the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 18 Mar 2024
The effect of ice rheology on shelf edge bending
Abstract. The distribution of water pressure on the vertical front of an ice shelf has been shown to cause downward bending of the edge if the ice has vertically uniform viscosity. Satellite lidar observations show upward bending of shelf edges for some areas with cold surface temperatures. A simple analysis shows that upward bending of shelf edges can result from vertical variations in ice viscosity. Such variations are an expected consequence of the temperature dependence of ice viscosity and temperature variations through a shelf. Resultant vertical variations in horizontal stress produce an internal bending moment that can counter the bending moment due to the shelf-front water pressure. Assuming a linear profile of ice temperature with depth and an Arrhenius relation between temperature and strain rate allows derivation of an analytic expression for internal bending moments. The effect of a power-law relation between stress difference and strain rate is also included analytically. The key ice rheologic parameter affecting shelf edge bending is the ratio of the activation energy, Q, and the power-law exponent, n. For cold ice surface temperatures and large values of Q/n, upward bending is expected, while for warm surface temperatures and small values of Q/n downward bending is expected. The amplitude of bending should scale with the ice shelf thickness to the power 3/2 and this is approximately consistent with a recent analysis of shelf edge deflections for the Ross Ice Shelf.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
(1175 KB)
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2024-557', Anonymous Referee #1, 22 Apr 2024
This manuscript presents a novel analysis of flexure at the terminus of a freely floating ice shelf. It addresses observations of upward flexure of the Ross Ice shelf near Roosevelt island. Whereas these were previously explained in terms of an eroded ice bench, this manuscript shows that a vertical variation in ice viscosity, arising from a linear variation in ice temperature and acting on a vertically uniform rate of extension, gives rise to a bending moment that can explain the observed flexure. This is an appealing hypothesis because ice benches are not observed at Ross (using the same sensor that has observed them elsewhere). The argument is supported by a clear and relatively simple mathematical theory that is consistent with a classical analysis of ice shelves (Weertman 1957). The manuscript is well written and well illustrated, easy to follow, and makes a novel and significant contribution to our understanding of ice-shelf dynamics. It has important implications for our understanding of ice-shelf calving. It should be published with minor revisions.
I see no major problems with the manuscript as written. The author has cleverly applied insights from plate flexure derived in the context of tectonics to ice shelves. Amusingly, the author highlights that his analysis was tee'd up by Reeh 1968, whose "mathematical troubles" are relieved by the simplifying assumption of a linear temperature variation through the ice shelf. This leads to a Taylor series expansion and truncation at leading order, making the moment integral tractable and unlocking a solution. This context prompts two relatively minor suggestions. The first is to better discuss and justify the linear temperature assumption, as this is crucial to progress. There are borehole measurements by Mike Craven et al (e.g., J. Glaciology, Vol. 55, No. 192, 2009) and likely others. Plotting their data in comparison to a linear fit might be nice.
The second is to use the second-order term in the Taylor series as a means to estimate the truncation error in equation (14). My quick calculation gives a multiplicative factor of exp[( T'/T_s z )2]. Taking z=h as an upper bound, this gives exp( (\delta T/T_s)2 ) ~ exp( (30/240)2 ) ~ 1.02. So a maximum 2% error in viscosity due to Taylor expansion. This could be propagated through the calculation to obtain the error on M_I (but in fact the linear temperature assumption must be a larger source of error).
My third suggestion is to more carefully discuss the time-dependence of viscoelastic flexure. Although the details will vary between problems, the scaling with time/(Maxwell time) should not. How does this affect the comparison with the Ross ice shelf? What is the age of that edge? Is it fresh (i.e., age/Maxwell << 1)? This relates to the approximate of stresses as, close to the shelf edge, they will be modified with time since calving. In this regard, the bi-metallic strip analogy is somewhat misleading, as it is in mechanical equilibrium at a fixed temperature.
Broadly, I think the author should draw more attention to the assumptions made and the caveats and cautions that they introduce. This would not detract from the importance of the manuscript, but would better promote further research to build and test the ideas introduced here.
Some detailed points, by line number in the manuscript:
[32] where ice shelf serves as an adjective, it should have a hyphen. E.g., ice-shelf edges
[Fig 1] expand the figure caption to explain the lines in these figures. Improve the resolution to clarify that the hashing are ascending and descending track lines.
[76--78] These two sentences say the same thing, which is confusing. Only one is needed.
[98--99] The sentence starting with "Imagine" is important but the reader hasn't yet been adequately informed about why. Somewhere above (maybe the introduction) there should be a brief discussion of how visco-elastic bending is time dependent.
[103] "To do this" grammatical issue here.
[163] The result here appear to be positive but represents downward flexure (line 124 states that upward bending corresponds to positive total applied moments). Please check signs.
[175] Spelling of MacAyeal.
[188] The assumption regarding stresses evaluated at large distance from the edge of the ice is somewhat sketchy so I think a bit more emphasis and discussion would be relevant here.
[210] A reference here to Weertman 1957 or similar would be appropriate and helpful.
[Fig 5b] I think that a version of this plot with a logarithmic x axis (and an expanded domain and range) would be helpful in seeing the asymptotic behaviour of M_I at large and small z_0/h.
[294] "illustrates shows"
[340] "places"
[throughout] mathematical notation should be italic but frequently appears as regular next.Citation: https://doi.org/10.5194/egusphere-2024-557-RC1 -
AC1: 'Reply on RC1', W. Roger Buck, 24 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-557/egusphere-2024-557-AC1-supplement.pdf
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AC1: 'Reply on RC1', W. Roger Buck, 24 May 2024
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RC2: 'Comment on egusphere-2024-557', Anonymous Referee #2, 28 Apr 2024
Summary:
The manuscript extends the theory first developed in Reeh. 1968 and convincingly proposes a new theory that can explain upward ice shelf bending without requiring the introduction of geometric features that aren’t always observed, as required the “bench” model. The paper is well-written and impactful while managing to remain brief. I do think some improvement in clarity could be offered by increasing the level of detail for some conceptual ideas. The paper, overall, is excellent and warrants prompt publication.
Suggestions:
- In the conceptual model section, bending generated by vertical variation in horizontal stress and associated internal moments is discussed. However, I am a little confused about the direction of the bending moment. If “there is a decrease in extensional stress with depth,” would this not correspond to a top out sense of bending? I am sure I am misunderstanding this in a perhaps amateurish way, but I am also sure that other readers will have this same confusion. I think the manuscript would benefit from a bit more detail explaining, from a conceptual perspective, the direction of internal moments generated by horizontal stresses that decrease in magnitude with depth, and the direction of the equivalent edge moment. Right now, it feels like clarity is sacrificed a bit in exchange for brevity.
Detailed comments:
[1] I feel that the introduction could flow slightly better if the first paragraph started broad and then narrowed (i.e. ice shelf breakup is important because it can enhance sea level rise -> one of the processes that cause breakup is bending -> bending causes breakup in the following two ways). This is just personal preference and does not need to be addressed unless the author agrees.
[40-41] Could be a good idea to mention a few ice shelves that show bending consistent with the typical downward bending and which ice shelves show the opposite sense of bending.
[51-57] This is written with panache and is fun to read!
[65] Presumably the gray lines are flow lines, but it might be good to mention this briefly in the caption.
[68] For context, it could be good to say something like “Ice shelves that are not heavily buttressed are under extension (Weertman, 1957). While ice shelves are typically assumed to have negligible vertical gradients in horizontal stress, significant vertical variation in viscosity generates vertical gradients in horizontal stress that cannot be neglected.”
[74] After this, we look at a couple examples in other geophysical settings. It could be good to end this paragraph, or start the next, with a sentence like “Similar physics are observed in several geophysical settings.” Right now, the jump is a little abrupt feeling.
[75-80] This paragraph is a little confusing. I would explicitly describe what causes vertical variation in horizontal stress in the lithospheric case. “Those authors note that gravity prevents lithospheric bending that is of much longer wavelength then the effective flexural wavelength of the layer. At very long length scales gravity prevents the layer from bending.” is redundant. Is gravity generating the vertical variation in horizontal stress? I would just explain this analogous case with a bit more detail to improve clarity.
[81] Remove “that that.”
[84] Should be “is equivalent” not “are equivalent.”
[84-85] I understand the general concept that uniform internal loading is equivalent to remote loading (common in fracture mechanics, for instance, when considering stresses acting on the edges of a cracked plate and stresses acting on the interior crack walls). It is not entirely clear to me why the equivalent on the end of a layer has opposite direction, however. A bit more explanation could be helpful. This ties into the broader comment above.
[110] This may just be how the draft version is typeset, but this figure is a little confusing to see before the reference horizontal stress has actually been defined.
[143] “water pressure” not “pressures”
[196] Quick half sentence explaining e-folding could be good. Most readers will probably already know, but clarity is always to be encouraged, especially because z_0 is an important parameter moving forward.
[340] “may allow new constraints to be placed” not “places.”
Citation: https://doi.org/10.5194/egusphere-2024-557-RC2 -
AC2: 'Reply on RC2', W. Roger Buck, 24 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-557/egusphere-2024-557-AC2-supplement.pdf
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AC3: 'Comment on egusphere-2024-557', W. Roger Buck, 03 Jun 2024
The attached figure is intended to replace figure 3 parts (b) and (c).
This is in response to Reviewer 2 who correctly noted that it was difficult to understand the signs of the stresses and bending moments in the old figures.
The new versions get rid of the reference horizontal stress and simply relate the distribution of the water pressure versus the total horizontal stress in the ice.
I hope it makes it clearer how a strong exponential dependence of ice viscosity on depth in a floating ice layer can lead to upward bending at a shelf edge.
In the revised paper I will have to define the e-folding length scale for viscosity with depth (z0) either in the caption and/or before the figure in inserted.
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2024-557', Anonymous Referee #1, 22 Apr 2024
This manuscript presents a novel analysis of flexure at the terminus of a freely floating ice shelf. It addresses observations of upward flexure of the Ross Ice shelf near Roosevelt island. Whereas these were previously explained in terms of an eroded ice bench, this manuscript shows that a vertical variation in ice viscosity, arising from a linear variation in ice temperature and acting on a vertically uniform rate of extension, gives rise to a bending moment that can explain the observed flexure. This is an appealing hypothesis because ice benches are not observed at Ross (using the same sensor that has observed them elsewhere). The argument is supported by a clear and relatively simple mathematical theory that is consistent with a classical analysis of ice shelves (Weertman 1957). The manuscript is well written and well illustrated, easy to follow, and makes a novel and significant contribution to our understanding of ice-shelf dynamics. It has important implications for our understanding of ice-shelf calving. It should be published with minor revisions.
I see no major problems with the manuscript as written. The author has cleverly applied insights from plate flexure derived in the context of tectonics to ice shelves. Amusingly, the author highlights that his analysis was tee'd up by Reeh 1968, whose "mathematical troubles" are relieved by the simplifying assumption of a linear temperature variation through the ice shelf. This leads to a Taylor series expansion and truncation at leading order, making the moment integral tractable and unlocking a solution. This context prompts two relatively minor suggestions. The first is to better discuss and justify the linear temperature assumption, as this is crucial to progress. There are borehole measurements by Mike Craven et al (e.g., J. Glaciology, Vol. 55, No. 192, 2009) and likely others. Plotting their data in comparison to a linear fit might be nice.
The second is to use the second-order term in the Taylor series as a means to estimate the truncation error in equation (14). My quick calculation gives a multiplicative factor of exp[( T'/T_s z )2]. Taking z=h as an upper bound, this gives exp( (\delta T/T_s)2 ) ~ exp( (30/240)2 ) ~ 1.02. So a maximum 2% error in viscosity due to Taylor expansion. This could be propagated through the calculation to obtain the error on M_I (but in fact the linear temperature assumption must be a larger source of error).
My third suggestion is to more carefully discuss the time-dependence of viscoelastic flexure. Although the details will vary between problems, the scaling with time/(Maxwell time) should not. How does this affect the comparison with the Ross ice shelf? What is the age of that edge? Is it fresh (i.e., age/Maxwell << 1)? This relates to the approximate of stresses as, close to the shelf edge, they will be modified with time since calving. In this regard, the bi-metallic strip analogy is somewhat misleading, as it is in mechanical equilibrium at a fixed temperature.
Broadly, I think the author should draw more attention to the assumptions made and the caveats and cautions that they introduce. This would not detract from the importance of the manuscript, but would better promote further research to build and test the ideas introduced here.
Some detailed points, by line number in the manuscript:
[32] where ice shelf serves as an adjective, it should have a hyphen. E.g., ice-shelf edges
[Fig 1] expand the figure caption to explain the lines in these figures. Improve the resolution to clarify that the hashing are ascending and descending track lines.
[76--78] These two sentences say the same thing, which is confusing. Only one is needed.
[98--99] The sentence starting with "Imagine" is important but the reader hasn't yet been adequately informed about why. Somewhere above (maybe the introduction) there should be a brief discussion of how visco-elastic bending is time dependent.
[103] "To do this" grammatical issue here.
[163] The result here appear to be positive but represents downward flexure (line 124 states that upward bending corresponds to positive total applied moments). Please check signs.
[175] Spelling of MacAyeal.
[188] The assumption regarding stresses evaluated at large distance from the edge of the ice is somewhat sketchy so I think a bit more emphasis and discussion would be relevant here.
[210] A reference here to Weertman 1957 or similar would be appropriate and helpful.
[Fig 5b] I think that a version of this plot with a logarithmic x axis (and an expanded domain and range) would be helpful in seeing the asymptotic behaviour of M_I at large and small z_0/h.
[294] "illustrates shows"
[340] "places"
[throughout] mathematical notation should be italic but frequently appears as regular next.Citation: https://doi.org/10.5194/egusphere-2024-557-RC1 -
AC1: 'Reply on RC1', W. Roger Buck, 24 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-557/egusphere-2024-557-AC1-supplement.pdf
-
AC1: 'Reply on RC1', W. Roger Buck, 24 May 2024
-
RC2: 'Comment on egusphere-2024-557', Anonymous Referee #2, 28 Apr 2024
Summary:
The manuscript extends the theory first developed in Reeh. 1968 and convincingly proposes a new theory that can explain upward ice shelf bending without requiring the introduction of geometric features that aren’t always observed, as required the “bench” model. The paper is well-written and impactful while managing to remain brief. I do think some improvement in clarity could be offered by increasing the level of detail for some conceptual ideas. The paper, overall, is excellent and warrants prompt publication.
Suggestions:
- In the conceptual model section, bending generated by vertical variation in horizontal stress and associated internal moments is discussed. However, I am a little confused about the direction of the bending moment. If “there is a decrease in extensional stress with depth,” would this not correspond to a top out sense of bending? I am sure I am misunderstanding this in a perhaps amateurish way, but I am also sure that other readers will have this same confusion. I think the manuscript would benefit from a bit more detail explaining, from a conceptual perspective, the direction of internal moments generated by horizontal stresses that decrease in magnitude with depth, and the direction of the equivalent edge moment. Right now, it feels like clarity is sacrificed a bit in exchange for brevity.
Detailed comments:
[1] I feel that the introduction could flow slightly better if the first paragraph started broad and then narrowed (i.e. ice shelf breakup is important because it can enhance sea level rise -> one of the processes that cause breakup is bending -> bending causes breakup in the following two ways). This is just personal preference and does not need to be addressed unless the author agrees.
[40-41] Could be a good idea to mention a few ice shelves that show bending consistent with the typical downward bending and which ice shelves show the opposite sense of bending.
[51-57] This is written with panache and is fun to read!
[65] Presumably the gray lines are flow lines, but it might be good to mention this briefly in the caption.
[68] For context, it could be good to say something like “Ice shelves that are not heavily buttressed are under extension (Weertman, 1957). While ice shelves are typically assumed to have negligible vertical gradients in horizontal stress, significant vertical variation in viscosity generates vertical gradients in horizontal stress that cannot be neglected.”
[74] After this, we look at a couple examples in other geophysical settings. It could be good to end this paragraph, or start the next, with a sentence like “Similar physics are observed in several geophysical settings.” Right now, the jump is a little abrupt feeling.
[75-80] This paragraph is a little confusing. I would explicitly describe what causes vertical variation in horizontal stress in the lithospheric case. “Those authors note that gravity prevents lithospheric bending that is of much longer wavelength then the effective flexural wavelength of the layer. At very long length scales gravity prevents the layer from bending.” is redundant. Is gravity generating the vertical variation in horizontal stress? I would just explain this analogous case with a bit more detail to improve clarity.
[81] Remove “that that.”
[84] Should be “is equivalent” not “are equivalent.”
[84-85] I understand the general concept that uniform internal loading is equivalent to remote loading (common in fracture mechanics, for instance, when considering stresses acting on the edges of a cracked plate and stresses acting on the interior crack walls). It is not entirely clear to me why the equivalent on the end of a layer has opposite direction, however. A bit more explanation could be helpful. This ties into the broader comment above.
[110] This may just be how the draft version is typeset, but this figure is a little confusing to see before the reference horizontal stress has actually been defined.
[143] “water pressure” not “pressures”
[196] Quick half sentence explaining e-folding could be good. Most readers will probably already know, but clarity is always to be encouraged, especially because z_0 is an important parameter moving forward.
[340] “may allow new constraints to be placed” not “places.”
Citation: https://doi.org/10.5194/egusphere-2024-557-RC2 -
AC2: 'Reply on RC2', W. Roger Buck, 24 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-557/egusphere-2024-557-AC2-supplement.pdf
-
AC3: 'Comment on egusphere-2024-557', W. Roger Buck, 03 Jun 2024
The attached figure is intended to replace figure 3 parts (b) and (c).
This is in response to Reviewer 2 who correctly noted that it was difficult to understand the signs of the stresses and bending moments in the old figures.
The new versions get rid of the reference horizontal stress and simply relate the distribution of the water pressure versus the total horizontal stress in the ice.
I hope it makes it clearer how a strong exponential dependence of ice viscosity on depth in a floating ice layer can lead to upward bending at a shelf edge.
In the revised paper I will have to define the e-folding length scale for viscosity with depth (z0) either in the caption and/or before the figure in inserted.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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