Tectonophysics 460 (2008) Contents lists available at ScienceDirect. Tectonophysics. journal homepage:

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1 Tectonophysics 460 (2008) Contents lists available at ScienceDirect Tectonophysics journal homepage: Continental rifting as a function of lithosphere mantle strength Frédéric Gueydan, Christina Morency 1, Jean-Pierre Brun Géosciences Rennes, UMR CNRS 6118, Campus de Beaulieu, Université de Rennes 1, France article info abstract Article history: Received 30 October 2007 Received in revised form 30 July 2008 Accepted 8 August 2008 Available online 28 August 2008 Keywords: Continental rifts Extensional tectonics Lithosphere rheology Numerical modelling The role of the uppermost mantle strength in the pattern of lithosphere rifting is investigated using a thermo-mechanical finite-element code. In the lithosphere, the mantle/crust strength ratio (S M /S C ) that decreases with increasing Moho temperature T M allows two strength regimes to be defined: mantle dominated (S M NS C ) and crust dominated (S M bs C ). The transition between the two regimes corresponds to the disappearance of a high strength uppermost mantle for T M N 700 C. 2D numerical simulations for different values of S M /S C show how the uppermost mantle strength controls the style of continental rifting. A high strength mantle leads to strain localisation at lithosphere scale, with two main patterns of narrow rifting: coupled crust mantle at the lowest T M values and deep crustal décollement for increasing T M values, typical of some continental rifts and non-volcanic passive margins. The absence of a high strength mantle leads to distributed deformations and wide rifting in the upper crust. These numerical results are compared and discussed in relation with series of classical rift examples Elsevier B.V. All rights reserved. 1. Introduction The so-called brittle ductile layering of the continental lithosphere is often modelled using frictional-type and viscous-type rheological behaviours. According to this conception, most of the lithosphere strength is located in the brittle upper crust and sub- Moho mantle (Brace and Kohlstedt, 1980; Carter and Tsenn, 1987; Molnar, 1992). Such strength profiles with peaks in the upper crust and sub-moho mantle were found to be in good agreement with depth location of earthquakes (Chen and Molnar, 1983; Sibson, 1983). Recent depth re-location of the earthquakes beneath Tibet has shown that deep earthquakes are more likely to nucleate in the lower crust (Maggi et al., 2000) than in the mantle (Chen and Molnar, 1983). The concentration of seismicity in the crust and the good relationship between elastic thicknesses calculated for foreland basins and the depth of seismicity lead Jackson (2002) to argue that most of the overall lithosphere strength is located in the seismogenic upper crust instead of within the uppermost mantle. This conception of lithosphere rheological layering is however in contradiction with the mechanics of lithosphere rifting that requires a high strength mantle (Handy and Brun, 2004; Ziegler and Cloetingh, 2004; Burov and Watts, 2006). According to Buck (1991), continental extension occurs according to three modes: narrow rift, wide rift and core complex. This comes from the balance between the three types of forces that contribute to rifting: buoyancy, thermal and strength Corresponding author. Tel.: ; fax: address: frederic.gueydan@univ-rennes1.fr (F. Gueydan). 1 Now at California Institute of Technology, Seismological Laboratory, Pasadena, CA, USA. forces i.e. softening the lithosphere during extension by the thinning of brittle layers and general heating. Narrow rifts occur when the total force change on rifting is negative, so that the ongoing thinning is kept in the same narrow region. On the contrary, wide rifts occur when the total force change is positive, so that ongoing thinning tends to spread out to accommodate stretching. Finally, core complexes form when the lower crustal flow dominates, allowed by the low viscosity of both deep crust and mantle. This provides an overall and simple explanation to the observed differences between continental rifts (e.g. Baikal, Rhine) and regions of distributed extension (e.g. Basin and Range, Aegean). This approach considers the bulk mechanical properties of the lithosphere but the mechanical behaviour of brittle and ductile layers can also be decisive. Lithosphere deformation is crust or mantle dominated for thick or thin crusts, respectively, and the coupling between the crust and mantle decreases with increasing crustal thickness (Behn et al., 2002). The presence of a high strength uppermost mantle exerts a direct control on the development of mantle shear zones and subsequent lithosphere necking (Allemand and Brun, 1991; Brun and Beslier, 1996; Frederiksen and Braun, 2001). At low strain rates, and therefore low strength of the ductile crust, the lower crust acts as a decoupling level i.e. décollement between the upper brittle crust and the high strength uppermost mantle (Brun, 1999, 2001) providing a simple explanation for the typical faulting pattern of passive margins (Brun and Beslier, 1996; Nagel and Buck, 2004). The development of rift asymmetry can be a function of the degree of strain softening in the brittle crust (Huismans and Beaumont, 2003) but, in nature, rift asymmetry often results from the presence of pre-existing structures (Ziegler and Cloetingh, 2004) /$ see front matter 2008 Elsevier B.V. All rights reserved. doi: /j.tecto

2 84 F. Gueydan et al. / Tectonophysics 460 (2008) The present paper aims to show that the strength of the lithospheric mantle, and more specifically the presence of a high strength sub-moho mantle, exerts a strong influence on the style of continental rifting on a lithosphere scale. We analyse lithosphere rifting as a function of the lithosphere thermal and rheological layering. Three patterns of deformation at lithosphere scale are obtained without imposing any weak zone or lateral thermal gradient to control the initial rift position (e.g, Behn et al., 2002; Huismans and Beaumont, 2003). The study focuses in particular on the wavelength of necking instabilities and on the deformation of the uppermost mantle and of the ductile crust. The development of the Metamorphic Core Complexes mode of lithosphere extension (Buck, 1991) is beyond the scope of the paper that is dedicated to the rifting proper and the transition between narrow and wide rifts. An analytical approach is first used to discuss the respective contributions of crust and mantle to lithosphere strength. Then, 2D numerical models are presented to analyse the dependence of continental rifting patterns on the rheological layering and more specifically to the presence/absence of a high strength sub-moho mantle. Finally, the results are tentatively compared to some natural examples of rifts. 2. Thermal and rheological layering In order to calibrate the range of rheological lithosphere layering for further 2D numerical modelling, lithosphere strength profiles are first computed analytically for different continental geotherms. The steady state temperature profiles (as a function of depths) T(z) are analytically computed knowing the imposed heat flux at the base of the lithosphere q m, the crustal thickness h C and the radiogenic crust thickness h RC. Within the crust, the temperature T(z) reads: z b h RC ; h C b z b h RC ; Tz ðþ¼ r z 2 þ q m þ r h RC z þ T 0 ; 2k c k c k c Tz ðþ¼ q m z þ r hrc 2 k c 2k þ T 0; c where r, k c, q m and T 0 are the radiogenic heat production (which occurred over a thickness h RC ), the crust conduction, the basal heat flux and the surface temperature set to 300 K, respectively. The values of these parameters are given in Table 1. It is commonly assumed that the radiogenic activity only exists in the upper part of the crust (Turcotte and Schubert, 1982). Thus, the radiogenic thickness h RC is assumed here to be equal to the brittle crust thickness. This assumption requires iterative computation to obtain a suitable strength profile, as discussed below. Temperatures in the mantle are defined by: z N h C ; Tz ðþ¼ q m ðz h C ÞþT M ; with T M ¼ q m h c þ r hrc 2 k m k c 2k þ T 0; ð2þ c where k m is the mantle conduction (Table 1). The thickness of the lithosphere h L can be derived from Eq. (2) since at z = h L, the lithosphere temperature is T L =1200 C: h L ¼ k m ðt L T M Þþh C ð3þ q m Table 1 Parameters used to compute the 1D continental geotherm Variable Unit Crust Mantle Density ρ c; ρ m kg m Diffusion k c, k m Wm 1 K Radiogenic heat production r μw m 3 1 (only brittle crust) 0.0 ð1þ Table 2 Ductile rheological parameters Mineral A [MPa n s 1 ] Q [kj mol 1 ] n Reference Wet quartz Ranalli (2000) Dry olivine Karato and Wu (1993) Wet olivine Karato and Wu (1993) Wet quartz and dry olivine were used for the entire 2D numerical simulation set. Wet olivine was used to construct the deformation map in Fig. 9. The lithosphere strength is computed from the temperature profile defined above using wet quartz and dry olivine rheological parameters (Table 2) as follows. The brittle strength τ B is equal to the Mohr Coulomb stress (Byerlee, 1978), assuming a non-cohesive material, τ B =μp, where p is the lithostatic pressure (p=ρgz, with varying density for the crust and mantle, Table 1). The friction coefficient is set to μ=0.6. The ductile strength τ D is derived from the power creep law which reads: : e ¼ A exp Q r τ n D RT ; ð4þ where ε, T and τ D are the strain rate (in s 1 ), the temperature (in K) and the shear stress (in MPa), respectively. Setting the value of the strain rate at /s, the strength at a given depth is taken as the lowest of the stress values calculated for brittle and ductile behaviours at this depth. Using an initial value of the radiogenic crust thickness of h RC =10 km leads to a first strength profile and thus gives an estimate of the brittle crust thickness. Then, setting h RC to the computed value of the brittle crust thickness provides a second estimate of temperature and strength profiles. Around four to five iterations are required to fit the radiogenic crust thickness to the brittle crust thickness and to thus obtain suitable strength and temperature profiles. In Fig. 1, the computed thicknesses of the lithosphere (solid lines) and of the brittle mantle (dashed lines) are plotted as functions of the Moho temperature T M with corresponding strength profiles for crustal thicknesses h C = 30 and 50 km. The lithosphere thickness that increases with increasing crustal thickness (Eqs. (2) and (3)), decreases with an increasing T M. The lithosphere strength decreases with an increasing Moho temperature T M. As an example, for h C =30 km, the lithosphere strength decreases from Model 1 to Model 3, together with the decrease of the thickness of the high strength mantle (Fig. 1). The critical Moho temperature above which the lithosphere mantle is wholly ductile can be defined as approximately 700 C for h C =30 and 50 km (Fig. 1) but this value depends on the chosen rheological parameters. For example, the use of wet olivine instead of dry olivine leads to a much lower critical Moho temperature of 600 C; above which the lithosphere mantle is entirely ductile. The above remarks show that the Moho temperature can be used as simplified indicator of the thermal and rheological lithosphere layering. 3. Contributions of crust and mantle to lithosphere strength Fig. 2 shows that the mantle crust strength ratio (S M /S C ) decreases with increasing Moho temperatures T M.AtT M ~700 C, S M is equal to S C. Above and below this critical temperature; the lithosphere strength is dominated by the strength of the crust S C and the mantle S M, respectively. The ratio S M /S C thus defines two distinct strength regimes: mantle dominated (S M NS C ) and crust dominated (S M bs C ). The transition between the two regimes corresponds to the presence/ absence of high strength sub-moho mantle for T M N700 C (Fig. 1). The strength of the lithosphere mantle is therefore strongly controlled by its brittle part (Molnar, 1992). Plotting S M /S C as a function of the Moho temperature, which depends on the crustal thickness, provides comparable values of that ratio, irrespective of the crustal thickness.

3 F. Gueydan et al. / Tectonophysics 460 (2008) Fig. 1. 1D analytical solution giving the lithosphere thickness h L (Eq. (3)) and the high strength uppermost mantle thickness as a function of the Moho temperature T M (Eq. (2)). Four strength profiles (Models 1 to 4) are used for a crustal thickness h C = 30 km. Two strength profiles (Models 5 and 6) are used for a crustal thickness h C =50 km. The thermal and rheological parameters used to construct strength profiles are given in Tables 1 and 2, respectively. Dislocation creep parameters for wet quartz and dry olivine are used for crust and mantle, respectively. The friction coefficient is μ=0.6. The strength ratio S M /S C is therefore also a good indicator of the thermal and rheological layering within the lithosphere. 4. Numerical modelling 4.1. Boundary conditions Fig. 3 presents an example of the model set-up and the rheological layering used for the 2D numerical simulations i.e. Model 1 (Figs.1 and 2): T M =356 Candq m =20mWm 2 and the mesh used for each model layer. ThesameprocedurewasusedforallmodelsinFigs. 1 and 2. Table 3 gives velocity, crustal thickness, basal heat flux and Moho temperature for the 6 selected models that are used for 2D simulations. The thicknesses of the brittle and ductile layers are computed as presented in the two previous sections. However, the thicknesses of brittle layers are slightly different from the Mohr Coulomb estimate, since we use Von Mises approximation for plastic material. The initial model width is 300 km. The thickness depends on the selected Moho temperature, which controls the lithosphere thickness (Eq. (3)). A velocity of V=1 cm/a is imposed at the right vertical boundary of the model, while the left vertical boundary is only allowed to move vertically. The isostatic equilibrium of the lithosphere base is achieved by the presence of an underlying asthenospheric layer whose viscosity is set to Pa s. A small geometrical perturbation of the crustal brittle ductile transition and of the Moho i.e. a deflection with an amplitude of 100 m is introduced at the model centre to initiate strain localization. The temperature profiles are the same as those used to compute strength profiles but are however partly modified by shear heating in the ductile layers i.e. thermo-mechanical coupling Brittle rheology: Von Mises approximation The brittle layers are modeled by a Von Mises associated elasto-viscoplasticity. Before yielding, the material is described by a classical linear elasticity law i.e. a Poisson ratio equal to 0.25 and a Young modulus equal to Pa and Pa for the crust and mantle, respectively (Turcotte and Schubert, 1982). Yielding occurs when the equivalent Von Mises shear stress τ v is greater than the yield stress σ y (ε p ), which is a function of the equivalent plastic strain ε p history. The equivalent Von Mises shear stress and the equivalent plastic strain are defined by: Fig. 2. Ratio of mantle strength S M to crustal strength S C as a function of Moho temperature T M for a crustal thickness of 30 km (solid line) and 50 km (dashed line) for Models 1 to 6 (Fig. 1, Table 3). The sinuous grey line indicates the position of T M =700 C that separates lithospheres with and without a high strength mantle (Fig. 1). Dislocation creep parameters for wet quartz and dry olivine are used for crust and mantle, respectively (Table 2). The friction coefficient is μ=0.6. rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 τ v ¼ 2 τ ijτ ij ; and e p 2 ¼ 3 Dp ij Dp ij ; where D p stands for the plastic strain tensor. The yield stress σ y is a three-linear function of the plastic strain ε p (Fig. 3; inset). The material ð5þ

4 86 F. Gueydan et al. / Tectonophysics 460 (2008) Fig. 3. Set-up of numerical modelling showing the initial model geometry and boundary conditions (left) and the initial strength profile (right). Brittle layers behave according to a Von Mises criterion corresponding to a pressure-independent elasto-viscoplastic material. Ductile layers behave as non-newtonian fluids sensitive to temperature and strain rate (See parameters in Table 2). The inset shows yield stress as a function of the plastic strain in brittle layers. remains elastic until its plastic stress reaches σ y 0. It then undergoes a hardening until the plastic strain ε p 1 =0.02 is reached for the peak stress σ y 1. Then the yield stress decreases towards a residual state with increasing plastic strain (until the plastic strain ε p 2 =0.12), defining the softening, during which localised plastic shear zones fully develop. For larger values of plastic strain, the material can reach an infinite strain with no increase in strength. In the present paper, fracturing will thus be modelled by regions where the plastic strain ε p is larger than ε p 2 =0.12. Modifications in the values of ε p 1 and ε p 2 will delay or speed up the development of faults. The viscosity of the plastic material η p, which has been introduced to avoid catastrophic fault propagation, defines a time-scale for the change of plastic deformation: : e p ¼ τ v σ y ðe p Þ η p ; ð6þ where ε p is the plastic strain rate. Increasing η p will thus decrease the plastic strain rate and therefore increase the time necessary to change the plastic strain. A value of Pa s for the plastic viscosity was found sufficient to stabilise the numerical development of the faults and has therefore been selected in the following computations. Note that this plastic viscosity was set to the same value in all of the simulations presented. Changing this value will not change the fracture pattern, but will solely modify the timing of its development (Schueller et al., 2005). The Von Mises plastic criterion used in this study, with the above selected values of hardening, softening and plastic viscosity, has been validated to model modes of faulting (Schueller et al., 2005). The thicknesses of the Von Mises-type layers are computed so that the strength is identical to a classical Mohr Coulomb prediction. This leads to greater depths of brittle ductile transitions in the crust and in the mantle. This assumption permits keeping the same value of the strength ratio S M /S C for the 2D modeling and the analytical calculation presented above. The value of σ y 0 is thus defined by the integral of the Mohr Coulomb strength for each brittle layer. The value of σ y 1 is set to 1.10 times σ y 0 (10% hardening) and σ y 2 is equal to 0.70 times σ y 1 (30% softening). The Von Mises criterion imposes that faults trend at 45 to the principal compression axis, since it is a pressure-independent failure criterion. Mohr Coulomb criterion has been previously used, at a crustal scale, to more accurately model the fault displacement as a function of brittle softening, with a particular emphasis on the reduction of the cohesion or of the friction coefficient (Lavier et al., 1999; Lavier et al., 2000; Huismans and Beaumont, 2003; Huismans et al., 2005). Our purpose differs from that of the above previous studies as it aims primarily to capture the role of mantle strength on the deformation patterns at lithosphere scale. The Von Mises criterion yields fault patterns at lithosphere scale that are rather similar to those produce by pressure-sensitive criteria (Huismans et al., 2005). The applicability of the Mohr Coulomb criterion to model mantle deformation is moreover still a matter of debate (Jackson, 2002). As proposed by Bassi (1995) under the term of power law breakdown stress it is probably more accurate to limit the mantle strength. This is what is implicitly done by the Von Mises criterion Ductile rheology: dislocation creep The ductile layers are modeled by an incompressible non-newtonian fluid such that the Kirchhoff stress tensor τ is given by τ P ¼ 2ηðT; e : Þ D P p I P ; ð7þ where p, η(t, ε ) and D are the pressure, the secant viscosity and the rate of deformation tensor, respectively. The material incompressibility constrains the trace of D at zero. Note that the tensors are underlined twice. Eq. (7) is the multi-dimensional generalization of the creep law (Eq. (4)). In Eq. (4), the equivalent shear stress and strain rate are derived from the stress and strain rate tensors: rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 τ ¼ 2 τ ijτ ij and e : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2D ij D ij : ð8þ Thus, the secant viscosity is simply η(t,ε )=τ/ε. The sensitivity of viscosity to strain rate and temperature could therefore be defined using the 1D creep law (Eq. (4)). Table 3 Velocity, crustal thickness, basal heat flux and Moho temperature for the 6 models used for the 2D numerical simulations Model Velocity Crustal thickness Basal heat flux Moho temperature V [cm/a] h c [km] q m [mw/m 2 ] T M [ C]

5 F. Gueydan et al. / Tectonophysics 460 (2008) Governing equations and numerical scheme Mechanical equilibrium and heat conduction are solved by numerical means using the thermo-mechanical finite-element code SARPP (Gueydan et al., 2004). The layered structure is discretised into Lagrangian elements with 9 nodes. Shear heating is accounted for in the heat conduction (source term τε ), inducing a thermomechanical coupling. Heat advection is disregarded in this study. 5. Three patterns of continental rifting 5.1. Total strain patterns as a function of mantle strength The strength profiles, Moho temperature and strength ratio for the three Models 1, 2 and 3 are defined in Figs. 1 and 2. The total strain for these three models, are plotted in Fig. 4, after a horizontal displacement of both 30 and 60 km, with an applied velocity of 1 cm/a. The two first models are typical of a mantle dominated strength regime, while the third one is marked by the absence of a high strength mantle and corresponds to a crust dominated strength regime (Figs. 1 and 2; Table 3). The Moho temperature T M increases and thus the strength ratio S M /S C decreases from Model 1 to Model 3. For low T M, the brittle mantle is affected by distributed faulting that gives numerous necks with a wavelength dictated by the depth of the brittle ductile transition in the mantle (Model 1; Fig. 4). This wavelength is indeed defined by an upward propagation along a 45 dipping mantle shear zone. A reflection occurs at the free surface leading to a 45 downward dipping propagation of the shear zone. This is responsible for a strain concentration and the nucleation of new faults in the brittle mantle. Thus, the distance between the two regions of intense faulting within the brittle mantle is simply 2(h C +h BM ), where h BM is the brittle mantle thickness. For Model 1, h C =30 km and h BM =40 km (Fig. 1), which leads to a mantle necking wavelength of approximately 140 km. This estimate is very consistent with the results after 30 km of horizontal displacement (Fig. 4). The distance between the two regions of intense strain is almost the same in both the brittle crust and the uppermost mantle i.e. the crust and mantle are strongly coupled. In this paper, the term coupling means that two layers undergo rather similar amounts of strain. For larger horizontal displacement, rift spacing increases due to large strain and fault motion. The presence of two major lithosphere scale normal fault zones in the central part of the model induces a strong Moho deflection at model centre. Rift spacing is controlled by the wavelength of mantle necking and is thus of the order of 2(h C +h BM ). For larger T M, the deformation is more localized on the whole lithosphere scale, leading to a more pronounced necking (Model 2; Fig. 4). After 30 km of horizontal displacement, the brittle mantle is still affected by distributed faulting. Here, the wavelength of mantle necking is of the order of 100 km. The analysis for the wavelength of mantle necking remains valid and gives a very consistent estimate of 2 (h C +h BM )=96 km i.e. h BM =18 km for Model 2; Fig. 1. With increasing horizontal displacement, a single zone of mantle necking localizes at model centre. This strain localization on a lithospheric scale is favored by a strong layer-parallel shearing i.e. décollement at the ductile crust base. This décollement zone tends also to distribute strain that is concentrated in a narrow mantle zone into a large upper crustal domain. Consistently, minor faults in the mantle are still accumulating strain at the tip of the décollement zone where the intensity of deep crustal shearing strongly decreases. Lower crust décollement was not observed in Model 1 as the Moho temperature was low; therefore the strength of the deep crust was high. Incipient layer-parallel shearing in the deep crust leading to lithosphere scale necking is therefore strongly controlled by the Moho temperature and the mantle/crust strength ratio S M /S C. Regions of intense strain in the upper crust are located aside the mantle neck above two deep crustal décollements with converging senses of shear (see red arrows; Fig. 4). As a consequence, the upper crust deforms through two sets of parallel faults dipping towards the rift axis. They can be seen as two branches of a large rift issued from a central mantle neck. The wavelength of mantle necking 2(h C +h BM ) thus defines the initial rift width. In the absence of a high strength sub-moho mantle, deformation is more distributed and the rift spacing is fully controlled by the brittle crust thickness (Model 3; Fig. 4). The above analysis for rift spacing and rift width is not relevant since the mantle lithosphere cannot localize strain. The brittle crust is the only layer that induces plastic strain localization and thus controls rift spacing. From the above results, three basic patterns of continental rifting can be defined (Fig. 4; right): coupled crust mantle for high mantle Fig. 4. Role of the initial Moho temperature T M. Patterns of total strain in Models 1 to 3 for an initial crustal thickness of 30 km, after 30 km and 60 km of boundary extensional displacement, with V =1 cm/a, and line drawings of corresponding lithosphere scale structures. See Figs. 1 and 2 and Tables 1, 2 and 3 for model parameters. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

6 88 F. Gueydan et al. / Tectonophysics 460 (2008) subsidence. Fig. 5 that displays the topographic profiles of Models 1, 2 and 3, after 60 km of extension, shows that rift topography increases with decreasing Moho temperature T M and thus with increasing mantle strength. When crust and mantle are coupled, rift flanks are well marked with a maximum uplift of 1.2 km while the tectonic subsidence is low, around m. When deep crustal décollement occurs, the tectonic subsidence reaches a large value at rift centre, around 2 km, close to the prediction of the pure shear model of extension (McKenzie, 1978). In that case, rift flanks are less pronounced. Finally, the topography of wide rift is low, with regularly spaced highs and lows. These results are consistent with previous numerical models (e.g. Braun and Beaumont, 1989) and show the control exerted by the high strength mantle on rift topography and subsidence. 6. The effects of variations in stretching rate Fig. 5. Rift topography in Models 1 to 3 after 60 km of boundary extensional displacement, with V =1 cm/a. strength, deep crustal décollement when the strength of the deep crust is low enough to decouple the crust from the mantle and, finally, wide rift when the lithosphere mantle is wholly ductile and low strength. The coupled crust mantle and deep crustal décollement both belong to the narrow rift mode of Buck (1991) Rift topography as a function of mantle strength Since our modelling approach accounts for elasticity, the topography of the free surface can be extracted from the numerical results. As we did not take into account sedimentation and erosion, our estimates of topography are maximum for reliefs and minimum for Fig. 6 presents the total strain in Models 1, 2 and 3 for a velocity of 1 cm/a and 1 mm/a, after a horizontal displacement of 40 km. At 1 cm/a, the results are identical to those seen in Fig. 4, but are shown at an earlier stage of extension (40 km instead of 60 km). A decrease in the applied velocity by one order of magnitude (1 mm/a) gives a more localized deformation at lithosphere scale. More specifically, at low T M, a decrease in the applied velocity drastically changes the pattern of continental rifting (Model 1; Fig. 6). For similar thermal conditions, V=1 cm/a and 1 mm/a give patterns of coupled crust mantle and deep crustal décollement, respectively. A decrease in the applied velocity, and therefore of strain rate, decreases the ductile crust strength and thus enhances the decoupling between the crust and mantle (Brun, 2001). The presence of a very high strain zone in the deep crust highlights this feature. For higher Moho temperatures, no significant change of continental rifting pattern is observed (Models 2 and 3; Fig. 6). The deformation in brittle layers is however more localized at lower velocities. This feature reflects brittle ductile coupling, as recently discussed in Schueller et al. (2005). In a brittle ductile material, ductile layers exert a viscous resistance to fault motion, which limits the maximum displacement rate along any fault connected to the ductile interface. Therefore the viscosity of Fig. 6. Role of the applied boundary velocity. Patterns of total strain after 40 km of boundary extensional displacement for V =1 cm/a (left) and V=1 mm/a (right) for Models 1 to 3. See Figs. 1 and 2 and Tables 1, 2 and 3 for model parameters.

7 F. Gueydan et al. / Tectonophysics 460 (2008) Fig. 7. Role of the crustal thickness. Patterns of total strain after 45 km of boundary extensional displacement for h C = 30 km (left) and h C =50 km (right) for Moho temperatures T M = 482 C (up) and T M =550 C (down), and with V=1 cm/a. See Figs. 1 and 2 and Tables 1, 2 and 3 for model parameters. ductile layers controls the fault displacement rate. Low/high viscosity induces high/low fault displacement rate, respectively. An increase of the viscosity therefore makes new fault nucleation necessary in order to accommodate the applied velocity. Low viscosities favour a localized mode of fracturing, while high viscosities favour a distributed mode of fracturing. Following this analysis, a decrease in the applied velocity decreases the overall strain rate and the ductile strength (Eq. (3)), leading to a more localized fracturing, as shown for Models 2 and 3. These results exemplify the role of the deep crust strength in the patterns of continental rifting. 7. The effects of crustal thickness The role of crustal thickness is documented in Fig. 7, which presents the total strain patterns for four models after a horizontal displacement of 45 km at a velocity V=1 cm/a. The crustal thickness for Models 2 and 4 is 30 km and 50 km for Models 5 and 6 whereas the Moho temperature for Models 2 and 5 is 482 C and 550 C for Models 4 and 6 (Table 3). For T M =482 C, the increase in the crustal thickness leads to a switch from crustal décollement to coupled crust mantle, marked by distributed faulting in the brittle mantle. This change corresponds to an increase in lithosphere strength and more specifically in mantle strength (Fig. 1). In Model 5, the mantle strength is significantly larger than in Model 2. This again highlights the crucial role of the uppermost mantle strength in the pattern of continental rifting. No significant change is observed for T M =550 C. The effect of crustal thickness (Fig. 7) is fully represented by the strength ratio S M /S C. At a constant Moho temperature T M, the increase in crustal thickness increases the strength ratio S M /S C, and leads to a transition from the coupled crust mantle to the deep crustal décollement patterns, as illustrated by Models 2 and 5 (Figs. 7 and 8). On the contrary, the effect of the boundary velocity is not well represented by the strength ratio S M /S C. A decrease in the overall strain rate, used to compute the strength profiles, decreases the ductile strength in both the crust and mantle and therefore the S M /S C value remains almost unchanged. 9. Deformation maps of continental rifting patterns In order to compare our modelling results with natural examples of rifts, the domains representing the three patterns of continental rifting i.e. coupled crust mantle, deep crustal décollement and wide rift are mapped as a function of the whole crust and brittle 8. Transitions between the three patterns of continental rifting The plot of the set of models computed in the S M /S C T M diagram according to the stretching patterns observed shows that three domains can be identified according to the S M /S C value (Fig. 8). Coupled crust mantle occurs when S M /S C N10 corresponding to the presence of a very high strength mantle that imposes its pattern of strain localization at lithosphere scale. Deep crustal décollement occurs when 1bS M /S C b10, with mantle stretching, localized into a single neck zone, transmitted to the crust through two layer-parallel shear zones located in the deep crust. Finally, wide rifting occurs when S M bs C, corresponding to the absence of a high strength brittle mantle. Fig. 8. Patterns of continental rifting as a function of the crust mantle strength ratio S M /S C and Moho temperature T M,forh C =30 km (squares) and h C =50 km (dots). The patterns coupled crust mantle and deep crustal décollement both belong to the narrow rift mode of Buck (1991).

8 90 F. Gueydan et al. / Tectonophysics 460 (2008) Fig. 9. Patterns of continental rifting as a function of brittle crust and whole crust thicknesses for dry olivine at V =1 cm/a (top) and 1 mm/a (middle) and for wet olivine at V =1 cm/a (bottom). The patterns boundaries are plotted using the strength ratio value S M /S C (See Fig. 8). Curves of initial Moho temperatures T M = 400, 600 and 800 C are calculated with Eq. (2). crust thicknesses (Fig. 9). The boundaries between the three patterns are first defined analytically by the strength ratio S M /S C that corresponds to the inception of a deep crustal décollement (S M /S C ~10) and to the disappearance of a high strength mantle (S M /S C ~1; Fig. 8). Numerical simulations are then carried out to validate the pattern boundaries that correspond to progressive transitions rather than to sharp boundaries. Contours of Moho temperatures are also reported on deformation maps, based on Eq. (2) (Fig. 9). For a boundary velocity of 1 cm/a, the transition between the coupled crust mantle and deep crustal décollement patterns occurs at T M ~450 C and at higher T M for larger crustal thicknesses. The transition between deep crustal décollement and wide rift occurs at T M ~700 C regardless of the crustal thickness, as it is controlled by the absence of a high strength mantle. These transitions between the continental rifting patterns also depend on the rheological parameters and boundary velocities chosen to compute the models. A decrease of the boundary velocity by an order of magnitude decreases the ductile crust strength and thus enhances the decoupling between the crust and mantle. Consequently, the boundary of the deep crustal décollement pattern shifts towards lower T M values (Fig. 9; top). The same effect occurs if wet olivine parameters are used instead of those for dry olivine (Table 2 and Fig. 9; bottom). In contrast, using Plagioclase instead of wet quartz increases the crustal strength and thus shifts the boundary between the rifting patterns towards higher T M values. 10. Discussion: comparison with natural examples of rifts The thicknesses of the whole crust and of the brittle crust i.e. taken here as an equivalent of the seismogenic layer for several well documented examples of continental rifts are given in Table 4, as well as the method used to obtain these thicknesses. These examples are classically referred as continental rifts (Rhine Graben, Baikal Rift, Main Ethiopian Rift and Rio Grande Rift), passive margins and regularly spaced rifts (Basin and Range, Aegean and the Tibet-type plateau extension). Fig. 10 presents the position of

9 F. Gueydan et al. / Tectonophysics 460 (2008) Table 4 Crustal thickness and brittle crust thickness for different natural examples of rifts Location Crustal thickness Brittle crust thickness Methods h C [km] h RC [km] Rhine Limagne Bresse 30 km km Reflection seismic for Rhine (Brun et al., 1992). Refraction seismic for Limagne Bresse (Zeyen et al., 1997) Baikal 40 km 20 km Seismic profiles (ten Brink and Taylor, 2002), gravity inversion (Tibéri et al., 2003) and earthquake depth distribution (Déverchère et al., 2001) East African Rift km km Receiver function (Dugda et al., 2005); gravimetry (Mahatsente et al., 1999; Petit and Ebinger, 2000) Rio Grande km km Local tomography (West et al., 2004; Wilson et al., 2005) and reflection seismic (Klemperer, 1987) Evvia Corinthe 40 km 7 8 km Gravimetric study (Tiberi et al., 2000, 2001) and local tomography study (Latorre et al., 2004; Rigo et al., 1996) Basin and Range 50 km 10 km poorly constrained Seismic profiles (Stewart, 1978) Tibet 60 km km poorly constrained Indepth-Reflection seismic, receiver (Haines et al., 2003) and discussion T M N700 C (partial melting in the middle crust) in (Armijo et al., 1986). Partial melting in the middle crust (Mechie et al., 2004; Unsworth et al., 2005) The methods used to estimate these thicknesses are also given. these natural rift examples on a synthetic map of continental rifting patterns. The crustal structure of the Baikal rift has been investigated using seismic profiles (ten Brink and Taylor, 2002), gravity inversion (Tibéri et al., 2003) and earthquake depth distribution (Déverchère et al., 2001). It yields consistent estimates of the crustal thickness at approximately 40 km and of the brittle crust thickness at 20 km. The Baikal rift likely belongs to the coupled crust mantle pattern. The same holds for the Main Ethiopian rift (Dugda et al., 2005; Mahatsente et al., 1999; Petit and Ebinger, 2000) and the Rio Grande rift (Klemperer, 1987; West et al., 2004; Wilson et al., 2005). Note that inherited structures could induce an asymmetric structure within the forming rift, as also invoked for the Rhine Graben (Brun et al., 1992), the Baikal Rift (Déverchère et al., 2001) and the East African Rift (Tommasi and Vauchez, 2001). The deep structure of the Rhine graben has been documented by two deep seismic lines (DEKORP-ECORS) in the northern and in the southern part of the graben (Brun et al., 1992). The rooting of master normal faults in the lower reflective crust at ~15 km provides an estimate of the brittle crust thickness. The crustal thickness was estimated to be approximately 30 km (Table 4). Below the graben, gently dipping reflectors cross-cut the Moho with a normal sense of offset, indicating the existence of a narrow mantle shear zone. The Rhine graben falls within the deep crustal décollement type of continental rifting. The Limagne Bresse rift Fig. 10. Location of some famous examples of rifts on a synthetic map of continental rifting patterns (see natural rifts data in Table 4). system in France is marked by two asymmetric branches separated by ~150 km (Michon and Merle, 2003). The crustal thickness is similar to that of the Rhine graben, although the brittle crust seems to be much smaller (~10 km). This smaller brittle crust, compared to the Rhine graben, is consistent with a thinner thermal lithosphere of approximately 80 km (Sobolev et al., 1997) instead of 100 km below the Rhine graben (Achauer and Masson, 2002). Like the Rhine Graben, the Limagne and Bresse rift systems fall within the deep crustal décollement. This continental rifting pattern induces efficient lithosphere necking leading to the formation of non-volcanic passive margins for increasing amount of stretching. As most non-volcanic passive margins lack off deep crustal partial melting, their Moho temperature is lower than 700 C. We thus propose to range them in the deep crustal décollement (Fig. 10). This pattern has been exemplified by analogue experiments (Brun and Beslier, 1996) and numerical models (Nagel and Buck, 2004). That also well explains the typical faulting pattern of modern passive margins (Nagel and Buck, 2004), as well as those exhumed in the Alps (Whitmarsh et al., 2001). As already pointed out, it must be recalled here that the coupled crust mantle and the deep crustal décollement patterns of lithosphere extension both belong to the narrow rift mode of Buck (1991). In volcanic passive margins, at variance with the mechanics of extension discussed here, the uppermost mantle is initially disrupted by the intrusion of magma bodies that anticipate and/or strongly control lithosphere necking (Callot et al., 2001). Finally, the typical examples of regularly spaced rifts (Aegean rift systems, Basin and Range and Tibet plateau-like extension) fall in the wide rift. The crustal thickness of the Aegean rifts prior to extension (e.g. Corinth and Evia) were estimated by reference to continental Greece using gravity inversion (Tiberi et al., 2000, 2001). The brittle ductile transition within the crust is well marked by the presence of microseismic clusters at depths of 7 8 km(rigo et al., 1996) and by local tomographic images (Latorre et al., 2004). Crustal thicknesses prior to extension in the Basin and Range and in the Tibet plateau were estimated using seismic profiles (Stewart, 1978), reflection seismic and receiver functions (Indepth program, Haines et al., 2003). For the Basin and Range, estimates were made taking into account the timing of rift development and the corresponding amount of stretching and crustal thinning. Brittle crust thicknesses are poorly constrained in these two rift systems. However, evidence of partial melting in the middle crust (Mechie et al., 2004; Unsworth et al., 2005) indicates that Moho temperature T M is significantly higher than 700 C (Table 4). The three examples of regularly spaced rifts quoted above have in common a flat lying Moho over large horizontal distances (Allmendinger et al., 1987; Tirel et al., 2004, Braitenberg et al., 2000). Consistently, in our models, the absence of the Moho offset is only obtained when the mantle is wholly ductile and low strength and the deep crust strength is low

10 92 F. Gueydan et al. / Tectonophysics 460 (2008) enough to permit rapid ductile flow, allowing the Moho to remain relatively flat during continuing extension. 11. Conclusion This study provides some mechanical hints on the consequences of mantle and deep crustal strengths on the pattern of continental rifting. 1. The presence of a high strength uppermost mantle is essential to promote lithosphere necking. On the contrary, homogeneous lithosphere stretching is observed if the lithosphere mantle is entirely ductile and low strength. 2. The Moho temperature T M, the applied velocity, and thus the strength of the deep crust control the amount of crust mantle coupling. For low T M, crust and mantle are strongly coupled, leading to the formation of normal sense shear zones on a lithosphere scale. For higher T M, the deep crust acts as a décollement between the brittle crust and the uppermost mantle. 3. Rift spacing in the upper brittle crust is a direct function of the mantle strength. Rift spacing is controlled by the high strength mantle thickness and decreases with decreasing mantle strength. If the mantle is wholly ductile and low strength, wide rifting occurs in the upper crust with small rift spacing. 4. Three patterns of continental rifting are defined: coupled crust mantle for high mantle strength, deep crustal décollement when the strength of the deep crust is low enough to decouple the crust from the mantle and, finally, wide rift when the lithosphere mantle is wholly ductile and low strength. The coupled crust mantle and deep crustal décollement both belong to the narrow rift mode of Buck (1991). These results, gathered with previous numerical and analogue models, demonstrate that lithosphere necking and thus continental break-up requires a high strength brittle mantle (Brun and Beslier, 1996; Brun, 1999; Nagel and Buck, 2004; Burov and Watts, 2006). However, some geophysical evidences question this mechanical prerequisite (e.g. Jackson, 2002). 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