Geoscience Reference
In-Depth Information
Since it was first proposed by
Hasselmann
(
1960
),
RTE
has been widely used in scien-
tific studies and practical applications relating to the evolution of wind-generated waves.
In water of finite depth, this equation takes the form (
Komen
et al.
,
1994
)
∂
∂
ω
∂
∂
∂
∂
t
+
(
c
g
+
U
c
)
·
x
−∇
·
=
S
in
+
S
nl
+
S
ds
+
S
bf
(2.61)
k
where the left-hand side represents the evolution of the wave action density
as a result
of the physical processes of atmospheric input from the wind
S
in
, nonlinear interactions
of various orders within the spectrum
S
nl
, dissipation due to 'whitecapping'/breaking
S
ds
(2.25)
, and decay due to bottom friction
S
bf
. Function
/ω
(
f
,
k
,θ)
is the directional wave
spectrum, i.e.
is the direction of wave propagation. All the source terms are also spec-
tra along the wavenumber-frequency-direction, as well as functions of other parameters.
The term
U
c
is the surface current and
θ
(
k
)
is the Doppler-shifted frequency
(
k
)
=
ω(
U
c
.
The dissipation-due-to-breaking source (sink) term
S
ds
(2.25)
was introduced and defined
above (
Section 2.7
) and will be mentioned many times throughout this topic. While the
waves would not be generated without the wind input
S
in
in the first place, the other
terms in
(2.61)
are not of the secondary importance they may seem. Without
S
ds
,the
wind-generated waves would keep growing which obviously does not happen. As soon
as the waves pass the infinitesimally small stage, the dissipation switches on. The dissipa-
tion due to breaking does not turn on until the spectral threshold value is overcome (see
Sections 5.2
,
5.3.2
), but once it is active it becomes as significant as the wind input and the
other two general terms mentioned in
(2.61)
as far as wave evolution is concerned. Many
more less general source/sink terms in
RTE
are possible, but the four introduced in
(2.61)
are
invariably important in the finite-depth wave environment, and the first three in deep water.
Knowledge of the terms other than dissipation, based on either experimental or ana-
lytical (or both) approaches, is incomplete but still rational (see
Komen
et al.
,
1994 and
The WISE Group
,
2007
, in general and
Donelan
et al.
(
2006
) and
Babanin
et al.
(
2007b
)
on recent developments on the wind input
S
in
). In contrast, understanding of the dissi-
pation term remains poor. A number of theoretical and conjectural approaches have been
attempted to predict the spectral dissipation function, but none of these have been validated
experimentally. It is generally assumed that
S
ds
is a function of the wave spectrum
k
)
+
k
·
:
n
S
ds
(
f
,
k
,θ)
∼
(
f
,
k
,θ)
,
(2.62)
but there is no agreement even on such basic ground as to whether the spectral dissipa-
tion
S
ds
(
f
,
k
,θ)
is linear in terms of the spectrum
(
f
,
k
,θ)
or not, i.e whether
n
=
1
or
n
1. On the other hand, such experimentally known features of wave-breaking dissi-
pation as threshold behaviour, cumulation of dissipation at smaller scales and dependence
of dissipation on the wind at extreme-forcing conditions have not been accounted for in
present-day dissipation terms in any way (see
Section 7.4
). See
Chapters 5
,
6
and
7
for
more detailed discussions on the dissipation-term issues and
and The WISE Group
(
2007
)
for an up-to-date discussion of the entire topic of state-of-the-art wave modelling.
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