We thank the reviewer for their positive assessment, interest in our work, and endorsement of our benchmarks. We have extensively revised and restructured the paper in response to the reviewer's comments, and in particular have shortened it considerably from 49 pages to 30 pages by removing unnecessary details and redundant results.

We appreciate the constructive line-by-line feedback, which we address below.

Comment 1: Terminology Confusion ("tracers" vs "markers")

"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... As such I suggest a change of names: markers for 'volume' points and tracers for additional surface points."

Response: We completely agree with the reviewer that our original choice of terminology caused unnecessary confusion, given the established naming conventions of the marker-in-cell technique. We have updated the text throughout the entire manuscript to follow the reviewer’s suggestion. The standard full-domain Lagrangian points are now consistently referred to as "volumetric markers". The specialized points introduced for the free surface are now referred to as "surface markers".

Comment 2: l83: calculating -> calculated

Response: Section removed

Comment 3: l88: C -> C's

Response: Section removed

Comment 4: Section 2.2: Concern about material entrainment

"Although the topic is addressed later with the surface markers techniques in section 4.2, I was wondering when reading this section about entrainment?"

Response: Section modified, material entrainment is addressed later

Comment 5: l111: Definition of 'surface mesh refinement' in FDM/FVM codes

"What is meant by 'surface mesh refinement' in the context of FDM codes here?"

Response: Section removed, but the refinement in question was referring to decreased vertical grid spacing near the surface.

Comment 6: l130: Reference for the 'divergence-free quadratic spline method'

"I think a reference for the 'divergence-free quadratic spline method' in this context would be welcome."

Response: We have added the appropriate references for this method, citing Gerya et al. (2021).

Comment 7: l157: Specify page/chapter for Gerya's textbook

Response: We have updated this citation to specify Chapter 18 of "Introduction to Numerical Geodynamic Modelling" (Gerya, 2019).

Comment 8: l169, 257, 265, 316, 321: Equation formatting and sentence structure

"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 subsequent lines."

Response: We have systematically updated the punctuation of all equations highlighted by the reviewer (as well as a general sweep of the rest of the manuscript). Mathematical equations are now treated as integral parts of the text, with correct trailing commas or periods, and subsequent explanatory lines properly starting with a lowercase "where".

Comment 9: l224: Reference for the Yin-Yang grid

Response: We have added a reference to Tackley (2008), which presents the explicit implementation and geometry of the Yin-Yang grid layout within StagYY.

Comment 10: l255: 'is' -> in

Response: Section removed

Comment 11: l376-380: Paragraph on numerical diffusion

"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."

Response: We have revised this paragraph to make its operational purpose explicitly clear. We now clarify that global reinitialisation inherently introduces a small amount of numerical diffusion, which functions as an implicit smoothing mechanism. We state clearly that this is actively used and implemented automatically within our bilinear method to maintain surface stability, and contrast it with the integral method which lacks this implicit diffusion and therefore requires explicit slope-limiting/smoothing.

Comment 12: l542: Semi-colon replacement

"The semi-colon should probably replaced by a comma + i.e. ?"

Response: Corrected as suggested to improve readability.

Comment 13: l670: Word duplication

"The word marker appears twice."

Response: We fixed this typographical error.

Comment 14: Section 7.3.1: Citations for geodynamical codes coupled to surface processes

"I think that some key publications of geodynamical codes coupled to surface processes could be cited in the context of this paragraph."

Response: We agree that expanding this context strengthens the section. We have incorporated key community references demonstrating the direct coupling of deep mantle/crustal geodynamics with surface process models (e.g., Salles et al., 2018; Yuan et al., 2019; Bahadori et al., 2022).

Comment 15: Reference to the DOUAR code

"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."

Response: We have added an explicit discussion of DOUAR (Braun et al., 2008) in our introduction of alternative techniques, noting that our 3D integral method builds upon a similar conceptual framework of a Delaunay-triangulated surface embedded within a fixed grid environment.

Sincerely,

Timothy Stephen Gray, Paul James Tackley, and Taras Gerya

Thank you for taking the time to review this paper. We have substantially restructured and revised the paper in response to the reviews, and have addressed your specific concerns as follows:

Major Concerns:

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Comment 1: Paper Length, Visual Bloat, and Structure

"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... by moving technical details to appendix, and trimming text. b) Work on the structure."

Response: We have significantly changed the paper's structure to trim it down to a more suitable size. Unnecessary introductory content has been trimmed, the original section 2 has been mostly removed, and standard mathematical derivations have been condensed or removed, leaving only the key content in the body.

Comment 2: Clarification of Novelty vs. Implementation

"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..."

Response: We have revised the Introduction and Discussion to explicitly demarcate our conceptual contributions from code-specific mechanics:

- Methodological Novelty: The novelty lies in the systematic development and rigorous comparative benchmarking of two distinct surface-to-grid mapping topologies (the node-mapped bilinear method vs. the direct unstructured Delaunay triangulation/integral method) designed specifically for preserving a sharp, sub-grid interface within parallelised Eulerian codes experiencing severe geodynamic deformation (e.g., localized subduction over geological timescales).

- Implementation Focus: We clarify how this is mapped efficiently onto a staggered-grid, parallelised finite-volume framework using MPI ghost/overlap regions.

- Literature Expansion: We have expanded our references to ground this work within a broader community context, referencing alternative formulations in other codes (such as LaMEM, DOUAR, and standard level-set/VOF models), highlighting exactly what distinct advantages our decoupled surface marker framework introduces.

Specific Comments:

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Comment 3: Line 24: Explain "sticky air"

Response: We have added a clear definition in the introduction: "sticky air" is a numerical approximation consisting of a low-viscosity, low-density layer placed above the solid planet boundary to simulate a true free surface condition on a rigid Eulerian grid.

Comment 4: Line 74: Novelty of the Bilinear and Integral methods

"“the bilinear method and the integral method, which differ in how they transfer surface information to the underlying grid” are these new methods?"

Response: Disussion of the novelty of these methods has been revised.

Comment 5: Overview of alternative approaches (e.g., Height Functions, Grid Deformation)

"Why is a height function formulation with simple vertical grid deformation not used in this field?"

Response: Such approaches have been used, e.g. by Duretz et al. 2016 and Kaus et al. 2016 - references have been added accordingly.

Comment 6: Line 90: Clarify vertical sub-grid resolution

"Clarify that you mean resolution in the vertical, not 100 m resolution in the horizontal direction..."

Response: We have rephrased this text to explicitly specify "vertical sub-grid level topographic resolution" to avoid any horizontal ambiguity.

Comment 7: Figure 1: Visual rotation

Response: This figure has been removed.

Comment 8: Figure 3: Discrepancy, noise, and convergence in the marker-in-cell method

"It looks like the surface topography itself is very different, not just that it is noisy, and it does not seem to converge?. Can you comment on that?"

Response: The marker-in-cell method is inherently noisy due to the randomness associated with the markers. As can be seen in higher marker-per-cell cases such as 500 markers per cell test, it does indeed slowly converge to the smooth results we see later on. In any case, this figure has been removed and its content moved to the results section.

Comment 9: Figures 4 and 5: Condensation

Response: Figure 4 and 5 have been removed.

Comment 10: Line 140: Prior work context

"Is this your method? What similar things has been done in other codes before? Some general introduction text is needed here before all the subsections."

Response: We have added introductory context at the head of this section, mapping out the architecture of our tracking routines and clearly cross-referencing how our approach relates to other's work such as LaMEM (Kaus et al., 2016) or Duretz et al. (2016).

Comment 11: Line 165 / Equations 5-6 / Moving equations to Appendix

"Since this is a method for which you have a reference, you can move some of the details to an appendix. Many of the following equations in section 3 are also standard can can be moved to appendix."

Response: We have eliminated many of the unnecessary standard equations, preferring to cite them.

Comment 12: Line 381: Reference to slope limiting

Response: This was a novel method - the section has in any case been restructured to hopefully better reflect this.

Comment 13: Line 392: Vague subsection title

Response: We have renamed this subsection to be much more descriptive of its technical content (e.g., updating it to clearly reflect the exact stabilisation or reinitialisation routine being detailed).

Comment 14: Line 420: Clarify "coupling" definition

Response: We have expanded the text to define exactly what coupling implies in this context: it denotes the algorithmic transfer of the sub-grid free-surface position to the mechanical Stokes solver to continuously modulate cell-averaged density and viscosity fields, ensuring that topography variations dynamically feed back into the model's driving buoyancies.

Comment 15: Figure 15: Structural location

Response: We have decided to retain the original location

Comment 16: Figures 22 & 24: Label accuracy

"Change 'topography with time' to 'maximum topography with time'"

Response: We have updated the figure captions to explicitly specify "maximum topography" rather than generic topography.

Comment 17: Line 503: Consolidating brief sections (6.1 & 6.5)

Response: We have removed these isolated, ultra-short sub-sections and integrated their contents directly into the introductory paragraphs of the Results section, framing our benchmarks with a cohesive rationale for why each numerical test was executed.

Comment 18: Section 6.2: Framing a fair comparison

"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."

Response: We have added an explanatory paragraph in Section 5 (Benchmarks) clarifying our comparative controls. To ensure a fair assessment, all tests are performed under identical spatial Eulerian grid resolutions, velocity fields, and time-stepping criteria. We explain how the volumetric marker densities were chosen to represent common community choices, demonstrating how our surface markers operate independently of full-domain marker counts to prove the scaling efficiency of our new approach.

Comment 19: Line 533: Value of the sticky air benchmark

"If the discrepancy could be due to the sticky air, what is the value of this benchmark? Clarify"

Response: We have added text clarifying the value of this benchmark. While the sticky air approximation is known to possess minor, predictable intrinsic limits, this test remains highly valuable because it is the primary benchmark utilized by the broader community (e.g., the Crameri et al., 2012a community comparison). Replicating this test allows us to validate that our tracking methods match established community results perfectly while highlighting the significant noise reduction our method achieves over standard marker-in-cell setups.

Comment 20: Figures 25, 30, 32, 34: Layout space reduction

Response: We have removed several figures and reduced the space taken by others to make for a less cluttered paper.

Comment 21: Expansion of the Reference List

"The reference list is quite short and with many papers written be the same authors."

Response: We have expanded the reference list where appropriate.

Sincerely,

Timothy Gray, Paul Tackley, and Taras Gerya