Geology Reference
In-Depth Information
the vector triple product in (
2.60
), this expression
can be rewritten as follows:
N
D
X
i
plate, ¨
ir
, it is possible to determine its absolute
Euler vector, ¨
i
, by adding the absolute angular
velocity of the reference plate: ¨
i
D
¨
ir
C
¨
r
.
Therefore, Eq. (
2.65
) can be viewed as a linear
system of three equations with respect to the three
unknown components of ¨
r
:
X
D
i
Z
S
i
.r
r/ ¨
i
dS
D
i
Z
S
i
C
X
i
.r
¨
i
/rdS
i
(2.61)
i
!
¨
r
D
X
i
i
D
i
Q
D
i
Q
¨
ir
(2.66)
i
If we assume that the Earth's radius is normal-
ized to unity, then
r
r
D
1, so that:
The total
Q
tensor for the whole lithosphere
can be obtained simply by summation of the
tensors
Q
i
D
i
Z
S
i
N
D
X
i
D
i
S
¨
i
C
X
i
associated with each plate. It results:
.r
¨
i
/rdS
i
(2.62)
Q
D
X
i
8
3
I
Q
i
D
(2.67)
Expression (
2.62
) can be further simplified
introducing a new tensor quantity, which is
strictly related to the inertial tensor of a tectonic
plate (Eq.
2.23
). Using the index notation (see
given by:
where
I
is the identity matrix. Further simplifi-
cation of Eq. (
2.66
) follows if we assume that
the drag coefficients
D
i
coincide for all plates:
D
i
D
D
. In this instance, using (
2.67
) we obtain an
immediate solution for ¨
r
in terms of the relative
Euler vectors of a velocity model:
Z
•
jk
x
j
x
k
dS
D
A
i
•
jk
Q
i
jk
8
X
i
3
S
i
i
¨
r
D
Q
¨
ir
(2.68)
Z
x
j
x
k
dS
I
j;k
D
1;2;3 (2.63)
This solution corresponds to a condition of
no-net-rotation
(NRR) for the whole lithosphere
(
L
D
0
). In fact, for
D
i
D
D
Eq. (
2.65
) can be
rewritten as follows:
S
i
where
A
i
is the area of the
i
th plate. Using
this new tensor quantity, which depends only
from the plate geometry, Expression (
2.62
) can
be rewritten as follows:
N
D
X
i
X
8
3
D
0
)
D
0
(2.69)
i
¨
i
Q
D
Q
i
i
D
i
Q
¨
i
(2.64)
where can be considered as the net rotation of
the whole lithosphere. It should be noted that the
solution (
2.68
) only holds in the unlikely event
that the unique torques exerted on the lithosphere
come from asthenospheric drag,
and
that the drag
coefficient
D
can be considered constant over
the entire lithosphere. Of course, none of these
two strong conditions is likely to be verified.
Slab pull forces are essential components of the
global torque balance, and the drag coefficient
along the irregular continental LAB cannot be
If this were the only torque exerted on the
lithosphere, the torque balance equation would be
written:
N
D
0
,thatis:
X
i
D
i
Q
¨
i
D
0
(2.65)
i
Let ¨
r
be the Euler vector of a reference plate,
for example the Pacific plate, with respect to the
top transition zone. Knowing the Euler vector
of any other plate with respect to the reference
