Of the above special cases, IV most nearly corresponds to the
situation of relative motion on the earth referred to a coordinate
system fixed relative to the earth. In this case
is the angular velocity of the earth about its
own axis. Other motions of the earth through space are neglected
for the time being. The relative coordinate system and the
vectors 
and 
are shown in
Fig. 2.13-1. The origin of the relative coordinate system
O' is located on the sea level geoid, the z' axis is
opposite to 
, and the x' and y' axes form a
horizontal plane tangent to the sea level geoid at O', with
the x axis being taken as towards the east and the y
axis towards the north. The point O' is at a distance
from the earth's center and has a fixed latitude
and longitude relative to the earth. Consequently,
the relative coordinate system has the same angular velocity
as that of the earth.
The quantity 
represents the peripheral
velocity of the point O' and is directed towards the east. It can be
readily
verified that is identical to
. The magnitude of
is 
or simply 
, but 
is equal to R.
Thus, the magnitude of is
equal to that of and the direction
is the same. Therefore, the expression
appearing in
Eq. (112-iv) is the same
as 
or 
.
However, 
is simply the absolute
radius vector 
from the center of the earth to the
point P which is at an arbitrary position in the
x', y', z' reference frame.
The absolute acceleration of a particle at point P can therefore be expressed as
neglecting the motion of the earth's center for the time being.
The vector 
has a
direction opposite that of the vector and has
the magnitude 
. Thus the vector is
simply the centripetal acceleration of the point P due to the angular
velocity of the earth.
The vector 
is the
geostrophic acceleration associated with the
relative motion 
of a particle with respect to
the rotating earth. This term is also referred to as the
Coriolis acceleration in honor of the French physicist
who established the mathematical transformation on a rigorous
basis. The term geostrophic acceleration (meaning related to
the earth's turning) is perhaps a more descriptive name.
The vector 
is actually the vorticity of the
earth. Its x', y' and z' components in the relative
reference frame are, repectively,
as can be seen from Fig. 2.13-2.
If the relative rectilinear coordinate system is taken such
that the y' axis is at an azimuth 
clockwise from the
north but x and y are still in the horizontal plane then
the x', y', z' components of the vorticity are
These expressions can be verified by reference to Figure 2.13-3.
The expressions 
and 
occur
frequently in dealing with relative motions on the earth and it
is therefore convenient to abbreviate the expressions by the
symbols 
and 
, i.e.,
The term is often referred to as the Coriolis parameter.
However, it is more appropriate to regard in terms of
its physical interpretation which is as the local
vertical component of the earth's vorticity.
In terms of the above definitions the geostrophic acceleration can be expressed by the determinant
where 
are the unit vectors in the
relative coordinate system which has the azimuth
previously defined. The components of relative velocity
u', v', w' and the x', y', z' components, respectively,
in this same system.
Expansion of the determinant of Eq. (117) gives the following components of the geostrophic acceleration
When 
(i.e., y' axis pointing to the north) then
the x', y', z' components of the geostrophic acceleration
reduce to
and for horizontal motions on the earth they reduce to
The above expressions for the components of geostrophic
acceleration apply only to the particular right-handed
coordinate system used here.