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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
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3.3 Nonlinear transfer
In Komen et al. (1994) the derivation of the source function
, describing the nonlinear energy transfer, was given from first
principles. For surface gravity waves the nonlinear energy transfer is caused
by four resonantly interacting waves, obeying the usual resonance conditions
for the angular frequency and the wave numbers. The evaluation of 
therefore requires an enormous amount of computation because a three
dimensional integral needs to be evaluated. In the past several attempts
have been made to try to obtain a more economical evaluation of the nonlinear
transfer. The approach that was most succesful to date is the one by Hasselmann et al. (1985). The reason
for this is that their parametrization is both fast and it respects the
basic properties of the nonlinear transfer, such as conservation of momentum,
energy and action, while it also produces the proper high-frequency spectrum.
To this end, Hasselmann et al. (1985) constructed a nonlinear interaction operator by considering only a small number of interaction configurations consisting of neighbouring and finite distance interactions. It was found that, in fact, the exact nonlinear transfer could be well simulated by just one mirror-image pair of intermediate range interactions configurations. In each configuration, two wavenumbers were taken as identical
. The wavenumbers 
and 
are of different magnitude and lie at an angle to
the wavenumber 
, as required by the resonance conditions.
The second configuration is obtained from the first by reflecting the wavenumbers
and 
with respect to the 
-axis. The scale and direction of the reference wavenumber are allowed
to vary continuously in wavenumber space. The simplified nonlinear operator is computed by applying the same symmetrical integration method as is used to integrate the exact transfer integral (see also Hasselmann and Hasselmann, 1985), except that the integration is taken over a two-dimensional continuum and two discrete interactions instead of five-dimensional interaction phase space. Just as in the exact case the interactions conserve energy, momentum and action. For the configurations
|
(3.15) |
where
, satisfactory agreement with the exact computations was achieved.
From the resonance conditions the angles 
, 
of the wavenumbers 
and 
relative to 
are found to be 
, 
. The discrete interaction approximation has its most simple form for the rate of change in time of the action density in wavenumber space. In agreement with the principle of detailed balance, we have
 ,
|
(3.16) |
where
, 
, 
are the rates of change in action at wavenumbers 
, 
, 
due to the discrete interactions within
the infinitesimal interaction phase-space element 
and 
is a numerical constant. The net source function
is obtained by summing equation (3.16) over all wavenumbers, directions
and interaction configurations. For a JONSWAP spectrum the approximate and exact transfer source functions have been compared in Komen et al. (1994). The nonlinear transfer rates agree reasonably well, except for the strong negative lobe of the discrete-interaction approximation. This feature is, however, less important for a satisfactory reproduction of wave growth than the correct determination of the positive lobe which controls the down shift of the spectral peak.
. All variables are made dimensionless
using 
and 
.
The usefulness of the discrete-interaction approximation follows from its correct reproduction of the growth curves for growing wind sea. This is shown in Fig. 3.2 where a comparison is given of fetch-limited growth curves for some important spectral parameters computed with the exact nonlinear transfer, or, alternatively, with the discrete-interaction approximation. Evidence of the stronger negative lobe of the discrete interaction approximation is seen through the somewhat smaller values of the Phillips constant
. The broader spectral shape corresponds with the smaller values
of peak enhancement 
for the parametrized case. On the other hand, the agreement of the
more important scale parameters, the energy 
and the peak frequency 
is excellent (note that, as always, an asterix denotes nondimensionalisation
of a variable through 
and the friction velocity 
).The above analysis is made for deep water. Numerical computations by Hasselmann and Hasselmann (1981) of the full Boltzmann integral for water of arbitrary depth have shown that there is an approximate relation between transfer rates in deep water and water of finite depth: for a given frequency-direction spectrum, the transfer for finite depth is identical to the transfer for infinite depth, except for a scaling factor
:
 ,
|
(3.17) |
where
is the mean wavenumber. This scaling relation holds in the
range 
, where the exact computations could be closely
reproduced with the scaling factor
 ,
|
(3.18) |
with
. This approximation is used therefore in the WAM model.Next Section
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23.04.2002 |
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