Geoscience Reference
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4.1.1 Simulating the evolution of nonlinear waves to breaking
For the purposes of studying wave breaking, the model's ability to reproduce wave evo-
lution without limitations in terms of steepness or duration of propagation is crucial. For
this reason, the CS model was chosen in
Babanin
et al.
(
2007a
,
2009a
,
2010a
) for detailed
numerical simulations of physical characteristics of strongly nonlinear waves leading to the
onset of breaking. Before, the CS model was extensively verified and tested (
Chalikov &
Sheinin
,
1998
;
Chalikov & Sheinin
,
2005
) and used in a number of strongly nonlinear
applications (e.g.
Chalikov
,
2005
,
2007
). It was then additionally checked in terms of its
capacity to model nonlinear wave features associated with wave breaking.
As mentioned above, one of the essential checks for a wave-breaking model is its ability
to describe wave asymmetry with respect to the vertical axis. Definitions of the asymmetry
A
s
(1.3)
and skewness
S
k
(which is asymmetry with respect to the horizontal axis
(1.2)
)
are given in
Figure 1.2
and
Section 1.2
. The capacity of the CS model in this regard is
demonstrated in
Figure 2.1
where a transient steep wave dynamically develops very large
asymmetry and skewness.
In numerical simulations of the fully nonlinear evolution of steep two-dimensional waves
to the point of breaking, we will concentrate on three physical properties featuring nonlin-
earity, i.e. wave steepness, skewness and asymmetry, and their inter-relationships. We will
then try to reproduce and investigate these properties in a laboratory experiment with two-
dimensional waves (
Section 5.1.1
). If these properties are indeed linked to wave breaking,
but the percentage of breaking waves is small, as it usually is (e.g.
Babanin
et al.
,
2001
),
then examination of average steepness, skewness or asymmetry is likely to be of little
use. Therefore, the numerical analysis here will be dedicated to nonlinear properties of
individual waves.
In
Figure 4.1
, development of an unforced wave (no wind) to the point of breaking
is shown. The wave shown had the initial monochromatic steepness IMS
16 and is
regarded as moderately steep in terms of the modulational instability. This moderate steep-
ness will allow a reasonably long evolution before breaking occurs. Therefore, the simu-
lation will produce a general, rather than detailed picture to begin with (the time scale is
expressed in wave periods, i.e. the wave breaks after 82 periods).
As seen in the figure, the steepness of individual waves stays reasonably constant for a
significant number of periods (
=
0
.
30), before it starts oscillating noticeably. The magnitude
of the oscillation increases significantly beyond the 60th period, and from this point grows
rapidly until the simulation ceases (wave breaks) after the 80th period mark.
Similar behaviour is exhibited by the skewness and asymmetry. The simulation starts
from a Stokes wave of
S
k
∼
0. It is informative to note that prior to
breaking, the magnitude of the skewness oscillation is so large that at times the wave
even becomes negatively skewed (i.e. the trough is deeper than the crest). At the termi-
nation of the simulation, however,
S
k
=
0
.
18 and
A
s
≈
1, that is the crest is twice as high as the trough
(1.2)
.In
Section 5.1.1
, it is shown that two-dimensional laboratory waves asymptote to
≈
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