the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 08 Feb 2024
Refinement of the fluidity parameter range with a stress exponent of four in Glen’s law: insights from Antarctic bed topography model
Abstract. Understanding the stress-dependent behaviour of the ice sheet is critical to projecting ice mass balance. Glen's law is used to calculate ice viscosity, conventionally with a stress exponent (n) equal to three. However, a stress exponent of four has recently been proposed for ice dynamics. The suggested range of the fluidity parameter (A) for n = 4 is of the order of six (i.e., 10-35 to 10-29 Pa-4 s-1), leading to a significant uncertainty in ice velocity than when n = 3. Here, we refined A to within one order, aligning with observed Antarctic ice velocities with simplified slope and Antarctic bed topography models. The Antarctic bed topography models, based on Antarctic BedMap2 data, specifically include the Ronne, Thwaites, and Ross Ice Shelves for West Antarctica, and the Amery, Shackleton Ice Shelves, and Mertz Glacial Tongue for East Antarctica. We found that the simplified model and the West and East Antarctic models share a common range of A values for matching observed Antarctic ice velocities. Refinement narrows the range of A to within one order for both the simplified and West/East Antarctic models. The narrowed range for A is from 4.0 × to 16.0 × Pa-4 s-1. Both models have common A values for representing Antarctic ice velocity, providing an insight into the applicable A values.
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Status: closed
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RC1: 'Comment on egusphere-2023-2987', Daniel Richards, 19 Mar 2024
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AC2: 'Reply on RC1', Byung-Dal So, 22 Apr 2024
Dear Editor and Reviewers,
We appreciate the reviewers' insightful and careful comments on our manuscript. We have done our best to address the reviewers' concerns.
The main text and all figures have been substantially revised. All authors agree with the modifications made to the manuscript.
Please see the attached response letter.
Best wishes,
Byung-Dal and SuJeong
-
AC2: 'Reply on RC1', Byung-Dal So, 22 Apr 2024
-
RC2: 'Comment on egusphere-2023-2987', Anonymous Referee #2, 22 Mar 2024
This study seeks to constrain the values of the fluidity parameter A when using a stress exponent value of n=4, rather than the canonical n=3 value. The authors use a full Stokes model in two configurations – a simplified, linear bed slope model and a model of Antarctic bed topography – with varying values of A and n=4 and compare the resulting velocities to those observed in Antarctica. The authors ultimately constrain the value of A to a small range of 4 x 10^-32 to 16 x 10^-32 Pa^-4 s^-1. The goal of this work is an important one, as it makes the use of n=4 accessible in ice sheet models by providing a calibrated value of A to use alongside n=4. Further, constraints on the fluidity parameter inherently provide constraints on many physical properties of ice sheets, stated by the authors, that can affect future flow. I have questions about the methodology that would benefit from further analysis and explanation in the paper for the readers to derive insight from the results in this study. I would recommend that those questions be explored prior to publication.
Methodology: The method of prescribing a constant fluidity value across the domain (of both the simplified model and the Antarctic models), and then varying this value to match observations seems to have some limitations. Firstly, the constant A field inherently limits the takeaways of this study, as fluidity is likely to vary spatially (and temporally) due to a number of ice properties, such as temperature, ice damage, ice crystalline fabric, etc. Deviations between the modeled velocities and the observed velocities could be due to other factors besides the average A field, such as small regions of elevated A values (due to, perhaps, damage and fracturing), or variations in temperature at depth. Secondly, the method is restricted by the specific values that the authors choose to evaluate. It makes me wonder why the authors didn’t apply a formal inversion method (such as Larour and others, 2005, among many others that have used such a technique) with n=4. Such a method would be able to capture spatial variations in A, at least in a two-dimensional sense, and would not be restricted by the values chosen by the authors. At the very least, the authors should add to the discussion section a description of these simplifications and the implications for these results.
The simplified slope model seems to be the most limited in comparisons to observations. How do we know that velocities that differ from observed Antarctic velocities are not due to the simplified nature of the bed geometry? I also wonder if the results from the simplified slope model add to the results that are produced from the Antarctic model, for which the comparison to observations can provide more insight.
Finally, the authors discuss in the Discussion section the effect of the sliding exponent m, which is a poorly constrained parameter and could also affect Antarctic velocities. However, the friction coefficient C is not discussed in this section and also has an effect on ice velocities. The authors should discuss how they chose the value of C, particularly in the Antarctic models, and whether this affects their results. In theory, one might imagine that you could obtain the velocities with one combination of A and C values and obtain the same velocities with a very different A and an appropriately tuned C.
Results: The range of A values ultimately constrained by these simulations is quite narrow, far more narrow than the expected spatial variation in A due to heating, fabric (which itself can affect A by 0.5-1 order of magnitude), damage (which can affect A by many orders of magnitude, in theory), water content, etc. I believe this result would be different if the authors used an inverse method, but if the authors choose not to, it’s important to put this range into context – in particular, that it is a range of average values, and in dynamical regions of ice sheets, this value may be significantly higher or (possibly) lower due to material and physical properties.
Other Items:
- Citations in lines 30-33 could be adjusted. Citations for n=3 could include Jezek et al. 1985, Martin and Sanderson 1980, Paterson et al. 1983, and the original Glen papers (Glen 1955). I also believe that Behn et al. 2021 was not primarily using geodetic data, they were applying ice core and laboratory data along with models. Further, the statement that n=3 reproduces surface ice velocity does not seem to be supported by the citations, as I believe most of the citations were looking at field observations, both at the surface and at depth. If any of this is not correct, please feel free to ignore, but it may be worth double-checking these citations.
- In the paragraph starting at line 34, I believe Millstein et al. 2022 and Bons et al. 2018 should be cited here, since they are both observational studies suggesting that n varies but tends to n=4 in their study regions. I know that the authors cite both these studies in the next paragraph about A, but as both of these studies are primarily about n, they fit into this paragraph as well.
- L41: The fluidity parameter A is also affected by fabric, damage, impurities, among others
- L45: what is the “previous ice sheet model”?
- L46: The sentence “The values of A recent inference…argues for n of approximately four” I had trouble understanding.
- L139: 2000 m/yr velocities are not higher than the maximum value in Antarctica (Pine Island Glacier has velocities at or near 4000 m/yr), but it is on the high side for the average of ice shelves
- Fig 4: the yellow and orange lines are a bit hard to see
- In general, I would recommend italicizing A and n, or using the Latex math environment, to distinguish them from the prose
Citations
Larour (2005), Rheology of the Ronne Ice Shelf, Antarctica, inferred from satellite radar inferometry data using an inverse control method, Geophysical Research Letters (32)5, doi: 10.1029/2004GL021693
Jezek, et al. (1985), Rheology of Glacier Ice, Science (227)4692, doi: 10.1126/science.227.4692.1335
Paterson (1983), Deformation within polar ice sheets: An analysis of the Byrd Station and Camp Century borehole-tilting measurements, Cold Regions Science and Technology, (8)2, doi: 10.1016/0165-232X(83)90007-1
Glen (1955), The creep of polycrystalline ice, Proceedings of the Royal Society A: Mathematical, Physical, and Engineering Sciences
Citation: https://doi.org/10.5194/egusphere-2023-2987-RC2 -
AC1: 'Reply on RC2', Byung-Dal So, 22 Apr 2024
Dear Editor and Reviewers,
We appreciate the reviewers' insightful and careful comments on our manuscript. We have done our best to address the reviewers' concerns.
The main text and all figures have been substantially revised. All authors agree with the modifications made to the manuscript.
Please see the attached response letter.
Best wishes,
Byung-Dal and SuJeong
Status: closed
-
RC1: 'Comment on egusphere-2023-2987', Daniel Richards, 19 Mar 2024
-
AC2: 'Reply on RC1', Byung-Dal So, 22 Apr 2024
Dear Editor and Reviewers,
We appreciate the reviewers' insightful and careful comments on our manuscript. We have done our best to address the reviewers' concerns.
The main text and all figures have been substantially revised. All authors agree with the modifications made to the manuscript.
Please see the attached response letter.
Best wishes,
Byung-Dal and SuJeong
-
AC2: 'Reply on RC1', Byung-Dal So, 22 Apr 2024
-
RC2: 'Comment on egusphere-2023-2987', Anonymous Referee #2, 22 Mar 2024
This study seeks to constrain the values of the fluidity parameter A when using a stress exponent value of n=4, rather than the canonical n=3 value. The authors use a full Stokes model in two configurations – a simplified, linear bed slope model and a model of Antarctic bed topography – with varying values of A and n=4 and compare the resulting velocities to those observed in Antarctica. The authors ultimately constrain the value of A to a small range of 4 x 10^-32 to 16 x 10^-32 Pa^-4 s^-1. The goal of this work is an important one, as it makes the use of n=4 accessible in ice sheet models by providing a calibrated value of A to use alongside n=4. Further, constraints on the fluidity parameter inherently provide constraints on many physical properties of ice sheets, stated by the authors, that can affect future flow. I have questions about the methodology that would benefit from further analysis and explanation in the paper for the readers to derive insight from the results in this study. I would recommend that those questions be explored prior to publication.
Methodology: The method of prescribing a constant fluidity value across the domain (of both the simplified model and the Antarctic models), and then varying this value to match observations seems to have some limitations. Firstly, the constant A field inherently limits the takeaways of this study, as fluidity is likely to vary spatially (and temporally) due to a number of ice properties, such as temperature, ice damage, ice crystalline fabric, etc. Deviations between the modeled velocities and the observed velocities could be due to other factors besides the average A field, such as small regions of elevated A values (due to, perhaps, damage and fracturing), or variations in temperature at depth. Secondly, the method is restricted by the specific values that the authors choose to evaluate. It makes me wonder why the authors didn’t apply a formal inversion method (such as Larour and others, 2005, among many others that have used such a technique) with n=4. Such a method would be able to capture spatial variations in A, at least in a two-dimensional sense, and would not be restricted by the values chosen by the authors. At the very least, the authors should add to the discussion section a description of these simplifications and the implications for these results.
The simplified slope model seems to be the most limited in comparisons to observations. How do we know that velocities that differ from observed Antarctic velocities are not due to the simplified nature of the bed geometry? I also wonder if the results from the simplified slope model add to the results that are produced from the Antarctic model, for which the comparison to observations can provide more insight.
Finally, the authors discuss in the Discussion section the effect of the sliding exponent m, which is a poorly constrained parameter and could also affect Antarctic velocities. However, the friction coefficient C is not discussed in this section and also has an effect on ice velocities. The authors should discuss how they chose the value of C, particularly in the Antarctic models, and whether this affects their results. In theory, one might imagine that you could obtain the velocities with one combination of A and C values and obtain the same velocities with a very different A and an appropriately tuned C.
Results: The range of A values ultimately constrained by these simulations is quite narrow, far more narrow than the expected spatial variation in A due to heating, fabric (which itself can affect A by 0.5-1 order of magnitude), damage (which can affect A by many orders of magnitude, in theory), water content, etc. I believe this result would be different if the authors used an inverse method, but if the authors choose not to, it’s important to put this range into context – in particular, that it is a range of average values, and in dynamical regions of ice sheets, this value may be significantly higher or (possibly) lower due to material and physical properties.
Other Items:
- Citations in lines 30-33 could be adjusted. Citations for n=3 could include Jezek et al. 1985, Martin and Sanderson 1980, Paterson et al. 1983, and the original Glen papers (Glen 1955). I also believe that Behn et al. 2021 was not primarily using geodetic data, they were applying ice core and laboratory data along with models. Further, the statement that n=3 reproduces surface ice velocity does not seem to be supported by the citations, as I believe most of the citations were looking at field observations, both at the surface and at depth. If any of this is not correct, please feel free to ignore, but it may be worth double-checking these citations.
- In the paragraph starting at line 34, I believe Millstein et al. 2022 and Bons et al. 2018 should be cited here, since they are both observational studies suggesting that n varies but tends to n=4 in their study regions. I know that the authors cite both these studies in the next paragraph about A, but as both of these studies are primarily about n, they fit into this paragraph as well.
- L41: The fluidity parameter A is also affected by fabric, damage, impurities, among others
- L45: what is the “previous ice sheet model”?
- L46: The sentence “The values of A recent inference…argues for n of approximately four” I had trouble understanding.
- L139: 2000 m/yr velocities are not higher than the maximum value in Antarctica (Pine Island Glacier has velocities at or near 4000 m/yr), but it is on the high side for the average of ice shelves
- Fig 4: the yellow and orange lines are a bit hard to see
- In general, I would recommend italicizing A and n, or using the Latex math environment, to distinguish them from the prose
Citations
Larour (2005), Rheology of the Ronne Ice Shelf, Antarctica, inferred from satellite radar inferometry data using an inverse control method, Geophysical Research Letters (32)5, doi: 10.1029/2004GL021693
Jezek, et al. (1985), Rheology of Glacier Ice, Science (227)4692, doi: 10.1126/science.227.4692.1335
Paterson (1983), Deformation within polar ice sheets: An analysis of the Byrd Station and Camp Century borehole-tilting measurements, Cold Regions Science and Technology, (8)2, doi: 10.1016/0165-232X(83)90007-1
Glen (1955), The creep of polycrystalline ice, Proceedings of the Royal Society A: Mathematical, Physical, and Engineering Sciences
Citation: https://doi.org/10.5194/egusphere-2023-2987-RC2 -
AC1: 'Reply on RC2', Byung-Dal So, 22 Apr 2024
Dear Editor and Reviewers,
We appreciate the reviewers' insightful and careful comments on our manuscript. We have done our best to address the reviewers' concerns.
The main text and all figures have been substantially revised. All authors agree with the modifications made to the manuscript.
Please see the attached response letter.
Best wishes,
Byung-Dal and SuJeong
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