84
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
where
Sn = S(crn, f3n), J-L = 0.25, C12 = C11, C13 = (1 + J-L)C11, C14 = (1- J-L)C11,
(32 = fJ1, (33 = (31 - 11.48°, (34 = (31 + 33.56°, f = cr j2n, C = 2.8 · 10 7 /2n.
Unfortunately, this approximation takes into account a very limited number of wave components in the interaction. It leads to large discrepancies
between these approximation results and the numerical estimations of the
exact non-linear energy transfer.
A more developed approximation of non-linear energy transfer, the socalled diffusion approximation was proposed by Zakharov and Pushkarev
(1999) and was more accurate than DIA. The diffusion approximation can
be written as follows:
where a is an indefinite constant. The approximation preserves total energy,
wave action and momentum. The disadvantage of this approximation is that
the value a varies by two orders for different wind wave spectrum approximations. The diffusion approximation produces a non-linear energy transfer
value with a narrower frequency form than the exact computation does.
Many qualitative properties of the integral were studied in further calculations (Hasselmann et al., 1985; Polnikov, 1988, 1989; Lavrenov & OcampoTorres, 1999, etc.).
There are two main extremes in the non-linear transfer function: one is
a positive G~i l and the other is a negative G~~) for a typical wind wave spectrum. Their location and values are determined by the spectrum shape. The
positive maximum G~i l is usually located in the general spectral direction at
the point where the non-dimensional frequency is iJ-( + l = cr / CYmax :::::; 0.94-1.00
(where CYmax is the spectral maximum frequency). The negative minimum
G~~) is usually located in the general direction at the point iJ-(- l :::::; 1.05-1.60.
There are two additional positive maximas symmetrical about the general direction. They are located at the point a-~+) :::::; 1.5-3.0, where the angle with
general direction comprises (3:::::; 25°-45°.
However, in spite of these results, the problem of the accuracy of the
calculation remained open in many papers. The estimations showed that
a typical numerical error of collision integral calculations within the spectral
maximum area was not less than 10-50 per cent and could be much higher
in the other frequency-angular bands. Such accuracy might be sufficient for
the usual practical wind wave evolution calculations taking into consideration
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
where
Sn = S(crn, f3n), J-L = 0.25, C12 = C11, C13 = (1 + J-L)C11, C14 = (1- J-L)C11,
(32 = fJ1, (33 = (31 - 11.48°, (34 = (31 + 33.56°, f = cr j2n, C = 2.8 · 10 7 /2n.
Unfortunately, this approximation takes into account a very limited number of wave components in the interaction. It leads to large discrepancies
between these approximation results and the numerical estimations of the
exact non-linear energy transfer.
A more developed approximation of non-linear energy transfer, the socalled diffusion approximation was proposed by Zakharov and Pushkarev
(1999) and was more accurate than DIA. The diffusion approximation can
be written as follows:
where a is an indefinite constant. The approximation preserves total energy,
wave action and momentum. The disadvantage of this approximation is that
the value a varies by two orders for different wind wave spectrum approximations. The diffusion approximation produces a non-linear energy transfer
value with a narrower frequency form than the exact computation does.
Many qualitative properties of the integral were studied in further calculations (Hasselmann et al., 1985; Polnikov, 1988, 1989; Lavrenov & OcampoTorres, 1999, etc.).
There are two main extremes in the non-linear transfer function: one is
a positive G~i l and the other is a negative G~~) for a typical wind wave spectrum. Their location and values are determined by the spectrum shape. The
positive maximum G~i l is usually located in the general spectral direction at
the point where the non-dimensional frequency is iJ-( + l = cr / CYmax :::::; 0.94-1.00
(where CYmax is the spectral maximum frequency). The negative minimum
G~~) is usually located in the general direction at the point iJ-(- l :::::; 1.05-1.60.
There are two additional positive maximas symmetrical about the general direction. They are located at the point a-~+) :::::; 1.5-3.0, where the angle with
general direction comprises (3:::::; 25°-45°.
However, in spite of these results, the problem of the accuracy of the
calculation remained open in many papers. The estimations showed that
a typical numerical error of collision integral calculations within the spectral
maximum area was not less than 10-50 per cent and could be much higher
in the other frequency-angular bands. Such accuracy might be sufficient for
the usual practical wind wave evolution calculations taking into consideration