GEOS3104-3804: GEOPHYSICAL METHODS Geodynamic modelling using Ellipsis Patrice F. Rey GEOS-3104 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. 1 SECTION 1 The rise of computer science During the mid-19th century, the laws of thermody- tle convection, and plate tectonic processes which also namics - which describe the relation between heat and involves the flow - brittle or ductile - of rocks. 21st cen- forces between contiguous bodies - and fluid dynam- tury computers have now the capability to compute in ics - which describes the flow of fluids in relation to four dimensions the flow of complex fluid with com- pressure gradients - reached maturity. Both theories plex rheologies and contrasting physical properties we underpin our understanding of many natural proc- typically encounter on Earth. For geosciences, this is a esses from atmospheric and oceanic circulation, man- significant shift. 2 Computational tectonics and geodynamics: There is a revolution Navier-Stokes equations: Written in Georges G. Stokes (1819-1903) unfolding at the moment in all corners of science, engineering, and the 1850’s, the Navier-Stokes equations other disciplines like economy and social sciences. This revolution are the set of differential equations that is driven by the growing availability of increasingly powerful describe the motion of fluid and associ- high-performance computers (HPC), and the growing availability ated heat exchanges. They relate veloc- of free global datasets and open source data processing tools (Py- ity gradients to pressure gradients and thon, R, Paraview, QGIS-GRASS, etc). Here in Australia, supercom- can be analytically solved only when puters such as Magnus (36,000 cores) at the Pawsey Supercomput- considering steady laminar flows. 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. 3 Some examples of simple laminar flow for which analytical solu- of their time. In short, in the second half of the 20th century com- tions exist include the problem free fall of a spherical object into a puters were not powerful enough to take advantage of these new newtonian fluid (the settling of crystal into a magma), the flow in- numerical methods. As com- duced by the motion of a rigid plate puters grew in power, so did above a newtonian fluid (Couette flow the complexity of fluid flow in relation to the motion of tectonic plate problems that one can tackle. above the asthenosphere), and the flow However, for quite some time, of newtonian fluid between two static computational tectonics in- plates (Poiseuille flow, for instance the volving a layered lithosphere 1950’s mainframe computers flow of the lower crust in orogenic pla- made of stronger (upper crust teaux). 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. 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 @ Yuen, Minneapolis same time, numerical methods and computer algorithms were pro- @ Beaumont, Halifax gressing so fast that they quickly overtook the computer capability 4 Particle-in-cell numerical methods. In the late 1990’s, computers Over the past decade, the growing availability of powerful high- become powerful enough to allow the implementation of a 1950’s performance computers has unleashed the power of computer- numerical method called particle-in-cell (PIC, developed at Los Ala- based modelling in all branches of science. Geoscientists are now mos National Laboratory) in which individual fluid elements, car- able to simulate in four dimensions, using realistic coupled ther- rying material properties and flow history, are advected through a mal and mechanical properties, lithospheric-scale deformation and fixed computational grid. For geologists, this progress meant that 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 PIC = Eulerian mesh + Lagrangian particles (or a physical experi- one could simulate the deformation of mechanically layered sys- ment) do not pretend to tems such as the Earth’s lithosphere; albeit in two dimensions. El- reproduce a natural lipsis, the code we will use in GEOS3104, is one of the earliest and process in a realistic most robust codes (along with Citcom) implementing an efficient manner. The aim fo- PIC method on the back of a robust multigrid solver. 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 Rey, Coltice, Flament, the very late 70’s) perfectly illustrates the concept of escape tecton- Nature 2014 ics, a process which accommodates convergence via the lateral ex- pulsion of continental blocks in front of a rigid indenter. Clearly, We have used Ellipsis to show that early proto-continent and thick oceanic plateaux this experiment bears very little resemblance with tectonics in Asia had enough gravitational power to slowly force adjacent oceanic lithospheres to sub- and South East Asia, but it does a nice job in illustrating a concept 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. which has changed our understanding of collisional tectonics. 5 Numerical modelling aims at under- simulations, in the context of geosciences, assumes that we are able standing a particular process within to implement a very broad range of coupled natural mechanical, a particular tectonic context. The ini- petrological, geochemical processes including the migration of par- tial and boundary conditions are tial melt and the associated advection of heat, the release and con- carefully thought through to describe sumption of water and other volatiles, evolving anisotropic rheolo- a geologically realistic setting. Re- gies due to grain reduction, dilatancy due to micro-cracking and sults of the modelling are detailed other softening mechanisms. It will take decades before we can enough to be able to be compared - to properly engage with numerical simulation in a meaningful man- the first order - to natural ner. geological exam- ples, without Multi-grid solver: The Finite Element Method (FEM) con- trying to match accu- sists in discretizing of a set of partial differential equations rately any- across a 2D or 3D domain (i.e. ∂# are transformed into ∆ # ). one of them. 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 Numerical simulations aim at reproducing with the greatest level design methodologies taking advantage of parallel comput- of detail and the greatest level of realism a particular process on a ing, as well as other techniques. One involves the use of a particular region. For most people this is what modelling is all stack of computational grids of increasing resolution. In- about, and therefore they are quick to point to the many shortcom- stead of solving the set of equations directly at the node of ings of Paul Tapponnier’s experiments and dismiss its relevance one single fine grid, a rough solution is computed on a since it neither account for the presence of the highest mountain coarser grid, and this solution is iteratively refined at the chain nor the highest plateau on Earth. Numerical simulation as- nodes of grids of increasing resolution. Ellipsis uses a multi- sumes that our models are able to include parameters with realistic grid solver.