Geoscience Reference
In-Depth Information
An alternative approach to deriving the full Coriolis force is to perform a
formal coordinate translation from the inertial to the rotating frame of refer-
ence. The physically motivated derivation provided here gives the exact same
results.
An approximate expression for the Coriolis force can be obtained from Eq.
6.25 by neglecting terms inversely proportional to the radius of the earth (
a
)
and terms involving vertical velocity, since
w
<<
u
and
v
(see
chapter 2
and
exercise 6.4). The vertical component of the full Coriolis force can also be ne-
glected in this approximation, since it is small compared with gravity (exercise
6.4). These approximations leave two dominant terms in Eq. 6.25, and
v
t
t
t
t
F
COR
.
2
Ω
v
sinf
φ
i
−
2
Ω
u
sinf
φ
j
=
f vi
(
−
uj
),
(6.26)
where
f
/
(6.27)
2 in
Ω
φ
is known as the
Coriolis parameter.
In vector form,
v
|
v
F
.
fkv
#
.
(6.28)
COR
In the approximation to the full Coriolis force given in Eq. 6.26, the meridi-
onal component derives from centrifugal forces and the zonal component from
conservation of absolute angular momentum.
Note that the Coriolis force couples the two horizontal directions of motion.
According to Eq. 6.26, if a parcel has a non-zero meridional velocity
v
(in the
rotating frame of reference), it will acquire a zonal acceleration due to conserva-
tion of absolute angular momentum. If the parcel has a non-zero zonal velocity
u
, it will be subjected to a meridional acceleration due to centrifugal forces.
6.2 PRESSURE GRADIENT FORCE
Pressure gradient forces are also extremely important in determining winds and
ocean currents. These forces arise when pressure—or geopotential height—
fields are not uniform. In that case, air and water parcels experience a pressure
gradient force directed to lower values, or down the gradient.
Consider a volume
V
of air or water with pressure
p
1
acting on the left face
located at
n
1
, and pressure
p
2
acting on the right face located at
n
2
, as depicted
in
Figure 6.5.
The area of each face of the volume is
A
, and the volume contains
a mass of material
m
with density r. The direction indicated by unit vector
t
can be any direction, horizontal or vertical or some combination.
The pressure on the left side of the volume,
p
1
, is the force per unit area ex-
erted by the surrounding air or water, and
p
2
is the pressure on the right face.
If
p
1
, the parcel will not move (accelerate) in the
t
direction. If
p
1
>
p
2
, the
parcel will accelerate to the right ( +
t
direction), and if
p
1
<
p
2
, the parcel will
accelerate to the left (
t
direction). The magnitude of the acceleration depends
on the difference in
p
1
and
p
2
, and the direction is determined by which is bigger.
To translate this concept into a mathematical expression for use in Eqs.
6.3- 6.5, we define
F
PG
v
as the force per unit mass acting on the parcel due to
pressure gradients. Then,
