Geography Reference
In-Depth Information
has important consequences for the influence of upper level disturbances on the
lower troposphere. The induction of geopotential changes in the lower troposphere
by an upper level potential vorticity change can be demonstrated in a simple
example.
Suppose that the flow in the upper troposphere is characterized by the geopo-
tential distribution of (6.20). The potential vorticity can then be expressed as
q
+
f
0
σ
∂
2
A
∂p
2
,
and σ is assumed to be constant.
4
If the latitudinal variation of f is neglected, the
quasi-geostrophic potential vorticity equation can then be expressed as
=−
k
2
l
2
A
=
f
+
Q (p) sin kx cos ly, where Q
+
∂q
∂t
=−
U
∂q
V
g
·∇
q
=−
∂x
=−
kUQ cos kx cos ly
∂
∂t,
so that again setting χ
=
∂
2
χ
∂p
2
1
f
0
∇
f
0
σ
2
χ
+
=−
kUQ cos kx cos ly
(6.30)
which can be solved for the geopotential tendency. Letting
χ (x, y, p, t)
=
X (p, t) cos kx cos ly
and substituting into (6.30) yields an equation for the vertical dependence of the
geopotential tendency:
d
2
X
dp
2
σ
f
0
λ
2
X
−
=−
kUQ
(6.31)
l
2
σf
−
0
. Equation (6.31) shows that potential vorticity advec-
tion at a given altitude will generate a response in the geopotential tendency whose
vertical scale (measured in pressure units) is λ
−
1
. Thus, for example, upper-level
vorticity advection associated with disturbances of large horizontal scale (small k)
will produce geopotential tendencies that extend down to the surface with little
loss of amplitude, whereas for disturbances of small horizontal scale the response
is confined close to the levels of forcing as shown in Fig. 6.11. In mathematical
terms, the differential operator in (6.31) spreads the response in the vertical so that
forcing at one altitude influences other altitudes.
≡
k
2
where λ
2
+
4
Actually, σ varies substantially with pressure even in the troposphere. However, the qualitative
discussion in this section would not be changed if we were to include this additional complication.
