Geography Reference
In-Depth Information
Fig. 4.4
Relationship between circulation and vorticity for an area element in the horizontal plane.
In more general terms the relationship between vorticity and circulation is given
simply by Stokes' theorem applied to the velocity vector:
U
·
d
l
=
(
∇
×
U
)
·
n
dA
A
Here A is the area enclosed by the contour and
n
is a unit normal to the area
element dA (positive in the right-hand sense). Thus, Stokes' theorem states that the
circulation about any closed loop is equal to the integral of the normal component of
vorticity over the area enclosed by the contour. Hence, for a finite area, circulation
divided by area gives the
average
normal component of vorticity in the region.
As a consequence, the vorticity of a fluid in solid-body rotation is just twice the
angular velocity of rotation. Vorticity may thus be regarded as a measure of the
local angular velocity of the fluid.
4.2.1
Vorticity in Natural Coordinates
Physical interpretation of vorticity is facilitated by considering the vertical compo-
nent of vorticity in the natural coordinate system (see Section 3.2.1). If we compute
the circulation about the infinitesimal contour shown in Fig. 4.5, we obtain
2
V
∂n
δn
δs
V
δs
d(δs)
−
∂V
δC
=
+
+
2
Recall that n is a coordinate in the horizontal plane perpendicular to the local flow direction with
positive values to the left of an observer facing downstream.
