Non-dissipative effects of breaking on the wave field - Breaking and Dissipation of Ocean Surface Waves - page 327

Here, B thr is the limiting value of steepness and B X = ( is the combined steepness

of wave components i at distance X along the tank. The threshold B thr =

a i k i )

0 . 32 was chosen

and perhaps it needs to be updated to B thr = 44 in the future (see Section 5.1.2 ).

Now, the dissipation/input are disallowed past distance X and ordinary nonlinear wave
evolution proceeds. The model, updated this way, correctly reproduces the entire evolution
of the wave train in the super tank. The waves break in the prescribed location, perma-
nent downshift occurs, followed by restabilisation and steepness oscillations with the new
carrier frequency. Hwung et al. ( 2009 ) and Hwung et al. ( 2010 ) conclude that the post-
breaking wave system 'forgets' its pre-breaking initial conditions and further evolution is
described by the nonlinear behaviour of the past-breaking three wave components.
Thus, we can conclude that downshift of the wave energy is a robust feature of the break-
ing due to modulational instability. If such instability is present in the real wave fields,
then the dissipation term in RTE (2.61) should be attributed with the downshifting. Recent
experiments have demonstrated that the modulational instability is indeed active in typi-
cal directional fields with typical background wave steepness ( Babanin et al. , 2011a ,see
Section 5.3.3 ). Since the breaking happens at the scale of tens of wave periods, whereas the
weak nonlinear interactions at the scale of thousands of wave periods (see Babanin et al. ,
2007a , Section 5.1.2 and discussion in Section 7.3.3 ), then S ds term in RTE perhaps can
be held responsible for the downshifting, in addition to that provided by S nl . Such models
have been attempted (e.g. Schneggenburger et al. 2000 ; WISE-meeting presentations by
Donelan and Meza), and this issue needs further research and clarification.

0 .

8.2 Role of wave breaking in maintaining the level of the spectrum tail

Mention of the role of wave breaking in maintaining the level of the spectrum tail is
scattered throughout the topic. We, however, thought that it is helpful to briefly summarise
and update the effects which this role imparts on the spectrum shape. As has been dis-
cussed in Sections 5.3.4 , 6.2 and 7.3.5 , the equilibrium level of this spectrum, even though
with some variations, remains remarkably stable given the very large magnitudes of differ-
ences in wind forcing. These differences occur both in absolute values, that is in terms of
the wind speed which in this topic varies from near-zero to some 30 + m / s, and in relative

terms across the wind-wave scales, that is if U 10 / c p = 1 for waves with f p = 0 . 1 Hz, then

at the 10 Hz tail end it will be U 10 / 100 (neglecting the surface-tension contribution

for clarity). Here, we will largely follow Babanin (2010) in discussing the role of breaking
at the tail of the wave spectrum.
The first analytical justification for the equilibrium range of the spectrum, based on
a dimensional argument, was proposed by Phillips ( 1958 ). The idea of the equilibrium
spectrum tail was already in the air, and a number of experimentalists had suggested
their dependences for such a spectral interval. The first parameterisation was by Neumann
( 1954 ) for a 'fully-developed spectrum',

c p ∼