Geography Reference
In-Depth Information
where
D
h
Dt
≡
∂
∂t
+
∂
∂x
+
∂
∂y
u
v
As mentioned earlier, (4.22a) is not accurate in intense cyclonic storms. For these
the relative vorticity should be retained in the divergence term:
f )
∂u
D
h
(ζ
+
f )
∂v
∂y
=−
(ζ
+
∂x
+
(4.22b)
Dt
Equation (4.22a) states that the change of absolute vorticity following the hori-
zontal motion on the synoptic scale is given approximately by the concentration
or dilution of planetary vorticity caused by the convergence or divergence of the
horizontal flow, respectively. In (4.22b), however, it is the concentration or dilu-
tion of absolute vorticity that leads to changes in absolute vorticity following the
motion.
The form of the vorticity equation given in (4.22b) also indicates why cyclonic
disturbances can be much more intense than anticyclones. For a fixed amplitude of
convergence, relative vorticity will increase, and the factor (ζ
f ) becomes larger,
which leads to even higher rates of increase in the relative vorticity. For a fixed
rate of divergence, however, relative vorticity will decrease, but when ζ
+
→−
f ,
the divergence term on the right approaches zero and the relative vorticity cannot
become more negative no matter how strong the divergence. (This difference in
the potential intensity of cyclones and anticyclones was discussed in Section 3.2.5
in connection with the gradient wind approximation.)
The approximate forms given in (4.22a) and (4.22b) do not remain valid, how-
ever, in the vicinity of atmospheric fronts. The horizontal scale of variation in
frontal zones is only
10 cm s
−
1
.For
these scales, vertical advection, tilting, and solenoidal terms all may become as
large as the divergence term.
∼
100 km, and the vertical velocity scale is
∼
4.5
VORTICITY IN BAROTROPIC FLUIDS
A model that has proved useful for elucidating some aspects of the horizontal
structure of large-scale atmospheric motions is the barotropic model. In the most
general version of this model, the atmosphere is represented as a homogeneous
incompressible fluid of variable depth, h(x,y,t)
z
1
, where z
2
and z
1
are the heights of the upper and lower boundaries, respectively. In this model, a
special form of potential vorticity is conserved following the motion. A simpler
situation arises if the fluid depth is constant. In that case it is absolute vorticity that
is conserved following the motion.
=
z
2
−
