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Computation of the distribution of A(λ, µ) for all grid points can be carried out
without need to introduce finite differences for derivatives by noting that we can
express the advection term in the form
1
µ 2 1
A λ j k =
[F 1 F 2 +
F 3 F 4 ]
(13.49)
where
1
µ 2 ∂ψ/∂µ,
2 ψ/∂λ,
F 1 =−
F 2 =
1
µ 2
2 ψ/∂µ,
F 3 =
F 4 =
∂ψ/∂λ
The quantities F 1
F 4 can be computed exactly for each grid point using the
spectral coefficients ψ γ and the known differential properties of the spherical
harmonics. For example,
imψ γ Y γ λ j k
F 4 =
∂ψ/∂λ
=
γ
Once these terms have been computed for all grid points, A(λ, µ) can be computed
at the grid points by forming the products F 1 F 2 and F 3 F 4 ; no finite difference
approximations to the derivatives are required in this procedure. The transform
(13.48) is then evaluated by numerical quadrature to compute the spherical har-
monic components A γ . Finally, (13.47) can be stepped ahead by a time increment
δt in order to obtain new estimates of the spherical harmonic components of the
streamfunction. The whole process is then repeated until the desired forecast period
is reached. The steps in forecasting with the barotropic vorticity equation using
the spectral transform method are summarized in schematic form in Fig. 13.4.
Fig. 13.4
Steps in the prediction cycle for the spectral transform method of solution of the barotropic
vorticity equation.
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