Preprints
https://doi.org/10.5194/npg-2018-35
https://doi.org/10.5194/npg-2018-35

  09 Aug 2018

09 Aug 2018

Review status: this preprint was under review for the journal NPG but the revision was not accepted.

Asymptotes of the nonlinear transfer and wave spectrum in the frame of the kinetic equation solution

Vladislav G. Polnikov1, Fangli Qiao2,3, and Yong Teng2 Vladislav G. Polnikov et al.
  • 1A.M. Obukhov Institute of Atmospheric Physics of RAS, Moscow, 119017, Russia
  • 2First Institute of Oceanography of SOA, Qingdao, 266061, China
  • 3Laboratory for Regional Oceanography and Numerical Modeling, Qingdao National Laboratory for Marine Science and Technology, Qingdao, 266061, China

Abstract. The kinetic equation for a gravity wave spectrum is solved numerically to study the high frequencies asymptotes for the one-dimensional nonlinear energy transfer and the variability of spectrum parameters that accompany the long-term evolution of nonlinear waves. The cases of initial two-dimensional spectra S(ω,θ) of modified JONSWAP type with the frequency decay-law S(ω) ~ ω−n (for n = 6, 5, 4 and 3.5) and various initial functions of the angular distribution are considered. It is shown that at the first step of the kinetic equation solution, the nonlinear energy transfer asymptote has the power-like decay-law, Nl(ω) ~ ω−p, with values p ≤ n − 1, valid in cases when n ≥ 5, and the difference, n-p, changes significantly when n approaches 4. On time scales of evolution greater than several thousands of initial wave periods, in every case, a self-similar spectrum Ssf(ω,θ) is established with the frequency decay-law of form S(ω) ~ ω−4. Herein, the asymptote of nonlinear energy transfer becomes negative in value and decreases according to the same law (i.e., Nl(ω) ~ −ω−4). The peak frequency of the spectrum, ωp(t), migrates in time t to the low-frequency region such that the angular and frequency characteristics of the two-dimensional spectrum Ssf(ω,θ) remain constant. However, these characteristics depend on the degree of angular anisotropy of the initial spectrum. The solutions obtained are interpreted, and their connection with the analytical solutions of the kinetic equation by Zakharov and co-authors for gravity waves in water is discussed.

Vladislav G. Polnikov et al.

 
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Vladislav G. Polnikov et al.

Vladislav G. Polnikov et al.

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Short summary
The Hasselmann kinetic equation for gravity waves (describing wave turbulence) was solved numerically with the aim of searching for features of the Kolmogorov turbulence. Two versions of the numerical algorithm are used, preserving values of total wave action and energy, because both of them are not preserved. In every case, the solutions result in formation of the same self-similar spectrum shape, with the frequency tail S(ω) ~ ω−4, what contradicts to applicability the Kolmogorov approach.