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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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5.1 Introduction
In this section we discuss the numerical aspects of the solution of the action balance equation as implemented in the ECMWF version of the WAM model.
Although, thus far, we have discussed the transport equation for gravity waves for the action density, because this is the most natural thing to do from a theoretical point of view, the actual WAM model is formulated in terms of the frequency-direction spectrum
of the variance of the surface elevation. The reason for
this is that in practical applications one usually deals with surface elevation
spectra, because these are measured by buoys. The relation between the action
density and the frequency spectrum is straightforward. It is given by
 ,
|
(5.1) |
where
is the intrinsic frequency (see also equation
(2.4)). This relation is in accordance with the analogy between wave
packets and particles, since particles with action 
have energy 
and momentum 
. The continuous wave spectrum is approximated in the numerical model by means of step functions which are constant in a frequency-direction bin. The size of the frequency-direction bin depends on frequency. A distinction is being made between a prognostic part and a diagnostic part of the spectrum. The prognostic part of the spectrum has KL directional bands and ML frequency bands. These frequency bands are on a logarithmic scale, with
, spanning a frequency range 
. The logarithmic scale has been chosen in order to have uniform
relative resolution, and also because the nonlinear transfer scales with
frequency. The starting frequency may be selected arbitrarily. In most global
applications the starting frequency 
is 0.042 Hz, the number of frequencies ML is 25
and the number of directions KL is 24 (
resolution). For closed basins, such as the Mediterranean Sea where
low-frequency swell is absent, a choice of starting frequency 
of 0.05 Hz is sufficient. The present version
of the ECMWF wave prediction system has 24 directions and 30 frequencies,
with starting frequency 
. Beyond the high-frequency limit
of the prognostic region of the spectrum, an 
tail is added, with the same directional distribution as the last
band of the prognostic region. The diagnostic part of the spectrum is therefore
given as
 .
|
(5.2) |
In the ECMWF version of the WAM model the high-frequency limit is set as
 .
|
(5.3) |
Thus, the high-frequency extent of the prognostic region is scaled by the mean frequency
. A dynamic high-frequency cut-off,
, rather than a fixed cut-off at 
is necessary to avoid excessive disparities in the
response time scales within the spectrum. A diagnostic tail needs to be added for
to compute the nonlinear transfer in the prognostic region and also
to compute the integral quantities which occur in the dissipation source
function. Tests with an 
tail show that (apart from the calculation of the wave-induced stress)
the results are not sensitive to the precise form of the diagnostic tail.
The contribution to the total energy from the diagnostic tail is normally
negligible. Because observations seem to favour an 
power law (Birch and Ewing, 1986, Forristall, 1981, Banner, 1990) this power law is used for the
high-frequency part of the spectrum. The prognostic part of the spectrum is obtained by numerically solving the energy balance equation. We will now discuss the different numerical schemes and time steps that are used to integrate the source functions and the advective terms of the transport equation.
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23.04.2002 |
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