Geography Reference
In-Depth Information
Fig. 13.3
Patterns of positive and negative regions for the spherical harmonic functions with n
=
5
and m
=
0, 1, 2, 3, 4, 5. (After Washington and Parkinson, 1986, adapted from Baer, 1972.)
13.5.2
Rossby-Haurwitz Waves
Before considering numerical solution of the barotropic vorticity equation, it is
worth noting that an exact analytic solution of the nonlinear equation can be
obtained in the special case where the streamfunction is equal to a single spherical
harmonic mode. Thus, we let
ψ
γ
(t) e
imλ
P
γ
(µ)
ψ (λ, µ, t)
=
(13.42)
Substituting from (13.42) into (13.36) and applying (13.39), we find that the non-
linear advection term is identically zero so that the amplitude coefficient satisfies
the ordinary linear equation
−
n
(
n
+
1
)
dψ
γ
/dt
=−
2imψ
γ
(13.43)
which has the solution ψ
γ
(t)
=
ψ
γ
(0)exp(iν
γ
t), where
ν
γ
=
2m/ [n (n
+
1)]
(13.44)
is the dispersion relationship for
Rossby-Haurwitz waves
, which is the name given
to planetary waves on a sphere. [This should be compared to (7.91), which is the
