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substantial shift of the energy to the lower sideband was observed at the peak of the
modulation, it was near-recurrent without breaking.
The breaking would change the dynamics in a most principal way. Transfer of the energy
to the lower sideband would increase, and, most essentially, it would become irreversible.
“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; the further evolution of this
almost two-wave system was not studied
...
”
Here, for more details, we will follow
Hwung
et al.
(
2009
) and
Hwung
et al.
(
2010
), who
combined the
Trulsen & Dysthe
(
1990
) and
Tulin & Waseda
(
1999
) theories with the most
recent observations of the wave breaking and dissipation in a two-dimensional super tank
of the National Cheng Kung University of Taiwan.
The super tank is 300m-long, with a cross-section of 5m
×
.
2m. In such a long tank,
there was no need to seed the instability sidebands and therefore the naturally developing
evolution of nonlinear groups was investigated, with the lower and upper sidebands grow-
ing from the background noise (e.g.
Reid
,
1992
;
Babanin
et al.
,
2007a
,
2010a
). The waves
were recorded with 66 high-resolution capacitance-wire probes along the fetch.
The long tank also gave the possibility of studying evolution of the wave system past
the breaking occurrence, and additional features of this subsequent evolution of the wave
trains were revealed (
Hwung
et al.
,
2004
;
Hwung & Chiang
,
2005
). Once the energy was
lost to breaking and nonlinearity could no longer lead to a new breaking, the train resta-
bilised asymmetrically with respect to the two-wave system which was the final stage of the
Tulin & Waseda
(
1999
) observations. The lower sideband grew up further and finally peri-
odic modulations were observed, without further energy loss and with the lower sideband
as the new carrier wave.
Measurements were then compared with the theory.
Hwung
et al.
(
2009
) and
Hwung
et al.
(
2010
) derived the initial evolution equations for energy density
E
5
ga
2
2 and
group velocity
c
g
, following the variational approach of
Tulin
(
1996
) and
Tulin & Li
(
1999
):
=
/
∂
E
∂
t
+
∂
c
g
E
∂
=
I
−
D
b
;
(8.2)
x
a
xx
a
c
g
8
k
2
∂
c
g
∂
c
g
∂
c
g
∂
1
4
k
∂
E
∂
∂
c
g
E
γ
t
+
x
=
x
+
+
D
b
.
(8.3)
∂
x
Here,
I
is the wind-input energy rate, and the nonlinear group velocity is used (see
(2.14)
-
(2.19)
):
1
2
c
2
−
(
ak
)
a
xx
8
ak
2
c
g
=
−
.
(8.4)
4
Dissipation rate
D
b
and momentum-loss rate
M
b
are such that
γ
D
B
=
cM
b
−
D
b
>
0
(8.5)
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