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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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4.2 Wave height analysis
First, an analysis of the significant wave height field
is created by optimum interpolation (cf. Lorenc, 1981):
 ,
|
(4.1) |
where
denotes the significant wave height field observed
by the altimeter and 
is the first-guess significant wave height field computed by the
WAM model. Since long-term statistics of the prediction and observational
error covariance matrices equation were not available, empirical expressions
were taken:
 .
|
(4.2) |
Good results were obtained for a correlation length
. This is consistent with the optimal scale length found by Bauer
et al. (1992) using a triangular interpolation scheme. However, at ECMWF
we use a much smaller value of 300 km.4.2.1 The analysed wave spectrum
In the next step, the full two-dimensional wave spectrum is retrieved from the analysed significant wave height fields. Two-dimensional wave spectra are regarded either as wind sea spectra, if the wind sea energy is larger than 3/4 times the total energy, or, if this condition is not satisfied, as swell.
In both cases an analysed two-dimensional wave spectrum
is computed from the first-guess wave spectrum 
and the optimally interpolated wave heights 
by rescaling the spectrum with two scale parameters 
and 
:
 .
|
(4.3) |
Different techniques are applied to compute the parameters
and 
for wind sea or swell spectra.4.2.2 Retrieval of a wind sea spectrum
The parameters
and 
in equation (4.3) can be determined from empirical
duration-limited growth laws relating, in accordance with
Kitaigorodskii's (1962) scaling laws, the nondimensional energy 
(where 
), mean frequency 
and duration 
. Specifically, we take the following relations (which deviate considerably
from the ones proposed by Lionello et al. (1992)):
 ,
|
(4.4) |
and
 .
|
(4.5) |
The mean frequency is preferred to the peak frequency because its computation is more stable. Since the first-guess friction velocity was used to generate the waves and the first-guess wave height is known, an estimate of the duration
of the wind sea can be derived from the duration-limited growth laws.
Assuming this estimated duration is correct, the analysed wave height yields
from the growth laws, equations
(4.4) and (4.5),
best estimates of the friction velocity 
and mean frequency 
. The analysed wave height and mean frequency determine then the
two parameters 
and 
:
 .
|
(4.6) |
The corrected best-estimate winds are then used to drive the model for the rest of the wind time step. In a comprehensive wind and wave assimilation scheme, the corrected winds should be also inserted into the atmospheric data assimilation scheme to provide an improved wind field in the forecast model.
4.2.3 Retrieval of a swell spectrum
A spectrum is converted to swell and begins to decay at the edge of a storm, before dispersion has separated the swell into spatially distinct frequencies. One can therefore distinguish between a nonlinear swell regime close to the swell source and a more distant linear regime, where dispersion has reduced the swell wave slopes to a level at which nonlinear interactions have become negligible. Because of these complexities, and also because of a lack of adequate data, there exist no empirical swell decay curves comparable to the growth curves in the wind sea case. However, Lionello and Janssen (1990) showed that for the WAM model swell spectra the average wave steepness,
|
(4.7) |
is approximately the same for all spectra at the same decay times, despite the wide range of significant wave heights and mean frequencies of their data set. Assuming that the effective decay time and therefore the wave steepness is not affected by the correction of the wave spectrum, the scale factors are then given by
|
(4.8) |
 .
|
(4.9) |
Intuitively, this approach appears reasonable, because a more energetic spectrum will generally also have a lower peak frequency, and increasing the energy without decreasing the peak frequency produces a swell of unrealistic steepness. Since the swell spectrum is not related to the local stress, and only the local wind field is corrected in the assimilation scheme, the wind field is not updated in the case of swell.
4.2.4 The general case
It was shown in Lionello et al. (1992) that the wind sea and swell retrieval scheme works well for simple cases or pure wind sea or swell. If the spectrum consists of a superposition of wind sea and swell, and the wind sea is well separated from the swell, the wind sea and swell correction methods can, in principle, still be applied separately to the two components of the spectrum. In this case, however, one needs to introduce additional assumptions regarding the partitioning of the total wave height correction between wind sea and swell.
The arbitrariness of the present and similar methods of distributing a single wave height correction over the full two-dimensional wave spectrum could presumably be partially alleviated by using maximum likelihood methods based on a large set of observed data, which is now becoming available through ERS-1 . However, a more satisfactory solution is clearly to assimilate additional data, such as two-dimensional SAR spectral retrievals, to overcome the inherently limited information content of altimeter wave height data.
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23.04.2002 |
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