5.5 Wave Element Transformations in Current, Varying Along its Direction
183
Similar calculations performed for a narrower angular distribution
[:::::: cos 4 (,B)] in the equilibrium spectrum area reveal the negative energy outflow at low frequencies. This becomes more intensive and exists not only in
the general direction ,B = 0°, but also in the direction ,B = 15°. Thus, the
statement of Leykin & Monin (1985) that the rip wave spectrum is formed
as a result of its small-scale excitation and subsequent enlargement due to
non-linear wave action transfer to low frequencies is not confirmed even for
the area of the second frequency maximum. As can be seen, non-linear transfer is of a more complicated character. It can also transfer wave action in the
opposite direction at some frequency ranges.
In conclusion it should be noted that the rip spectrum can be conventionally divided into two areas, taking into account the peculiarities of the
non-linear energy transfer, namely, the areas of the first and second maxima.
The intensity of the non-linear energy transfer in the area of the energycarrying frequency range of the second maximum is greater than its value
in the vicinity of the low-frequency one. For this reason non-linear energy
transfer can be considered as one of the most efficient spectrum formation
mechanisms in the vicinity of the second maximum. Practically, it does not
influence low-frequency spectrum formation. The spectrum evolution can be
considered to be a simple superposition of independent harmonics subjected
to the influence of a horizontally non-uniform current. The rip formation
in this frequency range can be described with the help of the "soliton gas"
notion. As soon as two frequency spectrum peaks are merged, a change of
character of the non-linear energy transfer takes place. Being still negligible
in the low-frequency area, a negative energy flux can be observed in the frequency range between the spectral peaks, testifying the energy being pumped
in the opposite direction, i.e. from the lower frequencies to the higher ones.
5.5 Wave Element Transformations in Current,
Varying Along its Direction
At present there are in fact no well-grounded practical recommendations for
estimating wave parameters in a non-uniform current. The results of LonguetHiggins and Stewart, obtained in the early 1960s (1960, 1961, 1962, 1964) are
often used to estimate the current effect on waves. It should be remembered
that the wave energy balance equation, taking into account the radiation
stresses due to the influence of current non-uniformity, is deduced in these
papers. A wave height change in the one-dimensional case of regular wave
propagation in deep water from the area without current (Vo = 0) to the
area with the countercurrent (V < 0) or the fair current (V > 0) is described
by the formula (Longuet-Higgins, Stewart, 1964):
h
ho
y'c(c + 2V) '
(5.37)
183
Similar calculations performed for a narrower angular distribution
[:::::: cos 4 (,B)] in the equilibrium spectrum area reveal the negative energy outflow at low frequencies. This becomes more intensive and exists not only in
the general direction ,B = 0°, but also in the direction ,B = 15°. Thus, the
statement of Leykin & Monin (1985) that the rip wave spectrum is formed
as a result of its small-scale excitation and subsequent enlargement due to
non-linear wave action transfer to low frequencies is not confirmed even for
the area of the second frequency maximum. As can be seen, non-linear transfer is of a more complicated character. It can also transfer wave action in the
opposite direction at some frequency ranges.
In conclusion it should be noted that the rip spectrum can be conventionally divided into two areas, taking into account the peculiarities of the
non-linear energy transfer, namely, the areas of the first and second maxima.
The intensity of the non-linear energy transfer in the area of the energycarrying frequency range of the second maximum is greater than its value
in the vicinity of the low-frequency one. For this reason non-linear energy
transfer can be considered as one of the most efficient spectrum formation
mechanisms in the vicinity of the second maximum. Practically, it does not
influence low-frequency spectrum formation. The spectrum evolution can be
considered to be a simple superposition of independent harmonics subjected
to the influence of a horizontally non-uniform current. The rip formation
in this frequency range can be described with the help of the "soliton gas"
notion. As soon as two frequency spectrum peaks are merged, a change of
character of the non-linear energy transfer takes place. Being still negligible
in the low-frequency area, a negative energy flux can be observed in the frequency range between the spectral peaks, testifying the energy being pumped
in the opposite direction, i.e. from the lower frequencies to the higher ones.
5.5 Wave Element Transformations in Current,
Varying Along its Direction
At present there are in fact no well-grounded practical recommendations for
estimating wave parameters in a non-uniform current. The results of LonguetHiggins and Stewart, obtained in the early 1960s (1960, 1961, 1962, 1964) are
often used to estimate the current effect on waves. It should be remembered
that the wave energy balance equation, taking into account the radiation
stresses due to the influence of current non-uniformity, is deduced in these
papers. A wave height change in the one-dimensional case of regular wave
propagation in deep water from the area without current (Vo = 0) to the
area with the countercurrent (V < 0) or the fair current (V > 0) is described
by the formula (Longuet-Higgins, Stewart, 1964):
h
ho
y'c(c + 2V) '
(5.37)