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skewness
S
k
(1.2)
, but as was argued in
Section 1.2
, this is not how breaking waves appear
(see
Figure 1.2
).
In this regard, at least as far as the visual appearance of surface waves is concerned, the
Lagrangian approach is more convincing (see also
Section 4.2
). Even at the second-order
expansion, it is capable of reproducing both skewness
S
k
and asymmetry
A
s
of waves with
respect to the vertical axis
(1.3)
in two dimensions (
Fouques
et al.
,
2006
) and horse-shoe
patterns in three dimensions (
Fougues & Stansberg
,
2009
).
Analytical and numerical means of describing the evolution of nonlinear wave trains can
be broadly subdivided into five major groups. The nonlinear Schrödinger equation (NLSE,
Benney & Newell
,
1967
;
Zakharov
,
1967
,
1968
) and its extensions (i.e.
Dysthe
,
1979
;
Sti-
assnie
,
1984
;
Shemer & Dorfman
,
2008
) work in the physical space and model the complex
wave envelope. The Zakharov equation (
Zakharov
,
1968
) and its modifications (
Gramstad
et al.
,
2010
) are for the amplitude spectrum. The Alber equation (
Alber
,
1978
;
Stiassnie
et al.
,
2008
) is a stochastic counterpart of the deterministic dynamic equations where the
evolving variable is the correlation function. The most frequently used stochastic model
is the kinetic equation for wave spectra (
Hasselmann
,
1962
;
Zakharov
,
1968
;
Krasitskii
,
1994
;
Janssen
,
2003
;
Annenkov & Shrira
,
2006
). The latter, as mentioned above, gives
the interesting possibility of estimating the dissipation indirectly if compared to evolution
simulated by means of dynamic equations (
Zakharov
et al.
,
2007
and see
7.1.2
).
In this chapter, however, we will only be interested in the fifth group, fully nonlinear
analytical theories for potential surface waves, as opposed to the above theories which are
based on first principles, but then involve further assumptions, like a 'small parameter'
in perturbation theories, usually wave steepness
(1.1)
in Eulerian approaches, or the ratio
of the particle displacement to wavelength in Lagrangian derivations, or narrow-banded
spectrum in NLSE, and so on. Theories that rely on a small steepness, or even on a finite
steepness are indeed also nonlinear, and they do depict some nonlinear effects as mentioned
above, including some very important nonlinear characteristics of wave shape, wave fields
and wave dynamics, but they can hardly be expected to adequately attend to the problem
of wave breaking where the steepness of the individual wave is by definition ultimately
extreme. Some techniques use a combination of solutions of fully nonlinear equations and
one of the approaches outlined above, and such a combination is another potentially very
promising direction which does not depend on limiting assumptions and at the same time
allows us to scrutinise and better understand the physics underlying specific analytical
methods (e.g.
Irisov & Voronovich
,
2011
).
Analytical solutions of fully nonlinear equations and boundary conditions have not so
far proved feasible, therefore the existing fully nonlinear approaches are numerical models.
In this regard, we can mention modelling by
Watson & West
(
1975
),
Longuet-Higgins &
Cokelet
(
1976
),
Dold & Peregrine
(
1986
),
Wes t
et al.
(
1987
),
Dold
(
1992
),
Craig & Sulem
(
1993
),
Tulin
et al.
(
1994
),
Dyachenko
et al.
(
1996
),
Chalikov & Sheinin
(
1998
,
2005
),
Landrini
et al.
(
1998
) and
Dyachenko & Zakharov
(
2005
). The model by
Landrini
et al.
(
1998
) is Lagrangian and will be described in
Section 4.2
. The Eulerian approach will be
considered next in
Section 4.1
.
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