Download Numerical models of subduction and forearc deformation

Geophys. J. R . astr. Soc. (1985) 8 0 , 4 1 9 - 4 3 7
Numerical models of subduction and
forearc deformation
Thomas M.Tharp Department of Geosciences, Purdue university,
West Lafayette, Indiana 47907, USA
Accepted 1984 August 15. Received 1984 August 15; in original form 1983 September 9
Summary. Finite element models for shallow subduction produce realistic
behaviour for a wide variety of mechanical strength and density distributions.
Characteristic displacements are found t o occur even without a discrete lowstrength megathrust if there is a high-density subducted plate t o localize
lithospheric compression. A high-density plate is itself unnecessary in the
presence o f a low-strength megathrust and regional compression.
Successful finite element models produce an outer arc at the top of the
trench slope, and forearc basin with geometry characteristic of natural
analogues. These structural features occur by upward inelastic bending o f
the lithospheric wedge overlying the megathrust. This mechanically unstable
behaviour may dissipate significant energy and cause the megathrust to
migrate continuously by accretion, tectonic erosion, or abandonment and
reinitiation farther offshore. Upward bending in the overriding plate is
promoted b y low megathrust dip, low megathrust shear strength, and high
horizontal compression in the overriding plate.
Introduction
The zone between the trench axis and magmatic arc is the locus of accumulation and
deformation of large masses of rock. It is also the ‘bearing’ which allows thrust of one plate
beneath another. In collisions with buoyant masses like continents or island arcs this zone is
telescoped and severely deformed (Nur & Ben-Avraham 1982). When only oceanic lithosphere is consumed, subduction gives the perhaps deceptive appearance of being stable and
mechanically efficient. It is tempting t o view the process through the attendant seismicity
(Chen, Frohlich & Latham 1982). This suggests a thin zone of intense shearing along the
megathrust, with little deformation in the competent part of the overriding plate. In this
model deformation in the overriding plate is restricted to the ‘incompetent’ accretionary
wedge. While useful, this picture may oversimplify the mechanical behaviour of the forearc
region,
420
T. M. Tharp
Structure of the arc-trench gap
The accretionary wedge is a common feature of the zone overlying the megathrust
(Dickinson & Seely 1979). The inner wall of the trench exhibits tightly folded sediments
separated by landward-dipping thrust faults (Chase & Bunce 1969; Karig & Sharman 1975;
Moore, Curray & Moore 1978) and locally seaward-dipping thrust faults (Moore & Allwardt
1980; Seely 1977). As material is added to the base of the accretionary wedge, upward and
landward rotation (Karig & Sharman 1975) is affected by displacement on imbricate thrust
faults (Moore & Karig 1976). Closely spaced faults are generally considered the dominant
agency o f penetrative deformation in rocks of low metamorphic grade, while folding and
continuum deformation may be more important at higher temperatures (Toriumi 1982).
Forearc basins are common at the top of trench slopes. They are structural as well as
bathymetric basins resulting from backward tilting in the overriding plate (Seely 1979).
Forearc basins are characterized by a landward limb dipping gently toward the trench, and
a n outer limb starting at the top of the accretionary wedge and dipping more steeply in a
landward direction. Deformation contemporaneous with deposition is indicated by synclinal
deformation of the basin sediments (Katili 1973), and by a progressive landward shift in
depocentre (Coulbourn & Moberly 1977). Forearc basins may be inconspicuous or absent
(Dickinson & Seely 1979), but some ancient examples are significant structure features.
Dickinson (1971, 1979) indicates a thickness of 12-15 km for the Great Valley sequence
o f California, 17.5 km for the Murihiku Supergroup of New Zealand, 10- 12.5 km for the
Onogawa Group and 5-7.5 km for the Izumi Group of Japan, and 15 km for the Mesozoic
forearc basin in central Oregon.
The forearc region is a zone of intense and rapid deformation. Tectonic response of the
overriding plate may take several forms, but the controlling parameters are' not well understood. Accumulation of a thick accretionary wedge and seaward migration of the trench axis
seems to occur in response to thick sediment cover on the subducting plate (Karig, Caldwell
& Parmentier 1976; Buffler & Watkins 1977). The widespread presence of accretionary
wedges suggests that accretion and seaward growth of the overriding plate is the dominant
mode of forearc evolution. However, geologic relationships imply episodes of tectonic
erosion and sediment subduction in some overriding plates. Data reported by Hussong
(1978) suggest that the forearc region of the Mariana Islands is being consumed rather than
accreted. Tectonic erosion is also suggested for the Japan Trench (Murauchi & Ludwig
1980), the western margin of South America (Coulbourn 1981; Hussong & Wipperman 1981;
Schweller, Kulm & Prince 1981; Shephard & Moberly 1981; Schweller & Kulm 1978;
Hussong et al. 1976; Scholl & Marlow 1974; Plafker 1972; Rutland 1971; Katz 1971) and
the Mesozoic margin of western North America (Page 1970). Karig (1974) and Karig,
Cardwell & Moore (1978) suggest strike-slip or spreading offsets as alternative truncation
mechanisms in some cases.
Modelling subduction and the forearc region
Cowan & Silling (1978) studied deformation in the accretionary wedge with a physical
model using clay t o represent the soft sedimentary material scraped from the subducting
plate. Cloos (1982) analysed deformation of flow melanges such as the Franciscan complex.
Other analyses summarized by Davis, Suppe & Dahlen (1983) model the shallow
sedimentary wedge as a plastically deforming mass overlying a decollement. Most subduction
models which consider the entire lithosphere have employed numerical methods t o represent
material inhomogeneities and complex geometries (Smith & Toksoz 1972). Sleep (1975)
studied near-surface manifestations of subduction, but used a non-linear viscous model for
Numerical subduction models
421
the lithosphere. More realistic viscoelastic and plastic flow laws were used by Bischke (1974)
and by Neugebauer & Brietmayer (1975) t o study transient stress changes preceding and
following seismic events, In these studies, the time intervals studied are too short t o allow
conclusions about long-term permanent deformation. Bird (1978a) and Bird & Toksoz
(1976) used finite element models t o study continent-continent collisions.
My analysis relies on the finite element method t o establish the kinematics of forearc
deformation and the mechanical relationship with other elements of the subduction system.
The finite element analysis also sewes to delimit the condition required for subduction.
The finite element programme
The two-dimensional, plane strain formulation employs 8-node isoparametric elements. The
displacement within each element and on its boundaries is quadratic as defined b y displacement of the nodes.
The subduction system is modelled as an elastic-plastic lithosphere overlying a viscous
asthenosphere. ‘Consistent’ gravity loading was employed (Zienkiewics 197 1). The von Mises
plastic flow criterion (Nadai 1950) governs elastic-plastic behaviour in the lithosphere
exclusive of the megathrust. This allows a constant yield stress, facilitating an assessment o f
overall lithospheric strength. The megathrust is modelled with an anisotropic yield criterion
representing a plane of weakness or a zone of parallel planes of weakness. Vertical
deflections of the plate result in an isostatic restoring force which is modelled as a stiffness.
To account for the path dependence of plasticity and time dependence of the viscous
asthenosphere, displacements are applied to the model incrementally. The principles of the
anisotropic megathrust, isostatic stiffness and time interation are described in the Appendix.
FlNITE ELEMENT MESH
The central part of the finite element mesh is shown in Fig. 1, the full mesh extends another
300 km to the left and 800 k m t o the right. A smaller mesh including only the lithosphere
was also used. It consists of the one-element-thick lithosphere of the subducting and
overriding plates, and the megathrust between.
The models represent shallow subduction, the plate not having penetrated much beyond
the probable low-viscosity region of the upper mantle. In all models, lithospheric plates were
assumed t o be 70 km thick, The mesh is coarse, but the quadratic displacement distribution
allows fairly accurate modelling of bending, even with a single layer of elements. The
quadratic element is capable of modelling an end-loaded cantilever exactly. Accuracy in
the presence of isostasy was checked for a beam lOOOkm long with the thickness and mesh
100 km
Figure 1. Finite element mesh,
422
T. M, Tharp
refinement used in subduction models. For end deflection of 1.5 km the location of the
highest point on the outer bulge (480 km from the' loaded end) was overestimated by only
15 per cent and average deflection over lOOOkm was in error by 17 per cent. Parabolic
distribution of displacement within elements combined with this reasonably accurate
representation of bending restricts inter-element discontinuities in the slope of the
deflection curve, and a smooth pattern of surface uplift and subsidence results. The coarse
mesh is able to model the dominant large-scale features of subduction and, most
importantly, it allows large displacements before element distortion becomes unacceptable.
Strength parameters
The geometry, boundary conditions, and body forces associated with subduction are
relatively well understood. The greatest uncertainty exists with respect to strength
parameters. The three discrete regions to be modelled are the mantle, megathrust and lithosphere. The mantle is characterized by an effective Newtonian viscosity, while the
megathrust and lithosphere are assumed to be plastic-elastic and characterized by a yield
stress.
M A N T L E VISCOSITY
The choice of viscosity is dictated by the subducted plate's relatively shallow depth of
penetration. Response to loads extending over large areas suggests a relatively uniform
mantle viscosity of 1OZ2P (Sabadini, Yuen & Boschi 1982; Hager & O'Connell 1981;
Peltier & Andrews 1976; Peltier 1976; Cathles 1975). However, there is strong evidence
for a low-viscosity zone a short distance below the lithosphere. Cathles (1975) suggests
that a 75 km thick zone with a viscosity of 4 x 1020Pwould explain rapid isostatic rebound
for loads of small area, without affecting loads the size of Fennoscandia. For Lake
Bonneville upper mantle viscosity estimates include 2 x 1OZoP(Passey 198 l), less than
10z'P (Cathles 1975), 6 x 2OZ0P(Walcott 1970) and 1OZ1P(Crittenden 1963). The depth of
the low-viscosity zone is not well constrained, but the thickness may exceed Cathles estimate
in tectonically active areas like Lake Bonneville. The upper mantle may also have lower
viscosity under oceanic lithosphere than under continental. Several numerical models of
global plate tectonics (which are strongly dominated by the oceanic asthenosphere) seem
most consistent with an upper mantle viscosity of less than 10z2P(Forsyth & Uyeda 1975;
Solomon, Sleep & Richardson 1975; Harper 1975). Forsyth & Uyeda (1975) suggest
viscosity eight times higher under continents than under oceans for an asthenosphere of
constant thickness.
The asthenosphere also seems t o exhibit low effective viscosity in its interaction with
subducting plates. Spence (1977) calculates mantle viscosity of 6 x lOI9P based on strain
diffusion associated with a thrust event in the Aleutians. To account for post-seismic
displacements associated with the Nankaido earthquake of 1946, viscosities of 5 x 1019
and loz1P are reported by Nur & Mavko (1974) and Thatcher & Rundle (1979) respectively.
For the 1923 Kanto earthquake Thatcher & Rundle compute a viscosity of 6 x lOZoP,while
Matsu'ura & Tanimoto (1980) report 1OZoP. The intraplate Riku-u earthquake of 1896
yields 1020P (Thatcher et al. 1980). There are several reasons to expect a reduced effective
viscosity for subduction. The long-term high strain rates imposed by the subducting plate
and attendant mantle circulation might allow easier flow by causing grain size and other
textural changes on a regional scale. The subducting plate also causes high stress concentrations which do not arise from the smooth deflections of surface loading. Given power law
flow, this will cause a reduction in apparent viscosity. Since relatively shallow subduction is
modelled, a viscosity of 1O2'P was assumed for the asthenosphere. This uniform value was
Numerical subduction models
423
chosen to he between the low-viscosity values of the upper asthenosphere and the higher
value of 1OZ2P appropriate to the mantle at greater depths. It is intended to provide a
reasonable magnitude of lithosphere-asthenosphere interaction in lieu of reliable estimates
of viscosity variation with depth.
STRENGTH O F MEGATHRUST
Bird (1978b, c) estimates shear stresses of 220 and 165 bar on the megathrusts associated
with the Tonga and Mariana arcs respectively, and shear stress on the Himalaya megathrust
of 200-300bar. Sleep’s (1975) subduction models imply shear stresses at island arcs of
200-300 bar. There is additional evidence for both higher and lower stresses on megathrusts, and shear strength may vary both spatially and temporally. Archambeau (1976)
indicates that large earthquakes initiate in zones of stress concentration where the tectonic
stress is cn the order of 1 kbar. Average stress over the whole area of the main quake drops
from 150 bar before the event to 50 bar after, but zones of 0.5-1 kbar result at the ends of
the rupture. These localized high stresses are consistent with stress drops of 890 and 650 bar
reported by House & Boatwright (1980) for small megathrust events in the Shumagin seismic
gap in the Aleutians.
Long-term shear stresses of at least 1kbar are suggested by investigations of shear heating
on faults (Scholz 1980). Palaeopiezometers (Christie & Ord 1980; Kohlstedt & Weathers
1980; Burg & Laurent 1978) suggest fault shear stresses ranging from a few hundred bars to
more than 1 kbar. The depth range to which these high stresses apply is poorly known. Nearsurface stress which is governed by the strength of fault gouge (Morrow, Shi & Byerlee
1982) will be less, and stresses should also drop rapidly at the base of the lithosphere or
perhaps above (Meissner & Strehlau 1982; Chen & Molnar 1983).
STRENGTH O F LITHOSPHERE
The high stresses cited above for lithospheric faults represent lower limits for lithospheric
strength. Studies of lithospheric flexure at trenches may provide the most reliable estimates
for strength of the lithosphere as a whole. Studies by Tharp (1980), McAdoo, Caldwell
Turcotte (1978) and Turcotte, McAdoo & Caldwell (1978) establish a strong case for
strength of at least several kilobars in the interior of the oceanic lithosphere. A constant
yield stress of 2 kbar would produce a flexural rigidity consistent with these studies. This
was the maximum value considered in the finite element analyses.
Short-term finite element results
Oscillatory movements with periods of tens to hundreds of years occur in seismically active
regions adjacent to subduction zones. Bischke (1974) used a finite element model to analyse
elastic deformations associated with megathrust seismicity in Shikoku, Japan. He concluded
that known deformations between major earthquakes are best explained by a vertical force
acting downward on the base of the lithosphere in the vicinity of the trench. Such a force
could be caused by pull of the dense subducted lithosphere. Stresses resulting from this force
system are illustrated in Fig. 2. Isostatic restoring forces due to vertical deflections and the
gravity force due to high-density subducted lithosphere are applied in this model.
Downward pull due to a high-density subducted plate is generated by imposing a gravity
load on the two lowest subducted elements. For a lithosphere 7 0 k m thick, a density
contrast of approximately 0.05 g ~ r n -is~appropriate (Livshits 1965 ;Hatherton 1969; Grow
1973; Segawa & Tomoda 1976; Sager 1980). This represents the positive density contrast
424
T. M. Tharp
Compressive Stress
Tensile Stress
M
500 Bars
Figure 2. Finite element principal stresses with megathrust locked for 1000 yr.
between the cold subducted plate and the surrounding asthenosphere. Asthenospheric
viscosity is assumed to be 102’P. The lithosphere is elastic, and its horizontal boundaries are
free, The stresses of Fig. 2 are those that exist after lOOOyr of load application with the
megat hrust locked.
When the megathrust is locked, the near-surface lithosphere deforms as a continuous
beam subjected t o a downward force. In a zone extending to both sides of the trench the
upper lithosphere is in horizontal compression, while the lower lithosphere undergoes a
horizontal tensile bending stress. The addition of lithostatic stress of course renders all net
stresses deep in the lithosphere compressive. As noted by Bischke (1974) the horizontal
compression in the upper lithosphere is sufficient to localize large-scale thrusting in the
vicinity of the trench. A maximum shear stress near the trench of 2.8 kbar is approached
asymptotically as stresses in the asthenosphere relax with time. In models with asthenospheric
viscosity of 1OZ1P,shear stresses reach 25 per cent of this value in 1000 yr and 50 per cent in
8000yr.
In the oceanic plate seaward of the trench axis this short-term bending is opposite in
sense (concave upward) to the long-term concave downward bending associated with subduction. As a result, the bending stresses which lead to normal faulting in the oceanic plate are
decreased, while at the same time increased compression enhances the likelihood of slip on
the megathrust. As a result, normal fault earthquakes in the oceanic plate seaward of the
trench are most likely to occur shortly after an earthquake on the megathrust has relieved
this horizontal compression in the upper lithosphere. This is the sequence of events observed
in the Rat Island earthquakes of 1964.
Tensile bending stress in the lower lithosphere reduces down-dip compression in the nearsurface part of the subducting plate, As a result, activity in the double seismic zone of the
subducting plate may be episodic at its upper end, particularly where it overlaps the megathrust (Kawakatsu & Sen0 1983). The zone of down-dip tension should show a peak of
activity shortly before megathrust events, while the zone of down-dip compression should
b e most active following movement on the megathrust. This predominance of down-dip
tensional over compressional events is reported by Kawakatsu & Sen0 (1983) for a locked
segment of the northern Honshu arc.
Long-term behaviour and conditions for subduction
The long-term behaviour of subduction zones depends strongly on inelastic modes of deformation. In this study, inelastic deformation occurs by plastic failure in the bulk of the
lithosphere and by anisotropic plastic failure in the megathrust. In all models a displacement rate of 4 c m yr-’ is imposed on the seaward end of the oceanic plate. This rate of
convergence continues for 600000 yr to produce a total horizontal motion of 24km. This
limit is imposed to prevent excessive distortion of elements near the megathrust. Isostatic
stiffness is imposed at the base of the lithosphere except in its subducted portion. Isostasy
Numerical subduction models
425
is imposed at the base because the vertical isostatic force is dominated by the change in
pressure exerted by the dense asthenosphere in response to deflection. The change in
pressure exerted on the surface by seawater is much less.
Fig. 3 is a contour plot of maximum principal stress magnitude in the asthenosphere
(ignoring the lithostatic contribution) for a convergence rate of 4 cm yr-'. The asthenosphere with a viscosity of 102'P exerts stresses of only several tens of bars on the lithosphere. With a short plate this was shown to have no significant effect on stresses in the
I
'--,
\
100 km
'P
Figure 3. Contours of maximum principal stress magnitude in asthenosphere with viscosity of 102'P
and plate convergence rate of 4 em yr-',
lithosphere. For this reason the viscous asthenosphere was not employed in the models
considered below. The asthenosphere was represented only by its isostatic effect. This
assumption would not be valid for a long subducted plate, with its greater surface area and
deeper penetration. The following analyses and conclusions are therefore restricted to the
case of a short subducted plate.
S T E A D Y -ST A T E C O N D IT 1 0 N S
A true steady-state stress distribution would probably only be achieved after hundreds of
kilometres of subduction. However, a near steady-state is reached very early in the megathrust, and convergence of 24 km is sufficient to cause plastic failure in all elements adjacent
to the trench. Since the subducting oceanic plate undergoes a monotonic increase in curvature as it approaches the trench axis, the path dependence of stresses in this part of the
lithosphere may not be strong. This lead to the possibility that a near steady-state may be
achieved in this part of the system. Curvature of the plate as it passes the megathrust may
continue to increase to a very high value, but it may decrease at depth (Engdahl & Scholz
1977). This being the case, a steady-state probably does not exist in the down-going plate
below the megathrust. For a short subducted plate and low asthenospheric viscosity this will
have little effect on the lithospheric interaction above. For reasons discussed later, the
forward part of the overriding plate may never reach a true steady-state.
MODEL PARAMETERS
Models were run to study the effect of an anisotropic megathrust, a high density subducted
plate, and different values of lithospheric strength. Lithospheric yield stresses of 1 or 2 kbar
were used for all models. Megathrust yield stress was assumed to fall between 200bar and
1 kbar, which is within the range cited above.
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T. M. Tharp
Table 1. Finite element subduction models, 24 km
convergence.
Model
Lithosphere
Yield ( k b )
Fault Yield
(kb)
Excess D e n s i t y
(g/cm3)
Push*
(kb)
1
1
0.5
0.05
0.59
2
1
0.5
0.10
0.70
3
2
no f a u l t
0.05
2.94
4
2
no f a u l t
0
4.00
5
2
0.5
0.05
0.91
6
2
0.5
0.10
0.62
7
2
1 .o
0.05
1.70
8
2
1 .o
0
2.90
9
2
0.2
0.05
0.57
*Horizontal stress pushing on the end of the 7 0 k m
thick lithosphere.
C O N D I T I O N S FOR REALISTIC S U B D U C T I O N BEHAVIOUR
Table 1 lists results for a number of subduction models. Models with maximum shear
strength of up to 2kbar (differential strength of 4kbar) were not found to inhibit subduction, and for a relatively strong megathrust the energy required to drive subduction seemed
t o be insensitive to shear strength outside the megathrust. This confirms previous results
based on power dissipation in the subduction system (Tharp 1980).
Subduction occurred for models with an anisotropic megathrust whether or not the
subducted plate exerted a pull. This is demonstrated by the similarity of deformations in the
top two models of Fig. 4. The same deformation pattern holds for models without a
mechanically discrete megathrust (megathrust region mechanically identical to other lithosphere) when the subducted plate exerts a downward force (pull). This is seen in the third
model of Fig. 4. The downward force of the dense subducted lithosphere concentrates deformation in the zone of the megathrust even if no fault-weakened region exists. With neither
anisotropic megathrust nor pull, no subduction occurs, as seen in the fourth model of Fig. 4.
In no model did subduction occur without horizontal compression. This is due at least
in part t o the absence of a long subducted slab and the significant down-dip pull that results
from interaction of a long slab with the viscous asthenosphere (Tharp 1978). Previous
estimates of the magnitude of net compression in the lithosphere have been based on
numerical models and magma pressures at spreading ridges. The global plate tectonic model
o f Forsyth & Uyeda (1975) allows a ridge push of several hundred bars, and net stresses of
this magnitude are also consistent with models by Richardson, Solomon & Sleep (1979) and
Solomon, Richardson & Bergman (1980). Bird (1978a) estimates a driving force for the Red
Sea Rift, which averaged over a lOOkm lithosphere gives a push of 295 bar. Net compressions of this magnitude are also consistent with megathrust shear stresses computed for the
Tonga and Mariana arcs (Bird 1978b, c). The horizontal compressions found in Table 1 are
slightly to greatly in excess o f the expected values of a few hundred bars. The implication
is that strengths of the megathrust and lithosphere are probably lower than the assumed
values. The two analyses with a 1 kbar megathrust shear strength required compressions of
1.7 and 2.9kbar. These values appear to be excessive and seem to require that the average
strength of the megathrust be substantially lower.
Those models without a distinct megathrust also required kilobar range compressions.
If the megathrust is not significantly weaker than surrounding lithosphere, then both the
Numerical subduction models
r
I
\
-
/ /
-
-
427
-
/
LITHOSPHERE YIELD
MEGATHRUST YIELD
DENSITY CONTRAST
10 cm/yr
-
-
-
LITHOSPHERE YIELD
MEGATHRUST YIELD
DENSITY CONTRAST
.
-
//
/
2 kb
1 kb
0.05 glcm’
2 kb
1 kb
-
- LITHOSPHERE YIELD
MEGATHRUST YIELD
DENSITY CONTRAST
LITHOSPHERE YIELD
MEGATHRUST YIELD
DENSITY CONTRAST
2 kb
0.05 g/cm’
2 kb
-
-
Figure 4. Finite element velocity vectors.
megathrust and overriding lithosphere must have average strengths of only several hundred
bars, rather than several kilobars. For other models horizontal compression is modest with a
strong lithosphere as long as the megathrust is weak. The most probable value for average
megathrust shear strength is 100-200 bar.
SURFACE DEFLECT10NS
Vertical deflection patterns are similar for models with a wide variety of mechanical
properties. Following the deflection curves from right to left in Figs 5 and 6, the subdued
positive deflection of the outer bulge is encountered first. This is succeeeded to the left by
the profound negative displacement of the trench, the sharp positive peak of the outer arc
at the top of the trench slope, and the synclinal deflection of the forearc basin.
The outer bulge amplitude of 500- 1000m is consistent with observation. The excessively
deep trenches are probably due to inadequate representation of isostasy near the trench. The
finite element models exhibit three features which bear directly on the forearc deformation
problem: (1) the outer arc, (2) the forearc basin, and (3) a high rate of energy dissipation in
the lithospheric wedge overlying the megathrust.
The peak of the outer arc occurs at the junction of the megathrust and overlying plate,
428
T. M. Thaip
OUTER ARC
,"
-5 k
FDREARC @ I S IbN
Z
TRENCH
*b
MEGATHRUST YIELD
0.5 hb
DENSITY CONTRAST
0.05 plsm'
LITHOSPHERE YIELD
Z hb
MEGATHRUST YIELD
0.5 kb
E
1
-5
2
P
I-
LITHOSPHERE YIELD
CONTRAST DENSITY
0.05 gICm'
0
-5
,
I
!
I
400
300
200
100
LITHOSPHERE YIELD
1 hb
MEGATHRUST WELD
0.5 hb
I
0
200
100
300
HORIZONTAL DISTANCE (km)
Figure 5. Finite element vertical deflections at the surface of the lithosphere. Low megathrust strength.
-5'
k
k
-E
MEGATHRUST
k
YIELD
1bkb
CONTRAST DENSITY
0.05 glcm'
z
2
0
0 -
P
'
LITHOSPHERE YIELD
2 kb
MEGATHRUST YIELD
-
CONTRAST DENSITY
0.05 glcm'
2
b k>
T { {-5
'
I
I
I
I
I
I
I
I
400
300
200
100
0
100
200
300
HORIZONTAL DISTANCE (km)
Figure 6 . Finite element vertical deflections at the surface of the lithosphere. High megathrust strength.
with a total positive deflection of 5-10km. The uplifted region is analogous to the bathymetric ridge (or outer arc) at the top of the inner trench slope in many subduction zones.
The seaward slope of outer arcs is characterized by active thrusting, with a general downward increase in activity. In the finite element models the seaward slope of the outer arc is
underlain by the megathrust, with the apex at the top of the megathrust zone. The tectonic
position of the outer arc seems therefore to be accurately represented by the models.
Numerical silbduction models
429
The forearc basin is located between the outer arc and the volcanic arc. In the models it is
generally a region with downward deflection of a few kilometres. The only model in which
the lithosphere did not deflect below its original level was one without excess weight in the
subducted plate (Fig. 6 , second model). Even in this case the synclinal structure is observed,
although net deflection is not downward. The bending and kilometre-scale regional uplift
observed in this case might result from subduction of a marginally buoyant oceanic platform
or chain of seamounts (Nur & Ben-Avraham 1982). In geologic terms, this model predicts
orogeny and angular unconformity when the overall negative buoyancy of the subducted
plate approaches zero.
Forearc basins in the models have an outer limb dipping steeply landward. The landward
limb dips gently toward the outer arc. The shape of the forearc basins generated by the finite
element models bears a striking resemblance to the forearc basin described by Katili (1973)
off west Sumatra (Fig. 7) and the Iquique and Arica Basins landward of the Peru-Chile
trench (Coulbourn & Moberly 1977). The large-scale deformation causing the outer arc and
forearc basin also has important implications for energy dissipation in the subduction zone.
Analyses of subduction power dissipation reported by Tharp (1980) included: (1) viscous
drag of the asthenosphere on the subducting plate, (2) frictional dissipation in the megathrust, and (3) plastic bending in the subducting plate. The finite element models were
found to absorb as much as 2 to 3 times the power predicted by power balance calculations
including these three mechanisms. Less than half of this excess input energy goes into
changes in elastic strain and isostatic potential energy due to non-steady-state behaviour. The
remaining excess is dissipated by additional plastic flow, almost all in the forearc region
above the megathrust.
Megathrust friction generally dominates energy dissipation in the region of subduction,
However, when the megathrust has low shear strength, more energy may be dissipated in the
overriding plate. A weak megathrust not only dissipates less energy, but indirectly increases
deformation in the fmearc region by producing a stress concentration there. The effect of
low megathrust strength is seen in the contrasting behaviour of models 7 and 9 (Table 1). In
model 7 with megathrust yield strength of lkbar and 4 c m yr-' convergence, power dissipation per centimetre of trench length is 12.7 x 108erg s-l in the megathrust and
9.5 x I08erg s-l in the forearc region. In model 9 with the same convergence rate but a
megathrust yield strength of 200bar, power dissipation is 2.5 x 108erg s-l in the megathrust
and 7.0 x 108erg s-l in the forearc region, Model 7 is stronger and requires a 1.7 kbar push to
drive subduction, but even in this case of high megathrust strength power dissipated in the
forearc region is comparable to dissipation on the megathrust. In model 9 reduced megathrust strength is reflected in a required push of only 0.6kbar. The reduced stress level
decreases yielding and power dissipation in the forearc region, but this effect is nearly
balanced by increased stress concentration in the fmearc region caused by the weak megathrust. This pattern of dissipation represents a significant departure from the generally
N.E
s.w
Meters
Figure 7. Section across arc-trench gap, West Sumatra. After Katili (1973) with permission of the
University of Western Australia Press. Trench is t o the right.
430
T. M. Tharp
accepted mode of subduction zone plate interaction. An examination of force equilibrium
and stress distribution in the forearc region provides a plausible explanation for this phenomenon and better defines the conditions under which it may arise.
Stress concentration in overriding wedge
T h e stress concentration responsible for plastic deformation in the wedge overlying the
megathrust could result from the combination of net horizontal compression in the overriding lithosphere, and a megathrust of low shear strength. Since the stress available to
cause failure is directly proportional to horizontal stress in the forearc region, it is necessary to estimate the sense, direction and magnitude of this stress. Sykes & Sbar (1973)
conclude that interior regions of both oceanic and continental plates are dominated by
compression. Magnitudes of 100-300 bar seem reasonable based on the work cited above.
Nakamura & Uyeda (1980) suggest that compression in the overriding plate near the trench
is a feature of many subduction plate boundaries. Fault plane solutions (Stauder 1975)
indicate maximum compression perpendicular t o the trench for a considerable distance into
t h e South American plate. A similar sense and orientation of stress is implied for the
Aleutian Islands by the location of volcanic eruptions (Nakamura, Jacob & Davies 1977).
The net regional compression up (mean value over the thickness of the lithosphere) is
equilibrated at the megathrust by mean tangential stress r and mean normal stress u
(Fig. 8). The relationship between stresses may be seen by a summation of forces in a direction parallel to the top of the plate (nearly horizontal) and perpendicular to the trench.
Forces resulting from these stresses must sum t o zero to preserve static equilibrium:
t
t
--(Jpt-t 7 7
cosa + u -sina = 0
sin CY
sin (Y
where t is the thickness of the lithosphere and a is the megathrust dip. Simplifying:
u = up-r/tana.
(1)
A n y combination of u and r satisfying this relationship will satisfy static equilibrium in the
direction parallel t o the plate. Megathrust shear and normal stresses which result in no stress
concentration are found by resolving up into shear r f and normal uf components on a plane
dipping at the angle of the megathrust a (Timoshenko & Goodier 1970, p. 19):
rf = (1/2) up sin2a
Of =
up cos2 ff.
Stresses uf and rf acting on the megathrust would not cause a stress concentration in the
forearc region. However, r cannot exceed the shear strength of the megathrust, which may
be less than Tf. To maintain static equilibrium, normal stress u acting upwards on the overriding lithospheric wedge must be increased for a decrease T . This stress system results in a
net upward force exerted on the overriding plate at the megathrust. This would tilt the plate
upward slightly, SO the upward force would be balanced regionally by a downward isostatic
Figure 8. Stresses for horizontal force equilibrium.
Numerical subduction models
43 1
Figure 9. Coordinates and boundary conditions for infinite wedge.
force. Tilting of the plate results in a small weight component acting parallel t o the plate.
This reduces the compression felt at the megathrust by up t o 1/3 for a wide range of values
for up and a . Tilting due t o other causes might have a similar or opposite sense. The effects
of tilting may be significant, but they are uncertain and are not incorporated in the analysis.
The stress concentration due t o r less than rf and the resulting increased u are approximated
by a closed-form solution due to Timoshenko & Goodier (1970). They state a general stress
function for an infinite 2-D wedge with stresses applied t o the upper and lower faces (Fig. 9).
The stress function @ in polar coordinates r and 0 has the form:
4 = bor2+ dor29 + b l r 3 cos 8 t d l r 3sin 0 + a2r2cos 219 + c 2 r 2sin 29
+ a3r3cos 38 + c3r3sin 39.
The stress function is related t o stresses at any point in the wedge through the relationships:
Ue
a 2@
=-
ar2
1
7r0
=-
r2
a@
ae
1
a2@
__
r arae
Constants bo, d o , b l , d l , a 2 , c 2 ,i13 and c j are determined to satisfy shear and normal
stress boundary conditions on the faces. Shear and normal stresses are zero on the upper face
of the wedge (0 = a ) which represents the surface of the Earth. For a stress concentration t o
develop, r less than rf is assumed t o act on the lower face o f the wedge (the megathrust).
Normal stress u on the lower face is computed by equation (1). For constant boundary
stresses, the stresses within the wedge are independent of r , the distance from the apex of
the wedge.
The effect of low r is to bend the apex of the wedge upward as observed in the finite
element models. This is attended by radial tension on the lower face of the wedge and
increased radial (horizontal) compression ob in the top of the wedge. The ratio u b / o p
(computed horizontal compressive stress q, in the wedge divided by regional compressive
stress up) may be considered a stress concentration. In Fig. 10 this stress concentration is
plotted versus r / r f for various subduction angles a . Even for subduction on a megathrust
dipping 30°, a stress concentration of 10.0 is possible. For shallower dips even higher stress
concentrations are expected. High stresses will be associated with shallow megathrust dip,
high horizontal compression, and a large difference AT between r and rf.This stress concentration may lead t o substantial regions of failure in the overriding wedge even if AT is only a
432
T. M. Tharp
loo
h
20
1
0
0.2
0.4
0.6
0.8
1.0
Figure 10. Compressive stress concentration at upper surface of infinite wedge versus 7 / T f .
few tens of bars. Stress departures of this magnitude are expected, and the observation that
they cause stresses in the kilobar range is consistent with the finite element results.
Stress concentration will occur as long as 7 is less than 71,but magnitude and location of
maximum stress are a function of the distribution of shear and normal stresses on the megathrust, and these stresses will surely depart from the constant stress assumption employed
here. Stresses are uncertain due to the lack of an adequate model for megathrust shear
strength or for interaction between the two plates. It may be speculated that low shear
strength near the surface would reduce shear stress on the megathrust without necessarily
reducing normal stress. This would increase stress concentration near the apex of the overthrust wedge. Increased strength and shear stress at depth along the megathrust would
reduce the stress concentration distant from the apex. The finite radial dimension of the
wedge would have the same effect.
Discussion and conclusions
Subduction is a very robust process. Once initiated, density contrasts generate localized high
stresses which are probably sufficient to overcome mechanical obstacles such as surface
irregularities on the subducting plate. Presence of a low-strength megathrust allows the
continuation of subduction even with zero negative buoyancy. While subduction itself is
likely to continue as long as plate kinematics are favourable and driving forces are positive,
stasis in the overriding wedge may be an exceptional condition. Where the subducted slab is
short, progressive upward bending and rotation of the overriding wedge is likely, resulting
in the generation of a deep forearc basin and the dissipation of significant energy. The forearc region is generally a zone of mechanical instability when subjected to horizontal
compression. Deformation and vertical tectonics should result not only from accretion, but
from regional compressive stress and variable buoyancy of the subducting plate.
If the overriding plate and megathrust bend upward near the surface, the megathrust will
in time become concave upward. Under these circumstances, maintenance of a mechanically
efficient subduction zone might lead t o one of three situations depicted in Fig. 11. Accre-
Numerical subduction models
New Megathrust
433
7
Figure 11. Possible responses Lo forearc bending.
tion of new material, or tectonic erosion, would allow the dip of the subduction zone t o
remain unchanged. In lieu of these two adjustment mechanisms, subduction might
eventually shift to a new megathrust. In any case, the locus of subduction would change,
either continuously by accretion or erosion, or discontinuously by reintiation at another
location. This is consistent with the progressive shifts in volcanic arc plutonism documented
b y Dickinson (1973).
Acknowledgments
This research was supported in part by the Geological Society of America's Penrose Research
Grant 1831-74. I am grateful t o Sandra M. Parks for her thoughtful comments and
suggestions on an earlier draft of this paper.
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Appendix: finite element procedures
ANISOTROPIC MEGATHRUST
The yield condition states that shear stress must not exceed some specified value on planes
of a particular orientation. Since 7 , y = T~,., this yield condition applies to an orthogonal
plane as well. When the yield condition is exceeded, out-of-balance stresses are integrated
over the element to generate nodal loads which are applied to the model in the next iteration. This so-called ‘initial stress’ approach allows iterative adjustments to the model without
reforming the stiffness matrix. For each time step this iteration is continued until the shear
stress on a plane in the megathrust orientation is close to the correct value. The convergence
criterion is a 2 per cent average error over all plastic elements.
ISOSTATIC STIFFNESS
The isostatic restoring stress results from a stiffness added to the overall stiffness matrix of
the body. The restoring stress to be imposed at the base of the lithosphere is:
0=
pgv,
where p is the difference in density between the seawater above the lithosphere and the
asthenosphere below, g is the gravitational acceleration, and u is the vertical deflection.
The stiffness is derived through a surface integral which becomes a line integral for a 2-D
vertical cross-section. The energy W due to deflection is:
Numerical subduction models
43 7
where 6 is the vector of vertical nodal displacements, and N is the matrix of shape functions.
The factor of (1/2) is cancelled by partial differentiation of W with respect to 6 and the
resulting stiffness is:
JNhVdl.
The line of integration follows element boundaries, and each such boundary is treated as a
three-node ‘line element’. Integration is by two-point Gauss quadrature. In models with
viscous elements, displacement rates du/dt replace displacements u , and the time incremental change in isostatic stress becomes u = pg(du/dt)At, where A t is the time step. The
resulting isostatic stiffness integral is:
h
p g A t Ndl.
TIME ITERATION
Models may have both elastic-plastic and purely viscous elements. Since elastic elements
deform instantaneously under stress and the instantaneous deformation of a viscous element
is zero, some accommodation must be made to allow a structure with both types of
elements. The solution is t o distribute elastic strain over a period of time. To illustrate, an
elastic strain e is equal to u/E in the uniaxial case, where u is the uniaxial stress and E is
Young’s modulus. In models with viscous elements, the solution yields strain rates rather
than strains. For elastic uniaxial loading the strain rate becomes: i. = u/AtE where A t is the
time step. The strain rate in the viscous plain strain elements remains simply: 6 = U / Q where
TI is the viscosity. To accommodate both elastic and viscous elements, Young’s modulus E
is replaced by AtE in the formulation of the stiffness matrix. This linearization of the viscoelastic system is made arbitrarily accurate with sufficiently small time steps.
Iteration in time requires that accumulated elastic strains be retained between iterations.
This is done in each iteration by imposing initial stresses equal t o stresses from the previous
cycle for elastic-plastic elements. Isostatic forces due to accumulated vertical deflections are
also imposed at each iteration.