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equivalent expression for a midlatitude β-plane.] Because the horizontal scale of a
spherical harmonic mode is proportional to n
−
1
, (13.44) shows that a single mode
propagates westward on a sphere at a speed that is approximately proportional
to the square of the horizontal scale. This solution also suggests why for some
problems the spectral method is superior to the finite difference method at coarse
resolution. A model containing even a single Fourier component can represent
a realistic meteorological field (the Rossby wave), while many grid points are
required for an equivalent representation in finite differences.
13.5.3
The Spectral Transform Method
When many spherical harmonic modes are present, the solution of (13.36) by
a purely spectral method requires evaluation of the nonlinear interactions among
various modes due to the advection term. It turns out that the number of interaction
terms increases as the square of the number of modes retained in the series (13.40)
so that this approach becomes computationally inefficient for models with the sort
of spatial resolution required for prediction of synoptic-scale weather disturbances.
The spectral transform method overcomes this problem by transforming between
spherical harmonic wave number space and a latitude-longitude grid at every time
step and carrying out the multiplications in the advection term in grid space so that
it is never necessary to compute products of spectral functions.
To illustrate this method it is useful to rewrite the barotropic vorticity equation
in the form
2
∂ψ
A (λ, µ)
2
ψ
∂t
∂
∇
1
a
2
=−
∂λ
+
(13.45)
where
2
ψ
∂λ
2
ψ
∂µ
∂ψ
∂µ
∂
∇
∂ψ
∂λ
∂
∇
A (λ, µ)
≡
−
+
(13.46)
Substituting from (13.40) into (13.45) then yields for the spherical harmonic coef-
ficients
1)]
−
1
=
iν
γ
ψ
γ
+
+
dψ
γ
/dt
A
γ
[n (n
(13.47)
where A
γ
is the γ component of the transform of A(λ, µ):
2π
+
1
1
4π
A (λ, µ)Y
γ
dλdµ
A
γ
=
(13.48)
0
−
1
(n, m) is taken over
a finite number of modes, the integral in the transform (13.48) can be evaluated
exactly by numerical quadrature [i.e., by summing appropriately weighted values
of A(λ, µ) evaluated at the grid points (λ
j
,µ
k
) of a latitude-longitude grid mesh.]
The transform method utilizes the fact that if the sum γ
=
