Geography Reference
In-Depth Information
ν<0so
that m>0, whereas for u<0wehavem<0. In the former situation the lower
signs apply on the right in (7.45a,b), whereas in the latter the upper signs apply. In
both cases the vertical phase propagation is downward relative to the mean flow,
and vertical energy propagation is upward.
When m
2
Here we see from (7.4
4
) that if we set k>0 then for u>0wehave
ˆ
im
i
is imaginary and the solution to (7.46) will have the
form of vertically trapped waves:
< 0,m
=
w
=ˆ
−
w exp (ikx) exp (
m
i
z)
, the magnitude of the
frequency relative to the mean flow, is less than the buoyancy frequency. Stable
stratification, wide ridges, and comparatively weak zonal flow provide favorable
conditions for the formation of vertically propagating topographic waves (m real).
Because the energy source for these waves is at the ground, they must transport
energy upward. Hence, the phase
s
peed relative to the mean zonal flow must have
a downward component. Thus if u>0, lines of constant phase must tilt westward
with height. When m is imaginary, however, the solution (7.43) has exponential
behavior in the vertical with an exponential decay height of µ
−
1
, where µ
Thus, vertical propagation is possible only when
|
uk
|
=|
m
|
.
Boundedness as z
requires that we choose the solution with exponential
decay away from the lower boundary.
In order to contrast the character of the solutions for real and imaginary m,we
consider a specific example in which there is westerly mean flow over topography
with a height profile given by
→∞
h (x)
=
h
M
cos kx
where h
M
is the amplitude of the topography. Then because the flow at the lower
boundary must be parallel to the boundary, the vertical velocity perturbation at the
boundary is given by the rate at which the boundary height changes following the
motion:
=
Dh
Dt
z
=
0
≈
u∂h
∂x
w
(x, 0)
=−
ukh
M
sin kx
and the solution of (7.46) that satisfies this condition can be written
uh
M
ke
−
µz
sin kx,
−
u
k > N
w (x, z)
=
(7.48)
−
uh
M
k sin (kx
+
mz) ,
uk < N
For fixed mean wind and buoyancy frequency, the character of the solution
depends only on the horizontal scale of the topography. The two cases of (7.48) may
be regarded as narrow ridge and wide ridge cases, respectively, since for specified
