Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T17:39:01.528Z Has data issue: false hasContentIssue false
  • English
  • Français

Crest speeds of unsteady surface water waves

Published online by Cambridge University Press:  17 July 2020

Francesco Fedele Open the ORCID record for Francesco Fedele [Opens in a new window] ,
Michael L. Banner Open the ORCID record for Michael L. Banner [Opens in a new window]  and
Xavier Barthelemy Open the ORCID record for Xavier Barthelemy [Opens in a new window]
Francesco Fedele*
Affiliation:
School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA30332, USA
Michael L. Banner
Affiliation:
School of Mathematics and Statistics, UNSW Australia, Sydney, NSW2052, Australia
Xavier Barthelemy
Affiliation:
School of Mathematics and Statistics, UNSW Australia, Sydney, NSW2052, Australia Water Research Laboratory, School of Civil and Environmental Engineering, UNSW Australia, Sydney, NSW2052, Australia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Intuitively, crest speeds of water waves are assumed to match their phase speeds. However, this is generally not the case for natural waves within unsteady wave groups. This motivates our study, which presents new insights into the generic behaviour of crest speeds of linear to highly nonlinear unsteady waves. While our major focus is on gravity waves where a generic crest slowdown occurs cyclically, results for capillary-dominated waves are also discussed, for which crests cyclically speed up. This curious phenomenon arises when the theoretical constraint of steadiness is relaxed, allowing waves to change their form, or shape. In particular, a kinematic analysis of both simulated and observed open-ocean gravity waves reveals a forward-to-backward leaning cycle for each individual crest within a wave group. This is clearly manifest during the focusing of dominant wave groups essentially due to the dispersive nature of waves. It occurs routinely for focusing linear (vanishingly small steepness) wave groups, and it is enhanced as the wave spectrum broadens. It is found to be relatively insensitive to the degree of phase coherence and focusing of wave groups. The nonlinear nature of waves limits the crest slowdown. This reduces when gravity waves become less dispersive, either as they steepen or as they propagate over finite water depths. This is demonstrated by numerical simulations of the unsteady evolution of two- and three-dimensional dispersive gravity wave packets in both deep and intermediate water depths, and by open-ocean space–time measurements.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 899 , 25 September 2020 , A5
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

Baldock, T. E., Swan, C. & Taylor, P. H. 1996 A laboratory study of nonlinear surface waves on water. Phil. Trans. 354 (1707), 649–676.Google Scholar
Banner, M. L., Barthelemy, X., Fedele, F., Allis, M., Benetazzo, A., Dias, F. & Peirson, W. L. 2014 Linking reduced breaking crest speeds to unsteady nonlinear water wave group behavior. Phys. Rev. Lett. 112, 114502.CrossRefGoogle ScholarPubMed
Banner, M. L. & Peirson, W. L. 2007 Wave breaking onset and strength for two-dimensional deep-water wave groups. J. Fluid Mech. 585, 93–115.CrossRefGoogle Scholar
Banner, M. L. & Young, I. R. 1994 Modeling spectral dissipation in the evolution of wind waves. Part I: assessment of existing model performance. J. Phys. Oceanogr. 24 (7), 1550–1571.2.0.CO;2>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, 463–488.CrossRefGoogle Scholar
Benetazzo, A., Fedele, F., Gallego, G., Shih, P.-C. & Yezzi, A. 2012 Offshore stereo measurements of gravity waves. Coastal Engng 64, 127–138.CrossRefGoogle Scholar
Berry, M. 1988 The geometric phase. Sci. Am. 259 (6), 46–52.CrossRefGoogle Scholar
Beyá, J. F., Peirson, W. L. & Banner, M. L. 2012 Turbulence beneath finite amplitude water waves. Exp. Fluids 52, 1319–1330.CrossRefGoogle Scholar
Boccotti, P. 2000 Wave Mechanics for Ocean Engineering. Elsevier Sciences.Google Scholar
Craciunescu, C. C. & Christou, M. 2019 a The effect of wave crest speed in breaking waves. IOP Conf. Ser. 586, 012039.CrossRefGoogle Scholar
Craciunescu, C. C. & Christou, M. 2019 b Identifying breaking waves from measured time traces. In The 29th International Ocean and Polar Engineering Conference, 16–21 June, Honolulu, Hawaii, USA. International Society of Offshore and Polar Engineers.Google Scholar
Dyachenko, A. I., Kachulin, D. I. & Zakharov, V. E. 2017 a Envelope equation for water waves. J. Ocean Engng Marine Energy 3, 409–415.CrossRefGoogle Scholar
Dyachenko, A. I., Kachulin, D. I. & Zakharov, V. E. 2017 b Super compact equation for water waves. J. Fluid Mech. 828, 661–679.CrossRefGoogle Scholar
Dysthe, K. B. 1979 Note on a modification to the nonlinear schrodinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369 (1736), 105–114.Google Scholar
Ewans, K. C. 1998 Observations of the directional spectrum of fetch-limited waves. J. Phys. Oceanogr. 28 (3), 495–512.2.0.CO;2>CrossRefGoogle Scholar
Fedele, F. 2008 Rogue waves in oceanic turbulence. Physica D 237 (14–17), 2127–2131.CrossRefGoogle Scholar
Fedele, F. 2014 a Geometric phases of water waves. Europhys. Lett. 107 (6), 69001.CrossRefGoogle Scholar
Fedele, F. 2014 b On certain properties of the compact Zakharov equation. J. Fluid Mech. 748, 692–711.CrossRefGoogle Scholar
Fedele, F. 2014 c On the persistence of breathers at deep water. J. Expl. Theor. Phys. Lett. 98 (9), 523–527.CrossRefGoogle Scholar
Fedele, F. 2015 On the kurtosis of ocean waves in deep water. J. Fluid Mech. 782, 25–36.CrossRefGoogle Scholar
Fedele, F., Abessi, O. & Roberts, P. J. 2015 Symmetry reduction of turbulent pipe flows. J. Fluid Mech. 779, 390–410.CrossRefGoogle Scholar
Fedele, F., Benetazzo, A. & Forristall, G. Z. 2011 Space-time waves and spectra in the Northern Adriatic Sea via a wave acquisition stereo system. In Proceedings of the ASME 2011 30th International Conference on Offshore Mechanics and Arctic Engineering, vol. 2, pp. 651–663.Google Scholar
Fedele, F., Benetazzo, A., Gallego, G., Shih, P.-C., Yezzi, A., Barbariol, F. & Ardhuin, F. 2013 Space-time measurements of oceanic sea states. Ocean Model. 70, 103–115.CrossRefGoogle Scholar
Fedele, F., Chandre, C. & Farazmand, M. 2016 Kinematics of fluid particles on the sea surface: Hamiltonian theory. J. Fluid Mech. 801, 260–288.CrossRefGoogle Scholar
Fedele, F. & Dutykh, D. 2012 a Hamiltonian form and solitary waves of the spatial dysthe equations. J. Expl. Theor. Phys. Lett. 94 (12), 840–844.CrossRefGoogle Scholar
Fedele, F. & Dutykh, D. 2012 b Special solutions to a compact equation for deep-water gravity waves. J. Fluid Mech. 712, 646–660.CrossRefGoogle Scholar
Fedele, F. & Tayfun, M. A. 2009 On nonlinear wave groups and crest statistics. J. Fluid Mech. 620, 221–239.CrossRefGoogle Scholar
Fenton, J. 1985 A fifth-order Stokes theory for steady waves. J. Waterway Port Coastal Ocean Engng 111 (2), 216–234.CrossRefGoogle Scholar
Fenton, J. D. 1979 A high-order cnoidal wave theory. J. Fluid Mech. 94, 129–161.CrossRefGoogle Scholar
Fochesato, C. & Dias, F. 2006 A fast method for nonlinear three-dimensional free-surface waves. Proc. R. Soc. A 462 (2073), 2715–2735.CrossRefGoogle Scholar
Fochesato, C., Grilli, S. & Dias, F. 2007 Numerical modeling of extreme rogue waves generated by directional energy focusing. Wave Motion 44 (5), 395–416.CrossRefGoogle Scholar
Gallego, G., Yezzi, A., Fedele, F. & Benetazzo, B. 2011 A variational stereo method for the three-dimensional reconstruction of ocean waves. IEEE Trans. Geosci. Remote Sens. 49 (11), 4445–4457.CrossRefGoogle Scholar
Gallego, G., Yezzi, A., Fedele, F. & Benetazzo, B. 2013 Variational stereo imaging of oceanic waves with statistical constraints. IEEE Trans. Image Process. 22 (11), 4211–4223.CrossRefGoogle ScholarPubMed
Gemmrich, J. R., Banner, M. L. & Garrett, C. 2008 Spectrally resolved energy dissipation rate and momentum flux of breaking waves. J. Phys. Oceanogr. 38 (6), 1296–1312.CrossRefGoogle Scholar
Grilli, S. T., Guyenne, P. & Dias, F. 2001 A fully non-linear model for three-dimensional overturning waves over an arbitrary bottom. Intl. J. Numer. Methods Fluids 35 (7), 829–867.3.0.CO;2-2>CrossRefGoogle Scholar
Grilli, S. T. & Horrillo, J. 1997 Numerical generation and absorption of fully nonlinear periodic waves. J. Engng Mech. 123 (10), 1060–1069.Google Scholar
Grilli, S. T., Skourup, J. & Svendsen, I. A. 1989 An efficient boundary element method for nonlinear water waves. Engng Anal. Bound. Elem. 6 (2), 97–107.CrossRefGoogle Scholar
Grilli, S. T. & Subramanya, R. 1994 Quasi-singular integrals in the modeling of nonlinear water waves in shallow water. Engng Anal. Bound. Elem. 13 (2), 181–191.CrossRefGoogle Scholar
Grilli, S. T. & Subramanya, R. 1996 Numerical modeling of wave breaking induced by fixed or moving boundaries. Comput. Mech. 17 (6), 374–391.CrossRefGoogle Scholar
Grilli, S. T. & Svendsen, I. A. 1990 Corner problems and global accuracy in the boundary element solution of nonlinear wave flows. Engng Anal. Bound. Elem. 7 (4), 178–195.CrossRefGoogle Scholar
Grue, J., Clamond, D., Huseby, M. & Jensen, A. 2003 Kinematics of extreme waves in deep water. Appl. Ocean Res. 25 (6), 355–366.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). Ergänzungshef, 8 (12).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 II: directional distribution. J. Phys. Oceanogr. 30 (11), 2768–2787.2.0.CO;2>CrossRefGoogle Scholar
Itay, U. & Liberzon, D. 2017 Lagrangian kinematic criterion for the breaking of shoaling waves. J. Phys. Oceanogr. 47 (4), 827–833.CrossRefGoogle Scholar
Jessup, A. T. & Phadnis, K. R. 2005 Measurement of the geometric and kinematic properties of microscale breaking waves from infrared imagery using a PIV algorithm. Meas. Sci. Technol. 16 (10), 1961.CrossRefGoogle Scholar
Johannessen, T. B. 2010 Calculations of kinematics underneath measured time histories of steep water waves. Appl. Ocean Res. 32 (4), 391–403.CrossRefGoogle Scholar
Johannessen, T. B. & Swan, C. 2001 A laboratory study of the focusing of transient and directionally spread surface water waves. Proc. R. Soc. Lond. A 457 (2008), 971–1006.CrossRefGoogle Scholar
Johannessen, T. B. & Swan, C. 2003 On the nonlinear dynamics of wave groups produced by the focusing of surface-water waves. Proc. R. Soc. Lond. A 459 (2032), 1021–1052.CrossRefGoogle Scholar
Katsardi, V. & Swan, C. 2011 The evolution of large non-breaking waves in intermediate and shallow water. I. Numerical calculations of uni-directional seas. Proc. R. Soc. Lond. A 467 (2127), 778–805.CrossRefGoogle Scholar
Khait, A. & Shemer, L. 2018 On the kinematic criterion for the inception of breaking in surface gravity waves: fully nonlinear numerical simulations and experimental verification. Phys. Fluids 30 (5), 057103.CrossRefGoogle Scholar
Kinsman, B. 1965 Wind Waves: Their Generation and Propagation on the Ocean Surface. Prentice-Hall.Google Scholar
Kleiss, J. M. & Melville, W. K. 2010 Observations of wave breaking kinematics in fetch-limited seas. J. Phys. Oceanogr. 40 (12), 2575–2604.CrossRefGoogle Scholar
Komen, G. J. 1994 Dynamics and Modelling of Ocean Waves. Cambridge University Press.CrossRefGoogle Scholar
Krasitskii, V. P. 1994 On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272, 1–20.CrossRefGoogle Scholar
Latifah, A. L. & van Groesen, E. 2012 Coherence and predictability of extreme events in irregular waves. Nonlinear Process. Geophys. 19 (2), 199–213.CrossRefGoogle Scholar
Liberzon, D. & Shemer, L. 2011 Experimental study of the initial stages of wind waves’ spatial evolution. J. Fluid Mech. 681, 462–498.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1957 On the velocities of the maxima in a moving wave-form. Math. Proc. Camb. Phil. Soc. 53 (01), 230–233.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1987 The propagation of short surface waves on longer gravity waves. J. Fluid Mech. 177, 293–306.CrossRefGoogle Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1960 Changes in the form of short gravity waves on long waves and tidal currents. J. Fluid Mech. 8, 565–583.CrossRefGoogle Scholar
Ma, Q. 2010 Advances in Numerical Simulation of Nonlinear Water Waves. World Scientific.CrossRefGoogle Scholar
Melville, W. K. 1983 Wave modulation and breakdown. J. Fluid Mech. 128, 489–506.CrossRefGoogle Scholar
Melville, W. K. & Matusov, P. 2002 Distribution of breaking waves at the ocean surface. Nature 417 (6884), 58–63.CrossRefGoogle ScholarPubMed
Melville, W. K., Veron, F. & White, C. J. 2002 The velocity field under breaking waves: coherent structures and turbulence. J. Fluid Mech. 454, 203–233.CrossRefGoogle Scholar
Miller, S. J., Shemdin, O. H. & Longuet-Higgins, M. S. 1991 Laboratory measurements of modulation of short-wave slopes by long surface waves. J. Fluid Mech. 233, 389–404.CrossRefGoogle Scholar
Phillips, O. M. 1985 Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J. Fluid Mech. 156, 505–531.CrossRefGoogle Scholar
Rapp, R. J. & Melville, W. K. 1990 Laboratory measurements of deep-water breaking waves. Phil. Trans. R. Soc. Lond. A 331 (1622), 735–800.Google Scholar
Shemer, L. 2013 On kinematics of very steep waves. Nat. Hazards Earth Syst. Sci. 13 (8), 2101–2107.CrossRefGoogle Scholar
Shemer, L. & Ee, B. K. 2015 Steep unidirectional wave groups – fully nonlinear simulations vs. experiments. Nonlinear Process. Geophys. 22 (6), 737–747.CrossRefGoogle Scholar
Shemer, L. & Liberzon, D. 2014 Lagrangian kinematics of steep waves up to the inception of a spilling breaker. Phys. Fluids 26 (1), 016601.CrossRefGoogle Scholar
Song, J.-B. & Banner, M. L. 2002 On determining the onset and strength of breaking for deep water waves. Part I: unforced irrotational wave groups. J. Phys. Oceanogr. 32 (9), 2541–2558.CrossRefGoogle Scholar
Stansell, P. & MacFarlane, C. 2002 Experimental investigation of wave breaking criteria based on wave phase speeds. J. Phys. Oceanogr. 32 (5), 1269–1283.2.0.CO;2>CrossRefGoogle Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Sullivan, P. P., Banner, M. L., Morison, R. P. & Peirson, W. L. 2018 Turbulent flow over steep steady and unsteady waves under strong wind forcing. J. Phys. Oceanogr. 48 (1), 3–27.CrossRefGoogle Scholar
Sutherland, J., Greated, C. A. & Easson, W. J. 1995 Variations in the crest kinematics of wave groups. Appl. Ocean Res. 17 (1), 55–62.CrossRefGoogle Scholar
Tayfun, M. A. 1986 On narrow-band representation of ocean waves: 1. Theory. J. Geophys. Res. 91 (C6), 7743–7752.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.CrossRefGoogle Scholar
Viotti, C., Carbone, F. & Dias, F. 2014 Conditions for extreme wave runup on a vertical barrier by nonlinear dispersion. J. Fluid Mech. 748, 768–788.CrossRefGoogle Scholar
Wang, D. W. & Hwang, P. A. 2001 Evolution of the bimodal directional distribution of ocean waves. J. Phys. Oceanogr. 31 (5), 1200–1221.2.0.CO;2>CrossRefGoogle Scholar
Wiegel, R. L. 1964 Oceanographical Engineering. Prentice-Hall.Google Scholar
Wilczek, F. & Shapere, A. (Ed.) 1989 Geometric Phases in Physics, Advanced Series in Mathematical Physics, vol. 5. World Scientific.CrossRefGoogle Scholar
Young, I. R., Verhagen, L. A. & Banner, M. L. 1995 A note on the bimodal directional spreading of fetch-limited wind waves. J. Geophys. Res. 100 (C1), 773–778.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. Sov. Phys. J. Appl. Mech. Tech. Phys. 4, 190–194.Google Scholar
Zakharov, V. E. 1999 Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid. Eur. J. Mech. B/Fluids 18 (3), 327–344.CrossRefGoogle Scholar