energy transfer. The Kolmogorov constants should be defined using numerical results in this range. The high
frequency damping is located in the domain ( ω
> ωp ). Values of the frequencies ω min ; ω f , ω p are defined in
such a way that the appropriate frequency ranges include sufficient number of grid points to obtain reliable integral
estimations defined by spectrum values within the corresponding frequency ranges. The following values are used:
ω f = 1.0 , ω p = 6.5
ωmin = 0.5;
In order to comply with the requirements of the kinetic equation (1) application, the coefficient C3 in (12) is defined
as C3 = 0.001 satisfying the conditions of smallness of the growth rate with respect to the co rresponding
f << ω .
The coefficients C1,2 as well as the appropriate frequencies ω min ;ω f , ω p are defined
experimentally. The conditions of effectiveness of fluxes absorption and minimization of the pumpin g and damping
intervals with respect to appropriate frequency are used. It should be noted that exact values of the constants C1,2
(satisfying the conditions C1,2 < 1.0 ) are not so principal to obtain a qualitative solution within the range
ωmin < ω < ωp . The values C1,2
are defined in such a way that the energy, wave action and momentum fluxes
are absorbed by dissipation within appropriate frequency ranges and by numerical accuracy of fluxes estimations.
Numerical results are estimated in 96 directions and 50 frequencies. Such a detailed angular resolution is used for
obtaining an accurate estimation for narrow angular distributions of external force approximation (12) considered in
paper. In this case the angular increment is equal to ∆β 2π ≈ 0.065 , being the same order of frequency increment:
∆ω ω ≈ 0.068 . Such discritization is optimal for numerical integration in general case, when solution is not
known before hand (Lavrenov, 1998).
As the solution is to be obtained for a large time scale (up to 100000 seconds), an optimal numerical algorithm of
non-linear energy transfer computation is used (Lavrenov 1998, 2001).
3. NUMERICAL RESULTS
3.1 Spectrum evolution
Now, the numerical results for the isotropic case with n = 0.0 and A=1.0 should be considered. The frequency
spectrum can be defined as a function of wave action as follows:
∫ S (ω , β ) d β = ∫
β ) 2ω 4
The spectrum (13) for the following time steps: 100, 300, 500, 1000, 3000, 5000, 10000, 30000 and 50000 sec.,
respectively, is shown in the logarithmic scale (Fig 1).
Two different stages can be defined in the wave spectrum evolution. On the first stage the spectrum growth is
observed within the range of the external force impact: 0.5 < ω < 1.0 . The spectrum is quickly increased in more
than 6 orders. Duration of this time interval is estimated as: t ≈ 10000 sec. The spectrum becomes almost stationary
at t ≥ 30000 sec. Its values coincide exactly with the spectrum at t ≈ 50000 sec. The initia l and final stages of
spectral evolution are presented in Fig. 1a. Within the range 0.8 < ω < 8.0 , the stationary spectrum is very close to
the Zakharov- Filonenko one:
S (ω) ≅ω-4.0 0.1
11000 < t < 15000 sec. is presented in Fig.1b.
Spectra evolution within the intermediate time interval