4.1 Non-Linear Energy Transfer in Wind Wave Spectrum
99
come much wider in this case. The non-linear transfer has the same positivenegative structure of the extremes. The positive maximum G~i) is located
in a general direction at the point Q-(+) ~ 0.95. Two additional maxima are
located at the points a-~+) = 1.9, where the angle f3 ~ ±37.5°. The main
negative minimum G~!) is located at the point a-<-) ~ 1.06. The area of
the negative non-linear transfer function extends from the main minimum to
the left semi-plane over a large distance. It passes round the centre of the
polar coordinate system {iT, /3}, decreasing with distance from the centre. In
close proximity to the beginning of the polar coordinate system {iT, /3} the
non-linear transfer is practically equal to zero. The non-linear transfer begins increasing at the opposite side of the zero area (at f3 ::::: ± 180°). A new
area with positive non-linear energy transfer values is observed, being surrounded with negative values. The maximum area is located at the point
iT~+) = 3.3, f3 ::::: ±180° comprising about 0.5 per cent of the main maximum
of the positive non-linear energy transfer.
Similar calculations, but for the JONSWAP spectrum with peakness
"f = 7.0 have been made for studying a new positive maximum. The relative non-linear transfer value is less in the new maximum area. The new
maximum comprises about 0.1 per cent of the main value.
Similar values for the spectrum peakness "( = 1.0 are shown in Fig. 4.8a,b.
The main details are the same. A new positive maximum is even more explicitly defined. The relative maximum value has been increased and it comprises
3.6 per cent of the main positive maximum of the non-linear energy transfer.
Similar calculations have been performed for the spectrum approximation
proposed by Donelan ( 4.17) with the spectrum peakness "( = 3.3, as well as
for the specified version of this approximation (4.18), suggested by Banner
(see Fig. 4.9a,b). The main details of the non-linear transfer function are the
same, i.e. they have positive and negative extremes. The area of negative
values extends to the high-frequency range along the general direction due
to the high-frequency "tail" of the spectrum approximation decreasing more
slowly as"' cr- 4 • This is contained in the angular sector f3::::: ±45°, starting
from the point iT ::::: 1.0. In the left-hand semi-plane, a new positive area is
located approximately in a similar symmetrical angular sector. Its maximum
is located at the point iT ::::: 1.74, f3 = 180° comprising 0.5 per cent of the
main positive extreme value.
The problem of high-frequency radar measurements of the sea surface is
discussed by Crombie et al. (1978). The existence of spectral components,
propagating against the wind, is shown. The angular energy distribution
function (4.15) with power index 2s = 4 is used to estimate the non-linear
interaction. In this study calculations are repeated for the similar spectral
approximation with peakness "f = 3.3. The obtained results are presented in
Fig. 4.10a,b. The main calculation details of the non-linear interaction are
similar to the previously obtained ones. Four clearly separate areas in the
plane {iT, /3} can be identified. Three of them are characterized by positive
99
come much wider in this case. The non-linear transfer has the same positivenegative structure of the extremes. The positive maximum G~i) is located
in a general direction at the point Q-(+) ~ 0.95. Two additional maxima are
located at the points a-~+) = 1.9, where the angle f3 ~ ±37.5°. The main
negative minimum G~!) is located at the point a-<-) ~ 1.06. The area of
the negative non-linear transfer function extends from the main minimum to
the left semi-plane over a large distance. It passes round the centre of the
polar coordinate system {iT, /3}, decreasing with distance from the centre. In
close proximity to the beginning of the polar coordinate system {iT, /3} the
non-linear transfer is practically equal to zero. The non-linear transfer begins increasing at the opposite side of the zero area (at f3 ::::: ± 180°). A new
area with positive non-linear energy transfer values is observed, being surrounded with negative values. The maximum area is located at the point
iT~+) = 3.3, f3 ::::: ±180° comprising about 0.5 per cent of the main maximum
of the positive non-linear energy transfer.
Similar calculations, but for the JONSWAP spectrum with peakness
"f = 7.0 have been made for studying a new positive maximum. The relative non-linear transfer value is less in the new maximum area. The new
maximum comprises about 0.1 per cent of the main value.
Similar values for the spectrum peakness "( = 1.0 are shown in Fig. 4.8a,b.
The main details are the same. A new positive maximum is even more explicitly defined. The relative maximum value has been increased and it comprises
3.6 per cent of the main positive maximum of the non-linear energy transfer.
Similar calculations have been performed for the spectrum approximation
proposed by Donelan ( 4.17) with the spectrum peakness "( = 3.3, as well as
for the specified version of this approximation (4.18), suggested by Banner
(see Fig. 4.9a,b). The main details of the non-linear transfer function are the
same, i.e. they have positive and negative extremes. The area of negative
values extends to the high-frequency range along the general direction due
to the high-frequency "tail" of the spectrum approximation decreasing more
slowly as"' cr- 4 • This is contained in the angular sector f3::::: ±45°, starting
from the point iT ::::: 1.0. In the left-hand semi-plane, a new positive area is
located approximately in a similar symmetrical angular sector. Its maximum
is located at the point iT ::::: 1.74, f3 = 180° comprising 0.5 per cent of the
main positive extreme value.
The problem of high-frequency radar measurements of the sea surface is
discussed by Crombie et al. (1978). The existence of spectral components,
propagating against the wind, is shown. The angular energy distribution
function (4.15) with power index 2s = 4 is used to estimate the non-linear
interaction. In this study calculations are repeated for the similar spectral
approximation with peakness "f = 3.3. The obtained results are presented in
Fig. 4.10a,b. The main calculation details of the non-linear interaction are
similar to the previously obtained ones. Four clearly separate areas in the
plane {iT, /3} can be identified. Three of them are characterized by positive