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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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5.2 Implicit integration of the source functions
An implicit scheme was introduced for the source function integration to enable the use of an integration time step that was greater than the dynamic adjustment time of the highest frequencies still treated prognostically in the model. In contrast to first and second generation wave models, the energy balance of the spectrum is evaluated in detail up to a high cut-off frequency. The high-frequency adjustment time scales are considerably shorter than the evolution time scales of the energy-containing frequency bands near the peak of the spectrum, in which one is mainly interested in modelling applications. Thus, in the high-frequency region it is sufficient to determine the quasi-equilibrium level to which the spectrum adjusts in response to the more slowly changing low-frequency waves, rather than the time history of the short time scale adjustment process itself. An implicit integration scheme whose time step is matched to the evolution of the lower frequency waves meets this requirement automatically: for low-frequency waves, the integration method yields, essentially, the same results as a simple forward integration scheme, while for high frequencies the method yields the (slowly changing) quasi-equilibrium spectrum (WAMDI, 1988).
The original WAM model used a time-centred implicit integration scheme, but Hersbach and Janssen(1999) found that numerical noise occurred which may be avoided by a two-time level, fully implicit approach. The fully implicit equations (leaving out the advection terms) are given by
 ,
|
(5.4) |
where
is the time step and the index 
refers to the time level. If
depends linearly on 
, equation (5.4) could be solved directly for the
spectrum 
at the new time step. Unfortunately,
none of the source terms are linear. We therefore introduce a Taylor expansion
 .
|
(5.5) |
The functional derivative in (5.5) (numerically a discrete matrix
) can be divided into a diagonal matrix 
and a nondiagonal residual 
,
 .
|
(5.6) |
Substituting (5.5) and (5.6) into (5.4), realizing, in addition, that the source term
may depend on the friction velocity 
at time level 
, we obtain
|
(5.7) |
with
. A number of trial computations indicated that the off
diagonal contributions were generally small if the time step was not too
large. Disregarding these contributions, the matrix on the left-hand side
can be inverted, yielding for the increment 
,
 .
|
(5.8) |
Nevertheless, in practice numerical instability is found in the early stages of wave growth. These are either caused by the neglect of the off diagonal contributions or by the circumstance that the solution is not always close to the attractor of the complete source function. Therefore a growth limitation needs to be imposed. In the ECMWF version of WAM a variant of the growth limiter of Hersbach and Janssen(1999) is used: the maximum increment in the spectrum,
, is given by
|
(5.9) |
For a typical test case, good agreement was obtained between an explicit integration with a time step of 1 minute and the implicit scheme with only diagonal terms for time steps up to about 20 minutes.
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23.04.2002 |
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