Geography Reference
In-Depth Information
Fig. 4.5
Circulation for an infinitesimal loop in
the natural coordinate system.
However, from Fig. 4.5, d(δs)
δβ δn, where δβ is the angular change in the wind
direction in the distance δs. Hence,
=
δn δs
∂V
∂n
+
V
δβ
δs
δC
=
−
or, in the limit δn, δs
→
0
δC
(δn δs)
=−
∂V
∂n
+
V
R
s
ζ
=
lim
δn,δs
(4.9)
→
0
where R
s
is the radius of curvature of the streamlines [Eq. (3.20)]. It is now apparent
that the net vertical vorticity component is the result of the sum of two parts: (1)
the rate of change of wind speed normal to the direction of flow -∂V /∂n, called the
shear
vorticity; and (2) the turning of the wind along a streamline V/R
s
, called the
curvature
vorticity. Thus, even straight-line motion may have vorticity if the speed
changes normal to the flow axis. For example, in the jet stream shown schematically
in Fig. 4.6a, there will be cyclonic relative vorticity north of the velocity maximum
and anticyclonic relative vorticity to the south (Northern Hemisphere conditions)
as is recognized easily when the turning of a small paddle wheel placed in the
flow is considered. The lower of the two paddle wheels in Fig. 4.6a will turn in a
clockwise direction (anticyclonically) because the wind force on the blades north
of its axis of rotation is stronger than the force on the blades to the south of the axis.
The upper wheel will, of course, experience a counterclockwise (cyclonic) turning.
Thus, the poleward and equatorward sides of a westerly jetstream are referred to
as the cyclonic and anticyclonic shear sides, respectively.
Conversely, curved flow may have zero vorticity provided that the shear vorticity
is equal and opposite to the curvature vorticity. This is the case in the example
shown in Fig. 4.6b where a frictionless fluid with zero relative vorticity upstream
flows around a bend in a canal. The fluid along the inner boundary on the curve
flows faster in just the right proportion so that the paddle wheel does not turn.
