lithospheric-scale stresses and shear localization induced ...563931/fulltext02.pdf · shear...

Examensarbete vid Institutionen för geovetenskaper ISSN 1650-6553 Nr 187

Christiane Heinicke

Lithospheric-Scale Stresses and Shear Localization Induced by

Density-Driven Instabilities

Copyright c© Christiane Heinicke och Institutionen for geovetenskaper, Geofysik, Uppsala universitet.

Tryckt hos Institutionen for geovetenskaper, Geotryckeriet, Uppsala 2010-02-23.

Abstract

The initiation of subduction requires the formation of lithospheric plates which

mostly deform at their edges. Shear heating is a possible candidate for producing

such localized deformation. In this thesis we employ a 2D model of the mantle with a

visco-elasto-plastic rheology and enabled shear heating. We are able to create a shear

heating instability both in a constant strain rate and a constant stress boundary con-

dition setup. For the first case, localized deformation in our specific setup is found

for strain rates of 10−15 1s

and mantle temperatures of 1300◦C. For constant stress

boundaries, the conditions for a setup to localize are more restrictive. Mantle mo-

tion is induced by large cold and hot temperature perturbations. Lithospheric stresses

scale with the size of these perturbations; maximum stresses are on the order of the

yield stress (1 GPa). Adding topography or large inhomogeneities does not result in

lithospheric-scale fracture in our model. However, localized deformation does occur for

a restricted parameter choice presented in this thesis. The perturbation size has little

effect on the occurrence of localization, but large perturbations shorten its onset time.

Keywords: Lithosphere, localization, numerical modeling, shear heating, stress,

subduction initiation.

Lithospheric-scale stresses and shear localizationinduced by density-driven instabilities

I

Contents

1 Introduction 3

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Earth’s internal structure . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Visco-elasto-plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.2 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.3 Visco-elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.4 Real visco-elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.5 Brittle and Plastic Behavior . . . . . . . . . . . . . . . . . . . . 9

1.3.6 Visco-elasticity with Mohr-Coulomb-plasticity . . . . . . . . . . 11

1.4 Lithosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Viscosity and Shear heating . . . . . . . . . . . . . . . . . . . . . . . . 12

1.6 Employed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Model Setup 17

2.1 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 General stress and temperature evolution in the standard model . . . . 18

2.3 Evolution of stress and strain rate in the standard model . . . . . . . . 21

3 Results 23

3.1 Shear Localization under Constant Strain Rate Boundary Conditions . 23

3.1.1 Dependency on viscosity . . . . . . . . . . . . . . . . . . . . . . 24

3.1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Constant Stress Boundary Conditions not resulting in Shear Localization 29

3.2.1 Dependency of maximum stress on perturbation size . . . . . . 29

3.2.2 Maximum stress dependency on perturbation size for a viscous

rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.3 Maximum stress dependency for different initial setups . . . . . 31

1

CONTENTS

3.2.4 Effects of viscosity . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.5 Subduction initiation by density difference . . . . . . . . . . . . 36

3.2.6 Maximum stress dependency when a heterogeneity is included in

the lithosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.7 Maximum stress dependency when topography is present . . . . 42

3.2.8 Effects of mantle temperatures . . . . . . . . . . . . . . . . . . . 43

3.2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Constant Stress Boundary Conditions resulting in Shear Localization . 45

3.3.1 Evolution Characteristics . . . . . . . . . . . . . . . . . . . . . . 48

3.3.2 Influence of Inhomogeneities and Topography . . . . . . . . . . 51

3.3.3 Influence of Material Properties . . . . . . . . . . . . . . . . . . 51

3.3.4 Influence of Perturbations . . . . . . . . . . . . . . . . . . . . . 51

3.3.5 Influence of Viscosity . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Discussion and Conclusion 55

4.1 Shear Localization under Constant Strain Rate Boundary Conditions . 55

4.2 Constant Stress Boundary Conditions not resulting in Shear Localization 56

4.3 Constant Stress Boundary Conditions resulting in Shear Localization . 57

4.4 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Acknowledgments 59

Bibliography 61

List of Figures 65

A Code ammendments - Initial Stress Field 69

B Code ammendments - Inhomogeneity 71

C Code ammendments - Phases 73

D Code ammendments - Topography 75

2 Lithospheric-scale stresses and shear localizationinduced by density-driven instabilities

Chapter 1

Introduction

1.1 Motivation

Earth is strikingly different from its siblings in the solar system. Besides the usually

named water- and life-supporting environment, the seemingly solid planet’s interior is

moving vigorously beneath a cover of rigid, plate-like blocks [Bercovici (2003), Turcotte

and Schubert (2002)]. Already hot from its formation process, heated from within by

radioactive decay, and from the center by a 3000 K-hot iron core, Earth’s mantle trans-

ports material and energy to the surface by convection. At the surface the lithospheric

cover conducts the heat received from the mantle to the atmosphere and thus into

outer space. This lithospheric cover is not an intact shell, or ”stagnant lid” [Bercovici

(2003)], but it is fractured into smaller plate-like pieces. More or less floating on the

mantle, these plates make up continents and oceanic basins. However, their configura-

tion on the planet’s surface is not fixed; as the plates move apart, oceanic ridges slide

past each other in a strike-slip manner and converge at subduction zones. Whereas

new plate material is created at the ridges by the uprise, cooling, and thus densifying,

of the underlying mantle, plates sink back into the mantle at subduction zones.

An important unresolved question is why and how these plates are moving. A very

simple but also very wrong picture depicts the convecting mantle as turning wheel that

drives the ’conveyor belt’-plates [Bercovici (2003), Turcotte and Schubert (2002), Stern

(2004)]. However, research [Forsyth and Uyeda (1975); Conrad and Lithgow-Bertelloni

(2002)] has shown that plates are mainly driven by slab-pull from the subducting slabs

attached to them and by a lesser degree by ridge-push. The slab-pull has been shown to

be a mixture of actual pull from the sinking slab at the connected plate and suction of

already-detached slab parts. Mantle convection is connected to the plates’ movement,

but certainly not its driving force. Therefore it is thought that cold and dense oceanic

3

CHAPTER 1. INTRODUCTION

lithosphere due to its negative buoyancy plunges into the mantle and pulls the attached

plate behind [e.g. Mueller and Phillips (1991), Mart et al. (2005)].

During the Archean (until 2.5 Ga BP) Earth’s mantle was too hot to be able to

produce a cold and therefore dense enough lithosphere that could sink into the mantle

due to its negative buoyancy [Stern (2004)]. Moreover, because of the higher heat

transport and higher temperature close to the surface inducing more melting, Archean

oceanic crust was thicker than today’s [Sleep and Windley (1982)]. A thicker, light

crust counteracts lithospheric gravitational instability and inhibits subduction. Adding

geologic evidence like ophiolites [Stern (2004)] - or rather their absence - it is thought

that plate tectonics with its driving motor plate subduction did not begin until very

recently, presumably only 1 Ga ago.

Now here is the problem: In Earth’s youth, its mantle is convecting vigorously and

subduction is impossible. Now, Earth’s surface shows fully developed plate tectonics

enhancing mantle convection. It is thought that to initiate subduction it is necessary to

(1) create weak zones in a formerly intact lithosphere, (2) have one side of the fracture

zone descend into the mantle, and (3) evolve the young subduction zone into a mature

subduction zone which is able to pull its tailing plate. While several mechanisms have

been proposed for subduction initiation in already-fractured and already-subducting

plates [Gurnis et al. (2004); Stern (2004), Mueller and Phillips (1991)], there must have

been a point at which the very first subduction zone evolved. Obviously, at some point

the formerly intact lithosphere must have fractured and one of the two resulting edges

must have sunken into the less dense (because hotter) mantle [Stern (2004), Mart et al.

(2005)].

Unfortunately, neither laboratory experiments nor numerical experiments [Bercovici

(2003)] can reproduce a sinking lithosphere into an underlying convecting mantle satis-

factorily. In laboratory experiments [e.g. Funiciello et al. (2004)] this hurdle is usually

taken by punching the lithosphere-representing material into the ”mantle” by hand af-

ter which the further evolution of the subduction zone is studied. In other experiments

[Mart et al. (2005)] one lithosphere can be forced underneath another (also compare

to section 3.2.5), but the distance achieved by this has not been the necessary distance

to start real, self-sustaining subduction, estimated to be ≈ 120 km.

Numerical experiments suffer from a similar problem: Either they need prescribed

or prefractured plates [as in Gerya et al. (2008)], or they use somewhat unrealistic

parameters of yield stress [Tackley (2000a,b)], or only consider a free-floating mid-

lithosphere [Regenauer-Lieb et al. (2001), Branlund et al. (2001)]. In the real Earth,

however, the lithosphere has evolved from a convecting mantle. The formerly intact

”stagnant lid” must have fractured somehow, most likely through a weakening of the



lithosphere by shear heating (section 1.5).

This thesis is dedicated to creating weak zones in an overlying lithosphere by

density-driven instabilities. These weak zones are sites of possible evolution into plate

boundaries, esp. subduction zones.

1.2 Earth’s internal structure

While the inner (1216 km) and outer (2270 km) core make up approximately half of

Earth’s radius, they are not directly involved in neither plate tectonics nor mantle

convection [Tarbuck et al. (2007)]. The mantle is usually divided into a lower and an

upper mantle; the boundary between them is a measurable discontinuity in density

and seismic velocity. For our purpose, a distinction between the mechanical layers is

appropiate: The asthenosphere as part of the upper mantle is mechanically weak, has

a low effective viscosity. Above the asthenosphere lies the lithosphere, which comprises

of the upper part of the upper mantle and the crust, as can be seen in fig. 1.1. The

boundary between these two layers (lithosphere and asthenosphere) is defined by tem-

perature, usually around 1600 K. A typical depth of the isotherm is 100 km, varying for

different ages of lithosphere. Rocks above this isotherm (lithosphere) are cold enough

to behave rather rigidly, rocks below this isotherm (asthenosphere) deform readily; a

more complex model will be introduced in section 1.4.

The crust (fig. 1.1) has a different composition from the underlying mantle, it is

less dense. Typical thicknesses for continental crust are 35 km and 6 km for oceanic

crust. The boundary between crust and mantle is the well-defined Moho-discontinuity.

Even though the crust is lighter than the mantle, it is comparably thin, so that the

entire lithosphere can be denser than the underlying asthenosphere and therefore grav-

itationally unstable. However, this alone is not enough to have the lithosphere sink

into the asthenosphere, as Mueller and Phillips (1991) showed.

1.3 Visco-elasto-plasticity

Visco-elasto-plasticity describes materials that exhibit elastic, viscous and plastic be-

havior, depending on the amount and time evolution of the applied stress.

1.3.1 Elasticity

A common example for elastic behavior is a rubber band: It expands instantaneously

when put under stress and relaxes as soon as the stress is removed, meaning that it


5


Figure 1.1: Internal structure of Earth. Taken from Tarbuck et al. (2007)

recovers its original length. At atomic level this reversible strain can be understood

by a stretching of the binding length between the atoms. School physics taught us

Hooke’s law, which describes exactly this linear behavior between stress σ and strain

ε for uniaxial stresses:

σ = Eε (1.1)

where E is Young’s modulus. It is important to note that elasticity is a characteristic

of solids. Seismic waves can propagate through the Earth due to such elastic behavior.



1.3.2 Viscosity

A fluid is anything that flows under stress. A viscous fluid (like honey, for example)

is a fluid that resists flow. That is, when this fluid is subjected to a constant stress,

it will - unlike an elastic solid - strain continuously. At nanometer-scale, the atoms

diffuse past each other.

For one-dimensional uniaxial stresses, viscosity µ is given by:

τ = µ∂v

∂x(1.2)

If strain rate is defined as εij = 12

(∂ui∂xj

+∂uj∂xi

), the above equation yields following

relation between stress and strain for viscous fluids:

τij = 2µεij (1.3)

Here, µ is the dynamic viscosity [unit Pas]; τij is the deviatoric stress tensor τij =

σij + δijP , with P being the pressure and δij Kronecker’s Delta (1 for i = j, else 0).

Eq. 1.3 makes clear that the total strain is not only dependent on the applied stress

but also on the time elapsed.

1.3.3 Visco-elasticity

In the Earth’s mantle, the response to stresses depends on the time scale of their

application. Seismic waves that travel on time scales of 1 . . . 104 s see the mantle as an

elastic medium [Turcotte and Schubert (2002)]. They can propagate through barely

attenuated due to the quasi-instantaneous reaction of the mantle. Scandinavia, on

the other hand experiences a viscous response of the mantle: it is uplifted by about a

centimeter per year after it had been depressed by the ice sheets of the last ice age. The

total duration of this postglacial rebound amounts to approx. 1014 s. Visco-elasticity

can be described by the Maxwell model (among others):

dε

dt= εvisc + εelast =

1

2µσ +

1

G

dσ

dt(1.4)

G, the shear modulus for an incompressible fluid is exactly one third of Young’s

modulus E. Equation 1.4 reveals some interesting characteristics about visco-elastic

materials:

1. If the stress is kept constant, εelast vanishes and the material strains continuously,

it creeps.


7


2. If on the other hand the strain is kept constant, eq. (1.4) becomes

dσ

σ= − G

2µdt (1.5)

which can be integrated with the initial condition σ = σ0 at t = 0 to give

σ = σ0 exp

(−Gt

2µ

). (1.6)

The stress is decreasing exponentially with time, i.e. the material is relaxing

under constant strain.

3. As we will not implement a constant strain, but a constant strain rate boundary

condition with dεdt

= const., eq. 1.4 becomes

ε =1

2µσ +

1

G

dσ

dt= const. (1.7)

of which eq. 1.6 is a homogeneous solution. Solving with the additional constant

(mathematical) inhomogeneity ε and using the same initial condition as under

(2) yields:

σ = 2µε+ (σ0 − 2µε) exp

(−Gt

2µ

)(1.8)

The stress will approach the value 2µε for long time spans.

4. A visco-elastic material that had been under stress and has crept accordingly,

will not return to its original shape before straining if the stress is relieved.

1.3.4 Real visco-elasticity

The Maxwell model needs to be adjusted for the ”real” Earth. Viscous deformation in

the Earth at high temperatures occurs by diffusion creep and dislocation (power-law)

creep. High temperatures are ∼ 0.5 Tm, with Tm being the melting temperature of the

rocks. The general creep law is according to Kameyama et al. (1999):

ε = Aa−mσn exp

(− H

RT

)(1.9)

with



A, m, n . . . constants

R . . . universal gas constant

σ . . . differential stress

H . . . activation enthalpy of creep

a . . . grain size

T . . . temperature.

Both diffusion creep and dislocation creep are sensitive to temperature and stress,

but the magnitude of this dependence differs between the two: The strain rate that

is caused by diffusion creep increases linearly with stress (n = 1), but significantly

decreases with grain size (m = 2 . . . 3). The strain rate controlled by dislocation creep

on the other hand increases nonlinearly with stress (n = 3 . . . 3.5), but is independent

of grain size (m = 0).

Diffusion creep is the predominant creep mechanism at low stress levels and low

temperatures (fig. 1.2). It results from the diffusion of atoms inside the grains of rock

crystals. The so-called Herring-Nabarro creep denotes the diffusion within the grain

boundaries, whereas Coble creep denotes atom diffusion along grain boundaries.

At intermediate stresses and high temperatures dislocation creep is the most impor-

tant creep mechanism. Dislocations are imperfections within the crystal lattice; these

can wander through the lattice principally by dislocation climb and by dislocation slip.

Both mechanisms are described in detail in Turcotte and Schubert (2002), therefore I

will not go into more detail here.

1.3.5 Brittle and Plastic Behavior

The plastic deformation common in rocks at low temperatures (T ≤ 0.3 Tm) is referred

to as Peierls mechanism. It is more sensitive to stress but less sensitive to temperature

than the dislocation creep [Kameyama et al. (1999)]:

ε = A exp

(− H

RT

[1− σ

σp

]q). (1.10)

σp is the so-called Peierls stress, the yield stress at which Peierls plastic deformation

onsets; q is a material constant.

Rocks at room temperature and atmospheric pressure show a brittle behavior: They

strain elastically under low stress, but fracture if the stress exceeds a critical value, the

yield stress. The rock’s brittle strength is described by the maximum stress it can

support without fracture, i. e. the yield stress.


9


Figure 1.2: Deformation mechanism map (grain size 0.1 mm) White area indicates theparameters for which Peierls plasticity is dominant, light gray area is dominated by diffusioncreep, dark gray by dislocation creep. Solid lines are lines of constant strain rate. From[Kameyama et al. (1999)]

If the rock is placed in conditions where the pressure approaches the rock’s brittle

strength, it will deform plastically instead of fracture. Ductile deformation is the same

as unlimited strain: Once and as long as the yield stress is applied, the rock deforms.

Upon stress relief, the rock does not regain its original shape.

It is worth to note here again the difference between viscous and plastic defor-

mation: Visco-elastic behavior describes the combination of solid-like behavior at low

time scales and fluid-like behavior at long time scales. Elastic-plastic behavior on the

other hand describes the combination of reversible deformation at low stresses and

irreversible deformation at the yield stress.

Jargon Box

elastic = instantaneous, reversible deformation

viscous = irreversible deformation

plastic = irreversible deformation above yield stress allowing localization

brittle = elastic behavior + fracture at yield stress

ductile = viscous, but typically used for solids



Figure 1.3: Mohr’s circle and strength envelope.

1.3.6 Visco-elasticity with Mohr-Coulomb-plasticity

Summarizing all the various deformation mechanisms so far we have:

εtotal = εvisc + εelast + εplastic (1.11)

of which εvisc can further be divided into

εvisc = εpc + εdc + εpp (1.12)

where the superscripts pc, dc and pp denote power-law creep (dislocation creep),

diffusion creep, and Peierls plasticity, respectively.

There exist several models for plastic yielding; the one used in our code is the

so-called Mohr-Coulomb failure criterion. The rock will deform plastically as soon as

the shear stress τ applied to it is equal to the mean normal stress determined by the

applied stress σ, the internal friction angle φ and the cohesion c:

τ = σn tan(φ) + c (1.13)

This is displayed in fig. 1.3; the circle represents Mohr’s circle at a given two-

dimensional stress state and the line represents the strength envelope. As soon as shear

stress τ and normal stress σn are sufficiently large, i.e. touch the line, the material

which is associated with that particular cohesion and friction angle will yield.

1.4 Lithosphere

Even though the lithosphere is composed of various materials and rock types, a general

behavior under stress can be described. While it is not the aim of this thesis to examine


11


the depth dependency of lithospheric stress, it is noteworthy that a purely visco-elasto-

plastic model neglects that the prevailing deformation mechanism depends on depth

and temperature. The classic strength envelope as developed by Goetze and Evans

(1979) and Kohlstedt (1995) roughly assumes three regimes in the lithosphere: brittle

behavior and frictional sliding at the top where low temperatures and low pressures

prevail, plastic flow towards the mantle which is dominated by high pressures and

temperatures, and an intermediate, semi-brittle transition zone where both mechanisms

are of importance. Stresses in the upper lithosphere are mostly controlled by pressure,

therefore stresses rise with increasing depth. Since the lower lithosphere is controlled

by temperature, stresses decrease again with further increasing depth.

Deviations from this overall distribution are a research area for themselves; one of

the arguably most important mechanisms affecting lithospheric strength with immedi-

ate relation to the formation of tectonic plates is shear heating and will be discussed

in the next section.

1.5 Viscosity and Shear heating

Viscosity is defined as the resistance to fluid flow by eq. 1.2. However, this is only true

for fluids with constant viscosity, so-called Newtonian fluids. In our model, viscosity is

dependent on strain rate ε and temperature T as given by eq. 1.14:

µ = µ0

(ε

ε0

) 1n−1

exp

(H

R

[1

T− 1

T0

])(1.14)

The subscript 0 on viscosity µ and T in eq. 1.14 denotes values at standard condi-

tions, ε0 is a characteristic strain rate. n is the power-law constant as in eq. 1.9. Note

the strong exponential dependence of viscosity on temperature.

Besides sources outside the system, a temperature rise can be caused by frictional,

or shear heating: Shear (a gradient in a velocity profile) causes internal friction inside

a fluid; friction produces heat. Melosh (1976) discussed a model of lithosphere moving

over asthenosphere, which acts as an insulator to the heat produced in the astheno-

sphere. This heat reduces viscosity which in turn allows for faster deformation and thus

an even stronger temperature increase if heat diffusion and conduction are too slow

processes to transport the heat away from the site of production. Since this thermal

runaway is limited to a small volume, it is commonly termed shear localization.

A small rise in temperature (e.g. 100 K) can lead to a strong decrease in viscosity

and thus to an (e.g. order-of-magnitude) increase in strain rate [Brun and Cobbold

(1979)]. Shear heating can be so efficient that lithosphere viscosity at localized defor-



mation sites may drop to values as low as asthenosphere viscosity, i.e. from typically

1024 Pas to 1020 Pas [Fleitout and Froidevaux (1979)]. Under some circumstances,

shear heating may have a significantly larger effect on lithospheric strength than e.g.

rock parameters and creep-law [Hartz and Podladchikov (2008)].

Melosh (1976) found that shear heating is especially intense for large stresses and/or

fast straining, where only non-elastic strain rates contribute:

SH = χτij

(εvisij + εplij

)ρcp

(1.15)

with εvisij being the viscous strain rate, εplij the plastic deformation rate and χ the

efficiency of shear heating (0 < χ < 1), which can be reduced if some of the produced

heat is used to change the physical state of the deformed material [Brun and Cobbold

(1979)]. If stress reduction due to the excess heat is too large, thermal runaway may

be inhibited and the shear heating process is stopped.

More recent work [Kaus and Podladchikov (2006)] has shown that there exist pa-

rameter spaces for which shear localization can be expected. These parameters include

(not exclusively) the initial stress (the larger, the stronger is the shear heating) and

heat capacity and thermal diffusivity (larger values lead to an increased heat transport

and therefore inhibit shear heating). As these results will be related to the findings of

this thesis in later chapters, we will not go into more detail here.

1.6 Employed Model

The model employed by us is based on the conservation of energy (with T di = T −T0):

(∂T di

∂t+ vi

∂T di

∂xi

)= κ

∂2T di

∂x2j+ χ

τij

(εvisij + εplij

)ρcp

(1.16)

and momentum:

��

��XXXXXXXXXX

ρ

(∂vi∂t

+ vj∂vi∂xj

)=∂σij∂xj

+ ρgi (1.17)

The left hand side in eq. 1.17 is zero because inertia is neglible in highly viscous,

geologic problems:

∂σij∂xj

+ ρgi = 0 (1.18)

The used variables are:


13


T0 . . . initial temperature

x . . . spacial coordinate

v . . . velocity

κ . . . thermal diffusivity

χ . . . efficiency of shear heating

cp . . . heat capacity

ρ . . . density

g . . . gravity.

The shear heating term χτij(εvisij +εplij)

ρcpon the right hand side of eq. 1.16 acts as a

source term for the energy equation. This system of equations 1.16 and 1.18 is solved

numerically for a cross section of the Earth’s mantle and lithosphere.

For our simulations we use a 2D finite element code called MILAMIN VEP, which is

based on the solver MILAMIN [Dabrowski et al. (2008)]. MILAMIN VEP is specialized

for slowly moving, incompressible ( ∂vi∂xi

= 0), visco-elasto-plastic rheologies which its

name is based on. We employ quadrilateral (Q1P0) elements; our mesh is refined in

regions of special interest (in our case the lithosphere), and coarser elsewhere. An

extension interesting for our problem is the ability of MILAMIN VEP to remesh the

domain and to thus allow the mesh to stay refined in the ’interesting’ areas [based on

B. J. P. Kaus (2008)].

MILAMIN is originally provided by Dabrowski et al.; the extension MILAMIN VEP

by B. Kaus. Both codes have been used in various previous studies and are therefore

thoroughly benchmarked. The simulations have been run on ”Brutus”, the 8548-core

and 1006-node-cluster of the ETH Zurich with a peak performance of 75 Teraflops

(though MILAMIN runs on a single processor).

1.7 Goal

This thesis is dedicated to creating a 2D mantle convection model that induces weak

zones in an overlying lithosphere through a shear-heating instability. These weak zones

develop into plate boundaries that are possible sites of future subduction. We will con-

duct systematic numerical experiments with different setups and boundary conditions

to show that plate forces are sufficient to produce lithospheric stresses of the order of

the plastic yield stress (≈ 1 GPa) [chapter 3.2].

Crameri (2009) has demonstrated that it is possible to create shear localization with

a constant strain rate boundary condition representing e.g. a constant ridge push. We



will transfer his findings and the ones of Kaus and Podladchikov (2006) into a constant

strain rate version of our model [chapter 3.1].

Finally, we will use our model to produce lithospheric failure due to shear heating

in a constant stress setup [chapter 3.3]. We will examine the restrictions necessary to

obtain the localization.


15

Chapter 2

Model Setup

Our objective is to find a convection model featuring a lithosphere that develops a

localized shear zone. For achieving this goal we will test various setups, like an ad-

ditional prescribed heterogeneity, added topography, various material parameters. To

have a point of reference we settled for a standard model which will be described in

the following.

2.1 Standard Model

The standard model box is 1900 km deep and 7600 km across (aspect ratio 4) rep-

resenting the mantle. The mantle is capped by a lithosphere 100 km thick. In the

standard model, the lithosphere differs from the mantle only in its temperature; as it

is the connection between the hot mantle and the cold atmosphere it is cold at its top

and warm at its bottom. To avoid numerical artifacts, a thin layer of “sticky air” is

placed on top of this lithosphere, with a very low density and low viscosity, but (for

air) a very high cohesion and shear modulus, making it effectively viscous.

All material parameters listed in table 2.1 can be adjusted, as well as the dimensions

of both box and lithosphere. According to Kaus (2005) it is possible to describe the

visco-elasto-plastic rheology with all its parameters for the different deformation modes

by a simplified rheology that uses effective parameters like the initial effective viscosity

µ0 and the effective activation energy Q = Ea/R which is a measure for the temperature

dependence of the viscosity:

µ = µ0 ·(εIIε0

)( 1n−1)· exp

(Q

[1

T− 1

T0

])(2.1)

where µ0 is the viscosity at standard laboratory conditions, i.e. T0 = 20◦C and

17

CHAPTER 2. MODEL SETUP

zero stress, n the powerlaw exponent (n = 1 denotes a Newtonian fluid), and T the

temperature at the position where the viscosity is to be calculated. εII/ε0 is the ratio

of actual to initial strain rate. The subscript II indicates that ε is the second invariant

of the strain rate tensor (in Einstein notation)

εII =

√1

2εijεij. (2.2)

Temperatures can be set at the top of the setup (air temperature), for the bulk

mantle (thus defining the temperature gradient in the lithosphere) and at the bottom

(core-mantle-boundary). The grid resolution is 450×150 points in the standard setup;

it will later be reduced (to 150 × 75 points) to save computing time. The grid points

are not spaced equally, but are further apart at the bottom and get closer together

towards the lithosphere; so each element in the lithosphere is about 15 km × 6.7 km

in size.

In order to initiate motion, we superimposed a cold temperature perturbation of

semicircular form extending 550 km downward from the top of the lithosphere. Upon

starting the simulation, this perturbation sinks towards the bottom dragging at the

lithosphere, and when reaching the bottom boundary representing the core-mantle-

boundary, it induces mantle plumes to rise up that further affect the lithosphere. Ad-

ditionally, we introduced small, random temperature perturbations (≤ 10 K at single

nodes) reflecting the natural inhomogeneity of the mantle. By doing so, we follow the

results of Mancktelow (2002), who found that the size of the inhomogeneities is not

important for obtaining localization, but that they are present at all.

The material parameters listed under “Temperatures” and “Material properties” in

table 2.1 are sought to represent as realistic values as possible.

2.2 General stress and temperature evolution in

the standard model

Figure 2.1 shows a typical time evolution of the stress and temperature field in the

mantle. These graphics are intended to give an overall view of how the model behaves.

The top row (fig. 2.1 a,e) shows the descending major perturbation of ∆T = −300 K

just before it reaches the core-mantle-boundary. Two hot plumes begin forming at

both sides of the drop, rise up, and reach the lithosphere as demonstrated in fig. 2.1

b,f. The third row, Fig. 2.1 c,g depicts cold secondary plumes that are driven from the

lithospheric bottom by the hot plumes and that are breaking through the ’lake’ of hot



Parameters for standard runBox dimensionswidth x 7600 kmDepth z 1900 kmThickness lithosphere 100 kmNumber of grid pointsHorizontal nx 450Vertical nz 150TemperaturesSticky air temperature 283 KMantle temperature 2250 KCMB-temperature 3200 KMaterial properties Mantle Sticky airElastic shear modulus 5×1010 Pa 5×1010 PaViscosity 1×1021 Pas 1×1021 PasPowerlaw exponent 3 1

Density 3300 kgm3 1 kg

m3

Cohesion 30×106 Pa 1×1012 PaFriction angle 35◦ 0◦

Shear heating efficiency 1 0Conductivity 3.3 W

m·K 1× 106 Wm·K

Heat capacity 1050 JkgK

1050 JkgK

Radioactive heat production 1×10−7 Wm3 1×10−7 W

m3

Thermal expansivity 3.2×10−5 1K

0 1K

Effective activation energy (Ea/R) 1×104 K 0 KPerturbation propertiesRadius 550 kmPosition center x 0 kmPosition center z 0 km (top of lithosphere)Temperature deviation -300 K

Table 2.1: Parameters used for standard setup.


19


(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 2.1: Stress and temperature fields for the standard model. Maximum stresses reachedin the lithosphere are just above 500 MPa. (a-d) Stress field. (e-h) Temperature field. (a,e)After 8.9 ka. (b,f) After 23.1 ka. (c,g) After 59.6 ka. (d,h) After 142.3 ka.

material below the lithosphere. The ’lake’ has increased in thickness in the last row of

pictures and creates few more small cold perturbations.

This evolution will always remain the same for all setups; a cold prescribed pertur-

bation is sinking towards the bottom of the modeled box and induces hot secondary

plumes. Both perturbations, primary and secondary take about the same amount of

time to reach bottom or top boundary, namely ≈ 10 ka.



2.3 Evolution of stress and strain rate in the stan-

dard model

Both plumes influence the stress and strain rate fields of the lithosphere. Here, we

will look at the maximum stress and strain rate fields of two different regions of the

lithosphere. The first region is situated directly above the primary, cold perturbation

and extends from −500 km to +500 km. The other region is outside the immediate

influence of the cold plume, but comprises the area affected by the hot uprising plumes,

[1000 km . . . 2000 km]. Both regions are considered for the entire depth of the litho-

sphere. The maximum stress in these areas occurs between −500 km < x < +500 km

and −100 km < z < 0 for the center region and 1000 km < x < +2000 km and

−100 km < z < 0 for the side regions. As it turns out, this analysis gives reasonable

results as long as the yield stress (1 GPa) is not reached.

Fig. 2.2a clearly shows two maxima during the stress evolution, which are more or

less distinct in each region. The first maximum is directly caused by the descending

plume and has a value of 430 MPa at the horizontal center of the lithosphere after

approximately 6 ka. The second stress maximum of 540 MPa is caused by the uprising

plumes after 23 ka. Interestingly, the resultant second stress at the center of the

lithosphere, i.e. at x = 0 and where the driving perturbation had been, only amounts

to 300 MPa. Approximately 1000 km outside this region, however, where the plume

actually reaches the lithosphere, the stress is not only larger than at the center, but

- for the standard model - larger than the maximum stress during the descend of the

perturbation, namely about 540 MPa.

A similar picture can be drawn for the strain rate evolution (fig. 2.2b); the two

maxima are very distinct for the hot plume in the outside area (red) and the cold

plume at the center area (blue), but comparably weak for the respective other area.

The maximum strain rates reached are about 2×10−11 1s

for both plumes.

Since strain heating is most effective if both stress and strain rate are large (cp.

section 1.5), both these values are plotted together in fig. 2.2c for the center re-

gion and in fig. 2.2d for the outside region. Remarkable is that whereas subfigure

d only exhibits one pronounced maximum in stress and strain rate (their product be-

ing 8.6×10−9 Wm3 ), subfigure c shows both plume events (the maximum shear heating

there being 10.8×10−9 Wm3 ).


21


(a) (b)

(c) (d)

Figure 2.2: (a) Evolution of maximum stress in the lithosphere. (b) Evolution of maximumstrain rate in the lithosphere. (c) Stress and strain rate development at the center of thelithosphere. (d) Stress and strain rate development off the center of the lithosphere.


Chapter 3

Results

As discussed in the introduction, like any solid material, the lithosphere will yield if

it is subjected to large differential stresses. Once the lithosphere is broken, it might

be subducted. An upper bound on the stress for which this failure occurs is given by

the Peierls plasticity (≈ 1 GPa) which is the point at which atomic bonds break. This

is typically at stresses that amount to ca. 10% of the elastic shear module which we

have (realistically) set to ≈ 1010 GPa. Thus, the absolute strength of our lithosphere

is roughly 1 GPa (Scholz, 2002; Katayama and Karato, 2008). The main questions

addressed here are therefore if we can produce such large stresses with our convection

model and if these result in localization.

In order to answer this question, we performed simulations with several setups.

First, we apply constant strain rate boundaries in section 3.1 to imitate pre-existing

plate movement. These simulations show that our model is indeed capable of creating

localizing shear zones. Then we will apply constant stress boundary conditions (3.2)

and investigate the effects of various additions to our setup on the maximum stress

found in the lithosphere. Finally, we present a setup that leads to localization in

section 3.3.

3.1 Shear Localization under Constant Strain Rate

Boundary Conditions

In a previous Master thesis, Fabio Crameri investigated shear heating leading to shear

localization in a constant strain rate boundary condition setup. In short, his standard

setup was a 120 km deep box consisting of 25 km strong upper crust, 10 km thick

lower crust, and 85 km upper mantle. The bottom temperature was fixed to a variable

value Tbot; the temperature distribution determined by cooling the top surface to 0◦C.

23

CHAPTER 3. RESULTS

Figure 3.1: The occurrence of localization depends on the combination of bottom temperatureand strain rate. Red shaded area: localization regime; gray area: no localization occurs. Takenfrom Crameri (2009)

Based on Kaus and Podladchikov (2006) he determined the parameters that result in

localization. Crameri (2009) found that localization is favored by low bottom temper-

atures and fast deformations (see fig. 3.1); for the Earth these parameters are in the

range of [900◦C . . . 1400◦C] and [10−16 1s

. . . 10−14 1s], respectively.

Transfered to our model, the mantle temperature should be 1300◦C; we set our

background strain rate to 10−15 1s. The top boundary is a free surface, i.e. its stress

state is always zero and no sticky air lies on top.

Snapshots of the simulation at three different time steps are shown on fig. 3.2. A

heterogeneity in the viscosity field is imposed to initialize localization, which forms 45◦

shear zones. The light green “x” in the viscosity field represents zones of low viscosity,

i.e. the shear zones. The small dent in the middle of the surface of the lithosphere in

fig. 3.2 (b,e,h) indicates already shortly after the beginning of the simulation that the

lithosphere is about to yield. Fig. 3.2 (c,f,i) shows the fully developed fracture.

3.1.1 Dependency on viscosity

One of the main parameter deciding whether or not localization will occur is viscos-

ity. For a given strain rate of 1×10−15 1s, the lithosphere localizes if the viscosity is

above 1×1023 Pas, below this value it does not. If viscosity is large enough to allow

localization at all, it further determines the time after which localization is starting.


CHAPTER 3. RESULTS

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 3.2: Evolution of the lithosphere in a constant strain rate setup with an inhomogeneity.(a-c) Stress field. (d-f) Temperature field. (g-i) Viscosity field. (a,d,g) After 0.3 Ma. (b,e,h)After 1.8 Ma. (c,f,i) After 12.4 Ma.


25

CHAPTER 3. RESULTS

Figure 3.3: Onset time of localization vs. viscosity. Blue: small lithospheric inhomogeneity(50 km × 30 km). Green: large lithospheric inhomogeneity (80 km × 50 km).

Moreover, the onset time is shortened when a large, extra inhomogeneity is introduced

in the lithosphere. Figure 3.3 shows the onset times for various viscosities; the blue

markers stand for a small inhomogeneity of 50 km × 30 km introduced in the center

of the lithosphere, the green for a large inhomogeneity of 80 km × 60 km. Note that

we are considering large inhomogeneities comprising several dozen nodes. Small noisy

perturbations as introduced in section 2.1 only consist of few nodes and always have

to be present, independent of the large inhomogeneity.

Low viscosities just above the localizing limit inhibit a fast development of a shear

zone; larger viscosities speed up the process, as well as a larger inhomogeneity.

Kaus (2005) derived a scaling law (eq. 3.1) that predicts the onset of shear local-

ization if the background strain rate εBG is larger than a value determined by material

setup and properties:

εBG =1.4

R

√κρcpµ0γ

. (3.1)

Here, κ is the thermal diffusivity (3.3 m2

s), ρ the lithospheric density (3300 kg

m3 ), cp

the heat capacity (1050 JkgK

), µ0 the initial viscosity, and γ is the effective activation

energy, which depends on the initial temperature (compare section 2.1). In the 2D setup

of Kaus (2005) R designated the radius of a there used circular heterogeneity; there is

some unclarity as to what exact length of our rectangular inclusion best resembles this

circular inclusion. We settled for the same-area-solution (the area of our rectangular

inclusion = the area of Kaus’ circular one) as opposed to chosing simply one side of the

rectangle, as it gives the best results for the smaller inhomogeneity. By rearranging

eq. 3.1, setting the strain rate as fixed, and calculating the viscosities necessary for


CHAPTER 3. RESULTS

localization, we obtain

µ0 =

(1.4

RεBG

)2κρcpγ

. (3.2)

µ0,min ≈ 2.3×1023 Pas for the small inclusion (50 km × 30 km) and µ0,min ≈8.5×1022 Pas for the large inclusion (80 km × 50 km). However, as mentioned before,

the lithosphere only yields for viscosities larger than 5×1023 Pas, no matter how large

the viscosity was. This discrepancy is far less than an order of magnitude and might

be attributed to the uncertainty involved in comparing a rectangular with a circular

shape. Considering that his scaling law was derived for Newtonian viscosities, whereas

we use non-Newtonian and depth-dependent rheology, the predictions made from this

scaling law are surprisingly correct.

Figure 3.4 shows the stress (a) and strain rate (b) evolutions in the lithospheric

center for various viscosities larger than the treshold viscosity. All of the plotted curves

represent localizing setups. The curves are smoothed to suppress large fluctuations

and make overall trends visible; note that not the overall stresses/strain rates are

considered here, but only their maxima. There is a tendency for stresses to be smaller

if the viscosity is larger; the opposite is true for strain rates, though the effect is less

pronounced. Possibly, shear heating is more efficient for larger viscosities and therefore

localization starts earlier, i.e. maximum stresses are reduced for these cases.

3.1.2 Summary

For a constant strain rate setup, the occurrence of localized deformation greatly de-

pends on the magnitude of the viscosity. The treshold value is around 5×1023 Pas for

a strain rate of 1×10−15 1s, localization is obtained for viscosities at or above this value.

While the temperature distrubtion has to be kept heterogeneous as introduced in

chapter 2 by small fluctuations, there is no large driving perturbation required to reach

sufficiently large stresses for the lithosphere to yield.

Moreover, the large extra inhomogeneity in the lithosphere is not necessary to

initiate localization. However, localization begins earlier, when the large inhomogeneity

is included; the bigger the inhomogeneity, the earlier the onset time of localization.

Small noisy heterogeneities, on the other hand, always have to be present.

If localization occurs, larger viscosities tend to decrease the maximum stress reached

and to slightly increase the initial maximum strain rate.


27

CHAPTER 3. RESULTS

(a)

(b)

Figure 3.4: Stress evolution vs. time for several initial mantle viscosities. All curves representsimulations of localization. (a) Tendency of stresses to be larger if viscosites are smaller. (b)Weak tendency for higher viscosities to cause higher strain rate during the simulation.


CHAPTER 3. RESULTS

3.2 Constant Stress Boundary Conditions not re-

sulting in Shear Localization

All boundaries in this model version have a free-slip boundary condition, i.e. additional

to the vanishing gradient in tangential velocity the normal velocities are zero at the

wall as well. Unlike the constant strain rate model, a driving perturbation as in the

standard model is needed to offset mantle motion. Material parameter values are as

in the standard model, unless stated otherwise. The layer of sticky air is added to

avoid numerical effects due to a free surface. Most of the simulations in this section

are run with a smaller box (2000 km × 1000 km) and lower resolution (1/6 of that in

the standard model), to save computing time.

3.2.1 Dependency of maximum stress on perturbation size

The characterizing parameters for the driving semicircular perturbation are its radius

and its temperature deviation compared to the temperature of the surrounding man-

tle. The resulting stress field is expected to scale with ∼∆ρgr, as follows from the

momentum equation (1.17) neglecting inertia. We assumed a linear dependency of the

density ρ on the temperature difference to standard conditions:

ρ = ρ0 + ∆ρ = ρ0 (1 + α∆T ) . (3.3)

α is the thermal expansivity, its value can be taken from table 2.1.

Figure 3.5 shows the dependence of maximum induced stresses on the driving stress;

three interesting features can be noted:

1. First, as expected, the maximum stress above the driving perturbation increases

(approximately) linearly with the size ∆ρgr of the perturbation. For our setup,

the desired stress of 1 GPa is reached for a perturbation of ∼ 500 MPa, which

corresponds to a radius of 1000 km and a temperature deviation of −500 K.

2. Second, the maximum stress in the region where the uprising mantle plume

reaches the lithosphere is basically independent of the size of the driving per-

turbation, it varies around 550 MPa. This indicates that this stress is due to the

bending of the lithosphere by the uprising plume. Vasilyev et al. (2001) have sim-

ulated a similar stress behavior of the lithosphere. Since the stress is restricted to

a small area of the lithosphere, they termed this plume-driven build-up of large

stresses (in their case ∼ 100− 300 MPa) “stress-focussing effect”.

Low driving stresses (< 200 MPa) result in lower stresses at the center of the


29

CHAPTER 3. RESULTS

(a) (b)

(c) (d)

Figure 3.5: Dependency of lithospheric stress on the size of the driving perturbation for avisco-elastic rheology. (a) Maximum stress versus perturbation size. (b) Time when maximumstress occurs. (a,b) At the center of the lithosphere. (c) Maximum stress versus perturbationsize. (d) Time when maximum stress occurs. (c,d) 1000 km off the center of the lithosphere.


CHAPTER 3. RESULTS

lithosphere than 1000 km off center. Only for high stresses (> 200 MPa) is

the center stress larger than the off center stress caused by the rising secondary

plume.

3. Third, in both regions, the time needed to reach the maximum stress decreases

with increasing perturbation size. That is, even though the magnitude of the

stress maximum 1000 km off the center of the lithosphere does not change, it oc-

curs earlier for a larger driving stress. The best fit found for a time - driving stress

- dependence is t ∝ µ0/∆ρgr.

3.2.2 Maximum stress dependency on perturbation size for a

viscous rheology

In addition, we performed simulations with the same setup and parameters for a viscous

rheology, in order to be able to understand the differences between a visco-elastic setup

and a viscous or visco-plastic setup ignoring elasticity, which is typically employed (e.g.

Tackley (2000a,b)). Despite two “outliers” (fig. 3.6), we see a similar behavior as for the

visco-elastic case. The absolute stresses are about 2.5 times higher; but the dependency

of the maximum lithospheric stress on the driving stress is approximately linear in the

center region and only slightly decreasing for the off-center region. The time point

when the stress maxima are reached show a similar exponential behavior.

3.2.3 Maximum stress dependency for different initial setups

Viscous start

The initial stress field of the standard model is zero everywhere. The “true” stress

field develops during the first time steps of the simulation. The so acquired stress field

does not change significantly from one time step to the next after less than 10 time

steps, which correspond to roughly 2 ka depending on the exact setup. In order to

inhibit elastic deformation during these first time steps we implemented a version that

employs a viscous rather than visco-elastic rheology, but only during the initial 10 time

steps. The exact code can be seen in appendix A.

A simulation with a viscous start (right column) is compared with the standard

model at their initial time steps in fig. 3.7. Considered are only the stress fields.

The next graph (fig. 3.8) shows the stress evolution of the modified model (blue) and

the standard model (green). The stress maximum for the sinking of the perturbation

(first peak) is by a factor 2 smaller for the model with the viscous start. The second


31

CHAPTER 3. RESULTS

(a) (b)

(c) (d)

Figure 3.6: Dependency of lithospheric stress on inhomogeneity size for a viscous rheology.(a) Maximum stress versus perturbation size. (b) Time after which the maximum stress oc-curs. (a,b) At the center of the lithosphere. (c) Maximum stress versus perturbation size. (d)Time after which the maximum stress occurs. (c,d) 1000 km off the center of the lithosphere.


CHAPTER 3. RESULTS

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3.7: Stress fields at the first, second, and fifth time step. Left column: for the standardmodel. Right column: Modified standard model with initially viscous rheology. (a,b) After0 ka. (c,d) After 0.2 ka. (e,f) After 0.27 ka.


33

CHAPTER 3. RESULTS

Figure 3.8: Stress evolutions for standard (green) and modified (blue) model.

peak, on the other hand, is twice as large as for the standard model. Apparently the

initial stress field does have an influence on the following evolution of stresses which

should be kept in mind. However, this finding is rather counterproductive for our quest

for high stresses.

Different arrangement of plumes

So far, we considered a cold temperature perturbation hanging below the lithosphere.

Shear heating has not yet been sufficient to cause the lithosphere to localize. This

section is dedicated to the idea that an uprising hot perturbation might bring more heat

to the lithosphere and so enhance shear heating. Although a temperature rise leads

to a stress relaxation, it also decreases viscosity and therefore increases strain rate.

Depending on which effect is stronger, a temperature rise can result in either thermal

runaway or inhibit shear heating. We will see in this section that a temperature rise in

our standard model inhibits shear heating by reducing stresses more drastically than

enhancing strain rates. The next section, however, will show that a temperature rise

can indeed result in shear heating strong enough to initiate shear localization.

The initial plume is put at the bottom of our box instead of the top. Since we want

to compare this modified setup to the standard model, the radius is set to 550 km

as in the standard model and the temperature perturbation to +300 K, where it was

−300 K in the standard model. Figure 3.9 shows the evolution of the stress (a) and

strain rate (b) field around the lithospheric center. The green curve represents the

standard model, the modified model is blue.

As expected, the initial stresses and strain rates for the modified model are sig-

nificantly smaller than for the standard model, since the hot plume does not affect

the lithosphere in the beginning. In this particular setup, the largest stress reached

in the meanfield amounts to only ca. 10% of those in the standard model. However,


CHAPTER 3. RESULTS

Figure 3.9: Initial driving perturbation is a hot plume at the bottom instead of a cold plumehanging at the lithosphere. Compared are stress and strain rate evolution at a fixed point closeto the center of the lithosphere. Blue: Standard model. Green: Model with uprising plume.


35

CHAPTER 3. RESULTS

Figure 3.10: Stress evolution versus elapsed time for standard model (blue) compared tomodels of higher mantle viscosity (green,red)

the maximum strain rate is notably higher than in the standard model. While we are

so far able to reach stresses that are as high as the yield stress of 1 GPa, this setup

is worth to be noted as a possible starting point for increasing the lithospheric strain

rate to values high enough for localization to occur.

3.2.4 Effects of viscosity

Varying viscosity by two orders of magnitude ([1×1021 Pas . . . 1×1023 Pas]) elicits only

minor changes in the magnitude of the maximum stress induced in the lithosphere. A

significant influence can however be seen on the time scale in which the evolutions take

place: An increase in viscosity by one order slows down the process by roughly two

orders of magnitude, as displayed in fig. 3.10.

Stresses are slightly higher for higher viscosities, which is also reflected in the strain

rate - stress - relationship τ = 2µε (eq. 1.3).

3.2.5 Subduction initiation by density difference

Mart et al. (2005) showed with a standard model that a simple lateral density difference

in the lithosphere may result in the creeping of the denser plate underneath the lighter

one. They used a centrifuge with enhanced gravity, and no more driving forces. Their

reasoning was that an initial plunge of lithosphere into the asthenosphere is necessary

to start subduction. McKenzie (1977) had calculated that ≈ 120 km initial dip of

a slab is necessary to start self-sustaining subduction. The creeping mechanism that


CHAPTER 3. RESULTS

Mart et al. found, might be one mechanism of achieving the required 120 km.

In a hope to find a mechanism for enhancing the strain rates in our model, we

implemented the density difference proposed by Mart et al. However, as the series in

fig. 3.11 and the maximum strain rate evolution in fig. 3.12 demonstrate, does the

maximum strain rate never rise above ≈ 6×10−10 1s. Thus, we can confirm the creep

deformation found in the experiments of Mart et al., but it is neither sufficient to start

subduction in our model nor even to initiate shear localization of our lithosphere.

3.2.6 Maximum stress dependency when a heterogeneity is

included in the lithosphere

In order to localize high stress regions and possibly produce a shear heating insta-

bility, we included a heterogeneity of considerable size and rectangular shape in the

lithosphere. This viscosity inhomogeneity (µ = 1020 Pas) was varied in its size from

30 km × 20 km to 300 km × 100 km as well as in its position: As depicted in fig-

ure 3.13, there is a jump in the stress field at a depth of 50 km. The first position,

represented by the blue squares in figure 3.14 and shown in figure 3.13 (a), puts the

center of the inhomogeneity at a depth of 50 km. This means that the inhomogeneity

is situated both in the high and low stress region. The second position, shown in figure

3.13 (b) and denoted by the red circles in 3.14, is entirely in the high stress area. This

implies that the maximum extent in the z-direction possible to implement is only about

50 km, in contrast to the 100 km possible extent for the position in figure 3.13. A

sample implementation of the inhomogeneity can be taken from appendix B.

Figure 3.13 shows the dependency of the maximum stress above the driving per-

turbation (a) and around the area of the rising plume (b). It is apparent that an

inhomogeneity totally included by the high stress area in the lithosphere elicits sig-

nificantly higher maximum stresses than an inhomogeneity of the same size in both

regions. Whereas the lithosphere away from the driving perturbation shows a linear

stress behavior for both inhomogeneity positions, the stresses reached are generally

smaller than in the center of the lithosphere (except for the standard model - inho-

mogeneity size = 0). The stress behavior above the descending perturbation is rather

interesting - for small inhomogeneity sizes the stress maximum seems to increase lin-

early with increasing sizes. For inhomogeneity sizes above 4000 km2 which corresponds

to 60 km × 60 km in our simulations, on the other hand, the stress in the lithosphere

reaches the desired 1 GPa. The two blue data points to the right in figure 3.14 (a)

are quite off; these points correspond to inhomogeneities extending through the entire

lithosphere. Therefore the lithosphere is no longer behaving as an intact plate (with a


37

CHAPTER 3. RESULTS

(a) (b)

(c) (d)

Figure 3.11: Simulation reflecting the results of Mart et al. (2005). The lithosphere is dividedinto two neighboring blocks of different density.(a) After 0.02 ka. (b) After 0.10 ka. (c) After0.42 ka. (d) After 7.80 ka.

Figure 3.12: Maximum strain rate evolution for a setup with density step in lithosphere.


CHAPTER 3. RESULTS

(a) (b)

Figure 3.13: Abrupt stress decrease around z = 50 km. (a) Heterogeneity centered in thelithosphere. Time = 3.3 ka, stress maximum = 1011.0 MPa. (b) Heterogeneity completelyincluded by the high stress region. Time = 3.3 ka, stress maximum = 634.7 MPa.

(a) (b)

Figure 3.14: (a) Evolution of maximum stress at the center of the lithosphere versus thesize of the inhomogeneity. (b) Evolution of maximum stress 1000 km off the center of thelithosphere versus the size of the inhomogeneity.

hole), as is the case for the smaller inhomogeneities.

Note that the lithosphere in some of these simulation is put under stresses of around

1 GPa, but does not yield. From eq. 1.15 we know that the onset of shear localization

through shear heating not only requires high stresses, but also high strain rates. How-

ever, in all of the aforementioned simulations of this subsection the maximum strain

rate does not exceed 2×10−11 1s.

In his dissertation, B. Kaus found a non-dimensionalization for the equation of en-

ergy conservation suitable for this problem that includes the use of two non-dimensional

numbers, the Brinkman number Br and the Peclet number Pe. The Brinkman number

is originally defined as the ratio of heat transported away by a fluid from a bordering

wall to the heat transmitted from the wall to the fluid. Adjusted for our case, the


39

CHAPTER 3. RESULTS

Brinkman number is the ratio of heat transported away from the site of production

(i.e. a region of higher shear heating) to the rate of production (eq. 3.4), it is thus

a measure for the efficiency of shear heating. The Peclet number is the ratio of a

heterogeneity length scale (R) to a diffusion length scale:

Pe =R√κµ0G

(3.4)

Br =σ0

2γ

ρcpG(3.5)

with ρ being the density (3300 kgm3 ), cp the heat capacity (1050 J

Kkg), G the elastic

shear modulus (5×1010 Nm2 ), σ0 the stress at the start of the simulation, κ = kρcp the

thermal diffusivity, and k the thermal conductivity of 3.3 Wm·K . γ Q

RT 20

in our case lies in

the range of [0.01 1K

. . . 0.1 1K

] (depending on the exact temperature in the region of

the inhomogeneity); µ0 is on the order of ≈ 1026 Pas, which also is the initial viscosity;

and σ0 ≈ 100 MPa for all cases. The logarithms of the Br and Pe numbers are thus

within [−3.0 . . .−1.8] and [−0.5 . . . 0.4], respectively, for the high stress area around

the inhomogeneity.

For localization to occur under constant stress boundary conditions, Kaus (2005)

found the condition (eq. 3.6) that the Peclet number has to be larger than some

constant c divided by the square root of the Brinkman number. If Pe is too small,

than diffusion removes all the heat generated by shear and thus inhibits localization.

The constant depends on the deformation mode, it is 2.3 for simple shear and 1.7

for pure shear. The difference is small; it is less than the uncertainties related to

determining Br and Pe (for example, remember that Kaus used a circular inclusion of

radius R whereas our inhomogeneity is rectangular); therefore we set c = 2.

Pe ≥ cBr−0.5 (3.6)

If eq. 3.6 is implemented, its graph represents the boundary between diffusion dom-

inated deformation and localized deformation: One can see clearly from fig. 3.15 that

all our simulations including an inhomogeneity are well in the non-localizing regime.

We thus cannot expect to see localization in these setups.

We will see later (in section 3.3) that a much larger strain rate must be achieved in

order to obtain localization.


CHAPTER 3. RESULTS

Figure 3.15: Peclet- vs. Brinkman numbers for different inhomogeneity sizes (blue dots).Black line indicates boundary between localization parameter range (above) and parameterrange not giving localization (below).

Figure 3.16: Topography density and maximum stress. Blue: µ = 1×1025 Pas. Red: µ =1×1023 Pas. Lines: Linear regression through data points.


41

CHAPTER 3. RESULTS

(a) (b)

Figure 3.17: Snapshot of (a) stress field and (b) composition of our model setup for a litho-sphere of variable thickness and a topography 15 km high at its maximum. Green: topography,here with same parameter values as the lithosphere (yellow). The mantle (red) has a lowerviscosity than the lithosphere.

(a) (b)

Figure 3.18: Snapshot of (a) stress field and (b) composition of our model setup for atopography 15 km high at its maximum. The hanging perturbation has been removed to showthe stresses induced solely by the topography. Composition colors as in fig. 3.17.

3.2.7 Maximum stress dependency when topography is present

Regenauer-Lieb et al. (2001) showed that it is possible to deform a lithosphere (of

uneven thickness) by simply adding a large pile of sediments. This we were not able

to reproduce; their lithosphere was free floating with springs as substitution for the

mechanical support of the asthenosphere and and a growing pile of sediments was

simulated by an adjustable stress boundary condition. Our “mountain” has maximum

dimensions of 15 km height and 150 km width, where the tip lies asymmetrically 100 km

from the left bottom end. An example implementation can be found in appendix D.

As with the perturbation size, the maximum stress depends linearly on the density

of the sediments (fig. 3.16). As in subsection 3.2.4, higher viscosity leads to higher

stresses. Curiously though, we were unable to find a clear dependency of the stress on

the width or height of the topography. Neither does the stress depend on topography


CHAPTER 3. RESULTS

Figure 3.19: Stress evolution of standard model (green) compared to a model with colder(blue) and hotter (red) mantle. All three setups have the same ratio of Tmantle : TCMB

shape. The independence of stress on the size of the topography was obtained even

though we used different implementations of topographies; a programming error can

thus be excluded.

Fig. 3.17 is a close-up of the composition and stress field around the lithosphere

and topography. Note that our simulations normally do not have a change in litho-

spheric thickness; the depicted particular setup was adjusted to resemble the setup of

Regenauer-Lieb and Yuen (2003), who induced a fracture in their lithosphere by simply

adding boundary stresses representing topography.

For comparison, snapshots of our standard lithosphere are shown in fig. 3.18. The

cold driving perturbation has been removed to reveal the effects solely induced by the

topography.

3.2.8 Effects of mantle temperatures

Our standard mantle temperature is based on the average mantle temperature as given

e.g. by Turcotte and Schubert (2002), which is roughly 2250 K. However, the use of the

mantle adiabat with a potential temperature (∼ 1600 K) might be more appropriate

(value can be found in Schubert et al. (2001)).

To test the effects of changing the mantle temperature, we modified our standard

model both towards a hotter mantle (Tmantle = 2750 K, TCMB = 3900 K, such that the

ratio between Tmantle and TCMB is the same as for the standard model) and a colder

mantle (Tmantle = 1750 K, TCMB = 2450 K).

The result of these simulations is shown in figure 3.19, where the cold mantle model


43

CHAPTER 3. RESULTS

is represented by the blue line, the hot by the red line, and the standard model by the

green line. As can be clearly seen, the colder the mantle is, the larger are the stresses.

An explanation for this can be found in the viscous behavior of our rheology (eq.

1.2) and the temperature dependence of viscosity (2.1): Larger temperatures lower the

viscosity in the lithosphere; since viscosity and shear stress are proportional this also

results in a lower stress state of the lithosphere.

Although resulting in lower stresses, the strain rates in the lithosphere are signif-

icantly larger for hotter mantles. The maximum strain rate for the 2750 K - mantle

model is about one order of magnitude larger (∼ 10−14 1s

than for the 1750 K - mantle

model (∼ 10−13 1s, with the standard model (2250 K) lying in between: hotter material

with a therefore lower viscosity deform more readily.

3.2.9 Summary

This section was dedicated to finding mechanisms that produce high stresses in the

lithosphere of the order of the yield stress 1 GPa. We varied the perturbation size

as well as its location; we examined the effects of a large heterogeneity included in

the lithosphere and a topography on top. Mantle temperature and viscosity, a viscous

rheology and a non-zero initial stress field were tested as well.

The maximum stress was found to scale linearly with perturbation size, which

is determined by the product ∆ρgr. For small perturbations, the maximum stress

off the center of the lithosphere caused by the uprising secondary plume is larger

than the maximum stress in the lithospheric center which is caused by the primary

sinking perturbation. For large perturbations, the center stress is larger. Even though

maximum stresses reach the yield stress in some simulations, localization does not

occur.

The viscous rheology simulations produce maximum stresses of twice as high as

the yield stress. Enforcing an initial viscous rheology that is later changed into a

viscoelastic rheology allows for the stress field to build up without elastic deformations

due to these stresses. This reduces the stresses near the center of the lithosphere.

A different arrangement of the initial driving perturbation - a hot plume instead of

a cold drop - causes lower stresses but higher strain rates in the lithosphere.

The inclusion of a heterogeneity easily allows for stresses ∼ 1 GPa; below this

value stresses scale linearly with heterogeneity size. Scaling laws show that our choice

of material parameters and heterogeneity size in combination with our initial stresses

is outside the domain of localization.

An added topography barely changes the stress field. Higher topography density


CHAPTER 3. RESULTS

elicits larger lithospheric stresses, but the height remains rather unaffecting.

Higher mantle temperatures decrease lithospheric stresses, but result in larger strain

rates. The initial viscosity, on the other hand, barely affects lithospheric stresses, but

significantly slows down the mixing process if increased.

A creep mechanism prompted by a lateral density difference in the lithosphere can

be reproduced but proves to be ineffective to start localization.

3.3 Constant Stress Boundary Conditions resulting

in Shear Localization

The previous section dealt with changing perturbation, inhomogeneity, and topography

size in order to increase stress or strain rate in the lithosphere. We are now going to

use the results obtained to find a model setup able of localization.

From eq. 3.6 we know that localization is impossible for the parameter range chosen.

Using the definitions eqs. 3.4 and 3.5 and rearranging such that values we have varied

so far are on the left hand side of the equation and unchanged material properties on

the right hand side, we obtain:

R

√σ20

µ0

≥ c

√ρcpκ

γ(3.7)

This section is based on the right hand side of the equation: another approach,

which is not the best of all choices but so far the only combination leading to success

is reducing κ, ρ and cp by a factor 10. Doing so, eq. 3.7 promises to allow for a

localization. However, more restrictions on the setup are necessary to localization to

occur. The entire parameter setup leading to localization is shown in table 3.1.

Another approach might be to reduce lithospheric viscosity such that κ, ρ and cp

can be kept at their natural values. For σ0 ∼ 100 MPa, R ∼ 100 km, γ ∼ 0.05 1K

and c = 2, κ, ρ and cp as in the standard (non-localizing) setup eq. 3.7 predicts that

localization will occur when viscosity is below roughly 4×1023 Pas. However, due to

the great uncertainty in the determination of R, σ0, and γ, this value can only be a

general orientation. Test runs revealed that localization does not occur for a lithosphere

viscosity of ≈ 1×1022 Pas. Only reducing the viscosity to ≈ 5×1020 Pas resulted in

localization. Thus, this seems to be a promising direction for future research aiming

at encorporating realistic parameter values.

The following sections will deal with influence of different setup variations on the

occurrence of localization, i.e. if, how fast, and at what stresses the lithosphere localizes


45

CHAPTER 3. RESULTS

when the setup is changed.

Parameters for run showing localizationBox dimensionswidth x 1000 kmDepth z 1000 kmThickness lithosphere 100 kmTemperaturesSticky air temperature 283 KMantle temperature 1573 KCMB-temperature 3200 KMaterial properties Mantle LithosphereElastic shear modulus 5×1010 Pa 5×1015 PaViscosity 0.6×1021 Pa s 0.6×1025 Pa sCohesion 30×106 Pa 30×105 Pa

Density 330 kgm3 330 kg

m3

Conductivity 0.33 Wm·K 0.33 W

m·KHeat capacity 105 J

kgK105 J

kgK

Perturbation propertiesRadius 300 kmPosition center x 0 kmPosition center z 0 km (top of lithosphere)Temperature deviation -500 KRadius 300 kmPosition center x ± 1000 kmPosition center z 0 km (top of lithosphere)Temperature deviation +500 K

Table 3.1: Parameters used in localizing setup.

A typical development of the lithosphere for a localizing setup is depicted in fig.

3.20. As before, the stress and temperature fields are shown. Additionally, the compo-

sition of the setup indicating the location of lithosphere and mantle material (appendix

C) at the chosen four time steps are shown in order to clearly demonstrate the defor-

mation of the lithosphere.

One can see that stresses are too small to fracture the lithosphere for more than

150 ka. It takes the uprising plume about 130 ka to reach the bottom of the lithosphere

(not shown), and another 20 ka to reach the hanging, cold perturbation. But only

when enough hot material has collected below the lithosphere and is hindered in its

spreading by the hanging perturbation, localization is onsetting after around 170 ka.

Testing several setups demonstrates that each side of the box is equally likely to be

the place of initiation of localization; there is no ab initio preference of the system for

either side.


CHAPTER 3. RESULTS

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

Figure 3.20: Evolution of the localizing constant stress boundary condition setup. (a-e) Stressfield. (f-j) Temperature field. (k-o) Mantle (red) and lithosphere (yellow). (1st row) After20 ka. (2nd row) After 150 ka. (3rd row) After 171 ka. (4th row) After 173 ka. (5th row)After 174 ka.


47

CHAPTER 3. RESULTS

Our standard localizing setup has a lithospheric viscosity of 1×1024 Pas. A close-

up on the lithosphere (not shown here) reveals small bands in which viscosity drops to

≈ 1×1021 Pas several ka before localized deformation visibly onsets. As stated earlier

in this section, a viscosity below ∼ 1023 Pas is probably necessary to enable shear

localization. This might explain why the deformation occurs at the edges of the box

- that is where viscosity locally drops to values sufficiently small (≈ 1×1021 Pas) to

enable shear localization, whereas it stays too large (1×1024 Pas) elsewhere.

Extra attention was paid to the possibility of the localization being the result of

numerical artifacts known as the “drunken seaman” effect which may occur in systems

with a free surface and too large timesteps. Test runs were made with a sticky air layer

of 50 km thickness and timesteps were controlled to be ∼ 1 a long before localization

starts. B. Kaus developed an amendment to MILAMIN VEP [Kaus et al. (2010)] that

averts this effect; even using this amendment localization could still be seen in the

simulations.

3.3.1 Evolution Characteristics

A close look at stress, strain rate, and temperature development of a localizing setup

compared to a non-localizing setup is provided in figure 3.21. The non-localizing setup

features the same parameters as the localizing setup; the difference between these two

is that the first has a wider box, while its hot perturbations are fixed at the same

positions as in the latter.

There is a distinct difference between localization and non-localization in stress,

strain rate, and temperature evolution, even before visible localization starts. In the

used simulation, localization first occured on the left side of the box.

If we look at the stress evolution in different parts of lithosphere first, we find:

1. The evolution is the same for both sides of the box, no matter if the simulation

shows localization or not; while the middle behaves differently. Initially, the

stress in the center of the lithosphere is lower than at the sides (remember that

the plumes are rising at the edges); but reach higher values once localization

started. Stresses in the localizing setup at the sides are larger (∼ ×5) than in

the non-localizing setup from the very start of the simulation.

2. For the non-localizing setup, stress in the center (fig. 3.21 d) rises slightly during

the first 100 ka. This is the period of formation and uprise of the initially

imposed plume. Stress rise is accelerated until 120 ka, when the plume reaches

the lithosphere. Stresses at the edges additionally show a marked stress decrease


CHAPTER 3. RESULTS

at 120 ka that is missing in the localizing simulation; it is not apparent what

causes this decrease. After that follows a stress increase until about 160 ka,

when hot material brought up by the plumes has filled the entire space between

the hanging perturbation and the box edges. There is another marked decrease

by 50% in stress when the cold perturbation finally sinks into the mantle at

around 400 ka; and another increase at 900 ka when secondary hot plumes begin

to rise. But these stresses in no simulation exceed the stresses reached during

the first few hundred ka, and therefore the further development is neglected.

3. The localizing setup shows the same lithospheric stress behavior as in the non-

localizing case up to 120 ka. However, beyond this time, lithospheric stresses

either increase further (middle) or at least remain approximately even (edges)

for a few ten thousand years until the lithosphere fractures close to the left side

of the box (in this run) after 160 ka. Therefore, the fracture of the lithosphere

can be predicted from the stress field several 10 ka before it becomes visible

by a distinct disruption in the compositional plot, for example. Curiously, the

higher stresses are not reached at the location of fracture, but at the center of the

lithosphere. The maximum stress close to the fracture is only about 200 MPa;

although the stress at the beginning is much higher close to the sides of the box

than at the center.

4. The strain rate evolution (third row in fig. 3.21) is very similar to the stress

evolution. The most important difference to the behavior in stress evolution is

the magnitude: strain rates increase by 7 orders compared to the not localizing

case (stress: 4 in the middle), and here the maximum is reached in the actual

area of fracture.

5. Maximum stress and strain rate (3.21 a,b) show clearly the divergence between

localizing and non-localizing simulations: though the fracture occurs much later,

both curves start to diverge already when the uprising plume reaches the litho-

sphere.

6. The temperature field (bottom row of 3.21) shows in both cases a local max-

imum when the plume reaches the lithosphere, followed by a decrease in the

non-localizing case and a strong rise in the localizing case. The difference here,

again, is the magnitude of this process: the point farthest away from the fracture

is heated much less than the point immediately besides the fracture side or the

center of the lithosphere, where deformations are large.


49

CHAPTER 3. RESULTS

(a) (b)

(c) (d) (e)

(f) (g) (h)

(i) (j) (k)

Figure 3.21: Evolution characteristics for localizing setup (blue) and non-localizing setup(green). (a) Development of maximum strain rate. (b) Development of maximum stress. (c-e) Lithospheric strain close to the left edge, the x-center, and the right edge of the box. (f-h)Lithospheric stress close to the left edge, x-center, and right edge of the box. (i-k) Lithospherictemperature evolution at the three mentioned points.


CHAPTER 3. RESULTS

3.3.2 Influence of Inhomogeneities and Topography

Leaving out both the large lithospheric inhomogeneity and the sediment on top of the

lithosphere does not at all affect the occurrence of localization. This is in sofar not

surprising, as the localization starts at the edges of the box, whereas both inhomogene-

ity and topography are placed in the middle. They might play a role in moving the

fracture to the center of the lithosphere, which we were unable to accomplish, but the

localization as described above occurs independently of inhomogeneity and topography.

3.3.3 Influence of Material Properties

A factor of 10 reduction for three material properties is dissatisfying and naturally pro-

voked testing. It turns out that a ’normal’ density ρ = 3300 kgm3 still allows localization

for an otherwise unchanged setup. However, setting back either the thermal conductiv-

ity or specific heat capacity back to their typical values (3.3 Wm·K and 1050 J

Kkg, resp.)

immediately inhibits localization. This appears to be logical, since a low thermal con-

ductivity in the lithosphere ensures that any produced heat is only poorly transported

out of the system into the atmosphere, and a low heat capacity allows the temperature

to rise by a small amount of energy input. It might be interesting to find out where

exactly is the border between leading to localization and not in the product of all three

values. Though this would depend on the initial setup (compare to eq. 3.7).

Important to note is that the three values of ρ, k, cp only enable localization if they are

reduced for the entire setup, i.e. not only for the lithosphere, but also for the mantle.

As we have seen in section 3.3.2, inhomogeneity and topography have no noticeable

effect on the localization, and therefore their material properties have no further effect.

3.3.4 Influence of Perturbations

Perturbations were varied in radius and temperature, as well as their position at the

start of each simulation. Whereas the standard model only has one hanging (cold)

perturbation beneath the center of the lithosphere, the ’sidestandard’ model giving

localization has two additional uprising (hot) perturbations in the bottom corners of

the simulation box (compare fig. 3.20 top row). Interestingly, if the x-position of these

perturbation is changed closer to the middle, no localization occurs. The box size

itself does not effect localization - as long as the perturbations are kept at the corners.

Attempts to induce localization at places other than the edge of the box by changing

the distribution and number of perturbations as well as the size, number, and location


51

CHAPTER 3. RESULTS

Figure 3.22: The onset time of localization in dependence on the size of the perturbations.Ranges are between [200 km, 200 K] and [500 km, 500 K]. Both the rising plumes and thehanging perturbation always have the same initial sizes.

of inhomogeneities have been unsuccessful so far.

Varying the size of the perturbations (while keeping them at their standard po-

sitions) does not greatly affect whether localization in the lithosphere occurs or not.

Radii and temperatures within a reasonable range give localization, as long as the rest

of the setup remains unchanged from the standard setup. The only exception are very

small perturbations (∼ [100 km, 100 K]). Here, the cold perturbation is heated up to

the temperature of its surroundings and is gone long before the hot perturbations reach

the lithosphere. By this one can see the importance of the middle (cold) perturbation:

while the hot perturbations obviously bring heat into the lithosphere that reduces its

viscosity and thus support the onset of strain heating, the cold perturbation acts like

an anchor or an impediment for the sublithospherical flow of the uprisen plumes. The

plumes push the hanging perturbation away from the box wall and thus drag along

the stiff lithosphere still tightly attached to the plume. This mechanism works until

the cold plume finally sinks into the mantle. Thus, any setup that allows its cold

perturbation to vanish before the arrival of the uprising plumes (due to its small size,

100 km, 100 K) or to sink into the mantle too early (too cold, too great lateral distance

between hot and cold plumes) can be expected to not show localization.

While barely affecting the ’if’ of the onset of localization in the system, the pertur-

bation size does influence the ’when’. Generally, the hotter and/or larger the pertur-

bation is, the earlier the fracture in the lithosphere occurs. This dependence [fig. 3.22]

is especially strong for small perturbations (obviously, too small perturbations take

infinitely long to enable shear heating), and becomes weaker for large perturbations.

The best approximation within the range of localization-yielding setups was found to


CHAPTER 3. RESULTS

be

lnt

tc=

0.089 GPa

∆ρgr− 29.9 (3.8)

where t is the onset time in s, tc = 1×1015 s a characteristic time scale of our

model, r the perturbation radius in km, g the gravity (9.81 ms2

) and ∆ρ the density

difference resulting from the perturbation temperature T , as in eq. 3.3.

Any effect of perturbation sizes on maximum or mean stresses could not be deter-

mined unequivocally.

3.3.5 Influence of Viscosity

Viscosities in our model are bounded by an upper and a lower cutoff. While the lower

cutoff was always kept at 1×1018 Pas, the upper cutoff was varied between 1×1024 Pas

and 1×1030 Pas. The upper cutoff viscosity determines the initial viscosity in the

lithosphere, which has the highest viscosity in the entire system.

Choosing the parameters as in table 3.1 at the beginning of this section, we found a

bimodal dependence of onset time of localization on the cutoff viscosity: For 1×1025 Pas

and lower, localization onsets around 160 ka; for 1×1026 Pas and higher, localization

onsets around 290 ka. The variations for these times are only ≈ 5 ka or less.

Unrelated to the cutoff viscosity, the mantle viscosity is of importance for the litho-

spheric localization as well: While a mantle viscosity of 0.6×1021 Pas is our standard

value for the localizing setup, slightly higher viscosities (1×1021 Pas) still result in

localization, but viscosities of 5×1021 Pas and higher not anymore. This viscosity is

consistent with values found in the literature (e.g. Fleitout and Froidevaux (1979)),

giving an average viscosity for the upper mantle of ≈ 1020 Pas.

3.3.6 Summary

The section showed that it is possible to obtain localization in a lithosphere placed on

top of a moving mantle.

Obvious drawbacks of our localization are the by a factor 10 too low heat capacity

and thermal diffusivity, and its position: In all simulations localization always occurred

at the sides of the box, never in the middle. It might be possible to restore realistic

values for heat capacity and thermal diffusivity on the cost of initial viscosity and still

reproduce localization.

The difference between localizing setups and nonlocalizing setups are the much

higher strain rates. The maximum strain rates are observed at the place of localiza-


53

CHAPTER 3. RESULTS

tion. Maximum stresses, on the other hand, are found outside the area of localized

deformation.

It is important to note that the onset of localization greatly depends on the position

of the temperature perturbations driving mantle motion. Minimum perturbation sizes

to obtain localization are around 200 km in radius and 200 K in temperature.

Larger perturbations accelerate the formation of a localized deformation zone. If a

localization will occur, it is visible in the stress and strain rate evolutions several tens

of ka before the localization begins.

Inhomogeneities and an added topography have neither an influence on whether

localization occurs nor on where it occurs.

High mantle viscosities inhibit the onset of localization; lithosphere viscosities only

influence the time of onset.


Chapter 4

Discussion and Conclusion

The results presented in the previous chapter lead to a better understanding of the

formation of localized shear zones, possible predecessors of subduction zones. We have

developed a model that can simulate lithospheric-scale shear localization both under

constant strain rate and under constant stress boundary conditions. Here we will

discuss some of the implications and limitations of our results.

4.1 Shear Localization under Constant Strain Rate

Boundary Conditions

We imposed a constant background strain rate of 10−15 1s, widening our domain. If

viscosities were 5×1023 Pas and higher, the deformation concentrated in the center of

the lithosphere; below this localization did not occur. The value of the treshold viscosity

is in very good agreement with the scaling law derived in Kaus and Podladchikov (2006)

(eq. 3.1).

Due to the imposed box boundary velocity, no driving perturbation was needed to

start mantle movement. Neither was a large inhomogeneity in the lithosphere needed

to initiate localization in its center. If present, however, it sped up the formation of

the shear zone.

Independent of this large inhomogeneity, small, randomly distributed temperature

perturbations were always present in the mantle. These small inhomogeneities are

necessary to start localization, as Mancktelow (2002) had shown.

Crameri (2009) had demonstrated that localization is possible for a constant strain

rate boundary setup comprising of the uppermost 120 km of lithosphere and upper

mantle, when temperatures at the model bottom are at roughly 1100◦C. We were able

to expand his model to include the entire mantle and to obtain deformation localization

55

CHAPTER 4. DISCUSSION AND CONCLUSION

for an extending instead of a shortening setup.

Constant strain rate boundary condition setups have the disadvantage of assuming

an a priori mantle movement. With the long-term goal of creating a mantle convec-

tion model that is able to produce large-scale fractures in the lithosphere and initiate

subduction, it is desirable to obtain straining as a result of acting forces due to temper-

ature gradients etc., instead of obtaining forces due to deformation. A step towards this

direction is the implementation of constant stress boundary conditions in our model.

4.2 Constant Stress Boundary Conditions not re-

sulting in Shear Localization

All boundaries are free slip boundaries in this model version. Stresses are therefore

zero at the boundaries, but may evolve freely within the bulk. A driving density per-

turbation is needed to offset mantle motion. As can be expected from the momentum

equation, maximum stresses in the lithosphere scale linearly with perturbation size:

σmax ∼ ∆ρgr.

For large perturbations (∆ρgr ≈ 500 MPa) the maximum stress in the lithosphere

reaches the yield stress. However, strain rate is independent of perturbation size and

too small to make shear heating (∼ τ ε) efficient enough to start localization. This is

in agreement with the scaling law from Kaus and Podladchikov (2006) which indicates

that localization should not be expected for our parameter choice, even if we include a

large inhomogeneity.

However, these results show that it is indeed possible to induce stresses on the

order of magnitude of the yield stress (1 GPa). Tackley (2000a,b) had employed a

3D convection model and successfully reproduced subduction - under the assumption

of a yield stress ∼ 100 MPa. One future goal could be to reconcile the convection-

controlled mantle model with the density-perturbation-controlled mantle model such

that a more realistic mantle movement can be combined with a realistic yield stress.

Somewhat irritating is the influence of the topography: While induced stresses in

the lithosphere are proportional to topography densities as is also expected from above

equation, neither width nor height of the topography have an unambiguous effect on

lithospheric stresses. As this result has been triple-checked with different topography

shapes and various ways of implementation, a programming error can be excluded

almost certainly.

However, adding a large topography without getting localizied deformation contra-

dicts the findings of Regenauer-Lieb et al. (2001); Regenauer-Lieb and Yuen (2003);



Branlund et al. (2001), who broke the lithosphere by simply adding a sedimentary

load as a boundary condition. The discrepancy might be a result of our uniformly

thick lithosphere; Regenauer-Lieb’s lithosphere is not only generally thinner but has a

crustal region and a thinner oceanic region.

A visco-plastic rheology produces maximum stresses twice as large as the yield stress

stress. This is expected, as visco-plastic materials cannot relax as visco-elasto-plastic

materials can. For our model typical relaxation times are τMaxwell ∼ µG∼ 1 ka [Turcotte

and Schubert (2002)], which is much smaller than the time intervals simulated.

Bringing more heat into the lithosphere by either raising mantle temperatures or

imposing hot plumes instead of cold hanging perturbations decreases stresses signifi-

cantly, and raises strain rates slightly. Whereas the stress decrease is predictable by

the strain rate - stress - relation (τn−1 ∝ exp(−H/RT )), the strain rate increase can

be attributed to the smaller viscosity.

4.3 Constant Stress Boundary Conditions resulting

in Shear Localization

We have found a constant stress setup in that our lithosphere deforms in a localized

manner. Different from the standard model, the localizing setup has a lithosphere with

an initially spatially constant viscosity, and a mantle of low viscosity. Heat capacity

and thermal diffusivity are reduced by a factor 10 in the entire system. Arranging

hot and cold temperature perturbations as depicted in fig. 3.20 f, results in localized

deformation after ∼ 100 ka.

The size of the driving pertubations only affects the time when localization onsets

if the cold ones are large enough to not have resolved (by diffusion, conduction) by the

time the hot plume reaches the lithosphere.

Changing the plume distribution or the above mentioned material parameters pre-

vents the lithosphere from fracturing. Another major drawback of our setup besides its

parameter restriction is the position of localization initiation: always at the box edge.

If a setup will localize is predictable ∼ 50 ka before localization onsets visibly.

Whereas stress and strain rates drop after the uprising plume has reached the bottom

of the lithosphere in the standard model, they continue to rise until shear heating is

efficient enough to fracture the lithosphere in the localizing setup. Localization occurs

even though the maximum stress reaches the yield stress only locally; stress at the

localizing site is found to always be significantly smaller than the maximum stress.

Maximum strain rates, on the other hand, are registered at the site of localization.


57


Even though inhomogeneities and topography add to the lithospheric stresses, they

have no influence on the occurrence of shear localization.

4.4 Conclusion and Outlook

On the way to self-consistently initiate subduction, we were able to make a step from

localizing constant strain rate boundary condition setups to a localizing constant stress

boundary condition setup. The geometrical restrictions and parameter ranges are tight

and unrealistic (too low heat capacity and thermal diffusivity; specific arrangement of

plumes); therefore the next goal should be to expand the parameter ranges such that

material properties can be set back to realistic values. A possible approach is to

significantly reduce lithospheric viscosity, such that the requirement in eq. 3.7 is met

for standard cp and κ.

It is also very desirable to control the position of localization. Since topography and

inhomogeneity failed to give such a control, another approach might be a non-uniform

thickness of the lithosphere or a different perturbation distribution. Our inhomogeneity

was of lower viscosity than the surrounding lithosphere; a change in heat capacity or

diffusivity might be of more success.

The next step is to make the model a “real” convection model. Motion in our

mantle was entirely driven by density perturbations. Although these perturbations

might occur in the Earth’s mantle, it is a well-established fact that the real mantle is

not still but convects, and thus convection should not only not be neglected but might

as well act as a driving force for fracturing the lithosphere.


Acknowledgments

To avoid the ever-present conditional that always makes you want and not do something

(’I would like to thank...’): I am, above all, thanking Boris for his encouraging, critical,

supportive, always helpful, and enviably fast (yes, all at the same time) responses to

all my questions and mails. I am impressed by your work spirit and feel very lucky to

have chosen you as my supervisor.

Knowing of the pathetic sound of this (and ignoring this knowledge, because oth-

erwise this page would stay empty), many thanks to Tobias for the always patient and

extensive explanations when I needed them, and the insightful discussions on the roof

of Building NO and below the too hot summer sun by the auxiliary heat generator

barbecueing our meat.

Chris, when not prevented by parallel lectures, I had the honor of being unknowingly

persuaded by you to write this thesis in Geodynamics. It was fun being in your class.

Well, and in Peter’s too. Sometimes. When he was not driving me insane.

Short time though we only shared our office - I am thanking Fabio and Cyrill

for their reliable comradeship in the battle against the invincible blinds outside our

window. Be ensured, Johanna is a worthy follow-up. Plus, she got bribes! - Thanks

for making me sweat both in your kitchen and the gym! Rebecca, and Katharina, and

Ylona, and all the rest of the Geo-groups: thanks for being nice to me.

Thank you, you unknown helpers and employees of the Studienstiftung, for making

the expensive decision to grant me a scholarship. I figure, without you I would never

have gotten into the trouble of explaining my cross-nationed student career.

Almost finally, I thank two guys, Roland and Hanjo (who else would still be read-

ing?!). The first, for letting me go and gypsy around; the latter - well, I sent the first

e-mail, so actually you should thank ME.

And last, though not least, I thank you, who are still reading and have thus probably

(ok, hopefully) read my entire thesis. Thank you for making me not the only one to

have ever made it to the last page.

59

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63

BIBLIOGRAPHY


List of Figures

1.1 Internal structure of Earth. Taken from Tarbuck et al. (2007) . . . . . 6

1.2 Deformation mechanism map (grain size 0.1 mm) White area indicates

the parameters for which Peierls plasticity is dominant, light gray area

is dominated by diffusion creep, dark gray by dislocation creep. Solid

lines are lines of constant strain rate. From [Kameyama et al. (1999)] . 10

1.3 Mohr’s circle and strength envelope. . . . . . . . . . . . . . . . . . . . . 11

2.1 Stress and temperature fields for the standard model. Maximum stresses

reached in the lithosphere are just above 500 MPa. (a-d) Stress field.

(e-h) Temperature field. (a,e) After 8.9 ka. (b,f) After 23.1 ka. (c,g)

After 59.6 ka. (d,h) After 142.3 ka. . . . . . . . . . . . . . . . . . . . . 20

2.2 (a) Evolution of maximum stress in the lithosphere. (b) Evolution of

maximum strain rate in the lithosphere. (c) Stress and strain rate de-

velopment at the center of the lithosphere. (d) Stress and strain rate

development off the center of the lithosphere. . . . . . . . . . . . . . . . 22

3.1 The occurrence of localization depends on the combination of bottom

temperature and strain rate. Red shaded area: localization regime; gray

area: no localization occurs. Taken from Crameri (2009) . . . . . . . . 24

3.2 Evolution of the lithosphere in a constant strain rate setup with an in-

homogeneity. (a-c) Stress field. (d-f) Temperature field. (g-i) Viscosity

field. (a,d,g) After 0.3 Ma. (b,e,h) After 1.8 Ma. (c,f,i) After 12.4 Ma. 25

3.3 Onset time of localization vs. viscosity. Blue: small lithospheric inho-

mogeneity (50 km × 30 km). Green: large lithospheric inhomogeneity

(80 km × 50 km). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4 Stress evolution vs. time for several initial mantle viscosities. All curves

represent simulations of localization. (a) Tendency of stresses to be

larger if viscosites are smaller. (b) Weak tendency for higher viscosi-

ties to cause higher strain rate during the simulation. . . . . . . . . . . 28

65

LIST OF FIGURES

3.5 Dependency of lithospheric stress on the size of the driving perturbation

for a visco-elastic rheology. (a) Maximum stress versus perturbation size.

(b) Time when maximum stress occurs. (a,b) At the center of the litho-

sphere. (c) Maximum stress versus perturbation size. (d) Time when

maximum stress occurs. (c,d) 1000 km off the center of the lithosphere. 30

3.6 Dependency of lithospheric stress on inhomogeneity size for a viscous

rheology. (a) Maximum stress versus perturbation size. (b) Time after

which the maximum stress occurs. (a,b) At the center of the lithosphere.

(c) Maximum stress versus perturbation size. (d) Time after which the

maximum stress occurs. (c,d) 1000 km off the center of the lithosphere. 32

3.7 Stress fields at the first, second, and fifth time step. Left column: for the

standard model. Right column: Modified standard model with initially

viscous rheology. (a,b) After 0 ka. (c,d) After 0.2 ka. (e,f) After 0.27 ka. 33

3.8 Stress evolutions for standard (green) and modified (blue) model. . . . . 34

3.9 Initial driving perturbation is a hot plume at the bottom instead of a cold

plume hanging at the lithosphere. Compared are stress and strain rate

evolution at a fixed point close to the center of the lithosphere. Blue:

Standard model. Green: Model with uprising plume. . . . . . . . . . . . 35

3.10 Stress evolution versus elapsed time for standard model (blue) compared

to models of higher mantle viscosity (green,red) . . . . . . . . . . . . . 36

3.11 Simulation reflecting the results of Mart et al. (2005). The lithosphere is

divided into two neighboring blocks of different density.(a) After 0.02 ka.

(b) After 0.10 ka. (c) After 0.42 ka. (d) After 7.80 ka. . . . . . . . . . 38

3.12 Maximum strain rate evolution for a setup with density step in lithosphere. 38

3.13 Abrupt stress decrease around z = 50 km. (a) Heterogeneity centered

in the lithosphere. Time = 3.3 ka, stress maximum = 1011.0 MPa.

(b) Heterogeneity completely included by the high stress region. Time =

3.3 ka, stress maximum = 634.7 MPa. . . . . . . . . . . . . . . . . . . 39

3.14 (a) Evolution of maximum stress at the center of the lithosphere versus

the size of the inhomogeneity. (b) Evolution of maximum stress 1000 km

off the center of the lithosphere versus the size of the inhomogeneity. . 39

3.15 Peclet- vs. Brinkman numbers for different inhomogeneity sizes (blue

dots). Black line indicates boundary between localization parameter range

(above) and parameter range not giving localization (below). . . . . . . . 41

3.16 Topography density and maximum stress. Blue: µ = 1×1025 Pas. Red:

µ = 1×1023 Pas. Lines: Linear regression through data points. . . . . . 41


LIST OF FIGURES

3.17 Snapshot of (a) stress field and (b) composition of our model setup for

a lithosphere of variable thickness and a topography 15 km high at its

maximum. Green: topography, here with same parameter values as the

lithosphere (yellow). The mantle (red) has a lower viscosity than the

lithosphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.18 Snapshot of (a) stress field and (b) composition of our model setup for

a topography 15 km high at its maximum. The hanging perturbation

has been removed to show the stresses induced solely by the topography.

Composition colors as in fig. 3.17. . . . . . . . . . . . . . . . . . . . . . 42

3.19 Stress evolution of standard model (green) compared to a model with

colder (blue) and hotter (red) mantle. All three setups have the same

ratio of Tmantle : TCMB . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.20 Evolution of the localizing constant stress boundary condition setup. (a-

e) Stress field. (f-j) Temperature field. (k-o) Mantle (red) and lithosphere

(yellow). (1st row) After 20 ka. (2nd row) After 150 ka. (3rd row) After

171 ka. (4th row) After 173 ka. (5th row) After 174 ka. . . . . . . . . 47

3.21 Evolution characteristics for localizing setup (blue) and non-localizing

setup (green). (a) Development of maximum strain rate. (b) Develop-

ment of maximum stress. (c-e) Lithospheric strain close to the left edge,

the x-center, and the right edge of the box. (f-h) Lithospheric stress close

to the left edge, x-center, and right edge of the box. (i-k) Lithospheric

temperature evolution at the three mentioned points. . . . . . . . . . . . 50

3.22 The onset time of localization in dependence on the size of the perturba-

tions. Ranges are between [200 km, 200 K] and [500 km, 500 K]. Both

the rising plumes and the hanging perturbation always have the same

initial sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52


67

LIST OF FIGURES


Appendix A

Code ammendments - Initial Stress

Field

%=====================================================================================================

% Create a non−z e ro s t r e s s f i e l d a t t h e b e g i nn in g o f t h e s imu l a t i o n by

% a l l ow i n g t h e s t r e s s in t h e box to a d j u s t v i s c o u s l y

%=====================================================================================================

%=====================================================================================================

% Mate r i a l p r op e r t y d e f i n i t i o n

%=====================================================================================================

MATERIALS.G1 = [5 e10 ; 5 e10 ; 5 e10 ; 5 e10 ; 5 e10 ; ] ; % E l a s t i c Shear Module [ Pa ]

MATERIALS.G0 = 1e100 ∗ [ 5 e10 ; 5 e10 ; 5 e10 ; 5 e10 ; 5 e10 ; ] ; % E l a s t i c Shear Module [ Pa ]

%=====================================================================================================

% Non−Dimens i ona l i z a t i on o f parameters

%=====================================================================================================

MATERIALS.G0 = MATERIALS.G0 . / ( S t r e s s cha r ) ;

MATERIALS.G1 = MATERIALS.G1 . / ( S t r e s s cha r ) ;

MATERIALS.G = MATERIALS.G0 ;

for i t ime=t ime s t a r t : 1 e4

TimeStep Start = cputime ;

%=================================================================================================

% COMPUTE THERMOMECHANICAL FEEDBACK

%=================================================================================================

cpu s t a r t = cputime ;

i f i t ime<=10


e l s e i f i t ime==11


end

% [ . . . ]

end

69

APPENDIX A. CODE AMMENDMENTS - INITIAL STRESS FIELD


Appendix B

Code ammendments -

Inhomogeneity

%=====================================================================================================


%=====================================================================================================

% Mantle St . Air Inhomogene i ty

MATERIALS. MeshPhase = [1 ; 2 ; 3 ; ] ;

MATERIALS.G = [5 e10 ; 5 e10 ; 5 e10 ; ] ; % E l a s t i c Shear Module [ Pa ]

MATERIALS.Mu = [1 e21 ; 1 e21 ; 1 e20 ; ] ; % Vi s c o s i t y [ Pa s ]

MATERIALS. n = [ 3 ; 1 ; 3 ; ] ; % Powerlaw exponent

MATERIALS.Rho = [ 3300 ; 1 ; 3300 ; ] ; % Dens i t y ( nor r e a l l y used ! ) [ kg /m3 ]

MATERIALS. Cohesion = [30 e6 ; 1 e12 ; 30 e6 ; ] ; % Cohesion [ Pa ]

MATERIALS. Phi = [ 3 5 ; 0 ; 35 ; ] ; % Fr i c t i o n ang l e [ deg ]

MATERIALS. ShearHeatEff = [ 1 ; 0 ; 1 ; ] ; % E f f i c i e n c y o f shear−h e a t i n g 0−1

MATERIALS. Conduct iv ity = [ 3 . 3 ; 1 . e6 ; 3 . 3 ; ] ; % Thermal c o n d u c t i v i t y [W/m/K ]

MATERIALS. HeatCapacity = [ 1050 ; 1050 ; 1050 ; ] ; % Sp e c i f i c hea t c a p a c i t y [ J/ kg /K]

MATERIALS. Radioact iveHeat = [ 1 . e−7; 1 . e−7; 1 . e −7 ; ] ; % Rad i oa c t i v e hea t p roduc t i on [W/m3 ]

MATERIALS. ThermalExpansivity = [ 3 . 2 e−5; 0 . e−5; 3 .2 e −5 ; ] ; % Thermal e x p a n s i v i t y [ 1/K ]

MATERIALS. Ef f ec t iveQ = [ 1 . e4 ; 0 ; 0 ; ] ;

MATERIALS. Plast icStra inWeakening . Stra in StartWeakening = [ 0 . 0 0 1 ; 0 . ; 0 . 0 0 1 ; ] ;

MATERIALS. Plast icStra inWeakening . Strain EndWeakening = [ 0 . 1 0 ; 0 . ; 0 . 1 0 ; ] ;

MATERIALS. Plast icStra inWeakening . C Start = [ 3 0 . e6 ; 10 . e12 ; 30 . e6 ; ] ;

MATERIALS. Plast icStra inWeakening . C End = [ 3 0 . e3 ; 10 . e12 ; 30 . e3 ; ] ;

MATERIALS. Plast icStra inWeakening . Ph i Star t = [ 3 0 ; 0 . ; 30 ; ] ;

MATERIALS. Plast icStra inWeakening . Phi End = [ 1 0 ; 0 . ; 10 ; ] ;

%[ . . . ]

%===================================================================================================

% INITIALIZE PARTICLES

%==================================================================================================

%[ De f i n i t i o n o f mant le ma t e r i a l ]

% Inhomogene i ty

per t x = 50 . e+3;

p e r t z = 30 . e+3;

p e r t z c en = −50. e+3;

ind = find ( P a r t i c l e s . z<=0 & . . .

abs ( P a r t i c l e s . x ).ˆ2< per t x ˆ2/4/ l c ˆ2 & . . .

abs ( P a r t i c l e s . z−pe r t z c en / l c ).ˆ2< pe r t z ˆ2/4/ l c ˆ 2 ) ;

P a r t i c l e s . phases ( ind ) = 3 ;

71

APPENDIX B. CODE AMMENDMENTS - INHOMOGENEITY


Appendix C

Code ammendments - Phases

%=====================================================================================================


%=====================================================================================================

% Mantle St . Air Inhom . L i t h . Topography

MATERIALS. MeshPhase = [ 1 ; 2 ; 3 ; 4 ; 5 ; ] ;

MATERIALS.G = [5 e10 ; 5 e10 ; 5 e10 ; 5 e15 ; 5 e10 ; ] ; % [Pa ]

MATERIALS.Mu = [1 e21 ; 1 e21 ; 1 e20 ; 1 e25 ; 1 e21 ; ] ; % [Pa s ]

MATERIALS. n = [ 3 ; 1 ; 3 ; 3 ; 3 ; ] ; %

MATERIALS.Rho = [ 3 3 0 ; 1 ; 330 ; 330 ; 330 ; ] ; % [ kg /m3 ]

MATERIALS. Cohesion = [30 e6 ; 1 e12 ; 30 e6 ; 30 e5 ; 30 e6 ; ] ; % [Pa ]

MATERIALS. Phi = [ 3 5 ; 0 ; 35 ; 35 ; 35 ; ] ; % [ deg ]

MATERIALS. ShearHeatEff = [ 1 ; 0 ; 1 ; 1 ; 1 ; ] ; %

MATERIALS. Conduct iv ity = [ 0 . 3 3 ; 3 . 3 ; 0 . 3 3 ; 0 . 3 3 ; 0 . 3 3 ; ] ; % [W/m/K ]

MATERIALS. HeatCapacity = [ 1 0 5 ; 105 ; 105 ; 105 ; 105 ; ] ; % [ J/ kg /K]

MATERIALS. Radioact iveHeat = [ 1 . e−7; 1 . e−7; 1 . e−7; 1 . e−7; 1 . e −7 ; ] ; % [W/m3 ]

MATERIALS. ThermalExpansivity = [ 3 . 2 e−5; 0 ; 3 .2 e−5; 3 .2 e−5; 3 .2 e −5 ; ] ; % [1/K ]

MATERIALS. Ef f ec t iveQ = [ 1 . e4 ; 0 ; 0 ; 1 . e4 ; 1 . e4 ; ] ;

MATERIALS. Plast icStra inWeakening . Stra in StartWeakening = [ 0 . 0 0 1 ; 0 . ; 0 . 0 0 1 ; 0 . 0 0 1 ; 0 . 0 0 1 ; ] ;

MATERIALS. Plast icStra inWeakening . Strain EndWeakening = [ 0 . 1 0 ; 0 . ; 0 . 1 0 ; 0 . 1 0 ; 0 . 1 0 ; ] ;

MATERIALS. Plast icStra inWeakening . C Start = [ 3 0 . e6 ; 10 . e12 ; 30 . e6 ; 30 . e6 ; 30 . e6 ; ] ;

MATERIALS. Plast icStra inWeakening . C End = [ 3 0 . e3 ; 10 . e12 ; 30 . e3 ; 30 . e3 ; 30 . e3 ; ] ;

MATERIALS. Plast icStra inWeakening . Ph i Star t = [ 3 0 ; 0 . ; 30 ; 30 ; 30 ; ] ;

MATERIALS. Plast icStra inWeakening . Phi End = [ 1 0 ; 0 . ; 10 ; 10 ; 10 ; ] ;

%[ . . . ]

%=====================================================================================================

% INITIALIZE PARTICLES

%=====================================================================================================

% Def ine l i t h o s p h e r e

ind = find ( P a r t i c l e s . z<=0 & Pa r t i c l e s . z>−d l i t h ) ;


% Def ine topography ma t e r i a l

ind = find ( P a r t i c l e s . z>0);


73

APPENDIX C. CODE AMMENDMENTS - PHASES


Appendix D

Code ammendments - Topography

%=====================================================================================================

% Adjustment o f Topography

%=====================================================================================================

%[ j u s t b e f o r e i n i t i a l i z i n g p a r t i c l e s ]

%=====================================================================================================

% Vers ion 1 , g i v e s smooth h i l l w i t h a max h e i g h t 15 km at x=0 ( comment V. 2 )

%=====================================================================================================

% Create Topography

Topo x = x vec ;

Topo z = .015∗ ( cos (pi /(x max−x min ) .∗ Topo x ) ) . ˆ 2 5 + z max ;

%=====================================================================================================

% Vers ion 2 , g i v e s an asymmetric t r i a n g u l a r h i l l w i t h a max h e i g h t 15 km

% at x=0 ( comment V. 1 )

%=====================================================================================================

% Create Topography

Topo x = x vec ;

for topo=1: s ize ( x vec , 2 )

i f x vec ( topo ) < −100e3/ l c

Topo z ( topo ) = 0 ;

e l s e i f x vec ( topo ) > −100e3/ l c & x vec ( topo ) <= 0

Topo z ( topo ) = 15/100∗ x vec ( topo ) + 15 e3/ l c ;

e l s e i f x vec ( topo ) > 0 & x vec ( topo ) < 50 e3/ l c

Topo z ( topo ) = −15/50∗ x vec ( topo ) + 15 e3/ l c ;

else

Topo z ( topo ) = 0 ;

end

end

%[ remember to a d j u s t d e f i n i t i o n s o f e i t h e r topography p a r t i c l e s or s t i c k y a i r

% p a r t i c l e s −− or s e t z max=0 at t h e box dimension d e f i n i t i o n s ]

75

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