Wave-breaking probability - Breaking and Dissipation of Ocean Surface Waves - page 144

from continuous noise. If the modes are not imposed, some sidebands naturally appear
from the background and are expected to be defined by the ratio of characteristic wave
steepness

ω 0 are some characteris-
tic wavenumber and angular frequency, respectively, and a is the mean amplitude at this
wavenumber:

= ak 0 to spectral bandwidth ω/ω 0 , where k 0 and

ω/ω 0 .

M I = (5.1)

This ratio was shown to be important in the original studies of instabilities of weakly
modulated trains of monochromatic carrier waves of small amplitude (e.g. Yuen & Lake ,
1982 ). Here, we denote this ratio as M I (modulational index), and in Section 5.3.3 we will
introduce directional modulational index M Id by analogy.
The evolution of wave trains described in this section mainly deals with slowly mod-
ulated two-dimensional monochromatic waves. Such wave trains are subject to modula-
tional instability which is commonly termed Benjamin-Feir instability after the work of
Benjamin & Feir ( 1967 ). Many authors point out, however, that it was first discovered by
Lighthill ( 1965 ) who established the growth rate for this instability in the limit of very long
modulation. The growth rate was proportional to the wavenumber of the modulational per-
turbation, a result which has obvious physical limitations if applied to short waves (large
wavenumbers). The general description for the behaviour of the growth rate was found by
Zakharov ( 1966 ) before Benjamin & Feir ( 1967 ), but in English-language literature the
papers were published independently in the same year ( Zakharov , 1967 ). Feir ( 1967 )was
the first to observe modulational instability in the experiment and Zakharov ( 1968 ) further
developed and summarised its theory.
The Benjamin-Feir instability was developed for nearly-linear waves, and in this topic,
dedicated to wave breaking, the waves even initially are not of small amplitude. Therefore,
the analogy of the observed empirical modulational interplay with the small-amplitude
near-monochromatic theoretical phenomenon should be treated with caution and we will
avoid the term of Benjamin-Feir instability. Here, M I signifies the fact that the wave
steepness and length of wave modulation (or number N of waves in the modulation), where

1 / N ∼ ω/ω 0 , (5.2)

are not independent quantities, i.e. steeper waves will correspond to fewer waves in a mod-
ulation (similarly, as far as wave breaking is concerned, wave steepness and directional
spread are not independent quantities in the directional modulational index (5.57) ). Thus,
if nonlinear waves are allowed to evolve naturally, they will form groups where N is not a
free parameter, but will be defined by the initial steepness

(1.1) .
Therefore, as mentioned above, in the experiment we expect the necessary resonant
modes to develop naturally from the background turbulent noise. These modes, however,
can be suppressed or even prohibited in a discretised numerical model. In such circum-
stance, the waves, even if they are steep Stokes waves, will propagate for an indefinitely
long period without breaking.