Geoscience Reference
In-Depth Information
4
Fully nonlinear analytical theories for surface waves
and numerical simulations of wave breaking
The previous chapter was dedicated to experimental methods of detecting wave break-
ing, quantifying the breaking probability and severity, and measuring effects related to
the breaking, including the wave energy dissipation. In the next chapter, it is logical to
describe theoretical methods of describing wave-breaking physics or phenomena leading
to the breaking.
While experimental oceanography has produced an abundant variety of techniques and
approaches to detect and measure breaking, the theories capable of dealing with wave
breaking are few. These should not be confused with analytical methods intended to detect
the breaking events in surface-wave records (
Section 3.7
) or with the statistical methods
of quantifying the breaking probability and strength (
Section 3.8
). Both such groups of
analytical techniques are placed into experimental
Chapter 3
for a good reason - they
principally rely on empirical criteria.
Another significant group of analytical approaches, dealing with the dissipation due to
breaking, rather than with the breaking as such is also not included in this chapter. Some of
these models are based on assumptions intended to interpret pre-breaking or post-breaking
properties of the waves, rather than on working with the physics leading to breaking or
driving the breaking and its consequences. Others attempt to deduce the dissipation from
differences between wave-evolution predictions done by means of kinetic and dynamic
equations. In any case, these are indirect techniques that do not depict the wave-breaking
event explicitly. They will be described in
Section 7.1
.
In this regard, here we will consider as a wave-breaking theory an analytical method that
is able to describe the evolution of nonlinear waves to the point/moment of breaking onset,
or even beyond, without relying on empirical criteria, or assumptions yet to be proved, or
some interpretations of wave properties that supposedly allow us to reveal wave-breaking
impacts provided that those would take place or have already taken place. In this section,
we will refer to wave theories based on first principles.
The first analytical theory that produced limiting steepness for two-dimensional steady
monochromatic water waves
(2.46)
was that by
Stokes
(
1880
). It is based on perturba-
tion expansion terms added to the solution of Eulerian linear wave theory, and can be
extended to three-dimensional irregular waves (
Laing
,
1986
). As a result the wave pro-
file appears much more realistic, it does exhibit such nonlinear features of wave shape as
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