Geography Reference
In-Depth Information
0 (nonrotating system) the vorticity and height perturbations are
uncoupled, and (7.72) yields a two-dimensional shallow water wave equation for
h[compare with (7.22)]:
For f
0
=
c
2
∂
2
h
∂x
2
∂
2
h
∂t
2
∂
2
h
∂y
2
−
+
=
0
(7.73)
which has solutions of the form
h
=
A exp
[
i(kx
+
ly
−
νt)
]
(7.74)
c
2
k
2
l
2
=
gH
k
2
l
2
. However, for f
0
=
with ν
2
0 the h
and ζ
fields
are coupled through (7.72). For motions with time scales longer than 1/f
0
(which
is certainly true for synoptic-scale motions), the ratio of the first two terms in (7.72)
is given by
=
+
+
∂
2
h
∂t
2
c
2
∂
2
h
∂x
2
<
f
0
L
2
gH
∂
2
h
∂y
2
+
which is small for L
1 km. Under such circum-
stances the time derivative term in (7.72) is small compared to the other two terms,
and (7.72) states simply that the vorticity is in geostrophic balance.
If the flow is initially unbalanced, the complete equation (7.72) can be used to
describe the approach toward geostrophic balance provided that we can obtain a
second relationship between h
and ζ
taking
∂ (7.70)
∂x
∼
1000 km, provided that H
∂ (7.69)
∂y
−
yields
f
0
∂u
∂ζ
∂t
+
∂v
∂y
∂x
+
=
0
(7.75)
which can be combined with (7.71) to give the linearized potential vorticity con-
servation law:
∂ζ
∂t
−
∂h
∂t
=
f
0
H
0
(7.76)
Thus, letting Q
designate the perturbation potential vorticity, we obtain from
(7.76) the conservation relationship
ζ
f
0
−
h
H
Q
(x, y, t)
=
=
Const.
(7.77)
Hence, if we know the distribution of Q
at the initial time, we know Q
for all
time:
Q
(x, y, t)
Q
(x, y, 0)
=
