Geography Reference
In-Depth Information
lower boundary is at the variable height h
T
(x, y) where
|
h
T
|
H. We also use
quasi-geostrophic scaling so that
|
ζ
g
|
f
0
. We can then approximate (4.26) by
H
∂
ζ
g
+
f
=−
f
0
Dh
T
Dt
∂t
+
V
·∇
(7.94)
Linearizing and applying the midlatitude β-plane approximation yields
∂
∂t
+
ζ
g
+
∂
∂x
f
0
H
u
∂h
T
βv
g
=−
u
(7.95)
∂x
We now examine solutions of (7.95) for the special case of a sinusoidal lower
boundary. We specify the topography to have the form
Re
h
0
exp (ikx)
cos ly
h
T
(x, y)
=
(7.96)
and represent the geostrophic wind and vorticity by the perturbation streamfunction
Re
ψ
0
exp (ikx)
cos ly
ψ(x, y)
=
(7.97)
Then (7.95) has a steady-state solution with complex amplitude given by
f
0
h
0
/
H
K
2
K
s
ψ
0
=
−
(7.98)
The streamfunction is either exactly in phase (ridges over the mountains) or
exactly out of phase (troughs over the mountains), with the topography depending
on the sign of K
2
K
s
. For long waves (K < K
s
) the topographic vorticity source
in (7.95) is balanced primarily by the meridional advection of planetary vorticity
(the β effect). For short waves (K > K
s
) the source is balanced primarily by the
zonal advection of relative vorticity.
The topographic wave solution (7.98) has the unrealistic characteristic that when
the wave number exactly equals the critical wave number K
s
the amplitude goes to
infinity. From (7.93) it is clear that this singularity occurs at the zonal wind speed
for which the free Rossby mode becomes stationary. Thus, it may be thought of as
a resonant response of the barotropic system.
Charney and Eliassen (1949) used the topographic Rossby wave model to
explain the winter mean longitudinal distribution of 500-hPa heights in North-
ern Hemisphere midlatitudes. They removed the resonant singularity by including
boundary layer drag in the form of Ekman pumping, which for the barotropic vor-
ticity equation is simply a linear damping of the relative vorticity [see (5.41)]. The
vorticity equation thus takes the form
∂
∂t
+
−
ζ
g
+
∂
∂x
f
0
H
u
∂h
T
βv
g
+
rζ
g
=−
u
(7.99)
∂x
τ
−
1
e
where r
≡
is the inverse of the spin-down time defined in Section 5.4.
