4.1 Non-Linear Energy Transfer in Wind Wave Spectrum
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(such as the location and value of the main positive and negative extremes,
etc.) are confirmed and specified by the numerical results. The strong influence of the frequency-angular spectrum form on the non-linear interaction
function is revealed.
In particular, the non-linear energy transfer is found to be limited by approximately the same frequency-angular interval for the typical cosine spectral approximations (4.14). Its non-zero values are located in the semi-plane
I.BI :::; n/2. In the case of a wider angular distribution function, having small
non-zero values at 1.81 ;:::: n/2, significant changes occur with the non-linear energy transfer function at I.BI ; : : : : n/2. Its value differs from zero over the entire
frequency-angular plane { u, ,B}. The calculation results show great sensitivity of the non-linear energy transfer towards the angular energy distribution
function, especially for the left-hand semi-plane {u,,B} (1.81;:::: n/2).
The non-zero values of the non-linear transfer in the direction opposite to
the general direction of the wave spectrum propagation (i.e. at ,B = 180°) are
of special interest. In spite of the fact that the angular distribution function
of the energy spectrum in this direction is practically equal to zero, the stable
existence of the area of positive non-linear energy transfer values is observed.
These values depend both on the angular distribution function of the energy
spectrum and its frequency approximation. As for the same angular distribution function, the relative value of the non-linear energy transfer becomes
much greater for a wider frequency spectrum. Thus, its value is changed by
more than an order of magnitude with the spectrum peakness being varied
from ' Y = 7.0 to ' Y = 1.0. This indicates an increase of the non-linear transfer
in the direction opposite to the general one with wave development.
It should be noted that the non-linear energy transfer function becomes
equal to zero near the origin of the polar coordinates { u, ,B}, i.e. for the small
frequency values. In other words, the non-linear energy transfer by-passes the
origin of the coordinate system {u,,B}, as soon as it is "prohibited" by the
resonance conditions (4.2).
The following interpretation of the spectral components generated in the
opposite direction to the wind can be suggested with the help of the aforementioned facts. At the initial stage of offshore wave development, when they
are formed under a uniform offshore wind, their spectrum is represented by
a sufficiently narrow frequency-angular approximation. Its general direction
coincides with the wind. The angular wave distribution becomes wider with
its further development as a result of wind speed variations and non-linear
transfer. At some moment, components whose direction is different from the
average wind direction by an angle greater than 90° (I,BI ; : : : : n/2), can appear
in the wave spectrum.
From this moment on, the generation of spectral components directed
against the wind occurs due to non-linear transfer.
It should be noted that Hasselmann (1962) interpreted the integral (4.1)
in terms of quadrupole interactions between three active wave components,
which define the interaction intensity, and a passive fourth component, re-
103
(such as the location and value of the main positive and negative extremes,
etc.) are confirmed and specified by the numerical results. The strong influence of the frequency-angular spectrum form on the non-linear interaction
function is revealed.
In particular, the non-linear energy transfer is found to be limited by approximately the same frequency-angular interval for the typical cosine spectral approximations (4.14). Its non-zero values are located in the semi-plane
I.BI :::; n/2. In the case of a wider angular distribution function, having small
non-zero values at 1.81 ;:::: n/2, significant changes occur with the non-linear energy transfer function at I.BI ; : : : : n/2. Its value differs from zero over the entire
frequency-angular plane { u, ,B}. The calculation results show great sensitivity of the non-linear energy transfer towards the angular energy distribution
function, especially for the left-hand semi-plane {u,,B} (1.81;:::: n/2).
The non-zero values of the non-linear transfer in the direction opposite to
the general direction of the wave spectrum propagation (i.e. at ,B = 180°) are
of special interest. In spite of the fact that the angular distribution function
of the energy spectrum in this direction is practically equal to zero, the stable
existence of the area of positive non-linear energy transfer values is observed.
These values depend both on the angular distribution function of the energy
spectrum and its frequency approximation. As for the same angular distribution function, the relative value of the non-linear energy transfer becomes
much greater for a wider frequency spectrum. Thus, its value is changed by
more than an order of magnitude with the spectrum peakness being varied
from ' Y = 7.0 to ' Y = 1.0. This indicates an increase of the non-linear transfer
in the direction opposite to the general one with wave development.
It should be noted that the non-linear energy transfer function becomes
equal to zero near the origin of the polar coordinates { u, ,B}, i.e. for the small
frequency values. In other words, the non-linear energy transfer by-passes the
origin of the coordinate system {u,,B}, as soon as it is "prohibited" by the
resonance conditions (4.2).
The following interpretation of the spectral components generated in the
opposite direction to the wind can be suggested with the help of the aforementioned facts. At the initial stage of offshore wave development, when they
are formed under a uniform offshore wind, their spectrum is represented by
a sufficiently narrow frequency-angular approximation. Its general direction
coincides with the wind. The angular wave distribution becomes wider with
its further development as a result of wind speed variations and non-linear
transfer. At some moment, components whose direction is different from the
average wind direction by an angle greater than 90° (I,BI ; : : : : n/2), can appear
in the wave spectrum.
From this moment on, the generation of spectral components directed
against the wind occurs due to non-linear transfer.
It should be noted that Hasselmann (1962) interpreted the integral (4.1)
in terms of quadrupole interactions between three active wave components,
which define the interaction intensity, and a passive fourth component, re-