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dependent variable. The solution proceeds by determining the evolution of a finite
number of Fourier coefficients. Because the distribution in wave number space
of the Fourier coefficients for a given function is referred to as its
spectrum
,itis
appropriate to call this approach the spectral method.
At low resolution the spectral method is generally more accurate than the grid
point method, partly because for linear advection the numerical dispersion dis-
cussed in Section 13.3.4 can be severe in a grid point model, but does not occur in
a properly formulated spectral model. For the range of resolutions used commonly
in forecast models the two approaches are comparable in accuracy, and each has
its advocates.
13.5.1
The Barotropic Vorticity Equation in Spherical Coordinates
The spectral method is particularly advantageous for solution of the vorticity equa-
tion. When the proper set of basis functions is chosen, it is trivial to solve the
Poisson equation for the streamfunction. This property of the spectral method not
only saves computer time, but eliminates the truncation error associated with finite
differencing the Laplacian operator.
In practice the spectral method is applied most frequently to global models.
This requires use of spherical harmonics, which are more complicated than Fourier
series. In order to keep the discussion as simple as possible, it is again useful to
consider the barotropic vorticity equation as a prototype forecast model in order
to illustrate the spectral method on the sphere.
The barotropic vorticity equation in spherical coordinates may be expressed as
D
Dt
(ζ
+
2 sin φ)
=
0
(13.32)
2
ψ , with ψ a streamfunction, and
where as usual ζ
=∇
D
Dt
≡
∂
∂t
+
u
a cos φ
∂
∂λ
+
v
a
∂
∂φ
(13.33)
sinφ as the latitudinal coordinate, in
which case the continuity equation can be written
It turns out to be convenient to use µ
≡
u
cos φ
1
a
∂
∂λ
1
a
∂
∂µ
(v cos φ)
+
=
0
(13.34)
so that the streamfunction is related to the zonal and meridional velocities accord-
ing to
u
cos φ
=−
1
a
∂ψ
∂µ
;
1
a
∂ψ
∂λ
v cos φ
=
(13.35)
