Validity of the thin viscous sheet approximation in models of continental collision

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2010jb007770, 2011 Validity of the thin viscous sheet approximation in models of continental collision Matthew C. Garthwaite 1 and Gregory A. Houseman 1 Received 8 June 2010; revised 5 October 2010; accepted 10 December 2010; published 8 February [1] The two dimensional thin viscous sheet approximation is widely used to describe large scale continental deformation. It treats the lithosphere as a fluid layer in which deformation results from a balance between buoyancy forces and tectonic boundary conditions. Comparisons between two dimensional thin sheet and full three dimensional solutions of a simple indenter model show that appreciable differences exist, especially when the indenter half width, D, is of the same order as the thickness of the deforming layer, L (i.e., D/L 1). These differences are amplified by increasing the power law exponent of the viscous constitutive law (n) but decrease as the Argand number (Ar) is increased. The greatest differences between two dimensional and three dimensional solutions are found at the indenter corner, where the thin sheet consistently overestimates vertical strain rates. Differences between strain rates at the corner may be 50% or greater for small Argand numbers. Other differences arise because a lithospheric root zone is formed in the three dimensional solutions and vertically averaged strain rate is decreased in regions close to the indenter. This effect is absent from thin sheet calculations since the thickness of the load bearing layer is assumed constant. In general, the thin viscous sheet approximation provides a reasonably accurate estimate of long wavelength deformation for D/L as low as 1 if n is less than 3. However, even at large D/L the solutions may be inaccurate close to strain rate concentrations at the indenter corners where horizontal gradients of deformation are large. Citation: Garthwaite, M. C., and G. A. Houseman (2011), Validity of the thin viscous sheet approximation in models of continental collision, J. Geophys. Res., 116,, doi: /2010jb Introduction 1 School of Earth and Environment, University of Leeds, Leeds, UK. Copyright 2011 by the American Geophysical Union /11/2010JB [2] The process of continental collision produces some of the most dramatic topographic features on Earth. The best example of this is the Tibetan Plateau, which has formed as a consequence of the India Asia collision. Northward indentation of the Eurasian continent by the Indian shield has resulted in 2500 km of shortening and major internal deformation of the Eurasian lithosphere [Houseman and England, 1993]. The resulting plateau is an uplifted topographic structure with low relief and thickened crust. In the case of Tibet, the plateau has an average elevation of 5 km above sea level and extends over a broad region. It is widely agreed that plate tectonic theory can not explain the broad patterns of deformation observed within continental regions [e.g., Molnar and Tapponnier, 1975]. An alternative description of the deformation represents the lithosphere as a continuum which deforms viscously in response to the balance between lateral and gravitational forces. Bird and Piper [1980] originally used the finite element method to model deformation of southern California under these conditions. They approximated the crust as a thin layer within which horizontal flow velocities are invariant with depth and upon which the mantle imparts no shear tractions. England and McKenzie [1982] applied this thin viscous sheet (TVS) approach directly to the continental collision problem. Further, they incorporated the major gravitational buoyancy forces arising from crustal thickness variations within a two dimensional formulation. Indentation of the TVS is represented by a velocity boundary condition on the periphery of the sheet. They used a depth averaged rheological model for the layer to describe the deformation of broad regions whose horizontal dimension is much greater than the layer thickness. Sonder and England [1986] show that the vertically averaged rheology can be well approximated by a single power law relationship. Thus England and McKenzie [1982] describe the average mechanical behavior of the lithosphere in terms of only two variable parameters: the power law exponent (n) and the Argand number (Ar). The Argand number describes the impact of buoyancy forces on deformation relative to the viscous stresses imparted by the indenting boundary condition. For greater Argand numbers, crustal thickness variations produce greater internal stresses, and consequently the formation of a topographic plateau and thickened crust adjacent to the indenter is resisted. However if the Argand number is 1of13

2 sufficiently small, very large crustal thicknesses may be attained [England and McKenzie, 1982]. Further studies used the TVS approximation to describe aspects of the observed deformation in the India Asia collision zone [e.g., Houseman and England, 1986, 1993; England and Houseman, 1985, 1986, 1989]. Broad consistency between observations of topography and model predictions was achieved for the cases where n =3,Ar = 1, and n = 10, Ar = 3 [England and Houseman, 1986]. In addition, a TVS model was used to describe the effect of a rigid inclusion within the sheet [England and Houseman, 1985], syn convergent extension within a thickened plateau by means of a locally increased Argand number [England and Houseman, 1989], and the lateral escape of material toward a stress free boundary [Houseman and England, 1993]. [3] One problem with the TVS approximation is that strain rates are assumed constant with depth. As a result, shortening processes such as folding or thrusting are not explicitly represented. Other proposed models address different aspects of deformation within the collision problem: lateral escape [e.g., Tapponnier et al., 1982; Davy and Cobbold, 1988; Avouac and Tapponnier, 1993], folding and thrusting [e.g., Davy and Cobbold, 1988; Braun, 1993; Burg and Schmalholz, 2008], flow of a weak lower crustal layer [e.g., Avouac and Burov, 1996; Royden et al., 1997; Cook and Royden, 2008], lower crustal exhumation [e.g., Beaumont et al., 2001], and gravity driven flow [e.g., Copley and McKenzie, 2007]. Medvedev and Podladchikov [1999] define the England and McKenzie [1982] TVS formulation as a pure shear approach and go on to summarize other TVS formulations that may be applied to geodynamic problems. In the simple shear approach, deformation is driven solely by shear tractions which act on the base of the layer. With this condition, gradients of gravitational potential energy are balanced against basal shear stress, and deformation varies with depth in the model. Ellis et al. [1995] compared deformation driven by either basal or lateral forces. They found that for laterally driven models the deformation length scale depends on the length scale of the applied velocity boundary condition, but in basally driven models it depends on the degree of crust mantle coupling. They concluded that both types of forcing are capable of reproducing the deformation observed in the India Asia collision. On a global scale, Ghosh et al. [2008] find that basal shear tractions imparted on a thin sheet by long wavelength mantle circulation are required to reproduce the present day deviatoric stress field in the presence of known topography. Therefore at very large length scales basal shear is an essential component of the force balance. [4] Further refinements of the TVS methodology are possible: Medvedev and Podladchikov [1999] included subhorizontal rheological layering within the layer while also allowing simultaneous application of velocity or traction conditions to all external model boundaries. This extended TVS formulation allows for more accurate description of three dimensional (3 D) deformation (such as folding) while maintaining the efficient computation of the two dimensional (2 D) formulation. Yet the classic pure shear 2 D TVS formulation continues to be widely used for modeling tectonic deformation [e.g., Flesch et al., 2001; Robl and Stüwe, 2005; Whitehouse et al., 2005; Jiménez Munt et al., 2005; England and Molnar, 2005; Vergnolle et al., 2007; Dayem et al., 2009]. The question how accurate is the TVS approximation? is therefore of continuing interest. In this paper we explore the differences between 2 D TVS calculations and analogous 3 D solutions of the viscous flow equations. We model the classic indentation problem in which the lithosphere deforms in response to gravitational body forces and a convergent boundary condition, a test problem inspired by continental collision. Here we restrict our attention to a system in which the rheological parameters are constant. This enables us to isolate and explain the basic differences between the TVS approximation and a full 3 D solution without additional complications, such as internal viscosity layering. [5] We use the 2 D Basil program, which is based on the pure shear TVS formulation of England and McKenzie [1982] but uses the Lagrangian frame finite element method implemented by Houseman and England [1986]. In 3 D we use the Oregano program described by Houseman and Gemmer [2007]. The Basil program runs on an ordinary desktop computer, whereas Oregano uses the Message Passing Interface (MPI) to run on a multiprocessor machine. Both programs use a velocity pressure formulation of the governing equations, which are solved using the conjugate gradient method on a triangular (2 D) or tetrahedral (3 D) Lagrangian mesh with quadratic interpolation of the velocity field and linear interpolation of the pressure field. Deformation is advanced through time using a two step Runga Kutta time stepping algorithm. 2. Model Formulation [6] The continental lithosphere is assumed to deform over geological time by creeping flow. Although different deformation mechanisms may apply at different levels within the lithosphere, we assume here that the overall effect of these different mechanisms (elastic and viscous) is represented over long time periods by continuum deformation. For deformation occurring at geological strain rates, change of momentum is negligible [England and McKenzie, 1982] and the full 3 D force balance j g iz ¼ 0 where s ij is the ijth component of the total stress tensor, r is density, g is the acceleration due to gravity, and d iz is the Kronecker delta (z is positive upward). The indices i and j refer to each of the cartesian coordinate directions x, y,andz and the convention of summation over repeated indices applies. The stress tensor can be expressed in terms of the deviatoric stress (t) and the pressure (or mean total stress, p = 1 3 s kk): ij ¼ ij þ p ij A non Newtonian (power law) viscous constitutive law is invoked to relate deviatoric stress to strain rate ( _"): ð1þ ð2þ ij ¼ B _E ð1 n 1Þ _" ij ð3þ 2of13

3 which depends on the power law exponent n and the viscosity coefficient B [England and McKenzie, 1982]. Strain rates are defined in terms of the velocity gradients: _" ij ¼ 1 i i and _E is the second invariant of the strain rate tensor (again assuming summation over repeated indices): _E ¼ ð4þ 1 _" ij _" 2 ij ð5þ Incompressibility of the continuum velocity field requires that " zz ¼ _" xx þ _" yy Our 2 D calculations make use of the TVS approximation as formulated by England and McKenzie [1982]. Assuming that there are no vertical gradients of horizontal velocity and no shear tractions on the upper and lower surfaces, (3) and (1) become ð6þ ¼ B _E ð1 n 1Þ zz where a and b only refer to the two horizontal coordinate directions x and y, overbars indicate vertical averages, and B represents the depth averaged effective viscosity of the layer. The coefficient B provides a convenient way of parameterizing lithospheric rheology without explicitly representing variation of temperature or material properties with depth. It is assumed that horizontal stresses are balanced within a layer of constant thickness L. Neglecting vertical shear tractions and assuming the lithosphere is locally in isostatic equilibrium, the horizontal gradients of vertically averaged vertical stress can be expressed zz ¼ g c 2L 1 c= ð8þ ð9þ terms of the indentation velocity U 0, mantle density r m, and layer thickness L: ðx; y; z; SÞ ¼ Lx ; ð y ; z ; S Þ u ¼ U 0 u t ¼ ðl=u 0 Þt _" ¼ ðu 0 =LÞ_" ¼ BU ð 0 =LÞ 1 n ¼ m ð11þ where primes indicate dimensionless variables. Substituting (9) into (8) and @ ¼ Ar ð12þ in which horizontal gradients of deviatoric stress are balanced by horizontal gradients of gravitational potential energy caused by variations in crustal thickness. The relative importance of buoyancy forces in the evolving deformation is determined by the Argand number, Ar [England and McKenzie, 1982]: Ar ¼ g clð1 c = m Þ BU ð 0 =LÞ 1 n ð13þ The formulation used in our 3 D calculations is conceptually simpler since the viscous constitutive law includes the full deviatoric stress tensor (3). Substituting (2) into the 3 Dforce balance (1) and nondimensionalizing i iz ¼ 0 ð14þ where x is the dimensionless ratio of buoyancy derived stress to indentation derived stress: ¼ g ml BU ð 0 =LÞ 1 n ð15þ The body force coefficient x plays a similar role in the 3 D formulation as the Argand number in 2 D. For consistent comparison of 2 D and 3 D experiments we require Ar ¼ c 1 c m m ð16þ where S is the thickness of a low density crustal layer, and r c and r m are the (assumed uniform) densities of crust and mantle layers, respectively. The crustal thickness evolves with time in a Lagrangian reference frame based on the local divergence of the horizontal flow field ¼ r h u ð10þ where u =(U x, U y ) represents the horizontal velocity field. In setting up numerical models of lithospheric deformation, it is convenient to nondimensionalize the physical parameters in 3. Numerical Experiments [7] We choose a simple geometry for the experiments, similar to that used by Houseman and England [1986] (Figure 1). Table 1 gives values of physical parameters used in the dimensionalization of the calculations. In the dimensionless length scale, the half width of the indenter is D = D/L. The undeformed model lithosphere is represented by a box with edges of dimensionless length 4D in both the x and y directions and unit thickness in the z direction. The solution domain is one half of a symmetrical area, with x =0 as a plane of symmetry bisecting the indenter. The indenter 3of13

4 Figure 1. (a) Plan view of 2 D model domain and boundary conditions (where U i and T i denote the velocity and traction components parallel to the i direction, respectively). The grey lines graphically describe the indenting velocity function U y imposedonthey = 0 boundary as described in the text. Hatching indicates a rigid boundary condition. (b) Perspective view showing additional boundary conditions for the 3 D model domain. The boundary conditions shown in Figure 1a also apply to the 3 D model and are invariant in the z direction. The surface at z = 0.65 corresponds to the crust mantle boundary (Moho). All quantities are dimensionless. is represented by the nondimensionalized velocity function on y =0: U y ðþ¼ x 8 4D 0 x D >< h 4D cos 2 i ðx D Þ D x 2D 2D >: 0 2D x 4D ð17þ The dimensionless indentation velocity is defined as proportional to D in order that experiments with different values of D/L have the same relative indentation at the same dimensionless time. With this choice the indenting boundary would travel across the whole solution domain in a dimensionless time of t = 1.0. Since the horizontal dimension of the solution region is proportional to the half width of the indenter D, the geometry of deformation in the horizontal plane is the same for all experiments, even as we change the ratio D/L in 3 D, and in 2 D allow D/L. Boundary conditions applied to the models are summarized in Figure 1. In 3 D models at t =0,z = 1.0 and z = 0.0 correspond to the top and bottom of the layer, respectively. We choose to use a free slip condition on the top surface (U z = s xz = s yz =0onz = 1.0) rather than a stress free condition. The zero vertical displacement condition is a simplifying assumption which enables approximate solutions to be obtained for any Argand number we consider without producing unrealistically large free surface displacements. Thickening of the layer accompanies downward movement of the bottom surface and forms a lithospheric root zone. Under a zero stress boundary condition on the top surface, a small fraction of the layer thickening would manifest as a topographic plateau above the location of the root in order to balance the vertical stress. In this situation the depth extent of the root would be marginally reduced. Nevertheless, the development of topography relative to the z = 1.0 surface can be calculated after the fact by balancing the vertical stress component on that surface against the weight of topography, a commonly used approximation in mantle convection studies [e.g., McKenzie, 1977]. We assume that the model lithosphere comprises a constant density crustal layer which sits above a denser mantle layer. The crust mantle boundary (Moho) is the surface initially at z =(1 S 0 /L), where S 0 is the initial crustal thickness (Figure 1b). For all experiments presented here, S 0 /L = We remove the lithostatic pressure gradient from the 3 D calculations by subtracting the mantle density from the lithospheric column, so that the Table 1. Nominal Dimensional Values Assumed for the Physical Parameters Parameter Value L 100 km S 0 35 km U 0 50 mm y 1 t 2Ma _" s 1 r m kg m 3 r c kg m 3 g 10 m s 2 for Ar >0 4of13

5 Figure 2. The x y distribution of vertical deflections of (a) the crust mantle boundary (Moho) originally at the level z = 0.65 and (b) the lithosphere asthenosphere boundary (LAB) originally at the level z =0.0,att = 0.2 from a 3 D calculation with n =3,Ar =0andD/L = 1. The surfaces are colored according to the z coordinate at the center of each triangular mesh element. (c) Deformed LAB surface of the 3 D finite element mesh at t = 0.2. All quantities are dimensionless. dimensionless densities in the mantle and crust are zero and (r c /r m 1), respectively. Removing the static part of the pressure field has no effect on the calculated velocity field, but it converts the hydrostatic stress condition on the deforming base of the lithospheric layer into a zero stress condition. In 2 D TVS calculations the same crust mantle density contrast is accounted for by the Argand number (13). In comparing the 2 D and 3 D experiments, equivalent parameters were used as far as possible. To avoid unnecessary complications and to isolate purely geometrical effects, we assume that the viscosity coefficient (B) is constant with depth in 3 D calculations. We varied the power law exponent (n) of the viscous constitutive law and the Argand number (Ar) or its 3 D equivalent (x). A zero Argand number implies that gravitational forces are negligible relative to the viscous stress induced by indentation, effectively g = 0. In this situation the density contrast across the Moho surface does not affect deformation. For D/L = 1, 48 elements span the x and y directions of the 2 D triangular mesh and 3 D tetrahedral mesh, while 12 elements span the z direction of the 3 D mesh. Figure 2c shows the x y configuration of the basal plane of the 3 D mesh following deformation. The 3 D calculations were parallelized in the y direction with 48 processors each calculating the solution for a single slice of mesh elements. We compare results at the dimensionless times t = 0.0, 0.1, and 0.2, at which the indenter has moved dimensionless distances of 0.0, 0.4D, and 0.8D, respectively. For both 2 D and 3 D solutions we interpolate solution fields on to a regular 401 by 401 grid (node spacing equal to D /100). To compare 2 D and 3 D solutions, an approximate vertical average of the 3 D vertical strain rate field was calculated using interpolated values on four horizontal surfaces at z = 1.0, 0.7, 0.4, and 0.1. A difference field is obtained by subtracting the average of the four horizontal surface fields from the 2 D field. In section 4 we describe the deformation of a 3 D indentation experiment with n =3,Ar = 0, and D/L =1. In section 5 we compare the calculated vertical strain rate fields from 2 D and 3 D models for 12 experiments in which n takes the values 1, 3, and 10, and Ar takes the values 0, 1, 3, and 10. In these experiments the ratio D/L is set to 1 in order to first examine a case where horizontal gradients of deformation are relatively large. In section 6 we Table 2. Maximum Values of Lithospheric and Crustal Thickness for 3 D Models and Measured Differences in Vertical Strain Rate Between 2 D and 3 D Models a n Ar D/L t max(l) b max(s) b g c c l c a At t = 0.2 for all 12 experiments with D/L = 1, and for the three experiments where D/L is varied. Refer to the text for descriptions of the measures g (18), (19), and l (20). All quantities are dimensionless. b Maximum values of lithospheric and crustal thickness for 3 D models. c Measured differences in vertical strain rate (D _" zz ) between 2 D and 3 D models. 5of13

6 Figure 3. Vertical sections through (left) the symmetry plane (x = 0.0) and (right) the indenter corner (x = 1.0) at t = 0.2 showing (a) vertical strain rate ( _" zz ), (b) horizontal strain rate (indenter parallel, _" yy ), (c) shear strain rate ( _" yz ), (d) vorticity component perpendicular to the plane (W x ), (e) vertical velocity component (U z ), and (f) indenter parallel horizontal velocity component (U y ) for an experiment with n =3,Ar =0, and D/L = 1. For normal components of strain rate, positive values indicate extension while negative values indicate shortening. Negative shear strain rate and positive vorticity indicate that the bottom of the layer moves faster than the top in the positive y direction. The black dotted line represents the intersection with the Moho surface. Scale bar end arrows indicate saturation of the color scale. All quantities are dimensionless. 6of13

7 Figure 4. Vertical profiles showing the evolution through time of dimensionless vertical strain rate ( _" zz ) in experiments with n =1,3,10,Ar = 0, and D/L = 1. The profile is located at the midpoint of the indenter (x = 0) and moves with the indenting boundary. Positive strain rates indicate thickening. All quantities are dimensionless. describe the effect of increasing the ratio D/L for the case where n =3,Ar =0. 4. Deformation in Three Dimensions [8] In this section we describe the deformation of a 3 D indentation experiment with n =3,Ar = 0, and D/L =1. Figure 2 shows that the layer deforms by a combination of shortening, thickening and shearing in response to indentation. Material immediately ahead of the indenting boundary is pushed downward, which displaces the Moho and the lithosphere asthenosphere boundary (LAB) to form a root zone (Figures 2a and 2b). For the case n =3,Ar =0att = 0.2, the maximum lithospheric and crustal thicknesses have increased by 44% and 53%, respectively (see Figures 2a and 2b and values in Table 2). The values in Table 2 confirm that rates of layer thickening increase sharply with n and decrease with Ar. Vertical sections through the 3 D solution domain (Figure 3) show that deformation varies laterally (in both x and y) and with depth. Thickening and shortening strain rates (the components _" zz and _" yy ) are greatest in the upper lithosphere (Figures 3a and 3b). These strain rate components decrease with depth in the root zone because displacement of the root is not resisted by the surrounding asthenosphere to the same extent as the upper lithosphere (which is constrained by the remote rigid boundary condition at y = 4). As the root develops, a localized region of high shear strain rate on horizontal and vertical planes ( _" yz ; Figure 3c) occurs toward the base of the layer due to gradients in the vertical (U z ) and horizontal (U y ) velocity components (Figures 3e and 3f, respectively). The shear strain rate is negative there because the rate of movement in the indentation direction at the bottom of the layer is greater than at the top. Thus the root is pushed ahead of the overlying layer while simultaneously growing in volume as material is forced downward ahead of the indenter. The sign of the x vorticity component (W x ) confirms a circulating flow in which the fastest downward velocities occur close to the indenter as the whole body translates in the indenter parallel direction (Figure 3d). Figure 4 shows the variation of vertical strain rate ( _" zz ) with depth and time for experiments with Ar = 0 and varying n. Vertical gradients of strain rate increase with n, but all three cases show a similar evolution with time. In the upper part of the layer, vertical strain rate generally increases marginally with time as the horizontal domain is shortened. Within the growing root zone, however, strain rates decrease with time. As the root increases in volume it becomes less effectively constrained by the boundary conditions, and strain rates decrease. Conversely, the upper lithosphere remains confined, so material is strained at ever greater rates. 5. Comparison of 2 D and 3 D Solutions [9] In this section we compare 2 D and 3 D indentation models in which we vary the parameters n and Ar for the case where D/L = 1. Since the predominant manifestation of the indenting boundary condition is thickening (e.g., Figure 2), we first examine contours of the 2 D and averaged 3 D vertical strain rate fields ( _" zz ) and their difference (D _" zz = _" 2D zz _" 3D zz )inx y plan view at t = 0.2 (Figure 5). The greatest difference between 2 D and average 3 D solutions occurs systematically near the indenter corner (or syntaxis) where the 2 D solutions consistently have greater strain rates. This strain rate concentration (which has been observed in previous studies involving 2 D TVS models, e.g., Houseman and England [1986, 1993]) is generally assumed to have the form of a weak singularity caused by the discontinuity in boundary conditions at the corner. The magnitude of the strain rate concentration increases with time in 2 D solutions but decreases with time in 3 D solutions (Figures 6a and 6b). Horizontal profiles of vertical strain rate within the 3 D solution (Figures 6c and 6d) show that at t = 0 the strain rate concentration exists to some degree at all levels within the lithosphere. Moreover, the strain rate magnitude increases with depth. At t = 0.2, however, the concentration is only evident toward the base of the layer. The diminution of this feature in 3 D calculations shows that accurately accounting for vertical shear tractions has a considerable damping effect 7of13

8 Figure 5. Plots of (left) 2 D and (middle) averaged 3 D vertical strain rate field ( _" zz ), and (right) their difference (D _" zz = _" 2D zz _" 3D zz )att = 0.2 for experiments with D/L = 1 and varying n and Ar. The plotted fields have been smoothed using a circular averaging filter with radius 0.05 units to reduce interpolation noise. Grey regions show the extent of the indenting boundary condition. Black indicates saturation of the color scale. All quantities are dimensionless. 8of13

9 Figure 6. Profiles along the indented boundary (y = 0.0 and 0.8) of vertical strain rate ( _" zz ) for a calculation with n =3,Ar = 0, and D/L = 1. (a and b) The 2 D solution, averaged 3 D solution, and their difference at t = 0.0 and 0.2, respectively. (c and d) The strain rates at the four depth levels used to construct the average of the 3 D solution (also shown) at t = 0.0 and 0.2, respectively. All quantities are dimensionless. on the strain rates near the indenter corner relative to the predictions of the 2 D TVS approximation. Unsurprisingly, the TVS approximation is least accurate where the horizontal gradients of strain rate are greatest. Away from the indenter corner, solutions with n 3 are characterized by differences of relatively low magnitude ( D _" zz 0.25; Figures 5a 5d). The difference between 2 D and averaged 3 D fields reduces as Ar is increased and buoyancy forces cause thickening to become more evenly distributed across the solution domain. In experiments where Ar = 0 (Figures 5a, 5b, and 5e) and n = 1 the differences are small ( D _" zz 0.05), but they increase significantly for n = 3, and further still for n = 10. At large n, the vertical strain rate is significantly greater in the 2 D solution immediately in front of the indenter but decays to lower strain rates over a shorter distance compared with the equivalent averaged 3 D solution (Figure 7a). The variation of strain rate with depth within the 3 D solution is also much greater when n is large; at the midpoint of the indenting boundary, there is a factor of 3 difference in the range of vertical strain rate with depth between the top surface (z = 1.0) and the surface at z = 0.1 (n = 10, Figure 7b). [10] In Table 2 we quantify the differences between 2 D and averaged 3 D solutions at t = 0.2 in terms of three single number measures: (1) g, the normalized mean of the absolute difference field: ¼ 1 _" avg 1 X N jd _" zz j ð18þ zz N i¼1 (2), the normalized standard deviation of the absolute difference field: ¼ 1 _" avg zz 1 X N!1 jd _" zz j _" avg 2 2 zz ð19þ n 1 i¼1 9of13

10 Figure 7. Profiles along the axis of symmetry (x = 0.0) of vertical strain rate ( _" zz ) for a calculation with n = 10, Ar = 0, and D/L =1att = 0.2. (a) The 2 D solution, averaged 3 D solution, and their difference. (b) The strain rates at the four depth levels used to construct the average of the 3 D solution (also shown). All quantities are dimensionless. (3) l, the normalized maximum of the absolute difference field (which systematically occurs close to the indenter corner; Figure 5): _" avg zz ¼ 1 2N X N i¼1! _" 3D zz þ XN _" 2D zz i¼1 ð21þ ¼ 1 _" avg zz maxðjd _" zz jþ ð20þ where N is the number of interpolated points and each measure is normalized by the mean of the averaged 2 D and 3 D vertical strain rate fields: [11] Consistent with mass conservation, variation of the normalization factor for the 12 experiments at t = 0.2 ( _" avg zz ) is attributable to discretization error. In Figure 8 we graphically display the variation of these three difference measures (Table 2) in the parameter space n versus Ar. For the experiments presented here, the greatest Figure 8. Contour plots of the three measures of difference (a) g, (b), and (c) l in the parameter space n versus Ar for D/L = 1 and t = 0.2. Interpolated contours are constrained by the irregularly spaced points plotted as stars, which represent the twelve experiments with D/L = 1. The contour interval is 0.05 in Figure 8a, 0.1 in Figure 8b, and 2.0 in Figure 8c. The grey shaded region indicates the parameter range which best describes the Tibetan Plateau in previous TVS calculations (i.e., n 3; Ar 3, England and Houseman [1986]). Refer to the text for descriptions of the measures g,, and l. All quantities are dimensionless. 10 of 13

11 Figure 9. Plots of difference between 2 D and averaged 3 D vertical strain rate fields (D _" zz = _" 2D zz _" 3D zz ) at (a) t = 0.0, (b) t = 0.1, and (c) t = 0.2, for D/L values of (left) 1, (middle) 2, and (right) 4 for the case n = 3, Ar = 0. The plotted fields have been smoothed using a circular averaging filter with radius 0.05 units to reduce interpolation noise. Grey regions show the extent of the indenting boundary condition at each time. Black indicates saturation of the color scale. All quantities are dimensionless. difference in vertical strain rate between 2 D and 3 D solutions is found when n = 10 and Ar = 0. For experiments with zero Argand number, the three difference measures increase approximately linearly with n. For Newtonian deformation (n = 1) the difference measures vary little with Ar. However, with greater values of n, the difference measures are more sensitive to Ar, and decrease rapidly as Ar is increased. 6. Variation of D/L Ratio [12] We expect the difference between 2 D and 3 D solutions to vanish as the ratio D/L approaches infinity because the 2 D TVS approximation assumes that D L. We therefore conduct further 3 D experiments in order to examine this convergence using D/L = 1, 2, and 4 for the case where n = 3 and Ar = 0. When changing the value of D/L for each 3 D mesh, the shape of the tetrahedral elements is unaltered while the number of elements in the x and y dimensions are 48, 96, and 192 for D/L ratios of 1, 2, and 4, respectively. The number of elements in the z dimension remains at 12 so that vertical resolution is constant for all experiments. In Figure 9 we plot contours of the difference between 2 D and averaged 3 D vertical strain rates (D _" zz ) in x y plan view at t = 0.0, 0.1, and 0.2. Additionally, in Table 2 we quantify the differences at these times using the three difference measures g,, and l defined in section 5. The time zero difference fields confirm that the 2 D and averaged 3 D solutions converge as D/L is increased (Figure 9a and g in Table 2). The largest difference occurs at the indenter corner as described in section 5, but this progressively reduces in magnitude with increasing D/L. Elsewhere in the solution domain the difference approaches zero with increasing D/L, indicated by the increased noisiness of the zero contour in a flat field. Differences near the indenter 11 of 13

12 corner continue to increase marginally with time as noted previously (Figures 6a and 6b). Directly in front of the indenter, the difference can be seen to increase with time and D/L. After the root zone has begun to develop in 3 D calculations (i.e., t > 0), the increased layer thickness in front of the indenter causes strain rates to be decreased in this region and increased further away where the layer has not thickened as much. This effect is amplified as deformation progresses since the contrast between the layer thickness close to and far from the indenting boundary only increases with time. Increasing the ratio D/L in 3 D reduces the layer thickness relative to the indenter half width. Consequently, the same finite indentation stress is distributed over a smaller depth interval. In the 2 D TVS approximation the same stress is assumed to be distributed over a layer of constant thickness [England and McKenzie, 1982]. For t > 0, horizontal gradients of the stress bearing layer thickness are no longer zero throughout the solution domain of a 3 D calculation, though the gradients are reduced in proportion to D/L. The reduction of horizontal gradients of vertical shear stress permits the most highly stressed regions to strain faster. Consequently, the rate of thickening at the indenter corner increases with D/L and is closer in magnitude to the 2 D solution. Another consequence of increasing D/L is that the maximum crust and layer thicknesses in the 3 D calculations also increase (Table 2). 7. Discussion and Conclusions [13] Our comparison between 2 D and 3 D calculations of indentation (in the context of continental collision) has highlighted differences between the TVS approximation and the full 3 D implementation. The magnitude of the differences depends on the parameters n, Ar, D/L and on the total convergence. Larger values of Ar result in a more accurate TVS estimate, while larger values of n cause the solutions to diverge. At time zero, the 2 D and 3 D solutions converge as D/L is increased. The greatest difference between 2 D and 3 D solutions occurs systematically at the indenter corner. The TVS approximation neglects vertical shear tractions, so errors that scale with the horizontal gradients of strain rate are introduced. These errors increase with n and decrease with Ar (Table 2). For small Argand numbers, vertical strain rates near the indenter corner may be in error by 50% or more. [14] The ability of the 3 D domain to vary in thickness through time is one important contributor to differences between 2 D and 3 D solutions. Since the TVS approximation assumes constant thickness, in plane stress is always distributed over a constant layer thickness (L). Consequently, the 2 D strain rate is overestimated compared with the 3 D solution after the thickened lithospheric root develops (i.e., when t > 0). Moreover, this difference amplifies with both time and increasing D/L. Because deformation in the 3 D implementation is described in terms of the full deviatoric stress tensor (3), vertical gradients of horizontal velocity develop (Figure 3e), which do not exist in the 2 D TVS approximation. Although the crustal thickness distribution can be estimated from 2 D TVS calculations, the variation of strain rate with depth is lost. In particular, in 3 D calculations we see that the lithospheric root may be forced ahead of the upper lithosphere in the collision zone. The TVS approximation is most accurate when the modeled layer is thin compared to the length scale (i.e., D L). The tendency of the 2 D and 3 D solutions to converge as D/L is increased confirms the validity of the TVS for larger values of D/L. However, when horizontal gradients of strain rate are large (i.e., D/L is small in our experiments) it is clearly preferable to use the full 3 D implementation. [15] Of present day orogens resulting from continentcontinent collision, the Tibet Himalaya system [e.g., Flesch et al., 2001; England and Molnar, 2005; Vergnolle et al., 2007; Dayem et al., 2009] and the European Alps [e.g., Robl and Stüwe, 2005; Jiménez Munt et al., 2005; Robl et al., 2008] have been modeled using the 2 D TVS approximation. The half width of the Indian indenter is approximately 1250 km. In contrast, the half width of the Adriatic indenter is approximately 250 km. Assuming a precollision lithospheric thickness of 100 km (as here; see Table 1), these half widths correspond to D/L ratios of 12.5 for the India Asia collision and 2.5 for the Alpine collision. Our observations in section 6 show that certainly the length scale of the Tibet Himalaya system satisfies the condition D L required by the TVS approximation. England and Houseman [1986] found that the best match between a TVS model and topography in the Tibetan Plateau occurred when using n = 3, Ar = 1, or n = 10, Ar = 3. For these parameter ranges we see significant differences between the 2 D and 3 D solutions close to the indenter when D/L = 1 (Figures 5c and 5f). While increasing D/L to 12.5 will reduce the errors in the 2 D solution attributed to the neglect of vertical shear tractions, increasing D/L will also increase those errors attributed to the 2 D assumption that the stress bearing layer is of constant thickness (Figure 9). Clearly, the TVS should be treated with some caution wherever large gradients of strain rate are encountered, for example at an indenter corner or when n is large. Despite this, we consider the TVS to be a valuable and useful approximation in parts of the solution not affected by large gradients of strain rate. [16] In this study we have not considered the impact of subhorizontal layering in the rheological properties of the models, but strong vertical variations in strength can cause important changes to the mode of lithospheric deformation. For instance, Schmalholz et al. [2002] argue that under the conditions favored by England and Houseman [1986] the lithosphere will deform by a combination of homogeneous thickening and folding and that folding should dominate at the indenter corners. Our use of rheological parameters that are invariant with both horizontal position and depth prevents deformation by thrusting or folding. Despite this, our calculations clearly reveal the physical processes that underlie the geometrical differences between 2 D and 3 D models without introducing additional complications. However, there is clearly scope for further investigation of the indentation problem in the presence of a layered rheological structure. [17] We have also neglected the impact of basal shear stress here. Ghosh et al. [2008, Appendix D] calculate the present day deviatoric stress field in a global thin sheet calculation and compare with the output of a 3 D mantle convection model. They find that two components are required in the force balance before the deviatoric stress fields of thin sheet and 3 D models are equivalent. The first is a gravitational potential energy term calculated by vertically integrating the vertical stress down to a reference level at 100 km depth, as used in the classic TVS approximation (8). The 12 of 13

13 second term relates to shear tractions which are applied at the base of the sheet. Without the basal tractions, Ghosh et al. [2008] conclude that the thin sheet model underestimates the total deviatoric stress field by approximately 50%. This highlights the importance of basal shear stress when D/L is very large. [18] A further consideration when choosing between 2 D and 3 D calculations is the much greater computational cost involved with 3 D. Even at D/L = 4, the cost of our 3 D calculations is high. So for any experiment with D/L >4it can be cost effective to use 2 D calculations for initial exploration of the parameter space. Subsequent 3 D calculations can then be used to focus on smaller areas of the parameter space, where high strain rate gradients are indicated. [19] In conclusion, our observations show that the TVS approximation is adequate to model long wavelength deformation for D/L as low as 1 providing that n is less than 3. 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Calais, and L. Dong (2007), Dynamics of continental deformationinasia,j. Geophys. Res., 112, B11403, doi: / 2006JB Whitehouse, P. L., P. C. England, and G. A. Houseman (2005), A physical model for the motion of the Sierra Block relative to North America, Earth Planet Sci. Lett., 237(3 4), , doi: /j.epsl M. C. Garthwaite and G. A. Houseman, School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK. (m.garthwaite@ see.leeds.ac.uk; g.houseman@see.leeds.ac.uk) 13 of 13