the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 23 Jan 2026
Lagrangian tracking methods applied to free surface boundaries in numerical geodynamic models
Abstract. A desirable characteristic of mantle convection models is the ability to determine surface topography evolution over time on a global scale. This capability is increasingly important due to growing interest in coupling planetary geodynamics with climate, landscape, habitats, and biological evolution systems. A common way of achieving this in numerical geodynamic models with a fixed Eulerian grid is through the implementation of a free-surface boundary condition using the sticky air method in conjunction with an appropriate method of tracking the free surface. Although existing methods for tracking the interface between the air and rock layers are available, they often struggle to provide high-resolution results on a global scale without incurring significant computational costs. We propose a method for representing surfaces directly using Lagrangian markers that can track the location of a free surface in 2D and 3D models and test it using the finite-volume mantle convection code StagYY. This approach offers a direct, high-resolution representation of the surface without the need for a large number of high-cost tracers throughout the model domain. Benchmarks demonstrate the effectiveness of this method compared to the commonly-used marker-in-cell method. The direct representation of the surface enables additional features such as the direct tracking of sea levels over time, potential for coupling with surface process models on the global scale, and enables the implementation of alternative discretisations of the Stokes equations to improve Stokes solver accuracy near the free surface boundary.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2025-6546', Anonymous Referee #1, 08 Mar 2026
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RC2: 'Comment on egusphere-2025-6546', Anonymous Referee #2, 13 Mar 2026
General comment:
This paper describes a method for representing free surfaces in mantle convection models and its implementation in the code StagYY. I appreciate that the method is thoroughly tested and the implementation seems carefully done. The new implementation appears to be more computational efficient which is of clear value. I have two major concerns that I think needs to be addressed before this paper is considered for publication:
1) The paper is very long (48 pages, 37 figures) and unstructured. It is therefore hard to get an overview of the paper. My suggestion is to
a) shorten the paper to <30 pages. This can partly be achieved by moving some of the technical details, basic equations and figures to appendix, and trimming the text. b) Work on the structure.The current structure now is:
1. Introduction2. "Limitations of the marker-in-cell method". This still is written like an introduction but there are numerical results mixed in
3 "Surface tracking" - it is not clear from the start of this subsection that this is the novelty. Thes subsections here have no hierachy.
4 "Stabilisation" -seems like this should be a subsection to surface tracking?
5. "Coupling". Write what you mean by coupling already in the title. Coupling between... ?
6. "Results" - Usually results would come after a section explaining what the experiment is.
7. "Summary", but it is actually summary, discussion and outlook. This is way to long and it is hard to get an overview.
Overall the subsection-structure must have a clearer hierachy and more informative headings. Numerical results are scattered throughout the paper which is unorthodox but could perhaps work but only if it is clearer from the beginning what the core numerical experiments will be and what experiments are used to explain the topic.
2) The level of novelty is not clear. How much of the novelty lies in the methodology itself, and how much lies only in its implementation in this particular code and the thorough testing? This must be clarified, and more references to similar earlier work is needed (the reference list is relatively short). What is special with the implemention in this particular code?
Specific comments.Line 24, explain what sticky air is to the reader
Line 74; "the bilinear method and the
integral method, which differ in how they transfer surface information to the underlying grid" are these new methods?The introduction is lacking an overview of similar methods introduced in other codes.
Why is a height function formulation with simple vertical grid deformation not used in this field?
Line 90: Clarify that you mean resolution in the vertical, not 100 m resolution in the horizontal direction: "While this method enables vertical sub-grid level topographic resolution, doing so accurately requires large numbers of
tracers per cell. For example, providing 100 m topography sensitivity on a grid with a vertical resolution of 10 km would require at least 100 tracers per cell."Figure 1: It would help if you rotate the figure so that the air is on top of the rock
Figure 3: It looks like the surface topography it self is very different, not just that it is noisy, and it does not seem to converge?. Can you commment on that?
Figure 4 and 5 are very large can be put together in one figure.
Line 140: is this your method? What similar things has been done in other codes before?
Line 140: Some general introduction text is needed here before all the subsections.
Line 165 - Since this is a method for which you have a reference, you can move some of the details to an appendix.
Equation 5- 6: Is it necesssary to write out these formulas? Can they be moved to appendix? Many of the following equations in section 3 are also standard can can be moved to appendix.
Line 381: Add some reference to slope limiting
Line 392 - The name of this subsection is very general
Line 420 - Explain exactly what you mean by coupling.Figure 15 - Should this be in the result section?
Figure 22: Change "topography with time" to "maximum topography with time"
Line 503: Is section 6.1 necessary? it is very short. Same with section 6.5 If you want to have summaries/overview, I think they should contain a reasoning for why you do all these tests.
Section 6.2. How do you choose the number of tracers to make a fair comparison? In general I would appreciate if you clarify how you ensure your experiments constitute a fair comparison.
Line 533 If the discrepency could be due to the sticky air, what is the value of this benchmark? Clarify
Figure 24: Topography -> maximum topography?
Figure 25: This figure takes too much space
Figure 30,32,34 Also takes too much space
The reference list is quite short and with many papers written be the same authors.
Citation: https://doi.org/10.5194/egusphere-2025-6546-RC2
Data sets
StagFreeSurface (StagFS): A testbed for free-surface methods in geodynamical simulations using the staggered-grid finite volume (difference) discretization. Paul Tackley and Timothy Gray https://doi.org/10.5281/zenodo.18096249
Model code and software
StagFreeSurface (StagFS): A testbed for free-surface methods in geodynamical simulations using the staggered-grid finite volume (difference) discretization. Paul Tackley and Timothy Gray https://doi.org/10.5281/zenodo.18096249
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I have read with great interest the manuscript "Lagrangian tracking methods applied to free surface boundaries in numerical geodynamic models" by Gray et al. The authors present two new and/or updated methods to allow for more accurate tracking of the free surface atop geodynamical models.
I found the article well written and structured and the figures clear and helpful. It is a detailed methodological paper which showcases many results obtained on benchmarks which I find convincing. As such I recommend publication with minor modifications.
I hereafter list all my comments:
- At line 35 we read: ``For clarity, the Lagrangian points already implemented in StagYY and distributed throughout the model domain are hereafter referred to as "tracers".''
I find this choice rather unfortunate: the standard method of tracking materials/fluids inside the entire domain in these codes is called the marker-in-cell technique (e.g. Gerya & Yuen, 2003, 2007) but in this work the authors choose to call such Lagrangian points 'tracers' while referring dozens of times in the text to the marker-in-cell method.
More confusion is added in my opinion through the choice of naming the additional set of points introduced to specifically track the surface 'markers' (line 37: ``In contrast, Lagrangian points introduced in this study specifically for tracking the free surface are referred to as “surface markers”.''
The next sentences at line 39 & 47 illustrate my point: ``In StagYY, as in many other codes, the standard technique for interface tracking employs Lagrangian tracers using a marker-in-cell method'' ; ``Using the marker-in-cell method, finer resolution can be achieved trivially by increasing the number of tracers.''
As such I suggest a change of names: markers for 'volume' points and tracers for additional surface points.
- l83: calculating -> calculated
- l88: C -> $C$'s
- Section 2.2: although the topic is addressed later with the surface markers techniques in section 4.2, I was wondering when reading this section about entrainment?
- l111: what is meant by 'surface mesh refinement' in the context of FDM codes here?
- l130: I think a reference for the 'divergence-free quadratic spline method' in this context would be welcome.
- l157: when citing Gerya's book please specify page or chapter
- l169: this is not a full sentence, I think a comma should be inserted in Eq 2, and the line should start with 'where ...'. Same at line 316 after Eq 16, line 321 after Eq 18, line 257 after Eq 11, line 265 after Eq 12.
- l224: the yin-yang grid is here mentioned but it has not been mentioned before. Most readers will know what it pertains to, but a ref could be added.
- l255: 'is' -> in
- l376-380: I am not sure of the point of the paragraph. Diffusion is mentioned, pros and cons are mentioned, but it is not clear to me whether this is implemented/used.
- l542: the semi-colon should probably replaced by a comma + i.e. ?
- l670: the word marker appears twice
- section 7.3.1: I think that some key publications of geodynamical codes coupled to surface processes could be cited in the context of this paragraph.
- somewhere in the text a reference to DOUAR (Braun et al, PEPI 171, 2008) should be added since the free surface was also tracked by means of a Delaunay-triangulated surface embedded in a Eulerian mesh.