176
5 Wave Evolution in Non-uniform Currents in Deep Water
the help of the Kolmogorov spectrum S ~ g 4 1 3 f7- 11 1 3 . This allows them to
put forward the suggestion that small-scale rip excitations and their further
enlargement are a result of non-linear wave action transfer to low frequencies
up to the limits of the Phillips breaking. However, this explanation of rip formation seems to be doubtful, when the Kolmogorov spectrum is obtained for
the transparency interval in the wind wave spectrum (Zakharov & Zaslavskii,
1982, 1983a,b). The influence of the non-uniform current on waves occurs over
the entire frequency range; that is why the suggestion about the existence of
the transparency interval in the rip spectrum can hardly be fulfilled in reality.
In addition, the asymmetrical form of rip waves ("sharpened crests and
gentle troughs" as described by observers) and their intensive breaking ("boiling water") indicate the role of strong non-linear effects and dissipation over
a wide range of spectral frequencies. These facts are confirmed by the derived
solution (5.25), with the parameter q, determining the spectrum in the highfrequency range. It should be remembered that the non-linear degree of the
initial equation (5.19) is described by the parameter q. As shown, the spectral
solution fh corresponds more to field data with increasing parameter q. This
indicates the significance of strongly non-linear effects on the formation of the
high-frequency spectrum range. But the role of non-linear wave interaction
in forming the rip spectrum remains open.
Formation of two-dimensional rip spectrum. It seems that the nonlinear wave energy transfer could be estimated using the frequency spectrum
approximation derived in the previous section. But in order to do this, it
would be necessary to get not only a one-dimensional frequency spectrum,
but also the spatial or frequency-angular spectrum S(f7, (J), which is not obtained in the experiment (Barenblatt et al., 1985). It is possible to obtain
its analytical solution using (5.19)-(5.22), with the parameter q tending to
infinity. Practically this means that the spectral density value is assumed to
be equal to the Phillips equilibrium interval in the high-frequency area.
Thus, using (5.19) of the wave action spectral density N(k) when N «: N 00 ,
the spectral density of the energy S, depending on the frequency f7 and the
angle (J = arctan(ky/kx) can be easily found:
f7k ak (ak0 ) - l
S(f7,(3) = So(f7o,f3o)-k -a -a
,
f7o o f7
f7o
(5.27)
where f7o = f7 (1 - y cos(f3) ); f3o = arcsin[(f7 / f7o) 2 sin(f3)]; and So(f7o, f3o) is the
initial wave energy spectral density in the area with no current. Its angular
energy distribution function is assumed to be proportional to cosine squared.
The frequency wand the wave vector component ky are preserved along
the ray in the case of wave propagation from the area without any current
towards a non-uniform stationary current, whose velocity is changed along
its direction. In this case, the solution (5.27) can be written in the form:
S(f7, (3) = So [f7 (1 - y cos(f3))] (1 - y cos(f3)) - 2 .
(5.28)
5 Wave Evolution in Non-uniform Currents in Deep Water
the help of the Kolmogorov spectrum S ~ g 4 1 3 f7- 11 1 3 . This allows them to
put forward the suggestion that small-scale rip excitations and their further
enlargement are a result of non-linear wave action transfer to low frequencies
up to the limits of the Phillips breaking. However, this explanation of rip formation seems to be doubtful, when the Kolmogorov spectrum is obtained for
the transparency interval in the wind wave spectrum (Zakharov & Zaslavskii,
1982, 1983a,b). The influence of the non-uniform current on waves occurs over
the entire frequency range; that is why the suggestion about the existence of
the transparency interval in the rip spectrum can hardly be fulfilled in reality.
In addition, the asymmetrical form of rip waves ("sharpened crests and
gentle troughs" as described by observers) and their intensive breaking ("boiling water") indicate the role of strong non-linear effects and dissipation over
a wide range of spectral frequencies. These facts are confirmed by the derived
solution (5.25), with the parameter q, determining the spectrum in the highfrequency range. It should be remembered that the non-linear degree of the
initial equation (5.19) is described by the parameter q. As shown, the spectral
solution fh corresponds more to field data with increasing parameter q. This
indicates the significance of strongly non-linear effects on the formation of the
high-frequency spectrum range. But the role of non-linear wave interaction
in forming the rip spectrum remains open.
Formation of two-dimensional rip spectrum. It seems that the nonlinear wave energy transfer could be estimated using the frequency spectrum
approximation derived in the previous section. But in order to do this, it
would be necessary to get not only a one-dimensional frequency spectrum,
but also the spatial or frequency-angular spectrum S(f7, (J), which is not obtained in the experiment (Barenblatt et al., 1985). It is possible to obtain
its analytical solution using (5.19)-(5.22), with the parameter q tending to
infinity. Practically this means that the spectral density value is assumed to
be equal to the Phillips equilibrium interval in the high-frequency area.
Thus, using (5.19) of the wave action spectral density N(k) when N «: N 00 ,
the spectral density of the energy S, depending on the frequency f7 and the
angle (J = arctan(ky/kx) can be easily found:
f7k ak (ak0 ) - l
S(f7,(3) = So(f7o,f3o)-k -a -a
,
f7o o f7
f7o
(5.27)
where f7o = f7 (1 - y cos(f3) ); f3o = arcsin[(f7 / f7o) 2 sin(f3)]; and So(f7o, f3o) is the
initial wave energy spectral density in the area with no current. Its angular
energy distribution function is assumed to be proportional to cosine squared.
The frequency wand the wave vector component ky are preserved along
the ray in the case of wave propagation from the area without any current
towards a non-uniform stationary current, whose velocity is changed along
its direction. In this case, the solution (5.27) can be written in the form:
S(f7, (3) = So [f7 (1 - y cos(f3))] (1 - y cos(f3)) - 2 .
(5.28)