Definitions for wave breaking - Breaking and Dissipation of Ocean Surface Waves - page 24

In the original JONSWAP formulation, γ , the right width σ right and the left width σ left were

chosen to be constant, but in reality depend on the wave development stage

and in general vary significantly (e.g. Donelan et al. , 1985 ; Babanin & Soloviev , 1998a ).
Based on comparisons of the count for dominant breaking conducted by two independent
acoustic techniques, the dominant frequency band was later reassigned as

α, σ and γ

f p =± 0 . 35 f p (2.8)

( Manasseh et al. , 2006 ). Other frequency bands, such as 10% are also often

employed (e.g. Banner et al. , 2002 ; Babanin & Young , 2005 ; Manasseh et al. , 2006 ;
Babanin et al. , 2007c ). This narrower band is usually needed when breaking probabilities
are defined for frequencies other than the spectral peak. Away from the peak, the spec-
tral density decays very rapidly which would make a count of the crests and the relative
importance of the breaking waves very uncertain if the spectral band were too broad.
Is there a physical meaning for the spectral band implied in the breaking-probability
definition (2.3) - (2.5) ? For the dominant waves, such a physical meaning can certainly
be pointed out. That is, the width of the spectral peak defines the characteristics of the
groups of dominant waves, and the wave-breaking frequency of occurrence depends on
these groups.
Indeed, for a narrow-banded spectrum, which the spectral peak of wind-generated waves
is, the width of the peak is related to modulational properties in the train of dominant waves.
This relation was investigated statistically by Longuet-Higgins ( 1984 ). He suggested a
nondimensional bandwidth parameter

f =±

ν to describe such modulational properties:

m 2 m 0
m 1

2

ν = − 1 (2.9)

where m i is the spectral moment of order i and in the general case is defined as

∞

f i F m i = ( f ) df . (2.10)

0

Longuet-Higgins ( 1984 ) found that integral limits in (2.10) have to be set to

f p =±

.

0 5 f p (2.11)

in order to explain experimentally observed properties of wave groups. This limit is some-
what larger than the width of the JONSWAP peak enhancement or the

f =± 30-35% in

the breaking statistics of dominant waves, but it is reasonably close.
Once they exist, the groups of dominant waves have a verified association with breaking
probability. Donelan et al. ( 1972 ) reported observing several consecutive waves breaking
at the peak of the group envelope before sufficient energy was lost from the group. In
another open-ocean study, Holthuijsen & Herbers ( 1986 ) found a significant influence of
wave groups on wave breaking. When breaking occurred, the position of the first breaker
in a group was slightly ahead of the centre of the group. They concluded that the overall
fraction of breaking occurring within wave groups was close to 70%.