Geoscience Reference
In-Depth Information
The most obvious feature of the simulations is the oscillations of the values of steepness,
asymmetry and skewness. These are not the waves that are shown in the figure, these are
the characteristics of wave nonlinearity evolving in physical space.
The top three panels correspond to a moderate wind-forcing condition of
U
5.
Under such a wind, it takes approximately 32 wave periods to reach the point of breaking.
Oscillations of wave steepness begin immediately and the wave reaches a steepness of
/
c
=
2
.
=
0
26 (first minimum)
within the next wave period. The period of the modulation is equal to twice the wave
period, which is consistent with the theoretical expectations for Benjamin-Feir instability
(
Longuet-Higgins & Cokelet
,
1978
). During each oscillation, the steepness relaxes back to
almost
.
3 (first maximum) within one period which then relaxes back to
=
0
.
269).
With the wind energy input imposed, however, the maxima of instantaneous steepness keep
growing and reach a value of
=
0
.
26 (magnitude of the last steepness trough before breaking is
=
0
.
=
0
.
34 at the point that is interpreted as incipient breaking
by the model
(2.1)
.
The skewness and asymmetry oscillate with the same double-wave period, but without
a noticeable increase in magnitude of the oscillation. For example, the value of
S
k
=
84
of the skewness at breaking is repeatedly reached by the wave in its progress without
breaking. Therefore it appears that the local steepness, if anything, defines the breaking.
Visually, skewness is in phase with the steepness oscillations, and it relaxes back to zero
when the steepness is minimal.
The oscillations of asymmetry are apparently shifted in phase with respect to steep-
ness and skewness. The asymmetry oscillates about zero in the range
0
.
45 which means
that the waves are periodically tilted backward and forward. When steepness (skewness)
is maximal, asymmetry is zero, i.e. the wave is symmetric with respect to the vertical.
If the point of maximum steepness (skewness) is passed without breaking, the asymme-
try becomes negative. That is, the wave begins to lean forward. If this point signifies the
breaking onset, the wave is apparently still continuing to tilt forward, and this explains
why all the breaking waves exhibit negative asymmetry. The negative asymmetry thus is
not an indication of breaking but is rather an indication of the modulation phase at which
breaking in progress may or may not occur.
The second set of three panels correspond to wind forcing of
U
±
0
.
0. Whilst such
forcing is quite strong, and therefore the steepness growth rate is much faster than above,
apart from the steepness growth almost all the other breaking and non-breaking properties
of nonlinear evolution remain similar to the previous test. The wave steepness, skewness
and asymmetry oscillate with the same period and their phase-shifting pattern is qualita-
tively the same, steepness (
/
c
=
5
.
82) values at breaking are
close to those of the above test, and asymmetry at breaking approaches zero.
It is interesting to note that, according to known results on wave amplification by wind,
the wave growth increment at non-extreme conditions should be approximately a quadratic
function of the wind (e.g.
Donelan
et al.
,
2006
). If indeed there is some critical steepness
signifying breaking onset, then doubling the wind speed in numerical tests should lead to
this limiting value being reached four times as fast. This conjecture produces a result close
=
0
.
36) and skewness (
S
k
=
0
.
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