5.4 Estimation of Non-Linear Wave Interaction in the Rip Spectrum
179
This expression presents the angular spectrum distribution narrowing in
a countercurrent and the presence of its two-peak structure connected with
the existence of straight and reverse waves reflected by blocking.
The estimation of wave breaking in the countercurrent can be introduced
using the Phillips equilibrium interval. If the spectral density is assumed to
be greater than the equilibrium interval as an ultimate value, the spectrum
is taken to be equal to the equilibrium interval value. However, the equilibrium interval is determined by the angular distribution accuracy. It is quite
natural to assume it to be equal to the similar value of ordinary wind waves.
The angular distribution is assumed to expand in the equilibrium area due
to the tendency for spectrum isotropy connected with wave breaking. According to Davidan et al. (1985), the equilibrium interval is approximated
by the quadratic cosine formula ( ~ cos 2 ({3)). Thus, a two-dimensional wave
spectrum in a current is described by the ratio:
where
S(a,{3, V) = min{S(a,{3, V); SR(a,{3)} ,
with l f31 ~ n/2
with l f31 > n/2
(5.33)
(5.34)
The evolution of the frequency-angular wave spectrum S(a, {3), integrated
over the directions and normalized by the maximum of its initial value in the
absence of current is shown in Fig. 5.11:
7[
S(a/amax) = S 0
1 (amax) j S(a,{3, V) d{3.
(5.35)
-7[
The values of the determining parameters in (5.35) are the same as in
the previous section. The graphs of the spectra along the stream at points
of different current velocities v (see Fig. 5.4 for the area x < xm) are given
in Fig. 5.11. The relative value a I a max = a is plotted along the horizontal axis, often used in the presentation of the spectrum rather than in the
non-dimensional frequency fl. There is no symmetrical form in the spectrum
relative to the central frequency fJ = 0.5 (see Fig. 5.7a), as soon as the point
fJ = 1 tends to infinity in these variables. In this case the spectrum S(a)
can be compared to the experimental value presented in Fig. 5.5. There is
a two-peak structure in the spectrum S(a), as shown in Fig. 5.11. An increase
of the wave energy spectral density takes place in waves propagating to the
larger countercurrent velocity area, whereas the spectral density in the vicinity of the high-frequency spectral maximum is increased to a greater extent.
It is shifted to the low-frequency area. At the same time the low-frequency
maximum is shifted to higher frequencies to a lesser extent. Both maxima
converge into one at v > 0.2.
179
This expression presents the angular spectrum distribution narrowing in
a countercurrent and the presence of its two-peak structure connected with
the existence of straight and reverse waves reflected by blocking.
The estimation of wave breaking in the countercurrent can be introduced
using the Phillips equilibrium interval. If the spectral density is assumed to
be greater than the equilibrium interval as an ultimate value, the spectrum
is taken to be equal to the equilibrium interval value. However, the equilibrium interval is determined by the angular distribution accuracy. It is quite
natural to assume it to be equal to the similar value of ordinary wind waves.
The angular distribution is assumed to expand in the equilibrium area due
to the tendency for spectrum isotropy connected with wave breaking. According to Davidan et al. (1985), the equilibrium interval is approximated
by the quadratic cosine formula ( ~ cos 2 ({3)). Thus, a two-dimensional wave
spectrum in a current is described by the ratio:
where
S(a,{3, V) = min{S(a,{3, V); SR(a,{3)} ,
with l f31 ~ n/2
with l f31 > n/2
(5.33)
(5.34)
The evolution of the frequency-angular wave spectrum S(a, {3), integrated
over the directions and normalized by the maximum of its initial value in the
absence of current is shown in Fig. 5.11:
7[
S(a/amax) = S 0
1 (amax) j S(a,{3, V) d{3.
(5.35)
-7[
The values of the determining parameters in (5.35) are the same as in
the previous section. The graphs of the spectra along the stream at points
of different current velocities v (see Fig. 5.4 for the area x < xm) are given
in Fig. 5.11. The relative value a I a max = a is plotted along the horizontal axis, often used in the presentation of the spectrum rather than in the
non-dimensional frequency fl. There is no symmetrical form in the spectrum
relative to the central frequency fJ = 0.5 (see Fig. 5.7a), as soon as the point
fJ = 1 tends to infinity in these variables. In this case the spectrum S(a)
can be compared to the experimental value presented in Fig. 5.5. There is
a two-peak structure in the spectrum S(a), as shown in Fig. 5.11. An increase
of the wave energy spectral density takes place in waves propagating to the
larger countercurrent velocity area, whereas the spectral density in the vicinity of the high-frequency spectral maximum is increased to a greater extent.
It is shifted to the low-frequency area. At the same time the low-frequency
maximum is shifted to higher frequencies to a lesser extent. Both maxima
converge into one at v > 0.2.