Geography Reference
In-Depth Information
useful to replace (6.11) by an equation for the evolution of the vorticity of the
geostrophic wind, in which case only the divergent part of the ageostrophic wind
plays a role in the dynamics. The quasi-geostrophic momentum equation (6.11)
can still be used diagnostically, however, to determine the nondivergent part of
V
a
,
for a known distribution of
V
g
(see Section 6.4.3).
6.2.2
The Quasi-Geostrophic Vorticity Equation
Just as the horizontal momentum can be approximated to O(Ro) by its geostrophic
value, the vertical component of vorticity can also be approximated geostrophi-
cally. In Cartesian coordinates the components of (6.7) are
∂
∂x
,
0
u
g
=−
∂
∂y
f
0
v
g
=
(6.14)
Thus, the geostrophic vorticity, ζ
g
=
k
·∇
×
V
g
, can be expressed as
∂
2
∂x
2
∂
2
∂y
2
∂v
g
∂x
−
∂u
g
∂y
=
1
f
0
1
f
0
∇
2
ζ
g
=
+
=
(6.15)
Equation (6.15) can be used to determine ζ
g
(x, y) from a known field (x, y).
Alternatively, (6.15) can be solved by inverting the Laplacian operator to determine
from a known distribution of ζ
g
, provided that suitable conditions on are
specified on the boundaries of the region in question. This
invertibility
is one reason
why vorticity is such a useful forecast diagnostic; if the evolution of the vorticity
can be predicted, then inversion of (6.15) yields the evolution of the geopotential
field, from which it is possible to determine the geostrophic wind and temperature
distributions. Since the Laplacian of a function tends to be a maximum where the
function itself is a minimum, positive vorticity implies low values of geopotential
and vice versa, as illustrated for a simple sinusoidal disturbance in Fig. 6.7.
The quasi-geostrophic vorticity equation can be obtained from the x and y
components of the quasi-geostrophic momentum equation (6.11), which can be
expressed, respectively, as
D
g
u
g
Dt
−
f
0
v
a
−
βyv
g
=
0
(6.16)
and
D
g
v
g
Dt
+
f
0
u
a
+
βyu
g
=
0
(6.17)
Taking ∂(6.17)/∂x
∂(6.16)/∂y, and using the fact that the divergence of the
geostrophic wind defined in (6.7) vanishes, immediately yields the vorticity
equation
−
f
0
∂u
a
D
g
ζ
g
Dt
∂v
a
∂y
=−
∂x
+
−
βv
g
(6.18)
which should be compared with (4.22b).
