Geology Reference
In-Depth Information
In the general case of two plates whose relative
motion occurs about a continuously changing
Euler pole, it is not possible to determine the
instantaneous rotation axis and angular velocity
starting from a stage rotation. Therefore, a more
general, although complicate, approach to the
problem becomes necessary. We know that all
the plates are represented in a plate circuit at
time
T
. For each node
i
, a finite reconstruction
matrix
R
ij
(
T
) exists, which allows to calculate the
position of the corresponding plate with respect
to the parent plate
j
. We also know that the
edges of a plate circuit attest the existence of
specific kinematic relations between plate pairs,
consisting in the fact that the relative motions
are stage rotations, thereby they link pairs of
conjugate plates
. Therefore, for
any
plate
i
not
coinciding with the root of the tree, it is possible
to determine a stage rotation matrix
S
ij
and the
relative velocity field
v
ij
with respect to the parent
plate
j
in the hierarchical representation using the
following simple expression, which combines Eq.
(
2.17
) with Eqs. (
2.48
)and(
2.49
):
two plates, it is always possible to calculate their
relative velocity by subtracting their absolute
velocities:
v
ij
.r;T/
D
v
i
.r;T/
v
j
.r;T/
(2.52)
Another important kinematic variable is the
relative or absolute acceleration of a tectonic
plate. Curiously, this kind of vector fields have
been seldom considered in plate kinematics stud-
ies, in spite of their importance for the geody-
namic assessment of the models. Probably this is
a consequence of the fact that accelerations have
been traditionally considered as point events that
only occur at stage boundaries, consistently with
the current description of the driving mechanism
of plate tectonics. However, recent research has
shown that phases of accelerated motion have
existed in the geologic past, possibly associated
with the action of mantle plumes (Cande and
Stegman
2011
). We shall consider in detail the
In order to obtain an acceleration field, it is
necessary to consider two successive times,
T
and
T
0
, close enough, and calculate a velocity field for
each of them. For example, Schettino and Scotese
(
2002
) used a time interval of 1 Myr for determin-
ing the acceleration across stage boundaries in the
Mediterranean region during the Mesozoic. The
acceleration is calculated simply by dividing the
velocity variation by the size of the time interval:
v
ij
.r;T/
D
R
j
.T/
n
ij
.0/
k
T
k
T
k1
(2.50)
for any
T
k
1
T
T
k
. In the case of the root
continent,
r
, the corresponding finite reconstruc-
tion matrix coincides with a total reconstruction
matrix,
R
r
(
T
), that represents the transformation
of
r
with respect to the paleotectonic reference
frame. Therefore, if (
i
,
j
,
k
, :::,
r
) is a path in the
tree structure from node
i
to the root, then we
can always determine the
absolute velocity field
of a plate in the selected reference frame by
composition of velocity vectors:
v
ij
.
r
;T
0
/
v
ij
.
r
;T/
T
0
T
a
ij
.r;T/
D
(2.53)
It is possible that in the previous discussion,
about the velocity field between non-conjugate
plate pairs, some readers wondered about the
effective number of situations characterized by
this kind of relative motion. We shall satisfy
the curiosity of these readers by proving an in-
teresting topological theorem, which will help
to clarify some key features of plate tectonic
configurations. We say that a plate boundary is
a
conjugate boundary
if it separates a pair of
conjugate plates. In this instance, any geological
structure associated with strike-slip motion, for
example a transform fault, will be aligned with
v
i
.r;T/
D
v
ij
.r;T/
C
v
jk
.r;T/
CC
v
r
.r;T/
(2.51)
where the absolute velocity field of the root node,
v
r
, is determined calculating stage rotations with
respect to the paleotectonic reference frame.
Now we are ready to solve the problem of
determining relative velocity fields between non-
conjugate plate pairs. In fact, if
i
and
j
are
any
