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where
is an empirical coefficient of the order of 1, which in numerical simulations
described here was varied in the range
γ
γ =
0
.
3-0
.
8(in Tulin & Waseda ( 1999 ), it was
chosen
4, see also (7.14) ).
The positive last term in the evolution equation for group velocity (8.3) simulates the
wave breaking. It signifies a continuous increase of the group velocity, that is the frequency
downshift at the rate which depends on the breaking energetics.
Hwung et al. ( 2009 ) and Hwung et al. ( 2010 ) combined (8.2) - (8.3) into a single equation
for the complex amplitude B :
γ
0
.
B I
2 dx
kD b
g
c g
B
B
D b
k 2
2 B
t +
x +
ω
|
|
=
γ
i
B
i 4
(8.6)
2
g
|
B
|
|
B
|
where
ae i + α)
B
=
(8.7)
and the phases are
θ =
k 0 x
ω 0 t
(8.8)
for the carrier waves, and
α
is the fluctuating phase such that (see (8.4) ):
2
2 k ∂α
1
x = (
ak
)
a xx
8 ak 2 .
+
(8.9)
4
2 , and therefore another empiri-
cal proportionality coefficient was introduced. The dissipation D b , following Tulin ( 1996 )
parameterisation based on the fetch laws, was adopted as
In the computations, input I was assumed I
g
|
B
|
2
D b =
.
ω · (
)
.
0
1 E
ak
(8.10)
To compare the theory with the experiments, a solution was sought for a wave system
consisting of the respective carrier wave and the two resonant sidebands. Simulations were
performed for the range of initial carrier steepness of
25.
The comparison was very good for the evolution leading to the breaking, including the
breaking event itself. Initial exponential growth rates of the resonant waves, asymmetri-
cal growth of the sidebands, distance to the location where the amplitudes of the carrier
waves and the sub-harmonics were level, redistribution of energy between the harmonics
both at the growing and breaking-in-progress stage - all of these features were correctly
reproduced quantitatively.
Past the breaking occurrence, however, the train stabilisation was not described by the
model. Rather, it continued to quench the wave amplitude as the wave-breaking term in the
model was still active.
To help this apparent shortcoming of the model, the Heaviside function of Trulsen &
Dysthe ( 1990 ) from (8.1) was incorporated into (8.6) :
0 =
a 0 k 0 =
0
.
15
0
.
B I
2 dx
H |
1
B
c g
B
D b
kD b
g
B X |
B thr
k 2
2 B
t +
x +
i
ω
|
B
|
=
i 4
γ
·
.
(8.11)
g
|
B
|
2
|
B
|
 
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