ω0
ω front ≅
(16)
(t 0 - t )
13
Numerical estimations obtained in our experiment are in agreement with these theoretical results (Fig.2). A self -
similar front propagating solution can be explained as a result of energy diffusion in frequency space in order to
reach a system stable state.
4. ESTIMATION OF KOLMOGOROV'S CONSTANTS
4.1 Definition of Komlogorov's constants
According to the weak turbulence theory the general Kolmogorov spectrum is defined by fluxes of the wave energy
P , wave action Q and momentum M (Zakharov et al., 1992). The spectrum is assumed to be symmetrical with
g 4 3 P1 3 ωQ g M
F
, cos β
S (ω, β) =
,
(17)
P ωP
ω4
In this paper only a case with no wave action flux at infinity: Q = 0 is considered. A similar solution was obtained
analytically (Zakharov and Zaslavskii, 1982) for the infinite frequency interval, when energy pumping was located in
the low frequency range and energy damping in the high frequency one.
For the large value ω , the function F (ξ , β ) can be expanded into the Taylor series for small ξ = g M ωP :
F (ξ, β) ≈ α0 + α1 cos β
(18)
S (ω, β) , it can be presented as follows:
For spectrum
g M cos( β )
g 4 3 P1 3
S (ω , β ) ≅
α 0 + α1
,
(19)
ωP
ω4
where α 0 and
α 1 are the first and the second Kolmogorov constants, which are coefficients of the spectral density
expansion:
π
ω4
1
∫ S (ω, β) dβ ;
α0 =
(20)
2 π g 4 3 P1 3
-π
π
5
23
1ω P
∫ S (ω, β) cos(β) dβ
α1 =
(21)
π g7 3M
-π
The energy flux P , directed into the high frequency range, is estimated as:
π
∞
∫ ∫ γ(ω, β) S (ω, β) dω dβ
P=
(22)
-π ωp
The momentum flux is estimated similarly:
π
∞
k
∫∫
M=
γ (ω,β) S (ω,β)
cos β dω dβ
(23)
ω
-π ω p
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