the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Wave-triggered breakup in the marginal ice zone generates lognormal floe size distributions
- Department of mathematics and Statistics, University of Otago, Dunedin, New Zealand
- Department of mathematics and Statistics, University of Otago, Dunedin, New Zealand
Abstract. Fragmentation of the sea ice cover by ocean waves is an important mechanism impacting ice evolution. Fractured ice is more sensitive to melt, leading to a local reduction in ice concentration, facilitating wave propagation. A positive feedback loop, accelerating sea ice retreat, is then introduced. Despite recent efforts to incorporate this process and the resulting floe size distribution (FSD) into the sea ice components of global climate models (GCM), the physics governing ice breakup under wave action remains poorly understood, and its parametrisation highly simplified. We propose a two-dimensional numerical model of wave-induced sea ice breakup to estimate the FSD resulting from repeated fracture events. This model, based on linear water wave theory and viscoelastic sea ice rheology, solves for the scattering of an incoming time-harmonic wave by the ice cover and derives the corresponding strain field. Fracture occurs when the strain exceeds an empirical threshold. The geometry is then updated for the next iteration of the breakup procedure. The resulting FSD is analysed for both monochromatic and polychromatic forcings. For the latter results, FSDs obtained for discrete frequencies are combined appropriately following a prescribed wave spectrum. We find that under realistic wave forcing, lognormal FSDs emerge consistently in a large variety of model configurations. Care is taken to evaluate the statistical significance of this finding. This result contrasts with the power-law FSD behaviour often assumed by modellers. We discuss the properties of these modelled distributions, with respect to the ice rheological properties and the forcing waves. The projected output will be used to improve empirical parametrisations used to couple sea ice and ocean waves GCM components.
Nicolas Guillaume Alexandre Mokus and Fabien Montiel
Status: final response (author comments only)
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RC1: 'Comment on tc-2021-391', Anonymous Referee #1, 16 Feb 2022
The authors use a 2D wave–ice model involving wave scattering, viscoelastic dissipation and a strain breaking threshold to conduct a detailed statistical analysis of the steady-state FSD produced by wave forcing. Strong evidence is given that the model predicts lognormal FSDs. The study is communicated clearly and the key outcome is potentially a valuable contribution towards modelling the marginal ice zone.
I recommend revisions before publication.
The Introduction is missing an overview of the considerable literature on modelling wave propagation in the MIZ. At present, readers could be led into thinking that the model used is accepted by the community, when, as the authors surely know, debate and open questions remain. There are, for example, different methods for modelling wave scattering and many different models of viscous damping. Certain models have been validated using experimental data. A similar comment applies to models of ice breakup caused by waves. It should be clear at the end of the overview why the particular wave propagation and ice breakup models have been chosen for the present investigation.
The two paragraphs starting from the bottom of page 2 are not particularly relevant for the study presented (e.g. the ideas are not picked up again later) and would be better in Sect 6, leading into a discussion on how the proposed model and findings could be implemented in CICE, etc. Sect 6 would also be strengthened by comments on possible implications of the reduced dimension of the model (e.g., in comparison to the 3D model of Montiel & Squire, 2017) and whether the predicted FSD properties are consistent with the ideas used by Dumont, Williams and co to parameterize power-law FSDs (such as the maximum floe size being half a wavelength).
At the beginning of Sect 5.1, the move from monochromatic to polychromatic forcing requires more explanation and justification. Presumably the definition of the FSD for polychromatic forcing in equation (23) is computationally efficient, but is it representative of the ensemble average FSD created by (random) irregular wave forcing that obeys the prescribed spectrum? Can examples be given to demonstrate this? Better understanding of this aspect of the model will improve interpretation of the results. Incidentally, I was unable to find f_L and tilde{f}_L when scanning back through the paper at this point. Perhaps the latter could be introduced in Sect 2.
A title that indicates the scope of the study would be better, e.g. Model predictions of lognormal floe size distributions in the marginal ice zone caused by wave forcing
Minor:
25: With thinner and weaker first-year ice becoming dominant in the Arctic
28: Elaborate on the sentence starting The individual description.
55: The sentence on short time scales for breakup appears to contradict the steady state model assumption.
Sec 3.1: Similar wave scattering models should be referenced at the beginning of the section [1,2, etc], and any notable differences identified.
149: travelling and evanescent …
170: For completeness, say that the complex roots can become purely imaginary for high frequencies and/or thick ice.
178: I think the phases are used to normalize rather than cancel out the exponential terms.
Eqn (13): Replace the full stop with a comma.
248+250: for every floe and none of the floes break
253: Give the distribution used to randomly redistribute the floes after breakup.
258: Give details on the local resonances plus references.
Figure 3d: The levelling off/decrease of the median floe size with increasing ice thickness for T=8s is interesting and worth discussing in the text.
Figure 4 caption: State the amplitude(s) used.
348: Space needed after the full stop.
428: Note that the value gamma=13.5 Pa s/m was derived from measurements in the Antarctic MIZ [3].
Mathematics needs a capital M in the institution name.
References
[1] Kohout+Meylan, 2008, Journal of Geophysical Research, 113
[2] Montiel et al, 2012, Journal of Fluids and Structures, 28
[3] Massom et al, 2018, Nature, 558
- AC1: 'Reply on RC1', Nicolas Mokus, 16 May 2022
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RC2: 'Comment on tc-2021-391', Anonymous Referee #2, 07 Mar 2022
First of all, I’d like to apologize to the Authors and the Editors for a very long delay in submitting this review. Unfortunately, I was not able to finish it at an earlier date. I am very sorry for that.
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The manuscript „Wave-triggered breakup in the marginal ice zone…” by Nicolas Mokus and Fabien Montiel describes a numerical study of wave propagation in sea ice and wave-induced sea ice breaking. The main focus of the paper are the properties of floe size distributions (FSDs) resulting from breaking of ice with different properties (strength, thickness) by waves of different periods and amplitudes.
Undoubtedly, the problems discussed in the study are important for the current research on sea ice–wave interactions. Our better understanding of the physical mechanisms underlying wave-induced sea ice breaking is crucial for developing better parameterizations of those processes for large-scale sea ice and climate models. Although I find the manuscript and the results interesting and valuable, and the model developed by the Authors well presented, I have some doubts, described below, regarding some parts of the analysis. I recommend the manuscript for publication in The Cryosphere after a major revision.
Major comments:
- The main point in my critics is related to the procedure described in Section 5.1: the whole algorithm is based on an assumption that the FSD resulting from sea ice breaking on irregular waves is “the weighted average of distributions resulting from monochromatic model runs”. Why?
I really can’t see the reason why it should be so simple.
Let’s consider a very simple example of a wave field composed of two monochromatic waves with very different wavelengths, and let’s assume that wave #1 does break the ice and produces very small floes, and wave #2 is very long and doesn’t break the ice at all (or produces very large floes). The ice sheet in that case would break into small floes, corresponding to that resulting from wave #1 anyway, so computing FSD from a weighted average would produce truly weird results!
It’s the part of the spectrum that leads to breakup that’s important, not the whole spectrum!
As the Authors rightfully demonstrate in their manuscript for monochromatic waves, the relationships between floe size, ice properties, and wave length are quite complex and nonlinear, so there is no reason why the FSD resulting from a wave energy spectrum should behave as the Authors assume.
I have the impression that the shapes of FSDs in Fig. 6 to a large degree simply reflect the shape of the wave frequency spectrum, and that this is an artefact of the algorithm (or, more precisely, its part related to the computation of weighted averages).
In my opinion, it is a very weak part of the analysis, but the Authors don’t even discuss those weaknesses.
Of course, as I have serious objections regarding the above-mentioned assumption, I have also doubts regarding the results presented in sections 5.2-5.4 of the manuscript.
Why can’t the model be forced by a superposition of monochromatic waves? The scattering model is linear, isn’t it, so it shouldn’t be difficult. All one needs to do is to add up the wave solutions for individual spectral components (assuming random phases) and use those to compute strain (as, e.g., in section 6 of Kohout & Meylan 2008). - Are the FSDs obtained for monochromatic waves lognormal as well?
Why is that pdf introduced first in Section 5.2 and not earlier? That would allow comparisons between FSDs obtained for regular and irregular wave forcing. - The algorithm, as described in Section 3.3, does not take into account the time evolution of breakup – in the sense that the breaking events during one “sweep” are all taking place at the same time instance, and a breaking event at one location does not influence what is going on in an immediate vicinity of that location (sudden stress release etc.).
I’m not criticizing it, I just wonder whether/how this limitation can influence the resulting FSDs. What is the Authors’ opinion about that? - Figure 4a,b shows the total number of floes for various combinations of the model forcing. How does the width of the MIZ (i.e., the total length of the broken ice) change? It is an important parameter for several reasons, so it would be interesting to see plots analogous to those in Fig.4ab, but showing the MIZ width. Or at least some comments on that in the text.
- I know it’s beyond the scope of this paper, but I’m just curious: Have the Authors analyzed the shape of the attenuation curves produced by their model? Are they approximately exponential, or are there deviations from the exponential curve (as in eq. 2.1 of Squire, Phil Trans A, 2018), especially close to the ice edge?
Minor, technical and other comments:
- Line 38: “Hence…” suggests this sentence follows from the previous one, but I don’t really see the connection. I think I know what is meant here, but I’d suggest formulating it more clearly.
- Lines 41-43: I’d suggest to add here that this technique not only leads to erroneous values of the power law exponents, but, in the first place, suggests the existence of power law tails even when there aren’t any and when the pdfs aren’t heavy-tailed at all.
- Line 93: The recent paper by Dumas-Lefebvre and Dumont (currently under discussion in TCD: https://tc.copernicus.org/preprints/tc-2021-328/) is worth citing here, as it describes a wonderful observational dataset of sea ice breaking by waves. (It’s not self-advertisement, I’m not an author of that paper.)
- Lines 256-258: I understand that those tests suggest that the details of how the floes are placed after breaking are not important.
Maybe it’s a naïve question, but are those empty spaces between floes necessary? Does the algorithm work for densely packed ice field, with zero spaces between floes? - Lines 265-266: “FSD dispersion”. Dispersion? As the term “dispersion” has a clearly defined meaning in the context of waves, I’d suggest replacing it here with “median floe size”.
- Lines 268-269: “a positive relationship between the ice mechanical resistance […] and the presence of larger floes”. But the skewness is larger for smaller strength and thinner ice, isn’t it? The presence of larger floes itself can result from a simple shift of the distribution to the right and is not directly related to the skewness, so this sentence is a bit misleading.
- And further: “Qualitatively, increasing epsilon_c has only a moderate effect on the FSD and seems to be only affecting its mode, shifting it towards larger floes, while its shape remains the same.” Is it really so? Are the shape parameters of the pdfs in Fig.3a really so similar? My impression from the figure is quite different. It might be wrong, of course, but please back up this statement by some numbers, e.g., skewness values (maybe you could add them to the panels in Fig.3a,b for those three cases presented?).
As far as the mode is concerned, in Fig. 3a it changes by ~100% between case 1 and 3, so I’d say it is a quite substantial change. - Line 274: “the dispersion in floe sizes”: again, it’s not clear what exactly is meant here. The range of floe sizes? (i.e. pdf width?)
- Line 277: crisp -> sharp? rapid?
- Line 349: “the definition of the ice edge is not clear, as it is period-dependent”. I don’t understand this statement, please clarify. And further: “the total length of ice in each period category”. Period category? Overall, I’d recommend rephrasing this whole paragraph, as it contains a lot of statements that are hard to follow (although the overall meaning is clear, of course).
- Lines 426-427: “scattering alone is not effective enough at dissipating wave energy”!!! Scattering does not dissipate energy at all! Moreover, in a 1D setting, scattering alone does not lead to wave energy attenuation within sea ice: even for an extremely long ice cover, the wave energy at its downwave end must be equal to the energy of the incomming wave minus the energy reflected from the from the upwave edge. In other words, if any attenuation is observed in the scattering-only model runs, it only results from numerical inaccuracies.
- AC2: 'Reply on RC2', Nicolas Mokus, 16 May 2022
- The main point in my critics is related to the procedure described in Section 5.1: the whole algorithm is based on an assumption that the FSD resulting from sea ice breaking on irregular waves is “the weighted average of distributions resulting from monochromatic model runs”. Why?
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RC3: 'Comment on tc-2021-391', Anonymous Referee #3, 07 Apr 2022
General Comments
This paper aims to develop an efficient model of wave breakup of sea ice floes including a random component of floe positioning that can be used to generate statistical descriptions of floe size (probability) distributions (FSD) that might emerge from wave breakup from sea ice and rapidly explore relevant parameter spaces within this setup (e.g. wave period, sea ice thickness). The study finds that the emergent FSD can be best characterised using a lognormal distribution and discusses implications of these results for finding the best fit to observations of floe size and for future parameterisations of floe breakup by waves in sea ice models. This work intersects two areas of research that have had significant focus in recent years: modelling the role of individual processes in determining the emergent FSD in sea ice models and modelling interactions between waves and sea ice and how sea ice can impact wave propagation. This study builds on earlier efforts to develop simple but accurate models of wave breakup of floes. The value of the model presented here is that it is efficient and can be used to rapidly explore relevant parameter spaces and include stochastic elements within the model to represent uncertainty / variability (in this case to capture variability in floe positioning without a full treatment of sea ice dynamics). I therefore believe this paper makes a useful contribution to both the sea ice and wave modelling communities, and also has potential value in understanding and characterising observations of floe size.
The scientific quality of the work presented is generally strong, with good associated analysis and discussion. The figures are of a very good quality and appropriate to the discussion. The structure of the paper seems fine and is easy to follow, though it would be good to see a more thorough overview of the paper structure at the end of the introduction. The paper reads well, is clear in its conclusions, and also has a representative abstract and title. I do have a couple of major concerns that would need to be addressed before I can recommend publishing. Firstly, I am not sure the methodology used has been sufficiently justified. Specifically, the choice to use monochromatic model runs and then taking the weighted average to determine the emergent FSD from a full wave spectrum is not properly justified / supported as a reasonable approximation. In addition, the study repeatedly refers to whether observations of the FSD should be fitted to a power law. Whilst this is an important discussion, I find the paper focuses too much on this point and insufficiently on other impacts / conclusions of the findings presented. Full details of these concerns are provided in the specific comments.
Overall, I believe that this paper is within the scope of The Cryosphere and, provided the above concerns can be adequately addressed, merits publishing.
Specific Comments
- General point: A key conclusion from this study is that the FSD that emerges from this model of wave breakup of sea ice is a lognormal distribution. The study uses this result to backup conclusions from other studies such as Stern et al. (2018) that other possible fits should be tested against observations of floe size, not just a power law. These conclusions are justified on the basis of the evidence presented. However, throughout the manuscript the authors question the validity of power law fits to FSD data. Whilst this is a reasonable and justified question to ask and one several previous papers have discussed as noted in the manuscript, I find this point is too frequently made within the manuscript, at the expense of other important results that emerge from this study, given this study does not appear to present any new evidence to suggest that a power law does not produce a valid fit to observed FSDs (as opposed to new evidence to support the testing of alternative fits to observations, which the study does present, as noted above). Even in regions of high wave activity, observed FSDs are not necessarily solely a result of wave breakup. Even if they are, there are physical features that may determine the FSD not considered within the model used here (e.g. variable ice thickness, existing weaknesses in the sea ice, fractures that are not perpendicular to the direction of wave propagation). The emergence of a lognormal distribution from this model does not necessarily tell us anything about the validity of a power law fit to observations of floe size unless this model can be validated using observations of an FSD under wave control, which has not been presented in this study.
- P2 L28-29: ‘The individual description of these, floating pieces of sea ice is not possible.’ What do you mean by this comment? Individual pieces of ice cannot presently be simulated in continuum models, but they can in discrete element models of sea ice.
- P3 L65-68: You should also describe / discuss the most recent study from Horvat and Roach (2022) that introduced a machine-learning-derived parameterization of wave breakup of floes that can be used within the prognostic model.
- P2 L57 – P3 L81: In this section you have described existing treatments of wave breakup of floes within sea ice models but there are other approaches that you have not described e.g. both Bateson et al. (2020) and Boutin et al. (2021) include treatments of wave breakup of sea ice within FSD models. It would be helpful to either briefly discuss these treatments or at least highlight that your discussion is not exhaustive.
- P3 L80-81: ‘Nevertheless, the model sensitivity analysis conducted by Zhang et al. (2016) revealed compelling improvement on ice extent simulation when considering their FSD formulation.’ What were the improvements? This statement is vague and should be clarified.
- P4 L97-105: It would be helpful to describe the overall structure of the paper at the end of the introduction i.e. describe how the paper proceeds, section by section.
- P8 L202-203: Why did you decide to use a fixed sea ice thickness in your simulations? Do you anticipate that a lognormal distribution would still emerge if the sea ice thickness was variable in a single evaluation of the model?
- P8 L205-206: ‘A sensitivity analysis (not shown here) proved Nv = 2 to be adequate in terms of convergence.’ Please provide more details on this. How are you assessing adequate convergence here?
- Section 4: In this section you provide a physical explanation / interpretation of the results presented in Fig. 4 but not Fig. 3. It would be good to see more discussion of the results in Fig. 3; in particular, can you explain the different trends in the variability / dispersion of floe size shown in panels (c) and (d) in Fig. 3?
- Figure 3: Why did you decide to use the median floe size to characterise the average floe size (rather than, for example, a linear-weighted mean)? An explanation in the text somewhere would be useful.
- Figure 3: Do you have any explanation for the oscillatory behaviour in panel (d) for the two shorter wave periods when the ice thickness exceeds 1.5 m?
- P13 L295-297: ‘To estimate the effect of a developed sea on the FSD fL, we take the weighted average of distributions resulting from monochromatic model runs,’. This appears to be a significant model assumption to only consider single amplitude-frequency pairs at once rather than the full wave spectrum since it ignores possible interactions between the different pairs in fracturing the sea ice. What is the justification for this model approach? There needs to be some evidence presented (e.g. test cases evaluating the model using full polychromatic forcing) to show that the error resulting from this approximation is not large enough to impact the conclusions.
- P13 L300: Can you comment on the sensitivity of your results to the choice of spectrum?
- P14 L310: What is the reason for drawing a single FSD fl at random rather than including all 50 realisations?
- Section 5.3: As it currently exists, I am not sure this section is adding much insight to the manuscript and it could be removed without detracting from the paper. All this section demonstrates is that the average floe size increases moving away from the ice edge, a behaviour several previous observational and modelling studies have identified. What might make this section more insightful would be if the results could be used to generate a mathematical description of how the emergent FSD changes with distance from the ice edge or plots to show how the parameters of the lognormal fit change with distance from the ice edge.
- P16 L364-365: ‘For simplicity, even though we did conduct multivariate simulations, we focus here’. Why mention this if you are not going to discuss the results? It would be beneficial to discuss some of these results - since in the results you present much of the parameter space is unexplored leaving open the potential for different behaviour elsewhere in the parameter space.
- P17 L382-384: ‘As the peak propagating wavelength is proportional to the significant wave height, this non-monotonic evolution does not support wave properties alone govern the dominant floe size,’. Can you provide a more precise explanation of why this happens? Given the simplified model treatment used, it should be possible to explain how this behaviour emerges.
- Section 6: It would be good to see more focus in the discussion / conclusions on what needs to be done to validate this model using observations of floe size i.e. what are the key emergent features of the FSD produced by this model that could potentially be identified in observations (not just the general lognormal shape, but how the distribution evolves with changes to key parameters such as the distance from the ice edge).
- P21 L446-447: ‘These results aim at being a step towards the parametrisation of wave action in FSD-evolving models.’ Working towards this parametrisation seems to be a key result of this study and merits more than a single line in the discussion / conclusion section. What more needs to be done to develop this parameterisation? How will this parameterisation compare to the alternative scheme developed by Horvat and Roach (2022)?
Technical Corrections
- P2 L30-31: ‘In particular, fragmentation caused by ocean waves makes the floes more sensitive to melt’. Maybe change ‘In particular’ to ‘Of particular interest here’ or something similar, since there exists other mechanisms of ice fragmentation that can drive the same feedback.
- P2 L36-37: Most studies listed fit the observed FSD to a simple power law (or combination of the two). I am not sure it is correct to describe these as Pareto distributions (see e.g. Herman, 2010).
- P2 L37-38: ‘However, a variety of processes such as failure from wind or internal stress, lateral melting or growth, ridging, rafting or welding, are susceptible to alter the FSD.’ Can you provide references for these processes having been observed to influence the FSD?
- P2 L48-49: ‘evaluate the impact of its introduction on other quantities such as ice thickness or concentration (Roach et al., 2018)’. There are other studies you should consider referencing here e.g. Bateson et al., 2020; Boutin et al. 2021.
- P3 L92: ‘ensuing’. Should this be ensuring?
- P4 L96: Reference is incorrect. Boutin et al. (2020b) should be Boutin et al. (2021).
- P9 L243-244: ‘Hence, the number of floes at most doubles, if all the floes break in a single simulation.’ If my understanding is correct, single iteration would be a clearer choice here rather than single simulation.
- P18 L394-395: ‘The prevalence of smaller floes, however, tends to build up slightly.’ Phrasing here is awkward.
- P19 L399: ‘shows’ / ‘points out’ rather than ‘point out’.
References
Bateson, A. W., Feltham, D. L., Schröder, D., Hosekova, L., Ridley, J. K. and Aksenov, Y.: Impact of sea ice floe size distribution on seasonal fragmentation and melt of Arctic sea ice, Cryosphere, 14, 403–428, doi:10.5194/tc-14-403-2020, 2020.
Boutin, G., Williams, T., Rampal, P., Olason, E., and Lique, C.: Wave–sea-ice interactions in a brittle rheological framework, The Cryosphere, 15, 431–457, https://doi.org/10.5194/tc-15-431-2021, 2021.
Herman, A.: Sea-ice floe-size distribution in the context of spontaneous scaling emergence in stochastic systems, Phys. Rev. E, 81, 1–5, https://doi.org/10.1103/PhysRevE.81.066123, 2010.
Horvat, C. and Roach, L. A.: WIFF1.0: a hybrid machine-learning-based parameterization of wave-induced sea ice floe fracture, Geosci. Model Dev., 15, 803–814, https://doi.org/10.5194/gmd-15-803-2022, 2022.
Stern, H. L., Schweiger, A. J., Zhang, J. and Steele, M.: On reconciling disparate studies of the sea-ice floe size distribution, Elem Sci Anth, 6(1), doi:10.1525/elementa.304, 2018.
- AC3: 'Reply on RC3', Nicolas Mokus, 16 May 2022
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CC1: 'Comment on tc-2021-391', Elie Dumas-Lefebvre, 08 Apr 2022
Hi Nicolas and Fabien,
It is a great paper you have here. I have some sugesstions , comments and questions about it, please see the attached file.
Best,
Elie
- AC4: 'Reply on CC1', Nicolas Mokus, 16 May 2022
Nicolas Guillaume Alexandre Mokus and Fabien Montiel
Data sets
Model code and simulation results for the investigation of a wave-generated floe size distribution Mokus, Nicolas; Montiel, Fabien https://doi.org/10.6084/m9.figshare.17303927
Model code and software
Model code and simulation results for the investigation of a wave-generated floe size distribution Mokus, Nicolas; Montiel, Fabien https://doi.org/10.6084/m9.figshare.17303927
Nicolas Guillaume Alexandre Mokus and Fabien Montiel
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