the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 11 May 2026
Consistent ridging and opening coefficients for multi-category sea ice models with modified viscous-plastic rheologies
Abstract. In multi-thickness category sea ice models, subgrid-scale ridging and the opening of leads are represented by a redistribution function. This function modifies the thickness distribution based on grid-scale strain rates. There is a physical link between sea ice rheology and redistribution by assuming that the work done by internal stresses in deforming sea ice is equal to the change in potential energy and frictional loss during the formation of ridges. Hence, modifications of the rheology require changes to the redistribution function to be consistent. For the special case of an elliptical yield curve and a non-normal flow rule, associated consistent ridging and opening coefficients can be formulated such that they reduce to the standard ones in the case of a normal flow rule. It is further demonstrated that the coefficients are independent of biaxial tensile strength. Satisfying specific criteria for the yield curve and plastic potential aspect ratios ensures that the ridging and opening coefficients are bounded by 0 and 1.
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2026-1362', Anonymous Referee #1, 25 Jun 2026
- AC1: 'Reply on RC1', J.-F. Lemieux, 01 Oct 2026
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RC2: 'Comment on egusphere-2026-1362', Anonymous Referee #2, 19 Sep 2026
This paper presents a correction to derivation of ridging coefficient for the elliptical yeild curve with non-normal flow rule that was presented Lemieux et al. (2025), also published in The Cryosphere.
This paper provides an errata to a previously published paper, however it has more details in the introduction which make me feel the authors want the paper to stand on it's own. Personally: I believe this paper should stand on its own. Whether this is an errata, or a stand alone paper, the introduction and discussion need to be tailored better to the results and should motivate why the model development work was done. There are details missing as to why one might want to use a non-linear flow rule with the elliptical rheology, what the new formulation provides for people using these models, and the context for why you only considered elliptical yield curves allowing variable eccentricity to describe the yield curve and a plastic potential that does not have to follow this. You can, of course, refer to the earlier Ringeisen et al. (2021) paper, but some context is needed in the introduction. Perhaps echo the point made in the 2025 paper that using an elliptical plastic potential requires only small modification to zeta and Delta to implement a non-normal flow rule. I can see that this is attractive to those who do not want to code up different rheological shapes and deal with the issues of convergence that other yield curve shapes can bring.
In the discussion it would really help to draw out for the reader why flexibility is needed in defining the flow rule. How would you use the flexibility this flow rule gives to obtain realistic results? You have made a start at understanding the sensitivity of the ice thickness to variations in e_F and e_G, what does this physically mean and how can someone use this information to tune a model? Here I would focus on the discussion of opening vs ridging and why it is so important to get this right, and how one can adjust this behaviour with this model. I realise we may not have data to validate this, but the sensitivity study it itself is useful information.
As this paper is an errata, it should be mentioned if the results in the 2025 paper are still valid. The 2025 paper focuses on LKF intersection angle. In this paper I appreciate your focus on ice thickness distribution, and sensitivity of this to the error and eccentricity of the plastic potential. It is apparent that the LKFs also change (figure 4, you can see changes in thickness along linear features), does this impact your previous reported result about LKF intersection angles for the idealised cases or pan-Arctic run? If there is no impact, that is also important to point out.
I very much appreciate the additional context this paper brings to the use of non-normal flow laws, and particularly mapping out ranges of realistic e_F and e_G. In my mind, this is worthy of a stand alone paper and so I hope the authors will take into account my concerns that the paper needs more context in the introduction and discussion.
Specific CommentsIn the interest of completeness, do you need to mention Babko's work to add rafting to the redistribution function. I realise that you may not want to do the math to include these terms, and your numerical model might not include rafting in it's redistribution function, but I feel you could be more comprehensive in the introduction.
Babko, O., Rothrock, D. A., & Maykut, G. A. (2002). Role of rafting in the mechanical redistribution of sea ice thickness. Journal of Geophysical Research: Oceans, 107(C8), 27-1.Make sure you have the correct reference for Lemieux et al. (2025), I think the preprint is referencing the discussions rather than the final version.
Line 248: "unridging" is actually not unrealistic. I have observed this. When ice diverges at the start of the melt season, ridges can fall apart and blocks create brash fields between flows. I agree that this is not something we consider in our models, and it is un-realistic in the ridging process. But I do want to point out that it is possible for pack ice to "extrude" into a thinner material in divergence. No need to fix anything here, unless you agree with me that we should not constrain future readers imagination as to how the thickness distribution evolves. I have no idea how large an effect this collapse of ridges is in the thickness distribution, it is only an anecdotal observation I (and others) have seen with our own eyes.
Figure 6: I find myself curious what the difference is between INC2a and CORRa
A concluding paragraph is missing from the paper. Perhaps you can use this to better provide context as to what your findings mean for Joe-modellor who might find themselves implementing your flow rule or questioning if they need to in order to represent ridging and lead opening correctly. Do we really need this added complexity in the EVP model? [Christian Haas: If the answer is no, we should still publish this paper because it is a theoretical advance that is very important to document].
Citation: https://doi.org/10.5194/egusphere-2026-1362-RC2 - AC2: 'Reply on RC2', J.-F. Lemieux, 01 Oct 2026
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This paper presents modifications to the standard ice thickness redistribution formulation to make it compatible with VP rheologies that use a non-normal flow rule. The authors also demonstrate that no modifications to the redistribution formulation are required when tensile strength is introduced. The paper is well written and organised, and the results are presented clearly and concisely. The work is interesting, if somewhat niche.
My only major concern is motivation and context for the results. Since the paper has been submitted for publication in The Cryosphere, I would expect a broader discussion of the potential impact these numerical changes may have on our understanding of the physical system. As it stands, the paper is a better fit for, e.g., GMD, where I would probably only have asked for minor revisions. However, with some not-very-major revisions, I think the paper would be fit for publication in The Cryosphere.
Introduction: The authors need to expand substantially on why using a non-normal flow rule is of interest. At the moment, only the use of tensile strength is properly motivated. Still, since the default formulation for it is already valid, I would even skip discussing it in the introduction (the paragraph starting at line 40). Also, a brief explanation of what the phrase "non-normal flow" means could be useful in the introduction (even if a more detailed explanation comes later).
Discussion and concluding remarks: There is no mention here of the impact on simulated volume and ice growth, but this is what a more general (The Cryosphere) audience is interested in. If I understand the paper correctly, your results show that the impact of using a non-normal flow rule is substantially smaller than that reported by Lemieux et al. (2025). Is that correct? This should be highlighted in this section. You should then also discuss the relevance of using a non-normal flow rule for large-scale modelling. If the conclusion is that using a non-normal flow rule has a very limited effect, then that may feel counterproductive, but it's still very important for the community to know. A strong conclusion like that would also make this paper much more interesting than it looks in its current version and significantly improve its impact.
Minor comments:
L65: I understand that you're implementing your ideas into CICE, but I'd always mention Lipscomb et al. (2007) first and only put CICE in parentheses.
Eq 14: I think it would help the reader to point out that this is the same equation as Eq (3) with \alpha_0 (\theta) P_o |\dot\varepsilon| added
L180: Why introduce the second incorrect formulation? You show INC1 because that's what Lemieux et al. (2025) did, but INC2 is not well motivated.
L248: Shouldn't the limit on e_F = e_G be \sqrt{3}/2 and not 1? As per eq (36). This is a nice result that deserves better highlighting in the discussion and conclusions. Does this also mean that in a model without an ITD, we should only use e > \sqrt{3}/2?