Definitions for wave breaking - Breaking and Dissipation of Ocean Surface Waves - page 46

based on field observations, by Bonmarin ( 1989 ) and Griffin et al. ( 1996 ) from laboratory
experiments, and by Tulin & Landrini ( 2001 ) through numerical simulations. In addition,
Kjeldsen & Myrhaug ( 1980 ) found that the front trough of the incipient breaker is shal-
lower compared to the rear trough, which is a persistent feature of wave breaking due to
modulational instability, observed in the laboratory ( Babanin et al. , 2010a ) and produced
by means of fully nonlinear numerical simulations ( Dyachenko & Zakharov , 2005 , see also
Chapter 4 ).
Tulin & Landrini ( 2001 ), however, point out that deformation of the wave shape prior to
breaking

“has to be viewed as a consequence of the breaking process and not the cause of it”,

and we fully agree with this. In particular, recent investigations of breaking onset brought
about by modulational instability of wave trains showed that the asymmetry at the break-
ing point is a rapidly changing characteristic, with a value close to zero and increasingly
becoming negative. Thus, it would be difficult to employ asymmetry as a certain geometric
breaking criterion. Skewness, on the other hand, indeed exhibits a robust asymptotic trend
to a limiting value of

S k limiting = 1 (2.57)

for two-dimensional waves ( Babanin et al. , 2007a , 2009a , 2010a ) and

S k limiting ≈

.

0 7 (2.58)

for three-dimensional waves ( Babanin et al. , 2011a ).
Returning to the dynamic criteria, the most significant difficulty of applying limiting
downward-acceleration (2.50) to real sea waves is the fact that the natural wave fields are
multi-scaled and therefore the surface elevation at any point and any instant of time is a
superposition of an unlimited number of wave components. Therefore, among other dynamic
criteria, we should mention the downward acceleration derived by Longuet-Higgins ( 1985 )
for a complicated sea, rather than for a monochromatic Stokes wave:

a downward = g . (2.59)

As a general approach, the acceleration criterion is treated in the sense that the wave surface
will break when its downward acceleration exceeds a limiting fraction

γ , which is a tuning

parameter, i.e.

2

a downward = a ω >γ g . (2.60)

When it is applied to a monochromatic wave, this approach is straightforward even
though there are uncertainties about the value of

. In theoretical/statistical studies of
the limiting-steepness Stokes wave, it has generally been assumed that

γ

5( (2.50) ,
Longuet-Higgins et al. , 1963 ; Snyder et al. , 1983 ). The latter paper also verified this limit
by observing the breaking of dominant waves in the field. Field observations of Ochi & Tsai

γ = 0 .