Short communication: Horizontal movement and deformation in large-scale landform evolution models

Hergarten, Stefan; Fraters, Menno

doi:10.5194/egusphere-2026-2060

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https://doi.org/10.5194/egusphere-2026-2060

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21 Apr 2026

| 21 Apr 2026

Status: this preprint is open for discussion and under review for Earth Surface Dynamics (ESurf).

Short communication: Horizontal movement and deformation in large-scale landform evolution models

Stefan Hergarten and Menno Fraters

Abstract. Coupled models of geodynamics and landform evolution are receiving growing interest. While landform evolution models typically describe the response of the topography to vertical uplift and subsidence, horizontal movement becomes an essential component in combination with geodynamics. This study compares Eulerian and Lagrangian schemes for including horizontal movement and deformation of the crust in fluvial landform evolution models. As a main result, Eulerian schemes do not allow rivers to move perpendicularly to their main flow direction. In turn, they generate a strong artificial increase in surface elevation at high velocities, which makes them unsuitable for scenarios with strong horizontal movement. The Lagrangian approach avoids these problems. In turn, it is technically more complicated if deformation is so strong that remeshing is necessary. Remeshing is challenging for the widely used D8 topology, which derives the flow pattern from the 8 nearest and diagonal neighbors. Furthermore, remeshing causes artifacts by damming rivers temporarily. While these artifacts can be reduced technically, the question remains under which conditions remeshing is necessary. Our results suggest that the simple D8 topology can be used without remeshing for a shear strain of up to about 2, which is already a quite strong deformation.

Received: 10 Apr 2026 – Discussion started: 21 Apr 2026

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Stefan Hergarten and Menno Fraters

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RC1:
'Comment on egusphere-2026-2060', Boris Gailleton, 17 Jun 2026 reply
This study explores the impact of numerical choices on simulating horizontal tectonic motions in large-scale landscape evolution models. The topic is important for the geomorphological and tectonic communities and is suitable for ESURF. The technical aspect is well demonstrated and developed. However, the scientific motivations and implications are largely missing—this is especially important for a journal oriented toward the broader geomorphological community. I do not think it requires major revisions, but the manuscript could strongly benefit from reasonably more context and discussion.
Please find below my main comments and some more detailed in-text comments.
Main Comments

Overall, the scientific motivations of this manuscript remain unclear. Do not get me wrong, I believe horizontal tectonics in Landscape Evolution Models is a worthy topic and many scientific questions lie behind it. I also believe the technical aspect of how the grid is managed is largely worth a short communication in ESURF. However in the current form, the motivations behind this development need to be outlined more clearly; I am a numerical modeller and geomorphologist but not a tectonicist, and the implications behind adding the horizontal tectonic forces “more or less correctly” remain unclear. For example, the introduction reads well, but falls a bit short on why this study matters. It ends with a dense paragraph about numerical diffusion and then switches directly to the modelling setup. It is also overall quite specific and does not really draw the big picture. I feel it lacks a paragraph outlining the main questions and impact of including horizontal tectonic movements in LEMs (beyond the technical reason—which is important of course—but may not reach all the readers of ESURF) and how it will explore them.

My first conceptual concern lies with l. 149-152: “[…] So the preliminary conclusion must be that Eulerian schemes are unsuitable for including horizontal movement in fluvial erosion models.” The main reason outlined (and well demonstrated) is that the coupled model on Eulerian schemes fails to move rivers perpendicularly to their flow direction even at high horizontal tectonic velocities, resulting in unrealistic relief and valley shapes hovering over quasi-fixed major rivers (if I understand correctly?). Is this not a matter of timescale? The erosion-deposition is not fully implicit as stated: drainage area and slope direction are explicit. So, one could expect that having differential time steps could leave more time for the river to respond to the horizontal advection, or at least show an effect of it. Erosion is forced on a D8 flow path and will artificially fight U on a static location for long time steps, whereas “in real life” flow could shift further during this time step and shift erosion, allowing for movement (even numerically with a more diffusive multiple flow routing for example). Another point not mentioned is the treatment of outlets: are they affected by horizontal motions too? SPIM schemes are highly sensitive to this, and a fixed outlet can quickly pin the whole river network spatially—especially at long time steps (in regard to the potential change in slope direction, so even in the non-dimensionalised framework).

My second conceptual concern is about processes and is more of a discussion point rather than a missing point in the study. I am not a tectonicist so I might be wrong on this, but it seems to me horizontal tectonics occur mostly by jumps (earthquakes). Also, erosion and sediment transport occur mostly during floods and go beyond vertical motions (lateral erosion and deposition). Finally, long-term averaged flow can diffuse beyond the D8 steepest descent. None of these effects are taken into account in the current framework—which is common for SPIM-like LEMs to reach large timescales—but could the inability of the Eulerian framework to simulate realistic advection actually reflect missing processes? For example, one could expect a river's lateral processes to fight back during floods and re-equilibrate more “realistically”. Lagrangian-based remeshing does not exactly produce realistic topography (Fig. 9) either, highlighting that some processes may be missing.

It is not entirely clear how the Lagrangian advection and the surface processes are coupled. Are they operating on the same irregular grid, or is the Lagrangian displacement projected back to the Eulerian grid where the modified SPIM is solved?

Finally, the reference list is composed of >30% of self-citation, on topics that have been researched by others as well. The manuscript would benefit from a more diverse bibliography with independent sources to strengthen the literature review.

Inline Comments

l. 23: While I acknowledge the model in Hergarten and Neugebauer (2001) was probably solving the SPIM with a similar upstream propagation of the implicit solution from a fixed/known outlet node, the original manuscript barely describes the numerical scheme beyond “an implicit time discretization of the slope gradients Delta_i in Eq. (1) can be performed if the discharges are once computed.” One could argue that Braun and Willet (2013) went beyond making the implicit method “popular,” but rather properly described the full numerical scheme, generalised it to the non linear case all alongside the use of graph theory (topological ordering “stack” and how to compute it efficiently for rivers) to ensure efficiency.

l. 23-25: The scheme is not fully implicit as drainage area and steepest descent are explicit—which can matter for long timesteps where drainage divide migration or flow path modification could occur sub step—or if deposition is enabled like in Hergarten (2020) columns of sediments in the deposition area.

l. 46: Add a reference to Howard and Kerby (1983) for the SPIM.

l. 50-60: One could also mention that the effect of n>1 and n<1 also causes non-numerical knickpoint attenuation or combination. It goes beyond the scope of numerical diffusion but could help frame the main story around the fact that horizontal tectonics, when not done right, can further obscure the reading of tectonic signals in landscapes and LEMs by adding uncertainties.

l. 63: Sediment transport in LEMs is not really new as implied (e.g., CHILD had it in 1999). Also it is not clear why it particularly matter for this specific study.

l. 64: Is it shown anywhere that the scheme is numerically the most efficient for sediment deposition? This is highly dependent on the actual numerical implementation. Using a quick test on my own numerical framework, my implementation of Hergarten (2020) is only noticeably faster than Yuan et al. (2019) if n=1 and G>1 (which is far from being systematically the case—Guerit et al., 2020). I am not questioning the relevance and elegance of the numerical scheme - but if the choice is motivated by numerical efficiency, it needs more support.

l. 86: Again, not fully implicit (slope direction and drainage area are not).

l. 87-90: Is the time step used in the tectonic model the same as for erosion, or are they treated separately? Also, can you use the CFL then to get an adaptive time step based on the advection?

Fig. 2 and l. 96-106: While it is discussed somewhat, what is the effect of dt? Can a dt close to the CFL with the upstream scheme reduce numerical diffusion, or does it happen no matter the dt value? What about grid spacing?

Reply
Citation: https://doi.org/10.5194/egusphere-2026-2060-RC1
- AC1:
  'Reply on RC1', Stefan Hergarten, 24 Jun 2026 reply
  Dear Boris,
  
  thank you for your constructive review! Let me, for the moment, just share my thoughts about your concerns and ideas and think about potential improvements after the second review has arrived.
  Overall, the scientific motivations of this manuscript remain unclear. Do not get me wrong, I believe horizontal tectonics in Landscape Evolution Models is a worthy topic and many scientific questions lie behind it. I also believe the technical aspect of how the grid is managed is largely worth a short communication in ESURF. However in the current form, the motivations behind this development need to be outlined more clearly; I am a numerical modeller and geomorphologist but not a tectonicist, and the implications behind adding the horizontal tectonic forces "more or less correctly" remain unclear. For example, the introduction reads well, but falls a bit short on why this study matters. It ends with a dense paragraph about numerical diffusion and then switches directly to the modelling setup. It is also overall quite specific and does not really draw the big picture. I feel it lacks a paragraph outlining the main questions and impact of including horizontal tectonic movements in LEMs (beyond the technical reason—which is important of course—but may not reach all the readers of ESURF) and how it will explore them.
  
  We would, of course, have written it in a different way as a research paper, and it would not be a huge problem to extend the introduction. In turn, I still feel that there should be a clear difference between research papers and short communications (or technical notes in other journals). And to be honest, I am not good in telling stories. The idea behind this paper was never to motivate anyone to start research on coupling landform evolution models with geodynamic convection models since some (or several) groups have started working on this topic intensely. They will explain quite well in their (upcoming) papers why considering coupled models is important. We just want to leave a message for them if they ever encounter problems with rivers in their models.
  
  My first conceptual concern lies with l. 149-152: "[…] So the preliminary conclusion must be that Eulerian schemes are unsuitable for including horizontal movement in fluvial erosion models." The main reason outlined (and well demonstrated) is that the coupled model on Eulerian schemes fails to move rivers perpendicularly to their flow direction even at high horizontal tectonic velocities, resulting in unrealistic relief and valley shapes hovering over quasi-fixed major rivers (if I understand correctly?). Is this not a matter of timescale? The erosion-deposition is not fully implicit as stated: drainage area and slope direction are explicit. So, one could expect that having differential time steps could leave more time for the river to respond to the horizontal advection, or at least show an effect of it. Erosion is forced on a D8 flow path and will artificially fight U on a static location for long time steps, whereas "in real life" flow could shift further during this time step and shift erosion, allowing for movement (even numerically with a more diffusive multiple flow routing for example).
  
  You understood it correctly, which means that the paper somehow conveys its message. However, it is neither a matter of timescale nor related to the explicit treatment of the flow directions or to the "non-diffusive" D8 scheme. Changes in flow direction of major rivers are rare under uplift. So a simple fixed-point iteration would even converge and make the flow directions formally implicit. And a multiple flow direction scheme would not make the rivers climb up the walls of the valleys just because the valley is slightly asymmetric. This would only have a notable effect in flat areas and potentially on hillslopes. The problem is inherent to the type of model, so on mapping rivers as linear objects on a discrete grid. It is the second fundamental weakness of this type of model, behind the still unsolved and serious scaling problem (dependence of the result on the spatial resolution) when coupling fluvial erosion with diffusion. The Eulerian treatment cannot keep track of gradual horizontal shift; erosion is erasing all information on this rapidly. However, I am not entirely sure whether I got your point fully. Are you just not fully convinced that the problem exists or would you suggest a way to fix it? In principle, you could disregard horizontal movement until it accumulates to one unit of grid spacing, then adjust the elevation instantaneously, recompute the flow directions and finally erode. I did not test this because it looks a bit weird compared to a Lagrangian scheme and I guess that it would end in a mess.
  
  Another point not mentioned is the treatment of outlets: are they affected by horizontal motions too? SPIM schemes are highly sensitive to this, and a fixed outlet can quickly pin the whole river network spatially—especially at long time steps (in regard to the potential change in slope direction, so even in the non-dimensionalised framework).
  
  I am also not entirely sure about the direction of your idea. To my experience, you can destroy your results easily be defining bad boundary conditions, but not fix inherent problems by defining specific boundary conditions. Here the northern and southern boundaries are defined by zero elevation, while outflow is allowed along the entire edge (as usual). Do you think about allowing outflow only at a few points and letting this points "jump" by one grid spacing after a certain time? This will generate dams and the model will fight against this by generating boundary-parallel channel segments. Without having tested it numerically, I am sure that this will not solve the problem.
  
  My second conceptual concern is about processes and is more of a discussion point rather than a missing point in the study. I am not a tectonicist so I might be wrong on this, but it seems to me horizontal tectonics occur mostly by jumps (earthquakes).
  
  Small-scale tectonics is complicated and represented in landform evolution models only in very specific ways to my knowledge. So let us better talk about large-scale geodynamics. I would request that a coupled geodynamic-landform evolution model must at least be able to translate and to rotate an island or a continent without destroying the topography.
  
  Also, erosion and sediment transport occur mostly during floods and go beyond vertical motions (lateral erosion and deposition). Finally, long-term averaged flow can diffuse beyond the D8 steepest descent. None of these effects are taken into account in the current framework—which is common for SPIM-like LEMs to reach large timescales—but could the inability of the Eulerian framework to simulate realistic advection actually reflect missing processes? For example, one could expect a river's lateral processes to fight back during floods and re-equilibrate more "realistically".
  
  I think that once having an efficient model in which valley floors and rivers consist of multiple grid points (and perhaps includes lateral erosion) would solve several problems (see above), but it would have to be shown that they are really compatible with Eulerian schemes. Anyway, we cannot tell those who are actually working on coupled models to stop and wait for such a model.
  
  Lagrangian-based remeshing does not exactly produce realistic topography (Fig. 9) either, highlighting that some processes may be missing.
  
  How do you see that it is not realistic? And if so, I would not be sure that a model with a better process representation would be better in such a situation.
  
  It is not entirely clear how the Lagrangian advection and the surface processes are coupled. Are they operating on the same irregular grid, or is the Lagrangian displacement projected back to the Eulerian grid where the modified SPIM is solved?
  
  Each point has a variable x- and y-coordinate and distances and cell sizes (and cell edges for diffusion in oceans, but not ret released) are computed accordingly (introduced in OpenLEM version 45). I guess that this is "operating on the same irregular grid" in your comment. However, I am not entirely sure about the alternatives you mention. Projecting back in the sense of interpolation would not bring any progress compared to the Eulerian approach. The other option would be not letting the LEM grid know that it has been deformed. In the
  
  shear example (Fig. 9), this would be the version "Lagrangian with frozen flow pattern" as there would be not motivation to change any flow direction. Not really good, but clearly better than the Eulerian approach.
  
  Finally, the reference list is composed of >30% of self-citation, on topics that have been researched by others as well. The manuscript would benefit from a more diverse bibliography with independent sources to strengthen the literature review.
  
  Just give me an allowed percentage and we will remove self-citations accordingly. Alternatively, I would also appreciate getting the references that you would like to see.
  
  l. 23: While I acknowledge the model in Hergarten and Neugebauer (2001) was probably solving the SPIM with a similar upstream propagation of the implicit solution from a fixed/known outlet node, the original manuscript barely describes the numerical scheme beyond "an implicit time discretization of the slope gradients Delta_i in Eq. (1) can be performed if the discharges are once computed." One could argue that Braun and Willet (2013) went beyond making the implicit method "popular", but rather properly described the full numerical scheme, generalised it to the non linear case all alongside the use of graph theory (topological ordering “stack” and how to compute it efficiently for rivers) to ensure efficiency.
  
  Researchers with a very deep background in numerics (not very likely in geomorphology) would probably recognize from the the words in my 2001 PRL paper that there cannot be a serious difference between the two approaches. Otherwise, we would need my 2002 book "Self-organized Criticality in Earth Systems" (one more unwanted self-citation) to recognize that:
  
  (1) My nonlinear version was implemented only for n = 2 by solving a quadratic equation. So restricted to n = 2, but in turn more efficient than the Newton scheme used by Braun and Willett.
  
  (2) Recursive function calls are the "natural" way to solve such a system (where either the donors or the flow target has to be treated prior to the respective points). Each recursive algorithm can be converted into a "list-based" classical scheme, which is what Braun and Willett did. Employing graph theory it just overkill in this context. "My" recursive implementation is a bit faster, while the list implementation of the stack is a bit more memory-efficient under some conditions and helpful for very old programming languages such as FORTRAN77 that do not allow recursion.
  
  (3) Of course, both schemes are O(n).
  
  (4) Braun and Willett indeed mentioned that completely separated catchments can be treated in parallel, which I found too trivial to be mentioned.
  
  And if I was allowed to tell the story behind: I was disappointed at first that Jean Braun did not reply to my mail. But after talking in person, everything was fine because it was clear
  
  that he really did not know that almost everything was already there 12 years earlier. Finally, I was a bit disappointed again because he never kept his promise to cite my work.
  
  l. 23-25: The scheme is not fully implicit as drainage area and steepest descent are explicit—which can matter for long timesteps where drainage divide migration or flow path modification could occur sub step—or if deposition is enabled like in Hergarten (2020) columns of sediments in the deposition area.
  
  True, but only relevant in quite flat areas or at hillslopes (see above).
  
  l. 46: Add a reference to Howard and Kerby (1983) for the SPIM.
  
  Ok if you are sure that this is really the oldest reference.
  
  l. 50-60: One could also mention that the effect of n>1 and n<1 also causes non-numerical knickpoint attenuation or combination. It goes beyond the scope of numerical diffusion but could help frame the main story around the fact that horizontal tectonics, when not done right, can further obscure the reading of tectonic signals in landscapes and LEMs by adding uncertainties.
  
  Right, but as mentioned in the introduction, Benjamin Campforts has demonstrated that knickpoints are preserved better with a suitable shock-capturing scheme, which would also improve the behavior for horizontal movement parallel to the rivers. So I would see knickpoints not as a clear argument against the Eulerian approach.
  
  l. 63: Sediment transport in LEMs is not really new as implied (e.g., CHILD had it in 1999). Also it is not clear why it particularly matter for this specific study.
  
  Perhaps a matter of the wording, but focus should have been on "while keeping the conceptual simplicity and the numerical efficiency of the SPIM."
  
  l. 64: Is it shown anywhere that the scheme is numerically the most efficient for sediment deposition? This is highly dependent on the actual numerical implementation. Using a quick test on my own numerical framework, my implementation of Hergarten (2020) is only noticeably faster than Yuan et al. (2019) if n=1 and G>1 (which is far from being systematically the case Guerit et al., 2020). I am not questioning the relevance and elegance of the numerical scheme - but if the choice is motivated by numerical efficiency, it needs more support.
  
  There is no doubt that it is also possible to implement my scheme in an inefficient way. I think it is also possible to construct scenarios where the simple fixed-point iteration around the explicit treatment of the sediment fluxes by Yuan et al. (2019) is a bit faster than my implicit scheme. In my other applications, the main aspect is that my scheme keeps its performance even for a completely transport-limited model, but this is not relevant here. Anyway, we will find another excuse for using OpenLEM.
  
  l. 86: Again, not fully implicit (slope direction and drainage area are not).
  
  True, but not really relevant here (see above).
  
  l. 87-90: Is the time step used in the tectonic model the same as for erosion, or are they treated separately? Also, can you use the CFL then to get an adaptive time step based on the advection?
  
  Here it is the same, but not necessarily in future applications involving numerical convection models. And a single adaptive dt for all points would not make much sense because it would only capture the fastest point well. Even for the rotation of a rigid body, it would not help much. Theoretically, we could hold back movement individually for each point and individually in the x- and y-direction until it reaches dx. However, I think that this will generate dams and violate the "mass balance". And at one point we should ask: Given that the present version of Fastscape indeed cannot handle Lagrangian coordinates, would it really make sense to build a new world around Fastscape?
  
  Fig. 2 and l. 96-106: While it is discussed somewhat, Fig. 2 and l. 96-106: While it is discussed somewhat, what is the effect of dt? Can a dt close to the CFL with the upstream scheme reduce numerical diffusion, or does it happen no matter the dt value? What about grid spacing?
  
  Fig. 2 is only the reference without erosion to illustrate that the central scheme and the Lax-Wendroff scheme would not be too bad without fluvial erosion. Here v*dt is 3 orders of magnitude smaller than dx, which means that the error from dx dominates over the error from dt. In this sense, it happens no matter the dt value, except for the very specific case of uniform translation into the x-direction at the CFL limit. And about grid spacing: Finally, it is a nondimensaional combination of velocity, erodibility and grid spacing that defines the behavior. However, we can just switch between rivers just not moving and destroying topography and not get rid of all problems easily.
  
  Best regards and thanks again,
  
  Stefan
  
  Reply
  
  Citation: https://doi.org/10.5194/egusphere-2026-2060-AC1
  - RC2: 'Reply on AC1', Boris Gailleton, 26 Jun 2026 reply
    
    Please see the attached pdf for my response (I wanted to attach a figure).
    
    Reply
    
    Citation: https://doi.org/10.5194/egusphere-2026-2060-RC2

Stefan Hergarten and Menno Fraters

Model code and software

Horizontal displacement and deformation in large-scale landform evolution models Stefan Hergarten https://doi.org/10.5281/zenodo.19365029

Stefan Hergarten and Menno Fraters

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Short summary

Coupled models of geodynamics and landform evolution are receiving growing interest. Such coupled models, however, typically involve strong horizontal movement and deformation of the crust, which likely cause problems if landform evolution is simulated on a fixed grid. These problems are discussed in this note and potential solutions are presented.


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