| 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.
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.