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

, cos *β *

,

(17)

*P ωP*

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 *:

(18)

For spectrum

*α * 0 + *α*1

,

(19)

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) *d*β ;

α0 =

(20)

2 π *g * 4 3 P1 3

-π

π

5

23

1ω *P*

∫ S (ω, β) cos(β) *d*β

α1 =

(21)

π *g*7 3M

-π

The energy flux *P *, directed into the high frequency range, is estimated as:

π

∞

∫ ∫ γ(ω, β) *S *(ω, β) *d*ω *d*β

(22)

-π ωp

The momentum flux is estimated similarly:

π

∞

∫∫

γ (ω,β) *S *(ω,β)

cos β *d*ω *d*β

(23)

ω

-π ω p