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Non-potential phase-resolvent models which have to treat the turbulence and deal with
the air explicitly, that is with bubbles in the water and spray above the surface, will also
not be considered in this chapter. They allow us to describe wave-breaking dissipation
directly and will be reviewed in
Section 7.2
together with other dissipation theories and
applications.
The Chalikov-Sheinin (CS) model (
Chalikov & Sheinin
,
1998
,
2005
, see page 131)
has been mentioned a number of times throughout the topic as a model based on a fully
nonlinear approach. In very simple terms, this model employs solution of the Euler equa-
tion, the fundamental equation of hydrodynamics and Newton's law applied to fluids in
the absence of friction force. The model does not involve any initial assumptions on wave-
steepness magnitude or any other physical/spectral conditions. As mentioned previously,
there are several models based on fully nonlinear equations, and it is their accuracy, stabil-
ity and ability to integrate the evolution equations in space and time without accumulating
numerical errors, that differentiate and distinguish them from one another. We will give a
brief overview of the fully nonlinear models, based primarily on the CS example, largely
following
Babanin
et al.
(
2007a
,
2009a
,
2010a
).
4.1 Free surface at the wave breaking
Numerical computations of nonlinear surface waves have previously been undertaken
based on solutions of the potential flow equations (e.g.
Watson & West
,
1975
;
Longuet-
Higgins & Cokelet
,
1976
;
Wes t
et al.
,
1987
) and with a Cauchy-type integral algorithm
(
Dold & Peregrine
,
1986
;
Dold
,
1992
). Both schemes have no limitation in terms of wave
steepness, and both are capable of simulating the initial phase of wave breaking (the later
stages are rotational and remain extremely difficult to simulate directly). More recently, a
method based on a Taylor expansion of the Dirichlet-Neumann operator was developed
by
Craig & Sulem
(
1993
). The capabilities of this method were illustrated by computing
the evolution of modulated wave packets and a low-order approximation of a Stokes wave
for relatively short periods of time. We should point out that this appears to be a principal
limitation of all the above schemes: for a steep wave field, they have only been used for
simulations of relatively short time/space evolution. These approaches could not be applied
to longer periods of time because none of them appear to provide conservation of integral
invariants (mass, energy, horizontal momentum).
A numerical scheme for direct hydrodynamical modelling of two-dimensional nonlin-
ear gravity and gravity-capillary waves was developed by
Chalikov & Sheinin
(
1998
)
(see also
Chalikov & Sheinin
,
2005
;
Chalikov
,
2005
,
2007
). This approach is based on
a non-stationary conformal mapping, which allows the equations of potential flow with the
inclusion of a free surface to be written in a surface-following coordinate system. This
transformation does not impose any restriction on the shape of the surface, except that
it has to be possible to represent this surface in terms of a Fourier series. An analogous
approach was developed by
Dyachenko
et al.
(
1996
) and
Dyachenko & Zakharov
(
2005
).
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