TECTONICS, VOL. 28, TC3002, doi: /2008tc002264, 2009

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1 Click Here for Full Article 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 Molnar 1 Received 17 January 2008; revised 31 December 2008; accepted 16 March 2009; published 19 May [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,, doi: /2008tc 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 /09/2008TC002264$12.00 TECTONICS, VOL. 28,, doi: /2008tc002264, 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- 1of15

2 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 a 1 to 50 mm a 1 [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 a 1 ) is estimated from offset uaternary glacial and fluvial landforms [Mériaux et al., 2004]. Relatively slow estimated rates include: 9 mma 1 determined from GPS measurements [Bendick et al., 2000; Shen et al., 2001; Wallace et al., 2004]; 11 ± 5 mm a 1 determined from repeated interferometric synthetic aperture radar measurements [Elliott et al., 2008]; 10 mm a 1 from uaternary terrace offsets and GPS data [Cowgill, 2007; Zhang et al., 2007]; mm a 1 from paleoseismic data [Washburn et al., 2001, 2003]; <10 mm a 1 from post early Miocene offsets [Yue et al., 2004]; and mm a 1 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 2of15

3 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 V o 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 : t ij ¼ B _E ð1=nþ 1 _e ij ; ð1þ where n is a rheological parameter relating strain rate to deviatoric stress, B is a spatially variable viscosity coefficient, and _E =(_e ij _e ij ) 1/2 is the second invariant of the strain rate tensor: _e ij ¼ i j ; ð2þ i Figure 2. Maximum shear strain rate in the horizontal plane in units of V o /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 V o /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. 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 where ~u is horizontal velocity and x i and x j measure distance in the x and y directions. If dislocation creep dominates deformation, the constitutive law can be written as _e ij ¼ Aq n 1 t ij exp ; ð3þ RT where q is the second invariant of the deviatoric stress tensor, A and (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 B ¼ A 1=n 1 L Z L 0 exp dz; nrt 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 V o. Strain rates are thus scaled by V o /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: ð4þ Ar ¼ gr cð1 ðr c =r m ÞÞL ; ð5þ B o ðv o =LÞ 1=n where g is gravitational acceleration, r c is crustal density, r m is mantle density, and B o 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. 3of15

4 [12] To represent the strong Tarim Basin block in the calculation space, we include an area in which the strength coefficient B l B o. 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 (B l ) and the rest of the sheet (B o ). 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_e ij 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 B l in the strong region to be 1000, 10, 4, and 2 times greater than B o in the surrounding regions for n = 1, 3, 5, and 10, respectively, so that the effective contrast in strain rate (which scales with (B l /B o ) 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, p ffiffiwith the e-folding distance inversely proportional to n. They also show that for a strike-slip shear boundary, the deformation field decays 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 l 0, 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 l 0 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 V o =50kmMa 1. Dimensional lengths, therefore, are estimated by multiplying nondimensional lengths by 2500 km. Dimensional strain rates are estimated by multiplying nondimensional rates by 3*10 16 s 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 B _E (1/n 1) 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 Decrease of Shear Zone Width With Indenter Penetration [21] As the indenter penetrates into the thin viscous sheet, the width of the shear zone (l 0 ) decreases, indicating that shear strain rate progressively localizes along the boundary 4of15

5 Figure 3. The perpendicular gradient of the component of velocity (in units of V o /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 V o =50km Ma 1 ). Labels on curves represent the distance of indenter penetration in units of W. The e-folding distance l 0 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, l 0 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 l 0 decreases until it levels out at a minimum, l 0 min (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 l 0 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 Dependence of Shear Zone Width on Strong Region Orientation, n, and Ar [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 (B o ) 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 (B _E (1/n 1) ) 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, enhances the value of _E, and hence leads to a decreased 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 l 0 min depends on H/W (Figures 6 and 7). Figure 6 shows l 0 min for H = 0.53W versus l 0 min for H = 0.27W over ranges of n and Ar. Ifl 0 min 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 l 0 min generally decreases with increasing H/W. [26] In Figure 7 we compare the measured values of l 0 min 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 and l 0 ¼ 2H=W p p ffiffi n l 0 ¼ H=W p 2p ffiffi ; n ð6aþ ð6bþ respectively. For the longer strong region (H = 0.53W) the measured l 0 min values fall just above those expected for a 5of15

6 Figure 4. Time evolution of the parameter l 0 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, l 0, decreases with increasing penetration and increasing n, and its minimum, l 0 min, decreases with decreasing Ar. sheared boundary (Figure 7b), but for the shorter strong region (H = 0.27W) the l 0 min values are close to, but generally somewhat smaller than, those predicted for an indented boundary (Figure 7a). Both cases show an approximate n 1/2 dependence consistent with (6a) and (6b), with exceptions when a steady state l 0 min is not reached (Ar =1). The apparent dependence of l 0 min on H/W thus suggests that deformation is accommodated more by flow around a small Figure 5. Minimum shear zone width l 0 min 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 l 0 min for a small strong region (H = 0.27W) versus l 0 min for a large strong region (H = 0.53W) forn = 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 p ffiffi l min decreases with n by a factor of approximately n, but l 0 min is larger for the smaller strong region for Ar >1. 6of15

7 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 (l 0 min 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 l 0 min 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. 7of15

8 Figure 9. Time evolution of l 0 in the strain-dependent B calculations with H/W = 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 l 0 min for larger H/W (Figure 7) 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 l 0 min 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 (l 0 min) 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 8of15

9 homogeneous thin viscous sheet with values of n = 10 and Ar =3orn = 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 =3orn = 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] FromtheH/W = 0.27 runs we use l 0 min =0.11forn = 3, Ar = 3, and l 0 min = 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 (Wl 0 min) 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 l 0 min = for n =3,Ar = 3, and l 0 min = 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]. 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 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 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: B ¼ A 1=n 1 bl Z TL T B exp dt; nrt where T L and T B 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 t d = L 2 /k where k, the thermal diffusivity, is 300 Ma (10 16 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 t d 1 :ifthe100-km-wide zone accommodates 10 mm a 1 displacement, _e 3*10 15 s 1. 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: ¼ X i;j _e ij t ij ; where r is density and C p 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 0 ðln BÞ ¼ G _E 0ð1þnÞ=n ; ð7þ ð8þ ð9þ 9of15

10 where the primes indicate dimensionless variables, and G ¼ 1 V 1=n 0 exp rc p bl AW nrt B 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. Notethat thedegreetowhichb 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 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 l 0 decreases with increasing n and decreasing Ar. Increased G leads to a greater decrease in l 0 (Figure 9). For n = 10 and small weakening parameter G, the shear zone width again reaches a minimum (l 0 min). In many cases, however, we are not able to run the calculations to a point in which a clear minimum in l 0 is reached (Figure 9). The existence of a finite value of l 0 min 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)), l 0 increases slightly after l 0 min 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 l 0 obtained 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, l 0 min = 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 l 0 is not reached (not shown), but l 0 = 0.02 is attained in some of these 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 ( ) 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 l 0 min < 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 for wet and dry olivine (n = 3.5) given by Hirth and Kohlstedt [1996]. Extrapolating values of the minimum shear zone width l 0 min from Figure 9, we estimate that for n 3 3.5, G = 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 = , r = kg m 3, C p = J kg 1 K 1, b =10Kkm 1, L = 100 km, V o =40mma 1, W = 2600 km, R =8.3JK 1 mol 1, and either A = /(MPa n s) and = 535 kj mol 1 for dry olivine or A = /(MPa n s) and = 515 kj mol 1 for wet olivine, we calculate T B 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 = 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 K, effective values of n between 8 and 26 are obtained (n is smaller at higher temperature). For this range of 10 of 15

11 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 K and a strain rate _e s 1 (given by 10 mm a 1 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. 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]. [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 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 ¼ B _E ð1þnþ=n 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 A 1=n 1 bl Z TL T B exp exp dt: nrðt þ dtþ nrt ða2þ [47] We use a Taylor Series expansion on the term containing the temperature increment, db ¼ A 1=n 1 bl Z TL T B exp nrt We then factor and simplify, db ¼ A 1=n 1 bl Z TL T B exp nrt 1 dt T exp nrt exp dt nrt 2 and again apply a Taylor series expansion: db dt ¼ A 1=n blnr Z TL T B 1 1 T 2 exp dt: nrt dt; dt: ða3þ ða4þ ða5þ 11 of 15

12 an indentation rate V o that _e = V o = W ðln BÞ ¼ rc p bl V o AW and a distance scale W, so 1=n E _ 0ð1þnÞ=n exp nrt B : ða10þ Thus the change in B is controlled by one dimensionless 0 ðln BÞ ¼ G _E 0ð1þnÞ=n ; ða11þ where G ¼ 1 V 1=n o exp : ða12þ rc p bl AW nrt 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. Appendix B [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/t 2 ), the integral above is db dt ¼ A 1=n blnr Z 1=TL 1=T B exp x dx ¼ A 1=n 1 nr bl x exp nr 1=TL 1=T B ða6þ or db 1 dt ¼ A 1=n bl exp exp : ða7þ nrt B nrt L We ignore the second term, which for sensible values of, n, and R is roughly 500 times smaller than the first term: db 1 dt ¼ A 1=n bl exp : ða8þ nrt B [48] Combining (A1) and ðln BÞ 1 ¼ A 1=n E rc p bl _ ð1þnþ=n exp : ða9þ nrt B 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 s 1 ): _e =10 15 (dashed black line), _e = (solid black line with plus symbols), _e =10 14 (dotted black line), _e =10 13 (solid gray line), and _e =10 12 (dashed gray line). Strain rate along the Altyn Tagh shear zone is approximately s of 15

13 Table B1. Apparent, A, and G Values for Given Strain Rate, Temperature Range, and Apparent n a Strain Rate (s 1 ) T (K) n (J mol 1 K 1 ) 1/A (1/n) (1 s 1 Pa 1 ) G 1E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E a 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 ¼ _e o exp H a RT 1 s # 11 s 2 33 ; ðb1þ s 0 where _e 0 = s 1, s o = 8.5 GPa, and H a = 525 kj mol 1 are experimentally derived parameters [Goetze, 1978], and s 11 and s 33 are maximum and minimum stresses. Molnar and Jones [2004] rewrite this as a constitutive law of the form of (1): " t ij ¼ _e ij p _E ffiffi s o 1 RT p ffiffi 1=2 # 3 _eo ln : ðb2þ 3 H a 2 _E [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 ( s 1 ), the apparent values of n in (1) range from n = 8 for T = 1000 K to n =26forT = 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 we eff = nrt calculate the effective values of the power law constant A = _e ij = t n o in (3) assuming that q = t ij and an effective activation energy eff 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 K, n = 18, and _e 3*10 15 s 1, G = 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 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 We note, however, that in their given temperature range, their flow law seems to apply to strain rates that are to s 1, 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) or time variable (section 5). B l strength parameter of strong region. B o strength parameter of calculation space outside of strong region. n rheological parameter, relates stress to a power of strain rate. G shear heating weakening parameter. l 0 width of shear zone. l 0 min minimum width of shear zone. Other Parameters and Variables A empirical constant. C p heat capacity. g gravitational constant. H length of strong region. L thickness of thin viscous sheet (or lithosphere). empirical activation energy. R universal gas constant. T temperature. T L temperature at base of lithosphere. T B temperature at top of rheologically strong part of lithosphere. t d diffusive timescale. V o 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. crustal density. r c 13 of 15

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