As it is commonly done in continuum mechanics, the applied numerical algorithm solves the continuity and momentum equations coupled to conservation of energy expressed with respect to temperature. We consider incompressible visco-elasto-plastic materials that slowly flow (inertia forces are negligible) under gravity and the influence of boundary tractions. Here, the term plastic refers to all rate-independent (instantaneous) irreversible deformation controlled by a Drucker–Prager yield function (Eq. ). The applied pressure-dependent yield function is motivated by laboratory experiments and is representative of rock deformation mechanisms at low temperatures (shear banding, faulting). The term viscous comprises all rate-dependent inelastic deformation, which is generally thermally activated. Material properties are advected using a marker-in-cell approach involving initially 16 Lagrangian markers per Eulerian finite-difference grid cell. Up to 56 million numerical markers are advected in total. We do not apply any frictional or viscous strain softening. However, thermal softening, caused by shear heating, is active, resulting from the conservation of energy and temperature-dependent viscous flow stress. The algorithm has already been used to model deformation processes at various scales including upper mantle convection coupled to lithospheric-scale deformation . Appendix  provides a detailed description of the algorithm.

We here follow the modelling approach of and we employ a 1600 km wide and 680 km deep (see Fig. ) model domain. The global model resolution is 1 km × 1 km in horizontal x and in vertical z direction, respectively. A stress-free surface is set initially at z=0 km and, to allow for dynamic build-up of topography, the topmost +20 km of the model domain is left free of material. We impose constant material inflow and outflow velocities at the lateral boundaries, and the mechanical boundary at the bottom of the domain is free to slip. Further, we assume that no heat flows laterally out of the model domain, and the top and bottom temperature is kept constant at 15 and 1613 ∘C, respectively. Viscous deformation is modelled leveraging a combination of rheological flow laws, namely dislocation, diffusion and Peierls creep. Diffusion creep flow laws (Eq. ) require knowledge of the grain size. Including an evolving grain size in lithospheric-scale models is usually achieved by employing additional parameterizations e.g. paleowattmeters; . As these parameterizations are commonly determined in laboratory experiments conducted at relatively fast deformation rates (compared to tectonic deformation), these parameterizations have to be extrapolated to natural conditions. This extrapolation introduces additional uncertainties. To keep the model simple, we here neglect grain size evolution and choose a constant average mantle grain size of 1 mm in Eq. (). Previous work has demonstrated that this choice, in combination with suitable flow law parameters, results in a viscosity structure that is consistent with geophysical constraints with respect to the convection dynamics of the mantle and the thermal thickness of the overlying, horizontal lithosphere . Similar average grain sizes of the order of 1 mm in the upper mantle have been predicted by whole mantle convection models that include state-of-the-art grain size evolution models .

The effective density of the materials is either directly calculated via a linearized equation of state or pre-computed based on equilibrium phase-diagram sections for specific bulk-rock composition (see Appendix ). We consider surface processes such as erosion (constant erosion rate of 0.5 mmyr-1 above z≥2 km of topographic elevation) and sedimentation (instantaneous basin fill below z≤-5 km of topographic elevation, alternating calcite and mica rheology every 2 Myr). For more detailed justifications of model assumptions and explanations for the implementation of boundary conditions, the reader is referred to the study of . Model units include an initially 25 km thick upper crust with two vertical levels of 11 elliptical inclusions each, whose rheology is different from the upper crustal matrix (weakened wet anorthite or Westerly granite rheology; see Table  for all material parameters). These elliptical inclusions represent structural and petrological units inherited from previous orogenic cycles and are motivated by field observations e.g.. Weak (wet quartzite) and strong (Maryland diabase) inclusions may represent metasedimentary units and mafic intrusions, respectively. Below the upper crust an 8 km thick lower crust (wet anorthite) is employed. The initial Moho is set to z=-33 km and the mantle lithosphere (dry olivine) extends down to z=-120 km. In total, 12 weak (wet olivine) elliptical inclusions over two vertical levels are included in the mantle lithosphere. These heterogeneities are motivated by geophysical observations e.g. and represent weaknesses owing to spatial lithological variations. Elongated or elliptical inclusions allow for the generation of asymmetric margin geometries . We include the asthenosphere and transition zone (grouped and termed upper mantle hereafter, dry olivine rheology) down to z=-660 km. The distribution and emplacement of the inclusions is described in detail in Appendix . The difference between mantle lithosphere and upper mantle is due to temperature and pressure only; i.e. all material parameters are the same.

Four types of simulations are performed: (1) the reference model (REF) generates a long-term (> 160 Myr) geodynamic cycle of subsequent extension–cooling–convergence in a single and continuous simulation. During a 50 Myr rifting period, an ≈ 360 km wide basin is formed. This basin is floored by exhumed continental mantle and bounded by two hyper-extended magma-poor rifted margins. We apply an absolute extension velocity of 1 cm yr-1 and we assume an ultra-slow to slow spreading rift system with minor melt production and therefore neglect decompressional melting of the peridotites. No far-field plate velocity is applied to the system for the following 60 Myr. This choice of applied deformation velocity and duration of the periods leads to margin geometries comparable to reconstructions from the ancient Alpine Tethys margin system and Piemont–Liguria ocean . In all models presented here, the subsequent convergence is driven by a kinematic boundary velocity of 1.5 cm yr-1 (absolute value). The convergence phase involves subduction initiation, closure of the basin floored by exhumed mantle, continental collision and orogenic wedge formation. Model REF simulates a scenario without serpentinization of the exhumed mantle peridotite. Except for the mantle lithology that utilizes a complex density (CD) model, the density of all other model lithologies is calculated using a LEOS (Eq. ).

The geodynamic evolution of REF at the end of the cooling stage (109 Myr) serves as a self-consistently generated initial configuration for three additional types of simulations. (2) We parameterize a serpentinization front propagating through the topmost layer of exhumed peridotites to test the impact of serpentinite strength on the convergent deformation. In this parametrization, we replace the dry olivine flow law parameters of the exhumed peridotites in the basin above z=-9 km with parameters for antigorite rheology . This approach results in an effective average thickness of the serpentinite layer of ca. 5.5 km. In order to investigate the effects of variable serpentinite strength, we gradually increase the prefactor in front of the dislocation creep flow law (ζ in Eq. ) for antigorite . We here report the modelling results for prefactors 1 and 18, because these models are end-member types for subduction and orogen dynamics. The background upper and lower crust are feldspar-dominated (see Table  for details). As in REF, density is computed with a LEOS except for the mantle where a CD model is employed. (3) We consider serpentinization and investigate the impact of (upward-directed) buoyancy forces on the collisional stage employing a CD model for all lithologies. Type (4) is identical to type (3), except that compared to a feldspar-dominated (wet anorthite) rheology a relatively weaker quartz-dominated rheology (Westerly granite) is employed for the upper crust. A summary of all simulations is given in Table . The applied material parameters are given in Table , and the bulk rock compositions as well as the solution models used for phase diagram calculations are given in Table .

Figure shows the geodynamic evolution of the reference model (REF). Crustal break-up occurs at ≈ 19 Myr in model history (Fig. a). At this stage, hot mantle material rises in the horizontal centre and diverges laterally below the plates (see arrows in Fig. a). The 1300 ∘C isotherm remains at a depth of ≈ 20–30 km below the rift centre (Fig. d). Two asymmetric continental margins have formed. The necking zone of the left margin is ≈ 20 km wide. In total, the width of the left margin is ca. 80–90 km. The right margin is in total ca. 140–160 km wide including an ≈ 60–80 km wide necking zone. At the end of the cooling period (109 Myr, Fig. b), an ≈ 360 km wide basin has opened, which is floored by exhumed mantle material. Convection in the upper mantle stabilizes the thermal and mechanical thickness of the lithosphere to ca. 90–100 km (no velocity glyphs in Fig. b and e; see also the discussion in for more detail). At 112 Myr subduction is initiated at the transition from the proximal to the distal continental margin in the right model side (see , for nomenclature). Partly, the necking zone of this margin has been subducted to ca. 40 km depth (Fig. c and f). At 144 Myr in model history, the continental crust of the left margin (blue colours in Fig. a and d) is subducted to ca. 60 km depth. Sediments originally deposited in the trench have been incorporated in between the subducting and the overriding plate. With ongoing convergence, the upper crust of the subducting plate is buried to a maximum depth of ca. 120 km. At 155 Myr, upper crustal units are sheared off the subducting plate at the transition to the lower crust and begin to form thrust sheets (see deflected isotherms at x≈-80 km and z≈-30 km in Fig. e). Major parts of continental crust are sheared off the subducting plate before entering the subduction zone. An orogenic wedge with several thrust sheets has formed at 164 Myr (Fig. f). At this stage, the lower part of the subducting slab detaches and the deeper subducted upper crustal material flows upward. This return flow is limited to depths between ca. 60 and 80 km (see arrows in Fig. f).

As in REF, subduction is initiated at the transition from the proximal to the distal right continental margin in AS1 and AS18. A large volume of serpentinite material is sheared off the subducting plate in AS1 and AS18 (Fig. a and e). In AS1, the serpentinites form a coherent, inclined channel down to ca. 100–120 km depth (Fig. a). At 148 Myr, the upper continental crust of the left margin (blue colours in Fig. ) has been subducted to ca. 90–100 km depth (Fig. a). Between 154 and 162 Myr, the subducted upper continental crust returns to the surface through the weak serpentinite channel in AS1. An ≈50 km wide block of the overriding plate's upper crust is separated from the right margin by subducted crustal material that is being exhumed to the surface. At 164 Myr, large parts of the subducted continental crust have been exhumed to the surface (Fig. d). Units containing weak inclusions are internally deformed during the exhumation. Strong units are folded but exhumed coherently (Fig. d). The serpentinite material is surrounding the exhumed upper crustal material and the separated continental upper crustal block of the overriding plate. In AS18, the serpentinite channel is intersected by the two colliding plates at z≈-30 km (Fig. e). In contrast to AS1, the continental upper crust is partly sheared off the subducting plate at z≈-25 km in AS18 (Fig. e). Maximum burial depth of upper continental crust in this model is ca. 80–90 km. Exhumation of subducted crustal material is limited to z≈-70 km (Fig. f and g). Instead, the upper continental crust is wedged at shallower depths, similar to REF (see deflected isotherms at x≈-125 km in Fig. g). Thrust sheets propagate along the upper continental crust of the subducting plate (Fig. h). The serpentinized mantle material forms a coherent unit emplaced in between the wedged continental upper crust of the subducting plate and the relatively undeformed continental upper crust of the overriding plate. Dimensions of the evolved orogenic wedge in AS18 are similar to the wedge dimensions evolved in REF (compare Fig. f to Fig. h).

Density models that include predicted metamorphic assemblages (CD models) lead to more variable density fields and larger density changes as a function of pressure and temperature compared to LEOS models (compare Fig. g to f). Similar to AS1, a serpentinite channel (low viscosity region in Fig. a) forms after subduction initiation in AC1 (Fig. a). The continental upper crust of the subducting plate (blue colours) is subducted to ca. 150 km depth, which is deeper compared to AS1 (compare Fig. b to Fig. b and c). In contrast to AS1, exhumation of the subducted upper continental crust has not occurred until 163 Myr (compare Fig. c to Fig. c) in model history. At 170 Myr, the subducted continental crust in AC1 has been exhumed to the surface along the weak subduction interface (Fig. d). In contrast to AS1, the continental upper crust of the overriding plate has not been separated by the returning continental crust of the subducting plate in AC1 (compare Fig. d to Fig. d). Figure a and b show the density field of models AS1 and AC1, respectively, computed by the algorithm at 170 Myr in model history. Compared to AS1, in AC1 the density of the material in the wedge is much more variable. The density of the subducted crustal units in AC1 is up to ca. 100 kgm-3 higher than in AS1. The resulting buoyancy contrasts are much smaller, leading to significantly less uplift of topography in AC1 compared to AS1. In AS1, the maximum elevation of topography exceeds 10 km; in AC1 the maximum elevation is locally ≈7 km, but high topographies are on average ≈5 km. The width of the wedge is ≈350 and ≈275 km in AS1 and AC1, respectively. Compared to AS18, only a minor volume of serpentinite material has been sheared off the subducting slab in AC18 (compare Fig. e to Fig. e). The majority of serpentinite material is coupled to the slab and subducted into the upper mantle (Figs. f and c). Subduction of continental upper crust reaches a depth of ca. 180 km after 163 Myr (Fig. g). In the absence of a weak, serpentinized subduction interface, the resistance of the subduction channel is increased in AC18 (compare Fig. a to c). The deeply subducted continental upper crust cannot be exhumed along the strong subduction interface and breaks through the less resistant mantle wedge (Fig. h), relaminating below the overriding plate (similar to the models presented in , and ).

Similar to AS1 and AC1, in GC1 the serpentinites form a channel along the subduction interface down to ca. 120 km depth (Figs. a and b). Continental upper crust of the subducting plate is buried to z≈-120 km (Fig. a and b). At ca. 153 Myr, the subducting slab detaches at a depth of ca. 400 km (see ). Between 156 and 162 Myr, the subducted continental upper crust flows back to the surface along the weak subduction interface breaking through the continental upper crust of the overriding plate (Fig. c). The serpentinite material is surrounding the exhumed crustal material (Fig. d) within the orogenic wedge. Exhumation of upper continental crust from z≈-80 km to z≈-30 km occurs between 162 and 164 Myr. At ca. 165 Myr, a second slab detachment occurs at z≈-250 km (see ). In contrast to AC18, less crustal volume is subducted in GC18 (Fig. e). Instead, the continental upper crust is largely sheared off the subducting lithosphere at z≈-40 km (Fig. f), and several thrust sheets form (Fig. f). Compared to GC1, the relatively stronger serpentinite in GC18 is largely subducted (Fig. d) and sediments originally deposited in the trench are incorporated into a growing orogenic thrust wedge (Fig. f–h).

In order to investigate the relative impact of buoyancy and shear forces on collision, we quantify the temporal evolution of the horizontal driving force per unit length (Fig. b, FD hereafter; see Appendix C) and of the (upward-directed) buoyancy forces of subducted crustal material (Fig. c, FB hereafter; see Appendix C). Buoyancy forces calculated here represent an upward-directed (positive) buoyancy force of the subducted crust acting against further subduction. The ratio FB/FD (Fig. a, ArF, hereafter) is a measure for the relative dominance of buoyancy or shear forces driving the deformation and exhumation within the orogenic wedge (see Appendix ). Points 1–5 in Fig. a–e represent important stages of model GC1 (thick dashed turquoise blue line) and are representative for the evolution of all presented models.

Subduction initiation occurs for FD≈22 TNm-1 in models of feldspar dominated continental upper crust (AS1, AS18, AC1 and AC18) and for FD≈18 TN m-1 in models of quartz dominated upper crust (GC1, GC18, 1 and grey area in Fig. b). Magnitudes of FD decrease after subduction initiation to ca. 10–15 TNm-1 and remain, on average, relatively constant during basin closure (between 1 and 2 in Fig. ). Until basin closure, ArF≈0.25 and remains relatively constant in all models. Magnitudes of FB increase with an increasing volume of crustal material involved in subduction (see Fig. c and d) between 140 and 153 Myr. In response, magnitudes of ArF≈1 are reached in AS1, AC1 and GC1. Between ca. 153 and 156 Myr, shallower buried crustal material is exhumed in GC1 (3 in Fig. ; see also Fig. a) coinciding with a deep slab detachment (see ). At ca. 162 Myr, the deeper subducted units are exhumed (see also Fig. c). This exhumation event is followed by a shallow slab detachment leading to a rapid increase in magnitude of FD at ca. 165 Myr. The buoyancy pull of the subducting slab is lost, and a larger horizontal driving force is needed to overcome the (upward-directed) buoyancy push of subducted crust and to continue subduction with the prescribed kinematic boundary velocity. While the buoyancy forces remain relatively constant, the increase in FD decreases the magnitude of ArF (between 4 and 5 in Fig. a–c). The maximum burial depth of continental crust is reached in AC1 and AC18 (feldspar-dominated upper crust) and is ca. 200 km. In REF, GC1, GC18, AS1 and AS18 the maximum burial depth varies between 120–150 km. During the formation of the collisional orogen, the magnitude of FD increases and eventually exceeds the magnitude that was necessary for subduction initiation (grey area in Fig. b). Magnitudes of FD and FB are on average higher in models of feldspar-dominated continental upper crust compared to models of quartz-dominated upper crust. The increased shear resistance in these feldspar-dominated models allows for deeper subduction of continental upper crust when the serpentinite is weak. High values for FD in AS18 and REF are explained by a more resistant subduction interface caused by either an increased shear resistance of the serpentinite material, or absence of serpentinization.

The geometry, kinematics and dynamics of the evolving collision zone and orogenic wedge varies significantly in the presented models (Fig. ). We refer here to these variations as different modes of orogenic wedge formation. These varying modes depend mainly on the shear resistance of upper crust and serpentinite, and its buoyancy contrast to the surrounding mantle material. The magnitude of ArF is used to classify the different modes of orogenic wedge formation.

For low shear resistance of serpentinites, generating a weak subduction interface, ArF≈1, indicating equal importance of buoyancy and shear forces building the orogenic wedge. Two modes are observed: (1) a diapir-like mode of exhumation due to either (i) increased buoyancy contrast (LEOS models) and high shear resistance of the upper crust (AS1; see Fig. a) or (ii) a decreased buoyancy contrast (CD models) and low shear resistance of the upper crust (GC1; see Fig. c); (2) a channel flow mode of exhumation occurs for a decreased buoyancy contrast (CD models) and increased shear resistance of the upper crust (AC1; see Fig. b).

In models of high serpentinite shear resistance, generating a strong subduction interface, two different modes are also observed: (3) a thrust-diapir mode of deformation for either (i) an increased buoyancy contrast and high shear resistance of the upper crust or (ii) a decreased buoyancy contrast and low shear resistance of upper crust. In both cases ArF≈0.5, indicating that shear forces dominate the building of the orogenic wedge (Fig. d and f). (4) A relamination mode due to relamination of deeply subducted continental crust is observed for a decreased buoyancy contrast and high shear resistance of the upper crust (ArF≈0.75; see Fig. e).

The shear resistance of (i) serpentinites and (ii) the upper continental crust directly impacts the shear forces (compare GC1 to AC1 in Fig. b). Determining the shear resistance, or effective viscosity, of the lithosphere deforming under geological timescales remains challenging . Rock deformation experiments are performed at deformation rates that are many orders of magnitude higher than tectonic deformation rates. Best-fitting curves have to be extrapolated to natural conditions, which introduces large uncertainties on the actual strength of rocks deforming under natural conditions . Serpentinite plays a crucial role in subduction zones and, ultimately, in the formation of Alpine-type collisional orogens . Despite its importance, the rheology of serpentinite at lithospheric-scale pressure and temperature conditions remains elusive and references therein. Several deformation mechanisms for serpentinite material, often based on experiments with antigorite, have been discussed in the literature, including dislocation creep , semi-brittle or plastic deformation behaviour , grain boundary sliding , and sliding on shear cracks . Numerical models are useful to test different end-member rock rheologies and to investigate the impact of rock strength on subduction and formation of collisional orogens. To first order, weak serpentinite material may indeed form a subduction channel and lubricate the subduction interface. The subduction channel in model AC1 has formed self-consistently, because subduction was initiated without a prescribed major weak zone in the lithosphere. Deep subduction and exhumation to the surface of crustal material is feasible in this kind of model. In contrast, strong serpentinite material leads to detaching and thrusting of crustal material already escaping subduction at shallow depths. In natural settings, the effective strength of serpentinites may vary along the upper regions of the subducting plate parallel and/or orthogonal (along-trench) to the subduction direction, for example due to varying degrees of serpentinization. Such spatial strength variation may cause temporal and/or along-trench alternations between channel mode and thrust-diapir mode when serpentinites of different strength enter the subduction zone. The shear resistance of the upper crust resulting from the rheological flow laws employed here, wet anorthite and Westerly granite , is similar and neither extremely low nor extremely high (see also Fig. 1 in ). However, employing these two different flow laws for the upper crust changes the mode of orogenic wedge formation significantly. Similar to the strength of serpentinites, in natural settings the effective strength of the upper crust may also vary and cause temporal and along-trench variations of the orogenic wedge modes during the evolution of continental collision.

Magnitudes of buoyancy forces in our models are enhanced by large density contrasts between the subducted material and the surrounding mantle. We tested end-member models of simple and complex density calculations. The precomputed density tables are based on calculated equilibrium phase diagram sections (Fig. ). In our calculations we have assumed H2O saturation and therefore our system is open with respect to its H2O content, but closed with respect to other elements. This is a valid approximation for the kilometre scale that we are considering. However, the preservation of high-grade metamorphic rocks at the near-surface environment indicates that the crustal metamorphic rocks are not always at thermodynamic equilibrium as is assumed here. The rate of metamorphic re-equilibration is strongly affected by temperature and by the availability of fluids . Coupling mineral-scale phase equilibria modelling to large-scale geodynamic models remains challenging. Although coupling of petrological and thermo-mechanical modelling via CD models as presented here is simplified, this approach has proven useful to explain observations that cannot be predicted by the commonly used LEOS models. These observations include (i) varying sediment thickness accumulated during basin subsidence , (ii) evolution of subducting slab dynamics and (iii) the exhumation of (U)HP rocks . In addition, our results demonstrate that CD models avoid unrealistically high topographic elevation during orogen formation.

Varying mechanical strength of (i) serpentinites and (ii) the upper crust in combination with varying buoyancy contrasts changes the mode of orogenic wedge formation modelled here (Fig. ). Hence, it is essential to (1) further refine our knowledge on the mechanical strength of crustal rocks and serpentinites under natural deformation conditions and (2) to account for realistic density structures including metamorphic reactions in numerical models of collisional orogen formation.

We follow here the modelling approach by . The margin geometry and thermal structure, prior to convergence, is generated during a modelled rifting and cooling period. This way, the generated passive margin, the marine basin system and the subsequent convergence are modelled in an internally consistent manner. Subduction initiation is horizontally forced , and a major lithospheric shear zone forms around the transition from the distal to the proximal margin (see also the discussion in ). The ad hoc parameterized layer of serpentinite is not relevant for subduction initiation, as subduction is initiated also in the reference model without serpentinite (see Fig. f). Instead, geometrical focusing of stresses below the margin together with thermal softening and a temperature-dependent viscosity leads to spontaneous formation of a shear zone transecting the lithosphere . In our models, magnitudes of FD between 18 and 22 TNm-1 are necessary to initiate subduction for quartz- and feldspar-dominated upper crust, respectively. These magnitudes are significantly lower compared to FD≈37 TNm-1 obtained by . They studied subduction initiation at an idealized, ad hoc constructed passive margin without mechanical heterogeneities in the form of a multi-layer or elliptical geometry (see also , for a detailed comparison). The Peierls flow law parameters describing the rheology of olivine used here have been elaborated by . Recent studies suggest that the olivine strength resulting from this parameterization is likely overestimated . If true, stresses in the mantle lithosphere would be lower than predicted by our models, which could further reduce the magnitude of shear forces necessary for subduction initiation. However, the minimum value of FD required for subduction initiation in our models should be determined in future studies.