State of the Art Numerical Subduction Modelling with ASPECT; Thermo

Home , Slab pull

State of the art numerical subduction modelling with ASPECT; thermo- mechanically coupled viscoplastic compressible rheology, free surface, phase changes, latent heat and open sidewalls

A MSc. thesis by C.A.H. Blom Supervisors: C.A.P. Thieulot1, A.C. Glerum1, M.R.T. Fraters1, W. Spakman1 1Utrecht University, Department of Earth sciences, Mantle Dynamics Group July 2016

Abstract. Subduction dynamics exert great influence on surface processes, plate tectonics and mantle convection. Feedback between these processes in combination with the many parameters involved in numerical subduction modelling makes it a chal- lenge to find out how a single variable affects the system. In this study we focus on three problems. Firstly, we try to find out how boundary conditions (particularly open boundary conditions) influence subduction evolution. Secondly, the effects of mantle phase transitions on subduction dynamics will be investigated. Thirdly, the role of compressibility in our models will be explored. Meanwhile, we try to find a relation between surface expressions and their causative mantle processes. Our research is performed in a two-dimensional domain with a free surface, either free slip, prescribed velocity or open sidewalls and a free slip bottom boundary. Up to seven compositional fields are involved which feature a thermo-mechanically coupled viscoplastic compressible rheology. Prescribed velocity boundaries are found to restrict the influences of the internal dynamics. As a result, the subduction geometry is largely defined by the boundary conditions. Open boundary conditions allow a multitude of new subduction geometries to evolve more naturally. The 410 km phase transition increases the slab pull force considerably and produces a surface depression. Open boundaries allow the slab to accelerate significantly after it has crossed the 410 km phase transition. Slabs in this study are probably too weak to penetrate through the 660 km phase transition. Instead they buckle on the phase transition and high stresses cause elevated surface expressions. Compressibility is found to have a second order effect on subduction evolution. Contents

Abstract

1 Introduction 1 1.1 Open boundary conditions...... 1 1.2 Slab morphology induced by mantle phase transitions...... 2 1.3 Compressibility...... 3 1.4 Surface expressions of subduction zones...... 4

2 Model description6 2.1 ASPECT...... 6 2.2 Model setup...... 6 2.2.1 Governing equations...... 8 2.3 Rheology...... 8

3 Model adaptations 11 3.1 Material properties...... 11 3.1.1 Viscosity...... 11 3.1.2 Density...... 13 3.2 Model extensions...... 14 3.2.1 The adiabatic mantle temperature...... 14 3.2.2 The phase transition function...... 14 3.2.3 The entropy derivative...... 16

4 Results 19 4.1 Why a maximum number of nonlinear iterations increases model efficiency.... 19 4.2 Reducing the wall-time by material averaging...... 19 4.3 How do free slip, free surface, prescribed velocity and open boundary conditions affect subduction evolution?...... 21 4.4 The combined effects of mantle phase transitions and sidewall boundary conditions on subduction evolution and surface topography...... 24 4.5 Testing the impact of compressibility on a model with phase transitions and a model without phase transitions...... 28

5 Discussion 30 5.1 Reducing the computational time...... 30 5.2 The effects of prescribed and open sidewalls...... 31 5.3 The combined effects of mantle phase transitions and sidewall boundary conditions on subduction evolution...... 32 5.4 Why compressibility can be ignored...... 34 5.5 The combined effects of mantle phase transitions and sidewall boundary conditions on surface topography...... 34 6 Conclusion 36

Bibliography 36

Appendix A Derivation of the adiabatic temperature equationi

Appendix B Benchmarking open boundaries in ELEFANT and ASPECT iii B.1 TEST OPENBC 1: 3D Sinking cube in ELEFANT and ASPECT ...... iii B.1.1 Compositional initial conditions...... iii B.1.2 Velocity diﬀerences invoked by viscosity averaging schemes...... iv B.1.3 Zero velocity boundary conditions (1,2,3,4,5,6)/(0,1,2,3,4,5)...... v B.1.4 Open boundary conditions (1,2)/(0,1), 48x48x48...... viii B.1.5 Open boundary conditions (1,2,3,4)/(0,1,2,3), 48x48x48, No material averaging...... x B.2 TEST OPENBC 2: 2D simple lithospheric model...... xii B.2.1 Compositional initial conditions...... xii B.2.2 Free slip on the sidewalls...... xiii B.2.3 Open boundary conditions on the sidewalls...... xv

Appendix C Code xvii 1. Introduction

Subduction zones are one of the most extensively studied terrestrial geodynamic areas because they affect human life by inducing seismicity and volcanic activity. Our inability to observe deep-subcrustal processes makes us dependent on numerical models to better understand what happens in subduction zones. Evermore complicated subduction models in combination with increasingly powerful computers allow us to study parameters that have been simplified or ignored in dated subduction setups. However, to effectively study the influences of individual parameters the total number of parameters should be limited, without over-simplification of the subduction physics (Gerya, 2011).

1.1 Open boundary conditions

Regional subduction zones are modeled with a limited spatial extent, therefore, boundary conditions are required at the edges of the domain. The selected boundary conditions represent the mechanical and thermal state of the surrounding system and influence the evolution of the model (Quinquis et al., 2011). Free-slip boundary conditions are applied in most older and recent subduction models (e.g. Keppie et al., 2009, C´ı˘zkováet˘ al., 2012, Tosi et al., 2016). Impermeable boundary conditions, such as free-slip, are found to restrict slab rollback at subduction zones because of strong return flows. In addition, the aspect ratio of a domain with impermeable sidewalls affects slab evolution while a domain with open traction boundaries is less sensitive to its aspect ratio. Therefore, models with impermeable boundary conditions require larger aspect ratios than models with open traction sidewalls (Chertova et al., 2012).

Permeable boundaries have proven to be a more realistic option than impermeable boundaries because they do not inhibit lateral ﬂow. However, most permeable boundaries are based on prescribed conditions such as prescribed velocities (e.g. Quinquis et al., 2011), prescribed stresses (other than the lithostatic pressure) or periodic boundaries (e.g. Ita and King, 1994, Capitanio et al., 2010, Crameri and Tackley, 2015). Prescribed conditions have been found to control trench motion and the geometry of the subducting slab. Quinquis et al.(2011) and van Hunen et al.(2001) for instance recognize that a slab subducting beneath a stationary overriding plate often results in the slab folding on the ∼660 km discontinuity whereas subduction under an advancing overriding plate results in trench retreat and the slab resting on the lower mantle. Also, Ita and King(1994) recognized that the symmetry induced by periodic boundary conditions reduces the maximum wavelength in the mantle compared to free slip boundary conditions. Consequently, more and smaller convecting cells developed in domains with periodic sidewalls

1 Introduction 2

Furthermore, subduction is regarded as the main driving force for plate motion and mantle convection (Forsyth and Uyedat, 1975, Richter and McKenzie, 1978). So, a subducting system should be driven by lithosphere subduction and not by prescribed conditions. Self consistent lithosphere subduction has been modelled in high width to depth ratio box models with free slip boundary conditions (e.g. C´ı˘zkováet˘ al., 2012, Androvi˘cováet al., 2013, Tosi et al., 2016). Self consistent lithosphere subduction in which the flow through the sidewalls was completely determined by the model interior has only been described a few times (e.g. Quinteros et al. (2010), Chertova et al.(2012)).

1.2 Slab morphology induced by mantle phase transitions

Tomographic studies show a diverse range of subducting slab types. Some slabs penetrate the phase transition zone almost vertically while others flatten out, bend or buckle on the 660 km phase transition and sink into the lower mantle after up to several 100 km of lateral movement (e.g. van der Hilst et al., 1997, Fukao et al., 2001, Zhao, 2004, Hayes et al., 2012). To distinguish between slab morphologies Garel et al.(2014) divided subducting slabs into four groups: (i) slabs which have a small dip in the upper mantle and flatten at the 660 km phase transition, (ii) slabs which are steep in the upper mantle and flatten at the 660 km transition zone, (iii) slabs with a constant dip from upper to lower mantle and (iv) near vertical slabs that get thicker at the base of the upper mantle.

Previous research has demonstrated that some parameters have a larger effect on slab morphology than other parameters. Factors that have a second order effect on subduction settings, such as water content, latent heat release and grain size, are ignored in most regional studies while they might locally result in strong viscosity variations (Billen, 2008). In contrast, the most influential parameters have been studied thoroughly and the morphology of a slab has been found to depend on four parameters. These are the phase transition properties (foremost the density and viscosity increase), slab age (expressed through strength and density), trench motion and the angle at which the slab reaches a transition zone (e.g. Kincaid and Olson, 1987, Gurnis and Hager, 1988, Guillou-Frottier et al., 1995, Behounkova and C´ı˘zková˘ , 2008, Alisic et al., 2012, C´ı˘zkováand˘ Bina, 2013). The effects of these model parameters should be considered carefully when modeling (self sustained) subduction.

A cold subducting slab is accelerated by the positive Clapeyron slope of the ∼410 km phase transitions. On the other hand, the ∼660 km phase transition has a negative Clapeyron slope and opposes thermally driven ﬂow (e.g. Christensen and Yuen, 1985, Tackley et al., 1993, van Hunen et al., 2001, Schubert et al., 2001, Billen, 2008). In rare cases the ∼410 km phase transition counteracts subduction. Young subducting slabs descending into the mantle at low angles force upper-mantle material through the ∼410 km phase transition. Latent heat released in this process can produce small thermal anomalies with negative buoyancies, counteracting the downward force of the subducting slab (van Hunen et al., 2001).

2 Introduction 3

The age of a subducting slab determines its strength and density (Billen, 2008). If given enough time, dense slabs will sink into the underlying mantle. However, their strength controls the timing and subduction angle. King(2001) and C´ı˘zkováet˘ al.(2002) found that most slabs in the upper mantle are strong enough to maintain shallow dip angles whereas weak slabs are more likely to sink vertically. Also, C´ı˘zkováet˘ al.(2002) recognized that stiffer slabs are less likely to get stuck at the ∼660 km phase transition compared to weak slabs. This finding is supported by a compilation study (Billen, 2008) that shows strong slabs subducting straight into the lower mantle when they are unaffected by trench roll-back.

The fourth important parameter affecting slab morphology is trench motion. Trench motion can be either a cause or effect of slab dynamics (Garel et al., 2014). If trench motion is imposed, strong slabs can be trapped in the transition zone (C´ı˘zkováet˘ al., 2002, Enns et al., 2005). In contrast, inhibiting slabs from subducting into the lower mantle causes high stresses and may result in trench motion (Schellart, 2004, Bellahsen et al., 2005). Other numerical modelling has shown that trenches are more likely to move if subducting slabs have a small thickness, a high density and/or a high viscosity (e.g. Bellahsen et al., 2005, Enns et al., 2005, Capitanio et al., 2007, Stegman et al., 2010). More recent models have demonstrated interaction between deep dynamics and the upper plates (e.g. Capitanio et al., 2010, Leng and Gurnis, 2011).

1.3 Compressibility

Studies that use a compressible fluid model are rare (e.g. Jarvis and McKenzie, 1980, Ita and King, 1994). Thus, the possible effects of compressibility are disregarded in most studies. Advances in geodynamic models and computational power allow increasingly complex rheologies. When implementing more realistic rheologies, it is important to do this consistently. Therefore, the effects of compressibility should be tested in new setups. In addition, earlier studies on the effects of compressibility show diverse and inconsistent results.

Austmann et al.(2014) found the effects of elastic compressibility to be significant for both gravity driven flow in a cylinder and near material boundaries in an oceanic subduction setting. Lee(2014) recognized that in a compressible model the ”vigor of thermally driven mantle convection weakens, especially in the lowermost mantle”. Consequently, the temperature in the upper mantle decreases, resulting in a thicker thermal lithosphere and a steeper mantle adiabat. According to Lee(2014), compressibility increased the mantle adiabat by 0.14-0.25 K/km, which is comparable to the 0.35 K/km mantle adiabat generated by adiabatic heating. Trubitsyn and Trubitsyna(2015) modeled purely thermally driven flow in a square domain with only a single composition. When only the effects of compressibility on the energy equation were considered, a minor increase of the mantle adiabat was found.

Still, compressibility is neglected in most recent models for computational reasons and because its effects are small and often do not affect the flow structure discernibly. Particularly, in systems with high velocities, such as subduction zones, the effects of compressibility are of

3 Introduction 4 second order. Lee and Scott(2009) investigated the effects of compressibility on thermal and flow structures of subduction zones and found no considerable dissimilarities between compressible and incompressible models. However, compressibility in the mantle wedge was observed through increased viscous dissipation. Still, the heat generated by viscous dissipation was small (< 10−10 J/kgs), what resulted in a trifling effect on the thermal and flow structures.

1.4 Surface expressions of subduction zones

Traditionally, the top boundary of a modeling domain is defined by a free slip boundary condition with a weak crust directly below it to allow the subducting lithosphere to detach from the top boundary (e.g. Gurnis and Hager, 1988). This type of top boundary does not allow for surface topography. The sticky-air method enables the build-up of topography. In this method a layer with a low viscosity and a very low density separates the top boundary from the crustal layers. Because of its low viscosity and density the layer does not affect the surface stresses and topography can build up until the shear and normal stresses between the sticky-air layer and the crust are zero (Schmeling et al., 2008). However, large viscosity contrasts give numerical problems. Also, solving for large viscosity contrasts is computationally expensive because high resolutions are required to prevent under- and overshoots in the numerical approximation (Crameri et al., 2012). A deformable grid which is not fixed along the top boundary allows the shear and normal stresses at the top of the domain to be zero. This boundary condition is called a free surface. By deforming the grid, topography can build up at lower computational costs compared to the sticky-air method (Gerya, 2010).

Surface topography is mostly explained by the interaction between moving tectonic plates. However, mantle density variations cause viscous stresses which are balanced by surface topography (Braun, 2010). The dynamic topography is often hidden by large amplitude, short wavelength topography produced by horizontal tectonic movements. Despite this ’noise’ dynamic topography can be observed at the surface. Some examples are mantle ﬂow induced large scale (intercontinental) sedimentary basins (e.g. Pysklywec and Mitrovica, 1998), subsidence at plate margins caused by subduction induced mantle ﬂow (e.g. Gurnis, 1992) and topographic highs as found in southern Africa (Gurnis and van Heijst, 2000).

The geoid is the sum of all density anomalies within the Earth. Density anomalies can drive mantle convection, such as buoyant plumes or subducting slabs. But density anomalies can also be the result of boundaries being deformed by convection, such as shifting of phase transition depths (King, 2002). A subducting plate is dense and contributes to the positive term of the geoid. In contrast, the down warping of the surface produces a mass deﬁcit of the same magnitude which contributes to the negative term (Richards and Hager, 1984). Still, subduction zones have long been associated with maxima in the long-wavelength component of the geoid (e.g. Runcorn, 1967, McKenzie, 1969, Hager, 1984).

4 Introduction 5

Obviously, processes other than isostasy are responsible for the high geoid above subduction zones. King(2002) recognized the effects of the ∼660 km phase transition, without viscosity increase, are insufficient to reproduce the local maxima in the geoid over subduction zones. This reinforces the conclusions from previous studies which state that a viscosity increase of a factor 30 or more causes a sufficiently large resistance to flow to produce a higher than expected topography above subducting plates (Richards and Hager, 1989, Zhong and Gurnis, 1992, King and Hager, 1994). Although the effects of a subduction zone on the geoid have been long resolved, the build-up of topography remains indefinite. In this study a free surface allows us to investigate the magnitude and timing of surface topography related to a subducting slab.

In this report we describe a thermo-mechanically coupled subduction model with a viscoplastic rheology using the finite element code ASPECT (short for Advanced Solver for Problems in Earth’s ConvecTion) (Bangerth et al., 2014). Mantle phase transitions have been added to the code and their functionality will be tested by answering the following research question: How do mantle phase transitions influence subduction dynamics? Another plug-in (Glerum et al., in prep) allows the utilization of open boundaries. Open boundary conditions are a novelty in geophysics. Less than a handful of papers have been published that feature both open boundary conditions and a subduction setting. In addition, the plug-in used in this study is the first plug- in ever to produce open boundaries in ASPECT. The functioning of the open boundary plug-in will be tested by investigating how boundary conditions (especially open boundary conditions) affect subduction evolution. For the first time open boundary conditions will be employed to- gether with several other types of boundary conditions and phase transitions in one model. Their combined effects will be investigated. Because phase transitions are associated with large density increases we will also test if the implementation of compressibility changes subduction evolution considerably. Meanwhile, we try to find a relation between surface expressions and their causative mantle processes.

5 2. Model description

2.1 ASPECT

ASPECT utilizes the latest numerical and computational methods such as mesh adaptation, accurate discretizations, efficient linear solvers, parallelization and modularity of software. Be- cause of its modularity, ASPECT consists of a small core which solves the basic equations. For other tasks it relies on external libraries and plug-ins (Fraters, 2014, Kronbichler et al., 2012). The DEAL.II software library (Bangerth et al., 2007) supports the creation of finite element codes. In turn, it depends on p4est (Burstedde et al., 2011) for adaptive mesh refinement and parallelization and on TRILINOS (Heroux, 2005) for solvers to solve the linear and non-linear systems of equations. Because each underlying software library has its own support community, improvements to numerical and computational techniques can be frequently implemented.

Besides applying external libraries, the core of ASPECT employs plug-ins. Plug-ins are pieces of code that can be attached to the core of ASPECT to execute a task and return the result to the core. Plug-ins are for example used to calculate material properties such as the viscosity and density, to determine latent, adiabatic, shear and radiogenic heating, to deﬁne the compositional initial conditions and to post-process the results. In this research we use an extended version of the viscoplastic material model created by Glerum et al.(in prep). In addition, another plug-in built by A. Glerum is used to implement open pressure boundaries. This plug-in has been benchmarked to good results with an open boundary implementation for ELEFANT (Thieulot, 2014) (see Appendix B). The initial temperature distribution and the compositional initial conditions of the subduction system are described in two plug-ins based on plug-ins by Fraters (2014). The changes made to the plug-ins above are illustrated in Appendix C.

2.2 Model setup

We model the subduction of an oceanic plate through the ∼410 km and ∼660 km phase transitions up to a depth of 1000 km. The 2D Cartesian geometry has a width of 1600 km and a depth of 1000 km (Figure 2.1). The isothermal top boundary is defined by either a free slip or a free surface boundary condition. The bottom boundary is isothermal and free slip. The sidewall boundaries can be described by either free slip, prescribed velocity or open boundary conditions. A prescribed velocity profile is shown at the right sidewall of the model setup. Pre- scribed velocity boundaries may not cause a net volume flux over a boundary. Thus, the amount of lithosphere and crust that flows into the domain is equal to the amount of mantle material

6 Model description 7

ﬂowing out of the domain. Open boundary conditions (shown at the left sidewall of the model setup) require the lithostatic pressure to prescribe the tractions asτ ¯ = −Plithnˆ. The vertical component of traction on the boundary is equal to zero and the horizontal normal component is equal to either −Plith or Plith (Glerum et al., 2016).

Figure 2.1: Model setup

The initial model comprises five compositional fields: an oceanic crust, continental crust, upper mantle, transition zone and lower mantle. Two more compositions form when the subducting slab crosses the ∼410 km and ∼660 km phase transitions. The continental lithosphere with a thickness of 100 km and the oceanic lithosphere with a thickness of 80 km are mainly thermally defined. The initial slab has a length of 75 km and a dip angle of 25 ◦. The subducting slab is assumed to be formed at a ridge outside of the domain and the prescribed velocities are caused by ridge push processes. Decoupling between the subducting slab and the continental plate is facilitated by a low viscosity oceanic crust with a viscosity of 1020 P as.

When a slab crosses the ∼410 km phase transition its density will increase. Androvi˘cov´a et al.(2013) investigated the eﬀect of the oceanic crust viscosity on slab velocity and recognized that a viscosity value of 1020 P as prevents plate velocities from increasing to unrealistically high velocities. Furthermore, we found that a minimum viscosity value of 1020 P as prevents slab necking when a slab crosses the ∼410 km phase transition. Slab necking has also been observed by Fraters(2014). Therefore, models without phase transition have a minimum viscosity of 10 19 P as whereas models with phase transitions have a minimum viscosity of 1020 P as.

The domain has been discretized into a grid with a variable resolution depending on local gradients of the density, viscosity and composition. The height of the grid cells varies from 1.6 km in the subducting and overriding plates and 3.2 km in the transition zones to 25 km in the lower mantle. The small grid sizes guarantee that the phase transitions and weak oceanic crustal layer are well resolved. 7 Model description 8

2.2.1 Governing equations

Equation 2.1 and Equation 2.2 describe the compressible Stokes equations as implemented in ASPECT to describe the motion of highly viscous fluids. Equation 2.2 is an approximation to the conservation of mass equation because the compressible system is not symmetric and therefore hard to solve. The Stokes system is coupled to the energy equation (Equation 2.3) by its temperature dependence. In turn, the energy equation is coupled to the Stokes system because it is dependent on velocity and pressure. Its right hand side terms correspond respectively to the frictional heating, adiabatic compression and latent heat contributions. If multiple compositional fields are defined, Equation 2.4 describes their advection and reactions (Bangerth et al., 2014).

h 1 i − ∇ · 2η (u) − (∇ · u) + ∇p = ρg (2.1) 3 1 δρ ∇ · u = − ρg · u (2.2) ρ δp δX δT 1 1 ρC − ρT ∆S + u · ∇T − ∇ · k∇T =2η (u) − (∇ · u)1 : (u) − (∇ · u)1 p δT δt 3 3 + αT (u · ∇p) δX + ρT ∆S u · ∇p δp (2.3) δc i + u · ∇c = q (2.4) δt i i

1 T δX δX In this set of equations (u) = 2 (∇u + ∇u ), ci is a compositional ﬁeld and δT and δp are the derivatives of the phase transition function with respectively temperature and pressure. The other parameters are described in Table 2.1 and Table 2.2.

2.3 Rheology

In numerical geodynamic modelling, highly viscous flow is typically described by a combination of viscous, plastic and elastic deformation mechanisms (Schubert et al., 2001). The plug-in by Glerum et al.(prep) implements a viscoplastic rheology. Two creep mechanisms cause viscous deformation: diffusion creep (Equation 2.6) and dislocation creep (Equation 2.5). Diffusion creep is the result of the migration of vacancies within crystals (Karato, 2008). Dislocation creep is caused by deforming crystals inside minerals and is stress dependent (Schubert et al., 2001). Plastic deformation has been implemented in the model through the Mohr-Coulomb criterion (Equation 2.7) and describes the brittle behaviour of rocks. Elastic deformation is recoverable and assumed to be only of short timescale effect, therefore, it is often ignored when modelling over timescales larger than 1 million years (Billen, 2008). The effect of elastic deformation on

8 Model description 9 materials exposed to high strain rates in a short time (< 104 years), such as magma intrusions and subducting slabs, is unknown (Gerya, 2010).

−1 1−ni 1 n Qdsi + Vdsi P i 0 ni ηds = νdsi Ads imax(˙II , min) exp (2.5) 2 niRT

1 1 Qdfi + Vdfi P ηdf = exp (2.6) 2 Adfi RT

P sin(φi) + Cicos(φi) ηplastic = 0 (2.7) 2.0max(˙II , min)

In the formulas above, df and ds abbreviate respectively diffusion creep dislocation creep. The subscript i indicates that the values are dependent on composition. The effective viscosity of one composition is calculated as the harmonic average of its diffusion creep viscosity, dislocation creep viscosity and plastic viscosity. The final viscosity is calculated as the weighted harmonic average of the effective viscosities of each composition and their compositional value (which varies between 0 and 1). Before the viscosity is returned to the core of ASPECT, the effective viscosity can be lowered by the introduction of weak zones (Fraters, 2014).

Symbol Meaning Value Units Various material properties and model parameters k Thermal conductivity 4 W m−1K−1 α Thermal expansion coefficient 3 × 10−5 K−1 3 −1 −1 cp Specific heat capacity 1 × 10 Jkg K β Compressibility 5.124 × 10−12 P a−1 η Viscosity - P a · s u Velocity field - m/s p Pressure - P a Strain rate - P as ∆S The change of entropy - m2s−2K−1 g Acceleration due to gravity 9.81 ms−2 R Gas constant 8.314 JK−1mol−1 −3 ρref Reference density 3300 kgm Tsurface Surface temperature 273.15 K Tpot Potential slab temperature at the surface 1553 K Tptsm Potential mantle temperature at the surface 1600 K Phase transition properties 6 −1 γ410 Clapeyron slope 410 km phase transition 2 × 10 P aK 6 −1 γ660 Clapeyron slope 660 km phase transition −1.5 × 10 P aK d Transition widths 10 × 103 m −3 δρ410 Density contrast 410 km phase transition 273 kgm −3 δρ660 Density contrast 660 km phase transition 342 kgm 10 P410 Reference pressure at 410 km phase transition 1.325 × 10 P a 10 P660 Reference pressure at 660 km phase transition 2.16 × 10 P a

Table 2.1: Model parameters and material properties that are independent of composition.

9 Symbol Meaning Units UM OC CC TZM LMM TZC LMC Reference T K 1800 400 400 1800 1800 1800 1800 ref temperature −3 ρref Reference density kgm 3300 3150 2900 3573 3915 3423 3765 C Cohesion P a 20 × 106 20 × 106 100 × 106 20 × 106 20 × 106 20 × 106 20 × 106 φ Angle of friction ◦ 30 0 0 30 30 0 0 Stress exponent n - 3 1 4 3 3 1 1 dislocation creep Prefactor A P a−ns−1 5.5 × 10−16 1.0 × 10−21 1.1 × 10−28 5.5 × 10−16 5.5 × 10−20 1.0 × 10−21 1.0 × 10−21 ds dislocation creep Activation energy Q Jmol−1P a−1 4.0 × 105 0 2.23 × 105 4.0 × 105 4.0 × 105 0 0 ds dislocation creep Activation volume V m3mol−1 14 × 10−6 0 0 14 × 10−6 14 × 10−6 0 0 ds dislocation creep Constant coefficient ν − 2 1 1 2 2 1 1 ds dislocation creep Prefactor A P a−1s−1 5.92 × 10−11 1.92 × 10−11 1.92 × 10−11 5.92 × 10−11 0.4 × 10−19 1.92 × 10−11 1.92 × 10−11 df diffusion creep Activation energy Q Jmol−1P a−1 3.35 × 105 3.35 × 105 3.35 × 105 3.35 × 105 1 × 105 3.35 × 105 3.35 × 105 df diffusion creep Activation volume V m3mol−1 4.0 × 10−6 4.0 × 10−6 4.0 × 10−6 4.0 × 10−6 1 × 10−6 4.0 × 10−6 4.0 × 10−6 df diffusion creep Constant coefficient ν − 1 1 1 1 1 1 1 df diffusion creep

Table 2.2: Model parameters and material properties that are dependent on composition. Model compositions are indicated as follows: UM is the upper mantle, OC is the oceanic crust, CC is the continental crust, TZM is the transition zone mantle, LMM is the lower mantle, TZC is the crust in the transition zone, LMC is the crust in the lower mantle. 3. Model adaptations

3.1 Material properties

A density and viscosity profile are needed to compute a model of mantle flow (Steinberger and Calderwood, 2006). The viscosity profile used in this research is based on Steinberger and Calderwood(2006) and C´ı˘zkováet˘ al.(2012) and the density profile is based on Dziewonski and Anderson(1981) and Tosi et al.(2016).

3.1.1 Viscosity

Steinberger and Calderwood(2006) computed the viscosity variations from mineral physics results and obtained the density model by converting S-wave anomalies from seismic tomography. The viscosities in the resulting mantle flow model were treated as free parameters and adjusted to minimize the difference between the predicted and observed geoid, to fit post-glacial rebound

200 C´ı˘zkov´aet˘ al., 2012 A-family C´ı˘zkov´aet˘ al., 2012 B-family Steinberger and Calderwood, 2006

) 400 Our viscosity km

Depth ( 600

800

1,000 1020 1021 1022 1023 1024 Viscosity (P as)

Figure 3.1: A comparison between the viscosity profile used in this study (red line), the viscosity profile found by Steinberger and Calderwood(2006) (black line) and the A family (blue line) and B family (green line) viscosity profiles described by C´ı˘zkováet˘ al.(2012).

11 Model adaptations 12 results and to minimize the misfit between the radial heat flux profile and observations. The resulting viscosity is illustrated in Figure 3.1 (black line). C´ı˘zkováet˘ al.(2012) investigated the viscosity profile from the inferred sinking speed of remnants of subducted lithosphere in the mantle. Two viscosity profile families were defined in this research; the A-family (Figure 3.1 blue line) and the B-family (Figure 3.1 green line). The A-family features a lower mantle viscosity profile that is almost constant with depth whereas the lower mantle viscosity increases gradually with depth in the B-family. C´ı˘zkováet˘ al.(2012) classified the viscosity model by Steinberger and Calderwood(2006) as an A-family model.

Our viscosity profile is located under a static continental crust where stresses are low and at a distance of 300 km from the closest sidewall. Because the viscosity profile of the Earth remains unknown, we can not choose one of the rheologies described above over the others. Therefore, our model has a crustal and mantle viscosity which resembles the viscosity values found by Steinberger and Calderwood(2006) and C´ı˘zkováet˘ al.(2012) as illustrated by the red line in Figure 3.1. The viscosity of the lower mantle is only constrained by a limited number of high-pressure experiments (Karato, 2008) and profiles obtained from joint inversions of post- glacial rebound data and the geoid (Forte and Mitrovica, 1996, Kaufmann and Lambeck, 2000, Mitrovica and Forte, 2004). Figure 3.1 shows a viscosity increase of a factor 5 over the width of the 660 km phase transition which produces a mantle viscosity comparable to the B-family rheology from C´ı˘zkováet˘ al.(2012).

Close to the surface, where pressures are low, the strength of a slab is dominated by plastic deformation. In the warm mantle, deformation is accommodated by diﬀusion and dislocation

Figure 3.2: Spatial distribution of the dominant deformation mechanisms after 20 Myr of subduction. Plastic, dislocation creep and diﬀusion creep deformation are indicated by respectively red, grey and blue ﬁelds. The yellow lines represent the mantle phase transitions.

12 Model adaptations 13 creep producing a composite rheology. Dislocation creep is the dominant deformation mechanism when strain rates are high and is not grain-size dependent whereas diﬀusion creep is temperature dependent and more active when grain sizes are larger (Billen, 2008, Garel et al., 2014). The dominant deformation mechanisms found in our study are illustrated in Figure 3.2. Plastic deformation is dominant close to the surface and in the subducting slab up to a depth of almost 300 km. Dislocation creep is the dominant deformation mechanism in the upper mantle while diﬀusion creep is most active in the lower mantle.

3.1.2 Density

The density proﬁle with depth obtained in this study resembles the Preliminary Reference Earth Model (PREM) (Dziewonski and Anderson, 1981) as can be seen in Figure 3.3. The density is dependent on composition, pressure and temperature. The pressure dependence is especially important for models with high density gradients, e.g. due to phase transitions (Bangerth et al., 2014). Likewise, Figure 3.3 shows that the total density increase over a phase transition is larger for a compressible model than for an incompressible model. The corresponding compressibility (β) is constant with depth and has a value of 5.124 ∗ 10−12 P a−1 (Bangerth et al., 2014).

0 PREM Incompressible 200 Compressible

) 400 km

600 Depth (

800

1,000 3,200 3,400 3,600 3,800 4,000 4,200 4,400 4,600 4,800 Density (kgm−3)

Figure 3.3: A comparison between the compressible density proﬁle (red line), the incompressible density proﬁle (blue line) and the PREM model (black line). The PREM density values were computed by Bormann(2002).

Compressibility (pressure dependence of the density) was implemented into the viscoplastic material model by modifying Equation 3.1 to Equation 3.2(Gerya, 2010). The density in a compressible medium is dependent on pressure, composition and temperature whereas the density in an incompressible medium is only dependent on composition and temperature. 13 Model adaptations 14

ρincomp = ρref (1.0 − α(T − Tref )) (3.1)

ρcomp = ρref (β(P − Psurface))(1.0 − α(T − Tref )) (3.2)

where ρref and Tref are the arithmetic averages of the reference densities and reference temperatures of the compositional ﬁelds.

3.2 Model extensions

3.2.1 The adiabatic mantle temperature

If a nonlinear solver starts with an unphysical pressure, it will have trouble converging. There- fore, ASPECT computes the adiabatic pressure and adiabatic temperature ﬁeld using Equa- tion 3.3 before calling the initial conditions plug-in (Bangerth et al., 2014). Both, Tosi et al. (2016) and Kronbichler et al.(2012) use Equation 3.3 to calculate the adiabatic temperature (see Appendix A). This formula will be used when calculating the mantle temperature ﬁeld in the initial conditions and when determining the depths of the phase transitions in the material model plug-ins.

αgz T = Tptsm exp (3.3) cp

The difference between the old adiabatic mantle temperature by Fraters(2014) and the mantle temperature described by the formula above is illustrated in Figure 3.4. The new mantle temperature is non-linear and has a slightly higher gradient than the old temperature. Because the new adiabatic temperature field is closer to the adiabatic temperature field computed by ASPECT before the iteration process started, the nonlinear solver will converge a bit faster.

3.2.2 The phase transition function

The upper mantle is separated from the lower mantle by three major phase transitions. At ∼410 km depth, olivine is being transformed into wadsleyite in an exothermic reaction (char- acterized by a positive Clapeyron slope). At the ∼520 km phase transition, wadsleyite changes exothermically into ringwoodite. Ringwoodite is transformed into perovskite at ∼660 km depth in an endothermic reaction (Shearer, 2000, Billen, 2008). The ∼410 km and ∼660 km phase transitions have the largest seismic velocity gradient, therefore, the necessity to include the diﬀuse ∼520 km discontinuity has been questioned (e.g. Bock, 1994). Hence, only the olivine- wadsleyite and ringwoodite-perovskite phase transitions are included in this study.

14 Model adaptations 15

100

200

300

400 Depth (km)

500

600 Old adiabatic temperature New adiabatic temperature 700 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 Temperature (K)

Figure 3.4: Old and new adiabatic temperatures. The proﬁle is located under the overriding plate.

The progress of the phase transition has been formulated by Richter(1973) and is represented by the phase function (Equation 3.4)

1h P − P − γ (T − T )i X = 1 + tanh t t t (3.4) 2 d

where X is a number between 0 and 1 representing the fraction of material that has changed into a diﬀerent phase, Pt is the reference pressure at the transition depth and Tt is the adiabatic temperature at the transition depth. d is deﬁned by Ita and King(1994) as half the transition width. The unit of the transition width should be P a for Equation 3.4 to be dimensionless.

Because the fixed composition boundaries are dependent on the initial (compositional) conditions, the phase transitions have to be defined in the compositional initial conditions. If not estimated accurately, the phase transition depths will migrate and no longer correspond to the depths used by the fixed boundary conditions. The temperature is accurately calculated by Equation 3.3. The pressure calculation is more complicated because of the different compositional fields and their nonlinear dependencies. Therefore, the phase transition function is changed to a formula which is depth dependent instead of pressure dependent. This results in Equation 3.5 which is independent of pressure. The units of the gradient of the Clapeyron slope P a m changed from K to K and the units of the transition width changed from P a to m. Another advantage of a pressure independent phase transition function is that it is not affected by the negative pressures which are sometimes found close to the surface.

15 Model adaptations 16

zt 1h z − zt − γt (T − Tt)i X = 1 + tanh Pt (3.5) 2 d zt pt

In this formula z is the depth of the transition zone and d zt is the distance from z at which t pt t 90% of the material has been transformed. The output of the phase transition function template can be seen in Figure 3.5. The crust has not been deﬁned by a phase transition function. It is included for completeness.

0 0

200 200

400 400

600 600 Depth (km) Depth (km)

800 800 Crust Upper mantle 1,000 1,000 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Compositional factor Compositional factor 0 0

200 200

400 400

600 600 Depth (km) Depth (km)

800 800 Transition zone Lower mantle 1,000 1,000 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Compositional factor Compositional factor

Figure 3.5: The compositional distribution of the crust, upper mantle, transition zone and lower mantle with depth after 2 × 105 years. The extend of the crust is not deﬁned by Equa- tion 3.5.

3.2.3 The entropy derivative

A material releases or consumes heat when it undergoes a phase transition. Consequently, its entropy changes. Because phase transitions are dependent on temperature and pressure the amount of latent heat released can be calculated from the total change of entropy ∆S and the

16 Model adaptations 17

δX δX pressure and temperature derivatives of the phase function ( δT and δp ). These coeﬃcients have to be provided to ASPECT to solve the latent heat contribution of Equation 2.3. ∆S can be calculated using Equation 3.6(Bangerth et al.(2014)):

δρ ∆S = γ t (3.6) t ρ2

Here, δρ is the increase in density over the phase transition.

The entropy derivative template in the material model returns ”the product of the change dX in entropy across phase transitions and the pressure derivative of the phase function (∆S dP ) (if this is the pressure derivative) or the product of minus the former two and the Clapeyron dX slope (−∆S dP γ) (if this is the temperature derivative)”(Bangerth et al.(2015)). The pressure derivative of the phase function is given by Equation 3.7.

dX 1h P − P − γ (T − T )i = 1 − tanh2 t t t (3.7) dP 2 d

In Equation 3.7 d is the pressure width in Pascal.

The derivative of the phase function for this subduction model has been drawn in Figure 3.6. 1 The derivative is largest where the phase transition function is 2 . Close to the surface, the phase function derivative ﬂuctuates because of local pressure deviations. These ﬂuctuations are trivial because of their small values.

0 410 km phase function derivative 660 km phase function derivative 200

400

600 Depth (km)

800

1,000 10−25 10−23 10−21 10−19 10−17 10−15 10−13 10−11 10−9 Phase transition function derivative (P a−1)

Figure 3.6: The derivatives of the 410 km and 660 km phase transition functions.

17 Model adaptations 18

The entropy derivative that is returned to the core of ASPECT is shown in Figure 3.7 for dX dX ∆S dP and in Figure 3.8 for −∆S dP γ. The entropy derivative dependent on temperature is approximately 6 orders of magnitude larger than the entropy derivative dependent on pressure because it is multiplied with the Clapeyron slope.

Figure 3.7: Entropy derivative when dependent on pressure.

Figure 3.8: Entropy derivative when dependent on temperature.

18 4. Results

In this chapter the model results will be explained. Firstly, we will show how the relative residual is dependent on the maximum number of nonlinear iterations. Secondly, the relationship between the wall-time and material averaging will be demonstrated. Thirdly, the eﬀects of top and sidewall boundary conditions on subduction evolution will be investigated with a focus on open boundaries. Fourthly, the combined eﬀects of sidewall boundary conditions and phase transitions will be studied. Next, we will look at the development of surface topography when a slab interacts with mantle phase transitions. At last we will consider if compressibility has any impact on the evolution of two of our models.

4.1 Why a maximum number of nonlinear iterations increases model eﬃciency

Nonlinearities introduced by a nonlinear rheology can be signiﬁcant and need to be iterated out. In this study we use the ”iterated Stokes” scheme to solve the energy equation (Equation 2.3) at the beginning of each time step and uses its result to iterate out the solution of the Stokes equations (Equation 2.1 and Equation 2.2). After each iteration the current residual is compared to the initial residual, giving the relative residual. A system with a low relative residual is likely to have a small error (Ismail-Zadeh and Tackley, 2010). The number of nonlinear iterations increases until the nonlinear solver tolerance or the maximum number of nonlinear iterations is reached.

Figure 4.1 shows the evolution of the relative residual with an increasing number of nonlinear iterations for several time-steps. At a low number of nonlinear iterations, the relative residual of the Initial Reﬁnement Steps (IRS) converges irregularly to the nonlinear solver tolerance. Only, IRS 3 and IRS 4 reach the nonlinear solver tolerance of 1 × 10−6 within 100 iterations. The residuals of time-steps after the IRS converge smoothly. Later time-steps require fewer iterations to reach the nonlinear solver tolerance. For time-step 50 the nonlinear solver tolerance is reached after 11 iterations. At time-step 100 the minimum residual of 1.00082 × 10−6 is reached after 8 iterations and does not decrease thereafter.

4.2 Reducing the wall-time by material averaging

For large viscosity and velocity gradients, averaging all quadrature points in a given element may increase the speed of computation in the Finite Element Method without changing the 19 Results 20

100 IRS 1 IRS 2 10−1 IRS 3 IRS 4 10−2 Timestep 1 Timestep 10 10−3 Timestep 50 10−4 Timestep 100

Relative residual 10−5

10−6

10−7 0 10 20 30 40 50 60 70 80 90 100 Number of nonlinear iterations

Figure 4.1: The residual after each nonlinear iterations for the Initial Refinement Steps (IRS) and several time-steps. solution significantly (Bangerth et al.(2014)). This averaging method reduces the magnitude of discontinuities, resulting in easier convergence. Commonly, the arithmetic, harmonic or geometric mean is used to average over the quadrature points. The efficiency results of the averaging methods applied to a non-compressible subduction model with a single phase transition are shown in Figure 4.2 and Figure 4.3.

Figure 4.2 shows the required wall time to get to a certain time-step for several material averaging methods. The harmonic and geometric material averaging methods attain a slightly lower wall time than the arithmetic material averaging. Most computational time is needed when no material averaging scheme is used. The material averaging scheme does not aﬀect the

·104

1 No averaging Arithmetic averaging Harmonic averaging Geometric averaging 0.5 Walltime (s)

0 0 20 40 60 80 100 120 140 160 180 200 Timestep

Figure 4.2: The cumulative walltime for each timestep for several averaging methods.

20 Results 21 size of the time step, so the computed geologic time is the same for all models. If the solver tolerance is not reached after 20 cheap solver steps and 50 nonlinear iterations, the time step is concluded. Figure 4.3 shows the residual at the end of each time-step for the diﬀerent material averaging methods. When no averaging method is used, the end residual of each time-step rapidly converges to the nonlinear solver tolerance. The ﬁnal residual is up to two orders of magnitude larger when a material averaging method is applied.

After averaging over its quadrature points, an element has only one value for each of the averaged parameters. Therefore, neighbouring elements can have very diﬀerent values which results in distinct elements when visualized. On the other hand, when no averaging method is applied the gradients within the grid cells generate diﬀuse boundaries.

100 No averaging Arithmetic averaging 10−1 Harmonic averaging Geometric averaging 10−2

10−3

10−4 End residual 10−5

10−6

10−7 0 20 40 60 80 100 120 140 160 180 200 Timestep

Figure 4.3: The end residuals of the ﬁrst 200 time-steps for diﬀerent material averaging methods. The maximum number of nonlinear iterations has been set to 50 and the nonlinear solver tolerance is 1 × 10−6.

4.3 How do free slip, free surface, prescribed velocity and open boundary conditions aﬀect subduction evolution?

Both models demonstrated in Figure 4.4 have prescribed velocities at the left sidewall (slab velocity is 4 cm/yr) and free slip boundary conditions at the bottom and right boundaries. The top boundary of the model shown in Figure 4.4a is deﬁned by a free-slip boundary condition whereas the other model has a free surface (Figure 4.4b).

A free slip top boundary condition appears to causes slightly more continental lithosphere delamination than a free surface. After 18 million years of subduction, the model with a free surface shows a vertically subducting slab while a bended slab has evolved in the model with a free slip top boundary condition. Just before 18 Myr a dense instability formed under the

21 Results 22 oceanic lithosphere in the free slip model. This instability may be the result of the pressure devia- tion which would, in case of a free surface, induce vertical movement of the oceanic and the continental plates.

Five different combinations of sidewall boundary conditions and initial compositions are demonstrated in Figure 4.5. Each produces a distinct subduction geometry, what emphasizes the importance of selecting sidewall boundary conditions carefully. Figure 4.5a and Figure 4.5b show the differences between models with a single and two a) Free slip b) Free surface prescribed sidewalls. In the domain Figure 4.4: Dissimilarities between a model with a free slip with only a single in- and outflow top boundary and a model with a free surface. boundary, the plate subducts almost vertically and a pronounced convection cell develops between the slab and the free-slip sidewall. In models featuring an advancing overriding plate (Figure 4.5b), trench migration results in an inclined subducting slab and higher subduction velocities.

Figure 4.5c features an open boundary at the left sidewall. The ﬁgure shows that self sustained subduction of a slab with a density of 3150 kgm3, a slab length of 200 km and a dip angle of 40 ◦ results in a slab velocity smaller than 1 cm/yr. After 15 Myr this slab has subducted less than 300 km. Also, the slab temperature has increased steadily over time resulting in smaller viscosity and density contrasts with the mantle. To facilitate the subduction of the slab even more, several changes were made to the initial temperature and compositional conditions of the model shown in Figure 4.5d. The density of the oceanic crust has been increased by 70 kgm−3 to 3220 kgm−3, the slab length has been increased to 300 km and the dip angle has been increased to 60 ◦. The resulting subduction models depicts a slab which subducts with an increasing speed. Velocities as high as 17 cm/yrhave been observed. Note that the maximum depth of the convection cell at the open boundary roughly follows the depth of the tip of the subducting slab.

Figure 4.5e depicts the evolution of a model with a prescribed velocity at the left boundary and a time dependent boundary on the right sidewall. Before 10 Myr the right sidewall was deﬁned by free-slip conditions to start oﬀ subduction and after 10 Myr the free-slip conditions were replaced by open boundary conditions. The oceanic crust has a density of 3220 kgm−3, just like the model shown in Figure 4.5d. In contrast, the oceanic crust has a density of 3150

22 a) vslab = 4 cm/yr b) vslab = 4 cm/yr c) vslab = open boundary d) vslab = open boundary e) vslab = 4 cm/yr Free-slip at the right sidewall vplate = -2 cm/yr from t = 3 Myr Free slip at the right sidewall Free slip at the right sidewall vplate = open boundary from t = 10 Myr Initial slab length = 76 km Initial slab length = 76 km Initial slab length = 200 km Initial slab length = 300 km Initial slab length = 76 km Initial dip angle = 25 ◦ Initial dip angle = 25 ◦ Initial dip angle = 40 ◦ Initial dip angle = 60 ◦ Initial dip angle = 25 ◦ Oceanic crust density = 3150 kgm−3 Oceanic crust density = 3150 kgm−3 Oceanic crust density = 3150 kgm−3 Oceanic crust density = 3220 kgm−3 Oceanic crust density = 3220 kgm−3

Figure 4.5: Diﬀerent combinations of sidewall boundary conditions and initial conditions result in disparate subduction models. vslab denotes the prescribed velocity of the subducting slab and vplate denotes the prescribed velocity of the continental plate. Results 24 kgm−3 in the ﬁrst three models. From 10 Myr to ±11 Myr the continental plate was pushed out of the domain. Then, trench advance gradually changed trench retreat. Over time, the trench retreat velocity increased producing low angle subduction.

The velocity arrows in the Figure 4.5 do not have the same scale. In most cases the scaling is approximately the same. However, in Figure 4.5d, the scale has been reduced signiﬁcantly because of the high subduction velocity.

4.4 The combined eﬀects of mantle phase transitions and sidewall boundary conditions on subduction evolution and surface topography

All subduction models depicted in Figure 4.6 show that slabs subduct almost vertically after crossing the 410 km phase transition. The viscosity increase at the 660 km phase transition opposes subduction and causes slabs to buckle. The rate of buckling depends on the viscosity increase. The models shown in Figure 4.6a and Figure 4.6b feature viscosity increases of respectively a factor 2 and a factor 5 over the 660 km phase transition. A small viscosity increase allows higher velocities in the lower mantle and allows the slab to penetrate further into the lower mantle compared to a larger viscosity increase.

Figure 4.6b and Figure 4.6c are identical, except for the prescribed velocities at the right sidewall. In Figure 4.6b the overriding plate has a velocity of -2 cm/yr starting at t = 8 Myr so that the overriding plate will not approach the left sidewall too closely. The overriding plate velocity in Figure 4.6c is 1 cm/yr and is constant over time. Until approximately 14 Myr the slab in Figure 4.6c has subducted considerably deeper than the slab in Figure 4.6b because of the distance the overriding plates have advanced. Figure 4.6b and Figure 4.6c also show that the angle of subduction is smaller if the overriding plate velocity is higher. After the slab in Figure 4.6c reached the 660 km phase transition it started to fold halfway down its length while the slab in Figure 4.6b developed two hinge points. Both slabs have penetrated roughly the same distance into the lower mantle, despite the diﬀerent angles at which they arrived at the 660 km phase transition.

As demonstrated before, self-sustained subduction is not easily produced without changing material properties or geometry parameters to unrealistic values. However, a combination of measures can be taken to induce self sustained subduction. Firstly, the slab pull force can be increased by pushing the slab through the 410 km phase transition using prescribed velocity boundary conditions. This is why the model shown in Figure 4.6d is a restart from the model shown in Figure 4.6a at 13 Myr. Secondly, buoyant oceanic and continental crustal material is dragged down into the mantle in all models, except the one demonstrated in Figure 4.6d. In this model we followed an approach by Androvi˘cov´aet al.(2013) in which crustal material is changed into mantle material at a depth of 200 km. Consequently, the density anomaly is

24 a) 2x viscosity increase b) 5x viscosity increase c) 5x viscosity increase d) 2x viscosity increase vslab = 4 cm/yr vslab = 4 cm/yr vslab = 4 cm/yr vslab = open boundary from 13 Myr vplate = -2 cm/yr from t = 8 Myr vplate = -2 cm/yr from t = 8 Myr vplate = -1 cm/yr from t = 0 yr vplate = -2 cm/yr from t = 8 Myr No crust at depths greater than 200 km

Figure 4.6: Different boundary conditions and different viscosity increases over the 660 km phase transition result in different subduction evolutions. The yellow lines represent the depths of the 410 km and 660 km phase transitions. Results 26 purely thermal. Most buoyant crustal material in our models does not reach a depth of 200 km because it collects under the lithosphere of the overriding plate. Also, crustal material sinking into the mantle gradually evolves into mantle material due to melting and minor phase transitions (such as the basalt/gabbro-eclogite phase transition). Furthermore, a model with this adaptation requires less computational resources because it contains only 5 compositions.

Open boundaries allow the in- and outﬂow velocities to change between time-steps. As a result, the plate velocity of the model in Figure 4.6d increased to 10.5 cm/yr just before the slab hit the 660 km phase transition. The slab velocity is deﬁned as the average velocity found in a box with a width of 300 km and a height of 44 km centered at (2.5×105; 9.72×105). Subduction did not cease after the slab started buckling on the 660 km phase transition. However, the slab velocity decreased to 4.89 cm/yr after 19 Myr because lower mantle material was pushed aside. Before the subducting slab hit the 660 km phase transition, hardly any material had crossed the open boundary in the lower mantle.

The surface topography of the model shown in Figure 4.6c is illustrated at several points in time in Figure 4.7. Notably, the elevation of the free surface at the sidewalls does not change because the sidewall grid cells are ﬁxed. At all time steps the left side of the graph has descended

5,000

4,000

3,000

2,000

1,000

0 t = 0 Myr −1,000 t = 1 Myr t = 2 Myr

Free surface elevation (m) −2,000 t = 4 Myr t = 8 Myr −3,000 t = 12 Myr t = 16 Myr −4,000 t = 20 Myr t = 24 Myr t = 28 Myr −5,000 0 200 400 600 800 1,000 1,200 1,400 1,600 x (km)

Figure 4.7: The evolution of the free surface over time. After 12 Myr the slab reaches the 410 km phase transition and after 20 Myr it starts subducting into the lower mantle. The arrows show the location of the slab tip at each time.

26 Results 27 below zero meters while the right side generally shows a high free surface elevation. Over time, the trench moves to the left because of the advancing continental plate entering the domain through the right sidewall. The amplitude of the topography varies between -5 km and 5 km which is not unlikely for a model which does not consider erosion.

During the ﬁrst 8 million years of subduction, the slab subducts at a low angle under the continental lithosphere as can be seen in Figure 4.6c. Figure 4.7 shows that the free surface elevation follows the tip of the advancing slab. Until the slab reaches the 410 km phase transition at 12 Myr, the average continental surface lowers land inwards. After the slab crossed the 410 km phase transition, its tip started to subduct almost vertically. As a result a depression in the continental surface formed just left of the tip of the slab. Locally, the continental surface elevation falls below 0 meters. However, close to the contact zone high elevations can be observed. When the slab approached the 660 km phase transition, after 20 Myr of subduction, it encoun- tered a strong subduction opposing force. As a result, surface topography above the subducting slab increased considerably because the constant inﬂow of oceanic crust and lithosphere from the left sidewall exceeded the rate at which the slab sank into the lower mantle.

The latent heat release and consumption over the 410 km and 660 km phase transitions are shown in Figure 4.8. Because the maximum latent heat transmitted at the 410 km phase transition is up to three orders of magnitude larger than the maximum latent heat transmitted at the 660 km phase transition two figures with different scales are shown. The difference in latent heat production may be attributed to the larger Clapeyron slope of the 410 km phase transition or to the larger amount of material that crosses the 410 km phase transition compared to the 660 km phase transition. The total latent heat rate throughout the domain is one order of magnitude smaller than the total shear heating rate and two orders of magnitude smaller than the total adiabatic heating rate.

a) Latent heat production over the 410 km phase transition. b) Latent heat production over the 660 km phase transition.

Figure 4.8: The latent heat (W ) produced at the 410 km and 660 km phase transitions diﬀers up to three orders of magnitude. a) The amount of latent heat produced at the 410 km phase transition. b) The amount of latent heat produced at the 660 km phase transition. The yellow lines represent the depths of the phase transitions.

27 Results 28

4.5 Testing the impact of compressibility on a model with phase transitions and a model without phase transitions

In Figure 4.9 we show the diﬀerence in density distribution between a compressible and an incompressible model. The density of a compressible medium will increase with depth because it is dependent on pressure. In contrast, the density of an incompressible rheology without phase transitions decreases with depth because of its inverse relationship with temperature. Hence, a compressible medium will have an increasingly higher density and viscosity with depth compared to an incompressible.

In our models, a slab subducting in an incompressible model subducts slightly deeper and is less curved at its tip compared to a slab in a compressible medium. Figure 4.10 illustrates that at 200 km depth the density increase due to compressibility is 100 kgm−3. The two high density peaks on each proﬁle represent the cold oceanic lithosphere and cooled down surrounding mantle material. In contrast, the oceanic crust produces the density lows. The density peaks of the incompressible and compressible models are approximately of the same magnitude resulting in comparable buoyancy forces. As the slabs subduct deeper into the mantle, the temperature anomalies become less pronounced due to the conductivity of heat. a a a a 3,600 a a a )

3 3,500 a a kgm a 3,400 a a Density ( a a 3,300 a a a 3,200 a 600 800 1,000 1,200 1,400 a a x (km)

Figure 4.9: Density fields of an incom- Figure 4.10: The density differences between an in- pressible and compressible model after 18 compressible model at 200 km (green line) and 500 km Myr using the same scale. The colored (black line) depth and a compressible model at 200 km horizontal lines indicate the depths of the (blue line) and 500 km (red line) depth. density profiles used in Figure 4.10.

28 Results 29 a a a 4,400 a a 4,200 a a ) 3 a 4,000 a kgm a 3,800 a a Density ( a 3,600 a a 3,400 a a 3,200 a 600 800 1,000 1,200 1,400 a x (km) a

Figure 4.11: Density fields of an in- Figure 4.12: The density differences between an incompressible and compressible model with compressible model at 200 km (green line), 500 km phase transitions after 25 Myr. The col- (black line) and 700 km (blue line) depth and a com- ored horizontal lines indicate the depths pressible model at 200 km (red line), 500 km (orange of the density profiles used in Figure 4.12. line) and 700 km (cyan line) depth.

Despite the larger density increases, the difference between an incompressible and a compressible model with phase transitions remains insignificant (Figure 4.11). Also, the density profiles in Figure 4.12 show similar size density anomalies as Figure 4.10.

The prescribed velocity boundary conditions used in this study assure a volume balance inside the domain. However, the inflow of light crustal rocks and dense lithospheric rocks combined with the outflow of varying density mantle rocks causes a mass flux over the boundaries. Table 4.1 shows the mass fluxes over a single prescribed velocity boundary for four models. A mass flux into the domain is found for an incompressible model without phase transitions because density decreases with depth. A compressible model without phase transitions results in a mass flux out of the domain. Adding phase transitions increases the density with depth and causes a larger mass flux out of the domain.

Incompressible Compressible Incompressible and Compressible and phase transitions phase transitions −1.875 ∗ 105 9.790 ∗ 105 1.041 ∗ 106 2.439 ∗ 106

Table 4.1: Outward mass ﬂux (kg/yr) over the left boundary.

29 5. Discussion

In this chapter the results will be interpreted and our research questions, as posed in the introduction, will be answered. Also, our results will be compared to the literature and the signiﬁcance of open boundaries in combination with phase transitions will be elaborated.

5.1 Reducing the computational time

The computational time required by the model can be lowered without reducing its accuracy signiﬁcantly by limiting the maximum number of nonlinear iterations, adjusting the tolerances of the solvers and averaging the material model output data when constructing the linear system for velocity/pressure, temperature and compositions (Fraters, 2014, Bangerth et al., 2015). The observations described below are only valid for this speciﬁc subduction setup.

The number of nonlinear iterations per time-step contributes to the accuracy and eﬃciency of the model. The maximum number of nonlinear iterations and nonlinear solver tolerance control the number of nonlinear iterations. The maximum number of nonlinear iterations terminates the iterative process during the IRS and at low time-steps when residuals are much larger than the nonlinear solver tolerance. Also, during some later time-steps the residual does not converge to the nonlinear solver tolerance and the iterative process is ceased by the maximum number of nonlinear iterations. The nonlinear solver tolerance generally limits the number of nonlinear iterations during later time-steps when residuals are low and converge easily. In our model, the convergence does not increase signiﬁcantly when more than 50 nonlinear iterations are allowed. The nonlinear solver tolerance is set to 10−6 and allows for convergence within 10 nonlinear iterations at later time-steps. Fraters(2014) found that a comparable model usually required 30 to 60 nonlinear iterations to converge at low time-steps and recommended a maximum number of nonlinear iterations of about 100. The ideal nonlinear solver tolerance was found to vary with time. The default Stokes tolerance of 10−7 is too strict for this system and a value of 10−6 has been adopted here.

Figure 4.2 and Figure 4.3 illustrate that material averaging schemes can be used to save computational time at the cost of convergence. The arithmetic, harmonic and geometric averaging schemes resulted in similar time reductions of up to a factor three during the ﬁrst 200 time-steps. However, the averaging of material model output increased the end residuals to values between 10−6 and 10−4. Despite the higher end residuals the subduction models were not discernibly altered. These results are in conformity with observations described in Bangerth et al.(2014).

30 Discussion 31

5.2 The eﬀects of prescribed and open sidewalls

Our results underscore the substantial effects of sidewall boundary conditions on subduction evolution. In models with small domain ratios and a single prescribed velocity boundary, strong return flows on one side of the domain in combination with outflow that is constant with depth on the other side impose vertical subduction. If two prescribed velocity sidewalls push the oceanic and continental plates into the domain, the slab subducts at a lower angle. At the same time, the overriding plate advances over the slab resulting in trench retreat. These findings are in agreement with results from van Hunen et al.(2001), Billen(2008) and Quinquis et al. (2011). The aforementioned observations demonstrate how well-known restrictions are imposed upon a model by using boundary conditions. In addition, these observations emphasize the importance of large domain ratios to minimize unwanted effects induced by boundary conditions as demonstrated by Chertova et al.(2012).

Androvi˘cováet al.(2013) and Tosi et al.(2016) have modeled self sustained subduction in high width to depth ratio domains with only free-slip boundary conditions. To initiate self- sustained subduction the authors prescribed a no slip condition at one half of the top boundary (above the static overriding plate) and either kinematic conditions (for t < n yr) or free slip conditions (for t > n yr) on the other half of the top boundary (n is the number of years for which the kinematic conditions were active). These models do not allow a free surface or a sticky-air layer to generate surface topography because of the force applied at the top boundary. Therefore, ridge push forces can’t be incorporated. In addition, the subducting slab is of finite length. In contrast, open boundaries can be combined with a free surface or a sticky-air layer because the initial driving force can be applied at a sidewall to simulate the ’ridge push force’. Because the ’ridge push force’ never ceases and doesn’t push at a constant velocity, the sidewalls would ideally consist of prescribed stress boundary conditions for the oceanic and continental plates and open (lithostatic pressure) boundary condition for the mantle. In this study we explored only the effects of a prescribed velocity sidewall in combination with an entirely open boundary.

Implementing open sidewalls allows a multitude of new subduction geometries to evolve. Because the velocity of the plates is no longer prescribed, the subduction location, angle and velocity are less confined. We found that substituting a prescribed velocity boundary, which forces an overriding plate into the domain, by an open boundary resulted in more complex evolutions of the oceanic and continental plates. In one of our models the direction of the overriding plate even reversed from outflow to inflow. After several million years, the inflow velocity of the overriding plate was doubled compared to the prescribed velocity in the reference model. Consequently, the trench retreated rapidly and the slab subducted at a lower angle. Thus, the employment of open boundary conditions makes the model evolution more sensitive to its internal dynamics.

31 Discussion 32

A prescribed velocity boundary condition is required in our models to drive subduction until the slab pull force overcomes the forces opposing subduction. We have shown that subduction can also be initiated by increasing the slab length, density and dip angle. However, these measures resulted in unrealistically high slab velocities when subduction continued for a longer time. Doubtfully high velocities were also found in a model by Androvi˘cov´aet al.(2013) in which the authors initiated subduction without external driving forces. In this model the oceanic crust had the same density as the mantle because the sole purpose of the oceanic crust was the mechanical decoupling of the oceanic and continental plates. The combination of a weak zone, lateral pressure diﬀerences due to the increase of plate thickness with age and the high crustal density allowed for self-initiated subduction.

5.3 The combined eﬀects of mantle phase transitions and sidewall boundary conditions on subduction evolution

In our open boundary models, the 410 km phase transition turns out to be essential for self sustained subduction. After the tip of the slab had crossed the 410 km phase transition the prescribed slab velocity of 4 cm/yr increased to 10.5 cm/yr. After the slab hit the 660 km phase transition, its velocity gradually decreased to 4.89 cm/yr and large scale buckling developed. Androvi˘cov´aet al.(2013) described similar observations. In their free slip models a slab with a yield stress of 108 Pa (corresponding to a viscosity of 1024 Pas) reached a velocity of 10 cm/yr just before it hit the 660 km phase transition. Then it buckled over the 660 km phase transition with a viscosity increase of a factor 10. Despite the buckling the velocity kept increasing to 18 cm/yr.

Another experimental setup by Androvi˘cov´aet al.(2013) featured a slab with a yield stress of 109 Pa (corresponding to a viscosity of 1026 - 1027 Pas). The velocity of this stiﬀ slab steadily increased to 17 cm/yr just before it hit the 660 km phase transition. After it penetrated through the 660 km phase transition, its velocity slowly decreased due to the high lower mantle viscosity. Models by Quinteros et al., 2010 also show subducting slabs with a viscosity varying between 1024 and 1026.5 Pas penetrating into the lower mantle. Their ability to cross the 660 km phase transition is dependent on the viscosity increase over the phase transition. Plate velocities exceeded 10 cm/yr in the upper mantle and dropped below 6 cm/yr after the slab crossed the 660 km phase transition.

All models described above stress the importance of a viscosity increase at the 660 km phase transition to limit the velocity of the gravitationally driven subducting slab. Likewise, Tosi et al.(2016) found that a viscosity increase of 30 at the 660 km phase transition results in low velocities (v < 4 cm/yr) at the trench whereas a viscosity increase of a factor 10 allows velocities of up to 5 cm/yr. The subducting slabs in models by Androvi˘cov´aet al.(2013) reached higher velocities than the slabs in our open boundary models because of the high crustal densities. This is because the onset of spontaneous nucleation subduction zones depends on gravitational

32 Discussion 33 instabilities while the onset of induced nucleation subduction zones requires an external force (Stern, 2004). To prevent unrealistically high slab velocities, self sustained subduction should be preceded by prescribed conditions to simulate far ﬁeld driving mechanisms.

Our models show that if a slab is only propelled by gravitational forces, its velocity decreases steadily upon approaching the 660 km phase transition until the slab stagnates at the phase boundary. Likewise, a slab with a prescribed velocity is impeded at the 660 km phase transition. Even a viscosity increase of a factor two does not allow the slab to cross the 660 km phase transition although a viscosity increase between one and two orders of magnitude is commonly described in the literature (e.g. van Hunen et al., 2001, Steinberger and Calderwood, 2006, Quinteros et al., 2010, C´ı˘zkováet˘ al., 2012, Androvi˘cováet al., 2013, Tosi et al., 2016). Studies by Billen, 2008, Quinteros et al., 2010 and Androvi˘cováet al., 2013 demonstrate that stiff slabs are more likely to cross the 660 km phase transition than weak slabs. We only studied weak slabs, which appear to lose momentum by buckling on the 660 km phase transition.

The slab pull force can be increased by changing the oceanic and continental crust into eclogite at a depth of 50 km. The eclogite needs to have a low viscosity to continue decoupling the oceanic crust from the continental lithosphere. When there is no more friction between the oceanic and continental plates (at ± 200 km depth), the eclogite can be transformed into upper mantle material. Additionally, this method is likely to reduce the accumulation of oceanic crust in the mantle wedge.

Because the viscosity increases with depth, the flow velocity in the lower mantle is less than the velocity in the upper mantle. Takeuchi and Sugi(1972) use a mantle convection velocity of 30 mm/yr to estimate the amount of material coming up at a ridge system. According to van der Meer et al.(2010) the average sinking speed of subducted slabs in the lower mantle is 12 ± 3 mm/yr. Prescribed velocity boundaries enforce a velocity profile and disregard the effect of the viscosity increase with depth. In contrast, the in- and outflow velocity over an open boundary is dependent on the mantle viscosity and the depth to which the slab has subducted. Our open boundary models show a very small flow velocity in the lower mantle before the slab has reached the 660 km phase transition. Hence, open boundaries are preferred over prescribed boundary conditions in models with viscosity increases.

From our models we can conclude that the latent heat contribution to the energy equation can be neglected. The total latent heat rate throughout the domain is one order of magnitude smaller than the total shear heating rate and two orders of magnitude smaller than the total adiabatic heating rate. The total latent heat rate is small because the process is only active at phase transitions. In addition, models including latent heat do not differ from models without latent heat in any discernible way. Previous research has also described that the effect of latent heat is a second order effect (e.g. Christensen, 1996, van Hunen et al., 2001, Sinha and Butler, 2009).

33 Discussion 34

5.4 Why compressibility can be ignored

Compressibility increases the density of our models to resemble the PREM accurately up to a depth of 1000 km. However, the resulting density increase does hardly change the absolute density diﬀerence between the slab and the mantle. Therefore, the slab pull force in an incompressible medium is comparable to the slab pull force in a compressible medium. Higher densities cause higher pressures which, in turn, lead to a viscosity increase with depth. Al- though the viscosity increase is perceptible, it does not have a considerable eﬀect on subduction evolution.

Phase transitions and compressibility increase the net mass flux over open or prescribed boundaries. This has no effect on the evolution or stability of the model because it doesn’t change the volume of the domain. The total mass inside the domain is only affected by changing the ratio between compositional fields, i.e. due to the continuous inflow of oceanic crust and lithosphere and outflow of mantle material.

Despite the recent interest in compressibility and the benchmarking of compressible Cartesian codes (King et al., 2010), we find the effects of compressibility in our subduction models to be negligible. Several other rheological parameters have a more pronounced effect on subduction evolution. Therefore, our findings support the widespread application of incompressible models in subduction studies.

5.5 The combined eﬀects of mantle phase transitions and sidewall boundary conditions on surface topography

Large scale surface expressions caused by mantle processes have been observed in our models. The low angle subduction of dense oceanic lithosphere causes a surface depression in the continental plate which is proportional in width to the lateral distance the slab has subducted. This depression is likely to be be caused by a combination of isostasy and viscous drag of the sinking slab. After crossing the 410 km phase transition the slab pull force increases. Because the prescribed velocities do not allow the slab velocity to increase, local depressions with large magnitudes form at the surface. Dynamic topography produced by the viscous drag of subducting slabs has also been observed by Husson(2006) in narrow retreating subduction systems such as the Scotia, Aegean and Mariana seas. These small-scale tectonic systems are largely independent of mantle scale circulations and viscosity variations. The authors suggest that this type of dynamic topography is also active in other locations where it is more diﬃcult to observe.

The 660 km phase transition reduces the sinking speed of subducting slab signiﬁcantly. Stresses transmitted vertically through the slab cause high surface topography and, probably, maxima in the geoid. Other studies have also shown that the 660 km phase transition below subduction zones causes local maxima in the geoid (e.g. King(2002), Richards and Hager,

34 Discussion 35

1989, Zhong and Gurnis, 1992, King and Hager, 1994). Because the dynamic topography is the diﬀerence between the mean surface elevation (which is equal to zero) and the geoid, the geoid should be equal to the dynamic topography when no erosion and sedimentation processes are included.

The surface elevation of models with open boundaries is expected to be less sensitive to phase transitions because their slab velocities will depend on mantle properties. E.g. if a slab encounters a viscosity increase and its sinking speed decreases, its inflow velocity will reduce correspondingly. Still, phase transitions have been shown to have a significant effect on surface topography. On the other hand, surface expressions are expected to affect mantle flow as well (Braun, 2010). Therefore, the implementation of surface processes, such as erosion and sedimentation, should be considered.

35 6. Conclusion

The models compared in this study demonstrate again the importance of high width to depth domain ratios and the merits of choosing the boundary conditions carefully. Free slip sidewalls cause strong return flows. Prescribed velocity boundaries are, in some cases, an improvement over free slip boundaries, but restrict the influences of the internal dynamics. Consequently, the subduction geometry is largely defined by the boundary conditions. Open boundary conditions allow a multitude of new subduction geometries to evolve more naturally. By changing a prescribed velocity sidewall to an open sidewall, overriding plate velocities have doubled and subducting slabs were no longer hindered by low velocities.

Mantle phase transitions have been found to exert great inﬂuence on subduction dynamics. The 410 km phase transition increases the slab pull force considerably. Because the slab velocity is limited, the slab will subduct vertically below 410 km. As a result, a high magnitude surface depression forms in the overriding plate. In contrast, at the 660 km phase transition the slab is obstructed and vertically transmitted stresses cause high surface elevations. In both this study and previous studies, weak slabs have been demonstrated to buckle on the 660 km phase transition. In contrast, stiﬀ slabs have been found to penetrate into the lower mantle.

A combination of open boundaries and mantle phase transitions generates promising results. The density increase at 410 km depth accelerates the subducting slab and may change the direction of trench motion from advancing to retreating. In contrast, the 660 km phase transition limits the subducting slab velocity to realistic values. To prevent the usage of unnatural material properties, open sidewalls should be preceded by prescribed boundary conditions until the slab has crossed the 410 km phase transition.

Compressibility is found to have a second order effect on subduction evolution. Our compressible density profile resembles the PREM model well up to a depth of 1000 km. But the resulting viscosity change has no considerable effect on subduction dynamics.

36 Bibliography

Alisic, L., Gurnis, M., Stadler, G., Burstedde, C., and Ghattas, O. (2012). Multi-scale dynamics and rheology of mantle ﬂow with plates. J. Geophys. Res., 117:19–34.

Androvi˘cová,A., C´ı˘zková,˘ H., and van den Berg, A. (2013). The effects of rheological decoupling on slab deformation in the earths upper mantle. Stud. Geophys. Geod., 57:460–481.

Austmann, W., Govers, R., and Burov, E. (2014). The role of elastic compressibility in dynamic subduction models. Geophysical Research Abstracts, 16:13,465.

Bangerth, W., Hartmann, R., and Kanschat, G. (2007). deal.iia general purpose object oriented ﬁnite element library. ACM Trans. Math. Softw., 33(4):24:1–24:27.

Bangerth, W., Heister, T., et al. (2014). ASPECT: Advanced Solver for Problems in Earth’s ConvecTion. Computational Infrastructure for Geodynamics.

Bangerth, W., Heister, T., et al. (2015). ASPECT: Advanced Solver for Problems in Earth’s ConvecTion. https://aspect.dealii.org/.

Behounkova, M. and C´ı˘zkov´a,H.˘ (2008). Long-wavelength character of subducted slabs in the lower mantle. Earth Planet. Sci. Lett., 275:43–53.

Bellahsen, N., Faccenna, C., and Funiciello, F. (2005). Dynamics of subduction and plate motion in laboratory experiments: insights into the plate tectonics behavior of the earth. J. Geophys. Res., 110:1–15.

Bevis, M. (1988). Seismic slip and down-dip strain rates inwadati-benioﬀ zones. Science, 240:1,317–1,319.

Billen, M. (2008). Modeling the dynamics of subducting slabs. Annu. Rev. Earth Planet. Sci., 36:325–356.

Bina, C. R. and Helﬀrich, G. R. (1994). Phase transition clapeyron slopes and transition zone seismic discontinuity topography. J. Geophys. Res., 99:15,853–15,60.

Bock, G. (1994). Synthetic seismogram images of the upper mantle structure: No evidence for a 520-km discontinuity. J. Geophys. Res., 99:15,843–15,851.

Bormann, P. (2002). Datasheet: Global 1-d earth models.

Braun, J. (2010). The many surface expressions of mantle dynamics. Nature Geoscience, 3:825–833.

Burstedde, C., Wilcox, L. C., and Ghattas, O. (2011). p4est: Scalable algorithms for parallel adaptive mesh reﬁnement on forests of octrees. SIAM J. Sci., 33(3):1103–1133.

Capitanio, F., Morra, G., and Goes, S. (2007). Dynamic models of downgoing plate-buoyancy driven subduction: Subduction motions and energy dissipation. Earth Planet. Sci. Lett., 262:284–297.

Capitanio, F., Stegman, D., Moresi, L., and Sharples, W. (2010). Upper plate controls on deep subduction, trench migrations and deformations at convergent margins. Tectonophysics, 483(1):80–92. 37 Conclusion 38

Chertova, M. V., Geenen, T., van den Berg, A., and Spakman, W. (2012). Using open sidewalls for modelling self-consistent lithosphere subduction dynamics. J. Geoph. Res: Solid Earth, 3:313–326.

Christensen, U. R. (1996). The inﬂuence of trench migration on slab penetration into the lower mantle. Earth Planet. Sci. Lett., 140:27–39.

Christensen, U. R. and Yuen, D. A. (1985). Layered convection induced by phase transitions. J. Geophys. Res., 90(B12):10,291–10,300.

Clark, S. R., Stegman, D., and Muller, R. D. (2008). Episodicity in back-arc tectonic regimes. Phys. Earth Planet. Inter., 171(1):265–279.

Crameri, F., Schmeling, H., Golabek, G. J., Duretz, T., Orendt, R., Buiter, S. J. H., May, D. A., K. B. J. P., Gerya, T. V., and Tackley, P. J. (2012). A comparison of numerical surface topography calculations in geodynamic modelling: an evaluation of the ’sticky air’ method. Geoph. J. Int., 189:38– 54.

Crameri, F. and Tackley, P. J. (2015). Parameters controlling dynamically self-consistent plate tectonics and single-sided subduction in globalmodels of mantle convection. J. Geoph. Res: Solid Earth, 10:3,680–3,706.

Dziewonski, A. M. and Anderson, D. L. (1981). Preliminary reference earth model. Phys. Earth planet. Inter., 25.

Enns, A., Becker, T. W., and Schmeling, H. (2005). The dynamics of subduction and trench migration for viscosity stratiﬁcation. Geophys. J. Int., 160:755–761.

Forsyth, D. and Uyedat, S. (1975). On the relative importance of the driving forces of plate motion. Geophys. J. R. astr. Soc., 43:163–200.

Forte, A. and Mitrovica, J. (1996). New inferences of mantle viscosity from joint inversion of long- wavelength convection and post-glacial rebound data. Geophys. Res. Lett., 23:1,147–1,150.

Fraters, M. (2014). Thermo-mechanically coupled subduction modelling with aspect. A Master thesis.

Fukao, Y., Widiyantoro, S., and Obayashi, M. (2001). Stagnant slabs in the upper and lower mantle transition region. Rev. Geophys., 39:291–323.

Garel, F., Goes, S., Davies, D. R., Davies, J. H., Kramer, S. C., and Wilson, C. R. (2014). Interaction of subducted slabs with the mantle transition-zone: A regime diagram from 2-d thermo-mechanical models with a mobile trench and an overriding plate. Geochem. Geophys. Geosyst., 15:1,739–1,765.

Gerya, T. (2010). Numerical Geodynamic Modelling. Cambridge University Press, The Edinburgh Build- ing, Cambridge CB2 8RU, UK.

Gerya, T. (2011). Future directions in subduction modeling. Journal of Geodynamics, 52:344–378.

Glerum, A., Thieulot, C., and Blom, C. (2016). Benchmarking open boundaries in aspect and elefant. Personal communications.

Glerum, A., Thieulot, C., van den Berg, A., and Spakman, W. (in prep). Benchmarking viscoplasticity implementations in aspect and application to 3d subduction modeling.

38 Conclusion 39

Guillou-Frottier, L., Buttles, J., and Olson, P. (1995). Alaboratory experiments on the structure of subducted lithosphere. J. Geophys. Res., 133:19–34.

Gurnis, M. (1992). Rapid continental subsidence following the initiation and evolution of subduction. Science, 255:1556–1558.

Gurnis, M. and Hager, H. (1988). Controls of the structure of subducted slabs. Nature, 335:317–321.

Gurnis, M., M. J. R. J. and van Heijst, H.-J. (2000). Constraining mantle structure using geological evidence of surface uplift rates: The case of the african superplume. Geochem. Geophys. Geosys., 1.

Hager, B. H. (1984). Subducted slabs and the geoid: Constraints on mantle rheology and ﬂow. J. Geophys. Res., 89:6,003–6,016.

Hayes, G. P., Wald, D. J., and Johnson, R. L. (2012). Global tomographic images of mantle plumes and subducting slabs: insight into deep earth dynamics. J. Geophys. Res., 117.

Heroux, M. A. (2005). An overview of the trilinos project. ACM Trans. Math. Softw., 31:397–423.

Husson, L. (2006). Dynamic topography above retreating subduction zones. Geology, 34:741–744.

Ismail-Zadeh, A. and Tackley, P. (2010). Computational methods for geodynamics. Cambridge University Press.

Ita, J. and King, S. D. (1994). Sensitivity of convection with an endothermic phase change to the form of governing equations, initial conditions, boundary conditions, and equation of state. J. Geoph. Res, 99:15,919–15,938.

Jarvis, G. T. and McKenzie, D. P. (1980). Convection in a compressible ﬂuid with inﬁnite prandtl number. J. Fluid Mech., 96:515–583.

Karato, S.-i. (2008). Deformation of earth materials. Cambridge University Press.

Kaufmann, G. and Lambeck, K. (2000). Mantle dynamics, postglacial rebound and the radial viscosity proﬁle. Phys. Earth Planet. Inter., 121:303–327.

Keppie, D. F., Currie, C. A., and Warren, C. (2009). Subduction erosion modes: Comparing ﬁnite element numerical models with the geological record. Earth Planet. Sci. Lett., 287:241–254.

Kincaid, C. and Olson, P. (1987). An experimental study of subduction and slab migration. J. Geophys. Res., 92:13,832–13,840.

King, S. D. (2001). Subduction zones: observations and geodynamic models. Phys. Earth Planet. Inter., 127:9–24.

King, S. D. (2002). Geoid and topography over subduction zones: The eﬀect of phase transformations. J. Geoph. Res: Solid Earth, 107:2,156–2,202.

King, S. D. and Hager, B. H. (1994). Subducted slabs and the geoid, 1, numerical experiments with temperature-dependent viscosity. J. Geophys. Res., 99:19,843–19,852.

King, S. D., Lee, C., van Keken, P. E., Leng, W., Zhong, S., Tan, E., Tosi, N., and Kameyama, M. C. (2010). A community benchmark for 2-d cartesian compressible convection in the earths mantle. Geophys. J. Int., 180:73–87.

39 Conclusion 40

King, S. D., Raefsky, A., and Hager, B. H. (1990). Conman: vectorizing a ﬁnite element code for incompressible two-dimensional convection in the earths mantle. Phys. Earth planet. Inter., 59:195– 207.

Kronbichler, M., Heister, T., and Bangerth, W. (2012). High accuracy mantle convection simulation through modern numerical methods. Geophys. J. Int., 191:12–29.

Lee, C. (2014). Eﬀects of radiogenic heat production and mantle compressibility on the behaviors of venus and earths mantle and lithosphere. Geosciences Journal, 18:13–30.

Lee, C. and Scott, D. (2009). Eﬀect of mantle compressibility on the thermal and ﬂow structures of the subduction zones. Geochemistry, Geophysics and Geosystems, 10:1–24.

Leng, W. and Gurnis, M. (2011). Dynamics of subduction initiation with diﬀerent evolutionary pathways. Geochem. Geophys. Geosyst., 12.

McKenzie, D. P. (1969). Speculations on the consequences and causes of plate motions. Geophys. J. R. Astron. Soc., 18:1–32.

McKenzie, D. P. (1970). Temperature and potential temperature beneath island arcs. Tectonophysics, 10:357–366.

Mitrovica, J. and Forte, A. (2004). A new inference of mantle viscosity based upon a joint inversion of convection and glacial isostatic adjustment data. Earth Planet. Sci. Let., 225:177–189.

Moresi, L., Zhong, S. J., and Gurnis, M. (1996). The accuracy of ﬁnite element solutions of stokes ﬂow with strongly varying viscosity. Phys. Earth planet. Inter., 97:83–94.

Nothard, A., Haines, J., Jackson, J., and Holt, B. (1996). Distributed deformation in the subducting lithosphere at tonga. Geophys. J. Int., 127:328–338.

Priestley, K. J., Cipar, A., Egorkin, and Pavlenkova, N. (1994). Upper mantle velocity structure beneath the siberian platform. Geophys. J. Int., 118:369–378.

Pysklywec, R. and Mitrovica, J. (1998). A mantle ﬂow mechanism in the long-wavelength subsidence of continental interiors. Geology, 26:678–690.

Pysklywec, R. N. and Shahnas, M. H. (2003). Time-dependent surface topography in a coupled crust- mantle convection model. Geophys. J. Int., 154(2):268–278.

Quinquis, M. E. T., Buiter, S. J. H., and Ellisc, S. (2011). The role of boundary conditions in numerical models of subduction zone dynamics. Tectonophysics, 497:57–70.

Quinteros, J., Sobolev, S. V., and Popov, A. A. (2010). Viscosity in transition zone and lower mantle: Implications for slab penetration. Geoph. Res. Letters, 37.

Richards, M. A. and Hager, B. H. (1984). Geoid anomalies in a dynamic earth. J. Geophys. Res., 89:5,987–6,002.

Richards, M. A. and Hager, B. H. (1989). Eﬀects of lateral viscosity variations on long-wavelength geoid anomalies and topography. J. Geophys. Res., 94:10,299–10,313.

Richter, F. and McKenzie, D. (1978). Simple plate models of mantle convection. J. Geophys., 44:441–471.

40 Conclusion 41

Richter, F. M. (1973). Dynamical models for the thermal evolution of the earth. Rev. Geophys. Space Phys., 11:223–287.

Runcorn, S. K. (1967). Flow in the mantle inferred from the low degree harmonics of the geopotential. Geophys. J. R. Astron. Soc., 42:375–384.

Schellart, W. P. (2004). Kinematics of subduction and subduction-induced ﬂow in the upper mantle. J. Geophys. Res., 109:1–19.

Schmeling, H., Babeyko, A., Enns, A., Faccenna, C., F. F., Gerya, T., Golabek, G., Grigull, S., Kaus, B., Morra, G., Schmalholz, S., and van Hunen, J. (2008). A benchmark comparison of spontaneous subduction models - towards a free surface. Phys. Earth. Planet. Inter., 171:198–223.

Schubert, G., Turcotte, D. L., and Olson, P. (2001). Mantle convection in the earth and planets. Cambridge University Press.

Shearer, P. M. (2000). Upper mantle seismic discontinuities. Geophysical monograph, 117:115–131.

Sheng, J., Liao, J., and Gerya, T. (2016). Numerical modeling of deep oceanic slab dehydration: Im- plications for the possible origin of far ﬁeld intra-continental volcanoes in northeastern china. J. of Asian Earth Sci., 117:328–336.

Sinha, G. and Butler, S. L. (2009). The combined eﬀects of continents and the 660 km-depth endothermic phase boundary on the thermal regime in the mantle. Phys. Earth planet. Inter., 173:354–364.

Stegman, D., Farrington, R., Capitanio, F., and Schellart, W. (2010). A regime diagram for subduction styles from 3-d numerical models of free subduction. Tectonophysics, 483(1):29–45.

Steinberger, B. and Calderwood, A. R. (2006). Models of large-scale viscous ﬂow in the earths mantle with constraints from mineral physics and surface observations. Geophys. J. Int., 167:1,461–1,481.

Stern, J. R. (2004). Subduction initiation: spontaneous and induced. Earth Planet. Sci. Lett., 226:275– 292.

Tackley, P. J., Stevenson, D. J., Glatzmaier, G. A., and Schubert, G. (1993). Eﬀects of an endothermic phase transition at 670 km depth in a spherical model of convection in the earths mantle. Nature, 361:699–704.

Takeuchi, H. and Sugi, N. (1972). Rotation of the Earth, 212-214. International Astronomical Union.

Thieulot, C. (2014). Elefant: a user-friendly multipurpose geodynamics code. Solid Earth, 6:1,949–2,096.

Tosi, N., Maierova, P., and Yuen, D. A. (2016). Inﬂuence of variable thermal expansion and conductivity on deep subduction. Subduction dynamics: from mantle ﬂow to mega disasters, Geophysical Monograph, 211:115–133.

Trubitsyn, V. P. and Trubitsyna, A. P. (2015). Eﬀects of compressibility in the mantle convection equations. Physics of the Solid Earth, 51:801–813.

C´ı˘zkov´a,H.˘ and Bina, C. (2013). Eﬀects of mantle and subduction-interface rheologies on slab stagnation and trench rollback. Earth Planet. Sci. Lett., 379:95–103.

C´ı˘zkov´a,H.,˘ van den Berg, A., Spakman, W., and Matyska, C. (2012). The viscosity of earths lower mantle inferred from sinking speed of subducted lithosphere. Phys. Earth Planet. Int., 200-201:56–62. 41 Conclusion 42

C´ı˘zkov´a,H.,˘ van Hunen, J., van den Berg, A. P., and Vlaar, N. J. (2002). The inﬂuence of rheological weakening and yield stress on the interaction of slabs with the 670 km discontinuity. Earth Planet. Sci. Lett., 199:447–457.

Uyeda, S. and Kanamori, H. (1979). Long-wavelength character of subducted slabs in the lower mantle. J. Geophys. Res., 84(B3):1,049–1,061. van der Hilst, R. D., Widiyantoro, S., and Engdahl, E. (1997). Evidence for deep mantle circulation from global tomography. Nature, 386:578–584. van der Meer, D., Spakman, W., van Hinsbergen, D., Amaru, M., and Torsvik, T. (2010). Towards absolute plate motions constrained by lower-mantle slab remnants. Nat. Geosci., 3:36–40.

van Hunen, J., van den Berg, A. P., and Vlaar, N. J. (2001). Latent heat eﬀects of the major mantle phase transitions on low-angle subduction. Earth Planet. Sci. Lett., 190:125–135.

Zhao, D. P. (2004). Global tomographic images of mantle plumes and subducting slabs: insight into deep earth dynamics. Phys. Earth Plan. Int., 146:3–34.

Zhong, S. and Gurnis, M. (1992). Viscous ﬂow model of a subduction zone with a faulted lithosphere: Long and short wavelength topography, gravity, and geoid. Geophys. Res. Lett., 19:1,891–1,894.

42 A. Derivation of the adiabatic temperature equation

Derivation of Equation 3.3:

ASPECT computes the temperature Tad(z) and pressure Pad(z) ﬁelds that satisfy the adiabatic conditions. The adiabatic condition that the temperature ﬁeld has to satisfy is given by Equation A.1 (Bangerth et al., 2014).

d δρ ρC T (z) = T (z)g (A.1) p dz ad δT ad

According to the Boussinesq approximation the heat capacity per unit volume, ρCp, is assumed constant??? This result in Equation A.1

d δρ ρ C T (z) = T (z)g (A.2) ref p dz ad δT ad

ρ is deﬁned as:

h i ρ = ρref 1 − α(T − Tref ) (A.3)

The partial derivative of ρ(z) over δT is

δρ = −ρ α (A.4) δT ref

Substituting the derivative of ρ(z) in Equation A.2 gives

d ρ C T (z) = −ρ αT (z)g (A.5) ref p dz ad ref ad

g is negative because it works in the -z direction: g=-g

d ρ C T (z) = ρ αT (z)g (A.6) ref p dz ad ref ad

which can be rewritten to

d αTad(z)g Tad(z) = (A.7) dz Cp

i Conclusion ii

which can be rewritten to

Z Tpot 1 Z b αg dTad(z) = dz (A.8) T Tad(z) a Cp

which can be rewritten to

T Tad(z) αgz [ln(Tad(z))]T pot = ln = (A.9) Tpot Cp

Then, take the exponential of both sides

T (z) αgz ad = exp( ) (A.10) Tpot Cp

This results in Equation 3.3, also used by Tosi et al.(2016)

αgz Tad(z) = Tpotexp( ) (A.11) Cp

ii B. Benchmarking open boundaries in ELEFANT and ASPECT

Here we will compare velocity values of similar set-ups in ASPECT and ELEFANT. Two model set-ups will be compared in which the boundary conditions will be varied between free slip and open lithostatic. By comparing the percentual velocity diﬀerences of the free slip boundary models with the percentual velocity diﬀerences of open boundary conditions, the implementation of the open lithostatic boundary conditions in both codes will be benchmarked.

B.1 TEST OPENBC 1: 3D Sinking cube in ELEFANT and AS- PECT

The domain is a cube of size L=1.123456789, gravity is set to g z=-1. It contains two materials. A cube of size L/3 centered in the cube has a density ρ=2.123456789 and a viscosity µ=10 while the surrounding material is characterised by ρ=1.123456789 and µ=0.1. Top boundary is no slip and bottom boundary is free slip.

B.1.1 Compositional initial conditions

ELEFANT: ASPECT: set Variable names = x, y, z if(abs(xq-Lx/2) < Lx/6 .and. set Function constants = L=1.123456789 abs(yq-Ly/2) < Ly/6 .and. set Function expression = if(x > L/3 & x < 2 ∗ abs(zq-Lz/2) < Lz/6) (L/3) & y > L/3 & y < 2 ∗ (L/3) & z > L/3 & z < 2 ∗ (L/3), 1, 0)

iii Conclusion iv

a) Density ELEFANT b) Viscosity ELEFANT c) Pressure ELEFANT

d) Density ASPECT e) Viscosity ASPECT f) Pressure ASPECT

B.1.2 Velocity diﬀerences invoked by viscosity averaging schemes

In this study all boundaries are free slip, except the top boundary which is no slip. The eﬀect of the material averaging scheme on sinking speed is tested here.

ELEFANT: ASPECT: fix bc1 u = .true. fix bc2 u = .true. fix bc3 v = .true. fix bc4 v = .true. set Tangential velocity boundary indicators = 0,1,2,3,4 fix bc5 w = .true. set Zero velocity boundary indicators = 5 fix bc6 u = .true. fix bc6 v = .true. fix bc6 w = .true.

The table below shows that in this model setup the maximum velocities in the x- and y-direction are always smaller in ASPECT than in ELEFANT. For both ASPECT and ELEFANT, the 32x32x32 grid shows the highest maximum velocities in the x- and y-direction. No pattern between ELEFANT and ASPECT is recognized for the maximum velocity.

The viscosity averaging schemes have little eﬀect on the velocity proﬁle. The arithmetic averaging scheme results in the lowest maximum velocities because higher viscosity values have higher weights. On iv Conclusion v

Viscosity Absolute max/ Absolute max/ Vel z (m/s) averaging scheme min vel x (m/s) min vel y (m/s) Min Max ELEFANT 24x24x24 No 1 .628 ∗ 10 −2 1 .628 ∗ 10 −2 −4 .020 ∗ 10 −2 ? ELEFANT 32x32x32 No 1 .632 ∗ 10 −2 1 .632 ∗ 10 −2 −3 .976 ∗ 10 −2 ? ELEFANT 48x48x48 No 1 .627 ∗ 10 −2 1 .627 ∗ 10 −2 −4 .104 ∗ 10 −2 ? ASPECT 24x24x24 Harmonic 1 .454 ∗ 10 −2 1 .454 ∗ 10 −2 −3.925 ∗ 10−2 1 .783 ∗ 10 −2 ASPECT 32x32x32 Harmonic 1 .559 ∗ 10 −2 1 .559 ∗ 10 −2 −4.216 ∗ 10−2 1 .900 ∗ 10 −2 ASPECT 48x48x48 Harmonic 1 .536 ∗ 10 −2 1 .536 ∗ 10 −2 −4.039 ∗ 10−2 1 .890 ∗ 10 −2 ASPECT 48x48x48 Arithmetic 1 .523 ∗ 10 −2 1 .523 ∗ 10 −2 −4.039 ∗ 10−2 1 .874 ∗ 10 −2 ASPECT 48x48x48 Geometric 1 .533 ∗ 10 −2 1 .533 ∗ 10 −2 −4.021 ∗ 10−2 1 .887 ∗ 10 −2 ASPECT 48x48x48 Maximum composition 1 .537 ∗ 10 −2 1 .537 ∗ 10 −2 −4.042 ∗ 10−2 1 .891 ∗ 10 −2

Table B.1: Comparison of maximum velocities of ELEFANT and ASPECT for several resolutions and viscosity averaging schemes. Density is arithmetically averaged and no material averaging is used. The downward velocity in the z-direction also represents the maximum velocity. The italic printed numbers were obtained from visualization ﬁles opened in Paraview.

the other hand, the harmonic averaging scheme gives higher maximum velocities. The geometric averaging scheme shows maximum velocities in between. The maximum composition ’averaging’ scheme shows marginally higher velocities than the harmonic averaging scheme. Therefore, the maximum composition viscosity averaging scheme will be used in the upcoming experiments in combination with arithmetic averaging for density and no material averaging.

B.1.3 Zero velocity boundary conditions (1,2,3,4,5,6)/(0,1,2,3,4,5)

In this section we study the differences in maximum velocities between ELEFANT and ASPECT for different resolutions. Dissimilarities may arise because ASPECT uses computational fields and ELEFANT uses markers.

ELEFANT: ASPECT: Zero velocity set Zero velocity boundary indicators = 0,1,2,3,4,5

v Conclusion vi

Viscosity Vrms (m/s) Absolute max/ Absolute max/ Vel z (m/s) averaging scheme min vel x (m/s) min vel y (m/s) Min Max ELEFANT 9x9x9 No 9.12 ∗ 10−3 1.1402 ∗ 10−2 1.1402 ∗ 10−2 −2.5818 ∗ 10−2 1.1532 ∗ 10−2 ELEFANT 15x15x15 No 9.8 ∗ 10−3 1.2129 ∗ 10−2 1.2129 ∗ 10−2 −2.8600 ∗ 10−2 1.2230 ∗ 10−2 ELEFANT 18x18x18 No 9.91 ∗ 10−3 1.2042 ∗ 10−2 1.2042 ∗ 10−2 −2.9248 ∗ 10−2 1.2367 ∗ 10−2 ELEFANT 21x21x21 No 1.00 ∗ 10−2 1.2117 ∗ 10−2 1.2117 ∗ 10−2 −2.9564 ∗ 10−2 1.2478 ∗ 10−2 ELEFANT 27x27x27 No 1.011 ∗ 10−2 1.2160 ∗ 10−2 1.2160 ∗ 10−2 −1.2160 ∗ 10−2 1.2160 ∗ 10−2 ELEFANT 30x30x30 No 1.014 ∗ 10−2 1.2094 ∗ 10−2 1.2094 ∗ 10−2 −3.0219 ∗ 10−2 1.2512 ∗ 10−2 ELEFANT 33x33x33 No 1.017 ∗ 10−2 1.2116 ∗ 10−2 1.2116 ∗ 10−2 −3.0318 ∗ 10−2 1.2575 ∗ 10−2 ELEFANT 39x39x39 No 1.02 ∗ 10−2 1.2063 ∗ 10−2 1.2063 ∗ 10−2 −3.0499 ∗ 10−2 1.2582 ∗ 10−2 ELEFANT 45x45x45 No 1.02 ∗ 10−2 1.2225 ∗ 10−2 1.2225 ∗ 10−2 −3.0625 ∗ 10−2 1.2583 ∗ 10−2 ELEFANT 57x57x57 No 1.03 ∗ 10−2 1.2220 ∗ 10−2 1.2220 ∗ 10−2 −3.0788 ∗ 10−2 1.2614 ∗ 10−2 ELEFANT 60x60x60 No 1.028 ∗ 10−2 1.2222 ∗ 10−2 1.2222 ∗ 10−2 −3.0823 ∗ 10−2 1.2643 ∗ 10−2 ELEFANT 75x75x75 No 1.031 ∗ 10−2 1.2230 ∗ 10−2 1.2230 ∗ 10−2 −3.0924 ∗ 10−2 1.2650 ∗ 10−2 ASPECT 9x9x9 Max composition 8.1550 ∗ 10−3 2 .7961 ∗ 10 −2 ASPECT 15x15x15 Max composition 8.9996 ∗ 10−3 9 .940 ∗ 10 −3 9 .940 ∗ 10 −3 −2 .9130 ∗ 10 −2 9 .000 ∗ 10 −3 ASPECT 21x21x21 Max composition 9.3726 ∗ 10−3 2 .9618 ∗ 10 −2 ASPECT 30x30x30 Max composition 9.6596 ∗ 10−3 1 .129 ∗ 10 −2 1 .129 ∗ 10 −2 −3 .0040 ∗ 10 −2 9 .660 ∗ 10 −3 ASPECT 39x39x39 Max composition 9.8178 ∗ 10−3 3 .0286 ∗ 10 −2 ASPECT 45x45x45 Max composition 9.8892 ∗ 10−3 1 .154 ∗ 10 −2 1 .154 ∗ 10 −2 −3 .0399 ∗ 10 −2 9 .889 ∗ 10 −3 ASPECT 57x57x57 Max composition 9.9880 ∗ 10−3 3 .0559 ∗ 10 −2 ASPECT 60x60x60 Max composition 1.0007 ∗ 10−2 1 .178 ∗ 10 −2 1 .178 ∗ 10 −2 −3 .0590 ∗ 10 −2 1 .001e − 02 ASPECT 69x69x69 Max composition 1.0267 ∗ 10−2 −3 .0649 ∗ 10 −2 ASPECT 75x75x75 Max composition 1.0079 ∗ 10−2 −3 .0710 ∗ 10 −2 ASPECT 90x90x90 Max composition 1.0127 ∗ 10−2 3 .0792 ∗ 10 −2

Table B.2: Comparison of the maximum velocities and root mean square velocities of ASPECT and ELEFANT for several resolutions. All boundaries have zero velocity boundary conditions. The velocities in the z-direction also represent the maximum velocity. The italic printed numbers were obtained from visualization ﬁles opened in Paraview.

vi Conclusion vii

Vrms (m/yr) Absolute max/ Vel z (m/s)

min vel (m/s) x Min Max

ELEFANT 45x45x45 1.02 ∗ 10−2 1.2225 ∗ 10−2 −3.0625 ∗ 10−2 1.2583 ∗ 10−2

ASPECT 45x45x45 Max Comp 9.8892 ∗ 10−3 1 .154 ∗ 10 −2 −3 .0399 ∗ 10 −2 9 .889 ∗ 10 −3

Diﬀerence (%) 3.14 5.94 0.74 2.72

Table B.3: Comparison of velocities in ELEFANT and ASPECT for a 45x45x45 domain with only no slip boundaries. The italic printed numbers were obtained from visualization ﬁles opened in Paraview.

Figure B.1 shows that the Vrms for ELEFANT and ASPECT converge at higher resolutions. For 45x45x45 and 57x57x57 grid resolutions, the Vrms of ELEFANT are resepctively 3.14% and 3.12% higher than the Vrms of ASPECT when all boundaries have no slip boundary conditions.

Vrms versus the number of grid cells 1.05 · 10−2 12

1 · 10−2 10

−3 9.5 · 10 8 Percent diﬀerence ELEFANT ASPECT −3 Vrms (m/s) 9 · 10 6

8.5 · 10−3 4

2 Diﬀerence between ELEFANT and ASPECT (%) 8 · 10−3 0 10 20 30 40 50 60 70 80 90 100 Nr. of grid cells in each dimension

Figure B.1: The number of cells on each axis vs the Vrms as calculated by ELEFANT (red line) and ASPECT (blue line). The lines converge at higher resolutions.

vii Conclusion viii

B.1.4 Open boundary conditions (1,2)/(0,1), 48x48x48

ELEPHANT: ASPECT: use openbc 1 = .true. use openbc 2 = .true. fix bc3 v = .true. set Tangential velocity boundary indicators = 2,3,4 fix bc4 v = .true. set Zero velocity boundary indicators = 5 fix bc5 w = .true. set Prescribed traction boundary indicators = 0: time and po- fix bc6 u = .true. sition dependent lithostatic pressure , 1: time and position de- fix bc6 v = .true. pendent lithostatic pressure fix bc6 w = .true.

a) Velocity X ELEFANT b) Velocity Y ELEFANT c) Velocity Z ELEFANT

d) Velocity X ASPECT e) Velocity Y ASPECT f) Velocity Z ASPECT

viii Conclusion ix

g) Velocity magnitude ELEFANT h) Strain rate ELEFANT i) Pressure ELEFANT

j) Velocity magnitude ASPECT k) Strain rate ASPECT l) Pressure ASPECT

Vrms (m/s) Absolute max/ Absolute max/ Vel z (m/s)

min vel x (m/s) min vel y (m/s) Min Max

ELEFANT 48x48x48 ? 3 .0819 ∗ 10 −2 1 .7762 ∗ 10 −2 −5 .4592 ∗ 10 −2 1 .1803 ∗ 10 −2

ASPECT 48x48x48 2.1738 ∗ 10−2 2 .9076 ∗ 10 −2 1 .6758 ∗ 10 −2 −5.3593 ∗ 10−2 1 .1300 ∗ 10 −2

Diﬀerence (%) ? 5.99 5.99 1.86 4.45

Table B.4: Comparison of velocities in ELEFANT and ASPECT in a 48x48x48 domain with two open boundaries. The velocities in the z-direction also represent the maximum velocity. The italic printed numbers were obtained from visualization ﬁles opened Paraview.

Maximum velocities between ASPECT and ELEFANT diﬀer up to almost 6 percent in this set-up.

ix Conclusion x

B.1.5 Open boundary conditions (1,2,3,4)/(0,1,2,3), 48x48x48, No material averaging

ELEPHANT: ASPECT: use openbc 1 = .true. use openbc 2 = .true. set Tangential velocity boundary indicators = 4 use openbc 3 = .true. set Zero velocity boundary indicators = 5 use openbc 4 = .true. set Prescribed traction boundary indicators = 0: time and po- fix bc5 w = .true. sition dependent lithostatic pressure , 1: time and position de- fix bc6 u = .true. pendent lithostatic pressure , 2: time and position dependent fix bc6 v = .true. lithostatic pressure , 3: time and position dependent lithostatic fix bc6 w = .true. pressure

a) Velocity X ELEFANT b) Velocity Y ELEFANT c) Velocity Z ELEFANT

d) Velocity X ASPECT e) Velocity Y ASPECT f) Velocity Z ASPECT

x Conclusion xi

g) Velocity magnitude ELEFANT h) Strain rate ELEFANT i) Pressure ELEFANT

j) Velocity magnitude ASPECT k) Strain rate ASPECT l) Pressure ASPECT

m) Velocity magnitude ELEFANT n) Velocity magnitude ASPECT

xi Conclusion xii

Vrms (m/s) Absolute max/ Absolute max/ Vel z (m/s)

min vel x (m/s) min vel y (m/s) Min Max

ELEFANT 48x48x48 ? 2 .7015 ∗ 10 −2 2 .7015 ∗ 10 −2 −6 .1973 ∗ 10 −2 7 .8507 ∗ 10 −3

ASPECT 48x48x48 2.5098 ∗ 10−2 2 .5496 ∗ 10 −2 2 .5496 ∗ 10 −2 −6.0756 ∗ 10−2 7 .5699 ∗ 10 −3

Diﬀerence (%) ? 5.96 5.96 2.00 3.71

Table B.5: Comparison of velocities in ELEFANT and ASPECT in a 48x48x48 domain with four open boundaries. The velocities in the z-direction also represent the maximum velocity. The italic printed numbers were obtained from visualization ﬁles opened Paraview.

Maximum velocities between ASPECT and ELEFANT diﬀer up to almost 6 percent in this set-up. The Vrms remains to be compared and is likely to give lower diﬀerences.

B.2 TEST OPENBC 2: 2D simple lithospheric model

This test is somewhat similar to a lithospheric model. The computational domain is 900x600km. It is filled with four materials of various densities and (linear) viscosities. Boundary conditions are NO slip at the top and FREE slip at the bottom. Resolution is 256x170 elements. The density and viscosity fields are defined as follows in the material model:

B.2.1 Compositional initial conditions

ELEPHANT: ASPECT: if (zq > 0.81*Lz) then rho=2800 mueff=1.d24 elseif (zq > 0.33*Lz) then rho=3100 mueff=1.d21 set Variable names = x, z else set Function constants = Lz=600000, Lx=900000 rho=3200 set Function expression = if(z > 0.81 ∗ Lz, 1, 0) ; mueff=1.d22 if(z > 0.33 ∗ Lz & z < 0.81 ∗ Lz &(x − Lx/2)2 + (z − end if 0.667 ∗ Lz)2 > (50e3)2, 1, 0) ; if ((xq-Lx/2)**2 + (zq-0.667*Lz)**2 if((x − Lx/2)2 + (z − 0.667 ∗ Lz)2 < (50e3)2, 1, 0) < 50.d3**2) then rho=2900 mueff=1.d23 end if

xii Conclusion xiii

a) Density ELEFANT b) Viscosity ELEFANT c) Lithospheric pressure ELEFANT

d) Density ASPECT e) Viscosity ASPECT f) Pressure ASPECT

B.2.2 Free slip on the sidewalls

The boundary conditions are then implemented as follows:

ELEPHANT: ASPECT: fix bc1 u = .true. fix bc2 u = .true. fix bc3 v = .true. set Zero velocity boundary indicators = 3 fix bc4 v = .true. set Tangential velocity boundary indicators = 0, 1, 2 fix bc5 w = .true. fix bc6 u = .true. fix bc6 w = .true.

xiii Conclusion xiv

a) Velocity X ELEFANT b) Velocity Y ELEFANT c) Strain rate ELEFANT

d) Velocity X ASPECT e) Velocity Y ASPECT f) Strain rate ASPECT

g) Velocity magnitude (arrows) and X velocity (background) ASPECT

xiv Conclusion xv

B.2.3 Open boundary conditions on the sidewalls

ELEPHANT: ASPECT: use openbc 1 = .true. set Zero velocity boundary indicators = 3 use openbc 2 = .true. set Tangential velocity boundary indicators = 2 fix bc3 v = .true. set Prescribed traction boundary indicators = 0: time fix bc4 v = .true. and position dependent lithostatic pressure , 1: time fix bc5 w = .true. and position dependent lithostatic pressure fix bc6 u = .true. fix bc6 w = .true.

a) Velocity X ELEFANT b) Velocity Y ELEFANT c) Strain rate ELEFANT

d) Velocity X ASPECT e) Velocity Y ASPECT f) Strain rate ASPECT

xv Conclusion xvi

g) Velocity magnitude (arrows) and X velocity h) Velocity magnitude (arrows) and X velocity (background) ELEFANT (background) ASPECT

Vrms (m/yr) Absolute max/ Vel z (m/yr) Max

min vel (m/yr) x Min Max velocity (m/yr)

ELEFANT freeslip ? 1 .320 ∗ 10 −2 −4 .876 ∗ 10 −3 1 .102 ∗ 10 −2 ?

ASPECT freeslip 2.9337 ∗ 10−3 1 .3552 ∗ 10 −2 −5 .2058 ∗ 10 −3 1 .2010 ∗ 10 −2 1.3598 ∗ 10−2

Diﬀerence (%) ? −2.60 −6.3 −8.2 ?

Table B.6: Comparison of velocities in ELEFANT and ASPECT for a simple lithospheric model. The italic printed numbers were obtained from visualization ﬁles opened Paraview.

Vrms (m/yr) Absolute max/ Vel z (m/yr) Max

min vel (m/yr) x Min Max velocity (m/yr)

ELEFANT open boundaries ? 1 .328 ∗ 10 −2 −4 .768 ∗ 10 −3 1 .108 ∗ 10 −2 ?

ASPECT open boundaries 2.9562 ∗ 10−3 1 .3595 ∗ 10 −2 −5 .1449 ∗ 10 −3 1 .2050 ∗ 10 −2 1.3664 ∗ 10−2

Diﬀerence (%) ? −2.31 −7.3 −8.05 ?

Table B.7: Comparison of velocities in ELEFANT and ASPECT for a simple lithospheric model. The italic printed numbers were obtained from visualization ﬁles opened Paraview.

Maximum velocities between ASPECT and ELEFANT in 2D differ up to over 8 percent in this set-up. The Vrms remains to be compared and is likely to give lower differences. The effect of open boundaries on the percentual difference between ASPECT and ELEFANT can not be determined.

xvi C. Code

Figure C.1: The reaction term template.

xvii Conclusion xviii

Figure C.2: The entropy derivative template.

xviii Figure C.3: The phase transition function.

Figure C.4: The pressure derivative of the phase transition function. Figure C.5: Compositional initial conditions with phase transitions.

Figure C.6: The pressure dependence of the density.