Editorial Type:
Article
Article Type:
Research Article

On the Role of Resonant Interactions in the Short-Term Evolution of Deep-Water Ocean Spectra

Mitsuhiro Tanaka Department of Mathematical and Design Engineering, Faculty of Engineering, Gifu University, Gifu, Japan

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Print Publication:
01 Apr 2007
DOI:
https://doi.org/10.1175/JPO3029.1
Page(s):
1022–1036
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Abstract

The temporal evolution of the energy spectrum of a field of random surface gravity waves in deep water is investigated by means of direct numerical simulations of the deterministic primitive equations. The detected rate of change of the spectrum is shown to be proportional to the cubic power of the energy density and to agree very well with the nonlinear energy transfer Snl as predicted by Hasselmann. Despite the fact that use of various asymptotic relations that are valid only for t → ∞ or integration with respect to time over a time scale much longer than O[period × (ak)−2] is necessary in the derivation of Hasselmann’s Snl, it is clearly demonstrated that the rate of change of the spectrum given by the numerical simulation agrees very well with Hasselmann’s Snl at every instance of ordinary time scale comparable to the period. The result implies that the four-wave resonant interactions control the evolution of the spectrum at every instant of time, whereas nonresonant interactions do not make any significant contribution even in a short-term evolution. It is also pointed out that the result may call for a reexamination of the process of derivation of the kinetic equation for the spectrum.

Corresponding author address: Mitsuhiro Tanaka, Department of Mathematical and Design Engineering, Faculty of Engineering, Gifu University, 1-1 Yanagido, Gifu 501-1193, Japan. Email: tanaka@cc.gifu-u.ac.jp

Abstract

The temporal evolution of the energy spectrum of a field of random surface gravity waves in deep water is investigated by means of direct numerical simulations of the deterministic primitive equations. The detected rate of change of the spectrum is shown to be proportional to the cubic power of the energy density and to agree very well with the nonlinear energy transfer Snl as predicted by Hasselmann. Despite the fact that use of various asymptotic relations that are valid only for t → ∞ or integration with respect to time over a time scale much longer than O[period × (ak)−2] is necessary in the derivation of Hasselmann’s Snl, it is clearly demonstrated that the rate of change of the spectrum given by the numerical simulation agrees very well with Hasselmann’s Snl at every instance of ordinary time scale comparable to the period. The result implies that the four-wave resonant interactions control the evolution of the spectrum at every instant of time, whereas nonresonant interactions do not make any significant contribution even in a short-term evolution. It is also pointed out that the result may call for a reexamination of the process of derivation of the kinetic equation for the spectrum.

Corresponding author address: Mitsuhiro Tanaka, Department of Mathematical and Design Engineering, Faculty of Engineering, Gifu University, 1-1 Yanagido, Gifu 501-1193, Japan. Email: tanaka@cc.gifu-u.ac.jp

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  • Annenkov, A. Y., and V. I. Shrira, 2006: Role of non-resonant interactions in evolution of nonlinear random water wave fields. J. Fluid Mech., 561 , 181–207.

    • Search Google Scholar
    • Export Citation

  • Canuto, C., M. Y. Hussaini, A. Q. Quarteroni, and T. A. Zang, 1988: Spectral Methods in Fluid Dynamics. Springer-Verlag, 557 pp.

  • Dommermuth, D. G., and D. K. P. Yue, 1987: A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech., 184 , 267–288.

    • Search Google Scholar
    • Export Citation

  • Donelan, M. A., J. Hamilton, and W. H. Hui, 1985: Directional spectra of wind-generated waves. Philos. Trans. Roy. Soc. London, 315A , 509–562.

    • Search Google Scholar
    • Export Citation

  • Hasselmann, K., 1962: On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory. J. Fluid Mech., 12 , 481–500.

    • Search Google Scholar
    • Export Citation

  • Janssen, P. A. E. M., 2003: Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr., 33 , 863–884.

  • Krasitskii, V. P., 1994: On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech., 272 , 1–20.

  • Masuda, M., 1980: Nonlinear energy transfer between wind waves. J. Phys. Oceanogr., 10 , 2082–2093.

  • Resio, D., and W. Perrie, 1991: A numerical study of nonlinear energy fluxes due to wave-wave interactions. Part 1. Methodology and basic results. J. Fluid Mech., 223 , 603–629.

    • Search Google Scholar
    • Export Citation

  • Tanaka, M., 2001a: Verification of Hasselmans’s energy transfer among surface gravity waves by direct numerical simulations of primitive equations. J. Fluid Mech., 444 , 199–221.

    • Search Google Scholar
    • Export Citation

  • Tanaka, M., 2001b: A method of studying nonlinear random field of surface gravity waves by direct numerical simulation. Fluid Dyn. Res., 28 , 41–60.

    • Search Google Scholar
    • Export Citation

  • Tanaka, M., and N. Yokoyama, 2004: On the effects of discretization of the spectrum in numerical study of water wave turbulence. Fluid Dyn. Res., 34 , 199–216.

    • Search Google Scholar
    • Export Citation

  • West, B. J., K. A. Brueckner, R. S. Janda, M. Milder, and R. L. Milton, 1987: A new numerical method for surface hydrodynamics. J. Geophys. Res., 92 , 11803–11824.

    • Search Google Scholar
    • Export Citation

  • Yokoyama, N., 2004: Statistics of gravity waves obtained by direct numerical simulation. J. Fluid Mech., 501 , 169–178.

  • Young, I. R., and Y. Eldeberky, 1998: Observations of triad coupling of finite depth wind waves. Coastal Eng., 33 , 137–154.

  • Yuen, H. C., and B. M. Lake, 1982: Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech., 22 , 67–229.

  • Zakharov, V. E., 1968: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys., 2 , 190–194.

    • Search Google Scholar
    • Export Citation

  • 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 , 326–344.

    • Search Google Scholar
    • Export Citation

  • Zakharov, V. E., V. S. L’vov, and G. Falkovich, 1992: Kolmogorov Spectra of Turbulence I—Wave Turbulence. Springer, 277 pp.

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