Lithospheric-Scale Stresses and Shear Localization Induced by Density-Driven Instabilities

Transcription

1 Examensarbete vid Institutionen för geovetenskaper ISSN Nr 187 Lithospheric-Scale Stresses and Shear Localization Induced by Density-Driven Instabilities Christiane Heinicke

2 Copyright c Christiane Heinicke och Institutionen för geovetenskaper, Geofysik, Uppsala universitet. Tryckt hos Institutionen för geovetenskaper, Geotryckeriet, Uppsala

3 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 condition setup. For the first case, localized deformation in our specific setup is found for strain rates of and mantle temperatures of s 1300 C. For constant stress boundaries, the conditions for a setup to localize are more restrictive. Mantle motion 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 GP a). 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 localization I

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5 Contents 1 Introduction Motivation Earth s internal structure Visco-elasto-plasticity Elasticity Viscosity Visco-elasticity Real visco-elasticity Brittle and Plastic Behavior Visco-elasticity with Mohr-Coulomb-plasticity Lithosphere Viscosity and Shear heating Employed Model Goal Model Setup Standard Model General stress and temperature evolution in the standard model Evolution of stress and strain rate in the standard model Results Shear Localization under Constant Strain Rate Boundary Conditions Dependency on viscosity Summary Constant Stress Boundary Conditions not resulting in Shear Localization Dependency of maximum stress on perturbation size Maximum stress dependency on perturbation size for a viscous rheology Maximum stress dependency for different initial setups

6 CONTENTS Effects of viscosity Subduction initiation by density difference Maximum stress dependency when a heterogeneity is included in the lithosphere Maximum stress dependency when topography is present Effects of mantle temperatures Summary Constant Stress Boundary Conditions resulting in Shear Localization Evolution Characteristics Influence of Inhomogeneities and Topography Influence of Material Properties Influence of Perturbations Influence of Viscosity Summary Discussion and Conclusion Shear Localization under Constant Strain Rate Boundary Conditions Constant Stress Boundary Conditions not resulting in Shear Localization Constant Stress Boundary Conditions resulting in Shear Localization Conclusion and Outlook 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 localization

7 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 transports 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 configuration 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

8 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 satisfactorily. 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 after 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 midlithosphere [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 4 Lithospheric-scale stresses and shear localization

9 CHAPTER 1. INTRODUCTION 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 The boundary between these two layers (lithosphere and asthenosphere) is defined by temperature, 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 gravitationally 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 behavior, depending on the amount and time evolution of the applied stress 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 Lithospheric-scale stresses and shear localization 5

10 CHAPTER 1. INTRODUCTION 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. 6 Lithospheric-scale stresses and shear localization

11 CHAPTER 1. INTRODUCTION 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 ( u i x j If strain rate is defined as ε ij = 1 2 relation between stress and strain for viscous fluids: (1.2) + u j x i ), the above equation yields following τ ij = 2µ ε ij (1.3) Here, µ is the dynamic viscosity [unit P as]; τ ij is the deviatoric stress tensor τ ij = σ ij + δ ij P, with P being the pressure and δ ij 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 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 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 s. Visco-elasticity can be described by the Maxwell model (among others): dε dt = ε visc + ε elast = 1 2µ σ + 1 dσ G 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, ε elast vanishes and the material strains continuously, it creeps. Lithospheric-scale stresses and shear localization 7

12 CHAPTER 1. INTRODUCTION 2. If on the other hand the strain is kept constant, eq. (1.4) becomes dσ σ = G dt (1.5) 2µ which can be integrated with the initial condition σ = σ 0 at t = 0 to give ( σ = σ 0 exp Gt ). (1.6) 2µ The stress is decreasing exponentially with time, i.e. under constant strain. the material is relaxing 3. As we will not implement a constant strain, but a constant strain rate boundary condition with dε dt = const., eq. 1.4 becomes ε = 1 2µ σ + 1 dσ G 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 = 2µ ε + (σ 0 2µ ε) exp Gt ) 2µ (1.8) The stress will approach the value 2µ ε 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 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 T m, with T m being the melting temperature of the rocks. The general creep law is according to Kameyama et al. (1999): with ( ε = Aa m σ n exp H ) RT (1.9) 8 Lithospheric-scale stresses and shear localization

13 CHAPTER 1. INTRODUCTION 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 = ). The strain rate controlled by dislocation creep on the other hand increases nonlinearly with stress (n = ), 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 important 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 Brittle and Plastic Behavior The plastic deformation common in rocks at low temperatures (T 0.3 T m ) 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 σσp ] q ). (1.10) σ p 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. Lithospheric-scale stresses and shear localization 9

14 CHAPTER 1. INTRODUCTION Figure 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)] 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 deformation: 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 viscous plastic brittle ductile = instantaneous, reversible deformation = irreversible deformation = irreversible deformation above yield stress allowing localization = elastic behavior + fracture at yield stress = viscous, but typically used for solids 10 Lithospheric-scale stresses and shear localization

15 CHAPTER 1. INTRODUCTION Figure 1.3: Mohr s circle and strength envelope Visco-elasticity with Mohr-Coulomb-plasticity Summarizing all the various deformation mechanisms so far we have: of which ε visc can further be divided into ε total = ε visc + ε elast + ε plastic (1.11) ε visc = ε pc + ε dc + ε pp (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 = σ n tan(φ) + c (1.13) This is displayed in fig. 1.3; the circle represents Mohr s circle at a given twodimensional stress state and the line represents the strength envelope. As soon as shear stress τ and normal stress σ n 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 Lithospheric-scale stresses and shear localization 11

16 CHAPTER 1. INTRODUCTION the depth dependency of lithospheric stress, it is noteworthy that a purely visco-elastoplastic 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 immediate 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 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: ( ) 1 ( [ n 1 H 1 µ = µ 0 exp ε ε 0 R T 1 ]) T 0 (1.14) The subscript 0 on viscosity µ and T in eq denotes values at standard conditions, ε 0 is a characteristic strain rate. n is the power-law constant as in eq 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 asthenosphere. 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- 12 Lithospheric-scale stresses and shear localization

17 CHAPTER 1. INTRODUCTION mation sites may drop to values as low as asthenosphere viscosity, i.e. from typically P as to P as [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: ) τ ij ( ɛ vis ij + ɛ pl ij SH = χ (1.15) ρc p with ɛ vis ij being the viscous strain rate, ɛ pl ij 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 parameter 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 T 0 ): and momentum: ( T di t ) ) T di + v i = κ 2 T di τ ij ( ɛ vis ij + ɛ pl ij + χ (1.16) x i x 2 j ρc p ( ) vi ρ t + v v i j x j = σ ij x j + ρg i (1.17) The left hand side in eq is zero because inertia is neglible in highly viscous, geologic problems: The used variables are: σ ij x j + ρg i = 0 (1.18) Lithospheric-scale stresses and shear localization 13

18 CHAPTER 1. INTRODUCTION T 0 x v κ χ c p ρ g... initial temperature... spacial coordinate... velocity... thermal diffusivity... efficiency of shear heating... heat capacity... density... gravity. The shear heating term χ τ ij( ɛ vis ij + ɛpl ij) ρc p on the right hand side of eq 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 ( v i x i = 0), visco-elasto-plastic rheologies which its name is based on. We employ quadrilateral (Q 1 P 0 ) elements; our mesh is refined in regions of special interest (in our case the lithosphere), and coarser elsewhere. 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). An 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 conduct 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 GP a) [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 14 Lithospheric-scale stresses and shear localization

19 CHAPTER 1. INTRODUCTION 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. Lithospheric-scale stresses and shear localization 15

20 CHAPTER 1. INTRODUCTION 16 Lithospheric-scale stresses and shear localization

21 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 additional 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) representing 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 µ 0 and the effective activation energy Q = E a /R which is a measure for the temperature dependence of the viscosity: µ = µ 0 ( εii ε 0 ) ( 1 n 1) ( [ 1 exp Q T 1 ]) T 0 (2.1) where µ 0 is the viscosity at standard laboratory conditions, i.e. T 0 = 20 C and 17

22 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. ε II / ε 0 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) ε II = 1 2 ε ijε ij. (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 points in the standard setup; it will later be reduced (to 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-mantleboundary, it induces mantle plumes to rise up that further affect the lithosphere. Additionally, 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 18 Lithospheric-scale stresses and shear localization

23 CHAPTER 2. MODEL SETUP Parameters for standard run Box dimensions width x 7600 km Depth z 1900 km Thickness lithosphere 100 km Number of grid points Horizontal nx 450 Vertical nz 150 Temperatures Sticky air temperature 283 K Mantle temperature 2250 K CMB-temperature 3200 K Material properties Mantle Sticky air Elastic shear modulus P a P a Viscosity P as P as Powerlaw exponent 3 1 Density 3300 kg 1 kg m 3 m 3 Cohesion P a P a Friction angle 35 0 Shear heating efficiency 1 0 Conductivity 3.3 W m K W m K Heat capacity 1050 J 1050 J kgk kgk Radioactive heat production W W m 3 m 3 Thermal expansivity K Effective activation energy (Ea/R) K 0 K Perturbation properties Radius Position center x Position center z Temperature deviation 550 km 0 km 0 km (top of lithosphere) -300 K Table 2.1: Parameters used for standard setup. Lithospheric-scale stresses and shear localization 19

24 CHAPTER 2. MODEL SETUP (a) (e) (b) (f) (c) (g) (d) (h) Figure 2.1: Stress and temperature fields for the standard model. Maximum stresses reached in the lithosphere are just above 500 MP a. (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 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 perturbation 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. 20 Lithospheric-scale stresses and shear localization

25 CHAPTER 2. MODEL SETUP 2.3 Evolution of stress and strain rate in the standard 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 km]. Both regions are considered for the entire depth of the lithosphere. 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 < 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 GP a) 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 MP a at the horizontal center of the lithosphere after approximately 6 ka. The second stress maximum of 540 MP a 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 M P a. 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 M P a. 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 for both plumes. s 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 region 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 being W ), subfigure c shows both plume events (the maximum shear heating m 3 there being W ). m 3 Lithospheric-scale stresses and shear localization 21

26 CHAPTER 2. MODEL SETUP (a) (b) (c) (d) Figure 2.2: (a) Evolution of maximum stress in the lithosphere. (b) Evolution of maximum strain rate in the lithosphere. (c) Stress and strain rate development at the center of the lithosphere. (d) Stress and strain rate development off the center of the lithosphere. 22 Lithospheric-scale stresses and shear localization

27 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 GP a) 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 GP a. Thus, the absolute strength of our lithosphere is roughly 1 GP a (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 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 T bot ; the temperature distribution determined by cooling the top surface to 0 C. 23

28 CHAPTER 3. RESULTS Figure 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) Based on Kaus and Podladchikov (2006) he determined the parameters that result in localization. Crameri (2009) found that localization is favored by low bottom temperatures and fast deformations (see fig. 3.1); for the Earth these parameters are in the range of [900 C C] and [ s ], respectively. s Transfered to our model, the mantle temperature should be 1300 C; we set our background strain rate to The top boundary is a free surface, i.e. its stress s state is always zero and no sticky air lies on top. Snapshots of the simulation at three different time steps are shown on fig 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 Dependency on viscosity One of the main parameter deciding whether or not localization will occur is viscosity. For a given strain rate of , the lithosphere localizes if the viscosity is s above P as, 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. 24 Lithospheric-scale stresses and shear localization

29 CHAPTER 3. RESULTS (a) (d) (g) (b) (e) (h) (c) (f) (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 M a. (b,e,h) After 1.8 Ma. (c,f,i) After 12.4 Ma. Lithospheric-scale stresses and shear localization 25

30 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 localization if the background strain rate ε BG is larger than a value determined by material setup and properties: ε BG = 1.4 κρcp R µ 0 γ. (3.1) Here, κ is the thermal diffusivity (3.3 m2 kg ), ρ the lithospheric density (3300 ), c s m 3 p the heat capacity (1050 J ), µ kgk 0 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 26 Lithospheric-scale stresses and shear localization

31 CHAPTER 3. RESULTS localization, we obtain ( ) κρc p µ 0 = R ε BG γ. (3.2) µ 0,min P as for the small inclusion (50 km 30 km) and µ 0,min P as for the large inclusion (80 km 50 km). However, as mentioned before, the lithosphere only yields for viscosities larger than P as, 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 Summary For a constant strain rate setup, the occurrence of localized deformation greatly depends on the magnitude of the viscosity. The treshold value is around P as for a strain rate of , localization is obtained for viscosities at or above this value. s 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. Lithospheric-scale stresses and shear localization 27

32 CHAPTER 3. RESULTS (a) (b) Figure 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 viscosities to cause higher strain rate during the simulation. 28 Lithospheric-scale stresses and shear localization

33 CHAPTER 3. RESULTS 3.2 Constant Stress Boundary Conditions not resulting 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 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 mantle. 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 = ρ 0 + ρ = ρ 0 (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 GP a is reached for a perturbation of 500 MP a, 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 perturbation, it varies around 550 MP a. This indicates that this stress is due to the bending of the lithosphere by the uprising plume. Vasilyev et al. (2001) have simulated 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 MP a) stress-focussing effect. Low driving stresses (< 200 MP a) result in lower stresses at the center of the Lithospheric-scale stresses and shear localization 29

34 CHAPTER 3. RESULTS (a) (b) (c) (d) Figure 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 lithosphere. (c) Maximum stress versus perturbation size. (d) Time when maximum stress occurs. (c,d) 1000 km off the center of the lithosphere. 30 Lithospheric-scale stresses and shear localization

35 CHAPTER 3. RESULTS lithosphere than 1000 km off center. Only for high stresses (> 200 MP a) 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 occurs earlier for a larger driving stress. The best fit found for a time - driving stress - dependence is t µ 0 / ρgr 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 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 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 Lithospheric-scale stresses and shear localization 31

36 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 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 Lithospheric-scale stresses and shear localization

37 CHAPTER 3. RESULTS (a) (d) (b) (e) (c) (f) Figure 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. Lithospheric-scale stresses and shear localization 33

38 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 significantly 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, 34 Lithospheric-scale stresses and shear localization

39 CHAPTER 3. RESULTS Figure 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. Lithospheric-scale stresses and shear localization 35

40 CHAPTER 3. RESULTS Figure 3.10: Stress evolution versus elapsed time for standard model (blue) compared to models 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 GP a, 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 Effects of viscosity Varying viscosity by two orders of magnitude ([ P as P as]) 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 Stresses are slightly higher for higher viscosities, which is also reflected in the strain rate - stress - relationship τ = 2µ ε (eq. 1.3) 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 36 Lithospheric-scale stresses and shear localization

41 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 and the maximum strain rate evolution in fig demonstrate, does the maximum strain rate never rise above Thus, we can confirm the creep s 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 Maximum stress dependency when a heterogeneity is included in the lithosphere In order to localize high stress regions and possibly produce a shear heating instability, we included a heterogeneity of considerable size and rectangular shape in the lithosphere. This viscosity inhomogeneity (µ = P as) was varied in its size from 30 km 20 km to 300 km 100 km as well as in its position: As depicted in figure 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 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 perturbation (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 significantly 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 - inhomogeneity size = 0). The stress behavior above the descending perturbation is rather interesting - for small inhomogeneity sizes the stress maximum seems to increase linearly with increasing sizes. For inhomogeneity sizes above 4000 km 2 which corresponds to 60 km 60 km in our simulations, on the other hand, the stress in the lithosphere reaches the desired 1 GP a. 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 Lithospheric-scale stresses and shear localization 37

42 CHAPTER 3. RESULTS (a) (b) (c) (d) Figure 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. Figure 3.12: Maximum strain rate evolution for a setup with density step in lithosphere. 38 Lithospheric-scale stresses and shear localization

43 CHAPTER 3. RESULTS (a) (b) Figure 3.13: Abrupt stress decrease around z = 50 km. (a) Heterogeneity centered in the lithosphere. Time = 3.3 ka, stress maximum = M P a. (b) Heterogeneity completely included by the high stress region. Time = 3.3 ka, stress maximum = MP a. (a) (b) Figure 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. 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 GP a, but does not yield. From eq we know that the onset of shear localization through shear heating not only requires high stresses, but also high strain rates. However, in all of the aforementioned simulations of this subsection the maximum strain rate does not exceed s In his dissertation, B. Kaus found a non-dimensionalization for the equation of energy 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 Lithospheric-scale stresses and shear localization 39

44 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: P e = R κµ0 G Br = σ 0 2 γ ρc p G (3.4) (3.5) with ρ being the density (3300 kg J ), c m 3 p the heat capacity (1050 ), G the elastic Kkg shear modulus ( N ), σ m 2 0 the stress at the start of the simulation, κ = kρc p the thermal diffusivity, and k the thermal conductivity of 3.3 W. γ Q in our case lies in m K RT0 2 the range of [ ] (depending on the exact temperature in the region of K K the inhomogeneity); µ 0 is on the order of P as, which also is the initial viscosity; and σ MP a for all cases. The logarithms of the Br and Pe numbers are thus within [ ] and [ ], 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. P e cbr 0.5 (3.6) If eq. 3.6 is implemented, its graph represents the boundary between diffusion dominated deformation and localized deformation: One can see clearly from fig 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. 40 Lithospheric-scale stresses and shear localization

45 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 parameter range not giving localization (below). Figure 3.16: Topography density and maximum stress. Blue: µ = P as. Red: µ = P as. Lines: Linear regression through data points. Lithospheric-scale stresses and shear localization 41

46 CHAPTER 3. RESULTS (a) (b) Figure 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. (a) (b) Figure 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 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 42 Lithospheric-scale stresses and shear localization

47 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 T mantle : T CMB 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 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 lithospheric 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 The cold driving perturbation has been removed to reveal the effects solely induced by the topography 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 (T mantle = 2750 K, T CMB = 3900 K, such that the ratio between T mantle and T CMB is the same as for the standard model) and a colder mantle (T mantle = 1750 K, T CMB = 2450 K). The result of these simulations is shown in figure 3.19, where the cold mantle model Lithospheric-scale stresses and shear localization 43

48 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 significantly larger for hotter mantles. The maximum strain rate for the 2750 K - mantle model is about one order of magnitude larger ( than for the 1750 K - mantle s model ( , with the standard model (2250 K) lying in between: hotter material s with a therefore lower viscosity deform more readily Summary This section was dedicated to finding mechanisms that produce high stresses in the lithosphere of the order of the yield stress 1 GP a. 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 GP a; 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 44 Lithospheric-scale stresses and shear localization

49 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: σ0 2 ρcp κ R c µ 0 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 c p 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 c p can be kept at their natural values. For σ MP a, R 100 km, γ K and c = 2, κ, ρ and c p as in the standard (non-localizing) setup eq. 3.7 predicts that localization will occur when viscosity is below roughly P as. However, due to the great uncertainty in the determination of R, σ 0, and γ, this value can only be a general orientation. Test runs revealed that localization does not occur for a lithosphere viscosity of P as. Only reducing the viscosity to P as 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 Lithospheric-scale stresses and shear localization 45

50 CHAPTER 3. RESULTS when the setup is changed. Parameters for run showing localization Box dimensions width x 1000 km Depth z 1000 km Thickness lithosphere 100 km Temperatures Sticky air temperature 283 K Mantle temperature 1573 K CMB-temperature 3200 K Material properties Mantle Lithosphere Elastic shear modulus Pa Pa Viscosity Pa s Pa s Cohesion Pa Pa Density 330 kg 330 kg m 3 m 3 Conductivity 0.33 W 0.33 W m K m K Heat capacity 105 J 105 J kgk kgk Perturbation properties Radius 300 km Position center x 0 km Position center z 0 km (top of lithosphere) Temperature deviation -500 K Radius 300 km Position center x ± 1000 km Position 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 As before, the stress and temperature fields are shown. Additionally, the composition 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 deformation 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. 46 Lithospheric-scale stresses and shear localization

51 CHAPTER 3. RESULTS (a) (f) (k) (b) (g) (l) (c) (h) (m) (d) (i) (n) (e) (j) (o) Figure 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. Lithospheric-scale stresses and shear localization 47