A Beniest
email: a.beniest@gmail.com

A Koptev

E Burov

1ϯ

Numerical models for continental break-up: implications for the South Atlantic

We propose a mechanism that explains in one unified framework the presence of continental break-up features such as failed rift arms and high-velocity and high-density bodies that occur along the South Atlantic rifted continental margins. We used 2D and 3D numerical models to investigate the impact of thermo-rheological structure of the continental lithosphere and initial plume position on continental rifting and breakup processes. 2D experiments show that break-up can be 1) "centred", mantle plume-induced and directly located above the centre of the mantle anomaly, 2) "shifted", mantle plume-induced and 50 to 250 km shifted from the initial plume location or 3) "distant", self-induced due to convection and/or slabsubduction/delamination and 300 to 800 km off-set from the original plume location. With a 3D, perfectly symmetrical and laterally homogenous setup, the location of continental break-up can be shifted hundreds of km's from the initial position of the mantle anomaly. We demonstrate that in case of shifted or distant continental break-up with respect to the original plume location, multiple features can be explained. Its deep-seated source can remain below the continent at one or both sides of the newly-formed ocean. This mantle material, glued underneath the margins at lower crustal levels, resembles the geometry and location of high velocity/high density bodies observed along the South Atlantic conjugate margins. Impingement of vertically up-welled plume material on the base of the lithosphere results in pre-break-up topography variations that are located just above this initial anomaly impingement. This can be interpreted as aborted rift features that are also observed along the rifted margins. When extension continues after continental break-up, high strain rates can relocalize. This relocation has been so far attributed to rift jumps. Most importantly, this study shows that there is not one, single rift mode for plume-induced crustal break-up.

Introduction

Over the last decades a large variety of rift features have been recognised and explained with different methods and different concepts. These features include for example aborted rift structures, anomalous topography or anomalously high velocity/high density bodies located in the lower crust. Explanations for anomalous features often link one mechanism with one observed rift feature. For example, on plume impingement, a stratified lithospheric rheology (e.g. D 'Acremont et al., 2003;[START_REF] Burov | Plume headlithosphere interactions near intra-continental plate boundaries[END_REF]) would result in topographic uplift, as has been modelled with thermo-mechanical modelling. Forward modelling shows that magmatic underplating can cause topographic variations [START_REF] Hirsch | Tectonic subsidence history and thermal evolution of the Orange Basin[END_REF]. Anomalously high velocity/high density bodies have been observed on tomographic images below the continents, implying that in some regions magmatic processes dominate rifting [START_REF] Cornwell | Northern Main Ethiopian Rift crustal structure from new high-precision gravity data[END_REF]. The latter is also suggested by gravity modelling that revealed the presence of anomalously high-density bodies in e.g. the South Atlantic domain, implying that volcanic activity played a key role in margin development [START_REF] Blaich | Crustal breakup and continent-ocean transition at South Atlantic conjugate margins[END_REF][START_REF] Maystrenko | Structural features of the Southwest African continental margin according to results of lithosphere-scale 3D gravity and thermal modelling[END_REF].

Review papers combine all these studies on one specific topic. Examples are the role of the Moho in extensional settings [START_REF] Cloetingh | The Moho in extensional tectonic settings: Insights from thermo-mechanical models[END_REF], the effect of volcanism in rifting and continental break-up [START_REF] Franke | Rifting, lithosphere breakup and volcanism : Comparison of magma-poor and volcanic rifted margins[END_REF] or the dynamic processes that control rifting (Ziegler and Cloetingh, 2004).

With this study we demonstrate how one break-up mechanism can explain a multitude of features. We use the South Atlantic break-up as our case study for plume-induced continental break-up. Since the South Atlantic developed diachronously and it is a very complex region requiring a 3D approach, we have not the intension to reproduce the South Atlantic evolution as such, including along-axis northward break-up propagation to close to the plume [START_REF] Franke | Rifting, lithosphere breakup and volcanism : Comparison of magma-poor and volcanic rifted margins[END_REF], but rather to address general observations on continental break-up. Our fully coupled lithospheric-grade 2D and 3D models have an explicit elasto-visco-plastic rheology that accounts for realistic deformation of the lithosphere and a slip free surface that can calculate vertical motions. The 2D model has proven to be very useful to investigate plume-lithosphere interactions (e.g. d'Acremont et al., 2003;[START_REF] Burov | Plume headlithosphere interactions near intra-continental plate boundaries[END_REF]. We take it one step further by developing one scenario to explain multiple anomalous features, such as high-velocity/high-density bodies and anomalous topographic variations with one single model. The 3D model is used to include the lateral component in a very simple, completely lateral homogeneous setting [START_REF] Koptev | Contrasted continental rifting via plume-craton interaction: Applications to Central East African Rift[END_REF].

Geological and geophysical setting

South Atlantic opening

Initial extension between Africa and South America was accommodated along a former fold-and-thrust belt (present-day location see Fig. 1), known as the Gondwana Fold Belt (GFB) or the Cape Fold Belt (CFB). This fold-and-thrust-belt was reactivated during the Early Mesozoic as a strike-slip system before the opening of the South Atlantic [START_REF] Cobbold | Permo-Triassic intracontinental deformation in SW Gondwana[END_REF]. During this reactivation it weakened the South American plate prior to the development of the Atlantic Mid-Oceanic Ridge, forming a first set of extensional basins [START_REF] Autin | Colorado Basin 3D structure and evolution, Argentine passive margin[END_REF], their axes oriented obliquely to the present-day orientation of the spreading centre (Fig. 1). Several extensional pulses caused the opening of the South Atlantic between 134 Ma and 113 Ma (e.g. [START_REF] Torsvik | A new scheme for the opening of the South Atlantic Ocean and the dissection of an Aptian salt basin[END_REF][START_REF] Moulin | A new starting point for the South and Equatorial Atlantic Ocean[END_REF]. Voluminous volcanic activity, recognised on seismic reflection profiles as 'Seaward Dipping Reflectors' (SDR's) in the form of aerial flood basalts (extrusive) and/or underplating (intrusive) accompanied an episode of extension that created the South Segment (Fig. 1), starting between 134 Ma and 132 Ma [START_REF] Moulin | A new starting point for the South and Equatorial Atlantic Ocean[END_REF]. Another pulse contributed to the formation of the Central Segment, starting around 112 Ma [START_REF] Moulin | A new starting point for the South and Equatorial Atlantic Ocean[END_REF] and is marked by massive salt deposits that have not been found along the margins of the South Segment (Fig. 1, [START_REF] Torsvik | A new scheme for the opening of the South Atlantic Ocean and the dissection of an Aptian salt basin[END_REF]. Only minor volcanic activity has been recorded in this segment as the typical SDR's are mostly absent, except just north of the Rio Grande Rise [START_REF] Franke | Rifting, lithosphere breakup and volcanism : Comparison of magma-poor and volcanic rifted margins[END_REF]. The opening of the South Atlantic and formation of the Mid-Atlantic Ridge is considered to be due to a combination of passive far-field forces [START_REF] Husson | Plate motions, Andean orogeny, and volcanism above the South Atlantic convection cell[END_REF] and the presence of different hotspots [START_REF] Torsvik | A new scheme for the opening of the South Atlantic Ocean and the dissection of an Aptian salt basin[END_REF]. A major far-field stress component that enhanced the growth of the South Atlantic domain during the Mesozoic is the subducting and 'pulling' Nazca plate to the west of the South American continent, which also resulted in the faster west-ward migration of the South American plate with respect to the almost stationary African plate (e.g. [START_REF] Husson | Plate motions, Andean orogeny, and volcanism above the South Atlantic convection cell[END_REF]. The South-African super plume rises from the core-mantle boundary (CMB) to below the mantle transition zone [START_REF] Hassan | Geochemistry, Geophysics, Geosystems[END_REF] which is reflected in present-day topography by a "superswell" at the margins of the south-west African continent [START_REF] Nyblade | The African Superswell[END_REF][START_REF] Davaille | Convective patterns under the Indo-Atlantic « box[END_REF]. As shown by a.o. [START_REF] Lithgow-Bertelloni | Dynamic topography, plate driving forces and the African superswell[END_REF], this excess of topography elevation is dynamically supported by upwelling flow of buoyant material through the mantle. From this large-scale, lower mantle low-velocity anomaly, the hotspots and their tracks (e.g. the Bouvet (Meteor), the Trinidad and St Helena Hotspots [START_REF] Torsvik | Large igneous provinces generated from the margins of the large low-velocity provinces in the deep mantle[END_REF][START_REF] Torsvik | A new scheme for the opening of the South Atlantic Ocean and the dissection of an Aptian salt basin[END_REF]) and the only deep rooted Tristan plume (Fig. 1, [START_REF] Torsvik | A new scheme for the opening of the South Atlantic Ocean and the dissection of an Aptian salt basin[END_REF] and references therein) might develop over a long period of time (~200 Myr).

Lithosphere structure margins

The selected profile for our 2D model connects the offshore southwest Africa Orange Basin and its conjugate with the Colorado basin on the South American side (pink line, Fig. 1). The Tristan hotspot lies actually in the middle of the two transects (Fig. 1). The present-day crustal and lithosphere structure of these margins is constrained by combining published work on deep seismic refraction data, tomography, gravity and magnetic studies (Fig. 2).

On the African side of the transect (Fig. 2a) the lithosphere thickness ranges from 120 km below the oceanic crust to 200 km below the continent [START_REF] Fishwick | Surface wave tomography: Imaging of the lithosphereasthenosphere boundary beneath central and southern Africa?[END_REF]. With gravity modelling and seismic interpretation the Moho-depth has been estimated to be less than 10 km below the oceanic crust of the Orange basin and over 40 km below the continent [START_REF] Maystrenko | Structural features of the Southwest African continental margin according to results of lithosphere-scale 3D gravity and thermal modelling[END_REF]. Even though crustal movements have been observed in central Africa, the southern African plate is considered relatively stable with a strong rheology [START_REF] Heine | Kinematics of the South Atlantic rift[END_REF].

On the South American side the lithosphere-asthenosphere boundary (LAB) reaches depths of 160km below the stable continent in Eastern Brazil and 120 km in below the continent in Central Argentina [START_REF] Heit | An S receiver function analysis of the lithospheric structure in South America[END_REF]. The Moho depth varies between 70 km below the plateau in the Andean orogeny to 25-35 km below the flat continent [START_REF] Van Der Meijde | Gravity derived Moho for South America[END_REF]. For the Colorado basin specifically, deep refraction seismic studies reveal a crustal thickness of the margins of 30 km [START_REF] Franke | Crustal structure across the Colorado Basin, offshore Argentina[END_REF].

We assume that before continental break-up, the lithosphere thickness of the South American plate was similar to that of the African plate. However, the South American plate underwent an earlier deformation phase prior to the formation of the South Atlantic domain [START_REF] Autin | Colorado Basin 3D structure and evolution, Argentine passive margin[END_REF]. Extensional deformation does result in lithospheric thinning and weakening (Ziegler and Cloetingh, 2004). We, therefore, adopt a weaker strength compared to the African plate and a thickness of 180 km, which is the mean between the 200 km of the African lithosphere and the present-day 160 km South American lithosphere, to account for this earlier deformation phase.

High velocity/high density bodies and aborted rift structures

Along the South Atlantic conjugate margins high velocity/high density bodies have been described at lower crustal depths below the continent and the margin (Fig. 1) using seismic data and gravity modelling.

Anomalous gravity and velocity bodies have been noted in the Central segment on the African side from Gabon [START_REF] Dupré | Tectonic history along the South Gabon Basin: anomalous early post-rift subsidence[END_REF] to the Lower Congo [START_REF] Contrucci | Deep structure of the West African continental margin (Congo, Zaïre, Angola), between 5°S and 8°S, from reflection/refraction seismics and gravity data[END_REF] and the Kwanza Basin [START_REF] Blaich | Crustal breakup and continent-ocean transition at South Atlantic conjugate margins[END_REF] and on the South American side from the Sergipe-Alagoas Basin [START_REF] Mohriak | Crustal architecture, sedimentation, and petroleum systems in the Sergipe-Alagoas Basin, Northeastern Brazil[END_REF], to the Camamu-Atmada basin [START_REF] Blaich | Crustal breakup and continent-ocean transition at South Atlantic conjugate margins[END_REF] and the Santos basin [START_REF] Blaich | Crustal breakup and continent-ocean transition at South Atlantic conjugate margins[END_REF]. In the South segment on the African side these bodies have been observed in the Walvis Basin [START_REF] Blaich | Crustal breakup and continent-ocean transition at South Atlantic conjugate margins[END_REF] and the Orange Basin [START_REF] Dressel | Reconstruction of the southwestern African continental margin by backward modelling[END_REF] and on the South American side in the Colorado basin, along the Uruguayan margin [START_REF] Clerc | Ductile extensional shear zones in the lower crust of a passive margin[END_REF] and in the deep Argentina Basin [START_REF] Franke | Crustal structure across the Colorado Basin, offshore Argentina[END_REF][START_REF] Schnabel | The structure of the lower crust at the Argentine continental margin, South Atlantic at 44°S[END_REF].

These bodies differ from the seaward-dipping reflectors (SDR) as they are situated at the base of the lithosphere or at lower crustal levels and do not necessarily have a magmatic origin. They could be serpentinized mantle or mafic and ultramafic crustal rocks (Fig. 2, [START_REF] Blaich | Crustal breakup and continent-ocean transition at South Atlantic conjugate margins[END_REF].

Graben structures or aborted rift structures onshore along the whole South American margin of the South Atlantic domain [START_REF] Burke | Development of graben associated with the initial ruptures of the Atlantic Ocean[END_REF], are located near the anomalously high velocity/high density bodies. In the South Segment, graben structures and failed rift structures are less-abundant along the African margin, where they appear mainly along the south South-African margin and in the Central segment along the Gabon and Benin margins. On the South American side of the South Segment, the basins oriented perpendicular to the present-day ridge extend onshore as aborted rift features [START_REF] Burke | Development of graben associated with the initial ruptures of the Atlantic Ocean[END_REF]. Another failed rift feature is observed in the southwestern part of the Santos Basin, where the now aborted Abimael ridge is located parallel to the present-day Mid Oceanic Ridge [START_REF] Heine | Kinematics of the South Atlantic rift[END_REF].

Model setup

The 2D thermo-mechanical numerical code FLAMAR, based on the FLAC-Para(o)voz algorithm [START_REF] Cundall | Numerical experiments on localization in frictional materials[END_REF][START_REF] Poliakov | Initiation of salt diapirs with frictional overburdens: numerical experiments[END_REF] has been used to investigate the effect of plume location on continental breakup using the South-Atlantic as an example of a fully developed rift-to-spreading system. We built our case on the continuation of earlier parametric studies on the rheology of the lithosphere and plume-continental lithosphere interactions [START_REF] D'acremont | Numerical modelling of a mantle plume: the plume headlithosphere interaction in the formation of an oceanic large igneous province[END_REF][START_REF] Burov | Plume headlithosphere interactions near intra-continental plate boundaries[END_REF]. Where needed, we adjust the parameters according to the geological and geophysical evidence described above. A symmetric simulation that is not area-specific are carried out with the 3D viscous-plastic numerical code 3DELVIS [START_REF] Gerya | Robust characteristics method for modelling multiphase visco-elastoplastic thermo-mechanical problems[END_REF]. All mechanical and thermo-rheological parameters are listed in Table 1. We have performed a series of 36 experiments. Controlling parameters and principal results are summarized in Table 2.

2D numerical model

The FLAMAR code has been updated and modified over the last 20 years [START_REF] Burov | The effective elastic thickness (Te) of continental lithosphere: What does it really mean?[END_REF][START_REF] Burov | Erosion and rheology controls on synrift and postrift evolution: Verifying old and new ideas using a fully coupled numerical model[END_REF][START_REF] Pourhiet | Rifting through a stack of inhomogeneous thrusts (the dipping pie concept)[END_REF][START_REF] Yamato | HP-UHP exhumation during slow continental subduction: Self-consistent thermodynamically and thermomechanically coupled model[END_REF]. For the sake of coherency with previously published papers we only describe the main features and essentials of the model used for this study. Detailed descriptions of the code can be found in studies that have tested the code for many different geological cases [START_REF] D'acremont | Numerical modelling of a mantle plume: the plume headlithosphere interaction in the formation of an oceanic large igneous province[END_REF][START_REF] Pourhiet | Rifting through a stack of inhomogeneous thrusts (the dipping pie concept)[END_REF][START_REF] Yamato | HP-UHP exhumation during slow continental subduction: Self-consistent thermodynamically and thermomechanically coupled model[END_REF]. FLAMAR is a fully explicit, finite element/finite difference code based on a Cartesian coordinate frame. It has a 2D strain formulation with a Lagrangian mesh that consists of quadrilateral elements consisting of two couples of triangular sub-elements containing tri-linear shape functions. It uses a large-strain, time-marching scheme. The code solves for full Newtonian equations of motions in a continuum mechanics approximation (3.1)

< 𝜌𝒖̈> -∇𝜎 -𝜌𝒈 = 0 (3.1)
where 𝜌, 𝒖̈, σ and g stands for density, acceleration of the object, stress and acceleration due to body forces or gravity, respectively.

It

𝜌𝐶 𝑝 𝐷𝑇/𝐷𝑡 -𝑘∇ 2 𝑇 -∑ 𝐻 𝑖 𝑛 𝑖 = 0; 𝜌 = 𝑓(𝑃, 𝑇) (3.3) 𝑑ℎ 𝑑𝑡 = 𝑎∇ 2 ℎ (3.4)
In this case, t stands for time, u is the displacement vector, and T is temperature. The heat transfer equation relies on C p for the specific heat, k for thermal conductivity respectively and H for the internal heat production, including radiogenic heat and frictional heat dissipation. P stands for pressure that become negative for compression. The linear diffusion equation uses a constant a and the height or thickness of the sediments h.

The code is capable of calculating realistic visco-elasto-plastic rheologies explicitly. Pressure-dependent deformation is maintained through the Mohr-Coulomb criteria for the plastic regime and the non-linear viscous flow law at depth. The free surface upper boundary condition calculates high-resolution topographic changes due to deformation of the lithosphere.

Model geometries

The model box is 2000 km wide and 400 km deep. The grid size is 400 x 80 elements, resulting in a resolution of 5 km x 5 km per element. We have tested three different lithospheric setups with diverse complexities (see 3.1.2, Fig. 3) and three different locations of a 1700 o C thermal and compositional mantle anomaly at 400 km depth [START_REF] D'acremont | Numerical modelling of a mantle plume: the plume headlithosphere interaction in the formation of an oceanic large igneous province[END_REF]. The initial locations vary laterally at the base of the model with the center of the anomaly positioned 1) at the centre of the model (i.e. plume location at 1000 km, see Table 2), 2) at 200 km to the right of the model box's centre (i.e. plume location at 1200 km) and

3) at 200 km to the left of the model box's centre (i.e. plume location at 800 km). Each location is tested in a separate calculation. Following previous studies the base of the anomaly lies at 400 km depth as the deeper mantle phase does not have a large impact on the crustal evolution [START_REF] D'acremont | Numerical modelling of a mantle plume: the plume headlithosphere interaction in the formation of an oceanic large igneous province[END_REF][START_REF] Ribe | Three-dimensional modeling of plume-lithosphere interaction[END_REF]. The anomaly has a simplified, symmetric, spherical shape since at depth viscous bodies take a spherical shape and this follows the line of 2D and 3D numerical modelling experiments on plumelithosphere interaction (a.o. D' Acremont et al., 2003;[START_REF] Burov | Asymmetric three-dimensional topography over mantle plumes[END_REF][START_REF] Koptev | Contrasted continental rifting via plume-craton interaction: Applications to Central East African Rift[END_REF]. In most of the experiments, it has a diameter of 230 km. The effects of a mantle anomaly with a diameter of 100 km were tested in a limited number of models. The composition of the mantle anomaly is olivine with a density of 3250 kg/m 3 (except for several models where it is 3310 kg/m 3 ) which has been determined to be an intermediate plume in previous studies [START_REF] Turcotte | Geodynamics[END_REF][START_REF] D'acremont | Numerical modelling of a mantle plume: the plume headlithosphere interaction in the formation of an oceanic large igneous province[END_REF]. No background density tests have been performed as the background density used for background calculations is the same for the plume as well as the surrounding mantle. The thermal contrast between the plume and the mantle varies as thermal exchanges happen between the plume and the mantle, decreasing the temperature of the plume. The mantle also cools as the plume rises to shallower depths.

Density and rheological structure

The 2D model consists of four horizontal rheological layers. For Setup 1 (Fig. 3a and Fig. 3b), a laterally homogeneous 40 km thick two-layered crust and a 160 km thick lithospheric mantle have been applied. We test the model's sensitivity for two different rheological properties of the crust. We use a "weaker" rheological strength envelope (Setup 1a; Fig. 3a), composed of 1) a wet quartz upper crust with a density of 2500 kg/m 3 and 2) a diabase lower crust with a density of 2750 kg/m 3 . Our second rheological strength envelope has the characteristics of a "strong", cratonic crust that consists of: 1) a dry quartz upper crust with a density of 2600 kg/m 3 and 2) a strong diabase lower crust with a density of 2850 kg/m 3 . The rheological differences of the two strength envelopes represent a "weaker" crust that has been subject to an earlier deformation phase, before the opening of the South Atlantic, which is the case for the South American side [START_REF] Autin | Colorado Basin 3D structure and evolution, Argentine passive margin[END_REF], and a "stronger" crust of cratonic nature that represents the stable southern African continent(after [START_REF] Burov | The effective elastic thickness (Te) of continental lithosphere: What does it really mean?[END_REF]. Dry olivine flow law has been assumed for both lithospheric and sub-lithospheric mantle in all our experiments. The initial density of the mantle decreases from 3330 kg/m 3 to 3310 kg/m 3 at the lithosphere-asthenosphere boundary. Specific values of the rheological parameters used are given in Table 1.

For Setups 2 and 3 we apply a laterally non-homogeneous crustal rheology: a "weak" crustal rheology for the left half of the model and a "strong" crustal rheology for the right one (Fig. 3c-f). The crustal and lithospheric thicknesses are laterally homogeneous in Setup 2a: 20 km for upper crust, 20 km for lower crust and 160 km for lithospheric mantle (Fig. 3c). Setups 2b and 3 are characterized by laterally varying lithospheric layers, based on the lithospheric scale structure described in section 2.2: the "weaker" left half has a 15 km-thick upper crust, 15 km-thick lower crust and a 150 km-thick lithospheric mantle, whereas the "stronger" right half has a 20 km-thick upper crust, a 20-km think lower crust and a 160 km-thick lithospheric mantle (Fig. 3c-f). Three different contacts between the rheological strengths are tested. Setup 2b has a straight vertical contact. For Setup 3 we have adopted a geometry resembling the old suture zone that is reactivated during the first extensional phase. The suture is dipping either 40 degrees towards the 'strong', African rheology (Setup 3a) or towards the 'weak' South American rheology (Setup 3b).

By the setups described above we have tested the following parameters: initial position of the plume, density of the mantle plume (limited to Setup 3a) and different half-rate velocity boundary conditions (see Table 1).

Mechanical and thermo-rheological boundary conditions

We simulate tectonic forcing by applying a constant, time independent, extension rate along the vertical side of the box of 25 mm/yr. An equal half-rate velocity is applied on both sides of the box (12.5 mm/yr) to one set of models and 5 mm/yr on the right side and 20 mm/yr on the left side is applied to a second set of models (Table 1). The half-rate velocities are adopted from [START_REF] Müller | Age, spreading rates, and spreading asymmetry of the world's ocean crust[END_REF]. The bottom of the box is defined by hydrostatic pressure with free slip in all directions. The upper side of the box is a free surface boundary, implying a free stress and a free slip condition in all direction, allowing the lithosphere to develop freely. A moderate erosion by diffusion is applied (a = 500 m 2 /yr).

The upper and bottom thermal boundary condition is a fixed temperature 10 o C and 1400 o C respectively to represent a 'cold' geotherm. An old lithosphere of 500 Ma [START_REF] Burov | The effective elastic thickness (Te) of continental lithosphere: What does it really mean?[END_REF] has been assumed

for the tectonic age used to represent the super-continent Pangea before break-up. The geotherm used for the models reaches 500 o C at Moho depth, 1330 o C at the lithosphere-asthenosphere boundary (LAB) after which it becomes adiabatic until it reaches 1400 o C at the base of the model at 400 km [START_REF] Ribe | Three-dimensional modeling of plume-lithosphere interaction[END_REF].

3D numerical model

A 3D model has been performed with the thermo-mechanical viscous-plastic 3DELVIS code [START_REF] Gerya | Dynamical instability produces transform faults at mid-ocean ridges[END_REF][START_REF] Gerya | Robust characteristics method for modelling multiphase visco-elastoplastic thermo-mechanical problems[END_REF] the 3D model setup and rheological and material properties used in our 3D experiments can be found in [START_REF] Burov | Asymmetric three-dimensional topography over mantle plumes[END_REF] and [START_REF] Koptev | Dual continental rift systems generated by plumelithosphere interaction[END_REF][START_REF] Koptev | Contrasted continental rifting via plume-craton interaction: Applications to Central East African Rift[END_REF].

Model results

2D model results

Three types of model scenarios result from our set of experiments. "Centred" break-up, when the mantle anomaly moves vertically upwards and break-up happens directly above the original location of the centre of the mantle anomaly. "Shifted" break-up, when the mantle anomaly first migrates vertically and, once it reaches the base of the lithosphere, migrates laterally until break-up occurs with a 50 to 250 km offset with respect to the initial anomaly position. "Distant" break-up, when a mantle anomaly rises to the base of the lithosphere and remains there, while the location of break-up takes place more than 300 km away from the initial site of the anomaly.

Experiment 6, characterized by a "strong" homogenous lithosphere, is an example of "centred" break-up (Fig. 4a). The mantle anomaly reaches the base of the lithosphere rapidly within 2 Myr, after which it penetrates into the lithosphere. The rising flow of plume material is strong enough to break apart the overlying lithospheric mantle and crust between 7 and 8 Myr. The surface reacts by uplift, then subsidence and alternating positive and negative vertical movements of the margins and the rift centre. Although the initial position of break-up centre is situated directly above mantle plume, the continuous extensional evolution, including strain relocation and changing temperature distribution, suggest a post-rift lateral shift of the spreading axis. Note that after continental break-up mantle plume material reaches the surface where it contributes to the formation of new oceanic lithosphere.

The "shifted" mode of continental break-up is illustrated by Experiment 12 where the mantle plume anomaly has been seeded below a stronger lithosphere composing the right half of the model domain (Fig. 4b). As in the case of Experiment 6, the onset of rifting starts with a rapid rise of the anomaly, impinging the lithosphere not later than 2 Myr. Surface topography associated with localized crustal strain is formed around 3-4 Myr with small offset (<50 km) from the point directly above mantle plume impingement.

Further upslope migration plume material leads to continental break-up between 7 and 10 Myr. A principal difference from the "centred" experiment 6 is the lateral shift (50 to 250 km) of the newly formed spreading axis with respect to the initial position of the mantle plume. Lateral migration of the plume head to this break-up axis leads to an asymmetrical distribution of the plume material: some of the material reaches the surface at the spreading centre, another part remains glued beneath the highly thinned lithosphere at depths between 200 and 10 km. Similar to Experiment 6, the final stage of the "shifted" system development is the strain relocation corresponding to 200 km-wide jump of the spreading axis.

Finally, experiment 16 illustrates the "distant" break-up mode that starts with a rapid uplift of the mantle plume to the bottom of the lithosphere, an observation typical for all performed models. This expectedly results in associated topography variations (Fig. 4c). In contrast with the two previously discussed break-up modes (experiments 6 and 12), mantle plume material remains glued beneath the base of the lithosphere without localized ascent towards the formed break-up centre. Lithosphere thinning that will result in breakup occurs at large distance (more than 500 km) from the plume impingement. This appears to be related to secondary mantle convection associated with plume-induced subduction of the lithospheric mantle that has developed upon plume upwelling to the lithosphere-asthenosphere boundary on both sides of the plume head. It is noteworthy that initial topographic changes created by the impingement of the plume remain visible throughout the model evolution. Given the lack of near-surface plume material, this "distant" mode cannot be considered as break-up directly induced by the impact of the mantle plume. Nevertheless, it might reflect the implicit influence of the upwelled plume on "distant" break-up processes via plume-induced subduction and mantle convection.

"Centred" break-up preferably takes place using initial Setup 1, where the crust and lithospheric mantle are laterally homogeneous and no inherited structures are given (Fig. 4a, Table 1), but other setups can also evolve according to this mode (Fig. 5a-c). Break-up occurs between 6 and 10 Myr, directly above the initial location of the mantle anomaly. Mantle material reaches the surface at the point of impingement that evolves into the break-up axis. Almost all plume material is involved in formation of new oceanic lithosphere. As a result, after continuous (more than 10 Myr) calculations, only little material remains below the thinned continental lithosphere. Note that, even though central located plumes are expected to develop to a symmetric or "central" mode, a central located plume evolves the least likely into "central" type of breakup (table 1). "Shifted" break-up is favoured by Setup 2a where the thickness of the lithospheric layers is laterally homogeneous but crustal rheology differs (Fig. 5d-f). The mantle anomaly rises and break-up also occurs between 6 and 10 Myr, but in this case it is shifted from the initial point of impingement. Most mantle material remains below the lithosphere, but through migration along the bottom of the lithosphere some material still reaches the surface. This mode of break-up only occurs when the lithosphere properties (rheology and/or thickness) varies laterally, but it does not completely control "shifted" break-up, because not all laterally varying rheology experiments result in "shifted" break-up. Plume location is also not a controlling factor for the model to result in "shifted" break-up as all three plume locations can result in "shifted" break-up. The "distant" break-up experiments have a preference for Setup 2 and 3, where both the lithospheric layers' thickness and the crustal rheology are laterally different (Fig. 5g-i). Crustal break-up happens slightly later compared to the "centred" and "shifted" experiments: between 9 and 12 Myr. Mantle anomaly material does not reach the surface, but remains completely glued to the bottom of the lithosphere.

Most of the continental break-up modelled with the "distant" experiments occurs in the lithospheric segment that is characterized by a strong crust.

Almost half (12 out of 25) of the equal half-rate velocities boundary condition results in "centred" breakup. More than half (6 out of 11) of the unequal half-rate velocity boundary conditions result in "distant" break-up mode. The different velocity parameters do have a preference for a certain break-up style, but it is not a controlling factor.

The models that resulted in "shifted" break-up have a mantle anomaly that rises to the base of the lithosphere and upon arrival, migrates, in most cases, towards the weaker lithosphere to break through this less strong segment. This is in contrast with the "distant" model results that develop crustal break-up in the stronger lithosphere (11 out of 11 models) when the plume remains glued below the weaker lithosphere and does not break through. In case of "distant" break-up mode, the rheology is very important and strongly controls this mode of break-up.

3D model results

Similarly to the 2D experiments, the 3D model shows a quick (<2 Myr) upwelling of the plume material up to lithosphere-asthenosphere boundary (Fig. 6a). After this, the plume head starts to spread laterally within lower part of lithospheric mantle (Fig. 6b-d). When the mantle plume impinges on the base of lithosphere, almost all plume material is partially melted (Fig. 6a). Following spreading and cooling expectedly leads to gradual solidification of the plume (Fig. 6b-c), which has been completed at 50 Myr (Fig. 6d).

The interplay between far-field forces and active mantle upwelling results in a "classical" single rift that crosses the entire model domain in the direction perpendicular to external extension (Fig. 6a). Continuous evolution shows the formation of a wide rift valley where localized brittle deformation is concentrated along the boundary fault (Fig. 6b-c). This rift basin opens rapidly (Fig. 6b) reaching a width of 600 km in less than 35 Myr while passive extension applied at the boundaries would result in only 200 km width (Fig. 6c). This highlights the important role of plume-related buoyancy forces in the context of studied "active-passive" rifting.

The next stage of the system evolution (65 Myr) is a quick switch of deformation localization from riftbounding faults to narrow zones inside the rift valley (Fig. 6e). This change in rifting style is caused by initiation of localized upwelling of plume material along stretched zone(s) highlighted at the surface by localized high strain rates (Fig. 6e). Further localized plume ascent associated with decompression melting of plume material increases the rate of lithospheric thinning leading to continental break-up along a spreading axis that has shifted laterally outwards from the centre of the plume head (Fig. 6f). This asymmetrical position of the spreading zone arises spontaneously within initially symmetrical and laterally homogenous lithosphere and is likely controlled by melting and cooling processes into head of mantle plume.

Thus, a lateral shift of plume-induced break-up centres with respect to initial plume impingement revealed in certain 2D experiments (see for example Experiment 6, Fig. 4a) appears to be an intrinsic characteristic of self-induced plume-related processes that do not necessary requires fast (>1 cm/y) external extension nor any lateral lithospheric heterogeneity (see also Experiment 2 for a 2D example of "distant" break-up in the context of laterally homogenous lithosphere).

Discussions

General aspects

The results of our models are important in the context of ongoing discussion on plume-induced continental break-up. We show that continental break-up can be initiated by just one single mantle plume under different initial and boundary conditions. In most of our 2D models (32 out of 36) continental break-up takes place as a result of the direct ("centred" or "shifted" modes) or indirect ("distant" mode) impact of the mantle plume (Fig. 8). Four remaining models do not result in break-up. On one hand, the models that develop according to "centred" (Fig. 8b) or "shifted" (Fig. 8c) modes are directly induced by the plume anomaly which results in penetration of plume material up to the surface. On the other hand, the "distant" mode is characterized by secondary mantle convection associated with plume-induced subduction and/or convection. In this case, the mantle plume is not involved directly in break-up processes and remains glued at the base of adjacent unbroken lithosphere (Fig. 8d).

In a very early phase, strain rate localizes and topographic variations develop directly above the initial plume impingement location (Fig. 8a). They remain visible only in the "shifted" and "distant" models and can be interpreted as aborted rifts. It was commonly accepted and almost self-evident that in the case of plumeinduced continental break-up, its axis should be situated directly above the initial plume impingement position [START_REF] D'acremont | Numerical modelling of a mantle plume: the plume headlithosphere interaction in the formation of an oceanic large igneous province[END_REF]. However, observations such as failed rifts and deep-seated mantle sources beneath a strong continent that are significantly remote from the mid-oceanic ridge, actually imply that continental rifting and break-up are not a purely symmetric and "plume-centred" processes. Our modelling demonstrates that symmetric development of mantle material ascent and subsequent continental break-up are not a definite outcome. More than half of our models (19 out of 32) result in "shifted" and "distant" break-up modes, suggesting that these modes should also be expected in a wide range of initial and boundary conditions. Even so, our perfectly symmetric and lateral homogeneous 3D model shows that in a purely symmetric setting, with no lithospheric rheological heterogeneities, continental break-up shifted from the original centre of the mantle anomaly is possible. We argue here that "centred" symmetric continental break-up developed directly above mantle plume is only one particular case of possible evolutions of plume-induced break-up systems.

The case of the South Atlantic

The South Atlantic is considered to be a good example of plume-induced continental break-up (e.g. [START_REF] Torsvik | A new scheme for the opening of the South Atlantic Ocean and the dissection of an Aptian salt basin[END_REF]. Some of the observations such as failed rifts [START_REF] Heine | Kinematics of the South Atlantic rift[END_REF] and high velocity bodies (e.g. [START_REF] Blaich | Crustal breakup and continent-ocean transition at South Atlantic conjugate margins[END_REF]) cannot be explained with conventional models, usually assuming a "centred"-like breakup style (e.g. [START_REF] Burov | Plume headlithosphere interactions near intra-continental plate boundaries[END_REF][START_REF] D'acremont | Numerical modelling of a mantle plume: the plume headlithosphere interaction in the formation of an oceanic large igneous province[END_REF]). Yet, our experiments showing "shifted" breakup mode (Fig. 7; Fig. 8c) can be used to explain these features. In these models, initial crustal deformation associated with mantle plume impingement (Fig. 7a-b; Fig. 8a) are formed within the first 5 Myr. Significant topography variations developed during this initial stage of rift evolution (i.e. before break-up) can be interpreted as very early failed rift features (e.g. the failed Abimael rift in the southwest of the Santos Basin).

The topographic plateaus that remain elevated long after break-up have also been observed with dynamic topography studies [START_REF] Nyblade | The African Superswell[END_REF]. Next, localized strain becomes concentrated close to the boundary between strong and weak lithosphere that is laterally offset (~400 km) from the area of initial plume impingement (Fig. 7c). Transition from wide rift stage to lithospheric break-up is marked by narrowing a broad rift region (over 1000 km width) down to narrow rift valley (Fig. 7c-d) with the width of 10's of kilometers between the two rift-shoulders. Associated lithospheric thinning leads to a separation of the two plates along a spreading centre corresponding to South Atlantic ridge (Fig. 7e; Fig. 8c). Part of strong crust that remains attached to the weaker lithosphere segment, could correspond to the Brazilian craton that was once bordering the African continent [START_REF] Heine | Kinematics of the South Atlantic rift[END_REF].

Simultaneously with thinning of the lithosphere below the future break-up centre, the plume material migrates along the base of the lithosphere and rises towards the deformed crust where it breaks through.

This migration can go as far as 300 km from the plume impingement point and only ceases when the material that is still at the base of the lithosphere (at depths between 200 km and 10 km) reaches thermal equilibrium (in this case after 22 Myr). This confirms the hypothesis that one mantle anomaly (or plume) can flow laterally over significant distances below a slow-moving continent, after being risen to the base of the lithosphere (e.g. [START_REF] Sleep | Mantle plumes from top to bottom[END_REF]. When thermal equilibrium is reached, the mantle material glued to the base of the lithosphere at shallower depths corresponds geometrically and location-wise to high-velocity/highdensity bodies observed on seismic data below the thinned continental lithosphere and the transition zone of the South Atlantic domain [START_REF] Clerc | Ductile extensional shear zones in the lower crust of a passive margin[END_REF]. During migration, products of partial melting of the mantle material can move vertically to (shallow) lower crustal levels. They might resemble high density bodies observed at lower crustal levels inside continental crust with similar geometries observed with gravity modelling [START_REF] Blaich | Crustal breakup and continent-ocean transition at South Atlantic conjugate margins[END_REF]. These processes cannot be reproduced by our 2D modelling, because no melt production and extraction have been simulated.

Note that our 2D study has not the intention to capture such 3D features like along-axis northward propagation of the break-up axis [START_REF] Franke | Rifting, lithosphere breakup and volcanism : Comparison of magma-poor and volcanic rifted margins[END_REF] up to the centre of the surface manifestation of Tristan plume activitythe Paraná-Etendeka continental flood basalts province.

After continental break-up, the mantle plume anomaly continues to play an important role in the spreading evolution of the system. Strain rate relocation takes place around 18 Myr, when the spreading axis shifts another 200 km towards the left from the original position of the break-up centre (Fig. 7f). This phenomenon could correspond to a rift-jump that have also been both observed and modelled [START_REF] Brune | Rift migration explains continental margin asymmetry and crustal hyper-extension[END_REF] in the South Atlantic domain.

The question we raised about the initial position of the mantle plume anomaly responsible for continental break-up in the South Atlantic remains open. On figure 8e, we show a reconstructed configuration of the slow velocity anomaly (corresponding to the South African Super Plume) at the CMB based on present-day seismic tomography model (after [START_REF] Davaille | Convective patterns under the Indo-Atlantic « box[END_REF]. The orange dots refer to the three possible principal locations of initial thermal anomaly at the upper/lower mantle boundary corresponding to the Tristan plume.

The central point refers to the most evident "centered" scenario (Fig. 8b) when the deep-seated thermal anomaly in the upper mantle is supposed to be located directly below its surface manifestation and hints to voluminous Paranà-Etendeka continental flood basalts province (Fig. 8e, e.g. [START_REF] Torsvik | A new scheme for the opening of the South Atlantic Ocean and the dissection of an Aptian salt basin[END_REF][START_REF] Heine | Kinematics of the South Atlantic rift[END_REF]. This scenario, however, is not consistent with the commonly considered concept that plumes emerge from the edges of the large low-velocity anomalies at the CMB that has been confirmed by both numerical modelling [START_REF] Hassan | Geochemistry, Geophysics, Geosystems[END_REF] and by empirically established correlation between downward projected plume-associated large igneous provinces and the margins of the large low shear velocity province beneath Africa [START_REF] Torsvik | Large igneous provinces generated from the margins of the large low-velocity provinces in the deep mantle[END_REF]. Moreover, with this 'centred' scenario we cannot explain additional features such as failed rift arms and anomalous bodies at lower crustal levels. The 'distant' break-up mode (Fig. 8d), where the initial plume centre is located below the South American section and remains there after "plume-independent" continental break-up, does not fit well with geological observations of the voluminous Paranà-Etendeka continental flood basalts that are supposed to be related with direct influence of the Tristan hot spot [START_REF] Torsvik | A new scheme for the opening of the South Atlantic Ocean and the dissection of an Aptian salt basin[END_REF]. Finally, initial plume position slightly moved to African side (right dot on Fig. 8e) refers to "shifted" scenario that seems to be preferable (Fig. 8c). The time length of the modelled rift phase (10 Myr +/-3 Myr) is much shorter than has been inferred from geological and geophysical observations (160 Ma to 134 Ma, [START_REF] Franke | Rifting, lithosphere breakup and volcanism : Comparison of magma-poor and volcanic rifted margins[END_REF]) in the South Segment of the South Atlantic. Despite this, with the eastward offset initial position of the mantle plume with respect to a heterogeneous thermorheological lithospheric structure we are able to explain not only plume induced flood basalts but also a set of anomalous features such as failed rift systems, and deep crustal bodies.

Conclusion

Different lithospheric strengths comparable to the South American and African continental crust, inherited structures, boundary velocity conditions corresponding to average spreading rates, and initial location of a thermal mantle anomaly (i.e. plume) have been tested to investigate the dynamics of plume induced continental break-up. A set of 36 models shows that with only one anomaly three very different scenarios for continental break-up can be realized depending on the rheological structure, anomaly location and inherited structures. Continental break-up does not necessarily occur above the centre of the initial location of a mantle anomaly. As mentioned above, our models show three types of break-up 1) "centred" break-up,

2) "shifted" break-up and 3) "distant" break-up.

"Centred" and "shifted" break-up types of models refer to plume-induced type of break-up. For the first mode, mantle material rises vertically towards the bottom of the lithosphere after which it breaks through the crust and reaches the surface directly above the initial plume position. The "shifted" type of break-up shows continental break-up that is 50 to 200 km shifted from the initial location of the mantle anomaly. In this case, the mantle plume rises and impinges the lithosphere, after that it migrates laterally and cuts through the lithosphere reaching the surface at a break-up point considerably shifted from the area of initial, prebreak-up impingement. Some material remains glued underneath the lithosphere at depths between 200 and 10 km. These deep-seated bodies, at depths of 200 km, are not situated directly below the break-up centre, but are spread over a large area below the continental margins. The shallower bodies geometrically resemble high density/high velocity bodies detected by seismic profiling and gravity modelling along the margins of the South Atlantic domain and at lower crustal levels.

The "distant" break-up mode refers to continental rupture that is indirectly induced by the presence of the mantle plume ponding at the bottom of the weaker continental lithosphere, when "plume-independent" break-up of adjacent stronger lithosphere appears to be considerably (from 250 and 800 km) displaced from the location of plume-lithosphere interaction. In this case, laterally widely spread plume material remains glued below unbroken segments of the lithosphere.

Topographic changes that occur very early during initial rifting stage remain visible for a long period and can possibly be interpreted as failed rift systems (in the cases of "shifted" and "distance" modes). Strain relocation after continuous post-break-up extension could be interpreted as rift jumps. A simple 3D model has been built to illustrate that even in a fully symmetric setup, rift-to-break-up processes are not by default symmetric and can very well evolve asymmetrically.

There is no controlling parameter for one of the three types of rifting, with a combination of parameters determining the outcome, but the location of the mantle anomaly with respect to the rheology is the most with application to the Western Alps. Earth Planet. Sci. Lett. 271, 63-74. doi:10.1016/j.epsl.2008.03.049 Ziegler, P.A., Cloetingh, S., 2004. Dynamic processes controlling evolution of rifted basins. Earth-Science

Rev. 64, 1-50. doi:10.1016/S0012-8252(03)00041-2 from 10 to 30 km on the South American side from ocean to continent [START_REF] Schnabel | The structure of the lower crust at the Argentine continental margin, South Atlantic at 44°S[END_REF]. On the African side the depth varies from less than 10 km to over 40 km, ocean to continent [START_REF] Maystrenko | Structural features of the Southwest African continental margin according to results of lithosphere-scale 3D gravity and thermal modelling[END_REF]. The lithosphere-asthenosphere boundary (LAB) varies from 50 km to 120 km, ocean to continent for Colorado basin on the Argentinean margin [START_REF] Heit | An S receiver function analysis of the lithospheric structure in South America[END_REF]. The Orange Basin on the South African margin has a LAB depth ranging between 80 km and 200 km from ocean to continent [START_REF] Fishwick | Surface wave tomography: Imaging of the lithosphereasthenosphere boundary beneath central and southern Africa?[END_REF]. The location of anomalous bodies is depicted (in green) for the Colorado Basin [START_REF] Schnabel | The structure of the lower crust at the Argentine continental margin, South Atlantic at 44°S[END_REF] and the Orange Basin [START_REF] Maystrenko | Structural features of the Southwest African continental margin according to results of lithosphere-scale 3D gravity and thermal modelling[END_REF]. The gravity profile has been extracted from the global marine gravity map of Sandwell and Smith v18.1 [START_REF] Sandwell | Global marine gravity from retracked Geosat and ERS-1 altimetry: Ridge segmentation versus spreading rate[END_REF]. between Africa and South America on the lower mantle low velocity zone (South African Super Plume) (from [START_REF] Davaille | Convective patterns under the Indo-Atlantic « box[END_REF]. The Parana-Etendeka flood basalts are depicted in green (after [START_REF] Torsvik | A new scheme for the opening of the South Atlantic Ocean and the dissection of an Aptian salt basin[END_REF]. The orange dots refer to the three possible principal locations of initial thermal anomaly at the upper/lower mantle boundary corresponding to the Tristan plume

  that combines the finite difference method with a marker-in-cell technique. The 3D model box has the horizontal dimensions 1500 × 1500 × 635 km and consists of 297 × 297 × 133 nodes offering spatial resolution of ca. 5 × 5 × 5 km per grid cell. Not area-specific initial setup consists of a stratified three-layer (upper/lower crust and lithospheric mantle) continental lithosphere underlain by an asthenosphere. The total thickness of the two-layer crust is 36 km; the depth of lithosphere-asthenosphere boundary is 150 km. The mantle plume has been seeded at the base of the modelled domain by a spherical thermal anomaly of 370 o C with a radius of 200 km. The initial geotherm is piece-wise linear with fixed temperatures at the surface (0°C), at the Moho (700°C), at the base of the lithosphere (1300°C), and at the bottom of the model box (1630°C). Weak tectonic forcing has been simulated by applying a constant ultraslow divergent horizontal velocity of 3 mm/year along the sides of the model. More detailed information on

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. Lithosphere scale cross-sections of present-day South Atlantic margins. The Moho depth varies

Figure 3 .

 3 Figure 3. Six tested numerical setups. a) Setup 1a: 4-layered weak rheology, crust 40 km thick, lithosphere

Figure 4 .

 4 Figure 4. Models with different rheology and plume location showing the most representative examples of

Figure 5 .

 5 Figure 5. Examples of models with different setup, plume location rheological structure showing the

Figure 6 .

 6 Figure 6. Evolution of 3D model a) rapid plume uplift leading to formation of linear extension-perpendicular

Figure 7 .

 7 Figure 7. Detailed display of experiment 12. Setup 2 is the base for this experiment with a laterally varying

Figure 8 .

 8 Figure 8. Schematic representation of the three modes of break-up. a) The very early phase of rifting (0 -

Table caption Table 1 .

 caption1 Summary of the thermal and mechanical parameters used for this study. 1)[START_REF] Turcotte | Geodynamics[END_REF], 2) Ranalli (1995); 3) d'Acremont et al. (2003) and references therein; 4)[START_REF] Tsenn | Upper limits of power law creep of rocks[END_REF]; 5)[START_REF] Burov | Erosion and rheology controls on synrift and postrift evolution: Verifying old and new ideas using a fully coupled numerical model[END_REF].

Table 2 .

 2 Controlling parameters and principal results of the experiments.

	Etendeka
	FB

essential. The most important result of this study is that there is not one single rift mode for plume-induced crustal break-up.

Acknowledgment

This study is co-funded by the Advanced ERC Grant 290864 RHEOLITH to A. Koptev and E.Burov. 3D numerical simulations were performed on the ERC-funded SGI Ulysse cluster of ISTEP. We thank William Sassi (IFPEn), Xavier Guichet (IFPEn) and Sylvie Leroy (UPMC) for fruitful discussions in the early phases of the project. We would also like to thank Sierd Cloetingh (Utrecht University) for constructive and thorough reading of the manuscript. The two anonymous reviewers are also thanked for their helpful remarks and critical points of view.