On the Dynamics of Plate Tilting an Analytical and Numerical Approach

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On the Dynamics of Plate Tilting An Analytical and Numerical Approach

ÖMER FARUK BODUR

A thesis submitted in fulfilment of the

requirements for the degree of Doctor of Philosophy

School of Geosciences

Faculty of Science

The University of Sydney

2020 “On the Dynamics of Plate Tilting, An Analytical and Numerical Approach”

SUPERVISORS

Assoc. Prof. Patrice F. Rey

Prof. Gregory A. Houseman

Prof. Dietmar Müller

TIME FRAME

September 2016 — October 2019

Sydney, NSW 2006, Australia

DECLARATION I declare that this thesis is less than 80,000 words in length and the intellectual content of this thesis is the product of my own work which has not been submitted for a higher degree at any other university or institution. I also certify that all the assistance received in preparing this thesis and sources have been acknowledged. Ömer Faruk Bodur

22nd Jun. 2020 PREFACE This thesis consists of four articles, the first article presented in Chapter 2 has been published by EGU-Solid Earth with open-access. Other chapters are prepared for submission. The thesis starts with an introductory chapter that retrospectively expands on the subject and briefly gives the aims of the thesis. The findings are presented in detail in the following chapters, and they form a coherent whole in the Discussion and Conclusion where I review their significance along with reflections about future studies.

No ethical approval was required during the completion of this study.

ii ACKNOWLEDGEMENTS

To begin with, I wish to express my deep gratitude to my advisor, Patrice Rey, one of the kindest people I know to this point in my life. In three years, I learned a lot from him. He became my role model in matters not only to do with science, but of in life. Now, I better understand why he gave me the freedom to make all the mistakes I made. I want to thank

Gregory Houseman to whom I am deeply indebted. He patiently listened to me whenever I asked questions, and shared an enormous amount of his time with me in discussing geodynamic problems that I could never address without his support. I am also grateful to Dietmar Müller, who did his very best to support my progress. His presence has always given me energy and motivation.

Since I stepped into this wonderful Australian continent on a very hot day on 30th

August 2016, walked up the stairs of the Gothic-style School of Geosciences building, was assigned a desk and a comfy chair to brood in, I met my colleagues: Sarah, Rakib, Sabin,

Nick, Xuesong, Wenchao, Luke, Gilles, Amy, Rhiannon, Amanda and Carmen to which I am thankful for their friendship. Later, my list of wonderful friends increased after I got to know Chang, Xianzhi, Annie, Calvin and Maelis. I am especially very thankful to Chang and his family for their support and encouragement. I am greatly indebted to MingMing for giving me motivation and tremendous support. I would like to thank Matthew da Silva for opening his house to me and sharing his ideas on life, also helping me to improve my knowledge of English grammar. I also would like to thank Cumhur, Hadi, Haluk and my other friends in Sydney, as they made me feel like I am at home. I am thankful to the members of Basin Genesis Hub at The University of Melbourne for their technical support on Underworld geodynamics modelling code. I would like to thank Equinor, Oil Search and Chevron for their financial support.

My flirtation with literature and my love of the beauty in the details of everyday life coloured my Ph.D. life here in Sydney. Of course, I had difficulties, at times, but my romantic

iii environment was good enough to overcome my struggles. I was very lucky that whenever

I needed, the Quadrangle of The University of Sydney was always there for me to open my

heart and listen my stories, the writings of Karl Ove Knausgård, Anton Chekhov,

Dostoevsky, Samuel N. Kramer and Marcel Proust were in my rucksack to improve my

understanding of the people and life, my audio playlist, including songs by Sezen Aksu,

Harry Gregson, Anouar Brahem, Yilmaz Erdogan ready to play and help me to relax, the ducks of the Victoria Park were next door to fill me with peace as I watch their unceasing feeding. At times, I couldn’t help myself watching the ibis bird, or “the bin chicken”, searching for food in desperation among people’s leftovers, and taking break from time to time in the treetops. I hope that I will be filming their daily lives one day. I was very lucky to experience the lively suburbs of this city that I found great enjoyment walking through. It was a great pleasure to walk to the city centre, stop by the Australian War Memorial in Hyde

Park where I pay my respect to the history of this wonderful country and my hometown, and end up at the Opera House, making my favourite circle, breath enough air filled with ocean flavour, and get my strength up to survive until the following Friday. On my way back home, I had happy moments when I stopped by Xian noodle for dinner, always having been offered the best seat available. There, I enjoyed the delicious hot noddles and pancakes with my friends. After submission of the thesis, another phase of my life initiated. The COVID-19 pandemic strongly impacted my life as it did to others, but I was very lucky that I was offered a postdoc position at the University of

Wollongong, which helped me in transitioning from being a PhD student, in this chaotic period, to a different one that I just started exploring in my post-PhD life coming with many responsibilities. In that regard, MingMing played a big role in supporting my transition by sharing her life experience with me. Last, but not the least, I wish to express my heartfelt thanks to my parents for their tremendous support. I felt very safe whenever I heard their voices. During my journey, my much-loved sister and my brother always made me laugh, hence they deserve a big hug and, of course, nice gifts to be received soon! iv Returning Home, Ardahan, 2004 © Nuri Bilge Ceylan

v ABSTRACT

The Earth’s topography is dynamically evolving under the forcing of convective motions in its mantle. As tectonic plates override subducted slabs and rising plumes, they are pulled down and pushed up, hundreds of meters over durations of tens to hundreds of Myr, and over lateral length scales of thousands of kilometres. These vertical motions of plates impact on the evolution of sedimentary basins which are the Earth’s billion-year history tablets, with records of sea-level variations, climate changes, and evolution of species. They are also the major source for oil, gas, geothermal energy and water. To unravel the Earth’s geodynamic history and make better predictions about the Earth’s natural resources, it is of critical importance to understand the coupling between tectonic plates and the underlying mantle. In that regard, predictions for amplitude of dynamic topography derived from global mantle convection models don’t agree well with dynamic topography amplitudes derived from investigations based on sedimentary basins and ocean bathymetry (i.e. residual topography).

Furthermore, the geodynamic models considering mantle density anomalies, hitherto, cannot explain the long-wavelength, episodic and rapid subsidence and uplift of sedimentary basins, such as the enigmatic Neogene tilting of India and rapid tilting events in Australia since the

Late Cretaceous. In this thesis, first, I propose that, at shorter wavelengths less than 1,000 km, predictions for dynamic topography amplitude driven by mantle density anomalies can be improved by considering the non-Newtonian rheology of the upper mantle. In the second chapter, a torque-balance analysis of relative horizontal rigid plate motion suggests that basal shear stress, on its own, induces dynamic plate-scale tilting, producing elevation changes of a few tens of metres, comparable to eustatic sea-level changes. This finding motivates the further analysis of horizontal plate motion via 2D thermo-mechanical numerical experiments considering progressive transition in viscosity between the plate and underlying mantle. These numerical experiments show that horizontal plate motions are associated with gradients in basal

vi shear stress and normal stress, the latter generally dominating the plate’s dynamic topography.

This process could give insights into the Neogene tilting of India. In the following chapter, to

estimate the tempo of this dynamic plate tilting, a suite of analytical models is presented by

assuming that plate has constant viscosity and density. The results suggest that, after a

change in plate motion, the associated basal stresses can tilt the plate in ~50 kyr for a stress

anomaly wavelength of 3,000 km and a plate viscosity of 1023 Pa s. This rapid plate-tilting can explain the anomalous vertical motions of Australia that followed sudden plate accelerations or decelerations since the Late Cretaceous. In the last chapter, 2D numerical models simulating relative horizontal plate motion, but now including conjugate passive margins and a connecting ocean basin show that, during plate motion, the trailing edges of the plates are being compressed. This process could shed light on the post mid-Eocene episodic basin inversions documented along the south-east Australian margin. In summary, this thesis provides a broad perspective on the coupling between the plates and the underlying mantle. It predicts the tempo and magnitude of vertical motions of plates via a suite of analytical and numerical models and applies them to the rapid long-wavelength vertical motions of India and Australia.

vii AUTHORSHIP ATTRIBUTION

This thesis is based on the following four articles:

ARTICLE 1: Bodur, Ö. F., Rey, P. F. (2019), “The Impact of Rheological Uncertainty on Dynamic Topography Predictions”, Solid Earth, 10, 2167–2178, https://doi.org/10.5194/se- 10-2167-2019, 2019. [Ö.F.B designed and run the numerical experiments, drafted the manuscript, and P.F.R. helped in analysing the experimental results and finalising the manuscript.]

ARTICLE 2: Bodur, Ö. F., Rey, P. F., Houseman, G. A.,“Fast Tilting of Plates Driven by Relative Horizontal Plate Motion”, (Prepared for Submission) [Ö.F.B performed the torque-balance analysis, designed and run the numerical experiments. Ö.F.B drafted the manuscript and all authors analysed the results and discussed on their significance and application and contributed on finalising the manuscript.]

ARTICLE 3: Bodur, Ö. F., Houseman, G. A., Rey, P. F., “On the Tempo and Amplitude of Tilting of Tectonic Plates by Basal Stresses” (Prepared for Submission) [G.A.H. and Ö.F.B performed the analytical calculations and analysed the results. P.F.R. contributed on analysing the significance of the results and their application to Australia.]

ARTICLE 4: Bodur, Ö. F., Rey, P. F., Houseman, G. A.,“Contrasting dynamic topography and inversion of passive margins” (Prepared for Submission) [Ö.F.B. designed and run the numerical experiments and drafted the manuscript. All authors analysed the results, discussed on their significance and contributed on finalising the manuscript.]

In addition to the statements above, in cases where I am not the corresponding author of a published item, permission to include the published material has been granted by the corresponding author. Ömer Faruk Bodur

As supervisor for the candidature upon which this thesis is based, I confirm that the authorship attribution statements above are correct.

Patrice F. Rey

viii TABLE OF CONTENTS

ABSTRACT……………………………………………...... v

AUTHORSHIP ATTRIBUTION……………………………………………...vii

1. INTRODUCTION……………………………………………………………1

2. ARTICLE 1………………………………………………………………….32 The Impact of Rheological Uncertainty on Dynamic Topography Predictions

3. ARTICLE 2………………………………………………………………….45 Fast Tilting of Plates Driven by Relative Horizontal Plate Motion

4. ARTICLE 3………………………………………………………………….64 On the Tempo and Amplitude of Tilting of Tectonic Plates by Basal Stresses

5. ARTICLE 4………………………………………………………………….83 Contrasting Dynamic Topography and Inversion of Passive Margins

6. DISCUSSION………………………………………………………………99

7. CONCLUSION…………………………………………………………... 103

APPENDIX A (Fundamentals of Geodynamics)……………………………108

APPENDIX B (Supplement to Article 2)…………………………………....113

APPENDIX C (Supplement to Article 3)……………………………………123

APPENDIX D (Supplement to Article 4)……………………………………131

BIBLIOGRAPHY……………………………………………………………141 1. INTRODUCTION

Over millions of years, the Earth’s surface topography changes by plate-tectonic processes such as rifting of continents, subduction of oceanic plates and collision of continental blocks. The formation of the east-African rift and the rise of the Tibetan plateau are pronounced examples of tectonic topography, which are mostly localized around plate boundaries. As tectonic plates move relative to each other, they also record widespread anomalous subsidence and uplift that produce sedimentary basins. However, the theory of plate-tectonics is insufficient to explain those observed anomalous vertical plate motions.

Stratigraphic analyses of North American Craton basins revealed a Late Cretaceous long-wavelength tilting of the continent (Sloss, 1963). The thickness of strata observed in the marginal and interior basins was higher than that predicted by eustatic sea-level changes (Bond,

1976). Cross and Pilger (1978) suggested that subduction of a cold slab beneath the craton could have driven this tilt. The increased subsidence to the western margin was attributed to the cooling of the base of the crust by conduction as the colder subducted slab might have replaced a hotter asthenosphere under the craton. However, this explanation was not consistent with the rate of the observed subsidence that is notably shorter than the timescale of thermal diffusion. A physically plausible dynamic explanation came after unification of physics of the convecting mantle (Bénard, 1901; Rayleigh, 1916; Chandrasekhar, 1961) with the recently established plate-tectonic theory at the time (Runcorn, 1964; Toksöz et al., 1971; McKenzie et al., 1974; McKenzie, 1977; Parsons and Daly, 1983). Henceforth, it became clear that it was not thermal conduction, but the dynamical effects of the subducting slab (Mitrovica et al., 1989) that tilted the North American Craton (Fig. 1.1).

1 Figure 1.1: Illustration for two distinct ways of producing dynamic topography, proposed for the formation of stratal sequences across the North American Craton. As the subduction induces platform subsidence to the west, the heated mantle by supercontinent insulation induces dynamic uplift to the east (adapted from Burgess, (2019)).

These show that Earth’s convecting mantle has a significant impact on the evolution of the Earth’s surface. Density anomalies of sinking subducted slabs and rising hot plumes (Fig.

1.2a) induce dynamic vertical motions of the surface, producing dynamic topography (Pekeris,

1935; Runcorn, 1964; Morgan, 1965a; Richards and Hager, 1984; Hager and Clayton, 1989;

Molnar et al., 2015). In contrast to tectonic topography, dynamic topography can be induced in continents and ocean floor at a wide spectrum of wavelengths reaching to tens of thousands of km. The wavelength of dynamic topography is determined by the planform of the convecting mantle (Fig. 1.2b,c) (Gurnis et al., 1996; Zhong et al., 1996, 2007; Hoggard et al., 2016;

Arnould et al., 2018).

A comprehensive understanding of the dynamics of the Earth’s mantle and its interaction with overlying tectonic plates is of critical importance, because the dynamic vertical motions of plates can induce the flooding of continents (Bond, 1978; DiCaprio et al., 2009;

Heine et al., 2010) and the formation of sedimentary basins (Bond, 1976, 1978; Mitrovica et

2 al., 1989; Gurnis, 1990; DiCaprio et al., 2009; Petersen et al., 2010; Flament et al., 2013;

Burgess, 2019) by which valuable earth resources such as mineral deposits (Rey, 2013), oil

and gas can build up (Blewett, 2012). In that regard, many authors have used dynamic

topography to explain the sequence stratigraphy and present-day topography of various

Figure 1.2: a) Illustration of dynamic topography over a dense and buoyant spherical body beneath the plate (adapted from Eakins and Lithgow-Bertelloni (2018)). Numerical models simulating the mantle convection dominated by b) small wavelengths with many disconnected downwellings (blue) and upwellings (yellow), and c) possible longest wavelength (degree 1) with marked density anomalies in the opposite hemispheres (adapted from Zhong et al. (2007)).

continental and oceanic basins (Morgan, 1965b; Burgess and Gurnis, 1995; Burgess et al., 1997;

Lithgow-Bertelloni and Gurnis, 1997; Pysklywec and Mitrovica, 1998; Müller et al., 2000;

Moucha et al., 2008; Braun, 2010; Flament et al., 2013; Liu, 2015; Hoggard et al., 2016; Eakins

and Lithgow-Bertelloni, 2018; Burgess, 2019).

Improvements in computational power, numerical methods and tools to solve mantle

convection equations in discrete form allowed for advanced models of plate subduction (e.g.

Farallon plate subduction underneath the North American Craton) coupled with forward

stratigraphic models, which can reproduce the major stratal sequences across continents

(Burgess and Gurnis, 1995; Burgess et al., 1997). These models are quite insightful, but due to

computational constraints, they often neglect the complex physical properties of the crust and

mantle, such as the variation of viscosity of rocks with pressure, temperature and strain-rate,

which is an empirical observation derived from rock deformation experiments (Post and Griggs,

1973; Karato and Wu, 1993; Hirth and Kohlstedt, 2003). In Chapter 2 of this thesis, 3D thermo-

mechanical numerical models which account for such complex viscosity function are presented.

These numerical models solve for the instantaneous mantle flow driven by a spherical density anomaly at depth. The induced total normal stress at the surface is converted to dynamic topography. The numerical models show that, at short wavelengths less than 1,000 km, the predictions for the amplitude of dynamic topography notably reduce if one uses a plausible viscosity function compared to models in which crust and mantle are isoviscous.

The list of mechanisms driving the Earth’s topography is not complete. There are many records of plates experiencing anomalous vertical motions that cannot be attributed to density anomalies or records of eustatic sea-level change. For example, the Australian Bight and Eucla

Basins recorded successive anomalous subsidence and uplift events in the mid-to-late Eocene

(Fig. 1.3a). These events cannot be explained by sea-level fluctuations (Li et al., 2003) or dynamic topography (DiCaprio et al., 2009). Likewise, the Atlantic margin of North America experienced rapid subsidence events at ~83 Ma and ~56 Ma (Heller et al., 1982) that cannot be attributed to density anomalies in the mantle or sea-level fluctuations. These observations create some confusion because the predicted direction of the tilting of the North America during these episodes of anomalous vertical motions is reverse (downward to the east) to what the Farallon slab has produced (downward to the south-west). Furthermore, along the same mid-ocean ridge spreading system, two major hiatuses were identified in Western Europe at

~55 Ma and ~23 Ma Vibe et al. (2018). These events also cannot be explained by the above- mentioned mechanisms.

4 What is remarkable is that those anomalous subsidence and uplift events in Southern

Australia and North Atlantic occurred when the sea-floor spreading increased or decreased at

the mid-ocean ridge of the boundary of those plates (Fig. 1.3b, 1.4). For example, when the

Figure 1.3: a) Illustration of mid-to-late Eocene successive anomalous subsidence and uplift events in the Eucla and Bight Basins of southern Australia (adapted from Li et al., (2003)). b) Magnetic anomalies around the southern Australia and Tasmania, and sea-floor half spreading rates shown inside the brackets (see Li et al., (2003) and references therein). The grey lines indicate four unconformities correlated with variations in spreading rate.

full-spreading rate between the Australia and Antarctica increased from ~0.8 cm yr-1 to ~3.0

cm yr-1, the Australian plate tilted downward to the south, resulting in flooding of the Eucla and Bight Basins of southern Australia (Shafik, 1990; Li et al., 2003). These suggest that there could be a dynamic link between change in spreading rates (or change in relative plate motion), and anomalous vertical motions. In Chapter 3 of this thesis, this link is addressed by investigating how different components of stresses produced at the base of the tectonic plates, driven by changes in the flowing mantle, can affect the topography of the plates. By using torque analysis, it is shown that the rigid plates can tilt by basal shear stress. The calculations show that induced topography amplitude is a few tens of metres, which is on the order of changes in sea-level. The 2D thermo-mechanical numerical models also suggest that a change

5 in the horizontal flow field of the mantle, therefore a change in relative horizontal plate motion, is associated with horizontal gradient in normal stress and shear stress acting on the base of the plate. The model results suggest that plates’ dynamic topography is generally dominated by the normal stress component, inducing hundreds of metres of topography. These findings are used to explain the Neogene tilting of India. In Chapter 4, using analytical models, it is aimed to reveal not just the amplitude of induced topography by tilt of the plate but also the time required for the tilt and its sensitivity to wavelength and basal stress amplitude. The analytical models suggest that for wavelengths less than 3,000 km, the induced topography ranges between ~10 m and ~100 m for harmonic spatial variation in shear stress amplitudes of 1 MPa and 10 MPa, respectively. For basal stress wavelength of 3,000 km and 1023 Pa s plate viscosity,

the relaxation of the topography takes ~50 kyr (same for shear and normal stresses). This shows

the rapid tilting of plates in response to changes in basal stress conditions. This analysis is used

in explaining the three distinct vertical motions of the Australian plate that are coeval to the

observations in Australian plate velocity changes.

Figure 1.4: Average see-floor spreading rates around the North Atlantic and its correlation with hiatus area in NE Europe (adapted from Vibe et al. (2018)).

In Chapter 5 of this thesis, the 2D thermo-mechanical numerical models are presented to

investigate the dynamic topography of passive margins. They show that tilting occurs in both

oceanic and continental plates, but at different amplitudes. They also suggest that the trailing

edges of plates are compressed by large-scale counter-flow induced beneath the oceanic

6 lithosphere during horizontal plate motion. The model results shed light on the dynamic topography of the Australian plate in the mid-Eocene and coeval episodic basin inversion events observed in the Otway Basin of south-east Australia.

It’s important to illuminate on the contribution of this thesis to the rapidly growing literature. In that regard, in the following section, discoveries in earth sciences that are of critical importance to introduce to the subject are presented, mostly in chronological order but in a way that maintains the logical flow throughout the text. In that section, in places consistent with the flow of the text, this thesis’ contribution to the existing literature is emphasized with more details. This section can also be regarded as a brief and focused review on plate tectonics and mantle convection, but does not aim to constitute an all-inclusive literature review on those subjects. As our current understanding of the geodynamics of the Earth is largely built on discoveries about mantle convection, related concepts and terminologies are widely cited throughout the thesis.

Appendix A of this thesis expands on some of the fundamentals in geodynamics. This section is mainly prepared for students and teaching purposes; therefore, the ideas expressed there are in a prose different from that used in the rest of the text. It briefly compares the difference in dynamics between slow processes in the Earth’s interior and fast motion of objects around us. In that regard, the idea of negligible momentum of plates is described with examples, and some fluid dynamical terminologies are used with explanations to clarify its meaning and significance.

1.1 Development of Ideas in Plate Tectonics and Mantle Convection

The purpose of this section is to review some of the developments in plate tectonics

and mantle convection with an emphasis on areas closely linked with the subject of this thesis.

The intention is to explore the progression of ideas about the interaction between the Earth’s

surface and the deep mantle.

Our knowledge about the Earth’s system is thanks to geologists’ endeavours, for

decades, aimed at understanding when and how the Earth’s landscape formed, and their

continuing investigations into the nature of forces driving the tectonic events in our planet. On

the shoulders of giants such as Galileo Galilei, Nicolaus Copernicus and René Descartes who

are the key drivers for the scientific revolution, in 1788, James Hutton proposed that the surface

of the Earth has been continuously exposed to change, and will continue to change with the

same laws as applied before. In his vision of the Earth, the past and the present are connected

through infinitesimal but everlasting changes, so-called uniformitarianism, and he predicted

that those changes should be governed by the same laws of physics (Hutton, 1788).

In the 19th century, Charles Lyell and Charles Darwin built their theories upon this

idea and introduced the principles of geology which made a big impact on our knowledge of

the origin of the Earth. In 1915, Alfred Wegener proposed that continents were together in the

past, and must have separated from one another in time (Wegener, 1915). He came up with

this idea with hindsight of the fitting of the South America and the South Africa, but also gave

evidence from similarity of plant and animal species between those continents (Wegener, 1915).

Figure 1.5: Illustration of subsidence of the surface by loading of the ice sheet and post-glacial rebound of the lithosphere in a viscous mantle subsequent to melting of the ice sheet, modified after Fowler et al. (2005, p. 224).

However, he was lacking a reasonable dynamic explanation for why continents should be

drifting away from each other and, unfortunately, he was misled by an intuition that a

centrifugal force must be forcing them to drift from the poles to the equator, mainly because of

the Earth’s rotation. His proposal was to be refuted by Harold Jeffreys, an exceptional

theoretical physicist of his era. However, Jeffreys was also mistaken by his intuition that the strength of the continents were too much to permit them to drift around the surface of the Earth

(Molnar, 2015, p.6).

In 1935, 20 years after Wegener’s proposal, Norman Haskell calculated that the

Earth’s mantle must have an average viscosity of ~1021 Pa s in order to explain the uplift rates in Fennoscandia after removal of the ice sheet, so-called post-glacial uplift (Haskell, 1935,

1936). When an ice sheet depressing the land starts to melt, the surface will start to rebound in order to equilibrate the pressures, by isostasy, at rates depending on the viscosity of the mantle underneath (Figure 1.5). This provided a glimpse into the tempo of the convection in the Earth’s mantle. From a retrospective look, one can confirm that vertical layering of the Earth’s interior

Figure 1.6: Illustration for statistical and geometrical fitting of South America and South Africa continental boundaries in Mercator projection, modified after Bullard et al. (1965). The blue and red colours show the gaps and overlaps, respectively.

9 as well as vertical motions of the Earth’s surface were recognized long before the confirmation

of the continental drift hypothesis. In 1962, Harry Hess, for the first time, proposed his idea of

Figure 1.7: Magnetic anomalies across the Reykjanes Ridge, nearby Iceland. The ridge axis is indicated by the straight lines. The figure is adapted from Heirtzler et al. (1966) and Peltier (1989).

how continents could drift. He outlined the mechanism of sea-floor spreading, the rise and

cooling of magma at mid-ocean ridges, forming new ocean floor (Hess, 1962). Subsequently,

the drift hypothesis has been taken more seriously, and it was Bullard and his colleagues who statistically tested the fit of South Africa and South America around the Atlantic as well as the fit of the other continents (Fig. 1.6). They concluded that the fit between those continents cannot be explained just by chance (Bullard et al., 1965). In 1963, the revolutionary evidence came from Vine and Mathews. They mapped the magnetic anomalies over the north Atlantic as well as northwest Indian Ocean, and recognized that the data support Hess’ sea-floor

spreading hypothesis (Vine and Matthews, 1963). It has been understood that as the lava

ascends to the surface it cools and produces a new oceanic crust, and more importantly, it

records the Earth’s ambient magnetic field as it cools. Because the Earth’s magnetic field

changes its polarity, about three reversals per million years for the last 50 Myr (Molnar, 2015,

p.17), the oceanic crust formed at different time periods can record magnetic fields aligned in

a reverse direction. Once we know the history of those reversals, we can determine the

spreading rate, therefore the magnitude of the relative motion of plates. An example of a

magnetic anomaly in Reykjanes ridge, near Iceland, is given in Figure 1.7 (Heirtzler et al.,

1966; Peltier, 1989). With the detailed mapping of the ocean floor, now available in a digital

Figure 1.8: The digitised version of the age of the oceanic crust (Müller et al., 2008) in Molleweide projection. The continents are coloured as light gray, and continental margins are medium gray. The black lines represent the plate boundaries (Bird, 2003). The figure is adapted from Müller et al. (2008).

version (Fig. 1.8), we have a very good knowledge about the age of the ocean floor younger

than ~280 Myr across the globe (Müller et al., 2008). Accompanied with the available open-

source software (e.g. GPlates), one can use age grids (i.e. digital version of the age of the ocean floor) in creating plate-reconstruction models (Boyden et al., 2011; Gurnis et al., 2012), which has proved to be quite useful in solving many geodynamic problems (Zahirovic et al., 2014;

Williams et al., 2016).

It was Tuzo Wilson who drew the big picture of plate tectonics. In 1965, he published a paper in Nature, in which he described the characteristics of plates and plate boundaries

(Wilson, 1965). He described the characteristics of plates and explained their motion relative to each other. However, his version of plate tectonics was imperfect, because he did his analysis assuming a flat Earth, rather than a spherical geometry, and later he mentioned his mistake in

11 a personal conversation to Kevin Burke (Burke, 2011). Thanks to those early works, the

foundations of the plate tectonic theory have been established. However, it was still unclear if

this theory can be valid for the times beyond Mesozoic in the Earth’s geological past. It was

already suggested, through vigorous work, by Alexander Du Toit, that continents should have

assembled in the Palaeozoic to form the big-but-not-super continents of Gondwana near the

south pole and Laurasia around the equator extending towards the north pole (Du Toit, 1937).

Tuzo Wilson, building his ideas upon Toit’s proposal, proposed that the north Atlantic margin

must have experienced more than one opening along the identified sutures near the margin (Fig.

Figure 1.9: The maps showing the closure and opening of the North Atlantic and prediction of the existence of a Proto-Atlantic Ocean along the suture near the margin indicated by red line. The figure is adapted from Wilson et al. (2019).

1.9), suggesting the closure of a proto-Atlantic Ocean before the most recent opening event

(Wilson, 1966). It should be noted that the existence of the proto-Atlantic Ocean was first

proposed by Émile Argand (Argand, 1924), (as highlighted in Wilson et al. (2019)), a primary

supporter of the theory of continental drift. Then it was realized that continents should have

experienced repeated assembly and breakup (i.e. Wilson Cycle), mostly in the proximity of the

same continental boundary. This finding was very important in the sense that it changed the

theory of plate tectonics from a limited concept to a geological and global phenomenon (Wilson

et al., 2019). These developments paved the way for the proposal of a super-continent cycle,

suggesting that these processes should be systematic, thereby occurring at every ~440 Myrs

12 rather than being random (Nance et al., 1988), with implications for the Earth’s heat budget

and for mantle dynamics in the Mesozoic (Anderson, 1982; Coltice et al., 2007). The proposal

of the ancient super-continent of Pangea put Alfred Wegener’s idea (Wegener, 1915) into a

global context in deep time.

Between 1960-1970’s, a nearly complete picture of the theory of plate tectonics was

drawn. In 1967, Dan McKenzie recognized the need to analyse the motion of plates as rotations

around poles (by Euler’s rotation theorem) across the spherical Earth (McKenzie and Parker,

1967). This analysis was very important in the sense that it allowed quantitative description of the relative motions of plates and created a framework for future plate reconstructions (Minster et al., 1974; Solomon and Sleep, 1974; Cox and Hart, 2009; Seton et al., 2012). The following year, a careful analysis of plate-tectonics, considering the sea-floor spreading history, was provided by Xavier Le Pichon (Le Pichon, 1968). The “new global tectonics” as defined by

Isacks, Oliver and Sykes, was convincingly complete (Fig. 1.10), and well supported by

seismological evidence (Isacks et al., 1968).

After these achievements, it was well accepted that the consumption of plates by

Figure 1.10: Schematic representation of the possible occurrences in the new global tectonics. Different plate boundaries, plate motion and mantle flow. The figure is adapted from Isacks et al. (1968). subduction and creation of new crust at the mid-ocean ridges were part of the same system.

However, the link between plate motion and mantle convection was still not clear, partly due

to a lack of understanding of the thermal and chemical structure of the plates as well as

planform of convection in the mantle which can be quite unconstrained by the uncertainties in

13 rheology (i.e. viscosity) of mantle rocks. Post and Griggs (1973) proposed that Earth’s mantle

should behave like a Non-Newtonian fluid, meaning that the deformation of rocks is not linear

with the applied stress. They proposed the following formula for the rate of deformation of

olivine (i.e. strain rate):

�̇ = �(�)� (1) where �̇ is the strain rate, B(z) is a factor varying with depth, � is shear stress and n is the stress exponent. They went a step further and showed that the uplift data from Fennoscandia fits to

Equation 1 when n=3.21. Likewise, high-pressure laboratory experiments of deformation of olivine had already shown that a change in phase to spinel occurs at pressures corresponding to ~400 km depth (Ringwood, 1956; Akimoto and Fujisawa, 1966), that could imply a viscosity contrast across this boundary, which put more complications on the rheology of the mantle.

Over the last few decades, much effort has been given on constraining the viscosity of mantle rocks (Karato and Wu, 1993; Ranalli, 1995; Hirth and Kohlstedt, 2003) at relatively higher pressures and temperatures. They revealed that it should have the following form:

� � [()/] �̇ = �( �) ( �) � (2)

,where A, �, b, d, m and n are pre-exponential factor, shear modulus, Burgers vector and grain

size, and exponents for grain size and stress respectively. E, V, T, P and R in the exponential

term are activation energy, activation volume, temperature, pressure and ideal gas constant,

respectively. Although it was of high interest to consider the non-linear rheology of the mantle

Figure 1.11: Geometry of the problem that Jason Morgan formulated in 1965. The Earth’s interior is assumed to be either a) a single layer, or b) is divided into two isoviscous layers, the upper representing the lithosphere and lower including a spherical density anomaly at depth. The figure is adapted from Morgan (1965a).

14 in the numerical mantle convection models, it was both computationally hard to solve the

equations and was quite expensive to afford for the computation (i.e. $1000 hr-1 as McKenzie

(2018) reports).

Thanks to the newly available satellite gravity data at the time, Runcorn (1964) postulated that Earth’s convective currents should be creating the long-wavelength (i.e. low- degree harmonics) of the Earth’s gravity anomalies. Motivated by this finding, a formulation between topography, gravity anomalies and mantle density anomalies has been made by Jason

Morgan via two successive papers in 1965, in first of which he presented an analytical solution to topography and gravity anomaly induced by a sinking or rising sphere (Morgan, 1965a), and in the latter he applied his hypothesis to the mid-Atlantic rise (Morgan, 1965b). Although his analytical solutions were valid only for isoviscous 1 layer (Fig. 1.11a) and 2-layers comprising of the lithosphere and underlying mantle (Fig. 1.11b), they gave much insight on the link between vertical variations in viscosity in the Earth’s interior and the magnitude of topography induced by density anomalies at depth. There followed spherical harmonics approach to the

Earth’s dynamic topography and long-wavelength geoid anomalies that expanded the discussion to the whole mantle with radial variations (Parsons and Daly, 1983; Hager et al.,

1985; Hager and Clayton, 1989) and harmonic lateral variations in viscosity (Richards and

Hager, 1989).

Over last few decades, seismically mapped density anomalies (e.g. (Ritsema et al.,

2004)) have been used in 3D global mantle convection models to infer the Earth’s present-day dynamic topography (Hager et al., 1985; Hager and Clayton, 1989; Richards and Hager, 1989;

Lithgow-Bertelloni and Silver, 1998; Steinberger, 2007; Moucha et al., 2008; Conrad and

Husson, 2009; Flament et al., 2013; Steinberger et al., 2017). Dynamic topography predictions

(for the present day) are commonly compared with residual topography which is calculated by removing the isostatic components from the Earth’s total topography, presenting only the

15 dynamic components (Crough, 1983; Cazenave et al., 1989; Davies and Pribac, 1993; Hoggard et al., 2016). Dynamic topography predictions and estimations of residual topography, in theory, must give the same amplitude for the same wavelengths, however they are increasingly at odds with increase in wavelength (Hoggard et al., 2016; Steinberger, 2016; Cowie and Kusznir,

2018). Recent works using more advanced Earth-like mantle convection models considering

Earth’s shallow and deep mantle addressed this problem (Arnould et al., 2018; Osei Tutu et al.,

2018; Bauer et al., 2019; Davies et al., 2019). Despite the significant progress, the problem remains to be unsolved and the amplitude of dynamic topography is predicted to be significantly higher than residual topography measurements for a wide range of wavelengths

(Davies et al., 2019; Flament, 2019; Steinberger et al., 2019).

There are also other aspects to be considered such as calculation method of spectral contributions to residual topography and the spatial variability of the admittance function

(Parsons and Daly, 1983; Colli et al., 2016; Yang and Gurnis, 2016; Hoggard et al., 2017; Yang et al., 2017; Davies et al., 2019). These hopefully will be better constrained with more available residual topography data and improved numerical models (Bauer et al., 2019; Davies et al.,

2019; Flament, 2019). In that regard, our current understanding of the Earth’s dynamic topography should be grounded on clearly defined terms and explanations, on which one can follow a step-by-step approach to build upon the currently existing ideas (Molnar et al., 2015).

Chapter 2 of this thesis addresses the Earth’s dynamic topography by making use of Morgan’s analytical works and 3D thermo-mechanical numerical models. These models are designed in a way that they can be compared with their analytical counterpart in the case for isoviscous mantle rheology. Our current knowledge about the viscosity of rocks depending on temperature, pressure and strain-rate in the Earth’s mantle is also considered. The numerical model results show that, at small wavelengths (<1,000 km), the misfit in amplitudes between observed

16 residual topography and dynamic topography predictions can partly be attributed to Non-

Newtonian viscosity of the mantle.

Significant progress has been made in the study of the Earth’s tectonic plates and its

connection to the deep mantle thanks to Dan McKenzie’s interest and efforts in understanding

the dynamics of plate creation at the mid-oceanic ridges and the thermal profile of an oceanic

lithosphere. In 1967, simply by solving the heat equation in 2-D (Equation 3), he calculated

the isotherms of an oceanic lithosphere (McKenzie, 1967, 2018):

! ! �� + � ∙ ∇� = � + + � (3) ! !

where r is density, CP is heat capacity at constant pressure, v is velocity in x-direction, and H

is radioactive heat generation. McKenzie simplified Equation 3 by neglecting the contribution

from radioactive heat generation (H) and term as they come out to be very small compared

to other terms. Therefore, he solved the following by also assuming that in a ridge system

! ! vertical diffusion of heat must be significantly higher than horizontal diffusion ≫ , ! !

therefore:

Figure 1.12: a) Isotherms in an oceanic lithosphere calculated by using equation 5, modified from McKenzie (1967). b) Comparison of heat flow measurements from the Pacific Ocean with analytical model prediction by assuming 50 km thick plate having 550 °C constant lower boundary temperature. The figure is adapted from McKenzie (1967).

17 ! �� = � (4) !

McKenzie’s result is plotted in Figure 1.12 (McKenzie, 2018). Despite the good fit between

his model prediction with the heat flow data in the Pacific Ocean, the scattering of the heat

flow at longer distances from the ridge or at older oceanic floor was confusing. His reasoning

was only based only on conduction of the heat, and it was to be discovered, especially from the

heat flow measurements in Galapagos Spreading centre, that heat transport by convection can

be quite significant in the Earth’s mantle (Williams et al., 1974). However, McKenzie’s finding gave insights on the thermal structure of plates and motivated further studies about the physical properties of the mantle. The heat flow problem has been solved later by Barry Parsons and

Dan McKenzie with the finding that convective instability, small-scale convection and removal

of lower part of mantle lithosphere (erosion of the base) could be causing the increase in heat

flow (Parsons and McKenzie, 1978). The convective thinning (large scale) and removal of

dense lithosphere from the upper part of mantle lithosphere and the crust has been used to

explain the rising of the Tibetan Plateau (Houseman et al., 1981). However, present-day surface

wave tomography observations suggests the existence of a thick (~200 km) lithospheric root

(not removal) beneath the Tibetan Plateau (McKenzie, 2018). Those show the revision of ideas

as our knowledge about the Earth’s interior increases. In that regard, the development of ideas

in continental tectonics continued at its own pace and much progress has been made on

understanding the formation of mountain belts.

Advances on geodynamics of the Earth’s mantle has much benefited from the theory

of Rayleigh-Bénard convection (Bénard, 1901; Rayleigh, 1916) which came to existence by

much interest in understanding heat transport and convection in planetary bodies. A

dimensionless parameter has been introduced (i.e. Rayleigh number) to designate the vigour of

convection in a viscous fluid:

" � = (5)

18 where a is thermal expansion coefficient, d is depth of the convective layer, �� is temperature variation across the boundaries, � is thermal conductivity, g gravitational acceleration and � is

kinematic viscosity (Brindley, 1967). The Rayleigh number has been widely used in Earth’s

mantle convection problems. In 1974, McKenzie and his colleagues published one of the first

2D thermo-mechanical numerical models of mantle convection in the context of understanding

plate motions, surface heat flow and topography (McKenzie et al., 1974). They used constant

viscosity with varying temperature boundary conditions, therefore at different Rayleigh

numbers. This work partly originated from the idea that horizontal gradients of temperature

should be driving the convection in the mantle (Pekeris, 1935).

Figure 1.13: The numerical model results in a dynamic quasi-steady state (McKenzie et al., 1974). The boxes represent a) gravity anomaly, b) surface heat flow, c) isotherms, d) horizontal average of temperature profile, e) surface topography, f) top boundary horizontal velocity, and g) streamlines. The figure is adapted from McKenzie et al. (1974).

One of the key result of those preliminary numerical models was that they showed the

formation of a plume, and an almost isothermal region for all scenarios. Furthermore, the

19 predicted gravity anomaly was positive over the upwelling mantle, because the contribution

from the deformation of the surface was dominant to that caused by temperature variations in

the mantle. The resulting topography was a few hundreds of meters, and the lateral speed of

top boundary (interpreted as induced plate velocity) was less than 2.0 cm yr-1 (Fig. 1.13).

However, three important characteristics of the Earth’s interior are considered in those models.

The first is the lateral size of the box, which requires large aspect ratios to comply with typical

plate lengths, second is that the temperature gradient is around a quarter of the real Earth, and

most importantly, viscosity is constant. After three years, McKenzie published another paper

in which he accounted for the variation of viscosity with temperature in the form of Equation

6 (McKenzie, 1977).

�(�) = �� (6)

where � = 7.4 × 10 �� · �, and C is constant but varied between the experiments. He

observed that the convective cells in the model couldn’t form in aspect ratio greater than 6,

which made him to conclude that 3-D factors could also play in this ratio, despite his models

Figure 1.14: The contribution to the surface topography (x) by pressure perturbations (P1, dashed lines) and deflection of the streamlines (dotted lines) a) for constant viscosity and b) viscosity depending on the temperature with C=0.04 and Rayleigh number of R=2.4 x 105 with heat supplied from the bottom boundary of a square box (McKenzie, 1977). The horizontal axis is dimensionless distance from the left side of the box. The topography in magnitude can be found by multiply the values on the y-axis 20.1. The figure is adapted from McKenzie (1977).

20 were 2-D. More importantly, he captured the pressure perturbation, which is variation in

pressure by convection, and its contribution to the surface topography (Fig. 1.14). His models

having viscosities depending on temperature showed formation of a cold, therefore high

viscosity, region near the top boundary of the models. One of the key results was that the

surface above a mantle upwelling may subside when the viscosity depends on temperature (Fig.

1.14-b), which is quite counter-intuitive. However, this has later been proved to be incorrect

due to a failure to recognise a loss of accuracy in McKenzie’s calculations as first numerically

shown (Fig. 1.15) by Jian and Parmentier (1985). Later, this has been confirmed by showing

that the inaccuracy is amplified when there is a large viscosity gradient in the numerical model

(Moresi and Parsons, 1995). In their models, although perturbation pressure tends to subside

Figure 1.15: The contribution to the surface topography by pressure perturbations (-P, dotted lines) and deviatoric normal stress (dashed lines) a) for constant viscosity and b) viscosity depending on the temperature with C=4 and R=3 x 105 , similar to McKenzie (1977), and c) C=5. Note the overall positive topography above the rising plumes (left side of the box in cases b and c) where viscosity depends on temperature. The figure is adapted from Jian and Parmentier (1985).

the surface, the total topography is still positive over an upwelling region (Moresi and Parsons,

1995). These suggest that variations in viscosity of the Earth’s interior had caused

complications for the interpretation of the early numerical models.

We know from rock deformation experiments that viscosity in the Earth’s interior

can be more complex due to simultaneous dependence on pressure, temperature and strain-rate

(Karato and Wu, 1993; Hirth and Kohlstedt, 2003). Therefore, the resulting pressures and

deviatoric stresses induced in the mantle can be best captured if one considers the complex

viscosity function with a resolution high enough to account for steep viscosity gradients in

21 thermo-mechanical numerical models. In that regard, Chapter 3 presents how normal stress

gradients can form during a plate motion and/or change in horizontal flow field of the mantle

which produce large-scale dynamic tilting of the lithosphere.

Since the inertia of plates are negligible (Appendix A), the plates must be in dynamic equilibrium, therefore sum of the torques applied on plates by the surrounding forces must vanish for each plate (i.e. equal to zero):

Figure 1.16: a) Torque balance in a spherical Earth and b) components of boundary forces acting on the plates, adapted from Forsyth and Uyeda (1975).

In many applications, geologists have used the centre of the Earth as the reference point in spherical coordinates (Fig. 1.16a), because in that reference frame vertical forces do not play any role in the computation of the radial component of the torque. It’s a way of considering only the balance of forces in the horizontal surface and is useful in addressing plate velocities.

Forsyth and Uyeda (1975) used this method to find the relative importance of plate-driving forces (Fig. 1.16b). Since then, authors have been typically using the better-known forces on one part of the plate boundary to constrain the less well-known forces on another part of the plate boundary using torque-balance (Copley et al., 2010; Warners-Ruckstuhl et al., 2010;

Iaffaldano and Bunge, 2015; Stotz et al., 2018). Chapter 3 of this thesis makes use of horizontal torque-balance, which is different from (radial) torque-balance by choice of reference point for calculation of the torque applied on plates. In horizontal torque-balance, centre of mass of

22 plate is used as a reference point (Chapter 3 and Appendix B), because the zero-torque principle is independent of the coordinate origin. The balance of horizontal torque can often be complicated because of inclusion of vertical forces, but it provides insights on investigating dynamic vertical motions of plates.

Dynamic topography and its temporal evolution at continental interiors and passive margins have received much attention in regard to their potential of preserving high value energy resources, especially for hydrocarbon explorations. The early models for thermo- mechanical evolution of sedimentary basins had already been established by McKenzie (1978) and Wernicke (1985) with a pure-shear and a simple-shear model, respectively. These paved the way for future works that can analyse the contributions of the convective mantle to passive margin morphology and the sedimentation history. In that regard, along with the quantitative understanding of the ocean floor and improvements in sequence stratigraphic methods as well as growing knowledge on the eustatic sea-level variations (Haq et al., 1987), it became possible to identify anomalous vertical motions of passive margins (Sloss and Speed, 1974; Heller et al., 1982; Burgess et al., 1997; Müller et al., 2000; Japsen et al., 2012; Green et al., 2018).

These revealed that many continental margins were exposed to anomalous subsidence and subsequent uplift and exhumation long after their formation (i.e. post-rift phase).

Preliminary models of post-rift evolution of passive margins generally focus on the effect of a steep thermal gradient at the base of continental margins, driving edge-driven convection (King and Anderson, 1995, 1998; Farrington et al., 2010; Ramsay and Pysklywec,

2010, 2011; van Wijk et al., 2010; Currie and van Wijk, 2016). Another proposed model is the impact of a mantle plume (Yamato et al., 2013). What is striking is that those passive margins were exposed to episodic basin inversion events long after their formation. The pronounced examples are the north-east Atlantic (Doré et al., 1997), Angola Basin of the Western Africa

(Hudec and Jackson, 2002) and Otway Basin of south-east Australia. The problem with

23 previously proposed models is that they cannot capture the basin inversion events and dynamic vertical motions at the same time. Furthermore, the lack of any plume-related magmatism at some of those continental margins, especially in south-east Australia (Meeuws et al., 2016), rules out the plume model as a driving mechanism. Chapter 5 of this thesis proposes an alternative dynamic explanation for the episodic vertical motions of passive margins and coeval basin inversion events. A suite of 2D thermo-mechanical numerical models are created to simulate the dynamics of conjugate passive margins formed by extension of the lithosphere for

~34 Myrs. The numerical models suggest that changes in horizontal mantle flow field tilts the overlying plate (including oceanic and continental parts), and results in subsidence of the trailing edge and uplift of the leading edge of the plate. The dynamically tilted plate has different topographic gradients in the oceanic and continental parts. As the plate moves over the asthenosphere, a large-scale convective cell forms beneath the oceanic domain, which applies compressive viscous stresses on the trailing margin, resulting in folding and inversion of the pre-existing faults. The model results are used to explain the anomalous subsidence and uplift events in Bight and Eucla Basins of the south Australia, and the episodic inversion events in Otway Basin of south-east Australia.

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31 2. ARTICLE 1

[Article, EGU Solid Earth]

The Impact of Rheological Uncertainty on Dynamic Topography Predictions

Ö. F. Bodur , P. F. Rey,

EarthByte Group, School of Geosciences, The University of Sydney, NSW 2006, Australia

32 Solid Earth, 10, 2167–2178, 2019 https://doi.org/10.5194/se-10-2167-2019 © Author(s) 2019. This work is distributed under the Creative Commons Attribution 4.0 License.

The impact of rheological uncertainty on dynamic topography predictions

Ömer F. Bodur1,a and Patrice F. Rey1 1EarthByte Group and Basin Genesis Hub, School of Geosciences, The University of Sydney, Sydney, NSW 2006, Australia anow at: GeoQuEST Research Centre, School of Earth and Environmental Sciences, University of Wollongong, Wollongong, NSW 2522, Australia

Correspondence: Ömer F. Bodur ([email protected])

Received: 12 April 2019 – Discussion started: 23 April 2019 Revised: 14 November 2019 – Accepted: 23 November 2019 – Published: 20 December 2019

Abstract. Much effort is being made to extract the dynamic 1 Introduction components of the Earth’s topography driven by density heterogeneities in the mantle. Seismically mapped density anomalies have been used as an input into mantle convection The Earth’s mantle is continuously stirred by hot upwellings models to predict the present-day mantle flow and stresses from the core–mantle boundary and by subduction of colder applied on the Earth’s surface, resulting in dynamic topog- plates from the surface into the deep mantle (Pekeris, 1935; raphy. However, mantle convection models give dynamic to- Isacks et al., 1968; Molnar and Tapponnier, 1975; Stern, pography amplitudes generally larger by a factor of ∼ 2, de- 2002). This introduces temperature and density anomalies pending on the flow wavelength, compared to dynamic to- that stimulate mantle flow and forces dynamic uplift or sub- pography amplitudes obtained by removing the isostatically sidence at the plates’ surfaces (Gurnis et al., 2000; Braun, compensated topography from the Earth’s topography. In this 2010; Moucha and Forte, 2011; Flament et al., 2013). Dy- paper, we use 3-D numerical experiments to evaluate the ex- namic topography can affect the entire planet’s surface with tent to which the dynamic topography depends on mantle varying magnitudes. Because it is typically a low-amplitude rheology. We calculate the amplitude of instantaneous dy- and long-wavelength transient signal, it is often dwarfed by namic topography induced by the motion of a small spher- isostatic topography associated with variations in the thick- ical density anomaly (∼ 100 km radius) embedded into the ness and density of sediments, crust and mantle lithosphere. mantle. Our experiments show that, at relatively short wave- For the present day, the observational constraints on dy- lengths (< 1000 km), the amplitude of dynamic topography, namic topography come from residual topography measure- in the case of non-Newtonian mantle rheology, is reduced by ments (Hoggard et al., 2016). Residual topography is cal- a factor of ∼ 2 compared to isoviscous rheology. This is ex- culated by removing the isostatically compensated topogra- plained by the formation of a low-viscosity channel beneath phy from the Earth’s topography (Crough, 1983; Cazenave the lithosphere and a decrease in thickness of the mechanical et al., 1989; Davies and Pribac, 1993; Steinberger, 2007, lithosphere due to induced local reduction in viscosity. The 2016). The comprehensive work from Hoggard et al. (2016) latter is often neglected in global mantle convection models. revealed that residual topography varies between ±500 m at Although our results are strictly valid for flow wavelengths very-long wavelengths (i.e. ∼ 10 000 km) and can increase less than 1000 km, we note that in non-Newtonian rheology up to ±1000 m at shorter wavelengths (i.e. ∼ 1000 km). all wavelengths are coupled, and the dynamic topography at However, these residuals depend on our knowledge of the long wavelengths will be influenced. thermal and mechanical structure of the lithosphere and therefore may not be an accurate estimation of the deeper mantle contribution to the Earth’s topography. Another approach to constrain present-day Earth’s dynamic topogra-

Published by Copernicus Publications on behalf of the European Geosciences Union.

33 2168 Ö. F. Bodur and P. F. Rey: The impact of rheological uncertainty on dynamic topography predictions phy involves numerical modelling of present-day mantle flow where g is the gravitational acceleration, δρ is density dif- using seismically mapped density anomalies as an input ference between the anomaly and the ambient material, r is (Steinberger, 2007; Moucha et al., 2008; Conrad and Hus- radius of the sphere, and D is distance from the surface to son, 2009). However, this method requires a detailed knowl- the centre of the anomaly (modified from Morgan, 1965a, edge of the viscosity structure in the Earth’s interior (Parsons see Fig. 1a). The dynamic topography e is given by the fol- and Daly, 1983; Hager, 1984; Hager et al., 1985; Hager and lowing: Clayton, 1989), and translating seismic velocities to physical σ properties (e.g. temperature) of the mantle introduces further e (x) = zz (x, 0) at z = 0, (2) uncertainties (Cammarano et al., 2003). The problem is that g 1ρ dynamic topography predictions derived from mantle con- where 1ρ is the density difference between the mantle and vection models are generally larger by a factor of 2 (more air (or water assuming a sea-load when e < 0) (Morgan, significant at the very-large scales) than estimates from resid- 1965a; Houseman and Hegarty, 1987). In Fig. 1a, we plot ual topography (Hoggard et al., 2016; Cowie and Kusznir, the dynamic topography induced by a sphere of 1 % density 2018; Davies et al., 2019; Steinberger et al., 2019). We hy- anomaly, whose centre is at 372 km depth (D = 372 km) be- pothesise that this could be related to an oversimplification low the free surface. We calculate the vertical total stress and of the mantle rheology. In this paper, we explore how, at convert it to dynamic topography by using Eq. (2) for differ- wavelengths < 1000 km, the magnitude of dynamic topogra- ent values of the radius of the sphere. The amplitude of dy- phy changes when we use a rheological model in which the namic topography shows an accelerating increase by cubic viscosity depends on strain rate, temperature, pressure and dependence on the radius of the spherical density anomaly fluid content. We first summarise the well-established ana- (Fig. 1a, solid black line). For the same problem, Molnar lytical solution for calculating dynamic topography induced et al. (2015) provided a solution by considering a higher- by a spherical density anomaly embedded into an isoviscous order term, resulting in a slight difference from the solution fluid (Morgan, 1965a; Molnar et al., 2015). Then, assuming of Morgan (1965a) (see Appendix A3 in Molnar et al., 2015), isoviscous rheology, we illustrate that the amplitude of dy- which allows the consideration of density anomalies of finite namic topography depends on the viscosity structure of the viscosity (ηsphere) (Eq. 3): Earth’s interior as shown by Morgan (1965a) and Molnar et al. (2015). Finally, we use 3-D coupled thermo–mechanical −δρr3D 3 − 2f 18(f − 1)r2 numerical experiments of the Stokes flow to assess the de- σzz (C,0) = + 3f C3 C5 pendence of dynamic topography on nonlinear rheology using viscosity which depends on temperature, pressure, strain 6f D2 30(f − 1)r2D2 + − , (3) rate and fluid content. We show that plausible nonlinear rhe- C5 C7 ologies can induce local variations in viscosity and result in √ dynamic topography of lower amplitude compared to those = 2 + 2 = + 3ηsphere + where C D x , and f (η1 2 )/(η1 ηsphere). derived from models using isoviscous rheology. One can find that f = 1.5 if the sphere is very viscous (ηsphere η1), and f < 1.5 for any other case. In Fig. 1a, we present two more plots of dynamic topography where 2 Dynamic topography driven by a rising sphere: f = 1.5 for hard sphere and f = 1.25 for ηsphere = η1 by us- analytical and numerical solutions ing Eqs. (2) and (3). Figure 1a shows that a rising deformable sphere creates higher dynamic topography compared to a 2.1 Analytical solution for one layer isoviscous fluid very-viscous sphere. These show that the viscosity contrast between the spherical anomaly and the surrounding mate- We assume here a simple 2-D model representing a very- rial can affect the dynamic topography. In the section that viscous spherical density anomaly embedded into a semi- follows, we explore how dynamic topography varies when infinite isoviscous fluid bounded by an upper free sur- there is layering in viscosity such as the presence of a strong face. Earliest analytical investigations revealed that, albeit lithosphere above the convective mantle. counter-intuitive, the magnitude of the induced surface deflection due to the rising sphere is independent of the vis- 2.2 The impact of layered viscosity structure on cosity of the fluid. The dynamic topography is a function of dynamic topography the vertical total stress (σ ) applied to the surface which is zz A more generalised solution has been put forward to accom- proportional to the size and depth of the density anomaly ac- modate the presence of a stronger upper layer representing a cording to Eq. (1) (Morgan, 1965a, b): lithosphere with viscosity η2 above a weaker layer with viscosity η , and with η < η representing the convective man- 3 1 1 2 h 3i D tle (Fig. 1b). In this case, Morgan (1965a) showed (Eq. 4) σzz (x,0) = 2gδρr , (1) D2 + x25/2 that the total normal stress induced by the density anomaly

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34 Ö. F. Bodur and P. F. Rey: The impact of rheological uncertainty on dynamic topography predictions 2169

Figure 1. Dynamic topography driven by a spherical density anomaly of radius r at depth D embedded in a fluid whose viscosity structure is varied. (a) Variation in dynamic topography by radius of a spherical 1 % density anomaly centred at 372 km depth in a single isoviscous fluid whose viscosity is η1. The normal total stresses are calculated by Eq. (1) taken from Morgan (1965a) (hard sphere) and Eq. (3) taken from Molnar et al. (2015) (hard and deforming spheres), and converted to dynamic topography by using Eq. (2). (b) The case where the fluid is no longer a single layer but is composed of two layers with viscosities η1 and η2 for the lower and upper layers, respectively. We plot the dynamic topography for the same density anomaly in (a) using Eq. (4), taken from Morgan (1965a), but with varying relative viscosities (R = η1/η2). The ratio of upper-layer thickness to depth to the centre of the anomaly (d/D) also affects the dynamic topography, and higher values correspond to shallow density anomalies or thicker lithosphere for constant depth (D).

is dependent on the mass anomaly per unit length (Mu, for Ch = cosh(nd), Sh = sinh(nd) and d is the upper-layer point sources integrated along a continuous line), the depth thickness (modiﬁed from Morgan, 1965a). Following Mor- of the centre of the sphere (D) and marginally on the ratio gan (1965a), Fig. 1b illustrates the relative importance of of the viscosity of the convective mantle to the viscosity of R as well as the ratio of the thickness of the upper layer the lithosphere (R = η1/η2). The 2-layer problem is treated to the depth of the anomaly (d/D). As long as the litho- in Fourier domain with the resulting total normal stress as sphere is more viscous than the asthenosphere, the vertical below: total stress at the surface has a minor dependence on the ∞ viscosity of the lithosphere (see solid lines with R = 1 and Z R = 0.01 in Fig. 1b). Figure 1b also shows that the magni- σzz (x,0) = σn cosnx dn, (4) tude of dynamic topography increases as the density anomaly 0 is brought closer to the surface (compare R = 1, the solid where black line and the dashed black line). Moreover, its sensitiv- −n(D−d) ity on the relative viscosity of the lithosphere also increases. Muge σn = 1 + n(D − d) Although an unrealistic proposition for the Earth, when the 2π(RS + C ) h h lithosphere is less viscous than the asthenosphere, the nor- 1 − nD + n(D − d)(RCh + Sh)/(RSh + Ch) mal stress is much reduced and is strongly dependent on the + nd , 2 1 + nd(1 − R )/(RSh + Ch)(RSh + Sh) www.solid-earth.net/10/2167/2019/ Solid Earth, 10, 2167–2178, 2019

35 2170 Ö. F. Bodur and P. F. Rey: The impact of rheological uncertainty on dynamic topography predictions viscosity of the lithosphere (Fig. 1b). These demonstrate that layering in viscosity can have a strong impact on the amplitude of dynamic topography (Sembroni et al., 2017). In the next section, we use the analytical solutions above to bench- mark a numerical model, which we will then extend to nonlinear viscosity.

2.3 Numerical solutions

For comparison with analytical solutions (Morgan, 1965a; Molnar et al., 2015), we consider 3-D numerical models involving 1, 2 and 3 isoviscous layers. These bench- mark experiments will be used as references for non- isoviscous models discussed in Sect. 3. We use the open- Figure 2. Three-dimensional numerical model of a spherical tem- source code Underworld which solves the Stokes equa- perature anomaly having 96 km radius and a density of 1 % less tion at insignificant Reynolds number (Moresi et al., 2003, dense than the ambient mantle embedded in a depth of 372 km. The 2007). The 3-D computational grid represents a domain model space is 3840 km long in x and y axes, and 576 km deep 3840 km × 3840 km × 576 km with a resolution of 6 km along the z axis. The dynamic topography is depicted as an exag- along the vertical z axis and 10 km along the x and y axes gerated surface on the top of the model and is also reflected on the (Fig. 2). In all experiments, we include a 42 km thick con- x − z plane. tinental crust above the upper mantle. The density structure is sensitive to the geotherm via a coefficient of thermal expansion and compressibility (see Table 1 for all parame- that increasing the depth of our model from 576 to 864 km ters). The geotherm is defined using a radiogenic heat pro- increases the dynamic topography from 114 to 122 m. There- duction in the crust, a constant temperature of 20 ◦C at the fore, we attribute the misfit in amplitude of dynamic topog- surface and a constant temperature of 1350 ◦C at 150 km. raphy to the finite space in our numerical experiments. Our We disregard the adiabatic heating, and the asthenosphere is numerical experiment using isoviscous material delivers a re- kept at 1350 ◦C. We embed a positive spherical temperature sult globally consistent with the analytical solutions of Mor- anomaly of +324 ◦C at a depth of 372 km below the surface, gan (1965a) and Molnar et al. (2015). which delivers a 1 % volumetric density difference. The radius of the sphere is 96 km. In all experiments, we impose 2.3.2 Dynamic topography on a strong lithosphere free slip velocity boundary conditions at all walls, such as Vx above an isoviscous asthenosphere −1 and Vy are set to be free, but Vz = 0 cm yr at the top wall. Taking advantage of the symmetry of the experimental setup, In Experiment 2, we assign to the lithosphere a constant vis- we extract viscosity and velocity fields along a 2-D cross sec- cosity 100 times larger (1023 Pa s) than that of the astheno- tion passing through the centre of the thermal anomaly, from sphere (1021 Pa s, Fig. 3b) between z = 150 km and base of which we derive the streamlines and vertical velocity profiles the model. The convective cells become narrower by the in- along the vertical axis at the centre of the models. We calcu- duced viscosity contrast (Fig. 3b). The streamlines are de- late the instantaneous dynamic topography from the normal flected across the lithosphere–asthenosphere boundary due stress computed at the surface. to the large viscosity contrast (Fig. 3b), and there is a sharp variation in vertical velocity at the base of the lithosphere 2.3.1 Dynamic topography due to a rising sphere in an (Fig. 4a, solid red line). The maximum vertical velocity isoviscous fluid ∼ 2.1 cm yr−1 is attained near the centre of the anomaly. When compared to Experiment 1, the dynamic topography In the first experiment (Fig. 3a Experiment 1), we assign the (Fig. 4b, solid red line) shows a significant increase from same depth-independent viscosity of 1021 Pa s to the crust, ∼ 114 to ∼ 174 m. This increase is consistent with analyti- mantle and the density anomaly. The streamlines for Exper- cal estimations showing an increase in dynamic topography iment 1 (Fig. 3a) show formation of two convective cells at when viscosity increases toward the surface (Fig. 1b, R < 1). the sides of the sphere covering the entire crust and man- In Experiment 2a (not shown here), we tested a different ra- tle. The vertical velocity profile indicates that the thermal tio of thickness of the lithosphere to the depth of the anomaly anomaly rises with a peak velocity of ∼ 2.4 cm yr−1, which (see d/D in Eq. 4) by increasing the lithospheric thickness is faster than the 2.0 cm yr−1 predicted by the analytical solu- from 150 to 200 km, while keeping all parameters identical tion (Fig. 4a). Experiment 1 predicts dynamic topography of to those of Experiment 2. As predicted by Eq. (4), Exp. 2 114 m (Fig. 4b) which is lower than 132 m predicted by the predicted dynamic topography of ∼ 191 m, being the largest analytical solution of Molnar et al. (2015). We have verified among all experiments (Fig. 4b, dashed red line). Overall,

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36 Ö. F. Bodur and P. F. Rey: The impact of rheological uncertainty on dynamic topography predictions 2171

Table 1. Thermal and rheological parameters.

Parameter Symbol Exp 4a–b,5a–b Exp 4a, 4b Exp 5a, 5b Crusta Mantleb Mantleb Pre-exponential factor (MPa−n s−1) A 6.7 × 10−6 1.1 × 105 1600 Activation energy (kJ mol−1) Q 156 530 520 Power-law exponent n 2.4 3.5 3.5 Water fugacity f N.A. N.A. 1000 Water fugacity exponent r N.A. N.A. 1.2 Activation volume (m3 mol−1) V 0.0 6 × 10−6 11 × 10−6 or or 27 × 10−6 33 × 10−6 −3 Reference density (kg m ) ρ0 2700 3370 3370 Reference temperature (K) T 293.15 293.15 293.15 Initial cohesion (MPa) C0 10 10 10 Cohesion after weakening (MPa) C1 2 2 2 Initial coefﬁcient of friction µ0 0.577 0.577 0.577 Coefﬁcient of friction after weakening µ1 0.017 0.017 0.017 Saturation strain max 0.2 0.2 0.2 Thermal diffusivity (m2 s−1) κ 1 × 10−6 1 × 10−6 1 × 10−6 Thermal expansivity (K−1) α 3 × 10−5 3 × 10−5 3 × 10−5 Compressibility (MPa−1) β 4 × 10−5 0 0 −1 −1 Heat capacity (J K kg ) CP 1000 1000 1000 Radiogenic heat production (W m−3) H 0.5 × 10−6 0.2 × 10−7 0.2 × 10−7

n/a – not applicable. We use the rheological parameters from a quartzite (Ranalli, 1995), b dry or wet olivine (Hirth and Kohlstedt, 2003). and perhaps counter-intuitively, the presence of a thick vis- solid orange line) from 114 to 88 m. This is due to the damp- cous lithosphere enhances the dynamic topography. Interest- ing effect of the low-viscosity channel that acts as a decou- ingly, in analogue experiments where density anomaly is al- pling layer, which reduces the deviatoric stress through its lowed to rise and interact with the lithosphere, the amplitude ability to flow. of the dynamic topography is inversely correlated with the Until now, the viscosities were assumed to be constant. thickness of the lithosphere (e.g. Griffiths et al., 1989; Sem- However, results from experimental deformation on mantle broni et al., 2017). rocks strongly suggest that the viscosity is highly nonlinear (Hirth and Kohlstedt, 2003). In what follows, we explore the 2.3.3 The impact of low-viscosity channel on the influence of more realistic viscosities on dynamic topogra- dynamic topography phy. In Experiment 3 (Fig. 3c), we introduce a third 60 km 19 thick low-viscosity layer (i.e. 10 Pa s) beneath the base of 3 The impact of nonlinear viscosity on dynamic the lithosphere. The existence of a low-viscosity layer has topography been discussed in several studies (e.g. Craig and McKenzie, 1986; Phipps Morgan et al., 1995; Stixrude and Lithgow- 3.1 Viscosity structure of the Earth’s interior Bertelloni, 2005; Becker, 2017). In this experiment, in order to prevent large viscosity contrast that can impede the numer- The Earth’s mantle is not isoviscous. Geological records ical convergence, the viscosities of the lithosphere and that of of relative sea level changes related to postglacial rebound, the asthenosphere are set to 1022 and 1021 Pa s, respectively. geophysical observations of density anomalies inferred from When compared to Experiment 1, streamlines indicate a fur- seismic velocity variations in the mantle and satellite mea- ther decrease in size of the convective cells and more im- surements of the longest wavelength components of the portantly, a strong horizontal divergence of the streamlines Earth’s geoid have been used to infer the radial viscosity within the low-viscosity layer (Fig. 3c). The vertical veloci- profile of the Earth’s interior (Hager et al., 1985; Forte and ties are also enhanced in the asthenosphere reaching up to ∼ Mitrovica, 1996; Mitrovica and Forte, 1997; Kaufmann and 2.8 cm yr−1 slightly above the centre of the anomaly (Fig. 4a, Lambeck, 2000). Henceforward, beneath the lithosphere, a solid orange line). When compared to Experiment 1, we ob- variation in viscosity up to 2 orders of magnitude has been serve a strong reduction in dynamic topography (Fig. 4b, proposed (e.g. Kaufmann and Lambeck, 2000). Investiga- www.solid-earth.net/10/2167/2019/ Solid Earth, 10, 2167–2178, 2019

37 2172 Ö. F. Bodur and P. F. Rey: The impact of rheological uncertainty on dynamic topography predictions

Figure 4. (a) Vertical velocity profiles (Vy) along the centre, and (b) analytical solution and numerical modelling results showing dynamic topography induced by a sphere of temperature anomaly in the mantle (r = 96 km, δρ/ρ = 1 %). The misfit between the numerical model for R = 1 and the analytical solution is due to finite space in the numerical model compared to semi-infinite space assumed in Figure 3. Predicted peak amplitudes of dynamic topography for the analytical solution (Morgan, 1965a). layered Earth models with isoviscous rheology. Centred at 372 km depth, the embedded spherical density anomaly (black circle) is 96 km in radius. It has a temperature anomaly of +324 ◦C, giving 1 % effective density difference with the background. The resulting In the case where mantle flow is driven by the temper- streamlines are shown in a 2-D cross section (x −z plane) along the ature difference at the boundary of the convective layer or centre of each numerical model (y = 0 km). by internal heating, the dominant strain mechanism is diffusion creep because low deviatoric stresses are expected in the weak convective mantle (Karato and Wu, 1993; Turcotte tions of the rheological properties of crustal and mantle rocks and Schubert, 2014). However, mantle flow in the vicinity via rock deformation experiments revealed a nonlinear de- of a moving density anomaly is likely driven by deviatoric pendence of viscosity on applied deviatoric stress, pressure, stresses that exceed the threshold for dislocation creep. In temperature, grain size and the presence of fluids (Post and this case, nonlinear viscosities lead to strong local variation Griggs, 1973; Chopra and Paterson, 1984; Karato, 1992; in viscosity. Are those local variations in viscosity important Karato and Wu, 1993; Gleason and Tullis, 1995; Ranalli, for dynamic topography? To answer this question, we need 1995; Hirth and Kohlstedt, 2003; Korenaga and Karato, reasonable constraints on the rheological parameters control- 2008). These experiments lead to the following relationship: ling the viscosity of mantle rocks. However, the extrapolation from laboratory strain rates typically in the range of 10−6 s−1 − − + 1 ( r ) 1 −1 Q PV −4 −1 ˙ = n n ˙ n nRT to 10 s to mantle conditions where strain rates are typ- ηeff (ε,P,T ) A fH O ε e , (5) 2 ically on the order of 10−13 s−1 results in significant uncer- where ε˙ and A stands for strain rate and pre-exponential fac- tainties on the activation volume, activation energy and stress tor; r and n are exponents for water fugacity (fH2O) and de- exponent (Hirth and Kohlstedt, 2003; Korenaga and Karato, viatoric stress, respectively; V and Q are the volume and en- 2008). In what follows, we explore how nonlinear viscosity ergy of activation. impacts the dynamic topography and address how the uncer-

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38 Ö. F. Bodur and P. F. Rey: The impact of rheological uncertainty on dynamic topography predictions 2173

Figure 5. Viscosity map and streamlines for experiments using nonlinear rheologies (dry or wet olivine) with various activation energies. The rising sphere is shown by the circles.

tainties on the activation volume can affect dynamic topog- where ρ0, T0, α and P0 signify the reference density, ref- raphy. erence temperature, thermal expansion coefficient and the In Experiments 4 and 5 (Fig. 5), the viscosity depends on compressibility, respectively. temperature, pressure and strain rate as indicated by Eq. (5), using published visco–plastic rheological parameters for the 3.2 Numerical results: the case of dry olivine crust and mantle. Specifically, we use quartzite rheology for the crust (Ranalli, 1995), and we test both dry and wet olivine In Experiments 4a and 4b, we consider dry dislocation creep rheologies for the mantle (Hirth and Kohlstedt, 2003). Other for olivine (n > 1, p = 0, r = 0). The reported activation vol- parameters are identical to those in Experiments 1–3. We ume for this rheology varies between 6 × 10−6 and 27 × give all the rheological and thermal parameters in Table 1. 10−6 m3 mol−1 (Hirth and Kohlstedt, 2003). In Experiment For a given olivine rheology (i.e. dry or wet) we vary the 4a (Fig. 4b), we test the lower value. The streamlines show activation volume by using the minimum and maximum re- a similar a pattern to Experiment 2. Interestingly, the max- ported values (Hirth and Kohlstedt, 2003). imum vertical velocity peaks at 75 cm yr−1, near the upper In the numerical models, the plastic (i.e. brittle) deforma- boundary of the sphere (Fig. 6a, dashed black line). This is tion is described via due to the formation of a low-viscosity region above the rising sphere (Fig. 5a, Experiment 4a). This experiment gives a ∼ τ = µσn + C0, (6) dynamic topography of 149 m (Fig. 6b, dashed black line). It confirms that a strong contrast in viscosity between the lithosphere and asthenosphere enhances the dynamic topog- where τ is the 2nd invariant of the deviatoric stress tensor, raphy signal. We note that the viscosity contrast is attained which varies with the coefficient of friction (µ), and depth by smoother transition between the lithosphere and astheno- via lithostatic pressure (σn), as well as the cohesion (C0). Due sphere (Fig. 7a, dashed black line). We infer the mechani- to strain weakening, the cohesion and coefficient of friction cal thickness of the lithosphere from the viscosity profiles = = = decrease from C0 10 MPa and µ0 0.577 to C0 2 MPa plotted in Fig. 7a, along which the lithosphere–asthenosphere = and µ1 0.017 at which the maximum plastic strain (max) is transition zone shows a rapid decrease in viscosity (Conrad reached (i.e. 0.2, Table 1). The effective density (ρ) of rocks and Molnar, 1997). We observe that the effective mechanical is determined by the pressure and temperature using the fol- thickness of the lithosphere is reduced to 140 km, compared lowing equation: to the thickness of the thermal lithosphere (Fig. 7c). When we increase the activation volume to 27 × −6 3 −1 ρ = ρ0[1 − α(T − T0)][1 + β(P − P0)], (7) 10 m mol , the convection cells grow much larger and www.solid-earth.net/10/2167/2019/ Solid Earth, 10, 2167–2178, 2019

39 2174 Ö. F. Bodur and P. F. Rey: The impact of rheological uncertainty on dynamic topography predictions

sitting above the rising anomaly. The dynamic topography is ∼ 110 m (Fig. 6b, dashed orange line). This is a bit surprising given the strong contrast in viscosity (3 orders of magnitude) between the lithosphere and asthenosphere. However, Fig. 7a shows that the thickness of the mechanical lithosphere is reduced by about 30 km in comparison to Experiment 4a (e.g. 10 km reduction from thermal thickness) which resulted in lower dynamic topography with similar viscosity contrast (Fig. 7b, c). In Experiment 5b, we increase the activation volume from 11 × 10−6 to 33 × 10−6 m3 mol−1. The vertical velocities show significant decrease from 140 to 0.34 cm yr−1 (Fig. 6a, solid orange line). This is due to an increase in viscosity above the rising sphere. Compared to Experiment 5a, the dynamic topography decreases from ∼ 110 to ∼ 90 m (Fig. 6b, solid orange line). Compared to Experiment 4b, we expect the dynamic topography to be higher due to slight increase in viscosity contrast (Fig. 7a, b). However, the increase in thickness of the low-viscosity channel (Fig. 7a, d) is more effective and thereby causes a greater reduction in magnitude of the dynamic topography. In summary, experiments using nonlinear rheology generally give lower amplitudes of dynamic topography compared to experiments using isoviscous rheology (Fig. 8). When we use dry olivine rheology for the upper mantle, the dynamic topography varies between ∼ 105 and ∼ 149 m, whereas un- Figure 6. (a) Vertical velocity profiles (Vy) along the centre der wet conditions, the dynamic topography varies between and (b) dynamic topography induced by a sphere of temperature ∼ 90 and ∼ 110 m (Fig. 8). These variations are due to un- anomaly (r = 96 km, δρ/ρ = 1 %) in the mantle that has nonlinear certainties in the activation volume as well as fluid content in rheology depending on temperature, pressure and strain rate. olivine rheologies. show continuity through the lithosphere (Fig. 5a, Experi- ment 4b). The sphere has a very-low rising speed of ∼ 4 Discussion and conclusion 0.25 cm yr−1 (Fig. 6a, solid black line). Compared to Exper- iment 4a, the dynamic topography shows a strong decrease Using coupled 3-D thermo-mechanical numerical experi- from ∼ 149 to ∼ 105 m (Fig. 6b, solid black line). This is an ments, we have modelled the dynamic topography driven by example where the system behaves nearly as a single layer a rising sphere of 1 % density anomaly, having 96 km radius with homogenous viscosity. The near absence of viscosity and emplaced at 372 km depth. In line with analytical studies contrast between the lithosphere and asthenosphere explains (Morgan, 1965a; Molnar et al., 2015), the experiments show the smaller magnitude of the dynamic topography. Moreover, that dynamic topography is sensitive to viscosity contrast the formation of moderately low-viscosity channel (Fig. 7a, between the lithosphere and asthenospheric mantle, and the solid black line) also contributes to the decrease of the dy- thickness of the lithosphere (Fig. 7). Higher viscosity con- namic topography. trasts amplify the dynamic topography (Fig. 7a, b), whereas formation of a low-viscosity channel just below the litho- 3.3 Numerical results: the case of wet olivine sphere has the opposite effect (Fig. 7a, d). The experiments using nonlinear rheologies show local variations in viscosity, In Experiments 5a and 5b, we consider dislocation creep which contribute to the dynamic thinning of the mechani- of wet olivine. The reported activation volume varies be- cal lithosphere and causes reduction in dynamic topography. tween 11×10−6 and 33×10−6 m3 mol−1 (Hirth and Kohlst- In addition, models using high-activation volume creates edt, 2003). In Experiment 5a, we test the lower value. The a low-viscosity channel above the density anomaly, which streamlines show a pattern similar to Experiment 4a but with contributes to decreasing the dynamic topography. Using a slightly larger convective cells (Fig. 5b, Experiment 5a). The larger viscosity range in the models (1018 Pa s ≤ η(P T ε)˙ ≤ rising velocity of the anomaly exceeds 140 cm yr−1 (Fig. 6a, 1023 Pa s) resulted in ∼ 5 % variation in the amplitude of dashed orange line), promoted by the low-viscosity region dynamic topography, indicating that the effects of nonlinear

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40 Ö. F. Bodur and P. F. Rey: The impact of rheological uncertainty on dynamic topography predictions 2175

Figure 7. Factors affecting the dynamic topography. (a) Vertical viscosity profiles at the centre of the models. Variation in dynamic topography (b) by viscosity contrast between the lithosphere and part of the asthenosphere above the anomaly, (c) by lithospheric thickness (d) and by thickness of low-viscosity channel. rheology are reasonably captured in our models with smaller (Morgan, 1965a; Hager and Clayton, 1989; Osei Tutu et al., viscosity range (1019 Pa s ≤ η(P T ε)˙ ≤ 1022 Pa s). 2018), and Newtonian viscosity may lead to higher misfits. Predictions of dynamic topography derived from mantle Our models suggest that for shallow density anomalies convection models are compared against residual topogra- in the mantle, nonlinear rheologies not only produce lateral phy which is the component of Earth’s topography that is variations in viscosity (Richards and Hager, 1989; Moucha not compensated by isostasy (Flament et al., 2013; Hoggard et al., 2007) but also additional vertical variations in viscos- et al., 2016). In a recent work (Cowie and Kusznir, 2018), ity that impacts a relatively large area compared to the size it has been argued that dynamic topography predictions re- of the anomaly in the mantle. We show that this impacts on quire scaling of amplitudes by ∼ 0.75 to match the resid- the thickness of the mechanical lithosphere and predictions ual topography, and when density anomalies shallower than of the amplitude of dynamic topography. 220 km are included, the misfit requires a scaling factor of As shown in Fig. 8, uncertainties on the activation vol- ∼ 0.35. It is also important to consider that this misfit de- ume result in variations in dynamic topography which are pends on the flow wavelength and is suggested to be high- higher in experiments using dry olivine rheology (i.e. 17 %) est at lowest spherical harmonic degrees (l = 2) or very-long compared to experiments using wet olivine rheology (10 %). wavelengths (Steinberger, 2016). Our numerical experiments The comparison between numerical experiments using dry show that amplitude of dynamic topography can be nearly olivine (Exp. 4a) and wet olivine (Exp. 5b) indicates that the halved (e.g. from ∼ 174 m in Exp. 2 to ∼ 90 m in Exp. 5b) variation in dynamic topography can be as much as 25 %. when we consider nonlinear mantle rheology. Therefore, we These variations can be lessened if we have better constraints propose that, at shorter wavelengths (i.e. less than 1000 km), on the mantle rheology, which will advance the dynamic to- part of the misfit between the dynamic topography extracted pography models as well as our understanding of the interac- from mantle convection models and dynamic topography es- tion between deep mantle and the Earth’s surface. timated from residual topography can be attributed to the Newtonian mantle viscosity used in convection models. If the density sources are shallower, the dynamic topography Code and data availability. In our experiments we used Under- becomes more sensitive to the viscosity and density structure world, a free open-source code developed under the Australian AuScope initiative. www.solid-earth.net/10/2167/2019/ Solid Earth, 10, 2167–2178, 2019

41 2176 Ö. F. Bodur and P. F. Rey: The impact of rheological uncertainty on dynamic topography predictions

Scheme supported by the Australian Government, the Pawsey Su- percomputing Centre with funding from the Australian Government and the government of Western Australia, and support from the Australian Research Council through the Industrial Transformation Research Hub grant ARC-IH130200012. We beneﬁted from dis- cussions with Gregory Houseman and Nicolas Flament about the Earth’s dynamic topography.

Financial support. This research has been supported by the Aus- tralian Research Council (grant no. IH130200012).

Review statement. This paper was edited by Lucia Perez-Diaz and reviewed by Bernhard Steinberger and Mark Hoggard.

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44 3. ARTICLE 2

[Manuscript in Preparation]

Fast Tilting of Plates Driven by Relative Horizontal Plate Motion

Ö. F. Bodur1, P. F. Rey1, G. A. Houseman2

1EarthByte Group, School of Geosciences, The University of Sydney, NSW 2006, Australia,

2School of Earth and Environment, University of Leeds, Leeds, UK,

Fast Tilting of Plates Driven by Relative Horizontal Plate Motion

Ö. F. Bodur1†*, P. F. Rey1, G. A. Houseman2

1EarthByte Group, Basin Genesis Hub, School of Geosciences, The University of Sydney,

NSW 2006, Australia,

2School of Earth and Environment, University of Leeds, Leeds, UK,

*E-mail: [email protected]

† Now at: GeoQuEST Research Centre, School of Earth and Environmental Sciences,

University of Wollongong, NSW 2522, Australia

Abstract

Temporal and spatial correlations of subsidence history of sedimentary basins indicate

episodic tilting of tectonic plates. This is often explained either by dynamic topography

driven by underlying mantle density anomalies or flexural loading. However, some

continents record tilting events that are at once too fast to be associated with gradually

evolving dynamic topography, and/or they are at long-wavelengths that plate flexure

cannot explain. Using a torque analysis, we show that basal shear stress due to horizontal

motion of a rigid plate relative to the underlying mantle can induce plate-scale tilting

which delivers subsidence or uplift between ±5 and ±55 metres, which can create ~100 m

relief across a plate longer than 1,000 km. This can explain flooding or emergence of

continental margins near sea-level. Using a suite of 2D thermo-mechanical numerical

experiments which account for non-Newtonian mantle rheology, we also show that the

relative horizontal motion of plates also induces a gradient in normal stress across the

plate, giving topography about 100’s of metres with a very fast vertical plate motion

(>1,000 m Myr-1). We tentatively apply our findings to the abrupt tilting of the Indian

46 peninsula at ~23 Ma that produced ~±1 km of long-wavelength residual topography along the NNE-SSW direction. We speculate that the Neogene tilting of India could be driven by an increase in horizontal motion of the plate relative to the underlying mantle.

Introduction

The surface of the solid Earth is sensitive to large-scale mantle flow imposing tractions at the base of the lithospheric lid. To balance these basal tractions, small-amplitude (<1 km), long-wavelength (>1,000 km) vertical deflections of the Earth’s surface, called dynamic topography, develop (Lithgow-Bertelloni and Gurnis, 1997; DiCaprio et al., 2009; Braun, 2010;

Flament et al., 2013). Dynamic topography is typically associated with vertical flow driven by lateral density variations in the mantle (Morgan, 1965; McKenzie et al., 1974; McKenzie,

1977). Nevertheless, it is also clear that the flow field induced by these density anomalies involves horizontal pressure gradients and stress perturbations that manifest as long- wavelength topographic variation (McKenzie, 1977; Richter and McKenzie, 1978; Jian and

Parmentier, 1985; Moresi and Parsons, 1995). Several authors have recently explored horizontal pressure gradients in the context of plate motion (Iaffaldano and Bunge, 2015;

Semple and Lenardic, 2018; Stotz et al., 2018), but there has been little attempt to investigate their contribution to the topography (Morgan and Smith, 1992; Colli et al., 2014). In this paper, we first perform a torque-balance analysis of a moving rigid plate to describe the link between normal stress, shear stress and the surface topography they drive. From this, we find that shear stress on its own can induce topographic changes comparable to changes in eustatic sea-level.

In the second part of the paper, to quantify the combined effect of shear and normal stresses on the topography of a plate the base of which transitions into the convective mantle. We use coupled thermo-mechanical numerical experiments and show that the magnitude of the topography induced by relative horizontal motion between a plate and the underlying mantle

is comparable to the dynamic topography produced by density anomalies but develops over a

much narrower time window. Finally, we tentatively apply this concept to the Neogene tilting

of the Indian continent.

Torque-Balance Around a Horizontal Axis

Figure 1. Torque balance analysis for a rigid plate moving with constant velocity (Vplate) relative to the underlying asthenosphere. A: Free body diagram of plate-asthenosphere

system in which the plate is exposed to basal shear stress (txz) and vertical normal stress

(szz) in relation to relative motion. The resulting torque vector (wy) passes through the plate’s centre of mass. The counterclockwise rotation indicates an increase in upward normal force towards the trailing edge of the plate. B: Free-body diagram showing other possible plate boundary forces acting on the plate such as slab-pull (Fsp) and ridge-push (Frp), which have negligible contribution to the torque balance around horizontal axis.

Discarding lateral changes in densities across a plate, Figure 1A shows the net torque arising from relative horizontal motion between a rigid plate and the underlying viscous mantle. This torque acts around a horizontal axis passing through the plate’s centre of mass. We note that the horizontal forces that may be applied at the edge of the plate (e.g. ridge-push or slab pull)

make very little contribution to this torque (Fig. 1B). In addition, we note that plate’s elastic

flexure (e.g. due to slab-pull) can only influence the topography to a distance of a few hundred

km from the plate edges (Turcotte and Schubert, 2002). Therefore, for a sufficiently long plate,

elastic flexure may also be neglected in the analysis of torque balance. Hence, in Figure 1A,

the torque T is due to the combination of shear stress (txz) and vertical normal stress (szz) acting

on the base of the plate, and may be expressed as the surface integral of vector product of the

distance (r) and total stress (Methods, Fig. M1A):

� (�) = ∫ � × [�̂ + ��] �� , (1)

For simplicity, we assume that the shear stress is constant across the base of the plate (txz=ta).

The normal stress may vary laterally due to a variety of sources (e.g., pressure gradient). Here

for simplicity, we assume that the normal stress linearly increases across the plate (szz=Wx),

where W is constant. For a plate of length L we can show (Fig. S1A in Appendix B) that the

magnitude of the torque per unit width (Tw) is:

! � = �� + � , (2)

where l is the distance between the base of the plate and its centre of mass. The first term

describes the contribution of the shear stress, and the second quantifies the contribution of the

gradient of normal stress. As the latter scales with the cube of the plate length, the gradient of

normal stress should dominate the torque. Furthermore, depending on the rheology, the

existence of a pressure gradient flow can also affect the shear stress at the base of the plate.

This torque must be balanced by another torque, which emerges from the tilting of the plate as

shown in Fig. S1B in Appendix B. Because of tilting, the plate will also be subject to a

buoyancy force (Serway and Jewett, 2003; Lima, 2012) by Archimedes’ principle (Appendix

1, Fig. S2). The condition that the sum of all torques applied on the plate must vanish gives:

! ! −�� (� + � ) + �� + � = 0, (3) _ where the tilt (�) is expressed in radian, �_ is the average density of the rigid lithospheric

block, � is the density of the asthenosphere, and g is the gravitational acceleration. A more

accurate calculation can be done by considering all density boundaries within the plate (i.e.

crust-air or crust-water, crust-mantle lithosphere and lithosphere-asthenosphere boundaries),

but for simplicity, we assume one representative density (i.e. �_ ) for the lithosphere. The relative magnitudes of normal and shear stress acting on the base of the plate are needed to calculate the tilt (�). However, equation 3 also points that in the absence of normal stress, shear stress on its own can produce topography. In this case, if we assume that the asthenosphere and the average lithosphere density are same, we can simplify the contribution of the basal shear stress to the plate’s topography (hBasalShear) as:

ℎ = " . (4)

Basal shear stresses of 2.5 MPa and 4 MPa have been proposed for the Canadian Shield

(Bokelmann and Silver, 2002) and North America (Humphreys and Coblentz, 2007) respectively. More generally, estimations on basal shear stress vary between ~ 0.3 MPa (van

Benthem and Govers, 2010) and ~ 30 MPa (Barba et al., 2008). This large variation arises from

assumptions on the rheology of the plate and asthenosphere (Melosh, 1977). For a conservative

shear stress value of 2 MPa acting on the base a 150-km-thick plate, we find that the induced

topography is ±7 m for a 3,000-km-long plate, and ±17 m for a 1,000-km-long plate (Fig. 2).

Figure 2. Basal shear driven topography based on torque-balance analysis. Here we derive from Equation 4 the dynamic topography due to shear stress alone acting on plate

lengths between 1,000 km and 5,000 km. We assume shear stress values in the range of 1 to 4 MPa acting on the base of a 150-km-thick plate (black lines), and 2.5 MPa acting on the base of a 350-km-thick plate (red dashed line).

Considering the Canadian Shield as an example (shear stress of 2.5 MPa, plate thickness 350

km), the induced range of elevation increases to ±19 m and ±56 m (Fig. 2). On continental margins close to sea level, such topographic variations will induce flooding or emergence, and can impact on the formation of stratigraphic sequences (Petersen et al., 2010).

To go further in our understanding of the dynamic topography due to plate motion, we need to consider that, in contrast to what is depicted in Figure 1, the lithosphere is not a rigid body floating on a weak asthenosphere. Current understanding of mantle rheology (Post and Griggs,

1973; Karato and Wu, 1993; Hirth and Kohlstedt, 2003) leads to constitutive laws explaining that the lithosphere transitions progressively into the asthenosphere, following an exponential decrease in viscosity with a decay constant of 5 to 12 km (Conrad and Molnar, 1997). This impacts on the coupling between the plate and the mantle, and therefore on the stress acting in between (Bokelmann and Silver, 2002). Therefore, in what follows, we use thermo-mechanical numerical experiments with plausible visco-plastic rheologies, to document the topography induced by horizontal relative motion.

Experimental Setup

We use 2D coupled thermo-mechanical numerical experiments with laterally

homogenous crust and upper mantle. We solve Stokes’ equations at minute Reynolds number

using Underworld open source code (Moresi et al., 2003, 2007).

51 Figure 3. Numerical experiment results for dynamic topography driven by horizontal motion of plate relative to the asthenosphere. A: Model geometry, velocity boundary conditions (i.e. yellow arrows), initial geotherm, steady-state viscosity field and velocities field (white arrows). B: Steady state horizontal component of velocities at the centre of the model (along y-y’ in A) and C) steady state surface topography for various basal velocities (Vx base).

Our 3,840-km long plate model represents the upper 700 km of the Earth including a 40-km- thick continental crust. The initial geotherm reaches 1350 ºC at 150 km depth and is kept constant at greater depths (Fig. 3A). The model has 68 km of compressible low-viscosity air- like material on top of the crust to accommodate the topography. We assume visco-plastic rheologies for the crust and mantle, with the viscous flow laws of dislocation creep of quartzite

(Ranalli, 1995) and dry olivine (Karato and Wu, 1993), respectively (Table S1). The viscosity range is restricted to between 1018 Pa·s and 1023 Pa·s. The tectonic plate is held stationary and

we impose velocity boundary conditions on the vertical walls and the base of the model to

simulate motion of the plate relative to the asthenosphere (i.e. yellow vectors in Fig. 3A). At

-1 the vertical walls, Vx = 0 cm yr from the surface (z = 0 km) to the base of the lithosphere (z =

150 km) and increases linearly with depth thereafter, reaching its peak of 10 cm yr-1 at the base of the model. The vertical component of velocity is set to zero on both vertical walls. We have checked that a free slip condition doesn’t affect the experiment outcomes. For this reference

-1 -1 experiment, conditions on the base of the model are Vx = 10 cm yr and Vz = 0 cm yr ,

consistent with the velocity on the vertical walls at 700 km depth. The top boundary is stress

free. We also run another experiment in which we impose a free-slip boundary at the top of the

crust (without air-like layer) to compute the induced stresses in the model (Fig. 3 and Fig. S4

in Appendix B). In what follows, we describe the variation of topography in the centre of the

model for the reference experiment, which is relatively free from boundary artefacts.

Results

In conjunction with the variable effective viscosity field, the linear velocity profiles

imposed on both vertical walls result in an internal velocity field in which the vertical gradient

of horizontal velocity is no longer constant (Fig. 3A-B). Down to ~300 km depth, we observe

a reduction of velocity and even a reversal in the direction of the flow, notably below ~130 km

depth. Horizontal velocities in the mechanical lithosphere can reach -2.0 cm yr-1 at the centre of the plate (Fig. 3B) vanishing at the vertical walls due to the fixed boundary conditions imposed at the both ends. The plate is being dragged in a direction opposite to the flowing asthenosphere, in a similar fashion than in the ‘plug flow’ model (Höink et al., 2011; Semple and Lenardic, 2018). In contrast, below ~300 km depth, the horizontal components of velocities are higher than those imposed on the vertical walls (see also Fig. S1 for the case with free-slip top wall). These lateral changes in velocity profile across the model are associated with lateral variation in shear stress, which peaks at ~5 MPa (Fig. 4A-B) around the base of the mechanical

lithosphere (~900 ºC). The magnitude of the shear stress associated with non-Newtonian

mantle rheology confirms previous estimates on shear stress at the base of the lithosphere.

Figure 4. Induced stresses in an experiment with similar boundary conditions at the base and side walls, but with free slip top wall. A) Shear stress map in the lithosphere, B) shear stress profiles at the centre of the model (x= 0 km) and 1,000 km away from the centre (x= 1,000 km) for the model with base velocity of 10 cm yr-1. C) Induced normal stresses at the top wall for models with base velocities of 1, 5 and 10 cm yr-1.

In our numerical experiments, the horizontal flow is modulated by the non-Newtonian rheology of the mantle, and the velocities vary across the model, converging to a steady-state solution

54 towards centre of the model (Fig. 3A-B). The lateral variation in velocity is associated with a

lateral gradient in normal and shear stress, which explains the tilting of the plate. It is worth

noting that when we extract the computed steady state velocity profile from the experiment and

apply this profile as side boundary conditions, no lateral change in velocity, no lateral gradient

in normal stress, and no tilting is observed. Also, the linear velocity imposed as boundary

conditions would remain laterally unchanged and give no tilting if we use iso-viscous mantle

rheology. In terms of application to nature, our experiments suggest that a far-field perturbation

to the horizontal flow in the mantle will always result in gradient in normal stress and have an

impact on the topography of the plate by long-wavelength tilting. In our experiments, the

induced normal stress at the surface (z = 0) reach ± 50 MPa at both ends of our 2,000 km-long plate (Fig. 4C). These drives dynamic tilting in counter-clockwise direction, and the plate is uplifted by ~1,600 m at x = 1,000 km and is subsided by the same amount at x = -1,000 km

(Fig. 3C). The induced topography in the numerical experiment with a free surface (Ref.

Experiment, Fig. 3C) agrees well with the induced normal stress in the numerical experiment where the top wall is free slip (Fig. 4C).

We ran several experiments to test the sensitivity of the tilt to the relative velocity. When the peak relative velocity decreases from 10 cm yr-1 to 5 cm yr-1, the topography due to tilt is reduced from ~1,600 m to ~1250 m (Fig. 3C). The change in the topography as well as induced gradient in normal stress is not linear with change in magnitude of the applied basal velocity in the models but has an apparent power-law dependence with an index of ~2.4 (Fig. S5 in

Appendix B).

Figure 5. Results for numerical experiment with different frame of reference. A: Model geometry, velocity boundary conditions (i.e. yellow arrows along the vertical walls), initial geotherm, viscosity field and velocities at equilibrium (white arrows). B: Equilibrium state horizontal component of velocities at the centre of the model (along y-y’ in A). C: Topography at the equilibrium state.

Changing the frame of reference didn’t not impact the induced variation in stress and the topography (Fig. 5). This shows the self-consistency of the numerical experiments and the unique dependence of the experiment results to the relative motion between the plate and underlying mantle. We conclude that the horizontal motion of plates relative to the asthenosphere drive dynamic tilting. The magnitude of the associated topography is on the order of that can be excited by density anomalies.

We know that the shortest time it takes for a plate to change its motion is on the order of 100 kyr (Iaffaldano and Bunge, 2015), but, it’s uncertain if the rate of uplift or subsidence is at

similar pace. The numerical experiments show that tilting occurs in a period less than 0.5 Myr.

This gives a rate of uplift or subsidence which is much faster (i.e. >1,000 m Myr-1) than those

associated to mantle density anomalies (i.e. <100 m Myr-1) (Gurnis et al., 1998; Moucha et al.,

2008; Flament et al., 2013; Austermann et al., 2017; Arnould et al., 2018; Davies et al., 2019).

Using analytical methods and assuming iso-viscous mantle, we confirmed that, at long-

wavelengths, the topographic response to a perturbation in basal stress can be quite rapid (e.g.

t=0.5 Myr at a wavelength of 1,500 km and plate viscosity of 1023 Pa·s in Fig. S6). In our

calculations, we find that the rate of uplift or subsidence is very sensitive to the wavelength

(Fig. S6). Also, both topography and rate of uplift or subsidence linearly scales with viscosity.

In what follows, we apply the insights gained from our numerical experiments and torque

analysis to the enigmatic tilting of India.

Insights on Abrupt Tilting of The Indian Plate

The Indian peninsula is tilted upward to the SSW with a peak elevation of ~1,500 meters (Fig. 6A) along the Western Ghats escarpment (Eakins and Division, 2009). Residual depth anomaly measurements suggest that the tilt includes the oceanic part of the plate adjacent to the Indian peninsula. These residual depths are lower to the east of India (i.e. Bay of Bengal) and higher to the west (Hoggard et al., 2016), suggesting a long-wavelength (>2,000 km) tilt

(Fig. 6A). The change in laterite formation pattern and variation in sedimentation type from carbonate to clastic at offshore areas of the peninsula (Raju, 2008) suggest that peninsular India started to tilt after ~23 Ma (Richards et al., 2017). It has been suggested that this tilt has been driven by ~200 km-thick ±100 K asthenospheric temperature anomaly (Richards et al., 2017), inferred from seismic shear wave velocity (Ritsema et al., 2010). Although this may be actively supporting the residual topography in the peninsular India, there is not much argument for the mantle upwelling or advective heating at the base of the plate that could drive the tilt at ~23

57 Ma. Conductive heating, which require hundreds of Myr to develop the thermal anomaly that

were supposed to be driving the tilting, cannot explain the abrupt initiation of the dynamic uplift of the peninsular India.

Figure 6. The tilted Indian plate. A: Relief of the Indian plate reproduced from ETOPO1 (Eakins and Division, 2009). Circles/triangles: positive/negative residual depth anomaly measurements (Hoggard et al., 2016). Inset: Continental elevation along cross sections A- A’ (dashed black) and B-B’ (black) with intersecting residual depths (circles and triangles) showing the SSW upward tilting of the southern India in analogy with Fig. 4 in Richards et al. (2017). Red sticks: station averaged SKS splits (Becker et al., 2012). Burma subduction zone is shown by red sawtooth dashed curve connected to the solid curve representing Sunda subduction zone after (Steckler et al., 2016). B: Subduction profile and topography along X-Y cross-section passing through Indo-Burma ranges, after

58 Steckler et al. (2016). C: A schematic representation for the formation of a return flow at the vicinity of a subduction zone, after Morgan and Smith (1992).

As an alternative, we propose that the long wavelength tilting of peninsular India might have been driven by changes in relative horizontal motion between the Indian plate and its

underlying mantle. Reconstructions on absolute motion of the Indian plate doesn’t reveal

significant change in plate motion at ~23 Ma. Hence, we suggest that an increase in the rate of

horizontal asthenospheric flow relative to the plate’s frame of reference, is a plausible driver

of the tilt.

Morgan and Smith (1992) and Russo and Silver (1994) have shown that asthenospheric

flow can be induced by return flow near a subducting slab (Fig. 6C), and Becker (2017) has

shown that such flow can extend to distances greater than 2,000 km away from the trench in a

low viscosity channel. We speculate that the asthenospheric flow underneath India could be

related to eastward subduction of the Indian plate underneath Burma plate (Steckler et al.,

2016), pushing the asthenospheric mantle to the SSW (Fig. 6B).

Beneath peninsular India, the flow trajectories in the asthenosphere may be inferred

from station averaged SKS splits (Becker et al., 2012), which indicate an almost E-W flow

direction at higher latitudes (i.e. between 21°-24° N) with an increase in N-S component at

lower latitudes (Fig. 6A), and change to SSW direction. The inferred flowlines roughly align

with the topographic gradient in peninsular India. The SKS splits have 180° ambiguity in

terms of providing the orientation of the flow. We propose that the flow orientation of the

asthenosphere should be towards the SSW relative to the peninsular India. This is

based on the subduction location, assumption for the existence of return flow and numerical

modelling result (Fig.3) The inferred change in the asthenospheric flow can change the

torque balance around plate’s axis perpendicular to the direction of flow. In this case, to

rebalance the torque, the plate must tilt

upward to the SSW, which is the present-day tilt in peninsular India. Accordingly, changes in the horizontal flow field of the asthenosphere could drive the Neogene tilting of India and also explain the shear wave splitting anomalies observed across the peninsula (Becker et al., 2012).

Conclusions

Our torque analysis on a rigid plate reveals that basal shear stresses, on their own, can induce subsidence-uplift on the order of 10’s of metres, comparable to the magnitude of sea- level changes. Our numerical experiments considering more plausible plate and mantle rheologies have also shown that horizontal gradient in normal stress can be induced by relative horizontal plate motion. In this case, the magnitude of the topography reaches 100’s of metres in a quite short period (< 0.5 Myr). We speculate that the initiation of the enigmatic long- wavelength tilt of India at ~23 Ma could be the result of change in horizontal asthenospheric mantle flow driven by the subduction beneath the Burma plate.

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63 4. ARTICLE 3

[Manuscript in preparation]

On the Tempo and Amplitude of Tilting of Tectonic Plates by Basal Stresses

Ö. F. Bodur1, G. A. Houseman2, P. F. Rey1,

1EarthByte Group, School of Geosciences, The University of Sydney, NSW 2006, Australia,

2School of Earth and Environment, University of Leeds, Leeds, UK,

On the Tempo and Amplitude of Tilting of Tectonic Plates by Basal Stresses

Ö. F. Bodur1*, G. A. Houseman2, P. F. Rey1,

1EarthByte Group, School of Geosciences, The University of Sydney, NSW 2006, Australia,

2School of Earth and Environment, University of Leeds, Leeds, UK,

*E-mail: [email protected]

Abstract

Sinking slabs and rising hot plumes have been invoked to explain the long-term evolution of sedimentary basins. However, there are numerous occurrences of long-wavelength rapid vertical motions in Australian sedimentary basins since the Late Cretaceous which cannot be explained by density anomalies. Here, we investigate a recently proposed idea that changes in relative horizontal motion of plates and associated long-wavelength basal stress promote dynamic plate tilting. We analytically solve for temporal evolution of topography in a viscous lid with harmonic variation in basal stress. We find that, at wavelengths less than 3,000 km, shear stresses between 1 and 10 MPa induce topography between ~10 and ~100 m, respectively. At a viscosity of 1023 Pa·s and wavelengths of 3,000 km and 1,500 km, the surface topography re-adjusts in ~50 kyr and 500 kyr, respectively, after an instantaneous change in basal stress condition. This rapid adjustment timescale of the topography can explain the rapid vertical motions of Australia following three episodes in plate velocity changes since the Late

Cretaceous.

65 Introduction

Mantle density anomalies (i.e. plumes and sinking slabs) induce anomalous normal

stress at the base of the plates which drive dynamic uplift and subsidence producing dynamic

topography (Morgan, 1965; McKenzie, 1977; Hager and Clayton, 1989; Gurnis et al., 1998;

DiCaprio et al., 2009; Flament et al., 2013; Molnar et al., 2015; Eakins and Lithgow-Bertelloni,

2018). In a recent paper (Bodur et al., in prep.), by using torque-balance analysis, we have

shown that shear stress, on its own, acting on the base of the rigid plates can also produce

dynamic topography in the form of plate tilting, with a magnitude comparable to eustatic sea-

level changes. This basal shear stress-driven topography can cause emergence or flooding of a

passive margin close to sea-level, which makes it equally important among other processes

impacting on the evolution of passive margins. Furthermore, we found that horizontal plate

motions are associated with gradients in normal stress that can dominate the plate’s dynamic

topography with a km-scale elevation for a few thousand km-long plate (Bodur et al., in prep.).

An important implication of these findings is that, as the horizontal relative motion of a plate

changes, the basal shear stress and gradient in normal stress should also change and induce

variation of the plate tilt. We know that changes in horizontal plate velocity can be as fast as

on the order of 100 kyr (Iaffaldano, 2014; Iaffaldano and Bunge, 2015) which is at the pace of geological processes, however, it is uncertain, yet, if the tempo of topographic change is on the same order. Here we establish a link between topographic changes and changes in relative plate motion to understand anomalous vertical motions of plates. By using stream function method

(Turcotte and Schubert, 2002), we first solve the problem of 2D viscous flow beneath a rigid lid of length L and thickness h with constant viscosity (�) and density (�). As boundary

conditions (Fig. 1A), the plate is exposed to harmonic variation in shear stress at the base,

which permits us to solve for different wavelengths if vertical displacements are small

compared to the other length scales L and h. The vertical walls are stress free and we assume

isostatic normal stress on the basal boundary. The top wall can deflect by a similar harmonic

function (with time dependency) to balance the applied basal stresses. For consistency, we also

solve for the case where base of the plate is exposed to harmonic variation in normal stress

with zero shear stress (Fig. 1B), and advance our analysis by comparing solutions for both

cases.

Figure 1. Model configuration and boundary conditions for 2D incompressible plane- strain problem in a plate with horizontal periodicity and constant viscosity and density. We solve for both harmonic variation in A) shear stress (Case 1) and B) normal stress (Case 2) acting on the base of the viscous plate.

Formulation for 2D Flow with Harmonic Variations in Basal Stresses

In 2-D, the momentum equation for an incompressible fluid with constant viscosity

includes terms for viscous dissipation, buoyancy and pressure gradients that are in balance:

�∇� + �� = ∇�, (1) where u is velocity, P is pressure and g is the gravitational acceleration. By Helmholtz theorem, the incompressibility condition permits expressing the velocity via curl of a stream function

(y), which is a vector perpendicular to the flow plane (Turcotte and Schubert, 2002). Then, for

a 2D flow field:

(�, �) = (− , ) , (2)

where u and v are horizontal and vertical components of velocities. If we take curl of equation

(1) by assuming constant density:

∇� = 0 , (3)

For the given boundary conditions the form of solution for the stream function is assumed to be of the form:

�(�, �) = �(�) sin(��), (4) where k is the wave number and k = . Substituting Equation 4 into Equation 3 and taking the

Fourier transform gives:

! + � �(�) = 0, (5) ! therefore,

�(�) = �� + �� + �� + ��, (6)

Then, the stream function is:

�(�, �) = [�� + �� + �� + ��]sin (��), (7)

If we take the divergence of the momentum equation (Eqn. 1) by assuming constant density, we get:

∇� = 0, (8)

We assume the following function for the pressure:

�(�, �) = �(�)cos (��), (9)

If we take Fourier transform of Equation 9:

! + � �(�) = 0, (10) !

therefore,

�(�) = �� + ��, (11)

Then, the pressure will be:

�(�, �) = [�� + ��]cos (��), (12)

From the stream function, we can find the velocities, velocity gradients and strain rates. The

incompressibility condition also requires that (�̇ + �̇ ) is zero everywhere (Supp. File).

By substitution of (7) and (12) into (1), the integration constants E and F can be expressed in

terms of the other four constants and the general solution of the plane-strain flow problem in a

layer with horizontal periodicity given in terms of velocity and stress is:

�(�, �) = −[(�� + � + ��)� − (�� − � + ��)�]sin(��) (13)

�(�, �) = �[�� + �� + �� + ��]cos(��) (14)

�(�, �) = 2��[−(�� + 2� + ��)� + (�� − 2� + ��)� ]cos(��) (15)

�(�, �) = 2��[(�� + ��)� − (�� + ��)� ]cos(��) (16)

�(�, �) = −2��[(�� + � + ��)� + (�� − � + ��)� ]sin(��) (17)

Results

The formulations in previous section constitute a compact form of equations which we solve

for the four unknowns (i.e. A, B, C, D) by applying the boundary conditions:

On z = 0: u = 0, � = −��(�)cos(��) (top: deflected rigid lid) (18)

On z = h: � = 0, � = �sin(��) (base: isostatic, sheared) (19)

On x = 0 and x = �/2, u = 0, � = 0 (sides: impermeable, stress free) (20)

The boundary conditions on Equations 18 and 19 indicate that top of the rigid lid deflects via a harmonic function to accommodate the applied stress on the base (z = h). Substitution of (13-

17) into (18) and (19) and evaluation at z = 0 and h allows the four boundary conditions to be

expressed as the following matrix equation:

1 0 −1 0 � ��(�) � 1 −� 1 � 0 = (21) �exp(�ℎ) �ℎexp(�ℎ) −�exp(−�ℎ) −�ℎexp(−�ℎ) � 0 �exp(�ℎ) (1 + �ℎ)exp(�ℎ) �exp(−�ℎ) (−1 + �ℎ)exp(−�ℎ) � −�!⁄(2��)

which can be solved by elimination and back-substitution. The analytic solution of the matrix

equation (21), using the normalizing factor J, is:

� = �[cosh(�ℎ)� + �ℎ(�ℎ + 1)]( )� − �cosh(�ℎ)(" ) (22) !

� = −�[cosh(�ℎ)� + �ℎ]()� + �sinh(�ℎ)( " ) (23)

� = �[cosh(�ℎ)� + �ℎ(�ℎ − 1)]( )� − �cosh(�ℎ)(" ) (24) !

� = �[cosh(�ℎ)� − �ℎ]()� − �sinh(�ℎ)( " ) (25)

where the normalizing factor is:

� = (26) ()

The rate of change of the surface deflection is given by the vertical component of v on z = 0:

cos(��) = [� + �]�cos(��) (27)

= 2�[cosh(�ℎ) + �ℎ]()� − 2�ℎ�cosh(�ℎ)( " ) (28)

To simplify the following, we write the preceding equation in the form

= �� − � (29)

where α and β are constants obtained by comparison of (28) and (29). The surface deflection

reaches a static value (the time derivative is zero) when:

� = = () ( " ) (30) [!()!!]

A general time-dependent solution of (29) may be obtained given an initial condition e = e0 at t = 0:

� = � + (� − �)exp(��) (31)

In general, α will be negative because of how we defined J, so (31) defines an exponential

change from initial state e0 to final state es, with a timescale on which the unrelaxed signal is diminished by the factor of e:

70 () � = = [ ]( ) (32) !()!!

Equation 32 shows that the time for relaxation of topography is independent of the magnitude

of the applied stress, it linearly increases with viscosity and depends strongly on the

geometry via the factor of kh.

One can follow the same procedure to find the solution for the case where boundary condition at the base of the layer has harmonic variation in normal stress. For compatibility with the side-wall conditions which are unchanged from (21) we use the cosine dependence on x for the applied harmonic normal stress variation on the base. This normal stress is relative to the depth-dependent lithostatic stress variation assumed laterally invariant:

On z = 0: u = 0, � = ��(�)cos(��) (top: deflected rigid lid) (33)

On z = h: � = −�cos(��), � = 0 (base: varied normal stress) (34)

The minus sign is included so that the uplift is positive on the right side of the layer. The solution for Case 2 is determined in a similar way; the matrix is unchanged in (21) but the vector on the right side includes the normal stress term in the third element in place of � in the 4th element. In this case, the surface deflection (eN) when topography reaches a static value is (Supp. File):

( ) � = − [ ()] ( " ) (35) [!()!!]

The characteristic timescale for the development of topography (Supp. File) is found to be same with we found for Case 1 (Eqn. 32). The solutions are plotted for the amplitude of topography (Fig. 2A) and time constant of the tilt (total time to reach z = es or eN, in Fig. 2B) for different wavelengths and viscosities (the amplitude of topography is independent of viscosity). The reported shear stress beneath the plates has been variously estimated as between

~0.3 MPa (van Benthem and Govers, 2010) and ~30 MPa (Barba et al., 2008) due to the

uncertainty on the lithosphere-asthenosphere transition rheology (Melosh, 1977; Bokelmann

and Silver, 2002). To cover a plausible range, we consider basal shear stress amplitudes of 1 and 10 MPa, similar to shear traction amplitudes predicted by global mantle convection models

(Conrad and Lithgow-Bertelloni, 2006). Our analytical modelling results show that the induced dynamic topography varies between ~15 and ~150 m at a wavelength of 750 km, where it reaches its maximum, and varies between ~10 and ~100 m at wavelengths lesser than 3,000 km (Fig. 2A). At longer wavelengths, it shows a gradual decrease (Fig 2A) giving amplitudes between 2 and 20 m at ~16,000 (i.e. harmonic degree l=2). These topography changes are on the order of eustatic sea-level changes, and they can induce emergence or flooding of low- elevation platforms and continental shelves that are close to sea-level.

Figure 2. Analytical results for the incompressible 2D plane-strain flow problem A) Amplitude of surface topographic variation as a function of wavelength for harmonic variation in shear stress (red lines) or normal stress (black lines) at the base of a viscous plate with constant viscosity and density. B) Time scale for exponential adjustment of topography, after a change to the basal stress, as a function of wavelength. The time constant is independent of the magnitude of the applied stress and is linearly proportional to viscosity of the layer.

Analytical results for variations in basal normal stress show that, depending on the magnitude of the applied stress, they can either have smaller contribution to the topography, such as ~30 m at 1 MPa stress amplitude, or they can dominate the topography at stress

magnitudes higher than 10 MPa. The induced dynamic topography, in both cases, is proportional to the applied stress but decreases rapidly with increasing wavelength for the applied shear stress. In the case where normal stress gradient is applied to the base, the amplitude of long-wavelength topographic variation is directly proportional to the applied normal stress and insensitive to the wavelength (Fig. 2A). Therefore, at equal shear and normal stress variation (e.g. compare 10 MPa for both in Fig. 2A), the dynamic topography induced by normal stress will dominate the total dynamic topography as the wavelength increases.

The exponential time dependence of the tilt following a change of the basal stress depends on the wavelength and linearly scales with viscosity (by Equation 32 and Fig. 2B). At a viscosity of 1023 Pa·s, re-adjustment of the tilt can occur in a period as short as ~100 year at a wavelength of ~16,000 km (Fig. 2B). Overall, for wavelengths less than 3,000 km, it requires at least ~50 kyr to develop. This shows how quickly the topography can re-adjust following a change in basal stress. Therefore, rapid vertical motion of a sedimentary basin may follow a change in relative plate velocity. However, we should note that the calculations don’t account for elastic behaviour of the plate. We can predict that elasticity of the plate would have the effect of removing short wavelength signals (i.e. < the flexural wavelength which depends on elastic layer thickness but is at most about 200 km (Turcotte and Schubert, 2002, p.386)), and therefore it would moderate the vertical movement at the ends of the plate. This aspect of the plate-tilt is worth considering in future models that account for elasticity. In what follows, we apply the insights we gained from the analytical results to the episodic dynamic tilting events of the Australian plate which have been occurring since the late Cretaceous.

Figure 3. Present-day map of the Australian plate in orthographic projection and paleogeographic reconstruction of the Australian plate since 118 Ma (Boyden et al., 2011; Müller et al., 2016) in True Polar Wander-corrected paleomagnetic reference frame before 100 Ma (Steinberger and Torsvik, 2008) and the Indo-Atlantic hotspot reference frame after 100 Ma (O’Neill et al., 2005). The major sedimentary basin boundaries are taken from Raymond (2018).

Insights on the Episodic Dynamic Tilting of the Australian Plate

In Figure 3, we show the paleo-positions (i.e. filled circles) of three geographical coordinates across the Australian plate since 118 Ma (Müller et al., 2016) plotted at every 2

Myr by using plate reconstruction tool pyGPlates (Boyden et al., 2011; Williams et al., 2017).

These motion trails are useful to observe changes in plate speed and direction with respect to a

75 reference point. We also show the Australian absolute plate speed over time in Figure 4 for

every Myr since 118 Ma. In these plots, the history of the motion of Australian plate is

presented in Indo-Atlantic hotspot reference frame for times between 0-100 Ma and in True-

Polar Wander-adjusted reference frame for older ages until 118 Ma. It’s important to note that

although the reconstruction model, or simply the rotation model, is created by combining many

datasets including seafloor spreading magnetic anomalies and hotspot trails, it is not a product of adjoint evaluation of different datasets, but is rather a nonunique hierarchical connection between the plates, therefore uncertainties are not straightforward to constrain. Reader is referred to visit Müller et al., (2016) for more details on the method. However, for the recent times in which the ocean floor magnetic anomaly data from each sides of the ridge flanks are preserved, the rotation model is relatively more reliable. According to the motion trails, the first phase of decrease in plate speed occurs at ca 100 Ma (Fig.3 and 4), when the

Australia was part of the eastward drifting Gondwana. This period also correlates with the break-up age (ca. 95 ± 5 Ma) between Australia and Antarctica (Veevers, 1986) which sustained ~1 cm yr-1 relative motion since then. Between ca 80 Ma and ca 60 Ma, the

Australian plate continues its motion by slowly drifting towards the west. By ca 60 Ma, it increases its speed from ~1 cm yr-1 to ~3 cm yr-1 in 5 Myr, then attains ~4 cm yr-1 speed at ca 50 Ma. At the same time, it’s direction of motion changes from west to northeast.

Between ca 50 Ma to ca 42 Ma, the plate gradually slows from ~4 cm yr-1 down to ~2 cm yr-1. Then, it records another phase of increase in speed at ca 42 Ma, which increases its speed from ~2 cm yr-1 up to ~4 cm yr-1, after which the Australian plate continues its relatively fast motion at an average speed of ~5 cm yr-1 towards the northeast. In summary,

Australian plate experiences three distinct episodes of relatively major plate velocity changes at ~100 Ma, ~60 Ma and ~42 Ma, of which we explore hereafter their signals for the dynamic plate tilting.

In the Albian, a large area of the Australian continent was below sea-level (see

Dettmann et al. (1992) and references therein). This subsidence and formation of epi-

continental seas in central Australia (Frakes et al., 1987) is interpreted as the effect of

Gondwana drifting over the subducted Pacific plate between ca 130 Ma and ca 97 Ma (Gurnis et al., 1998). In the mid-to-late Albian, the Eromanga sea covering 60% of the continent (Frakes et al., 1987) started to retreat and disconnect from the east-coast of Australia (Dettmann et al.,

1992). This gradual retreat can be partly associated with migrating long-term dynamic topography (Gurnis et al., 1998). In the Cenomanian, between ca 100 Ma and ca 90 Ma, the analysis of wells on the central and western Eromanga Basin indicates a phase of accelerated tectonic subsidence (Gallagher et al., 1994). On the other hand, the wells in the east-most

Eromanga Basin, Surat Basin and further eastern Australia suggests coeval rapid uplift indicated by the lack of Cenomanian sediments in the stratigraphic record (Gallagher et al.,

1994). Analysis of the thermal history of the sediments in the region also suggests an increase in thickness of the eroded sections of sediments towards the east (Gallagher et al., 1994).

Therefore, we can conclude that the Australian plate tilted downward to the west in the

Cenomanian. This tilt has been attributed to subduction of the Pacific plate beneath the

Australian plate on the eastern margin of Australia, however, the development of the topography cannot be captured in dynamic models considering this subduction (Matthews et al., 2011). One primary reason is that the tilt developed and recovered quite rapidly. This is illustrated by the onset (ca 97 Ma) and end (ca 93 Ma) of sedimentation in Eromanga Basin, which resulted in the ~1,2000 m-thick Winton Formation typified by fluvial-lacustrine sediments with volcanogenic detritus signatures.

Figure 4. Australian plate absolute velocity variation over time since 118 Ma (Boyden et al., 2011; Müller et al., 2016) in True Polar Wander-corrected paleomagnetic reference frame before 100 Ma (Steinberger and Torsvik, 2008) and the Indo-Atlantic hotspot reference frame after 100 Ma (O’Neill et al., 2005).

Moore and Pitt (1984) and Dettmann et al. (1992) suggested the transport of sediments from the eastern volcanic margin to the west. The long-wavelength (> 1,000 km) character of the tilt rules out elastic flexure caused by surface loading as an explanation for the tilt. The duration of the tilt accords with the rapid slowing speed of east Gondwana (Boyden et al., 2011;

Müller et al., 2016). We propose that this tilt was driven by decrease in speed of the plate at ca 100 Ma (Fig. 3). This change in plate speed could have tilted the plate upward in the direction of decrease in speed (towards the east). According to our calculations, for a plate viscosity of 1023 Pa s and a wavelength of 1,000 km, stress perturbations gives less than 1 Myr

for the tilt to develop (Fig. 2B).

After ~30 Myr (from ca 93 Ma to ca 60 Ma) of quiescent non-deposition in the

Eromanga Basin, the Eyre Formation (i.e. Eyre Basin, Fig 3C) records the onset of fluvial-

lacustrine sedimentation in the Great Artesian Basin from ca 60 Ma. The deposition continues

until ca 42 Ma reaching a thickness of up to 120 m, though its top section has been eroded in

the Oligocene. We propose that the significant increase in speed of the Australian plate from 1 cm yr-1 to 3 cm yr-1 and its change in direction from west to northeast (Fig. 3) could have

initiated tilting of the plate down to the north, driving the deposition of sediments to the Great

Artesian Basin and Eyre Basin. We note that this period of tilting is more gradual, and it could

be reasoned that the dynamic topography driven by mantle density anomalies could be equally

important in the formation of the basin. However, the correlation between rapid change in plate

velocity vector and initiation of the sedimentation is well described by this model of plate

tilting driven by changes in basal shear stress.

In the mid-to-late Eocene (at ca 43 Ma), the Bight and Eucla basins of southern

Australia record rapid flooding (Shafik, 1990) with max 250 m of subsidence (DiCaprio et al.,

2009) also indicated by rapid change in deposition from siliciclastic to carbonate type (Li et al., 2003). These basins re-emerge before ca 39 Ma, suggesting ~4 Myr episode of flooding.

The initiation of the flooding temporally correlates with the initiation of accelerated subsidence in the Otway Basin of the south-east Australia (Müller et al., 2000), suggesting that south and south-eastern margins are subsided during this period. The northern and north-western margins

(i.e. Arafura, Browse, Canning, McArthur and Bonaparte Basins), based on apatite-fission track analysis, record uplift of up to ~1,500 m (Duddy et al., 2003) initiating at mid-to-late

Eocene, suggesting that the plate tilted upward to the north-northwest at that period. There is lack of data if the northern basins also subsided at ~39 Ma to confirm the recovery of the tilt of plate. However, the initiation of these events is coeval with increase in speed of the

Australian plate to ~5 cm yr-1 (Veevers et al., 1991; Müller et al., 2008). In this case, the upward motion of the plate is towards the direction to which the plate is increasing its speed, which is the opposite for previous two episodes of plate tilt. It can be related to the horizontal motion of the margin with respect to the flowing mantle underneath, which could be either passively

79 dragged or faster moving and dragging the plate which give different relative directions.

Therefore, the change in relative velocity vector is what determines a change in tilt, not the absolute velocity vector.

Conclusions

After a change in stress condition associated with a change in relative plate velocity, a plate-scale tilt is induced. Accordingly, long-wavelength basal shear stress anomaly with an amplitude of less than 10 MPa can induce elevation changes of tens of metres. This can cause flooding of a continental margin close to sea-level and induce formation of sedimentary basins.

The analytical calculations show that the adjustment time scale of the topography is quite rapid, giving ~50 kyr at 3,000 km wavelength and 1023 Pa·s. This can explain the three distinct episodes of rapid tilting of the Australian plate, which are coeval with changes in plate velocity vector.

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82 5. ARTICLE 4

[Manuscript in preparation]

Contrasting Dynamic Topography and Inversion of Passive Margins

Ö. F. Bodur1, P. F. Rey1, G. A. Houseman2

1EarthByte Group, School of Geosciences, The University of Sydney, NSW 2006, Australia,

2School of Earth and Environment, University of Leeds, Leeds, UK,

83 Contrasting Dynamic Topography and Inversion of

Passive Margins

Ö. F. Bodur1*, P. F. Rey1, G. A. Houseman2

1EarthByte Group, School of Geosciences, The University of Sydney, NSW 2006, Australia,

2School of Earth and Environment, University of Leeds, Leeds, UK,

*E-mail: [email protected]

ABSTRACT

Long after their formation, passive margins can experience enigmatic episodes of dynamic

uplift and compressional inversion. Here we show that trailing and leading margins of moving

continents can record contrasting dynamic topography and deformation history when exposed

to lateral asthenospheric flow. Dynamic horizontal pressure gradients in the mantle sustained

by relatice horizontal motion of plate and underlying mantle can result in tilting of an entire

plate, thereby causing dynamic subsidence at the margin trailing the flow (i.e. trailing margin),

and dynamic uplift at the margin leading the flow (i.e. leading margin). The bathymetry of the

modelled ocean basin indicates a max. +/-500 m dynamic tilt, hence asymmetric subsidence of

the ocean floor nearby the mid-ocean ridge. Furthermore, a large-scale counter-flow (i.e. shear

induced upwelling) is induced beneath the ocean basin. This convective flow applies

compressional stress to the trailing margin that may result in basin inversion. The modelling

results reveal strikingly different dynamic topography and mechanical evolution of conjugate

passive margins, and can explain the mid-Eocene NW upward tilting of the Australian plate as

well as coeval inversion of Otway Basin in SE Australia.

INTRODUCTION

Detailed investigations of passive margins has revealed that a majority of them have experienced, long after the rift phase, fluctuating vertical motions (Hoggard et al., 2016; White,

2016), the magnitude of which is often higher than glacio-eustatic sea-level changes (Pedoja et al., 2011; Japsen et al., 2012; Green et al., 2018). In addition, many passive margins have recorded enigmatic compressional inversion despite the absence of nearby collisional plate boundaries (Glennie, 1981; Cooper et al., 1989; Hill et al., 1995; Lowell, 1995; Withjack et al.,

1998; Lundin and Doré, 2011; Pedoja et al., 2011; Holford et al., 2014). These margins can be easily inverted by compressive stresses due to the weakness in their rheological strength which is modulated by density and thickness of sediments filling the basins (Sandiford and Hand,

1998; Buiter et al., 2009). Although episodes of vertical motion and tectonic inversion can develop independently, observations suggest a potential unresolved link between them

(Meeuws et al., 2016). Mantle processes have been put forward to explain anomalous vertical motions in terms of dynamic topography, motivated by the idea that Earth’s convecting mantle is coupled with tectonic plates (Pekeris, 1935; McKenzie, 1969; Houseman, 1983). Dynamic topography is mainly driven by motion of mass anomalies in the convective mantle (Gurnis et al., 1998; Braun, 2010; Flament et al., 2013), but it is also possible that, the development of stratigraphic sequences may reflect small scale convection at the lithosphere-asthenosphere boundary (Petersen et al., 2010), a mechanism that has been put forward to explain the observed basal horizontal isotherm in oceanic lithosphere (Parsons and McKenzie, 1978). The steep variation in lithospheric thickness at continental edges is particularly favourable to small-scale convection (Buck, 1986; King and Anderson, 1998; Shahnas and Pysklywec, 2004; Ramsay and Pysklywec, 2010). It is also favourable to the build-up of deviatoric stresses in relation to the relative motion between the continental edge and the underlying asthenosphere (Bodur et al., in prep.). Here we use 2D numerical experiments to investigate the impact of horizontal

85 asthenospheric flow on the evolution of a conjugated set of passive margins separated by an

oceanic plate. Our results show that plates can dynamically tilt as a response to lateral

asthenospheric flow, which is also observed in only continent-asthenosphere case (Bodur et

al., in prep.). The model results show that the magnitude of the tilt is lesser in the oceanic

domain, but significant. We also find that horizontal asthenospheric flow induces a counter-

flow underneath the oceanic domain, which can potentially explain compressional deformation

and basin inversion at the trailing margins of plates such as Otway Basin of the south-east

Australia.

NUMERICAL MODEL CONFIGURATION

The geometry of our reference model at time to derives from a forward passive extension experiment of a continental lithosphere that is laterally homogeneous and in thermal equilibrium. Although various complications can arise in rifting process and create asymmetric features in conjugate margins (Sengör and Burke, 1978; Huismans et al., 2001; Buiter et al.,

2009; Huismans and Beaumont, 2011), we aimed at creating symmetric conjugate margins to fairly compare their dynamics during plate motion. In our numerical experiments, we use laboratory-based temperature, stress and strain-rate dependent visco-plastic rheologies for the crust and mantle, as well as realistic thermal properties including radiogenic heating (see Table

S1 for all parameters). After 34 Myr of symmetric extension at a total extension velocity of 1.5 cm yr-1 (i.e. 0.75 cm yr-1 from both sides), we get a pair of nearly symmetric continental

margins (Fig. S1). Because our thermo-mechanical code doesn’t account for the formation of

oceanic crust, once continental separation is reached, we insert a 1,000-km-wide ocean basin,

which includes a 7-km-thick oceanic crust in between the conjugate margins (Fig. 1A and S1).

The temperature structure of the oceanic lithosphere is based on plate cooling model of ~50

Myr, leading to a maximum thickness of 80 km at the margins of the ocean (Parsons and

Sclater, 1977; Turcotte and Schubert, 2002). Our composite continent-ocean-continent plate is

then exposed to horizontal asthenospheric flow to model the horizontal motion of plate relative

to the underlying mantles. We impose a horizontal velocity condition at the vertical walls of

-1 the model consisting in a linear velocity increase from Vx = 0 cm yr at 150 km, to Vx = 10 cm

-1 yr at 700 km depth. A zero-horizontal-velocity boundary condition (Vx = 0) is imposed to the

vertical edges of the lithosphere down to 150 km depth, whereas the vertical components of

the velocities (Vy) are let free. The bottom wall has free-slip boundary condition. We record the topography and bathymetry of the model through time, and we document and analyse the deviatoric stress field, the temperature and viscosity fields, and the accumulated plastic strain at both continental margins.

Figure 1. Reference model results. A: Zoomed view of the model geometry and isotherms. B: Velocities and viscosities down to 300 km depth at t = 0.5 Myr. C: Horizontal component of the velocities t = 0.5 Myr along a vertical transect at the centre of the model. When the mantle flows to the right, velocities are positive, and negative otherwise. D: Dynamic topography across the model at t = 0.5 Myr. The red dashed line represents the topographic gradient in the oceanic domain.

MODEL RESULTS

The linear increase in velocity applied to the walls of the asthenosphere evolves into a curved velocity profile (Fig. 1B), in which much of the velocity increase becomes confined between the base of the lithosphere ~150 km and ~350 km, at which depth the velocity remains nearly constant down to the base of the asthenosphere (Fig. 1B,C). The relative horizontal motion of the plate induces normal stress gradient which tilts the entire plate by ~+/-1,400 metres upward in the direction of the imposed flow (Fig. 1D). In the oceanic part of the lithosphere the tilt has a lower slope (Fig. 1D). The passive margin trailing the prescribed flow

(i.e. trailing margin between x = -750 and x= -500 km) subsides by ~500 m, whereas the margin facing the flow (i.e. leading margin between x = 500 and x = 750 km) is uplifted by ~500 m

(Fig. 1D).

Underneath the oceanic domain of the plate, a counter flow is induced with a peak velocity of ~2.3 cm yr-1 (Fig. 2A). This counter flow, previously defined as shear-driven

upwelling by Conrad et al. (2010) and Ballmer et al. (2015), is induced by advection the

asthenosphere to the cavities in the upper mantle. In our case, the cavity is not an intra-

continental lithospheric gap, but is the space due to lithospheric thickness difference beneath

the ocean and the continent.

During the relative motion of plate, the trailing margin experiences a deviatoric stress

(i.e. 2nd invariant of the deviatoric stress tensor) more than 35 MPa, whereas the deviatoric

stress acting on the leading margin is relatively low (Fig. 2B). This is due to the induced

counter-flow, which is continuously applying compressive stress on the trailing margin. By t =

20 Myr, ongoing convergence at the trailing margin can result in 6% horizontal finite

shortening (Fig. 2C), creating topographic variations about 1,000 m (Fig. S2). Both margins

show convergence near the base of the crust related to lower crustal flow (Fig. 2A). However,

most of the plate hosting the leading margin is under extension (Fig. 2A).

Reference model results show that post-rift evolution of passive margins can be

controlled by their relative position with the underlying horizontally flowing asthenosphere.

During a relative motion between the plate and asthenosphere, continental margins residing

opposite to the asthenospheric flow can record contrasting mechanical history and dynamic

topography.

We tested key parameters in the evolution of topography and the formation of the

convective cell beneath the ocean basin. We varied relative horizontal asthenospheric flow

velocity from 10 cm yr-1 to 5 cm yr-1 (Fig. S3a), and observed that the amount of dynamic

tilting decreased from ~1,400 m to ~1,200 km (Fig. S3c). In that case, the induced counter flow

velocity decreased from 2.3 cm yr -1 to ~1 cm yr-1 (Fig. S3b). We also varied the width of the oceanic gap, keeping all other parameters unchanged. A larger (1.5x) oceanic domain (Fig.

S4a) resulted in +/-1,200 metres of tilting at t = 0.5 Myr (Fig. S4c). This decrease in tilt with respect to our reference experiment could be a consequence of the reduced plate thickness of the lithosphere. A wider ocean gap provides larger convective cell beneath the ocean basin which accommodates more vigorous counter-flow (i.e. ~3 cm yr-1 in Fig. S4b). In the next

section, we apply the understandings we gained from our models to the Cenozoic evolution of

Australian passive margins.

Figure 2. Passive margin compression and asymmetric dynamic subsidence of conjugate margins and the ridge flanks separating Australia and Antarctica. A: Horizontal strain rate map in the lithosphere above 1273 K isotherm at t = 0.5 Myr. Red and blue colours indicate contraction and extension, respectively. C) Cumulative horizontal shortening of the margins after 20 Myr of model evolution. D) Mean bathymetry profiles vs. square root of oceanic crust age across the ridge flanks (with max. and min. ranges) between Australia and Antarctica, derived by using the transects shown in the map.

DISCUSSION

Mid-Eocene NW Upward Tilting of Australian Plate

In the Oligocene, the Australian plate was exposed to a broad-scale progressive uplift in the central and southwest basins, and gradual subsidence in the northeast margins, implying a NE downward tilting (DiCaprio et al., 2009). This is explained as an effect of dynamic topography associated with the positive mass anomalies (i.e. buried slabs) underneath the

Australian plate (DiCaprio et al., 2009, 2011). However, there are other anomalous subsidence and uplift events that cannot be explained by mass anomalies. For example, between ~50 Ma and ~35 Ma the well data from the Bass Strait shows accelerated subsidence (Müller et al.,

2000). Furthermore, the Eucla and Bight basins record notable flooding (i.e. up to 250 metres of anomalous subsidence) initiating at ~43 Ma (Shafik, 1990; DiCaprio et al., 2009) (Fig. 3).

This suggests that there is a broad accelerated subsidence in the south and south-east margin of Australia at the mid-Eocene.

However, in the northern and north-western Australia, apatite fission track data and vitrinite reflectance analysis of data from Keep River (Bonaparte Basin), Torres (Arafura

Basin), White Hills (Canning Basin), Rob Roy (Browse Basin) and Broadmere (McArthur

Basin) wells indicate a broad-scale uplift up to ~1,500 metres initiated between 55 Ma and 45

Ma (Fig. 3) (Duddy et al., 2003). The co-occurrence of anomalous uplift in the NW margin, and accelerated subsidence in the SE margin of the continent implies a mid-Eocene NW upward tilting of the plate. These vertical motions of the margins are distinct from a progressive northeast downward tilting of the Australian plate since the Eocene or mid-Miocene at the latest (Sandiford, 2007; DiCaprio et al., 2011). We propose that the NW upward tilting of the

Australian plate could be driven by relative horizontal motion of the plate and underlying asthenosphere, which increased at ~43 Ma, which could be driven by the subduction along the

NNE boundary of the Australian plate (Müller et al., 2000). The acceleration of the Australian

plate (Fig. 3), in Indo-Atlantic hotspot reference frame (O’Neill et al., 2005), from a speed of

~1.2 cm yr-1 up to ~5.0 cm yr-1 at ~43 Ma (Müller et al., 2016) could be interpreted as an increase in relative motion of the plate with underlying asthenosphere, which could have driven the tilting of the entire continent.

Dynamic Tilt of the Ocean Basin between Australia and Antarctica

The numerical models predict tilting in the ocean basin with lesser slope that should be observed between Australia and Antarctica. In order to show that the present-day ocean basin between the Australia and Antarctica is tilted downward towards the Australian side, we take

8 different swath profiles with 1 km cross and along-track intervals of the ocean floor bathymetry between Australia and Antarctica (Fig. 2D). To avoid the possible effect of asymmetric sea-floor spreading on the subsidence pattern, we take the square root of age of the ocean floor and plot it against the bathymetry. The plot is limited for a certain age of ocean crust (at least 1,000 km away from the thick sedimentary basins of the Australia and Antarctica passive margins) which helps to avoid the flexural uplift of the ocean floor with older ages

(Whittaker et al., 2013). We show that the Antarctic side of the ocean basin is elevated compared to the Australian side, which is similar to previous residual topography estimations

(Flament et al., 2013; Hoggard et al., 2016). It is interesting that this tilt is reverse to the ~300 m dynamic downward NE tilting of the Australia which is driven by density anomalies in the mantle. We propose that this tilt could be due to the present-day relative motion between the plate and underlying asthenosphere.

Australian-Antarctic discordance is often attributed to a broad subsidence, whose formation has been explained by sinking slab underneath (Gurnis et al., 1998). The modelled dynamic topography due to this anomaly predicts more subsidence towards the Antarctica side of the ocean basin, therefore cannot explain the present-day tilt in the ocean basin as it is reverse

to what numerical models of Gurnis et al. [1998] predicts. Furthermore, a recent study suggest that asymmetries in ocean bathymetry can be driven by convective flow produced by density anomalies in the mantle (Watkins and Conrad, 2018). In the mentioned work, they propose that

~500 m of dynamic topography explains tilting of ocean basins at Mid-Atlantic ridge as well as the East-Pacific Rise. However, the tilt in our case is not driven by thermal anomalies, but instead by the relative motion between the plate and asthenosphere.

Figure 3. Australian plate paleo-positions and contrasting evolution of NW and SE margins. Paleo-positions of the Australian and Antarctic plate relative to Atlantic-Indian hotspots reference frame (O’Neill et al., 2005) created by using pygplates (Müller et al., 2016; Williams et al., 2017). Orange coloured plus signs denote the uplifted basins. The

size of the symbol is proportional to the amount of uplift (max. 1,500 metres). The time for accelerated subsidence for Eucla and Bight Basins derived from nannofossil records is shown as blue region in geological time scale bar (Shafik, 1990). The red arrows in the Otway Basin show the compression direction responsible for basin inversion. Top right: uplift apatite fission track ages for wells in the Arafura, Bonaparte, Browse, Canning and McArthur Basins (modified from Duddy et al. (2003)). Bottom right: tectonic subsidence of Durroon and Nerita wells (green and yellow circles on the map) in the Bass Strait (modified from Müller et al. (2000)).

Inversion of the Otway Basin in the Southeast Australian Margin

Coeval with the predicted NW upward tilting of the plate, Otway Basin in SE Australia

has experienced compression dominated basin reactivation events starting at the mid-Eocene

(Holford et al., 2014). The reactivation of the pre-existing structures was interpreted as the

consequence of long-lasting compressive stresses arising from plate-boundary forces.

However, models of paleo-stresses for the Australian plate in the Early Eocene (Müller et al.,

2012) shows that stresses due to plate-boundary forces were markedly small along the

southeast margin (i.e. <1 MPa), hence cannot explain the observed inversion events. In the northern margin, there is no significant basin inversion event relative to the southern margin for the same period, except a mild inversion in the Adele province of Browse Basin in the

Eocene to Miocene (Symonds et al., 1994). Our models indicate that continuous compression of the Otway Basin, documented by a 200-km wide low amplitude fold structure, can be explained by compression of the trailing margin driven by the convective flow (Fig. 1d, 2c).

As soon as the relative motion continues, the margin will be continuously compressed.

CONCLUSION

Passive margins can record strikingly different thermo-mechanical history when they are in relative horizontal motion with underlying asthenosphere. The dynamic tilting arising from relative horizontal motion results in subsidence at the margin trailing the flow, and uplift at the margin leading the flow. During plate motion, the long-wavelength counter-flow is

induced beneath the ocean impact on contrasting long-term effects on the conjugate passive margins. The trailing margin can be inverted by enhanced compressional stress applied by the counter-flow. However, the elevated leading margin shows opposite behaviour, and is under extension. The results reasonably apply to the contrasting evolution of NW and SE margins of

Australian plate since the mid-Eocene, when the Australian plate accelerated to speeds up to 5 cm yr-1 relative to the Antarctica. The models also explain the asymmetric subsidence of the ocean floor at either side of the mid-ocean ridge between Australia and Antarctica, which cannot be explained by density anomalies in the mantle.

Acknowledgements

This research was undertaken with the assistance of resources from the National Computational

Infrastructure (NCI), through the National Computational Merit Allocation Scheme supported by the Australian Government; the Pawsey Supercomputing Centre with funding from the

Australian Government and the Government of Western Australia, and support from the

Australian Research Council through the Industrial Transformation Research Hub grant ARC-

IH130200012.

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98 1. DISCUSSION

In the preceding chapters, the dynamics of plate tilting is investigated by a suite of

analytical and numerical models. The analysis starts with investigations of dynamic topography

induced by a spherical density anomaly in the mantle. It is emphasized that the solutions for

dynamic topography induced by a rising sphere at depth with constant density and viscosity

(Morgan, 1965a, 1965b) stands as a reference for more complex dynamic topography models.

The solution for dynamic topography amplitude for layered viscosity (two layers, the top represents the lithosphere, Chapter 2, Fig 1. and Eq. 4), provides sensitivity of the solution to the thickness of layers and ratios of viscosities (Morgan, 1965a). That level of understanding is sufficient to interpret more complex models of dynamic topography arising from nonlinear rheology of the upper mantle. There is also a spherical harmonics approach to derive the Earth’s

dynamics topography for various radial variations in viscosity for the whole mantle (Hager and

Clayton, 1989). If the rheology of the mantle is non-Newtonian, this analysis limits the

investigation to simplified lateral viscosity variations (Richards and Hager, 1989; Moucha et

al., 2007). For simplicity, Morgan (1965a)’s solution is used for comparison with numerical

models of upper mantle density anomaly and resulting dynamic topography. The numerical

models solve for instantaneous flow and stresses in response to embedded thermal anomaly,

therefore they don’t capture time dependent characteristics. The top wall is free slip (Vz = 0),

which forces it to be stationary along the vertical axis, results in accumulation of non-zero

normal stress from which we predict the dynamic topography. At one hand, this method doesn’t

require considerable usage of computational resources, it’s time efficient and it enables the

users to perform numerical experiments at high resolution of model geometry and complex

rheologies. On the other hand, this method is not suitable to capture time-dependent

characteristics of the geodynamic problem being investigated partly because there is no

feedback from the arising topography back to the mantle.

99 The discrepancy between predictions of dynamic topography amplitude (relatively

higher) and estimations of residual topography (relatively lower) is addressed by 3D thermo-

mechanical numerical experiments. The numerical experiments are designed to be comparable

with the analytical solutions, which make them easier to interpret at increasing complexity.

The first few experiments use constant viscosities for the lithosphere and asthenosphere.

Because the dynamic topography solutions depend on the viscosity contrast, not on the

magnitude, a Haskel value of 1021 Pa×s used for the asthenospheric viscosity is not meant to

represent “low” viscosity asthenosphere. We model the impact of low viscosity layer on

dynamic topography in a separate experiment (Exp. 3, Chapter 2), by describing a low viscosity

channel between the lithosphere and deep asthenosphere, which has a viscosity of 1019 Pa×s.

Later, we increase complexity of the models by introducing nonlinear mantle rheology. The

experimental results show that, at wavelengths <1,000 km, predictions for amplitude of

dynamic topography are significantly reduced by considering nonlinear rheology driven by

local variations in viscosity in the upper mantle. This local variation could be either in the form

of a low viscosity channel underneath the lithosphere, or via decreasing the mechanical

thickness of the lithosphere by changing the viscosity within the lithosphere. The latter cannot

be captured in mantle convection models if one defines the lithospheric thickness a priori. It

should be noted that because the numerical experiments are designed to minimize the boundary

effects, the radius of the embedded density anomaly (~100 km) is forced to be relatively small

compared to the dimensions of the box (3,840 km × 3,840 km × 576 km) and distance to the walls (372 km away from the top, 204 km away from the bottom). However, this limits our

analysis of dynamic topography to wavelengths less than 1,000 km. In that regard, it is of our

interest, in the future, to uncover the effect of nonlinear rheology on dynamic topography

amplitudes at larger wavelengths (>1, 000 km), especially at degree 2 at which the largest misfit

was reported (Steinberger, 2016). We acknowledge that our dynamic topography solutions are

100 based on the assumption that the spherical anomaly in the mantle is rigid. This assumption fails

if the spherical anomaly were allowed to rise (or sink if denser), because the shape of the anomaly would change in time according to the velocity gradients around the sphere (Sleep,

1997; Burov and Gerya, 2014). Therefore, the anomaly cannot be considered as rigid in more realistic scenarios. Such deformation in the sphere can affect the amplitude and wavelength of the dynamic topography. We consider, in the future, to investigate the same problem with time- dependency.

In the Chapters 3 to 5 of this thesis, it is proposed that dynamic vertical motions of plates are driven not only by density anomalies, but also by relative horizontal plate motions.

Both produce long-wavelength dynamic topography in the form of plate tilting. This finding brings up the following question: Is it possible to differentiate one signal from another? First, based on temporal correlations between subsidence history of sedimentary basins and relative horizontal plate motions, one can constrain the changes in dynamic topography due to changes in plate motion. However, this is not enough to say that the dynamic tilting was driven by changes in plate motion. One may use information about, if available, the time it takes for the dynamic uplift or subsidence to develop in order to differentiate the signals (i.e. rate of uplift/subsidence). A recent compilation of shoreline data indicates very fast uplift of passive margins (~200 m Myr-1). However, in the case of mantle density anomalies, for the convecting

mantle, it requires tens of Myr to introduce significant change in normal stress and dynamic

topography at the base of plates. These drive long-wavelength vertical motions, but they

typically develop at relatively slow rate, typically below 100 m Myr-1. For example, at

relatively more stable platforms, Moucha et al. (2008) finds dynamic vertical motions by ~3.3

m Myr-1 for the New Jersey and its conjugate African margin. Conrad and Gurnis (2003)

reports an uplift rate of ~10 m Myr-1 for the southern Africa. Likewise, Austermann et al. (2017)

finds less than ~40 m Myr-1 global mean change in passive margin dynamic topography for the

101 last 125 kyr. To put it in the context of mantle dynamics, it should take ~100 Myr to change

the buoyancy structure of the mantle (Bunge et al., 1998; Iaffaldano and Bunge, 2015), which

makes the mantle density anomalies to be less likely to drive very rapid long-wavelength vertical motions of plates.

On the other hand, changes in plate motion can occur as fast as 100 kyr (Iaffaldano,

2014). These, depending on the change in relative speed, may be associated with significant

change in basal shear and normal stresses. Our analytical models show that dynamic

topography can develop in ~50 kyr at a basal stress anomaly wavelength of 3,000 km and a

plate viscosity of 1023 Pa s, suggesting a relative rapid development of topography, and allowing uplift rates of about a few 100 m Myr-1, consisten with observations such as (Pedoja et al., 2011; Gurnis et al., 2020) However, our analysis also shows at relatively smaller wavelengths, this period could be extended to a few Myr. Therefore, it’s important to focus on long-wavelength components of dynamic topography when differentiation the source for the dynamic topography signal. Also, at flexural wavelengths (< 300 km), vertical motions could be affected by elasticity of the plate. We conclude that rapid long-wavelength vertical motions of plates can be linked to coeval changes in horizontal plate motion. These arguments motivate us to further investigate, in the future, the dynamic topography of plates by considering both density anomalies and changes in plate motions (with viscoelasticity) acting together in geodynamic models.

The findings about dynamic topography due to horizontal plate motion were derived from relatively simple models, consisting a single laterally homogenous continent. In the last chapter, the models are modified to mimic a typical passive margin with smooth transition in isotherms between the continental and oceanic lithosphere. The similar boundary conditions are used to model the relative horizontal motion of plate with respect to the underlying mantle.

The models show that a large-scale counter-flow forms beneath the ocean basin. This impacts

102 on the topography, and therefore, the ocean basin and continental interior is tilted by differing amounts. Furthermore, the passive margin on the trailing edge is being compressed as it is facing the induced counter-flow. It is argued that this can cause episodic inversion of the basin, similar to the basin inversion events since the mid-Eocene identified in the Otway Basin of southeast Australia. As long as there is a relative horizontal plate motion the passive margins on either side of the mid-ocean ridge should record different mechanical histories. This finding motivates us to further investigate the dynamics of rifting under relative horizontal plate motion.

Previous models mimicking rifting near an Euler pole suggest that asthenospheric doming and associated gravitational stresses can exert mild compressive stresses (Mondy et al., 2018).

Similarly, numerical models investigating the inversion of rifted margins show the importance of thermal structure of the basin which is related to period of cooling (Buiter et al., 2009). The origin of compressive stresses for the inversion of sedimentary basins could be due large-scale counter-flow beneath the ocean lithosphere. The strength of the inversion will certainly depend on the density structure, pre-existing shear zones and sediment thickness (Sandiford and Hand,

1998; Hansen and Nielsen, 2003; Buiter et al., 2009), which could be implied in future models with significant resolution.

2. CONCLUSION

This thesis addresses the Earth’s dynamic topography driven by mantle density anomalies and investigates the dynamics of plate tilting due to horizontal plate motion. This is done by investigating the coupling between the plates and underlying mantle. In that regard, a suite of analytical and numerical models is used to predict the tempo and amplitude of vertical motions of plates. The results are applied to explain the rapid and long-wavelength vertical motions of India and Australia. The findings of the thesis can be summarized in five following points:

103 1) The predictions for magnitude of dynamic topography via global mantle

convection models can be improved by considering the non-linear rheology of the

mantle. The 3D thermo-mechanical numerical models are implemented to predict the

amplitude of dynamic topography due to a rising sphere in the mantle. They show that,

at wavelengths smaller than 1,000 km, the predictions for amplitude of dynamic

topography are nearly half of what models with isoviscous rheology predict. This is

driven by the induced local variations in viscosity which manifests itself as a low

viscosity channel beneath the lithosphere. It can also be due to variation in lithospheric

thickness determined by the viscosity profile. It is important to re-emphasize that our

numerical models are currently limited to dynamic topography wavelengths less than

1,000 km. However, we seek to further improve them to investigate the longer

wavelengths (at least a few thousands of km) at which the misfit increases (Hoggard et

al., 2016; Steinberger, 2016).

2) Basal shear can induce large-scale tilting of a rigid plate, inducing topography

amplitudes on the order of a few tens of metres, comparable to sea-level variations.

Applying the torque-balance method for a rigid plate in horizontal motion, it is found

that up to a few MPa shear stress acting on the base of the plate is balanced by tilting

of the plate (at a few thousand-km length) by a few tens of metres. The induced

topography increases with increase in distance between the plate’s centre of mass and

its base (l) but lessens by increase in plate length (L): ℎ = )*!$ . !"#"$%&'"( +,-

3) Horizontal motions of plates can be associated with horizontal gradients in shear

stress and normal stress which together induce plate-scale tilting. The analysis of

104 horizontal plate motion is investigated via 2D thermo-mechanical numerical

experiments with non-linear variation in viscosity between the plate and underlying

mantle. They show that plate-scale tilting is enhanced and dominated by the induced

gradient in normal stress, creating topography on the order of hundreds of metres. This

finding is tentatively applied to enigmatic Neogene tilting of India.

4) Plates can rapidly tilt after a change in relative horizontal plate motion. The 2D

analytical calculations by assuming constant plate viscosity and density show that basal

stresses (associated with change in plate motion) can tilt the plate in ~50 kyr for a stress

anomaly wavelength of 3,000 km and a plate viscosity of 1023 Pa s. This rapid plate-

tilting, with uplift/subsidence rates about a few 100 m Myr-1, can explain the

anomalous vertical motions of the Australian plate which followed relative plate

motion changes since the Late Cretaceous.

5) The edges of plates trailing a horizontally flowing mantle can be exposedto

compressional stresses that induce inversion of the passive margins. 2D numerical

models simulating relative horizontal platemotion, with conjugate passive margins

connected by an ocean basin show that large-scale counter-flow is induced beneath the

ocean lithosphere. This compressesthe trailing edges of plates which can induce

inversion of the pre-existing faults. This mechanism is applied to basin inversion events

since the mid-Eocene documented along the south-east Australian margin.

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107 APPENDIX A – Fundamentals of Geodynamics

I. Momentum of Plates

It is important to understand that the dynamics of the Earth’s interior largely differs

from the mechanics of visible objects around us. Our intuitions about the latter are derived

mainly from daily experiences which may mislead us when we contemplate geological

processes such as the motion of rigid plates relative to each other. Newtonian physics can

satisfactorily explain the motion of a tennis ball stroke by a racquet or the track of a space

shuttle launched for a mission to outer space. Newton’s first law states that a moving object

will keep its motion unless acted upon by an external force (Newton, 1833). The tendency of

objects to keep their motion is called inertia and is quantified by multiplying their speed and

mass, thereby finding their momentum. The heavier and faster the object, the harder to stop it,

which complies with our intuition. We assume a tectonic plate 1,000 km wide and long, 100

km thick and moving at 3 cm yr-1 relative to the deep mantle underneath. The plate will have a

momentum about 3 x 1011 kg m s-1 in the direction of its motion. This amounts to nearly 35,000

times the momentum of a Hexie (和谐号) super-fast train in China with 1,000 people on board

and moving at a speed of 350 km per hour. It looks like tectonic plates have quite a large

momentum, so it is tempting to think that it must be very hard to stop a moving plate.

Surprisingly, this is not the case. Plates have very large basal areas that resistive forces are

acting on. This is analogous to the air drag for a moving tennis ball, or water resistance for a

swimmer in a pool. For the case of plates, the resistance is much higher (i.e. ~1021 Pa s

compared to ~0.001 Pa s for water) and applies to a very large basal area which results in a

significant amount of resistive force.

108

In order to have a sense of the magnitude of forces in action, one can calculate how much time

it takes to stop the plate by basal friction. By using Newton’s second law stating that a change

in momentum of an object is proportional to the applied net force, in this case the basal friction,

one can find that it takes only around one millisecond to stop our moving plate. That is certainly

notably smaller than what one might predict. This result carries a significant meaning. Tectonic

plates are in a continuous force balance, and they are not accelerating like a tennis ball, at least

in a period that we can identify. Therefore, it is safe to ignore the inertia or momentum of plates

in the analyses of plate motions (this example is modified from Randall Richardson’s lecture

notes). This is one of the fundamental assumptions used by those studying the dynamics of the

Earth’s crust and mantle. This idea has been quantified by Osborne Reynolds, and provides

great simplifications when we describe properties of a viscous fluid, which is discussed in the

next section.

II. A Glimpse of Dynamics of Plates through Reynolds Number In the realm where momentum is negligible, the rocks behave as a very viscous fluid,

and for now, we neglect elastic and brittle behaviour and ignore volcanic eruption dynamics

where viscosity can be extremely low (Giordano et al., 2008). A typical rate of deformation of

those rocks (i.e. strain rates) is about 10-15 s-1. The properties of such conditions are explained

by a simple dimensionless parameter first proposed by Osborne Reynolds (Reynolds, 1883),

later to be recognized as Reynolds number (Re) in 1908 thanks to Arnold Sommerfeld (Rott,

1990).

When Reynolds put forward his ideas based on laboratory experiments, Sir George Stokes, one

of the well-known physicists of his era, commented that “...there is absolutely nothing to prove

that he (Osborne Reynolds) has discovered the necessity of an additional constant (Reynolds

number) to define a fluid.” when he reviewed Reynolds’ paper (Jackson and Launder, 2007).

109 However, due to the uniqueness of the work, he recommended the paper to be printed in The

Phil. Trans in 1883. It was fortunate that Reynolds’ work, providing a simple method to

illuminate the fluids’ properties, has found its place in the literature.

The Reynolds number basically signifies the strength of inertial forces relative to viscous forces

for a system of particles or a fluid. Now, we will compare the Reynolds number for our moving

rigid plate with the Reynolds number of another reference object. If we assume that the plate

is moving at a speed of 3 cm yr-1 and it has a viscosity of 1021 Pa·s, then the Reynolds number is:

, ×. ×, [] �� = = = 2.85 × 10 (1) [ ]

,where r is the average density, h is the viscosity and l is the characteristic length of the plate.

The resulting minute Reynolds number (Re <<1) shows that inertial forces (nominator in

Equation 1) can be neglected because they are very small compared to viscous forces

(denominator) acting in the system. This enables us to neglect the inertial terms in the momentum equation given below:

⃗ = ∇ ∙ � + �⃗ (2)

,where s is stress tensor and g is gravitational acceleration vector. Because the momentum is negligible, the left-hand side of Equation 2 can be replaced with zero, therefore:

0 = ∇ ∙ � + �⃗ (3)

If we expand the stress tensor, we get:

0 = ∇� + ∇ ∙ � + �⃗ (4)

,where p is the pressure and t is deviatoric stress tensor (see Pedlosky, [1979] for a full derivation of Stokes’ equations and McKenzie et al., [1974] for the inertial terms and other constraining equations).

110

The Re is also useful in predicting if a fluid will show flow instabilities (e.g. formation of

eddies and vortex street when Re>>1), therefore it can give insights on dynamics of different physical environments. For example, a swimmer in a pool of water will have a Re of about 104

(Fig. A1) and can easily move by thrust, but bacteria living in a low Re (~10-4) needs more sophisticated movements to make a displacement (Purcell, 1977). This difference is due to a change in relative strength of viscous and inertial forces in different scales and environments.

We can predict that the dynamics of plates that have very low Reynolds number will be quite different than those of a swimmer or a scorpion fish (Fig. 1.5). The Reynolds number is also iterated as the ratio of momentum advection to the momentum diffusion. The low Re in the

Figure A1: Illustration objects (not to scale) with different Reynolds number (Re), modified after (Purcell, 1977). Earth’s interior means that advection of momentum is negligible. Therefore, convective motions in the mantle cannot be impacted by daily rotation of the Earth around it’s spin axis

(McKenzie, 2018). This analysis of plate motion through Reynolds number shows that constraining the components of viscous forces in the system, therefore viscosities in the Earth’s interior are necessary to analyse the dynamics of plates and their interaction with the underlying mantle. Furthermore, as we can see from Equation 4 that gravitational forces are also components of this system and it requires an accurate knowledge on the temperature and density structure of the Earth’s interior to investigate its dynamics. These information are an input to the momentum equation which is coupled with the continuity equation and

111 conservation of thermal energy of the system (McKenzie et al., 1974; Pedlosky, 1979). Now, we are in the stage of going through some of the major findings on thermal, density and viscosity structure of plates and underlying mantle, as well as a sequence of ideas on the link between the deep Earth and the surface.

112 APPENDIX B Supplement to Article 2

Supplementary Material for

“Fast Tilting of Plates Driven by Horizontal Plate Motion”

Ö. F. Bodur1*, P. F. Rey1, G. A. Houseman1,2

1EarthByte Group, School of Geosciences, The University of Sydney, NSW 2006, Australia,

2School of Earth and Environment, University of Leeds, Leeds, UK,

*Email: [email protected]

This file includes:

Numerical modelling method

Derivation of torque applied by basal stresses

Derivation of counter-balancing torque by tilt of the plate

Rheologies and thermal parameters (Table S1)

Figures S1-S5

Other Supplementary Materials for this manuscript include the following:

Input files for numerical models (xml scripts)

113

Numerical modelling method

We used the open source code Underworld (Moresi et al., 2003, 2007) to solve 2D

Stokes equation at small Reynolds number using a rectangular grid of 2 km resolution in both

lateral and vertical extent. Lagrangian particles (i.e. particles in cells) are advected through the

grid and record temporal properties of rocks. Pressures and velocities are computed at the nodes

of the grid. We used a high-density of passive tracers to track the elevation of the continent

through time. We disregard the lower mantle (i.e. below ~670 km) because mixing of the lower

and upper mantle is slow and intermittent due to the much greater viscosity of the lower mantle.

Hence, our model represents the upper 700 km of the Earth, and includes a 150 km thick

continental lithosphere with a 40 km thick continental crust above which there is a 68 km thick

“sticky air” layer to accommodate the evolution of topography. Our reference model has a

lateral extent of 3,840 km. The viscosity of the sticky air is 1x1018 Pa s, which ensures the

convergence of the solution. We also verified that a less viscous sticky air (i.e. 1x1017 Pa s) does not affect the results. Assuming an incompressible flow,

� ∙ � = 0 (1) the momentum is conserved via:

�� + �� = 0 (2) where v,s,r and g denote velocity, stress tensor, density and gravitational acceleration, respectively. Temperature (T) is determined by advection-diffusion interaction and is modulated by radioactive heating (H), heat capacity (cp), and thermal conductivity (k).

Altogether, they take part in the conservation of internal energy of the system via the following

equation:

�� + � ∙ ∇� = �∇� + �� (3)

114

The initial geotherm is steady-state, and typical of the average continental geotherm with a

Moho temperature at 823 K and the isotherm 1626 K, which defines the lithosphere-

asthenosphere boundary (LAB), at 150 km. We set the initial temperature everywhere below

the LAB to 1626 K (including the boundary temperature at 700 km) in order to avoid upwelling

from the base.

As the densities vary by temperature and pressure, the effective densities are calculated using:

r = r0 [1- a(T-T0)] [1+ b(P-P0)] (4) where r0 ,T0,a, and P0 denote reference density, reference temperature, thermal expansion

coefficient, and compressibility respectively. Melting and surface processes (i.e. erosion and

sedimentation) are not activated in the models.

In the reference model, we used temperature and strain rate dependent visco-plastic

rheologies for the crust (quartzite) (Ranalli, 1995) and mantle (dry olivine) (Karato and Wu,

1993). They yield at the lower value of either viscous stress (determined by viscous rheology

parameters) or depth dependent frictional plastic yielding (determined by coefficient of friction

and cohesion). The viscosity range was limited for both the crust and mantle to be between

1x1018 Pa s to 1x1023 Pa s to facilitate the convergence of the solution. The viscous flow law is implemented via (Karato and Wu, 1993):

�(�̇, �, �) = � � �̇ � (5) where �̇ is the 2nd invariant of the strain rate, (A) is pre-exponential factor, (n) is exponent for the power law, (d) is grain size, (p) is exponent of grain size, (Q) is activation energy (see Table

1), (P) is pressure, (V) is activation volume, (R) is ideal gas constant, (T) is temperature in

Kelvin. A linear dependence on depth is used for the frictional plastic deformation using the following formula:

� = �� + � (6)

115

where t is the 2nd invariant of the deviatoric stress tensor (we call as strain hereinafter in this

section), depending on coefficient of friction (µ), lithostatic pressure at the node of interest (sn),

and cohesion of the rock (C0). As plastic strain accumulates, the cohesion and coefficient of friction linearly decrease from their initial values (C0 = 10 MPa and µ = 0.577) to their defined limits (C0 = 2 MPa and µ = 0.017) when the saturated plastic strain is reached (i.e. 0.2, see

Table S1). The parameters (Table S1) that are necessary to calculate viscous and frictional plastic rheologies are taken from laboratory experiments (Karato and Wu, 1993; Ranalli, 1995).

We impose velocity boundary conditions at side and bottom walls to model the relative motion of plate and asthenosphere. For the reference model, the horizontal component of the velocity increases from 0 cm yr-1 at 150 km depth to a maximum velocity of 10 cm yr-1 at 700

-1 km depth. We imposed Vz=0 cm yr at the side and bottom walls.

Derivation of torque due to basal stresses

Figure S1. Free body diagrams to calculate the torque due to A) applied basal stresses and B) due to tilt of the plate.

116

The torque, � (�), applied by basal stresses to the lithosphere is calculated by the surface integral of vector product of the distance vector (r) and total stress vector (Fig. S1A):

� (�) = ∫ � × [� ̂ + � �] �� (7)

where (̂) and (�) are unit normal vectors in x and z directions, respectively. The torque per unit width is:

�� (�) = ∫ � × (� ̂ + � �) �� = ∫ � × (� ̂) �� + ∫ � × (� �) �� (8)

� = ∫ �� (�) �� + ∫ �� (∅) �� (9)

The plate extends from –L/2 to L/2, and we replace the sines with their corresponding

parameters by using Fig. S1A:

/ / / / � = ∫ �� (�)�� + ∫ �� (∅)�� =∫ �� + ∫ �� (10) / / / /

/ / � = � �� + � �� (11) ∫/ ∫/

where l is the half plate thickness. We assume that shear stress at the base of the plate is constant

(txz =ta), but normal stress shows linear gradient across the plate such as szz=Wx + c, where W

and c are constants. For simplicity, we assume c=0. Then the horizontal torque per unit width

is:

/ / � = � �� + � � �� (12) ∫/ ∫/

� = �� + � (13)

Derivation of counter-balancing torque by tilt of the plate

We assume that the plate is tilted with a slope of l in radians upward to the right and downward to the left. The increased lithostatic stress will apply a torque in –y direction (Fig.

S1B). The torque per unit width of the plate is:

117

��/� (�) = − ∫ � × (� �) �� (14)

�/ = − ∫ ��sin (�) �� (15)

/ � = − �ℎ(�)� ��in (�) �� (16) / ∫/ _

where h(x) is the topography varying with x coordinate. By geometry in Fig. S1B, h(x)=sin(l)x

»lx, and sin(q)=x/r. Therefore:

/ _ / �/ = − ∫ �ℎ(�)�_��in (�) ��=− ∫ �� = − ∫ ��_�� (17) / /

� = −�� (18) / _ where r is the average density of the lithosphere and g is gravitational acceleration. By

Archimedes’ principle, the tilt of the plate will induce a buoyant force applied at the centre of

the displaced volume of asthenosphere (Fig. S2).

Figure S2. Free body diagram to calculate the torque applied by buoyancy force due to tilting.

The torque per width of plate around the centre of mass of the plate can be calculated by the

integral below:

118 / / � = −� �� = −� �� (19) / ∫/ ∫/

� = −� �� (20) /

The condition that sum of torques (by Eqn. 13, 18 and 20) must vanish gives:

−�� (� + � ) + �� + � = 0 (21) _

If we consider a tilted plate moving horizontally (Fig. S3), then the applied torque may vary in response to a slight change in geometry. For a 1,000 km-long plate and 1.5 km topography at one edge, the tilt of the plate will decrease the applied shear stress by 0.000001125 �, which can be considered as a minor effect.

Figure S3. Free body diagrams to calculate the torque for a tilted plate.

Rheologies and thermal parameters

In our models, we use rheological flow law and values from laboratory experiments

given below:

Parameters Crust Upper Mantle

Pre-exponential factor (MPa-n s-1) 6.7 x 10-6 2.41 x 10-16

119

Activation energy (kJ mol-1) 156 540

Power law exponent 2.4 3.5

Activation volume (m3 mol-1) 0 15 x 10-6

Reference density (kg m-3) 2,700 3,370

Reference temperature (K) 293.15 293.15

Initial cohesion (MPa) 10 10

Cohesion after weakening (MPa) 2 2

Initial coefficient of friction 0.577 0.577

Coefficient of friction after weakening 0.017 0.017

Saturation strain 0.2 0.2

Thermal diffusivity (m2 s-1) 1x10-6 1 x 10-6

Thermal expansivity (K-1) 3x10-5 3 x 10-5

Compressibility (MPa-1) 4x10-5 0

Heat capacity (J K-1 kg-1) 1,000 1,000

Radiogenic heat production (W m-3) 0.5 x 10-6 0.02 x 10-6

Table S1. Rheologies and thermal parameters. Rheological and thermal parameters for crust (i.e. quartzite) (Ranalli, 1995) and mantle (i.e. dry olivine). Saturation strain refers to the maximum possible plastic strain (second invariant of the deviatoric strain tensor) beyond which strain weakening is not effective.

Other Supplementary Figures and Captions

120

Figure S4. A) Viscosity map and velocity field for model having same properties and boundary conditions with reference model, except it doesn’t have air on top of the crust and the top wall is free-slip. B) Horizontal component of velocities in the centre of the Model STRESS (in black) and Reference Model (in grey).

Figure S5. Variation in induced normal stress and imposed basal velocity in the reference model with three different boundary velocities. The graph shows an apparent power-law

121

with an exponent of n=2.4, which is the power-law exponent for crustal rheology. In the above plot, the scaling factor is a= 19.2.

Figure S6. The time it takes to dynamically tilt the plate by harmonic basal shear stress. The plot is derived from analytical calculations assuming an isoviscous mantle. The tilting episode of the plate is mostly sensitive to the wavelength, and linearly scales with plate viscosities, as shown by comparison of curves for 1021 Pa·s and 1023 Pa·s plate viscosities.

122 APPENDIX C Supplement to Article 3

Supplementary File for

“On the Tempo and Amplitude of Tilting of Tectonic Plates by

Basal Stresses”

Ö. F. Bodur1*, G. A. Houseman2, P. F. Rey1,

1EarthByte Group, School of Geosciences, The University of Sydney, NSW 2006, Australia,

2School of Earth and Environment, University of Leeds, Leeds, UK,

*E-mail: [email protected]

Stream function formulation for 2-D flow with harmonic variation in basal

stress, and constant viscosity and density

From the stream function that we derived in the main text (Equations 8) we can find

the velocities and velocity gradients as follows (by using Equations 3 and 8 in the main text):

�(�, �) = −[(�� + � + ��)� − (�� − � + ��)�]sin(��) (1)

�(�, �) = [�� + �� + �� + ��]�cos(��) (2)

= −�[(�� + � + ��)� − (�� − � + ��)�]cos(��) (3)

= −�[(�� + 2� + ��)� + (�� − 2� + ��)�]sin(��) (4)

= −�[�� + �� + �� + ��]sin(��) (5)

= �[(�� + � + ��)� − (�� − � + ��)�]cos(��) (6)

By using Equations above, strain rates can be derived as follows:

�̇ (�, �) = + = −�[(�� + � + ��)� + (�� − � + ��)�]sin(��) (7)

123 �̇ (�, �) = −�[(�� + � + ��)� − (�� − � + ��)� ]cos(��) (8)

�̇ (�, �) = �[(�� + � + ��)� − (�� − � + ��)� ]cos(��) (9)

We can confirm that the sum of normal strain rates (�̇ + �̇ ) is zero everywhere as required for the incompressible flow.

�̇ + �̇ = −�[(�� + � + ��)� − (�� − � + ��)� ]cos(��) + �[(�� + � +

��)� − (�� − � + ��)�]cos(��) = 0

Now, we can get the deviatoric stresses:

�(�, �) = −2��[(�� + � + ��)� − (�� − � + ��)� ]cos(��) (10)

�(�, �) = 2��[(�� + � + ��)� − (�� − � + ��)� ]cos(��) (11)

�(�, �) = −2��[(�� + � + ��)� + (�� − � + ��)� ]sin(��) (12) and total stresses (relative to lithostatic contribution):

�(�, �) = [[� − 2��(�� + � + ��)]� + [� + 2��(�� − � + ��)]� ]cos(��) (13)

�(�, �) = [[� + 2��(�� + � + ��)]� + [� − 2��(�� − � + ��)� ]]cos(��) (14)

�(�, �) = −2��[(�� + � + ��)� + (�� − � + ��)� ]sin(��) (15)

From the x-component of conservation of momentum,

+ = 0 (16)

+ + = 0 (17)

2� + � ( + ) + = 0 (18)

because the flow is incompressible (� = −�):

124

� () = � () = −� () (19)

�� + = 0 and �� + = −�� (20)

= −[(�� + � + ��)� − (�� − � + ��)�]�cos(��) (21)

= �[(�� + � + ��)� − (�� − � + ��)�]sin(��) (22)

= −�[(�� + 2� + ��)� + (�� − 2� + ��)�]sin(��) (23)

= −�[(�� + 3� + ��)� − (�� − 3� + ��)�]sin(��) (24)

�� = −2�[�� + ��]sin(��) (25)

= −�[�� + ��]sin(��) (26)

Therefore,

−2��[�� + ��] − [�� + ��] = 0 or [(� + 2��)� + (2�� + �)�] = 0

(� + 2��) = 0(2�� + �) = 0

Hence:

� = −2�� = −2�� (27)

The total stresses can be restated as:

�(�, �) = 2��[−(�� + 2� + ��)� + (�� − 2� + ��)� ]cos(��) (28)

�(�, �) = 2��[(�� + ��)� − (�� + ��)� ]cos(��) (29)

�(�, �) = −2��[(�� + � + ��)� + (�� − � + ��)� ]sin(��) (30)

We can check that the divergence of the stress is zero:

+ = 0 (31)

125 −2��[−(�� + 2� + ��)� + (�� − 2� + ��)� + [(�� + 2� + ��)� − (�� −

2� + ��)�]]sin(��) = 0

+ = 0 (32)

−2��[(�� + � + ��)� + (�� − � + ��)� − (�� + � + ��)� − (�� − � +

��)�]cos(��) = 0

We note that we have subtracted the lithostatic part of the stress field from above expressions

which give the anomalous part of the stress field associated with the forced flow. Then, the

general solution of the plane-strain flow problem in a layer with horizontal periodicity given

in terms of velocity and stress is:

�(�, �) = −[(�� + � + ��)� − (�� − � + ��)�]sin(��) (33)

�(�, �) = �[�� + �� + �� + ��]cos(��) (34)

�(�, �) = 2��[−(�� + 2� + ��)� + (�� − 2� + ��)� ]cos(��) (35)

�(�, �) = 2��[(�� + ��)� − (�� + ��)� ]cos(��) (36)

�(�, �) = −2��[(�� + � + ��)� + (�� − � + ��)� ]sin(��) (37)

And boundary conditions representing flow driven by shear on base of the layer:

On z = 0: u = 0, � = ��(�)cos(��) (top: deflected rigid lid) (38)

On z = h: � = 0, � = �sin(��) (base: isostatic, sheared) (39)

On x = 0 and x = �/2, � = 0 (sides: stress free) (40)

Conditions from z = 0 (top):

[(�� + �) − (�� − �)] = 0 (41)

126 � − � = �(�) = ��(�) (42)

Conditions from z = h (base):

(�� + ��ℎ)� = (�� + ��ℎ)� (43)

[(�� + � + ��ℎ)� + (�� − � + ��ℎ)�] = − (44)

As a matrix equation:

1 0 −1 0 � ��(�) � 1 −� 1 � 0 = (45) �� ℎ� −�� −�ℎ� � 0 �� (1 + �ℎ)� �� (−1 + �ℎ)� � −�⁄(2��)

which can be solved by elimination and back-substitution, and confirmed by verification that

Equations (41) to (44) are satisfied. The analytic solution of the matrix equation (45), using the normalizing factor J, is:

� = �[cosh(�ℎ)� + �ℎ(�ℎ + 1)]( )� − �cosh(�ℎ)( ) (46)

� = −�[cosh(�ℎ)� + �ℎ]()� + �sinh(�ℎ)( ) (47)

� = �[cosh(�ℎ)� + �ℎ(�ℎ − 1)]( )� − �cosh(�ℎ)( ) (48)

� = �[cosh(�ℎ)� − �ℎ]()� − �sinh(�ℎ)( ) (49)

where the normalizing factor is:

� = (50) ()

The rate of change of the surface deflection is given by the vertical velocity component v on z

= 0:

127 cos(��) = [� + �]�cos(��) (51)

= 2�[cosh(�ℎ) + �ℎ]()� − 2�ℎ�cosh(�ℎ)( ) (52)

To simplify the following, we write the preceding equation in the form:

= �� − � (53)

where α and β are constants dependent on the physical parameters and the geometry (hk). The

surface deflection reaches a static value (the time derivative is zero) when:

� = = () ( ) (54) [()]

A general time-dependent solution of (81) can be obtained given an initial condition e = e0 at

t = 0:

� = � + (� − �)exp(��) (55)

In general, α will be negative because of how we defined J. so the prediction is for an

exponential relaxation from initial state e0 to final state es, with a timescale on which the

unrelaxed signal is diminished by the factor of e:

() � = = [ ]( ) (56) ()

The time for relaxation clearly increases with viscosity, but depends on geometry via the factor kh.

Now, we consider the problem in which flow is driven by a normal stress perturbation acting on the base of the layer:

On z = 0: u = 0, � = ��(�)cos(��) (top: deflected rigid lid) (57)

On z = h: � = 0, � = −�sin(��) (base: normal stress) (58)

128 The minus sign is included so that the uplift is positive on the right side of the layer. The

form of the solution (Eqns. 1 to 5 in the main text). The conditions from z=0 (top):

[(�� + �) − (�� − �)] = 0 (59)

� − � = �(�) = ��(�) (60)

Conditions from z=h (base):

(�� + ��ℎ)� − (�� + ��ℎ)� = (61)

[(�� + � + ��ℎ)� + (�� − � + ��ℎ)�] = 0 (62)

As a matrix equation, the matrix is unchanged from (45), but the vector on the right side

changed:

1 0 −1 0 � ��(�) � 1 −� 1 � 0 = (63) �� ℎ� −�� −�ℎ� � −�/(2��) �� (1 + �ℎ)� �� (−1 + �ℎ)� � 0 which can be solved by elimination and back-substitution, and confirmed by verification that

Equations 59 to 62 are satisfied. We find that:

( ) � = () + �� (64) () ()

( ) � = () �� + () () (65) () ()

( ) ( ) � = − − �� (66) () ()

( ) � = () �� + () () (67) () ()

To summarize the four coefficients, using the normalizing factor J:

� = (68) ()

� = �[cosh(�ℎ) � + �ℎ(�ℎ + 1)] ∆ � + �[�ℎ sinh (�ℎ) + cosh (�ℎ)] (69)

� = −�[cosh(�ℎ) � + �ℎ] ∆ � − ��ℎ(�ℎ) (70)

129 � = −�[cosh(�ℎ) � + �ℎ(�ℎ − 1)] ∆ � + �[cosh(�ℎ) + �ℎ sinh (�ℎ)] (71)

� = �[cosh(�ℎ) � − �ℎ] ∆ � + � cosh (�ℎ) (72)

The rate of change of the surface deflection is given by the vertical velocity component v on

z=0:

cos(kx) = [� + �]� cos (kx) (73)

= 2�[��ℎ(�ℎ) + �ℎ] � + 2�[ℎ� sinh(�ℎ) + cosh (�ℎ)] (74)

To simplify the following, we write the preceding equation in the form:

= �� − � (75)

where � and � are constants dependent on the physical parameters and the geometry (hk). The surface deflection reaches a static value (the time derivative is zero) when:

[ ( ) ] � = = − () (76) [()]

A general time-dependent solution of (76) can be obtained given an initial condition e = e0 at t

= 0:

� = � + (� − �)� (77)

In general, � will be negative because of how we defined J. So, the prediction is for an exponential relaxation from initial state � to final state � , with a timescale on which the unrelaxed signal is diminished by the factor e:

( ) � = = (78) ()

The time for relaxation clearly increases with viscosity, but depends on geometry via kh. This also shows that the timescale of exponential relaxation is the same for both shear and normal tractions.

130 APPENDIX D Supplement to Article 4

Supplementary File for

“Contrasting Dynamic Topography and Inversion of Continental

Margins”

Ö. F. Bodur1*, P. F. Rey1, G. A. Houseman2

1) EarthByte Group, School of Geosciences, The University of Sydney, NSW 2006, Australia

2) School of Earth and Environment, University of Leeds, Leeds, UK

*Correspondence to: [email protected]

Methods

In our numerical models, we have used Underworld (Moresi et al., 2003, 2007) code

(open source) which solves Stokes equation in 2D at small Reynolds number. The particles in cells (i.e. Lagrangian particles) advect through the grid and trace properties of rocks varying in time. Velocities as well as pressures (lithostatic and total) are computed on the nodes of the rectangular grid with 2 km resolution in both axes. The lateral extent of the reference model is

3,840 km and the base is at 700 km depth. In the models, the lower mantle is disregarded as mixing of with upper mantle is gentle and irregular due to notably higher viscosity contrast.

The models include up to 135 km thick continental lithosphere with 35 km thick continental crust at the sides. On top the crust, there is a 68-km thick “sticky air” layer with constant viscosity (i.e. 5x1018 Pa s) to accommodate the evolution of topography (Fig. 1a). We also

tested that the results are not affected if we use a less viscous sticky air (i.e. 5x1017 Pa s). We

make use of passive tracers to track the vertical motion of the surface through time. An

incompressible viscous flow is described as follows,

� ∙ � = 0 (1)

131 with conservation of momentum:

�(�) + �� = 0 (2)

where s, v, r and g signify stress tensor, velocity, density and gravitational acceleration, respectively. The internal energy of the system is conserved via the following equation:

�� + � ∙ ∇� = �∇� + �� (3) where T,H,cp and k represents temperature, radioactive heating, heat capacity, and thermal

conductivity.

As an initial condition, the temperature at Moho is 550 °C, and LAB (lithosphere-

asthenosphere thermal boundary at 130 km depth) is defined by the 1350 °C isotherm. The

temperature of the asthenosphere is set to 1350 °C to evade formation of plume at the base, but the temperatures can evolve by diffusion-advection during model run. However, the temperature at the bottom wall is kept constant. The variation of temperature and pressure results in effective densities of rocks calculated via:

r = r0 [1- a(T-T0)] [1+ b(P-P0)] (4) where r0 ,T0, P0, b and a stands for reference density, temperature, and pressure, compressibility and thermal expansion coefficient respectively. Surface processes (i.e. sedimentation and erosion) as well as melting are not implemented in the models.

We used strain rate and temperature dependent visco-plastic rheologies (Table S1) for the crust (quartzite) (Ranalli, 1995) and mantle (dry olivine) (Karato and Wu, 1993). The rocks deform at the lower value of either frictional plastic yield stress (coefficient of friction and cohesion) or viscous stress. The viscosity range is limited between 1x1019 Pa s and 1x1023 Pa s to assist the convergence of the solution. Dislocation creep is used for the viscous component of the rheology as follows:

( ) �̇ = �� (5)

132 where �̇, is the strain rate depending on the pre-exponential factor (A), applied stress (s) and

its exponent -power law exponent- (n), grain size (d) and its exponent (p), and exponential of the activation energy (Q), pressure (P), activation volume (V), power law exponent, ideal gas constant (R), and temperature (T). A linear dependence on depth is used for the frictional plastic deformation using the following formula:

� = �� + � (6) where t is the 2nd invariant of the deviatoric stress tensor, depending on coefficient of friction

(µ), lithostatic pressure at the node of interest (sn), and cohesion of the rock (C0). The cohesion and coefficient of friction are decreased during deformation by a described strain weakening process.

We impose sidewall velocity boundary conditions to force asthenospheric flow relative to the stationary lithosphere. The horizontal component of the velocity increases from 0 cm yr-

1 at 160 km depth to a maximum velocity (between 1 and 10 cm yr-1, varying for each experiment) at 700 km depth. We imposed zero vertical traction on the vertical walls. A free- slip boundary condition (zero vertical velocity, zero horizontal traction) is imposed at the base of the model (i.e. at 700 km depth).

Rheologies and thermal parameters

In our models, we use nonlinear rheologies for both crust and mantle. All parameters are given below:

Parameters Crust Mantle

Pre-exponential factor (MPa-n s-1) 6.7 x 10-6 2.41 x 10-16

Activation energy (kJ mol-1) 156 540

133 Power law exponent 2.4 3.5

Activation volume (m3 mol-1) 0 15 x 10-6

Reference density (kg m-3) 2,700 3,370

Reference temperature (K) 293.15 293.15

Initial cohesion (MPa) 10 10

Cohesion after weakening (MPa) 2 2

Initial coefficient of friction 0.577 0.577

Coefficient of friction after weakening 0.017 0.017

Saturation strain 0.2 0.2

Thermal diffusivity (m2 s-1) 1x10-6 1 x 10-6

Thermal expansivity (K-1) 3x10-5 3 x 10-5

Compressibility (MPa-1) 4x10-5 0

Heat capacity (J K-1 kg-1) 1,000 1,000

Radiogenic heat production (W m-3) 0.5 x 10-6 0.02 x 10-6

Table S1. Rheological and thermal parameters for crust (i.e. quartzite) (Ranalli, 1995) and mantle (i.e. dry olivine) (Karato and Wu, 1993). Saturation strain refers to the maximum possible total strain beyond which strain weakening is not effective.

Creating conjugate margins by numerical modelling

We create 2D thermo-mechanical numerical models of extension with 0.75 cm yr-1 of extension velocity from each side of the continent. To keep the volume of material constant, we also impose influx from either side of the asthenosphere. The boundary conditions are summarised in Fig. S1a. The melting is not implemented in the model and they accumulate as the surface subsides below 1 km depth. After 34.5 Myr of model evolution, the conjugate margins form on either side of the centre of the model (Fig. S1c). Then, by keeping the margin

134 geometries unchanged, we introduce 1,000-km-long ocean basin and oceanic lithosphere

between the passive margins (Fig. S1c). For that purpose, we describe a 7-km-thick oceanic

crust. The ocean lithosphere thermal structure is described based on the following: we first

calculate the interval of time between the break-up of Australia from Antarctica at ~95 Ma

(Veevers, 1986) and the acceleration of the Australian plate towards NE (~43) (Powell et al.,

1988; Müller et al., 2016). We calculate that ~50 Myr of cooling results in ~80 km thick oceanic lithosphere based on plate-cooling model (Parsons and Sclater, 1977; Turcotte and Schubert,

2002). Then, we apply a smoothly varying LAB (thermal lithosphere-asthenosphere boundary) increasing in depth from the centre of the model to the continents at sides (i.e. down to 80 km depth and then gradually increases to 135 km towards the continent). Before we initiate the relative horizontal motion of plate or horizontal mantle flow, we ensure that the models are in dynamic near-steady-state condition at t=0, which is ensured by minute fluctuations due to isostatic unbalance (i.e., the magnitude of velocities before we initiate inflow/outflow is <0.3 mm yr-1). We follow the same procedure for three different aspect ratios (i.e., 3x1,5x1,9x1).

Supplementary Figures and Captions

135

Figure S1. 2D thermo-mechanical numerical model of continental extension and formation of conjugate passive margins. a) Model configuration and boundary conditions, b) model result at t = 34.5 Myr and c) manually separating the conjugate margins 1,000 km away and introducing oceanic crust and mantle lithosphere, which forms the model geometry of the reference model described in the main text. The entire model is also thermally equilibrated based on assumptions on the cooling of oceanic lithosphere.

136 Figure S2. Topography after model evolution of 5 Myr and 20 Myr, showing strong and rapid variation in topography with magnitude up to 1,000 m, attributable to compressive stresses at the trailing edge of the plate. On the other hand, the leading edge records mild inversion and lesser amount of uplift.

137 Figure S3. Model results for imposed velocity of 5 cm yr-1 at the base of the model. a) Velocities, b) 2nd invariant of the deviatoric stress tensor on the lithosphere, c) horizontal components of the velocities at the centre of the model (along the cross section shown in (a) ), and d) dynamic topography at t= 0.5 Myr.

138 Figure S4. Model results with larger (2x) oceanic gap and imposed velocity of 10 cm yr-1 at the base of the model. a) Velocities with isotherms b) horizontal components of the velocities at the centre of the model (along the cross section shown in (a) ), and c) dynamic topography at t= 0.5 Myr.

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