Geography Reference
In-Depth Information
where A is the area enclosed by the contour and the unit normal
n
is defined by the
counterclockwise sense of the line integration using the “right-hand screw rule.”
Thus, for the contour of Fig. 4.1,
n
would be directed out of the page. If the line
integral is computed in the horizontal plane,
n
is directed along the local vertical
(see Fig. 4.2). Now, by a vector identity (see Appendix C)
∇
×
U
e
=
∇
×
(
×
r
)
=
∇
×
(
×
R
)
=
∇·
R
=
2
so that (
f is just the Coriolis parameter. Hence, the
circulation in the horizontal plane due to the rotation of the earth is
∇
×
U
e
)
·
n
=
2 sin φ
≡
C
e
=
2
sin φ
A
=
2A
e
where
denotes an average over the area element A and A
e
is the projection
of A in the equatorial plane as illustrated in Fig. 4.2. Thus, the relative circulation
may be expressed as
sin φ
C
=
C
a
−
C
e
=
C
a
−
2A
e
(4.4)
Differentiating (4.4) following the motion and substituting from (4.3) we obtain
the Bjerknes circulation theorem:
DC
Dt
=−
dp
ρ
−
2
DA
e
Dt
(4.5)
For a barotropic fluid, (4.5) can be integrated following the motion from an initial
state (designated by subscript 1) to a final state (designated by subscript 2), yielding
the circulation change
C
2
−
C
1
=−
2(A
2
sin φ
2
−
A
1
sin φ
1
)
(4.6)
Fig. 4.2
Area A
e
subtended on the equatorial plane by horizontal area A centered at latitude φ.
