Geoscience Reference
In-Depth Information
"
Ω
k
^
"
F
SB
"
F
GRAV
"
g
EFF
"
F
SB
φ
Figure 6.3. Centrifugal acceleration for solid body
rotation.
around the same circle as the parcel in solid body rotation, but at a faster
rate—it is “super rotating.” The centrifugal force acting on this parcel,
F
CEN
v
,
is greater than that acting on the parcel in solid body rotation,
F
S
v
, and can be
written
t
v
v
v
2
v
F
=+ =+
F
F
l
Ω
rnF
l
,
(6.19)
CENT
SB
CENT
CENT
v
where
F
CENT
l
is the difference in centrifugal acceleration that arises when
u
!
0
v
and Eq. 6.17 has been used. If
u
> 0, then
F
CENT
l
points in the
+
t
direction, and
l
v
points in the
t
direction.
By analogy with Eq. 6.17, the full centrifugal acceleration acting on the
parcel can also be written
if
u
< 0 (the case of “sub-rotation”),
F
CENT
==
+
=++
U
2
(
Uu
)
2
2
u
v
ABS
t
ROT
t
t
t
2
F
n
n
Ω
rn
d
2
Ω
u
n
n
.
(6.20)
CENT
r
r
r
Comparing Eq. 6.20 with Eq. 6.19, we see that
u
2
v
t
F
l
=+
d
2
u
n
n
.
(6.21)
CENT
r
v
y
CENT
t
, and a vertical
F
CENT
l
can be decomposed into a meridional component,
Fj
t
, as diagrammed in
Figure 6.4.
In local Cartesian coordi-
nates, the unit vector
t
is
z
CENT
component,
Fk
t
t
t
n
=−
sin
φ
j
+
cos
φ
k
.
(6.22)
With
r
a
cos f, Eq. 6.21 becomes
