For the tectonic modelling, we assume that the Earth's lithosphere and mantle deform like the incompressible viscous fluid on geological time scales. The behaviour of the fluid follows a set of equations covering momentum, mass : 1a∇⋅σ-∇p=f,1b∇⋅u=0,1cρCp∂T∂t+u⋅∇T=∇⋅(k∇T)+ρH, where σ is the deviatoric stress tensor, p is the pressure, 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.

The following boundary conditions are considered here: 2No slip: u=0,3Free slip: u⋅n=0,4Free surface: σ⋅n=0.

Several of our experiments employ an approximation of Earth's surface using the “sticky air” method in the Eulerian scheme for comparative analysis. This approach allows the modelling of topographic variations within a purely Eulerian framework by introducing an upper layer of sticky air. The density of this layer is set close to zero for ensuring it exerts no pressure on the actual free surface (the interface between the air and rock), or it is set to 1000 kgm-3 to approximate a water-loaded free surface . investigated the influence of the viscosity contrast and the thickness of the sticky air layer and concluded that, for this method to produce reliable results, certain conditions must be satisfied. One such condition is that the isostatic compensation factor must be considered: 5Cisost=316π3Lhst3ηstηch, where L is the box width, hst and ηst denote the thickness and viscosity of the sticky air layer, respectively, and ηch represents the characteristic viscosity controlling relaxation, typically approximated by the mantle viscosity. Cisost is a nondimensional combination of geometric and material parameters that quantifies the ratio of dynamic stresses to the static pressure scale set by the system. When Cisost≪1, the surface exhibits near-isostatic behavior over the relevant timescale. This means the pressure field can adjust effectively to balance loads, resulting in minimal residual traction on the surface. Consequently, the error introduced by the “sticky air” method is small.

The upper boundary condition over the air layer can be modeled as either free-slip or open (zero stress). As discussed in and , an open boundary condition can suppress the return flow of sticky air, which is usually generated under a free-slip boundary condition, thereby reducing the velocity of the air layer. For an open top boundary, the thickness of the sticky air layer does not need to be sufficiently large as indicated by Eq. (). However, for the purpose of consistent comparison with previous studies , all our experiments employ a free-slip boundary condition at the top of the air layer and utilize a relatively thick air layer.

We implement the true free surface simulation in ALE-IB scheme. Generally, the mesh undergoes regridding to align with the free surface through the following steps, similar to ALE scheme (See Fig. ):

Free Surface Advection

The mesh nodes along the internal boundary represent the discrete free surface of the domain. Their location coordinates, denoted as X, is advected forward in time using displacements determined by the forward Euler scheme: 6Xn+1=Xn+Δtun on Γfs, where Γfs indicates the location of the time-dependent free surface. When coupled with surface processes, X will also be influenced by these processes.

Next, we update the vertical mesh coordinates forward in time using displacements determined by the forward Euler scheme: 8Xzn+1=Xzn+Dz.

The loading of the Earth's surface can be described as the initial periodic surface displacement of an isoviscous fluid within the infinite half-space . The setup is shown in Fig. a. The initial free surface displacement is given by: 10w(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* : 11w(x,t)=w(x,0)e-t/t*, with the relaxation time t*: 12t*=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 500km×500km for the ALE scheme, and 500km×600km for ALE-IB and Eulerian schemes. A constant time step of 10-2t* here was employed, with Q1dQ0 finite elements and with a mesh of 51×51 nodes (or 51×61 nodes for the larger domain with the air layer). Material properties are: ρ=3300 kgm-3 and η=1021 Pas for the lithosphere layer, ρ=0 kgm-3 and η=1018 Pas for the air layer. Gravitational acceleration is g = 9.81 ms-2. The side boundaries are free-slip, the bottom is no-slip, and the top boundary is either a free surface or free-slip (over sticky air).

The Rayleigh–Taylor instability model is adapted from and (See Fig. b). A dense and more viscous layer (ρ=3300 kgm-3, η=1021 Pas) is sinking through a less dense fluid (ρ=3200 kgm-3, η=1020 Pas). Side boundaries are free slip, the bottom boundary is no-slip and the top boundary is a free surface or free-slip (sticky air). The domain size is 500km×500km for ALE scheme and 500km×600km for ALE-IB and Eulerian scheme, with Q1dQ0 elements and 51×51 nodes (or 51×61 nodes). The initial perturbation has an amplitude of 5 km. A constant time step of 2500 years was employed in the simulations.

This experiment builds upon the models developed in to examine conditions leading to triggered dripping and lithospheric delamination (See Fig. c). The model domain includes a layered crust and mantle with the following parameters: upper crust (20 km thick, ρ=2800 kgm-3, η=1023 Pas), lower crust (20 km thick, ρ=3300 kgm-3, η=1019 Pas), lithosphere (100 km thick, ρ=3300 kgm-3, η=1021 Pas), and mantle (ρ=3250 kgm-3, η=1018 Pas). Side boundaries are free slip, the bottom boundary is no-slip and the top boundary is a free surface or free-slip, depending on the simulation scheme. For free surface simulations in ALE-IB and Eulerian scheme, there is a sticky air layer with ρ=0 kgm-3, and viscosity of η=1019 Pas, 150 km thickness, bordered with free-slip top boundary condition. The computational domain is 900km×600km in size for the Eulerian scheme with free-slip top boundary and free surface within ALE scheme, 900km×750km in size for free surface in ALE-IB and Eulerian schemes). The mesh employs Q1dQ0 elements and 193×129 nodes (or 193×161 nodes).

The rising sphere model is adapted from Case 2 in for validating the sticky air approach (See Fig. d). A plume with a radius of rp = 50 km, a density of ρp=3200 kgm-3, and viscosity of ηp=1020 Pas, is initially centred at (0, -400 km) of the mantle with ρm=3300 kgm-3, and viscosity of ηm=1021 Pas. The lithosphere, with ρl=3300 kgm-3 and viscosity of ηl=1023 Pas, has a thickness of 100 km. For simulations in ALE-IB and Eulerian schemes, there is a sticky air layer with ρ=0 kgm-3, and viscosity of η=1019 Pas, bordered with free-slip top boundary condition. Side boundaries are free slip, the bottom boundary is no-slip. The model domain is 2800km×700km in size for ALE scheme (2800km×850km in size for ALE-IB and Eulerian schemes), discretized with Q1dQ0 elements and 561×281 nodes (or 561×341 nodes).

Models with a free surface boundary condition produce more realistic slab bending, dip angles, and stress states compared to free-slip models, as shown in . The free surface approach more accurately captures topographic features, whereas free-slip models tend to exhibit more short- and intermediate-wavelength components in the simulated topography .

The subduction model is modified from (See Fig. e). It is a thermo-mechanical model designed to simulate the subduction of a visco-plastic slab into the mantle and generate realistic topography signals. The simulation here is run over a short duration with side boundaries are free slip. In contrast to , where the driving force is based on the temperature-dependent Rayleigh number, here the body force is driven by the same density contrast used in the previous experiments. The materials are assigned a temperature-dependent density, expressed as: 13ρ=ρ0(1-αΔT), where α is the thermal expansion coefficient, ΔT=T-T0 with T being the temperature, and ρ0 is the reference density at the reference temperature T0=300 K.

To simulate the deformation of the subducted lithosphere and surrounding mantle, a visco-plastic rheology is employed. The model uses the Drucker–Prager yield criterion with a pressure-dependent yield stress based on Byerlee's law, which approximates brittle behavior. Frictional-plastic deformation occurs when the stress exceeds the frictional-yield stress σy: 14σy=C+Pμ, where P, C, and μ are the pressure, cohesion and friction coefficient respectively.

The effective plastic viscosity is given by: 15ηeffpl=σy2ϵ˙, Where ϵ˙ is the strain rate.

Viscous deformation is modeled with a thermally activated power-law rheology, expressed by: 16ηeffvcreep=12A-1nexp⁡EnRT, where A is the pre-factor set as the effective viscosity giving the reference viscosity at T = 1600 K, E is the activation energy, n is the stress exponent, R is the gas constant and T is the temperature.

The effective viscosity combines brittle and ductile rheologies as: 17ηeff=min⁡(ηeffvcreep,ηeffpl), and is limited within nine orders of magnitude by applying upper and lower bounds: ηmax=105η0 and ηmin=10-4η0. where η0 is a reference viscosity.

An initial weak hydrated crustal layer of 7.5 km thickness is included on top of the subducting plate. Additionally, a sticky air layer with ρ=0 kgm-3, and viscosity of η=1019 Pas, is implemented, bordered by a free-slip top boundary in the Eulerian and ALE-IB schemes. The model assumes ongoing subduction, represented by a finite-length initial slab. An initial divergent plate boundary is specified at the tail of the subducting plate, with the boundary layer thickness WBL increasing away from this spreading centre toward the subduction zone according to the age-law : 18WBL(x)=WBL,0(x)⋅ΔXsc, where WBL,0(x) controls the maximum boundary layer thickness, here set as 100 km, x is the horizontal coordinate, and ΔXsc is the distance from the spreading centre at any given position x. The radial component of the initial temperature is related to plate thickness as Tz(x)=T0+(T1-T0)(erf(d/2WBL(x))), where T0=300 K is the temperature at the surface (and the top of the model domain), T1=1600 K is the temperature at the model base, d is depth below the surface.

The initial slab is approximately 500 km long, straight from trench to tip, inclined at θ=30° via an abrupt kink, which relaxes during the evolution. All materials share the same heat production rate. The top boundary (and the air layer, if present) is maintained at 300 K, while the bottom boundary is insulated with a zero heat-flux boundary condition. The domain size is 3000km×800km for ALE scheme (3000km×1000km in size for ALE-IB and Eulerian schemes). It employs Q1dQ0 elements and 601×161 nodes (or 201 nodes). Physical and numerical parameter details are given in Table .

In Experiment 1, the initial topography relaxes toward equilibrium over approximately 100 ka. Figure  compares the topography obtained from free-surface simulations across three different numerical schemes with the analytical solution Eq. (). Discrepancies among the schemes are illustrated in Fig. a, which shows the temporal evolution of the topography. Figure c presents the topography at Time=2t*, approximately equal to 49.21 ka. Both the ALE and ALE-IB schemes demonstrate good agreement with the analytical solution, whereas the Eulerian scheme exhibits fluctuations that reduce accuracy.

When the free surface is not explicitly tracked using additional tracers, the surface becomes unidentifiable, as shown in Fig. e. In such cases, the surface must be tracked via particles representing the top of the solid or the interface between rock and air, or through an averaged interface based on volume ratios. Using extra particles to trace the surface, common in this study, often results in a rough interface with undesired spatial fluctuations as discussed in . These fluctuations arise because the distance between markers and the interface is finite and irregular, leading to small velocity variations during advection.

Such fluctuations can be mitigated by employing finer vertical spacing in the computational mesh or by utilizing marker chains or level-set methods to more accurately assign viscosity and density to nodal points. Another strategy is the volume of fluid method , which involves interpolating material types (0=Air, 1=Rocks) from Lagrangian markers to Eulerian mesh nodes using a distance-dependent average. This is followed by computing the position of the topography surface, defined as the isosurface of 0.5, based on the Eulerian nodal material type values. The ALE-IB scheme introduced here provides an alternative way to inherently suppress these fluctuations, achieving accuracy comparable to the ALE scheme while maintaining robust surface tracking.

Additionally, the convergence of the Stokes solver within the ALE and ALE-IB scheme with the FSSA (see Eq. ), as well as in all schemes without the FSSA, was tested over a range of time steps. This was assessed using the L2-norm of the error in the topography at Time=2t* from the numerical modelling compared to the analytical solution. The convergence study involved a sequence of seven time steps: [1,1/2,1/4,1/8,1/16,1/32,1/64]t*. Figure  illustrates how the FSSA effectively reduces topography errors even at relatively larger time steps. When using FSSA, both ALE and ALE-IB achieve relatively small errors. It is important to note that FSSA introduces a conditional time step constraint. Therefore, we recommend using it primarily when calculations have to be performed with large time steps.

Following the methodologies outlined in and , we continuously monitored the evolution of the lithosphere-asthenosphere interface, defined here as the boundary between denser and less dense materials, and tracked the position of the free surface over time. The results (shown in Fig. b), demonstrate that all three simulation schemes: ALE, ALE-IB, and Eulerian are capable of accurately reproducing the results reported in when employing sufficiently small time steps. Notably, the time step used in these simulations is smaller than the Courant criterion, fixed at 2.5 ka, to prevent numerical instabilities such as the “drunken sailor” oscillations commonly encountered in free surface simulations. Both the ALE and ALE-IB schemes exhibit excellent agreement in tracking the evolution of the interfaces and the free surface. In contrast, the Eulerian scheme displays significant fluctuations in both the free surface and the lithosphere/asthenosphere interface, along with asymmetric features, especially in the interface's depth profile (Fig. d).

The fluctuations observed in the Eulerian approach are likely attributable to the inherent numerical diffusion and irregularities associated with fixed-grid advection, which can cause the interface to oscillate and distort over time. Conversely, the ALE and ALE-IB schemes, with their moving mesh and improved interface tracking strategies, maintain more stable and physically consistent interface evolutions, underscoring their robustness for long-term geodynamic simulations.

For the chosen model configuration, delamination of the denser lithosphere occurs progressively over time. Comparing the model from with a free-slip boundary condition at the top, the free-surface simulations within the ALE-IB and Eulerian schemes exhibit relatively faster delamination (Fig. a, c, and d). In this context, the free-slip top boundary can be interpreted as a very rigid layer over the upper crust, whereas the free surface in the ALE schemes effectively represents a weak, deformable upper boundary.

However, the ALE scheme exhibits strong counter-clockwise deformation patterns, even with small time steps, primarily due to asymmetry in the model geometry. This is particularly evident with the presence of a denser lithosphere confined to one half of the domain (see Fig. b). In contrast, the ALE-IB scheme offers advantages over the traditional ALE approach in such scenarios, providing more stable simulations of free-surface evolution when dealing with asymmetric geometries. Similar cases are discussed in , where slab bending is triggered by asymmetrical lithospheric thicknesses. Additionally, in models requiring an open bottom boundary, the ALE-IB and Eulerian schemes with a free-slip top boundary condition over the sticky air layer can handle such situations more effectively, whereas the standard ALE scheme tends to exhibit strong numerical instabilities under these conditions.

In the rising sphere model, the plume ascends and approaches the lithosphere over time. Figure  displays the surface topography at 4 and 8 Ma. The results from the ALE and ALE-IB schemes remain in good agreement with each other, demonstrating consistent plume evolution and corresponding topographic signals. In contrast, the Eulerian method exhibits strong fluctuations, with the topography reaching approximately a 7 % difference compared to the ALE and ALE-IB results.

The topography generated by the ALE-IB and ALE schemes is similar, displaying smooth and physically plausible surface features. In contrast, the Eulerian scheme produces a basin with a sharper angle on the left side of the island arc, resulting in less realistic surface morphology. Our ALE and ALE-IB scheme results are more comparable to the free-surface case in the Eulerian scheme reported in , as illustrated in their Fig. 4, where a shape-function averaging method was employed in Eulerian staggered grid scheme on all the uppermost rock tracers and the lowermost air tracers. This approach, combined with the sticky air method, yields more accurate surface representations than the pure Eulerian scheme we used here.

The ALE-IB scheme can produce realistic, single-sided subduction features similar to those obtained with the shape-function averaging method. It also achieves reasonably accurate topography and effectively overcomes mesh distortion issues common in the standard ALE scheme (see Fig. a). This demonstrates that our approach not only maintains topographic accuracy but also enhances numerical stability during complex subduction simulations.

While the proposed ALE-IB scheme offers significant advancements in simulating true free surface dynamics, several limitations should be acknowledged:

Computational cost: Although the internal boundary approach reduces certain numerical artifacts, it introduces additional complexity in mesh management and boundary condition implementation. This can result in increased computational expense, particularly for large-scale or high-resolution simulations.

Future research should focus on optimizing mesh management algorithms, incorporating more comprehensive physical processes, and validating results against observational data. These steps are essential for enhancing the applicability, accuracy, and overall robustness of this scheme in realistic geodynamic modelling.