Geography Reference
In-Depth Information
In large-scale dynamic meteorology, we are in general concerned only with the
vertical components of absolute and relative vorticity, which are designated by η
and ζ , respectively.
η
k
·
(
×
U a ),
ζ
k
·
(
×
U )
In the remainder of this topic, η and ζ are referred to as absolute and relative
vorticities, respectively, without adding the explicit modifier “vertical component
of.” Regions of positive ζ are associated with cyclonic storms in the Northern
Hemisphere; regions of negative ζ are associated with cyclonic storms in the
Southern Hemisphere. Thus, the distribution of relative vorticity is an excellent
diagnostic for weather analysis. Absolute vorticity tends to be conserved following
the motion at midtropospheric levels; this conservation property is the basis for
the simplest dynamical forecast scheme discussed in Chapter 13.
The difference between absolute and relative vorticity is planetary vorticity ,
which is just the local vertical component of the vorticity of the earth due to its
rotation; k
·∇ ×
U e =
2 sin φ
f . Thus, η
=
ζ
+
f or, in Cartesian coordinates,
∂v
∂x
∂u
∂y ,
∂v
∂x
∂u
∂y +
ζ
=
η
=
f
The relationship between relative vorticity and relative circulation C discussed in
the previous section can be clearly seen by considering an alternative approach
in which the vertical component of vorticity is defined as the circulation about a
closed contour in the horizontal plane divided by the area enclosed, in the limit
where the area approaches zero:
d l A 1
ζ
lim
A
V
·
(4.8)
0
This latter definition makes explicit the relationship between circulation and vor-
ticity discussed in the introduction to this chapter. The equivalence of these two
definitions of ζ is shown easily by considering the circulation about a rectangular
element of area δxδy in the (x, y ) plane as shown in Fig. 4.4. Evaluating V
·
d l
for each side of the rectangle in Fig. 4.4 yields the circulation
v
∂x δx δy
u
∂y δy δx
∂v
∂u
δC
=
uδx
+
+
+
vδy
∂v
∂x
δxδy
∂u
∂y
=
Dividing through by the area δA
=
δxδy gives
∂v
∂x
δC
δA =
∂u
∂y
ζ
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