Geography Reference
In-Depth Information
13.6
PRIMITIVE EQUATION MODELS
Modern numerical forecast models are based on a formulation of the dynamical
equations, referred to as the primitive equations , which is essentially the formu-
lation proposed by Richardson. The primitive equations differ from the complete
momentum equations (2.19)-(2.21) in that the vertical momentum equation is
replaced by the hydrostatic approximation, and the small terms in the horizontal
momentum equations given in columns C and D in Table 2.1 are neglected. In most
models some version of the σ -coordinate system introduced in Section 10.3.1 is
used, and the vertical dependence is represented by dividing the atmosphere into a
number of levels and utilizing finite difference expressions for vertical derivatives.
Both finite differencing and the spectral method have been used for horizontal dis-
cretization in operational primitive equation forecast models. Excellent examples
of such models are the grid point and spectral models developed at the European
Centre for Medium-Range Weather Forecasts (ECMWF). The numerical forecast-
ing system at ECMWF, and several other centers, includes both a deterministic
model and an ensemble prediction system. The deterministic model is typically
run at very high resolution with the best estimate of the initial conditions. The
ensemble prediction system consists of a number of parallel runs of a reduced
resolution model in which the initializations (and sometimes the model physics)
are slightly perturbed. The spread of forecasts given by members of the ensemble
can then be used to assess confidence in the forecasts.
13.6.1
The Ecmwf Grid Point Model
The earliest version of the ECMWF operational model was a global grid point
model with second-order horizontal differencing on a uniform grid with intervals
of 1.875˚ in both latitude and longitude. For computational efficiency and noise
control the grid was staggered in space, as shown in Fig. 13.5, so that not all
variables were carried at the same points in λ and φ. Variables were also staggered
vertically, as shown in Fig. 13.6, and there were a total of 15 unequally spaced
σ layers, arranged to provide the finest resolution near the ground, the coarsest
resolution in the stratosphere, and an uppermost layer near the 25-hPa level.
The difference equations on the staggered grid can be expressed most conve-
niently by defining a special operator notation as follows:
u λ
=
[u (λ
+
δλ/2)
+
u (λ
δλ/2)] /2
(13.50)
δ λ u
=
[u (λ
+
δλ/2)
u (λ
δλ/2)] /δλ
These represent, respectively, the averaging and differencing of adjacent grid point
values of the field u on a latitude circle. Similar expressions can be defined for the
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