Geography Reference
In-Depth Information
Fig. 4.1
Circulation about a closed contour.
where
l
(s) is a position vector extending from the origin to the point s(x, y, z) on
the contour C, and d
l
represents the limit of δ
l
=
+
−
→
0. Hence,
as indicated in Fig. 4.1, d
l
is a displacement vector locally tangent to the contour.
By convention the circulation is taken to be positive if C>0 for counterclockwise
integration around the contour.
That circulation is a measure of rotation is demonstrated readily by considering
a circular ring of fluid of radius R in solid-body rotation at angular velocity
l
(s
δs)
l
(s) as δs
about the z axis. In this case,
U
R
, where
R
is the distance from the axis of
rotation to the ring of fluid. Thus the circulation about the ring is given by
=
×
2π
R
2
dλ
2π R
2
C
≡
U
·
d
l
=
=
0
In this case the circulation is just 2π times the angular momentum of the fluid
ring about the axis of rotation. Alternatively, note that C/(πR
2
)
2 so that the
circulation divided by the area enclosed by the loop is just twice the angular speed
of rotation of the ring. Unlike angular momentum or angular velocity, circulation
can be computed without reference to an axis of rotation; it can thus be used to
characterize fluid rotation in situations where “angular velocity” is not defined
easily.
The circulation theorem is obtained by taking the line integral of Newton's
second law for a closed chain of fluid particles. In the absolute coordinate system
the result (neglecting viscous forces) is
=
D
a
U
a
Dt
∇
p
·
d
l
·
d
l
=−
−
∇
·
d
l
(4.1)
ρ
where the gravitational force
g
is represented as the gradient of the geopotential
, defined so that -
∇
=
g
=−
g
k
. The integrand on the left-hand side can be
