the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 17 Oct 2024
Calving from horizontal forces in a revised crevasse-depth framework
Abstract. Calving is a key process for the future of our ice sheets and oceans, but representing it in models remains challenging. Among numerous possible calving parameterisations, the crevasse-depth law remains attractive for its clear physical interpretation and its performance in models. In its classic form, however, it requires ad-hoc and arguably unphysical modifications to produce crevasses that are deep enough to result in calving. Here, we adopt a recent analytical approach accounting for the feedback between crevassing and the stress field and varying the density of water in basal crevasses, and show that it removes the need for such ad-hoc modifications. After accounting for ice tensile strength and basal friction, we show that the revised formulation predicts that full-thickness calving should occur at flotation when the calving front ice thickness is greater than around 400 m. It also predicts no calving for ice thinner than around 400 m, suggesting that calving at such glacier fronts is not driven purely by horizontal forces. We find good observational support for this analysis. We advance the revised crevasse-depth formulation as a step towards understanding differing calving styles and a better representation of calving in numerical models.
Competing interests: The contact author has declared that neither of the authors has any competing interests.
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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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Interactive discussion
Status: closed
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CC1: 'Comment on egusphere-2024-2927', Yue Meng, 22 Nov 2024
Publisher’s note: this comment is a copy of RC1 and its content was therefore removed on 25 November 2024.
Citation: https://doi.org/10.5194/egusphere-2024-2927-CC1 -
RC1: 'Comment on egusphere-2024-2927', Yue Meng, 22 Nov 2024
General comments
Following Buck (2023), The authors modified the crevasse depth law with a simple method to account for the stress concentration under crevassing, a variable density of water in basal crevasses, and non-zero ice tensile strength. The framework has the potential to understanding differing calving styles and a better parameterization of calving in numerical models. However, there are major concerns that need to be resolved before considering publication.
- The structure of the analytical model can be confusing. The heart of the analytical model – the horizontal force balance (Eqn11) – involves many hidden assumptions that the authors prove to be invalid in sections afterwards. For instance, Eqn11 assumes no basal friction (proved to be inaccurate in Sec.3.5), static condition (proved to be wrong when there is no real solution to Eqn15,16, line 335 “the forces on the nascent calving block cannot be balanced, so the block accelerates relative to the glacier…”). To improve readability of the paper, it might be better to start with the most general form of force balance by taking basal friction and acceleration into account in Eqn.11.
- I’m confused by the authors’ definition of calving: when there is no real solution to surface/basal crevasse depth (Eqn15,16), which, to me, implies invalid assumptions underlying current analytical model. For instance, in Fig4, an increase in rho_c will give real solutions to Eqn15,16. Does this mean rho_c=1000 kg/m^3 is just not physical? In line 85, the authors state that “it is assumed there is an open hydraulic connection from the bed below the glacier to the calving front”, does this contradict with the later assumption of rho_c=1000kg/m^3? If we incorporate an inertia term in Eqn. 11, will it be guaranteed to have real solutions instead?
As a side note, calving was previously defined as “total fractional crevassing f = 1” in literature (Buck, 2023, Coffey et al., 2023 Theoretical stability of ice shelf basal crevasses with a vertical temperature profile), which seems more intuitive and physical to me. I’m curious whether there are plausible ways to modify the analytical model (i.e., relax some assumptions), to avoid the “undefined crevasse fraction” issue.
- Section 3.2, shall the tensile strength be nondimensionalized by rho_i*g*H_i? Does it vary with ice thickness? The authors might want to justify why in many figures tensile strength is fixed while ice thickness is varying across a wide range? (Fig.5, 6, 7, 8, 9)
- Figure 7, 9: The original paper for the observational data also has a stability phase diagram of terminus configuration (Figure 3 in Ma et al., 2017), and it seems to explain the data better. It is reasonable that the authors (no shear failure) have different predictions than Ma et al., 2017 (including shear failure). However, in Ma et al., 2017, ice tensile strength is 0 (Figure 1), which is claimed to be invalid in this manuscript instead? Why is there a contradiction?
Specific comments
- Figure 5: A question comes to mind that “why happens for thick termini that are at floatation? (commonly observed across Greenland)” Does the figure imply termini thicker than 300m can never be at floatation stably?
- Line 245, “when we do assume zero tensile strength, the total fractional crevassing is either 1 or undefined for all possible calving fronts, hence this is not a useful calving law”. The sentence could be rephrased. This conclusion is drawn based on assumptions made in this paper but might not be general?
- Line 285: “if we use Eq.22 in the modified crevasse sizes Eqs. 15& 16 then for grounded glaciers, calving occurs…” Eqs. 15 & 16 is derived by assuming no basal friction in Eqn. 11, right? Is it equivalent between: 1) what the authors did here; and 2) including tau_b in Eqn. 11 -> rederive Eqn. 12 -> combining 12,13,14 -> expression for ds’/H, db’/H by considering basal friction. Does 2) seem easier to understand?
- Figure 9: the observations have different crevasse spacing, L, right? If the calving criterion is very sensitive to L, maybe the authors should include the information of L for the observation data as well, before comparing with the analytical model?
Citation: https://doi.org/10.5194/egusphere-2024-2927-RC1 - AC1: 'Reply on RC1', Donald Slater, 09 Jan 2025
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RC2: 'Comment on egusphere-2024-2927', Anonymous Referee #2, 30 Nov 2024
GENERAL COMMENT:
The manuscript “Calving from horizontal forces in a revised crevasse-depth framework” by Slater and Wagner modifies the ‘classic’ crevasse-depth calving law by introducing feedback with the background stress field and the water density within basal crevasses. This work builds on recent analytical formulations to improve the ‘classic’ crevasse-depth law in a largely theoretical exercise. The manuscript is of immediate interest of folks in the cryosphere community and is appropriate for the readership of the Cryosphere journal.
While the manuscript provides a thorough introduction to calving laws, I found it quite difficult to follow. The structure poses some challenges, particularly in the introduction. Although the discussion of calving laws is detailed, it took me several readings to fully understand it. For instance, it wasn’t immediately clear that Nye (1955) is considered the 'classic' crevasse-depth law (is that correct?). There have been many modifications since Nye’s original stress-balance formulation, and the authors do a good job to list them. However, later in the methodology sections (i.e. 2.2), the manuscript includes a detailed formulation of this approach, but it’s unclear where these equations originate from—Nye? Weertman? Furthermore, in line 111, the 'line of argument' from Buck (2023) is introduced, which raises the question: are we still discussing the original formulation? While I’m not suggesting the formulas are incorrect, these sections are quite confusing, and the definition of the 'classic' formulation remains ambiguous. Following this, Buck (2023)’s modifications are presented in Section 2.3, but the integration of equations (1–2) with the classic formulation (equations 3–8) is unclear. After reading these sections, I got completely lost on the novelty of this paper leaving me with this question: What is the main difference with Buck 2023?
Finally, in the results section (i.e. 3.5), the equations are further edited with the inclusion of basal friction. I appreciated this part, and I think it is an important aspect of the manuscript given the ongoing discussion in the community surrounding the importance of sliding laws. However, this section mixes methods and results, which had to bring me back to the initial equations, further slowing down the reading process. Wouldn’t it be perhaps more efficient to introduce the novel formulation at the beginning of the methodology section and move the classic laws in the appendix? I know this would be a major edit for this manuscript and ultimately it is the authors’ personal choice, but it is a scenario that is perhaps worth considering.
To be clear, I am not suggesting that this work lacks novelty, and I commend the authors for the detailed mathematical analysis presented—it is evident that this required significant time and effort. However, the structure in which the methods and results are presented makes the manuscript occasionally slow and difficult to follow. These issues should be addressed to improve clarity and accessibility before publication.
MINOR COMMENTS:
Line 20: There is also a large body of literature which analyze the how individual rifts and the modulation of their infill (ice melange) can lead to calving, whilst in complete absence of the factors listed here (e.g. Borstad 2017 GRL and Larour 2021 PNAS). This aspect is important to mention here.
Line 192: I am not sure what is meant with ‘an example of the modified crevasse sizes’.
Line 316: Unfortunately, this sentence at this point of the manuscript leaves me questioning the novelty again. What is the main difference with Buck 2023?
Line 320: See my general comment above.
Citation: https://doi.org/10.5194/egusphere-2024-2927-RC2 - AC1: 'Reply on RC1', Donald Slater, 09 Jan 2025
Interactive discussion
Status: closed
-
CC1: 'Comment on egusphere-2024-2927', Yue Meng, 22 Nov 2024
Publisher’s note: this comment is a copy of RC1 and its content was therefore removed on 25 November 2024.
Citation: https://doi.org/10.5194/egusphere-2024-2927-CC1 -
RC1: 'Comment on egusphere-2024-2927', Yue Meng, 22 Nov 2024
General comments
Following Buck (2023), The authors modified the crevasse depth law with a simple method to account for the stress concentration under crevassing, a variable density of water in basal crevasses, and non-zero ice tensile strength. The framework has the potential to understanding differing calving styles and a better parameterization of calving in numerical models. However, there are major concerns that need to be resolved before considering publication.
- The structure of the analytical model can be confusing. The heart of the analytical model – the horizontal force balance (Eqn11) – involves many hidden assumptions that the authors prove to be invalid in sections afterwards. For instance, Eqn11 assumes no basal friction (proved to be inaccurate in Sec.3.5), static condition (proved to be wrong when there is no real solution to Eqn15,16, line 335 “the forces on the nascent calving block cannot be balanced, so the block accelerates relative to the glacier…”). To improve readability of the paper, it might be better to start with the most general form of force balance by taking basal friction and acceleration into account in Eqn.11.
- I’m confused by the authors’ definition of calving: when there is no real solution to surface/basal crevasse depth (Eqn15,16), which, to me, implies invalid assumptions underlying current analytical model. For instance, in Fig4, an increase in rho_c will give real solutions to Eqn15,16. Does this mean rho_c=1000 kg/m^3 is just not physical? In line 85, the authors state that “it is assumed there is an open hydraulic connection from the bed below the glacier to the calving front”, does this contradict with the later assumption of rho_c=1000kg/m^3? If we incorporate an inertia term in Eqn. 11, will it be guaranteed to have real solutions instead?
As a side note, calving was previously defined as “total fractional crevassing f = 1” in literature (Buck, 2023, Coffey et al., 2023 Theoretical stability of ice shelf basal crevasses with a vertical temperature profile), which seems more intuitive and physical to me. I’m curious whether there are plausible ways to modify the analytical model (i.e., relax some assumptions), to avoid the “undefined crevasse fraction” issue.
- Section 3.2, shall the tensile strength be nondimensionalized by rho_i*g*H_i? Does it vary with ice thickness? The authors might want to justify why in many figures tensile strength is fixed while ice thickness is varying across a wide range? (Fig.5, 6, 7, 8, 9)
- Figure 7, 9: The original paper for the observational data also has a stability phase diagram of terminus configuration (Figure 3 in Ma et al., 2017), and it seems to explain the data better. It is reasonable that the authors (no shear failure) have different predictions than Ma et al., 2017 (including shear failure). However, in Ma et al., 2017, ice tensile strength is 0 (Figure 1), which is claimed to be invalid in this manuscript instead? Why is there a contradiction?
Specific comments
- Figure 5: A question comes to mind that “why happens for thick termini that are at floatation? (commonly observed across Greenland)” Does the figure imply termini thicker than 300m can never be at floatation stably?
- Line 245, “when we do assume zero tensile strength, the total fractional crevassing is either 1 or undefined for all possible calving fronts, hence this is not a useful calving law”. The sentence could be rephrased. This conclusion is drawn based on assumptions made in this paper but might not be general?
- Line 285: “if we use Eq.22 in the modified crevasse sizes Eqs. 15& 16 then for grounded glaciers, calving occurs…” Eqs. 15 & 16 is derived by assuming no basal friction in Eqn. 11, right? Is it equivalent between: 1) what the authors did here; and 2) including tau_b in Eqn. 11 -> rederive Eqn. 12 -> combining 12,13,14 -> expression for ds’/H, db’/H by considering basal friction. Does 2) seem easier to understand?
- Figure 9: the observations have different crevasse spacing, L, right? If the calving criterion is very sensitive to L, maybe the authors should include the information of L for the observation data as well, before comparing with the analytical model?
Citation: https://doi.org/10.5194/egusphere-2024-2927-RC1 - AC1: 'Reply on RC1', Donald Slater, 09 Jan 2025
-
RC2: 'Comment on egusphere-2024-2927', Anonymous Referee #2, 30 Nov 2024
GENERAL COMMENT:
The manuscript “Calving from horizontal forces in a revised crevasse-depth framework” by Slater and Wagner modifies the ‘classic’ crevasse-depth calving law by introducing feedback with the background stress field and the water density within basal crevasses. This work builds on recent analytical formulations to improve the ‘classic’ crevasse-depth law in a largely theoretical exercise. The manuscript is of immediate interest of folks in the cryosphere community and is appropriate for the readership of the Cryosphere journal.
While the manuscript provides a thorough introduction to calving laws, I found it quite difficult to follow. The structure poses some challenges, particularly in the introduction. Although the discussion of calving laws is detailed, it took me several readings to fully understand it. For instance, it wasn’t immediately clear that Nye (1955) is considered the 'classic' crevasse-depth law (is that correct?). There have been many modifications since Nye’s original stress-balance formulation, and the authors do a good job to list them. However, later in the methodology sections (i.e. 2.2), the manuscript includes a detailed formulation of this approach, but it’s unclear where these equations originate from—Nye? Weertman? Furthermore, in line 111, the 'line of argument' from Buck (2023) is introduced, which raises the question: are we still discussing the original formulation? While I’m not suggesting the formulas are incorrect, these sections are quite confusing, and the definition of the 'classic' formulation remains ambiguous. Following this, Buck (2023)’s modifications are presented in Section 2.3, but the integration of equations (1–2) with the classic formulation (equations 3–8) is unclear. After reading these sections, I got completely lost on the novelty of this paper leaving me with this question: What is the main difference with Buck 2023?
Finally, in the results section (i.e. 3.5), the equations are further edited with the inclusion of basal friction. I appreciated this part, and I think it is an important aspect of the manuscript given the ongoing discussion in the community surrounding the importance of sliding laws. However, this section mixes methods and results, which had to bring me back to the initial equations, further slowing down the reading process. Wouldn’t it be perhaps more efficient to introduce the novel formulation at the beginning of the methodology section and move the classic laws in the appendix? I know this would be a major edit for this manuscript and ultimately it is the authors’ personal choice, but it is a scenario that is perhaps worth considering.
To be clear, I am not suggesting that this work lacks novelty, and I commend the authors for the detailed mathematical analysis presented—it is evident that this required significant time and effort. However, the structure in which the methods and results are presented makes the manuscript occasionally slow and difficult to follow. These issues should be addressed to improve clarity and accessibility before publication.
MINOR COMMENTS:
Line 20: There is also a large body of literature which analyze the how individual rifts and the modulation of their infill (ice melange) can lead to calving, whilst in complete absence of the factors listed here (e.g. Borstad 2017 GRL and Larour 2021 PNAS). This aspect is important to mention here.
Line 192: I am not sure what is meant with ‘an example of the modified crevasse sizes’.
Line 316: Unfortunately, this sentence at this point of the manuscript leaves me questioning the novelty again. What is the main difference with Buck 2023?
Line 320: See my general comment above.
Citation: https://doi.org/10.5194/egusphere-2024-2927-RC2 - AC1: 'Reply on RC1', Donald Slater, 09 Jan 2025
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Till J. W. Wagner
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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