Geoscience Reference
In-Depth Information
Table 2.3
Notation used in addition of rotation vectors
Rotation vector
Magnitude
Latitude of pole
Longitude of pole
A
ω
A
λ
BA
φ
BA
B
B
B
ω
B
λ
CB
φ
CB
C
C
A
ω
A
λ
CA
φ
CA
C
C
Figure 2.12.
Relative-rotation vectors
B
A
and
C
A
for the
plates A, B and C. The dashed line is the great circle on
which the two poles lie. The resultant rotation vector is
C
A
(Eq. (2.19)). The resultant pole must also lie on the same
great circle because the resultant rotation vector has to lie in
the plane of the two original rotation vectors.
ω
C A
ω
C B
ω
B A
lie; the resultant pole must also lie on that same great circle (Fig. 2.12). Note that
this relationship (Eqs. (2.19) and (2.20)) should be used only for infinitesimal
movements or angular velocities, not for finite rotations. The theory of finite
rotations is complex. (For a treatment of the whole theory of instantaneous and
finite rotations, the reader is referred to Le Pichon
et al.
(1973).)
Let the three vectors
B
A
,
C
B
and
C
A
be written as shown in Table 2.3.Itis
simplest to use a rectangular coordinate system through the centre of the Earth,
with the
x-y
plane being equatorial, the
x
axis passing through the Greenwich
meridian and the
z
axis passing through the North Pole, as shown in Fig. 2.13.
The sign convention of Table 2.2 continues to apply. Then Eq. (2.19) can be
written
x
CA
=
x
CB
+
x
BA
(2.21)
y
CA
=
y
CB
+
y
BA
(2.22)
z
CA
=
z
CB
+
z
BA
(2.23)
where
x
BA
,
y
BA
and
z
BA
are the
x
,
y
and
z
coordinates of the vector
B
A
, and so
on. Equations (2.21)-(2.23) become
x
CA
=
C
ω
B
cos
λ
CB
cos
φ
CB
+
B
ω
A
cos
λ
BA
cos
φ
BA
(2.24)
y
CA
=
C
ω
B
cos
λ
CB
sin
φ
CB
+
B
ω
A
cos
λ
BA
sin
φ
BA
(2.25)
z
CA
=
C
ω
B
sin
λ
CB
+
B
ω
A
sin
λ
BA
(2.26)
