• Barenblatt, G. I., 1984: Asymptotics and Intermediate Self-similarity (in Russian). Hydrometeoisdat, 256 pp.

  • Davidan, I. N., L. I. Lopatukhin, and V. A. Rozhkov, 1985: Wind Sea in the World Ocean (in Russian). Gidrometeoizdat, 248 pp.

  • Hasselmann, K., 1962: On the non-linear energy transfer in a gravity wave spectrum. Part 1. J. Fluid Mech., 12 , 481500.

  • Hasselmann, K., 1963: On the nonlinear energy transfer in a gravity-wave spectrum: Conservation theorem, wave particle correspondence, irreversibility. J. Fluid Mech., 15 , 273281.

    • Search Google Scholar
    • Export Citation
  • Hasselmann, K., and Coauthors. 1973: Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Dtsch. Hydrogr. Z., 8 , (Suppl. A),. 195.

    • Search Google Scholar
    • Export Citation
  • Hasselmann, S., and K. Hasselmann, 1981: A symmetrical method of computing the nonlinear transfer in a gravity wave spectrum. J. Hamburger Geophys. Einzelschrifte, A52 , 138.

    • Search Google Scholar
    • Export Citation
  • Hasselmann, S., and K. Hasselmann, 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 , 13691377.

    • Search Google Scholar
    • Export Citation
  • Komatsu, K., and A. Masuda, 1996: A new scheme of nonlinear energy transfer among wind waves: RIAM method—algorithm and performance. J. Oceanol., 52 , 509537.

    • Search Google Scholar
    • Export Citation
  • Komen, G. J., L. Cavaleri, M. Donelan, K. Hasselmann, S. Hasselmann, and P. A. E. M. Janssen, 1994: Dynamics and Modelling of Ocean Waves. Cambridge University Press, 532 pp.

    • Search Google Scholar
    • Export Citation
  • Lavrenov, I. V., 1991a: Non-linear interaction in rips spectrum. Izv. Acad. Sci. USSR., ser. Phys. Atmos. Ocean, 27 , 438447.

  • Lavrenov, I. V., 1991b: Non-linear evolution of wave spectrum in shallow water area. Izv. Acad. Sci. USSR., ser. Phys. Atmos. Ocean, 27 , 13731378.

    • Search Google Scholar
    • Export Citation
  • Lavrenov, I. V., 1998: Mathematical Modelling of Wind Waves at Non-Uniform Ocean (in Russian). Gidrometeoìzdat, 500 pp.

  • Lavrenov, I. V., 2001: Effect of wind wave parameter fluctuation on the nonlinear spectrum evolution. J. Phys. Oceanogr., 31 , 861873.

    • Search Google Scholar
    • Export Citation
  • Lavrenov, I. V., and T. A. Pasechnik, 1989: Swell propagation calculation for ocean taking into account the earth's surface sphericity. Russ. Meteor. Hydrol., 6 , 7381.

    • Search Google Scholar
    • Export Citation
  • Lavrenov, I. V., and F. J. Ocampo-Torres, 1999: Non-linear energy generation of waves opposite to the wind direction—the wind-driven air–sea interface. Proc. Symp. on the Wind-Driven Air–Sea Interface, Sydney, Australia, School of Mathematics, The University of New South Wales, 141–150.

    • Search Google Scholar
    • Export Citation
  • Masuda, A., 1980: Nonlinear energy transfer between wind waves. J. Phys. Oceanogr., 10 , 20822093.

  • Polnikov, V. G., 1989: Calculation of non-linear energy transfer by surface gravitational waves spectrum. Izv. Acad. Sci. USSR, ser. Phys. Atmos. Ocean, 25 , 12141225.

    • Search Google Scholar
    • Export Citation
  • Polnikov, V. G., 1990: Numerical study of the kinetic equation for surface gravity waves. Izv. Acad. Sci. USSR, ser. Phys. Atmos. Ocean, 26 , 168176.

    • Search Google Scholar
    • Export Citation
  • Polnikov, V. G., 1993: Numerical formation of flux spectrum of surface gravity waves. Izv. Acad. Sci. USSR, ser. Phys. Atmos. Ocean, 29 , 12141225.

    • Search Google Scholar
    • Export Citation
  • Polnikov, V. G., 1999: Numerical study of the equation of non-linear swell in the directional approximation. Izv. Acad. Sci. USSR, ser. Phys. Atmos. Ocean, 35 , 364370.

    • Search Google Scholar
    • Export Citation
  • Resio, D., and W. Perrie, 1991: A numerical study of non-linear energy flux due to wave–wave interaction. Part 1. Methodology and basic results. J. Fluid Mech., 223 , 603629.

    • Search Google Scholar
    • Export Citation
  • Snyder, R. L., W. C. Thacker, K. Hasselmann, S. Hasselmann, and G. Barzel, 1993: Implementation of an efficient scheme for calculating nonlinear transfer from wave–wave interactions. J. Geophys. Res., 98 , 1450714525.

    • Search Google Scholar
    • Export Citation
  • Snodgrass, F. E., G. W. Groves, K. Hasselmann, G. R. Miller, W. H. Munk, and W. H. Powers, 1966: Propagation of ocean swell across the Pacific. Philos. Trans. Roy. Soc. London, 259A , 256271.

    • Search Google Scholar
    • Export Citation
  • Tolman, H. L., 1992: Effects of numerics on the physics in a third-generation wind-wave model. J. Phys. Oceanogr., 22 , 10951111.

  • WAMDI Group, 1988: The WAM model—a third generation ocean wave prediction model. J. Phys. Oceanogr., 18 , 17751810.

  • Webb, D. J., 1978: Non-linear transfer between sea waves. Deep-Sea Res., 25 , 279298.

  • Zakharov, V. E., 1968: Stability of periodic waves of final amplitude on a deep liquid surface. Prikl. Mekh. Tekh. Fiz., 2 , 8694.

  • Zakharov, V. E., and N. N. Filonenko, 1966: The energy spectrum for stochastical oscillation of fluid surface. Dokl. Akad. Nauk SSSR, 170 , 12921295.

    • Search Google Scholar
    • Export Citation
  • Zakharov, V. E., and A. V. Smilga, 1981: About quasi-one–dimensional spectrum of weak turbulence. J. Exp. Tekh. Fiz., 18 , 13181326.

    • Search Google Scholar
    • Export Citation
  • Zakharov, V. E., and M. M. Zaslavskii, 1982: The kinetic equation and Kolmogorov's spectra in the weakly turbulent theory of wind waves. Izv. Acad. Sci. USSR, ser. Phys. Atmos. Ocean, 18 , 970979.

    • Search Google Scholar
    • Export Citation
  • Zakharov, V. E., and M. M. Zaslavskii, 1983: Dependence of wave parameters on wind speed, duration of its action, and fetch in a weakly-turbulent theory of wind waves. Izv. Acad. Sci. USSR., ser. Phys. Atmos. Ocean, 19 , 406416.

    • Search Google Scholar
    • Export Citation
  • Zaslavskii, M. M., 1989: About narrow directional approach of the kinetic equation for wind wave. Izv. Acad. Sci. USSR, ser. Phys. Atmos. Ocean, 25 , 402410.

    • Search Google Scholar
    • Export Citation
  • Zaslavskii, M. M., 2000: Non-linear evolution of swell spectrum. Izv. Acad. Sci. USSR, ser. Phys. Atmos. Ocean, 36 , 275283.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 119 40 3
PDF Downloads 32 24 0

A Numerical Study of a Nonstationary Solution of the Hasselmann Equation

View More View Less
  • 1 State Research Center of the Russian Federation, Arctic and Antarctic Research Institute, St. Petersburg, Russia
Restricted access

Abstract

The Hasselmann kinetic equation describing nonlinear spectrum evolution of gravity surface waves is investigated for their large periods of time. To solve the problem, the most optimal numerical algorithm of the nonlinear energy transfer computation is used. The nonlinear spectrum evolution is described by various features. The numerical results reveal some general details common for all cases of the spectrum evolution. There are two main domains in the spectrum: a sharply defined peak and a slowly decreasing high-frequency tail. Using the numerical results, an intermediate self-similar frequency spectrum asymptotic approximation is proposed. It is confirmed by field observations of swell spectrum propagating at large distances. This approximation describes the main features of the nonlinear spectrum evolution and provides a preservation of the total energy and wave action.

Corresponding author address: Igor V. Lavrenov, State Research Center of the Russian Federation, Arctic and Antarctic Research Institute, Bering 38, St. Petersburg 199397, Russia. Email: lavren@aari.nw.ru

Abstract

The Hasselmann kinetic equation describing nonlinear spectrum evolution of gravity surface waves is investigated for their large periods of time. To solve the problem, the most optimal numerical algorithm of the nonlinear energy transfer computation is used. The nonlinear spectrum evolution is described by various features. The numerical results reveal some general details common for all cases of the spectrum evolution. There are two main domains in the spectrum: a sharply defined peak and a slowly decreasing high-frequency tail. Using the numerical results, an intermediate self-similar frequency spectrum asymptotic approximation is proposed. It is confirmed by field observations of swell spectrum propagating at large distances. This approximation describes the main features of the nonlinear spectrum evolution and provides a preservation of the total energy and wave action.

Corresponding author address: Igor V. Lavrenov, State Research Center of the Russian Federation, Arctic and Antarctic Research Institute, Bering 38, St. Petersburg 199397, Russia. Email: lavren@aari.nw.ru

Save