Gerya T. Introduction to Numerical Geodynamic Modelling

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3D visualisation challenges 317

Fig. Outlook.3 Different 3D projections of thermal-chemical plumes

(Fig. Outlook 1) growing atop of the subducting slab. The numerical

model corresponds to a spontaneous retreating oceanic subduction (Fig. 17.2,

17.3) and explores the effects of mantle wedge hydration and melting in 3D

(Zhu et al., 2009). The temperature iso-surface of 1350 K is shown. Note that

rising plumes are colder than the mantle wedge and move upward due to their

compositional buoyancy (Gerya and Yuen, 2003b). Numerical experiments are

performed with the code I3ELVIS (Gerya and Yuen, 2007) based on multigrid

method (Chapters 14, 15) at the resolution 405 ×101 ×101 nodes with 50 million

markers. The experiment was computed on the shared-distributed memory

supercomputer BRUTUS at ETH-Zurich.

et al., 1994; Golabek et al., 2008a, b; Samuel and Tackley, 2008;Linet al., 2009; Ricard

et al., 2009; also see Fig. 17.14). A possible numerical solution of this challenging

problem may be based on a combination of various types of codes into one accretion-

impact-differentiation tool (Fig. Outlook.2).

3D visualisation challenges

As discussed in the introduction, modellers spend much, or even most of their

time on visualising and understanding numerical models. Since these models

are likely to become bigger and more complex in the future, the role of efﬁ-

cient visualisation technologies will be growing. Visualisation of large numerical

models is a non-trivial task in 2D already (Rudolf et al., 2004; Gorczyk et al.,

2006) since the amount of graphical information exceeds the resolutions of even

the most powerful graphic screens by a factor of thousands (e.g. snapshots from

40 gigapixel database shown in Fig. Outlook.1). It is even more complicated with

318 Epilogue: outlook

3D models (e.g. Damon et al., 2008;Kadlecet al., 2008;Chenet al., 2008)

because the structure even for a single ﬁeld cannot be seen at once, and requires

processing of many views for proper understanding (Fig. Outlook.3, also see ﬁlms

Cold_Plumes_1.mpeg and Cold_Plumes_2.mpeg associated with this chapter).

Future challenges in this respect are again quite obvious: geology-friendliness,

ultrahigh resolution, multiple-scales, multiple-ﬁelds, interactive visualisation etc.

Conceptual warning

Discussion on the forthcoming technical advances above may give the impression

that the only thing that we have to do is to write larger, more complex 3D codes

which run on parallel supercomputers and have efﬁcient ways of visualising results

and compress the data. Obviously, this is only part of what needs to be done. More

important is that we obtain an in-depth physical understanding of the dynamics of

geological and planetary processes. This understanding should not only be based

on the ‘powerful numerics’, but also on using scaling laws based on simpliﬁed

theories and on comparison of predictions of such theories with the much more

complex numerical simulations and with nature. If we ﬁnd consistency, we have

likely learned something essential, and we did not have to perform 2

simulations

to understand how nature works.

Conclusion

In conclusion: The future of numerical geodynamic modelling looks bright and

there is a lot of exciting work and challenging research to do. Just go on!

It was fun to write all this. Thank you for reading.

Appendix

MATLAB program examples

The following resources can be found here:

www.cambridge.org/gerya

Introduction

Program 1: Visualisation_is_important.m (Exercise Introduction.2) – visualisation of

‘sin’ and ‘cos’ functions with ‘plot’, ‘pcolor’, ‘contour’ and ‘surf’.

Chapter 1

Program 2: Divergence.m (Exercise 1.2) – computation and visualisation of velocity,

divergence of velocity and time derivatives of density with ‘pcolor’ and ‘quiver’.

Chapter 2

Program 3: Periclase_EOS.m (Exercise 2.2) – computation and visualisation of density,

thermal expansion and compressibility for periclase (MgO) using external Gibbs free

energy function G_periclase.m.

Program 4: Density_map.m (Exercise 2.3) – loading from data ﬁles (m895_ro, morn_ro)

and visualising density maps for pyrolite (m895_ro) and MORB (morn_ro) and

density difference between pyrolite and MORB.

Chapter 3

Program 5: Poisson1D.m (Exercise 3.1) – solution of 1D Poisson equation with ﬁnite

differences on a regular grid using direct solver ‘\’.

Program 6: Poisson2D_direct.m (Exercise 3.2) – solution of 2D Poisson equation with

ﬁnite differences on a regular grid using direct solver ‘\’.

Program 7: Poisson2D_Gauss_Seidel.m (Exercise 3.3) – solution of 2D Poisson equation

with ﬁnite differences on a regular grid using Gauss–Seidel iteration.

Program 8: Poisson2D_Jacobi.m (Exercise 3.4) – solution of 2D Poisson equation with

ﬁnite differences on a regular grid using Jacobi iteration.

Chapter 4

Program 9: Strain_rate.m (Exercise 4.2) – computation and visualisation of velocity

ﬁeld, strain rate, deviatoric strain rate, and second strain rate invariant.

319

320 Appendix

Chapter 5

Program 10: Streamfunction2D.m (Exercise 5.2) – solution of 2D Stokes and continuity

equations with ﬁnite differences on a regular grid using stream function – vorticity

formulation for a medium with constant viscosity.

Chapter 6

Program 11: Viscosity_proﬁle.m (Exercise 6.1) – computation and visualisation of

viscosity proﬁle across the lithosphere.

Program 12: Viscosity_map.m (Exercise 6.2) – computation and visualisation of

viscosity map in temperature – log stress coordinates.

Program 13: Viscosity_comparison.m (Exercise 6.3) – computation and visualisation of

viscosity maps in temperature – log stress coordinates for a combination of

dislocation and diffusion creep; comparison of wet and dry olivine rheology.

Chapter 7

Program 14: Stokes_continuity_constant_viscosity.m (Exercise 7.1) – solution of 2D

Stokes and continuity equations with ﬁnite differences on a regular grid using

pressure–velocity formulation for a medium with constant viscosity.

Program 15: Stokes_continuity_variable_viscosity.m (Exercise 7.2) – solution of 2D

Stokes and continuity equations with ﬁnite differences on a regular grid using

pressure–velocity formulation for a medium with variable viscosity.

Chapter 8

Program 16: Upwind_1D.m (Exercise 8.1) – comparison of upwind, downwind and

central differences for 1D advection of a square density wave in a constant velocity

ﬁeld.

Program 17: FCT_1D.m (Exercise 8.2) – using FCT algorithm for 1D advection of a

square density wave in a constant velocity ﬁeld.

Program 18: Markers_1D.m (Exercise 8.2) – using marker-in-cell algorithm with regular

Eulerian grid for 1D advection of a square density wave in a constant velocity ﬁeld.

Program 19: Markers_1Dirregular.m (Exercise 8.3) – using marker-in-cell algorithm

with irregular Eulerian grid for 1D advection of a square density wave in a variable

velocity ﬁeld; using bisection algorithm.

Program 20: Stokes_Continuity_Markers.m (Exercise 8.4) – solution of 2D Stokes

continuity and advection equations with ﬁnite-differences and marker-in-cell

technique on a regular grid using pressure–velocity formulation for a medium with

variable viscosity; use of the ﬁrst-order accurate in space and time marker advection

scheme.

Program 21: Stokes_Continuity_Markers_Runge_Kutta.m (Exercise 8.5) – solution of

2D Stokes continuity and advection equations with ﬁnite-differences and

marker-in-cell technique on a regular grid using pressure–velocity formulation for a

medium with variable viscosity; using of the fourth-order accurate in space

ﬁrst-order accurate in time Runge–Kutta marker advection scheme.

Appendix 321

Chapter 9

Program 22: Shear_heating.m (Exercise 9.3) – solution of 2D Stokes and continuity

equations with ﬁnite differences on a regular grid using pressure–velocity

formulation for a medium with variable viscosity; computation and visualisation of

shear heating distribution.

Program 23: Shear_adiabatic_heating.m (Exercise 9.4) – solution of 2D Stokes and

continuity equations with ﬁnite differences on a regular grid using pressure–velocity

formulation for a medium with variable viscosity; computation and visualisation of

shear and adiabatic heating distribution.

Chapter 10

Program 24: Explicit_implicit_1D.m (Fig. 10.2) – solution of 1D temperature equation

on a regular grid for a non-moving medium with constant conductivity; comparison

of implicit and explicit method.

Program 25: Explicit_Implicit2D.m (Exercise 10.1) – solution of 2D temperature

equation on a regular grid for a non-moving medium with constant conductivity;

comparison of implicit and explicit method.

Program 26: Variable_conductivity.m (Exercise 10.2) – solution of 2D temperature

equation on a regular grid for a non-moving medium with variable conductivity;

comparison of implicit and explicit method.

Program 27: Conduction_advection2D.m (Exercise 10.3) – solution of 2D Eulerian

temperature equation with advective terms on a regular grid for a moving medium

with constant conductivity; use of upwind differences for advection of temperature;

comparison of implicit and explicit method.

Program 28: Variable_conductivity_advection2D.m (Exercise 10.3) – solution of 2D

Eulerian temperature equation with advective terms on a regular grid for a moving

medium with variable conductivity; use of upwind differences for advection of

temperature; comparison of implicit and explicit method.

Program 29: Variable_conductivity_markers2D.m (Exercise 10.4) – solution of 2D

Lagrangian temperature equation on a regular grid with implicit ﬁnite differences for

a moving medium with variable conductivity; use of marker-in-cell approach for

advection of temperature.

Chapter 11

Program 30: i2vis.m (Exercise 11.1) – 2D thermomechanical viscous code; solution of

2D Stokes, continuity, temperature and advection equations with ﬁnite-differences

and marker-in-cell technique on a regular grid using pressure–velocity formulation

for a deforming incompressible medium with variable viscosity and thermal

conductivity; taking into account radiogenic, shear and adiabatic heating.

Chapter 12

Program 31: Viscoelastic_stress.m (Exercise 12.2) – computation of visco-elastic stress

build-up/relaxation with time.

Program 32: Viscoelastoplastic_strain_rate.m (Exercise 12.3) – computation of

visco-elasto-plastic stress build-up and associated viscous, elastic and plastic strain

rate evolution with time.

322 Appendix

Program 33: Peierls_creep.m (Exercise 12.4) – computation and visualisation of

viscosity maps in temperature – log stress coordinates for a combination of

dislocation, diffusion and Peierls creep; comparison of wet and dry olivine rheology.

Chapter 13

Program 34: Viscoelastic2D.m (Exercise 13.1) – 2D thermomechanical visco-elastic

code; solution of 2D Stokes, continuity, temperature and advection equations with

ﬁnite-differences and marker-in-cell technique on a regular grid using

pressure–velocity formulation for a deforming incompressible medium with variable

viscosity, shear modulus and thermal conductivity; taking into account radiogenic,

shear and adiabatic heating.

Program 35: i2elvis.m (Exercise 13.2) – 2D thermomechanical visco-elasto-plastic code;

solution of 2D Stokes, continuity, temperature and advection equations with

ﬁnite-differences and marker-in-cell technique on a regular grid using

pressure–velocity formulation for a deforming incompressible medium with variable

viscosity, shear modulus, plastic strength and thermal conductivity; taking into

account radiogenic, shear and adiabatic heating.

Chapter 14

Program 36: Gauss_Seidel_iterations_Poisson.m (Figs. 14.1, 14.2) – solution of 2D

Poisson equation with Gauss–Seidel iteration.

Program 37: Poisson_Multigrid.m (Fig. 14.5) – solution of 2D Poisson equation with

multigrid based on V-cycle and external functions Poisson_smoother.m,

Poisson_restriction.m, Poisson_prolongation.m; resolution between multigrid levels

changes by factor of two.

Program 38: Poisson_Multigrid_planet_arbitrary.m – solution of 2D Poisson equation

for the case of a circular planetary body embedded in a mass-less like medium with

multigrid based on V-cycle and external functions Poisson_smoother_planet.m,

Poisson_restriction_planet.m, Poisson_prolongation_planet.m; resolution between

multigrid levels changes in an arbitrary way.

Program 39: Stokes_Continuity_Multigrid.m – solution of 2D Stokes and continuity

equations for a constant viscosity medium with multigrid based on V-cycle and

external functions Stokes_Continuity_smoother.m , Stokes_Continuity_

restriction.m, Stokes_Continuity_prolongation.m; resolution between multigrid

levels changes by factor of two.

Program 40: Variable_viscosity_MultiMultigrid_arbitrary.m (Fig. 14.13) – solution

of 2D Stokes and continuity equations for a variable viscosity medium with

multi-multigrid based on V-cycle and external functions Viscosity_restriction.m,

Stokes_Continuity_viscous_smoother.m.

Program 41: Poisson_Multigrid_planet.m (Exercise 14.1) – solution of 2D Poisson

equation for the case of a circular planetary body embedded in a mass less-like

medium with multigrid based on V-cycle and external functions Poisson_smoother_

planet.m, Poisson_restriction_planet.m, Poisson_prolongation_planet.m;

resolution between multigrid levels changes by factor of two.

Appendix 323

Program 42: Constant_Viscosity_Multigrid_ghost.m (Exercise 14.2, Fig. 14.10) –

solution of 2D Stokes and continuity equations for a constant viscosity medium with

multigrid based on V-cycle, ghost-node-based smoother Stokes_Continuity_

smoother_ghost.m and external functions Stokes_Continuity_restriction.m,

Stokes_Continuity_prolongation.m; resolution between multigrid levels changes by

factor of two.

Program 43: Variable_viscosity_Multigrid_arbitrary.m (Exercise 14.3, Fig 14.12) –

solution of 2D Stokes and continuity equations for a variable viscosity medium with

multigrid based on V-cycle and external functions Viscosity_restriction.m,

Stokes_Continuity_viscous_smoother.m, Stokes_Continuity_viscous_restriction.m,

Stokes_Continuity_prolongation.m; resolution between multigrid levels changes in

an arbitrary way.

Chapter 15

Program 44: Temperature3D_Gauss_Seidel.m (Exercise 15.1, Fig. 15.9) – solution of

3D temperature equation on a regular grid for a non-moving medium with variable

conductivity; the solution is based on Gauss–Seidel iteration with the use of external

function Temperature3D_smoother.m.

Program 45: Poisson3D_Multigrid_planet_arbitrary.m (Exercise 15.2, Fig. 15.10) –

solution of 3D Poisson equation for the case of a spherical planetary body

embedded in a mass-less like medium with multigrid based on V-cycle and external

functions Poisson3D_smoother_planet.m, Poisson3D_restriction_planet.m,

Poisson3D_prolongation_planet.m; resolution between multigrid levels changes in

an arbitrary way.

Program 46: Stokes_Continuity3D_Multigrid.m (Exercise 15.3, Fig. 15.11a) – solution

of 3D Stokes and continuity equations for a constant viscosity medium with multigrid

based on V-cycle and external functions Stokes_Continuity3D_smoother.m,

Stokes_Continuity3D_restriction.m, Stokes_Continuity3D_prolongation.m;

resolution between multigrid levels changes by factor of two.

Program 47: Variable_viscosity3D_Multigrid.m (Exercise 15.3, Fig. 15.11b) –

solution of 3D Stokes and continuity equations for a variable viscosity medium with

multigrid based on V-cycle and external functions Viscosity_restriction3D.m,

Stokes_Continuity3D_viscous_smoother.m, Stokes_Continuity3D_viscous_

restriction.m, Stokes_Continuity3D_prolongation.m; resolution between multigrid

levels changes by factor of two.

Program 48: Variable_viscosity3D_MultiMultigrid.m (Fig. 15.12) – solution of

3D Stokes and continuity equations for a variable viscosity medium with

multi-multigrid based on V-cycle and external functions Viscosity_restriction3D.m,

Stokes_Continuity3D_viscous_smoother.m, Stokes_Continuity3D_viscous_

restriction.m, Stokes_Continuity3D_prolongation.m; resolution between multigrid

levels changes by factor of two.

Chapter 16

Program 49: Variable_viscosity_Ramberg.m (Fig. 16.1) – mechanical benchmark for a

two-layer Rayleigh–Taylor problem; solution of 2D Stokes, continuity and advection

324 Appendix

equations with ﬁnite-differences and marker-in-cell technique using external

function Stokes_Continuity_solver_ghost.m.

Program 50: Variable_viscosity_block.m (Fig. 16.3, Exercise 16.1) – mechanical

benchmark for a falling square block; solution of 2D Stokes, continuity and

advection equations with ﬁnite-differences and marker-in-cell technique using

external function Stokes_Continuity_solver_ghost.m.

Program 51: Variable_viscosity_channel.m (Fig. 16.4) – mechanical benchmark for a

channel ﬂow with a non-Newtonian rheology; solution of 2D Stokes, continuity and

advection equations with ﬁnite-differences and marker-in-cell technique using

external function Stokes_Continuity_solver_channel.m.

Program 52: Constant_viscosity_channel_T.m (Fig. 16.5) – thermomechanical

benchmark for a non-steady temperature distribution in a Newtonian channel;

solution of 2D Stokes, continuity, temperature and advection equations with

ﬁnite-differences and marker-in-cell technique using external functions

Stokes_Continuity_solver_channel.m, Temperature_solver.m.

Program 53: Variable_viscosity_Couette_T.m (Fig. 16.6) – thermomechanical

benchmark for a steady Couette ﬂow with viscous heating and

temperature-dependent viscosity; solution of 2D Stokes, continuity, temperature and

advection equations with ﬁnite-differences and marker-in-cell technique using

external functions Stokes_Continuity_solver_Couette.m, Temperature_solver.m.

Program 54: Solid_Body_Rotation_T.m (Fig. 16.7) – thermal benchmark for advection

and diffusion of sharp temperature fronts in a prescribed rigid-body rotation velocity

ﬁeld; solution of 2D temperature and advection equations with ﬁnite-differences and

marker-in-cell technique using external function Temperature_solver.m.

Program 55: Variable_conductivity_channel.m (Fig. 16.8) – thermomechanical

benchmark for a steady Newtonian channel ﬂow with variable thermal conductivity;

solution of 2D Stokes, continuity, temperature and advection equations with

ﬁnite-differences and marker-in-cell technique using external functions

Stokes_Continuity_solver_Couette.m, Temperature_solver.m.

Program 56: Variable_viscosity_convection_irregular_grid.m (Figs. 16.9, 16.10) –

thermomechanical benchmark for thermal convection with constant and temperature-

and depth-dependent viscosity; solution of 2D Stokes, continuity, temperature and

advection equations with ﬁnite-differences and marker-in-cell technique on

regular/irregular grid using external functions Stokes_Continuity_solver_grid.m,

Temperature_solver_grid.m; nearly steady-state temperature distribution for 1a, 1c

and 2a cases can be loaded from data ﬁles data_1a_regular.txt, data_1c_regular.txt,

data_2a_regular.txt, data_1a_irregular.txt, data_1c_irregular.txt, data_

2a_irregular.txt.

Program 57: Stress_buildup.m (Fig. 16.11) – mechanical benchmark for stress build-up

in a visco-elastic incompressible Maxwell body; solution of 2D Stokes, continuity

and advection equations with ﬁnite-differences and marker-in-cell technique using

external function Stokes_Continuity_solver_grid.m.

Program 58: Slab_deformation.m (Fig. 16.12, Exercise 16.2) – mechanical benchmark

for recovery of the original shape of an elastic slab; solution of 2D Stokes, continuity

and advection equations with ﬁnite-differences and marker-in-cell technique using

external function Stokes_Continuity_solver_grid.m.

Program 59: Sandbox_shortening_ratio.m (Fig. 16.14) – mechanical

visco-elasto-plastic benchmark for numerical sandbox shortening experiment;

solution of 2D Stokes, continuity and advection equations with ﬁnite-differences and

Appendix 325

marker-in-cell technique using external function Stokes_Continuity_solver_

sandbox.m.

Chapter 17

Program 60: Subducting_slab_bending.m (Fig. 17.1) – thermomechanical

visco-elasto-plastic numerical model for spontaneous bending of subducting oceanic

slab; the model uses external functions Stokes_Continuity_solver_sandbox.m,

Temperature_solver_grid.m.

Program 61: Subduction.m (Figs. 17.2, 17.3) – thermomechanical visco-elasto-plastic

numerical model for spontaneous retreating oceanic subduction; the model uses

external functions Stokes_Continuity_solver_sandbox.m, Temperature_

solver_grid.m.

Program 62: Extension.m (Fig. 17.4) – thermomechanical visco-elasto-plastic numerical

model for oceanic lithosphere extension; the model uses external functions

Stokes_Continuity_solver_sandbox.m, Temperature_solver_grid.m.

Program 63: Collision.m (Figs. 17.5, 17.6) – thermomechanical visco-elasto-plastic

numerical model for post-subduction continental collision; the model accounts for

erosion/sedimentation processes and uses external functions

Stokes_Continuity_solver_sandbox.m, Temperature_solver_grid.m.

Program 64: Collision_and_breakoff.m (Fig. 17.7) – thermomechanical

visco-elasto-plastic numerical model for slab breakoff during continental collision;

the model accounts for erosion/sedimentation processes and uses external functions

Stokes_Continuity_solver_sandbox.m, Temperature_solver_grid.m.

Program 65: Intrusion_emplacement.m (Fig. 17.9) – thermomechanical

visco-elasto-plastic numerical model for trans-lithospheric maﬁc-ultramaﬁc intrusion

emplacement into the crust; equilibrium melt fraction for different rocks is computed

with external function Melt_fraction.m; the model also accounts for

erosion/sedimentation processes and uses external functions

Stokes_Continuity_solver_sandbox.m, Temperature_solver_grid.m.

Program 66: Mantle_convection.m (Fig. 17.12, 17.13) – thermomechanical

visco-elasto-plastic numerical model for mantle convection with phase changes; the

phase changes are treated based on Gibbs free energy minimisation approach with

pre-computed density and enthalpy maps in P–T space; these maps are loaded with

external function loading_database.m from data ﬁles m895_ro, m895_hh, morn_ro,

morn_hh; pre-computed non-steady temperature distribution can be loaded from

data ﬁle convection.txt; the model also uses external functions

Stokes_Continuity_solver_sandbox.m, Temperature_solver_grid.m.

Program 67: Core_formation.m (Fig. 17.14) – thermomechanical visco-elasto-plastic

numerical model for the deformation of a self-gravitating iron–silicate planetary

body; gravity ﬁeld is computed with external function Poisson_solver_

planet_grid.m; phase changes in the silicate component are treated based on Gibbs

free energy minimization approach with pre-computed density and enthalpy maps in

P–T space; these maps are loaded with external function loading_database.m from

data ﬁles m895_ro, m895_hh, morn_ro, morn_hh; the model also uses external

functions Stokes_Continuity_solver_sandbox.m, Temperature_solver_grid.m.

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