Article
Volume 13, Number 1
31 January 2012
Q01012, doi:10.1029/2011GC003927
ISSN: 1525-2027
Deformation and seismicity associated with continental rift
zones propagating toward continental margins
V. Lyakhovsky and A. Segev
Geological Survey of Israel, 30 Malkhei Israel Street, Jerusalem 95501, Israel ([email protected])
U. Schattner
Dr. Moses Straus Department of Marine Geosciences, Charney School of Marine Sciences,
Faculty of Natural Science, University of Haifa, Haifa 31905, Israel
R. Weinberger
Geological Survey of Israel, 30 Malkhei Israel Street, Jerusalem 95501, Israel
[1] We study the propagation of a continental rift and its interaction with a continental margin utilizing a 3-D
lithospheric model with a seismogenic crust governed by a damage rheology. A long-standing problem in
rift-mechanics, known as the tectonic force paradox, is that the magnitude of the tectonic forces required
for rifting are not large enough in the absence of basaltic magmatism. Our modeling results demonstrate
that under moderate rift-driving tectonic forces the rift propagation is feasible even in the absence of magmatism. This is due to gradual weakening and “long-term memory” of fractured rocks that lead to a significantly lower yielding stress than that of the surrounding intact rocks. We show that the style, rate
and the associated seismicity pattern of the rift zone formation in the continental lithosphere depend not
only on the applied tectonic forces, but also on the rate of healing. Accounting for the memory effect provides a feasible solution for the tectonic force paradox. Our modeling results also demonstrate how the
lithosphere structure affects the geometry of the propagating rift system toward a continental margin. Thinning of the crystalline crust leads to a decrease in the propagation rate and possibly to rift termination
across the margin. In such a case, a new fault system is created perpendicular to the direction of the rift
propagation. These results reveal that the local lithosphere structure is one of the key factors controlling
the geometry of the evolving rift system and seismicity pattern.
Components: 12,100 words, 14 figures, 1 table.
Keywords: continental margin; continental rifting; damage rheology; numerical simulations; tectonic force.
Index Terms: 8002 Structural Geology: Continental neotectonics (8107); 8020 Structural Geology: Mechanics, theory, and
modeling.
Received 17 October 2011; Revised 1 December 2011; Accepted 2 December 2011; Published 20 January 2012.
Lyakhovsky, V., A. Segev, U. Schattner, and R. Weinberger (2012), Deformation and seismicity associated with continental rift
zones propagating toward continental margins, Geochem. Geophys. Geosyst., 13, Q01012, doi:10.1029/2011GC003927.
Copyright 2012 by the American Geophysical Union
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1. Introduction
[2] During the last decades considerable efforts
were dedicated to modeling of continental and
oceanic rifting [e.g., Keen, 1985; Parmentier and
Schubert, 1989; Houseman and England, 1986,
Sonder and England, 1989; Dunbar and Sawyer,
1989; Melosh, 1990; Negredo et al., 1995; Buck
and Poliakov, 1998]. Many authors have discussed how tectonic processes could produce relative tension that initiates and propagates rift zones
[e.g., Forsyth and Ueda, 1975; Solomon et al.,
1980] and estimated the magnitude of the tectonic
rift-driving forces. Spohn and Schubert [1982] and
Bott [1991] related the tectonic rift-driving forces to
the uplift over a region of abnormally hot mantle
and provided the scaling relations between extensional forces, uplift magnitude and the depth of the
low density compensation level. Both analytic and
semi-analytic models [e.g., McKenzie, 1978; Buck,
1991] as well as numerical simulations [e.g., Braun
and Beaumont, 1989; Bassi, 1991; Dunbar and
Sawyer, 1989] assume that the tectonic force
required to initiate rifting is available. Following these
ideas, most recent thermomechanical rifting models
combine asthenosphere doming with passive extension of the lithosphere with a given rate of remote
plate motion (e.g., Huismans et al. [2001], Ziegler and
Cloetingh [2004], Rosenbaum et al. [2010], and
others). These models do not consider the available
tectonic forces and energy required for the extension.
[3] Lithospheric strength profiles typically include
a brittle upper crust overlying a viscous lower crust
and a mantle. This viscous part can behave as a
ductile layer or change from brittle to ductile
behavior with depth [Goetze and Evans, 1979;
Brace and Kohlstedt, 1980; Kohlstedt et al., 1995].
The resulting curve is known as the Brace-Goetze
strength profile. It provides a general and simple
framework for estimating lithosphere strength. This
strength is defined as an integrated differential stress
over the whole lithosphere under a constant strain
rate and given depth-dependent temperature. Based
on these considerations, Vink et al. [1984] noted that
a continental lithosphere, with its granitic upper crust
and deep Moho interface, should be weaker than an
oceanic lithosphere. The mechanical behavior of the
oceanic lithosphere, with a relatively shallow upper
mantle, is dominated by a stronger olivine rheology.
Therefore, rift propagating through a continental
lithosphere might cease before approaching a transition to an oceanic lithosphere. Buck [2004, 2006,
2009] estimated the minimum tectonic force needed
to allow slip on optimally oriented normal faults in a
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typical continental lithosphere (i.e., 30 km thick
crust) with a thermal profile corresponding to surface
heat flow below 60 mW/m2. He concluded that the
available forces are not large enough for rifting in the
absence of basaltic magmatism (the “tectonic-force”
paradox). It was suggested that continental rifting is
always associated with simultaneous dike emplacement and tectonic forces.
[4] The above conclusions are based on the
assumption that tectonic stress has to overcome the
yielding stress integrated over the whole lithosphere
thickness ignoring any weakening mechanism,
either in the brittle upper crust or in the olivinedominated upper mantle. In spite of intensive
experimental and theoretical studies, mantle weakening processes are difficult to constrain [e.g.,
Karato, 2010, and references therein]. Among these
processes are water weakening, transition from diffusion to dislocation creep in olivine, and perhaps
the most important effect is the influence of the
change in grain-size. Bercovici et al. [2001] developed a phenomenological two-phase damage theory
that has been extended to consider the contribution
of grain size reducing damage to the development of
strain localization and strength reduction of the
lithosphere [Bercovici and Ricard, 2005; Landuyt
et al., 2008; Ricard and Bercovici, 2009; Landuyt
and Bercovici, 2009]. The two-phase damage theory as well as a new mathematical formulation of
continuum damage mechanics [Karrech et al., 2011]
are mostly relevant for deep lithospheric layers.
However, they are not appropriate for a detailed
analysis of brittle failures in the seismogenic zone
where elastic deformation and brittle fracturing play
the dominant role.
[5] At low temperatures and low pressures, typical
for the seismogenic zone, a material is deformed by
brittle failure rather than viscous flow. The pervasive damage in the form of micro-cracks and voids
develops as part of rock formation, and it usually
increases during tectonic loading leading to degradation of strength and rock elasticity. Weakening of
the brittle rocks with plastic strain accommodated
by evolving fault zones in models of rifting has
been considered through the decrease of the cohesion [e.g., Lavier et al., 2000] and the angle of
internal friction [e.g., Behn et al., 2002; Huismans
and Beaumont, 2007, 2011]. Buck [2009] noted
that the reduction in strength of the brittle layer due
to cohesion loss is fairly small compared to the
strength that remains due to rock friction. Based on
rate-dependent frictional strength change observed
in laboratory studies, Behn et al. [2002] suggested
the model with a degree of strain rate softening
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Figure 1. (a) A schematic 3-D diagram of the simulated volume. White dash line in the middle of the crystalline
basement marks the expected depth of the seismogenic zone. (b) Temperature distribution with depth.
proportional to the logarithm of the sliding velocity.
The adopted velocity weakening of fault zones
results in efficient strain localization, but has a
minor effect on the overall strength of the brittle
layer. Significant strength reduction of the brittle
layer was introduced by Huismans and Beaumont
[2007, 2011] and others. They assumed an unrealistically low friction angle of 15° (friction coefficient 0.27) for the intact rock, which is not
supported by laboratory experiments, and further
decreasing its value with accumulated strain up to
1 or 2° (friction coefficient below 0.03). This
mathematical model predicts formation of essentially frictionless fault zones.
[7] The geology of the Eastern Mediterranean
motivated this study providing two examples of
continental rifts, the Suez rift [Steckler and ten
Brink, 1986] and the Azraq-Sirhan rift [Sharland
et al., 2001; Schattner et al., 2006; Segev and
Rybakov, 2010, 2011]. Both rifts propagated during the Oligocene toward the northwest and terminated at the Levant continental margin. Some
aspects concerning these rift zones and the Andaman
Sea back-arc extensional basin will be shortly discussed in the light of the present results.
[6] We study the initiation of a non-magmatic
continental rift and the interaction of a propagating
rift with a continental margin. We utilize 3-D
simulations of the tectonic motion and faulting
process in a layered model of the lithosphere. These
layers consist of a brittle upper crust governed by a
nonlinear continuum damage rheology, overlying a
visco-elastic upper mantle of wet olivine. The
employed damage rheology constrained by various
observations from laboratory friction and fracture
experiments provides an adequate tool for studying
the evolution of geometrical and material properties
of crustal fault zones, along with the evolution of
various types of seismicity patterns. Results of the
simulations demonstrate that gradual weakening
and healing of the brittle rocks during earthquake
cycle allows rifting under moderate tectonic stress
and provides a feasible solution for the tectonicforce paradox. Results also demonstrate how
changes in the Moho depth affect the geometry of a
propagating rift toward continental margin, the
associated strain field, and its seismicity pattern.
[8] The numerical model simulates propagation of
a continental rift through a 3-D layered lithosphere
structure and its interaction with a continental
margin. The model consists of three different layers
representing a weak sedimentary cover, a crystalline
basement (crust) down to the Moho, and an upper
mantle (Figure 1). We first discuss the rheology of
the layers and main factors controlling their
mechanical behavior. Then we discuss regional
lithosphere stresses, their connection with forces
driving the rifting process and their implementation
in the numerical model through boundary conditions. A brief review of the numerical method and
geometry of the simulated volume is also presented.
2. Model Formulation
2.1. Mechanics of Rock Deformation
2.1.1. Brittle Deformation
[9] Hey et al. [1980] considered rift propagation
analogous to mode-I crack propagation in an elastic
material, where the crack propagation is controlled
by the stress concentration at its tip. Linear elastic
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Table 1. Material Properties
Material Unit
Sedimentary layer
Crust
Upper mantle
Densitya
(kg/m3)
2600
2800
3200
Seismic Wave
Velocitiesa (km/s)
Elastic Moduli for
Damage-Free
Materialb (GPa)
VP
l0
4.5
6.2
7.9
VS
2.5
3.3
4.2
23
46
86
m0
15
31
57
Ductile Flow Parametersc
A (Pan/s)
22
110
6.31020
1.91015
n
Q (kJ/mol)
1
3.05
3.0
276
420
a
Geophysical literature.
Calculated according to the density and seismic wave velocity (a) using linear elasticity.
c
Carter and Tsenn [1987], Tsenn and Carter [1987], Hirth and Kohlstedt [2003], and Karato [2010].
b
fracture mechanics postulates that an isolated crack
propagates at velocities approaching the speed of
sound in a medium, once a critical stress intensity
factor is reached [e.g., Freund, 1990]. At lower
values of the stress intensity factor the crack
remains stable. Therefore, the classical crack theory
introduces no limiting physical mechanisms to
determine the rate of rift propagation. Several
studies [e.g., Phipps Morgan and Parmentier,
1985; Schubert and Hey, 1986; Parmentier and
Schubert, 1989] suggested that viscous suction
near the tip of a propagating rift segment causes it to
resist propagation and could control its rate. Rocks
and rock-like materials, subjected to long-term
loading, show an accumulation of micro-cracks in
the process zone around the tip of a macroscopic
crack. The micro-cracks are localized in a narrow
weak zone and their coalescence leads to a slow
extension of the macroscopic crack at the stress
intensity factor values significantly lower than its
critical value. This phenomenon, known as subcritical crack growth [e.g., Swanson, 1984; Atkinson and
Meredith, 1987; Cox and Scholz, 1988], is another
mechanism that controls the rate of rift propagation.
A realistic rheological model of rifting processes
should include a subcritical crack growth from very
early stages of the loading, material degradation due
to increasing micro-crack concentration, macroscopic brittle failure, post failure deformation, and
healing. Continuum damage mechanics based on
pioneering works by Robinson [1952], Hoff [1953],
Kachanov [1958, 1986], Rabotnov [1959, 1988] and
further developed in engineering [e.g., Hansen and
Schreyer, 1994; Kachanov, 1994; Krajcinovic,
1996; Lemaitre, 1996; Allix and Hild, 2002] and
earth sciences [e.g., Newman and Phoenix, 2001;
Shcherbakov and Turcotte, 2003; Turcotte et al.,
2003; Bercovici et al., 2001; Bercovici, 2003;
Ricard and Bercovici, 2003; Bercovici and Ricard,
2003, Karrech et al., 2011] provide a framework
for the rheological model of the faulting and rifting
processes.
[10] In this study we use the damage rheology
model of Lyakhovsky et al. [1997] and Hamiel et al.
[2004], further developed by Lyakhovsky et al.
[2011] and Hamiel et al. [2011]. The model
accounts for the following general aspects of brittle
rock deformation: (1) nonlinear elasticity that connects the effective elastic moduli to a damage variable and loading conditions; (2) Evolution of the
damage state variable as a function of the ongoing
deformation and gradual conversion of elastic strain
to permanent inelastic deformation during material
degradation; (3) Macroscopic brittle instability at a
critical level of damage and related rapid conversion of elastic strain to permanent inelastic strain.
This approach was adopted by Ben-Zion et al. [1999],
Lyakhovsky [2001], Ben-Zion and Lyakhovsky [2002,
2006], Lyakhovsky and Ben-Zion [2009], BenAvraham et al. [2010], and Finzi et al. [2009, 2011]
to study the evolution of geometrical and material
properties of crustal fault zones, along with the evolution of various types of seismicity patterns.
[11] The damage rheology framework allows cal-
culating the simultaneous evolution of damage, a,
and its localization into narrow highly damaged
zones (faults), earthquakes, and associated deformation fields. Synthetic earthquake catalogs generated during the model runs enable analyzing the
coupled evolution of faults and seismicity pattern.
The elastic moduli of the damage-free rocks (a = 0)
are estimated from typical values of the rock density and seismic wave velocities (Table 1) widely
accepted in the geophysical literature. The condition for the onset of damage accumulation is
derived from the basic thermodynamic balance
equations [e.g., Lyakhovsky et al., 1997] and is
formulated
in terms of the strain invariants ratio,
pffiffiffiffi
I1/ I2 (I1 = "kk and I2 = "ij "ij are two invariants
of the elastic strain tensor "ij). The critical value of
the strain invariants ratio at the onset of damage
accumulation is connected to the internal friction
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Figure 2. Strength profile for the damage free upper brittle layer (a = 0) and yielding profile for the highly damaged
zone (a = 1). Colored lines show recovered strength profile hundred years after the seismic event for various kinetic
coefficients (see Figure 3 for values and details).
angle of the Byerlee’s law [Byerlee, 1978]. For most
rocks the coefficient of friction varies between 0.5
and 0.8; a value of 0.6 is a good approximation for
the general use, corresponding
to the critical strain
pffiffiffiffi
invariants ratio, I1/ I2 0.8. The tensile stress
(vertical minus horizontal components) required for
the onset of damage for the intact rock (a = 0;
Figure 2) is identical to the Brace-Goetze strength
profile for the upper brittle layer. With the onset of
damage the rate of its accumulation is controlled by
the applied loading and the model kinetic coefficient, Cd. Its value is associated with the duration of
the sample loading from the onset of acoustic
emission till its macroscopic failure in the rock
mechanics laboratory experiment. The laboratoryconstrained values Cd = 1–10 s1 are important for
understanding the damage accumulation process
leading to failure, but have a minor effect on the
geometrical properties of the new-created fault
zones and earthquake statistics in regional-scale
models [e.g., Finzi et al., 2009; Lyakhovsky and
Ben-Zion, 2009]. Damage accumulation leads to
significant material weakening toward the seismic
event. Lyakhovsky and Ben-Zion [2008] developed
a mathematical procedure for the local stress drop
that utilizes the Drucker-Prager model, which generalizes the classical Coulomb yield condition for
cohesion-less material. They use scaling relations
between the rupture area and seismic potency values
established in the seismology and typical range of
the stress drop (1–10 MPa) during earthquakes to
calibrate parameters of their local stress drop procedure and connected them to the dynamic friction
(0.2) of simpler models with planar faults.
A depth-dependent yielding function for the highly
damaged zone (a = 1) corresponding to the frictional sliding during the simulated seismic event is
shown in Figure 2. When the slip associated with
the macroscopic brittle failure is arrested, the postfailure material recovery starts. The recovery of
elastic moduli and material strengthening is associated with healing of microcracks and is facilitated
by a high confining pressure, a low shear stress and
high temperatures. Motivated by the observed logarithmic increase of the static coefficient of friction
[e.g., Dieterich, 1978, 1979], Lyakhovsky et al. [1997]
used an exponential damage-dependent function
with coefficients C1, C2 for the kinetics of the
healing. The rate of the damage (a), which decreases under a 3-D compaction strain ("), is
da
a
⋅ "2 :
¼ C1 ⋅ exp
dt
C2
ð1Þ
Lyakhovsky et al. [2005] showed that the damage
model with exponential healing (1) reproduces the
main observed features of rate- and state-dependent
friction and constrained the coefficients C1, C2 by
comparing model calculations with laboratory frictional data. A field-based constrain of the healing
parameters is given by Finzi et al. [2009, 2011], but
leaves room for a relatively wide range of possible
values. Figure 3 shows the material recovery starting from a total failure (a = 1) and lasting up to a
thousand years under a constant compaction strain
corresponding to 200 MPa confining pressure
(8 km depth in the earth crust). For every set of
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Figure 3. Damage decrease (material healing) under 200 MPa confining pressure (8 km depth) for various kinetic
coefficients C1 and C2. LH = Low Healing; NH = Normal Healing; FH = Fast healing.
the model parameters, the rate of healing is the
highest during the initial stage of healing (see
Figure 3b for one year zoom in) and becomes slower
afterwards, keeping the strength of the damaged rock
well below the strength of the intact rock (a = 0).
We demonstrate the effect of healing on the simulation results with three different sets of the model
parameters corresponding to fast and more efficient
healing (FH), relatively slow and low healing (LH)
and an intermediate case, called here “normal healing” (NH). Colored lines in Figure 2 show the
recovered strength profile hundred years after the
seismic event for three sets of the healing coefficients
adopted in this study.
[12] Brittle failure is apparently irrelevant for the
upper mantle accounting for the lack of seismicity
below the Moho interface. However, Lindenfeld
and Rumpker [2011] recently detected several
earthquakes located well within the mantle (53–
60 km depth) beneath the East African Rift. Keeping this option in mind, we also allow damage
accumulation in the upper mantle using the same
conditions for onset of damage accumulation as in
the upper brittle crust. We note that the damage
healing is much more efficient at the upper mantle
than the brittle crust due to a high confining pressure and high temperatures.
2.1.2. Ductile Flow
[13] Ductile deformation can be described by empir-
ical constitutive relations that express the strain rate,
"_ , as a function of stress and temperature for the
dominant lithologies comprising the continental and
oceanic lithospheres [e.g., Karato, 2010, and references therein]. The common lithologies of the continental lithosphere are quartz-diorite (upper crust),
diabase (lower crust) and dunite/olivine (mantle).
Most of the experimental constitutive relations are
established for minerals, but it is assumed that the
weakest among the dominant minerals controls the
flow rate of the entire rock [Brace and Kohlstedt,
1980]. Steady state plastic flow of rocks can take
place by a variety of mechanisms including dislocation
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precipitation processes [e.g., Kohlstedt et al., 1995].
Direct observations of ductile shear zones and
indirect analyses of seismic anisotropy in mantle
rocks demonstrate that dislocation creep dominates
at stresses above 10 MPa [e.g., Karato, 2010]. A
power law rheology is widely accepted for expressing the strain rate of the dislocation flow [e.g.,
Weertman, 1978]
Q þ PV "_ ¼ Asn exp ;
RT
ð2Þ
The parameters A and n are empirical constants, Q is
the activation energy, V* is the activation volume,
P is pressure, T is temperature, and R is the gas
constant. A, n and Q values are presented in Table 1.
For relatively low pressures corresponding to depths
less than 100 km, the PV* term in (2) is negligible.
For the lower crust we use the material constants
appropriated for diabase rocks. Since our study is
mostly focused on the brittle processes in the crust,
we utilize the common approach of water-weakening
of the upper mantle adopting wet olivine rheology
below the Moho interface.
[14] The ductile flow pattern in the lower crust and
upper mantle is strongly affected by the depthdependent temperature distribution, corresponding
to a regional heat flow. Following Turcotte and
Schubert [2002], we assume that the heat production due to radioactive elements decreases exponentially with depth and use the depth-dependent
temperature T(z):
T ðz Þ ¼ T 0 þ
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LYAKHOVSKY ET AL.: PROPAGATION OF CONTINENTAL RIFT ZONES
q ⋅ z rc ⋅ Hr ⋅ h2r 1 ez=hr ;
þ
k
k
ð3Þ
where T0 = 300 K is the surface temperature; k =
3.35 Wm1 K1 is the coefficient of thermal conductivity; constant heat flux q = 50 mW/m2 is used
for a relatively cold lithosphere; Hr is the surface
radiogenic heat production and hr is the length scale
of its exponential decay. We take rcHr = 3.7 103 mW/m3 and hr = 10 km for the radiogenic
component of the continental heat flux. The depthdependent temperature distribution (Figure 1b) is
kept constant during the simulation.
2.2. Rift-Driving Forces and Model
Boundary Conditions
[15] The regional stresses in the lithosphere are
commonly estimated using a thin plate approximation that treats the velocity and stress distributions
in terms of values averaged over the plate thickness
[e.g., Elsasser, 1969; England and McKenzie,
1982]. For the problem of a long-term deformation, the England-McKenzie model uses a viscous
plate riding over an invicid substrate, assuming a
complete decoupling between plate and substrate.
This approach lacks an appropriate description for a
relatively short-term elastic behavior of the lithosphere and mantle. The Elsasser model uses elastic
plate riding over a viscous substrate and therefore,
accounts for the drag forces transmitted from the
large-scale convection currents in the upper mantle
to the base of the plate. Rice [1980], Lehner et al.
[1981] and Li and Rice [1987] provided a generalized Elsasser model replacing the viscous rheology
of the substrate with a visco-elastic one. A few
attempts have been made to calculate crustal
deformation in a 3-D model with a depth-dependent
rheology [e.g., Ben-Zion et al., 1993; Reches et al.,
1994]. The results from those studies show that
outside the space-time domain of a large earthquake, the 3-D calculations do not differ significantly from those obtained by the Elsasser and
generalized Elsasser models. Reches et al. [1994]
presented a direct comparison between a faultparallel velocity averaged over the thickness of
the elastic layer from 3-D calculations and the
analytical solution of Li and Rice [1987] for the
generalized Elsasser model. After about 50 years
into the earthquake cycle the difference between
the 2-D and 3-D models is negligible and both
models fit the available geodetic data with similar
accuracy. Lyakhovsky et al. [2001] proposed some
modifications to the generalized Elsasser model
following the common seismological assumption
[e.g., Kasahara, 1981; Stein et al., 1994] that the
instantaneous response of the Earth to sudden
stress redistribution is accommodated by an elastic
half-space. This assumption is realistic, since the
strain relaxation is not instantaneous even for a
ductile lower crust. Their modified model provides
improvements over the generalized Elsasser model
in simulating the elastic component of the geodetic
signals, especially close to the master fault.
[16] Following the framework of the original
Elsasser model, the horizontal components of a
stress tensor, skm, averaged over a plate thickness
H are
1
skm ðx; yÞ ¼
H
Z0
skm ðx; y; zÞdz:
ð4Þ
H
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In this case, gradual recovery of the local stress
field to its regional level, t (r)
k , calculated using (6),
can be approximated by
∂t k
1 ðrÞ
þ
t k t k ¼ 0;
∂t
tload
ð7Þ
where tload is the characteristic time of the postseismic relaxation. Substituting (6) for the regional
stress into (7) leads to the equation for the drag
forces applied at the vertical edges of the simulated
volume:
∂t k
1 h
∂uk
ðk Þ
¼
tk :
Vplate ∂t
∂t
tload h
Figure 4. Schematic representation of the applied
boundary conditions.
Substituting (4) into the 3-D equation of equilibrium gives
∂skm t k
¼ ;
∂xm
H
ð5Þ
where t k is the basal drag force. This drag force
should satisfy an equation of motion that accounts
for the ductile strain components. The viscous
response of a layer with different velocities on its
upper and lower boundaries is
tk ¼
h
∂uk
ðk Þ
;
Vplate ∂t
h
ð6Þ
where h and h are the thickness and viscosity of
the layer; V (k)plate and uk are the k-th components
of the steady mantle velocity and displacement of
the upper layer.
[17] Co-seismic displacements associated with an
earthquake occurring in the seismogenic zone
(i.e., in the upper crust) leads to a sudden stress
re-distribution or stress-drop. Because of a very
short duration of the seismic event, the lithosphere
and the upper mantle respond elastically. Based on
the solution for a mode III crack in an elastic halfspace, Lehner et al. [1981] and Li and Rice [1987]
suggested an approximation of an elastic strain
component using a linear proportion between the
drag force and displacement (t k = mauk/b; ma is
rigidity and b is a scalar). This relation connects the
drag force applied to a certain point only with the
displacement of the same point. Relations based on
the Green functions for the elastic half-space,
adopted by Lyakhovsky et al. [2001], improved
this local relation, but still provide vertically averaged solutions. Displacements associated with postseismic relaxation can be adequately described by
an exponential function [e.g., Savage et al., 2003].
ð8Þ
Equation (8) assumes that the forces driving the
rifting process depend on a mismatch between the
mantle flow velocity and the rate of extension of the
simulated volume. The rate of the stress accumulation is defined by the characteristic time of the
post-seismic relaxation, tload. In a case of a strong
upper layer and lack of a rifting process, the driving
forces gradually approach their maximum value
h
t max ¼ Vplate :
h
ð9Þ
Following this formulation, variable boundary forces are applied to the vertical edges of the simulated
volume. Figure 4 shows a schematic plain-view
structure of the boundary conditions. A free slip
condition is applied to the edges oriented perpendicular to the direction of the propagating rift zone.
Virtual springs connected to the edges oriented
parallel to the rift trace schematically represent
variable tensile boundary forces. Using (8), force
values for every boundary node are proportional to
the velocity difference between the prescribed constant far-field plate motion and calculated rate of the
boundary node motion. These boundary conditions
account for the evolution of the elastic properties
and accumulated plastic strain in the model region
[Lyakhovsky and Ben-Zion, 2009]. A free surface
boundary condition is applied to the upper surface,
which is the top of the simulated volume. A free slip
in the horizontal direction combined with a force
balance of the vertical component is applied to the
bottom surface. The vertical force component acting
on the bottom surface is calculated assuming depthdependent lithostatic pressure in the upper mantle.
2.3. Numerical Approach and Model
Setup
[18] The numerical simulations were done using the
Fast Lagrangian Analysis of Continua (FLAC)
algorithm [Cundall and Board, 1988; Cundall, 1989;
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Figure 5. Rift opening rates with different t max and healing values.
Poliakov et al., 1993; Ilchev and Lyakhovsky, 2001].
This fully explicit numerical method relies on a
large-strain explicit Lagrangian formulation originally developed by Cundall [1989] for elasto-plastic
rheology and implemented in the well-known FLAC
2D software produced by ITASCA. Poliakov et al.
[1993] developed an adaptive time stepping and
applied the FLAC algorithm for visco-elasto-plastic
rheology. Their adaptive scheme does not require
iterating, which makes the numerical model stable even
for highly nonlinear damage rheology [Lyakhovsky
et al., 1993]. Physical instability is modeled without numerical instability since inertial terms are
included in the equilibrium equations. A modified
version of this code incorporating heat transport is
known as PAROVOZ (locomotive in Russian) and
is widely used by many researchers. Ilchev and
Lyakhovsky [2001] developed their own 3-D version of the code for visco-elastic damage rheology
[see also Hamiel et al., 2004]. This 3-D code was
utilized for several studies of a long-term tectonic
motion and faulting process in a layered model of
the lithosphere (Figure 1); among them recently
reported results of Lyakhovsky and Ben-Zion
[2009], Ben-Avraham et al. [2010], and Finzi et al.
[2009, 2011].
[19] Initially we simulated rifting processes in hor-
izontally layered models (i.e., “flat” structure),
corresponding to typical continental or oceanic
lithospheres. The simulated volume of 150 150 km and 70 km depth was divided into tetrahedral elements of variable sizes that increase gradually from 1 km in the seismogenic zone to 5 km
in the ductile zone. The model consists of three
different layers representing a weak sedimentary
cover, a crystalline basement (crust) down to the
Moho, and an upper mantle. In the case of the continental lithosphere, the thickness of the sedimentary
cover is chosen at 3 km and that of the crystalline
basement at 30 km (depth to Moho). In the oceanic
lithosphere, the thickness of the sedimentary cover is
8 km, and that of the crystalline basement is reduced
to 20 km (depth to Moho). These thicknesses mimic
the structure of the lithosphere in the Levant area
[Segev et al., 2006]. At the next stage, we constructed
a model composed of an oceanic lithosphere on the
west (50 km width), continental lithosphere on the
east (50 km width) and a gradual transition zone
(50 km width) between them, representing the continental margin. The simulated rift zone is expected
to propagate perpendicular to the continental margin
from east to west.
3. Results
3.1. Rifting of Lithosphere With “Flat”
Structure
[20] The rate of the mantle flow in all the simula-
tions was Vplate = 50 mm/y. Rifting of a continental
lithosphere was simulated for low (t max = 50 MPa)
and moderate (t max = 100 MPa) tectonic loading,
without any additional heating. The simulation
starts with an isostatically compensated structure
without any tectonic stress except for a depthdependent lithostatic pressure. Low damage (a < 0.2)
is randomly distributed through the model region
except for a small (two numerical elements) highly
damaged (a = 1) notch (i.e., pre-defined weak zone)
in the middle of the right edge of the simulated
volume, which serves as a rift initiation zone. At
the initial stage, with the onset of the mantle flow,
the tensile stress is accumulated in the upper crust.
The total opening rate of the simulated region
(Figure 5) steeply decreases from about 100 mm/y,
which is close to double the Vplate value, to 2–3
orders of magnitude lower values corresponding to a
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[21] Figures 6, 7, and 8 present results after
300,000 years of simulation. Figures 6a, 7a, and 8a
show maps of the tensile strain component ("yy) in
the brittle part of the crystalline basement (8 km
depth); and Figures 6b, 7b, and 8b show the
deformed grid together with damage distribution
along sections crossing the center of the simulated
volume in the y-direction. Animations in the
auxiliary material present the evolution of strain
and damage distribution associated with the rifting
process.1 Presented results include two end-member case studies corresponding to the formation of a
well-localized straight rift zone (Figure 6) and the
termination of the rift zone (Figure 7). An intermediate case corresponding to a relatively wide rift
zone is also shown in Figure 8.
Figure 6. (a) Tensile strain (ɛyy) distribution in the
seismogenic zone - map view at 8 km depth and (b) vertical profile across the center of the simulated volume.
Note 20 times exaggeration of the grid deformation.
The simulation with t max = 100 MPa and low healing.
continuous rift opening. A failure in rift propagation
(zero opening rate) occurs under a low drag force
t max = 50 MPa and a normal (violet line in Figure 5)
or faster (not shown) rate of healing. The damage
starts accumulating in the pre-defined weak zone at
the right edge of the simulated volume. This zone
serves as a high stress concentrator and enables rift
initiation even under low tectonic loading. In the
case of the successive rifting, the rift zone propagates into the simulated area and may cross the
whole model or terminate inside the region,
depending on the tectonic forcing and the rate of
healing.
[22] Relatively high forcing (t max = 100 MPa)
together with low healing initially leads to the formation of a 50 km wide zone of deformation
(Animation S1) and 70–80 km wide damage zone
(Animation S2). After 200,000 years (2/3 of the
total simulated time) the damage is localized into a
narrow regular zone and most of the deformation is
accumulated within a 20 km wide straight zone
crossing the simulated area (Figure 6a). The opening rate of the rift zone is 1.5 mm/y (Figure 5)
leading to thinning of the simulated structure
mostly accommodated by a vertical motion of the
lower boundary (Animation S3). Maximum uplift of
the mantle material below the center of the rift zone
is 1 km (Figure 6b), while the surface subsidence
is 200 m. The ongoing uplift of the mantle material in long-term simulations (not produced in this
study) supplies additional heat beneath the rift zone,
changing the temperature profile. Several recent
studies [Benallal and Bigoni, 2004; Regenauer-Lieb
and Yuen, 2004; Kaus and Podladchikov, 2006;
Regenauer-Lieb et al., 2006] noted the role of the
thermomechanical feedback and demonstrated that
spontaneous strain localization in the mantle occurs
during the extension of the continental lithosphere
[e.g., Weinberg et al., 2007; Rosenbaum et al.,
2010]. The thermomechanical coupling provides
an additional localization mechanism, accelerating
the rift propagation at low stresses.
[23] Reduced forcing and more efficient healing
prevent a localization of the rift zone and may even
lead to its termination. Such a scenario is obtained
for t max = 50 MPa and normal healing (Figure 7).
Starting from the right edge of the model, the rift
propagates 50 km into the simulated area, and
1
Auxiliary materials are available in the HTML. doi:10.1029/
2011GC003927.
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reduces the ability of the rift zone to develop, while
decreasing the efficiency (low healing) leads to the
formation of a wide rift zone with a very slow rate of
opening 0.2 mm/y (both cases are not shown
here). A higher and more realistic rate of opening
(0.6–0.7 mm/y) is obtained for fast healing and
t max = 100 MPa (green line in Figure 5). Efficient
healing prevents strong localization of the strain
(Animation S7) and the damage (Animation S8)
into a narrow rift zone. After 300,000 years of
simulation, the width of the rift zone is 70 km
(Figure 8a), and instead of a single localized
deformation zone, opening is distributed between
Figure 7. (a) Tensile strain (ɛyy) distribution in the
seismogenic zone - map view at 8 km depth and (b) vertical profile across the center of the simulated volume.
Note 20 times exaggeration of the grid deformation.
The simulation with t max = 50 MPa and normal healing.
then forms a wide distributed strain (Animation S4),
associated with a widening and a stagnation of the
damage zone (Animation S5). The opening rate of
the simulated volume along the section crossing its
center gradually decreases and practically stops
(<0.1 mm/y) after 150,000 years (Figure 5). This
minor stretching is accommodated by a small (up
to 100 m, Figure 7b) uplift of the lower boundary
(Animation S6), which remains flat, except for
some small boundary effect at the model corners.
Surface subsidence is negligible. Increasing the
healing efficiency (fast healing instead of normal)
Figure 8. (a) Tensile strain (ɛyy) distribution in the
seismogenic zone - map view at 8 km depth and (b) vertical profile across the center of the simulated volume.
Note 20 times exaggeration of the grid deformation.
The simulation with t max = 100 MPa and fast healing.
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Figure 9. Tensile strain (ɛyy) distribution along the profile crossing the center of the simulated volume at 8 km depth
in the y-direction. The strain is accumulated during 300,000 years with different t max and healing values.
three parallel segments with a spacing of 10–
15 km between them. Thinning of the simulated
structure is mostly accommodated by a vertical
motion of the mantle material below the rift zone
(Animation S9). The uplift of the lower boundary
(700 m) and surface subsidence (150 m) are
distributed within a wide zone with clear undulations corresponding to the different segments
(Figure 8b).
[24] Figure 9 shows the distribution of the tensile
strain component ("yy) along the section crossing
the center of the simulated volume in the y-direction
after 300,000 years for all the analyzed simulations.
A comparison between modeling results with different forcing and different rates of healing
demonstrates that more regular and localized structures are expected under higher forcing and slow
(less efficient) healing; fast healing leads to a widened and more disordered rift zone.
onset of fracturing (i.e., rifting) of the crystalline
rocks increase with depth proportionally to the
confining pressure produced by the overburden
sediments. Hence, fracture nucleation in the crystalline basement covered by a thick sedimentary
layer (8 km) in the oceanic lithosphere requires
higher stresses than that covered by a thin sedimentary layer in the continental lithosphere. This
explains why rifting fails to propagate in the oceanic lithosphere even with t max = 100 MPa and
succeeds only when the force is increased to t max =
150–200 MPa, depending on the rate of healing.
General features of the rifting process in a thin and
flat oceanic lithosphere are similar to those in the
continental lithosphere discussed above and are not
shown here.
3.2. Propagating Rift Approaching
a Continental Margin
[25] In the oceanic lithosphere (Figure 1a), with a
[26] A series of modeling with flat lithosphere
shallow depth to Moho (20 km), the temperature
below the crust is significantly lower than the
temperature below the continental crust with the
same temperature profile (Figure 1b). Under these
conditions, the olivine-rich upper mantle rocks are
much stiffer and could hardly be deformed without
additional heating, even accounting for their waterweakening. Moreover, the stresses needed for the
structure revealed that significantly lower forcing is
required for successive rifting of thick continental
lithosphere than that of a thin oceanic lithosphere
with the same healing parameters and temperature
regime. Hence, there are a wide range of conditions
leading to a successive rifting of a continental lithosphere, and to ceasing of rift propagation at the
continental margin. Similar considerations lead
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a 50 km wide transition zone between a typical
continental (on the right) and oceanic (on the left)
lithosphere. In map view, the distribution of the
tensile strain component, "yy, in the middle part of
the seismogenic zone (down to 10 km depth) is
shown for t max = 100 MPa and normal healing
(Figure 10b). A 50 km long zone of high strain
values (red zone in Figure 10b) spreads from the
right edge of the model, where the rift zone was
initiated, to the beginning of the continental slope.
The rift zone successively propagated only till the
onset of the thinning of the crystalline crust (Moho
uplift). Similarly, the distribution of the shear strain
component, "xy, shows that the rift zone fails to
propagate further along its own plane. Instead, new
shear zones, spanning parallel to the continental
slope, developed (Figure 10c). The location of these
shear zones depends on the applied tectonic load
and temperature profile (Figure 1b). An increase of
the regional heat flux would weaken the upper
mantle and enhance rift propagation across the
continental margin. The symmetry of the simultaneously formed left and right-lateral shear zones
stems from the idealized structure of the continental
margin and the right angle between the propagating
rift and the continental slope (Figure 10c). In any
realistic structure a certain asymmetry is expected
leading to a more pronounced development of one
of these zones than the other.
3.3. Seismic Activity Associated With
Propagating Rift Zone
[27] During the simulation of rifting using damage
Figure 10. (a) A cross section representing the geometry of the simulated volume. (b and c) distribution of tensile (ɛyy) and shear (ɛxy) strain components at 10 km
depth for rifting terminated at the continental margin;
t max = 100 MPa and normal healing. 50 km long zone
of high tensile strain, eyy, values (Figure 10b) spreads
from the right edge of the model to the beginning of
the continental slope. The distribution of the shear strain
component, exy (Figure 10c), shows that the rift zone
fails to propagate further along its own plane, and new
shear zones spanning parallel to the continental slope
are developed.
Vink et al. [1984], Steckler and ten Brink [1986]
and others to discuss rift termination at the continental margin. Figure 10 shows one of the possible
scenarios obtained in the course of this study,
demonstrating the deformation pattern associated
with the termination of rifting at the continental
margin. The model structure (Figure 10a) includes
rheology, the numerical model produces a catalog
of synthetic seismic events. The numerical procedure [e.g., Lyakhovsky and Ben-Zion, 2008] allows
storing information about every numerical element
involved in each event and calculating the magnitude of the synthetic earthquake. The seismic
activity starts around the notch at the right edge of
the model where the rift is initiated, propagates
along the new formed rift zone, and continues with
ongoing rift opening or ceases when the rift is terminated. The last scenario happens in the continental lithosphere with flat structure under t max =
50 MPa and normal healing (Figure 11a). In this case
some moderate activity was generated during the first
60,000 years; this activity lasted up to about
150,000 years, when the rift zone becomes stagnant
(see also Figures 5 and 7). The model with t max =
50 MPa and low healing generates significantly more
seismic events during the first 10,000–20,000 years
(Figure 11b), when the rate of opening is above
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Figure 11. A synthetic record of the seismic events for the simulation with t max = 50 MPa and (a) normal or (b) low
healing. The orange rectangle marks the stage with characteristic frequency-size statistics.
1 mm/y (Figure 5). After 50,000–60,000 years, when
the rate of opening decreases below 0.5 mm/y, seismic activity decreases, but as oppose to the case
with normal healing (Figure 11a), the rifting process is permanently associated with seismic activity. The magnitude range of the simulated seismic
events at this stage is relatively narrow around M 6
(orange rectangle in Figure 11b). The frequencysize earthquake statistics significantly deviates from
the power-law Gutenberg-Richter distribution and
resemble the characteristic one.
[28] Some reduction in the level of seismic activity
between the initial stage of the formation of the rift
zone and later stage of the ongoing opening was
also obtained for higher tectonic forcing (t max =
100 MPa), but this change is not so pronounced as
in the case with t max = 50 MPa. The overall activity
in the model with t max = 100 MPa is significantly
higher especially at the initial stage of rifting
(Figure 12). The model continuously generates lots
of relatively small events with a magnitude range
between M = 4.5 (lower model cut-off according to
the grid size) and M = 5.5. Strong events (M > 6)
are concentrated in a time with an outstanding main
shock and form earthquake swarms (Figure 12a).
Five swarms were generated during the very initial
stages of rifting from 5,000 to 15,000 years of simulation. During this period the depth of the simulated events (Figure 12b) gradually increases from
3 km (top basement) to 17 km, corresponding to
Figure 12. (a) Magnitude and (b) depth of the synthetic seismic events for the simulation with t max = 100 MPa and
fast healing. The dash green line shows a deepening of the simulated seismic events at the initial stage and depth of the
seismogenic zone.
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Figure 13. Frequency-size statistics of the simulated seismic events obtained from the model runs.
the typical depth of the seismogenic zone. The seismic activity starts at depths corresponding to the top
of the crystalline basement (seismicity in a weak
sedimentary layer is not simulated) and then deepens
gradually. At the later stage of rifting, swarms are
more seldom and generated every 5,000 or even
10,000 years. During the swarms, some events occur
as deep as 20 km and are located below typical
depths of the seismogenic zone where most of the
events occur. The overall log number of the simulated seismic events, including swarms, fits well the
linear Gutenberg-Richter frequency-size distribution
with b-value b = 1 (green dots in Figure 13). The
model run with normal healing and t max = 100 MPa
forcing (red makers in Figure 13) leads to some
deviation of the statistics from the linear trend with
b = 0.8. The large amount of events, generated
during the initial stage of rifting, exhibit a wide range
of magnitudes and an almost linear distribution (not
shown here). At later stages, the rift zone becomes
more regular and strong seismic events with magnitudes above M = 6.0 slightly dominate. Therefore, the
overall log number of the simulated seismic events
shown in Figure 13 with magnitudes M = 6.0 and
M = 6.5 are slightly above the values expected for the
Gutenberg-Richter statistics. Significant deviation
from the linear frequency-size distribution is obtained
in model runs with low healing generating the most
regular rift zone.
4. Discussion
[29] Previous studies have discussed how tectonic
processes produce relative tension to initiate and
propagate rift zones and estimated the magnitude of
rift-driving forces. Buck [2009] concluded that the
rift-driven tectonic force allowing continental rifting due to extension is well above that available for
rifting of a thick and strong lithosphere in the
absence of basaltic magmatism (the “tectonic
force” paradox). This conclusion is based on the
assumption that the tectonic stress has to simultaneously overcome the yielding stress of the whole
lithosphere thickness, and ignore the gradual rock
weakening under long-term loading. Nevertheless,
gradual accumulation of rock damage under ongoing, even moderate, tectonic extension, could lead
to a considerable weakening of the heavily fractured brittle rocks. These rocks could remain significantly weaker than the surrounding intact rock
(long-term memory), which can make rifting feasible. In the case of a flat layered structure of the
lithosphere, the formation of a rift system is controlled not only by remote (tectonic) loading, but
also by the rate of healing (strengthening) of the
rock mass.
[30] The numerical simulations of the plate motion
and rifting process adopted water weakening in the
upper mantle using dislocation creep parameters for
wet olivine and a continuum damage rheology
approach for brittle rocks. The damage mechanics
employed in this study for brittle failure in the
crystalline crust models the effects of distributed
cracks in terms of a single scalar damage parameter.
Representative elementary volumes with a sufficiently large number of cracks corresponding to
given damage values are assumed to be uniform and
isotropic. While most of the material parameters
controlling ductile flow and damage accumulation
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in a brittle regime are well constrained by field and
laboratory observations, values controlling the rate
of post-failure material healing are non unique and
may vary within a relatively wide range.
[31] Results of the modeling of a flat-layered
structure demonstrate a gradual formation of the rift
zone in the continental lithosphere. The successive
propagation of the rift system, the localization of
the rift zone and its final width, and the associated
seismicity pattern strongly depend on the applied
tectonic force, thickness of the crystalline crust
(depth to the top basement and Moho interfaces),
and the rate of the crustal rock healing. The presented results for a typical continental lithosphere
structure demonstrate various scenarios including a
formation of a well-localized straight rift zone
(Figure 6), stagnation of the rift zone (Figure 7),
and an intermediate case corresponding to a relatively wide rift zone (Figure 8). Passive continental
rifting may occur under tectonic forcing as low as
t max = 50 MPa, but only if the rate of healing is
low. Under elevated forcing of t max = 100 MPa the
rift is successively formed for every rate of healing
adopted in this study. With increased rates of
healing, the evolving damage zone becomes more
disordered and wide. Narrow, localized rift zones
form with low rates of healing and remain weak
for a very long time. This general tendency for
rifting processes is similar to previously studied
structural properties and deformational patterns of
evolving strike-slip faults [Lyakhovsky et al., 2001;
Lyakhovsky and Ben-Zion, 2009; Finzi et al., 2009,
2011].
[32] The present results of modeling demonstrate
how the lithosphere structure, in particularly the
depth to Moho interface, affects the force needed
for successive rifting and the geometry of the
propagating rift zone. Modeling of a flat-layered
structure typical for the oceanic lithosphere with a
20 km depth to Moho and a 8 km top basement
depth, show that rifting succeeds only when the force
is increased to t max = 150–200 MPa, depending on
the rate of healing. With the same temperature profile, the olivine-rich upper mantle material is uplifted and has much lower temperatures than in the
continental lithosphere. Under these temperatures,
the upper mantle rocks are very stiff and could
hardly be deformed without additional heating. In
addition, fracture nucleation in the crystalline basement covered by a thick sedimentary layer in the
oceanic lithosphere requires higher stresses than for
that cover by a thin sedimentary layer in the continental lithosphere. This feature explains why rifting
fails to propagate in the oceanic lithosphere even
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with t max = 100 MPa and requires significantly
larger tectonic forces.
[33] The tectonic evolution of the Middle East region
(paleo-geographical reconstruction in Figure 14)
shows that the Oligocene Suez [Steckler and ten
Brink, 1986; Bosworth et al., 2005] and AzraqSirhan rift zones [Schattner et al., 2006; Segev and
Rybakov, 2010, 2011] are propagating toward
northwest and terminate at the Levant continental
margin. The Suez rift is 70 km wide and 500 km
long. It consists of the northwestern branch of the
Red Sea rift separating the Arabian and the African
(Nubian) plates [McKenzie et al., 1970]. The AzraqSirhan rift, about 40 km wide and 700 km long, is
located on the northwestern edge of the Arabian
continent. The initial stage of rifting occurred before
the intense magmatism in the northeastern AfroArabia plate at 30 Myr [Avni et al., 2011]. Similarly, Whitmarsh et al. [2001] suggested that magmatism was essentially absent during the initial
stage of rifting of the Iberian-Newfoundland margins. Another possible example of non-magmatic
rifting was discussed by Lie and Husebye [1994] in
relation with the north Skagerrak graben of the Oslo
rift system.
[34] Under our modeling conditions, a rift propa-
gating through a continental lithosphere may terminate before a transition to an oceanic lithosphere
at a continental margin (t max = 150 MPa, normal
healing, Figure 10). This termination induces the
formation of a new fault system perpendicular to
the direction of the rift propagation. A similar
deformational pattern was discussed by Steckler
and ten Brink [1986] who suggested that the
northwest continuation of the Red Sea-Suez continental rift system stops propagating toward the
Levant Basin oceanic lithosphere, leading to the
initiation of the Dead Sea Transform. During
the same period, the gradual thinning of the crystalline basement and the lithosphere strengthening
of the Levant continental margin in the direction of
the propagating Azraq-Sirhan rift system, might
explain the rift termination. A further detailed study
of the Azraq-Sirhan continental rift evolution will
be presented in a forthcoming paper.
[35] The presented results address a relatively short
geological time period (up to 300,000 years) of an
early stage of rift initiation and propagation. The
study is mostly concentrated on the gradual weakening of fractured brittle rocks and ignores effect of
regional heating, shear heating and other mechanisms of heat generation. The time scale of the
weakening effect is associated with the seismic
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Figure 14. Reconstruction to Early Oligocene time (30 Myr) modified from Sharland et al. [2001] and Schattner
et al. [2006].
cycle (102 to 103 years) while the thermal effect is
essential at time scale of 106 years. Model results
demonstrate that rifting of a relatively cold continental lithosphere is possible even under moderate
to low tectonic forcing. Temperature increase due
to an elevated regional heat flow as well as heat
generation in a localized ductile shear zone below
Moho provide an additional mechanism of strain
localization and acceleration of the rifting process. The role of the thermomechanical feedback
at a long geological time scale was discussed in
several recent studies [Benallal and Bigoni, 2004;
Regenauer-Lieb and Yuen, 2004; Kaus and
Podladchikov, 2006; Regenauer-Lieb et al., 2006;
Weinberg et al., 2007; Rosenbaum et al., 2010]
demonstrating spontaneous strain localization during the extension of a cold continental lithosphere.
[36] An important advantage of the numerical
modeling using damage rheology approach is its
ability to produce a catalog of synthetic seismic
events. This allows us to study simultaneously
deformational and seismicity patterns of evolving
fault systems and to connect the style of fault zone
localization and earthquake frequency-size statistics. Previous studies [e.g., Ben-Zion et al., 1999;
Lyakhovsky et al., 2001; Lyakhovsky and Ben-Zion,
2009] show that if new created damage zones
remain weak for a very long time, their geometry
is regular and frequency-size statistics of the simulated seismic events is characteristic. With increased
rates of healing, the evolving damage zone becomes
more disordered and wide; frequency-size statistics
become more of a Gutenberg-Richter type. A similar tendency is obtained here for the rifting process;
event statistics for the model with fast healing fits
well a linear Gutenberg-Richter distribution with
b = 1 (Figure 13). Less efficient healing changes
the frequency-size statistics to a characteristic distribution with event magnitude falling within a relatively narrow range. Another remarkable feature of
the simulated seismicity is earthquake swarms, i.e.,
seismic activity concentrated in time without a distinct main shock (Figure 12). The occurrence of
earthquake swarms is most common in areas of
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active volcanism, but is also abundant in other
regions with extensional tectonics, including continental rifting [e.g., Ibs-von Seht et al., 2008]. Several large swarms were observed in the Andaman
Sea, a back-arc extensional basin in the Indian
Ocean [ Mukhopadhyay and Dasgupta, 2008]. The
frequency-magnitude distribution of earthquakes
during the largest 2005 swarm (651 registered
events) is centered around M = 4.5. We speculate
here that healing in this region is low relatively to
the extremely high tectonic forcing leading to
repeated swarms every several years. Although the
presented modeling is far from representing a realistic structure and loading conditions in the Andaman Sea, the general features of the seismic activity
are expected to be similar. Frequency-magnitude
distribution of earthquake swarms in continental
rifts follow the Gutenberg-Richter law with b-value
between 0.8 and 1 for the Rio Grande, Kenia,
and Eger rift zones [Ibs-von Seht et al., 2008]. This
behavior resembles the simulation with fast healing
and t max = 100 MPa (Figures 12 and 13) generating
earthquake swarms and a linear frequency-size
distribution.
5. Conclusions
[37] A rifting process under moderate tectonic
extension is feasible due to gradual damage accumulation and long-term rock memory. These lead
to a considerable rock weakening within the rift
zone relatively to the surrounding intact rock and
provide a possible solution for the tectonic force
paradox. Results of the flat-layered model demonstrate a gradual formation of a rift zone in a continental lithosphere. A successive formation of the
rift system and the associated seismicity pattern
strongly depends on the applied tectonic force and
the rate of healing of the crustal rocks. The geometry of the propagating rift zone, its associated
faulting and seismicity patterns are dictated by the
local lithosphere structure, and particularly by the
depth to Moho interface. Such a rift zone terminates
at the continental margins and a new fault system
forms perpendicular to the rift propagation.
Acknowledgments
[38] We thank the editor Thorsten Becker, the reviewer Taras
Gerya and anonymous reviewers for constructive comments.
The study was supported by the Israel Science Foundation
(ISF 753/08).
10.1029/2011GC003927
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