the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Revisiting the role of vertical shear in analytic ice shelf models
Chris Miele
Timothy C. Bartholomaus
Ellyn M. Enderlin
Abstract. Analytic modeling of ice shelf flow began when Weertman derived an expression for the strain rates within an unconfined shelf, of uniform thickness, extending only in one direction. Nearly two decades later, Thomas generalized Weertman's analysis to ice shelves of nonuniform thickness, deriving one of the most wellknown analytic models in glaciology: . However, despite the prevalence of this model in both historical and contemporary texts, there remain persistent miscommunications regarding the role of vertical shear in its construction. In Thomas' original approach, vertical shear stress was considered negligible in the stress balance; in a significant contrast, the same model is typically derived in contemporary texts by the neglect of basal resistance. These two approaches are not equivalent, and yet, it remains common to misinterpret vertical shear stress as typically neglected in current ice shelf modeling studies. This manuscript provides clarification on this pervasive misconception. We emphasize that vertical shear stress should not be interpreted as negligible in the construction of general shallow shelf models. However, we also demonstrate that the vertical shear stress inherent in Thomas' expression does not give rise to a welldefined vertical shear strain rate. For situations in which vertical shear stress in shallow ice shelf models is of interest, we provide guidance on how to best calculate it.

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negligiblein ice shelf models. We address this miscommunication, providing conceptual guidance regarding this often misrepresented stress. Fundamentally, vertical shear is required to balance thickness gradients in ice shelves.
Chris Miele et al.
Interactive discussion
Status: closed

RC1: 'Comment on egusphere20221310', Anonymous Referee #1, 26 Jan 2023
General comments:
I think that I reviewed this manuscript for another journal, and if so, I was supportive of its publication at that time, and still am. I do note some relatively minor, but important corrections to the “narrative” being presented, and I describe them now:
The abstract uses the words “miscommunications”, “misinterpretation” and “misconception”. I think that these words are sort of unfair to the early scientists who developed the initial modes of thinking about, doing analysis with and modeling ice shelves. These early scientists were well aware of the fact that shear stress in the vertical was prevalent in ice shelves at amplitudes that could be large (e.g., at an ice front or when there are large thickness gradients); however, their intention was to develop strategic simplifications and approximations which would allow glaciological science to make progress. Their pioneering work leading to the “shallow shelf approximation” was fundamental to the progress of glaciology through the 1960’s onward to the present day. It is thus not only unfair to their legacy to imply that they were “misleading”, but it is a kind of cheap writer’s trick to introduce the substance of the present paper. I strongly object to this tone and think that it detracts from the paper by setting up a false “combative” tone that completely misleads the reader.
I see that this tone that I object to is not present in the Introduction, and the authors very correctly laud the initial development of one of the most effective approximations in glaciological history (the shallow shelf approximation). This is important. And I compliment the authors for having done so. But again: I see words like (line 65) “persistent mischaracterizations”. This is a false and incorrect statement: approximation is not a mischaracterization.
A challenge: After recently attending the AGU and also reading a paper by Catherine Walker:
Walker, C. C. and Gardner, A. S. (2019). Evolution of ice shelf rifts: Implications for formation mechanics and morphological controls, Earth and Planetary Science Letters, 526,115764, doi:10.1016/j.epsl.2019.115764.
I became aware of the fact that many rifts on the Antarctic ice shelves are not vertical, but slightly offset from vertical, and that they have an interesting, notfullyunderstood asymmetry of the rift shoulders associated with bending moments. I wonder if this phenomena (also described in one of Walker’s papers on ice shelled planets) is an observable phenomena that is directly related to the subject of this paper. If the authors think that it is, then they might find that their paper is made even stronger by including references to the Walker study, and also to (not sure if this is as relevant):
Walker, C. C., J. N. Bassis, J. N. and Schmidt, B. E. (2021). Propagation of vertical fractures through planetary ice shells: The role of basal fractures at the iceocean interface and proximal cracks. The Planetary Science Journal, Vol. 2, No. 4, doi:10.3847/PSJ/ac01ee.
Specific comments:
line 2 of abstract “…, extending in only one direction,…”. I’m pretty sure that Weertman’s 1957 paper also gives the solution for spreading in two horizontal directions.
I was not able to find other errors or edits to make, and I commend the authors for doing a fine job of proof reading.
Citation: https://doi.org/10.5194/egusphere20221310RC1 
AC1: 'Reply on RC1', Chris Miele, 27 Jan 2023
Thank you (again) for the favourable review.
And thank you especially for pointing out that the tone in the abstract is at odds with that of the introduction and the rest of the paper. As you noted from the introduction, our position isn't that early papers by pioneering authors were misleading in any sense. The "miscommunication" that we mean to identify is more on the part of contemporary discussion, which can sometimes seem to imply that the historical treatment of vertical shear is still used (e.g., the footnoted examples on page 3. Presumably those cited statements are mostly shorthand for a more nuanced understanding, but, speaking from experience, an earlycareer glaciologist probably won't pick up on that subtlety).
Our hope is that it'll be beneficial, especially for beginning modelers, to have this potential misstep addressed directly. Thank you also for the additional feedback, and referral to the other references. We look forward to reviewing. Following additional reviews, we'll post a thorough revision.
Citation: https://doi.org/10.5194/egusphere20221310AC1

AC1: 'Reply on RC1', Chris Miele, 27 Jan 2023

RC2: 'Review of “Revisiting the role of vertical shear in analytic ice shelf models” by C. Miele et al.', Anonymous Referee #2, 16 Mar 2023
The manuscript is concerned with the mathematical modeling of ice shelf flow. In particular, the authors focus on the role of vertical shear in thinfilm approximations of the ice flow problem that are valid for the case of negligible basal friction. The authors state that their goal is to clarify what is in their view a misunderstanding in the main stream glaciological literature, i.e., that “it remains common to misinterpret vertical shear stress as typically neglected in current ice shelf modeling studies”.
As much as I understand that the intention behind this manuscript is constructive, and as much as I share the authors’ opinion that the published literature is somewhat confusing, I am afraid I have to express a negative opinion about the content and originality of this work. In fact, in my view this problem has been solved already in rigorous mathematical terms in publications concerning the asymptotic structure of the Stokes problem in the limits of thin flows (low aspect ratio) and extensional stresses much larger than the vertical shear stresses.
Two main strands of research support my statement: Doug MacAyeal’ s seminal shelfystream paper (JGR 1989, Largescale ice flow over a viscous basal sediment: theory and application to Ice Stream B, Antarctica, see model derivation in appendix A), and more recently Schoof and Hindmarsh, 2010 (ThinFilm Flows with Wall Slip: An Asymptotic Analysis of Higher Order Glacier Flow Models https://academic.oup.com/qjmam/article/63/1/73/1843730, esp. sec. 3.4, the limit of lambda << epsilon). For completeness, and purely as a side note, the only difference between these two derivations being that Schoof and Hindmarsh retain the stress ratio lambda <<1 (see eq. 2.18, lambda is the ratio between shear stress and extensional stress, whereas epsilon<<1 is the aspect ratio, both taken to be small; then lambda<<1 describe the case of ice flow dominated by extensional stresses, as is the case in an ice shelf), whereas MacAyeal assumes the distinguished limit lambda ~ epsilon^2 with (lambda, epsilon) <<1 (or, in MacAyeal’s notation, epsilon ~ delta^2), which is one peculiar case of the broader model category obtained in Schoof and Hindmarsh’s more general limit of lambda << epsilon. Yet, despite this minor formal difference, they obtain the same leadingorder model, as noted by Schoof and Hindmarsh.
To illustrate my point, I will refer to MacAyeal’s derivation, which is simple and clear; in his derivation, e_{xz}, e_{yz} are the vertical shear stresses therein. I now use epsilon to mean stress ratio, as per MacAyeal’s notation.
The first key point is about the zerothorder momentum balance: Yes, to leading order the solution reads e_{xz}^{0} = 0, e_{yz}^{0}=0 (eqs. A25 and A26), but all that means, as the derivation clearly explains, is that these stresses are of order epsilon (epsilon being the aspect ratio) compared to the extensional stresses and the lateral shear stress. This is very different from stating that the vertical shear stresses are zero  rather, it only says that extensional stresses are small compared to extensional and lateral shear stresses.
To find the leading order expressions for the vertical shear stresses, one needs to carry on with the expansions to the next (first) order. As expected, the order epsilon corrections e_{xz}^{1}, e_{yz}^{1} appear in the first order (order epsilon) momentum conservation equations (A30  A31), as they do in the dynamic boundary conditions that determine the horizontal gradients of the ice shelf surface and bottom (A34A35).
The misconception pointed out by the authors arises, if I understand correctly, by the fact that thanks to careful algebraic manipulations e_{xz}^{1}, e_{yz}^{1} disappear from the final momentum conservation equations (A 36 and A37) that must be solved to determine e_{xx}^(0), e_{yy}^(0), e_{xy}^(0). Yet, that does not mean that vertical shear stresses are zero  in fact, they can be computed diagnostically by vertical integration of the constitutive relation with appropriate boundary conditions, once the problem for e_{xx}^(0), e_{yy}^(0), e_{xy}^(0) has been solved.
It is my view that this work, published over 30 years ago, already clarifies the issue raised by the authors without any possible doubt, while Schoof and Hindmarsh 2010 provide a further generalization. In light of this, it is my opinion that the manuscript as is lacks the level of originality that is required to warrant publication.
Citation: https://doi.org/10.5194/egusphere20221310RC2 
AC2: 'Reply on RC2', Chris Miele, 16 Mar 2023
We thank the reviewer for their careful read, with which we largely agree. Clearly, the reviewer's knowledge on this topic demonstrates that they fall outside of the audience we hope to engage with our article. Additionally, we concur that we present no foundational "discovery" on which we can claim novelty. Instead, the novelty of our present work is motivated directly by our concurrence with the reviewer: "I share the authors' opinion that the published literature is somewhat confusing." We aspire to contribute to the existing, foundational literature (cited by the reviewer, as in our draft manuscript) in a novel way by shining a bright light on this confusion.
The reviewer argues that work published over 30 years ago preempts the need for our manuscript, yet the reviewer agrees that confusion persists. Given that many of the recent misrepresentations we cite (footnote 1 on p. 3) postdate the 1989 and 2010 papers discussed by the reviewer, we hope we can all agree that more work is necessary to lay this issue to rest. Neither reviewer 1 nor reviewer 2 has identified any material defect in our analysis. Nor have the contrasting interpretations of the SSA's vertical shear yet received the focused attention we provide with our manuscript. Therefore, we believe that we are wellposed to contribute positively to this topic in a unique, original way.
To the extent that we have fallen short of this goal, we accept responsibility. With the guidance and consideration of this peer review, we hope to better craft our manuscript so that, going forward, all students of glaciology will know that negligible vertical shear is not a necessary component of a shallow shelf model. We look forward to guidance from the editor and to a more thorough revision and response in the weeks ahead.
Citation: https://doi.org/10.5194/egusphere20221310AC2

AC2: 'Reply on RC2', Chris Miele, 16 Mar 2023

RC3: 'Comment on egusphere20221310', Josefin Ahlkrona, 21 Mar 2023
This paper is aiming to clarify to the broader glaociology community some assumptions made in models for ice shelfs, specifically the relation between vertical shearing and surface gradients. I think such a topic/disucssion is welcome, but the presentation of this needs to be improved in order to reach the inteded audience. I suggest the following edits:
1) Formalize the arguments using perturbation expansions and scalings. The vertical shearing is not "absent", it is just of a higher order in the perturbation expansion I presume (i.e. "neglected"). Is it of the same order as the surface gradients perhaps? I recommend connecting your arguments to the analysis by Schoof and Hindmarch (mentioned here by another reviewer). A large part of the glaciological community is familiar with perturbation expansions and scalings, and for the ones who are not, they could learn it from this paper. As the papers intent seems to be educate this should fit in well. You don't need to erase your other arguments, but I recommend adding perturbation expansions as an supporting, more formal, explanation.
2) The exeriment in Figure 2 is nice but in my opinion it could be made more interesting. For the purpose of showing that this is actually important in real ice shelf simulaitons, I would extend this experiment, make it more realstic  maybe some variation in the ice surface.
3) Overall it is sometimes a bit unnecessary difficult to follow the derivations, think about if there is a clearer way to present it. Make the two derivations side by side perhaps?
4) As mentioned by another reviewer, make sure that its not possible to percieve the tone in a bad way, although I am sure this is not the intent.Citation: https://doi.org/10.5194/egusphere20221310RC3 
AC3: 'Reply on RC3', Chris Miele, 21 Mar 2023
We appreciate the support of this topic, and especially the constructive suggestions presented in this review. We'll look forward to addressing these in the next iteration of the manuscript. We especially agree that the inclusion of perturbation expansions/scalings would fit in well here, and should prove useful in this educationoriented manuscript.
One followup question, in response to the reviewer's third suggestion: we'd like to clarify which two derivations the reviewer suggests presenting together.
Again, we thank the reviewer for these suggestions, which will be addressed fully in the next revision.
Citation: https://doi.org/10.5194/egusphere20221310AC3

AC3: 'Reply on RC3', Chris Miele, 21 Mar 2023

RC4: 'Comment on egusphere20221310', Anonymous Referee #4, 21 Mar 2023

AC4: 'Reply on RC4', Chris Miele, 21 Mar 2023
Thank you for the detail offered in this review. This review emphasizes the asymptotic analysis of Schoof and Hindmarsh (2010), which fits well with the comments of two other reviewers. The suggestion that our manuscript could be used to translate asymptotic analysis into "more digestible language" was also brought up by Reviewer #3, and we see that the inclusion of asymptotic analysis will be an important addition to our manuscript. We particularly appreciate the time taken by the reviewer to express the work of Schoof and Hindmarsh in our own notation, which is very helpful for us. We also concur with the suggestions for improving readability, which is of primary importance in a paper meant to clarify and educate. At the discretion of the editor, we look forward to the opportunity to more fully address this review.
Citation: https://doi.org/10.5194/egusphere20221310AC4

AC4: 'Reply on RC4', Chris Miele, 21 Mar 2023

EC1: 'Editor Comment on egusphere20221310', Reinhard Drews, 23 Mar 2023
Dear Authors and Reviewers,
thanks for a vivid discussions phase, which I have now ended after receiving four reviews. In order to consider publication in The Cryosphere, the manuscript requires substantial revisions likely including a change of the article type from “research article” to “brief communication”. My more detailed comments are as follows:
[1] I would like to thank all reviewers. Their comments demonstrate a strong expertise in the foundations of ice dynamic modelling including the asymptotic analysis required in modelling iceshelf flow. Many constructive suggestions are provided, including explicit derivations linking earlier works to the results of this study.
[2] There is no consensus between reviewers as to whether or not this paper satisfies the criteria of a research article. The main concern synthesizing from the more critical reviews (RC2, RC4) is a lack of novelty, e.g., compared to earlier works such as Schoof and Hindmarsh (2010). Here, the authors agree that there is no “discovery on which we claim novelty.” Instead they argue that “..all students of glaciology will know that negligible vertical shear is not a necessary component of a shallow shelf model”. This point, however, is confronted by multiple comments (RC3 & RC4) that the paper is hard to read even for some invited experts. It is therefore doubtful if this objective will be achieved.
[3] In summary, I believe that the lack of novelty precludes classification of this article as a “research article”. Going forward, I suggest that the authors provide a significantly revised version that takes the reviewer’s comments into account and delivers a clear message to the intended target audience in form of a “brief communication”. This is difficult to do for such a topic, and it will require substantial shortening and reframing. I also understand that there might be different understandings for the terms “novelty” & “research articles etc.” between different authors, reviewers, and editors. However, given the reviews and the publication criteria of The Cryosphere, I believe this will be the best option to go for.
Thank you again for all your work. Although this discussion contains some contraversial points, I surely appreciate this form of scientific exchange.
Kind regards,
Reinhard Drews
Citation: https://doi.org/10.5194/egusphere20221310EC1
Interactive discussion
Status: closed

RC1: 'Comment on egusphere20221310', Anonymous Referee #1, 26 Jan 2023
General comments:
I think that I reviewed this manuscript for another journal, and if so, I was supportive of its publication at that time, and still am. I do note some relatively minor, but important corrections to the “narrative” being presented, and I describe them now:
The abstract uses the words “miscommunications”, “misinterpretation” and “misconception”. I think that these words are sort of unfair to the early scientists who developed the initial modes of thinking about, doing analysis with and modeling ice shelves. These early scientists were well aware of the fact that shear stress in the vertical was prevalent in ice shelves at amplitudes that could be large (e.g., at an ice front or when there are large thickness gradients); however, their intention was to develop strategic simplifications and approximations which would allow glaciological science to make progress. Their pioneering work leading to the “shallow shelf approximation” was fundamental to the progress of glaciology through the 1960’s onward to the present day. It is thus not only unfair to their legacy to imply that they were “misleading”, but it is a kind of cheap writer’s trick to introduce the substance of the present paper. I strongly object to this tone and think that it detracts from the paper by setting up a false “combative” tone that completely misleads the reader.
I see that this tone that I object to is not present in the Introduction, and the authors very correctly laud the initial development of one of the most effective approximations in glaciological history (the shallow shelf approximation). This is important. And I compliment the authors for having done so. But again: I see words like (line 65) “persistent mischaracterizations”. This is a false and incorrect statement: approximation is not a mischaracterization.
A challenge: After recently attending the AGU and also reading a paper by Catherine Walker:
Walker, C. C. and Gardner, A. S. (2019). Evolution of ice shelf rifts: Implications for formation mechanics and morphological controls, Earth and Planetary Science Letters, 526,115764, doi:10.1016/j.epsl.2019.115764.
I became aware of the fact that many rifts on the Antarctic ice shelves are not vertical, but slightly offset from vertical, and that they have an interesting, notfullyunderstood asymmetry of the rift shoulders associated with bending moments. I wonder if this phenomena (also described in one of Walker’s papers on ice shelled planets) is an observable phenomena that is directly related to the subject of this paper. If the authors think that it is, then they might find that their paper is made even stronger by including references to the Walker study, and also to (not sure if this is as relevant):
Walker, C. C., J. N. Bassis, J. N. and Schmidt, B. E. (2021). Propagation of vertical fractures through planetary ice shells: The role of basal fractures at the iceocean interface and proximal cracks. The Planetary Science Journal, Vol. 2, No. 4, doi:10.3847/PSJ/ac01ee.
Specific comments:
line 2 of abstract “…, extending in only one direction,…”. I’m pretty sure that Weertman’s 1957 paper also gives the solution for spreading in two horizontal directions.
I was not able to find other errors or edits to make, and I commend the authors for doing a fine job of proof reading.
Citation: https://doi.org/10.5194/egusphere20221310RC1 
AC1: 'Reply on RC1', Chris Miele, 27 Jan 2023
Thank you (again) for the favourable review.
And thank you especially for pointing out that the tone in the abstract is at odds with that of the introduction and the rest of the paper. As you noted from the introduction, our position isn't that early papers by pioneering authors were misleading in any sense. The "miscommunication" that we mean to identify is more on the part of contemporary discussion, which can sometimes seem to imply that the historical treatment of vertical shear is still used (e.g., the footnoted examples on page 3. Presumably those cited statements are mostly shorthand for a more nuanced understanding, but, speaking from experience, an earlycareer glaciologist probably won't pick up on that subtlety).
Our hope is that it'll be beneficial, especially for beginning modelers, to have this potential misstep addressed directly. Thank you also for the additional feedback, and referral to the other references. We look forward to reviewing. Following additional reviews, we'll post a thorough revision.
Citation: https://doi.org/10.5194/egusphere20221310AC1

AC1: 'Reply on RC1', Chris Miele, 27 Jan 2023

RC2: 'Review of “Revisiting the role of vertical shear in analytic ice shelf models” by C. Miele et al.', Anonymous Referee #2, 16 Mar 2023
The manuscript is concerned with the mathematical modeling of ice shelf flow. In particular, the authors focus on the role of vertical shear in thinfilm approximations of the ice flow problem that are valid for the case of negligible basal friction. The authors state that their goal is to clarify what is in their view a misunderstanding in the main stream glaciological literature, i.e., that “it remains common to misinterpret vertical shear stress as typically neglected in current ice shelf modeling studies”.
As much as I understand that the intention behind this manuscript is constructive, and as much as I share the authors’ opinion that the published literature is somewhat confusing, I am afraid I have to express a negative opinion about the content and originality of this work. In fact, in my view this problem has been solved already in rigorous mathematical terms in publications concerning the asymptotic structure of the Stokes problem in the limits of thin flows (low aspect ratio) and extensional stresses much larger than the vertical shear stresses.
Two main strands of research support my statement: Doug MacAyeal’ s seminal shelfystream paper (JGR 1989, Largescale ice flow over a viscous basal sediment: theory and application to Ice Stream B, Antarctica, see model derivation in appendix A), and more recently Schoof and Hindmarsh, 2010 (ThinFilm Flows with Wall Slip: An Asymptotic Analysis of Higher Order Glacier Flow Models https://academic.oup.com/qjmam/article/63/1/73/1843730, esp. sec. 3.4, the limit of lambda << epsilon). For completeness, and purely as a side note, the only difference between these two derivations being that Schoof and Hindmarsh retain the stress ratio lambda <<1 (see eq. 2.18, lambda is the ratio between shear stress and extensional stress, whereas epsilon<<1 is the aspect ratio, both taken to be small; then lambda<<1 describe the case of ice flow dominated by extensional stresses, as is the case in an ice shelf), whereas MacAyeal assumes the distinguished limit lambda ~ epsilon^2 with (lambda, epsilon) <<1 (or, in MacAyeal’s notation, epsilon ~ delta^2), which is one peculiar case of the broader model category obtained in Schoof and Hindmarsh’s more general limit of lambda << epsilon. Yet, despite this minor formal difference, they obtain the same leadingorder model, as noted by Schoof and Hindmarsh.
To illustrate my point, I will refer to MacAyeal’s derivation, which is simple and clear; in his derivation, e_{xz}, e_{yz} are the vertical shear stresses therein. I now use epsilon to mean stress ratio, as per MacAyeal’s notation.
The first key point is about the zerothorder momentum balance: Yes, to leading order the solution reads e_{xz}^{0} = 0, e_{yz}^{0}=0 (eqs. A25 and A26), but all that means, as the derivation clearly explains, is that these stresses are of order epsilon (epsilon being the aspect ratio) compared to the extensional stresses and the lateral shear stress. This is very different from stating that the vertical shear stresses are zero  rather, it only says that extensional stresses are small compared to extensional and lateral shear stresses.
To find the leading order expressions for the vertical shear stresses, one needs to carry on with the expansions to the next (first) order. As expected, the order epsilon corrections e_{xz}^{1}, e_{yz}^{1} appear in the first order (order epsilon) momentum conservation equations (A30  A31), as they do in the dynamic boundary conditions that determine the horizontal gradients of the ice shelf surface and bottom (A34A35).
The misconception pointed out by the authors arises, if I understand correctly, by the fact that thanks to careful algebraic manipulations e_{xz}^{1}, e_{yz}^{1} disappear from the final momentum conservation equations (A 36 and A37) that must be solved to determine e_{xx}^(0), e_{yy}^(0), e_{xy}^(0). Yet, that does not mean that vertical shear stresses are zero  in fact, they can be computed diagnostically by vertical integration of the constitutive relation with appropriate boundary conditions, once the problem for e_{xx}^(0), e_{yy}^(0), e_{xy}^(0) has been solved.
It is my view that this work, published over 30 years ago, already clarifies the issue raised by the authors without any possible doubt, while Schoof and Hindmarsh 2010 provide a further generalization. In light of this, it is my opinion that the manuscript as is lacks the level of originality that is required to warrant publication.
Citation: https://doi.org/10.5194/egusphere20221310RC2 
AC2: 'Reply on RC2', Chris Miele, 16 Mar 2023
We thank the reviewer for their careful read, with which we largely agree. Clearly, the reviewer's knowledge on this topic demonstrates that they fall outside of the audience we hope to engage with our article. Additionally, we concur that we present no foundational "discovery" on which we can claim novelty. Instead, the novelty of our present work is motivated directly by our concurrence with the reviewer: "I share the authors' opinion that the published literature is somewhat confusing." We aspire to contribute to the existing, foundational literature (cited by the reviewer, as in our draft manuscript) in a novel way by shining a bright light on this confusion.
The reviewer argues that work published over 30 years ago preempts the need for our manuscript, yet the reviewer agrees that confusion persists. Given that many of the recent misrepresentations we cite (footnote 1 on p. 3) postdate the 1989 and 2010 papers discussed by the reviewer, we hope we can all agree that more work is necessary to lay this issue to rest. Neither reviewer 1 nor reviewer 2 has identified any material defect in our analysis. Nor have the contrasting interpretations of the SSA's vertical shear yet received the focused attention we provide with our manuscript. Therefore, we believe that we are wellposed to contribute positively to this topic in a unique, original way.
To the extent that we have fallen short of this goal, we accept responsibility. With the guidance and consideration of this peer review, we hope to better craft our manuscript so that, going forward, all students of glaciology will know that negligible vertical shear is not a necessary component of a shallow shelf model. We look forward to guidance from the editor and to a more thorough revision and response in the weeks ahead.
Citation: https://doi.org/10.5194/egusphere20221310AC2

AC2: 'Reply on RC2', Chris Miele, 16 Mar 2023

RC3: 'Comment on egusphere20221310', Josefin Ahlkrona, 21 Mar 2023
This paper is aiming to clarify to the broader glaociology community some assumptions made in models for ice shelfs, specifically the relation between vertical shearing and surface gradients. I think such a topic/disucssion is welcome, but the presentation of this needs to be improved in order to reach the inteded audience. I suggest the following edits:
1) Formalize the arguments using perturbation expansions and scalings. The vertical shearing is not "absent", it is just of a higher order in the perturbation expansion I presume (i.e. "neglected"). Is it of the same order as the surface gradients perhaps? I recommend connecting your arguments to the analysis by Schoof and Hindmarch (mentioned here by another reviewer). A large part of the glaciological community is familiar with perturbation expansions and scalings, and for the ones who are not, they could learn it from this paper. As the papers intent seems to be educate this should fit in well. You don't need to erase your other arguments, but I recommend adding perturbation expansions as an supporting, more formal, explanation.
2) The exeriment in Figure 2 is nice but in my opinion it could be made more interesting. For the purpose of showing that this is actually important in real ice shelf simulaitons, I would extend this experiment, make it more realstic  maybe some variation in the ice surface.
3) Overall it is sometimes a bit unnecessary difficult to follow the derivations, think about if there is a clearer way to present it. Make the two derivations side by side perhaps?
4) As mentioned by another reviewer, make sure that its not possible to percieve the tone in a bad way, although I am sure this is not the intent.Citation: https://doi.org/10.5194/egusphere20221310RC3 
AC3: 'Reply on RC3', Chris Miele, 21 Mar 2023
We appreciate the support of this topic, and especially the constructive suggestions presented in this review. We'll look forward to addressing these in the next iteration of the manuscript. We especially agree that the inclusion of perturbation expansions/scalings would fit in well here, and should prove useful in this educationoriented manuscript.
One followup question, in response to the reviewer's third suggestion: we'd like to clarify which two derivations the reviewer suggests presenting together.
Again, we thank the reviewer for these suggestions, which will be addressed fully in the next revision.
Citation: https://doi.org/10.5194/egusphere20221310AC3

AC3: 'Reply on RC3', Chris Miele, 21 Mar 2023

RC4: 'Comment on egusphere20221310', Anonymous Referee #4, 21 Mar 2023

AC4: 'Reply on RC4', Chris Miele, 21 Mar 2023
Thank you for the detail offered in this review. This review emphasizes the asymptotic analysis of Schoof and Hindmarsh (2010), which fits well with the comments of two other reviewers. The suggestion that our manuscript could be used to translate asymptotic analysis into "more digestible language" was also brought up by Reviewer #3, and we see that the inclusion of asymptotic analysis will be an important addition to our manuscript. We particularly appreciate the time taken by the reviewer to express the work of Schoof and Hindmarsh in our own notation, which is very helpful for us. We also concur with the suggestions for improving readability, which is of primary importance in a paper meant to clarify and educate. At the discretion of the editor, we look forward to the opportunity to more fully address this review.
Citation: https://doi.org/10.5194/egusphere20221310AC4

AC4: 'Reply on RC4', Chris Miele, 21 Mar 2023

EC1: 'Editor Comment on egusphere20221310', Reinhard Drews, 23 Mar 2023
Dear Authors and Reviewers,
thanks for a vivid discussions phase, which I have now ended after receiving four reviews. In order to consider publication in The Cryosphere, the manuscript requires substantial revisions likely including a change of the article type from “research article” to “brief communication”. My more detailed comments are as follows:
[1] I would like to thank all reviewers. Their comments demonstrate a strong expertise in the foundations of ice dynamic modelling including the asymptotic analysis required in modelling iceshelf flow. Many constructive suggestions are provided, including explicit derivations linking earlier works to the results of this study.
[2] There is no consensus between reviewers as to whether or not this paper satisfies the criteria of a research article. The main concern synthesizing from the more critical reviews (RC2, RC4) is a lack of novelty, e.g., compared to earlier works such as Schoof and Hindmarsh (2010). Here, the authors agree that there is no “discovery on which we claim novelty.” Instead they argue that “..all students of glaciology will know that negligible vertical shear is not a necessary component of a shallow shelf model”. This point, however, is confronted by multiple comments (RC3 & RC4) that the paper is hard to read even for some invited experts. It is therefore doubtful if this objective will be achieved.
[3] In summary, I believe that the lack of novelty precludes classification of this article as a “research article”. Going forward, I suggest that the authors provide a significantly revised version that takes the reviewer’s comments into account and delivers a clear message to the intended target audience in form of a “brief communication”. This is difficult to do for such a topic, and it will require substantial shortening and reframing. I also understand that there might be different understandings for the terms “novelty” & “research articles etc.” between different authors, reviewers, and editors. However, given the reviews and the publication criteria of The Cryosphere, I believe this will be the best option to go for.
Thank you again for all your work. Although this discussion contains some contraversial points, I surely appreciate this form of scientific exchange.
Kind regards,
Reinhard Drews
Citation: https://doi.org/10.5194/egusphere20221310EC1
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negligiblein ice shelf models. We address this miscommunication, providing conceptual guidance regarding this often misrepresented stress. Fundamentally, vertical shear is required to balance thickness gradients in ice shelves.
Chris Miele et al.
Chris Miele et al.
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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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