the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 27 Jan 2026
Linking ridge shapes to the ice thickness distribution via discrete element simulations
Abstract. Ridges significantly increase the sea-ice thickness compared to the level ice surrounding them. In continuum sea-ice models, this increase is either represented by an increase in mean ice thickness or by changes in the ice thickness distribution (ITD). The implementation of ITDs requires a sub-grid parametrization of ridging by using a redistribution scheme. In contrast, the discrete element method (DEM) enables explicit simulations of ridge formation process, including ice fragmentation into rubble and its subsequent redistribution to ridges. Here, we use a DEM model to simulate ridging across a sea ice domain of size 6 km x 6 km. The DEM simulations yield deformed ice cover with ridges of varying shapes, namely triangular and trapezoidal ridges; the trapezoidal ridges notably affect the ITD of the deformed ice cover by creating a bump in the ITD towards thicker ice. We find that the ITD of the deformed ice field from DEM simulations differs from those from the continuum model, that uses only mean thickness, and from two commonly used ridging functions within redistribution schemes used as sub-grid parametrizations. Further, we show how to formulate an analytical redistribution function that captures the effect of various ridge shapes and discuss when it could replace existing ridging schemes. Our results demonstrate that an improved representation of ridging is needed within continuum models to resolve ridges both with their depth and shape within the ITD, especially in high spatial resolutions. Additionally, we formulate open questions in need of answers to allow implementation of our new distribution of ridged ice into continuum models, which connect to the ridging process itself.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2025-6421', Anonymous Referee #1, 02 Mar 2026
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RC2: 'Comment on egusphere-2025-6421', Anonymous Referee #2, 14 Mar 2026
Dear Marek Muchow and co-authors, dear Qinghua Yang,
I am delighted to present my review of the manuscript “Linking ridge shapes to the ice thickness distribution via discrete element simulations”. I have enjoyed reading the study, and I have certainly learned a lot. I hope you find my (observational) perspective equally helpful, and I wish you the best of success in further refining the manuscript.
Best regards,
Referee #2
General comments
In this study, a Discrete Element Model is used to assess how triangular and trapezoidal sea ice ridges contribute to the ice thickness distribution (ITD). Results are compared to the output of a continuum sea ice model and two commonly used ridging functions. The study consequently suggests how the new learnings can be translated into an analytical redistribution function which captures the effect of ridge shapes on the ITD.
Overall, the manuscript gives relevant, interesting and new insights into modeling of sea ice ridges, which makes it well suited for publication in The Cryosphere. The manuscript is for the most part well structured and clearly presents the new findings. The English language is used in a proficient and efficient way, with only minor mistakes or typos. I do, however, have some criticism regarding the interpretations and conclusions of the study.
I recommend publishing the manuscript once the issues mentioned below have been addressed, in particular the following three:
1. There are several instances throughout the manuscript where you claim that the DEM improves representation of ridges compared to neXtSIM or the two ridging functions. This claim is missing its foundation because it is never defined what the baseline for such an improvement would be. How do you determine which is better than the other? The hypothesis that the much more realistic and detailed model setup and the ability to simulate different ridge shapes improves overall ridge representation is of course very valid and logical, but needs to be proven or at least properly discussed.
Are the results from the DEM closer to observations (see also my comment #2)?
Similarly, how do you determine that the new redistribution function n(h) is “better” than previous solutions?
This comment relates to several instances in the paper:
- Line 11-12: “Our results demonstrate that an improved representation of ridging is needed within continuum models …”: In the abstract, such a statement is acceptable if backed by argumentation in the paper’s results or discussion and a definition of what “needed” means. However, I don’t think this is sufficiently demonstrated in the manuscript.
- Line 193-195, in particular the words “underestimate” and “benefits” are evaluative without an argumentation and with the implicit assumption that HiDEM / the sub-grid parameterizations do a better job. Please convince me that they do.
- Implicitly, the same assumption is made in lines 340-349 and 359-361.
2. It is clear that an extensive comparison of your results to observations is neither feasible nor the point of your study. However, the manuscript in its current state feels quite detached from reality and would strongly benefit from at least a better qualitative overview of the difference between simulated and observed ridges and ITDs. A suitable partner for comparison would be observations of landfast young / first-year ice which is constrained in its motion and has not experienced many deformation events.
Such a comparison would be useful in several instances throughout the paper:
- Figure 3 shows the cross-sectional shapes of the simulated ridges. How close are these to observed ridge shapes?
- The beginning of section 3.1 / lines 153-160: Please describe qualitatively which of those better meet your expectations in relation to observations. Optionally, you could include an comparable observational example, e.g. from airborne laser scanning, in Fig. 2.
- Line 256: “Field observations have shown …”: This is a sentence that I would welcome much earlier in the manuscript, ideally already in the introduction, or where you introduce triangular and trapezoidal ridges.
- Section 4.1 includes some limited comparison to observations. You bring up the difference between the modeled ITD and observations and give reasons for these differences. However, the comparison is rather shallow and I am missing an evaluation of your result against observations that are better comparable than the mentioned studies:
- If possible, please show the ITD of a comparable observational study in one of your figures (this could, for example, be the one by von Albedyll which you already mention in line 285) and describe and explain the difference using this direct comparison.
- Please also have a look at Fig. 1b in Sumata et al. (2023). Their ITDs in the Fram Strait seem to be closer to your results than the mentioned negative exponential function in other observations. Can you comment on this?
- In discussion and conclusion, I invite you to suggest that a more extensive and also quantitative comparison to observations would be an interesting topic for further research (if you agree).
3. The introduction: The pure summary of current knowledge is relatively short (lines 17-45). I don’t mind this, as long as relevant background information is addressed and literature is cited, which is well done here. Starting from line 46, further background information, including important references, is mixed with a short summary of the current study. Although a bit unconventional, I enjoy the way this introduces the reader to the research done instead of simply listing one reference after the other. However, this section now also contains results and interpretations which do not belong in the introduction because they are based on methodology, assumptions and limitations that have not been mentioned in details so far and thus cannot be comprehended by the reader at this point. Rather than mentioning these results and interpretations, I encourage you to focus more clearly on your motivation and the aim of your study here, which are both only implicitly mentioned. Hypotheses and a description of how you will approach the research questions are of course welcome in the introduction, as well as the manuscript outline (currently lines 70-76).
In particular, I am referring to the following passages:
- Lines 58-63: Presenting results.
- Lines 63-66: Interpretation and conclusions (see also comment #1).
- Lines 68-69: Interpretation, but this could easily be converted into motivation.
Specific comments
- Line 19-20 “Overall, ridging has a higher influence on the ice volume via the thickness compared to the ice area.” I understand what you mean, but it is not very clearly phrased, especially as it sits right at the beginning of the introduction, where the reader is just getting started and warming up their brain. Please consider rephrasing.
- Line 45/131: “… assuming an exponential distribution of smaller ridges compared to deeper ridges”: In one of these instances, please mention that such an exponential decay is known from observations. The references you use later (line 280) are also sufficient if mentioned here. There are of course many more, see e.g. Wadhams (1980) or more recently Rabenstein et al. (2010), ...
- Line 51: “ … demonstrated its capability to simulate …” : Demonstrated how? In relation to observations? This is very interesting because it has the potential to justify some of your later statements, e.g., regarding your comparison of HiDEM and neXtSIM, and the discussion on which one performs better in certain tasks, see General comment #1.
- Line 103: “Nevertheless, …” I suggest to change this to “Therefore, …” or something alike. Averaging is well justifiable because the ITDs are so similar.
- Line 111: “ … via a mean thickness”: Please explain how this mean thickness is defined (over what area, …?) I would appreciate some more detail on how this works.
- Line 136: “the sum of A(h)”: Why do you here write “the sum of A(h)”, and later just “A(h)” when in both times you refer to the same area?
- Line 143-144: Please include a definition for the empirical thickness H*, either its physical interpretation (if possible), or describe what role it plays. In this paragraph and throughout the manuscript, please make sure to either consistently indicate the unit for H* (m, according to you and Hibler, 1980) or explain if you have a good reason to omit it. The unit is missing in lines 144, 239 and 309, and possibly other instances.
- Line 160: “the number of ridges … increases with decreasing hi”: Can you clarify whether you mean the total number of ridges or just the number of ridges above a certain threshold (e.g. 3hi)? In Figure 2 it looks like the total number of ridges is pretty similar across all hi.
- Line 162: “y = 1.0, 2.0, …”: Do you mean x? Same for “along the y-direction” in the caption of Fig. 3.
- Line 170: “Overall, the frequency of trapezoidal ridges increases with hi.”: In Figure 3 it looks like the total frequency of trapezoidal ridges decreases with increasing hi. Do you mean that the fraction of ridges with trapezoidal shape increases?
- Figure 4: Can you please clarify whether this shows the ITD based on area or volume?
- Line 179-181 “HI80 is, ….”: I don’t understand this sentence. Please rephrase, maybe split it up.
- Line 203-204: And this constant value is the value kmax has at t=2h? Please mention explicitly.
- Figure 7: Even though this is just a conceptual figure, please label the axes to give the reader an easy access to the understanding of their physical meaning.
- Line 215-217: “Thus, this boundary…” Please consider rephrasing this sentence to make it better understandable.
- Line 225: “a higher α results in a more pronounced "bump", signaling more trapezoidal ridges”: Shouldn’t it be the other way around?
- Line 282: “ice deformation that has occurred in multiple stages over longer periods”: Obviously it is much beyond the scope of this manuscript to implement anything alike, but can you comment on the feasibility of modeling such multiple ridging events over longer periods?
- Line 321: “the model developed by Salganik et al. (2020)”: Please remind us what this model does and why it is relevant here. I am also not sure if “therefore” is the right word here, this depends on what exactly you would like to say.
Technical corrections
- Line 50: “code” → model
- Line 71: insert “the” before “sea ice redistribution model”
- Line 90: “being due” → just “due”?
- Caption Fig 3: “… ranged of the axis” → range of the axes
- Line 186: “more smaller ridges” → a higher density of smaller ridges
- Line 191: insert “the” before HiDEM
- Caption Fig. 6: “Therefore” is the wrong word here. Maybe you mean something like “For this purpose, …”?
- Caption Fig. 6, last sentence: “limi” → limit
- Caption Fig. 7, first sentence: “the the” → the
- Line 240: Remove “thus”
- Line 249: “highlighting” → suggesting, or similar
- Line 256: “a triangular shapes” → decide for “a triangular shape” or “triangular shapes”. Same for trapezoidal shapes later in the sentence.
- Line 258: Insert “a” before “variety”
- Line 259: “DEM” → DEMs
- Line 260: Insert “The” before “number”
- Line 305: Insert “a”/”the” before “distribution”
- Line 305: Remove “Thus”
- Line 334: “land fast” → landfast
- Line 337: “are” → being. Or split sentences and remove “with” in line 336.
- Line 362: “… captures the triangular and trapezoidal ridge shapes” → captures the effect of triangular …
- Line 366: remove “of”
- Line 384 “compared to” → consider “as opposed to” or similar
- Line 455: Please include DOI https://doi.org/10.1175/1520-0493(1980)108<1943:MAVTSI>2.0.CO;2
- Please review the use of commas throughout the manuscript. I am not good with commas either, but I suspect there are mistakes in lines 256, 260, 375, 382, 404, 405 (possibly more).
References:
Rabenstein, L., Hendricks, S., Martin, T., Pfaffhuber, A., and Haas, C.: Thickness and surface‐properties of different sea‐ice regimes within the Arctic Trans Polar Drift: Data from summers 2001, 2004 and 2007, Journal of Geophysical Research: Oceans, 115, 2009JC005846, https://doi.org/10.1029/2009JC005846, 2010.
Sumata, H., de Steur, L., Divine, D.V. et al. Regime shift in Arctic Ocean sea ice thickness. Nature 615, 443–449 (2023). https://doi.org/10.1038/s41586-022-05686-x
Wadhams, P.: A Comparison of Sonar and Laser Profiles along Corresponding Tracks in the Arctic Ocean, Sea Ice Processes and Models, University of Washington Press, Seattle, Washington, pp. 283–299, 1980.
Citation: https://doi.org/10.5194/egusphere-2025-6421-RC2
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- 1
General Comments
This manuscript aims to use a Discrete Element Model (DEM) to explore how the formation of sea ice ridges impacts the sea ice thickness distribution. The manuscript then compares this to how different continuum sea ice models simulate changes in the ice thickness distribution resulting from ridging. In addition, the results from the DEM are used to motivate a new parameterisation of sea ice ridging to use in continuum models. The research presented here is an important and valuable contribution to both improving understanding of an important sea ice process and sea ice model development. In particular, the improved representation of ridge formation and ice thickness distribution in sea ice models has clear potential impacts on sea ice rheology, momentum exchange between the sea ice-ocean and atmosphere, and the sea ice mass-balance. The use of a DEM to supplement observations of sea ice, develop new physical understanding of sea ice, and motivate new parametrisations for continuum models is something that has been applied successfully before e.g. Wilchinsky et al., (2010), Tsamados et al. (2013). Whilst previous studies have applied DEMs to explore sea ice ridging (as acknowledged in the manuscript), the novelty here emerges from the use of a three-dimensional DEM and the application of the results to motivate a new ridging parameterisation.
The manuscript is mostly well written with a sensible structure, however at times I find the manuscript difficult to follow with key terms not defined and explanations missing. The figures are generally of a good quality but the axes could be labelled more clearly. The manuscript makes an effective use of references both in the discussion and conclusions to provide a clear context for this research. The manuscript also reaches a clear set of conclusions.
I do have some concerns about the methodology. This relates in particular to the limited number of simulations used in the analysis despite the stochastic method to introduce inhomogeneity into the model and the choice to only consider ridging under uniaxial convergent forcing. Addressing the former point at least requires an additional figure and ideally additional simulations to demonstrate limited model sensitivity to how the inhomogeneities are introduced. Whilst the decision to focus on uniaxial convergence to simplify the analysis and reduce the number of simulations can be justified, there does need to be more discussion about the limitations of this for the conclusions reached and ridging parameterisation proposed. In addition, the manuscript would be benefit from a more detailed description and comparison of the different models being used in this study.
Overall, I believe that this paper is within the scope of the journal and makes a valuable contribution to the literature. Whilst I do have significant concerns about the manuscript, I believe the paper should be accepted, provided these concerns can be adequately addressed. I provide a more detailed explanation regarding the points made above in the specific comments below.
Specific Comments
Abstract and Introduction
P1L1: The abstract would benefit from an introductory sentence explaining what a ridge is in the context of sea ice.
P2L29: ‘observed fields (10 km)’. It is not clear what the value in the brackets refers to, and it should also be clarified what the term ‘observed fields’ is referring to here.
P2L46-47: What is meant by the term ‘detailed’ in this section? This term should be replaced by something along the lines of ‘explicitly resolve ridge formation’ or ‘with greater physical fidelity’.
P2L54-P3L69: It is unconventional to summarise the key results in the introduction. This section does an effective job of explaining why this research is both important and novel, but the authors could consider presenting these points in the context of what the manuscript aims to do rather than the results.
Methods
P3L78-L86: The methods section would be improved by briefly summarising what the section will cover (e.g. a brief summary of the models that will be used). The two paragraphs currently at the start of the ‘Methods’ section could be a sub-section labelled ‘Simulation Setup’ or equivalent.
P3L88-P4L92: I find the description of the DEM insufficient, given the purpose of the manuscript is to use this DEM to produce insights into ridge formation to apply in continuum models. In particular, an overview should be provided of the equations governing beam failure and ridge formation.
P4L96-L98: What is the significance of each 10 m x 10 m grid cell? Is it that each grid cell is assigned a different single value between 0.4 and 1.0 to determine how many beams were removed? If so, this should be made clearer. Is there any physical justification for this approach to introducing inhomogeneity over other potential approaches (e.g. modifications to beam properties such as the failure threshold)? Were any sensitivity studies conducted to explore the impact of the different choices made here (e.g. changing the size of the grid cells, or the value of the beam probability outside the observation area)?
P4L99-L103: What was the computational cost of these simulations? Given the stochastic method used to introduce inhomogeneity, the authors should ideally produce a small ensemble (of around 10 members) for each value of hi tested for subsequent analysis.
P4L102-L104: ‘the resulting ITDs were practically identical.’. There should be a plot included in the manuscript to compare the results for simulations with the same hi (for at least one value of hi) to demonstrate this point.
P4L105: What was the motivation for choosing 4hi (rather than a fixed resolution for all values of hi)? Could a value of 50 m not have been used, to ensure consistency with neXtSIM output?
P5L107 – L112: As with the DEM, more details about the model are required here. In particular it would be helpful to describe the equations that govern the mean thickness in the model. In addition, there needs to be a summary of the parameter choices used for both neXtSIM and the DEM that are relevant to this study such as parameters that determine ice strength and failure thresholds. This should also include some discussion about whether the parameter choices used across the two models are consistent.
P5L114-L116: The variability introduced to neXtSIM to account for non-uniform ice strength was between 50 – 100 % of the maximum value, whereas in the DEM it is between 40 % - 100 %. What is the reason for this difference?
P6L134: Whilst this is covered in the Appendix, it would be helpful to explain how Δ is calculated here.
P6L143-150: As per previous comments, it would be helpful to discuss whether the parameter choices made for each method of simulating ridge formation are equivalent. I note that the two ridging functions have been compared in this way, but not against the DEM or neXtSIM.
Results
Figure 2: Given that ridging will generally occur at locations of joint failure, it would be interesting to compare the ridging pattern produced here to other studies that have used a DEM to simulate sea ice failure under uniaxial convergence e.g. Wilchinsky et al. (2010).
P7L157: ‘a task for future studies’. Given the motivation of this study is to motivate a parameterisation of ridging for use in continuum models of sea ice, there should be some discussion of how more complex forcing scenarios may impact the results and the proposed parameterisation.
P7L162-P8L163: What method was used to classify the ridges into different shapes?
P8L172: ‘bell curved-shaped’. I do not agree with this description of the ITD for HiDEM shown in Fig. 4. There is a clear asymmetry in the peak being described here. In addition, there are very sharp peaks in the distribution for a hi of 2.0 m, which seem worthy of note and discussion.
Figure 4: The truncation of the plot on the y-axis means it is not possible to see the full shape of the ITD for neXtSIM. Could these plots be arranged over two rows to allow more vertical space for each plot?
P12L209-L221: I find the explanation here tricky to follow. Formal definitions should be provided for atri and atra (and it would help to label both axes in Fig. 7). In addition, it would be helpful to provide a more detailed explanation of how the boundaries between the different parts have been determined (particularly since they are calculated from other parameters and not uniquely determined). The manuscript should also refer to previous sections to explain the decisions that have been made to determine the parameterisation.
P13L224: Atri and Atra are not defined.
P13L222-L233: The overall derivation here is difficult to follow as there are several steps implicit in the text. It would be helpful to provide a full derivation in the Appendix.
P14L240: Why were different values of α selected for each hi?
P14L243: ‘closer fit than the HI80 approach’. The HI80 approach seems to perform better for hi = 2.0 m.
Discussion, Conclusions, and Appendix
P16L324-P17L339: Whilst there are clear benefits to splitting g(h) into two separate distributions, there are also disadvantages to this e.g. increase in model complexity, potential conflict with other model parameterisations such as the floe size-thickness distribution used in ICEPACK (Hunke et al., 2024). Could the new ridging parameterisation be implemented within the existing ice-thickness distribution, as per previous ridging parameterisations?
P17L366-P18L368: Here and / or in the discussion, there should be some acknowledgement and discussion of the limitations of the methodology used in this study (e.g. from only considering uniaxial convergence) and the potential implications of this for the proposed parameterisation.
Appendix A: The information presented in Appendix A is not particularly technical or long, so it is not clear why it cannot be included in Sect. 2.1.
Technical Comments
P1L13: ‘We formulate open questions in need of answers to allow implementation of’. I find this phrasing awkward. Consider something along the lines of, ‘We discuss remaining challenges in implementing’.
P3L73: ‘with a notion on’. I do not understand the phrasing used here. Would ‘accounting for’ be better here?
P4L103: ‘below’. It would be better to reference the section or figure here.
P6L131-L132: Sentence beginning ‘Overall’. There is a missing ‘is’ in this sentence.
P8L167: ‘Large’. Should be ‘larger’.
P9L180: ‘triangular, that’. There is a missing ‘and’ here.
P12L210: Here and elsewhere, Figure 7 should be Fig. 7 unless at the start of a sentence (see journal submission guidelines). The same applies to any uses of Equation (Eq.) and Section (Sect.). In addition, the number after Equation should be in brackets e.g. Equation 1 on L218 should be Eq. (1).
P14L261: Missing ‘The’ at start of sentence.
P14L262: Sentence from ‘This observation’ to ‘elastic foundation’. I find this sentence difficult to understand. Consider rephrasing.
P17L366: ‘an interesting next steps’. There is a typo here.
Figure 5 caption: ‘Development of the 99th percentile is P99 of the ice thickness’. There is a typo / error here.
Figure 6: The units are missing on the y-axis. ‘limi’ should be ‘limit’ in the caption. In addition, here and in other figures, make sure axes are clearly defined either within the figure or in the caption.
References
Hunke, E., Allard, R., Bailey, D. A., Blain, P., Craig, A., Dupont, F., DuVivier, A., Grumbine, R., Hebert, D., Holland, M., Jeffery, N., Lemieux, J.-F., Osinski, R., Rasmussen, T., Ribergaard, M., Roach, L., Roberts, A., Steketee, A., Turner, M., and Winton, M.: CICEconsortium/Icepack: Icepack 1.5.0, Zenodo, https://doi.org/10.5281/zenodo.14188409, 2024.
Tsamados, M., Feltham, D. L., and Wilchinsky, A. V.: Impact of a new anisotropic rheology on simulations of Arctic sea ice, Journal of Geophysical Research: Oceans, 118, 91–107, https://doi.org/https://doi.org/10.1029/2012JC007990, 2013.
Wilchinsky, A. V., Feltham, D. L., and Hopkins, M. A.: Effect of shear rupture on aggregate scale formation in sea ice, Journal of Geophysical Research: Oceans, 115, https://doi.org/https://doi.org/10.1029/2009JC006043, 2010.