Geography Reference
In-Depth Information
Given an initial distribution of and suitable boundary conditions, (6.24) and
(6.25) can be used to forecast the evolution of the geopotential field. For simplicity
it is often assumed that the lower boundary corresponds to p
0
=
1000 hPa and that
ω vanishes at this surface so that from (6.13b) the lower boundary condition for
adiabatic motion is simply that temperature (or thickness) is conserved following
the geostrophic motion on the 1000-hPa surface:
∂
∂t
+
p
=
p
0
=
−
∂
∂p
V
g
·∇
0
(6.26)
6.3.3
Potential Vorticity Inversion
Equations (6.24)-(6.26), together with suitable lateral and upper boundary con-
ditions on , completely prescribe the evolution of the geostrophic flow. Thus,
under adiabatic conditions the evolution of large-scale midlatitude meteorologi-
cal systems is completely determined by the twin constraints of quasi-geostrophic
potential vorticity conservation following the geostrophic flow in the interior and
temperature conservation following the geostrophic flow along the lower boundary.
If boundary effects can be neglected, the distribution of potential vorticity alone
determines the geopotential distribution, and hence the distributions of temperature
and geostrophic wind. The process of recovering the geopotential field from the
potential vorticity field is referred to as potential vorticity inversion. Because is
related to q by a second-order differential operator, the geopotential disturbance
associated with a localized anomaly in q will extend horizontally and vertically
beyond the region of anomalous q. This remote influence can be illustrated by a
simple example.
For mathematical simplicity we assume that the static stability parameter, σ is
constant in height and introduce a stretched vertical coordinate defined as
z
ˆ
≡
p
σ
1
/
2
f
0
. Then (6.25) can be expressed as
∂
2
∂x
2
∂
2
∂y
2
∂
2
∂
1
f
0
q
=
q
=
f
+
f
+
+
+
(6.27)
z
2
ˆ
where q
and
designate the potential vorticity and geopotential anomalies,
respectively.
We now suppose that the potential vorticity anomaly consists of a compact ball
of constant positive potential vorticity, q
0
, with spherical shape in the
x, y,
z
coordinate system. Then, if the ball is centered at the origin, q
can be represented
in terms of the radial coordinate r
ˆ
≡
x
2
z
2
1
/
2
y
2
+
+ˆ
as
r
2
∂
∂r
1
f
0
r
2
∂
∂r
q
=
(6.28)
