Dual continental rift systems generated by plume ... · The numerical code I3ELVIS solves momentum, continuity and heat conservation equations for 3D creeping flow26,31. Its numerical

1

Dual continental rift systems generated by plume-lithosphere interaction

A. Koptev, E. Calais, E. Burov, S. Leroy, and T. Gerya

Supplementary Methods

Governing equations. The numerical code I3ELVIS solves momentum, continuity and heat

conservation equations for 3D creeping flow26,31. Its numerical schema is based on finite-

differences with a marker-in-cell technique, which allows for non-diffusive numerical

simulation of multiphase flow in a rectangular fully staggered Eulerian grid32. The momentum

equations are solved in the form of Stokes flow approximation. Conservation of mass is

approximated by the continuity equation. The components of the deviatoric stress tensor are

calculated using the viscous constitutive relationship between stress and strain rate for a

compressible fluid. The mechanical equations are coupled with heat conservation equations.

The code is fully thermo-dynamically coupled and accounts for mineralogical phase changes

(Perple_X33), adiabatic, radiogenic, and frictional internal heating sources. The free surface

topography is reproduced using the sticky air technique enhanced by the introduction of a

high-density marker distribution in the near-surface zone. The implicit parallel numerical

scheme uses an indirect multigrid solver31, which allows for considerable acceleration of 3D

calculations.

Viscous-plastic rheology. The material deforms according to Newtonian diffusion creep and

power law dislocation creep. In addition, Peierl’s plasticity applies when stress reaches a

specific limit (“Peierl’s stress”) that characterizes transition to plastic failure. At this moment

the dominant creep mechanism becomes dislocation glide and climb. The constitutive

equation for Peierl’s plasticity is linked to Dorn’s law34,35. The ductile rheology is combined

with a brittle/plastic rheology to yield an effective visco-plastic rheology with a Drucker-

1

Dual continental rift systems generated by plume-lithosphere interaction

A. Koptev, E. Calais, E. Burov, S. Leroy, and T. Gerya

Supplementary Methods

Governing equations. The numerical code I3ELVIS solves momentum, continuity and heat

conservation equations for 3D creeping flow26,31. Its numerical schema is based on finite-

differences with a marker-in-cell technique, which allows for non-diffusive numerical

simulation of multiphase flow in a rectangular fully staggered Eulerian grid32. The momentum

equations are solved in the form of Stokes flow approximation. Conservation of mass is

approximated by the continuity equation. The components of the deviatoric stress tensor are

calculated using the viscous constitutive relationship between stress and strain rate for a

compressible fluid. The mechanical equations are coupled with heat conservation equations.

The code is fully thermo-dynamically coupled and accounts for mineralogical phase changes

(Perple_X33), adiabatic, radiogenic, and frictional internal heating sources. The free surface

topography is reproduced using the sticky air technique enhanced by the introduction of a

high-density marker distribution in the near-surface zone. The implicit parallel numerical

scheme uses an indirect multigrid solver31, which allows for considerable acceleration of 3D

calculations.

Viscous-plastic rheology. The material deforms according to Newtonian diffusion creep and

power law dislocation creep. In addition, Peierl’s plasticity applies when stress reaches a

specific limit (“Peierl’s stress”) that characterizes transition to plastic failure. At this moment

the dominant creep mechanism becomes dislocation glide and climb. The constitutive

equation for Peierl’s plasticity is linked to Dorn’s law34,35. The ductile rheology is combined

with a brittle/plastic rheology to yield an effective visco-plastic rheology with a Drucker-

Dual continental rift systems generated by plume–lithosphere interaction

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NGEO2401

NATURE GEOSCIENCE | www.nature.com/naturegeoscience 1

© 2015 Macmillan Publishers Limited. All rights reserved

http://dx.doi.org/10.1038/ngeo2401

2

Prager yield criterion31. The visco-plastic rheology is assigned to the model by means of a

“Christmas tree”-like criterion, where the rheological behaviour depends on the minimum

viscosity (or differential stress) attained between the ductile and brittle/plastic fields36,37,38.

The upper surface of the crust is treated as internal free surface, which is implemented by

inserting a 30 km thick low-viscosity “sticky air” layer between the upper interface of the

model box and the surface of the model crust. The viscosity of the “sticky air” is 1018 Pa s and

its density is 1 kg/m3, according to optimal parameters derived in previous studies32,39.

Thermodynamic processes and partial melting. Petrological phase changes are

implemented using thermodynamic solutions for density and other physical rock properties

obtained from optimization of Gibbs free energy for a typical mineralogical composition of

the mantle, plume and lithosphere material. With that goal, the thermodynamic Perple_X33

algorithm and the associated petrological data33 has been coupled with the main code to

introduce progressive density and other material properties changes. Perple_X minimizes free

Gibbs energy for a given chemical composition to calculate an equilibrium mineralogical

assemblage for given pressure-temperature conditions. Partial melting is taken into account

using the most common parametrization40,41 of hydrous mantle melting processes.

Thermodynamic coupling applies to the mantle rocks and plume material. Melt is assumed to

be transported together with the mineral matrix. Thermo-mechanical effects of melt

percolation through the matrix are neglected. For crustal rocks we here use the simple

Boussinesq approximation since phase transformations in these rocks are of minor importance

for the geodynamic settings explored here.

Model setup. The Eulerian regularly-spaced rectangular numerical grid has a spatial

resolution of 3×3×3 km and comprises 500×500×211 nodes filled with a half billion of

randomly distributed Lagrangian markers. The spatial dimensions of the model are,

accordingly, 1500×1500×635 km. The model comprises a stratified continental lithosphere


3

composed of an upper crust (18 km), lower crust (18 km), and lithospheric mantle (150 km,

250 km in the case of a craton) overlaying the upper mantle (485 km). The upper crust has

ductile properties of granite, the lower crust is simulated as granulite, the mantle lithosphere

(dry olivine) uses olivine dislocation and Peierels flow properties, and the sub-lithospheric

mantle (dry olivine as well) deforms by diffusion creep42,43,44 (Supplementary Table 1). We

obtain the initial thermal structure by combining a conductive lithospheric geotherm with an

adiabatic mantle geotherm. The boundary conditions at the upper surface and the bottom of

the model are 0°C and 1700ºC, respectively. The initial temperature at the bottom of the

lithosphere is 1300ºC (150 km depth for normal lithosphere and 250 km depth for craton).

The resulting adiabatic thermal gradient in the mantle is 0.5-07°C/km. We impose zero

outflow lateral thermal boundary conditions. We initiate a mantle plume at the bottom of the

model as a 200 km-diameter sphere centred below the craton with an initial temperature of

2000ºC. We apply a constant divergent velocity boundary condition to the east and west sides

of the model. North and south boundary conditions are free slip. We compensate vertical

influx velocities through the upper and lower boundaries to ensure mass conservation in the

model domain.

In-depth description of the computer code I3ELVIS that may be necessary to reproduce our

results is provided in the book by T. Gerya31.


4

REFERENCES

31. Gerya, T.V. Introduction to Numerical Geodynamic Modelling. Cambridge University

Press, 345 pp (2010).

32. Duretz, T., May D.A., Gerya T. V. & P. J. Tackley. Discretization errors and free surface

stabilization in the finite difference and marker-in-cell method for applied geodynamics:

A numerical study. Geochemistry, Geophysics, Geosystems 12, Q07004 (2011).

33. Connolly, J.A.D. Computation of phase equilibria by linear programming: a tool for

geodynamic modeling and its application to subduction zone decarbonation. Earth and

Planetary Sci. Lett. 236, 524-541 (2005).

34. Kohlstedt, D.L., Evans, B. & Mackwell, S.J. Strength of the lithosphere: constraints

imposed by laboratory experiments. J. Geophys. Res. 100, 17 587-17 602 (1995).

35. Ranalli, G. Rheology of the Earth. Chapman and Hall, London, p. 413 (1995).

36. Burov, E. & Cloetingh, S. Controls of mantle plumes and lithospheric folding on modes of

intra-plate continental tectonics: differences and similarities. Geophys. J. Int. 178, 1691-

1722 (2010).

37. Burov, E. Rheology and strength of the lithosphere. Marine and Petroleum Geology 28,

1402-1443 (2011).

38. Bürgmann, R. & Dresen, G. Rheology of the lower crust and upper mantle: evidence from

rock mechanics, geodesy, and field observations. Annual Review of Earth and Planetary

Sciences 36, 531-567 (2008).

39. Crameri, F. et al. A comparison of numerical surface topography calculations in

geodynamic modelling: an evaluation of the ‘sticky air’ method. Geophys. J. Int. 189, 38-

54 (2012).

40. Gerya, T. Initiation of Transform Faults at Rifted Continental Margins: 3D Petrological–

Thermomechanical Modeling and Comparison to the Woodlark Basin. Petrology 21, 550-


5

560 (2013).

41. Katz, R.F., Spiegelman, M. & Langmuir, C.H. A new parameterization of hydrous mantle

melting. Geochemistry, Geophysics, Geosystems 4, 1073 (2003).

42. Durham, W.B., Mei, S., Kohlstedt, D.L., Wang, L. & Dixon, N.A., New measurements of

activation volume in olivine under anhydrous conditions. Phys. Earth and Planet. Int. 172,

67-73 (2009).

43. Karato, S.I. & Wu, P. Rheology of the Upper Mantle. Science 260, 771-778 (1993).

44. Caristan, Y. The transition from high temperature creep to fracture in Maryland diabase.

J. Geophys. Res. 87, 6781-6790 (1982).


6

Supplementary Figures

Supplementary Figure 1. Model setup. UC = upper crust; LC = lower crust; ML = mantle

lithosphere; M = mantle; P = plume; CM = craton mantle. The strength envelope and thermal

gradient of the lithosphere are shown on the right side. Grey arrows show the velocity

boundary conditions, applied in a direction perpendicular to the model domain. The initial

diameter of the plume is 200 km.


7

Supplementary Figures 2 to 9. Series of 3D numerical experiments showing plume

impingement beneath a craton. Each column corresponds to a set of experimental

parameters, listed in the Supplementary Table 2, with three successive evolution stages.

Plume material is shown in dark red, the craton is the dark blue quasi-rectangular volume

embedded in the lithosphere in the middle of the model domain. Blue to red colours at the

surface of the model indicate cumulative strain due to faulting. Experiments without a plume

(not shown here) result in small-offset distributed faulting, in part because of the very slow

plate divergence rate, while in the absence of a craton the plume leads to focusing of the

small-offset distributed faults in a single classical rift zone (R1). Experiment R8 represents a

case of central plume and oblique (NW-SE or SW-NE) far-field stretching. Experiment R9

refers to a weak (quartz-like rheology) sub-vertical rheological interface of 5 km width what

separates craton from normal lithosphere. This experiment tests the inferences of the

hypothesis that the boundary between the craton and the embedding lithosphere may be

mechanically weakened. None of these experiments is able to reproduce the first-order

observations of the CEAR described in the text as well as the experiment R3 shown in Fig. 3.


8

Supplementary Figure 2. Experiment R1. (See Supplementary Table 2 for details)


9



10



11



12



13



14



15



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Supplementary Figure 10. Surface topography and horizontal surface velocity

distribution illustrating counter-clockwise rotation of the cratonic micro-continent

between two rift branches in case of the best fitting experiment of Fig. 3.


17

Supplementary Figure 11. Comparison of the model-predicted and observed

topography. Top: Topography map showing the position of representative SW-NE

topography profile. Bottom: Comparison between the smoothed topography profile and mode

predicted EW topography profile (experiment from Fig. 3). Because our model ignores small-

scale processes such as erosion and the subsequent sedimentation in the rift basins, short

wavelengths on the topographic cross section shown in top panel had to be (low pass) filtered

to be compared to model results. Due to the difference in projection (SW-NE versus WE) and

the geometry between the model craton and the real one, the model-predicted profile has been

cut in the middle and its “western” and “eastern part” were slightly shifted by 3% (50 km) in

both directions.


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Supplementary Tables

Supplementary Table 1. Rheological and material properties.

The effective viscosity in ductile regime is computed as26:

11

12

nE PV nnRT

powl D IIA eη ε−+

=

& , where ηpowl is the effective viscosity, ε& is strain rate, AD is

the material constant (in [Pan s]): AD = A-1, where A is material constant of power flow law35: E PV

n RTA eε σ+−

=& , where E is activation energy, V is

activation volume, n is power law stress exponent. For the diffusion creep (mantle), creep parameters43,44 are: activation energy E = 300 kJ mol-1,

A = 1.92×10-10 MPa-1 s-1, n = 1 (grain size dependence is included in the material constant A). T and P are temperature and pressure,

respectively.

Material

0ρ

kg/m3

Rheological parameters

Thermal parameters

Flow law / reference

Ductile Brittle k W/(m×K)

rH µW/m3 E

kJ/mol n

DA Pan × s

V J/(MPa×mol)

C (MPa) )sin(ϕ ε

C0 C1 0b 1b 0ε

1ε

Upper crust 2750 Wet quartzite35

154 2.3 1.97×1017 0 10 3 0.6 0.0 0.0 0.25 0.64+807/(T+77) 2.00

Lower crust

1 2950 Wet quartzite35

154 2.3 1.97×1017 0 10 3 0.6 0.0 0.0 0.25 - // - 1.00

2 3000 Plagioclase35, An75

238 3.2 4.80×1022 0 10 3 0.6 0.0 0.0 0.25 1.18+474/(T+77) 0.25

Lithosphere-sublithosphere

mantle

3300

Dry olivine34

532

3.5

3.98×1016

1.6

10

3

0.6

0.0

0.0

0.25

0.73+1293/(T+77)

0.022

Plume mantle 3200 Wet olivine37 470 4.0 5.01×1020 1.6 3 3 0.1 0.0 0.0 0.25 - // - 0.024


19

Here 0ρ is reference density (at 0P = 0.1 MPa and 0T = 298 K), ε is strain, C0, C1 are maximal and minimal cohesion (linear softening law), 0ε ,

1ε are minimal and maximal strains (linear softening law), 0b , 1b are maximal and minimal frictional angles (linear softening law), ϕ is friction

angle, k is thermal conductivity, rH is radiogenic heat production. Mantle densities, thermal expansion, adiabatic compressibility, and heat

capacity are computed as function of pressure and temperature in accord with a thermodynamic petrology model by Conolly (Perple_X33). For

the crustal rocks we used simple Boussinesq approximation [ ][ ])(1)(1 000 PPTT −+−−= βαρρ , where α = 3×10-5 K-1 is thermal expansion

coefficient, β = 1×10-3 MPa-1 is adiabatic compressibility and pC = 1000 J×kg-1K-1 is heat capacity. For all types of rocks the transition

stress crσ , defining transition from diffusion creep to dislocation creep26 is 3×104 Pa.


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Supplementary Table 2. Controlling parameters of the experiments displayed in Fig. 2-4 and Supplementary Figures 2-9.

(N – north, S – south, W – west, E – east)

Experiment

title

Controlling parameters

Presence of

craton

Presence of a

weak rheological

interface craton

borders

Craton

thickness

(km)

Initial plume position Horizontal extension velocity

(mm/year)

R1 No No - Centre WE, 3

R2 Yes No 250 Centre WE, 3

R3 Yes No 250 NE (250km) shift WE, 3

R4 Yes No 250 NE (250km) shift WE, 1.5

R5 Yes No 250 E (100km) shift WE, 3

R6 Yes No 250 N (200km) shift WE, 3

R7 Yes No 200 NE (250km) shift WE, 3

R8 Yes No 250 Centre NW-SE, 3 (oblique)

R9 Yes Yes 250 NE (250km) shift WE, 3


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Dual continental rift systems generated by plume ... · The numerical code I3ELVIS solves momentum, continuity and heat conservation equations for 3D creeping flow26,31. Its numerical