Definitions for wave breaking - Breaking and Dissipation of Ocean Surface Waves - page 25

Support for the connection between frequency bandwidth, wave groups and breaking
occurrence also comes from hydrodynamic, rather than just statistical studies of modula-
tional instabilities of wave trains. Theoretical approaches, started by Zakharov ( 1966 , 1967 ,
1968 ), Benjamin & Feir ( 1967 ) and Longuet-Higgins & Cokelet ( 1978 )(see Yuen & Lake
( 1982 ) for an essential review, but progress on this topic is continuing), numerical models
(e.g. Dold & Peregrine , 1986 ; Chalikov & Sheinin , 1998 , 2005 ) and laboratory experi-
ments (e.g. Babanin et al. , 2007b , 2009a , 2010a ; Stiassnie et al. , 2007 ) all point out that
conditions for the onset of breaking, i.e. a very steep individual wave, can result from the
evolution of nonlinear wave groups. Initial conditions for such an evolution consist of steep
monochromatic waves with sidebands, i.e. this also involves some characteristic bandwidth
defined by the sidebands.
Interpretation of the applicability of this kind of modulational instability to field waves
has been a subject for debate for quite some time. First of all, the field waves have a
continuous spectrum and therefore the notion of primary waves and sidebands is uncertain:
waves at every frequency can be treated as both primary waves and sideband disturbances.
There have been attempts to draw analogies between the monochromatic wave trains with
sidebands and spectral waves. It has been shown ( Onorato et al. , 2001 ; Janssen , 2003 )
that the modulational properties in a spectral system of nonlinear waves depend on the
ratio of wave steepness = ak 0 to spectral bandwidth ω/ω 0 , where k 0 and ω 0 are some

characteristic wavenumber and radian frequency, respectively ( ω = 2 π f ), a is the mean

ω ν

amplitude at this wavenumber, is a characteristic width of the spectral peak, is a

dimensionless characteristic depth, and N m is the number of waves in the modulation:

ω/ω 0 = ν

M I = = N m . (2.12)

This same ratio, if the properties of the primary wave and sidebands are used, was shown to
be important in the original studies of instabilities of weakly modulated trains of monochro-
matic carrier waves of small amplitude. Here, we will denote this ratio as M I (Modulational
Index). The physical applicability and interpretation of the analogy is still a subject for
discussion. Note, however, that even if applicable in the case of a continuous spectrum,
definition (2.12) has a physical meaning for the modulational properties of dominant waves
only, as there are no characteristic k 0 ,

ω 0 and ω

away from the spectral peak.
The second major uncertainty is the essentially two-dimensional nature of the hydro-
dynamic modulational instabilities discussed. The theories, numerical simulations and
laboratory experiments mentioned above are two-dimensional or quasi-two-dimensional.
Against direct extrapolation of the outcomes of these studies into field conditions were
the known experimental and theoretical results on limitations which the nonlinear mod-
ulational mechanism has in broadband, and particularly in three-dimensional fields (e.g.
Brown & Jensen , 2001 ; Onorato et al. , 2002 , 2009a , b ; Waseda et al. , 2009a ). Eventually
evidence appeared that showed that in directional wave fields it is active ( Babanin et al. ,
2010a , 2011a ). This will be further discussed in Section 5.3.3 .