94
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
Banner (1990) specified the formula ( 4.17), using high-frequency stereophotography data, and obtained a new approximation for the parameter B
in the high-frequency band a~ 1.60:
B =lOY'
(4.18)
where y = -0.4 + 0.8393 exp [ -0.567 ln( a 2 )].
In the calculations below the aforementioned frequency-angular spectrum
approximations are used.
First the results of the non-linear energy transfer calculation are given
for the JONSWAP spectral approximation (4.13) with the cosine angular
distribution (4.14). The results are presented on the plane {a,,B} as isolines
of the normalized non-linear energy transfer and frequency-angular spectrum
(see Fig. 4.5a,b):
Gni(a,.B) = Gni(a,.B)f(a;);axS"!axfg 4 ),
S(a, ,8) = S(a, .8)/ Bmax ,
where Bmax is the frequency-angular spectral maximum.
( 4.19)
The JONSWAP normalized spectrum for the peakness "( = 3.3 and the
cosine angular energy distribution, with nf3 = 12 are shown in Fig. 4.5a. The
non-linear transfer values for the same case are shown in Fig. 4.5b. As can be
seen, the non-linear energy transfer value, different from zero, is localized in
the area along the horizontal axis and limited by the angles ±45°. The nonlinear transfer function has two main extremes of the plus-minus type G~~).
The maximum function G~i) is located in the general direction at the point
&(+) = afamax ~ 0.95. The negative extreme G~)) is located at the point
a<-) ~ 1.08.
The local maxima and minima can be found in the area of the main
extremes. T;hus, for a sufficiently narrow angular distribution "" cosnr> (.8),
where nf3 ~ 10, the main maximum is divided into two symmetrical directions relative to the general direction (the results are shown in Fig. 4.5c at
a smaller scale). This is probably evidence of the stabilizing effect of the nonlinear interaction on the angular energy distribution. A sufficiently narrow
angular distribution becomes wider, and vice versa. Moreover, in a general
direction the non-linear transfer becomes negative at frequencies less than
the maximum of the function G~i) with respect to the frequency for the narrow angular distribution. Its extreme is located at the frequency a~-) ~ 0. 7 4,
and its value is two orders less than the main maximum function G~i) (see
Fig. 4.5c).
There is a second extreme at the frequency a~-) ~ 1.35 in the main negative extreme area. It is about 52 per cent of the main negative extreme value.
There are also two additional symmetrical maxima relative to the general
direction. They are located at larger frequencies at the point a~+) ~ 1.62 at
the angle ,8 ~ 20.5°comprising 35 per cent of the main maximum function.
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
Banner (1990) specified the formula ( 4.17), using high-frequency stereophotography data, and obtained a new approximation for the parameter B
in the high-frequency band a~ 1.60:
B =lOY'
(4.18)
where y = -0.4 + 0.8393 exp [ -0.567 ln( a 2 )].
In the calculations below the aforementioned frequency-angular spectrum
approximations are used.
First the results of the non-linear energy transfer calculation are given
for the JONSWAP spectral approximation (4.13) with the cosine angular
distribution (4.14). The results are presented on the plane {a,,B} as isolines
of the normalized non-linear energy transfer and frequency-angular spectrum
(see Fig. 4.5a,b):
Gni(a,.B) = Gni(a,.B)f(a;);axS"!axfg 4 ),
S(a, ,8) = S(a, .8)/ Bmax ,
where Bmax is the frequency-angular spectral maximum.
( 4.19)
The JONSWAP normalized spectrum for the peakness "( = 3.3 and the
cosine angular energy distribution, with nf3 = 12 are shown in Fig. 4.5a. The
non-linear transfer values for the same case are shown in Fig. 4.5b. As can be
seen, the non-linear energy transfer value, different from zero, is localized in
the area along the horizontal axis and limited by the angles ±45°. The nonlinear transfer function has two main extremes of the plus-minus type G~~).
The maximum function G~i) is located in the general direction at the point
&(+) = afamax ~ 0.95. The negative extreme G~)) is located at the point
a<-) ~ 1.08.
The local maxima and minima can be found in the area of the main
extremes. T;hus, for a sufficiently narrow angular distribution "" cosnr> (.8),
where nf3 ~ 10, the main maximum is divided into two symmetrical directions relative to the general direction (the results are shown in Fig. 4.5c at
a smaller scale). This is probably evidence of the stabilizing effect of the nonlinear interaction on the angular energy distribution. A sufficiently narrow
angular distribution becomes wider, and vice versa. Moreover, in a general
direction the non-linear transfer becomes negative at frequencies less than
the maximum of the function G~i) with respect to the frequency for the narrow angular distribution. Its extreme is located at the frequency a~-) ~ 0. 7 4,
and its value is two orders less than the main maximum function G~i) (see
Fig. 4.5c).
There is a second extreme at the frequency a~-) ~ 1.35 in the main negative extreme area. It is about 52 per cent of the main negative extreme value.
There are also two additional symmetrical maxima relative to the general
direction. They are located at larger frequencies at the point a~+) ~ 1.62 at
the angle ,8 ~ 20.5°comprising 35 per cent of the main maximum function.