Plate Motions - Quantitative Plate Tectonics - page 33

This quantity is a measure of the inertial re-
sistance that a rigid plate opposes to variations of
its angular velocity about a rotation axis. Another
quantity that can be expressed in terms of Euler
vectors is the angular momentum, L . Starting
from expression ( 2.5 ), we can write:

where e i ( i D 1,2,3) are the base versors of the
coordinate system. In general, it is possible to
show that the momentum of inertia of a tec-
tonic plate about an arbitrary rotation axis n
can be expressed as a linear combination of the
components of the inertial tensor, thereby this
tensor contains all the relevant information for
the determination of the moment of inertia about
any rotation axis. In fact, using Eqs. ( 2.20 )and
( 2.21 ) we see that the component of the angular
momentum in the direction of n is given by:

Z

L D ¡.r/ r v .r/ dV

R

Z

D ¡.r/ r .¨ r/ dV

Z

¡.r/ h r 2 ¨ .n r/ 2 ¨ i dV D ¨I .n/

R

Z

L n D

¡.r/ r 2 ¨ .¨ r/r dV

D (2.21)

R

(2.25) R

A more compact expression can be determined
introducing the index notation (see Appendix 1 ) ,
x 1 x , x 2 y , x 3 z , and Einstein's summation
convention. With this notation, it is easy to prove
that the angular momentum has the following
simple expression in terms of Euler vectors:

Using Eq. ( 2.22 ), and taking into account that
¨ j D ¨ n j , we can also write:

L n D L i n i D n i I ij ¨ j D ¨n i I ij n j

Therefore, a comparison with Eq. ( 2.25 )fur-
nishes:

L i D I ij ¨ j (2.22)

I.n/ D n i I ij n j (2.26)

where the quantities I ij form a rank 2 symmetric
tensor, which is known as inertial tensor :

This expression proves our statement. The pre-
vious equations represent the basic framework for
the description of the instantaneous kinematics of
any rotating rigid plate, independently from the
choice of a reference frame. In the next section,
we shall consider the specific frames of reference
used in plate tectonics.

Z

¡.r/ r 2 • ij x i x j dV I i;j D 1;2;3

I ij D

R

(2.23)

In this expression, the quantity • ij represents
the Kronecker delta (• ij D 1 f i D j , zero
otherwise). The components of the inertial
tensor depend from the mass distribution and the
plate geometry, just like the moments of inertia
(Eq. 2.20 ). Therefore, we expect that a relation
exists between these quantities. It is quite evident
from ( 2.23 ) that the diagonal components of I
coincide with the moments of inertia about the
three coordinate axes:

2.3 Reference Frames

Two broad classes of reference frames are used
in plate tectonics. Geocentric reference frames
are global frames that are built assuming that
the Earth's centre of mass, R , coincides with the
origin of a Cartesian system of coordinates, so
that R D 0 . The best known of these reference
frames is the usual geographic coordinate system ,
in which the z axis coincides with the Earth's spin
axis, and the x and y axes are in the Equatorial
plane and point, respectively, to the Greenwich

Z

¡.r/ r 2 x i dV D I.e i / I

I ii I i D

R

i D 1;2;3 (2.24)