Geophys. J. Int. (1998) 132, 701–711
Flexure of subducted slabs
A. M. Marotta and F. Mongelli
Department of Geology and Geophysics, University of Bari, Bari, Italy
Accepted 1997 October 31. Received 1997 September 22; in original form 1996 June 20
SU MM A RY
The subducted lithosphere is regarded as a thin elastic plate that bends as a consequence
of slab pull, the pressure of the asthenospheric flow induced by the subduction motion
and the pressure exerted by the asthenospheric motion relative to the lithosphere. In
westward subductions the latter factor enhances the slab pull, but in eastward subductions it opposes it. As a result, the subduction angle changes continuously with depth,
following an elastic profile: it is smaller in eastward subductions and larger in those
having a westward direction. The application of the model to 13 subducted slabs shows
a good fit between the observed and the calculated shapes of the slabs.
Key words: slab flexure, subduction.
I NT R O DU C TI O N
In most subduction zones, the Benioff plane is well defined by
the distribution of earthquake hypocentres. Focal mechanisms
indicate that the hypocentres originate within the top 30 km
of the subducted slab rather than at its boundaries (Isacks &
Molnar 1971), whereas the deep earthquakes occur inside
the slab.
The subducted slab geometry can be defined essentially by
its length and dip. In an attempt to understand better the
mechanism of subduction, many authors have studied the
relationships between these and the other characteristic subduction parameters, such as slab convergence rate, subducted
slab age, absolute motion of slabs, etc.
The direct dependence of the slab length on the slab convergence rate and the subducted slab age is quite well verified;
conversely, the relationships between the slab dip and the other
parameters are not clear [Jarrard (1986) and references therein].
Slab dip differs from one region to another and along the
entire length of the subducted lithosphere (Fig. 1). A gradual
verticalization of the Benioff planes is observed in almost all
subduction regions (Fig. 1); the dip increases gradually from
the trench to a depth of about 80–130 km and stays almost
constant below this depth.
Jarrard (1986) considers three dip values: an average shallow
dip (DipS) from the trench to 60 km depth; an average
intermediate dip (DipI) from the trench to 100 km depth; and
an average, practically constant, deep dip (DipD) from 100 to
400 km depth. The greatest change in dip, at a depth of 60 km,
is ascribed to the fact that the basalt/eclogite transition occurs
at that level, resulting in increased density of the slab and thus
enhancing its gravitational instability.
Several authors have attempted to correlate these slab dips
and the other subduction parameters. Jarrard (1986) reviewed
the results and carried out new multiple correlations between
© 1998 RAS
the different parameters. Most of the correlations discussed
are not fully satisfactory: despite the fact that they are supported by correct physical principles, they are occasionally
contradictory and/or show low correlation coefficients.
It may well be that some factor or other has not been
considered and causes scatter of data. Special attention should
be given to (a) the possible correlation between slab length
and dip and ( b) an explanation of the major differences in dip
in the deeper portions of the slab. Fig. 1 clearly shows that the
Andean Benioff planes, characterized by a highly compressive
environment, exhibit the shallowest dips of all modern subduction zones. In particular, the slab flattens at about 100 km
depth along two segments of the Andean region (Peru and
central Chile) and moves on in a seemingly horizontal direction
along the lower surface of the South American lithosphere.
Conversely, subduction regions with back-arc spreading
(e.g. Tonga, Kermadec and Marianas) comprise some of the
steeper-dipping slabs. Moreover, a tentative separation between
subduction regions with comparatively low average deep dips
(about 30°) such as Chile, Peru and Macran and regions with
comparatively high average deep dips (more than 45°) (e.g.
Kermadec, Marianas and Kuriles) indicates that the direction
of subduction in the former trends predominantly east or
northeast and predominantly west in the latter, although there
are exceptions both in the first (e.g. Java and northern Chile)
and in the second group (e.g. northwest Japan). A seemingly
non-random relation can also be observed between slab dip
and length: in predominantly westward subductions, greater
lengths show correspondingly steeper average dips (e.g. the
dips in the Marianas are more ‘vertical’ than those in
Kermadec), while in eastward subductions greater lengths
show correspondingly lower average depths (central Chile
shallower than southern Chile).
The relationship between average deep dips and direction
of subduction could be explained by the global average
701
702
A. M. Marotta and F. Mongelli
Figure 1. Subduction profiles [according to Isacks & Barazangi (1977)]. NH=New Hebrides, CA=Central America, ALT=Aleutian,
ALK=Alaska, M=Marianas, IB=Izu-Bonin, KER=Kermadec, NZ=New Zealand, T=Tonga, KK=Kurile–Kamchatka, NC=North Chile,
P=Peru.
west-northwest-trending motion (a few centimetres per year)
of the lithosphere with respect to the underlying mantle
(Le Pichon 1968; Ricard, Doglioni & Sabadini 1991; Doglioni
1992, 1993). As a matter of fact, it would seem that in addition
to being subject to their own weight and to induced flow pressure,
subducted slabs also undergo an additional ‘horizontal’
thrust due to the relative motion of the mantle with respect
to the lithosphere; it seem that the thrust causes a further
verticalization of the slabs in westward subductions while
keeping the average dip quite low in eastward subductions.
Recently, Scholz & Campos (1995) stated that two forces
are important in this problem in order to explain the mechanics
of back-arc spreading and seismic decoupling: a small component of the slab pull force and ‘a sea anchor force exerted
on the slab that resists its lateral motion, assumed to occur at
the upper plate velocity’.
Fig. 2 (Doglioni 1993) shows the flow lines describing the
average trend of global motion versus subduction direction in
different regions. The deepest dips occur precisely in the
westward-subducting regions where the flow lines are normal
to the trench (e.g. Marianas, Kermadec and Kuriles), while the
lowest average deep dips are associated with the eastwardsubducting regions, again with flow lines running normal to
the trench.
Figure 2. Global flow pattern of plate motion [after Doglioni (1993), modified)].
© 1998 RAS, GJI 132, 701–711
Flexure of subducted slabs
In order to explain the above observations, a model based
upon the hypothesis that the slab retains an elastic behaviour
even at great depth is proposed. A possible behaviour of an
elastic nature at greater depth was already referred to by
Isacks & Molnar (1971), who spoke of ‘elastic return’ to
explain compression-type mechanisms of intermediate earthquakes. Recent studies on subduction slab rheology (Rubie
1984; Ji & Zhao 1994) confirm this hypothesis.
T HE O R Y
In order to study the shape of the deeper portion of a subducted
slab, this portion is assumed to be a thin elastic plate with a
uniform thickness along its entire length. This model has been
applied successfully in a large number of works on lithosphere
surface bending (e.g. Nunn & Aires 1988; Egan 1992; Royden
& Karner 1984; Royden 1988; Kruse & Royden 1994; Levitt
& Sandwell 1995).
The model we are now proposing assumes the deepest
part of the subducted lithosphere to be a thin plate of finite
length embedded, at its initial point of immersion, into the
asthenospheric mantle. The flexural equation is then
D
d4w
=q(x) ,
dx4
(1)
where D=Eh3/[12(1−s2)] is the flexural rigidity of the slab,
E=Young’s modulus, s=Poisson’s ratio, h=elastic thickness
of the slab, w=slab flexure calculated with respect to a
reference level and q(x) is the distribution of loads acting
vertically upon the slab.
The reference level for which flexure is to be determined is
established according to the direction defined by the lithosphere intermediate dip (DipI), namely the direction along
which the lithosphere begins to sink completely into the fluid
asthenospheric mantle.
The reference system selected to integrate eq. (1) is shown
in Fig. 3. The origin of the axes has been set at the base of the
overriding lithosphere, at the point of contact with the subducting lithosphere: the x-axis (i.e. the flexure reference level
mentioned above) is defined by the angle h . The slab would
a
remain in the same direction as the x-axis if all the forces
acting upon it were at equilibrium. The y-axis is perpendicular
to the x-axis and is directed inwards.
The distribution of loads q(x) can be resolved into two
terms, q (x) and q (x), where q (x) is the net balance between
g
f
g
the gravitational force and the hydrostatic pressure of the
fluid mantle; q (x) represents the induced flow pressure due to
f
the fact that the subducting slab induces an asthenospheric
drag flow.
Figure 3. Reference system for the flexure calculation.
© 1998 RAS, GJI 132, 701–711
703
Induced flow pressure in the asthenosphere in relative
motion (with respect to the lithosphere)
Mantle flow induced around itself by the subducting slab was
first investigated by McKenzie (1969), who found that the
resulting pressure lifts the slab, thus opposing the gravitational
force, and determines the value of DipD. The same problem
was further developed by Stevenson & Turner (1977), Tovish
et al. (1978), Molnar et al. (1979), Turcotte & Schubert (1982)
and Hsui & Tang (1988).
We now consider the case of an asthenosphere in relative
motion with respect to the lithosphere. Let us consider two
portions of lithosphere having different velocities relative to
the underlying mantle. Let V be the velocity of the non1
subducting, and V that of the subducting lithosphere. This is
2
equivalent (Fig. 4) to a condition where the velocity of the
non-subducting lithosphere is U=0, the subduction lithosphere velocity is U=U −U , and that of the asthenospheric
2
1
mantle is U =−V .
0
1
To determine the load component q(x) acting upon the
subducting lithosphere, consider the 2-D corner-flow model in
Fig. 4 (see Turcotte & Schubert 1982) for a highly viscous fluid
with constant viscosity. The origin O of the reference system
is set in correspondence with the trench, the x-axis is horizontal,
at the base of the non-subducting lithosphere, and the y-axis
is perpendicular to the x-axis, towards the interior of the Earth.
The subducting lithosphere is taken to be like a semi-infinite
line running downwards from the trench, according to the
direction h defined by the mean subduction angle, at a rate U.
This line divides the 2-D region investigated here into two
regions, called the ‘arc region’ and ‘oceanic corner region’
respectively. In order to determine the induced flow pressure,
the induced fluid flow is determined in both regions by the
motion of the semi-infinite line. The upper edges of the region
(x-axis) are subjected to the condition that, very far away from
Figure 4. Arc-corner and oceanic-corner regions: velocities referred
(a) to the mantle, ( b) to the non-subducting slab.
A. M. Marotta and F. Mongelli
704
the trench (x  ±2), the asthenospheric relative velocity
coincides with the relative velocity of the undisturbed
asthenosphere.
Assuming we have a Newtonian fluid that is both
incompressible and in steady-state conditions, the induced flow
pressure is determined by solving the Navier–Stokes equations
in the form
V4Y=0
(2)
with a wave function such that u, v=components of induced
flow velocity.
The solution of eq. (2), with the appropriate boundary
conditions, allows us to obtain the values of the velocity
components
∂Y
.
∂x
(3)
The pressure due to the asthenospheric flow induced by
subduction is obtained from the general expressions for the
balance of forces,
0=
A
B
∂2u ∂2u
∂P
+m
+
,
∂x
∂x2 ∂y2
0=−
Calculation of flexure
On the basis of the expressions obtained for the different load
components, eq. (1) is transformed into
D
1
d4w
=DrgS cos h +b .
a
dx4
x
(8)
If we consider the subducted slab to be a plate embedded
at its origin and whose length is L , then the boundary
conditions are as follows:
∂Y
,
u=−
∂y
v=
with increasing depth, in each subducted slab, though it does
change from one slab to another. We assume Dr(x) cos h(x)=
Dr cos h , where Dr is the mean value to be determined
a
(see below) and h is the angle at which the lithosphere enters
a
into the asthenosphere, which is almost equal to DipI.
A
B
∂2v ∂2v
∂P
+m
+
,
∂y
∂x2 ∂y2
(no flexure at the subduction hinge) ,
dw
(x=0)=0
dx
(plate embedded at the subduction hinge) ,
d2w
(x=L )=0 (external torque at the tip equal to zero) ,
dx2
d3w
(x=L )=0 (concentrated load at the tip equal to zero) .
dx3
(4)
where P is the induced flow pressure and m is the constant
viscosity of the asthenospheric mantle.
It is found that (Appendix A) the induced flow pressure in
the reference system described in Fig. 4 can be expressed by
1
P=b ,
x
w(x=0)=0
(9)
To integrate, let us divide both sides of eq. (8) by D:
1
d4w
=q+a ,
dx4
x
(10)
where
(5)
where
b=m(UDP +U DP ) .
(6)
1
0 0
DP and DP are two non-dimensional terms that depend
1
0
solely on h , the angle of entry into the asthenosphere. While
a
DP always produces an effect of lifting, DP produces an effect
1
0
of lifting or sinking depending on whether subduction is
favoured or is opposed by the mantle’s undisturbed relative
motion (eastward or westward subduction respectively).
q=
DrgS cos h
a,
D
a=
b
,
D
(11)
then, by integrating eq. (10) four times successively, we get
A
B
x4
x 11 x3
x2
+ A +a ln − a
+q ,
w(x)=A +A x+A
4
1
2
32
L
6
6
24
(12)
‘Relative’ weight
Concerning the term q (x), for each unit section of the subg
ducted lithosphere along the vertical direction y (Fig. 3), the
relative weight can be expressed as
q (x)=Dr(x)gS cos h(x) ,
(7)
g
where Dr(x)=difference in density between the lithosphere
and the mantle, g=gravitational acceleration, S=slab
thickness and h(x)=dip of the lithosphere.
Recent work on the thermal evolution of a subducting
lithosphere has demonstrated that, generally, the difference in
density is a function of subduction rate, subduction age and
subduction angle (e.g. Minear & Toksoz 1970; Hsui & Tang
1988), which have opposing effects. However, Minear & Toksoz
(1970) assume that the difference in density stays constant,
with A , A , A and A to be obtained through the boundary
1 2 3
4
conditions expressed by eqs (9).
From the first and second equations of eqs (9) we get
A =A =0; from the third and fourth equations it follows
1
2
that
A =−qL ,
4
L2
A =aL +q .
3
2
(13)
The solution is
A
B A
B
x4
x2
q
x 11 x3
+ −qL +a ln − a
+q .
w(x)= L a+ L 2
2
2
L
6
6
24
(14)
© 1998 RAS, GJI 132, 701–711
Flexure of subducted slabs
Fig. 5(a) shows the flexure curves for an ‘average’ subduction
condition. The solid line indicates the geometry obtained at
a zero relative mantle flow velocity, extrapolated up to the
surface by averaging the configurations of the Benioff planes
as they are represented by Isacks & Barazangi (1977).
The following values were adopted for the various parameters:
L =400 km; h=30 km; S=80 km; h =35°; Dr=30 kg m−3;
a
U=8 cm yr−1; m=1020 Pa s.
In order to test whether the distribution of loads might
justify the difference observed in the subduction angles, the
subduction rate U was fixed and U , the rate of the astheno0
spheric mantle undisturbed relative motion, was allowed to
range between 0 and 2 cm yr−1 under different subduction
conditions, either with or against the current.
Compared with the condition in which subduction occurs
in an asthenosphere at rest, a relative mantle motion favouring
subduction has a lifting effect whereas a relative mantle motion
opposing subduction causes the slab to sink further down.
Obviously, the stronger the contrast between subduction rates
U and U , the stronger the lifting effect will be.
0
Figs 5( b) and (c) show examples of flexure curves obtained
only by varying the length of the subducted slab. It can be
seen that the dip increases with length at any depth in the case
of a westward subduction. In the case of an eastward subduction, taking the other parameters constant, the dip decreases
until a certain value of the length and then increases for higher
values. This is due to the increased effect of gravity.
705
(a)
(b)
H O W T O AP P LY T HE M O D EL
For the 2-D model to be applied correctly, it is essential for
the subduction regions considered in the study to be large
enough and to present a gentle curvature of the trench, so that
the hypothesis that we are dealing with an infinite system in
the direction in which the trench extends can be taken as valid.
Complex regions either with several subductions developing
in different directions (e.g. Sangihe and Sulawesi) or with flow
lines cutting the trench crosswise (e.g. Cascades) are excluded:
estimating their thrust component by means of the proposed
2-D model would be not correct.
The model has been applied to 13 subduction zones, six
westward and seven eastward (Table 1). For each of them the
angle at which the lithosphere enters into the asthenosphere
h , the elastic thickness h and the total thickness S of the
a
subducted slab were obtained from the appropriate literature,
while the subduction rate and the local mantle velocity in the
central point of the associated arc were obtained from the
plate tectonic models Nuvel 1 and HS2-Nuvel 1 (DeMets et al.
1990; Gordon 1995). Since the difference of density increases
with the rate of subduction, we have simply assumed, after
many attempts, that Dr=30 kg m−3 for rates less than
6–7 cm yr−1 and Dr=40 kg m−3 for greater rates.
Fig. 6 shows the flexure curves and the subduction profiles;
it can be seen that the fit is rather satisfactory. Further
improvements can be obtained by varying Dr by few kg m−3.
Fig. 7 shows the relationship between the calculated pressure
exerted by the mantle horizontal motion only and the observed
difference DipD−h , for eastward and westward subductions.
a
This generalizes the finding of Scholz & Campos (1995) that
the slab dip varies linearly with horizontal force.
© 1998 RAS, GJI 132, 701–711
(c)
Figure 5. (a) Variation with depth of the subduction angle for typical
eastern- and western-dipping subduction slabs. See text for the
parameters used. (b) and (c) Variation with depth of dip for different
slab lengths in westward ( b) and eastward (c) subductions.
706
A. M. Marotta and F. Mongelli
Figure 6. Calculated flexure curves (****) and observed subduction profiles (----) of slabs listed in Table 1. The difference in dip between North
Chile and Central Chile (a) is associated with the different values of DipD−ha (30° and 25° respectively).
Flexure of subducted slabs
Figure 6. (Continued.)
© 1998 RAS, GJI 132, 701–711
707
708
A. M. Marotta and F. Mongelli
Table 1. Parameters of subducted slabs.
Name
Direction of
subduction
Length of
subducted slab
(km)
Total
thickness
(km)
Elastic
thickness
(km)
Intermediate
dip
(°)
CChile
NChile
New Zealand
SE Mexico
Nicaragua
Marianas
Kermadec
Aleutian
New Hebrides
Kurile
Colombia
Sumatra
Alaska
E
E
N-W
N-E
N-E
W
W
N-W
N-E
N-W
E
N-E
N-W
500
460
310
150
120
420
330
200
200
400
200
180
100
80
80
70
50–80
80
70
70
70
80
70
80
80
70
30
30
30
30
30
30
30
30
30
30
30
30
30
25
30
40
55
50
35
50
53
70
33
40
35
50
only a little greater than DipI. Conversely, in westward subductions, the asthenospheric thrust combined with the slab
pull prevails over the induced flow pressure and the angle
DipD is much greater than DipI.
All the above results are within the range of approximations
assumed in the model. The results could be improved further
by considering the possibility that the density anomaly might
not be constant with increasing depth.
In addition to explaining the values of the observed angles,
the model proposed here gives further confirmation of the
lithospheric flow relative to the asthenosphere. Lastly, the
model suggests that orientation could be regarded as a further
parameter correlating the various characteristic subduction
factors.
A CKN O W LE DG M ENT S
We thank S. King and an anonymous reviewer for helpful
comments on and suggestions about the manuscript.
Figure 7. Relationship between calculated pressure exerted by the
mantle horizontal motion only and the observed difference DipD−ha
(see text); solid circles=westward subduction, open circles=eastward
subduction.
CON CLU SION S
The distinction between eastward and westward subductions is marked by low eastward and high westward angles,
particularly in the deepest portion, below 100 km.
As a consequence of the relative motion between the lithosphere (westward) and the asthenosphere, the westward-sinking
slab is subject to a horizontal thrust acting from a depth of
80–100 km downwards. The slab reacts elastically, by continuous bending, to the action of its own weight, the induced
flow pressure—which depends on the subduction rate—and
the thrust induced by the mantle’s relative motion. The thrust
due to the eastward relative motion either favours or opposes
the sinking of the slab, according to whether subduction is
directed eastwards or westwards.
The three pressures are comparable with one another; no
one of the three can prevail over the other two. In eastward
subductions, the asthenospheric thrust plus the induced flow
pressure opposes most of the slab pull and the angle DipD is
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A P P EN DI X A: D ET ER M IN AT IO N OF
I ND U CED F LO W PR E S SU R E
(A1)
is of the type
v=U .
0
(2) The induced flow velocity at the surface of the subducted
slab coincides with the subduction rate:
v=U .
Arc-corner region
(1) At h=0°, the flow velocity is equal to the velocity of
the undisturbed relative asthenospheric flow:
u=U ,
0
v=0 .
(A4)
(2) At h=h , the flow velocity is equal to the subduction
a
rate:
u=U cos h ,
a
v=U sin h .
a
(A5)
Oceanic-corner region
(1) At h=h , the induced flow velocity is equal to the
a
subduction rate; therefore
u=U cos h ,
a
v=U sin h .
(A6)
a
(2) At h=p, the flow velocity is equal to the undisturbed
asthenospheric flow velocity:
u=U ,
0
v=0 .
(A7)
From eqs (A3) we obtain the following.
For h=0,
−B+(Cx)
A B
−x
=U ,
0
x2
A=0 .
Y=Ax+By+(Cx+Dy) arctan
x
.
y
(A2)
For h=h ,
a
A
A
B
−x
x
,
u=−B−D arctan +(Cx+Dy)
y
x2+y2
A
(A8)
−x
−B−Dh +(Cx+Dx tanh )
a
a x2+x2 tan2 h
Following eq. (3), we get
B
−y
x
.
v=A+C arctan +(Cx+Dy)
y
x2+y2
A
−x tan h
a
A+Ch +(Cx+Dx tan h )
a
a x2+x2 tan2 h
(A3)
Two distinct regions are recognized in the problem: the
left-hand one (arc-corner region) and the right-hand one
(oceanic-corner region).
The constants A, B, C, D are determined by imposing the
following boundary conditions in the different regions.
(1) Far away from the subduction region, the flow maintains the same velocity as the asthenosphere undisturbed
© 1998 RAS, GJI 132, 701–711
motion:
Arc-corner region
The general solution to this problem of the Navier–Stokes
equations
V4Y=0
709
a
B
a
B
=U cos h ,
a
=U sin h .
a
(A9)
Hence
B−C=U ,
0
A=0 ,
−B−C cos h −D(h +sin h cos h )=U cos h ,
a
a
a
a
a
C(h −sin h cos h )−D sin2 h =U sin h .
(A10)
a
a
a
a
a
The pressure due to the asthenospheric flow induced by
subduction is obtained from the general equations for the
A. M. Marotta and F. Mongelli
710
balance of forces:
0=
A
components. Let
B
∂2u ∂2u
∂P
+m
+
,
∂x
∂x2 ∂y2
0=−
A
B
∂2v ∂2v
∂P
+m
+
,
∂y
∂x2 ∂y2
(A11)
where P is the induced flow pressure and m is the constant
viscosity of the asthenospheric mantle.
By substituting the first of eqs (A3) into the first of eqs (A11)
and integrating with respect to x, we get
m
P =−2 (C cos h +D sin h ) ,
a
a
a
a
r a
m
P =−2 (C cos h +D sin h ) ,
a
b
a
b
r b
(A12)
with r being the distance measured along the direction of
subduction.
The same result is obtained by substituting the second of
eqs (A3) into the second of eqs (A12) and integrating with
respect to y.
Since, for the purpose of calculating the flow pressure, we
only want to know coefficients C and D, only these coefficients
will be determined below. After a few steps we get, from the
system (A10),
Uh sin h
U sin2 h
a
a +
0
a ,
C =
a (h2 −sin2 h ) (h2 −sin2 h )
a
a
a
a
U(sin h −h cos h ) U (h −sin h cos h )
a
a
a + 0 a
a
a .
D =
a
(h2 −sin2 h )
(h2 −sin2 h )
a
a
a
a
A
A
a
B
(A16)
(A17)
m
P =−2 (C cos h +D sin h )
a
ai
a
a
r ai
m
−2 (C cos h )+D sin h ) ,
a
a0
a
r a0
(A13)
m
P =−2 (C cos h +D sin h )
a
bi
a
b
r bi
m
−2 (C cos h )+D sin h ) .
a
b0
a
r b0
Oceanic-corner region
−x
−B−Dh +(Cx+Dx tan h )
a
a x2+x2 tan2 h
Uh sin h
a
a ,
C =
ai h2 −sin2 h
a
a
U sin2 h
a ,
C = 0
a0 h2 −sin2 h
a
a
U sin h (h −p)
a a
,
C =
bi (p2+h2 −2ph −sin2 h )
a
a
a
U sin2 h
0
a
,
C =
b0 (p2+h2 −2ph −sin2 h )
a
a
a
U(sin h −h cos h )
a
a
a ,
D =
ai
h2 −sin2 h
a
a
U (h −sin h cos h )
a
a ,
D = 0 a
a0
h2 −sin h
a
a
U(sin h −h cos h +p cos h )
a
a
a
a ,
D =
bi
p2+h2 −2ph −sin2 h
a
a
a
U (h −p−sin h cos h )
a
a .
D = 0 a
b0
p2+h2 −2ph −sin h
a
a
a
We obtain
=U cos h ,
a
B
−x tan h
a
=U sin h .
A+Ch +(Cx+Dx tan h )
a
a
a x2+x2 tan2 h
a
(A14)
As for the arc-corner region, we now get
U sin2 h
U sin h (h −p)
0
a
a a
+
,
C =
b (p2+h2 −2ph −sin2 h ) (p2+h2 −2ph −sin2 h )
a
a
a
a
a
a
U(sin h −h cos h +p cos h )
a
a
a
a
D =
b
(p2+h2 −2ph −sin2 h )
a
a
a
U (h −p−sin h cos h )
a
a .
+ 0 a
(A15)
(p2+h2 −2ph −sin2 h )
a
a
a
In the expressions for C , C , D and D we can clearly see
a b a
b
the separation between a first term that depends only upon
the subduction rate U and a second term that depends only
upon the velocity U of the ‘undisturbed’ asthenospheric
0
motion.
The expressions thus obtained for the constants apply both
to westward and eastward subductions, provided that the
angles considered are higher or lower than 90° respectively.
On the basis of the separation made in the expressions for
the constants C , C , D and D , a similar separation can also
a b a
b
be made in the equations for the induced flow pressure
(A18)
By substituting the expressions for the coefficients C and D
given by eqs (A16) and (A17) into eq. (A18), we obtain, after
a few simple steps,
A
A
B
B
U
U
DP + 0 DP
,
P =m
ai U
a0
a
r
U
U
P =m
DP + 0 DP
,
b
b0
bi U
r
(A19)
where
CA
CA
CA
A
CA
A
DP =−2
ai
DP =−2
a0
DP =−2
bi
+
DP =−2
b0
+
B A
B A
BD
BD
h sin h cos h
sin2 h −h sin h cos h
a
a
a +
a
a
a
a
,
h2 −sin2 h
h2 −sin2 h
a
a
a
a
sin2 h cos h
h sin h −sin2 h cos h
a
a + a
a
a
a
,
h2 −sin2 h
h2 −sin2 h
a
a
a
a
sin h cos h (h −m)
a
a a
h2 +p2−2ph −sin2 h
a
a
a
sin h (sin h −h cos h +p cos h )
a
a
a
a
a
,
h2 +p2−2ph −sin2 h
a
a
a
sin2 h cos h
a
a
h2 +p2−2ph −sin2 h
a
a
a
sin h (h −p−sin h cos h )
a a
a
a
.
(A20)
h2 +p2−2ph −sin2 h
a
a
a
B
BD
B
BD
© 1998 RAS, GJI 132, 701–711
Flexure of subducted slabs
Hence, the net flow pressure due to induced flow and to the
velocity of asthenospheric relative motion U , acting at each
0
point on the subducted slab, can be expressed as
m
P= (UDP +U DP ) ,
I
0 0
r
(A21)
where
DP =DP +DP ,
I
ai
bi
DP =DP +DP .
0
a0
b0
© 1998 RAS, GJI 132, 701–711
(A22)
711
Given that the quantities m, U, U and r are always positive
0
(owing to the way in which the reference system was selected),
the ‘sign’ of the pressures P , P , P and P will be
bi
b0
ai
a0
determined by the sign of the non-dimensional quantities
DP , DP , DP and DP . In particular, a positive sign for
ai
a0
bi
b0
DP indicates a pressure exerted towards the surface upon
jk
which it is acting, and hence a lifting or plunging effect
depending on whether we are considering an arc-corner or an
oceanic-corner region.