Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 258

According to Wikipedia (http://en.wikipedia.org/wiki/Kurtosis),

“In probability theory and statistics, kurtosis ... is a measure of the “peakedness” of the probability
distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to
infrequent extreme deviations, as opposed to frequent modestly sized deviations”.

For a Gaussian-distributed variable, which the surface-wave elevations are often assumed
to be, there should be

K = 3 . (7.17)

This is the case for the broken waves in (7.15) ,butthe incipient-breaking trains obvi-
ously exhibit the extreme deviations, which again highlights the connection of the breaking
events with the wave-group structure.
The non-breaking waves in this strongly forced steep-wave situation have quite high
skewness (1.2) , but the breaking waves are even more skewed. Note that these are ensemble-
average values which makes them particularly high. There was a possibility that, because
of the difference in skewness between breaking and non-breaking waves, there will be a
difference in surface orbital velocities and therefore in Doppler shifts between the breaking
and non-breaking segments. These effects were examined and found to be negligible.
The difference for the vertical asymmetry (1.3) is remarkable: non-breaking waves are,
approximately, symmetric, whereas their breaking counterparts show very large mean neg-
ative asymmetry (i.e. these waves are on average tilted forward). Since, as has been dis-
cussed in Section 5.1.1 , at the very point of the breaking onset the individual wave-to-break
is nearly symmetric, the negative asymmetry indicates the obvious fact that the waves mea-
sured in the incipient-breaking trains are already breaking (they were detected through
their whitecapping in the first place). Since, however, the average asymmetry here is an
ensemble-average, its distinctly negative value for the trains with many breaking waves
has an important implication.
Indeed, if the waves do not break, their asymmetry still goes through the oscillations, that
is the waves periodically tilt forwards, then recover the symmetric shape and tilt backwards
(see Sections 4 , 5.1 , Agnon et al. ( 2005 )). On average, however, the asymmetry of such
trains should be zero, as it certainly is in the post-breaking train (7.15) . When the waves
break, they tilt forwards and then do not appear to recover their symmetric form, at least not
until the breaking is ceased and a new cycle of the wave re-development starts. Therefore,
wave trains which contain the breaking waves will have negative-on-average asymmetry,
depending on how many breakers are embedded into the record. Thus, proper calibration
of the breaking probability in terms of ensemble-average wave-train asymmetry should
allow us to estimate wave-breaking rates in such trains without actually having to observe
and detect the breaking events, at least for the dominant waves. This has been a long-
standing problem, i.e. judging on the wave-breaking rates in widely available records of
surface-wave elevations. Based on linear interpolation of (7.15) , we can suggest that
A s 0%breaking

= 0

(7.18)

A s 100%breaking =− 0 . 2 ,