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On weakly turbulent scaling of wind sea in simulations of fetch-limited growth

Published online by Cambridge University Press:  05 January 2011

ELODIE GAGNAIRE-RENOU ,
MICHEL BENOIT  and
SERGEI I. BADULIN
ELODIE GAGNAIRE-RENOU*
Affiliation:
Saint-Venant Laboratory for Hydraulics, Université Paris Est (joint research unit EDF R&D, CETMEF, Ecole des Ponts Paris Tech), c/o EDF R&D, 6 quai Watier, BP 49, 78401 Chatou, France Laboratoire National d'Hydraulique et Environnement (LNHE), EDF R&D, 6 quai Watier, BP 49, 78401 Chatou, France Laboratoire de Sondages Electromagnétiques de l'Environnement Terrestre (LSEET), Université du Sud Toulon-Var, BP 132, 83957 La Garde, France
MICHEL BENOIT
Affiliation:
Saint-Venant Laboratory for Hydraulics, Université Paris Est (joint research unit EDF R&D, CETMEF, Ecole des Ponts Paris Tech), c/o EDF R&D, 6 quai Watier, BP 49, 78401 Chatou, France Laboratoire National d'Hydraulique et Environnement (LNHE), EDF R&D, 6 quai Watier, BP 49, 78401 Chatou, France
SERGEI I. BADULIN
Affiliation:
P.P. Shirshov Institute of Oceanology of Russian Academy of Sciences, 36, Nakhimovskii pr., Moscow 117997, Russia Laboratory of Nonlinear Wave Processes, Novosibirsk State University, 630090, Novosibirsk, Russia
*
†Email address for correspondence: [email protected]
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Abstract

Extensive numerical simulations of fetch-limited growth of wind-driven waves are analysed within two approaches: a ‘traditional’ wind-speed scaling first proposed by Kitaigorodskii (Bull. Acad. Sci. USSR, Geophys. Ser., Engl. Transl., vol. N1, 1962, p. 105) in the early 1960s and an alternative weakly turbulent scaling developed recently by Badulin et al. (J. Fluid Mech.591, 2007, 339–378). The latter one uses spectral fluxes of wave energy, momentum and action as physical scales of the problem and allows for advanced qualitative and quantitative analysis of wind-wave growth and features of air–sea interaction. In contrast, the traditional approach is shown to be descriptive rather than proactive. Numerical simulations are conducted on the basis of the Hasselmann kinetic equation for deep-water waves in a wide range of wind speeds from 5 to 30 m s −1 and for the ideal case of fetch-limited growth: permanent wind blowing perpendicularly to a straight coastline. Two different wave input functions, Sin, and two methods for calculating the nonlinear transfer term Snl (Gaussian quadrature method, or GQM, a quasi-exact method based on the use of Gaussian quadratures, and the discrete interaction approximation, or DIA) are used in the simulations. Comparison of the corresponding results firstly shows the relevance of the analysis of wind-wave growth in terms of the proposed weakly turbulent scaling, and secondly, allows us to highlight some critical points in the modelling of wind-generated waves. Three stages of wind-wave development corresponding to qualitatively different balance of the source terms, Sin, Sdiss and Snl, are identified: initial growth, growing sea and fully developed sea. Validity of the asymptotic weakly turbulent approach for the stage of growing wind sea is determined by the dominance of nonlinear transfers, which results in a rigid link between spectral fluxes and wave energy. This stage of self-similar growth is investigated in detail and presented as a consequence of three sub-stages of qualitatively different coupling of air flow and growing wind waves. The key self-similarity parameter of the asymptotic theory is estimated to be αss = 0.68 ± 0.1.

Further prospects of wind-wave modelling in the context of the presented weakly turbulent scaling are discussed.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows Waves/Free-surface Flows: Wind-wave interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 669 , 25 February 2011 , pp. 178 - 213
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Alves, J. H. G. M. & Banner, M. L. 2003 Performance of a saturation-based dissipation-rate source term in modeling the fetch-limited evolution of wind waves. J. Phys. Oceanogr. 33, 1274–1298.2.0.CO;2>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, 181–207.CrossRefGoogle Scholar
Ardhuin, F., Chapron, B. & Collard, F. 2009 Observation of swell dissipation across oceans. Geophys. Res. Lett. 36, L06607.CrossRefGoogle Scholar
Ardhuin, F., Collard, F., Chapron, B., Queffeulou, P., Filipot, J.-F. & Hamon, M. 2008 Spectral wave dissipation based on observations: a global validation. In Proceedings of Chinese–German Joint Symposium on Hydraulics and Ocean Engineering, Darmstadt, Germany, pp. 391–400.Google Scholar
Babanin, A. N. & Soloviev, Y. P. 1998 a Field investigation of transformation of the wind wave frequency spectrum with fetch and the stage of development. J. Phys. Oceanogr. 28, 563–576.2.0.CO;2>CrossRefGoogle Scholar
Babanin, A. N. & Soloviev, Y. P. 1998 b Variability of directional spectra of wind-generated waves, studied by means of wave staff arrays. Mar. Freshwater Res. 49, 89–101.CrossRefGoogle Scholar
Badulin, S. I., Babanin, A. V., Resio, D. & Zakharov, V. 2007 Weakly turbulent laws of wind-wave growth. J. Fluid Mech. 591, 339–378.CrossRefGoogle Scholar
Badulin, S. I., Babanin, A. V., Resio, D. & Zakharov, V. 2008 a Numerical verification of weakly turbulent law of wind wave growth. In Proceedings of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, Moscow, Russia, 2006 (ed. Borisov, A. V., Kozlov, V. V., Mamaev, I. S. & Sokolovskiy, M. A.), IUTAM Book Series, vol. 6, pp. 175–190. Springer.Google Scholar
Badulin, S. I. & Caulliez, G. 2009 Significance of laboratory observations for modeling wind-driven seas. In Geophysical Research Abstracts, EGU General Assembly 2009, vol. 11, EGU2009-12694.Google Scholar
Badulin, S. I., Korotkevich, A. O., Resio, D. & Zakharov, V. E. 2008 b Wave–wave interactions in wind-driven mixed seas. In Proceedings of the Rogue Waves 2008 Workshop, Brest, France (ed. Olagnon, M. & Prevosto, M.), pp. 77–86. IFREMER.Google Scholar
Badulin, S. I., Pushkarev, A. N., Resio, D. & Zakharov, V. E. 2005 Self-similarity of wind-driven seas. Nonlinear Proc. Geophys. 12, 891–946.CrossRefGoogle Scholar
Banner, M. L., Babanin, A. V. & Young, I. R. 2000 Breaking probability for dominant waves on the sea surface. J. Phys. Oceanogr. 30, 3145–3160.2.0.CO;2>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, 1550–1570.2.0.CO;2>CrossRefGoogle Scholar
Benoit, M. 2005 Evaluation of methods to compute the non-linear quadruplet interactions for deep-water wave spectra. In 5th International Symposium on Ocean Wave Measurement and Analysis (WAVES'2005), Madrid, Spain, Paper 52.Google Scholar
Benoit, M. 2006 Implementation and test of improved methods for evaluation of nonlinear quadruplet interactions in a third generation wave model. In Proceedings of the 30th International Conference on Coastal Engineering, San Diego, CA, pp. 526–538.Google Scholar
Benoit, M. & Gagnaire-Renou, E. 2007 Interactions vague–vague non-linéaires et spectre d'équilibre pour les vagues de gravité en grande profondeur d'eau. In Proceedings of the 18th Congrès Français de Mécanique, Grenoble, France (in French).Google Scholar
Benoit, M., Marcos, F. & Becq, F. 1996 Development of a third generation shallow-water wave model with unstructured spatial meshing. In Proceedings of the 25th International Conference on Coastal Engineering, Orlando, FL, pp. 465–478.Google Scholar
Booij, N., Ris, R. C. & Holthuijsen, L. H. 1999 A third-generation wave model for coastal regions. Part I. Model description and validation. J. Geophys. Res. 104 (C4), 7649–7666.CrossRefGoogle Scholar
Bottema, M. & vanVledder, G. P. Vledder, G. P. 2008 Effective fetch and non-linear four-wave interactions during wave growth in slanting fetch conditions. Coast. Engng 55, 261–275.CrossRefGoogle Scholar
Cavaleri, L., Alves, J.-H. G. M., Ardhuin, F., Babanin, A., Banner, M., Belibassakis, K., Benoit, M., Donelan, M., Groeneweg, J., Herbers, T. H. C., Hwang, P., Janssen, P. A. E. M., Janssen, T., Lavrenov, I. V., Magne, R., Monbaliu, J., Onorato, M., Polnikov, V., Resio, D., Rogers, W. E., Sheremet, A., McKee Smith, J., Tolman, H. L., van Vledder, G., Wolf, J. & Young, I. 2007 Wave modelling: the state of the art. Program. Ocean. 75, 603–674.CrossRefGoogle Scholar
Cavaleri, L. & Malanotte-Rizzoli, P. 1981 Wind wave prediction in shallow water: theory and applications. J. Geophys. Res. 86 (C5), 10 961–10 975.CrossRefGoogle 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
Donelan, M. A. & Pierson, W. J. Jr 1987 Radar scattering and equilibrium ranges in wind-generated waves with application to scatterometry. J. Geophys. Res. 92 (C5), 4971–5029.Google Scholar
Donelan, M. A., Skafel, M., Graber, H., Liu, P., Schwab, D. & Venkatesh, S. 1992 On the growth rate of wind-generated waves. Atmos. Ocean 30 (3), 457–478.CrossRefGoogle Scholar
Filipot, J.-F., Ardhuin, F. & Babanin, A. V. 2010 A unified deep-to-shallow water wave-breaking probability parameterization. J. Geophys. Res. 115, C04022.CrossRefGoogle Scholar
Gagnaire-Renou, E., Benoit, M. & Forget, P. 2010 Ocean wave spectrum properties as derived from quasi-exact computations of nonlinear wave–wave interactions. J. Geophys. Res. (in press), doi:10.1029/2009JC005665.CrossRefGoogle Scholar
Günther, H., Hasselmann, S. & Janssen, P. A. E. M. 1992 The WAM model Cycle 4 (revised version). Tech Rep. 4. Deutsch. Klimatol. Rechenzentrum, Hamburg, Germany.Google Scholar
Hara, T. & Belcher, S. E. 2002 Wind forcing in the equilibrium range of wind-wave spectra. J. Fluid Mech. 470, 223–245.CrossRefGoogle Scholar
Hashimoto, N., Tsuruya, H. & Nakagawa, Y. 1998 Numerical computations of the nonlinear energy transfer of gravity-wave spectra in finite water depths. Coast. Engng J. 40, 23–40.CrossRefGoogle Scholar
Hasselmann, D. E., Dunkel, M. & Ewing, J. A. 1980 Directional wave spectra observed during JONSWAP 1973. J. Phys. Oceanogr. 10, 1264–1280.2.0.CO;2>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. 1974 On the spectral dissipation of ocean waves due to white capping. Boundary-Layer Meteorol. 6, 107–127.CrossRefGoogle Scholar
Hasselmann, K., Ross, D. B., Müller, P. & Sell, W. 1976 A parametric wave prediction model. J. Phys. Oceanogr. 6, 200–228.2.0.CO;2>CrossRefGoogle Scholar
Hasselmann, S. & Hasselmann, K. 1981 A symmetrical method of computing the nonlinear transfer in a gravity wave spectrum. Hamburger Geophysikalische Einzelschriften, Reihe A, Heft 52, Max-Planck-Institut für Meteorologie, 138pp.Google Scholar
Hasselmann, S. & Hasselmann, K. 1985 Computations and parametrizations 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, 1369–1377.2.0.CO;2>CrossRefGoogle Scholar
Hasselmann, S., Hasselmann, K., Allender, J. H. & Barnett, T. P. 1985 Computations and parametrizations of the nonlinear energy transfer in a gravity-wave spectrum. Part II. Parameterizations of the nonlinear energy transfer for application in wave models. J. Phys. Oceanogr. 15, 1378–1391.2.0.CO;2>CrossRefGoogle Scholar
Hersbach, H. & Janssen, P. A. E. M. 1999 Improvement of the short-fetch behavior in the wave ocean model (WAM). J. Atmos. Ocean. Tech. 16, 884–892.2.0.CO;2>CrossRefGoogle Scholar
Hsiao, S. V. & Shemdin, O. H. 1983 Measurements of wind velocity and pressure with a wave follower during MARSEN. J. Geophys. Res. 88 (C14), 9841–9849.CrossRefGoogle Scholar
Janssen, P. A. E. M. 1989 Wave-induced stress and the drag of air flow over sea waves. J. Phys. Oceanogr. 19, 745–754.2.0.CO;2>CrossRefGoogle Scholar
Janssen, P. A. E. M. 1991 Quasi-linear theory of wind-wave generation applied to wave forecasting. J. Phys. Oceanogr. 21, 1631–1642.2.0.CO;2>CrossRefGoogle Scholar
Kahma, K. K. & Calkoen, C. J. 1992 Reconciling discrepancies in the observed growth of wind-generated waves. J. Phys. Oceanogr. 22, 1389–1405.2.0.CO;2>CrossRefGoogle Scholar
Kitaigorodskii, S. A. 1962 Applications of the theory of similarity to the analysis of wind-generated wave motion as a stochastic process. Bull. Acad. Sci. USSR, Geophys. Ser., Engl. Transl. N1, 105–117.Google Scholar
Kitaigorodskii, S. A. 1983 On the theory of the equilibrium range in the spectrum of wind-generated gravity waves. J. Phys. Oceanogr. 13, 816–827.2.0.CO;2>CrossRefGoogle Scholar
Komatsu, K. & Masuda, A. 1996 A new scheme of nonlinear energy transfer among wind waves: RIAM method. Algorithm and performance. J. Oceanogr. Soc. Japan 52, 509–537.CrossRefGoogle 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
Komen, G. J., Hasselmann, S. & Hasselmann, K. 1984 On the existence of a fully developed wind-sea spectrum. J. Phys. Oceanogr. 14, 1271–1285.2.0.CO;2>CrossRefGoogle Scholar
Korotkevich, A. O., Pushkarev, A. N., Resio, D. & Zakharov, V. E. 2008 Numerical verification of the weak turbulent model for swell evolution. Eur. J. Mech. B Fluids 27, 361–387.CrossRefGoogle 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
Lavrenov, I. V. 2001 Effect of wind wave parameter fluctuation on the nonlinear spectrum evolution. J. Phys. Oceanogr. 31, 861–873.2.0.CO;2>CrossRefGoogle Scholar
Lavrenov, I. V. 2003 Wind-Waves in Oceans: Dynamics and Numerical Simulations. Springer.CrossRefGoogle Scholar
Masuda, A. 1986 Nonlinear energy transfer between random gravity waves. In Waves Dynamics and Radio Probing of the Ocean Surface (ed. Phillips, O. & Hasselmann, K.), pp. 136–149. Plenum Press.Google Scholar
Pettersson, H. 2004 Wave growth in a narrow bay. PhD thesis, University of Helsinki, Helsinki, Finland.Google Scholar
Phillips, O. M. 1981 The dispersion of short wavelets in the presence of a dominant long wave. J. Fluid Mech. 107, 465–485.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
Pierson, W. J. & Moskowitz, L. A. 1964 A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii. J. Geophys. Res. 69, 5181–5190.CrossRefGoogle Scholar
Plant, W. J. 1982 A relationship between wind stress and wave slope. J. Geophys. Res. 87 (C3), 1961–1967.CrossRefGoogle Scholar
Polnikov, V. G. 1990 Numerical solution of the kinetic equation for surface gravity waves. Izv. Atmos. Ocean. Phys. 26 (2), 168–176.Google Scholar
Polnikov, V. G. 1997 Nonlinear energy transfer through the spectrum of gravity waves for the finite depth case. J. Phys. Oceanogr. 27, 1481–1491.2.0.CO;2>CrossRefGoogle Scholar
Pushkarev, A. N., Resio, D. & Zakharov, V. E. 2003 Weak turbulent theory of the wind-generated gravity sea waves. Phys. D Nonlinear Phenom. 184, 29–63.CrossRefGoogle Scholar
Resio, D. T., Long, C. E. & Vincent, C. L. 2004 Equilibrium-range constant in wind-generated wave spectra. J. Geophys. Res. 109, C01018.CrossRefGoogle Scholar
Resio, D. T. & Perrie, W. 1991 A numerical study of nonlinear energy fluxes due to wave–wind interactions. J. Fluid Mech. 223, 603–629.CrossRefGoogle Scholar
Snyder, R. L., Dobson, F. W., Elliot, J. A. & Long, R. B. 1981 Array measurements of atmospheric pressure fluctuations above surface gravity waves. J. Fluid Mech. 102, 1–59.CrossRefGoogle Scholar
Stewart, R. W. 1974 The air–sea momentum exchange. Boundary-Layer Meteorol. 6, 151–167.CrossRefGoogle Scholar
SWAMP Group 1985 Ocean Wave Modelling. Plenum Press.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.CrossRefGoogle Scholar
Toba, Y. 1978 Stochastic form of the growth of wind waves in a single-parameter representation with physical implementations. J. Phys. Oceanogr. 8, 494–507.2.0.CO;2>CrossRefGoogle Scholar
Toba, Y. 1997 Wind-wave strong wave interactions and quasi-local equilibrium between wind and wind sea with the friction velocity proportionality. In Nonlinear Ocean Waves (ed. Perrie, W.), Advances in Fluid Mechanics, vol. 17, pp. 1–59. Computational Mechanics Publications.Google Scholar
Tolman, H. L. 1991 A third generation model for wind on slowly varying, unsteady and inhomogeneous depth and currents. J. Phys. Oceanogr. 21, 766–781.2.0.CO;2>CrossRefGoogle Scholar
Tolman, H. L. 1992 Effects of numerics on the physics in a third-generation wind-wave model. J. Phys. Oceanogr. 22, 1095–1111.2.0.CO;2>CrossRefGoogle Scholar
Tolman, H. L. 2002 User manual and system documentation of WAVEWATCH-III version 2.22. Tech Rep. 222. NOAA/NWS/NCEP/MMAB.Google Scholar
Tolman, H. L. 2004 Inverse modeling of discrete interaction approximations for nonlinear interactions in wind waves. Ocean Model. 6, 405–422.CrossRefGoogle Scholar
Tracy, B. & Resio, D. 1982 Theory and calculation of the nonlinear energy transfer between sea waves in deep water. WES Rep. 11. US Army, Engineer Waterways Experiment Station.Google Scholar
Tsagarali, K. N., Babanin, A. V., Walker, D. J. & Young, I. R. 2010 Numerical investigation of spectral evolution of wind waves. Part I. Wind-input source function. J. Phys. Oceanogr. 40, 656–666.CrossRefGoogle Scholar
vanVledder, G. P. Vledder, G. P. 2006 The WRT method for the computation of non-linear four-wave interactions in discrete spectral wave models. Coastal Engng 53, 223–242.Google Scholar
WAMDI Group 1988 The WAM model: a third generation ocean wave prediction model. J. Phys. Oceanogr. 18, 1775–1810.2.0.CO;2>CrossRefGoogle Scholar
Webb, D. J. 1978 Non-linear transfers between sea waves. Deep Sea Res. 25, 279–298.CrossRefGoogle Scholar
van der Westhuysen, A. J., Zijlema, M. & Battjes, J. A. 2007 Nonlinear saturation-based whitecapping dissipation in SWAN for deep and shallow water. Coastal Engng 454, 151–170.CrossRefGoogle Scholar
Young, I. R. 2006 Directional spectra of hurricane wind waves. J. Geophys. Res. 111, C08020.CrossRefGoogle Scholar
Young, I. R. & Babanin, A. V. 2006 Spectral distribution of energy dissipation of wind-generated waves due to dominant wave breaking. J. Phys. Oceanogr. 36, 376–394.CrossRefGoogle Scholar
Young, I. R., Hasselmann, S. & Hasselmann, K. 1987 Computations of the response of a wave spectrum to a sudden change in the wind direction. J. Phys. Oceanogr. 17, 1317–1338.2.0.CO;2>CrossRefGoogle Scholar
Young, I. R. & van Vledder, G. 1993 A review of the central role of nonlinear interactions in wind-wave evolution. Phil. Trans. R. Soc. Lond. 342, 505–524.Google Scholar
Zakharov, V. E. 1966 Problems of the theory of nonlinear surface waves. PhD thesis, Budker Institute for Nuclear Physics, Novosibirsk, Russia.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, 327–344.CrossRefGoogle Scholar
Zakharov, V. E. 2009 On domination of nonlinear wave interaction in the energy balance of wind-driven sea. In Proceedings of the 11th International Workshop on Wave Hindcasting and Forecasting, Halifax, Canada, 2009.Google Scholar
Zakharov, V. E., Falkovich, G. & Lvov, V. 1992 Kolmogorov Spectra of Turbulence. Part I. Springer.CrossRefGoogle Scholar
Zakharov, V. E. & Filonenko, N. N. 1966 Energy spectrum for stochastic oscillations of the surface of a fluid. Sov. Phys. Dokl. 160, 1292–1295.Google Scholar
Zakharov, V. E., Korotkevich, A. O., Pushkarev, A. N. & Resio, D. 2007 Coexistence of weak and strong wave turbulence in a swell propagation. Phys. Rev. Lett. 99 (164501).CrossRefGoogle Scholar
Zakharov, V. E. & Zaslavsky, M. M. 1982 a Kinetic equation and Kolmogorov spectra in the weak-turbulence theory of wind waves. Izv. Atmos. Ocean. Phys. 18 (9), 970–980.Google Scholar
Zakharov, V. E. & Zaslavsky, M. M. 1982 b Integrals of input and dissipation in the weak-turbulence theory of wind-generated waves. Izv. Atmos. Ocean. Phys. 18 (10), 1066–1076.Google Scholar
Zakharov, V. E. & Zaslavsky, M. M. 1983 a Shape of spectrum of energy carrying components of a water surface in the weak-turbulence theory of wind waves. Izv. Atmos. Ocean. Phys. 19 (3), 207–212.Google Scholar
Zakharov, V. E. & Zaslavsky, M. M. 1983 b 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 (4), 300–306.Google Scholar
Zaslavsky, M. M. 1984 Wind-wave forecasting as a problem of the weak turbulence theory. Doctor of Science thesis, P. P. Shirshov Institute of Oceanology of the Academy of Sciences of the USSR, Moscow, Russia (in Russian).Google Scholar