Geography Reference
In-Depth Information
symmetric flow (no x dependence). More general purely geostrophic flows can-
not satisfy both these equations simultaneously, as there are then two independent
equations in a single unknown so that the system is over-determined. Thus, it
should be clear that the role of the vertical motion distribution must be to maintain
consistency between the geopotential tendencies required by vorticity advection
in (6.21) and thermal advection in (6.22).
6.3.1
Geopotential Tendency
If for simplicity we set J
0 and eliminate ω between (6.21) and (6.22), we obtain
an equation that determines the local rate of change of geopotential in terms of the
three-dimensional distribution of the field:
=
f
0
σ
1
f
0
∇
f
∂
∂p
∂
∂p
2
2
∇
+
χ
=−
f
0
V
g
·∇
+
B
A
(6.23)
f
0
σ
∂
∂p
∂
∂p
−
−
V
g
·∇
−
C
This equation is often referred to as the
geopotential tendency equation
. It pro-
vides a relationship between the local geopotential tendency (term A) and the
distributions of vorticity advection (term B) and thickness advection (term C). If
the distribution of is known at a given time, then terms B and C may be regarded
as known forcing functions, and (6.23) is a linear partial differential equation in
the unknown χ .
Although (6.23) appears to be quite complicated, a qualitative notion of its
implications can be gained if we note that term A involves second derivatives
in space of the χ field, and is thus generally proportional to minus χ . Term B
is proportional to the advection of absolute vorticity; it is usually the dominant
forcing term in the upper troposphere. As the discussion in Section 6.2.2 indicated,
for short waves term B is negative in region I of Fig. 6.7 (upstream of the upper level
trough). Thus, because the sign of the geopotential tendency is opposite to that of
term B in this case, χ will be positive and a ridge will tend to develop. This ridging
is, of course, necessary for the development of a negative geostrophic vorticity.
Similar arguments, but with the signs reversed, apply to region II downstream
from the upper level trough where falling geopotential heights are associated with
a positive relative vorticity advection. It is also important to note that the vorticity
advection term is zero along both the trough and ridge axes, as both
ζ
g
and v
g
are zero at the axes. Thus, vorticity advection cannot change the strength of this
∇
