For the tectonic modelling, we use the particle-in-cell and finite element method (FEM-PIC) code Underworld 2 to simulate the long-term deformation of the lithosphere by solving a set of equations covering momentum, mass and heat conservation:

1a∇⋅σ=f,1b∇⋅u=0,1cρCp∂T∂t+u⋅∇T=∇⋅(k∇T)+ρH, where σ is the stress tensor that is the sum of a deviatoric part τ and the pressure p (σ=τ-pI, where I is the identity tensor), f=ρg is the force term, ρ is the density and g is the gravity acceleration, u is the velocity, Cp is the heat capacity at constant pressure, T is the absolute temperature, k is thermal conductivity, and H is the (radiogenic) heat production per unit mass. Materials have a constant or temperature-dependent density, given by: 2ρ=ρ0(1-αΔT), where α is the thermal expansion coefficient, ΔT=T-T0, T is the temperature, ρ0 is the reference density at the reference temperature T0.

We consider visco-plastic materials to simulate the long-term deformation of the lithosphere layer. We employ the Drucker–Prager yield criterion model to approximate the brittle behaviour of the material. Frictional-plastic deformation occurs when the stress is above the frictional-plastic yield stress σy: 3σy=Ccos⁡ϕ+sin⁡ϕP(2D),

where P, C, and ϕ are the pressure, cohesion, and angle of friction, respectively.

Here, we also include the plastic strain weakening of the crust: 4C=C0,ε≤εmin⁡C0+C0-Cwεmin⁡-εmax⁡,(ε-εmin⁡),εmin⁡<ε<εmax⁡Cw,ε≥εmax⁡5φ=φ0,ε≤εmin⁡φ0+φ0-φwεmin⁡-εmax⁡,(ε-εmin⁡),εmin⁡<ε<εmax⁡φw,ε≥εmax⁡ where the cohesion C and internal friction angle φ are reduced linearly between plastic strain ε values of εmin⁡ and εmax⁡ before reaching the weakened value, where the C0 and φ0 are the initial cohesion and friction angle respectively.

The effective plastic viscosity is given by: 6ηeffpl=σy2ϵ˙, Where ϵ˙ is the second invariant of the strain rate tensor defined as ϵ˙=12ϵ˙ijϵ˙ij.

Nonlinear viscous deformation is modeled using a strain rate-dependent, thermally activated viscosity. This rheological behavior is represented by a power-law relationship, expressed by the following nonlinear equation: 7ηeffvcreep=12A-1nϵ˙(1-n)ndmnexp⁡E+PVnRT, where A is the prefactor, ϵ˙ is the square root of the second invariant of the deviatoric strain rate tensor, d is the grain size, m is the grain size exponent, E is the activation energy, P is the lithostatic pressure, V the activation volume, n is the stress exponent, R is the Gas Constant and T is the temperature.

The effective viscosity is calculated by comparing the plastic and viscous rheologies as: 8ηeff=min⁡ηeffvcreep,ηeffpl.

The surface of the lithosphere is subjected to a surface processes model, Badlands , which includes the effects of fluvial and hillslope processes. In Badlands, the continuity of mass is governed by the interaction of three process types: tectonics, hillslope processes, and fluvial processes. This relationship is expressed by the standard equation: 9∂h∂t=U-∇⋅qr-∇⋅qd, where h is the surface elevation, U is a source term representing tectonic uplift (myr-1), qr and qd are the depth-integrated bulk volumetric sediment flux per unit width (m2yr-1) that represent the processes of transport by channel flow and diffusion, respectively.

The sediment transport rate per unit width due to flowing water. qr is modeled by the Stream Power Incision Model (SPIM) as a power function of topographic gradient S (S=∇z) and contributing drainage area A, 10∇⋅qr=κfAmSn, where κf is a dimensional constant of erosional efficiency that lumps information related to lithology, climate, channel geometry, and perhaps sediment supply . A is related to surface water discharge per unit width qw through net precipitation P, which can be uniform or spatially variable. The coefficients m and n are generally positive constants that depend on the specific erosion process being simulated. Default formulation in Badlands assumes m=0.5 and n=1, which are derived for the unit stream power law for considering stream power per unit bed area (m≈0.5, n≈1, ).

Downslope simple creep is typically considered to operate within a shallow superficial layer , and is expressed as: 11∇⋅qd=κd∇2h, where κd is scale-dependent and its values depend on lithology and mean precipitation rate, channel width, flood frequency, channel hydraulics, and potentially other parameters and processes .

In this research, we implement the coupling modelling framework within the ALE-IB scheme using Underworld 2. To couple the tectonic and surface process models, we developed an algorithm inspired by the approaches described in . This algorithm integrates the FANTOM and Cascade models within the ALE scheme, facilitating the transfer of elevation information between the C-surface (surface generated by surface process modelling in Cascade) and F-surface (surface derived from FANTOM), which we denote here as the S-surface (surface from Badlands) and T-surface (surface from tectonic modelling in Underworld 2) in our framework (see Fig. ).

For each tectonic time step, the Stokes equations are solved to obtain the velocity field across the entire domain by Underworld 2. The velocity on the S-surface is evaluated from the velocity field in Underworld 2 using the cubic spline method, and then transferred to Badlands, where the S-surface is further advected by solving the surface processes equation (with the evaluated velocity representing the uplift rate). Subsequently, the S-surface elevation is interpolated onto the T-surface, which is then remeshed accordingly. Particles are further adjusted based on changes in the T-surface by updating material indices, distinguishing between sediment and air (see Fig. a). More details about the remeshing algorithm within the ALE-IB scheme applied to free surface simulations can be found in .

The coupling modelling framework within the Eulerian scheme is implemented in the UWGeodynamics module . We use this module to build models that investigate the interactions and feedback mechanisms between tectonic processes which primarily governed by lithospheric rheology, and surface processes responsible for material erosion and deposition. This module integrates the geodynamic code Underworld 2 with surface process codes such as Badlands . UWGeodynamics is inspired by the Lithospheric Modelling Recipe (LMR) , available on GitHub: https://github.com/LukeMondy/lithospheric_modelling_recipe (last access: 30 March 2026), originally developed for Underworld 1 . The LMR, built on Underworld 1.8, facilitates model customization by providing easy access to boundary conditions, a library of rheologies, and modules for processes such as partial melting and surface processes .

The coupled modelling framework provided by UWGeodynamics features a two-way, thermo-mechanical coupling that incorporates surface processes. Here, the velocity on the S-surface rectangle is derived from velocity field in Underworld 2 using the cubic spline method, and further advects the S-surface within the surface process model. As erosion and deposition alter the S-surface, the distribution of surface materials is dynamically updated (see Fig. b). The free surface simulation employs the “sticky air” method within the Particle-In-Cell Finite Element Method (PIC-FEM), utilizing an Eulerian scheme. In this approach, the surface is tracked by particles, and the “surface velocity” is interpolated from the mesh node velocities. In Badlands, the surface processes model typically requires smaller time intervals than the tectonic model. Therefore, the tectonic time step dttec is divided into multiple subtimesteps dtsp to accommodate the surface processes model.

The loading of the Earth's surface can be described as the initial periodic surface displacement of an isoviscous fluid within an infinite half-space . The setup is shown in Fig. a. The initial free surface displacement is given by: 12w(x,0)=w0cos⁡(kx), where w0=10 km is the initial load amplitude, k=2π/λ is the wave number, with λ=D (the wavelength). D=500 km is the depth of the model domain.

The analytical solution for the decay of topography is characterized by the relaxation time t* : 13w(x,t)=w(x,0)e-t/t*, with the relaxation time t*: 14t*=Dk+sinh⁡(Dk)cosh⁡(Dk)sinh⁡2(Dk)t0*,t0*=2kηρg, where η is the viscosity, ρ is the density. When λ≪D, t*≈t0*.

The computational domain is 320km×200km for the ALE-IB and Eulerian schemes. A constant time step of 2.5 ka was employed here, using Q1dQ0 finite elements and a mesh of 129×81 nodes, with 16 particles per cell. Material properties are: ρ=3300kgm-3 and η=1021Pas for the lithosphere layer, ρ=0kgm-3 and η=1018Pas for the air layer, ρ=2700kgm-3 and η=1019Pas for the sediment layer. Gravitational acceleration is g=9.81ms-2. The side boundaries are free-slip, the bottom is no-slip, and the top boundary is free-slip over sticky air.

The second experimental set couples a 2D thermo-mechanical model with surface processes to study the integrated influence of tectonics and surface processes on continental collision. The setup is shown in Fig. b and c. The tectonic models part is adapted from and . In their work, they discuss how convergence rate, crustal thickness, and rate-dependent viscous rheology control the strength of the wedge and the resulting structural style.

The computational domain measures 1280km×320km for both the ALE-IB and Eulerian schemes, discretized with Q1dQ0 finite elements on a 513×129 node grid, with 30 particles per cell. The initial configuration consists of a homogeneous crust with a 45° dipping weak zone within the mantle lithosphere at x=640km, representing a pre-existing subduction zone. The crust is 25 km thick and is overlain by a 40 km-thick sticky-air layer. Tectonic and surface-process time steps are 10 and 5 ka, respectively.

A constant surface temperature of T=0°C (and for air material) is imposed, and the side boundaries are adiabatic (no heat flux). The initial temperature distribution follows a geotherm: 25 °Ckm-1 from surface down to 10 km depth, then 12 °Ckm-1 to the lithosphere–asthenosphere boundary (LAB) at a depth of 97.5 km, where T=1300°C. The velocity boundaries are configured as follows: free-slip conditions are applied on the right and top boundaries; the bottom boundary is unconstrained to allow inflow/outflow, effectively placing the model above an infinite space with an inviscid fluid at a depth of 280 km (Gerya, 2009). Convergence is imposed at the left boundary with a fixed velocity of 5 cmyr-1, applied throughout the crust and mantle lithosphere. Below the lithosphere, the velocity decreases linearly from the convergence value at the base of the lithosphere to zero at the bottom boundary. Above the crust, an inflow/outflow is prescribed across the sticky-air layer to permit surface topography development.

The surface processes considered include hillslope and fluvial processes. The hillslope process is modeled using the linear diffusion equation as shown in Eq. (). In this study, the diffusion coefficient is set as a constant kd=0.1m2yr-1 . Typically, diffusion coefficients range from 0.001 to 10 in various environments, but the model evolution usually is not sensitive to variations in hillslope parameters . For our simulations, we use a relatively high, realistic value across aerial, marine, and river settings to produce smoother topography. This helps to prevent excessive surface distortion and simplifies the landscape modelling.

The fluvial process is modeled using the stream power law equation as shown in Eq. (), with landscape evolution simulated under a uniform rainfall rate of 1 myr-1. The fluvial erodibility coefficient kf exhibits significant uncertainty and spans a wide range, as it depends on various factors, including climate, rock type, channel width, hydraulics, and others . In this study, kf is assumed to be spatially uniform across the entire region. For sensitivity analysis, we explore a range of values from 10-4 to 10-6m1-2myr-1. The exponents m and n are set to 0.5 and 1, respectively, based on the unit stream power model . Alternative values for m and n can be found in .

In the models without the surface processes from Experiment 1, the relaxation of an initial topography toward equilibrium occurs over approximately 300 ka. Figure a and b compare the topography results obtained from free-surface simulations conducted with and without surface processes (where kd=0.1m1-2myr-1, kf=10-4m1-2myr-1) across two different numerical schemes: ALE-IB and Eulerian at Time=150ka (approximately 2τ*). The results from the models without surface processes (TR-TM and TR-TME) demonstrate that the simulation employing the ALE-IB scheme exhibits excellent agreement with the analytical solution, highlighting its accuracy in capturing topographic features. Conversely, the Eulerian scheme displays fluctuations that diminish the overall accuracy. The L2-norm error between the ALE-IB scheme and the analytical solution is 0.00040, while the error for the Eulerian scheme is 0.00113.

As illustrated in Fig. e and f, in coupling models, erosion predominantly accumulates in the mid-section of the hill above sea level, driven by fluvial erosion processes which positively correlated with slope. Sediment deposition, on the other hand, tends to occur near sea level. The relative difference in the average depth of erosion and deposition between these two schemes is 24.7 %.

The superior performance of the ALE-IB scheme can be attributed to its more precise velocity estimation and the ability to track topographic changes more accurately. Consequently, sediment transport and distribution are represented more realistically, as shown in Fig. c. In contrast, the Eulerian scheme results in some nonphysical sediment accumulation along the downhill slope below sea level, shown in Fig. d, likely due to less accurate velocity interpolation and topography tracking.

Raising the fluvial erodibility coefficient substantially enhances incision and sediment transport. By 150 ka, landscapes with higher kf exhibit deeper valley incision and a more convex longitudinal profile, consistent with increased valley widening and stronger hillslope–channel coupling (Fig. ). A small topographic peak forms near sea level at the hills' toe, likely reflecting a transient balance between base-level lowering and localized deposition. The drainage network becomes more expansive as channel incision intensifies (Fig. g). Across the 10–300 ka time series (Fig. ), the elevated-erodibility regime yields larger changes in relief and in channel geometry, indicating that fluvial processes dominate landscape evolution under stronger erodibility.

Models CC-CM2 and CC-CME employ the coupling framework within the Arbitrary Lagrangian–Eulerian Immersed Boundary (ALE-IB) and Eulerian schemes separately, with identical settings for tectonic and surface processes. The deformation evolution in the crust and lithosphere resembles that of Model CC-CM0, but features a narrow shear zone in the middle of the wedge due to sedimentation accumulating on both sides of the wedge (Figs. , , and ). This sedimentation mass enhances wedge uplift, ultimately forming a narrower and higher wedge. This outcome aligns with the perspective in that erosion can drive the growth of intracontinental mountains.

The deformation patterns do not differ substantially between the ALE-IB and Eulerian schemes (Fig. ), though differences emerge in the late stage viscosity field due to slight variations in strain rate, stemming from minor differences in sediment distribution. The ALE-IB scheme shows more sedimentation in the upper plate and less in the lower plate compared to the Eulerian scheme (Figs. c and d, c and d). The Eulerian scheme exhibits fluctuations that reduce overall accuracy, as evidenced by sediment distributions far from the wedge. On the left side of the sediments in the lower plate, small topographic peaks appear in the material field but are absent from the river patterns in the results from the Eulerian scheme (Fig. f). The relative difference in the average depth of erosion and deposition between these two schemes is 93.3 %. These discrepancies may arise from inaccurate velocity evaluations and surface tracing. Topographic gradients exhibit local changes in high-strain areas, which clearly link to river system patterns: rivers evolve along the sides of small peaks or become diverted in these regions, as prominently shown in the ALE-IB scheme results (Fig. e).

Models CC-CM1 and CC-CM3 employ the coupling framework within the Arbitrary Lagrangian–Eulerian Immersed Boundary (ALE-IB) scheme, with varying fluvial erosion parameters. In Model CC-CM3, the deformation evolution (Fig. ) closely resembles that of CM0, owing to the relatively low fluvial erosion parameters, which result in subdued surface processes and preserved crustal thickening. In contrast, Model CM1 exhibits markedly different crustal deformation due to intense erosion and deposition driven by high fluvial erosion parameters. This strong erosional regime disrupts the typical crustal stacking patterns, yielding a thin, wide wedge with low topography and relief, characterized by two localized bands of high strain (Fig. ).

Rather than forming multiple mountain ranges, as seen in models with weaker erosion, CC-CM1 develops only a single prominent mountain peak (Fig. a and b), accompanied by simpler river patterns (Fig. e and f). This outcome highlights how enhanced fluvial incision can promote efficient mass removal from the orogen, leading to isostatic rebound and lateral spreading of the wedge, consistent with critical taper theory where erosion reduces the wedge's taper angle and stabilizes a broader, lower-relief structure . For instance, similar dynamics are observed in natural settings, such as the eastern Tibetan Plateau margins, where high erosion rates flatten topography and simplify drainage networks . These variations highlight the sensitivity of coupling models to surface process parameters: in high-erosion scenarios, such as CC-CM1, fluvial systems act as a feedback mechanism, channelling sediment transport and influencing lithospheric strain localization.

Our models show the advantages of the ALE-IB coupling scheme over a purely Eulerian one, although no comparisons were made to a pure ALE implementation . The implementation of the ALE-IB method allows for straightforward handling of particles belonging to different materials (such as air, sediment and crust) across the free surface interface (the boundary between air and rock). There is no need to consider void spaces or other issues that arise when using the ALE scheme. This method is more efficient in the study of surface processes (e.g. erosion and sedimentation) because the interfaces between two phases are implicitly tracked and less numerical diffusion is present, as shown by Model CC-CM2 vs. Model CC-CME.

In nature, the evolution of orogenic wedges is inherently three-dimensional and involves a broad spectrum of complexities, including laterally and vertically varying rheological heterogeneities (e.g. due to compositional differences or thermal gradients) and structural inheritance (e.g. pre-existing faults or basement fabrics), all of which profoundly influence deformation patterns and topographic development. For instance, in real-world systems like the Himalayas or the Andes, these factors lead to asymmetric wedge growth, localized uplift, and complex drainage networks that cannot be fully captured in 2D simulations. Our model tests here remain limited to 2D due to the consideration of affordable computational costs, which restrict the resolution and scale of simulations. However, the ALE-IB coupling framework we propose is inherently scalable and supports 3D implementation. It leverages modern high-performance computing techniques, such as parallel processing and adaptive mesh refinement, both of which are capabilities of Underworld 2, to efficiently manage computational costs.

This framework's extensibility opens avenues for future research, including direct comparisons with the pure ALE scheme to quantify trade-offs in computational efficiency vs. accuracy, as well as 3D extensions of Models to explore erosion-driven feedback in more realistic geometries. Integrating field observations or geophysical data (e.g. seismic tomography for rheological constraints) could further help validate these models, enhancing their predictive power for understanding intracontinental orogeny and landscape evolution.