Geography Reference
In-Depth Information
The vorticity equation can then be expressed as
∂ψ
∂µ
2
ψ
∂t
2
ψ
∂λ
2
ψ
∂µ
∂
∇
1
a
2
∂
∇
∂ψ
∂λ
∂
∇
2
a
2
∂ψ
∂λ
=
−
−
(13.36)
where
∂
∂µ
1
µ
2
∂ψ
∂µ
∂
2
ψ
∂λ
2
1
a
2
1
2
ψ
∇
=
−
+
(13.37)
µ
2
1
−
The appropriate orthogonal basis functions are the
spherical harmonics
, which
are defined as
P
γ
(µ) e
imλ
Y
γ
(µ, λ)
≡
(13.38)
where γ
(n, m) is a vector containing the integer indices for the spherical
harmonics. These are given by m
≡
=
0,
±
1,
±
2,
±
3, ..., n
=
1, 2, 3, ..., where it
is required that
n. Here, P
γ
designates an associated Legendre function of
the first kind of degree n. From (13.38) it is clear that m designates the zonal wave
number. It can be shown
2
that n
|
m
|≤
−|
m
|
designates the number of nodes of P
γ
in the
interval
1 <µ<1 (i.e., between the poles), and thus measures the meridional
scale of the spherical harmonic. The structures of a few spherical harmonics are
shown in Fig. 13.3.
An important property of the spherical harmonics is that they satisfy the rela-
tionship
−
n (n
+
1)
2
Y
γ
∇
=−
Y
γ
(13.39)
a
2
so that the Laplacian of a spherical harmonic is proportional to the function itself,
which implies that the vorticity associated with a particular spherical harmonic
component is simply proportional to the streamfunction for the same component.
In the spectral method on the sphere, the streamfunction is expanded in a finite
series of spherical harmonics by letting
ψ (λ, µ, t)
=
ψ
γ
(t) Y
γ
(µ, λ)
(13.40)
γ
where ψ
γ
is the complex amplitude for the Y
γ
spherical harmonic and the sum-
mation is over both n and m. The individual spherical harmonic coefficients ψ
γ
are related to the streamfunction ψ(λ,µ) through the inverse transform
1
4π
Y
γ
ψ (λ, µ, t) dS
ψ
γ
(t)
=
(13.41)
S
dµdλ and Y
γ
where dS
=
designates the complex conjugate of Y
γ
.
2
See Washington and Parkinson (1986) for a discussion of the properties of the Legendre function.
