Geology Reference
In-Depth Information
Fig. 11.6
Alternative kinematic models of lithospheric
bending at a subduction zone. (
a
) Classic “ping-pong
ball” model (Frank
1968
). During the subduction the
trench curvature is progressively reduced and the trench
length increases. Isodepth lines of the Wadati-Benioff
zone are small circle arcs having a common pole, the slab
geometry is concavo-concave, and the Gaussian curvature
is conserved. (
b
) In the model proposed by Schettino and
Tassi (
2012
), the radius of curvature of the trench does not
change through time. In this instance, the slab geometry
is concavo-convex, thereby the Gaussian curvature is no
longer invariant. The subducted trench lines are isodepth
lines (
small circle arcs
) having constant curvature but
different poles. (
c
) Predicted relative velocity field in the
“ping-pong ball” model. (
d
) Typical velocity field along a
subduction zone.
Dashed lines
represent flow lines about
the Euler pole of relative motion (From Schettino and
Tassi
2012
)
aligned along a small circle arc and form an
isodepth line having depth
z
D
0 and length ` at
time
t
0
. In Frank's model, such a small circle arc
would sink vertically while conserving its length,
thereby no lateral deformation would occur
during the subduction in so far as the equilibrium
condition is satisfied. In the model of Schettino
and Tassi (
2012
), the points which entered the
trench at time
t
0
also form isodepth lines at any
time
t
>
t
0
, thereby the Wadati-Benioff zone
can be still represented by isodepth lines that
are small circle arcs. However, in this instance
a condition of lateral mechanical equilibrium
cannot exist at
any
time
t
, so that subduction
requires
some amount of lateral shortening
and extension during bending. Furthermore, in
conditions of lateral mechanical equilibrium,
each isodepth line conserves at some depth its
initial length ` and curvature, but differently
from Frank's model has a continuously changing
pole. Therefore, in the model of Schettino
and Tassi (
2012
) the trench geometry can be
conserved, at least in principle, for million years
and the relative velocity field, which is shown in
Fig.
11.6
d, is compatible with plate kinematics.
An interesting feature of the Wadati-Benioff zone
geometry predicted by Schettino and Tassi (
2012
)
is that the
radial
curvature of the unsubducted
lithosphere, ›
1
Š
1/
R
, can be potentially
conserved, when the subduction angle does
not change substantially, although the Gaussian
curvature of the subducting plate would be no
longer conserved, because it would assume a
negative value. Conversely, in Frank's model the
slab dip must decrease with depth and
K
is always
conserved. Therefore, in the model of Schettino
& Tassi a subducting plate cannot be considered
as a flexible-inextensible spherical cap.
To understand the theoretical grounds of this
model, let us start from the simple example of a
planar lamina that has been bent along a hinge
line coinciding with the
x
axis (Fig.
11.7
a) to
simulate the effect of the Earth's curvature. If we
bend the lamina again by rotating progressively
the margin about the hinge versors
n
0
and
m
0
(Fig.
11.7
b), it can be easily proved, for example
through a sheet of paper, that such a new bending
will cause lateral shortening, until the dip angle
