ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2016

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1354

Strain quantifications in differenttectonic scales using numericalmodelling

LUKAS FUCHS

ISSN 1651-6214ISBN 978-91-554-9513-8urn:nbn:se:uu:diva-280759

Dissertation presented at Uppsala University to be publicly examined in Hambergsalen,Geocentrum, Villavägen 16, Uppsala, Wednesday, 4 May 2016 at 10:00 for the degree ofDoctor of Philosophy. The examination will be conducted in English. Faculty examiner: Dr.Susanne Buiter (Geodynamics Team, Geological Survey of Norway, Trondheim, Norway).

AbstractFuchs, L. 2016. Strain quantifications in different tectonic scales using numerical modelling.Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science andTechnology 1354. 58 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9513-8.

This thesis focuses on calculation of finite and progressive deformation in different tectonicscales using 2D numerical models with application to natural cases. Essentially, two majortectonic areas have been covered: a) salt tectonics and b) upper mantle deformation due tointeraction between the lithosphere and asthenosphere.

The focus in salt tectonics lies on deformation within down-built diapirs consisting of a sourcelayer feeding a vertical stem. Three deformation regimes have been identified within the salt:(I) a squeezing channel flow underneath the overburden, (II) a corner flow underneath the stem,and (III) a pure channel flow within the stem. The results of the model show that the deformationpattern within the stem of a diapir (e.g. symmetric or asymmetric) can reveal information ondifferent rates of salt supplies from the source layer (e.g. observed in Klodowa-diapir, Poland).Composite rock salt rheology results in strong localization and amplification of the strain alongthe salt layer boundaries in comparison to Newtonian rock salt. Flow and fold structures ofpassive marker lines are directly correlated to natural folds within a salt diapir.

In case of the upper mantle, focus lies on deformation and resulting lattice preferredorientation (LPO) underneath an oceanic plate. Sensitivity of deformation and seismicanisotropy on rheology, grain size (d), temperature (T), and kinematics (v) has been investigated.The results of the model show that the mechanical lithosphere-asthenosphere boundary isstrongly controlled by T and less so by v or d. A higher strain concentration within theasthenosphere (e.g. for smaller potential mantle temperatures, higher plate velocities, or smallerd) indicates a weaker coupling between the plate and the underlying mantle, which becomesstronger with the age of the plate. A Poiseuille flow within the asthenosphere, significantlyaffects the deformation and LPO in the upper mantle. The results of the model show, thatdeformation in the upper mantle at a certain distance away from the ridge depends on theabsolute velocity in the asthenosphere. However, only in cases of a driving upper mantle basedoes the seismic anisotropy and delay times reach values within the range of natural data.

Keywords: Deformation modelling, progressive and finite deformation, salt tectonics, down-built diapir, upper mantle, lithosphere-asthenosphere boundary, seismic anisotropy, plate-mantle (de)coupling

Lukas Fuchs, Department of Earth Sciences, Mineralogy Petrology and Tectonics, Villav. 16,Uppsala University, SE-75236 Uppsala, Sweden.

© Lukas Fuchs 2016

ISSN 1651-6214ISBN 978-91-554-9513-8urn:nbn:se:uu:diva-280759 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-280759)

To my family

List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Fuchs, L., Koyi, H., Schmeling, H. (2015). Numerical modeling

of the effect of composite rheology on internal deformation in down-built diapirs, Tectonophysics (2015).

II Fuchs, L., Koyi, H., Schmeling, H. (2014). Numerical modeling on progressive internal deformation in down-built diapirs, Tec-tonophysics (632), 111-122.

III Fuchs, L., Schmeling, H., Koyi, H. Thermo-mechanical model-ling of progressive deformation and seismic anisotropy at the lithosphere-asthenosphere boundary. Manuscript in review. Ge-ophysical Journal International, submitted.

IV Fuchs, L. Schmeling, H., Koyi, H. Thermo-mechanical model-ling of progressive deformation at the lithosphere-asthenosphere boundary: The effect of a horizontal pressure gradient. Manu-script in preparation.

Reprints were made with permission from the respective publishers.

Personal contribution

All of the manuscripts in this thesis are the result of the combined efforts of all listed authors. My personal contribution to each paper is described below.

Paper I: 65 %. The numerical method and part of the model concept used in this paper is based on the work during my undergraduate studies. The nu-merical code (FDCON) was provided by Prof. Schmeling and I implemented the finite deformation routine prior to this study. The model concept was de-veloped together with Prof. Koyi. I run the models and analysed the data. In addition to contributing to the text, Prof. Schmeling helped regarding numer-ical/physical issues. Prof. Koyi helped in interpreting geological structures within the model.

Paper II: 65 %. Under supervision of Prof. Schmeling I implemented the routine to calculated composite rock salt rheology within FDCON. I run the models and analysed the data using the amplification factor. Prof. Koyi was very helpful in interpreting flow and fold structures in the salt layer.

Paper III: 65 %. The idea of the upper mantle deformation models was initiated by Prof. Koyi. I developed the 1D MATLAB code to calculate the shear deformation within the upper mantle and collected the natural seismic delay time data. Prof. Schmeling was very helpful in implementing the tem-perature and grain size calculations and in getting familiar with D-Rex and the routine to calculate seismic delay times. Prof. Koyi helped to interpret the shear strain profiles regarding the mechanical (de)coupling at the LAB.

Paper IV: 70 %. In extension to paper III, I developed the 2D staggered grid finite difference code to calculate the shear deformation in the upper man-tle for different flow models. In addition to contributing to the text, Prof. Schmeling helped in developing the code and the model. Prof. Koyi helped in analysing and interpreting the shear deformation pattern in the upper mantle.

Contents

1.  Introduction ......................................................................................... 11 1.1  Deformation modelling ................................................................... 11 1.2  Finite and progressive deformation ................................................. 12 1.3  Finite deformation and fabric .......................................................... 12 1.4  Salt tectonics ................................................................................... 14 1.5  Upper mantle deformation .............................................................. 16 1.6  Objectives ....................................................................................... 17 

2.  Numerical modelling ........................................................................... 18 2.1  Fundamental principals ................................................................... 18 2.2  Conservation equations ................................................................... 18 2.3  Deformation gradient tensor and finite deformation ....................... 20 2.4  Heat equation .................................................................................. 21 2.5  Grain size calculations .................................................................... 22 

3  Rheology .............................................................................................. 23 3.1  Basic concepts ................................................................................. 23 3.2  Rock salt .......................................................................................... 24 3.3  Olivine ............................................................................................. 25 

4  Paper summary .................................................................................... 27 4.1  Paper I: Progressive internal deformation in down-built diapirs .... 27 

4.1.1  Modelling concept ................................................................. 27 4.1.2  Summary ................................................................................ 28 4.1.3  Conclusions ............................................................................ 31 

4.2  Paper II: The effect of composite rock salt rheology ...................... 32 4.2.1  Modelling concept ................................................................. 32 4.2.2  Summary ................................................................................ 33 4.2.3  Conclusions ............................................................................ 36 

4.3  Paper III: Deformation and anisotropy in the upper mantle ............ 37 4.3.1  Modelling concept ................................................................. 37 4.3.2  Summary ................................................................................ 38 4.3.3  Conclusions ............................................................................ 41 

4.4  Paper IV: The effect of a horizontal pressure gradient ................... 42 4.4.1  Modelling concept ................................................................. 42 4.4.2  Summary ................................................................................ 43 4.4.3  Conclusions ............................................................................ 46 

6.  Sammanfattning på svenska ................................................................ 49 

Acknowledgements ....................................................................................... 51 

References ..................................................................................................... 53 

5  Conclusions ......................................................................................... 47 

Abbreviations

1D 2D A C/P/PC-flow CPO DLC dfc FD HSCM LAB LPO RTI umb WH

One-dimensional Two-dimensional Asse rock salt Couette/Poiseuille/Poiseuille-Couette flow Crystallographic preferred orientation Dislocation creep Diffusion creep Finite difference Half-space cooling model Lithosphere-asthenosphere boundary Lattice preferred orientation Rayleigh-Taylor instability Upper mantle base West Hackberry rock salt

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1. Introduction

1.1 Deformation modelling Finite and progressive deformation is one of the main features in the fields of geology and geodynamics and can provide significant information to under-stand the formation and development of geological structures. In the field, such deformation can be determined by structural geological measurements (e.g. Ramsay, 1967; Ramsay and Huber, 1983; Talbot, 2014). However, field measurements only represent the final state of a deformed body, whereby re-construction of the complete history of deformation is not always straight for-ward. Thus, analogue and numerical models are useful to simulate finite and progressive deformation. While the mechanical response of rocks to applied stresses is rather complicated (e.g. viscous, non-linear, elastic or anisotropic), modelling is seen as a method to test the hypotheses of geodynamic processes. A main problem of such models is the proper scaling of physical quantities (kinematics and dynamics), which demands proportionality between variables characterizing the model and nature (Ranalli, 2001).

Besides such complexities, deformation analyses, both qualitative and quantitative, have always been of high importance in geodynamic modelling. Deformation calculation and analysis has thereby evolved critically from a purely qualitative analysis of analogue compressional tectonic regimes (e.g. Cadell, 1888; Willis, 1893), to more complex homogeneous and inhomogene-ous analytical deformation analysis (e.g. Ramsay, 1967; Malvern, 1969; Ram-berg, 1975a, b; Ramsay and Huber, 1983; Weijermars, 1992; Davis and Titus, 2011) and advanced analogue (e.g. Dixon 1974, 1975; Schwerdtner and Troёng, 1978; Cruden, 1988; Warsitzka et al., 2013) and numerical (e.g. McKenzie, 1979; Schmeling et al., 1988; Weijermars and Poliakov, 1993; Becker et al., 2003; Schmid and Podladchikov, 2005; Cardozo and All-mendinger, 2009; Burchardt et al., 2011; Fuchs and Schmeling, 2013) models. Deformation analysis in these approaches is thereby based on different con-cepts and techniques suitable and applicable for each by itself. Depending on the rheology, the material behaves differently under similar applied stresses. Thus, deformation of, for example, an elastic material is confined by a yield criterion, while viscous material deforms over a fairly long range of stresses and undergoes ductile deformation. Under high temperature and confining pressure conditions (e.g. the Earth’s mantle), or in the case of a mechanically weak material (e.g. rock salt), the material behaves as a plastic material and

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deforms in a ductile manner, i.e. a continuous, large deformation without frac-ture. Thus, the material’s motion in geodynamics can often be described as that of a high viscous fluid, resulting in large finite deformation.

1.2 Finite and progressive deformation In general, deformation in fluid dynamics can be described by three different mechanisms governed by a velocity field and its corresponding displacement, i.e. translation, rotation, and strain (normal and shear). A body displacement consisting of a simultaneous translation and rotation without a change in size and shape is referred to as a rigid-body displacement. In contrast, changes in length (extension or shortening) or changes of angles between two initially orthogonal lines are referred to as normal or shear strain, respectively (e.g. Ramsay and Huber, 1983; Truesdell and Rajagopal, 2010).

Mathematically, deformation can be described by the mapping of a vector between two reference points in a body on a new vector (e.g. Malvern, 1969; Ramsay and Huber, 1983; Truesdell and Rajagopal, 2010). Finite deformation describes the finial state of an undeformed body that has undergone progres-sive deformation of each particle in the deforming body over time along its path, i.e. the sum of incremental deformation (e.g. Schmeling et al., 1988; Da-vis and Titus, 2011). In case of a homogeneous, i.e. space independent, and steady, i.e. time independent, flow field, finite deformation can be determined directly by the given velocity field and vice versa. However, in case of an inhomogeneous and time-dependent flow field, finite deformation of a body has to be calculated by integration of incremental deformation along its flow path. The more advanced geodynamic modelling techniques, both analogue and numerical, enable the analysis of inhomogeneous and time dependent large deformation in more detail. Finite deformation, both in models and in nature, can be best described by the so-called finite strain ellipse, describing the deformation of an initially circular body by a certain velocity field (e.g. McKenzie, 1979; Ramsay and Huber, 1983).

1.3 Finite deformation and fabric One of the most fundamental indicators in deformation analysis of natural rocks is passive markers (e.g. fossils, mineral inclusions) of known initial shape and size within a homogeneous matrix (e.g. Ramsay, 1967; Ramsay and Huber; 1983; Talbot, 2014). Although they only represent the final state of deformation and thus might limit a full reconstruction of the kinematical his-tory, they can provide information on current and past deformation processes.

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Furthermore, deformation can lead to cleavage and lineation in rocks, indicat-ing past and present stress conditions governing the deformation (e.g. Ramsay and Huber, 1983).

Besides such macroscopic indicators, deformation directly affects the tex-ture of rocks on a microscopic level, e.g. within mantle rocks. It is well-known that crystallographic orientations of crystals within a macroscopic displace-ment field reorient due to the dislocation creep (plastic) deformation along activated intracrystalline slip systems and due to dynamic recrystallisation by subgrain rotation and grain-boundary migration (e.g. Carter, 1976; Nicolas and Christensen, 1987; Mainprice et al., 2000). It has been found, empirically (e.g. Zhang and Karato, 1995; Skemer et al., 2012; Hansen et al., 2014) and theoretically (e.g. Kaminski and Ribe, 2001; Tommasi et al., 2000; Kaminski et al., 2004), that the crystallographic symmetry axes rotate during defor-mation and align in a preferred orientation, so called lattice preferred or crystal preferred orientation (LPO/CPO), almost parallel to the principle strain orien-tation. Many crystals, such as olivine, have anisotropic elastic properties, e.g. different elastic velocities along the crystallographic axes. Thus, an alignment of such crystals along a preferred orientation causes seismic anisotropy to be detectable in large tectonic scales, e.g. by SKS shear wave splitting or surface wave anisotropy. In this way, seismic anisotropy within the Earth’s mantle correlates directly with its internal deformation, either past or present, and provides direct information on flow within the Earth’s mantle (e.g. Becker et al., 2003; Conrad and Behn, 2010).

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1.4 Salt tectonics

Salt tectonics occurs on a rather global scale, on both oceanic and continental tectonic plates, resulting in a variety of different salt structures, from immature rollers over mature diapirs to effusive glaciers at the surface (Figure 1.1). However, some structures, like a diapir, are rather well confined within a small geological area of a few kilometres. Moreover, due to its relatively low vis-cosity (1017-1019 Pa s), rock salt flows as a viscous fluid within geologically short time periods feeding the diapir. As a result of their formation, all struc-tures, except the immatures, show at least three similar main geometrical fea-tures, namely, a source layer at the bottom, a stem fed by the source layer, and a head on top of the stem, either at the surface or covered by a mechanically stronger overburden (e.g. Jackson and Talbot, 1986; Koyi, 1998).

Since the early 19th century, formation of salt diapirs has been described by two different processes: a) as an active upbuilding of a less dense salt layer piercing through a denser overburden (e.g. Berner et al., 1972; Bateman, 1984) or b) as a passive downbuilding due to differential loading of sediments on top of the salt (e.g. Barton, 1933; Jackson and Talbot, 1986; Koyi, 1998; Hudec and Jackson, 2007). From the very beginning, upbuilding has been studied extensively as an application of the classical Rayleigh-Taylor Insta-bility (RTI) (e.g. Ramber, 1967, Woidt, 1978; Schmeling, 1987), while down-built salt diapirs have been studied in detail only during the past 50 years (e.g. Biot and Ode, 1965; van Keken et al., 1993; Poliakov et al., 1993; Chemia et al., 2008). Unlike magmatic diapirs, which are interpreted to form as a result

Figure 1.1. Schematic illustration of a variety of different salt structures, rising from line sources (a) and point sources (b). Besides the rheology and additional external feature (e.g. sedimentation rate), internal deformation of a salt diapir is strongly gov-erned by its shape. However, the majority of those structures show a similar geomet-rical feature, i.e. a source layer, a stem, and a head/bulb [Hudec and Jackson, 2007].

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of RTI or diking, salt diapirs are now thought to be formed by differential loading (e.g. Koyi et al., 1993; Koyi, 1998), while buoyancy is only of sec-ondary importance in many tectonic regimes (e.g. Hudec and Jackson; 2007). Salt within the source layer is driven from high to low pressure (more pre-cisely hydraulic head) areas by overburden sediments, resulting in the rise of salt in the stem. The ratio of sediment accumulation and rate of salt supply, can thereby define different diapiric shapes (Figure 1.2), consequently, result-ing in different internal deformation patterns. In case they are equal, a colum-nar diapir is formed.

While the development of salt diapirs has been studied using field data and analogue and numerical models, including external features not directly re-lated to the rheology or structure of the diapir itself (e.g. sedimentation or ero-sion as well as syn-diapiric extension or shortening), progressive and finite deformation within a down-built diapir has not been analysed in detail. Inter-nal deformation pattern of salt diapirs and their surrounding overburden can thus provide new information on formation of diapirs, as well as their internal structure. Regarding the many different purposes related to salt structures (e.g.

Figure 1.2. Shaping of a down-built salt diapir depending on the sedimen-tation rate and rate of salt supply. In case they are equal, a columnar diapiris formed [Koyi, 1998].

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hydrocarbon traps or nuclear waste storages), diapiric development, past, pre-sent, and future, is very important regarding their utilisation.

1.5 Upper mantle deformation Relative motion between the Earth’s plates, i.e. plate tectonics, and the Earth’s mantle, i.e. global mantle convection, results in strong deformation within the upper mantle, i.e. the upper ~400 km of the Earth. Thus, deformation pattern within the upper mantle could provide important data on the interaction and coupling between plates and the mantle. The plates are thereby driven by forces acting on their boundaries (e.g. Forsyth and Uyeda, 1975; Höink et al., 2011), while the sum of all forces must be in equilibrium (e.g. slab pull, ridge push, mantle driving/resisting), and motion within the mantle is governed mainly by thermo-chemical convection (e.g. Tackley, 2000; Ogawa, 2008). However, the exact contribution of each mechanism on the Earth’s dynamic, and the dominant driving mechanisms within certain tectonic regimes, re-mains unclear. Furthermore, flow within the upper mantle seems to be more active as assumed (e.g. Höink and Lenardic, 2010; Natarov and Conrad, 2012; Morgan et al., 2013), indicating a horizontal pressure gradient driven (i.e. Poiseuille/P-) flow within the upper mantle. A P-flow component would sig-nificantly affect deformation pattern in the upper mantle alternating from a standard shear deformation, driven mainly by the relative motion between the plate and the upper mantle base (umb).

Deformation is most likely concentrated close to the boundary between the high viscous mechanically strong lithosphere (i.e. the crust and upper most solid mantle) and the low viscous mechanically weak asthenosphere below (e.g. Podolefsky et al., 2004; Behn et al., 2009; Eaton et al., 2009). The litho-sphere-asthenosphere boundary (LAB) has been defined in different ways re-garding different physical properties, i.e. as a thermal (boundary between heat conduction and advection), compositional/chemical (due to melding and de-hydration along the oceanic ridges) and mechanical/elastic boundary (bound-ary between solid, elastic and weak, low viscous layer). Deformation along the boundary is thereby governed mainly by the kinematics and each layer’s viscosity, which strongly depends on properties such as temperature, water content, grain size, and composition. Thus, the viscosity within the astheno-sphere can vary by several orders of magnitude (e.g. Karato et al., 1986; Karato and Wu, 1993; Hirth and Kohlstedt, 2003; Karato, 2010).

The LAB has been identified initially by seismic measurements, relating the boundary to a sharp seismic shear velocity contrast in the upper hundred kilometres of the Earth’s upper mantle (e.g. Gaherty et al., 1999; Fischer et al. 2010; Rychert et al., 2012). Thus, one observes a strong anisotropy of surface waves and SKS shear wave splitting due to the preferred orientation of min-

17

erals governed by the upper mantle flow. Different flow fields within the up-per mantle, i.e. either driven mainly by the relative motion of the plates and the umb or including an active horizontal pressure gradient driven flow in within the asthenosphere, result in significant differences in the seismic ani-sotropy patterns.

1.6 Objectives Regarding the importance of large deformation in geodynamic models and its correlation to present and past motions, quantification of deformation in dif-ferent tectonic settings is important in understanding their formation and de-velopment. Different tools to measure deformation in the field, directly with strain markers or indirectly with geophysical data, provide fundamental con-straints on the deformation, observable in geodynamic models. Qualitative (pattern) and quantitative (strain markers and seismic anisotropy) deformation provides information on kinematics and dynamics of geodynamical systems. As deformation is mainly inhomogeneous and time dependent, finite defor-mation varies within different tectonics scales providing different constraints to size and time scales of numerical and analogue models. Therefore, two sig-nificantly different tectonic settings, on a local and global scale, are used to analyse finite and progressive deformation within geodynamic flows. A sys-tematic analysis of model parameters governing deformation (e.g. geometry, rheology, temperature, grain size, and kinematics) provides information on the sensitivity of finite and progressive deformation as well as overall deformation pattern on said parameters. As analysis of large deformation is applicable to a broad area within the field of geology and geodynamics, it provides an im-portant and significant tool in understanding formation and future develop-ment of geological structures, which is always important regarding their utili-sation or in understanding the evolution of the Earth. Consequently, a more detailed deformation analysis leads to a better constraint of model deformation data comparable to natural structures and their physical properties.

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2. Numerical modelling

2.1 Fundamental principals The fundamental basics of computational fluid dynamics are the principals governing equations in fluid dynamics, continuum, momentum, and energy equations. They mathematically describe three fundamental physical princi-pals upon which all fluid dynamics is based on.

2.2 Conservation equations Motion of a purely viscous flow for an incompressible fluid with no inertial forces, i.e. Stokes flow, can be described by the conservation equations of mass, momentum, and composition (e.g. Weinberg and Schmeling, 1992; Gerya, 2010):

0 v , (2.1)

0 ij

jx

P g , (2.2)

0kk

CC

t

v , (2.3)

where v is the velocity of the fluid, P is the pressure, τij is the deviatoric stress tensor, ρ is the density, g is the gravitational acceleration, t is the time, and Ck is the material concentration of the kth composition. At any location, the con-centrations of the different components add up to 1, where Ck is given either by 0 or 1 in case of an immiscible fluid. The deviatoric stress τij is given in terms of velocities by using the constitutive law

2 iij k j , (2.4)

where ηk is the effective viscosity of the kth composition and ij the strain rate tensor, defined as

1

2ji

ijj i

vv

x x

. (2.5)

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The density in the buoyancy term of the momentum equation is defined by the mean of the densities of the different materials weighted by the concentra-tions

k kk

c , (2.6)

where ck and ρk is the concentration and density of the kth composition, respec-tively. Following the discussion by Schmeling et al. (2008), as a best approx-imation of shear and due to the least error in finite and large viscous defor-mation at material interfaces (Fuchs and Schmeling, 2013), harmonic averag-ing scheme is used to define the viscosity of a mixture

1 k

k k

c

. (2.7)

Equations (2.1) and (2.2) are solved numerically using different approaches depending on the rheology and boundary conditions of the model. In case of the high viscosity contrast between the salt and the overburden (5 order of magnitudes), a stream function formulation approach provides a numerical stable solution of the flow field in salt tectonics, driven by differential loading induced by compositionally varied densities. Therefore, the two-dimensional (2D) Eulerian finite difference (FD) code FDCON is used to solve both equa-tions (Weinberg and Schmeling, 1992; Schmeling et al., 2008). As the stream function approach eliminates pressure, a 2D staggered FD approach (devel-oped after Gerya, 2010) in combination with PETSc (Portable, Extensible Toolkit for Scientific Computation) (Balay et al., 2012a, b) is used to directly solve equations (2.1) and (2.2) for flow fields within the upper mantle, ap-proximated by a channel flow governed by pressure and kinematic boundary conditions (e.g. Poiseuille-Couette/PC- flow). Flow within the upper mantle is assumed to be steady state, governed mainly by temperature and rheology. As we are mainly interested in the effect of different flow fields in the upper mantle on deformation pattern, density is kept constant in the upper mantle flow models.

The conservation equation of composition (2.3) is solved using a marker approach, which prevents numerical diffusion. Along a flow path of any fluid particle, equation (2.3) may be written in a Lagrangian reference system

0k

particle

c

t

. (2.8)

Equation (2.8) implies that we have to solve for the flow path equation

particle

x

t

v (2.9)

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and have to ensure a constant ck value at x(t)particle. Therefore, the model do-main is filled with closely spaced markers (100 markers per FD grid cell) with small random initial displacements of half a marker grid size. Each marker has its own chemical composition ck(xm), where its position at each time is defined by xm(t). Equation (2.9) is then solved for all markers using a fourth-order Runge-Kutta integration with a predictor-corrector step. Although the flow within the upper mantle models is assumed to be steady state and thus does not require a compositional conservation (2.3), the marker approach is used within the 2D staggered FD code to calculate finite deformation within the upper mantle, which must be integrated along a particle’s flow path.

2.3 Deformation gradient tensor and finite deformation

Finite deformation within a fluid dynamic field can be calculated using the deformation gradient or finite deformation tensor F. This tensor describes how a vector y between two points at time t0 is mapped on a deformed vector y’ between the same points (Ramsay and Huber, 1983). For inhomogeneous and time dependent deformation, the chosen distance between the two points has to be infinitesimal. The deformed vector y at time t can be written as (e.g. McKenzie, 1979)

0'( ) ( ) ( )t t t y F y , (2.10)

where F is initially described by the unit tensor I. The deformation gradient tensor for each marker is calculated within the model domain, using the method described by Fuchs and Schmeling (2013) based on the algorithm de-fined by McKenzie (1979). The tensor is calculated by integration, properly centred in time and space, of

t ij ik kjD F L F (2.11)

along each particle’s flow path, where Lij is the velocity gradient tensor (Mal-vern, 1969, p. 146). If the tensor is known, deformation of an arbitrary body can directly be calculated.

Assuming an initially circular body, which will be deformed by a certain velocity field, one can determine the parameters of the resulting so called fi-nite strain ellipse (i.e. the major and minor half axis a and b, and its orienta-tion) by knowing the deformation gradient tensor F at each position and time (e.g. McKenzie, 1979; Ramsay and Huber, 1983; Schmeling et al., 1988). The strain ellipse describes the deformed state of a circular body at a certain point in progressive deformation. In the field of structural geology, the finite strain

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ellipse is often used to calculate the finite deformation or logarithmic strain, which is given by (e.g. Ramsay and Huber, 1983)

10logf R with a

Rb

, (2.12)

where R is the ellipticity of the strain ellipse. The shear strain can be defined by the difference between the maximum and minimum principal strains (i.e. the strain ellipse axes) as (e.g. Ragan, 2009)

a b . (2.13)

In case of a simple shear deformation, γ directly correlates with the integral of the second invariant of the strain rate tensor over time by

0

2t

II t dt . (2.14)

As open boundary conditions are used for flow models of the upper mantle, i.e. deviatoric normal stress free and lithostatic pressure conditions with the option of adding a constant pressure (P1) at x = 0, passive markers transporting the tensor F can leave the model domain. Therefore, they re-enter the model domain on a flow path, corresponding to their stream function value while leaving the model domain. As the markers re-enter, deformation is set to zero, i.e. F=I.

2.4 Heat equation To calculate deformation within the upper mantle underlying an oceanic plate, its temperature field has to be defined. As hot mantle rocks upwell underneath oceanic ridges, rigid oceanic lithosphere is formed at their crest under certain temperatures and cools down while the plates move laterally away from the ridges due to the loss of heat to the cold seawater. Neglecting horizontal heat conduction in the lithosphere, temperature of an oceanic plate can be approx-imated by a half-space cooling model (HSCM) and calculated by solving the one-dimensional (1D) heat conduction equation (e.g. Turcotte, 1987; McKen-zie et al., 2005)

p

T TT c T k T

t z z

, (2.15)

where T is the absolute temperature, t is the age of the plate, z is the depth, k is the thermal conductivity, ρ is the density, and cp is the specific heat capacity.

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For constant thermal parameters (i.e., k, cp, and ρ), temperature can be de-scribed by a HSCM depending only on the square root of the age of the plate and the potential mantle temperature (e.g. Stein and Stein, 1992). In case of variable thermal parameters, heat equation needs to be solved numerically, starting with an initial temperature condition at the age tplate = 0 Ma (e.g. McKenzie et al., 2005). Constant thermal parameters are taken from Stein and Stein (1992), and variable thermal parameters are given by analytical expres-sions (McKenzie et al., 2005, and references therein). Given a constant plate velocity, this t-z-dependent temperature can be mapped onto a spatial x-z-de-pendent temperature field. Thus, for the temperature field, differential hori-zontal velocities near the LAB are neglected.

2.5 Grain size calculations Grain size evolution can be described by two different mechanisms: (a) grain growth and (b) grain size reduction (e.g. Bercovici and Ricard, 2012). Grain growth is governed mainly by temperature, whereas grain size reduction oc-curs due to dynamic recrystallisation and depends mainly on the amount of deformational work, governed mainly by dislocation creep (e.g. Ricard and Bercovici, 2009). In case grain growth and reduction are equal, grain size can be described as a steady state (Rozel et al., 2011)

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112

1 3

3

np

pII

Gd

pf A

, (2.16)

where d is the grain size (in µm), τII is the second invariant of the stress tensor (in MPa), G is a grain growth constant depending on T, p is the grain evolution power exponent, f is an energy partitioning factor depending of T, λ2 and λ3 are experimental constants for the grain size distribution, γ is the surface ten-sion, and A1 is the dislocation creep strain rate factor. Within the astheno-sphere, steady state grain size is reached faster in comparison to the advection of material (e.g. Behn et al., 2009) and thus, it is an appropriate approximation for the grain sizes in the upper mantle of an oceanic plate. The parameters for the grain size calculations are taken from Rozel et al. (2011). Although mantle rocks do not only consist of olivine, variable grain size is assumed by a for-mulation of a single phase grain size evolution model including damage theory (e.g. Rozel et al., 2011). However, secondary phases, resulting in pinning of minerals, involve more experimentally undetermined quantities and their ef-fect can be mimicked by variation of the grain evolution power exponent p.

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3 Rheology

3.1 Basic concepts In geology and geophysics, rheology refers to the study of flow or creep of materials, i.e. strain rates, under applied stresses (e.g. Turcotte and Schubert, 1982). As defined by the constitutive equation (2.4), the viscosity describes the proportionality constant between the strain rate and the stress. The rheol-ogy of Earth materials depends strongly on material (intrinsic) and environ-mental (extrinsic) properties. Using steady state deformation experiments of rock samples (e.g. triaxial deformation) within a certain temperature and pres-sure range, one can define the material parameters. Thereby, the steady-strain creep rate is related to the stress difference between the maximum and min-imum principal stress σ by

exp ntr

QA

RT

, (3.1)

where Atr and Q are material parameters, T is the absolute temperature, R is the universal gas constant (8.32 J mol-1), and n is the stress exponent. To obtain a generalised expression for the viscosity in fluid dynamic models, equation (3.1) needs to be expressed by the second invariant of the deviatoric stress and strain rate tensor (e.g. Ranalli, 1995)

exp nII II

QA

RT

, with

1 23

2

n

trA A

. (3.2)

Under certain conditions, viscous creep of Earth materials can be described mainly by diffusion (dfc) and dislocation (DLC) creep (e.g. Karato, 2008). The former describes a linear relation between strain rate and stress (i.e. a Newtonian fluid; n = 1) mainly dominant at low stress levels and/or small grain sizes and is significantly grain size sensitive. The latter describes a non-linear relation between strain rate and stress (i.e. a non-Newtonian fluid; n > 1) mainly dominant at high stress levels and/or large grain sizes and is grain size insensitive. Furthermore, both deformation mechanisms strongly depend on temperature and pressure. Thus, the dominant deformation mechanism may also change with depth (e.g. Karato and Wu, 1993). Keeping the stress con-stant, viscosity for each mechanism is defined by

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11 1

,

1exp

2 RTinin

i i II ii

QA

n

, for i = DLC and dfc. (3.3)

Composite rheology describes the combination of both deformation mech-anisms, assuming they occur in parallel, and its effective viscosity is described by

1

1 1

dfceff

DLC

. (3.4)

The rheology of each material used in the numerical calculations is de-scribed below.

3.2 Rock salt Under the assumption of the presence of a small amount of water (>0.05 wt%) at grain boundaries (e.g. Spiers et al., 1990), rock salt can be assumed to flow as a Newtonian fluid with a constant, relatively low viscosity (1017-1019 Pa s) (e.g. van Keken et al., 1993). However, within the range of certain parameters (e.g. temperature and grain size), the creep flow behaviour can be described as non-Newtonian (e.g. Wawersik and Zeuch, 1986; van Keken et al., 1993; Ter Heege et al., 2005). Therefore, a composite rheology, including both New-tonian and non-Newtonian flow laws, is simulated in order to quantify its ef-fect on finite deformation pattern within a salt diapir.

Diffusion creep (dfc) (here defined as pressure solution) occurs due to the presence of a small amount of water, e.g. due to a very thin film surrounding the grains. The second invariant of the dfc strain rate depends on temperature and grain size and is described by (e.g. Spiers et al., 1990; van Keken et al., 1993)

, 3

exp3

2dfc II

II dfc dfc

Q RTA

T d

, (3.5)

where Adfc and Qdfc are material constants and d is grain size. The second invariant of the dislocation creep (DLC) strain rate, here de-

scribed by cross slip of screw dislocations, is described by (e.g. Wawersik and Zeuch, 1986; van Keken et al., 1993)

( )( ) 1 2

( )0,

0

3

2

n Tn Tn T

II DCL IID

, (3.6)

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where D, τ0/µ0 and n=QCS/RT are material constants, Qcs is the activation en-ergy for cross slip controlled creep, and μ is the average shear modulus (1.4 GPa). DLC described by cross slip (3.6) is not exactly a power law model euqation as described in (3.1). However, cross slip mechanism, fitted to the data of West Hackberry (WH) and Asse (A) rock salt (Wawersik and Zeuch, 1986), has been chosen, as it provides a convenient empirical description em-bodying the mechanical behaviour of rock salt at temperatures below 200 °C and yields a creep law closely similar to (3.1) (van Keken et al., 1993). Thus, both rock salt types represent reasonable lower and upper bounds of a compo-site rheology within a natural parameter range and are used in the salt diapir models with a composite rheology. The parameters for each mechanism are taken from van Keken et al. (1993).

3.3 Olivine Upper mantle rocks mainly consist of olivine, pyroxene, and garnet, of which olivine is the most abundant (~70%) and probably the weakest under a wide range of temperature and pressure conditions (e.g. Karato and Wu, 1993; Karato, 2008). Thus, rheology of the upper mantle is most likely dominated by that of polycrystalline olivine. The main deformation mechanisms of oli-vine, confirmed in natural samples and by experimental observations, are DLC and dfc. Following the description given in section 3.1, the creep law of oli-vine can be defined by a power law relation between the second invariant of the stress and strain rate tensor. The second strain rate invariant for each mech-anism can be defined by (e.g. Hirth and Kohlstedt, 2003)

, , expi i ir m ni iII i i tr OH II

E PVA C d

RT

, for i = DLC and dfc, (3.7)

where COH is the water content, r is the water content exponent, d is the grain size, m is the grain size exponent, n is the stress exponent, E is the activation energy, V is the activation volume, and P is the lithostatic pressure. For the sake of convenience, a constant amount of water, i.e., in hydrous phases within the minerals (e.g. Hirschmann and Kohlstedt, 2012), is assumed (810 ppm H/Si), which represents the average water content of a mantle source rock of MORB (Hirth and Kohlstedt, 1996). For each deformation mechanism, wet olivine rheology has been assumed (Hirth and Kohlstedt, 2003). Using the viscosity ratio ηr between the DLC and dfc viscosity, one can determine the dominating deformation mechanism at each point within the upper mantle.

The elastic tensor Cijkl and LPO in the upper mantle flow models are calcu-lated using D-Rex (e.g. Browaeys and Chevrot, 2004; Kaminski et al., 2004). Thereby, only DLC strain rate and DLC shear strain are used to calculate LPO, as it is formed mainly by DLC deformation (e.g. Becker et al., 2003; Behn et

26

al., 2009). The maximum strain to integrate LPO over time along a particle’s flow path is set to the DLC shear strain from the upper mantle flow models or a maximum 12, depending on which value is smaller. In this way we assure that the formed LPO is a result of the deformation accumulated within the models. As hexagonal anisotropy is a usual approximation in seismic studies (Kaminski et al., 2004), only hexagonal anisotropy is shown, which is the norm of the hexagonal elasticity tensor relative to the norm of the full elastic tensor (Browaeys and Chevrot, 2004).

Knowing the elastic tensor, one can calculate the seismic parameters for a vertical incoming quasi S-wave underneath an oceanic plate of a certain age (Rümpker et al., 1999). For the seismic delay time calculations, the full elastic tensor with a standard elasticity at room temperature and pressure and an av-erage density of 3300 kg/m3 is used. The incoming S-wave is split into two different oscillating waves, i.e., one parallel to the plate motion, the other or-thogonal, resulting in different arrival times at the surface, where the differ-ence is known as seismic delay time δt. The total delay time at the surface is calculated by integration of the incremental δt, given by the local elastic tensor at certain points along a vertical profile.

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4 Paper summary

This chapter presents a brief summary of the papers included in this thesis. Each paper is summarised, describing the main objectives, the system of meth-ods used, and the main results and conclusions. Papers I and II address defor-mation patterns in down-built salt diapirs, as an example for a local tectonic scale. Papers III and IV are related to deformation patterns in the upper mantle underneath an oceanic plate, representing a global tectonic scale. Besides the different tectonic scales, different data (e.g. field or geophysical) can be used as a constraint in each application of the described deformation modelling method.

4.1 Paper I: Progressive internal deformation in down-built diapirs

Paper I addresses the general issue of internal deformation pattern in a down-built diapir with a linear (Newtonian) rock salt viscosity. In this paper, the results of 2D numerical calculations are used to describe the progressive and finite deformation within a salt diapir consisting of a source layer feeding a vertical diapiric stem.

4.1.1 Modelling concept The development of down-built salt diapirs has been studied by many inves-tigators using field data and analogue and numerical models (e.g. Poliakov et al., 1993; Talbot and Aftabi, 2004; Fuchs et al., 2011; Warsitzka et al., 2013). However, the internal progressive and finite deformation of a down-built salt diapir has not been analysed so far.

The model domain is defined by a 2D quadratic box (4x4 km) with a grid size of 201 x 201 in x and z-direction. As finite deformation within a diapir is strongly affected by its geometry (e.g. Dixon, 1975; Schwerdtner and Tröeng, 1978; Fuchs and Schmeling, 2013), a constant model geometry has been cho-sen to minimise the geometry effect on progressive and finite deformation (Figure 4.1). The model consists of a source layer with an initial thickness of hs (800 m), a density of ρs (2246 kg/m3) and a viscosity of ηs (1018 Pa s). On top of this layer, we added two rectangular shaped stiff overburden blocks, representing sedimentary overburden units, with a thickness of ho (800 m), a density of ρo (2600 kg/m3) and a constant viscosity of ηo (1023 Pa s). The source layer is connected to a vertical channel in the middle of the model with a width of 800 m and a height of ho. During the model run, the denser and high viscous

28

overburden blocks, flanking the stem, sink into the source layer driving the material with a constant load towards the middle of the model domain.

4.1.2 Summary Following passive markers at characteristic positions within the source layer, three distinctive deformation regimes within the salt layer have been identified (Figure 4.1): first, a squeezing channel-flow deformation within the source layer underneath the overburden (regime I); second, a corner-flow defor-mation directly underneath the vertical stem (regime II); and third, a pure channel-flow deformation within the stem (regime III).

Deformation within regime I is mainly governed by the vertical compres-sion in the middle of the source layer, due to the sinking overburden, and shear deformation along the overburden/salt interface and the lower boundary, due to the horizontal movement of the salt towards the stem (Figure 4.2). In regime II, deformation is dominated by shear deformation and rotation along the vi-cinity of the regime’s corners. In the centre of regime II, horizontal compres-sion results in high finite (Figure 4.2). The rotation in regime II leads to finite strain ellipses slightly tilted towards the vertical symmetry axis of the diapir before leaving regime II (Figure 4.3). Regime III is only dominated by shear deformation along its lateral boundaries, resulting in high finite deformation at the flanks of the stem and close to zero deformation within its middle (Fig-ure 4.2). All three regimes are directly connected to each other and kinematics

Figure 4.1. LEFT: Model setup for a down-built salt diapir consisting of a source layer and a vertical stem. RIGHT: Initial position for additional passive markers and identified deformation regimes in the salt layer.

29

overlap at their boundaries. Thus, finite deformation in regimes II and III is slightly affected by the previous regime(s).

Progressive deformation within the salt layer varies significantly depending on the initial position of a passive marker (Figure 4.3). Inherited strain from any previous regime affects the finite deformation in the following regimes. Thus, f is actually the largest within the middle of the stem, as particles accu-mulate strains within regimes I and II, most likely dominated by the horizontal compression in the centre of regime II (Figure 4.3). In conclusion, finite de-formation in the stem deviates significantly from the expected simple channel-flow deformation with no finite deformation in its middle. Moreover, as ori-entation of the finite strain ellipse is slightly tilted towards the vertical axis when leaving regime II, f decreases while the particle rises vertically through the stem (blue ellipse and line in Figure 4.3). Thus, inherited strains from re-gimes I and II result in a smaller finite deformation in the centre of each half

Figure 4.2: Individually accumulated deformation of each regimewithin the salt layer.

30

of the stem, in comparison to deformation accumulated only within regime III.

Although quantitative strain analysis in natural salt diapirs is rather difficult due to a small strain memory of rock salt and healing processes (e.g. Ter Heege et al., 2005), qualitative descriptions of high and low strain areas in a diapiric stem provide direct information about the rate of salt supply from the source layer. For example, an asymmetric sedimentation at the sides of the diapir, results in two separated episodes of salt supply, leading to a shift of the high finite deformation zone in the centre of the stem. This has been observed in natural diapirs (e.g. Burliga, 1996) and confirmed by a numerical model with different overburden thicknesses, simulating an asymmetric sedimenta-tion on top of the salt layer (Figure 4.3).

Figure 4.3: Top: Finite deformation (color scale) in the entire salt layer and velocityvector field (at the end of the model run). Middle: Progressive deformation of parti-cles flowing from regime I, through regime II and into regime III. Bottom: Asymmet-ric sedimentation at the sides of the diapir, results in a shift of the high deformationband in the middle of the stem, which provides direct information on rate of salt supplyfrom the source layer.

31

4.1.3 Conclusions Numerical model results show significant characteristics in deformation pat-tern and progressive and finite deformation within the salt layer of a down-built diapir. Thereby, the salt layer can be divided into three distinctive defor-mation regimes:

I A squeezing channel-flow deformation within the source layer underneath the overburden,

II A corner-flow deformation within the source layer underneath the stem, and

III A pure channel-flow deformation within the stem. Finite and progressive deformation within the stem (regime III) strongly de-pends on the inherited strain from the source layer, leading to

a) a high finite deformation within the centre of the stem, mainly gen-erated by the horizontal compression in regime II.

b) a low finite deformation zone in the centres of each half of the stem, mainly generated by the squeezing deformation in regime I.

c) a decrease in progressive deformation within each half of the stem, due to inherited strain and tilted strain ellipse orientation while leaving regime II.

Finite deformation pattern within the stem of a down-built diapir with sym-metric or asymmetric overburden reveals significant information about its his-tory and development and thus on the rate of salt supply from the source layer.

The numerical models of finite deformation in salt tectonics provide a very good example of the application of the described deformation modelling method. The results provide new and important information to better under-standing the dynamics and progressive deformation within a salt layer and, moreover, could directly be correlated to natural deformation pattern.

32

4.2 Paper II: The effect of composite rock salt rheology

Paper II addresses the issue of a composite rock salt rheology and different overburden geometries of a down-built salt diapir. While the creep behaviour of rock salt under certain experimental conditions can be approximated to be Newtonian (e.g. Spiers et al., 1990), it has been described as non-Newtonian within certain temperature and grain size ranges (e.g. Wawersik and Zeuch, 1986; van Keken et al., 1993; Ter Heege et al., 2005). In many salt diapir models, Newtonian rheology has been used for rock salt, for the sake of con-venience (e.g. Poliakov et al., 1993; Chemia et al., 2008; Burchardt et al., 2011). However, this approximation may not always be appropriate to analyse internal deformation patterns and flow structures in a salt diapir. In this study, 2D numerical calculations are used to describe how a composite rock salt rhe-ology (WH and A) with the most dominant non-Newtonian behaviour affects finite deformation and internal flow structures of a salt diapir in comparison to a purely Newtonian rheology (standard model; presented in paper I).

4.2.1 Modelling concept The model setup used in this paper, is based on the setup described in paper I, where the same parameters are used for the overburden. A constant tempera-ture (T = 160°C) and grain size (d = 0.03m) combination within a natural pa-rameter range was chosen, to obtain a rock salt rheology with the strongest non-Newtonian behaviour. By doubling the initial thickness of the overbur-den, a higher stress regime was applied in an additional model series, in order to amplify the non-Newtonian effect (models A2 and WH2). The local differ-ence in finite deformation f at each grid point within the salt layer between the composite rheology models (comp) and the standard (st) model is used to cal-culate an amplification factor of the ellipticity of the finite strain ellipse due to a composite rheology. The amplification factor is defined by

10 compf

st

RC

R , (4.1)

where Δf is the local difference in f and R the ellipticity of the strain ellipse. In contrast to overburdened blocks with sharp corners, an additional model

series using overburdened blocks with curved corners has been used to analyse the effect of overburden geometry on internal deformation pattern and flow structures.

Initial vertically and horizontally oriented passive marker lines within the source layer reveal additional information on flow folds, that form while the

33

salt flows from the source layer into the stem. The deformed marker lines re-sult in characteristic folds within the stem, which could be correlated with natural folds in salt diapirs.

4.2.2 Summary Due to the non-Newtonian component in the composite rheology, effective viscosity in the salt layer varies by up to two orders of magnitude and flow within the salt layer is a plug flow. Thereby, non-Newtonian creep dominates within most parts of the salt layer, where it is more dominant in WH rock salt than in A rock salt.

Figure 4.4: Amplification factor C for each composite rheology model and the WHrock salt rheology model with curved overburden corners. (A – Asse rock salt; WH –West Hackberry rock salt; (A2; WH2) – twice the overburden thickness

34

The amplification factor C indicates regions within the salt, where the el-lipticity of the strain ellipse is either amplified (C>1) or damped (C<1) in comparison to the standard model, due to the non-Newtonian component in the composite rheology (Figure 4.4). In general, finite deformation increases in regions of high strain rates, i.e. in the centre of lateral boundaries, at the overburden/salt interface, along the lower boundary of the model, and in the horizontal compression in regime II (C > 1; Figure 4.4). Furthermore, the plug flow leads to a broader zone of low deformation in the middle of each half of the stem and within the middle of the source layer, but still finite due to the vertical compression by the overburden in regime I. Therefore, the decrease in f, in comparison to the standard model, is maximum along the outer ends of the plug and not within its middle (C < 1; Figure 4.4). In the middle of each half of the stem, finite deformation seems to be enhanced due to the composite rheology. However, finite deformation in the composite rheology models is higher in comparison to the standard model due to the decreasing finite defor-mation in the standard model, while particles rise vertically through the stem (see blue finite strain ellipses in Figure 4.3). The amplification and damping of finite deformation is enhanced for a model with WH rock salt in comparison to A rock salt and for models in a higher stress regime (WH2 and A2).

A curved cornered overburden, results in a wider base of the stem, which, in turn, facilitates salt supply from the source layer towards the stem. There-fore, finite deformation decreases overall within the salt layer and becomes smoother with broader deformation areas in comparison to models with a sharp cornered overburden. In case of a composite rheology, overburden ge-ometry directly affects the mechanical behaviour of the salt. Close to the curved corners of the overburden, the areas of high viscosity are much wider. Thus, the base of the stem behaves more stiffly, which widens the low defor-mation zones. The decrease in finite deformation in comparison to a Newto-nian rheology (C<1) is more significant in models with a curved cornered overburden than with a sharp cornered overburden (Figure 4.4).

Horizontally and vertically oriented passive marker lines within the source layer form two sets of upright folds within the stem with parallel vertical axial surfaces (Figure 4.5). The upright folds within the middle of the stem, are in-itially formed out of horizontally oriented lines due to the horizontal compres-sion in regime II. The second set of upright folds is located in each half of the stem. These folds are initially formed out of vertically oriented lines within the source layer, which form first generation folds with sub-horizontal axial surfaces in the source layer. These first generation folds are refolded by the flowing of salt into the stem around the overburden corners at the base of the stem. Moreover, the hinge of the refolded folds shows a migration, both spatial and temporal, while flowing around the overburden corner into the stem. Composite rheology results in slightly more open folds, in both the source

35

layer and each half of the stem. In addition, an upright synformal and two antiformal folds are formed in the middle of the stem (Figure 4.5).

Folds with sub-horizontal axial surface are expected to form within the source layer of natural salt diapirs, either formed as compressional or diapiric folds or shear folds (e.g. Burliga, 1996; Burliga, 2014; Sans et al., 1996; Talbot and Jackson, 1987). These sub-horizontal folds within the source layer might show the same evolution while flowing from the source layer into the stem as ob-served within the numerical models. In case of an axisymmetrical salt diapir (d), or even a salt wall, these folds might form as upright sheath fold (s) or concentric ringfold (r) within the stem (Figure 4.6).

Figure 4.5. Flow and fold structures for the standard and WH2 model with a corneredor sharp curved overburden. The red lines originate from the lower half of the source layer and the black from its upper half.

36

4.2.3 Conclusions Composite rock salt rheology, in comparison to Newtonian rheology, signifi-cantly affects finite and progressive deformation within a salt layer of a down-built diapir, given a geometrically equal down-building scenario. Following main features are observed in the salt diapir models with a composite rheol-ogy:

a) Strong localised deformation along the overburden/salt interface and the lower model boundary within the salt layer.

b) Deformation within the middle of the source layer and each half of the stem decreases strongly.

c) The ellipticity of the strain ellipse can be amplified by a factor of up to 3 in high deformation regions (f > 2).

Finite deformation and deformation patterns within the salt diapir are strongly affected by the overburden geometry. An overburden with smooth corners re-sults in a generally smaller finite deformation and more homogeneous defor-mation pattern.

The deformed marker lines result in characteristic folds within the stem, which could be correlated with natural folds in down-built salt diapirs.

The application of numerical deformation modelling shows, how finite de-formation and deformation patterns are affected by a non-Newtonian rheol-ogy. As finite deformation is strongly affected by the rheology, deformation profiles within a natural salt diapir, might help to constraint non-Newtonian rheology. Furthermore, the application shows, how internal flow structures might be associated with natural fold structures due to fluid dynamical defor-mation.

Figure 4.6. Sketch of fold structures formed in the source layer and the stem.

37

4.3 Paper III: Deformation and anisotropy in the upper mantle

In contrast to a local tectonic scale as presented in papers I and II, this paper, as an example of a global tectonic scale, addresses the issue of deformation pattern and seismic anisotropy in the upper mantle underneath an oceanic plate. Finite deformation within the upper mantle is strongly controlled by its effective viscosity (e.g. Karato, 2010), which in turn depends on various pa-rameters (e.g. Karato and Wu, 1993; Hirth and Kohlstedt, 2003). Thus, the focus of paper III is to analyse the sensitivity of deformation patterns and its corresponding seismic anisotropy on parameters mainly governing the viscos-ity within the upper mantle (i.e., temperature, grain size, and plate velocity).

4.3.1 Modelling concept

To analyse the sensitivity of deformation patterns and seismic anisotropy within the upper mantle underneath an oceanic lithosphere, a 1D model ap-proach has been used (Figure 4.7). Assuming that motion within the upper mantle is driven mainly by the overriding plate and neglecting volume forces and a horizontal pressure gradient, horizontal velocity can be approximated by a 1D description of the Stokes equation (2.2). Boundary conditions are constant horizontal velocity at the top and zero at the bottom. The 1D descrip-tion of (2.2) is solved iteratively for the horizontal velocity along vertical pro-files of a certain depth H (400 km) using a 1D staggered finite difference scheme with a vertical resolution of nz = 1001. Temperature along each profile is defined by a cooling oceanic plate of a certain age (0-120 Ma) with a time stepping dt of 0.1 Ma between each profile. Effective viscosity of the upper mantle is described as a composite rheology of dfc and DLC deformation for wet olivine (Hirth and Kohlstedt, 2003).

Figure 4.7. Model approach to determine the sensitivity of deformation pattern and seismic anisotropy on variations in temperature, plate velocity, and grainsize.

38

Systematically, the effect of different temperature fields, plate velocities, and grain sizes on deformation patterns and seismic delay times for a plate of a certain age has been analysed. A reference model has been defined with a HSCM temperature field (Tpot

= 1315 °C) with variable thermal parameters, a plate velocity of 5 cm/a, and a constant grain size of 1 cm. Further temperature fields have been defined by a HSCM with constant thermal parameters as well as different Tpot (1215, 1315, and 1415 °C). Plate velocity is set to 1, 5, and 10 cm/a, and steady state grain sizes are calculated for different grain evolution power exponents p (2-4). Shear strain along each profile is directly calculated by (2.14), where the total shear strain can be separated in a DLC and dfc part. Maximum shear strain is most likely located within the asthenosphere, closely below the mechanical LAB, which defines the depth of no further deformation with time. Propagation of shear strain into the upper mantle, more precisely the strain localization within the asthenosphere, is used as a direct indicator for mechanical coupling between the lithosphere and asthenosphere.

4.3.2 Summary The constant plate velocity at the top results in a shear flow deformation within the upper mantle, where cooling of the plate with time results in thick-ening of the high viscosity zone at the top. Total shear strain increases with time due to a continuous simple shear deformation (Figure 4.8A). In the upper part of the model, incremental deformation decreases with time due to the cooling of the plate. Thus, deformation does not increase further with time at a certain depth and beyond a certain age, i.e. above the mechanical LAB (Fig-ure 4.8B).

In comparison to the reference model, variation of model parameters results in variation of the effective viscosity and thus in a variation of deformation pattern within the upper mantle. Total shear strain is thereby less or more sen-sitive to certain model parameters. For example, different temperature fields due to variable or constant thermal parameters used in (2.15) lead to slightly local temperature variations within the upper mantle, i.e. a hotter or colder asthenosphere. Thus, variation of the total shear strain within the upper mantle is only minor (Figure 4.8C). In contrast, a higher (smaller) Tpot, in comparison to the reference model, results in a more (in)homogeneous deformation of the upper mantle. Therefore, shear strain is less (more) concentrated within the asthenosphere (Figure 4.8D). Plate velocity, more precisely the relative veloc-ity between the plate and the umb, strongly governs the maximum shear strain within the upper mantle (Figure 4.8E). Steady state grain sizes with depth do not significantly affect the total shear strain in comparison to the reference model (Figure 4.8F). However, in case of the smallest grain sizes (p = 4), total shear strain becomes more concentrated within the asthenosphere. A higher concentration of shear strain within the asthenosphere, indicates a weaker me-chanical coupling between the plate and the underlying mantle. However, in

39

each model, mechanical coupling strengthens with time, as the viscosity in the asthenosphere increases with time and shear strain propagates deeper into the upper mantle (Figure 4.8A).

The characteristic depths of the mechanical LAB, the minimum viscosity, and the maximum shear strain, however, vary only slightly in case of local temperature variations due to the temperature difference between a HSCM with constant and variable thermal parameters. In this case, all three depths

Figure 4.8. Total shear strain and characteristic depths for the reference model andthe effect of parameter variations on the total shear strain. (A) Reference model. (B)Characteristic depths for the reference model and for a model with a HSCM usingconstant thermal parameters. (C) Effect of HSCM with variable or constant thermalparameters on the total shear strain γxz. (D) Effect of different Tpot. (E) Effect ofdifferent vx0 on γxz. (F) Effect of different steady state grain sizes on γxz.

40

are shifted upwards (~3-18 km) due to a hotter asthenosphere (TcHSCM, Figure 4.8B).

DLC shear strain within the upper mantle results in LPO and hexagonal seismic anisotropy. Anisotropy increases with time and reaches its maximum value (~8.5%) when DLC shear strain reached the critical value of γDLC = 12. The LPO results in an anisotropic layer of maximum anisotropy, in which thickness increases as the plate ages (up to ~200 km). Thus, total seismic delay time at the surface of the plate is controlled mainly by the maximum DLC shear strain in the asthenosphere and the thickness of the anisotropic layer.

Due to different model parameters, the dominant deformation mechanism within the upper mantle varies in comparison to the reference model. Total shear strain might vary significantly, e.g. due to a different Tpot, while varia-tions in δt are only minor due to an equally thick anisotropic layer. In contrast, total shear strain varies in a similar manner as in Tpot due to different grain sizes (Figure 4.8F), but δt is significantly smaller due to a smaller DLC shear strain (Figure 4.9B). Similarly, due to a smaller plate velocity (1 cm/a), DLC shear strain as well as δt is significantly smaller in comparison to the reference model (Figure 4.9A).

In comparison to natural seismic data of a fast oceanic plate (δt ~ 1-2 s for the Pacific, collected from Wüstefeld et al. (2009)), the numerical models pre-dict a larger delay time (~4 s), assuming deformation within the upper mantle is governed only by simple shear due to the motion of the plate and a stationary umb (Figure 4.9D). However, the amount of DLC is strongly controlled by

Figure 4.9. Most sensitive seismic delay times results (A-C). Natural data of a fast oceanic plate, i.e. the Pacific (D), might be explained by a moving umb or due to atotal smaller amount of DLC (~25-40 %).

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the rheology parameters, which are not very well constrained. Thus, a varia-tion of rheology parameters, within their experimental error range, shows that a small amount of DLC within the asthenosphere of ~25-40 %, would lead to a significant decrease (~50 %) in δt (Figure 4.9D), if LPO is not disturbed by dfc. In return, deformation within the upper mantle could also be altered due to further processes, which might lead to a decrease in δt, e.g. secondary con-vections, or P- flow within the asthenosphere. Thus, assuming DLC shear strain might be destroyed by dfc deformation or secondary convection, this analysis shows how much DLC shear strain would be necessary to observe a seismic anisotropy and delay time within the range of natural data. Further-more, including a mobile umb (e.g. 9 cm/a) would result in a smaller relative velocity between the plate and the umb, and thus, in a significant decrease of δt (Figure 4.9D).

4.3.3 Conclusions The results of the model show the sensitivity of deformation pattern and seis-mic anisotropy within the upper mantle on temperature, plate velocity, and grain size. Local temperature variations result in a shift of characteristic depths, like the depths of no further deformation (i.e. the mechanical LAB), maximum shear strain, and minimum viscosity. However, strain concentration within the asthenosphere remains the same. In contrast, a smaller Tpot, a higher plate velocity, and a smaller grain size lead to a higher shear strain concentra-tion within the asthenosphere and vice versa. This implies a weaker (stronger) mechanical coupling between the plate and the underlying upper mantle. Fur-thermore, our model results predict an increase in the coupling with plate age.

The seismic delay times at the surface of an oceanic plate are mainly gov-erned by the maximum anisotropy and the thickness of the DLC shear zone, i.e. the thickness of the anisotropic layer. The results of the model show, that δt is more sensitive to the plate velocity and grain size than to temperature. Seismic delay times within the model can be reduced significantly by smaller grain sizes or relative velocity between the plate and the umb to obtain delay times within the range of natural data.

This study provides a good example of how deformation calculations can be used to identify the effect of certain model parameters on the models kine-matics and how to constrain numerically calculated deformation with geo-physical data. Furthermore, it shows how shear strain profiles can provide sig-nificant information on the mechanical coupling between two layers. The re-sults show how strong intrinsic and extrinsic properties affect the partitioning of dfc and DLC in the upper mantle, which could lead to a strong variation in strain partitioning between dfc and DLC. Due to the fact, that seismic anisot-ropy strongly depends on the amount of DLC shear strain, this study shows the importance of correlating numerical deformation data with seismic anisot-ropy.

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4.4 Paper IV: The effect of a horizontal pressure gradient

This paper addresses the issue of a horizontal pressure gradient driven flow in the asthenosphere, i.e. a PC-flow. 2D/3D numerical convection models with a low viscosity zone representing an asthenosphere equivalent showed, that flow within the asthenosphere is a PC-flow (e.g. Höink et al., 2011; Shiels and Butler, 2015). The asthenospheric P-component was also confirmed in a quan-titative inversion method of azimuthal seismic anisotropy (Natarov and Con-rad, 2012). However, a PC-flow within the upper mantle, both in direction of and opposite to the plate’s motion, and its effect on the deformation pattern and seismic anisotropy has not been analysed in detail so far. Active flow within the asthenosphere would lead to significant differences in seismic ani-sotropy. Thus, a systematic analysis would provide new information and con-straints regarding the interaction between plates and the upper mantle.

4.4.1 Modelling concept

Motions within the upper mantle driven by kinematic and pressure boundary conditions, are calculated using a 2D channel flow model approach. The model domain is defined by a depth H (400 km) and length L (6000 km), where the aspect ratio is governed by the plate velocity and age (Figure 4.10). Equations (2.1) and (2.2) are solved iteratively using a 2D FD staggered grid method for variable viscosity (developed after Gerya, 2010) until steady state is reached. Boundary conditions are prescribed velocities at the top (vx0 = 5 cm/a) and bottom (vx1) and open lateral boundaries, with the option of adding a constant pressure P1 at x = 0. Viscosity is described by a composite rheology, and temperature within each model is a HSCM with variable thermal param-eters.

Figure 4.10. Model setup for the 2D numerical models of different channel flow fieldswithin the upper mantle.

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In this way, flow defined by Couette(C-) and PC-flow boundary conditions including a mobile umb with rheological and thermal conditions representing an oceanic plate of a certain age can be calculated. Systematically, the net horizontal pressure gradient dP/dxnet of the entire model and umb velocity (vx1) have been varied to analyse their effect on deformation patterns and seismic anisotropy in comparison to a reference model (C-flow, vx1 = 0 cm/a, P1 = 0 MPa). Shear strain within the model domain is calculated using the marker approach over a time of 120 Ma, i.e. the age of the plate. As the material can leave the channel on the right side, it re-enters it on the left side and defor-mation is set to zero.

4.4.2 Summary Similar to the 1D model approach, deformation within the reference model is governed by simple shear deformation, which is concentrated underneath the lithosphere due to the temperature, pressure, and rheology of the upper mantle. However, due to the no slip lower boundary condition, i.e. a stationary umb, and thus slowly moving material, maximum shear strain is located in deeper depths underneath the younger part of the plate. The thickening of the plate due to cooling creates a higher net flux within the upper part of the model, resulting in a slight vertical upwelling close to the left boundary of the model. Deviatoric horizontal extension and vertical compression due to the vertical upwelling, however, do not significantly affect finite deformation within the asthenosphere. More importantly, the thickening of the plate induces a partial horizontal pressure gradient pointing from the middle towards its lateral model boundaries. Therefore, horizontal velocity as well as the deviatoric shear stress varies significantly in comparison to the 1D C-flow, and shear strain becomes more concentrated within the low viscosity zone of the asthenosphere.

As velocities are prescribed along the horizontal model boundaries, flow within the asthenosphere does not essentially drive or resist motion of the plate. However, the applied total horizontal pressure gradient leads to a flow within the asthenosphere not controlled by the motion of the plate and thus referred to as ‘active’ flow. Furthermore, depending on the pressure’s ampli-tude, flow with in the asthenosphere is enhanced, slowed down or even re-versed in comparison to the plate’s motion. Thus, active flow within the as-thenosphere is referred to as a ‘driving’ or ‘resisting’ flow, with respect to the motion of the plate.

Active flow within the asthenosphere, both driving (e.g. PC03; P1 = 12 MPa) and resisting (e.g. PC06; P1 = -12 MPa), leads to characteristic results in the flow field, effective viscosity, deformation pattern, and seismic anisot-ropy of the upper mantle. Due to the composite rheology, flow within the as-thenosphere is a plug flow (Figure 4.11A and B). Effective viscosity within the asthenosphere varies significantly in comparison to the reference model, resulting in a rheological layering. Below the lithosphere, a more confined low

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viscosity zone is formed, followed by a high viscous layer in the middle of the asthenosphere, and a decrease in effective viscosity towards the lower bound-ary of the model.

In comparison to the reference model, deformation pattern within the as-thenosphere varies significantly, dominated by two high shear zone, one closely below the mechanical LAB and one at the lower boundary of the model, and a zone of low to no finite deformation within middle of the asthen-osphere (Figure 4.11C and D). Due to the driving (resisting) asthenosphere flow, shear sense varies with depth going from sinistral (dextral) in the shal-low to dextral (sinistral) in the deep shear zone (Figure 4.11C and D). There-fore, orientations of the strain ellipse, as well as LPO, vary significantly with depth (Figure 4.11E and F). Due to the different flow directions of the asthen-osphere relative to the plate’s motion, shear deformation within each zone is

Figure 4.11. Flow field (A and B), total shear strain (C and D), and hexagonal ani-sotropy (E and F) for a model with a driving asthenosphere flow (PC03) and a resist-ing asthenosphere flow (PC06).

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significantly higher for a resisting in comparison to a driving asthenosphere flow. In case of a stronger driving (resisting) asthenosphere flow, thickness of the low deformation zone in the middle of the asthenosphere, i.e. the high viscous layer, decreases, while the thickness of the upper shear zone, i.e. the low viscosity layer, increases (decreases). However, due to the stagnant umb, shear strain, and thus anisotropy within the upper mantle, is very high. Alt-hough flow in the asthenosphere consist of a plug flow with a low zone of deformation within its middle, seismic delay times do not decrease in com-parison to the reference model.

Assuming the umb may flow as part of a large scale convection cell, flow, deformation pattern, and anisotropy within the upper mantle vary significantly in comparison to the reference model. However, only in the case of a driving

Figure 4.12. Flow field, shear deformation, and hexagonal anisotropy for a C-flow model with a resisting (C02) and driving (C03) umb.

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umb, shear strain and seismic anisotropy are significantly reduced due to a smaller relative velocity between the plate and the umb (Figure 4.12A and B). As material within the entire model is advected laterally, the lower half of the model does not accumulate a large amount of strain (Figure 4.12C and D). Therefore, seismic delay times are significantly reduced in comparison to the reference model and fall within the range of natural data. Furthermore, de-pending on the relative motion between the plate and the umb, shear sense and LPO within the asthenosphere changes (Figure 4.12D and E). It has been found, that the kinematics (except for the sign) and rheology are exactly the same for the same relative velocities between the plate and the umb and same additional absolute pressures (Figure 4.12A and B). However, due to different absolute velocities and flow paths within the asthenosphere, shear strain and seismic anisotropy at a certain distance away from the ridge are significantly different, depending on the actual difference between the absolute velocities of the asthenosphere. Thus, shear strain within the upper mantle underneath an oceanic plate, does not merely depend on the relative velocity between the plate and the umb, but also on the actual velocity in the asthenosphere ad-vecting the deformed material.

4.4.3 Conclusions Using 2D numerical models, this study shows the effect of a horizontal pres-sure gradient driven asthenosphere, both in direction of and opposite to the plate’s motion, and the effect of a mobile umb on finite deformation and seis-mic anisotropy within the upper mantle. Deformation modelling provides new insights in kinematics of the upper mantle and how the plates and the upper mantle could be connected mechanically.

The thickening of the lithosphere due to cooling significantly affects the kinematics and stresses within the upper mantle, which vary in comparison to a 1D simple shear deformation with depth dependent viscosity.

Due to active flow within the asthenosphere, both driving and resisting with respect to the plate’s motion, and the composite rheology, kinematics, viscos-ity, finite deformation, and seismic anisotropy vary significantly in compari-son to a C-flow model. However, shear strain within the upper mantle, and thus anisotropy, is relatively large due to the high relative velocity between the plates and the upper mantle. Only in case of a driving umb base, does the seismic anisotropy and seismic delay time reach values within the range of natural data. Furthermore, the results of the model show, that the deformation in the upper mantle at a certain distance away from the ridge significantly depends on the absolute velocity in the asthenosphere. Therefore, assuming LPO is not destroyed by any secondary mechanisms, deformation might be significantly affected by the motion of the umb and not only by the plate’s velocity and active flow within the asthenosphere.

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5 Conclusions

In this thesis, calculation and quantification of large finite and progressive de-formation of geodynamical flows with application to the field of salt tectonics and upper mantle deformation is presented. Both fields provide natural data, primary and secondary, to constrain numerical deformation models, quantita-tively (strain) and qualitatively (pattern). The numerical models provide new insights on the kinematics and dynamics of the two different tectonic settings that the deformation modelling method has been applied to. The main findings of each tectonic system are summarised as follows.

(1) In case of a down-built salt diapir, three distinctive deformation regimes have been identified within the salt layer. Deformation in the salt layer can be divided into a squeezing channel flow deformation underneath the sediment overburden, a corner flow deformation directly underneath the stem, and chan-nel flow deformation within the stem itself. Qualitative analysis of finite de-formation pattern within the stem of a down-built salt diapir provides direct information on the rate of salt supply from the source layer and thus on its history and development.

(2) Finite and progressive deformation within the diapiric stem strongly depends on the inherited deformation from the source layer. Thereby, defor-mation patterns in the stem, i.e. a zone of high finite deformation within the middle of the stem and a degreasing progressive deformation within each half of the stem, deviate significantly from the deformation pattern individually expected within the stem, i.e. a pure channel flow.

(3) A non-Newtonian rheology of rock salt leads to a strong localization of deformation close to the salt/overburden interface and thus to a less deformed interior. The ellipticity of the strain ellipse can be amplified by a factor of up to 3 in high deformation regions.

(4) Flow and fold structures of initial horizontally and vertically oriented passive marker lines within the source layer could directly be correlated to natural folds in down-built salt diapirs.

(5) Shear deformation of the upper mantle mainly driven by the motion of an oceanic plate is strongly governed by temperature, kinematics, and rheol-ogy of the upper mantle. While local temperature variations lead to a shift in the depth of the mechanical LAB, the overall pattern remains the same. In contrast, a smaller potential mantle temperature, a higher plate velocity, or smaller grain sizes lead to a stronger shear strain concentration within the as-thenosphere. This implies a weaker mechanical coupling between the plate

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and the underlying upper mantle, which becomes stronger with the age of the plate.

(6) Seismic delay times at the surface of an oceanic plate are mainly gov-erned by the maximum anisotropy and thickness of the maximum anisotropy layer within the mantle. Thereby, delay times are more sensitive to the plate velocity and grain size than to temperature. Assuming anisotropy is not de-stroyed by any secondary mechanism, a small grain size (< 3 mm) or relative velocity between the plate and the upper mantle base is required to obtain seismic delay times within the range of natural data.

(7) A horizontal pressure gradient driven flow within the asthenosphere significantly affects the flow, effective viscosity, finite deformation and seis-mic anisotropy of the upper mantle. Furthermore, finite deformation at a cer-tain distance away from the ridge significantly depends on the absolute veloc-ity of the upper mantle. However, only in the case of a driving upper mantle base, seismic anisotropy and delay times reach values within the range of nat-ural data. This might indicate, a stronger connection between the motions within the deeper mantle and the plates within certain tectonic regions.

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6. Sammanfattning på svenska

Denna avhandling fokuserar på beräkningen och kvantifieringen av finit och progressiv deformation i geodynamiska flöden genom att använda 2D nume-riska modeller på olika tektoniska skalor för applikation på naturliga fall. Av-handlingen behandlar främst två tektoniska forskningsområden: a) salttekto-nik och b) deformation i övre manteln orsakad av interaktion mellan litosfären och astenosfären. Inom de båda områdena finns naturlig data (primär och se-kundär) som begränsar deformationsmodellerna kvantitativt (töjning) och kvalitativt (deformationsmönster). De numeriska modellerna ger nya insikter om kinematik och dynamik i de två olika tektoniska miljöerna som modelle-ringsmetoden har tillämpats på.

I delen om salttektonik ligger fokus på deformationsanalys av förenklade diapirer som består av ett saltlager (källa) som matar en vertikal stam. Tre deformationsregimer har identifierats inom ett saltlager: (I) ett pressat kanal-flöde i kontakt med det överliggande berget (II) ett hörnflöde under stammen, och (III) ett rent kanalflöde inom stammen. Modellresultaten visar att kvalita-tiv analys av finita deformationsmönster i diapirstammen (t.ex. symmetriska eller asymmetriska) kan direkt avslöja information om olika hastigheter av salttransport från saltlagret (t.ex. som observerats i Klodawa diapiren, Polen). Inom stammen beror dock den finita och progressiva deformationen till stor del på den ärvda deformationen från saltlagret (källan). Deformationsmönstret i stammen kan därigenom summeras som en zon med hög finit deformation i mitten av stammen med minskande progressiv deformation i varje stamhalva. Deformationsmönstret avviker därför väsentligt från det förväntade mönstret, dvs. rent kanalflöde. Kompositbergsalts reologi orsakar en stark lokalisering och amplifiering av töjning längs saltlagrets gränser och en mindre deforme-rad interiör i jämförelse med Newtonskt bergsalt. Flödes och veckstrukturerna som i början var horisontellt och vertikalt orienterade passiva markeringslinjer i saltlagret, kan direkt korreleras till fysiska veck i en saltdiapir.

Den andra delen av avhandlingen som behandlar den övre manteln fokuse-

rar på den finita och progressiva deformationen och resulterande gitter orien-tering (LPO) under en oceanplatta. Här undersöktes hur känslig deformation och seismisk anisotropi är för påverkan av reologi, kornstorlek, temperatur och kinematik. Modellresultaten visar att den mekaniska litosfär-asteno-sfärgränsen kontrolleras till stor del av temperaturen och i mindre skala av kinematik eller kornstorlek. En lägre potentiell manteltemperatur, en högre

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platthastighet eller mindre kornstorlekar kan däremot bidra till en högre kon-centration av skjuvtöjning i astenosfären. Den höga koncentrationen av skjuvtöjning innebär en svagare mekanisk koppling mellan plattan och den underliggande övre manteln. Kopplingen blir emellertid starkare med plattans åldrande, vilket gör att skjuvtöjningen utbreder sig djupare in i manteln. Seismiska fördröjningstider vid ytan av en oceanplatta styrs huvudsakligen av den maximala anisotropin och tjockleken av det maximala anisotropilagret in-uti manteln. Därigenom är fördröjningstider mer känsliga för plattans hastig-het och kornstorlek än dess temperatur. En liten kornstorlek (<3 mm) eller en relativ hastighet på ~ 1 cm / år mellan plattan och den övre mantelns bas krävs för att erhålla seismiska fördröjningstider inom intervallet för naturliga data (~ 1-2 s), förutsatt att anisotropin inte förstörts av någon sekundär mekanism. Ett Poiseuilleflöde i astenosfären påverkar avsevärt den övre mantelns flöde, effektiva viskositet, finita deformation och seismiska anisotropi. Modellresul-taten visar att på ett visst avstånd från mittoceanryggen beror deformationen i övre manteln i hög grad på den absoluta hastigheten i astenosfären. Dock, endast i fallet med en drivande övre mantelbas nås värden inom intervallet för naturliga data av seismisk anisotropi och fördröjningstider. Detta indikerar en starkare förbindelse mellan den djupare mantelns rörelse och plattorna.

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Acknowledgements

Since the beginning of my studies at Uppsala University I have meet many friends and colleagues whom I will always remember. I am especially grateful for all those who have been involved in this study and those who made my live in Uppsala more comfortable.

First and foremost, I would like to thank my supervisor, Hemin Koyi, who offered me the possibility to this research. Since the first time I met him in Frankfurt during my undergraduate studies, he led my research path quite early towards salt tectonics and I couldn’t let lose the salt since. During the past years he shaped my knowledge in geology and geodynamic and led me from salt diapirs to global mantle dynamics. I will be always more than grate-ful for all the things you have taught me. He guided me through my PhD stud-ies, not only as my supervisor but also as a friend, and I will always be im-pressed by his collection of anecdotes.

I would also like to thank my co-supervisor Harro Schmeling, who made me study geodynamics in the first place and who has been a colleague and friend for many years. I would like to express my deepest gratitude to all the things you have done and taught me through my career, that you have awaken my interests in numerical modelling, the many scientific and general discus-sions we had and the fact that you introduced me to Hemin.

I am thankful for Steffi, who welcomed me and shared her office with me the first time I visited Uppsala.

I would like to thank Jarek and Alicia for providing an accommodation, not only in the beginning of my study but also towards the end. Jarek, you have also been always helpful and generous throughout my stay in Uppsala and I am very thankful for that!

I would like to thank my former PhD colleagues, Ester, Jochen, Maria, Da-vid, Börje, Hongling, Sylvia, Zhina, Lara, and Peter who have been always helpful and considerate during the last years and with some I have shared the aquarium in the beginning. A special thanks to Kirsten who I shared my office with during the last years and who was always helpful and open for discussion.

Thanks as well to all my current PhD colleagues, Iwonna, Harri, Tobias, Sara, Muftha, Franz, and Lei. Special thanks to Tobias for the Swedish sum-mary of this thesis.

I would like to thank all my colleagues at the entire department, who always had an open door and were helpful and considerate. Thank you Abi (Thanks for all your help), Fran, Hans, Karin, Michael, Faramarz, Håkan, Bjarne, Bojan, Chris H. (thanks for all the help), Chris T. (thanks for the new insights

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on salt diapirs), Alireza, Rebekka, and Remi and a special thanks to all who I have forgotten!

I would like to thank my colleagues at Goethe University in Frankfurt. It was always a pleasure to come back to Germany for a couple of weeks during the year and to work with you. You made the time very enjoyable. A special thanks to Jan-Philipp (thanks for all your help), Meysam (you are one of the funniest people I know), Herbert, Yannick, Linda, Georg (thanks for the rou-tine), and Michael.

Thanks as well to my closest friends and band colleagues in Germany, who supported my choice to go to Uppsala and we are still able to enjoy some live music together!

My deepest gratitude to my entire family, Klaudia, Guntram, Paul, Jo-hanna, Anna-Maria, Elmar, Michael, Sybille, Olli, Sandra, Lena, Steve, and Noah who supported me during the entire time and especially during the last weeks. Thanks for all the parcels over the years!

Last but not least I would like to thank my girlfriend, Anne, who supported me and encouraged me the most during the entire time in Uppsala. Thanks for all the things you have done for me and your help and support during all those years!

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