Appendix Numerical method: extensive description.
Appendix A. Numerical model
We use our thermo-mechanical code Flamar v12 to assess the response of multilayered
visco-elasto-plastic lithosphere. The code is based on the FLAC [Cundall, 1989] and Parovoz
algorithm [Poliakov et al., 1993] and is described in many previous studies [e.g., Burov and
Poliakov, 2001; Burov et al., 2001; Burov et al., 2003; Burov and Guillou-Frottier, 2005;
Burov and Cloetingh, 2009]. Here we limit the description of the code to most essential
features: the ability to handle (1) large strains and multiple visco-elastic-plastic rheologies
(EVP) including Mohr-Coulomb failure (faulting) and non-linear pressure-temperature and
strain-rate dependent creep; (2) strain localization; (3) thermo-dynamic phase transitions; (4)
internal heat sources; (5) free surface boundary conditions and surface processes.
A.1. Basic equations
As its prototypes FLAC [Cundall, 1989] and Parovoz [Poliakov et al., 1993], Flamar has a
mixed finite-difference/finite element numerical scheme, with a Cartesian coordinate frame
and 2D plane strain formulation. The Lagrangian mesh is composed of quadrilateral elements
subdivided into 2 couples of triangular sub-elements with tri-linear shape functions. Flamar
uses a large strain fully explicit time-marching scheme. It locally solves full Newtonian
equations of motion in a continuum mechanics approximation:
𝜌𝑢̈ − 𝑑𝑖𝑣𝜎 − 𝜌𝑔 = 0
(A.1)
coupled with constitutive equations:
𝐷𝜎
𝐷𝑡
= 𝐹(𝜎, 𝑢, 𝑣, ∇𝑣, … 𝑇 … )
(A.2)
and with equations of heat transfer, with heat advection term included in the lagrangian
derivative DT/Dt, are:
n
CpDT/Dt –(kT)-  H i = 0
(A.3)
i
 = f(P,T)
(A.4)
Here u, , g, k are the respective terms for displacement, stress, acceleration due to body
forces and thermal conductivity. P is pressure (negative for compression). The inertial term
(first parameter in the equation A.1) is negligible for geodynamic applications. It is retained
since FLAC employs an artificial inertial dampening density allowing to slow-down the
elastic waves and hence advance with considerably larger time steps (Cundall, 1989) than
would be required in a fully inertial mode. The triangular brackets in Eq. (A.1) specify
conditional use of the related term (in quasi-static mode inertial terms are damped using
inertial mass scaling [Cundall, 1989]. The terms t, , Cp, T, Hi designate respectively time,
density, specific heat capacity, temperature, internal heat production per unit volume,
respectively. The symbol  means summation of various heats sources Hi. The expression  =
f(P,T) refers to the formulation, in which phase changes are taken into account and density is
computed by a thermodynamic module that evaluates the equilibrium density of constituent
mineralogical phases for given P and T as well as latent heat contribution Hl to the term (
n
åH
i
= Hr + Hf + Hl + Ha…), witch also accounts for radiogenic heat Hr , frictional
i
dissipation Hf and adiabatic heating Ha. The terms D/Dt, F is the objective Jaumann stress
time derivative and a functional, respectively. In the Lagrangian method, incremental
displacements are added to the grid coordinates allowing the mesh to move and deform with
the material. This allows for the solution of large-strain problems while using locally a smallstrain formulation: on each time step the solution is obtained in local coordinates, which are
then updated in a large-strain mode, as in a standard finite element framework. The adiabatic
𝛼𝑇𝜌𝑣𝑦 𝑔
heating term is specified as [Gerya, 2010]: 𝐻𝑎 = 1−𝛼(𝑇−𝑇 ) where To=298 K and α is the
0
coefficient of thermal expansion
Solution of Eq. (A.1) provides velocities at mesh points used for computation of element
strains and of heat advection
. These strains are used in Eq. (A.2) to calculate element
stresses and equivalent forces used to compute velocities for the next time step. Due to the
explicit approach, there are no convergence issues, which is rather common for implicit
methods in case of non-linear rheologies. The algorithm automatically checks and adopts the
internal time step using 0.1 – 0.5 of Courant’s criterion of stability, which warrants stable
solution.
A.2. Phase changes
Direct solution for density (Eq. (A.4),  = f(P,T)) is obtained from optimization of Gibbs
free energy for a typical mineralogical composition (5 main mineralogical constituents) of the
mantle and lithosphere. With that goal, we coupled Flamar with the thermodynamic codes
Perplex or PERPLE_X [Connolly, 2005] and Theriak [De Capitani, 1994]. Perplex was used
for mantle and most crustal rocks and Theriak – for granites and sediments [Yamato et al.,
2007, 2008, Figure sf06]. These codes minimize free Gibbs energy G for a given chemical
composition to calculate an equilibrium mineralogical assemblage for given P-T conditions:
n
G   Ni
(A.5)
i 1
where  i is the chemical potential and Ni the moles number for each component i constitutive
of the assemblage. Given the mineralogical composition, the computation of density is
straightforward [Yamato et al., 2007, 2008]. The thermodynamic and solid state physics
solutions included in PERPLE_X also yield estimations for elastic and thermal properties of
the materials, which are integrated in the thermo-mechanical kernel Flamar via Eq. (A.1),
(A.2), (A.3) and (A.4).
A.3. Explicit elastic-viscous-plastic rheology
We use a serial Maxwell-type solid valid for isotropic matarial, in which the total
strain increment in each numeric element is defined by the sum of elastic, viscous and brittle
strain increments. In contrast to fluid dynamic approaches, where non-viscous rheological
terms are simulated using pseudo-plastic and pseudo-elastic viscous terms [e.g., Bercovici et
al., 2001; Solomatov and Moresi, 2000], Flamar explicitly treats all rheological terms. The
parameters of elastic-ductile-plastic rheology laws for crust and mantle are derived from rock
mechanics data (Table 2) [Kirby and Kronenberg, 1987 and Kohlstedt et al., 1995].
A.4. Plastic (brittle) behavior
The brittle behavior of rocks is described by Byerlee’s law [Byerlee, 1978; Ranalli,
1995] which corresponds to a Mohr-Coulomb material with friction angle  = 30° and
cohesion C0 < 20 MPa:
 = C0 + n tan
(A.6)
where n is normal stress n = ⅓I + II dev sin, ⅓I = P is the effective pressure (negative
for compression), II dev is the second invariant of deviatoric stress, or effective shear stress.
The condition of the transition to brittle deformation (function of rupture f) reads as: f = II dev
+ P sin - C0 cos = 0 and f/t= 0. In terms of principal stresses, the equivalent of the yield
criterion (8) reads as:
1 - 3 = - sin (1 + 3 – 2C0/tan)
(A.7)
A.5. Elastic behavior
The elastic behavior is described by the linear Hooke’s law:
ij = iiij + 2Gij
(A.8)
where  and G are Lame’s constants. Repeating indexes mean summation and  is the
Kronecker’s operator.
A.6. Viscous (ductile) behavior
Within deep lithosphere and underlying mantle regions, creeping flow is highly
dependent on temperature and is non-linear non-Newtonian since the effective viscosity also
varies as function of differential stress [Kirby and Kronenberg, 1987; Ranalli, 1995]:
e d = A (1 - 3) n exp (-Q R-1T-1)
(A.9a)
where e d is effective shear strain rate, A is a material constant, n is the power-law
exponent, Q = Ea + PV is the activation enthalpy , Ea is activation energy, V is activation
volume, P is pressure, R is the universal gas constant, T is temperature in K, and are the
principal stresses. In case of the diffusion creep, A=amAd where a is grain size, m is
experimental parameter for diffusion creep. The effective viscosity eff for this law is defined
as:
eff = e (1-n)/n A-1/n exp (Q (nRT)-1)
(A.10a )
The dominating creep mechanism in the lithosphere is dislocation creep while in deeper
sublithosphere mantle this role is played by the diffusion creep. The composition viscosity is
calculated according to the generally adopted mixing rule [e.g., Burov, 2011]:
eff -1=(1-1 + 2-1 + …)-1
(A.10b)
The commonly referred value of the activation volume V for mantle rock is 9.5×10-6 m3
mol-1 [Durham et al., 2009]. Dislocation creep parameters are provided in the Table 2. For the
diffusion creep the activation energy Ea=300 KJ mol-1, Ad=1.92×10-1 MPa-1 s-1, a=1 and m=1
[Karato and Wu, 1993]. Due to the uncertainties of diffusion creep parameters, we have run
several numerical tests (supplementary Figure sf04) in which we have varied the activation
volume in order to obtain several commonly inferred viscosity profiles.
For non-uniaxial deformation, the law (Eq. (A.10)) is converted to a triaxial form, using
the invariant of strain rate and geometrical proportionality factors:
eff = e dII (1-n)/n (A*)-1/n exp (Q (nRT)-1 )
where e dII = (InvII ( e ij))½ and A* = ½A·3(n+1)/2
(A.11)
The parameters A, n, Q are the experimentally determined material parameters (Table 2).
Using olivine parameters, we verify that the predicted effective viscosity just below the
lithosphere is 1019 – 5  1019 Pa s matching post-glacial rebound data [Turcotte and Schubert,
2002]. Due to temperature dependence of the effective viscosity, the viscosity decreases from
1025 - 1027 Pa s to asthenospheric values of 1019 Pa s in the depth interval 0-250 km. Within
the adiabatic temperature interval in the convective mantle (250 km – 650 km), the
dislocation flow law (Eq. (A.10)) is dominated by nearly Newtonian diffusion creep. In this
interval, temperature increases very slowly with depth while linearly growing pressure starts
to affect viscosity resulting in its slow growth from 1019 Pa s in the asthenosphere to a value
ranging from 1020 Pa s to 1022 Pa s at the base of the upper mantle [e.g., Turcotte and
Schubert, 2002]. Due to the uncertainties on the viscosity at the base of the upper mantle we
have tested several assumptions on creep parameters (Supplementary Figure fs04). Based on
the test (Figure fs04) we have finally adopted the value of activation volume (V=4.5×10-6 m3
mol-1) yielding viscosity on the order of 1020 Pa s at the base of the model.
A.7. Surface processes
Short range erosion. A simple law is used to simulate erosion and sedimentation at
the scale of a mountain range. The evolution of a landscape results from a combination of
weathering processes that prepare solid rock for erosion, and transportation by hillslope and
stream processes [see Carson and Kirkby, 1972 for a review]. Although many factors,
depending on the lithologies and on climate may control this evolution, quite simple
mathematical models describing the geometrical evolution of the morphology at the small
scale have been proposed and tested successfully [e.g., Kirkby et al., 1993]. For example, the
evolution of a scarp-like landforms can be modelled assuming that the rate of downslope
transport of debris, q, is proportional to the local slope, h [e.g., Culling, 1960; Braun and
Sambridge, 1997].
q=-kh,
(A.12)
where k is the mass diffusivity coefficient, expressed in units of area per time [e.g.,
m2/y]. Assuming conservation of matter along a 2-D section and no tectonic deformation, h
must obey:
dh/dt= - q .
(A.13)
The equations (12) and (13) lead to the linear diffusion equation:
dh/dt=(kh) .
(A.14)
At the larger scale, hillslope and stream processes interact and the sediment transport
then depend non-linearly on the slope and on other factors such as the slope gradient, the area
drained above the point, the distance from the water divide, so that the simple 2-D linear
diffusion does not apply in general. In spite of these limitations, we have chosen to stick to a
linear diffusion law to model erosion. This model does not accurately mimic the spatial
distribution of denudation in the mountain range but it leads to a sediment yield at the
mountain front that is roughly proportional to the mean elevation of the basin relative to that
point (a rough approximation to the sediment yield resulting from a change of elevation h
over a horizontal distance d is k h/d) and therefore accounts for the apparent correlation
between elevation and denudation rates [Avouac and Burov, 1996].
We considered values for k of 500 m2/y that yield denudation rates of the order of
those predicted for by previous parametric models [e.g., Burov, 2011] for convergence rates
characterizing Zagros collision.
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