Volume 19, Issue 3 p. 642-666
Research Article
Free Access

Evaluating the Influence of Plate Boundary Friction and Mantle Viscosity on Plate Velocities

A. Osei Tutu

Corresponding Author

A. Osei Tutu

GFZ German Research Centre for Geosciences, Potsdam, Germany

Institute of Earth and Environmental Science, University of Potsdam, Potsdam, Germany

Correspondence to: A. Osei Tutu, [email protected]Search for more papers by this author
S. V. Sobolev

S. V. Sobolev

GFZ German Research Centre for Geosciences, Potsdam, Germany

Institute of Earth and Environmental Science, University of Potsdam, Potsdam, Germany

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B. Steinberger

B. Steinberger

GFZ German Research Centre for Geosciences, Potsdam, Germany

Centre for Earth Evolution and Dynamics, University of Oslo, Oslo, Norway

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A. A. Popov

A. A. Popov

Institute of Geosciences, Johann Gutenberg University, Mainz, Germany

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I. Rogozhina

I. Rogozhina

GFZ German Research Centre for Geosciences, Potsdam, Germany

Marum Centre for Marine Environmental Sciences, University of Bremen, Bremen, Germany

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First published: 01 February 2018
Citations: 11

Abstract

Lithospheric plates move over the low-viscosity asthenosphere balancing several forces, which generate plate motions. We use a global 3-D lithosphere-asthenosphere model (SLIM3D) with visco-elasto-plastic rheology coupled to a spectral model of mantle flow at 300 km depth to quantify the influence of intra-plate friction and asthenospheric viscosity on plate velocities. We account for the brittle-ductile deformation at plate boundaries (yield stress) using a plate boundary friction coefficient to predict the present-day plate motion and net rotation of the lithospheric plates. Previous modeling studies have suggested that small friction coefficients ( urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0001, yield stress urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0002 MPa) can lead to plate tectonics in models of mantle convection. Here we show that in order to match the observed present-day plate motion and net rotation, the frictional parameter must be less than 0.05. We obtain a good fit with the magnitude and orientation of the observed plate velocities (NUVEL-1A) in a no-net-rotation (NNR) reference frame with urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0003 and a minimum asthenosphere viscosity of urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0004 Pas to 1020 Pas. Our estimates of net rotation (NR) of the lithosphere suggest that amplitudes urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0005 ( urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0006/Ma), similar to most observation-based estimates, can be obtained with asthenosphere viscosity cutoff values of urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0007 Pas to urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0008 Pas and friction coefficients urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0009.

Key Points

  • We model plate motions using a global model that combines a 3-D visco-elasto-plastic rheology in the upper 300 km and a radial viscosity structure in the mantle
  • We estimate interplate friction and upper mantle viscosity to reproduce observed plate velocities as well as the magnitude and location of the Euler pole of lithosphere net rotation
  • Key parameters required to fit observations are a friction coefficient of less than 0.05 at plate boundaries and a minimum mantle viscosity cut-off of 1–5 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0010 Pas

1 Introduction

Compared to the other planets in our solar system the uniqueness of the Earth is related to the presence of plate tectonics, which is also believed to have influenced the development of the only known habitable environment (Nesbitt & Young, 1982; Raymo & Ruddiman, 1992; Veevers, 1990). This fact has enhanced a general interest in the concept of plate tectonics and has made it a focus of numerous studies in the past few decades. For example, Davies (1978) showed that the force balance contribution due to whole-mantle convection is non-negligible compared to other driving and resistive forces acting on the lithospheric plates. The total net torque on the lithosphere plates from plate boundary forces should be balanced by the basal shear stresses from the mantle beneath (Hager & O'Connell, 1981; Turcotte & Oxburgh, 1967). The stress that builds up in the crust and lithosphere (Bird, 1989; Bird et al., 2008; Zoback, 1992, Zoback et al., 1989) is shown to originate from both shallow and deep sources, with a larger part of these stresses coming from subducting plates (Forsyth & Uyeda, 1975). Previous studies have enabled a relatively good understanding of the factors that cause plate motion over a low-viscosity asthenosphere. Through geological observations (DeMets et al., 2010), combined with spectral (Conrad & Lithgow-Bertelloni, 2002; Hager & O'Connell, 1981) and finite element/finite difference numerical approaches (Conrad & Lithgow-Bertelloni, 2002; Minster et al., 1974; Moresi et al., 2000; Zhong & Gurnis, 1995), the knowledge about some of the major contributing factors was developed. Among such factors are the basal driving force acting on the lithosphere due to mantle convection (i.e. internal buoyancy force), plate boundary forces such as slab pull, ridge push, inter-plate friction and trench resistance, and the cratonic root resistance. Each of these forces is either a driving or a resisting force contributing to the total force balance of global plate motions. While one branch of studies has used kinematic analysis to quantify individual forces (Becker & O'Connell, 2001; Becker et al., 2015; Davies, 1978; Hager & O'Connell, 1979; Lithgow-Bertelloni & Richards, 1998; Torsvik et al., 2010) that come into play for the plates to move, other studies (Alisic et al., 2012; Becker, 2006; Lithgow-Bertelloni & Richards, 1995; Schellart, 2004; Stadler et al., 2010) have sought to explain rheological implications of the moving plates for both the upper and lower mantle. Nonetheless, studies that have tried to identify and quantify these individual driving and resisting forces have also arrived at non-unique conclusions regarding some force contributions. For example, Lithgow-Bertelloni and Richards (1998) suggested that subducting slabs acting at the margin of a moving plate at shallow depths together with deep mantle slab-driven flow contribute more than 90% of the total driving forces while the remaining driving forces, coming from the push due to thickening of oceanic plates at the mid-ocean ridges, contribute less than 10%. In contrast, lower values of about 50% to 70% as the slab pull contribution to the total force balance of global plate motions were reported in other studies (Becker & O'Connell, 2001; Conrad & Lithgow-Bertelloni, 2002). Also, these studies often ignore the influence of slab bending resistive force, which has been the focus of Wu et al. (2008), and the effect of non-convective forces due to topography in oceans and continents (Bercovici et al., 2000), both of which we take into account using our numerical method including an upper boundary free surface and the effect of self gravitation (see section 2).

Although it is generally acknowledged that subduction influences global plate velocities (Forsyth & Uyeda, 1975; Hager & O'Connell, 1981), the contribution of a subducting plate is also influenced by factors such as plate boundary friction, slab viscosity and the asthenosphere viscosity (Alisic et al., 2012; Davies, 1978; Tan et al., 2012; van Summeren et al., 2012), making a quantification of the slab pull contribution challenging. Geodynamic modeling studies of global plate motions often ignore the frictional resistance to the slab pull force at shallow depths (e.g., Conrad & Lithgow-Bertelloni, 2004; van Summeren et al., 2012) or quantify it with the aid of slab bending force (e.g., Wu et al., 2008), and only few studies have explored the importance of friction or plastic stress yielding at plate boundaries (e.g., Crameri & Tackley, 2015; Stadler et al., 2010). In this study, we explore the combined influence of plate boundary friction (plastic yielding) and asthenosphere viscosity on plate motion. The stress distribution along subducting plates due to slab pull and/or deep slab suction is influenced by the frictional force at shallow depths (Davies, 1978). The frictional resistance between subducting and overriding plates is of the order of tens of MPa (Lamb & Davis, 2003; Sobolev & Babeyko, 2005), giving rise to crustal and/or lithospheric deformation such as observed in the Andes, Himalayas and San Andreas faults. Sobolev and Babeyko (2005) explored the effect of frictional strength of the subduction interface on surface topography at active margins. They concluded that forming such tectonic features as observed in nature requires low friction, with a coefficient of static friction urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0011 within the range of 0.01 to 0.1 (friction angles urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0012 to urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0013) compared to a much higher value of urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0014 according to Byerlee's Law (Byerlee, 1978) for dry rocks and also values in the plate interior (Zhong & Watts, 2013). Similarly, low values of friction coefficient have been suggested for the San Andreas Fault systems and the Andean subduction zone (Bird & Kong, 1994; Carena & Moder, 2009; Humphreys & Coblentz, 2007; Iaffaldano et al., 2006; Mount & Suppe, 1987; Townend & Zoback, 2004; Zoback et al., 1987). Previous numerical models (Crameri & Tackley, 2015; Moresi & Solomatov, 1998; Richards et al., 2001; Tackley, 2000a) have shown as well that the use of smaller coefficients of friction urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0015 (yield stress 100 MPa) would lead to plate tectonics in models of mantle convection.

Furthermore, numerical modeling of subduction (Becker & Faccenna, 2009; Gurnis et al., 2004; Hall et al., 2003; Hassani et al., 1997; Kaus et al., 2008; Sobolev & Babeyko, 2005; Sobolev et al., 2006) has demonstrated that low coefficients of friction at the subduction interface are required to sustain subduction ( urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0016). At shallow depths ( urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0017 km), where brittle deformation occurs (Oleskevich et al., 1999; Schwartz & Rokosky, 2007), friction plays an important role at subducting margins. At greater depths with high temperatures, viscosity becomes an important parameter controlling plate motions (Bercovici & Richards, 2000). Using a 2D self-consistent dynamic model, the study of Crameri and Tackley (2015) has shown that treating plate boundaries with visco-elasto-plastic rheology enables subduction initiation. We treat our plate boundaries in a similar fashion to explore the influence of the friction coefficient on global plate motions. Such combined frictional and viscous treatment of plate boundaries is more realistic compared to previously suggested approaches using low-viscosity/weak zones (e.g., Becker, 2006; Gérault et al., 2012; Zhong et al., 2000) at plate boundaries.

Below the brittle layer, we have implemented a complex upper mantle rheology due to temperature increase with depth and the presence of cold slabs and cratons. To date, many numerical modeling studies have employed an over-simplified treatment of the complex upper mantle rheology and its effects on the magnitudes and directions of plate velocities using either a simplified (i.e. layered/radial) viscosity structure of the upper mantle or a parameterization of the other rheological/force contributions to match the surface observables such as plate motion (e.g., Becker & O'Connell, 2001; Conrad & Lithgow-Bertelloni, 2002; Lithgow-Bertelloni & Silver, 1998; Steinberger et al., 2001). In contrast, the nonlinear viscosity of the upper mantle arising from cold dense subducting plates (Burov, 2011; Faccenna et al., 2007), deeply penetrating cratonic roots of continents (Karato, 2010), and a wide range of temperature regimes has been the focus of other more advanced modeling studies of global mantle convection in relation to plate motions (Alisic et al., 2012; Becker, 2006; Ghosh & Holt, 2012; Stadler et al., 2010; van Summeren et al., 2012; Zhong & Gurnis, 1996; Zhong et al., 2000). For example, Stadler et al. (2010) and Alisic et al. (2012) have found that the slab pull force in the upper mantle is better transmitted to the subducting oceanic plate, if non-linear rheology of slabs is considered. Although these numerical studies have analyzed the rheological implications of each driving or resisting force on the overall plate motions, they often simplified or even neglected some critical aspects of the upper or lower mantle dynamics. To fully account for all the plate motion driving forces, a modeling approach must include visco-elasto-plastic rheology of the upper mantle.

The total balance of plate motion forces arising from subducting slabs, continental keels, viscous shearing of the asthenosphere, plate boundary yielding and/or strain-weakening may also cause the development of net rotation (NR) of the lithospheric plates with respect to the deep mantle (Becker & Faccenna, 2009; O'Connell et al., 1991; Ricard et al., 1991; Zhong, 2001). The NR of the lithospheric plates originates from toroidal flow due to pertaining lateral viscosity variations (LVVs) in the mantle (Bercovici & Richards, 2000; O'Connell et al., 1991; Ricard et al., 1991). Previous numerical studies that used NR to constrain mantle rheology (Becker, 2006; Zhong, 2001) have shown that the largest contribution to lithospheric NR comes from the uppermost urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0018 km of the mantle. For example, Zhong (2001) has shown that NR is mainly influenced by the continental keels ( urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0019 %), the deepest of which are placed by modern seismic imaging techniques and xenolith studies in the depth range of 200–300 km (Artemieva & Mooney, 2002; Carlson et al., 2005; Griffin et al., 2004; Gung et al., 2003; Yuan & Romanowicz, 2010). A similar conclusion was obtained by Becker (2006) who explored relative contributions of different mantle rheologies at different depths to lithosphere NR in global 3-D mantle flow models with both free-slip and prescribed dynamic upper mantle conditions. We explore the influence of LVVs within the top 300 km of the upper mantle on the NR of the lithospheric plates with a free surface at the top boundary in our numerical model.

The decoupled motion of the rigid lithospheric plates over the weak asthenosphere is also believed to be a consequence of the presence of water (Hirth & Kohlstedt, 2004; Karato & Wu, 1993) giving rise to a weakening of the asthenosphere by more than 2 orders of magnitude compared to dry mantle materials. Presence of water in the asthenosphere has been documented by experimental studies using magma samples from mid-ocean ridges. They have shown that at spreading plate boundaries magma contains about 0.1 - 0.3 wt % of water (Danyushevsky et al., 2000; Michael, 1995; Sobolev & Chaussidon, 1996) that corresponds to water content in the mantle of 0.01–0.03 wt %. At convergent boundaries the water content in the mantle is higher, exceeding 0.1 wt % (Sobolev & Chaussidon, 1996). Here we explore the effect of asthenospheric creep viscosity on plate motions, using wet and dry olivine parameters inferred from laboratory experiments and our setup for a 3-D thermal structure of the upper mantle (see section 3). We do this using a global Earth model, where deformation in the upper 300 km is computed with the 3-D finite-element numerical technique SLIM3D (Popov & Sobolev, 2008) with visco-elasto-plastic rheology, coupled to a spectral mantle flow code (Hager & O'Connell, 1981) at 300 km depth. We show that matching the observed plate motion requires the use of much lower friction coefficients at plate boundaries than suggested in the majority of mantle convection modeling studies, namely urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0020 (Crameri & Tackley, 2015; Moresi & Solomatov, 1998; Richards et al., 2001; Tackley, 2000a). We also explore the combined influence of plate boundary friction and asthenosphere viscosity on the magnitude and location of the net rotation (NR) Euler pole of the lithospheric plates.

2 Methods

2.1 Model Description

In this section we give a brief description of our coupled numerical model of mantle flow and deformation within the crust and lithosphere. The upper component is modeled through the particle-in-cell finite element, thermo-mechanical numerical code SLIM3D (3-dimensional lithosphere-asthenosphere code) of Popov and Sobolev (2008) that solves the energy and momentum equations using an arbitrary-Lagrangian-Eulerian grid with a Winkler boundary condition at the base and a true free surface upper boundary condition. This code has been extensively benchmarked by Popov and Sobolev (2008) with respect to various deformational processes within the crust and lithospheric mantle and also has been used in other studies focusing on lithospheric deformation processes (Brune et al., 2012, 2014, 2016; Quinteros & Sobolev, 2013). For this study we use an extention of the SLIM3D code to a global scale (Sobolev et al., 2009). In that modeling technique the SLIM3D code is coupled to the spectral code of Hager and O'Connell (1981) at some prescribed depth, typically 300–400 km (Figure 1c). The top domain (SLIM3D), has 3 different layers (phases) representing the crust, lithosphere and sub-lithospheric mantle. In our simulations we assume the thickness of the upper domain to be 300 km. In this way, the upper domain consists of the crust and mantle lithosphere including all major cratonic roots, a significant portion of the sub-lithospheric mantle and at least the upper portio of the subducting slabs. Thus, it operates within the depth range, where the consideration of nonlinear upper mantle rheology and the use of high spatial resolution are most important for resolving small scale deformation processes, which may influence global plate motions.

Details are in the caption following the image

(a) Depth-dependent scaling profile of s-waves velocity to density; (b) radial mantle viscosity structure (Steinberger & Calderwood, 2006) and (c) a schematic diagram of the numerical method coupling the upper mantle 3-D -lithosphere-asthenosphere code SLIM3D (Popov & Sobolev, 2008) to a lower mantle spectral mantle flow code (Hager & O'Connell, 1981) at a depth of 300 km.

The coupled model does not allow for a material exchange between the two model components but requires continuity of tractions and velocities at the interface. Tractions in the lower mantle are first computed using the spectral mantle convection model component with a 3-D density structure inferred from the hybrid seismic tomography model of Becker and Boschi (2002) using a velocity-to-density conversion profile (Figure 1a) and a radial mantle viscosity profile (Figure 1b), both from Steinberger and Calderwood (2006). Tractions computed by the spectral model, both the normal τrr and the shear urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0021 and urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0022 components, are then passed to the lithosphere-asthenosphere model as a dynamic boundary condition within a Newton-Raphson iteration procedure. At each iteration step, we prescribe mantle flow-derived tractions as a lower boundary condition in SLIM3D, and compute the flow velocities in the upper (SLIM3D) domain. Computed flow velocities at the lower boundary of the upper domain are in turn passed back to the spectral model component as an upper boundary condition.

We run the coupled model forward in time for half a million years with a time step of urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0023 years. The convergence is evaluated by comparing the velocity and traction norms of two successive iterations at 300 km depth up to spherical harmonic degree 63. For every Newton-Raphson iteration step, we estimate the displacement residual ratio and compare with a tolerance value set at urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0024. We consider our solution converged when the estimated ratio is less than the tolerance for a time step. Since our numerical method does not account for material exchange across the coupled interface, time evolution longer than 2 Ma creates numerical instability; for example, sinking slabs in the top 300 km cannot cross the coupled interface. Hence, we restrict our simulations to urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0025 years time-dependent evolution to predict present-day plate velocities in contrast to instantaneous calculations (e.g., Alisic et al., 2012). The iterations between both coupled components within a time step are considered converged if the norms between successive velocities remain almost the same. At each time step we calculate the net rotation of the lithosphere and plate velocities in the no-net-rotation reference frame. This study's modeling technique is also complemented with the calculation of the lithosphere stress field and dynamic surface topography (Osei Tutu et al., 2017), which are used to evaluate the influence of the crustal and upper mantle heterogeneities on the observed lithosphere stress field and the corresponding topography beneath air.

2.2 Upper Mantle Rheology and Thermal Structure

The coupling between the lithosphere and the mantle beneath in our model allows for an implementation of realistic rheological parameters in both model domains. In SLIM3D, the stress- and temperature-dependent rheology is implemented according to an additive strain rate decomposition into the viscous, elastic and plastic components:
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0026(1)
where G denotes the elastic shear modulus, urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0027 is the plastic potential function, urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0028 is the objective stress rate, urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0029 denotes the plastic multiplier, urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0030 is the Cauchy stress deviator, urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0031 is the pressure, urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0032 stands for the effective deviatoric stress, and ηeff is the effective creep viscosity derived by combining the diffusion and dislocation creep mechanisms, as follows:
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0033(2)
The effective scalar creep strain rates are given by Kameyama et al. (1999):
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0034(3)
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0035(4)
where the symbols A, E and V denote the experimentally prescribed pre-exponential factor, the activation energy and the activation volume, respectively, R denotes the gas constant, T is the temperature, n is the power law exponent, d is the grain size, and p is the grain size exponent. C urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0036 is water content in H/106Si, and rdiff and rdisl are the water content exponents.
Within the top 300 km (SLIM3D) of the coupled model, we self-consistently account for the plastic deformation according to a Drucker-Prager criterion, based on the full dynamic pressure as opposed to using the lithostatic pressure:
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0037(5)
where c is the cohesion and μ the coefficient of friction. Along the weak plate boundaries, we evaluate the influence of plastic (brittle) deformation on global plate motions by varying μ. Following Sobolev et al. (2009) we use the reduced friction coefficient values at the predefined plate boundaries (Bird, 2003) treated as narrow vertical zones in the crustal and lithospheric layer in the depth range 0–80 km, and high friction coefficient of 0.6 in all lithospheric materials outside of the plate boundaries.

The upper mantle creep viscosity is calculated using olivine parameters from the axial compression experiments of Hirth and Kohlstedt (2004). Crustal rheology is taken from Wilks and Carter (1990). All rheological parameters used in this study, except for the water content which is varied in section 6 (also see supporting information for the varying water content), are summarized in Table 1. For more details regarding the formulation of the physical model and numerical implementation the reader is referred to Popov and Sobolev (2008).

Table 1. Parameters for the Thermo-Mechanical Modeling of the Upper Mantle Modified After Hirth and Kohlstedt (2004)
Parameter Unit Crust

Lithosphere

(strong mantle)

Asthenosphere

(weak mantle)

urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0038 GPa 6.3 12.2 12.2
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0039 GPa 4.0 7.40 7.40
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0040 gcm– 3 2.85 3.27 3.30
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0041 MPa 5.0 5.0 5.0
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0042 - 0.6 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0043 0.6 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0044 0.6 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0045
Diffusion creep parameters (p =3, d=10 mm, r = 1)
Adiff Pa– 1s– 1 - 10 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0046 10 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0047
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0048 KJ/mol - 375 335
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0049 cm– 3/mol - 6.0 4.0
Dislocation creep parameters Dislocation (p = 0, r = 1.2)
Adisl Pa urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0050s– 1 10 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0051 10 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0052 10 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0053
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0054 KJ/mol 445 530 480
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0055 cm– 3/mol 10.0 17.0 14.0
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0056 - 4.2 3.0 3.0
  • Note. Olivine water content used for our viscosity profiles 100, 200, 500, 800, 1000 H/106Si in the weak mantle (asthenosphere), which is included in the pre-exponential factor for olivine, with dry lithospheric parameters. The creep parameters for the crust are given by Wilks (1990]. (*Away from plate boundaries).
In oceans, the 3-D thermal structure for the upper 300 km is inferred from sea floor age (Müller et al., 2008). We have implemented a half-space cooling model to determine temperature Tocean as a function of age and depth according to
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0057(6)
where urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0058 m2s– 1 is thermal diffusivity, τ is the age of the oceanic lithosphere, Ts is the reference surface temperature, Tm is the reference mantle temperature and z is depth below the surface. It is combined with the TC1 model (Artemieva, 2006) in continents, which is based on a compilation of heat flow data. We account for slabs in the upper mantle (300 km), using slab locations and depths from Steinberger (2000) and the slab age with equation 6 to infer a corresponding temperature field. In Figure 2, slices of the inferred thermal structure for the upper mantle at depths of 50, 100, 150 and 250 km show cratons and slabs as the coldest structures and ridges with the highest temperatures. For our modeling setup, we used reference crustal, lithospheric and asthenosphere densities (Table 1) and accounted for lateral density variations from our 3-D thermal structure (Figure 2) using the relation:
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0059(7)
where α is the thermal expansivity coefficient, chosen to be urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0060 K–1 within the lithospheric and asthenospheric mantle, K (Table 1) is the bulk modulus and ρref the reference density at reference temperature ( urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0061C) and zero pressure. In the crustal layer, we use urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0062 K–1 with a thermal conductivity of 2.5 W/m/K and a heat production rate of 0.5 nW/kg. In both the crust and lithospheric mantle, we use the value of 1.2 kJ/kg/K for the specific heat capacity and neglect the heat production in the lithospheric mantle for our short simulation time (0.5 Ma).
Details are in the caption following the image

3-D thermal structure at depths of (a) 50 km, (b) 100 km, (c) 150 km, and (d) 280 km, showing (a) high temperature due to mid-ocean-ridges, (b and c) cold cratonic roots in continents and d) cold subducting slabs along most convergent plate boundaries.

2.3 Validation of the Modeling Technique

We test our modeling technique by comparing our modeling results with the results of well-established modeling techniques. First we consider models of mantle convection with radial viscosity distribution in the entire mantle, 3-D density distribution from scaling of the SMEAN tomographic model (Becker & Boschi, 2002) and free slip boundary condition at the surface and core-mantle boundary. We compute velocities using the spectral code based on Hager and O'Connell (1981) for two different viscosities in the upper 100 km and the same viscosity structure in the rest of the mantle. The resulting velocities at the surface and at depth 300 km are shown in supporting information Figures S1b and S1c (left column) and corresponding vertical cross sections of viscosity and velocities are shown in supporting information Figures S2 and S3 (left column). We then compute velocities for the same viscosity and density distributions in the mantle and the same boundary conditions but with our modeling technique that combines the SLIM3D finite element code in the upper 300 km and the spectral code deeper than 300 km (supporting information Figures S1b, S1c, S2, and S3, right column). In this case we use a finite element size of 2 angular degrees obtaining the lowest RMS misfit 2% for surface velocities with lithosphere viscosity 1023 Pa s and the largest misfit of 8% for velocities at 300 km depth (supporting information Figure S1c). The difference between velocities computed using combined and single conventional modeling techniques is small for both viscosity structures. The vertical profiles over the Nazca and Sumatra subduction zones for both numerical techniques are very similar (supporting information Figures S2–S3).

To further test our numerical model with a 3-D viscosity structure in the upper mantle, we attempted to reproduce the model by Becker (2006) with temperature and strain-rate dependent viscosity, computed with the CitcomS code (Zhong et al., 2000). We used the same 3-D density distribution in the mantle and 3-D temperature and viscosity structure in the upper 300 km as Becker (2006) including the same low viscosity at plate boundaries. In the mantle deeper than 300 km we used a radial viscosity distribution calculated by laterally averaging the 3-D viscosity distribution of Becker (2006). Supporting information Figure S4 shows a comparison of surface velocities in the NNR reference frame of the model by Becker (2006) (red arrows) and our combined-technique model (blue color). Our model was calculated with finite element size of 1 angular degree and we ran the model for a model time of 250 thousand years. Supporting information Figure S4 shows that the surface velocities computed with two different modeling techniques are very similar. This is also true for the location of the pole of net rotation and the angular velocity of net rotation of the lithosphere (red and blue symbols and numbers in supporting information Figure S4). Interestingly, surface velocities and net rotation are almost the same in both models, although Becker (2006) has a 3-D viscosity distribution in the mantle below 300 km, while our model has only radially dependent viscosity below 300 km. From that we infer that relatively smooth lateral variations of viscosity deeper than 300 km, as in Becker (2006), affect neither the integrated tractions at the base of large lithospheric plates nor net rotation much. This justifies the assumption of radial-only dependence of viscosity in our model below 300 km depth. Another conclusion is that our combined modeling technique produces very similar results as the CitcomS code.

Finally, we test our numerical method by considering the modeling parameters from Alisic et al. (2012), with their yield stresses 100 MPa and 800 MPa and stress exponents n = 3.0, 3.5 and 3.75. The most significant observation is that as we increase the stress exponent from n = 3.0 to n = 3.5, the plates move faster, more than three fold, giving RMS velocities of 2.79 cm/yr (supporting information Figure S5, left column) and 9.57 cm/yr (supporting information Figure S6, left column) respectively, due to large strain rate weakening (Alisic et al., 2012). Resulting vertical cross sections show regions of continental keels (supporting information Figures S5 and S6, right column). However, when we increase the yield stress from 100 MPa to 800 MPa only for plate interiors without influencing plate boundaries, we observe almost no change in either magnitudes or directions of the lithospheric plate motions with RMS 2.79 cm/yr (n = 3.0, urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0063) and 2.78 cm/yr (n = 3.0, urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0064). These results suggest that plate velocities are less dependent on the strength of the plate interior, but largely on how easily the boundaries can deform, as they move relative to each other. We believe that the tests described above demonstrate that our combined modeling technique is robust and is able to reproduce modeling results of well-established modeling techniques.

3 Results and Discussion

3.1 Average Creep Viscosities and Corresponding Basal Tractions

First we compute a global dynamic geoid model and basal shear tractions at 300 km (Figures 3b and 3d) to ascertain that our upper mantle lateral viscosity variations yield realistic results. Then we compare the predicted geoid to an estimate from the layered/radial viscosity (ηr) model of Steinberger and Calderwood (2006) and the observed GRACE geoid model (supporting information Figures S7a–S7c). In Figure 4b, we show profiles of the resulting creep effective viscosity for continents and oceans within the top 300 km using the creep parameters given in Table 1 (see supporting information Table S1–S5) and our 3-D upper mantle thermal structure (Figure 2). The corresponding diffusion and dislocation creep estimates are shown in Figure 4a. We have conducted a series of experiments by varying the water content in the asthenosphere (C urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0065) within the following set: 100, 200, 500, 800, 1,000 ppm H/10 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0066Si. Figure 4b, shows how the average asthenospheric viscosity decreases with water content (Green et al., 2010; Karato, 2010).

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(a) Predicted basal tractions at 300 km with a layered viscosity structure (Steinberger & Calderwood, 2006) and (b) lateral viscosity variations from olivine parameters consistent with the depth averaged creep viscosity profile in Figure 4 (the pink line 500 H/10 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0067Si) with urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0068 along plate boundaries. (c and d) The same with one order of magnitude lower viscosity in the asthenosphere, with urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0069 at ocean-continent subducting regions and urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0070 at the remaining plate boundaries. The predicted geoid models corresponding to a) and b) are shown in supporting information Figure S7.

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(a) Calculated average strain rate versus depth for diffusion and dislocation creep across continents and oceans. (b) The corresponding average creep viscosity versus depth in the upper mantle with olivine parameters. (c) Reference temperature profile within the crust and lithospheric layers and (d) Yield strength profile using frictional coefficient urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0071 as inferred for dry rocks from laboratory experiments (Byerlee, 1978) and smaller frictional coefficients with urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0072 as suggested by numerical simulations (Moresi & Solomatov, 1998) used as an upper limit for the plate boundaries.

In general, all oceanic average viscosity profiles lie below their respective continental average effective viscosity (ηeff), within the urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0073 km depth range, which is broadly consistent with the mean depth of the oceanic asthenosphere. This depth range of low oceanic viscosity (Figure 4b) corresponds to a seismic wave velocity drop (∼ 5–10%) in most recent seismological studies showing the transition between lithosphere and asthenosphere (e.g., Fischer et al., 2010; Kawakatsu et al., 2009; Rychert et al., 2005; Schaeffer & Lebedev, 2013). The difference between the averaged ocean and continent viscosity profiles in the depth range of urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0074 km is urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0075 order of magnitude. We obtain the lowest average viscosity of urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0076 Pas for 1000 ppm H/10 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0077Si olivine beneath oceans. Figure 3 shows basal tractions causing plate motions and stresses in the lithosphere for some representative cases; the resulting geoid with prescribed rheology is shown in supporting information Figure S7b. In Figures 3a and 3c only radial viscosity variation (Steinberger & Calderwood, 2006) is considered, whereas in Figures 3b and 3d the creep parameters corresponding to the effective viscosity profile in Figure 4b (blue profile) are used, with yield stress adjusted with the coefficient of friction urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0078 (Figure 4d) within the crust and lithospheric layer above depth 80 km for all plate boundaries. The influence of the frictional parameter at plate boundaries will be discussed in the following sections with regards to global plate motions. The predicted geoid with LVVs (supporting information Figure S7b) when compared with the observed geoid (Reigber et al., 2005; supporting information Figure S7a) yields a correlation of 0.85, which is higher than the correlation of 0.82 obtained with the radial viscosity profile (Figure 1b). High geoid signal above the Andes and low signal in continental Europe and North America were reproduced quite well with the LVVs model.

3.2 Plate Motions

We use our coupled model to predict global plate motions (Figure 5), following the upper mantle creep parameters derived with lithospheric basal tractions (examples are shown in Figure 3) and a free surface as the top boundary condition. We seek to understand how plate boundary frictional deformation and asthenosphere viscosity influence global plate velocities. To compute global plate motions in a NNR reference frame, we explore both a layered/radial viscosity structure and LVVs for the upper mantle. For LVVs we first test olivine parameters (Table 1) corresponding to the blue average creep viscosity profile (Figure 4b). We use a fixed maximum and variable minimum viscosity cutoff in the top 300 km (see section 9) and a prescribed yield stress (coefficient of friction) (Figure 4d) at plate boundaries. The predicted global plate motions (Figure 5) are compared with the observed plate motion model NUVEL-1A (DeMets et al., 2010) in a NNR reference frame. We compare the global root mean square (rms) velocity (Figures 6a and 6b) and the angular misfits (Figures 6c and 6d) between the modeled and observed plate motions for different values of the plate boundary friction coefficient due to plastic yielding and asthenospheric cutoff viscosity. Here we used a variable lithosphere thickness (Table 2) with our stress and strain-rate dependent viscosity and a 3-D temperature structure for the upper mantle (discussed above in section 3) to predict our first set of plate velocities. Our second set of predicted plate motions is based on a Newtonian/layered viscosity structure (Steinberger & Calderwood, 2006) at all depths. Our calculation with layered/radial viscosity for the upper mantle is similar to previous geodynamic modeling studies of plate motions (Becker & O'Connell, 2001; Lithgow-Bertelloni and Richards, 1998), but in contrast to either imposing surface plate velocity (Becker, 2006; Becker & O'Connell, 2001) or using weak zones with a viscosity drop (Becker, 2006; Conrad & Lithgow-Bertelloni, 2006) we have imposed plastic yielding at the plate boundaries for a dynamically self-consistent generation of plate-like surface motion.

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(a) Set of predicted plate motions with an increasing friction coefficient (plastic yielding) at plate boundaries for different minimum viscosity cutoffs of the asthenospheric LVVs with olivine parameters corresponding to the blue profile in Figure 4. We show here plate motions with coefficient of friction up to 0.06. The rectangular boxes in Figure 5a show the best fit to the observed plate motion data shown in (b) (DeMets et al., 2010).

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Estimates of global root mean square (RMS) of predicted plate motions with (a) LVVs and (b) layered viscosity in the top 300 km for different coefficients of friction compared to the observed rms velocity of NNR-NUVEL-1A (DeMets et al., 2010), (c and d) their respective global angular misfits. e and f) The root mean square error (RMSE) for the velocity in percent. Viscosity cutoff η in Pa s.

Table 2. Summary of the Upper Mantle Viscosity for Different Simulations
Upper mantle Lateral viscosity variation (Pas) Layered viscosity variation (Pas)
1. Lithosphere layer
a. urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0079 ηeff max cutoff = 1024 1024
b. urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0080 ηeff max cutoff = 1024
urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0081
2. Asthenospheric layer ηeff min cutoffs = 1018 – 1021 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0082 = 1018 – 1021

3.3 Influence of Plate Boundary Friction on Plate Velocities

The lithospheric plate driving force coming from slab pull within the upper mantle (Conrad & Lithgow-Bertelloni, 2002, 2004; Forsyth & Uyeda, 1975; Harper, 1975; van Summeren et al., 2012) and deep mantle slab suction are broadly evident in all subducting surface plates, when we use a very low plate boundary friction urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0083 (Figure 5) and a relatively low asthenosphere viscosity. The Pacific plate, Nazca plate, Indo-Australian plate and Cocos plate are all moving with a maximum speed up to ∼ 20 cm/yr. On the other hand, non-subducting oceanic plates are moving with a relatively low velocity urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0084 cm/yr, and are mainly driven by either ridge push or gravitational sliding of the oceanic plate (Artyushkov, 1973; Conrad & Lithgow-Bertelloni, 2002; Harper, 1975; McKenzie, 1972) or slab suction from below 300 km, with the same low coefficient of friction urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0085 at plate boundaries and low asthenospheric viscosity. When we increase the resistive force due to friction by increasing the coefficient of friction at plate boundaries while keeping the same asthenospheric viscosity, we observe a corresponding decrease in lithospheric plate speed for all plates, which is more pronounced in subducting plates. For example, with a minimum asthenosphere viscosity cutoff 1018 Pas rms plate motion speed decreases by about 2 cm/yr for an increase of the coefficient of friction by 0.01 leading to an increase in yield stress (Figures 5 and 6a). This brings lithospheric plates to an almost stagnant lid regime as we move toward higher coefficients of friction urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0086 (Crameri & Tackley, 20152014, 2015), thereby resisting almost all of the total driving force due to slabs and basal traction. With higher coefficient of friction urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0087 the yield stress at plate boundaries is mostly not reached, even with the lowest asthenospheric viscosity 1018 Pas. This corresponds to a general decrease in the predicted rms global plate velocities (Figure 6a) as we increase the coefficient of friction from 0.01 (friction angle urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0088) to 0.1 (friction angle urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0089) for all minimum viscosity cutoffs. A similar gradual decrease in global rms velocities is obtained with layered viscosity models (Figure 6b).

Previous studies (e.g., Wu et al., 2008) have considered the slab bending force at subduction zones, and have arrived at similar conclusions on the effect of the resistive forces between subducting and overriding plates. The older is the subducting plate, the thicker it is and the stronger is the slab pull force. Hence, at intermediate coefficients of friction (e.g., 0.02, 0.03, and 0.04 for cutoff viscosity of 1018 Pas, Figure 5) young, thin subducting plates (e.g., Nazca and Cocos) move much slower, since the smaller slab pull does not tend to cause plastic yield. In contrast, it is easier for older, thicker plates (such as the Pacific) to achieve yielding and thus to move relatively fast. On the other hand, older plates become comparatively slower as we increase the asthenospheric viscosity while keeping the same coefficient of friction (see section 9).

Our findings that low friction coefficient urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0090 (low friction angle urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0091) is required at plate boundaries to generate global plate motions in a realistic fashion is consistent with previous studies (Crameri & Tackley, 2015; Faccenda et al., 2009; Hall et al., 2003; Hassani et al., 1997; Sobolev & Babeyko, 2005; Tan et al., 2012) that have considered the influence of frictional parameters on the subduction interface and numerical studies of global plate motions with plastic yielding at plate boundaries (Crameri & Tackley, 2014; Stadler et al., 2010). In contrast, the coefficient of friction urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0092 at plate boundaries from our preferred model is much lower than the value of ∼ 0.1 suggested by previous numerical studies of mantle convection (Moresi & Solomatov, 1998; Richards et al., 2001; Tackley, 2000a). The weak nature of plate boundaries that is required for plate motions (Iaffaldano et al., 2006), as well as subduction and fault systems (Bird & Kong, 1994; Carena & Moder, 2009; Humphreys & Coblentz, 2007; Mount & Suppe, 1987; Townend & Zoback, 2004; Zoback et al., 1987), compared to high values of yield stresses observed in the plate interior (Zhong & Watts, 2013), is also supported by observational constraints from heat flow data at plate boundaries (Gao & Wang, 2014) showing the different behavior of the lithospheric plates in their interior and at their boundaries. Our estimated friction coefficients are close to values (0.01 - 0.05) derived from models of subduction orogeny in the Andes (Sobolev & Babeyko, 2005).

3.4 Influence of the Asthenosphere Viscosity Structure on Plate Velocities

The second quantity that is varied in our simulations (Figure 5) is the viscosity contrast between the lithosphere and the asthenosphere, with minimum asthenosphere viscosity cutoff values for the LVVs ranging from 1018 Pas to a much higher value 1021 Pas. The distinct nature of the asthenosphere's mineral composition and thermal structure, probably due to the process of partial melting (Karato, 2012; Schmerr, 2012) and its water content (Hirth & Kohlstedt, 1996) results in a viscosity drop with respect to the overlying lithospheric plates (Dziewonski & Anderson, 1981). The high temperatures below mid-ocean-ridges and oceanic plates contribute to partial melting leading to low values in viscosity (e.g., Figure 4b, ocean profiles) compared to the cold continental cratonic regions (e.g., Figure 4b, continent profiles) resulting in high magnitudes of lateral viscosity variations (Karato, 2012). We test the plausible minimum viscosity values of the asthenosphere to study their influence on global plate motions. The use of higher cutoff values leads to a gradual decrease in the modeled plate velocities. At low cutoffs urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0093 Pas, and low plate boundary friction ( urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0094) both subducting and non-subducting plates move with a relatively high speed, with a predicted global rms often greater than 7 cm/yr (Figure 6a), exceeding the observed rms of ∼ 3.7 cm/yr for global plate motion model NNR-NUVEL-1A (DeMets et al., 2010). For the lowest viscosity cutoffs of 1018 Pas and 1019 Pas and μ = 0.01, the modeled average speed is more than twice the observed speed, giving estimates of root mean square errors (RMSE) ∼ urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0095 % (Figure 6e, red and green lines), due to the weak mechanical coupling between the lithosphere and asthenosphere (i.e. viscous drag) and between the plates. This results in a smaller drag force exerted by the mantle on plates, allowing them to move relatively faster, even in continental regions with high resistance due to keels. Note that with the weak viscous drag at the base of oceanic plates, the strong driving force due to slab pull and mantle suction (Conrad & Lithgow-Bertelloni, 2004; Forsyth & Uyeda, 1975; Harper, 1975; Stoddard & Abbott, 1996) results in a rapid motion of the Pacific, Indo-Australian, Nazca and Cocos plates with a predicted speed of more than ∼ 15 cm/yr.

Nonetheless, for low asthenosphere viscosity cutoffs of urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0096 Pas and low urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0097, the predicted plate motion directions are rather similar to the observed plate motion directions for NNR-NUVEL-1A (DeMets et al., 2010) with an angular misfit urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0098 (Figure 6e). However, as we increase the plate boundary friction coefficient to urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0099 while maintaining the same low asthenosphere viscosity cutoffs urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0100 Pas, the angular misfit increases and the corresponding rms velocity decreases to ∼ 1 cm/yr or less, as the resistance at plate boundaries increases and plates become slower. Also, increasing the minimum asthenosphere viscosity cutoffs to urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0101 Pas deteriorates the fit for both direction (angular misfit) and magnitude (RMS) of plate motion (blue, cyan and purple profiles in Figure 6), for all coefficients of friction at plate boundaries with either LVVs or layered viscosity calculations.

We have quantified the contribution of slabs in the top 300 km of the upper mantle by removing the slabs, and obtained a decrease in the rms velocity of ∼ 20% (Figure 7a) as well as an increase in the angular misfit (Figure 7b). The largest contribution comes from the mantle buoyancy below our coupling depth of 300 km providing about 70% of the driving force. The angular misfit is lowest if all contributions from the upper and lower mantle are considered (Figure 7b). In particular, angular misfit can reach 50 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0102 when we set the mantle buoyancy force to zero, thus setting all the tractions coming from the deep mantle, below our coupling interface at 300 km depth, to zero.

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(a) Estimated global rms velocity and (b) angular misfit for the modeled plate motion from the simulaton including the contributions of the upper and lower mantle (Total), simulation excluding the contribution of slabs in the upper 300 km (No SL in 300 km), and those excluding the contribution of the mantle below 300 km (No LM drive). Viscosity cutoff η in Pa s.

Our preferred model was obtained with minimum asthenosphere viscosity cutoff =1020 Pas and μ = 0.02 giving a global rms of 3.58 cm/yr and angular misfit of ∼ 8° (Figures 6a and 6c). In Figure 8 we have qualitatively compared this result to the NUVEL-1A plate motion model (DeMets et al., 2010) in a NNR reference frame. There is generally a good fit in terms of both magnitude and orientation, but an almost ∼ 30 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0104 deviation is found in Australia and more than 40 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0105 in parts of North America (Figure 8c). To achieve a better fit between the model and observations in these regions we have experimented with different coefficients of friction for subducting boundaries, transform and divergent boundaries. In Figure 8d, we have used a minimum asthenosphere viscosity cutoff of 1020 Pas and μ = 0.0124 for subducting plate boundaries with half the cohesion value and μ = 0.0219 for all other plate boundaries. This has resulted in an improved fit to the observed plate motions in almost all regions for both magnitude and orientation. The better fit with smaller friction coefficient in subduction zones could be due to additional lubrication from sediment and water between the subducting and overriding plates as observed in nature (Lamb & Davis, 2003; Schlaphorst et al., 2016; von Huene & Ranero, 2003). However, using different friction coefficients for subducting plate boundaries and other plate boundaries has resulted in a westward motion of North and South America at a higher speed of ∼ 5 cm/yr and also a fit degradation for the Nazca plate.

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(a). The observed plate motion NUVEL-1A in a no-net-rotation (NNR) frame (DeMets et al., 2010). (b) Our preferred modeled plate motions (min cutoff =1020 Pas and friction μ = 0.02). (c) Direction of NUVEL-1A (blue arrows) and our preferred (red) plate motion as in (b). (d) Direction of NUVEL-1A (blue arrows) plotted on predicted (red) plate motion with min cutoff 1020 Pas but with μ = 0.0142 (cohesion c= 2.5 MPa) for subducting plate boundaries and μ = 0.0219 for all remaining plate boundaries.

On average, the predicted plate motions are faster (Figure 6a) with LVVs and a given asthenosphere viscosity cutoff than with the layered/radial viscosity (Figure 6b). This is due to relatively faster lateral asthenospheric flow that develops under oceanic plates (Becker, 2006), resulting from the stress and strain-rate dependent viscosity implemented together with a 3-D thermal structure of the upper mantle. The low viscosity channel at ca. 100 km depth is seen in Figure 4b and discussed in section 6 Plate motion predicted with either LVVs or layered viscosity fits the observed plate velocities best when the asthenosphere viscosity is within the range of ∼ 5 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-01061019–1020 Pas underlying a lithosphere of variable thickness. This viscosity range is consistent with the values estimated by post-glacial rebound and geoid studies to constrain the upper mantle viscosity structure (Mitrovica, 1996; Mitrovica & Forte, 1997). Similar values of the asthenospheric viscosity were obtained by Cerpa et al. (2014), who studied the effect of mantle viscosity on the cyclicity of Andean slab folding.

3.5 Impact of Lithospheric Thickness on Plate Velocities

We then test our modeling setup with different values of uniform lithospheric thickness characterized by the radial viscosity of 1024 Pas and constant density within the lithosphere (Table 2). Our approach is similar to other modeling studies predicting global plate motions (Becker & O'Connell, 2001; Conrad & Lithgow-Bertelloni, 2002; Lithgow-Bertelloni & Richards, 1995). In the asthenosphere below each nominal lithospheric depth (60, 80, 100 and 150 km), we explore two scenarios: (I) with the presence of cratonic roots and slabs (LVVs, Figure 9a) and (II) without cratons and slabs but instead assuming a constant temperature of 1,375 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0107C (Figure 9b). In the first scenario (I), due to the presence of the LVVs beneath the nominal depth of the lithospheric base, the patterns of predicted plate motions with different uniform lithosphere thicknesses are quite similar to the patterns with variable lithospheric thickness in most regions, since the basal tractions due to mantle flow are not significantly altered (Conrad & Lithgow-Bertelloni, 2006). This is especially evident when we compare the angular misfit from model predictions with variable lithosphere thickness (Figures 6c) to predictions with different uniform lithosphere thicknesses (Figures 9a, second line) showing a similar minimum at friction coefficients 0.02 for LVV cutoff η = 1020 Pas. However, in models with constant temperature in the asthenosphere, we have obtained larger angular misfits, even with RMS similar to those in Figure 9a. This suggests that the presence of cratons and potentially slabs is necessary to correctly predict the plate motion orientations. In summary, thinner lithospheric plates (60 km and 80 km) are moving relatively fast with the largest rms velocities ∼ 13 cm/yr and ∼ 11 cm/yr respectively, compared to a thicker lithosphere (100 km and 150 km) with the largest rms velocities of ∼ 9 cm/yr and ∼ 8 cm/yr, respectively. As we increase the strength of the plate boundaries, all rms estimates for the predicted plate velocities decrease monotonically to ∼ 1 cm/yr or less, as in the case of the models using variable lithosphere thickness.

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Modeled plate motion global rms and angular misfits for uniform lithosphere thickness of 60, 80, 100 and 150 km, (a) with and (b) without the effect of cratons and slabs below each nominal depth using plate boundary friction ranging from 0.01 to 0.1. Viscosity cutoff ηeff in Pa s.

On average we obtain faster plate velocities using lithosphere thicknesses 60 km and 80 km than for the variable lithosphere thickness, whereas simulations with thicknesses 100 km and 150 km lead to a relatively lower speed with or without the effect of cratons and slabs. Significantly higher plate velocities of the thin lithospheric plates (60 km and 80 km) are due to thicker asthenosphere, that decreases the internal mantle traction by almost one third (Conrad & Lithgow-Bertelloni, 2006; Stoddard & Abbott, 1996), resulting in a lower resistance from the viscous/mantle drag to the slab pull or mantle suction forces. On the other hand, velocities with the thicker lithosphere plates are slower, because the internal mantle drag force through the thin asthenospheric layer at the base of the lithosphere provides more resistance to the driving force (Conrad & Lithgow-Bertelloni, 2006; van Summeren et al., 2012). The thinner the asthenospheric layer is the stronger is the driving force required to match observations. van Summeren et al. (2012) showed that, with no asthenosphere (i.e. using the reference mantle viscosity value in the asthenosphere) 100% of slab pull is required to match the observed plate velocity. Among the four simulations using uniform lithospheric thickness, the one using a thickness of 100 km yields the results most comparable with our simulations using a variable thickness of the lithosphere plates. Both were able to approximately match the observed plate motions from NUVEL-1A, with similar settings of the asthenosphere viscosity cutoff and plate boundary friction giving rms of 3.58 cm/yr for a variable lithosphere thickness and 3.79 cm/yr for a uniform lithosphere thickness of 100 km.

3.6 Resulting Net Rotation

Here we estimate the excitation of net rotation (NR) of the lithosphere by the LVVs, and the influence from varying plastic yielding at the plate boundaries. We do this for different coefficients of friction at the plate boundaries and for different asthenospheric viscosity cutoffs as in the previous sections. In Figures 10 and 11, we plot the amplitudes of NR and Euler pole locations for different asthenospheric viscosity cutoff values against the strength at plate boundaries varied through the coefficient of friction, for variable and uniform lithospheric thicknesses. We have also tested how large NR can be generated with layered viscosity structure (Steinberger & Calderwood, 2006) combined with different friction coefficients at plate boundaries (Figures 10b and 10d). Theoretically, mantle buoyancy structure combined with radial/layered viscosity causes no NR, and most numerical studies have shown it to be less than 0.1% of the observed NR (e.g., Becker, 2006; Ricard et al., 1991). With the introduction of plastic yielding at the plate boundaries, however we obtain larger, but still relatively low amplitudes of NRs which are mostly urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0108/Ma.

Details are in the caption following the image

Net rotation amplitudes and Euler pole locations of the lithospheric plates with respect to the lower mantle for (a and c) LVVs and (b and d) layered viscosity structure. Published NR amplitudes and corresponding Euler pole locations from observation-based and previous geodynamic studies SB04 (Steinberger et al., 2004), HS3 (Gripp & Gordon, 2002), GJ86 (Gordon & Jurdy, 1986), TV10 (Torsvik et al., 2010), T22 (Wang & Wang, 2001), Z01 (Zhong, 2001) and TB6a, TB6b, TB6c and TB6 (Becker, 2006) are plotted as yellow circles and squares, respectively, with size (in Figures 10c and 10d) corresponding to amplitude. For yellow circles and squares in Figures 10a and 10b only the y-coordinate is relevant, while the x-coordinate is arbitrary. Numbers at colored circles in c and d indicate the coefficient of plate boundary friction when multiplied by 0.01. Viscosity cutoff η in Pa s.

Details are in the caption following the image

Net rotation amplitudes and Euler pole locations for uniform lithosphere thickness 60, 80, 100 and 150 km without the effect of cratons and slabs below each nominal depth. Numbers at colored circles in the right column, indicate the coefficient of plate boundary friction when multiplied by 0.01. Viscosity cutoff η in Pa s.

Net rotation is close to zero when the asthenosphere viscosity is larger than 1020 Pas. The corresponding Euler pole locations (Figure 10c) with coefficients of friction urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0109 are placed in the Indian Ocean, close to most of the observation-based Euler poles, but as we increase the friction at the plate boundaries the pole location moves further away, similar to the findings of Becker (2006) who used strain-rate dependent rheology. Using layered viscosity structure tends to yield Euler poles further away from observation-based pole locations. Taken together with the low NR amplitudes, this confirms the importance of the LVVs in generating NR as previously suggested in several studies (e.g., Alisic et al., 2012; Becker & Faccenna, 2009; O'Connell et al., 1991; Ricard et al., 1991; Stadler et al., 2010; Zhong, 2001). Figure 10a shows that simulations with a variable lithosphere thickness and a free surface, using stress and strain-rate dependent viscosity and a 3-D thermal structure of the upper mantle are able to realistically reproduce a NR comparable to observations in a hotspot reference frame (Gordon & Jurdy, 1986; Steinberger et al., 2004; Torsvik et al., 2010; Wang & Wang, 2001) only for a low asthenosphere viscosity cutoff of urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0110 Pas. NR amplitudes constrained by azimuthal seismic anisotropy (e.g., Becker, 2008) and most of the observation-based estimates - with the exception of HS3 (Gripp & Gordon, 2002) fall within the range of 0.1 - 0.2°/Ma. Similarly, large NR amplitudes were reported by Alisic et al. (2012), who used a self-consistent global mantle flow model which also accounted for nonlinear slab rheology and plastic yielding at plate boundaries.

On the other hand, earlier 3-D numerical studies of global mantle flow (Becker, 2006; Zhong, 2001), with weak zones at plate boundaries and a free-slip top boundary condition estimated smaller NR magnitudes between 0.023 and 0.092°/Ma. This wide variation among different numerical predictions of NR amplitudes and also the deviations from observations (Čadek & Fleitout, 2003; Wen & Anderson, 1997; Zhang & Christensen, 1993) is due to different rheological models in the upper mantle and the choice of plate boundary deformation mechanism either through plastic yielding (e.g., Alisic et al., 2012; Bercovici, 1995; Tackley, 2000b) or imposed weak zones (e.g., Becker, 2006; Zhong, 2001). With larger lithosphere-asthenosphere viscosity contrast, for example with a minimum cutoff of 1018 Pas or 1019 Pas for the asthenosphere viscosity and urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0113 we obtain NR amplitudes that are larger than most observation-based estimates. For cutoff values below 1020 Pas, there is a monotonic decrease in the NR amplitudes when we increase the coefficient of friction from 0.01 to 0.1. For example, the NR amplitude will decrease from 0.365 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0114/Ma to 0.103 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0115/Ma if an asthenosphere viscosity cutoff of 1018 Pas and variable lithosphere thickness are used. A cutoff 5 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-01161019 Pas combined with low plate boundary friction urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0117 or less gives a net rotation which is similar to but with somewhat lower amplitudes than most of the observation-based values, and also similar to the results of Becker (2006) with temperature and strain rate dependent rheology. However, when we reduce the lithosphere-asthenosphere viscosity contrast by assigning cutoff values of 1020 Pas or greater, the predicted average NR amplitudes fall below 0.05 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0118/Ma which is similar to NR in our layered viscosity model (Figure 10b) and the prediction of Becker (2006) with only strain-rate dependent rheology. A similar conclusions also follows from theoretical results (Conrad & Behn, 2010; O'Connell et al., 1991; Ricard et al., 1991) that showed that at least one order of magnitude viscosity variation is needed to excite NR comparable to observations.

The gradual decline in the NR amplitudes with increasing plate boundary friction and asthenosphere viscosity cutoff urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0119 Pas supports the idea that a significant weakening at the plate boundaries is required to initiate lithospheric net rotation (Bercovici & Richards, 2000; Zhong et al., 1998). However, the largest effect is induced by the upper mantle viscosity. This is in accord with the notion that NR of the lithosphere is largely controlled by the LVVs resulting from the sub-continental and sub-oceanic structures, mainly continental keels (Zhong, 2001), which was also supported by Becker (2006) who used a model with strain-rate and temperature-dependent LVVs, and Gérault et al. (2012) using low-viscosity plate boundaries. The 2D-cylindrical-axisymmetric study of Gérault et al. (2012) gives a convenient way to estimate the influence of slab dip angle, slab rheology and trench properties on lithosphere NR, since there is no edge toroidal flow around slabs. In 3-D geometry these additional complexities arise, and it is furthermore possible to study the effect of depth and lateral extent of continental roots. Using 3-D geometry, Becker and Faccenna (2009) found that the slab contribution to NR is lower than that of the continental keels.

Becker (2006) showed that models with strain-rate and temperature-dependent rheology yield NR amplitudes close to observations in a hotspot reference frame, while the use of only strain-rate dependent rheology resulted in an amplitude underestimation. Our resulting Euler pole locations (Figure 10c) are mainly confined to the Indian Ocean and surroundings, with the locations moving southward toward the Antarctic continent as we increase the plate boundary friction to 0.1. This robust result occurs over a broad range of asthenospheric mantle viscosity cutoff values, even those that generate smaller NR amplitudes. With friction coefficients urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0120 and asthenosphere viscosity cutoff of urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0121 Pas, we are able to predict Euler pole locations in the vicinity of the observed present-day NR Euler pole locations (Gordon & Jurdy, 1986; Gripp & Gordon, 2002; Steinberger et al., 2004; Wang & Wang, 2001), and those obtained from previous numerical studies (Becker, 2006; Stadler et al., 2010; Zhong, 2001).

Next, we test how the choice of a constant asthenosphere temperature (no cratons and slabs) and a constant lithospheric thickness influences the predicted net rotation. With the lowest plate boundary friction urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0122, for every uniform lithosphere thickness some net rotation is excited for low asthenosphere viscosity cutoff of 1018 Pas and 1019 Pas. For urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0123, and without the presence of continental keels, the predicted NR amplitude falls around or below 0.05 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0124/Ma (Figure 11, left column), compared to higher values with cratons (see supporting information Figure S10). As an important component of the upper mantle structure continental keels are known to generate lithospheric NR (Becker, 2006; van Summeren et al., 2012; Zhong, 2001). For example, Zhong (2001) varied the thickness of the cratonic root and suggested that continental keels contribute ∼ 80% to lithosphere NR. This is also supported by Rudolph and Zhong (2014), who studied the influence of continental keels extending to the transition zone on NR for the past 200 Ma. However, none of our models with assumed uniform lithosphere thickness with or without cratons and slabs below the nominal lithosphere was able to predict the NR amplitude by Gripp and Gordon (2002). With the presence of continental keels in the asthenosphere, the 60 km lithosphere thickness gave about 85% of the HS3 amplitude (see supporting information Figure S10) for viscosity cutoff 1018 Pas with a gradual decrease, as we increase the friction coefficient, while the corresponding calculation without continental keels shows ∼ 60% of HS3 for μ = 0.01 and suddenly falls to ∼ 10% of HS3 for urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0125 (Figure 11a). Also increasing the uniform lithosphere thickness with/without cratons and slabs beneath the nominal lithosphere gives a general decrease in the NR amplitudes for all the coefficients of friction (Figure 11 and supporting information Figure S10), with 60 km thickness tending to give larger NR than 150 km.

3.7 Water Content in the Asthenosphere

Lastly, we study how a weak asthenosphere due to high olivine water content (see section 6, Figure 4b and supporting information Tables S1–S5) influences the lithospheric net rotation. Here we do not apply an asthenosphere viscosity cutoff with the natural minimum viscosity in our model occurring at 1017 Pas. Modeled plate velocities with 1,000 H/10 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0126Si in the asthenosphere exceed ∼ 20 cm/yr leading to the largest net rotation amplitude of up to ∼ 0.8349°/Ma (Figure 12a, red profile) and giving a global rms urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0127 cm/yr for urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0128 (see supporting information Figure S11). A twofold decrease in the olivine water content (i.e. 500 H/10 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0129Si) in the asthenospheric layer above 300 km depth, resulted in lower net rotations, similar to most of the observation-based estimates of around 0.1-0.2 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0130/Ma, even for low coefficients of plate boundary friction μ = 0.03 or less. When we increase the plastic yield at plate boundaries, we observe monotonic decrease in NR amplitudes similar to our previous calculations. The Euler pole locations (Figure 12b) for different olivine water content in the asthenosphere do not deviate significantly from each other, but all show a similar southward drift for higher values of plate boundary friction.

Details are in the caption following the image

(a) Predicted net rotation (NR) for different water content in the asthenosphere compared with observation-based estimates and previous numerical predictions (as in Figure 10) and (b) Corresponding Euler pole locations. Numbers at colored circles in c and d indicate coefficient of plate boundary friction when multiplied by 0.01.

4 Conclusions

We have studied effects of plastic yielding at plate boundaries and lateral viscosity variations (LVV) beneath lithospheric plates (due to deep cratonic roots, strong subducting slabs, and cooling of oceanic plates), on present-day plate velocities and net rotation of the lithosphere. From our results it is evident that frictional deformation at the plate boundaries influences both magnitude and orientation of plate velocities. To match the observed plate velocities in a no-net rotation reference frame, the frictional coefficient for the plate boundary yield stress should be less than 0.04 and the asthenospheric viscosity within the range of ∼ urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0131 Pas. To additionally match lithosphere net rotations, asthenosphere viscosity ∼ urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0132 Pas is preferred. Our inferred frictional parameters are similar to those used in previous studies to initiate and sustain subduction (Becker & Faccenna, 2009; Gerbault et al., 2009; Hall et al., 2003; Hassani et al., 1997; Sobolev & Babeyko, 2005; Tan et al., 2012). However, high values of frictional coefficient along plate boundaries lead to an almost stagnant lithospheric lid instead of moving lithosphere plates, as has also been reported by Crameri and Tackley (2015), hence lithospheric plates have strong interior (Zhong & Watts, 2013) and weak boundaries (Carena & Moder, 2009; Iaffaldano et al., 2006; Sobolev & Babeyko, 2005). We conclude that to attain the observed velocity of plate motions due to slab pull and slab suction, the inter-plate frictional force has to be low, with coefficient of friction urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0133 to 0.04. Previous geodynamic studies have imposed viscosity drop at plate boundaries (e.g., Becker, 2006) to allow the motion of the individual plates relative to each other or prescribed velocities as a boundary condition in mantle flow studies (Becker & O'Connell, 2001; Steinberger et al., 2004)

Also, plate boundary friction must be low to allow plates to obtain the observed magnitude of net rotation (NR) and Euler pole location, as high values of friction coefficient tend to cause the location of the Euler pole to drift southward to Antarctica. This was also shown with layered viscosity structure with plastic yielding at plate boundaries, which excites little to no net rotation of the lithosphere. NR amplitude is strongly influenced by viscosity in the asthenosphere and by lateral viscosity variations due to the presence of cratonic roots (Zhong, 2001). We are able to match observation-based NR amplitudes in the range of 0.1 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0134/Ma to 0.2 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0135/Ma and the location of the Euler pole (Doubrovine et al., 2012; Gordon & Jurdy, 1986; Steinberger et al., 2004; Torsvik et al., 2010; Wang & Wang, 2001) with urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0136 and asthenosphere minimum viscosity cutoff 1019 Pas to 5 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-01371019 Pas. With larger lithosphere-asthenosphere viscosity contrast of ∼ 4 – 5 orders of magnitude and low friction coefficients urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0138 we are able to predict a larger NR amplitude in close agreement with the observations reported by Gripp and Gordon (2002) (HS3) and some results of previous numerical studies (Alisic et al., 2012; Conrad & Behn, 2010; Gérault et al., 2012; Stadler et al., 2010), while with urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0139 we obtain too low NR amplitudes. Our dynamically self-consistent coupled lithosphere-mantle flow model includes plastic yielding at plate boundaries, as almost all successful numerical models (Alisic et al., 2012; Stadler et al., 2010) that have been able to match observed NR amplitudes, instead of a weak zone/viscosity drop, and we treat our upper boundary as a free surface. Our preferred NR with magnitude 0.10°/Ma and Euler pole location 49.31 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0141S and 74.88 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0142E was obtained with a minimum viscosity cutoff of 5 · 1019 Pas and μ = 0.02, which also gives a plate velocity prediction that fits the observed plate motion in a NNR reference frame. Calculations with a uniform lithosphere thickness and constant temperature below each nominal lithosphere thickness (60, 80, 100 and 150 km) excite little or no NR for friction coefficients μ = 0.02 or greater. Hence, we suggest that to match observed plate motion and NR with numerical models it is important to consider a combined visco-elasto-plastic rheology in the upper mantle structure (here we considered only the top 300 km), with a friction coefficient less than 0.04 at plate boundaries (plastic yielding) and minimum mantle viscosity (LVVs) cut-off of 1–5 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0144 Pas. Simulations with olivine water content 200 H/10 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0145Si with μ = 0.03 gave an rms plate velocity of 3.90 cm/yr close to the observed rms plate velocities (DeMets et al., 2010; Torsvik et al., 2010), and also gave an NR amplitude of 0.14 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0146/Ma with the pole location 56.62 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0147S and 89.85 urn:x-wiley:15252027:media:ggge21498:ggge21498-math-0148E, again close to observational-based values.

Acknowledgments

This work was funded by the Helmholtz graduate school GeoSim together with the Helmholtz Centre Potsdam-GFZ German Research Centre for Geosciences. Irina Rogozhina has been supported by the BMBF German Climate Modeling Initiative PalMod. The coupled global numerical code that was used to generate the results in this study builds on the SLIM3D (Popov & Sobolev, 2008) code and the spectral code of Hager and O'Connell (1981) and are available upon request from the corresponding authors. The rheological parameters used for the upper mantle are provided in the main and supporting information as tables. We have uploaded the 3-D temperature model, which were used in the top 300 km of the upper mantle for our calculations supporting information supporting information. Also the velocity-to-density scaling profile used to convert the Smean seismic topography (Becker & Boschi, 2002), used below the 300 km depth with the corresponding viscosity profile. Lastly, we thank the two anonymous reviewers for the constructive comments to improve our manuscript.