5.3 Spectral Model of Rips
173
spectrum 81 varies during the evolution within a considerably greater range
from 6.5arn to 2arn.
The dot-and-dash line (lines 1'-5') denotes the spectrum 82 of waves
passing from the area of maximum current speed ("barrier") to the area with
current speed being the same as in case of the spectrum 81. Unlike 81, there
is only one peak in the spectrum 82 • This coincides completely with the spectrum 81 being then cut off to zero in the small-frequency area. This cut-off
effect follows from the kinematic conditions for waves being blocked by the
current speed and not able to overcome the "barrier". But a sharply expressed
cut-off effect is hardly possible in field conditions. It appears due to neglecting
the effect of non-linear energy transfer in the wave spectrum and the wind
input. The wind input would have necessarily resulted in additional generation of high-frequency wave components and, accordingly, in a smoother
spectrum decrease at these frequencies. However, rapid spectrum decrease
is really observed within the given frequency range (curve 1 in Fig. 5.5) in
the experiment (Barenblatt et al., 1985). It confirms the obtained solution
qualitatively.
It should be noted that Curves 1-5 for the spectra 81 correspond to
the consecutive stages of spectra evolution with a wave propagating in the
direction of the countercurrent speed increase, while Curves 1'-5' for the
spectrum 82 reflect the reverse situation, i.e. the spectrum 5' precedes the
spectrum 1'. In this case the waves move in the area with decreasing countercurrent speed. As a result, the wave energy is absorbed by the radiation
tensions, and the spectrum 82 is decreased. The wave intensity behind the
barrier (at x > Xrn) is sharply decreased attaining a more regular character.
The waves become longer and gentler.
The spectrum cut-off location due to the blocking of the high-frequency
component is marked by the point A in the spectrum 81, following from
the solution (5.25). The spectrum solution follows from the initial assumptions, as described for the aforesaid spectrum 82• The point A moves to the
high-frequency area with increasing maximum current speed Vrn· It results in
"filling up" the spectrum with components, without being blocked at smaller
values Vrn.
The Phillips equilibrium interval normalized by the maximum of the initial spectrum SR = (v/f))- 5 exp (n~ 1 ) j80 is marked with a dashed line in
Fig. 5.7a (Curves 1"-5"). As can be seen, the second spectral maximum
and the spectrum of high frequencies are 2-3 times higher than the equilibrium interval value. It should be noted that the notion of the equilibrium
interval was introduced by Phillips (1980) as some ultimate spectrum state,
controlled by wave breaking in the presence of wind input. If the wave generation mechanism were connected not with the wind, but with the wave
interaction with a horizontally non-uniform current, the ultimate spectrum
state would not be supposed to coincide precisely with the Phillips equilibrium interval. The level of the spectral high-frequency components can be
diminished significantly in the framework of this model. In order to do this,
173
spectrum 81 varies during the evolution within a considerably greater range
from 6.5arn to 2arn.
The dot-and-dash line (lines 1'-5') denotes the spectrum 82 of waves
passing from the area of maximum current speed ("barrier") to the area with
current speed being the same as in case of the spectrum 81. Unlike 81, there
is only one peak in the spectrum 82 • This coincides completely with the spectrum 81 being then cut off to zero in the small-frequency area. This cut-off
effect follows from the kinematic conditions for waves being blocked by the
current speed and not able to overcome the "barrier". But a sharply expressed
cut-off effect is hardly possible in field conditions. It appears due to neglecting
the effect of non-linear energy transfer in the wave spectrum and the wind
input. The wind input would have necessarily resulted in additional generation of high-frequency wave components and, accordingly, in a smoother
spectrum decrease at these frequencies. However, rapid spectrum decrease
is really observed within the given frequency range (curve 1 in Fig. 5.5) in
the experiment (Barenblatt et al., 1985). It confirms the obtained solution
qualitatively.
It should be noted that Curves 1-5 for the spectra 81 correspond to
the consecutive stages of spectra evolution with a wave propagating in the
direction of the countercurrent speed increase, while Curves 1'-5' for the
spectrum 82 reflect the reverse situation, i.e. the spectrum 5' precedes the
spectrum 1'. In this case the waves move in the area with decreasing countercurrent speed. As a result, the wave energy is absorbed by the radiation
tensions, and the spectrum 82 is decreased. The wave intensity behind the
barrier (at x > Xrn) is sharply decreased attaining a more regular character.
The waves become longer and gentler.
The spectrum cut-off location due to the blocking of the high-frequency
component is marked by the point A in the spectrum 81, following from
the solution (5.25). The spectrum solution follows from the initial assumptions, as described for the aforesaid spectrum 82• The point A moves to the
high-frequency area with increasing maximum current speed Vrn· It results in
"filling up" the spectrum with components, without being blocked at smaller
values Vrn.
The Phillips equilibrium interval normalized by the maximum of the initial spectrum SR = (v/f))- 5 exp (n~ 1 ) j80 is marked with a dashed line in
Fig. 5.7a (Curves 1"-5"). As can be seen, the second spectral maximum
and the spectrum of high frequencies are 2-3 times higher than the equilibrium interval value. It should be noted that the notion of the equilibrium
interval was introduced by Phillips (1980) as some ultimate spectrum state,
controlled by wave breaking in the presence of wind input. If the wave generation mechanism were connected not with the wind, but with the wave
interaction with a horizontally non-uniform current, the ultimate spectrum
state would not be supposed to coincide precisely with the Phillips equilibrium interval. The level of the spectral high-frequency components can be
diminished significantly in the framework of this model. In order to do this,