GEOS3104-3804: GEOPHYSICAL METHODS Geodynamic modelling ... · Geodynamic modelling using Ellipsis...

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Geodynamic modelling using Ellipsis GEOS3104-3804: GEOPHYSICAL METHODS Patrice F. Rey

Transcript of GEOS3104-3804: GEOPHYSICAL METHODS Geodynamic modelling ... · Geodynamic modelling using Ellipsis...

Geodynamic modellingusing Ellipsis

GEOS3104-3804: GEOPHYSICAL METHODS

Patrice F. Rey

GEOS-3104

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Computational Geodynamics

Computational tectonics/geodynamics provides a robust plat-form for hypotheses testing and exploring coupled tectonic and geodynamic processes. It delivers the unexpected by re-vealing previously unknown behaviors.

SECTION 1

During the mid-19th century, the laws of thermody-namics - which describe the relation between heat and forces between contiguous bodies - and fluid dynam-ics - which describes the flow of fluids in relation to pressure gradients - reached maturity. Both theories underpin our understanding of many natural proc-esses from atmospheric and oceanic circulation, man-

tle convection, and plate tectonic processes which also involves the flow - brittle or ductile - of rocks. 21st cen-tury computers have now the capability to compute in four dimensions the flow of complex fluid with com-plex rheologies and contrasting physical properties we typically encounter on Earth. For geosciences, this is a significant shift.

The rise of computer science

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Computational tectonics and geodynamics: There is a revolution unfolding at the moment in all corners of science, engineering, and other disciplines like economy and social sciences. This revolution is driven by the growing availability of increasingly powerful high-performance computers (HPC), and the growing availability of free global datasets and open source data processing tools (Py-thon, R, Paraview, QGIS-GRASS, etc). Here in Australia, supercom-puters such as Magnus (36,000 cores) at the Pawsey Supercomput-ing Center in Perth, and Raijin (58,000 cores) at NCI (National Com-putational Infrastructure) in Canberra, have enabled and democra-tised the art of numerical modelling.

Broadly speaking, numerical modelling is a discipline which en-ables the exploration of the behavior of complex systems. It gives scientists, engineers and economists the capacity to extract new knowledge and understanding from big data. In the context of geol-ogy and geophysics for instance, numerical modelling allows us to build a model of lithosphere and explore how this lithosphere be-haves when submitted to tectonic forces. It also enables geoscien-tists to build spherical models of the Earth to explore mantle con-vection. These models can acount for a very broad range of petro-physical properties including radiogenic heat, heat diffusivity, den-sity, heat capacity, rheology (brittle and ductile), solidus and liqui-dus, etc. The behavior of the lithosphere and that of the convective mantle is governed by the laws of thermodynamics which de-scribes exchange of energies within the system, and fluid dynamics which relates deformation (i.e. flow) to pressure gradients. Both thermodynamics and fluid dynamics are fully developed theories from 19th century physics. Yet, it is only over the past decade that HPC became powerful enough to efficiently solve in 3D the rele-vant equations at a reasonable spatial and temporal resolution.

Navier-Stokes equations: Written in the 1850’s, the Navier-Stokes equations are the set of differential equations that describe the motion of fluid and associ-ated heat exchanges. They relate veloc-ity gradients to pressure gradients and can be analytically solved only when considering steady laminar flows.

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Georges G. Stokes (1819-1903)

Some examples of simple laminar flow for which analytical solu-tions exist include the problem free fall of a spherical object into a newtonian fluid (the settling of crystal into a magma), the flow in-

duced by the motion of a rigid plate above a newtonian fluid (Couette flow in relation to the motion of tectonic plate above the asthenosphere), and the flow of newtonian fluid between two static plates (Poiseuille flow, for instance the flow of the lower crust in orogenic pla-teaux).

One hundred years later (1950’s), with the advent of mainframe computers, the Navier-Stokes equations could be discretised and solved at the nodes of a numerical grid to explore the type of com-plex, time-dependent fluid flow we encounter in nature. At the same time, numerical methods and computer algorithms were pro-gressing so fast that they quickly overtook the computer capability

of their time. In short, in the second half of the 20th century com-puters were not powerful enough to take advantage of these new

numerical methods. As com-puters grew in power, so did the complexity of fluid flow problems that one can tackle. However, for quite some time, computational tectonics in-volving a layered lithosphere made of stronger (upper crust

and upper mantle) and weaker layers (lower crust and lower litho-spheric mantle) was limited to 2 dimensional models in which the node of the computational grid had to follow the model deforma-tion (Eulerian grid). Computational geodynamics could afford 3D models because models of mantle convection could make use of a more efficient fixed grid (Lagrangian grid) as long as the convec-tive mantle was made of one single newtonian fluid.

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1950’s mainframe computers

@ Beaumont, Halifax@ Yuen, Minneapolis

Particle-in-cell numerical methods. In the late 1990’s, computers become powerful enough to allow the implementation of a 1950’s numerical method called particle-in-cell (PIC, developed at Los Ala-mos National Laboratory) in which individual fluid elements, car-rying material properties and flow history, are advected through a fixed computational grid. For geologists, this progress meant that

one could simulate the deformation of mechanically layered sys-tems such as the Earth’s lithosphere; albeit in two dimensions. El-lipsis, the code we will use in GEOS3104, is one of the earliest and most robust codes (along with Citcom) implementing an efficient PIC method on the back of a robust multigrid solver.

Over the past decade, the growing availability of powerful high-performance computers has unleashed the power of computer-based modelling in all branches of science. Geoscientists are now able to simulate in four dimensions, using realistic coupled ther-mal and mechanical properties, lithospheric-scale deformation and mantle geodynamics. 21st century HPC have finally caught-up with 19th century physics, and computer science from the mid-50s.

Numerical experiments vs numerical modelling vs numerical simulation: Before going further we need to clarify the difference between experiment, modelling and simulation. For most people these concepts are interchangeable, but not for the experts.

Numerical experiments (or a physical experi-ment) do not pretend to reproduce a natural process in a realistic manner. The aim fo-cusses on trying to illus-trate a concept, or try-ing to understand the few most important pa-rameters involved in a particular process. The famous analogue experiment from Paul Tapponnier (performed for the first time in the very late 70’s) perfectly illustrates the concept of escape tecton-ics, a process which accommodates convergence via the lateral ex-pulsion of continental blocks in front of a rigid indenter. Clearly, this experiment bears very little resemblance with tectonics in Asia and South East Asia, but it does a nice job in illustrating a concept which has changed our understanding of collisional tectonics.

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Rey, Coltice, Flament, Nature 2014

We have used Ellipsis to show that early proto-continent and thick oceanic plateaux had enough gravitational power to slowly force adjacent oceanic lithospheres to sub-duct. The model above is 700 km x 2800 km, include continental crust (red), litho-spheric mantle (pink), partially molten mantle (bright blue), the rest is the mantle.

PIC = Eulerian mesh + Lagrangian particles

Numerical modelling aims at under-standing a particular process within a particular tectonic context. The ini-tial and boundary conditions are carefully thought through to describe a geologically realistic setting. Re-sults of the modelling are detailed enough to be able to be compared - to

the first order - to natural geological exam-

ples, without trying to match accu-rately any-one of them.

Numerical simulations aim at reproducing with the greatest level of detail and the greatest level of realism a particular process on a particular region. For most people this is what modelling is all about, and therefore they are quick to point to the many shortcom-ings of Paul Tapponnier’s experiments and dismiss its relevance since it neither account for the presence of the highest mountain chain nor the highest plateau on Earth. Numerical simulation as-sumes that our models are able to include parameters with realistic value distributions and time-dependencies, as well as nested het-erogeneities on a broad range of scales. In addition, numerical

simulations, in the context of geosciences, assumes that we are able to implement a very broad range of coupled natural mechanical, petrological, geochemical processes including the migration of par-tial melt and the associated advection of heat, the release and con-sumption of water and other volatiles, evolving anisotropic rheolo-gies due to grain reduction, dilatancy due to micro-cracking and other softening mechanisms. It will take decades before we can properly engage with numerical simulation in a meaningful man-ner.

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Multi-grid solver: The Finite Element Method (FEM) con-sists in discretizing of a set of partial differential equations across a 2D or 3D domain (i.e. ∂# are transformed into ∆ # ). This allows solving efficiently the Navier-Stokes equations at the nodes of a grid covering the model, but at the price of a small error. To minimize this error, users can use a very fine grid. The finer the grid the higher the resolution but the longer the compute time. Hence, the need to balance accu-racy and compute time. To solve the set of equations efficiently, computer scientists design methodologies taking advantage of parallel comput-ing, as well as other techniques. One involves the use of a stack of computational grids of increasing resolution. In-stead of solving the set of equations directly at the node of one single fine grid, a rough solution is computed on a coarser grid, and this solution is iteratively refined at the nodes of grids of increasing resolution. Ellipsis uses a multi-grid solver.

SECTION 2

Ellipsis solves the Navier-Stokes equations at the nodes of a computational grid using a Finite Element Methods (FEM). Solving the Navier-Stokes equations provides the velocity fields from which a range of other field properties can be com-puted (pressure, viscosity, stress etc). Lithologies are repre-sented by particles distributed inside the grid’s cells (hence

the acronym PIC: Particle In Cell). In this section, you will learn to compile Ellipsis on your own Mac or Linux computer, and you will learn the command to launch and stop Ellipsis from a Terminal window. Some basic Unix skills are neces-sary, but involve knowing a dozen Unix commands at most.

Ellipsis: FEM-PIC code

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Introduction: Ellipsis is a child of CitcomS, a code dedicated to mantle convection modelling. Ellipsis has a long his-tory of development starting in the late 90‘s at Caltech by Louis Moresi (now Melbourne University). It is designed to explore 2 dimensional coupled thermal and mechanical tectonic processes. It can simulate both brittle and ductile deformation where brittle rheologies evolve with accumulated strain, and ductile rheologies evolve with tempera-ture, stress, strain rate and melt fraction. Although more powerful parallel open source codes exist (e.g. GALE, Un-derworld), their installation require a higher level of computational expertise, and access to high performance com-puters. The main advantages of Ellipsis is that its installation is “relatively” simple, it can run on a laptop computer, and it is powerful enough to produce innovative research outcomes publishable in Nature.

Installation: Ellipsis is a free open source code that runs on unix-based computer operating systems (e.g. Linux, Mac OSX, etc). The source code (a bunch of files written in C) can be downloaded from the GeoFramework site at the California Institute of Technology. To compile Ellipsis source code on your machine you need access to an appro-priate gcc compiler and a Terminal window. On Mac, the Terminal window comes with the Xcode.app (free from the Mac App store). If you already have access to a Terminal check if you have an appropriate compiler: open a Terminal window and execute the following command: gcc --version. If no gcc compiler is installed you need to install one. Go to http://hpc.sourceforge.net/ and download the gcc compiler for your OS, then open a Terminal window and execute: sudo tar -xvf /path_to/gcc-5.1-bin.tar.gz -C /. On Linux, simply download from the Internet a gcc compiler for your version of Linux. To compile the source code, open a Terminal window, navigate to the Ellipsis source (use cd command and drag&drop), then execute the following three lines (you may have to specify the path to your com-piler by adding the option CC=/path_to/gcc-5.1 after CFLAGS=’-m32’ ):

An executable called ellipsis3d will be created in the source directory. Copy this ex-ecutable (its size is only a few 100s bytes) and paste it in a folder in which your input files will also be located (good practice is to keep a copy of the Ellipsis executable with each model, so you can keep track of which version of Ellipsis you have been

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./configure CFLAGS='-m32'make cleanmake

running). To run (i.e. execute) an input file, simply open a Terminal window, navigate to the folder holding ellipsis3d and the input file (here mycoolmodel.input) and enter the following command:

To stop the model, click the sequence control-C. More often than not, the reference model described in the input file requires access to a file laying out the temperature at each node of the computational grid. This file is called into the input file at previous_temperature_file=”mytempfile”. We will see later how this file can be generated. This temperature file must be present in the folder in which you are running the experiment, as shown on the figure on the right.

As Ellipsis runs, it dumps at each time step a bunch of files into the folder. These files, specified by the users in the input script, store grid and particles information (temperature, pressure, velocity, stress, strain rate, viscosity, ...). #.ppm0: These files are graphics ouput showing various lithologies, melt fraction, stress, viscosity etc.#.node_data:  These files store parameters such as temperature and pressure at the node of the computational grid. The in-formation in node_data can be used for data mapping (mapping of isotherms, super-solidus temperature, melt fraction, ...). For this you will need to process the data stored in .node_data using Matlab, R, Python or any other appropriate tools.#.profiles:  These files give access to profiles (horizontal or vertical) passing through "sampling tracers".  In the input file one can specify the initial position  of up to 50 sampling tracers (this position can remain fixed to the computational grid, or can move with the rocks through the grid), the direction of the profile (horizontal or vertical), and the type of data to be profiled (temperature, melt, depletion, pressure, velocity, strain rate, stress ... )#.sample : These files are updated at each time step to add the new value of parameters attached to one of the sampling tracers.  Hence, there is one .sample file per sampling tracer.   These files record the evolution through time of data attached to a tracer (x, z, Data), where data can be temperature, pressure, etc. These can be used to collect data to plot PTt paths.

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./ellipsis3d mycoolmodel.input

Resolution of the computational grid and particle density: Solving the Navier-Stokes equations is handled via a multigrid solver which uses a stack of increasingly finer computational grids. A rough solution of the velocity field is obtained rapidly on a coarser grid. This solution is then used as input on the next finer grid, and so on. In the in-put file, the number of levels refers to the number of grids in the multigrid stack (see fig. below). The number of grid cells is multiplied by 4 when moving from a grid level to the next. The larger the levels the better the resolution of the model, but the longer the running time. The pressure field is calculated using a sub-set of the multigrid stack (pmg_levels), the maximum depth of which is levels-1. The finer grid also contains Lagrangian tracers. They carry information about the material properties. The more tracers there are, the better defined are the density interfaces. There is no fast rule for how many grid levels and tracers one should use. Consequently, a model may take any time between a few minutes to several months… In most cases, levels=4 is sufficient when testing lithospheric-scale models. Once the model has been properly tested at levels=4 to ensure that geometry and boundary conditions are properly set, it is a good practice to let it run at levels≥4 (up to level=6).

Figure. A grid stack (Levels=3: grid 0, grid 1 and grid 2) for the multigrid solver. The number of cells in the finest grid is determined by the Levels (here 3) and the parameters mgunitx (number of cells along the x direction in grid 0, here 5) and mgunitz (number of cells along the z direction in grid 0, here 3). On the right, tracer_density is defined on the finest grid level. In this example tracer_density=4, meaning there are four tracers/elements both in the x and z direction (i.e. 4x4=16 trac-ers per element).

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Initial temperature field: The temperature field associated with an input file is computed by running the input script without any kinematic of dynamic boundary conditions (i.e. the model is not tectonically extended or short-ened). For maximum computational efficiency one can increase all viscosities and turn plastic rheologies off (so the model doesn’t deform while the geotherm evolves), this is achieved using the following over-riding toggles (at the end of the input script): TDEPV=off # visc_max=5e31" "" visc_min=1e31 " YIELD=off

… and increasing the fixed_timestep to 1 million years, since we use second for unit of time ...:# fixed_timestep=3.1536e13

Then, any #.node_data output can be used as a previous_temperature file by setting up in the Steady-States and Bound-ary Conditions section of the input file:previous_temperature_file="A_My_Previous_Temperature_File.node_data"

Time step: Unless when running thermal equilibration, time step should be kept below 6.3072e12 sec (that’s 200,000 years). If weird things happen (i.e. unexpected short-term fluctuation) try to reduce the time step.

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Installing Ellipsis on Windows:In terms of programming platform, Windows can’t compete with Unix-based operating systems. One solution for Window users is to partition the hard drive and install Ubuntu (or another Linux version like Debian) on one of them, giving you access to both Windows and Linux environments. If this is too intimidating, then the next best thing is the Cygwin environment. Cygwin gives Windows users a Unix-like environment. Cygwin is a collection of the most common programming tools and compilers (including the bash shell) for Windows. 1. Install Cygwin (https://www.cygwin.com/). Download the 32-bit or 64-bit setup executable depending on

which variant of Windows you are using. 2. From within Cygwin’s setup.exe program install the base (give access to bash), devel (give access to gcc compilers

and make), util and web (gives access to wget tools) packages. Some advices on installing Cygwin: http://preshing.com/20141108/how-to-install-the-latest-gcc-on-windows/

There are number of alternative C compilers and development environment available for windows, MinGW is one of the most popular providing access to gcc (http://www.mingw.org). MinGW can be installed alongside Cygwin. By editing the Cygwin path, calls to gcc or g++ can point to the MinGW bin directory.

Then in Cygwin execute the following:

./configure CFLAGS='-m32'make cleanmake

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Using Ellipsis on Earthbyte’s server via PC lab:In this solution users are completely dependent on the infrastructure provided. Once this course is finished you won’t be able to use Ellipsis. A great solution of teachers, but a not so great solution for students.

# 1# Log in normally using your unikey. # 2# Launch PuTTy from the Start Menu. # 3# Set the Host Name to be “129.78.236.27” and leave everything else as default. # 4# Click Open to open the connection, and use geos3104 as username.# 5# Type “goto_geos3104"# 6# Create a directory named after your unikey (mkdir ${unikey})# 7# Copy Ellipsis in that directory (cp ellipsis3d ${unikey}/)# 8# You are now ready to use ellipsis.

To make file transfer easier: # 1# Open “My Computer”# 2# Click “Map Network Drive” # 3# Folder: \\129.78.236.27\geos3104# 4# Enter the login credentials

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Using Ellipsis on a multi-platform VirtualBox:In this is a Universal solution as this Virtual Box can run any operating systems (including Mac OS X) inside any platform (OS X, Windows or Linux). A great advantage for teachers is that this is a one-size-fits-all approach. There is however a penalty to pay by the users as this solution is slower and - as some of you have experienced - Virtual-Box can be unstable (quit randomly) on some platforms.

We have prepared a Debian 8 VirtualBox (~2 Gb...) which includes the Ellipsis source code ready for compilation.

To download this VirtualBox go to: https://cloudstor.aarnet.edu.au/plus/index.php/s/OtVXCxN7MZH8bDIUsername: geophysicsPassword: usyd

In the VirtualBox, open a Terminal window, navigate into the Ellipsis source folder and compile the source code:

./configure CFLAGS='-m32'make cleanmake

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What have we learned so far?

1/ Laptop computers come in various flavors. For most people a laptop running Microsoft Windows, Microsoft Ex-plorer, and other Microsoft softwares is more than enough. Creating and/or installing non-Microsoft applications on Windows can be a challenge but most people do not need to do that. Apple enthusiasts are only a notch better off. For most of them a Macbook running Mac OS X, Safari, and other friendly Apple apps is also more than enough. However, because Mac OS X is based on Unix, Apple computers can be transformed into production ma-chine for the purpose of scientific programming, scientific modeling, data processing and data vizualisation. This is achieved by unleashing the full potential of Unix via the installation of Xcode, generic gcc compilers and other piece of codes (called libraries or packages) which extend Mac OS X’s Unix capability. Finally, Linux-based PCs have UNIX in their DNA. Linux platforms offer the most complete and native Unix system. It is the platform of choice of most Computer Scientists.

Science graduate students should all be able to tell what laptop they use, which operation system it is running, what version, the size of its hard drive and how mush space is left on it, and how much ram it has. If you don’t, then I suggest you pay a visit to your friend Google ...

2/ With the appropriate tools, any laptop can compile into an executable application a source code written in C, C++, Fortran, etc. For Windows-based laptop, there are quite a few solutions for doing this: One can install a Virtu-alBox on which a Linux OS has been installed. Another option is to partition the hard drive of your PC in two and install a Linux OS on one of the partition. Finally, one can install Cygwin and a bunch of Unix-like packages.

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3/ The process of compilation is never easy, and always frustrating. But the more you engage with this process the easier it becomes. For most people (i.e. non computer scientists) a painful trial and error learning process is the only avenue ...

4/ You/we have learn that we should read very carefully the instruction. I personally made the mistake of not fol-lowing the instruction to the letter ... download the gcc compiler for your OS, then open a Terminal window and execute: sudo tar -xvf /path_to/gcc-5.1-bin.tar.gz -C / ... instead of executing the command I recklessly double clicked on the zipped tar file and compilation was automatically done but not if the correct folder (i.e. usr/local).

Let’s have a closer look at the following command: sudo tar -xvf /path_to/gcc-5.1-bin.tar.gz -C /tar uncompressed tar file.-xvf means:-x : Extract to disk from the archive.-v : Produce verbose output i.e. show progress and file names while extracting files.-f : Read the archive from the file located on the end of this path ...The last command -C / means the files are extracted at the root directory. This requires root privilege, hence the Unix command sudo in front.

5/ Hopefully you have learn some new lingo (compile, path, libraries, configure, platform...) and some Unix com-mands:cd: means change directorypwd: path to working directory shows the address (i.e. folder) the Terminal is attached to.ls: lists files sitting at pwdls -la: lists all files including those starting with “.” and give useful information about permission, size etc

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mkdir: make a directory in pwd, e.g. mkdir folder1 will create a folder named folder1ff: find files anywhere on the system, e.g. ff AAMCC.Geology.input ff -p: You don't even need the full name, just the beginning, e.g. ff -p AAMCCps -u yourusername : lists your processes and their ID.kill ID : Ends the processes with the ID you gave.tar and sudo: see point 4/ rsync: this synchronize two folder, for instance rsync -avz /path_to_folder/folder1/ /path_to_folder/folder2will copy the content of folder 1 into folder 2. Pay attention to the backslash “/” after folder1, without it the folder 1 will be copied into folder 2. Hence, in Unix all character counts ...The option -avz means the following:-a: archive mean your want to “recursively” copy the content of folder 1. This will include all folders in folder1. This option also means that the copy procedure will preserve all attributes of the source files (e.g. permission). -v: this option means “verbose” ... increases the amount of information you are given during the transfer.-z: With this option, rsync compresses the file data as it is sent to the destination machine, useful over a slow connec-tion.

6/ You have learn you can use cd and drag-and-drop instead of writing the path.

7/ With Unix command, one is exactly 4 characters away from erasing an entire hard drive ... with no turning back. So before executing a Unix command think twice.

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SECTION 3

Ellipsis can be run from the jupyter environment. Ju-pyter offers a collaborative environment in which one can run codes, scripts and calculations, and process data using the python ecosystem, as well as other com-puter languages, to produce new data and visual out-puts. The jupyter environment available to GEOS3104

students can be accessed at the following web address: https://jupydw00308.srv.sydney.edu.au

Ellipsis is located in: /usr/local/bin/ellipsis3d

Ellipsis on jupyter

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The volume of most materials expand when they become warmer, leading to a decrease in density. The coefficient of thermal expansion of most rocks, as a volume fraction per degree, is about 3e-5 ºC-1. This means that close to the core-mantle boundary, where temperature reaches ~4000ºC, rocks are less dense by about 12% (3e-5x4000=0.12). This drives mantle convection, i.e. the buoyant rise of deep hot mantle rocks, and the sinking of an equivalent volume of colder, denser rocks.

CHAPTER 2

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Mantle Convection

SECTION 1

The purpose of this section is to understand mantle convection, understand the contribu-tion of this process in cooling the interior of planets, and develop through numerical ex-periments how mantle convection, and plate tectonics relate to one another. We will test dif-ferent modes of convection driven by various boundary and internal conditions. We will

also illustrate what role lithospheric plates play on the pattern of mantle convection. We start here by introducing the Rayleigh num-ber. This is a dimensionless scalar which repre-sents the ratio of the forces that drive convec-tion over the forces that resist convective mo-tion. In short the Rayleigh number is a meas-ure of the vigor of convection.

Summary

1. Why mantle convection

2. Rayleigh number

3. Mantle heating and Rayleigh number

Figure 2.1 Example of mantle flow underneath an extending lithosphere.

The Rayleigh number

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Mantle Convection

Upon heating rocks expand and consequently their density de-creases. Since temperature increases with depth one concludes that the density of rock decreases with depth as well. However, this is not ideal, as a system in which density decreases with depth has a higher internal gravitational potential energy compared to that of a system in which density increases with depth. Indeed, a funda-mental principle of physics states that a system naturally evolves to minimize its internal energy. Mantle convection emerges from the necessity of energy minimization: it forces deep, hot and less dense material up, and pushes shallow, cold and denser material down. However, because of the relationship between density and

temperature the Earth's mantle is maintained in a state of gravita-tionally unstability, indeed hot material rising toward the surface progressively cool down and become more dense. Heat (a form of energy) is consumed by mantle convection which acts as a giant heat exchanger, to cool down very efficiently the deep Earth’s inte-rior.

Rayleigh Number And The Strength Of ConvectionThe Rayleigh number measure the vigor of the convection. It is a di-mensionless number that expresses the ratio of the forces that drives convective motion and the forces that oppose it.

Exercise: Figure out which physical parameters are relevant to convec-tion? Which of these parameters promote convection, and which resist con-vection?

The buoyancy force - that arises from volumetric expansion (ther-mal expansion) - drives mantle convection. In the other hand, the effective viscous force (due to the viscosity of the mantle) and the thermal diffusivity (which smooth out thermal gradient) oppose mantle convection. The ratio between the buoyancy force to the vis-cous force is the Rayleigh number. For convection to occur, the Ray-leigh number must be larger than a critical value, which depends on whether the surface is free of fixed (.e. zero velocity at the sur-face).

The Rayleigh number takes slightly different forms depending on how heat is delivered to the mantle (e.g. internal heating, basal heating or a mix of both).

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Temperature increase with depth, but why?

In the Earth’s interior, temperature increases with depth for 3 main reasons:

1/The radioactive decay of radiogenic elements (mainly K40, U235, U238, U232 and Th) involves exothermic reactions that lib-erate thermal energy. 2/ During its formation, the Earth has trapped a finite amount of accretional energy (gravitational energy trans-formed into heat).3/ Last but not least, the Earth is cooled at the surface since it is suspended into the chilling ~0 K Universe.

The Rayleigh number driven by a hot base (at temperature T1) and a cold surface (at temperature T0):

RabT =ρ0.g.α.zm3

η.κ.(T1−T0)

Where ρ0 is the density are room temperature, g is the gravitational acceleration, α is the thermal expansivity, zm is the thickness of the convective layer, η is the average viscosity of the mantle, and κ is the thermal diffusivity.

The Rayleigh number driven by a hot base (at constant basal heat flow Qm) and a cold surface (at temperature T0):

Rabhf =ρ0.g.α.zm3

η.κ.(

Qm.zm

K−T0)

where K is the thermal conductivity.

The Rayleigh Number driven by internal heating:

Raih =ρ0.g.α.zm3

η.κ.(

H.zm2

K−T0)

where H is the radiogenic heat production per cubic meter.

The Rayleigh number driven by both basal heat flow and internal heating:

Ramh =ρ0.g.α.zm3

η.κ.(

Qm.zm+H.zm2

K−T0)

Clearly, the second term in the Rayleigh numbers describes the heat-ing mode.

For convection to occur, the Rayleigh number must be larger than a critical value, which varies from 600 to 3000 depending on the boundary conditions (e.g. stagnant lid, plate tectonics, etc).

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SECTION 2

Exercise 1: Basal heating vs internal heating vs mixed heating

We want to compare mantle convection under various mode of mantle heating, i.e. basal heat-ing (constant basal temperature or constant heat flow), internal radiogenic heating, or a combina-tion.

To ensure we compare apple with apple we need to compare models that are at once thermally and convectively equivalent. We will consider that two models are 1/ thermally equivalent if, in their non-convective state (infinite viscosity), they have the same steady-state average temperature,

Summary

1. Learning outcomes: To develop a deeper understanding of mantle convection and its dependence to the mode of heating of the convective mantle

2. Generic skills: Problem solving ability, computational skills, analytical skills

3. Assumed knowledge: high-school math, some notion of Tectonophysics relevant to the Earth geotherm.

4. Tools: R, Python, Matlab5. Reading: Turcotter &

Schubert: Geodynamics

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Id malesuada lectus. Suspendisse potenti. Etiam felis nisl, cursus bibendum tempus nec. Aliquam at turpis tel-lus. Id malesuada lectus. Suspendisse potenti.

Exercise: Keeping it cool ...

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and 2/ convectively equivalent if they have the same Rayleigh number.

Find the relationships that set up the conditions for thermal equiva-lence and convective equivalence and comment on your results. We give the following clues:

• The steady state conductive geotherm for constant basal tempera-ture is:

TbT(z) =(T1−T0)

zm.z+T0

• The steady state conductive geotherm for constant basal heat flow is: !

Tbhf (z) =Qm

K.z+T0

• The steady state conductive geotherm for internal heating is:

Tih(z) = −H.z2

2K+

H.zm

K.z+T0

• The steady state conductive geotherm for mixed basal heat flow and internal heating is:

Tmh(z) = −H.z2

2K+(

Qm

K+

H.zm

K).z+T0

where Qm, K and T0 are the basal heat flow entering the base of the convective mantle at the core-mantle boundary, the thermal con-ductivity and the surface temperature respectively; H is the volu-metric rate of radiogenic heat production.

You may need to know that:

α

∫β

a xn+b  dz = ( a.xn+1

n+1+b.x+c)

β

α

Exercise 2: What’s the Earth’s Rayleigh number?

The Earth is currently releasing heat into space at the rate of ~44 TW. About 20% of this amount (~8TW) originates from radiogenic isotopes trapped in the plates. This plate component of heat doesn’t contribute powering mantle convection. The Earth is left with a convective heat flow of 36 TW to drive mantle convection.

If mantle convection is driven by basal heating alone, then the 36 TW must come from the Earth’s core in the form of a basal heat flow. In the other hand if mantle convection is powered by inter-nal heating alone then the 36 TW must be partitioned into the con-vective mantle as radiogenic heat production. In nature, we expect the convective heat flow to be split into both internal and basal heating.

There is however a debate about the partitioning of the heat flow between the core and the mantle. This partitioning is described by the Urey ratio, the ratio between internal heating in the convective mantle to total convective heat flow (36 TW). Estimates varies from 0.10 to 0.90, not very helpful…

1/ Calculate the basal heat flow (in W.m-2) in the case where the convective heat flow is delivered to the convective mantle in the form of basal heating at the core-mantle boundary.

2/ Calculate the rate in radiogenic heat production (in W.m-3) in the case where the convective heat flow is in the form of internal heating.

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Please note that: The Earth’s radius is: 6371 km, the lithosphere av-erage thickness is 150 km, the depth of the core-mantle boundary is

at 2890 km. Volume of a sphere: 43

.π.R3 ; surface of a sphere: 4.π.R2

(R is the sphere radius).

3/ Calculate the Rayleigh Number for each heating modes (inter-nal heating, basal heating, and mixed heating) using the following values:

ρ0 = 4000 kg.m-3; g = 10 m.s-2; α = 3e-5; zm = 2890 km; K = 4 Wm-1 K-1; κ = 1e-6 m2 s-1; H = “your value from question 2” W.m-3; Qm = “your value from question 1” W.m-2; η = 5e21 Pa.s; T0 = 273 K

nb1: For the mixed heating mode, use 24 TW for the internal heating and 12 TW for basal heating.nb2: For the case involving a constant basal temperature one can calculate the temperature at the core-mantle boundary by extrapolating the linear conductive geotherm in the lithospheric plate (273 K at the surface and1603K at 150km depth).

4/ Assuming no mantle convection, estimate the temperature at the core-mantle boundary in both basal heating (constant heat flow) and internal heating models using the conductive geotherm equations given earlier. Compare these values with the estimated range of temperature of the core-mantle boundary in the convec-tive Earth (3000 to 4000 K).

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SECTION 3

Ellipsis is a finite-elements code that solves com-plex 2D fluid dynamic problems. It solves the relevant equations for incompressible fluids (Na-vier Stokes equations, equations of conservation of mass and energies and constitutive equations) at the nodes of a fixed computational grid using a multigrid solver (solutions are iteratively re-fined by using an increasingly finer grid). Parti-cles, representing different materials, move

through the grid carrying relevant information to solve the equations. The combination of a fixed grid (Eulerian grid) and free particles (Lagran-gian particles) allows numerical experiments in-volving large deformation of rheologically com-plex models.

The Ellipsis input script (a simple text file) de-scribes the model including the boundary and in-

Summary

1. Learning outcomes: To develop a deeper understanding of mantle convection and its relationship to plate tectonics

2. Learn to use non-dimensional approach

3. Generic skills: Problem solving ability, computational skills, analytical skills

4. Assumed knowledge: high-school math.

5. Tools: Ellipsis, R, Python6. Reading: Turcotter &

Schubert: Geodynamics

Exercise: Convective patterns

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ternal conditions. Ellipsis then executes the script. Since mantle convection depends only on the Rayleigh number, the values of all parameters entering its definition are of little importance. Hence, in Ellipsis we will set to 1 all parameters (gravity, diffusivity, viscos-ity, mantle thickness, radiogenic heat production etc), excepted for the coefficient of thermal expansion whose value will be that of the Raleigh number.

With this in mind, let’s build our first model in the case of basal heating. We first define the dimension of the model. The model horizontal length must be a few times longer than it is depth. Since the depth of the convecting mantle is 1, we choose 6 for the length for the x dimension.

# Dimensions of the model:dimenx=6.0dimenz=1.0

For the initial conditions, we have gravitational acceleration set to 1 and the initial geotherm is linear with the top surface at 0 and bot-tom surface at 1.

# Initial conditionsgravacc=1toptbcval=0 bottbcval=1

We fill our box with one material representing the mantle. This box is defined by the coordinates of its top-left corner (x1, z1) and bottom-right corner (x2, z2). nb: (0,0) is the top-left corner.

# Material distributionsMaterial_rect=1 # number of rectangular regions Material_rect_property=0 # Material id Material_rect_x1=0.0 # coordinates of tracer regionsMaterial_rect_x2=6.0 # Add rectangles by adding columnsMaterial_rect_z1=0.0Material_rect_z2=1.0

Then we define the boundary conditions. We use a constant tem-perature of 0 at the top of the model, and a constant temperature of 1 at its base.

# Boundary conditionsTemp_bc_rect=2 # number of bc Temp_bc_rect_norm=Z,Z# Z/X for horiz./vert. boundariesTemp_bc_rect_icpt=0,0# Coordinate of boundary planeTemp_bc_rect_aa1=0,0 # lateral coordinate extent Temp_bc_rect_aa2=6,6 # lateral coordinate extentTemp_bc_rect_hw=0,0 # half-width of bc smoothing edgeTemp_bc_rect_mag=0,1 # magnitude of bc

Heat_flux_z_bc_rect=0# no basal heat flow  

In what follow, we define the thermal and mechanical properties of the model materials. Again, because all relevant parameters are in-cluded in the Rayleigh number, density, viscosity, diffusivity and heat capacity are equal to 1. Because we consider a basally heated model we set the internal heating to 0, and the coefficient of ther-mal expansion is the Rayleigh Number.

# MATERIAL PROPERTIES Different_materials=1 # Number of different materials listed

## Mantle Material_0_density=1

# Rheological model Material_0_rheol_T_type=1 # model visc(T): visc=N0*exp(-T1*T) Material_0_viscN0=1 # N0 in viscosity model Material_0_viscT1=0 # T1 in viscosity model

# Thermal parameters Material_0_therm_exp=7.34e7 # coeff. of thermal exp./ Rayleigh Nb Material_0_therm_diff=1 # thermal diffusivity

Material_0_Cp=1.0 # isobaric heat capacity Material_0_Qt=0.0 # internal heating rate by mass

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# Colouring (Try to experiment with this.)We now consider the output files we would like to get from Ellip-sis:

# OUTPUT FILES datafile="ConvMod1" # root name of output files datatypes="Temp,Pres" # desired output variables at the grid nodes averages="Temp" # horizontally averaged values, timelog="Shfl,Bhfl" # time record of large-scale averages, # Shfl, Bhfl= av. surf. and basal heat fluxes storage_timesteps=1 # data writing interval (based on av. timestep) checkpt_timesteps=1 # PPM (graphics) file writing interval storage_timing=1.e-8 # absolute timestep data writing interval

checkpt_timing=1.0 # absolute timestep PPM file writing interval # Specifications for graphical output files PPM_files=1 # number of PPM files at each output step PPM_height=128 # vertical size of output PPM file PPM_coloring=1 # variable upon which to base colouring

# 1=temperature, 2=viscosity, 3=stress# 4=solid pressure, 5=grainsize, 6=compression,

PPM_coloring_autorange=1 # automatically scale colour PPM_coloring_min=0.0 # min value for color scale PPM_coloring_max=1.0 # max value for color scale  

Finally, we may want to extract some temperature (or other vari-ables) profiles across our model:

# for extracting profiles/historiesSampling_tracers=10 # number of sampling tracers

Tracers (fixed to the grid or to the material) can record information such as Temperature, Pressure, etc. They can also record parameter profiles (vertical or horizontal).

One last thing, to run this model over 400 time steps…

# ADVECTION-DIFFUSION PARAMETERS maxstep=400 # maximum number of adv-diff time steps  

Exercise 3: 2D Experiments of Mantle Convection using Ellipsis

1/ Run over 400 steps the previous script (convection with basal heating, constant temperature).

2/ Build and run a model with internal heating (Ra=2.62e8). For this model, you will need to 1/ remove the constant temperature at the base of the model, 2/ set to zero the magnitude of the basal heat flow (insulation condition, no heat is lost at the base of the model), and 3/ set the internal heating rate to 1.

# Boundary conditionsTemp_bc_rect=1 # only the first bc is taken into account.Temp_bc_rect_norm=Z,Z # Z/X for horiz./vert. boundariesTemp_bc_rect_icpt=0,0 # Coordinate of boundary planeTemp_bc_rect_aa1=0,0 # lateral coordinate extent Temp_bc_rect_aa2=6,6 # lateral coordinate extentTemp_bc_rect_hw=0,0 # half-width of bc smoothing edgeTemp_bc_rect_mag=0,1 # magnitude of bc

Heat_flux_z_bc_rect=1 # number of rectangular bc ranges (surfaces) Heat_flux_z_bc_rect_norm=Z # normal to plane of surfaceHeat_flux_z_bc_rect_icpt=1 # normal-axis intercept of bc planeHeat_flux_z_bc_rect_aa1=0 # lateral coord. extent in 1st dimensionHeat_flux_z_bc_rect_aa2=6Heat_flux_z_bc_rect_hw=0 # half-width of bc smoothing edgeHeat_flux_z_bc_rect_mag=0 # magnitude of bc

3/ Run the models 1 and 2 using a temperature dependent viscos-ity:

with:

Material_0_viscN0=5 # N0 in viscosity modelsMaterial_0_viscT1=5 # T1 in viscosity models

Verify that the average viscosity of the convective mantle is still 1.

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4/ Analyze and comment your results. Give particular attention to the shape of the averaged geotherm for both internally heated and basal heated models. How do the geotherms compare?

5/ Design a model with both internal heating and basal heating us-ing constant basal heat flow instead of constant basal temperature (use the Rayleigh number from exercise 2). Run this model and compare it with a model in which a lithospheric plate is added. The plate has a thickness of 0.1 and extends over 50% of the model. Its density must be 20% lower than the mantle and its viscosity (constant with temperature) must be 1000 times larger than that of the mantle. The plate can be maintained fixed or let to move around. You may want to experiment with both options.

For a fixed plate, use the following (change X1 and X2 by the hori-zontal coordinates for your plate):

Velocity_z_bc_rect=1 # nb of “rectangular” bc (surfaces)Velocity_z_bc_rect_norm=X # normal to plane where bc appliesVelocity_z_bc_rect_icpt=0 # coord. of the bc planeVelocity_z_bc_rect_aa1=X1 # lateral coordinate extentVelocity_z_bc_rect_aa2=X2 # lateral coordinate extentVelocity_z_bc_rect_hw=0 # half-width of bc smoothing edgeVelocity_z_bc_rect_mag=0 # magnitude of bc (0 for 0 velocity)

6/ How plates influence the convection, the average temperature and the temperature distribution in the mantle?

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The mode of continental lithospheric extension, and the formation of sedimentary basins along continental margins and in the interior of continents, depend on the rheological stratification of the lithosphere. We know that this mechanical layering strongly depends on the temperature distribution with depth. We will use the particle-in-cell finite element code Ellipsis to create a two-dimensional model of the continental lithosphere under extension. We will explore the impacts various geotherms and extensional velocities have on lithospheric deformation.

GEOS-3104

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Lithospheric Extension

SECTION 1

The lithosphere is the stronger outer shell of the Earth. In contrast to the hotter asthenosphere which flows under small deviatoric stresses (a few MPa at the most), the litho-sphere can sustain up to a few 100’s MPa of deviatoric stresses. At the base of the lithosphere, the abrupt de-crease in shear-wave velocities is often used to map the lithosphere-asthenosphere boundary (LAB), i.e. the depth at which the temperature reaches 1330ºC (give or take a

few 10s ºC). In the asthenosphere, the temperature gradi-ent is low (about 0.3ºC per km) due to convective mixing. This section summarizes the rheological and thermal prop-erties of the continental lithosphere, and how these prop-erties are implemented in Ellipsis. These are the petro-physical or “internal” conditions that one must consider when designing the reference model (model at time t0).

1. Rheology2. Geotherm3. Density4. Modelling Lithospheric

Extension

The continental lithosphere

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RHEOLOGY-

For more detailed information on the rheology of the lithosphere please consult: geos3101.html

The lithosphere displays a strong rheological layering. It comprises a succession of stronger plastic layers where brit-tle deformation and strongly localizing fault zones dominates, and weaker more viscous layers where ductile flow dominates. Plastic behavior implies that rocks can sustain a finite deviatoric stress (the yield stress) before strain can accumulate. In contrast, viscous behavior implies that strain accumu-lates as soon as a deviatoric stress is applied, though strain rate can be infini-tesimally small. The yield strength envelope of the continental lithosphere documents the rheological layering. Because reverse faults are stronger than normal faults, this envelope varies with the tectonic regime (compression, extension or strike slip).

Depending on the geotherm and/or composition of the lower crust, and/or imposed strain rates, the crust can be either mechanically coupled or decou-pled from the underlying mantle depending on the rheological contrast be-tween the lower crust and the upper mantle.

Brittle deformation: At low temperatures, and/or high strain rates, and/or high-pore pressures, rocks behave elastically until the yield stress is reached, at which point they fail by fracturing. A frictional law (Mohr-Coulomb criteria or alternatively Byerlee law when Co=0) with a maximum shear stress yield (lim_yield_stress) approximates the brittle behaviour :

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τ yield =Min{ c0 + cpp( ) f ε( ), lim_ yield _ stress}

where c0 is the cohesion (yield stress at zero pressure), and cp is the pres-sure dependence of the yield stress (known as the coefficient of friction), and p is the pressure. The yield stress is reduced by a strain weakening factor (0 < f(ε) < 1) which expresses the dependence between the fault’s strength and the accumulated strain. At a certain depth, the yield stress reaches a limiting yield stress value, and the yield stress no longer in-creases with pressure. Recent estimates of the limiting yield stress for the continental lithosphere suggests a value between 100 and 300 MPa.

Strain Weakening: During brittle deformation, faults become weaker. This process, called strain weakening, is implemented in Ellipsis using the following power-law function (see figure):

where ε is the accumulated plastic (i.e. brittle) strain, ε0 is the saturation strain beyond which no further weakening takes place, εn is an expo-nent which controls how weakening evolves with plastic strain (a value of 0.1 will trigger a drastic weakening very early in the strain history, a value of 10 will delay strain weakening until accumulated plastic strain is several 100%), and εa is a maximum value of strain weakening factor (εa of 0.1 means that the coefficient of friction is reduced by a factor of 10). The figure on the right shows map views of experiments illustrat-ing the role of εn and ε0.

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Ductile deformation: When temperatures reach a significant fraction of the melting temperature (typically 0.3-0.4 x solidus temperature) defects in crystal lattice (i.e. dislocations) become mobile enough for stable steady-state creep to occur over long time scales. It is through this process that macroscopic tectonic fabrics (i.e. foliations and linea-tions) develop.

Newtonian viscosity is characterized by a simple linear relationship between the imposed deviatoric stress and the resulting strain rate. The constant of proportionality is the viscosity, a measure of the resistance of material to flow. However, viscosity is commonly strongly dependent on temperature and deviatoric stress; the flow is no longer strictly Newtonian. This behaviour can be approximated by the so-called power-law rheology (see geos3101.html).

Two different temperature-dependent viscosity models can be used in Ellipsis: The complex Arrhenius viscosity which depends on temperature, strain rate, and a few material-specific constants (A, n, Ea), and the simpler Frank-Kamenetskii approximation in which the viscosity depends on the temperature only. The Arrhenius viscosity (ηarr) is defined as:

where A is a scaling factor, n is the stress exponent, Ea is the activation en-ergy, R is the universal gas constant (8.314 J/mol.K), T is the temperature (K) and ε̇ is the strain rate.

# In RStudio Arrhenius viscosity can be plotted using:z<-seq(0, 60000, 1000)n<-3A<-5*10^-6*(10^6)^-nR<-8.314TempGrad<-900/40000edot<-1e-15plot(z, log(((A^-(1/n))/2*exp(190000/(n*R*TempGrad*z))*edot^((1-n)/n)), 10), xlim=c(1, 60000), ylim=c(18,25))

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ηarr =1A!

"#

$

%&

1n⋅Exp Ea

R×T!

"#

$

%&

1n⋅ ε

1−nn

Here however, we will only use the Frank-Kamenetskii viscosity approximation:

where η0 is the viscosity at 0 K, T is the temperature (in kelvin) and T1 is the sensitivity constant. Deviatoric stress σ (force per unit area) is what drives deformation. Strain ε measures the change in length. In a vis-cous fluid, the rate of strain ( .ϵ=dε/dt) is related to the applied deviatoric stress σ and the viscosity η (a measure of the resistance of material to flow): # # # # # # # # # # ## In RStudio the Frank-Kamenetskii viscosity can be plotted using:z<-seq(0, 60000, 1000)No<-2.5e28T1<-0.015TempGrad<-900/40000plot(z, log(No*exp(-T1*( TempGrad *z)), 10), xlim=c(0, 60000), ylim=c(18,25))

NB: To plot both the Arrhenius and the Frank-Kamenetski viscosities on the same graph use the command “par(new=TRUE)” between both plots

GEOTHERMFor more detailed information on the Earth geotherm please consult: geos3101.htmlThe temperature at the surface of the model is usually maintained at 20ºC. The 35-45 km thick crust has a depth in-dependent radiogenic heat production H (Qt in Ellipsis) typically between 0.7 to 1.5 10-6 W.m-3; we often approxi-mate that there is no heat production in the mantle. A more realistic heat production profile assumes an exponential decrease in radiogenic element with depth, with a depth-scale of about 10 km (i.e. radiogenic content is divided by 2.71 every 10 km). In Ellipsis, such a profile can only be approached through a multilayered model. The mantle heat flow Qm is on average 20 to 30 10-3 W.m-2. Thermal diffusivity (κ) of most rocks is about 1x10-6 m2.s-1 and heat

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η T( ) =η0 ⋅e−T1⋅T

σ =η ˙ ε

capacity Cp varies between 800 to 1000 J.kg-1.K-1. nb1: In Ellipsis, the parameter Heat_flux_z_bc_rect_mag is in fact Qm/(Cp*Density)nb2: In Ellipsis, the parameter Material_#_Qt is the radiogenic heat production per kg.

# To plot the Geotherm in RStudio:z<-seq(0, 60000, 1000)H<-0.7*1e-6Diff<-1e-6Density<-2720Cp<-1000K<-Diff*Density*CpTo<-293Qm<-0.0275zc<-45000plot(z, (-H/(2*K))*z^2+((Qm+H*zc)/K)*z+To)

DENSITIESThe average density of crustal rocks increases from about 2500 kg m-3 at the surface, to 3000 kg m-3 at the Moho. This increase reflects a change in lithologies from felsic to mafic, and increase in metamorphic grades with depth. In Ellipsis one can build a multilayered crust with layers of increasing densities with depth, or one can impose a nega-tive coefficient of thermal expansion (e.g. -5.54 10-5), a trick that ensures that densities increase towards the Moho. The consequence is that as deep crustal rocks are exhumed and cooled, their densities decrease as well. This compo-sitional increase in density is mitigated by a decrease in density due to temperature. The coefficient of thermal ex-pansion for most rocks is about 3 10-5, which imposes a drop in density up to ~100 kg m-3 at Moho temperatures. For the mantle, we typically assume a coefficient of thermal expansion 3 10-5, and a density at room temperature and pressure of 3370 kg m-3 for continental lithospheric mantle, and 3395 kg m-3 for continental asthenospheric man-tle.

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MODELLING LITHOSPHERIC EXTENSIONPlease read carefully the assessable Chapter 16: Rifting, Seafloor Spreading and Extensional Tectonics from the textbook Earth Structure: An Introduction to Structural Geology and Tectonics

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SECTION 2

In this exercise, we will modify an Ellipsis input file de-scribing a continental lithosphere under extension. We aim to explore the role temperature, extensional veloc-ity and rheology play on continental extension.The reference lithosphere is 270 km long and 90 km deep. It includes from top to bottom: 7.5 km of com-pressible air, 7.5 km of incompressible air, 15 km of up-per crust, 25 km of lower crust, and 35 km of mantle.

In this model the upper and lower crusts have exactly the same physical properties.

The temperature field is described in the file AA_GEOS3108_Temp_###.node_data. The geotherm is such that the Moho (at 40 km depth) is close to ###ºC.The exercise aims at evaluating the effect of changing the geotherm, extension velocity and rheology on the

1. Exercise2. Report Format3. Some useful R Scripts

Exploring continental extension

38

style of extension using a Ellipsis scripts provided to you. You will work in pairs and share the modelling workload with the rest of the class.

Group A: Run a model with a Moho at 610ºC, at total extensional velocity of 2mm per year.Group B: Same as Group A but with a much stronger crust and mantle (viscosity x 50).Group C: Run a model with a Moho at 610ºC, at total extensional velocity of 2cm per year.Group D: Same as Group C but with a much stronger crust and mantle (viscosity x 50). Group E: Run a model with a Moho at 720ºC, at total extensional velocity of 2mm per year.Group F: Same as Group E but with a much stronger crust and mantle (viscosity x 50). Group G: Run a model with a Moho at 720ºC, at total extensional velocity of 2cm per year. Group H: Same as Group G but with a much stronger crust and mantle (viscosity x 50). Group I: Run a model with a Moho at 830ºC, at total extensional velocity of 2mm per year.Group J: Same as Group I but with a much stronger lower and mantle (viscosity x 50).Group K: Run a model with a Moho at 830ºC, at total extensional velocity of 2cm per year. Group L: Same as Group K but with a much stronger crust and mantle (viscosity x 50).In the Ellipsis script, the initial background temperature file is called in section steady states and boundary conditions: ## STEADY STATES AND BOUNDARY CONDITIONS ## previous_temperature_file="AALithoExt-900_720.node_data"

The velocity of both vertical walls (nb: negative velocities means that the wall moves to the left) is set through:# Moving grid BC" BCmoveX0v=-3.171e-10" # BC: Boundary Condition, X: motion in the X direction, BCmoveX1v=3.171e-10"" # 0v: left vertical wall, 1v: right vertical wall

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Summarize your experiments by assembling - in a one-page PDF - graphic outputs at the following time intervals:Slow extension experiments: 0, 5, 10, 20, 40 myr. Fast extension experiments: 0, 0.5, 1, 2, 4 myr. nb1: Use the #.timelogs file to determine timing of each graphical output. nb2: Before sending the PDF reduce its size.

Send your result (1 page pdf) to [email protected] before Saturday 5 pm. The results will be collated and send to all students before the following Monday.Final report (1 per group of 2): Analyse the set of experiments and explain how and why geotherm, extension veloc-ity and rheology affect extensional strain patterns. Write your analysis into a report no longer than 4 pages long ex-cluding figures and appendices (appendices include your input files). Your report should be saved in a PDF format only. The Portable Document Format was invented to share documents across various platforms (PC, Mac, Linux etc). Before sending it as an email attachment, show you are smart by reducing its size, aim at a document less than 3-4 Mbytes.

Your report must include your name, an informative title, an introduction presenting the objective of the modelling and the model setup to achieve it, a section presenting the results, a section on the interpretation of the results, a con-clusion, and if appropriate a reference list.

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Ellipsis input file. In a scrolling window (eBook compatible) ...

41

################################################################## Ellipsis Input Script: Continental extension (Units: m, kg, sec, K (kelvin))## # ## Comments follow a '#' character and continue to end of line. # # # # ## No white space is allowed in comma-delimited lists.# # # # # # ## Boolean variables can be {on,off}={1,0} # # # # # # # # ## The model include 4 rock-types (upper and lower crust, fault, mantle# # # ## with brittle-ductile transition # # # # # # # # # ## Based on Rey et al., Geology, 2011. # # # # # # # # # ##################################################################

#--------------------------------------------------------------------------------# In this input files:#--------------------------------------------------------------------------------# Time limits and time steps# Computational grid, multigrid and particles# Output files# Initial conditions# Boundary conditions# Material properties# Over-ridding toggles

#--------------------------------------------------------------------------------# TIME LIMITS AND TIME STEPS#--------------------------------------------------------------------------------# maxtotstep=1000000# # Allowed max timesteps (default=1000000)# maxadvtime=9.5e15# # Alowed max elapsed time (default=1.0e18=30 Byr)

# fixed_timestep=3.0879e13# # overridden if > calculated max (default=0.0=ignored)# Geometry=cart2d # # # # problem geometry

#--------------------------------------------------------------------------------# COMPUTATIONAL GRID, MULTIGRID AND PARTICLES#--------------------------------------------------------------------------------# Cartesian grid dimension# dimenx=270000## dimenz=90000

# Particle (aka tracer) density## Tracer_rect=1# # # # # nb of rect.regions of different tracer densities (default=0)# Tracer_rect_density=9,10## # # tracer density (N x N per finest element, <=12)# Tracer_rect_x1=0.0000,0.0000# # # coordinate extent of region# Tracer_rect_x2=270000,270000 #

Ellipsis input file. In a print-friendly version ...

################################################################## Ellipsis Input Script: Continental extension (Units: m, kg, sec, K (kelvin))## # ## Comments follow a '#' character and continue to end of line. # # # # ## No white space is allowed in comma-delimited lists.# # # # # # ## Boolean variables can be {on,off}={1,0} # # # # # # # # ## The model include 4 rock-types (upper and lower crust, fault, mantle# # # ## with brittle-ductile transition # # # # # # # # # ## Based on Rey et al., Geology, 2011. # # # # # # # # # ##################################################################

#--------------------------------------------------------------------------------# In this input files:#--------------------------------------------------------------------------------# Time limits and time steps# Computational grid, multigrid and particles# Output files# Initial conditions# Boundary conditions# Material properties# Over-ridding toggles

#--------------------------------------------------------------------------------# TIME LIMITS AND TIME STEPS#--------------------------------------------------------------------------------# maxtotstep=1000000# # Allowed max timesteps (default=1000000)# maxadvtime=9.5e15# # Alowed max elapsed time (default=1.0e18=30 Byr)# fixed_timestep=3.0879e13# # overridden if > calculated max (default=0.0=ignored)# Geometry=cart2d # # # # problem geometry

#--------------------------------------------------------------------------------# COMPUTATIONAL GRID, MULTIGRID AND PARTICLES#--------------------------------------------------------------------------------# Cartesian grid dimension# dimenx=270000## dimenz=90000

# Particle (aka tracer) density## Tracer_rect=1# # # # # nb of rect.regions of different tracer densities (default=0)# Tracer_rect_density=9,10## # # tracer density (N x N per finest element, <=12)# Tracer_rect_x1=0.0000,0.0000# # # coordinate extent of region# Tracer_rect_x2=270000,270000 # # Tracer_rect_z1=0.0000,0.0000# Tracer_rect_z2=90000.,15000.

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# Multigrid stack# mgunitx=6# # # coarsest multigrid cell number, min =2# mgunitz=2# # # min=2# levels=5# # # number of multigrid levels (default=1)# pmg_levels=3# # number of pressure multigrid levels (default=1), pmg_levels (>=1)## Solver=multigrid# # multigrid# VERBOSE=on# # on the input values as they are read in (default=off)# verbose=on# # # verbose behaviour for the code (debugging) (default=off)# ## Solver and related matter# mg_cycle=1## # # style of multigrid cycle: 1 = V cycle, 2 = W cycle, ... (default=1)# vel_relaxations=2# # # maximum number of velocity loops (default=2)# piterations=100# # # maximum Uzawa iteration loops (default=100)# viterations=25# # # number of velocity iterations before checking convergence (default=251)# accuracy=1.e-4# # # desired accuracy of Uzawa algorithm (default=1.e-4)# delta_accuracy_factor=1## change in accuracy level->level (>~1 for nonN, <~1 for Newt)

#--------------------------------------------------------------------------------# OUPUT FILES#--------------------------------------------------------------------------------# datafile="MCC-Patrice"# # root name of output files, MCC for Metamorphic Core Complex# datatypes="Temp,Pres,depl"# # data at grid nodes, possible keywords:# # # # # Velx, Vely, Velz = x, y, z velocity# # # # # Temp = temperature, Pres = Pressure, depl = melt# particle_data=""# # particle output variables, possible keywords for particle_data:# # # # # Temp = temperature# # # # # Pres= pressure# # # # # Visc = viscosity# # # # # Edot = strain rate# # # # # StrP, StrT = integrated plastic/total strain# averages=""## # horizontally averaged values for output, possible keywords for averages:# # # # # Temp = temperature# # # # # Visc = viscosity# # # # # Velo = magnitude of velocity# timelog="Shfl,Bhfl"# # # time record of large-scale averages (ascii)# # # # # # # possible keywords for timelog:# # # # # # # Shfl, Bhfl = average surface and basal heat fluxes# # # ## storage_timesteps=1# # data writing interval (default=50)# checkpt_timesteps=1# # PPM (graphics) file writing interval (default=10)

# Specifications for graphical output files# PPM_files=1# # # # number of PPM files at each output step (default=1)# # # # # # # first PPM file is *.ppm0, etc.# PPM_height=512,512,512## # vertical size of output PPM file (default=256)# PPM_coloring=1,15,8# # # variable upon which to base colouring# # # # # # possible choices for PPM_coloring: (default=1)

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# # # # # # 1=temperature, 2=viscosity, 3=stress=visc*edot,# # # # # # 4= pressure, 7=strain rate, 8=accumulated strain# # # # # # 15=melt# PPM_coloring_autorange=1,1,1# # automatically scale colour (default=1)# PPM_coloring_min=0.0,0.0,0.00# # min value for color scale (default=0.0)# PPM_coloring_max=1.0,1.0,1.00# # max value for color scale (default=1.0)# PPM_show_strain=0.5,0.35,0.35# # colour according to actual strength change (default=0.0)# # # # # # (as opposed to current strain rate) NB: if YIELD is on,# # # # # # one of these two is always shown, using "strained" colour.# for extracting profiles/histories# Sampling_tracers=5# # # # number of sampling tracers (default=0)# Sampling_lagrangian=1,0,1,1,1## # (fixed) Eulerian=0, Lagrangian=1 (default=0)# Sampling_plot_num=0,0,0,0,0# # # # # PPM file in which profile is stored (default=0)# Sampling_x=67500,150000,120000,121000,122000# Sampling_z=0.000,00000.,0.0000,0.0000,0.0000# # # ## Sampling_field=1,1,1,6,13# # field to sample (default=0)# # 1=temperature, 2=x velocity, 3=z velocity, 4=nodal pressure, 5=strain rate, 6=stress=visc*edot 14=depl 15=melt# Sampling_dirn=2,2,2,2,2 # # profile direction (1=x, 2=z, 3=y) (default=0) # Sampling_normalize=1,1,1,1,1# # 0=unnormalized, 1=normalized (default=0) # Sampling_plot_min=0,0,0,0,0# # If not autoranging, then need a max/min for the scale# Sampling_plot_max=1.e4,1.e4,1.e4,20,20

# Sampling_R=0.8,1.0,1.0,0.15,0.0# Sampling_G=0.0,0.0,0.0,1.00,0.0## Sampling_B=0.0,0.0,0.0,0.15,0.0

#--------------------------------------------------------------------------------# INITIAL AND INTERNAL CONDITIONS # Distribution of lithologies, temperature and temperature anomalies ...#--------------------------------------------------------------------------------# gravacc=9.81# # gravitational acceleration (default=9.81)## Initial linear temperature gradient (Overwritten when previous temperature file is called)# toptbcval=293 # bottbcval=801#

# Any combination of rectangular, triangular, and circular regions may# be specified. Initial conditions are specified as follows:## class_shape_subclass=value# # where class is one of Material, Temp, Strain## shape is rect, circ, trgl## subclass is e.g. mag(nitude),x1(coordinate), or sometimes empty## value is a (comma-delimited list of) numerical assignment(s)

# Material distributions# Material_rect=6 # # # # # # Nb of rect. regions with different properties# Material_rect_property=0,1,2,3,4,1# # # tracer group name (properties/colour)# Material_rect_x1=0.0000,0.0000.,0.0000,0.0000,0.0000,120000# # # coordinates of tracer regions

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# Material_rect_x2=270000,270000.,270000,270000,270000,140000## # (successively overwritten if overlap)# Material_rect_z1=0.0000,5000.00,15000.,30000.,55000.,2500# Material_rect_z2=15000.,15000.0,30000.,55000.,90000.,7500 # Material_trgl=4 # Material_trgl_vert=3,3,3,3# Material_trgl_property=1,1,5,5 ## tracer group name (properties/colour)# Material_trgl_x1=105000,140000,120000,120000# # coordinates of tracer regions# Material_trgl_z1=7500.0,2500.0,15000.,15000.# Material_trgl_x2=120000,140000,142500,123750# # (successively overwritten if overlap)# Material_trgl_z2=2500.0,7500.0,37500.,15000.# Material_trgl_x3=120000,155000,146250,146250# # (successively overwritten if overlap)# Material_trgl_z3=7500.0,7500.0,37500.,37500.

# Temperature field distributions# Temp_rect=1# # # # # Number of rectangular temperature regions (default=0)# Temp_rect_x1=0.0000# # # # coordinates of region# Temp_rect_x2=270000# Temp_rect_z1=0.0000# Temp_rect_z2=15000.# Temp_rect_hw=0.000.# # # # half-width of smoothed edge# Temp_rect_mag=293.0# # # # magnitude of initial condition# Temp_rect_ovl=R# # # # # A/M/R = add/multiply/replace overlaps

# Temp_circ=0# # # # # Number of rectangular temperature regions (default=0)# Temp_circ_x=131250.# # # # coordinates of region# Temp_circ_z=50625.0# # # ## Temp_circ_rad=19825# # # # radius# Temp_circ_hw=0.0000# # # # half-width of smoothed edge# Temp_circ_mag=1.120# # # # magnitude of initial condition(1.12)# Temp_circ_ovl=M## # # # A/M/R = add/multiply/replace overlaps

# Initial condition files# Read in a file (ascii) containing nodal values of a field, in the form: previous_name_file=string# where `name' is the field and `string' is the data file name. Allowed fields are# `temperature', `pressure', `velocity', `plastic_strain', `pore_pressure', `porosity'.

previous_temperature_file="AAMCC-Temp.node_data"

#--------------------------------------------------------------------------------# BOUNDARY CONDITIONS#--------------------------------------------------------------------------------# Boundary conditions are specified as follows:## class_bc_shape_subclass=value# where class is one of Temp, Stress_x/z/y, Velocity_x/z/y, Heat_flux_x/z/y## bc indicates a boundary condition## shape is rect, circ

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## subclass is e.g. mag(nitude),aa1(coordinate), or sometimes empty## value is a (comma-delimited list of) numerical assignment(s)# Default conditions are tangential stress=0 and normal velocity=0 for all boundaries,# T=toptbcval on top and T=bottbcval on bottom, heat flux=0 on sides for all boundaries.# Note that Velocity, Temp respectively override Stress, Heat_flux on coincident boundaries. # Temp_bc_rect=1# # # # # number of rectangular bc ranges (surfaces) (default=0)# Temp_bc_rect_norm=Z,Z,Z# # # normal to plane of surface# Temp_bc_rect_icpt=0.000,15000.# # normal-axis intercept of bc plane# Temp_bc_rect_aa1=0.0000,0.0000# # lateral coordinate extent in 1st dimension# Temp_bc_rect_aa2=270000,270000# Temp_bc_rect_hw=0.00000,0.0000# # half-width of bc smoothing edge# Temp_bc_rect_mag=293.00,293.00# # magnitude of bc # Heat_flux_z_bc_rect=1# # # # number of rectangular bc ranges (surfaces) (default=0)# Heat_flux_z_bc_rect_norm=Z# # # normal to plane of surface# Heat_flux_z_bc_rect_icpt=90000# # normal-axis intercept of bc plane# Heat_flux_z_bc_rect_aa1=0.0000# # lateral coordinate extent in 1st dimension# Heat_flux_z_bc_rect_aa2=270000# Heat_flux_z_bc_rect_hw=0.00000# # half-width of bc smoothing edge# Heat_flux_z_bc_rect_mag=9.99e-9# # magnitude of bc ## Velocity_z_bc_rect=0# # # # number of rectangular bc ranges (surfaces) (default=0)# Velocity_z_bc_rect_norm=X# # # normal to plane of surface# Velocity_z_bc_rect_icpt=2# # # normal-axis intercept of bc plane# Velocity_z_bc_rect_aa1=0# # # lateral coordinate extent in 1st dimension# Velocity_z_bc_rect_aa2=1# Velocity_z_bc_rect_hw=0.0# # # half-width of bc smoothing edge# Velocity_z_bc_rect_mag=0.0# # # magnitude of bc## free_upper=on# # # free upper surface (default=off)# free_lower=on# # # free lower surface (default=off)

#--------------------------------------------------------------------------------# Fixed velocity boundary conditions#--------------------------------------------------------------------------------# BCmoveX0v=-3.171e-11 # # X0 left vertical wall, X1 right vertical wall. <0 velocity to left# BCmoveX1v=3.171e-11 # #

#--------------------------------------------------------------------------------## MATERIAL PROPERTIES#--------------------------------------------------------------------------------# The first material is material 0, other materials count upwards (material 0, material 1, ... ).# Properties are assigned to tracer groups according to Material_rect_property=0,1,2,... etc.

# Different_materials=6# # Number of different materials listed# # # #

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#--------------------------------------------------------------------------------## Air

# Material_0_density=2# # # density (default=1.0)# Material_0_Bulk_visc=2.0# # >0 compressible material

# Rheological model# Material_0_rheol_T_type=1# # rheological temperature-dependence model (default=2)# # # # # # # (1) visc=N0*exp(-T1*T)## (Frank-Kamenetski)# # # # # # # (2) visc=N0*exp{ (E+Z*z)/(T1*(T+T0)) } (Arrhenius)

# Material_0_viscN0=5e18## # N0 is reference viscosity (default=1.0)# Material_0_viscT1=0.0# # # T1 (default=1.0): (1) sensitivity to temp OR (2) Boltzman constant

# Thermal parameters# Material_0_therm_exp=0.0# # thermal expansion coefficient (default=0.0)# Material_0_therm_diff=22e-6# # thermal diffusivity (default=0.0)# Material_0_Cp=1000# # # isobaric heat capacity (default=1.0)# Material_0_Qt=0.0# # # # internal heating rate by mass (default=0.0)

# Colouring# Material_0_Red=0.8,1.0# # # # RGB values for "cold" material (list one per PPM file)# Material_0_Green=0.8,1.0# # # ("hot" and "cold" are determined from T extremes)# Material_0_Blue=1.0,1.0# Material_0_Opacity=1.4,1.4# # # opacity for "cold" material (negative=off)# Material_0_Red_hot=0.76,1.0# # # values for "hot" material# Material_0_Green_hot=0.92,1.0# Material_0_Blue_hot=1.0,1.0# Material_0_Opacity_hot=1.4,1.4# Material_0_Red_strained=0.76,0.76# # values for strained material# Material_0_Green_strained=0.92,0.92# Material_0_Blue_strained=1.0,1.0# Material_0_Opacity_strained=1.4,1.4

## Hard air # # Material_1_density=2# # # density (default=1.0)# Material_1_Bulk_visc=-1e3# # >0 compressible material

# Rheological model# Material_1_rheol_T_type=1# # rheological temperature-dependence model (default=2)# # # # # # # (1) visc=N0*exp(-T1*T)## (Frank-Kamenetski)# # # # # # # (2) visc=N0*exp{ (E+Z*z)/(T1*(T+T0)) } (Arrhenius)# # # # # # # where z=depth, T=temperature, T0=0 because all temperature are in Kelvin# Material_1_viscN0=5e19## # N0 in viscosity models (default=1.0)# Material_1_viscT1=0.0# # # T1 in viscosity models (default=1.0): (1) sensitivity to temp, (2) Boltzman constant

# Thermal parameters

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# Material_1_therm_exp=0.0# # thermal expansion coefficient (default=0.0)# Material_1_therm_diff=22e-6# # thermal diffusivity (default=0.0)# Material_1_Cp=1000# # # isobaric heat capacity (default=1.0)# Material_1_Qt=0.0# # # # internal heating rate by mass (default=0.0)

# Colouring# Material_1_Red=0.6,1.0# # # # RGB values for "cold" material (list one per PPM file)# Material_1_Green=0.6,1.0# # # ("hot" and "cold" are determined from T extremes)# Material_1_Blue=1.0,1.0# Material_1_Opacity=1.4,1.4# # # opacity for "cold" material (negative=off)# Material_1_Red_hot=0.76,1.0# # # values for "hot" material# Material_1_Green_hot=0.92,1.0# Material_1_Blue_hot=1.0,1.0# Material_1_Opacity_hot=1.4,1.4# Material_1_Red_strained=0.76,0.76# # values for strained material# Material_1_Green_strained=0.92,0.92# Material_1_Blue_strained=1.0,1.0# Material_1_Opacity_strained=1.4,1.4

#--------------------------------------------------------------------------------## Upper crust Material_2_phases=1# # # number of unique phases (first phase is phase 0) (default=1)# # # # # # then visc = { sum(1/visc_n) }^(-1) Material_2_density=2720# # density (default=1.0)# Material_2_Bulk_visc=-1e3# # <0 incompressible material

# Rheological modelMaterial_2_rheol_T_type=2,2# # rheological temperature-dependence model (default=2)# # # # # # # (1) visc=N0*exp(-T1*T)## (Frank-Kamenetski)# # # # # # # (2) visc=N0*exp{ (E+Z*z)/(T1*(T+T0)) } (Arrhenius)

Material_2_viscN0=2.5e22,2.777e7# # N0 reference viscosity (default=1.0); Arrhenius: N0=1/(2 A) Material_2_viscT1=8.314,1# # # T1 (default=1.0): (1) sensitivity to temp OR (2) Boltzman constant Material_2_viscT0=0.0,0.0# # # T0=0 when all temperature are in Kelvin, otherwise 273 Material_2_viscZ=0.0,0.0# # # Z: Volume of activation Material_2_viscE=1.9e5,22853# # # E: activation energy Material_2_viscTmax=1273,1273# # Clipping maximum and minimum T to use in calculating viscosity Material_2_viscTmin=523,523# # # (default=1.e32,0.0) Material_2_sdepv_expt=3,1# # # exponent "s" in stress dependance of viscosity; Arrhenius: n# # # # # # visc = edot^[(1-s)/s] * (N0*exp{ (E+Z*z)/(T1*(T+T0)))^(1/s) (default=1.0)# # # # # # ## # # # # ## Thermal parameters Material_2_therm_exp=0.0# # # # thermal expansion coefficient (default=0.0) Material_2_therm_diff=1e-6# # # # thermal diffusivity (default=0.0) Material_2_Cp=1000# # # # # isobaric heat capacity (default=1.0) Material_2_Qt=5.64e-10# # # # # internal heating rate by mass (default=0.0)#Material_2_melt_on=0# # # # #Melt model, 1 turns melting on

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#Material_2_depl_T_type=1# # # Depletion model, 1=pressure, 2=depthMaterial_2_depl_exp=0.13# # # Density change of depleted materialMaterial_2_latent_heat=250# # # Latent heat typically 200 to 450 kJ kg-1 K-1Material_2_melt_extract=1.0# # # Melt fraction at which melt migratesMaterial_2_depl_visc_change=0# #Increase in viscosity of residue due to depletion # # # # # # # (visc = visc + depl*depl_vis_change) ... # # # # # #Material_2_melt_total_visc_change=1000 # #Viscosity reduction factor due to presence of partial melt # # # # # # # #This is difference in viscosity at solidus and liquidus # # # # # # # #- assumes linear decreaseMaterial_2_Melt_frac_start_visc_change=0.20# #Melt fraction at which viscosity starts to decreaseMaterial_2_Melt_frac_end_visc_change=0.30# #Melt fraction at which viscosity stops decreasing

# # # # # # Need when adiabat gradient is important# # # # # # Not so much for crustal melting# # # # # # Only for depletion model 2 i.e. depth-dependent solidus/liquidus# # # ## # # # #Material_2_dim_d0=0# # #Starting depth: depth(dimen) = d0 + d1*z(nondim) Material_2_dim_d1=1# # # d1 is aboveMaterial_2_dim_T0=0# # # Starting temp: T(dimen) = T0 + T1*T(nondim) + ag*depth(dimen) Material_2_dim_T1=1# # # ~ Potential temperature (if T0=0)Material_2_dim_ag=0.0# # # Adiabatic gradient, 0.3-0.5 for earth (deep-shallow resp).#Material_2_sp0=993# # # Solidus of material, based on third order, 0.619Material_2_sp1=-1.2e-7# # # polynomial fit of data:Material_2_sp2=1.2e-16# # # s=sp0+sp1*p+sp2*p^2+sp3*p^3Material_2_sp3=0.0

Material_2_lp0=1493# # #Liquidus, defined same wayMaterial_2_lp1=-1.2e-7Material_2_lp2=1.2e-16Material_2_lp3=0.0## Plastic behavior: Mohr-coulomb# stress = visc * (strain rate)# post-yield visc = (yield stress)/(strain rate)# yield stress = max{ (B0 + Bz*z + Bp*p) * f(e) , minimum yield stress }# yield stress = min{ (B0 + Bz*z + Bp*p) * f(e) , maximum yield stress }# yield stress = yield stress * 0.001, if (-p) > Bc# e=strain, z=depth, p=pressure# f(e) = decreasing power law for 0 < e < E0, f(e) = constant for e > E0# f(e) = 1 - (1-Ea)*(e/E0)^En **** strain localisation ***

Material_2_yield_stress_minimum=1.e-32# # # minimum yield stress for plastic deformation (default=1.e-32) Material_2_yield_stress_maximum=2.5e8# # # maximum yield stress for semi-brittle effect (default=1.e32) Material_2_yield_stress_B0=1.6e7# # # # "cohesion" B0 in above eqn (default=1.e32) [MPa] Material_2_yield_stress_Bz=0.0# # # # "friction coefficient" Bz in above eqn (default=0.0)

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Material_2_yield_stress_Bp=0.44# # # # "friction coefficient" Bp in above eqn (default=0.0) Material_2_yield_stress_Bc=1.e32# # # # tension cutoff Bc in above law (default=1.e32) Material_2_yield_stress_Ea=0.2# # # # ratio Ea = f(0,0)/f(E0,0) (default=1.0,range=[0,1])0 Material_2_yield_stress_E0=0.5 # # # # strain weakening E0 (default=1.e32) Material_2_yield_stress_En=1# # # # # exponent En in f(e), e<E0 (default=0.0)# # # # # ## Colouring Material_2_Red=0.7,1,0.13# # # RGB values for "cold" material (list one per PPM file) Material_2_Green=0.6,1,0.18# # # ("hot" and "cold" are determined from T extremes) Material_2_Blue=0.1,1,0.40 Material_2_Opacity=1.4,1.4,1.4## # opacity for "cold" material (negative=off) Material_2_Red_hot=0.0,1.0,0.25# # values for "hot" material Material_2_Green_hot=0.0,0,0.61 Material_2_Blue_hot=0.0,0,0.90 Material_2_Opacity_hot=1.4,1.4,1.4 Material_2_Red_strained=0.0,1.0,0.97# # # values for strained material Material_2_Green_strained=0.0,0.0,0.93 Material_2_Blue_strained=1.0,0.0,0.18 Material_2_Opacity_strained=1.4,0.4,1.4 Material_2_Red_depl=1.00,1.00,0# # values for depleted material Material_2_Green_depl=0.15,1.00,0 Material_2_Blue_depl=0.00,1.00,1 Material_2_Opacity_depl=1.4,1.4,1.4

#--------------------------------------------------------------------------------## Lower crust # Material_3_density=2720# # # density (default=1.0) # Material_3_Bulk_visc=-1e3# # # <0 incompressible material

# Rheological modelMaterial_3_rheol_T_type=2,2# # rheological temperature-dependence model (default=2)# # # # # # # (1) visc=N0*exp(-T1*T)## (Frank-Kamenetski)# # # # # # # (2) visc=N0*exp{ (E+Z*z)/(T1*(T+T0)) } (Arrhenius)

Material_3_viscN0=2.5e22,2.777e7# # N0 in viscosity (default=1.0); Arrhenius: N0=1/(2 A) Material_3_viscT1=8.314,1# # # T1 (default=1.0): (1) sensitivity to temp OR (2) Boltzman constant Material_3_viscT0=0.0,0.0# # # T0=0 when all temperature are in Kelvin, otherwise 273 Material_3_viscZ=0.0,0.0# # # Z: Volume of activation Material_3_viscE=1.9e5,22853# # # E: activation energy Material_3_viscTmax=1273,1273# # Clipping maximum and minimum T to use in calculating viscosity Material_3_viscTmin=523,523# # # (default=1.e32,0.0) Material_3_sdepv_expt=3,1# # # exponent "s" in stress dependance of viscosity; Arrhenius: n# # # # # # # visc = edot^[(1-s)/s] * (N0*exp{ (E+Z*z)/(T1*(T+T0)))^(1/s) (default=1.0)# # # # # #

# Thermal parameters Material_3_therm_exp=0.0# # # thermal expansion coefficient (default=0.0) Material_3_therm_diff=1e-6# # # thermal diffusivity (default=0.0) Material_3_Cp=1000# # # # isobaric heat capacity (default=1.0)

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Material_3_Qt=5.64e-10# # # # internal heating rate by mass (default=0.0)#Material_3_melt_on=0# # # #Melt model, 1 turns melting on#Material_3_depl_T_type=1# # #Depletion model, 1=pressure, 2=depthMaterial_3_depl_exp=0.13# # # Density change of depleted materialMaterial_3_latent_heat=250# # # Latent heat typically 200 to 450 kJ kg-1 K-1Material_3_melt_extract=1.0# # # Melt fraction at which melt migratesMaterial_3_depl_visc_change=0# #Increase in viscosity of residue due to depletion # # # # # # # (visc = visc + depl*depl_vis_change)Material_3_melt_total_visc_change=1000 # #Viscosity reduction factor due to presence of partial melt # # # # # # # #This is difference in viscosity at solidus and liquidus # # # # # # # #- assumes linear decrease

Material_3_Melt_frac_start_visc_change=0.20# #Melt fraction at which viscosity starts to decreaseMaterial_3_Melt_frac_end_visc_change=0.30# #Melt fraction at which viscosity stops decreasing

# # # # # # Need when adiabat gradient is important# # # # # # Not so much for crustal melting# # # # # # Only for depletion model 2 i.e. depth-dependent solidus/liquidus# # # # # # # # # ## Material_3_dim_d0=0# # #Starting depth: depth(dimen) = d0 + d1*z(nondim)# Material_3_dim_d1=1# # # d1 is above# Material_3_dim_T0=0# # # Starting temp: T(dimen) = T0 + T1*T(nondim) + ag*depth(dimen)# Material_3_dim_T1=1# # # Potential temperature (if T0=0)# Material_3_dim_ag=0.0# # # Adiabatic gradient, 0.3-0.5 for earth (deep-shallow resp).## Material_3_sp0=993# # # Solidus of material, based on third order, 0.619# Material_3_sp1=-1.2e-7# # # polynomial fit of data:# Material_3_sp2=1.2e-16# # # s=sp0+sp1*p+sp2*p^2+sp3*p^3# Material_3_sp3=0.0

# Material_3_lp0=1493# # #Liquidus, defined same way# Material_3_lp1=-1.2e-7# Material_3_lp2=1.2e-16# Material_3_lp3=0.0

# Plastic behavior: Mohr-coulomb# stress = visc * (strain rate)# post-yield visc = (yield stress)/(strain rate)# yield stress = max{ (B0 + Bz*z + Bp*p) * f(e) , minimum yield stress }# yield stress = min{ (B0 + Bz*z + Bp*p) * f(e) , maximum yield stress }# yield stress = yield stress * 0.001, if (-p) > Bc# e=strain, z=depth, p=pressure# f(e) = decreasing power law for 0 < e < E0, f(e) = constant for e > E0# f(e) = 1 - (1-Ea)*(e/E0)^En **** strain localisation ***

# Material_3_yield_stress_minimum=1.e-32# # minimum yield stress for plastic deformation (default=1.e-32) # Material_3_yield_stress_maximum=2.5e8# # # maximum yield stress for semi-brittle effect (default=1.e32)

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# Material_3_yield_stress_B0=1.6e7# # # # "cohesion" B0 in above eqn (default=1.e32) # Material_3_yield_stress_Bz=0.0# # # # "friction coefficient" Bz in above eqn (default=0.0) # Material_3_yield_stress_Bp=0.44# # # # "friction coefficient" Bp in above eqn (default=0.0) # Material_3_yield_stress_Bc=1.e32# # # # tension cutoff Bc in above law (default=1.e32) # Material_3_yield_stress_Ea=0.2# # # # ratio Ea = f(0,0)/f(E0,0) (default=1.0,range=[0,1])0 # Material_3_yield_stress_E0=0.5 # # # # strain weakening E0 (default=1.e32) # Material_3_yield_stress_En=1# # # # # exponent En in f(e), e<E0 (default=0.0)

# Colouring# Material_3_Red=1.0,1.0,0.13# # # # # RGB values for "cold" material (list one per PPM file)# Material_3_Green=0.6,0.6,0.18# # # # # ("hot" and "cold" are determined from T extremes)# Material_3_Blue=1.0,1.0,0.40# Material_3_Opacity=1.4,1.4,1.4## # # # opacity for "cold" material (negative=off)# Material_3_Red_hot=0.9,1,0.25## # # # values for "hot" material# Material_3_Green_hot=0.2,1,0.61# Material_3_Blue_hot=0.0,1,0.90# Material_3_Opacity_hot=1.4,1.4,1.4# Material_3_Red_strained=0.05,1,0.97# # # # values for strained material# Material_3_Green_strained=0.15,1,0.93# Material_3_Blue_strained=0.9,1,0.18# Material_3_Opacity_strained=1.0,1.4,0.18# Material_3_Red_depl=1.00,1.00,-1# # # # values for depleted material # Material_3_Green_depl=0.15,1.00,-1 # Material_3_Blue_depl=0.00,1.00,-1 # Material_3_Opacity_depl=1.4,1.4,-1

#--------------------------------------------------------------------------------## Upper mantle lithosphere # Material_4_density=3370# # # density (default=1.0) # Material_4_Bulk_visc=-1e3# # # <0 incompressible material

# Rheological model # Material_4_rheol_T_type=2# # # rheological temperature-dependence model (default=2)# # # # # # # # (1) visc=N0*exp(-T1*T)## (Frank-Kamenetski)# # # # # # # # (2) visc=N0*exp{ (E+Z*z)/(T1*(T+T0)) } (Arrhenius)

# Material_4_viscN0=1.78e12# # # N0 in viscosity (default=1.0); Arrhenius: N0=1/(2 A) # Material_4_viscT1=8.314## # # T1 (default=1.0): (1) sensitivity to temp OR (2) Boltzman constant # Material_4_viscT0=0.0# # # # T0=0 when all temperature are in Kelvin, otherwise 273 # Material_4_viscZ=0.0# # # # Z: volume of activation # Material_4_viscE=520000# # # E: activation energy # Material_4_viscTmax=2093# # #Clipping maximum and minimum T to use in calculating viscosity # Material_4_viscTmin=873# # # (default=1.e32,0.0) # Material_4_sdepv_expt=3.0# # # exponent "s" in stress dependance of viscosity; Arrhenius: n# # # # # # # # visc = edot^[(1-s)/s] * (N0*exp{ (E+Z*z)/(T1*(T+T0)))^(1/s) (default=1.0)# Thermal parameters # Material_4_therm_exp=2.8e-5# # # thermal expansion coefficient (default=0.0) # Material_4_therm_diff=1e-6# # # thermal diffusivity (default=0.0)

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# Material_4_Cp=1000# # # # isobaric heat capacity (default=1.0) # Material_4_Qt=0.0# # # # # internal heating rate by mass (default=0.0)

# Plastic behavior: Mohr-coulomb# stress = visc * (strain rate)# post-yield visc = (yield stress)/(strain rate)# yield stress = max{ (B0 + Bz*z + Bp*p) * f(e) , minimum yield stress }# yield stress = min{ (B0 + Bz*z + Bp*p) * f(e) , maximum yield stress }# yield stress = yield stress * 0.001, if (-p) > Bc# e=strain, z=depth, p=pressure# f(e) = decreasing power law for 0 < e < E0, f(e) = constant for e > E0# f(e) = 1 - (1-Ea)*(e/E0)^En **** strain localisation ***

# Material_4_yield_stress_minimum=1.e-32# # minimum yield stress for plastic deformation (default=1.e-32) # Material_4_yield_stress_maximum=4e8# # maximum yield stress for semi-brittle effect (default=1.e32) # Material_4_yield_stress_B0=2e8# # # "cohesion" B0 in above eqn (default=1.e32) # Material_4_yield_stress_Bz=0.0# # # "friction coefficient" Bz in above eqn (default=0.0) # Material_4_yield_stress_Bp=0.577# # # "friction coefficient" Bp in above eqn (default=0.0) # Material_4_yield_stress_Bc=1.e32# # # tension cutoff Bc in above law (default=1.e32) # Material_4_yield_stress_Ea=0.2# # # ratio Ea = f(0,0)/f(E0,0) (default=1.0,range=[0,1])0 # Material_4_yield_stress_E0=0.5 # # # strain weakening E0 (default=1.e32) # Material_4_yield_stress_En=1.0# # # exponent En in f(e), e<E0 (default=0.0)# # # # # ## Colouring # Material_4_Red=0.8,1,0.13# # # # RGB values for "cold" material (list one per PPM file) # Material_4_Green=0.3,1,0.18# # # # ("hot" and "cold" are determined from T extremes) # Material_4_Blue=0.8,1,0.40 # Material_4_Opacity=0.4,0.4,1.4## # # opacity for "cold" material (negative=off) # Material_4_Red_hot=0.4,1.0,0.25# # # values for "hot" material # Material_4_Green_hot=0.0,0.0,0.61 # Material_4_Blue_hot=0.0,0.0,0.90 # Material_4_Opacity_hot=1.4,1.4,1.4 # Material_4_Red_strained=0.0,0.0,0.97## # values for strained material # Material_4_Green_strained=0.0,0.0,0.93 # Material_4_Blue_strained=1.0,1.0,0.18 # Material_4_Opacity_strained=1.4,0.2,1.4

#--------------------------------------------------------------------------------### Fault Material_5_density=2720,2720## # # density (default=1.0) Material_5_Bulk_visc=-1e3,-1e3# # # <0 incompressible material

# Rheological modelMaterial_5_rheol_T_type=2,2# # # # rheological temperature-dependence model (default=2)# # # # # # # # (1) visc=N0*exp(-T1*T)## (Frank-Kamenetski)# # # # # # # # (2) visc=N0*exp{ (E+Z*z)/(T1*(T+T0)) } (Arrhenius) Material_5_viscN0=2.5e20,2.777e7# # # N0: reference viscosity (default=1.0); Arrhenius: N0=1/(2 A) Material_5_viscT1=8.314,1# # # # T1 (default=1.0): (1) sensitivity to temp OR (2) Boltzman constant

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Material_5_viscT0=0.0,0.0# # # # T0=0 when all temperature are in Kelvin, otherwise 273 Material_5_viscZ=0.0,0.0# # # # Z: volume of activation Material_5_viscE=1.9e5,22853# # # # E: activation energy Material_5_viscTmax=1273,1273# # # Clipping maximum and minimum T to use in calculating viscosity Material_5_viscTmin=523,523# # # # (default=1.e32,0.0) Material_5_sdepv_expt=3,1# # # # exponent "s" in stress dependance of viscosity; Arrhenius: n# # # # # # # # visc = edot^[(1-s)/s] * (N0*exp{ (E+Z*z)/(T1*(T+T0)))^(1/s) (default=1.0)# # # # # ## # # # # # ## Thermal parameters Material_5_therm_exp=0.0# # # thermal expansion coefficient (default=0.0) Material_5_therm_diff=1e-6# # # thermal diffusivity (default=0.0) Material_5_Cp=1000# # # # isobaric heat capacity (default=1.0) Material_5_Qt=5.64e-10# # # # internal heating rate by mass (default=0.0) #Material_5_melt_on=0# # # #Melt model, 1 turns melting on#Material_5_depl_T_type=1# # #Depletion model, 1=pressure, 2=depthMaterial_5_depl_exp=0.13# # # Density change of depleted materialMaterial_5_latent_heat=250# # # Latent heat typically 200 to 450 kJ kg-1 K-1Material_5_melt_extract=1.0# # # Melt fraction at which melt migratesMaterial_5_depl_visc_change=0# # Increase in viscosity of residue due to depletion Material_2_melt_total_visc_change=1000 # #Viscosity reduction factor due to presence of partial melt # # # # # # # #This is difference in viscosity at solidus and liquidus # # # # # # # #- assumes linear decrease

Material_5_Melt_frac_start_visc_change=0.20# #Melt fraction at which viscosity starts to decreaseMaterial_5_Melt_frac_end_visc_change=0.30# #Melt fraction at which viscosity stops decreasing

# # # # # # Need when adiabat gradient is important# # # # # # Not so much for crustal melting# # # # # # Only for depletion model 2 i.e. depth-dependent solidus/liquidus# # # # # # # # # ## Material_5_dim_d0=0# # #Starting depth: depth(dimen) = d0 + d1*z(nondim) # Material_5_dim_d1=1# # # d1 is above# Material_5_dim_T0=0# # # Starting temp: T(dimen) = T0 + T1*T(nondim) + ag*depth(dimen) # Material_5_dim_T1=1# # # Potential temperature (if T0=0)# Material_5_dim_ag=0.0# # # Adiabatic gradient, 0.3-0.5 for earth (deep-shallow resp).

# Material_5_sp0=993# # # Solidus of material, based on third order, 0.619# Material_5_sp1=-1.2e-7# # # polynomial fit of data:# Material_5_sp2=1.2e-16# # # s=sp0+sp1*p+sp2*p^2+sp3*p^3# Material_5_sp3=0.0

# Material_5_lp0=1493# # #Liquidus, defined same way# Material_5_lp1=-1.2e-7# Material_5_lp2=1.2e-16# Material_5_lp3=0.0## Plastic behavior: Mohr-coulomb

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# stress = visc * (strain rate)# post-yield visc = (yield stress)/(strain rate)# yield stress = max{ (B0 + Bz*z + Bp*p) * f(e) , minimum yield stress }# yield stress = min{ (B0 + Bz*z + Bp*p) * f(e) , maximum yield stress }# yield stress = yield stress * 0.001, if (-p) > Bc# e=strain, z=depth, p=pressure# f(e) = decreasing power law for 0 < e < E0, f(e) = constant for e > E0# f(e) = 1 - (1-Ea)*(e/E0)^En **** strain localisation ***

# Material_5_yield_stress_minimum=1.e-32# # minimum yield stress for plastic deformation (default=1.e-32) # Material_5_yield_stress_maximum=2.5e8# # maximum yield stress for semi-brittle effect (default=1.e32) # Material_5_yield_stress_B0=1.6e6# # # "cohesion" B0 in above eqn (default=1.e32) [MPa] # Material_5_yield_stress_Bz=0.0# # # "friction coefficient" Bz in above eqn (default=0.0) # Material_5_yield_stress_Bp=0.044# # # "friction coefficient" Bp in above eqn (default=0.0) # Material_5_yield_stress_Bc=1.e32# # # tension cutoff Bc in above law (default=1.e32) # Material_5_yield_stress_Ea=0.2# # # ratio Ea = f(0,0)/f(E0,0) (default=1.0,range=[0,1])0 # Material_5_yield_stress_E0=0.5 # # # strain weakening E0 (default=1.e32) # Material_5_yield_stress_En=1# # # # exponent En in f(e), e<E0 (default=0.0)# # # # ## Colouring# Material_5_Red=0.0,1.0,0.13# # # # RGB values for "cold" material (list one per PPM file)# Material_5_Green=0.0,0.0,0.18# # # # ("hot" and "cold" are determined from T extremes)# Material_5_Blue=2.0,0.0,0.4# Material_5_Opacity=1.4,1.4,1.4## # # opacity for "cold" material (negative=off)# Material_5_Red_hot=2.0,1.0,0.25# # # values for "hot" material# Material_5_Green_hot=0.0,1.0,0.61# Material_5_Blue_hot=0.0,0.0,0.90# Material_5_Opacity_hot=1.4,1.4,1.4# Material_5_Red_strained=0.52,.52,0.97# # values for strained material# Material_5_Green_strained=0.90,0.9,0.93# Material_5_Blue_strained=1.5,1.5,0.18# Material_5_Opacity_strained=1.4,1.4,1.4

#--------------------------------------------------------------------------------# OVER-RIDING TOGGLES #--------------------------------------------------------------------------------# TDEPV=on# # # use temperature-dependent rheological parameters (default=on)# # # # # (off is faster than turning all viscosity values to 1)# VMAX=on# # # use maximum viscosity (default=off)# VMIN=on# # # use minimum viscosity (default=off)# visc_max=5e23# # maximum, minimum viscosity cut-offs (no defaults)# visc_min=5e18# SDEPV=on# # # use stress dependence of viscosity (default=off)# YIELD=on# # # yield stress parameters on/off (default=off)# ELASTICITY=off# GRDEPV=off# # # use grain size dependence of viscosity (default=off)

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#===============================================================# This R script calculates and plot the continental geotherm and rheological profiles#===============================================================# ====== Geotherm =========library(mosaic)# Radiogenic heat productionH<-0.7*10^-6 # Thermal conductivity (diffusivity x density x heat capacity)K<-(10^-6)*2650*1000#Mantle heat flowQm<-0.020#Surface TemperatureTo<-293 #Thickness of the continental crustzc<-42000 #Temperature defining the base of the lithosphere Tl<-1603

#Crustal geothermTemp_crust<-function(z){(-H/(2*K))*z^2+(((Qm+H*zc)/K))*z+To}#Geotherm in the lithospheric mantleTemp_lithos_mantle<-function(z){(Qm/K)*(z-zc)+Temp_crust(zc)} #Depth of the base of the lithosphere zl<-findZeros(Temp_lithos_mantle(z) - Tl ~ z, z.lim=range(100000, 200000)) #Geotherm in the asthenosphere (called the adiabat because it follows the adiabatic gradient)Temp_asthenos<-function(z){Tl+0.0003*(z-zl)} #Geotherm in the first top 200 km …Geotherm<-function(z){ifelse(z<0, To, ifelse(z<=zc, Temp_crust(z), ifelse(z<=zl, Temp_lithos_mantle(z), Tl)))}#Plot of the geotherm aboveplotFun(Geotherm(z) ~ z, z.lim=range(0, 200000))

# ====== Rheology =========#Reference strain rate edot<-1e-15#Coehesion and Coefficient of friction (include pore pressure)Co<-20e6

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Bp<-0.268#Limiting yield stress (pseudo plasticity: at a certain depth, the yield stress no longer increases)limiting_yield_stress_sclm<-300e6limiting_yield_stress_crust<-150e6

Boltzman<-8.314g<-9.81#Arrhenius stress exponents (uc: upper crust, mant: mantle)n_mant<-3.0 n_uc<-3.2

#Arrhenius pre-stress constantsA_uc<-5*10^-6*(10^6)^-n_ucA_mant<-7*10^7*(10^6)^-n_mant

#Arrhenius activation energiesE_mant<-520000 E_uc<-244000

# Reference viscosity before applied strain rateNo_mant<-1/(2^(n_mant)*A_mant) No_uc<-1/(2^(n_uc)*A_uc)

# Densitiesrhocrust_o<-2700rhobasalt_o<-2900rhosclm_o<-3370 rhoasthe_o<-3395

#Depth of the base of the oceanic lithosphere (useful to calculate the average density of MOR column)zlol<-150000

# Coefficient of thermal expansion alpha<-0.00003

# Average densities corrected for temperaturerhocrust<-(rhocrust_o+(rhocrust_o-rhocrust_o*(alpha*Geotherm("zc"))))/2

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rhosclm<-((rhosclm_o-rhosclm_o*(alpha*Geotherm("zc")))+(rhosclm_o-rhosclm_o*(alpha*Geotherm("zl"))))/2 rhoasthe<-(rhoasthe_o-rhoasthe_o*(alpha*Geotherm("zlol")))

# Density profile of continent and oceanic lithospheresDensCont<-function(z){ifelse(z<=0, 0,ifelse(z<=zc, rhocrust, ifelse(z<=zl, rhosclm, ifelse(z>zl, rhoasthe, rhoasthe))))}

# Lithostatic pressure profile along continentPressure<-function(z){ifelse(z<=zc, rhocrust*g*z, ifelse(z<=zl, rhocrust*g*zc+rhosclm*g*(z-zc), rhocrust*g*zc+rhosclm*g*(zl-zc)+rhoasthe*g*(z-zl)))}

#Arrhenius viscosity profilesN_uc<-function(z, edot){No_uc*exp(E_uc/(Boltzman*Geotherm(z)))}Visc_uc<-function(z, edot){(edot^((1-n_uc)/n_uc))*N_uc(z)^(1/n_uc)}

N_mant<-function(z, edot){No_mant*exp(E_mant/(Boltzman*Geotherm(z)))}Visc_mant<-function(z, edot){(edot^((1-n_mant)/n_mant))*N_mant(z)^(1/n_mant)}

Plastic<-function(z){Co+Pressure(z)*Bp}

RheoLitho<-function(z, edot){ifelse(z<=zc, min(Plastic(z), Visc_uc(z, edot)*edot, limiting_yield_stress_crust), min(Plastic(z),Visc_mant(z, edot)*edot, limiting_yield_stress_sclm))}

plotFun(RheoLitho(z, edot) ~z, edot=1e-14, z.lim=range(0, zl), type='h', alpha=.5, lwd=4, col='red', xlab="Depth (m)", ylab="Deviatoric Stress")plotFun(RheoLitho(z, edot) ~z, edot=1e-15, z.lim=range(0, zl), type='h', alpha=.5, lwd=4, col='navy', xlab="Depth (m)", ylab="Deviatoric Stress", add=TRUE)plotFun(RheoLitho(z, edot) ~z, edot=1e-16, z.lim=range(0, zl), type='h', alpha=.5, lwd=4, col='green', xlab="Depth (m)", ylab="Deviatoric Stress", add=TRUE)

======================

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# Integrated strength of the lithosphereStrength=antiD(RheoLitho(z, 1e-15) ~z, z.lim=range(0, zl))Strength(150000)

# Brittle-Ductile transitionslim1<-findZeros(Plastic(z)-limiting_yield_stress_crust ~z, z.lim=range(15000, 55000))lim2<-findZeros((Visc_uc(z, 1e-15)*1e-15)-limiting_yield_stress_crust ~z, z.lim=range(0, 55000))lim3<-findZeros((Visc_mant(z, 1e-15)*1e-15)-limiting_yield_stress_sclm ~z, z.lim=range(zc, zl))

LithoStrength1=antiD(Plastic(z) ~z, z.lim=range(0, 29054.82)) LithoStrength1(0)-LithoStrength1(29054.82)

LithoStrength2=limiting_yield_stress_crust*(lim2-lim1)

LithoStrength3=antiD(Visc_uc(z, 1e-15)*1e-15 ~z) LithoStrength3(lim2)-LithoStrength3(55000)

LithoStrength4= limiting_yield_stress_sclm*(lim3-55000)

LithoStrength5=antiD(ViscMant(z, 1e-15)*1e-15 ~z) LithoStrength5(65231.56)-LithoStrength4(150000)

LithoStrength1+LithoStrength2+LithoStrength3+LithoStrength4+LithoStrength52.282934e+13+3.581046e+12+7.743365e+12+9.4124e+11+4.077892e+12=3.917288e+13

LithoStrength=antiD(RheoLitho(z, RefEdot) ~z) LithoStrength(z=0, RefEdot=1e-15)-LithoStrength(z=125000, RefEdot=1e-15)[1] -4.138011e+12

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Partial melting impacts on the mechanical behavior of both the mantle and the continental crust. Since melt has a much lower density and viscosity than its source, the production of melt impacts significantly on the dynamic evolution of both the mantle and the continental crust. It also affects in the long term their mechanical properties, since upon melt extraction, the residual depleted rock has a viscosity and density larger than the source rock before melting. Melting is therefore a fundamental process controlling the evolution of Earth’s main envelops.

GEOS-3104

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Partial Melting

SECTION 1

Melt ProductionPartial melting impacts on the mechanical behavior of both the mantle and the continental crust. Melting changes both the density and the viscosity of partially melted rocks. Upon melt extraction, the residual depleted rock has a viscosity and density that differ from the rock before melting. Melting is therefore a fundamental process controlling the differentia-tion of the Earth at all scales. In nature melt is produced when a material passes its solidus. In Ellipsis, the solidus and liquidus are approximated by polynomial functions:

where ρ.g.z is the pressure, ai and bi are polynomial coefficients.The melt fraction F is a simple function of Tss the supersolidus temperature:

Hence -0.5 < Tss < 0.5

Summary

1. Solidus and liquidus2. Effect on density and

viscosity3. Exercise

Crustal Crustal partial melting

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Tsol(z) = a0+a1(ρ.g.z)+a2(ρ.g.z)2+a3(ρ.g.z)3

Tliq(z) = b0+b1(ρ.g.z)+b2(ρ.g.z)2+b3(ρ.g.z)3

Tss(z) =T(z)−(Tliq(z)−Tsol(z))/2

Tliq(z)−Tsol(z)

Ellipsis uses the McKenzie and Bickle (1988) relationship as:

Density and viscosity changesDensity of the partially melted material changes with both temperature T and melt fraction F:

In Ellipsis, β is “Material_#_depl_exp”.

The viscosity changes due to the presence of a melt fraction F. It does so within a critical melt window (~0-5% for the mantle, ~20-40% from crustal melt). Hence, when F_min < F < F_max, then :

Where, γ is “Material_#_melt_total_visc_change”.In-situ melt also affects the behavior rocks in which it resides. Its primary effect is to decrease the viscosity. However, there is only so much melt a rock can trap before melt starts to migrate. The exact melt fraction at which melt is extracted varies from less than % in the mantle, up to 35% in the crust depending whether or not deformation occur during partial melting. Therefore, in Ellipsis there is an upper melt fraction limit beyond which the melt is extracted. The melt fraction at which melt migrates out of the system is: “Material_#_melt_extract “. To keep the melt in the system one will impose: Material_#_melt_extract=1.

The decrease in viscosity due to the presence of melt is not linearly proportional to F. In fact, the viscosity of the magma changes sharply over a relatively small range of melt fraction (i.e. the critical melt fraction) over which the magma evolves from a solid supported (viscosity of the solid matrix) to liquid supported (viscosity of the melt) magma. The

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F(Tss) = 0.3936+0.8936Tss+0.4256Tss2+2.988Tss

3

ρ(T,F) = ρ0.(1−ΔT−βF)

η(F) = η0 /(1+γF)

drop in viscosity can be as much as 8 orders of magnitude within 10% increase on the melt fraction, only the first 3 orders of magnitude are mechanically significant. We parameterize this evolution by defining the melt fraction interval (here from 20% and 25%) over which the drop in viscosity (/1000) occurs: Material_#_Melt_frac_start_visc_change=0.20Material_1_Melt_frac_end_visc_change=0.25Material_1_melt_total_visc_change=1000

When melt is extracted it advects heat out of the system, modulating further melting. We assume the fusion entropy to be 250 J kg-1K-1. When melt is extracted from the system, it leaves the temperature of the residue at the solidus. Upon melt ex-traction, volatiles are also removed which can only increase the viscosity of the residue following :

In Ellipsis, τ is “Material_#_depl_visc_change”.

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η(Depl) = η0.(1+τ.Depl))

Exercise: Lower crust diapirThe aim of this exercise is to model a rising diapir in the continental crust which undergoes decompression melting. We ex-plore the isentropic case (no heat loss) in which the melt stays in the source (Material_1_melt_extract=1). We build a 40 km x 40 km model with a linear geotherm with: T(z=0) = 20ºC and T(z=Moho) = 800ºC. The crust is made of a material that cannot melt and whose density at atmospheric condition is 2720 kg.m-3 increasing with temperature (therefore depth) ac-cording to a negative coefficient of thermal expansion of -5.4e-5 K-1. In the lower crust, we define a second material in the shape of a semi circle to simulate a region with lower density (ρ). This material has a lower density (2500 kg.m-3) and its solidus and liquidus are (in Kelvin): Tsol (z) = 773+ 3.33e-7⋅(ρ ⋅ g⋅ z ) and Tliq(z) = 1073+ 3.33e-7⋅(ρ ⋅ g⋅ z )This second material is centered in x=20 km z=40 km with a of radius 10 km. It has also a lower thermal diffusivity.The rheology of the crust and magma is viscous (i.e. no plasticity: YIELD=off).

Run the model, analyze the result, and summarize your finding into a 4-page report (less than 3-4 Mbyte). For this you may want to consider the information stored in the following files (all readable with a simple text editor):

#.node_data:  These files store at each time step data (temperature, pressure, ... ) at the node of the computational grid. The information in the node_data can be used for data mapping. For instance one can map isotherms, super-solidus temperature, melt fraction, etc. For this you will need to process the information stored in .node_data using Matlab, R, Python or any other appropriate tools.#.profiles:  These files give, at each time step, access to data profile (horizontal or vertical) passing through "sampling tracers".  In the input file one can choose the initial position  of up to 50 sampling tracers (this position can remain fixed to the computational grid, or can move with the rocks through the grid), the direction of the profile (horizontal or vertical), and the type of data to be profiled (temperature, melt, depletion, pressure, velocity, strain rate, stress ... )#.sample : These files are updated at each time step to add the new value of a parameter attached to one of the sampling tracers.  Hence, there is one .sample file per sampling tracer.   These files record the evolution through time of data (X, Z, Data), where data can be tempera-ture, pressure, etc

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It will help your analysis if you plot the geotherm, solidus and liquidus, and the super-solidus temperature, as well as map the melt Fraction F between the solidus and the liquidus. You may want to also plot the data stored into the files called #.profiles. These files store at each time step data profiles (vertical profiles or horizontal profiles) from sampling tracers whose coordinates are given in the section “## OUTPUT REQUIREMENTS ##.

Extra question: Should you design a model in which the vertical velocity of the diapir is faster, what parameter(s) would you change and why?

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SECTION 2

Basaltic volcanism is the result of partial melting of the Earth’s mantle (Fig. 1A). Volcanic activity in arcs and along mid-ocean ridges and continental rifts is related to plate tectonics, through decompression melting of the shallow sublithospheric mantle in regions of lithospheric extension and thinning, and/or hydra-tion of the mantle wedge above subducting slabs. In contrast, intraplate volcanism, responsible for thick continental flood ba-salts, has usually been linked to upwelling and decompression

melting of large volumes of deep, hot mantle (e.g., mantle plume; Morgan, 1983; Richards et al., 1989). More recently, de-compression melting in small-scale convection cells along thick lithospheric keels has been proposed to explain minor intraplate volcanism (Davies and Rawlinson, 2014).

One of the main differences between plate-tectonics–related melting of the sublithospheric mantle and partial melting in hot

The geodyamics of mantle melting

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upwelling mantle material is the pressure (P) and temperature (T) at which the adiabat (i.e., geotherm in the convective mantle) inter-sects the mantle solidus (Fig. 1B). These cannot be obtained di-rectly from the composition of basalts because they are affected by fractional crystallization and interaction with surrounding rocks, as well as the progressive evolution of the composition of the source. The geochemistry of the first-formed batch of melt as fertile mantle crosses its solidus (i.e., the primary melt) is a function of the mantle potential temperature, Tp , as shown by forward thermo-dynamic modeling (Asimow, 1999; Asimow et al., 2001; Herzberg and O’Hara, 2002; Herzberg, 2004). Tp is the temperature measured at the Earth’s surface from extrapolation of the mantle adiabat (McKenzie and Bickle, 1988), a convenient way to express the inter-section temperature of adiabat and solidus. The composition of the elusive primary melt of a volcanic province derives from process-ing the most magnesian liquids (i.e., parental melts) inferred from a suite of basalts (Herzberg et al., 2007). Comparison of the calcu-lated and modelled primary melt constrains the Tp from which the geodynamic setting of the partial melting can be inferred. By target-ing primary melts, petrologists mitigate geochemical alteration of parental magmas through fractional crystallization and interaction with surrounding rocks during melt ascent (e.g., Herzberg et al., 2007). Keeping in mind the many pitfalls (Herzberg et al., 2007; Herzberg, 2011), data from present continental rifting and spread-ing ridges derive a Tp of less than 1400 °C (Falloon et al., 2007; Herzberg et al., 2007). Hole (2015) shows that data from plume-related flood basalts point to a Tp of more than 1500 °C, in agree-ment with, for example, Thompson and Gibson (2000), Herzberg and O’Hara (2002), and Herzberg et al. (2007). The range of adia-bats during tectonic extension and exhumation of the shallow man-

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Fig. 1. A: Geodynamic settings of mantle melting. B: Phase diagram, contoured for MgO, of mantle melting. 1—lithospheric thinning; 2— mantle upwelling; 3—mantle warming. During (1) and (2), the convective mantle is exhumed adiabatically (blue and red dashed lines in B) intersecting the solidus at ~70 km and 180 km, respec-tively, with Tp (1330 ºC and 1620 ºC) shown by the blue and red dots at Earth’s sur-face. Following mantle warming (3), the adiabat (black dashed line) intersects the solidus at ~120 km depth, with Tp (~1470 °C) shown by the gray dot. Primary melts are produced at solidus temperatures (blue, gray, and red stars). Should these melts be extracted as soon as they are produced, they would follow their respective melt adiabat (solid blue, gray, and red arrows). In the case of continental rifting leading to a mid-ocean ridge, melt is usually extracted at melt fraction of <1% (i.e., fractional melting). As latent heat escapes with the melt, the source rock cools and follows the solidus during exhumation (path a-a’). As the source becomes more residual, it pro-duces parental melts with a range of compositions (empty blue arrows). In plumes, melt continuously reacts with the residue, with which it remains in equilibrium. As latent heat remains with the melt in the source, the upwelling mantle keeps follow-ing the adiabat (path b-b’). As the plume head slows down, melt segregates toward the top of the partially melted column, before escaping to the surface (i.e., batch or equilibrium melting) following the parental melt adiabat (empty red arrow). The pressure at which melt is extracted is the final melting pressure. Both fractional melt-ing and equilibrium melting can occur in succession depending on strain rate, perme-ability, and the relative upwelling velocities of the residue and melt.

tle (blue region in Fig. 1B) and during deep mantle upwelling (red region in Fig. 1B), do not overlap. The concept of Tp can thus be used to constrain the geodynamic setting of mantle melting. Let’s now consider the origin of the ca. 200 Ma Central Atlantic mag-matic province (CAMP).

The CAMP extends over Africa, South America, Europe, and North America, covering ~107 km2 (Marzoli et al., 1999). It post-dates by 50 m.y. the final stages of Pangea’s assembly. The peak igneous ac-tivity at 199–200 Ma pre-dates by ~10 m.y. the initial opening of the central Atlantic Ocean (Marzoli et al., 1999; Sahabi et al., 2004). The attribution of the CAMP to a mantle plume-head impinging the base of the lithosphere (Hill, 1991; Wilson, 1997; Leitch et al., 1998; Courtillot et al., 1999; Janney and Castillo, 2001; Ernst and Buchan, 2002) has been challenged on the basis that chemical and isotopic compositions of basalts point to shallow-mantle sources, including lithospheric ones, enriched by earlier subduction (Bertrand et al., 1982; Bertrand, 1991; Puffer, 2001; Deckart et al., 2005; Verati et al., 2005). However, the origin of the CAMP is still strongly debated be-cause basaltic magmas change on their way to the surface.

Hole (2015) processes the composition of parental magmas of the CAMP to recover that of their primary melts, and concludes most CAMP primary magmas point to a Tp between 1400 °C (olivine nor-mative basalts) and 1500 °C (quartz normative basalts). This tem-perature range is too hot for sublithospheric decompression melt-ing related to lithospheric thinning, and too cold for a source in a hot upwelling mantle.

Continental lithosphere is thicker and relatively more stable than oceanic lithosphere, and thus impedes the removal of heat from the

hotter convective mantle. As continents aggregate into superconti-nents, they enlarge the wavelength of the convective pattern, fur-ther impeding heat loss, thus forcing large-scale warming of the sublithospheric mantle (Grigné et al., 2005). Modulation of mantle temperature below continents depends on the size, number, and distribution of continental plates (Coltice et al., 2007), and thus de-pends on plate tectonics: continents amalgamate to create supercon-tinents through the Wilson cycle, then break into smaller continen-tal plates. Two- and three-dimensional numerical studies show that the Tp of the convective mantle can increase 10% to 15% (~130–200 °C) as plates aggregate into a supercontinent covering 20% to 35% of Earth’s surface (Coltice et al., 2009), thus the Tp of partial melt-ing following the formation of supercontinents is between that of plate tectonic processes (<1400 ºC) and deep mantle upwelling (>1500 ºC). The potential temperature determined by Hole from the CAMP’s primary magmas using the most advanced model (PRIMELT2), combined with the observation that magmatism fol-lowed supercontinent aggregation, confirms that the CAMP was the result of large-scale mantle melting following the aggregation of the last supercontinent (Coltice et al., 2009).

REFERENCES CITED

Asimow, P.D., 1999, A model that reconciles major- and trace-element data from abyssal peridotites: Earth and Planetary Science Letters, v. 169, p. 303–319, doi:10.1016/S0012-821X(99)00084-9.

Asimow, P.D., Hirschmann, M.M., and Stolper, E.M., 2001, Calculation of peri-dotite partial melting from thermodynamic models of minerals and melts, IV. Adiabatic decompression and the composition and mean properties of mid-ocean ridge basalts: Journal of Petrology, v. 42, p. 963–998, doi:10.1093/petrology/42.5.963.

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Bertrand, H., 1991, The Mesozoic tholeiitic province of northwest Africa: A volcanotectonic record of the early opening of the central Atlantic, in Kam-punzu, A.B., and Lubala, R.T., eds., Magmatism in Extensional Structural Set-tings: The Phanerozoic African Plate: Berlin, Springer-Verlag, p. 147–188.

Bertrand, H., Dostal, J., and Dupuy, C., 1982, Geochemistry of early Mesozoic tholeiites from Morocco: Earth and Planetary Science Letters, v. 58, p. 225–239, doi:10.1016/0012-821X(82)90196-0.

Coltice, N., Phillips, B.R., Bertrand, H., Ricard, Y., and Rey, P.F., 2007, Global warming of the mantle at the origin of flood basalts over supercontinents: Ge-ology, v. 35, p. 391–394, doi:10.1130/G23240A.1.

Coltice, N., Bertrand, H., Rey, P.F., Jourdan, F., Phillips, B.R., and Ricard, Y., 2009, Global warming of the mantle beneath continents back to the Archaean: Gondwana Research, v. 15, p. 254–266, doi:10.1016/j.gr.2008.10.001.

Courtillot, V., Jaupart, C., Manighetti, I., Tapponnier, P., and Besse, J., 1999, On causal links between flood basalts and continental breakup: Earth and Planetary Science Letters, v. 166, p. 177–195, doi:10.1016/S0012-821X(98)00282-9.

Davies, R., and Rawlinson, N., 2014, On the origin of recent intraplate volcan-ism in Australia: Geology, v. 42, p. 1031–1034, doi:10.1130/G36093.1.

Deckart, K., Bertrand, H., and Liegeois, J.P., 2005, Geochemistry and Sr, Nd, Pb isotopic composition of the Central Atlantic Magmatic Province (CAMP) in Guyana and Guinea: Lithos, v. 82, p. 289–314, doi:10.1016/j.lithos.2004.09.023.

Ernst, R.E., and Buchan, K.L., 2002, Maximum size and distribution in time and space of mantle plumes: Evidence from large igneous provinces: Journal of Geodynamics, v. 34, p. 309–342, doi:10.1016/S0264-3707(02)00025-X.

Falloon, T.J., Danyushenvsky, L.V., Ariskin, A., Green, D.H., and Ford, C.E., 2007, The application of olivine geothermometry to infer crystallization tem-peratures of parental liquids: Implications for the temperature of MORB mag-mas: Chemical Geology, v. 241, p. 207–233, doi:10.1016/j.chemgeo.2007.01.015.

Grigné, C., Labrosse, S., and Tackley, P.J., 2005, Convective heat transfer as a function of wavelength: Implications for the cooling of the Earth: Journal of Geophysical Research, v. 110, B03409, doi:10.1029/2004JB003376.

Herzberg, C., 2004, Geodynamic information in peridotite petrology: Journal of Petrology, v. 45, p. 2507–2530, doi:10.1093/petrology/egh039.

Herzberg, C., 2011, Basalts as temperature probes of Earth’s mantle: Geology, v. 39, p. 1179–1180, doi:10.1130/focus122011.1.

Herzberg, C., and O’Hara, M.J., 2002, Plume-associated ultramafic magmas of Phanerozoic age: Journal of Petrology, v. 43, p. 1857–1883, doi:10.1093/petrology/43.10.1857.

Herzberg, C., Asimow, P.D., Arndt, N., Niu, Y., Fitton, J.G., Cheadle, M.J., and Saunders, A.D., 2007, Temperatures in ambient mantle and plumes: Con-straints from basalts, picrites, and komatiites: Geochemistry Geophysics Geo-systems, v. 8, doi:10.1029/2006GC001390.

Hill, R.I., 1991, Starting plumes and continental break-up: Earth and Plane-tary Science Letters, v. 104, p. 398–416, doi:10.1016/0012-821X(91)90218-7.

Hole, M.J., 2015, The generation of continental flood basalts by decompres-sion melting of internally heated mantle: Geology, v. 43, p. 311–314, doi:10.1130/G36442.1.

Janney, P.E., and Castillo, P.R., 2001, Geochemistry of the oldest Atlantic oce-anic crust suggests mantle plume involvement in the early history of the cen-tral Atlantic Ocean: Earth and Planetary Science Letters, v. 192, p. 291–302, doi:10.1016/S0012-821X(01)00452-6.

Leitch, A.M., Davies, G.F., and Wells, M., 1998, A plume head melting under a rifting margin: Earth and Planetary Science Letters, v. 161, p. 161–177, doi:10.1016/S0012-821X(98)00147-2.

Marzoli, A., Renne, P., Piccirillo, E., Ernesto, M., Bellieni, G., and De Min, A., 1999, Extensive 200-million-year-old continental fl ood basalts of the Central Atlantic Magmatic Province: Science, v. 284, p. 616–618, doi:10.1126/science.284.5414.616.

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McKenzie, D., and Bickle, M.J., 1988, The volume and composition of melt generated by extension of the lithosphere: Journal of Petrology, v. 29, p. 625–679, doi:10.1093/petrology/29.3.625.

Morgan, W.J., 1983, Hotspot tracks and the early rifting of the Atlantic: Tec-tonophysics, v. 94, p. 123–139, doi:10.1016/0040-1951(83)90013-6.

Puffer, J.H., 2001, Contrasting high field strength element contents of conti-nental flood basalts from plume versus reactivated-arc sources: Geology, v. 29, p. 675–678, doi:10.1130/0091-7613(2001)029<0675:CHFSEC>2.0.CO;2.

Richards, M.A., Duncan, R.A., and Courtillot, V., 1989, Flood basalts and hot-spot tracks, plume heads and tails: Science, v. 246, p. 103–107, doi:10.1126/science.246.4926.103.

Sahabi, M., Aslanian, D., and Olivet, J.L., 2004, A new starting point for the history of the central Atlantic: Comptes Rendus Geoscience, v. 336, p. 1041–1052, doi:10.1016/j.crte.2004.03.017.

Thompson, R.N., and Gibson, S.A., 2000, Transient high temperatures in man-tle plume heads inferred from magnesian olivines in Phanerozoic picrites: Na-ture, v. 407, p. 502–506, doi:10.1038/35035058.

Verati, C., Bertrand, H., and Féraud, G., 2005, The farthest record of the Cen-tral Atlantic Magmatic Province into West Africa craton: Precise 40Ar/39Ar dating and geochemistry of Taoudenni basin intrusives (northern Mali): Earth and Planetary Science Letters, v. 235, p. 391–407, doi:10.1016/j.epsl.2005.04.012.

Wilson, M., 1997, Thermal evolution of the Central Atlantic passive margins: continental break-up above a Mesozoic super-plume: Journal of the Geologi-cal Society, v. 154, p. 491–495, doi:10.1144/gsjgs.154.3.0491.

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SECTION 3

The mantle solidus and liquidus are relatively well constrained. For instance, according to Katz et al., (2003):

Tsol(z): P<15GPa = 1358.7 + 132.899 (ρ.g.z)/(1e9)- 5.104 ((ρ.g.z)/(1e9))2 Tsol(z) P≥15GPa = 1510.76 + 46.27 (ρ.g.z)/(1e9)- 0.8036 ((ρ.g.z)/(1e9))2

Tliq(z): P<15GPa = 2053 + 45 (ρ.g.z)/(1e9)- 2.00 ((ρ.g.z)/(1e9))2 Tliq(z) P≥15GPa = 1470.3025 + 55.53 (ρ.g.z)/(1e9)- 0.9084 ((ρ.g.z)/(1e9))2

Melt fraction is calculated using the concept of super-solidus (see page 28) and eq. 21 of McKenzie and Bickle (1988):F(Tss) = 0.3936 + 0.8936 Tss + 0.4256 Tss 2 + 2.988 Tss 3

Basalts can be classified based on their MgO content: Komatiite (MgO>18%), komatiitic basalts (12%<MgO<18%), tholei-itic basalts (6%<MgO<12%). MgO content depends on the pressure at which melt is extracted (Herzberg and Gazel, 2009):

Tc(MgO, z) = 935 + 33 MgO - 0.37 MgO2 + 54 (ρ.g.z) - 2 (ρ.g.z)2

Summary

1. Solidus/liquidus2. Basalts MgO content3. Melting during fast

decompression4. Melting during slow

decompression5. Exercise: Melting in a mantle

plume

Modelling mantle melting

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Mantle melting during fast exhumation: The graph below shows a phase diagram with melt fraction mapped between the solidus and the liquidus. This graph describes what happened during a phase of fast homogeneous continental rifting back in the Archean when the mantle was ~200ºC hotter than present. The continental lithosphere is thinned rapidly (225 km to 110 km). The thinning of the continent triggers the decompression of the mantle below the continent. When exten-sion and thinning is rapid each point on the geotherm evolves following the adiabatic gradient (i.e. no conductive cooling). As the geotherm becomes warmer, it intersects the soli-dus at two locations (A0’ and A3’) which define the top and base of a partially molten mantle column. The base of the column remains close to 150 km throughout the volcanic history. Because extension is rapid, the par-tially molten column (i.e. solid matrix together with the melt) also follows the adiabat since there is no melt ex-traction, and therefore no loss of heat. After ~100% ex-tension, points on the initial geotherm located at A0 to A3 have moved upward to location A0’ to A3’. The par-tially molten mantle column consists of partially mol-ten fertile sub-lithospheric mantle (from ~150 to ~110 km), and partially molten SCLM (from ~110 to ~55 km). Because the melt accumulates in the column, the buoy-ancy of the molten column increases and its viscosity decreases, until melt escapes to the surface to form a ba-saltic crust. The maximum thickness of this crust is given by the area between the solidus and the geo-therm (purple triangle), here it is ~ 16 km (i.e. ~ 0.5 x (depth(A0’) - depth(A3’)) * maximum melt fraction).

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The phase diagram on the right shows the MgO content between the solidus and the liquidus, according to Herzberg and Gazel (2009). The MgO content of melts ex-tracted at various locations in the partially molten column ranges from 9 to 21 %.

Mantle melting during slow exhumation: When extension is slow (graphs below), the solid matrix flows upwards very slowly, and the melt - whose buoyancy in-creases as the pressure at which it is produced decreases - flows faster through the source and ponds at the top of the partially molten column, before escaping to the surface via diking. In the molten column, melt forms an interconnected network for a critical melt fraction as low as 0.1 %, limiting the amount of melt in the col-umn to less than 1%. While the flowing melt follows the melt adiabat, the depleted residue follows the solidus as latent heat escapes with the melt. The total amount

of latent heat con-sumed de-pends only on the volume of rock advected above the solidus, not on the history of melt extraction. Hence, the melt productivity, and therefore the de-pletion of the source, is not significantly affected by the residence time of the melt in the source be-fore its extraction. Hence, assuming 100% exten-sion, the thickness of the basaltic crust also reaches ~16 km. The MgO content of the melt var-ies with the pressure at which it is produced and varies between 9 and 17%.

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The small fraction of melt remaining in the source has only a modest impact on the buoyancy of the partially molten re-gion. Its impact on the rheology is, however, more significant. On one hand, the viscosity of the partially molten mantle strongly decreases because melt lubricates grain boundaries. On the other hand, partial melting drains water out of the solid matrix and increases the olivine content. Although the impact of dehydration on the viscosity may not be as signifi-cant as previously thought (Fei et al., 2013), both processes should contribute to increase the viscosity of the depleted resi-due once its temperature drops below the solidus.

Exercise - Melting in a mantle plume: We model the rise of a mantle plume in a box 700 km deep and 4200 km wide (i.e. simulating the upper mantle). The plume is modelled using a positive thermal anomaly 300ºC higher than the surround-ing mantle (1347ºC). We take advantage of the symmetry of the problem and embed the anomaly on the lower left corner of the box. The mantle is modelled as a visco-plastic material with temperature, stress, strain-rate and melt dependent vis-cosity. The viscous rheology of the mantle is based on dry olivine (Brace & Kohlstedt, 1980). Its plastic rheology is approxi-mated with a Coulomb failure criterion with cohesion 40 MPa, a coefficient of friction (μref ) 0.268, and a weakening factor dependent on the accumulated plastic strain (εp). The coefficient of friction (μ) evolves as a function of the accumulated plastic strain as:

for εp < 0.15: μ = μref ×(1-(1-0.0373)×( εp /0.15)0.25)for εp > 0.15: μ = μref × 0.0373

This weakening leads to a strong strain localisation simulating faults with nominal viscosity ~25 times weaker than that of the surrounding rocks when plastic strain reaches 15%. This weakening, which is a key factor for the operation of plate tec-tonics on present Earth, simulates the formation of phyllosilicates (serpentine, talc, micas) during the strain-induced hydra-tion of mantle rocks and basalts. For semi-brittle deformation independent of pressure, we impose an upper limiting yield stress of 300 MPa (Ord & Hobbs, 1980; Escartin & Hirth, 1997).

The top 15 km of the model consists of a thick layer of basalts, which simulates the oceanic crust. Because these basalts are emplaced below sea level, they are strongly hydrothermally altered. For simplicity, we assume that this layer of basalt has

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a nominal viscosity 1000 times weaker that the underlying mantle, a weak cohesion of 1 MPa, a reference coefficient of fric-tion of 0.134 and the same weakening properties as the rest of the mantle.

In our modelling of a rising plume, we assume that melt in the partially molten column is extracted when the melt fraction reaches 1%. In this case, latent heat escapes with the melt, and the ascending depleted residue follows the solidus closely. We assume the fusion entropy to be 400 J kg-1 K-1.

The small fraction of melt (<1%) has only a modest impact on the buoyancy of the partially molten region. However, the density of the residue decreases as it becomes more depleted. We assume a maximum density decrease of 1.5% upon full depletion. Because even a small fraction of melt lubricates grain boundaries, it affects the viscosity of the partially molten column. Hence, we impose a linear viscosity drop to a maximum of one order of magnitude when the melt fraction reaches 1%. On the other hand, partial melting drains water out of the solid matrix and reduces the number of phases. Al-though the impact of dehydration on the viscosity may not be as significant as previously thought (Fei et al., 2013), both processes should contribute to increase the viscosity of the depleted residue once its temperature drops below the solidus. In our experiment we impose an increase in viscosity proportional to depletion, assuming a viscosity increase of two or-ders of magnitude for 100% depletion.

The geotherm is linear in the lithosphere (T(0km)=293 K, T(150km)=1620K). In the convective mantle, the geotherm fol-lows the adiabatic gradient (~0.3 ºC per km).

Run the model (A_Plume100.input), analyze the result, and summarize your findings into a 4-page report (less than 3-4 Mbyte). For this you may want to consider the information stored in the following files (all readable with a simple text edi-tor): .node_data, .profiles, .samples and .timelogs.

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ReferencesBrace, W. F., Kohlstedt, D. L. Limits on lithospheric stress imposed by laboratory experiments. Journal of Geophysical Research 85, 6248-6252 (1980).

Escartin, J. & Hirth, G. Nondilatant brittle deformation of serpen-tinites: Implications for Mohr-Coulomb theory and the strength of faults. Journal of Geophysical Research 102, 2897-2913 (1997).

Fei, H., Wiedenbeck, M., Yamazaki, D. & Katsura, T. Small effect of water on upper-mantle rheology based on silicon self-diffusion co-efficients. Nature 498, 213-216 (2013).

Herzberg, C. & Gazel, E. Petrological evidence for secular cooling in mantle plumes. Nature 458, 619-623 (2009).

Katz, R. F., Spiegelman, M. & Langmuir, C. H. A new parameteriza-tion of hydrous mantle melting. Geochemistry, Geophysics Geosys-tems 4 , doi:10.1029/2002GC000433 (2003).

McKenzie, D. & Bickle, M. J. The volume and composition of melt generated by extension on the lithosphere. Journal of Petrology 29, 625-629 (1988).

Ord, A., Hobbs, B. E. The strength of the continental crust, detach-ment zones and the development of plastic instabilities. Tectono-physics 158, 269- 289 (1989).

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m)

1400 1600 1800 2000 2200Temperature (K)

Solidus

Liquidus20 40 60 80

20

46

8Pressure (GPa)

melt %100

150

200

250

300

Dep

th (

km

)

1400 1600 1800 2000 2200

Temperature (K)

Solid

us

Liq

uid

us

10 20 30

MgO %

20

46

8Pressure (GPa)

77

Solidus and liquidus according to Katz et al., 2003. Melt fraction from McKenzie & Bickle (1988)

Solidus and liquidus according to Katz et al., 2003. MgO content from Herzberg & Gazel (2009)

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CHAPTER 5

78

Subduction

Aliquam turpis tellus. Id malesuada lectus. Suspendisse potenti. Etiam felis nisl, cursus bibendum tempus nec. Aliquam at turpis tellus. Id malesuada lectus. Suspendisse potenti. Etiam felis nisl, cursus bibendum tempus nec, aliquet ac magna. Pellentesque a tellus orci. Pellentesque tellus tortor, sagittis ut cursus vitae, adipiscing id neque.

CHAPTER 6

79

Continental Collision

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CHAPTER 7

80

Vizualisation in Paraview