Geography Reference
In-Depth Information
potential vorticity is always in some sense a measure of the ratio of the absolute vor-
ticity to the effective depth of the vortex. In (4.12), for example, the effective depth
is just the differential distance between potential temperature surfaces measured
in pressure units (
∂θ/∂p).
In a homogeneous incompressible fluid, potential vorticity conservation takes a
somewhat simpler form. In this case, because density is a constant, the horizontal
area must be inversely proportional to the depth, h, of the fluid parcel:
−
M(ρh)
−
1
δA
=
=
Const/h
where h is the depth of the parcel. Substituting to eliminate δA in (4.11) yields
(ζ
+
f)/h
=
η/h
=
Const
(4.13)
where ζ is here evaluated at constant height.
If the depth, h, is constant, (4.13) states that absolute vorticity is conserved
following the motion. Conservation of absolute vorticity following the motion
provides a strong constraint on the flow, as can be shown by a simple example.
Suppose that at a certain point (x
0
, y
0
) the flow is in the zonal direction and the
relative vorticity vanishes so that η(x
0
, y
0
)
f
0
. Then, if absolute vorticity is
conserved, the motion at any point along a parcel trajectory that passes through
(x
0
, y
0
) must satisfy ζ
=
f
0
. Because f increases toward the north, trajectories
that curve northward in the downstream direction must have ζ
+
f
=
=
f
0
−
f<0,
whereas trajectories that curve southward must have ζ
f>0. However,
as indicated in Fig. 4.8, if the flow is westerly, northward curvature downstream
implies ζ>0, whereas southward curvature implies ζ<0. Thus, westerly zonal
flow must remain purely zonal if absolute vorticity is to be conserved following
the motion. The easterly flow case, also shown in Fig. 4.8, is just the opposite.
Northward and southward curvatures are associated with negative and positive
relative vorticities, respectively. Hence, an easterly current can curve either to the
north or to the south and still conserve absolute vorticity.
When the depth of the fluid changes following the motion, it is potential vorticity
that is conserved. However, again (4.13) indicates that westerly and easterly flows
behave differently. The situation for westerly flow impinging on an infinitely long
=
f
0
−
Fig. 4.8
Absolute vorticity conservation for curved flow trajectories.
