Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-30T20:34:25.916Z Has data issue: false hasContentIssue false
  • English
  • Français

Large-time evolution of statistical moments of wind–wave fields

Published online by Cambridge University Press:  11 June 2013

Sergei Y. Annenkov  and
Victor I. Shrira
Sergei Y. Annenkov
Affiliation:
Department of Mathematics, EPSAM, Keele University, Keele ST5 5BG, UK
Victor I. Shrira*
Affiliation:
Department of Mathematics, EPSAM, Keele University, Keele ST5 5BG, UK
*
†Email address for correspondence: v.i.shrira@keele.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the long-term evolution of weakly nonlinear random gravity water wave fields developing with and without wind forcing. The focus of the work is on deriving, from first principles, the evolution of the departure of the field statistics from Gaussianity. Higher-order statistical moments of elevation (skewness and kurtosis) are used as a measure of this departure. Non-Gaussianity of a weakly nonlinear random wave field has two components. The first is due to nonlinear wave–wave interactions. We refer to this component as ‘dynamic’, since it is linked to wave field evolution. The other component is due to bound harmonics. It is non-zero for every wave field with finite amplitude, contributes both to skewness and kurtosis of gravity water waves and can be determined entirely from the instantaneous spectrum of surface elevation. The key result of the work, supported both by direct numerical simulation (DNS) and by the analysis of simulated and experimental (JONSWAP) spectra, is that in generic situations of a broadband random wave field the dynamic contribution to kurtosis is small in absolute value, and negligibly small compared with the bound harmonics component. Therefore, the latter dominates, and both skewness and kurtosis can be obtained directly from the instantaneous wave spectra. Thus, the departure of evolving wave fields from Gaussianity can be obtained from evolving wave spectra, complementing the capability of forecasting spectra and capitalizing on the existing methodology. We find that both skewness and kurtosis are significant for typical oceanic waves; the non-zero positive kurtosis implies a tangible increase of freak wave probability. For random wave fields generated by steady or slowly varying wind and for swell the derived large-time asymptotics of skewness and kurtosis predict power law decay of the moments. The exponents of these laws are determined by the degree of homogeneity of the interaction coefficients. For all self-similar regimes the kurtosis decays twice as fast as the skewness. These formulae complement the known large-time asymptotics for spectral evolution prescribed by the Hasselmann equation. The results are verified by the DNS of random wave fields based on the Zakharov equation. The predicted asymptotic behaviour is shown to be very robust: it holds both for steady and gusty winds.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 726 , 10 July 2013 , pp. 517 - 546
Copyright
©2013 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

Abdalla, S. & Cavaleri, L. 2002 Effect of wind variability and variable air density on wave modeling. J. Geophys. Res. 107 (C7), 17.Google Scholar
Annenkov, S. Y. & Shrira, V. I. 2001 On the predictability of evolution of surface gravity and gravity-capillary waves. Physica D 152–153, 665–675.CrossRefGoogle Scholar
Annenkov, S. Y. & Shrira, V. I. 2006a Role of non-resonant interactions in the evolution of nonlinear random water wave fields. J. Fluid Mech. 561, 181–207.CrossRefGoogle Scholar
Annenkov, S. Y. & Shrira, V. I. 2006b Direct numerical simulation of downshift and inverse cascade for water wave turbulence. Phys. Rev. Lett. 96, 204501.CrossRefGoogle ScholarPubMed
Annenkov, S. Y. & Shrira, V. I. 2009a ‘Fast’ nonlinear evolution in wave turbulence. Phys. Rev. Lett. 102, 024502.CrossRefGoogle ScholarPubMed
Annenkov, S. Y. & Shrira, V. I. 2009b Evolution of kurtosis for wind waves. Geophys. Res. Lett. 36, L13603.Google Scholar
Annenkov, S. Y. & Shrira, V. I. 2011 Evolution of wave turbulence under ‘gusty’ forcing. Phys. Rev. Lett. 107, 114502.Google Scholar
Babanin, A. V. 2011 Breaking and Dissipation of Ocean Surface Waves. Cambridge University Press.CrossRefGoogle Scholar
Badulin, S. I., Babanin, A. V., Resio, D. & Zakharov, V. E. 2007 Weakly turbulent laws of wind-wave growth. J. Fluid Mech. 591, 339–378.Google Scholar
Badulin, S. I., Pushkarev, A. N., Resio, D. & Zakharov, V. E. 2005 Self-similarity of wind-driven seas. Nonlinear Process. Geophys. 12, 891–946.Google Scholar
Caulliez, G. & Guerin, C.-A. 2012 Higher-order statistical analysis of short wind wave fields. J. Geophys. Res. 117, C06002.Google Scholar
Connaughton, C., Nazarenko, S. & Newell, A. C. 2003 Dimensional analysis and weak turbulence. Physica D 184, 86–97.Google Scholar
Donelan, M. A., Babanin, A. V., Young, I. R. & Banner, M. L. 2006 Wave-follower field measurements of the wind-input spectral function. Part II. Parameterization of the wind input. J. Phys. Oceanogr. 36, 1672–1679.CrossRefGoogle Scholar
Fedele, F. 2008 Rogue waves in oceanic turbulence. Physica D 237, 2127–2131.Google Scholar
Fedele, F. & Tayfun, M. A. 2009 On nonlinear wave groups and crest statistics. J. Fluid Mech. 620, 221–239.Google Scholar
Fedorov, A. V. & Melville, W. K. 1998 Nonlinear gravity-capillary waves with forcing and dissipation. J. Fluid Mech. 354, 1–42.CrossRefGoogle Scholar
Gabor, D. 1946 Theory of communication. J. Inst. Elect. Eng. 93, 429–457.Google Scholar
Gagnaire-Renoud, E., Benoit, M. & Badulin, S. I. 2011 On weakly turbulent scaling of wind sea in simulations of fetch-limited growth. J. Fluid Mech. 669, 178–213.Google Scholar
Goda, Y. 2000 Random Seas and Design of Maritime Structures. World Scientific.CrossRefGoogle Scholar
Hasselmann, K. 1962 On the nonlinear energy transfer in a gravity-wave spectrum. Part 1. General theory. J. Fluid Mech. 12, 481–500.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., Meerburg, A., Müller, P., Olbers, D. J., Richter, K., Sell, W. & Walden, H. 1973 Measurements of wind wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). In Deutsche hydrographische Zeitschrift: Ergaenzungsheft: Reihe A, 12. Deutsches Hydrographisches Institut.Google Scholar
Hasselmann, K., Ross, D. B., Müller, P. & Sell, W. 1976 A parametric wave prediction model. J. Phys. Oceanogr. 6, 200–228.Google Scholar
Hsiao, S. V. & Shemdin, O. H. 1983 Measurements of wind velocity and pressure with a wave follower during MARSEN. J. Geophys. Res. 88, 9841–9849.CrossRefGoogle Scholar
Janssen, P. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33, 863–884.2.0.CO;2>CrossRefGoogle Scholar
Janssen, P. A. E. M. 2004 The Interaction of Ocean Waves and Wind. Cambridge University Press.CrossRefGoogle Scholar
Janssen, P. A. E. M. 2007 On the probability density function of wave height and wave period. ECMWF Research Department Memorandum, 22 June 2007.Google Scholar
Janssen, P. A. E. M. 2008 Progress in ocean wave forecasting. J. Comput. Phys. 227, 3572–3594.CrossRefGoogle Scholar
Janssen, P. A. E. M. 2009 On some consequences of the canonical transformation in the Hamiltonian theory of water waves. J. Fluid Mech 637, 1–44.CrossRefGoogle Scholar
Janssen, P. A. E. M. & Bidlot, J.-R. 2009 On an extension of the freak wave warning system and its verification. In ECMWF Technical Memorandum No. 588. European Centre for Medium-Range Weather Forecasts, Reading, UK.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.Google Scholar
Krasitskii, V. P. 1994 On reduced Hamiltonian equations in the nonlinear theory of water surface waves. J. Fluid Mech. 272, 1–20.CrossRefGoogle Scholar
Kudryavtsev, V. N., Makin, V. K. & Meirink, J. F. 2001 Simplified model of the airflow above waves. Bound.-Layer Meteor. 100, 63–90.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1957 The statistical analysis of a random, moving surface. Phil. Trans. R. Soc. Lond. A 249, 321–387.Google Scholar
Longuet-Higgins, M. S. 1983 On the joint distribution of wave periods and amplitudes in a random wave field. Proc. R. Soc. Lond. A 389, 241–258.Google Scholar
Longuet-Higgins, M. S. 1995 Parasitic capillary waves: a direct calculation. J. Fluid Mech. 301, 79–107.CrossRefGoogle Scholar
Leykin, I. A., Donelan, M. A., Mellen, R. H. & McLaughlin, D. J. 1995 Asymmetry of wind waves studied in a laboratory tank. Nonlinear Process. Geophys. 2, 280–289.Google Scholar
Lvov, Y. V., Nazarenko, S. & Pokorni, B. 2006 Discreteness and its effect on water-wave turbulence. Physica D 218, 24–35.Google Scholar
Mori, N. & Janssen, P. A. E. M. 2006a Freak wave prediction from directional spectra. In Coastal Engineering 2006, Proceedings of the 30th International Conference (ed. Smith, J. M.), pp. 714–725. World Scientific.Google Scholar
Mori, N. & Janssen, P. A. E. M. 2006b On kurtosis and occurrence probability of freak waves. J. Phys. Oceanogr. 36, 1471–1483.CrossRefGoogle Scholar
Nazarenko, S. V. 2011 Wave Turbulence. Springer.CrossRefGoogle Scholar
Onorato, M., Osborne, A. R., Serio, M., Cavaleri, L., Brandini, C. & Stansberg, C. T. 2006 Extreme waves, modulational instability and second-order theory: wave flume experiments on irregular waves. Eur. J. Mech. (B/Fluids) 25, 586–601.Google Scholar
Pushkarev, A., Resio, D. & Zakharov, V. E. 2003 Weak turbulent approach to the wind-generated gravity sea waves. Physica D 184, 29–63.Google Scholar
Pushkarev, A. N. & Zakharov, V. E. 2000 Turbulence of capillary waves – theory and numerical simulation. Physica D 135, 98–116.CrossRefGoogle Scholar
Rice, S. O. 1954 Mathematical analysis of random noise. In Reprinted of the 1944 original in Selected Papers on Noise and Stochastic Processes (ed. Wax, N.), pp. 133–294. Dover.Google Scholar
Stansell, P. 2004 Distributions of freak wave heights measured in the North Sea. Appl. Ocean Res. 26, 35–48.Google Scholar
Tayfun, M. A. 1980 Narrow-band nonlinear sea waves. J. Geophys. Res 85, 1548–1552.Google Scholar
Toba, Y. 1972 Local balance in the air–sea boundary processes. Part I. On the growth process of wind waves. J. Oceanogr. Soc. Japan 28, 109–121.Google Scholar
Toba, Y. 1973 Local balance in the air–sea boundary processes. Part III. On the spectrum of wind waves. J. Oceanogr. Soc. Japan 29, 209–220.Google Scholar
Young, I. R. 1999 Wind Generated Ocean Waves. Elsevier.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. (USSR) 9, 86–94.Google Scholar
Zakharov, V. E. 2005 Theoretical interpretation of fetch limited wind-driven sea observations. Nonlinear Proc. Geophys. 12, 1011–1020.CrossRefGoogle Scholar
Zakharov, V. E. & Zaslavsky, M. M. 1983 Dependence of wave parameters on the wind velocity, duration of its action and fetch in the weak-turbulence theory of water waves. Izv. Atmos. Ocean. Phys. 19, 300–306.Google Scholar
Zakharov, V. E., L’vov, V. S. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence I: Wave Turbulence. Springer.Google Scholar
Zavadsky, A., Liberzon, D. & Shemer, L. 2013 Statistical analysis of the spatial evolution of the stationary wind-wave field. J. Phys. Oceanogr. 43, 65–79.Google Scholar