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contributions (F
e
). We might then express the particle acceleration in response to the
net force F as:
a
¼
F
=
m
¼ð
F
p
þ
F
f
þ
F
e
Þ=
m
:
ð
3
:
4
Þ
There is, however, an important qualification to the use of Newton's second law if we
want to apply it to the ocean or the atmosphere. Strictly, Newton's second law is a
statement about the relation between acceleration and force in a non-accelerating
reference frame. Since the Earth is rotating, our usual reference frame is accelerating
and
Equation (3.4)
works only if we choose an external non-rotating reference frame,
for example a frame related to the 'fixed' stars.
3.2.1
Coriolis force (
F ¼ ma
on a rotating Earth)
To overcome this reference frame difficulty, we divide the acceleration in a fixed
reference frame into the acceleration as measured relative to coordinates fixed in the
Earth (a
rel
) and an additional component, termed the geostrophic acceleration (a
geo
),
which arises from the fact that the Earth is rotating about its axis with an angular
velocity
O ¼
2
p
radians/day:
a
¼
a
rel
þ
a
geo
:
ð
3
:
5
Þ
The dynamical statement (
3.4
) can then be rewritten to read:
F
0
=
a
rel
¼ð
F
ma
geo
Þ=
m
¼
m
:
ð
3
:
6
Þ
So the simple proportionality of acceleration to net force can be retained for Earth
coordinates if we add an additional 'force' term F
c
¼
ma
geo
. This is the Coriolis
force. It is strictly a mass
acceleration which has been moved from one side of the
equation to the other, but there is no reason why we should not treat it as a force
on equal terms with others (an example of a result in classical mechanics called
D'Alembert's principle). To proceed further, we need to find out how to calculate the
geostrophic acceleration. A rigorous derivation by a transformation of coordinates,
given in many texts on geophysical fluid dynamics (e.g. Gill,
1982
; Houghton,
2002
),
enables us to obtain all three components of the Coriolis force. In the ocean, vertical
velocities are usually much less than horizontal velocities (w
u, v) so we only need
to include the two horizontal components of the Coriolis force:
F
cx
¼
2
v sin f
L
¼
fv
;
F
cy
¼
2
u sin f
L
¼
fu
ð
3
:
7
Þ
is the angular velocity of the Earth's rotation
1
where
O
and f
L
is latitude. By
convention we define the Coriolis parameter as f
sinf
L
. The Coriolis parameter
determines the influence of latitude on the Coriolis force. The magnitude of f is
greatest at the poles. It is positive in the northern hemisphere, decreases to zero at the
equator, and is negative in the southern hemisphere. In the northern hemisphere,
the Coriolis force acts to the right of the direction of motion; in the southern
¼
2
O
1
2
T
E
with T
E
the rotational period of the Earth (24 hours), so
¼
2
24
3600
¼
7
:
27
10
5
s
1
¼
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