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Figure 7.1 Evolution of the wave modes for cases with and without breaking events: (a) the carrier,
(b) the lower and (c) the upper sideband waves are presented. The solid line is the evolution of the
case with breaking, and the broken line is the evolution of the non-breaking case. Breaking case is
=
133, δ =
1, δ =
.
.
=
.
.
0
0
785; non-breaking case is
0
0
894. Figure is reproduced from Tulin &
Waseda ( 1999 ) by permission
high-frequency spectrum tail. This includes discussion of differences between their obser-
vations, which concluded that the newly produced tail spectrum was discrete, and those
of Melville ( 1982 ) who, also for modulational-instability breaking, obtained a rise in the
continuous-spectrum tail.
Here, we will mostly be interested in their conclusions on what happened to the initially
existing primary-wave train and two sidebands, as these accounted for the bulk of energy
exchange and dissipation (note that this behaviour will be altered in the presence of the
wind, i.e. Waseda & Tulin , 1999 ). Tulin & Waseda ( 1999 ) write:
“The end state of the evolution following strong breaking is an effective downshifting of the spectral
energy, where the lower and the carrier wave amplitudes nearly coincide”
(see also Reid , 1992 ). This is illustrated in Figure 7.1 : after the breaking, the carrier-wave
amplitude a became approximately 60% of the original amplitude a 0 , the lower-sideband
b
initially rose
higher if compared with the recurrence case, but then continued to decrease towards its
pre-breaking magnitude.
The most important outcome here for further discussions is that, as a result of the break-
ing caused by the modulational instability, the primary wave lost the energy, and this loss
was substantial. The impact of the breaking was spectral, that is - not only the wave energy
was lost from the wave system, but it was also redistributed along the spectrum. The fact
that the energy is passed on to lower spectral scales is very instructive too. If such breaking
happens in real wave fields, it may be responsible, at least partially, for the spectrum peak
downshifting. Since the wind input S in in the radiative transfer equation (2.61) becomes
grew up and stayed at some 70% of a 0 , while the upper-sideband b
+
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