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short-duration processes (as indicated by
Dold
(
1992
) and
Chalikov
(
2011
)). In addition,
compared to the CS scheme, the surface integral method is cumbersome: a complete set of
its equations occupy several pages. For the CS scheme, the equations take three lines and
the core of the numeric scheme takes 11 lines in Fortran90.
The accuracy of this scheme was demonstrated by a long-term simulation of very steep
Stokes waves (
ak
42). The stability of Stokes waves has been the subject of much
speculation. The reality is quite simple: Stokes waves are always unstable to any distur-
bances, but the rate of development of the instability depends on the amplitudes of the
perturbations and their phases. In general terms, a Stokes wave is always unstable if it has
any perturbation from the pure Stokes form. In the CS case, 11 decimal places of precision
and a fourth order Runge-Kutta scheme were sufficient to simulate the propagation of a
virtually undisturbed Stokes wave for up to one thousand periods (e.g.
Chalikov
,
2007
).
The conformal mapping even made it possible to reproduce the initial stages of the
breaking process where the surface ceases to be a single-valued function. It should be
mentioned that the
Dold
(
1992
) scheme is also capable of achieving this, but with spe-
cial smoothing and regularisation (for capabilities of Lagrangian models in this regard, see
Tulin & Waseda
(
1999
),
Tulin & Landrini
(
2001
) and
Section 4.2
). The CS model, how-
ever, has a number of important advantages: (1) comparison with an exact solution showed
that the scheme has extremely high accuracy; (2) it preserves integral invariants; (3) it is
very efficient: its computation time scales as
M
=
0
.
where
M
is the number of modes,
whereas the Dold scheme scales as
M
2
; (4) the scheme demonstrates stability over millions
of time steps (thousands of periods of the dominant wave). This scheme is able to repro-
duce a nonlinear concentration of energy in physical space resulting in wave breaking and
potentially in the appearance of freak waves.
In the CS model, the wave model is also coupled with an atmospheric boundary-layer
model (see
Chalikov & Rainchik
,
2011
):
·
log
(
M
)
2
J
−
1
−
1
2
2
ζ
τ
=
ξ
−
ξ
−
−
F
z
p
(4.14)
where
p
is surface pressure, which describes the exchange of momentum and energy
between the air and water. In the wind-influence investigation described later in this sec-
tion, in order to speed up the computations, the coupling was conducted by means of a
β
-function which parameterised the connection of the surface pressure and the surface
shape on the basis of an exhaustive set of numerical simulations by means of the coupled
model. Real and imaginary Fourier amplitudes of pressure
p
r
and
p
i
are calculated as
linear functions of amplitudes of water elevation
η
r
and
η
i
:
p
r
+
ip
i
=
(β
r
+
i
β
i
)(η
r
+
i
η
i
).
(4.15)
The real and imaginary parts of this
β
-function are functions of non-dimensional frequency
n
=
u
(
l
k
/
2
)ω
where
u
(
l
k
/
2
)
is wind velocity at height equal to half of the wave length
1
/
2
(here, both
l
k
,
ω
=|
k
|
ω
and
k
are nondimensional variables).
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