Geography Reference
In-Depth Information
rewritten as
1
D
a
U
a
Dt
D
Dt
(
U
a
·
D
a
Dt
(d
l
)
·
d
l
=
d
l
)
−
U
a
·
≡
or after observing that since
l
is a position vector, D
a
l
/Dt
U
a
,
D
a
U
a
Dt
D
Dt
(
U
a
·
·
d
l
=
d
l
)
−
U
a
·
d
U
a
(4.2)
Substituting (4.2) into (4.1) and using the fact that the line integral about a closed
loop of a perfect differential is zero, so that
∇
·
d
l
=
d
=
0
and noting that
U
a
·
1
2
d
U
a
=
d(
U
a
·
U
a
)
=
0
we obtain the circulation theorem:
DC
a
Dt
D
Dt
ρ
−
1
dp
=
U
a
·
d
l
=−
(4.3)
The term on the right-hand side in (4.3) is called the solenoidal term. For a
barotropic fluid, the density is a function only of pressure, and the solenoidal term
is zero. Thus, in a barotropic fluid the absolute circulation is conserved following
the motion. This result, called
Kelvin's circulation theorem
, is a fluid analog of
angular momentum conservation in solid-body mechanics.
For meteorological analysis, it is more convenient to work with the relative
circulation C rather than the absolute circulation, as a portion of the absolute
circulation, C
e
, is due to the rotation of the earth about its axis. To compute C
e
,
we apply Stokes' theorem to the vector
U
e
, where
U
e
=
×
r
is the velocity of
the earth at the position
r
:
(
C
e
=
U
e
·
=
∇
×
·
d
l
U
e
)
n
dA
A
1
Note that for a scalar D
a
/Dt
=
D/Dt (i.e., the rate of change following the motion does not depend
on the reference system). For a vector, however, this is not the case, as was shown in Section 2.1.1.
