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TECTONICS, VOL. 28, TC3002, doi:10.1029/2008TC002264, 2009
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Localization of shear along a lithospheric strength discontinuity:
Application of a continuous deformation model to the boundary
between Tibet and the Tarim Basin
Katherine E. Dayem,1 Gregory A. Houseman,2 and Peter Molnar1
Received 17 January 2008; revised 31 December 2008; accepted 16 March 2009; published 19 May 2009.
[1] A marked contrast in strength (or viscosity)
within a continuously deforming zone can lead to
concentration of shear strain in the weaker material
adjacent to the boundary between them, but
localization comparable to the width of the Altyn
Tagh shear zone requires an additional weakening
process. During numerical experiments on a thin
viscous sheet indented by a rigid object, a shear zone
develops adjacent to a strong region mimicking the
Tarim Basin, when the boundary between the weak
and strong regions is oblique to the orientation of
convergence. The width of this shear zone narrows
with increased strain, and for comparable penetration
by the indenter, the strain is more concentrated for
larger values of n, the exponent that relates strain rate
to a power of stress, and for smaller values of the
Argand number Ar, a measure of buoyancy-induced
stress relative to viscous stress. Increasing
concentration of shear occurs as the indentation
develops without weakening because of a change in
material properties. Additional localization develops
with the inclusion of strain-dependent weakening
associated with, for instance, a temperature increase
due to shear heating. For such localization to scale to
the width of the Altyn Tagh fault zone of Tibet, the
initial temperature near the Moho must be relatively
low (600°C), and a large value of n (10) is
required. This suggests that deformation there is
described by a high-strength flow law, such as that
proposed by Evans and Goetze (1979), in which the
lithosphere would deform approximately plastically.
Citation: Dayem, K. E., G. A. Houseman, and P. Molnar (2009),
Localization of shear along a lithospheric strength discontinuity:
Application of a continuous deformation model to the boundary
between Tibet and the Tarim Basin, Tectonics, 28, TC3002,
doi:10.1029/2008TC002264.
1
Department of Geological Sciences and Cooperative Institute for
Research in Environmental Sciences, University of Colorado, Boulder,
Colorado, USA.
2
School of Earth and Environment, University of Leeds, Leeds, UK.
Copyright 2009 by the American Geophysical Union.
0278-7407/09/2008TC002264$12.00
1. Introduction
[2] Localized strain is observable at and near the surface
of the earth as faulting. The Altyn Tagh fault zone of
northern Tibet, one such example of localized strain, forms
a major part of the boundary between the Tibetan plateau
and the Tarim basin and accommodates strain resulting from
convergence between India and Asia (Figure 1). Structural
[Cowgill et al., 2000] and GPS [Zhang et al., 2007] data
suggest that near-surface strain is localized on faults that
span a width of 100 km. Localized strain appears to exist
in the lithosphere beneath the fault zone as well; tomographic data suggest that a narrow zone of low P wave
speeds may extend to a depth of 140 km [Wittlinger et al.,
1998]. We examine how such localized strain may develop
beneath a fault zone.
[3] The deforming Tibetan lithosphere has been described in several ways. In one view, a small number of
major faults with large offsets and slip rates separate
essentially rigid blocks so that deformation throughout the
lithosphere is concentrated in narrow zones [e.g., Avouac
and Tapponnier, 1993; Replumaz and Tapponnier, 2003;
Tapponnier et al., 1982]. A second view explicitly treats
large-scale deformation of continents as continuous deformation; faulting may occur in the brittle upper crust, but the
ductile strength of the lower crust and upper mantle governs
the distribution of regional deformation. These layers deform as continuous viscous media, if with spatially varying
properties [e.g., England and Houseman, 1986; England
and McKenzie, 1982; Molnar and Tapponnier, 1975;
Tapponnier and Molnar, 1976]. In a third view, the crust is
considered to be the stronger layer [e.g., Flesch et al., 2001],
but the same thin viscous sheet is used to simulate the
deforming layer, and flow in the crust and mantle are
essentially identical [Wang et al., 2008]. Finally, a fourth
view includes vertical heterogeneity of the lithosphere, in
which the lower crust can be relatively weak. Where the
lower crust is weak, upper crustal deformation can be
independent of strain in the deeper crust and mantle [e.g.,
Clark and Royden, 2000; Clark et al., 2005; Royden, 1996].
[4] We examine how strain may have become localized
in the lithosphere beneath a fault zone, which has implications for the localization of strain in the Altyn Tagh fault
zone or along other continental tectonic boundaries. In
particular, we test the degree to which strain in a continuous
material may localize along a strength discontinuity within
the lithosphere analogous to that between the Tibetan
plateau and the Tarim basin. In the context of the theories
outlined in the previous paragraph, we test whether local-
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Figure 1. (a) Map of the Altyn Tagh fault and surrounding structures from Tapponnier et al. [2001].
(b) Tibetan Plateau and surrounding area with box showing location of Figure 1a.
ized zones of deformation, as postulated in the rigid block
view of lithospheric deformation, may develop as a consequence of the continuous deformation, postulated in the last
three views. To do so, we first test the degree of localization
that occurs without any change in material properties as
deformation occurs. Finding that the resulting region of large
strain rates is insufficiently localized to explain the Altyn
Tagh fault zone, we introduce a simple form of straindependent weakening. Several possible strain-weakening
processes may contribute to localization of deformation
within the lithosphere, but we consider here a simplified
formulation based on the idea that mechanical work done by
the deformation heats the lithosphere, and the rise in temperature lowers the temperature-dependent viscosity coefficient.
We then assess the plausibility of such a form of weakening by
comparing the apparent viscous flow law parameters that
account for the localization of strain along the Altyn Tagh
fault with those determined by laboratory experiments.
2. Geological Setting of the Altyn Tagh Fault
[5] Continental collision between India and Asia began
50 Ma [e.g., Rowley, 1996, 1998; Searle et al., 1987; Zhu
et al., 2005] when the convergence rate slowed from greater
than 100 mm a1 to 50 mm a1 [e.g., Molnar and Stock,
2009; Molnar and Tapponnier, 1975; Patriat and Achache,
1984]. Some of the >2000 km of postcollision convergence
has been accommodated by strike-slip shear along the Altyn
Tagh Fault zone along the northern margin of the plateau.
Slip along the Altyn Tagh fault had begun by Oligocene
time [Ritts et al., 2004; Yue et al., 2005], and possibly as
early as 49 Ma [Yin et al., 2002].
[6] A broad range of slip rates on the Altyn Tagh fault
zone has been inferred from several techniques. Rapid slip
(27 mm a1) is estimated from offset Quaternary glacial
and fluvial landforms [Mériaux et al., 2004]. Relatively
slow estimated rates include: 9 mm a1 determined from
GPS measurements [Bendick et al., 2000; Shen et al., 2001;
Wallace et al., 2004]; 11 ± 5 mm a1 determined from
repeated interferometric synthetic aperture radar measurements [Elliott et al., 2008]; 10 mm a1 from Quaternary
terrace offsets and GPS data [Cowgill, 2007; Zhang et al.,
2007]; 10– 20 mm a1 from paleoseismic data [Washburn et
al., 2001, 2003]; <10 mm a1 from post –early Miocene
offsets [Yue et al., 2004]; and 12– 16 mm a1 from older
displaced geologic features [Yue et al., 2001].
[7] In addition to accommodating strain, the Altyn Tagh
Fault zone marks a lithospheric strength discontinuity. Steep
topographic gradients across the zone suggest the presence
of contrasting strength north and south of it [e.g., Clark and
Royden, 2000; Royden, 1996; Shen et al., 2001; Stüwe et al.,
2008]. In fact, the lithosphere of the Tarim Basin, north of
the Altyn Tagh fault, appears to be strong relative to the
surrounding lithosphere [e.g., Avouac and Tapponnier,
1993; Molnar and Tapponnier, 1981]. Whereas lithosphere
beneath the Tarim Basin does not seem to have undergone
deformation since Precambrian time, the crust of Tibet
consists of fragments accreted to stable Eurasia during
Paleozoic and Mesozoic time, and many of those fragments
have been deformed after being accreted [Chang and
Cheng, 1973]. Reconstruction of the tectonic evolution of
this region indicates that lateral strength variations may play
a significant role in the distribution of continental deformation. Cowgill et al. [2003] relate southward thrusting in the
western Kunlun in part to the strength contrast between the
relatively rigid Tarim basin and the weaker Songpan-Ganzi
flysch terrain. Paleozoic sutures and faults [e.g., Sobel and
Arnaud, 1999] may contribute additional weakness to the
Tibetan lithosphere.
3. Thin Viscous Sheet Equations
[8] We use a thin viscous sheet formulation to simulate
lithospheric deformation in a setting analogous to that of the
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Houseman [1997]. The viscous layer that corresponds to the
lithosphere is deformed by indentation of a boundary
segment of length W (Figure 2). The indentation rate is Vo
in the middle half of the indenter, and that speed tapers to
zero on either side over a distance W/4 (parameters and
variables are listed in the notation section).
[10] A vertically averaged constitutive law relates the
strain rate e_ ij, assumed constant with depth, to the deviatoric
stress t ij:
ð1=nÞ1
e_ ij ;
t ij ¼ BE_
ð1Þ
where n is a rheological parameter relating strain rate to
deviatoric stress, B is a spatially variable viscosity coefficient, and E_ = (_eij e_ ij)1/2 is the second invariant of the strain
rate tensor:
e_ ij ¼
1 @ui @uj
;
þ
2 @xj @xi
ð2Þ
where ~
u is horizontal velocity and xi and xj measure distance
in the x and y directions. If dislocation creep dominates
deformation, the constitutive law can be written as
Figure 2. Maximum shear strain rate in the horizontal
plane in units of Vo/W calculated for (a) no penetration and
(b) penetration of 40% of the width W of the indenter for n =
3, Ar = 3. The length of the strong region H is 0.53W, and
the initial obliquity of the strong region is 41° to the
indenter. Contour interval is 0.5 Vo/W. Note the localization
of strain rate along the near boundary of the strong region in
Figure 2b. The white line indicates the profile along which
we calculated the velocity gradient parallel to the strong
region.
Q
;
e_ ij ¼ Aqn1 t ij exp
RT
where q is the second invariant of the deviatoric stress
tensor, A and Q (the activation energy) are experimentally
derived constants for rock-forming minerals, and R is the
universal gas constant [e.g., Goetze, 1978; Karato et al.,
1986]. The depth-averaged lithospheric strength is then
quantified by
1=n
B¼A
Altyn Tagh fault in northern Tibet. Previous studies have
shown that strength discontinuities can alter a strain field.
Concentration of strain along a linear strength discontinuity
has been documented for plane strain [Bell et al., 1977] and
thin viscous sheet calculations [Robl and Stüwe, 2005;
Vilotte et al., 1984, 1986]. These studies show that the
strain rate gradient and width of the shear zone depend on
the size and rigidity of the strong region relative to the
surrounding material.
[9] Assuming continuous deformation, we examine the
effect of an obliquely oriented discontinuity of strength on
the localization of strain in a continental collision zone
analogous to the Northern Tibet – Tarim Basin region and
quantify the width of the shear zone as a function of the
parameters that control deformation using thin viscous sheet
calculations. Following previous studies [e.g., Bird and
Piper, 1980; England and McKenzie, 1982; Houseman
and England, 1986], we assume that the thin viscous sheet
behaves as a continuously deforming, incompressible, viscous fluid. The velocity field is continuous, but the physical
properties of the deforming sheet may be discontinuous, as
described by England and Houseman [1985] and Neil and
ð3Þ
1
L
ZL
exp
Q
dz;
nRT
ð4Þ
0
where L is the thickness of the sheet (or lithosphere).
[11] We define the unit length scale as W (Figure 2) and
the unit velocity scale as Vo. Strain rates are thus scaled by
Vo/W. We examine different constitutive relations by varying
the two parameters that govern the flow: n, which relates
strain rate to a power of deviatoric stress in (1), and the
Argand number Ar:
Ar ¼
grc ð1 ðrc =rm ÞÞL
Bo ðVo =LÞ1=n
;
ð5Þ
where g is gravitational acceleration, rc is crustal density,
rm is mantle density, and Bo is the viscosity coefficient that
applies in most parts of the viscous sheet [England and
McKenzie, 1982]. Thus Ar is a measure of the stress
produced by gravity acting on internal crustal thickness
variations relative to the viscous stress due to deformation
of the sheet.
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[12] To represent the strong Tarim Basin block in the
calculation space, we include an area in which the strength
coefficient Bl Bo. The strong region is obliquely oriented
to the indenter and is of length H, measured along its
southeastern boundary (Figure 2).
[13] The calculated deformation of the thin viscous sheet
is thus defined by two dimensionless numbers, n and Ar,
along with parameters that define relative shape and position of the indenter and strong region, and the initial
contrast in values of the strength coefficient B between
the strong region (Bl) and the rest of the sheet (Bo). A
triangular finite element mesh with approximately 41,000
node points suffices to produce results that are indistinguishable from those of higher resolution. We use this mesh
in subsequent calculations.
4. Development of Strain Localization in
Constant Strength Calculations
[14] With the strong region initially oriented 41° oblique
to the indenter and the length of the strong region scaled to
the width of the indenter H = 0.53W (Figure 2a), we
calculate the evolution of strain and strain rate for 16
combinations of n = 1, 3, 5, 10 and Argand number Ar =
1, 3, 10, 30. We choose these parameters to cover a broad
range of fluid properties. Note that if n = 1, (1) becomes t ij
= B_eij and stress and strain rate are linearly related. If n > 1,
the relationship is nonlinear; n 3 is commonly used to
describe an olivine rheology, and n > 10 is used to
approximate plastic behavior. Small (large) values of Ar
represent a fluid whose viscous resistance is relatively large
(small) compared to stresses arising from crustal thickness
variations.
[15] We set the strength coefficient Bl in the strong region
to be 1000, 10, 4, and 2 times greater than Bo in the
surrounding regions for n = 1, 3, 5, and 10, respectively,
so that the effective contrast in strain rate (which scales with
(Bl/Bo)n) is about 1000 for each case [Houseman and
England, 1996; Neil and Houseman, 1997]. In our initial
experiments B is held constant in time within each region.
The maximum shear strain rate field from one of these
calculations (Figure 2) demonstrates the development of a
zone of high shear strain rate along the southern margin of
the strong region.
[16] To test the effect of the initial obliquity of the strong
region on strain localization, we calculate eight additional
solutions in which we vary the obliquity of the strength
discontinuity but keep all other variables constant (n = 3, Ar
= 3, and H = 0.53W). We also test the effect of the length of
the strong region by decreasing the length of the strong
region by half (H = 0.27W).
[17] England et al. [1985] show that for a homogeneous
thin viscous sheet with a harmonically varying indenting
boundary velocity, the deformation field decays exponentially with a length scale of 2/3 the length of the driven
boundary for n = 1. For larger n (n = 3 to 10) the decay is
approximately exponential,
pffiffiffi with the e-folding distance
inversely proportional to n. They also show that for a
strike-slip shear boundary, the deformation field decays
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with a length scale approximately 1/4 that of the indenting
field of the same n. These results, based on the instantaneous strain rate fields, do not take into account the effect of
crustal thickening and buoyancy.
[18] For the case of a sheet with an obliquely oriented
strong region, the primary deformation field decays away
from the indenter. Superimposed is a secondary component
of the deformation field, which decays away from the
boundary of the strong region. Because the boundary of
the strong region is oblique to the principal convergence
direction, we expect the length scale of this secondary
component to be intermediate between those estimated by
England et al. [1985] for indentation and for shear on the
boundary. To quantify the decay of deformation with
distance from the strength discontinuity, we plot the gradient of the boundary parallel component of velocity (the
major component of horizontal shear strain rate) versus
distance along a profile perpendicular to the discontinuity
and passing through its midpoint (Figure 2). We then
estimate an approximate e-folding distance l0, the dimensionless distance from the strong region to the point where
the velocity gradient reaches 1/e times its value at the
boundary (Figure 3). We refer to l0 as the width of the
shear zone, but we emphasize that this measure is based on
the distribution of current strain rate, and not on the
distribution of cumulative strain.
[19] We include dimensional values in Figures 3, 4, 5, 6,
7, 8, and 9 by assuming the width of the Indian plate W =
2500 km, and the convergence rate Vo = 50 km Ma1.
Dimensional lengths, therefore, are estimated by multiplying nondimensional lengths by 2500 km. Dimensional strain
rates are estimated by multiplying nondimensional rates by
3*1016 s1.
4.1. Development of Strain Localization Along the
Strong Region Boundary
[20] Stress, which in general varies gradually over the
region, can be understood as the product of an effective
viscosity BE_ (1/n1) and the strain rate e_ ij. Because the strong
region has a relatively high effective viscosity, it deforms at
a low rate, and both thickening and shear of the sheet must
occur largely outside of the strong region. Increased thickening and high strain rates first develop on the side facing
the indenter (Figure 2). The increased strain rate near the
strong region leads to a concentrated zone of decreased
effective viscosity, and more strain occurs there than in the
surrounding fluid as the indenter penetrates further. As
strain rates increase, the continued reduction of effective
viscosity allows deformation to develop in a narrowing
zone. Thus the area of high strain rates becomes increasingly concentrated in a narrow zone along the strength
discontinuity, but the relevant material property, B, does
not change.
4.2. Decrease of Shear Zone Width With Indenter
Penetration
[21] As the indenter penetrates into the thin viscous sheet,
the width of the shear zone (l0) decreases, indicating that
shear strain rate progressively localizes along the boundary
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4.3. Dependence of Shear Zone Width on Strong Region
Orientation, n, and Ar
Figure 3. The perpendicular gradient of the component of
velocity (in units of Vo/W) parallel to the strong region as a
function of distance from the strong region for different
amounts of penetration of the indenter, where n = 3 and Ar =
3, and geometrical parameters as shown in Figure 2.
Dimensionless distances (x axis) and times (y axis) are
labeled above distance and time scaled to India and the
Tarim Basin (assuming W = 2500 km and Vo = 50 km
Ma1). Labels on curves represent the distance of indenter
penetration in units of W. The e-folding distance l0 is
calculated by finding the distance between the strong region
and the point where the gradient is 1/e times the maximum
gradient. Note the log scale on the y axis.
of the strong region (Figure 3). As we describe below, l0
depends on the rheological exponent n, the Argand number
Ar, the length of the strong region H/W, and its initial
obliquity to the indenting boundary. In most calculations l0
decreases until it levels out at a minimum, l0min (Figure 4).
The existence of a minimum shear zone width suggests that
the strain rate fields caused by boundary stresses and
buoyancy come into some sort of equilibrium after a certain
amount of indentation has taken place. The shear zone
narrows more for larger n and, eventually, for smaller Ar
(Figure 4). Early in the convergence history, the shear zone
width decreases more quickly for large Ar values than for
small ones, but ultimately the reduction of shear zone width
l0 with time leads to narrower shear zones for small Ar
(Figure 4). For n = 3, the minimum width eventually
attained by the shear zone is approximately half of its initial
value. For n = 5 and 10, the reduction in width is even
greater. By examining the dependence of the minimum
width of the shear zone on geometrical and physical
parameters, we focus on the limit that is relevant to the
present-day Tibet-Tarim boundary.
[22] For given rheologic exponent n and Argand number
Ar, the narrowest shear zone develops for a strong region
oriented 45° oblique to the indenter (Figure 5). For a
strong region oriented 41° oblique to the indenter, the final
width of the shear zone is 13% larger than for the 45°
case.
[23] Greater values of n lead to greater strain rate localization along the strength discontinuity in front of the
indenter (Figure 4) [e.g., Bell et al., 1977]. Because stress
varies gradually over the sheet and the strength coefficient
(Bo) is constant outside the strong region, strain rates for
large n vary more with distance from the discontinuity
than those for small n. Lateral gradients of effective
viscosity (BE_ (1/n1)) are therefore greater for large n, and
localization of strain is enhanced, as observed in the
numerical experiments.
[24] For Argand numbers Ar > 1, the minimum shear
zone width increases with Ar (Figure 4). A strong buoyancy
force resisting thickening (large Ar) should spread the
thickening rates more uniformly over the area between the
indenter and the strong region than should a small Ar. The
high thickening rate, and therefore high compressional
strain rate, near the strong region, permitted by small Ar,
_ and hence leads to a decreased
enhances the value of E,
effective viscosity there. Thus, eventually, more localization of deformation develops for small Ar than for large
Ar. (Calculations using Ar = 1 do not show the relationship described above because the width of the shear zone
has not reached a distinct minimum over the course of
those experiments.)
4.4. Dependence of Shear Zone Width on H/W
[25] The minimum shear zone width l0min depends on
H/W (Figures 6 and 7). Figure 6 shows l0min for H = 0.53W
versus l0min for H = 0.27W over ranges of n and Ar. If l0min
increased with H/W, the data would plot below the 1:1 line,
but all data for Ar > 1 lie above the 1:1 line, indicating that
l0min generally decreases with increasing H/W.
[26] In Figure 7 we compare the measured values of l0min
with those expected from a simple application of the theory
described by England et al. [1985], assuming that the theory
applies to deformation near the strong region. For a boundary of length H, the expected length scales for the decay of
deformation away from indented and sheared boundaries
are approximately
l0 ¼
2H=W
pffiffiffi
p n
ð6aÞ
l0 ¼
H=W
pffiffiffi ;
2p n
ð6bÞ
and
respectively. For the longer strong region (H = 0.53W) the
measured l0min values fall just above those expected for a
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Figure 4. Time evolution of the parameter l0 defined in Figure 3 for initial indenter obliquity 41°; H =
0.53W; n = 3, 5, and 10; and Ar = 1 (solid line with squares), 3 (dashed line with triangles), 10 (dashed
line with circles), and 30 (dot-dashed line with diamonds). Axis labels show dimensionless distances
above distance scaled to India and the Tarim Basin (W = 2500 km) in kilometers. The e-folding distance,
l0, decreases with increasing penetration and increasing n, and its minimum, l0min, decreases with
decreasing Ar.
sheared boundary (Figure 7b), but for the shorter strong
region (H = 0.27W) the l0min values are close to, but
generally somewhat smaller than, those predicted for an
indented boundary (Figure 7a). Both cases show an approximate n1/2 dependence consistent with (6a) and (6b), with
exceptions when a steady state l0min is not reached (Ar = 1).
The apparent dependence of l0min on H/W thus suggests that
deformation is accommodated more by flow around a small
Figure 5. Minimum shear zone width l0min as a function
of strong region obliquity for n = 3, Ar = 3, and H = 0.53W.
The y axis labels show dimensionless distances above
distance scaled to India and the Tarim Basin (W = 2500 km)
in kilometers. The obliquity angle is the initial angle
between the southern boundary of the calculation space and
the southeast boundary of the strong region.
Figure 6. Minimum shear zone width l0min for a small
strong region (H = 0.27W) versus l0min for a large strong
region (H = 0.53W) for n = 3 (gray symbols), n = 5 (white
symbols), and n = 10 (black symbols) and Ar = 1 (squares),
Ar = 3 (triangles), Ar = 10 (circles), and Ar = 30
(diamonds). Axis labels show dimensionless distances
above distance scaled to India and the Tarim Basin (W =
2500 km) in kilometers. Here
pffiffiffi lmin decreases with n by a
factor of approximately n, but l0min is larger for the
smaller strong region for Ar > 1.
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horizontal shear strain rate, and shortening strain takes up
more deformation than shear does. As the indentation
progresses, the horizontal shear strain rate becomes the
greater, and some deformation previously taken up by
thickening is now taken up by horizontal shear (Figure 8).
Thus with increasing indentation we expect the length scale
to be increasingly better represented by (6b) rather than
(6a), as horizontal shear encourages more localized deformation adjacent to the strong region. For maximum indentation, the ratio of horizontal shear strain rate to vertical
strain rate is greater for the larger than smaller H/W
Figure 7. Minimum e-folding distance (l0min in dimensionless length (top number) and kilometers (bottom
number)) versus n for Ar = 1 (squares), Ar = 3 (triangles),
Ar = 10 (circles), and Ar = 30 (diamonds) with indenter
obliquity 41° and (a) H = 0.27W or (b) H = 0.53W. Expected
decay distances based on equations (6a) and (6b) and
confirmed in thin viscous sheet calculations are shown by
the solid lines. Note that l0min is smaller when n is large and
Ar is small.
strong region than by shear localization along the indenterfacing boundary. Conversely for a large strong region,
deformation is accommodated more by localized shear
strain along the indenter-facing boundary and less by flow
around the strong region.
[27] To test this hypothesis, we examine the evolving
ratio of shear deformation rate to thickening rate in the shear
zone as the indentation progresses (Figure 8). At first the
vertical strain rate near the strong region is larger than the
Figure 8. Ratio of maximum shear strain rate to vertical
strain rate as a function of distance from the strong region
(in dimensionless length W (top number) and kilometers
(bottom number)) at successive times in experiments with
n = 10, Ar = 3, and (a) H = 0.27W and (b) H = 0.53W. Labels
on solid curves represent distance of indenter penetration in
units of W, and each dashed curve is for indentation 0.05W
greater than the solid curve below it. No curve for
indentation of 0.4W is shown in Figure 8b because the
calculation fails to converge before this point.
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Figure 9. Time evolution of l0 in the strain-dependent B calculations with H/W = 0.27. Axis labels
show dimensionless distances above distance scaled to India and the Tarim Basin (W = 2500 km) in
kilometers. (top left) The n = 3, Ar = 1 calculation; (top right) n = 3, Ar = 3; (bottom left) n = 10, Ar = 3;
and (bottom right) n = 10, Ar = 6. Symbols represent values of G, the parameter used to modify B in the
calculations: G = 0.01 (squares), G = 0.03 (triangles), G = 0.1 (circles), and G = 0.3 (diamonds).
(Figure 8), consistent with the observed decrease in l0min for
larger H/W (Figure 7).
4.5. Scaling to the Altyn Tagh Shear Zone
[28] To quantify the extent to which localization adjacent
to the strong region captures the essential physical aspects
of deformation in the earth, we compare our results to
observed deformation between northern Tibet and the Tarim
Basin, where strike-slip motion along the Altyn Tagh fault
accommodates shear between a deforming Tibet and a
strong Tarim Basin [e.g., Avouac and Tapponnier, 1993;
Molnar and Tapponnier, 1981]. We predict l0min for the
Altyn Tagh region by scaling the results of our numerical
experiments to the earth. Because the geometry of the
indenter in our calculations is not easily scaled to the shape
of the Indian plate, we use the minimum shear zone widths
(l0min) from both H/W = 0.27 and 0.53 calculations. In the
first, we use the H/W = 0.27 calculations and scale the
length of the strong region to the length of the Tarim Basin
(1400 km), and the width of the leading edge of the
indenter (W/2) to the width of India between the syntaxes of
the Himalayan arc for which we use 2600 km, the great
circle distance between 35°N, 72°E and 28°N, 98°E. In the
second estimate, we use the H/W = 0.53 calculations and
scale the full indenter width (W) to the width of India.
England and Houseman [1986] found that indentation of a
8 of 15
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DAYEM ET AL.: LOCALIZED SHEAR ALONG A DISCONTINUITY
homogeneous thin viscous sheet with values of n = 10 and
Ar = 3 or n = 3 and Ar = 1 replicates the hypsometric curve
of eastern Asia. Neil and Houseman [1997], following
England and Houseman [1989], however, suggest that Ar
increases locally in areas where the lithosphere may have
been convectively thinned (e.g., beneath Tibet), and they
use n = 3, Ar = 3 or n = 10, Ar = 6. Although we will use
these sets of parameters to scale the width of the calculated
shear zone to that of the Altyn Tagh fault zone, we
acknowledge that the high elevations of Tibet may be a
result of some other process such as lower crustal flow [e.g.,
Clark and Royden, 2000; Royden, 1996].
[29] From the H/W = 0.27 runs we use l0min = 0.11 for n =
3, Ar = 3, and l0min = 0.075 for n = 10, Ar = 6, to estimate
the width of the Altyn Tagh fault zone. Using W/2 =
2600 km, the scaled width of the calculated shear zone
(Wl0min) is 580 km for n = 3, Ar = 3 and 380 km for n = 10,
Ar = 6. For W = 2600 km we use the H/W = 0.53 runs, with
l0min = 0.096 for n = 3, Ar = 3, and l0min = 0.063 for n = 10,
Ar = 6. In this case, the scaled width of the calculated shear
zone is 230 km for n = 3, Ar = 3 and 160 km for n = 10, Ar
= 6. Thus, for a homogeneous sheet with a strong region
mimicking the Tarim Basin, we estimate a shear zone that
could be as narrow as 160 km or as wide as 580 km.
[30] Several types of data allow comparison with these
predicted shear zone widths. Wittlinger et al. [1998] showed
a two-dimensional tomographic image with a 40-km-wide
low-speed zone in the mantle beneath the Altyn Tagh fault.
Structural mapping by Cowgill et al. [2000] indicates that
strike-slip shear is distributed over a zone at least 100 km
wide, but less than 150 km [Cowgill et al., 2003]. Shear
wave splitting measurements suggest straining of the mantle
lithosphere over a zone 100 km wide [Herquel et al.,
1999].
[31] Our prediction of a shear zone that is up to six times
too wide suggests that localization of strain within a
homogeneous thin viscous sheet adjacent to a strong region
fails to describe the actual development of the Altyn Tagh
shear zone. Although vertical heterogeneity of the crust and
upper mantle could play a significant role in modifying
these predictions, it seems likely that one or more processes
that lead to strain- or temperature-dependent weakening,
such as the one considered below, is required to explain the
enhanced strain localization along the Altyn Tagh shear
zone [e.g., Matte et al., 1996; Peltzer and Tapponnier, 1988;
Tapponnier et al., 1986, 2001].
of lithospheric weakening related to deformation, but we
stress that it need not be the only, or even most important,
weakening process.
[33] Concentrated strain has previously been described in
calculations that include shear heating on a fault, and hence
a localized thermal anomaly at the outset of slip on the fault
[Fleitout and Froidevaux, 1980; Thatcher and England,
1998; Yuen et al., 1978]; here we allow the thermal anomaly
to develop as a result of deformation. In the case of shear
heating on a fault, the shear zone broadens over time until
diffusion of heat leads to thermal equilibrium. In our case,
the shear zone narrows over time and would reach thermal
equilibrium if we allowed for diffusion of heat, and if the
heat diffused equaled that produced by shear heating.
5.1. Dissipative Heating in Thin Viscous Sheet
Calculations
[34] To incorporate temperature-dependent strain weakening in the thin viscous sheet calculations, we derive an
expression in which the strength parameter B varies in time
as strain develops. Assuming a constant geothermal gradient
b we write (4) in terms of temperature:
B¼A
1=n
1
bL
ZTL
Q
exp
dT;
nRT
ð7Þ
TB
where TL and TB are the temperature at the base of the
lithosphere and the top of its rheologically strong section,
perhaps at the Moho if the mantle resists deformation
more than the crust [e.g., England, 1983], or the brittleductile transition if the lower crust also strongly resists
deformation.
[35] Heat transfer by conduction may be ignored if the
diffusive timescale is much smaller than the shear heating
timescale. The diffusive timescale td = L2/k where k, the
thermal diffusivity, is 300 Ma (1016 s) for a lithosphere of
thickness L 100 km. We estimate that strain rates in the
Altyn Tagh fault zone are at least an order of magnitude
larger than t1
d : if the 100-km-wide zone accommodates
10 mm a1 displacement, e_ 3*1015 s1. Consequently
we expect that the rate at which the lithosphere warms by
shear heating greatly exceeds the rate at which it cools by
diffusion. Accordingly, the rate of change of temperature T
is related to the work done by deformation:
rCp
5. Strain Localization With Strain-Dependent
Strength Parameter
[32] Increasing temperature caused by shear heating is
one process that leads to locally increased strain rates. Other
processes such as the development of anisotropic crystalline
fabric or the creation of faults in otherwise strong crust, the
reduction in grain size [Karato et al., 1986], or damage in
the form of voids and microcracks [e.g., Bercovici et al.,
2001; Ricard et al., 2001; Ricard and Bercovici, 2003] also
could affect the strength parameter B. We choose to include
shear heating in our calculations as one plausible example
TC3002
@T X
¼
e_ ij t ij ;
@t
i;j
ð8Þ
where r is density and Cp is heat capacity. Differentiating
(7) by time assuming that only temperature changes with
time, substituting (8), and integrating across the lithosphere
leads to an expression relating the time rate of change of B
to the strain rate (Appendix A):
9 of 15
@
ðln BÞ ¼ GE_ 0ð1þnÞ=n ;
@t 0
ð9Þ
DAYEM ET AL.: LOCALIZED SHEAR ALONG A DISCONTINUITY
TC3002
where the primes indicate dimensionless variables, and
G¼
1
V0 1=n
Q
exp
rCp bL AW
nRTB
ð10Þ
is a dimensionless number that describes weakening by
shear heating in our calculations. Strain-dependent weakening
thus is represented in our calculations by implementing (9)
to produce a temporally and spatially varying B. Note that
the degree to which B depends on strain is controlled by G,
so that the calculated deformation field is then controlled by
n, Ar, G, and geometric parameters. In the following
sections we test the effect of strain weakening (by varying
G), apply results to the Altyn Tagh shear zone, and suggest
that a low Moho temperature and a large n are required to
produce a localized shear zone like the Altyn Tagh.
5.2. Calculation Results With Time-Varying Strength
[36] Because the strength parameter B decreases more in
areas of greater strain, areas of high strain rates (namely
those along the southern and western boundaries of the
strong region and adjacent to the corners of the indenter)
deform more than in the constant B calculations. Consequently, strain localization adjacent to the strong region
develops early in these calculations.
[37] As in the constant B runs, shear zone width l0
decreases with increasing n and decreasing Ar. Increased
G leads to a greater decrease in l0 (Figure 9). For n = 10 and
small weakening parameter G, the shear zone width again
reaches a minimum (l0min). In many cases, however, we are
not able to run the calculations to a point in which a clear
minimum in l0 is reached (Figure 9). The existence of a
finite value of l0min is not guaranteed in this formulation
because it would imply that strain weakening ceases after
sufficient deformation.
[38] In several experiments (e.g., n = 10, G = 0.1 and 0.3,
and Ar = 3 and 6 (Figure 9)), l0 increases slightly after l0min
is reached. In these cases, measurable strain rates begin to
develop in the strong region, and correspondingly, strain
rates just outside the strong region decrease. At this stage,
our use of an e-folding distance to describe the width of the
shear zone becomes inapplicable, and in comparisons with
the Altyn Tagh fault zone, we use the smallest value of l0
obtained.
5.3. Scaling to the Altyn Tagh Shear Zone
[39] Recall that we use two different geometries to scale
the calculations to the Tarim Basin and India. The H/W =
0.27 calculations are scaled by the length of the Tarim Basin
(H = 1400 km) and the width of the leading edge of the
indenter (W/2 = 2600 km). The H/W = 0.53 calculations are
scaled by the full indenter width (W = 2600 km). For the
calculations using W = 2600 km, H/W = 0.53, n = 10, and
Ar = 3, for weakening parameter G = 0.01 and 0.03, l0min =
0.04 and 0.03, and we obtain scaled shear zone widths of
100 km and 75 km. In the calculations using W =
5200 km and H/W = 0.27, a minimum l0 is not reached
(not shown), but l0 = 0.02 is attained in some of these
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calculations (e.g., G = 0.3), for which the calculated shear
zone width scales to the Altyn Tagh shear zone width of
100 km. Using Ar = 6 yields slightly wider zones. Thus, if
n = 10 described the rheology of Tibetan lithosphere, relatively small values of the weakening parameter G (0.01–
0.3) would allow for a localized shear zone of width
comparable to that of the Altyn Tagh fault zone. Larger n
presumably implies smaller shear zone width for similar G,
even if these values cannot be attained in our numerical
simulations. If the rheology of Tibet were better described
by n 3, however, larger values of G must be used so that
the shear zone narrows sufficiently during penetration of
the indenter. For values of G between 0.01 and 0.3, values
of l0min < 0.05 could not be reached with our calculation
method (Figure 9).
[40] Assuming that dissipative heating is the primary
process that leads to weakening, we use our estimates of
the weakening parameter G and equation (10) to estimate
the temperature at the top of the strong part of the lithosphere (the Moho or perhaps the brittle-ductile transition),
using values of empirical coefficients A and Q for wet and
dry olivine (n = 3.5) given by Hirth and Kohlstedt [1996].
Extrapolating values of the minimum shear zone width l0min
from Figure 9, we estimate that for n 3– 3.5, G = 0.1 – 0.3
leads to localized strain over a scaled width that may
become comparable to that of the Altyn Tagh shear zone
within a scaled time of less than 40 or 50 Ma. Using (10)
with G = 0.1– 0.3, r = 3.3 103 kg m3, Cp = 1.2 103 J
kg1 K1, b = 10 K km1, L = 100 km, Vo = 40 mm a1,
W = 2600 km, R = 8.3 J K1 mol1, and either A = 4.85 104 1/(MPans) and Q = 535 kJ mol1 for ‘‘dry’’ olivine or
A = 4.89 106 1/(MPans) and Q = 515 kJ mol1 for
‘‘wet’’ olivine, we calculate TB 900 – 950 K (650°C) at
the top of the mantle lithosphere. Note, however, that
Goetze [1978] and Evans and Goetze [1979] point out that
for T 1000 K, dislocation creep, which is described with n
3 – 3.5, does not apply. Although power law creep with n
= 3– 3.5 may lead to a situation where strain localization
occurs in our calculations, it requires a temperature at the
top of the strong layer that is so low that it is inconsistent
with the assumption of dislocation creep.
6. An Argument for n > 3 Rheology
[41] Because large values of G are required for strain to
localize sufficiently during indentation, and hence temperatures lower than those at which n = 3 is applicable to
olivine are required, we reject constitutive laws with n 3.
Significant strain localization in the temperature-dependent
thin viscous sheet calculations is achieved with n 10,
however, and we infer that a more plastic (than viscous)
deformation (approximated by large n) can account for the
deformation of northern Tibet. Goetze [1978] and Evans
and Goetze [1979] presented a flow law, calibrated with
data from laboratory experiments, that applies to highstress, low-temperature deformation. Using their formula,
we show in Appendix B that for temperatures of 700–
1000 K, effective values of n between 8 and 26 are obtained
(n is smaller at higher temperature). For this range of
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DAYEM ET AL.: LOCALIZED SHEAR ALONG A DISCONTINUITY
temperatures G calculated from the high-stress flow law
(Appendix B) is within the range of magnitudes required for
strain localization in the thin sheet calculations with n = 10
(Figure 9). For a temperature at the top of the mantle
lithosphere of T 750 – 900 K and a strain rate e_ 3 1015 s1 (given by 10 mm a1 of shear across a 100 km
wide zone), n is approximately 18 and G = 0.11 (Table B1),
which is within the range that leads to the required degree of
strain localization in our thin viscous sheet calculations.
[42] On the basis of a derivation similar to the one we
show in Appendix A, Stüwe [1998] finds that the temperature change due to shear heating is effectively independent
of the initial temperature, suggesting that an initially cold
lithosphere can accommodate greater total strain than an
initially warm one. Thus the Tarim basin may promote
strain localization not only because it is relatively strong,
but also because it is relatively cold compared to the
lithosphere surrounding it.
[43] Thus we argue that deformation in the upper lithosphere should be more plastic (high n) than it is deeper in
the upper mantle, allowing the development of narrow
zones of shear strain. Because the thin sheet calculations
apply to depth-averaged properties of the lithosphere, we
cannot distinguish between strength in the lower crust and
the upper mantle. We do, however, emphasize that plastic
deformation may arise in the context of a ductile deformation process, and need not be interpreted as strength in the
seismogenic layer [e.g., Flesch et al., 2001]. Stüwe et al.
[2008] also suggest that large n may describe the lithosphere
of northern Tibet. In studying topographic profiles using a
coupled thin viscous sheet and erosion model, they suggest
that large n or rheological contrasts may explain the steep
topographic gradients between Tibet and the Tarim basin.
[45] Strain-dependent weakening, as would develop because of shear heating, does allow the zone of localized
strain to narrow to a width that is comparable to that of the
Altyn Tagh shear zone. The calculations suggest that for
such strain localization to occur sufficiently rapidly to apply
to the Altyn Tagh shear zone, the lithosphere must deform
more plastically than is described by dislocation creep of
olivine: n in (1) must be much larger than 3 – 3.5. Values of
n on the order of 10 are consistent with low-temperature
plasticity laws determined for olivine, as described by
Evans and Goetze [1979]. The rate of thermal dissipation
calculated for this law provides a process by which the
observed localization on the Altyn Tagh shear zone may be
predicted.
Appendix A
[46] We derive an expression for the time rate of change
of the strength parameter B for use in the thin viscous sheet
calculations. Using the constitutive relationship (1) and the
expression for the temperature change due to shear heating
(8), we write
rCp
@T
¼ BE_ ð1þnÞ=n
@t
ðA1Þ
to compute the effect of thermal dissipation on temperature
from one time step to the next. We now consider the effect
of a change in temperature on B using (7) and assuming a
constant increment to the temperature over the relevant
depth interval (in fact the temperature increase would be
concentrated where the stresses are largest):
dB ¼ A1=n
7. Conclusions
[44] Calculations of continuous deformation of a thin
viscous sheet show that shear localization develops along
a discontinuity in strength that is obliquely oriented with
respect to an indenter. Shear strain rate near a strong region
is more localized for larger rheologic exponent n and
smaller Argand number Ar. In addition, as the indenter
penetrates into the viscous material, strain rates become
more localized near the strong region as deformation is
progressively taken up more by horizontal shear and less by
thickening. Maximum localization is produced when the
strength boundary is approximately 45° oblique to the
indenter, but depends only weakly on the orientation for
orientations between 35° and 55°. A narrower shear zone
develops near a large strong region than a small one,
indicating that deformation is taken up more by horizontal
shear in front of a large strong region, but by thickening in
front of and flow around a small one. The width of the shear
zone for a homogeneous thin viscous sheet with a strong
region embedded in it, however, does not approach the
narrowness of the Altyn Tagh zone. Hence some additional
process must facilitate further localization of strain within
the lithosphere [e.g., Matte et al., 1996; Peltzer and
Tapponnier, 1988; Tapponnier et al., 1986, 2001].
TC3002
ZTL Q
Q
exp
dT :
exp
nRðT þ dT Þ
nRT
1
bL
TB
ðA2Þ
[47] We use a Taylor Series expansion on the term
containing the temperature increment,
dB ¼ A1=n
1
bL
ZTL Q
dT
Q
exp
1
exp
dT :
nRT
T
nRT
TB
ðA3Þ
We then factor and simplify,
dB ¼ A1=n
1
bL
ZTL
Q
QdT
1
dT ;
exp
exp nRT
nRT 2
ðA4Þ
TB
and again apply a Taylor series expansion:
11 of 15
dB
Q
¼ A1=n
dT
bLnR
ZTL
TB
1
Q
exp
dT :
T2
nRT
ðA5Þ
DAYEM ET AL.: LOCALIZED SHEAR ALONG A DISCONTINUITY
TC3002
TC3002
an indentation rate Vo and a distance scale W, so
that e_ = Vo =W :
@
1
Vo 1=n _ 0ð1þnÞ=n
Q
E
:
ðln
BÞ
¼
exp
@t0
rCp bL AW
nRTB
ðA10Þ
Thus the change in B is controlled by one dimensionless
number G:
@
0ð1þnÞ=n
ðln BÞ ¼ GE_
;
@t 0
ðA11Þ
1
Vo 1=n
Q
:
exp
rCp bL AW
nRTB
ðA12Þ
where
G¼
Appendix B
Figure B1. Stress-strain rate relationship from equation
(B2) for temperatures of 700, 800, 900, and 1000 K (solid
lines) and linear fits (dashed lines). The inverse of the slope
is equal to the apparent n value.
[49] We suggest that a low, precollision Moho temperature of Northern Tibet shifted deformation from the commonly assumed power law creep regime, as in (1) or (3), to
the high-strength regime described by Goetze [1978] and
Evans and Goetze [1979]. From their laboratory experi-
Making the substitution x = 1/T (so that dx = dT/T2), the
integral above is
dB
Q
¼ A1=n
dT
bLnR
1=TL
Z
exp
1=TB
1=TL
Qx
1
Qx
dx ¼ A1=n
exp
nR
bL
nR 1=TB
ðA6Þ
or
dB
1
Q
Q
¼ A1=n
exp
exp
:
dT
bL
nRTB
nRTL
ðA7Þ
We ignore the second term, which for sensible values of Q,
n, and R is roughly 500 times smaller than the first term:
dB
1
Q
¼ A1=n
exp
:
dT
bL
nRTB
ðA8Þ
[48] Combining (A1) and (A8) gives
@
1 _ ð1þnÞ=n
Q
E
exp
:
ðln BÞ ¼ A1=n
@t
rCp bL
nRTB
ðA9Þ
Using the dimensionless time units of the finite element
code of Houseman and England [1986], on the basis of
Figure B2. Stress calculated from the high-strength flow
law (B2) versus 1/T for five chosen strain rates (in units of
s1): e_ = 1015 (dashed black line), e_ = 3 1015 (solid
black line with plus symbols), e_ = 1014 (dotted black line),
e_ = 1013 (solid gray line), and e_ = 1012 (dashed gray
line). Strain rate along the Altyn Tagh shear zone is
approximately 3 1015 s1.
12 of 15
DAYEM ET AL.: LOCALIZED SHEAR ALONG A DISCONTINUITY
TC3002
Table B1. Apparent Q, A, and G Values for Given Strain Rate,
Temperature Range, and Apparent na
Strain Rate
(s1)
T
(K)
n
Q
(J mol1 K1)
1/A(1/n)
(1 s1 Pa1)
G
1E-15
3E-15
1E-14
1E-13
1E-12
1E-15
3E-15
1E-14
1E-13
1E-12
1E-15
3E-15
1E-14
1E-13
1E-12
850 – 1000
850 – 1000
850 – 1000
850 – 1000
850 – 1000
750 – 900
750 – 900
750 – 900
750 – 900
750 – 900
650 – 800
650 – 800
650 – 800
650 – 800
650 – 800
12
12
12
12
12
18
18
18
18
18
26
26
26
26
26
9.97E+5
8.18E+5
6.68E+5
4.99E+5
3.99E+5
5.53E+5
4.94E+5
4.49E+5
3.89E+5
3.29E+5
4.11E+5
3.89E+5
3.67E+5
3.24E+5
3.02E+5
4.34E+4
3.57E+5
2.39E+6
1.46E+7
8.89E+7
6.05E+7
5.70E+7
5.33E+7
1.27E+8
3.05E+8
2.48E+8
2.38E+8
2.27E+8
5.64E+8
5.17E+8
0.03
0.03
0.05
0.04
0.09
0.19
0.11
0.07
0.11
0.16
0.22
0.19
0.16
0.29
0.23
a
Q and A values are determined by best fit straight lines of 1/T and ln s in
Figure B2 from equation (B2), and apparent n is estimated from Figure B1.
G is estimated from equation (10).
ments of deformation of olivine they show that for large
differential stress, deformation is predicted well by
"
e_ 11
Ha
¼ e_ o exp RT
#
s11 s33 2
1
;
s0
ðB1Þ
where e_ 0 = 5.7 1011 s1, so = 8.5 GPa, and Ha =
525 kJ mol1 are experimentally derived parameters
[Goetze, 1978], and s11 and s33 are maximum and minimum
stresses. Molnar and Jones [2004] rewrite this as a constitutive law of the form of (1):
"
pffiffiffi 1=2 #
e_ ij
3e_ o
RT
t ij ¼ pffiffiffi so 1 ln
:
Ha
2E_
E_ 3
ðB2Þ
[50] Approximating conditions in a shear zone, we let
e_ ij = E_ (which introduces but a small error) and plot the
stress calculated from (B2) versus strain rate for T = 700 K,
800 K, 900 K, and 1000 K (Figure B1). The empirical
relationships defined by (B2) may also be interpreted using
_
the constitutive equation (1) and the assumption that e_ ij = E.
At high strain rates, log t ij and log e_ ij are approximately
linearly related with slope 1/n (Figure B1). For temperatures
up to 1000 K and strain rates appropriate for the Altyn
Tagh shear zone (3 1015 s1), the apparent values of n
in (1) range from n = 8 for T = 1000 K to n = 26 for T =
700 K (Figure B1).
[51] The empirical law (B2) thus may be approximated
using a power law with a relatively large n value. We test
the implications of using this high-stress constitutive law
(B2) in our thin sheet calculations by estimating
G. Defining
a deviatoric stress scale t o by t ij t o exp Qeff =nRT we
TC3002
calculate the effective values of the power law constant
A = e_ ij =t no in (3) assuming that q = t ij and an effective
activation energy Qeff from the best fit line for the highstress flow law, for the strain rate and temperature range
indicated in Figure B2 and Table B1. For the same temperatures and strain rates, we estimate G from (10) (Table B1).
Calculated values of G are similar to those needed to
produce the degree of strain localization observed in the
Altyn Tagh shear zone. For example, for T 750– 900 K, n =
18, and e_ 3*1015 s1, G = 0.11. Note that for temperatures
below 1000 K stress is relatively insensitive to strain rate
(apparent values of n are large) (Figure B2).
[52] Raterron et al. [2004] present a more complex version
of (B1), which they find is valid for temperatures 800–
1000 K. We perform the same analysis as described above
using their high-stress flow law and find apparent values
of n ranging from 11 for T = 1000 K to 53 for T = 700 K,
and values of G 0.6 –1.0. We note, however, that in their
given temperature range, their flow law seems to apply to
strain rates that are 1015 to 1010 s1, which are larger
than those typically found in the earth. Thus we use the
Goetze [1978] and Evans and Goetze [1979] formula and
parameterization to place constraints on values of n and G.
Notation
Key Calculation Parameters
Ar Argand number.
B strength parameter, time constant (section 4)
time variable (section 5).
Bl strength parameter of strong region.
Bo strength parameter of calculation space outside
strong region.
n rheological parameter, relates stress to a power
strain rate.
G shear heating weakening parameter.
l0 width of shear zone.
0
l min minimum width of shear zone.
Other Parameters and Variables
A empirical constant.
Cp heat capacity.
g gravitational constant.
H length of strong region.
L thickness of thin viscous sheet (or lithosphere).
Q empirical activation energy.
R universal gas constant.
T temperature.
TL temperature at base of lithosphere.
TB temperature at top of rheologically strong part
lithosphere.
td diffusive timescale.
Vo indentation rate.
W length of indenter.
b geothermal gradient.
e_ ij strain rate tensor.
E_ second invariant of strain rate tensor.
k thermal diffusivity.
q second invariant of stress tensor.
rc crustal density.
13 of 15
or
of
of
of
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DAYEM ET AL.: LOCALIZED SHEAR ALONG A DISCONTINUITY
rm mantle density.
t ij vertically averaged deviatoric stress.
TC3002
[53] Acknowledgments. We thank Todd Ehlers, Leigh Royden, and
Kurt Stüwe for especially thorough reviews. This research was supported in
part by the National Science Foundation under grant EAR-0337509 and by
a NASA Earth Systems Science Graduate fellowship (K.E.D.).
References
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plate deformation, Tectonophysics, 39, 501 – 514,
doi:10.1016/0040-1951(77)90150-0.
Bendick, R., R. Bilham, J. Freymueller, K. Larson, and
G. Yin (2000), Geodetic evidence for a low slip
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K.
E.Dayem
and P. Molnar, Department of
Geological Sciences, University of Colorado, Campus
Box 399, Boulder, CO 80309, USA. (dayem@colorado.
edu; [email protected])
G. A. Houseman, School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK.
([email protected])