CrossRefGoogle Scholar
Latheef, M., Swan, C. & Spinneken, J. 2017 A laboratory study of nonlinear changes in the directionality of extreme seas. Proc. R. Soc. Lond. A 473, 20160290.Google ScholarPubMed
Leckler, F., Ardhuin, F., Peureux, C., Benetazzo, A., Bergamasco, F. & Dulov, V. 2015 Analysis and interpretation of frequency-wavenumber spectra of young wind waves. J. Phys. Oceanogr. 45 (10), 24842496.CrossRefGoogle Scholar
Lindgren, G. 1970 Some properties of a normal process near a local maximum. Ann. Math. Statist. 41 (6), 18701883.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1976 On the nonlinear transfer of energy in the peak of a gravity-wave spectrum: a simplified model. J. Fluid Mech. 347, 311328.Google Scholar
Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics. Proc. R. Soc. Lond. A 360, 489505.Google Scholar
Ma, Y., Dong, G., Perlin, M., Ma, X., Wang, G. & Xu, J. 2010 Laboratory observations of wave evolution, modulation and blocking due to spatially varying opposing currents. J. Fluid Mech. 661, 108129.CrossRefGoogle Scholar
McLean, J. W. 1982 a Instabilities of finite-amplitude gravity waves on water of finite depth. J. Fluid Mech. 114, 331341.CrossRefGoogle Scholar
McLean, J. W. 1982 b Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315330.CrossRefGoogle Scholar
Melville, W. K. 1982 The instability and breaking of deep-water waves. J. Fluid Mech. 115, 165185.CrossRefGoogle Scholar
Meylan, M. H., Bennetts, L. G., Mosig, J. E. M., Rogers, W. E., Doble, M. J. & Peter, M. A. 2018 Dispersion relations, power laws, and energy loss for waves in the marginal ice zone. J. Geophys. Res. 123 (5), 33223335.CrossRefGoogle Scholar
Mori, N., Onorato, M. & Janssen, P. A. E. M. 2011 On the estimation of the kurtosis in directional sea states for freak wave forecasting. J. Phys. Oceanogr. 41, 14841497.CrossRefGoogle Scholar
Onorato, M., Osborne, A. R., Serio, M. & Bertone, S. 2001 Freak waves in random oceanic sea states. Phys. Rev. Lett. 86 (25), 58315834.CrossRefGoogle ScholarPubMed
Onorato, M., Osborne, A. R., Serio, M., Resio, D., Pushkarev, A., Zakharov, V. E. & Brandini, C. 2002 Freely decaying weak turbulence for sea surface gravity waves. Phys. Rev. Lett. 89 (14), 144501.CrossRefGoogle ScholarPubMed
Onorato, M. & Suret, P. 2016 Twenty years of progresses in oceanic rogue waves: the role played by weakly nonlinear models. Nat. Hazards 84 (2), 541548.CrossRefGoogle Scholar
Paulsen, B. T., Bredmose, H., Bingham, H. B. & Jacobsen, N. G. 2014 Forcing of a bottom-mounted circular cylinder by steep regular water waves at finite depth. J. Fluid Mech. 755, 134.CrossRefGoogle Scholar
Perlin, M., Choi, W. & Tian, Z. 2013 Breaking waves in deep and intermediate waters. Annu. Rev. Fluid Mech. 45, 115145.CrossRefGoogle Scholar
Peureux, C., Benetazzo, A. & Ardhuin, F. 2018 Note on the directional properties of meter-scale gravity waves. Ocean Sci. 14 (1), 4152.CrossRefGoogle Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9 (2), 193217.CrossRefGoogle Scholar
Rapizo, H., Waseda, T., Babanin, A. V. & Toffoli, A. 2016 Laboratory experiments on the effects of a variable current field on the spectral geometry of water waves. J. Phys. Oceanogr. 46 (9), 26952717.CrossRefGoogle Scholar
Segur, H., Henderson, D., Carter, J., Hammack, J., Li, C., Pheiff, D. & Socha, K. 2005 Stabilizing the Benjamin–Feir instability. J. Fluid Mech. 539, 229271.CrossRefGoogle Scholar
Shemer, L., Sergeeva, A. & Liberzon, D. 2010 Effect of the initial spectrum on the spatial evolution of statistics of unidirectional nonlinear random waves. J. Geophys. Res. 115, C12039.CrossRefGoogle Scholar
Simanesew, A., Krogstad, H. E., Trulsen, K. & Nieto Borge, J. C. 2016 Development of frequency-dependent ocean wave directional distributions. Appl. Ocean Res. 59, 304312.CrossRefGoogle Scholar
Simanesew, A., Krogstad, H. E., Trulsen, K. & Nieto Borge, J. C. 2018 Bimodality of directional distributions in ocean wave spectra: a comparison of data-adaptive estimation techniques. J. Atmos. Ocean. Technol. 35 (2), 365384.CrossRefGoogle Scholar
Socquet-Juglard, H., Dysthe, K., Trulsen, K., Krogstad, H. E. & Liu, J. 2005 Probability distributions of surface gravity waves during spectral changes. J. Fluid Mech. 542, 195216.CrossRefGoogle Scholar
Steer, J. N., Borthwick, A. G. L., Onorato, M., Chabchoub, A. & van den Bremer, T. S. 2019 Hydrodynamic X waves. Phys. Rev. Lett. 123 (18), 184501.CrossRefGoogle ScholarPubMed
Stiassnie, M. 1984 Note on the modified nonlinear Schrödinger equation for deep water waves. Wave Motion 6, 431433.CrossRefGoogle Scholar
Stiassnie, M. 2001 Nonlinear interactions of inhomogenous random water waves. In ECMWF report of the Workshop on Ocean Waves Forecasting, pp. 39–52. European Centre for Medium-Range Weather Forecasts.Google Scholar
Stiassnie, M. 2017 On the strength of the weakly nonlinear theory for surface gravity waves. J. Fluid Mech. 810, 14.CrossRefGoogle Scholar
Stiassnie, M & Shemer, L. 2005 On the interaction of four water-waves. Wave Motion 41, 307328.CrossRefGoogle Scholar
Stuhlmeier, R. & Stiassnie, M. 2017 Evolution of statistically inhomogenous degenerate water wave quartets. Proc. R. Soc. Lond. A 376, 20170101.Google Scholar
Su, M. Y., Bergin, M., Marler, P. & Myrick, R. 1982 Experiments on nonlinear instabilities and evolution of steep gravity-wave trains. J. Fluid Mech. 124, 4572.CrossRefGoogle Scholar
Toffoli, A., Bennetts, L. G., Meylan, M. H., Cavaliere, C., Alberello, A., Elsnab, J. & Monty, J. P. 2015 Sea ice floes dissipate the energy of steep ocean waves. Geophys. Res. Lett. 42 (20), 85478554.CrossRefGoogle Scholar
Toffoli, A., Benoit, M., Onorato, M. & Bitner-Gregersen, E. M. 2009 The effect of third-order nonlinearity on statistical properties of random directional waves in finite depth. Nonlinear Process. Geophys. 16 (1), 131.CrossRefGoogle Scholar
Toffoli, A., Onorato, M., Bitner-Gregersen, E. M. & Monbaliu, J. 2010 Development of a bimodal structure in ocean wave spectra. J. Geophys. Res. 115, C03006.Google Scholar
Trulsen, K. 2018 Rogue waves in the ocean, the role of modulational instability, and abrupt changes of environmental conditions that can provoke non equilibrium wave dynamics. In The Ocean in Motion (ed. Velarde, M. G., Tarakanov, R. Y. & Marchenko, A. V.). pp. 239247. Springer.CrossRefGoogle Scholar
Trulsen, K. & Dysthe, K. B. 1990 Frequency down-shift through self modulation and breaking. In Water Wave Kinematics (ed. Tørum, A. & Gudmestad, O. T.). pp. 561572. Springer.CrossRefGoogle Scholar
Trulsen, K. & Dysthe, K. B. 1996 A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24 (3), 281289.CrossRefGoogle Scholar
Trulsen, K. & Dysthe, K. B. 1997 Frequency downshift in three-dimensional wave trains in a deep basin. J. Fluid Mech. 352, 359373.CrossRefGoogle Scholar
Tulin, M. P. & Waseda, T. 1999 Laboratory observations of wave group evolution, including breaking effects. J. Fluid Mech. 378, 197232.CrossRefGoogle Scholar
Waseda, T., Kinoshita, T., Cavaleri, L. & Toffoli, A. 2015 Third-order resonant wave interactions under the influence of background current fields. J. Fluid Mech. 784, 5173.CrossRefGoogle Scholar
Waseda, T., Toba, Y. & Tulin, M. P. 2001 Adjustment of wind waves to sudden changes of wind speed. J. Oceanogr. 57, 519533.CrossRefGoogle Scholar
Whitham, G. B. 1967 Non-linear dispersion of water waves. J. Fluid Mech. 27 (2), 399412.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley and Sons.Google Scholar
Xiao, W., Liu, Y., Wu, G. & Yue, D. K. P. 2013 Rogue wave occurrence and dynamics by direct simulations of nonlinear wave-field evolution. J. Fluid Mech. 720, 357392.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190194.CrossRefGoogle Scholar
5
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Rapid spectral evolution of steep surface wave groups with directional spreading
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Rapid spectral evolution of steep surface wave groups with directional spreading
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Rapid spectral evolution of steep surface wave groups with directional spreading
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *