Definitions for wave breaking - Breaking and Dissipation of Ocean Surface Waves - page 45

of breaking onset, as (2.49) , but rather a point of no return: i.e. if the orbital velocity at
the crest exceeds such a value, it may still be increasing (and perhaps even reach values
comparable with those given by (2.49) ), but inevitably, the wave will eventually break:

d dk =

u orbital = c g . (2.54)

If the crest-particle velocity is below (2.54) -value, the wave will not proceed to breaking.
According to Tulin & Landrini ( 2001 ), the criterion is true not only for deep-water waves,
for which the ratio of phase and group velocities is determined by (2.19) , but also for
modulating trains in finite depths where this ratio is smaller.
Analytically, the criterion is justified by Tulin & Landrini ( 2001 ) through considering the
propagation of Stokes waves through a wave group. In the case of a uniform wave train, the
trajectory of the crest is a horizontal line. In the case of the crest passing through the peak of
a concave envelope, it must decelerate which, for gravity waves, is impossible unless values
of the crest orbital velocities are below those defined by (2.54) . This convincing derivation
is supported by substantial experimental observations outlined further by Tulin & Landrini
( 2001 ). Potentially, this criterion can serve to separate observed wave breakings occurring
due to different physical causes, e.g. modulational-instability breaking from directional-
focusing breaking, which is not connected with the modulation of wave trains ( Fochesato
et al. , 2007 ).
Since most wave measurements are conducted as time series of the surface elevation,
where the wave period (frequency) rather than wavelength (wavenumber) is known, the
Stokes steepness (2.47) can be further converted into its height/period version. This is
usually done by means of a linear dispersion relationship (see 2.17 ). Then, the criterion
suggests that the waves will break if ( Ramberg et al. , 1985 ; Ramberg & Griffin , 1987 )

027 gT 2 H ≥ 0 . . (2.55)

A variety of other criteria, both related and unrelated to the Stokes limit, have also
been proposed. Among the geometric limiters, Longuet-Higgins & Fox 's ( 1977 ) maxi-
mal inclination of the surface, which the Stokes wave can reach before breaking, should be
mentioned:

37 ◦ ,

θ critical = 30 . (2.56)

as it has been employed extensively in experimental studies (see Section 3.3 ).
Other widely used geometric properties of the pre-breaking wave shape are skewness
S k (1.2) and A s (1.3) (statistically, they are the third moment of the surface elevation
and the third moment of the Hilbert transform of surface elevation, respectively). They
were introduced by Kjeldsen & Myrhaug ( 1978 ), along with front and rear crest steepness.
These are empirical, rather than theoretically justified criteria based on the common per-
ception of the breaking wave as one with a sharp crest and front face leaning forward (see
Section 1.2 and Figure 1.2 ). Quantitative criteria for the asymmetry, for example, were sug-
gested by Kjeldsen & Myrhaug ( 1980 ), Myrhaug & Kjeldsen ( 1986 ) and Kjeldsen ( 1990 )