Detection and measurement of wave breaking - Breaking and Dissipation of Ocean Surface Waves - page 113

density function is a reasonable approximation across a broad range of statistical wave
applications.
For a Gaussian process with a narrow-banded spectrum, which is what the spectrum of
surface waves is, the probability density function of wave heights p ( H ) has a Rayleigh

distribution:

exp

H 2
8 m 0

H
4 m 0

p ( H ) = − . (3.43)

Thus, the probability that wave height H is greater than, for example, some limiting height
H lim is

exp

∞

H lim
H rms

P ( H > H lim ) = p ( H ) dH = − (3.44)

H lim

where

H rms = 8 m 0 . (3.45)

Now, for any given wave field with significant wave height H s (2.39) , the probability
of occurrence of waves exceeding, for example, the Stokes limiting wave steepness (2.47)
can be estimated. In a similar fashion, in terms of the wave amplitude a 2, the

probabilities of occurrence of waves exceeding a threshold acceleration (2.60) or limiting
orbital velocity (2.49) can be estimated.
Longuet-Higgins ( 1969a ) investigated the probability of waves having an acceleration
greater than the limiting downward acceleration for Stokes waves (2.50) . Such waves were
assumed to break until the wave height is reduced back to the limiting value, and the
difference was attributed to dissipation. Deductions were made for a narrow spectrum in
order to obtain the dissipation of wave energy as a function of the spectrum, which is
effectively an essential component of this probability model if assumptions are made with
respect to the breaking severity.
Yuan et al. ( 1986 ) extended the theory of Longuet-Higgins ( 1969a ) by removing the
restriction of a narrow-banded spectrum. Hua & Yuan ( 1992 ) further argued that breaking
does not stop once the wave steepness reaches down to the Stokes limit (2.47) , and applied
a similar probability model to investigate wave-breaking dissipation by assuming that the
lower limit of wave height in the course of breaking is determined by the mean value at
a particular frequency derived from the Phillips ( 1958 ) equilibrium spectrum. In all cases,
the dissipation was found to be a linear function of the wave spectrum.
More recently, as was discussed in Sections 3.2 and 3.7 , it was argued that break-
ing waves do not necessarily have the (2.50) acceleration ( Holthuijsen & Herbers , 1986 ;
Hwang et al. , 1989 ; Liu & Babanin , 2004 ). In addition, once they are breaking they do not
stop at the Stokes limiting steepness but may keep losing energy until their steepness is well
below the Stokes limit and even below the wave mean steepness ( Liu & Babanin , 2004 ;
Babanin et al. , 2009a , 2010a ; Figure 2.3 ) Therefore, even though conceptually attractive,
the probability models, as they have been derived, are so far not well justified quantitatively.

= H /