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Chapter 2. The kinematic part of the energy balance equation
Chapter 3. Parametrization of source terms and the energy balance in a growing wind sea
Chapter 4. An optimal interpolation scheme for assimilating altimeter data into the WAM model
Chapter 5 Numerical scheme
Chapter 6 The WAM-model software
Chapter 7 Wind-wave interaction at ECMWF
REFERENCES
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3.2 Wind input and dissipation
Results of the numerical solution of the momentum balance of air flow over growing surface gravity waves have been presented in a series of studies by Janssen (1989), Janssen et al. (1989a) and Janssen (1991). The main conclusion was that the growth rate of the waves generated by wind depends on the ratio of friction velocity and phase speed and on a number of additional factors, such as the atmospheric density stratification, wind gustiness and wave age. So far systematic investigations of the impact of the first additional two effects have not been made, except by Janssen and Komen(1985) and Voorrips et al. (1994). It is known that stratification effects observed in fetch-limited wave growth can be partly accounted for by scaling with
(which is consistent with theoretical results). The remaining
effect is still poorly understood, and is therefore ignored in the standard
WAM model. In this section we focus on the dependence of wave growth on
wave age, and the related dependence of the aerodynamic drag on the sea
state, which effect is fully included in the WAM model. A realistic parametrization of the interaction between wind and wave was given by Janssen (1991), a summary of which is given below. The basic assumption Janssen (1991) made, which was corroborated by his numerical results of 1989, was that even for young wind sea the wind profile has a logarithmic shape, though with a roughness length that depends on the wave-induced stress. As shown by Miles (1957), the growth rate of gravity waves due to wind then only depends on two parameters, namely
 .
|
(3.2) |
As usual,
denotes the friction velocity, 
the phase speed of the waves, 
the wind direction and 
the direction in which the waves propagate. The so-called profile
parameter 
characterizes the state of the mean
air flow through its dependence on the roughness length 
. Thus, through 
the growth rate depends on the roughness
of the air flow, which, in its turn, depends on the sea state. A simple
parametrization of the growth rate of the waves follows from a fit of numerical
results presented in Janssen(1991). One finds
 ,
|
(3.3) |
where
is the growth rate, 
the angular frequency, 
the air-water density ratio and 
the so-called Miles' parameter. In terms of the dimensionless critical
height 
(with 
the wavenumber and 
the critical height defined by 
) Miles' parameter becomes
 ,
|
(3.4) |
where
is the von Kármán constant and 
a constant. In terms of wave and wind quantities 
is given as
 ,
|
(3.5) |
and the input source term
of the WAM model is given by
 ,
|
(3.6) |
where
follows from (3.3) and with 
the action density spectrum. The stress of air flow over sea waves depends on the sea state and from a consideration of the momentum balance of air it is found that the kinematic stress is given as (Janssen, 1991)
 ,
|
(3.7) |
where
 .
|
(3.8) |
Here,
is the mean height above the waves and 
is the stress induced by gravity waves (the `wave stress')
 .
|
(3.9) |
The frequency integral extends to infinity, but in its evaluation only an
tail of gravity waves is included and the higher level of capillary
waves is treated as a background small-scale roughness. In practice, we
note that the wave stress points in the wind direction as it is mainly determined
by the high-frequency waves which respond quickly to changes in the wind
direction. The relevance of relation (3.8) cannot be overemphasized. It shows that the roughness length is given by a Charnock relation (Charnock, 1955)
 .
|
(3.10) |
However, the dimensionless Charnock parameter
is not constant but depends on the sea state through the wave-induced
stress since
 .
|
(3.11) |
Evidently, whenever
becomes of the order of the total stress in the
surface layer (this happens, for example, for young wind sea) a considerable
enhancement of the Charnock parameter is found, resulting in an efficient
momentum transfer from air to water. The consequences of this sea-state-dependent
momentum transfer will be discussed in Chapter 7. 
This finally leaves us with the choice of two unknowns namely
from (3.11) and 
from (3.4). The constant
was chosen in such a way that for old wind
sea the Charnock parameter 
has the value 0.0185 in agreement with observations collected by
Wu (1982) on the drag over sea
waves. It should be realised though, that the determination of 
is not a trivial task, as beforehand the ratio of
wave-induced stress to total stress is simply not known. It requires the
running of a wave model. By trial and error the constant 
was found to be 
.The constant
is chosen in such a way that the growth rate 
in (3.3) is
in agreement with the numerical results obtained from Miles' growth rate.
For 
and a Charnock parameter 
we have shown in Fig. 3.1 the comparison between Miles' theory
and (3.3). In addition observations as compiled
by Plant (1982) are shown. Realizing that the
relative growth rate 
varies by four orders of magnitude it is concluded that there is
a fair agreement between our fit (3.3), Miles' theory and observations. We
remark that the Snyder et al. (1981) fit to their field
observations, which is also shown in Fig. 3.1 , is in perfect accordance with
the growth rate of the low-frequency waves although growth rates of the
high-frequency waves are underestimated. Since the wave-induced stress is
mainly carried by the high-frequency waves an underestimation of the stress
in the surface layer would result. We conclude that our parametrization of the growth rate of the waves is in good agreement with the observations. The next issue to be considered is how well our approximation of the surface stress compares with observed surface stress at sea. Fortunately, during HEXOS (Katsaros et al. 1987) wind speed at 10 m height,
, surface stress 
and the one-dimensional frequency spectrum
were measured simultaneously so that our parametrization of the surface
stress may be verified experimentally. For a given observed wind speed and
wave spectrum, the surface stress is obtained by solving (3.7) for the stress 
in an iterative fashion as the roughness length 
depends, in a complicated manner, on the stress. Since the surface
stress was measured by means of the eddy correlation technique, a direct
comparison between observed and modelled stress is possible. The work of
Janssen (1992) shows that the
agreement is good. It is, therefore, concluded that the parametrized version of quasi-linear theory gives realistic growth rates of the waves and a realistic surface stress. However, the success of this scheme for wind input critically depends on a proper description of the high-frequency waves. The reason for this is that the wave-induced stress depends in a sensitive manner on the high-frequency part of the spectrum. Noting that for high frequencies the growth rate of the waves (3.3) scales with wavenumber as
 ,
|
(3.12) |
and the usual whitecapping dissipation scales as
 ,
|
(3.13) |
an imbalance in the high-frequency wave spectrum may be anticipated. Eventually, wind input will dominate dissipation due to wave breaking, resulting in energy levels which are too high when compared with observations. Janssen et al. (1989b) realized that the wave dissipation source function has to be adjusted in order to obtain a proper balance at the high frequencies. The dissipation source term of Hasselmann (1974) is thus extended as follows:
 ,
|
(3.14) |
where
and 
are constants, 
is the total
wave variance per square metre, 
the wavenumber and 
and 
are the mean
angular frequency and mean wavenumber, respectively. In practice, we take
and 
. The choice of the above dissipation source term may be justified
as follows. In Hasselmann (1974), it is argued that whitecapping
is a process that is weak-in-the-mean, therefore, the corresponding dissipation
source term is linear in the wave spectrum. Assuming that there is a large
separation between the length scale of the waves and the whitecaps, the
power of the wavenumber in the dissipation term is found to be equal to
one. For the high-frequency part of the spectrum, however, such a large
gap between waves and whitecaps may not exist, allowing the possibility
of a different dependence of the dissipation on wavenumber. This concludes the description of the input source term and the dissipation source term due to whitecapping. Although the wind input source function is fairly well-known from direct observations, there is relatively little hard evidence on dissipation. Presently, the only way out of this is to take the functional form for the dissipation in (3.14) for granted and to tune the constants
and 
in such a way that the action balance equation (2.24) produces results
which are in good agreement with data on fetch-limited growth and with data
on the dependence of the surface stress on wave age. In addition, a reasonable
dissipation of swell should be obtained. It was decided to follow this method
and, after an extensive tuning exercise, the constants 
and 
were given the values 4.5 and 0.5 while the constant
in the Charnock parameter was given the value 0.01.Next Section
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23.04.2002 |
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