Geoscience Reference
In-Depth Information
where the radius of the circle is the magnitude of the moment arm,
r
. For a
parcel in the troposphere or in the ocean,
z
<<
a
, and it is reasonable to neglect
the height of the parcel above the surface of the earth compared with the radius
of the earth and let
(6.9)
ra
φ
=
cos
.
This is known as the
thin atmosphere (or ocean) approximation.
Equation 6.8
indicates that the tangential velocity of a parcel in solid body rotation at the
equator is 463 m/s. This is a huge speed, an order of magnitude greater than
velocities observed relative to the rotating earth (
Figs. 2.10
-2.14)
.
Now, imagine that the parcel drawn in
Figure 6.2
has a zonal velocity
u
rela-
tive to the rotating earth, that is, as measured by an observer on the ground at
the same latitude as the parcel, but still
vw
In the diagram, instantaneous
westerly velocity (
u
> 0) would be motion into the page and easterly velocity
(
u
< 0) would be motion out of the page. We define the following:
.
v
Ω=
angular velocity of the earth, with magnitude
2 rad
π
−−
51
Ω=
=
7.29
#
10
s.
(6.10)
24 hr
U
ROT
is the instantaneous tangential velocity of the earth's surface in the
absolute frame of reference at the same latitude as the parcel (i.e., the velocity
of solid body rotation). Using the thin atmosphere/ocean approximation and
Eq. 6.10, we have
2
π
r
2
πφ
φ
Ω
a
cos
U
== =
Ω
a
cos
=
Ω
r
,
(6.11)
ROT
24 hr
2
π
where the circumference of the circle with radius
2 os
a
πφ is the distance trav-
eled in one day. Defining
U
ABS
as the instantaneous tangential velocity of the
parcel in the absolute frame of reference, it is equal to the sum of the velocity
of the rotating earth and the velocity relative to the rotating earth:
RO
=+
(6.12)
UUu
ABS
.
Then, the magnitude of the absolute angular momentum per unit mass of the
parcel (Eq. 6.7) is
(6.13)
MUr
==+++
[( )
Ω
a
z
cos
φ
u az
](
)
cos
φ
ABS
or, with the thin atmosphere/ocean approximation,
(6.14)
MUr
== +
(
Ω
a
cos
φ
ua
)
cos
φ
.
ABS
According to Eq. 6.14, if f changes, that is, if the parcel moves to the north
or south, then
u
must also change to keep
M
constant (to conserve absolute
angular momentum). If a parcel in the Northern Hemisphere, for example,
moves to a higher latitude, then cos f will decrease and
u
must increase. This
apparent zonal velocity that results from meridional motion is referred to as
the
Coriolis effect
.
To evaluate the importance of the Coriolis effect, consider a parcel of air
that moves from the equator to 30°N latitude due to some impulsive meridi-
onal force. In its initial position on the equator, the parcel has no zonal velocity
