Geology Reference
In-Depth Information
Therefore, if we take the limit as œ
!
0and
¨
!
0 in the expression (
11.23
)for
b
, we obtain
an expression for the
lateral strain
©
D
db
/
d
` at
downdip distance — from the hinge line:
©
D
—
"
.1
cos'/
1
approximation used to obtain (
11.26
)and(
11.27
)
is acceptable.
More problematic is the unrealistic hypothesis
that the rate of lithospheric bending along the
trench zone is compensated at shallow depths by
an equal rate of lithospheric unbending, so that
the subduction angle remains approximately con-
stant along the Wadati-Benioff zone. Real sub-
duction zones exhibit a more complex behaviour,
characterized by a general increase of the dip an-
gle ' and eventually by zones of upward bending,
especially when an old slab reaches the 670 km
discontinuity (e.g., Goes et al.
2008
). Considering
a vertical cross-section, we can ideally reproduce
the real slab geometry through a sequence of
small rotations about horizontal hinge lines. The
method is illustrated in Fig.
11.8
.Foraslab
having total downdip length —
max
, we start with
a small rotation of a straight segment having
length —
max
about the surface hinge line by an
angle '
0
. At the next step, we rotate a straight
segment having length —
max
—
0
about a hinge
line at downdip distance —
0
by an angle '
1
.
Then, the procedure is repeated iteratively. If the
slab geometry has been correctly reproduced as
far as downdip distance —, then the next step
is to rotate by a small angle '(—)asegment
having length —
max
— about a hinge line at
downdip distance —. At each step, the incremental
strain can be calculated by differentiating (
11.26
).
Therefore, we can generalize (
11.26
)and(
11.27
)
to an arbitrary downdip geometry of the Wadati-
Benioff zone. Le
x
be the horizontal offset from
the surface hinge line and '
D
'(
x
) a dip func-
tion. Then, generalized formulae for the finite
lateral strain and the lateral strain rate at offset
x
are:
R
sin '
#
(11.26)
1=2
1
R
2
1
¡
2
For ¡
D
R
sin'/2 (Frank's formula), the lateral
strain vanishes. Therefore, although the model
of Frank (
1968
) does not correctly describe the
subduction process, the corresponding relation
between subduction angle and trench curvature
holds as a condition of lateral mechanical equi-
librium at any depth. Expression (
11.26
) holds
when the lithosphere bends along a single hinge
line and differs substantially from the formula
proposed by Strobach (
1973
). It shows that the
lateral strain state increases linearly with the
downdip distance —.From(
11.26
) we easily ob-
tain the strain rate,
P
©, which is independent from
the downdip distance.
If
v
is the subduction velocity, then:
P
©
D
v
"
.1
cos '/
1
R
sin '
#
(11.27)
1=2
1
R
2
1
¡
2
The previous formulation started from the
simple bending of a lamina whose initial
geometry simulates the effect of the Earth's
curvature along one direction only, because
the radial curvature is assumed to be zero in
the starting state (the lamina is flat in the
x
direction). In order to estimate the effect of
the radial curvature ›
e
D
1/
R
of a tectonic
plate, we note that this is equivalent to assume
an initial radial bending of the lamina by a
small angle •', which is not accompanied by
lateral strain. However, for a small patch of
lithosphere having width —
D
1 km we would
have: •'
D
180/(
R
)
D
0.008993
ı
,wh ch s
effectively a negligible quantity. Therefore, the
Z
x
—.x/
—
x
0
©.x/
D
0
"
1
¡
2
R
cos '
x
0
#
d'
x
0
(11.28)
R
2
1=2
sin '
x
0
1
1
