Role of water in the tectonics of Earth and Venus

Master Science de la matiere Internship 2013Ecole Normale Supérieure de Lyon BOUFFARD MathieuUniversité Claude Bernard Lyon I M2 Physique

Role of water in the tectonics of Earth and Venus

Abstract: Earth and Venus are two planets very similar in terms of size, density and probablychemical composition. Yet they have very different dynamical regimes since the Earth is presentlyexperiencing plate tectonics whereas Venus’ surface is stagnant. This is commonly attributed tothe absence of water on Venus but the question is still an open debate. In this work, Paul Tackley’scode Stag was used in 2D-spherical geometry. In a first phase, we ran 22 simulations to show thatmodeling the lithospheric rheology simply by a constant plastic yield stress is sufficient to producethree different regimes (stagnant, sluggish and mobile lids) that we characterise. A Matlab routineperforming plate detection was developed to enrich this study. Some aspects of the dynamicsof mobile lids were also explored qualitatively. Finally, various effects of water on viscosity andsolidus as well as water diffusion were implemented in Stag and two simulations were run to studythe dehydration of a mantle by diffusion. The results are provisional but tend to prove that localdehydration of the lithosphere can be sufficient to modify the planet’s dynamical regime.

Key words: mantle convection, plate tectonics, water, Venus

Internship monitored by:Stéphane [email protected] / tél. 06 20 28 30 66

Laboratoire de Géologie de Lyon : Terre, Planète, Environnement UMR CNRS 5276(CNRS, ENS, Université Lyon1) Ecole Normale Supérieure de Lyon 69364 Lyon cedex07 Francehttp://lgltpe.ens-lyon.fr/

mailto: [email protected].

http://lgltpe.ens-lyon.fr/

Stage M2 DSM - ENS de Lyon - 2013 Bouffard Mathieu

Acknowledgement

First and foremost, I would like to express my warmest thanks to Stéphane Labrosse for his sinceregenerosity, kindness and patience. I learned a lot with him during this internship which has been botha pleasant and extremely enriching time for me.

A special thanks also goes to Paul Tackley whose spontaneous and generous help has been decisivethroughout this project.

I finally express my gratitude to Frédéric Chambat and to my friends Laure Chevalier and PolGrasland-Mongrain for their useful comments on this report.

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Contents

1 Plate tectonics : a complex problem addressed by numerical simulations 41.1 What is plate tectonics ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 A very complex problem - the debated role of water . . . . . . . . . . . . . . . . . . . 6

1.2.1 Comparison between Earth and Venus . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 The water cycle in the Earth’s mantle . . . . . . . . . . . . . . . . . . . . . . . 71.2.3 The complex interaction between water, viscosity, melting and plate tectonics . 7

1.3 Plate tectonics and numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.1 The utility of a numerical approach for problems of mantle convection . . . . . 81.3.2 Generating plate tectonics in numerical simulations . . . . . . . . . . . . . . . . 81.3.3 Boussinesq approximation, boundary conditions and presentation of STAG . . 9

2 Regimes of convection with Arrhenius-type rheology and plastic yield stress 102.1 Methodology and characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.2 Characterization - plate detection . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Different regimes of convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 Stagnant lid, sluggish and mobile lid . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Dynamics of mobile lid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Water and its role in plate tectonics, discussion and provisional results 183.1 Effect of water on mantle’s properties - diffusion of water . . . . . . . . . . . . . . . . 18

3.1.1 Effect of water and melt on mantle viscosity . . . . . . . . . . . . . . . . . . . . 183.1.2 Effect of water on the solidus temperature . . . . . . . . . . . . . . . . . . . . . 183.1.3 Partial melting: water partitioning and eruption . . . . . . . . . . . . . . . . . 183.1.4 Hydration of the lithosphere by water diffusion . . . . . . . . . . . . . . . . . . 19

3.2 Dehydration of a wet mantle (provisional results) . . . . . . . . . . . . . . . . . . . . . 193.2.1 Dehydration in mobile lid regime . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.2 Dehydration in stagnant lid regime . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Discussion and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

A Presentation of Stag 25A.1 Numerical methods - Staggered grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25A.2 Geometry, 2D spherical annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25A.3 Compositional fields and tracers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

B Plate detection 27

C Temperature, viscosity and convective heat flux profiles 28

D Intermediary regime 29

E Useful physical quantities for Earth and Venus 30


Introduction

The work started during this internship is in line with the debate related to the origins of platetectonics on Earth and thus its absence on other planets like Venus. Although there exists an obviouslink between plate tectonics and convection theory, the precise mechanisms by which surface platesare generated and sustained remain unclear. In particular, the comparison between Earth and Venushas raised the question of the role of water in plate tectonics and this has been at the heart of manydebates for several decades. In a first part, the context of this work is thoroughly presented. Weshow why plate tectonics is a very complex problem and why numerical simulations appear to be aninteresting approach. The physical equations and approximations are briefly presented as well as thecode Stag used in the rest of this work. In a second part, we show that modeling the lithosphericrheology in a simple way by a constant plastic yield stress is sufficient to generate three differentregimes of convection with a very rich dynamics. Eventually, various effects of water on rheology andsolidus temperature as well as a treatment for water diffusion are implemented in Stag to assess therole of water in plate tectonics.

1 Plate tectonics : a complex problem addressed by numerical simu-lations

1.1 What is plate tectonics ?

The Earth is composed of an internal metallic core surrounded by a mantle (cf fig. 1a). The mantleis a highly viscous solid layer of silicates rocks. However, on long time scales (over millions of years),this solid layer can undergo slow, creeping, viscous-like deformation so that it can be described by theequations of fluid dynamics for highly viscous flows.

Because of its secular cooling and the presence of radiogenic heating inside its mantle, the Earthmust evacuate a lot of heat. This heat can be efficiently transported by thermal convection in themantle, that is to say, by large scale motions of mantle materials. The strength of this convectiveprocess can be described by the Rayleigh number, defined as:

Ra =ρgα∆TD3

ηκ(1)

where ρ is the density, g the gravity coefficient, α the thermal expansion coefficient, ∆T the temperaturedrop between the Core-Mantle Boundary (CMB) and the surface, D the thickness of the mantle, η itsviscosity and κ its thermal diffusivity. The Rayleigh number evaluates the ratio of the buoyancy forcedriving convection over the viscous force which tends to slow the motion.

In some planets, like the Earth, the highly viscous top layer of the mantle (called lithosphere) canparticipate in this convective motion producing large-scale motions of the lithosphere: this is calledplate tectonics. The lithosphere can in this case be divided into several ‘blocks’ called plates, eachone moving with a homogeneous velocity. Plates diverge from ridges where they are formed and haveconvergent motion along subduction zones where one plate is carried into the mantle (cf fig. 1a). Forinstance, the Earth’s lithosphere can be divided into 7 or 8 major plates plus several minor plates (cffig. 1b).

Plate tectonics has been an active resurfacing process in Earth’s history and has important con-sequences on the nowadays geology of the Earth. Volcanic activity and earthquakes are frequentlyobserved along subduction zones; ridges are at the origins of magmatic and hydrothermal processes inthe oceanic lithosphere. Plate tectonics is a cause of the formation of mountains where plates collide.To finish, plate tectonics also plays a part in the evolution of the Earth’s biosphere by participatingin the regulation of the amount of atmospheric carbon dioxide (Franck et al., 1999) and by modifyingthe environment of living populations in the medium and long terms.

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(a)

(b)

Figure 1: (a): Simplified cartoon showing the relation between plate tectonics and mantle convection.Lithospheric plates are located at the top cold boundary layer of the mantle’s convective cells. Atsubduction zones, a plate is carried into the mantle generating local trench and ‘pulling’ the rest ofthe plate from the ridge. (source: web page of USDA http:/www.fs.usda.gov/). (b): the divergenceand the vertical vorticity of the surface velocity are represented to localize plates’ limits on Earth(source: Dumoulin et al. (1998)). These limits are very well delimited in space. 7 or 8 major platescan be identified on Earth as well as a several other minor plates.

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Because plates participate in the convective motions, there is a close relation between plate tectonicsand convection theory. Plates can in fact be seen as the top cold boundary layer of the convective cellsof the mantle (Bercovici et al., 2000) and the downwellings of these plates coincide with subductionzones (fig. 1a).

One might hence deduce that the stronger the convection, the more active plate tectonics. Actually,this is not so clear. For example, although the Rayleigh number only differs by roughly a factor2 between Earth and Venus, the latter is not experiencing Earth-like plate tectonics and resurfacingevents on this planet probably happen only episodically (Nimmo and McKenzie, 1998). In fact, whetherlithospheric motion is possible or not depends on what stress the convective forces can generate inthe lithosphere and how easily this latter can ‘break’ (Mian and Tozer , 1990). Plate tectonics istherefore controlled both by the strength of mantle convection and also crucially by the rheology ofthe lithosphere.

1.2 A very complex problem - the debated role of water

Beyond this simple relation between surface motion and mantle convection, understanding platetectonics precisely is far from easy. Indeed, lots of physical properties such as the compressibility of themantle, the variety of the different mineralogic phases that compose the mantle and the lithosphere, thecomplex rheology of the lithosphere which varies over short length-scales, are all likely to play a part inthis problem. In addition to these, the role of water in generating and maintaining plate tectonics hasbeen at the heart of a debate for several decades and remains unsolved. The next paragraphs explainin details what this debate is about and why the influence of water on plate tectonics is so difficult toassess.

1.2.1 Comparison between Earth and Venus

The question of the role of water in plate tectonics was mainly raised by the comparison betweenEarth and Venus. These planets are indeed similar in size, density, mantle temperature, and probablychemical composition (Nimmo and McKenzie, 1998) (see also table E in Appendix for some physicalcharacteristics). Yet the Earth is currently experiencing a regime of plate tectonics whereas Venus isin a very different state and shows apparent lack of active plate motion. The Earth’s oceanic surface iscontinuously recycled and is less than 200 Ma old everywhere. In contrast, the last resurfacing eventon Venus would be as old as 300 - 600 Ma and the present-day surface heat flux is less than the inferredradiogenic heat rate which tends to prove that the mantle of Venus has been heating up since this lastresurfacing event (Nimmo and McKenzie, 1998). Consequently, though Venus could have been in anactive-lid regime in the past (O’Neill et al.), it is likely to be presently in a stagnant lid regime (rigidsurface shell) and episodically experience catastrophic brittle mobilizations of its lithosphere (Moresiand Solomatov , 1998).

This difference is often attributed to a visible compositional dissimilarity: the surface of the Earthis hydrated by oceans whereas Venus’ surface is extremely dry (Moresi and Solomatov , 1998; Nimmoand McKenzie, 1998; Mian and Tozer , 1990). The absence of water on Venus might be explained bya different impact history during planetary formation (Nimmo and McKenzie, 1998). Other studiessuggest that Venus originally possessed at least a few tenths of a percent of a terrestrial ocean, whichevaporated because of the high temperatures on Venus’ surface, so that the dihydrogen was lost tospace (Donahue and Hodges, 1992).

This dryness is often evoked to explain Venus’ different regime because water is known to modifythe viscosity of olivine rocks as well as their fusion temperature (Hirth and Kohlstedt , 1996) and tofacilitate mechanical failure of the lithosphere (Mian and Tozer , 1990). However, other works tendto see the presence of water on Earth rather as a consequence of active plate tectonics (Sandu et al.,2011). Moreover, the exact mechanisms by which water may influence plate tectonics are not preciselyelucidated. To get further into the complexity of the alleged role of water in plate tectonics, the watercycle on Earth and its interaction with the mantle’s viscosity and partial melting are depicted in thenext two paragraphs.

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1.2.2 The water cycle in the Earth’s mantle

The total mass of oceans on Earth is approximately 1.4 · 1021 kg. The total quantity of waterstored in the Earth’s mantle is poorly constrained and has been estimated to be comprised between aquarter to four times the mass of oceans (Hirschmann, 2006). The portions of plates newly fabricatedat ridges are hydrated by circulation of water (hydrothermalism). The older hydrated lithosphere thensinks into the mantle at subduction zones and can release water to the mantle (triggering local partialmelting at the origins of subduction zones’ volcanism). Because it lowers the solidus temperature,water tends to generate partial melting of the mantle. This melted material can then erupt in themagmas at oceanic ridges, and the water contained in these magmas can eventually be released in theocean and atmosphere. The typical time of this cycle is comprised between 2 and 6 Ga (Fujita andOgawa, 2013), though it is difficult to say whether a steady-state has been reached on Earth or if thetotal content of the water stored in the mantle is slowly increasing or decreasing. This water cycle isillustrated in figure 2.

Figure 2: Water cycle in the Earth’s mantle. The oceanic lithosphere is hydrated by hydrothermalism.When the lithosphere subducts, this water is progressively released into the convecting mantle. Watercan decrease the solidus (dashed curve) and favor partial melting right in the upper regions of themantle. The melted material can be brought to the surface at ridges. Source: Sandu et al. (2011).

1.2.3 The complex interaction between water, viscosity, melting and plate tectonics

Water decreases both the mantle’s viscosity by more than a factor 100 at saturation and its solidustemperature by as much as 400 K depending on pressure (Hirth and Kohlstedt , 1996). As a result,water tends to generate partial melting in the mantle’s regions where the temperature is close to thedry solidus.

On the other hand, the presence of melt also decreases viscosity. A few percents of melt is sufficientto divide the viscosity by a factor 10 (Kohlstedt , 1992). Moreover, when partial melting occurs, anotherimportant phenomenon is partitioning: water preferentially moves to the liquid phase. If the producedmelt is erupted, a solid phase depleted in water is thus left. As a result, the combined effect of waterand melt is subtle: water tends to decrease viscosity by its own hydrolytic weakening effect and bygenerating partial melting, but if the melt erupts, a solid, dry and thus more viscous rock is left becauseof the partitioning of water into the erupted liquid phase.

Some works have tried to study these effects separately: simulation of a low viscosity zone (Richardset al., 2001), characterisation of the water cycle with an imposed regime of tectonics (Sandu et al., 2011),

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but few consensus exist. For example, some studies suggest that, on Earth, water could accumulate ina region located at about 100-300 km depth which would be at the origins of the LVZ (Low ViscosityZone) over which the plates would glide more easily (Tackley , 2000b; Richards et al., 2001). But,whether this LVZ is directly due to the presence of water alone or also to the presence of melt inducedby water is debated.

1.3 Plate tectonics and numerical simulations

1.3.1 The utility of a numerical approach for problems of mantle convection

The previous section showed that one may face numerous difficulties when studying mantle convec-tion and plate tectonics due to the diversity of the inter-connected phenomenons which may influenceit. Scaling laws might be found but the high non linearity and the coupling of the physical equationsmake a complete analytical solving out of reach. Experimental approach has given interesting insightinto convection with high viscosity contrast (Davaille and Jaupart , 1993), but it remains difficult toincorporate and control lots of ingredients experimentally.

Numerical simulations therefore appear as an interesting tool to study this kind of problem. Nev-erthless, even with numerical simulations, we still have to manage a large number of parameters andthe usual approach in physics consisting in studying the effect of each parameter separately might bea bit too simplistic in this case because of the complexity and interconnection between the differentphenomenons whose individual effect cannot just be added. Consequently, we always need to look fora compromise: finding the minimal number of parameters which does not over-simplify the problem.

In the next paragraphs, we show how it is possible to generate plate tectonics in numerical simula-tions. The physical approximations and equations used in the context of this work are then presentedas well as the code Stag.

1.3.2 Generating plate tectonics in numerical simulations

Self-generation of plate tectonics in numerical models has long been a challenge. Iso-viscous con-vection triggers surface motion but does not generate well delimited Earth-like plates.

A step forward in terms of ‘realism’ consists in modeling the lithosphere which forms a highlyviscous top layer on the mantle. Since viscosity strongly decreases with temperature, a way to describethis is to use an Arrhenius-type law of rheology for the mantle:

η(T ) = η0 exp(

EetaT + T0

)(2)

where η0 is a viscosity constant, Eeta an activation energy and T0 a constant of (dimensionless) temper-ature. However, both experimentally (Davaille and Jaupart , 1993) and numerically, this formulationgenerally leads to extremely high viscosity contrast between the CMB and the surface and to a toplayer that is too viscous to move (‘stagnant lid’). In fact, so high viscosity values may not representthe real behaviour of the lithosphere. Indeed, for sufficient stress, lithospheric material rather showbrittle and semi-brittle behaviours, partly due to the existing faults. The lithospheric rheology is inreality complex since it includes ductile, brittle and semi brittle processes which exist on small depthscale (a few kilometers). This level of detail is difficult to achieve on a numerical grid. Consequently,an easy way to take globally this brittle behaviour into account is to introduce a plastic yield stress σyabove which the deformation becomes plastic. The effective viscosity can then be expressed as:

ηeff = min[η(T ),σy

2.ε] (3)

where.ε is the second invariant of the strain rate tensor :

.ε=√

.εij

.εij (4)

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The advantage of this formulation is that the complex details of lithospheric rheology are ‘sum-marised’ by a unique parameter. This choice is similar to the one proposed by Tackley (2008) andwas adopted in the rest of this work. The viscosity and temperature constants in η(T ) (eq. (2)) wereadjusted so that η = 1 at the CMB (T = TCMB = 1 and η0 = exp(−Eeta

2 ). Eeta was given thedimensionless value 23.03 giving a variation of 5 orders of magnitude between CMB and surface. Thisvariation is far less than what is observed in reality but avoids numerical difficulties associated withhigh viscosity contrast.

1.3.3 Boussinesq approximation, boundary conditions and presentation of STAG

Non-dimensionalisation of the Boussinesq equations In order to reduce the problem to a smallnumber of dimensionless parameters, the equations are non-dimensionalised to the depth of the mantleD, the thermal diffusion timescale (D

2

κ , where κ is the thermal diffusivity) and the temperature drop∆T between the surface and the core-mantle boundary (CMB). Hence the time, velocities and stressesare respectively non-dimensionalised by D2

κ , κD and η0

κD2 . The pressures (thus the plastic yield stress)

should be non-dimensionalised by η0κD2 = 9 · 10−25 Pa which is not a convenient scale to use. Moreover,

most of the authors who used a plastic yield stress worked with dimensional values. We thus decidedto keep the yield stress dimensional in this work so as to avoid problematic scaling and be able tocompare directly with the values used in previous works.

This study was conducted in the infinite Prandtl approximation. Prandtl number is defined as Pr =νκ and measures the relative importance of kinematic viscosity and thermal diffusivity. Infinite Prandtlapproximation is in this case fully justified (Pr ∼ 1023) and allows to neglect the time total derivativeof velocity in the equation of momentum conservation. The flow is supposed to be incompressiblewhich naturally leads to the Boussinesq approximation in which the variations of density are neglectedin all terms except in buoyancy forces. The usual conservation equations can then be written:

1) Conservation of mass:∇ · v = 0 (5)

where v is the velocity. We take the following convention: underlined quantities are vectors; tensorsare underlined twice.

2) Conservation of momentum:∇ · σ −∇p = RaT er (6)

in which T is the temperature, p the pressure, σ is the deviatoric stress tensor defined as σij =η(T, σy)(vi,j + vj,i) = 2η

.εij , with η the viscosity defined in equation (3) and ε the strain rate tensor.

Ra is the Rayleigh number and appears in the right hand side term which represents buoyancy forces,with er a unit vertical vector.

3) Conservation of energy:∂T

∂t+ v · ∇T = ∇2T +H (7)

where H is the internal radiogenic heating rate.

Boundary conditions and other parameters At its base, the mantle is in contact with a hotplanetary core. The mantle’s top coincides with the planet’s cold surface. Consequently, for allsimulations, the temperature was imposed at the top of the mantle (surface, T = 0) and at the core-mantle boundary (CMB, T = 1). At both boundaries the radial velocity is zero and a free-slip conditionis set at the surface. In addition, the internal radiogenic heating was kept constant and no decay withtime was considered. The parameters that were kept constant in all the simulations of this work canbe found in the following table:

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Quantity non-Dim. to non-Dim. Value Dim. Value MeaningD - 1 2.8·106 m Mantle thickness

∆T - 1 2600 K ∆T between surface and CMBκ - 1 7.6·10−7 m2s−1 Thermal diffusivity

TCMB ∆T 1 2600 K CMB temperatureH Cpκ∆T

D2 2 2.85 · 10−13 Wkg−1 Internal heatingrCMB D 1.19 3500 km Radius of the CMB

Presentation of STAG In this work we used the code STAG developed in Fortran 90 by PaulTackley (Professor at ETH Zurich Institute fuer Geophysik). STAG has been widely used in the last10 years to address problems of mantle convection and plate tectonics. A detailed description of thiscode is available in appendix A. Stag solves the equations of highly viscous flows on a staggeredgrid. Several geometries are available and 2D spherical geometry is adopted for this work as it gives agood compromise between catching some aspects of 3D spherical geometry while remaining reasonablein terms of calculation times. One of the particularities of this code is the Lagrangian treatment ofcompositional fields using tracers (see appendix A.3) which allows to get rid of numerical diffusion.We use tracers in this work for the treatment of water (section 3).

2 Regimes of convection with Arrhenius-type rheology and plasticyield stress

In this section, we explore and characterise the different regimes of convection observed whenvarying the Rayleigh number and the plastic yield stress without partial melting or water.

2.1 Methodology and characterization

2.1.1 Methodology

With this mathematical formulation of viscosity (eq. (3)), two free parameters control the physicalproblem: the Rayleigh number, which characterizes the convection’s strength and the plasticity yieldstress which describes the rheology of the lithosphere. In this work, 22 simulations were run withdifferent values of Ra and σy to approximately cover the range 103 to 108 for Ra and 105 to 109 Pafor σy.

A first constraint is that the number of grid points in the radial direction must be chosen so as to getsufficient resolution of the thermal boundary layer. At least three points are theoretically needed fora correct numerical description of this structure. Because the thickness of the thermal boundary layerscales as Ra−

13 , multiplying Ra by a factor 10 means that the resolution should be doubled in both

directions (to keep the square shape of the cells). But the time step is also reduced when increasingthe number of grid points and more time steps are therefore necessary to reach a statistically steadystate. Consequently, running simulations at high resolution quickly becomes time-consuming, that iswhy we chose to proceed as follows:

• runs were launched at a resolution of 256x32 (256 points along circumference and 32 points alongthe radius) for Ra ≤ 105 and 512x64 for Ra > 105 until a statistically steady-state was reached.This choice provides a supposedly sufficient resolution for all simulations (three points in thethermal boundary layer or more).

• the simulations were then continued at 512x64 for Ra ≤ 105 and 1024x128 for Ra > 105 until anew statistically steady-state was reached.

Indeed, we found that this problem was strongly dependent on resolution and that 512x64 and 1024x128points were necessary for Ra respectively smaller and larger than approximately 105. This methodallows to reach this resolution without starting with too many grid points which would result inprohibitive calculation times.

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2.1.2 Characterization - plate detection

For each simulation, several physical quantities were computed with time and used to determinewhether a statistically steady-state was reached and to characterize the thermal regime:

• The average temperature and viscosity.

• The Nusselt number here defined as the dimensionless top heat flux. The Nusselt number in-creases as the convection becomes more active.

• The root mean square (RMS) surface velocity to be compared with RMS velocity of the wholesystem.

• The profile of average convective heat flux along the radius. The convective heat flux correspondsto the heat transported by convection. It can be constructed from the energy conservationequation:

∂T

∂t+ v · ∇T = ∇2T +H (8)

We then write T as the sum of its average plus a variation δT = T−T and integrate this equationover a spherical shell of volume V (r) comprised between a sphere of radius r and the surface:∫∫∫

V (r)

∂T

∂tdV +

∫∫∫V (r)

v · ∇δTdV =∫∫∫

V (r)∇2δTdV +

∫∫∫V (r)

HdV (9)

Using the incompressibility of the flux (∇ · v = 0) and the Green-Ostrogradski formula we get:

∫∫∫V (r)

∂T

∂tdV + qconv(rsurf )− qconv(r) = −qcond(rsurf ) + qcond(r) +

43πH[r3

surf − r3] (10)

where qconv is the convective heat flux: qconv(r) =∫∫S(r)wδTdS and qcond(r) = −

∫∫S(r)∇T ·erdS

the conductive heat flux. S(r) is the surface of a sphere of radius r and w is defined by: w = v ·er;er being a unitary radial vector. Because there is no vertical velocity at the surface (to conservethe global mass), qconv(rsurf ) = 0. Moreover, profiles of temperature (cf fig. 15 in Appendix)show that the internal temperature is nearly constant in the mantle except in the top andbottom boundary layers. Consequently, for a radius not to close to the surface or the CMB weget: qcond(r) ≈ 0 and equation (10) becomes:

qcond(rsurf ) = qconv(r) +

(43πH[r3

surf − r3]−∫∫∫

V (r)

∂T

∂tdV

)(11)

meaning that the surface heat flux is due to the heat transported internally by convection atradius r plus the supplementary radiogenic heat created in the shell comprised between r andthe surface and to the average temperature variation in this shell.The convective heat flux is an interesting quantity to look at especially to define stagnant lidregimes. Indeed, in this case, the heat is transported only conductively in the rigid surface shelland the conductive heat flux equals zero in this portion.

• The number of plates

To detect and count the number of plates in the system a Matlab code was developed during thisinternship. This code uses the anomalies of vertical velocities below ridges and subduction zones aswell as the derivative of the surface velocity to detect plate boundaries. It was tested visually ona series of images and, though some refinements could be brought in a near future, this gave goodenough results for the purpose of this work. This plate detection method is described in more detailsin Appendix B.

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2.2 Different regimes of convection

2.2.1 Stagnant lid, sluggish and mobile lid

With this rheology involing a constant yield stress and in the absence of water and partial melting,three different regimes have already been observed and described in 2D cartesian geometry by Richardset al. (2001); Moresi and Solomatov (1998) in 3D cartesian geometry by Tackley (2000a) and in 3Dspherical geometry by (Heck and Tackley , 2008). These are:

Stagnant lid This is the result of a too weak convection on a lithosphere which does not breakeasily, that is to say, for low Rayleigh numbers and/or high values of yield stress. In this case, thesurface of the planet does not take part in the convective motion and remains stagnant. The heatis thus transported conductively through this rigid shell. In this study we therefore define stagnantlid regimes as cases in which the convective heat flux reaches zero in at least two points below thesurface (that is to say a rigid shell actually exists, cf fig. 15 in Appendix C). The ratio V surf

rmsVrms

(whereV surfrms is the surface root mean square velocity and Vrms the global root mean square velocity) is also

an interesting quantity that typically tends to 0 in cases of stagnant lid (cf fig. 5). Moreover, sincethe heat is transported conductively through the lid, the heat flux at the surface has a smooth aspectbecause the deeper spatial and temporal variations in heat flux are ‘smoothed’ by diffusion. Most ofthe solid bodies of the Solar system are in a stagnant lid regime of convection. This is typically thecase for Mars, Mercury and the Moon.

Among the 22 simulations of this work, 8 stagnant lid regimes were obtained (cf fig. 3).

Mobile lid For sufficiently high Ra and if the plastic yield stress is low enough to induce failure of thelithosphere, this latter can take part in the convective motion and be recycled in the mantle throughsubducting downwellings. The lithosphere can then be divided into several blocks of material (plates)each one moving with a homogeneous velocity. The surface velocity profile resembles a piecewiseconstant function and the RMS surface velocity is comparable to the global RMS velocity (cf fig. 5).Moreover, the Nusselt shows more temporal fluctuations than in the stagnant lid case because of theevolution of the plates. The Earth is presently experiencing a particular type of mobile lid convection.In this work, we defined mobile lids as cases in which at least 3 plates were observed. This regimeis in fact extremely rich and covers a great variety of different subregimes which do not all resembleEarth-like plate tectonics. This is analysed in closer details in section 2.2.2.

A total of 8 mobile lid regimes were observed in this work (cf fig. 3).

Intermediary states This is an intermediary case between stagnant and mobile. It is characterizedby a weak mobility of the surface (a case refered to as ‘sluggish lid’ by Armann and Tackley (2012))associated with one large subducting downwelling which periodically stops functioning, leading to astagnant lid regime. Because the heat is then not evacuated efficiently enough, the internal temperatureincreases. As a result, the average viscosity diminishes which increases the Rayleigh number and thusthe strength of the convection. The lid is then broken in a point forming a subduction and thecycle continues. Intermediary states thus oscillate between stagnant and sluggish lids. This regimewas obtained in 2D cartesian geometry by Moresi and Solomatov (1998), in 2D-spherical by Armannand Tackley (2012) and in 3D cartesian geometry by Tackley (2000a). However, in this work, 3simulations with intermediary regimes were obtained (cf fig. 16 in Appendix D) but they alldisappeared when the resolution was increased, meaning that the simulations evolved to a caseno longer periodic but which had the characteristics of a weakly mobile lid (sluggish lid) functioningwith only 2 plates, one ridge and one subduction (cf fig. 5). The absence of intermediary case is bothinteresting and puzzling because Venus is expected to belong to this category. This problem appears tobe strongly resolution-dependent. Armann and Tackley (2012) were able to produce Venus-like regimeswith a yield stress of 100 MPa in the same geometry, but with a low resolution (256x32) which mightbe an explanation.

12


To sum-up, three different regimes were observed in this work:

• stagnant lid (8 simulations)

• sluggish lid (6 simulations). Among these, 3 simulations were at intermediary states at lowresolution and evolved to sluggish lid regime when the resolution was increased.

• mobile lid (8 simulations)

Figure 5 compares different characteristics of stagnant, sluggish and mobile lids. A visual repre-sentation of the domains for these 3 regimes as a function of Ra and σy is given in figure 3. Figure 16in Appendix D gives some characteristics of an intermediary state. The simulation corresponding tothis intermediary state was obtained at a resolution of 256x32. When the resolution was increased to512x64, this simulation evolved to the sluggish lid case that is shown on figure 5.

log10(Sy)4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.50

1

2

3

4

5

6

7

8

log 10

(Ra)

Stagnant

Sluggish

Mobile

4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5log10(Sy)

Figure 3: Domains for the three main regimes observed in the simulations as a function of Ra (inordinate) and plasticity yield stress (in abscissa). Each point corresponds to a simulation. The limitsbetween the different domains have been drawn qualitatively. The critical Ra for the onset of convectionwas not calculated here.

3.5 4 4.5 5 5.5 6 6.5 7 7.5 80

5

10

15

20

25

30

35

log10(Ra)

Ave

rag

e n

um

ber

of

pla

tes

σy = 105 Pa

σy = 106 Pa

σy = 3.106 Pa

σy = 107 Pa

σy = 108 Pa

σy = 109 Pa

3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

0.4

0.5

0.6

0.7

0.8

log10(Ra)

Ave

rag

e te

mp

erat

ure

StagnantSluggishMobile

(a) (b)

Figure 4: The average number of plates (a) and temperature (b) at steady-state are plotted as afunction of Ra (logarithmic scale) and for different values of σy. The different regimes are indicatedby circles (stagnant), squares (sluggish) and crosses (mobile). For a fixed Ra, the number of platesincreases when the plastic yield stress decreases since the lithosphere breaks more easily. Similarly,the lower σy, the lower the final average temperature: the surface is broken and participates activelyin convection resulting in an efficient evacuation of the planet’s heat.

13


0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.960

0.5

1

1.5

2

Time

Vrm

s surf

/Vrm

s

0 50 100 150 200 250 300 350−1.5

−1

−0.5

0

0.5

1

1.5

Angle (in degrees)

Vsu

rf/V

rms

Ra = 4,5.1e3Sy = 1e8 Pa

STAGNANT

0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.960

5

10

15

20

25

Time

Nu

1.3 1.35 1.4 1.450

0.5

1

1.5

2

Time

Vrm

s surf

/Vrm

s

0 50 100 150 200 250 300 350−1.5

−1

−0.5

0

0.5

1

1.5

Angle (in degrees)

Vsur

f/Vrm

s

Ra= 2,7.1e4Sy = 1e5 Pa

MOBILE

1.3 1.35 1.4 1.45 1.50

5

10

15

20

25

TimeN

u

1.5

0.41 0.42 0.430

5

10

15

20

25

Nu

Vrm

s surf

/Vrm

sVs

urf/V

rms

0.41 0.42 0.43 0.44 0.45 0.46 0.470

0.5

1

1.5

2

0.44 0.45 0.46 0.47

Ra= 1,74.1e5Sy = 1e7 Pa

0 50 100 150 200 250 300 350−1.5

−1

−0.5

0

0.5

1

1.5

Time

Time

Angle (in degrees)

SLUGGISH

Figure 5: For each column, from top to bottom: snapshot of the typical aspect of temperature profiletaken randomly at steady-state (coloration: red for T = 1 at the CMB up to blue for T = 0 atthe surface), Nusselt number as a function of time, surface rms velocity (normalized by volume rmsvelocity) as a function of time, and variation of surface velocity (normalized by volume rms velocity)as a function of angle (in degrees). In the case of stagnant lid regime, the Nusselt number is generallysmaller for the same Ra and the surface velocity can be neglected with respect to the global rmsvelocity. For mobile lid regime, the surface velocity profile is typically a piecewise constant function(each piece corresponding to a plate). Sluggish lid is a weakly mobile lid with only two plates. Thesurface velocity is less a clear piecewise constant function than for mobile lids.

14


In a second phase, we tried to analyse how the characteristics of these different regimes (averagetemperature, number of plates etc.) were related to the parameters which control the problem (Ra andσy). On figure 4, the number of plates and the average temperature at steady-state are plotted for thedifferent simulations. Note that for one simulation (Ra = 6.68 · 107, σy = 107), more than 30 plateswere obtained. This corresponds to an extremely mobile lid which is the result of high constraints ona lithosphere that breaks everywhere. Figure 4 suggests that the dynamics is mainly controlled by theratio σy

Ra (in Pa). This can be understood intuitively: a high ratio means either that the lithosphere isdifficult to break or that the Rayleigh number is too low so that the convection does not put strongconstraint on the lithosphere. Stagnant and sluggish lids are therefore expected for high ratios whereasmobile lids will be observed for low ratios. Furthermore, a scaling law can even be intuited fromfigure 4a since all the slopes ( N

log10(Ra)) seem to be equal for mobile lids, suggesting a law of the form:N(Ra, σy) = −c1ln(Ra) + c2(σy). So as to test this dependence a bit more quantitatively, the numberof plates was represented as a function of the logarithm of the ratio σy

Ra on figure 6. A linear regressioncould be performed on the mobile lid regimes and indeed suggests a scaling of the form:

N = α+ βln(σyRa

) (12)

where the values of α and β were determined: α = 20.28 and β = −5.22. It would be interesting tosee what this scaling law becomes in different geometries (2D cartesian, 3D-spherical).

−2 0 2 4 6 8 100

5

10

15

20

25

30

ln(σy/Ra)

Num

ber o

f pla

tes

sluggish and stagnant mobileN = 20.28 − 5.22ln(σy/Ra)

Figure 6: The number of plates is represented in the y-axis with the logarithm of σy

Ra in the x-axis. Themobile lid regimes are colored in green. The sluggish and stagnant lid are in red. A linear regressionwas performed for the mobile lid regimes and gives a good agreement with the scattered points, thoughtwo points seem to be off the line.

2.2.2 Dynamics of mobile lid

As mentioned before, the mobile lid regime is in fact very rich and encompasses a variety ofdifferent dynamics. In particular, in some cases (especially those close to the slugglish lid domain) aquasi-periodic behaviour could be observed. This is depicted in the next paragraph.

a) Periodicity In some of the simulations leading to mobile lid regime, particularly those whichare close to the sluggish domain, a quasi-periodic behaviour was observed. The quantities describingthe system (average temperature, Nu, number of plates...) were found to oscillate quasi-periodicallywith time (cf fig. 7a). Visual observation reveals the following cycle: one subduction initially existsand a second one forms and tends to merge with the first one. To follow this phenomenon more

15


precisely, the derivative of the surface velocity has been calculated along the circumference every 500time steps and plotted as a function of time on figure 7b. The ridges (corresponding to a high positivevariation of velocity) were colored in red whereas the blue color is associated to the high negativevariations of velocity in subduction zones. One can see that the newly formed subduction appearswere a ridge previously existed. This may be because the lithosphere is thinner in this region, thuseasier to break. Similarly, two subductions always merge at the point where a ridge previously existedand which therefore behaves as an attractor. In a non published work, Stéphane Labrosse obtainssimilar results in 2D-cartesian geometry, but the subductions’ positions merge following a paraboliccurve, which could be interpreted as ‘negative diffusion’. In this work the blue curves are not quadraticand show an inflexion point. This difference might be attributed to the geometry. This is an interestingdynamics and a more precise characterisation linking a Fourier analysis to the parameters that controlthe system is currently work in progress.

0 1 2 3 4 50.41

0.42

0.43

0.44

0.45

0.46

0.47

0.48

Avera

ge t

em

pera

ture

An

gle

(in

deg

ree)

Time

(a)

(b)

Figure 7: (a): Average temperature as a function of time for Ra = 6, 8.103 and σy = 105. (b):The derivative of the surface velocity has been calculated and represented with time by a range ofcolors. High values of derivatives appear either in red (positive values corresponding to ridges) orin blue (negative values corresponding to subductions). The blue ‘curve’ thus follows the position ofsubductions zones with time. The same is true for ridges and the red ‘curve’.

16


b) Heat flux coefficient in spherical geometry Another interesting point is to look at what heatflux scaling law becomes in spherical geometry. Indeed, a calculation on a steady-state loop model in2D cartesian geometry (Grigné et al., 2005) shows that the average heat flux per surface unit on aplate scales as:

q = C(L)Ra13T

43m (13)

where Ra is the local Rayleigh number (calculated with the average viscosity below the plate), Tm theaverage temperature of the portion of mantle delimited by the plate and C a coefficient which dependson the size L of the plate and on the global geometry. This coefficient has already been calculatedanalytically by a steady-state loop model in 2D cartesian geometry by Grigné et al. (2005) and wasfound to be equal to:

C(L) = (2π

)23

1

(L2 + L8λ3 )

13

(14)

but has never been calculated in spherical geometry. The Matlab routine built to detect and countthe number of plates was extended to compute this coefficient. Every 500 time steps, the heat fluxcoefficient C is calculated for each plate and plotted as a function of the plate’s length. This methodis illustrated on one simulation in figure 8. Although all plates sizes are not equally represented in thissimulation (causing regions of low points density), the aspect of the curve seems to be qualitativelyclose to what was observed in 2D cartesian geometry (Grigné et al., 2005), even if the decreasing partseems to have a softer slope.

A similar work must still be conducted for all the simulations that have an average number ofplates comprised between 2 and 10 (to avoid problems related to too small plates, cf fig 8) so as toequally cover the range of plates sizes. An analytical or numerical solving of the heat flux coefficient inspherical geometry could also be an interesting future work to compare with the computed coefficient.

0 50 100 150 2000

1

2

3

4

5

Plates size0 50 100 150 2000

1

2

3

4

5

Plate size (in degrees)

Hea

t fl

ux

coef

fici

ent

Hea

t fl

ux

coef

fici

ent

(a) (b)

Figure 8: This figure was obtained from a single simulation with Ra = 6, 8 · 108 and σy = 105Pa.(a): every 500 time steps at stationnary steady-state, the heat flux coefficient was calculated for eachplate detected. Each point represents a plate whose angular length (in degrees) was represented inthe x-axis. The corresponding heat flux coefficient is in the y-axis. (b): an average of the heat fluxcoefficient was performed using all points in intervals of 10 degrees. The signal is noisy for small plates(angular length smaller than 30 degrees) because these are not well stabilized plates and are eithergrowing of shrinking. Irregularities appear in the averaged curve where the density of points is toolow.

To conclude on this part, a single-parameterized model for lithospheric rheology can generate threedifferent regimes of convection with a very rich dynamics whose precise characterization still requiresmore work. Once the domains of each regime have been identified, they can be used as a reference anda base for more complex simulations investigating the role of water in plate tectonics.

17


3 Water and its role in plate tectonics, discussion and provisionalresults

To treat water as a compositional field, we used the method of tracers already implemented inStag. However, only the advection of tracers was initially treated. Part of the work of this internshipthus consisted in adding a series of different effects of water on rheology, solidus temperature and incoding a routine treating water diffusion in the grid using tracers.

3.1 Effect of water on mantle’s properties - diffusion of water

3.1.1 Effect of water and melt on mantle viscosity

As said previously, water locally decreases the viscosity of the mantle by a bit more than 2 ordersof magnitude at saturation (factor 100 to 180 at a confining pressure of 300 MPa according to Hirthand Kohlstedt (1996)), due to hydrolytic weakening (Karato et al., 1986). We neglect the fact that thesolubility of water depends on the type of mineral and also on the pressure thus on the depth. Wedefine cH2O = 1 as the concentration at which the olivine is saturated in water. To model the effect ofwater on viscosity, an exponential function is proposed (eq. (15)).

Similarly, a few percents of melt can decrease the viscosity by a factor 10 (Kohlstedt , 1992). Theeffect of melt on viscosity was explained by local stress enhancement due to the replacement of part ofeach grain with melt and by enhanced transport kinetics resulting from rapid or short-circuit diffusionthrough the melt (Cooper and Kohlstedt , 1986). An exponential function was also used so that theeffects of water and melt were implemented in Stag by:

η(T, σy, cH2O, f) = η(T, σy)e−ln(a)cH2Oe−ln(b)f (15)

where η(T, σy) is the viscosity defined in equation (3), cH2O is the non-dimensional concentration ofwater defined such that 1 corresponds to the concentration of saturation in olivine, f is the meltfraction, a is the factor by which the viscosity is multiplied at saturation, b is the correction factor dueto the melt. We set a = 0.01 to get the right order of magnitude. b was chosen equal to 5 · 10−15 soas to cut the viscosity by a factor 10 for 7 percents of melt (Kohlstedt , 1992). With this formulation,we must make sure not to generate too much melt (less than 10 percents) to avoid dramatic viscosityreduction.

3.1.2 Effect of water on the solidus temperature

Water decreases the temperature of the solidus by up to 400 K at saturation (Hirth and Kohlstedt ,1996). The work of Hirth and Kohlstedt (1996) suggests that a linear dependence with concentrationis reasonable. We thus added an extra linear term to the expression of the solidus temperature:

Tsolidus(z, cH2O) = Tsolidus(z)− γcH2O (16)

where Tsolidus is the non-dimensional solidus temperature, z the non-dimensional depth (ranging be-tween 0 at the surface and 1 at the CMB), γ a coefficient set to 0.01 to give the right order of magnitudeat saturation (cH2O = 1).

3.1.3 Partial melting: water partitioning and eruption

When partial melting occurs, water is not partitioned equally into melt and the remaining solid.In fact, water moves preferentially to the liquid phase. To evaluate the relative proportion of waterbetween melt and solid, we can define a partition coefficient PH2O:

18


PH2O =csolidH2O

cliquidH2O

(17)

i.e. the ratio of the concentrations of water respectively in the solid and liquid phases. We neglectthe variation of this coefficient with pressure and set it to a value of 4 · 10−4 which is obtained from acomparison of the solubility of water in experimentally annealed olivine and basaltic melt at a pressureof 300 MPa (Hirth and Kohlstedt , 1996).

As mentioned before, melt-induced reductions in viscosity occur due to a decrease in the area ofgrain to grain contact with increasing melt fraction. At the same time, melt-induced increases inviscosity can occur due to preferential partitioning of water into the melt (Hirth and Kohlstedt , 1996).

3.1.4 Hydration of the lithosphere by water diffusion

Hydrothermalism and hydration of the crust cannot be easily implemented but they can be globallytaken into account by a process of diffusion. This has already be done by Fujita and Ogawa (2013).However, water must diffuse only through the upper part of the lithosphere (hydrothermalism processonly exists down to a certain depth below the surface). A way to achieve this is to take a depth-dependent coefficient of diffusion DH2O. A rountine was added in Stag to treat the diffusive term ofthe non-dimensionalised convection-diffusion equation:

∂cH2O

∂t= ∇(

D(x, y, z)κ

∇c)−∇(vcH2O) (18)

where κ is the thermal diffusivity. The effective diffusion coefficient is thus DH2O(x, y, z) = D(x,y,z)κ .

This effective diffusion coefficient of water was given the form:

DH2O(z) = D0e−z/zd (19)

where D0 is the diffusion coefficient at the surface (T=0) and zd a typical depth which must beadjusted to be close to the lithosphere’s thickness. This way, the diffusion coefficient quickly decreaseswith depth and the diffusion is efficient only in the top cold regions corresponding to the lithosphere.Because the diffusion of water is expected to be less important than thermal diffusion, we must chooseD0 < 1.

However, we are currently facing problems with water diffusion in the code. In particular, the waterbudget is not verified when DH2O is not constant. We have not yet identified the precise cause of thisproblem. A discussion with Paul Tackley and a series of tests are in progress. In the meantime, somesimulations have been tried with a constant diffusion coefficient. In the limited available time, onlyone approach was considered: the progressive dehydration of a wet mantle by diffusion. Provisionalresults are discussed in the next section.

3.2 Dehydration of a wet mantle (provisional results)

3.2.1 Dehydration in mobile lid regime

A simulation of the dehydration of a mantle diffusion and eruption was run with the conditions:σy = 107 Pa, cmantleH2O

(t = 0) = 1, ctopH2O(t) = 0, DH2O = 0.1 (constant). The solidus was adjusted so

that very little partial melting and eruption were observed. The dehydration is thus mainly diffusive.The evolution of the system is illustrated in figures 9 and 10. On figure 9, one can observe two

antagonist phenomenons:

• The surface diffusion of water which tends to dehydrate the surface on a small layer. This layerbecomes more viscous and tends to slow plates motion.

19


• Plate tectonics which recycles the surface material into the mantle. The dehydrated portion ofthe lithosphere is thus carried inside the mantle and replaced by hydrated material.

The dehydration is thus rather global and not limited to a superficial layer. However, as the mantleprogressively dehydrates, the Rayleigh number diminishes so that the RMS velocity (thus the velocityof the plates) also decreases. One can imagine that if the run is continued, the diffusive time scalewill become comparable to the typical time of plate recycling. A dehydrated layer could then formnear the surface and might become sluggish or stagnant. Unfortunately, since we were lacking time,the simulation could not be pursued long enough to see whether the mobile lid becomes stagnant orsluggish in the long term.

t1 t2 t3

TEM

PERA

TURE

WAT

ER C

ON

CEN

TRAT

ION

Figure 9: Images of temperature (top three images) and water concentration (bottom three images) atdifferent times t1, t2 and t3 (represented by dashed lines in figure 10). For the pictures showing waterconcentration, a darker blue corresponds to a smaller concentration. A layer of dehydrated materialtends to appear at the surface which becomes more viscous, leading to fewer plates. Nevertheless,this is not sufficient to stop plate motion in this case and plates continue to be recycled, carrying thedehydrated lithosphere inside the mantle.

3.2.2 Dehydration in stagnant lid regime

A second simulation was run in the same conditions except σy = 109 Pa. The simulation thusdirectly started in a stagnant lid regime and the mantle progressively dehydrated by diffusion. Thestagnant lid could quickly dehydrate, contrary to the previous case in which plate tectonics was coun-teracting the formation of a dry surface layer. Figure 11 shows the thickness of the stagnant lid as afunction of time. The time evolution of the Rayleigh number is also plotted on the same figure. Onecan see that although the Rayleigh is nearly constant after t = 0.01 (which would normally impose a

20


fixed stagnant lid thickness), the stagnant lid continues to thicken. Indeed, the mantle is still fully hy-drated but the lithosphere becomes drier on a thickness which progressively increases as water diffusesoutside the planet. This shows that the effect of water only needs to be local (in the superficial part)to produce changes in the dynamical regime.

70

0 0.5 1 1.5 2 2.5 3x 10−3

10

20

30

40

50

60

Time

Nu

t1 t2 t30 0.5 1 1.5 2 2.5 3x 10−3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Average temperatureAverage water concentration

t1 t2 t3 0 0.5 1 1.5 2 2.5 3x 10−3

10

15

20

25

30

35

40

45

50

Time

Avera

ge p

late

s n

um

ber

(a) (b) (c)

t1 t2 t3

Figure 10: Temperature and water concentration (a), Nusselt number (b) and number of plates (c)as a function of time. As long as water diffusion is on, no steady-state can be reached (except whenthe planet is totally dehydrated). However, after a short period of rapid dehydration, a ‘pseudo-equilibrium’ is observed: the average temperature, Nusselt and the number of plates decrease in a firstphase and then tend to stabilize. This is due to the opposite actions of diffusion (which dehydrates thelithosphere, slowing the motion), and tectonics (which replace old dry lithosphere by newly hydratedmaterials).

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020.08

0.1

0.12

0.14

0.16

0.18

0.2

Time

Stag

nant

lid’

s th

ickn

ess

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

0.5

1

1.5

2

2.5

3

3.5x 104

Time

Ra

(a)

(b

Figure 11: (a): Thickness of the stagnant lid as a function of time (the resolution is limited by thevertical number of grid points). (b): Ra as a function of time. The lid’s thickness increases with timelike the typical diffusive length and thus probably scales as the square root of time. After t = 0.01 theRayleigh is nearly constant while the lid keeps thickening.

21


3.3 Discussion and perspectives

Generally speaking, as it took me significant time to get familiarised with Stag, to develop sometools (like a plate detection routine) and to implement water diffusion and water’s effects on viscosityand solidus, this work remained mainly descriptive in the time left. Different aspects of the dynamicsof mobile lid regimes were explored, but a more quantitative work aiming at obtaining scaling lawsmust be conducted.

Concerning the study on water, the 2 simulations run in this work to investigate the relationbetween water and plate tectonics only offer partial and provisional qualitative results suggesting thata relation does exist between water and plate tectonics. Indeed, plate tectonics modifies the distributionof water in the mantle and tends to counteract the formation of a dry surface layer. On the otherhand, dehydration of the surface creates a drier layer which slows down plate tectonics. A local effectof water (dehydration of the lithosphere) can be sufficient to trigger changes in the dynamical regime.

Future work needs to move in the direction of modeling more realistic effects. The diffusion ofwater is meaningful for the hydration of the lithosphere but not really for the dehydration of themantle which is done rather by eruption of melted materials. In addition, as mentioned earlier, aproblem still persists with the diffusion as currently implemented in the code and needs to be solvedin order to use more realistic depth-dependent diffusion coefficients.

Moreover, the effects of water on rheology, solidus, and the parameters related to the eruption ofmelted material need to be better constrained. A discussion with geochemists could cast some lightupon the precise mechanisms involved in the water cycle (in particular hydration and dehydrationprocesses). Furthermore, other effects of water have not yet been taken into account and could beintegrated in the model. For example, it was shown that water can speed up the rate of transitionbetween basalt and eclogite and thus increase the density of the crust, favoring subduction (Nimmoand McKenzie, 1998).

Finally, a more global and sophisticated model could be built in the long term, including sur-face topology and real oceans. But to reach this target, a better comprehension of all the involvedphenomenons is necessary.

22


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Karato, S., M. Paterson, and J. FItzGerald, Rheology of synthetic olivine aggregates : influence ofgrain size and water, J. Geophys. Res., 91, 453,473, 1986.

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O’Neill, C., A. Jellinek, and A. Lenardic, Conditions for the onset of plate tectonics on terrestrialplanets and moons.

Richards, M., W.-S. Yang, J. Baumgardner, and H.-P. Bunge, Role of a low-viscosity zone in stabilizingplate tectonics : Implications for comparative terrestrial planetology, Geochem. Geophys. Geosyst.,2 (2000GC000115), 2001.

Sandu, C., A. Lenardic, and P. McGovern, The effects of deep water cycling on planetary thermalevolution, J. Geophys. Res., 116 (B12404), 2011.

Tackley, P., Self-consistent generation of tectonic plates in time-dependent, three-dimensional mantleconvection simulations (a), Geochem. Geophys. Geosyst., 1 (2000GC000036), 2000a.

Tackley, P., Self-consistent generation of tectonic plates in time-dependent, three-dimensional mantleconvection simulations (b), Geochem. Geophys. Geosyst., 1 (2000GC000036), 2000b.

Tackley, P., Modelling compressible mantle convection with large viscosity contrasts in a three-dimensional spherical shell using the yin-yang grid, Phys. Earth Planet. Int., 171, 7,18, 2008.

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Appendix

A Presentation of Stag

A.1 Numerical methods - Staggered grid

Stag solves the equations of highly viscous flow in 2D and 3D geometry using a finite volumemethod on a staggered grid. The principle of a staggered grid is illustrated in figure 13: pressure andtemperature are computed at cells’ centers while velocities and fluxes are calculated at cells’ borders.This device allows convenient calculation of all first-order derivatives and makes boundary conditionsfor fluxes easier to implement. Stag uses an iterative solver working in either fine (basic) grid modeor multigrid mode. Fine grid solving always converges but roughly doubles the time of calculation.Multigrid solving is much more efficient but convergence problems may appear with high viscositycontrasts. Stag is portable between parallel computers or clusters and single-processor computers. Inthis work, Stag was run on single processor computers as well as parallel clusters on the PSMN serverof the ENS de Lyon.

Figure 12: A two-dimensional version of the staggered grid. Pressure (p) and temperature (T ) aredefined at the cell centers, horizontal velocity (u) at the center of the cell walls perpendicular to thex-direction, and vertical velocity (w) at the center of the cell walls perpendicular to the z-direction.Vertical and sides boundaries coincide with cell walls which makes boundary conditions on fluxes easierto implement.

A.2 Geometry, 2D spherical annulus

Geometry plays an obvious part in problems of convection. Different geometries can be modeled inStag: 2D or 3D, both in cartesian, cylindrical, axisymmetric and spherical coordinates. 2D-sphericalgeometry was developed by Hernlund and Tackley (2008) and consists in a 2D slice bisecting a sphericalshell at its equator. The variations in latitude are then neglected in the governing equations but animplied second degree of curvature remains in this formulation. 2D spherical annulus is consequentlyan interesting compromise because it captures several characteristics of a fully 3D-spherical geometrywhile remaining a 2D model requiring more reasonable times of calculation. In a first phase, we thuschose to work in 2D spherical geometry. 3D simulations can be the target of future works.

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Figure 13: Comparison between the 2D approximations for rigid body translations on the surface ofa sphere (left) and a cylinder (right) with the circular 2D slice of interest indicated by dashed lines.Arrows are shown to indicate a divergent motion such as that along mid-ocean ridge and subductionzones. The primary difference between these descriptions is that motions on a sphere are projectedonto a surface with two degrees of curvature, while a cylinder has only one degree of curvature.Source: Hernlund and Tackley (2008).

A.3 Compositional fields and tracers

Compositional fields (radioactive elements, melt, phases, water etc.) are also treated in Stagthanks to method of tracers. At the beginning of each simulation, a large number of particules calledtracers (at least 10 per grid cell) are spread randomly through the grid and are given some information(position, melt concentration etc.). Throughout the simulation, tracers are advected and the informa-tion they convey can be modified and exchanged between them. This Lagrangian formulation bringsa supplementary advantage to the Eulerian solving of the physical equations by eliminating numericaldiffusion. To finish, Stag can also deal with numerous refinements including treatments for severalmineralogic phases, partial melting, eruption, degassing, atmosphere, plus a large panel of differentavailable physical laws for rheology, core cooling etc.

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B Plate detection

A Matlab code was developed during this internship to detect plate boundaries using the temper-ature and velocity fields. Plates can be defined as ‘blocks’ of lithosphere moving with respect to oneanother with homogeneous velocity. At the boarder between plates, there thus is a discontinuity ofthe surface velocity. This first element can be used to detect plates and might be sufficient for manycases. However, it sometimes happens that small discontinuities exist within a plate and these can bedetected if the threshold is not properly adjusted. To avoid this difficulty, an idea is to check whetherthese discontinuities are actually associated with a downwelling (subduction) or upwelling (ridge) ofmantle material. An average of the vertical velocity is thus calculated in the code in the third of themantle near the surface and compared to the global root mean square velocity. A second threshold isthus introduced.

The code was tested on a series of images. An example is shown in figure 14. Difficulties are raisedwhen the number of plates increases because the lithosphere tends to break everywhere and to manyplates are detected. The problem also consists in giving a precise definition of plates, particularly interms of size.

Figure 14: Illustration of plate detection. The central figure is a snapshot of temperature profile chosenrandomly at steady state for a simulation with Ra = 2.66 · 104 and σy = 105 Pa. The surroundingcircle illustrates the plate detection: each plate detected has been given a color, alternatively blue andred (because in this case the number of plates is even). The detected plates seem to match correctlythe visible plates that can be identified by the subductions and ridges on the figure.

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C Temperature, viscosity and convective heat flux profiles

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

log10(eta)

stagnantmobilesluggish

0 2 4 6 8 10 12 14 16 180

0.2

0.4

0.6

0.8

1

Conductive heat flux per surface unit


Stagnant lid

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Temperature

Rad

ius

(1 a

t the

sur

face

, 0 a

t the

CM

B)


(a) (b) (c)

Figure 15: Respectively average temperature (a), viscosity (logarithmic scale)(b) and conductive heatflux per surface unit (c) along the radius (0 at CMB, 1 at surface), for stagnant lid regime withRa = 4, 28.103 and σy = 108 Pa; sluggish lid with Ra = 104 and σy = 106 Pa and mobile lidRa = 2, 66.104 and σy = 105Pa at statistically steady-state. The temperature and viscosity havealmost constant values inside the mantle and show important variations in the bottom hot and topcold boundary layers. For stagnant lid, the convective heat flux becomes zero in a portion of mantlebelow the surface (stagnant lid). A thickness of this shell can be approximated by drawing a tangentat the inflexion point (figure c).

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D Intermediary regime

INTERMEDIARY : observed only at low resolution.High resolution : becomes sluggish lid

0 50 100 150 200 250 300 350−1.5

−1

−0.5

0

0.5

1

1.5

Angle (in degrees)

Vsur

f/Vrm

s

t = 0,283 t = 0,265

0.255 0.26 0.265 0.27 0.275 0.28 0.2850

0.5

1

1.5

2

Time

Vrm

s surf

/Vrm

s

Ra = 1,74e5Sy = 1e7 Pa

0.255 0.26 0.265 0.27 0.275 0.28 0.2850

5

10

15

20

25

Time

Nu

SLUGGISH

STAGNANT

Figure 16: Characteristics of intermediary regime. For the graphs, from top to bottom, Nusseltnumber as a function of time, surface rms velocity (normalized by volume rms velocity) as a functionof time, and variation of surface velocity (normalized by volume rms velocity) as a function of angle(in degrees). The pictures are snapshots taken at the times designated by the letters A (t = 0.265)and B (t = 0.283). The system periodically oscillates between stagnant and sluggish lid regimes. Thisintermediary regime was only observed at a resolution of 256x32. The grid points were kept on thesnapshot corresponding to stagnant lid and it is interesting to notice that this resolution seems to besufficient (there are 3 points in the boundary layer). Yet, when the resolution was increased to 512x64,the periodicity was lost and the system evolved to a new steady-state: that of a sluggish lid regimethat is shown on figure 5. The same evolution was observed for all the intermediary regimes obtainedat too low resolutions.

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E Useful physical quantities for Earth and Venus

Quantity Earth VenusRadius 6371 km 6052 km

Mantle thickness 2890 km 3000 kmSurface temperature 293 K 735KCMB temperature 3950 +/- 200 K ? probably similar to Earth’s

Gravity 9,81 N/kg 8.87 N/kgMantle’s thermal diffusivity 7.10−7 m2/s ? probably similar to Earth’sMantle’s average viscosity 1019 Pa.s ? probably similar to Earth’s

Density 5,5 5.26Mantle’s density 3,4-5,6 ? probably similar to Earth’s

Mantle’s thermal expansion 2, 5.10−5K−1 ? probably similar to Earth’s

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Documents

Role of water in the tectonics of Earth and Venus