Definitions for wave breaking - Breaking and Dissipation of Ocean Surface Waves - page 43

In a wave of such limiting steepness, the crest takes the form of a

120 ◦ θ limiting = (2.48)

corner flow (see also McCowan , 1894 ; Packham , 1952 ; Lenau , 1966 ; Price , 1970 , 1971 ).
The Stokes limit can also be translated into kinematic limit, i.e. the surface wave orbital
velocity becomes greater than the wave's phase speed:

u orbital =

ω = ,

a c (2.49)

or into downward surface acceleration, i.e. a dynamic limit of

1
2 g

a downward = . (2.50)

Because of their significance for wave-breaking studies, these criteria have been exten-
sively revisited in the modern literature, both in theoretical and experimental studies, and
we refer the reader to Longuet-Higgins et al. ( 1963 ), Longuet-Higgins ( 1969a , 1974 ),
Brown & Jensen ( 2001 ), Stansell & MacFarlane ( 2002 ), Wu & Nepf ( 2002 ), Babanin et al.
( 2007a ) and Toffol i et al. ( 2010a ), among many others.
For two-dimensional waves, Brown & Jensen ( 2001 ) confirmed the steepness (2.47) as
the breaking limit for wave breaking induced by linear wave focusing, and Babanin et al.
( 2007a ) for breaking due to modulational instability. For directional waves, Toffol i et al.
( 2010a ) investigated the probability distribution of the steepness of individual waves in
three-dimensional wave fields, based on two different field data sets (Black Sea and Indian
Ocean, including waves generated by tropical cyclones), and two sets obtained in different
directional wave tanks (Marintek's ocean basin in Trondheim, Norway and the directional
tank of the Kinoshita Laboratory/Rheem Laboratory of the University of Tokyo, Japan)
and found a threshold for the ultimate steepness as
H front k
2

= 0 . 55 (2.51)

for the front steepness, and

H rear k
2

= 0 . 45 (2.52)

for the rear steepness. Here, account was taken of the fact that the front and rear troughs of
breaking waves were not symmetric (for example, Figures 1.3 , 5.5 , 5.14 , see also Soares
et al. , 2004 ), i.e. wave height H front is estimated as the vertical distance between the crest
and the front trough and H rear is for the rear trough. Since the probability function has a
cutoff at the threshold value, i.e. does not extend into the higher values of steepness, this
result should be interpreted as the ultimate steepness beyond which the directional waves
will certainly break.
Babanin et al. ( 2011a ), based onmeasurements of pre-breaking three-dimensional waves,
showed that their steepness at the breaking onset is more like
Hk
2 =

0 . 46 − 48 . (2.53)