Geography Reference
In-Depth Information
(z) at height z is just the work required to raise a unit mass to height z from
mean sea level:
z
=
gdz
(1.8)
0
Despite the fact that the surface of the earth bulges toward the equator, an object
at rest on the surface of the rotating earth does not slide “downhill” toward the pole
because, as indicated above, the poleward component of gravitation is balanced
by the equatorward component of the centrifugal force. However, if the object
is put into motion relative to the earth, this balance will be disrupted. Consider
a frictionless object located initially at the North pole. Such an object has zero
angular momentum about the axis of the earth. If it is displaced away from the
pole in the absence of a zonal torque, it will not acquire rotation and hence will
feel a restoring force due to the horizontal component of true gravity, which, as
indicated above is equal and opposite to the horizontal component of the centrifugal
force for an object at rest on the surface of the earth. Letting R be the distance from
the pole, the horizontal restoring force for a small displacement is thus
2
R,
and the object's acceleration viewed in the inertial coordinate system satisfies the
equation for a simple harmonic oscillator:
−
d
2
R
dt
2
2
R
+
=
0
(1.9)
The object will undergo an oscillation of period 2π/ along a path that will
appear as a straight line passing through the pole to an observer in a fixed coordinate
system, but will appear as a closed circle traversed in 1/2 day to an observer rotating
with the earth (Fig. 1.7). From the point of view of an earthbound observer, there
is an apparent deflection force that causes the object to deviate to the right of its
direction of motion at a fixed rate.
1.5.3
The Coriolis Force and the Curvature Effect
Newton's second law of motion expressed in coordinates rotating with the earth
can be used to describe the force balance for an object at rest on the surface of the
earth, provided that an apparent force, the centrifugal force, is included among the
forces acting on the object. If, however, the object is in motion along the surface
of the earth, additional apparent forces are required in the statement of Newton's
second law.
Suppose that an object of unit mass, initially at latitude φ moving zonally at
speed u, relative to the surface of the earth, is displaced in latitude or in altitude by
an impulsive force. As the object is displaced it will conserve its angular momentum
in the absence of a torque in the east-west direction. Because the distance
R
to
the axis of rotation changes for a displacement in latitude or altitude, the absolute
angular velocity,
+
u/R, must change if the object is to conserve its absolute
