The relationship between surface kinematics and deformation of the whole lithosphere

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1 University of Montana, Missoula From the SelectedWorks of Rebecca Bendick 2012 The relationship between surface kinematics and deformation of the whole lithosphere Rebecca Bendick, University of Montana - Missoula Available at:

2 The relationship between surface kinematics and deformation of the whole lithosphere L. Flesch 1 * and R. Bendick 2 * 1 Department of Earth and Atmospheric Sciences, Purdue University, 550 Stadium Mall Drive, West Lafayette, Indiana , USA 2 Department of Geosciences, University of Montana, 32 Campus Drive #1296, Missoula, Montana , USA ABSTRACT The variation of mechanical properties with depth in the lithosphere determines the relationship between surface deformation and whole-lithosphere deformation, hence between surface deformation and whole-lithosphere dynamics. Where viscosity (or elastic strength) is a continuous function with depth, surface deformation can be used to constrain both force balance and rheological parameters. Where viscosity is discontinuous, but the upper crust and mantle lithosphere have comparable maximum values, surface deformation can be used to approximate force balance and rheological parameters, but tradeoffs mean that estimates of stress and viscosity are effective equivalent values rather than actual values. Where viscosity is both discontinuous and differs by much more than an order of magnitude between the upper crust and mantle lithosphere, information about both force balance and rheology are absent from the surface deformation, so surface observations alone are insufficient to estimate either the dynamic or mechanical state of the lithosphere. INTRODUCTION The theory of plate tectonics began as a kinematic theory (McKenzie and Parker, 1967; Morgan, 1968) describing the geometric constraints on the motions of rigid spherical caps. It expanded to include tectonic dynamics, the effort to develop equations of state, and to identify the relevant forces that excite displacements of plates (e.g., Elsasser, 1969; Richter and Mc Kenzie, 1978). The field benefited from the fact that oceanic lithosphere has a continuous function of strength (or stiffness in the viscous case) with depth. Stiffness is the viscosity of a linear viscous fluid, analogous to the strength of a linear elastic material (they are interchangeable through the Stokes-Rayleigh analogy). In this case, there is a simple relationship between surface displacement, deformation, and the state of stress, such that tectonic effects can be understood as the consequence of a small number of boundary and body forces acting on materials of finite viscosity or strength. As new geodetic data measuring tectonic deformation in continents became available, this approach was transferred to the continental setting (England and Molnar, 1997), often through a thin viscous sheet approximation (Bird and Piper, 1980; England and McKenzie, 1982; Houseman and England, 1986; Wdowinski et al., 1989; Flesch et al., 2001; Garthwaite and Houseman, 2011). The thin viscous sheet model contains the same assumption that surface deformation is indicative of deformation through the whole lithosphere, thus it expresses all of the information about body and boundary forces and torques (Medvedev and Podladchikov, 1999). In contrast, numerical and analytic solutions for laminated or composite elastic and viscous systems where mechanical properties vary strongly with depth show surface deformation that is completely (Savage, 2000), mostly (Zatman, 2000), partly (Hetland and Hager, 2004), or not at all (Bourne et al., 1998) insensitive to deformation at depth. This range of possibilities reoccurs in studies of tectonically active regions, especially in continents, where surface displacements are interpreted as unrelated to deformation at depth, such as in the Basin and Range or Tibet (Meade and Loveless, 2009; Meade, 2007), partially related to deformation at depth in Tibet (Royden, 1996), or perfectly related to deformation at depth in East Africa (Stoddard and Abbott, 1996), the Basin and Range (Humphreys * s: lmflesch@purdue.edu; Bendick@mso.umt.edu. and Coblentz, 2007), and Tibet (Flesch et al., 2001). Assuming that none of these large data sets or numerical solutions is flawed in their construction, their differing conclusions must be due to real differences among them, illuminating relationships among surface and whole-lithosphere kinematics and dynamic state that vary according to the mechanical properties of the lithosphere. To test this, we create a set of forward numerical models with identical boundary and body conditions, but with end-member lithospheric strength profiles (Figs. 1 and 2). We show that varying lithospheric vertical strength alone produces large variations in the pattern of surface deformation and only for specific configurations do standard approximations hold true. METHODS We construct five three-dimensional numerical simulations using the commercial software COMSOL Multiphysics ( to solve the instantaneous equations for incompressible Newtonian Stokes flow. We consider the instantaneous case to compare with the information inferred from instantaneous GPS observations. All simulations (Fig. 1) consist of an Airy compensated generic mountain, assuming constant crustal and mantle densities, to excite buoyancy forces, and a moving wall along the southern model wall to excite boundary forces. The northern model wall is fixed, and the eastern and western walls and model base are free slip. The upper model surface is stress free. Thus, all of the solutions have identical dynamics and Figure 1. Numerical model consists of pushing (19 mm/yr) southern side wall, fixed northern wall, rollers on east and west sidewalls, free slip base, and stress-free surface. Maximum topography is 5000 m supported Airy isostatically with constant crustal (2750 kg/m 3 ) and mantle (3300 kg/m 3 ) densities. Colored surfaces represent boundaries at surface, between upper and lower crust, lower crust and upper mantle, and base of lithosphere. Vertical variations in strength are shown in Figure 2. GEOLOGY, August 2012; v. 40; no. 8; p ; doi: /g ; 3 fi gures. Published online 15 June GEOLOGY 2012 Geological August Society 2012 of America. For permission to copy, contact Copyright Permissions, GSA, or editing@geosociety.org. 711

3 Figure 2. Five vertical strength profiles used with dynamics shown in Figure 1 to generate velocity fields at surface (blue), 25 km depth (red), and base of model (green). Pushing wall is at 0. Surface topography is represented by gray shading. Comparing velocities in rows shows variation of deformation with depth in single model; comparing velocities in columns shows differences in deformation that arise as result of mechanical differences from model to model. Coherence of deformation through entire lithosphere in types 1 and 2a implies that surface information can be used to infer dynamic state. Strong vertical variation in deformation in types 2b and 3 means that surface information does not represent whole lithosphere and cannot be used to infer lithospheric dynamic state. Different numerical methods should be used to approach each type. See text for further discussion. we vary only the vertical strength profile (Fig. 2). These strength profiles are constructed so that all of the simulations have similar integrated effective strengths, η = Pa s (Fig. 2), thus have equivalent effective Argand numbers (England and McKenzie, 1982). Profiles represent limiting cases for continental lithosphere: type 1, where the viscosity profile is continuous (no abrupt stiffness contrasts); type 2, where the viscosity profile is discontinuous but the maximum viscosities of the crust and mantle lithosphere are of the same magnitude; and type 3, where the viscosity profile is both discontinuous and asymmetrical, so the crust and mantle lithosphere differ in maximum viscosity. RESULTS Figure 2 shows the lithospheric velocity fields for each of the numerical solutions. These solutions allow the identification of three distinct families of tectonic behavior, each with substantially different relationships between the surface kinematics and dynamics. Type 1 lithosphere, where the vertical viscosity is a continuous function of depth, is completely described by a thin elastic or viscous sheet. The resulting velocity field is also vertically continuous (Fig. 2). Hence, the surface deformation is representative of deformation throughout the lithosphere and contains information about both boundary tectonic forces and gravitational potential gradients. This is the most important simplifying assumption of the thin viscous sheet approximation, and Garthwaite and Houseman (2011) showed that it is true regardless of the specific boundary conditions. As a result, comparisons between observed and expected deformation can be used to estimate realistic values either for the flexural rigidity in the elastic approximation or the viscosity in the thin viscous sheet approximation. Type 2 lithosphere, where the upper crust and mantle lithosphere have similar maximum viscosities but are separated by much weaker lower crust, can also be approximated by an equivalent elastic plate or viscous sheet. This condition differs from the simply continuous case (type 1), because the effective viscosity or flexural rigidity of the model sheet is not representative of the actual mechanical properties of any of the materials involved. Instead, it represents the integrated property of an equivalent ideal material, which is not required to have values known for real Earth materials. The viscosity contrast of the competent layers with the weak lower crust further influences the information content of the surface velocity field. Specifically, if the weaker lower crustal layer is able to couple the vertical stresses (type 2a; Fig. 2), then the surface deformation is a harmonic of the individual layer displacements (Cerda and Mahadevan, 2003). The thin viscous sheet assumption that deformation is not depth dependent holds because the upper crustal and mantle lithosphere layers have the same set of imposed forces and the same maximum viscosity (Lechmann et al., 2011) (Fig. 2), so both have the same harmonic content. However, if the lower crustal layer is so weak that it mechanically decouples the lithosphere (type 2b), the surface deformation is instead a linear sum of the individual layer displacements. Furthermore, stresses due to topography are not transferred from the crust to the mantle lithosphere (Bendick and Flesch, 2007), so the mantle lithosphere responds mostly to boundary forces and contributes little to the surface deformation, only at very long wavelengths. For type 3 lithosphere, where the strength or stiffness profile of the lithosphere is discontinuous and the upper crust and mantle lithosphere have strongly different viscosity (or strength), the surface displacement field is a highly nonlinear combination of the deformation of different parts of the lithosphere, themselves dissimilar. Deformation through the lithosphere is strongly depth dependent, so surface displacements do not have a clear relationship to displacements at depth. The surface velocity field for the case of a strong upper crust (type 3a) is dominated by the boundary forces, whereas the surface velocity field for the strong mantle lithosphere (type 3b) is dominated by buoyancy forces, even though both boundary and buoyancy forces are important to the model dynamics. No equivalent elastic or viscous sheet that completely reproduces the surface deformation can be found for these strongly heterogeneous cases, exactly because information about the dynamic state of the lithosphere at depth is not transferred to the surface displacement. This agrees with results from Lechmann et al. (2011), who pointed out that discrepancies between surface deformation and deformation at depth are especially great when bending, buckling, and differential lateral flow occur within discrete lithospheric layers, and with Hetland and Hager (2004), who demonstrated that geodetic velocities are sensitive to the contrast between crustal and mantle viscosities. DISCUSSION This classification of modes of continental tectonics helps to explain the range of published results on whether surface kinematics provide August 2012 GEOLOGY

4 useful information about deformation of the whole lithosphere. At one extreme, elastic or viscous sheets (e.g., England and McKenzie, 1982; Houseman and England, 1986) are cases of type 1 mechanics, where the surface deformation provides complete information about the whole lithosphere. At the other extreme, elastic lids over viscous sheets or half spaces (e.g., Savage, 2000; Hetland and Hager, 2004) are cases of type 3a mechanics, where the surface deformation provides no information about the whole lithosphere. For a homogeneous, isotropic elastic lid with no brittle yield failure, surface displacements are determined only by boundary forces and torques on that lid (Turcotte and Schubert, 2002; Zatman, 2000). For a homogeneous, isotropic elastic lid with a brittle yield strength, once the yield stress is reached, the lid should fail into regular tiles or blocks, the length scale of which is determined by the elastic thickness (Hutchinson, 1967; Mahadevan et al., 2010). These tiles then dominate the surface deformation as strain becomes highly localized to block boundaries. Internal blocks are advected by basal tractions if coupling across block boundaries is low and by block-to-block interactions if coupling is high (Zatman, 2000). For an elastic lid with preexisting weaknesses or dislocations, these become the block boundaries, which again localize the surface strain as blocks dominate the surface deformation field. This behavior is illustrated in sandbox models by Sokoutis and Willingshofer (2011) with a priori weak zones. We can identify examples of each type both in the tectonic modeling literature and in currently active tectonic settings (Fig. 3) based on an evaluation of both the qualitative characteristics of surface velocity observations and information, mostly seismic, about likely vertical viscosity variations. In doing so, several of the important characteristics of these families are further illustrated. The plate tectonics of the oceanic lithosphere is an example of type 1, where the information transfer from dynamics to kinematics is nearly perfect. The equivalent analytic model consists of forces and torques applied to rigid plates (Turcotte and Schubert, 2002) The Tibetan Plateau is an example of type 2a, where a reasonable equivalent thin sheet can be found (Houseman and England, 1986; Flesch et al., 2001; Garthwaite and Houseman, 2011; Copley et al., 2011), but where that sheet may not have the same mechanical properties of any actual part of the Tibetan lithosphere. The deformation of the mantle lithosphere is similar to that at the surface (Wang et al., 2008; Bendick and Flesch, 2007). Type 3 lithosphere includes two distinct possibilities, one where the crust hosts most of the lithospheric stiffness, and one where the mantle lithosphere does. The best example of the stronger crust member is the Basin and Range Province, which has been inferred to have a very thin or missing mantle lithosphere and is instead underlain by a weak asthenosphere (Li et al., 2007). In this case blocks or microplates with length scales less than the lithospheric thickness dominate the surface deformation field and mainly express the contribution of tectonic boundary forces. In the Basin and Range, the surface strain is associated with individual normal faults and upper crustal blocks rotating in response to Pacific shear (Hammond et al., 2011; Hammond and Thatcher, 2007, 2004), and below 20 km depth there is very little contribution to the depth-integrated deviatoric stresses (Klein et al., 2009). None of the dynamic contributions from either the sublithospheric mantle or the mantle lithosphere are transferred to the surface kinematics, so the kinematics cannot be used to infer those dynamics (Zandt and Humphreys, 2008; Silver and Holt, 2002). The surface kinematics are best simulated with elastic lid over viscous half-space methods. The best example of the stronger mantle lithosphere case of type 3 is the Ethiopian Rift system. The kinematics of a weak material on a strong foundation were given by Huppert (1982). In this case, gravitational relaxation of the upper material dominates the surface deformation field. Geodetic results for Ethiopia (Bendick et al., 2006; Kogan et al., 2012) demonstrate that only 50% 70% of the surface deformation is bound within the structural Ethiopian Rift, with broadly distributed deformation due to buoyancy forces extending far into the continental interior. As in the case of the Basin and Range, information about the lithospheric dynamics in the Ethiopian Rift system is not transferred efficiently into the surface kinematics, so the underlying dynamics, in this case the boundary conditions, cannot be fully extracted from kinematic observations. Because of the range of possibilities, comparisons of numerical simulations to real kinematic data sets should be preceded by judicious use of a priori information about the likely vertical mechanical properties of a region. In particular, common approximations such as thin viscous sheets or elastic plates can be used robustly only when the vertical strength profile of a region is known to be continuous; blocks over fluids can be used only when the maximum competence of the system is in the upper crust. Numerical tools should be used with particular caution where strength profiles are probably discontinuous, and the resulting values for rigidity or viscosity should not be interpreted as diagnostic of particular earth materials. ACKNOWLEDGMENTS We thank three anonymous reviewers. Flesch was supported by National Science Foundation (NSF) grant EAR , Bendick by NSF grant EAR Figure 3. 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