In 2D models, the surface can be tracked by a marker chain which is initialised as an ordered array of Lagrangian markers, each of which stores its position xi and a type variable to facilitate manipulation of the chain (Fig. ). The position vector for each marker xi may be represented in either Cartesian coordinates xi=(xi,zi), or polar coordinates xi=(ri,ϕi), and may be dimensional or dimensionless. The height of each marker (zi or ri) is set by an initial surface height function zinit(x) or rinit(ϕ).

In 3D models the surface representation extends to a mesh. The initial surface consists of markers arranged in a grid with coordinates xi=(xi,yi,zi) in Cartesian geometry, or xi=(ri,θi,ϕi) in spherical geometry. As with the 2D case, the initial height of the markers zi or ri is given by an initial surface height function zinit(x,y) or rinit(θ,ϕ).

Surface markers are advected with the velocity field, determined from the Stokes solver, using a spatially fourth-order Runge-Kutta scheme, similar to the treatment of the markers used in StagYY. Velocities, which are face-centred in the staggered-grid finite-volume discretisation used by StagYY, are interpolated to surface markers using a second-order quadratic spline interpolation method .

It is necessary to be able to determine the location of the surface as a function of position through interpolation, i.e. finding z(x) or r(ϕ) in 2D, or z(x,y) or r(θ,ϕ) in 3D as a function of the position of the Lagrangian surface markers. A straightforward approach is to construct a piecewise continuous linear or bilinear representation of the surface using a first-order accurate distance-weighted linear interpolation Chap. 18. This method was employed with a 2D marker chain in the context of modelling free surfaces in numerical geodynamic models by , and a similar method is used in 3D models using the regional scale geodynamic code LaMEM . The method requires two steps: the interpolation of surface height data from Lagrangian surface markers to nodal points using distance-weighted linear interpolation, and the linear or bilinear interpolation of nodal values to arbitrary locations within the domain. Once surface heights are interpolated to nodes, determining the surface height at any point can be accomplished in constant O(1) time, independently of the number of surface markers.

For consistency, this approach is referred to throughout as the bilinear method, reflecting the bilinear interpolation used in 3D models. In 2D, the process is technically linear, but the term bilinear method is retained for uniformity.

Another method of determining the surface location at arbitrary points is through direct interpolation from the marker chain or mesh, without first mapping to the Eulerian grid. In 2D, this is achieved by locating the markers on the chain immediately to the left and right of the interpolation point and linearly interpolating between them (Fig. ). Higher order methods such as piecewise quadratic or cubic spline interpolation are possible, but in practice simple linear interpolation produces adequate results.

As advection allows markers to move laterally, the chain must be re-sorted by horizontal coordinate after each timestep. Since the chain is typically nearly sorted after advection, adaptive algorithms such as insertion sort are efficient for this purpose, and an O(log⁡n) binary search can then be used to locate neighbouring markers for interpolation.

In 3D, direct interpolation is more involved since the surface is represented by planes rather than line segments. As the surface markers are generally unstructured, a Delaunay triangulation of the horizontal marker coordinates is used (Fig. ). The triangulation is computed using GEOMPACK2 in O(nlog⁡n) time and must be recomputed at each timestep; adaptive methods exist to reduce this cost . Methods for performing Delaunay triangulations directly on spheres also exist , though for the yin-yang grid the errors introduced by projecting the spherical patch into a Cartesian plane are negligible. Each Delaunay triangle defines a plane through three markers a,b,c, with normal vector n=(b-a)×(c-a), from which the surface height at any point within the triangle can be recovered. An efficient O(log⁡n) triangle location method is available in GEOMPACK2; if the point lies outside the triangulation, the nearest triangle is used for extrapolation.

In order to couple the surface tracking method to the Stokes solver and to output results, the surface heights of the marker chain or mesh must be interpolated to the Eulerian grid. When using the bilinear method, this step is handled automatically by interpolating surface heights to the nodes of the Eulerian grid. For the integral method, a cellwise integration approach is used in both 2D and 3D.

The cellwise integration method computes the area (in 2D) or volume (in 3D) enclosed by the surface within each computational cell, and derives the equivalent topographic height that preserves this area or volume. This is physically motivated by the need to maintain conservation of mass of both rock and air within the model. In 2D the integral is evaluated using the trapezium rule by looping over marker chain intervals, which is accomplished in linear time O(n) as the chain is already sorted. In 3D, Delaunay triangles are intersected with the footprint of each cell column using a polygon intersection algorithm ; the volume beneath each intersection polygon is then computed and summed to obtain the surface height per column (Fig. ). Both 2D and 3D operations scale as O(n) with respect to the number of surface markers.

When solving on multiple CPU cores, the domain is split into parallel subdomains. Surface markers near subdomain boundaries are exchanged via MPI to form a one-cell-wide overlap region, ensuring the surface is defined at all locations for all subdomains (Fig. ).

Minor numerical errors in the Stokes solution and/or marker advection can cause the surface to drift upward or downward over time. To correct this, the height of all surface markers is adjusted at each timestep to preserve the total area or volume of rock and air in the model, and therefore mass under the assumption of near-incompressibility. This is achieved by integrating under the marker chain or mesh using the same trapezium rule or Delaunay-based approach described in Sect. , with corrections for the spherical area element dA=r2sin⁡θdθdϕ applied where necessary.

Although reinitialisation is essential for preserving the integrity of the marker chain over multiple timesteps, it does come with trade-offs: the process may degrade information about the surface geometry and introduce artificial numerical diffusion. Therefore, reinitialisation should only be applied when needed.

Two reinitialisation strategies are implemented. The first is periodic global reinitialisation, in which the entire surface marker chain/mesh is reinitialised to align with the current surface level at every timestep. This is computationally efficient and maintains a consistent number of surface markers, but introduces numerical diffusion that can smooth surface features over time (Fig. ). It does not require the chain to be sorted or the Delaunay triangulation to be computed, making it compatible with the bilinear method. Despite the diffusion it introduces, some smoothing is beneficial for stability, and the effect is relatively mild in practice.

The second strategy is interval reinitialisation, in which markers are added or removed locally based on the spacing between adjacent markers (in 2D) or the area of Delaunay triangles (in 3D). In 2D, a marker is added at the midpoint of any interval exceeding 2Δxinit, and two markers are merged if their spacing falls below 0.2Δxinit; empirical testing confirmed these thresholds give good results. The 3D analogue uses triangle area ratios A/Ainit with an upper threshold of 3 and a lower threshold of 0.2. This approach avoids the numerical diffusion inherent in global reinitialisation by only modifying the chain where necessary. However, it requires either sorting or Delaunay triangulation of the surface markers, which precludes its use with the bilinear method.

A fundamental problem with the Lagrangian surface marker method is the undesirable entrainment of markers into the mantle or air layer instead of remaining at the surface. The effect is most pronounced when using the integral method, and results in the formation of unrealistically deep trenches or high mountains that can cause numerical instabilities (Fig. ). Solving this problem requires a method of smoothing the Lagrangian markers.

A natural choice for smoothing is diffusion, which is conservative and physically motivated. However, it can smooth model features unnecessarily, and periodic global reinitialisation, if used, already introduces some numerical diffusion. The method implemented here for the integral method is instead a novel slope limiting approach.

The slope limiting method is physically motivated by the behaviour of loosely packed granular materials: a pile of sand under gravity forms a cone with a maximum angle of repose ϕ, related to the internal coefficient of static friction by tan⁡(ϕ)=μs. Slopes below this angle are stable; those above erode until the angle is reached. This behaviour is expressed as a steady-state PDE based on the diffusion equation: 1∇2zH||∇z||-scrit=0, where H is the Heaviside step function and scrit is the critical slope. The Heaviside function ensures that only regions where the slope exceeds scrit are modified, limiting the smoothing effect to regions of high deformation such as subduction or collision zones. The nonlinearity introduced by the Heaviside function is handled using an iterative solver in both 2D and 3D. Applied to sinusoidal test surfaces (Figs. and ), the method flattens steep wave flanks while leaving shallower regions unaffected.

A key advantage of slope limiting over diffusion is that it is timestep independent, producing consistent results across different temporal and spatial scales. A limitation is its relatively poor computational performance (see Sect. ), and the fact that it only modifies the surface location and not the underlying markers, which can become unmixed as a result. Solutions to this are discussed in Sect. .

During advection, rock markers can end up above the surface and air markers below it. This can also occur when the surface location is directly manipulated by a smoothing algorithm. The resulting erroneous marker distribution leads to numerical instability and must be corrected. Two methods were implemented (Fig. ).

In the bouncing method, erroneous markers are reflected back to their correct side of the surface, with the displacement equal to their distance from the surface. No markers are created or destroyed, making this method mass-conserving and applicable to models with many marker types. A downside is that bouncing can create gaps in the marker field in convergent regions where the surface is modified by smoothing but the underlying markers are not, which over time can cause numerical instability.

In the exchange method, erroneous markers are directly converted to the appropriate type: air markers below the surface become rock markers, and vice versa. This avoids the gap-forming tendency of the bouncing method. For the conversion of air markers to rock markers, a random template marker is selected from the cell immediately below the surface. This works well for simple compositional setups, but becomes unsuitable when markers carry physical properties such as temperature or viscosity, or when many distinct compositions are present. The method is not strictly volume conserving, though on average the number of conversions in each direction remains roughly equal.

StagYY typically determines the density and viscosity fields from the marker-in-cell marker distribution. With Lagrangian surface tracking, greater accuracy can be achieved by coupling the surface location directly to these fields through volume fractions. The volume fraction ϕ of a control volume is defined as the fraction of that volume occupied by rock, ranging from 0 (completely empty) to 1 (completely full), with partially filled cells near the surface having 0<ϕ<1 (Fig. ).

StagYY uses a fully staggered grid (Fig. ), a standard layout in geodynamic modelling , in which the density field is defined at cell centres and vz points, and viscosity at up to four locations (P, σxz, and in 3D also σxy and σyz points). Volume fractions are computed for each of these control volumes offset from the main cells.

As an example, the effective density used in the z-momentum equation is given by: 2ρeff=ρrefϕvz, where ρref is a reference near-surface rock density and ϕvz is the volume fraction for the vz control volume (Fig. ). A similar approach is applied to the viscosity field at each of the relevant staggered grid locations.

This coupling scheme requires that the density and viscosity fields not already account for the sticky air layer; otherwise, its contribution would be counted twice, once through the field values themselves and once through ϕ. We avoid this by setting the air's density and viscosity to the same reference values used in the coupling. One limitation of this is that, because a single reference value is assumed, the approach loses accuracy where surface density or viscosity varies strongly along the surface. In that case, the reference value can instead be taken locally from the cell immediately below the surface at each point.

Volume fractions are computed geometrically in 2D using the marker chain, and in 3D using the same Delaunay-based approach described in Sect. , both in linear time O(n). In all benchmark tests, computing volume fractions incurred negligible cost compared to other surface marker operations.

The 2D surface relaxation benchmark is based on Case 1 as presented by . It considers the time dependent relaxation of an initially non-flat surface with initial topography zinit(x) given by the function: 3zinit(x)=7cos⁡2πx2800kmkm.

This problem is simple enough to have an analytical solution for surface position as a function of time as derived using a three-layer model . Starting from an initial maximum of 7 km, the maximum topography as a function of time zmax(t) is given by: 4zmax(t)=7exp⁡t14.825kyrkm, where t is time and 14.825 kyr is the analytical relaxation timescale derived from the model viscosity structure.

The model domain is a 2800×800 km Cartesian box consisting of three compositional layers: a 600 km thick mantle layer, a more viscous 100 km thick lithosphere, and a 100 km thick air layer. The resolution is 512×128, with 100 markers per cell for advecting material properties and tracking the surface in the case of the marker-in-cell method. The sticky air layer has a viscosity of 1018 Pa s. Symmetric boundary conditions are imposed at the horizontal edges of the model, while a no-slip boundary condition is imposed at the bottom boundary, and a free slip condition is imposed at the top. Acceleration due to gravity g is set to 10 m s⁻². For consistency with the benchmarks run by , models were run until t=100 kyr to show the system reaching equilibrium. Timesteps were limited to 500 years in order to give high temporal resolution for plots of model evolution.

The three surface tracking methods can be compared against the analytical solution when plotting maximum topography with time, as in Fig. .

All methods are qualitatively able to track the analytical solution closely in agreement with the results presented in . Minor divergence between the methods becomes more apparent after 30 kyr, with the marker-in-cell method failing to match the analytical solution as closely as Lagrangian tracking methods after this point. At 100 kyr, while the other methods have converged to a maximum topography near zero, the marker-in-cell method resulted in around 200 m of topography, a result that can also be seen in Fig. 2 of . This result can be attributed to the maximum topography being affected by the noise inherent in the marker-in-cell method when using a finite number of randomly initialised markers throughout the model. Meanwhile, the Lagrangian surface marker-based tracking methods all converge to near zero at 100 kyr, indicating that Lagrangian surface tracking methods are less susceptible to this noise.

It should be noted that part of the remaining discrepancy between the methods presented here and the analytical solution can also be explained by the limitations of the sticky air method (i.e. finite viscosity of air) used to produce the solution to the Stokes equations rather than the surface tracking method itself. The primary value of this benchmark is therefore in the relative comparison between methods: the Lagrangian tracking methods converge more closely to zero at 100 kyr than the marker-in-cell method, demonstrating a clear reduction in marker-density-dependent noise that is independent of the Stokes solver used.

The 2D plume benchmark is based on Case 2 presented in , considering a buoyant plume (radius 100 km, density 3200 kg m⁻³, i.e. 100 kg m⁻³ less than the surrounding mantle and lithosphere) rising through a 2800×850 km domain at 1024×256 resolution over 20 Myr. The sticky air layer has a viscosity of 1019 Pa s. Free slip boundary conditions are imposed at the horizontal edges and top of the model, while a no-slip boundary condition is imposed at the bottom. All other parameters follow .

Analytical solutions for this benchmark are not available. However, results from multiple numerical geodynamic models, including finite element models with true deformable free surfaces, were run as part of the study of and consistently show a maximum surface deformation of 800–850 m after 20 Myr.

The results of the plume benchmark further demonstrate the strengths of Lagrangian tracking methods in isolating the topographic response of the model without the noise associated with the marker-in-cell method. The marker-in-cell method introduces random noise, which results in a ≈100 m discrepancy in topography at the centre of the plume. Lagrangian tracking methods closely matched the results demonstrated by for both the maximum topography at 20 Myr and the evolution of topography in the intervening time.

Scaling tests used a non-dimensional model in spherical annulus geometry with constant viscosity (Ra = 107), run for 5000 timesteps to ensure statistically steady-state convection (Fig. in the Appendix). Figure shows performance as a function of surface marker density for both methods. The bilinear method consistently accounts for approximately 3 % of total runtime across all tested MPC values, while the integral method peaks at 11 % at 64 MPC. In both cases the dominant cost is marker unmixing, which requires looping through all markers in the model to determine their position relative to the surface: a procedure taking O(1) time per marker for the bilinear method but O(log⁡n) for the integral method. Tests across four resolutions (128×16 to 1024×128) at a fixed density of 8 MPC confirm this pattern: unmixing scales with marker count rather than surface marker count, leaving the bilinear method below 3 % of total runtime at all resolutions tested.

The 2D plume benchmark was extended to 3D on a 1400×1400×850 km Cartesian domain at 256×256×64 resolution, with the plume geometry modified to a sphere of initial diameter 100 km. A no-slip boundary condition is imposed at the lower boundary, while free slip conditions are imposed on all other boundaries. All other physical parameters are identical to the 2D case. As no prior 3D results exist for comparison, results are assessed qualitatively.

The performance of different tracking methods, as illustrated in Fig. , highlights the inherent noise challenges of the marker-in-cell method in 3D. This method struggles with noise generation due to the inherent dependence on marker density and positioning. Nonetheless, it is still able to capture the main features of the mantle plume, particularly the central topographic uplift created by the buoyant upwelling. Both the bilinear and integral methods eliminate this noise, providing visually similar results that isolate the plume's influence on topography, with minor differences between the two methods visible in the cross-sectional profile (Fig. ), possibly a side effect of the differing smoothing methods used.

A simple Cartesian box setup (Fig. ) with a relatively coarse 64×64×32 resolution was used to compare the three methods under constant-viscosity convection with non-dimensional parameters (Ra=107). Visual inspection of Fig. confirms that both the bilinear and integral methods eliminate the marker-density-dependent noise of the marker-in-cell method, producing visually smooth topography that isolates the underlying plume and downwelling structure.

The same Cartesian box model was used to test scaling of surface tracking methods with respect to model resolution and marker density.

This study implemented and benchmarked two Lagrangian surface marker tracking methods, the bilinear method and the integral method, in the mantle convection code StagYY, comparing them against the existing marker-in-cell approach. Both new methods produce high-resolution, low-noise surface topography that is independent of volumetric marker density, unlike the marker-in-cell method, allowing significant reductions in volumetric marker count and associated computational cost. The bilinear method offers the best computational performance, consistently accounting for less than 3 % of total runtime, and is straightforward to implement, though it introduces some artificial numerical diffusion due to periodic global reinitialisation. The integral method produces topography of comparable quality but at greater computational cost, due primarily to the O(log⁡n) surface lookups required during marker unmixing and the overhead of slope limiting. Both methods can be used to obtain volume fractions for coupling the surface location to the density and viscosity fields used by the Stokes solver (ρ,η coupling), and this coupling is found to be essential for smooth and accurate surface tracking results.