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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-
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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
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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
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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.
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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-
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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).
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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.
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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.
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Supplementary Figure 2. Experiment R1. (See Supplementary Table 2 for details)
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9
Supplementary Figure 3. Experiment R2. (See Supplementary Table 2 for details)
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Supplementary Figure 4. Experiment R4. (See Supplementary Table 2 for details)
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Supplementary Figure 5. Experiment R5. (See Supplementary Table 2 for details)
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Supplementary Figure 6. Experiment R6. (See Supplementary Table 2 for details)
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Supplementary Figure 7. Experiment R7. (See Supplementary Table 2 for details)
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Supplementary Figure 8. Experiment R8. (See Supplementary Table 2 for details)
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Supplementary Figure 9. Experiment R9. (See Supplementary Table 2 for details)
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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.
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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
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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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