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

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  • 1 Department of Mathematical and Design Engineering, Faculty of Engineering, Gifu University, Gifu, Japan
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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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