(1) should be developed with reliable accuracy. That is why it is necessary to use sufficiently accurate and fast
numerical algorithm, not available at that time.
It should be noted that a full -scale experiment on numerical simulation of the equation (1) is performed recently
(Pushkarev et al., 2002). The approach, based on another numerical algorithm (Webb 1978; Resio&Perrie, 1991),
existing for more than two decades, is used. As this problem seems to be rather complicated for numerical
simulations and the results are obtained for the first time, there appears a necessity to produce independent
estimations in order to verify the above-mentioned results with the help of the another algorithm.
This is a motivation for writing present paper. Numerical results of the equation (1) solution taking into account
wave energy forcing and damping are obtained in this paper with the help of the numerical algorithm elaborated
recently by one of the authors (Lavrenov 1998, 2001). The algorithm is based on numerical methods of highest
precision. It is of high accuracy and calculation speed.
2. PROBLEM FORMULATION
The equation of spectral wave action evolution can be written as:
N
+γ
=G +F
N
(9)
nl
t
Gnl is the non -linear energy transfer function (1);
γ is a damping increment depending on the frequency
where
ω as follows :
C ω sin 2 ((1 - ω ω ) (π 2)), if ω < ω
1 min
min
min
γ = 0, if ωmin < ω < ω f
(10),
(
)
2
C 2ω p ω ω p -1 , if ω > ω p
where C1,2 are positive constants.
=
f = f ( ω, β) is a value not equal to zero
The value F is an external active force:
. The function
F
f
N
ωmin < ω < ωf
within the frequency range:
. It is equal to the following angular function:
Q cos n ( A β ), if cos( A β ) > 0 and ωmin < ω < ω f
f =
,
(11)
0, in another cases
where Q is a normalizing function, providin g the same integral value for the various parameters n and A:
π
∫ f (β, ω)dβ = C ω
3
(12)
f
-π
where C3 is a constant.
So, a problem is posed in such a way that the whole frequency range is divided into sub -ranges with
different energy sources. The low-frequency damping domain is located in the range (ω < ω in ) in order to
m
stabilize fluxes directed into the low frequency range. The energy pumping domain is located within the range
( ωmin < ω < ωf ). One of the most interesting frequency ranges is the domain (ω f < ω < ω p ) without any
damping or pumping, i.e. it is the so-called "transparency window", where the spectrum is formed only by non-linear
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