Weak turbulent approach to the wind-generated gravity sea waves

Abstract

We performed numerical simulation of the kinetic equation describing behavior of an ensemble of random-phase, spatially homogeneous gravity waves on the surface of the infinitely deep ocean. Results of simulation support the theory of weak turbulence not only in its basic points, but also in many details. The weak turbulent theory predicts that the main physical processes taking place in the wave ensemble are down-shift of spectral peak and "leakage" of energy and momentum to the region of very small scales where they are lost due to local dissipative processes. Also, the spectrum of energy right behind the spectral peak should be close to the weak turbulent Kolmogorov spectrum which is the exact solution of the stationary kinetic (Hasselmann) equation. In a general case, this solution is anisotropic and is defined by two parameters-fluxes of energy and momentum to high wave numbers. Even in the anisotropic case the solution in the high wave number region is almost proportional to the universal form ω −4 . This result should be robust with respect to change of the parameters of forcing and damping. In all our numerical experiments, the ω −4 Kolmogorov spectrum appears in very early stages and persists in both stationary and non-stationary stages of spectral development. A very important aspect of the simulations conducted here was the development of a quasi-stationary wave spectrum under wind forcing, in absence of any dissipation mechanism in the spectral peak region. This equilibrium is achieved in the spectral range behind the spectral peak due to compensation of wind forcing and leakage of energy and momentum to high wave numbers due to nonlinear four-wave interaction. Numerical simulation demonstrates slowing down of the shift of the spectral peak and formation of the bimodal angular distribution of energy in the agreement with field and laboratory experimental data. A more detailed comparison with the experiment can be done after developing of an upgraded code making possible to model a spatially inhomogeneous ocean.

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

“…After that, Polnikov (1990) has performed a numerical solution of KE (1), where it was shown the fact of establishing the self-similar shape of wave (ω,θ), in the course of long-term evolution of nonlinear waves. This result was later confirmed in a series of papers by the Zakharov's group (see references in Pushkarev et al, 2003;Badulin et al, 2005) and others (e.g., Lavrenov, 2004;Gagnaire-Renou and Benoit, 2007).…”

“…In this interpretation, the mentioned spectra acquire the proper Kolmogorov-type representation of the form (Zakharov and Zaslavskii, 1982; see also Pushkarev et al, 2003;Badulin et al, 2005) 4 3 4 3 1 1 …”

“…Soon, Zakharov (1966) rederived KE (1) using the technique of the Hamiltonian formalism (see also, Zakharov et al 1992;Badulin et al, 2005). Since late 60s, the KE became a separate subject of studying (e.g., Zakharov and 20 Filonenko, 1966;Zakharov and Zaslavskii, 1982;Webb, 1978;Masuda, 1980;Hasselmann and Hasselmann, 1981;Hasselmann et al, 1985;Polnikov, 1989Polnikov, , 1990Polnikov, , 1996Polnikov, , 2007Resio and Perrie, 1991;Pushkarev et al, 2003;Badulin et al, 2005;Young and Van Vledder, 2003;Van Vledder, 2006;Badulin and Zakharov (2017); Geodjaev and Zakharov 2017;Zakhariov, 2017, and references therein). Among them, analytical results found by Zakharov and co-authors (see below) play the crucial role in these investigations.…”

Quasibreathers in the MMT model

“…Functions T R kk 1 k 2 k 3 from [19] and T kk 1 k 2 k 3 , given by Eq. (5), are essentially different.…”

Wave modelling – The state of the art

“…In recent numerical simulations of equation (3.4), Pushkarev et al (2003) have shown that nonlinear interactions generate an  wave spectrum also in anisotropic conditions. Moreover, they have also shown the formation of the bimodal angular distribution of energy, in agreement with field and laboratory experiments.…”