The influence of firn-layer material properties on surface crevasse propagation in glaciers and ice shelves

Clayton, Theo; Duddu, Ravindra; Hageman, Tim; Martinez-Paneda, Emilio

doi:https://doi.org/10.5194/egusphere-2024-660

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https://doi.org/10.5194/egusphere-2024-660

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28 Mar 2024

| 28 Mar 2024

The influence of firn-layer material properties on surface crevasse propagation in glaciers and ice shelves

Theo Clayton, Ravindra Duddu, Tim Hageman, and Emilio Martinez-Paneda

Abstract. Linear elastic fracture mechanics (LEFM) models have been used to estimate crevasse depths in glaciers and to represent iceberg calving in ice sheet models. However, existing LEFM models assume glacier ice to be homogeneous and utilise the mechanical properties of fully consolidated ice. Using depth-invariant properties is not realistic, as the process of compaction from unconsolidated snow to firn to glacial ice is dependent on several environmental factors, typically leading to a lesser density and Young's modulus in upper surface strata. New analytical solutions for longitudinal stress profiles are derived, using depth-varying properties based on borehole data from the Ronne ice shelf, and used in an LEFM model to determine the maximum penetration depths of an isolated crevasse in grounded glaciers and floating ice shelves. These maximum crevasse depths are compared to those obtained for homogeneous glacial ice, showing the importance of including the effect of the upper unconsolidated firn layers on crevasse propagation. The largest reductions in penetration depth ratio were observed for shallow grounded glaciers, with variations in Young's modulus being more influential than firn density (a maximum difference in crevasse depth of 46 % and 20 % respectively); whereas, firn density changes resulted in an increase in penetration depth for thinner floating ice shelves (95 %–188 % difference in crevasse depth between constant and depth-varying properties). Thus, our study shows that the firn layer can increase the vulnerability of ice shelves to fracture and calving, highlighting the importance of considering depth-dependent firn-layer material properties in LEFM models for estimating crevasse penetration depths and predicting rift propagation.

Received: 05 Mar 2024 – Discussion started: 28 Mar 2024

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Journal article(s) based on this preprint

03 Dec 2024

The influence of firn layer material properties on surface crevasse propagation in glaciers and ice shelves

Theo Clayton, Ravindra Duddu, Tim Hageman, and Emilio Martínez-Pañeda

The Cryosphere, 18, 5573–5593, https://doi.org/10.5194/tc-18-5573-2024,https://doi.org/10.5194/tc-18-5573-2024, 2024

Short summary

Theo Clayton, Ravindra Duddu, Tim Hageman, and Emilio Martinez-Paneda

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-660', Anonymous Referee #1, 18 Apr 2024
Clayton et al. derived analytical equations and conducted LEFM analysis to study the influence of firn-layer material properties (depth-varying density and modulus) on surface crevasse propagation in glacier and ice shelves. They found that the firn layer has a stabilizing effect on grounded glaciers (free slip boundary condition), whereas a destabilizing effect on ice shelves, with regard to fracturing and calving. The study has important implications for assessing the stability of ice sheets or ice shelves. However, there are two major limitations in the assumptions of the models: i) Poisson ratio is assumed to be depth-invariant; ii) firn is assumed to be impermeable when evaluating the depth of meltwater-driven hydrofracture, neglecting the fact that meltwater will penetrate the porous firn layer instead of fracturing it. I suggest the authors reconsider the model assumption, or at least highlight the limitations, before it can receive further consideration.
Why do the authors neglect depth-variations in Poisson ratio? Shouldn’t Poisson ratio and Young’s modulus both strongly depend on the density? Will a depth-varying Poisson ratio (which is more realistic) significantly affect the results? Below attach some references on the depth variations in Poisson ratio [1, 2, 3]. One possible way to represent the depth varying mechanical properties could be developing empirical relationships between Poisson ratio/Young’s modulus, and the firn density.

The longitudinal stress was derived for compressible linear elasticity (Eqn.1 in the manuscript), why not viscous model? I think that a common approach, when looking at calving for example (e.g. Benn et al 2007, Ann [4]), is to calculate the background stresses from a viscous model (associated with long- term creep of the ice, and estimated perhaps from satellite-derived estimates of strain rate) instead of using an elastic model to calculate that background state. The authors might need some explanation justifying why they use linear elasticity to calculate the longitudinal stress.

Once the authors start to consider meltwater within the crevasse, it confuses me that the porous nature of firn is completely ignored. LEFM no longer holds for porous material and poromechanics [5] should be considered. Could the authors at least highlight the limitations of current results (Figure 5&7 in the main text)?

[1] Schlegel, R., Diez, A., Löwe, H., Mayer, C., Lambrecht, A., Freitag, J., ... & Eisen, O. (2019). Comparison of elastic moduli from seismic diving-wave and ice-core microstructure analysis in Antarctic polar firn. Annals of Glaciology, 60(79), 220-230.
[2] Smith, J. L. (1965). The elastic constants, strength and density of Greenland snow as determined from measurements of sonic wave velocity (Vol. 167). US Army Cold Regions Research & Engineering Laboratory.
[3] King, E. C., & Jarvis, E. P. (2007). Use of shear waves to measure Poisson's ratio in polar firn. Journal of Environmental and Engineering Geophysics, 12(1), 15-21.
[4] Benn, D. I., Cowton, T., Todd, J., & Luckman, A. (2017). Glacier calving in Greenland. Current Climate Change Reports, 3, 282-290.
[5] Coussy, O. (2004). Poromechanics. John Wiley & Sons.
Citation: https://doi.org/10.5194/egusphere-2024-660-RC1
- AC2: 'Reply on RC1', Emilio Martinez-Paneda, 04 Jun 2024
  
  Please see the attached document.
  
  Citation: https://doi.org/10.5194/egusphere-2024-660-AC2
RC2:
'Comment on egusphere-2024-660', Anonymous Referee #2, 03 May 2024

  The idea that it is worth considering how the properties of firn layers could affect the stresses that control surface crevasse opening is very compelling. However, this analysis makes the radical assumption that the stresses in an ice sheet or shelf are controlled solely by the elastic deformation of compressible ice. This is fine if one simply wants to go through a mathematical exercise, but the title, abstract and body of the paper imply that the results of this analysis applies to real ice sheets and shelves. I was particularly disturbed by the fact that the abstract does not make clear that this is an exercise based on ignoring viscous flow of ice. The fact that the Maxwell time of ice is on the order of days means that an ice sheet or shelf would have to have formed in less than a day for this analysis to be applicable.
   The major conclusion of the paper is that inclusion of low-density firn produces opposite effects for idealized ice sheets versus floating ice shelves. The abstract and a cursory reading of the paper makes this seem like a general conclusion. Upon closer reading it is clear that the ice shelf result is only for a particular region close to the edge of the shelf. The authors correctly note that assumption of perfect elasticity results in compression everywhere far from the shelf edge so that no surface crevasses should result for any assumed firn densities or Young’s Moduli! This is confusing because the paper only discusses analytic solutions for the stress field far from a shelf edge. To get surface crevassing on a compressible ice shelf with infinite viscosity requires bending stresses close to the edge of the shelf. The authors then use a finite element model to compute those stresses at a fixed position (250 m) from the shelf edge. At that position the predicted crevasse depth is increased by a decreasing firn density and Young’s Modulus. I assume that this is a robust result but it is hard to evaluate given the information in this paper. More importantly, the paper makes it seem that this is general result based on the analytical results derived in the paper, as is clear from the opening of the “Conclusions” section:
“In this paper, we derived analytical equations for the far field longitudinal stress including the effects of surface firn layers, described by depth-varying density and Young’s modulus profiles based on field data. These analytic expressions were used to perform fracture propagation studies on isolated air/water-filled surface crevasses in grounded glaciers and ice shelves …”
This certainly gave me the wrong idea when I first read the paper.
   The other major result of the paper is that low-density firn results in smaller crevasse depths for a grounded glacier compared to a uniform ice case. The authors note that this result contradicts the previous Linear Elastic Fracture Mechanics analysis of van der Veen (1998). I suspect that the difference with the previous study is caused by the assumption of purely elastic horizontal stresses which are less extensional at the ice sheet surface than the stresses assumed by van der Veen (1998). Thus, again I am not convinced that the results of the new analysis apply to the real world.
It is incumbent on these authors to make a case that the assumption of perfect elasticity gives insight into the opening of surface crevasses on real ice sheets and shelves.

Citation: https://doi.org/10.5194/egusphere-2024-660-RC2
- AC1: 'Reply on RC2', Emilio Martinez-Paneda, 04 Jun 2024
  
  Please see the attached document
  
  Citation: https://doi.org/10.5194/egusphere-2024-660-AC1

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-660', Anonymous Referee #1, 18 Apr 2024
Clayton et al. derived analytical equations and conducted LEFM analysis to study the influence of firn-layer material properties (depth-varying density and modulus) on surface crevasse propagation in glacier and ice shelves. They found that the firn layer has a stabilizing effect on grounded glaciers (free slip boundary condition), whereas a destabilizing effect on ice shelves, with regard to fracturing and calving. The study has important implications for assessing the stability of ice sheets or ice shelves. However, there are two major limitations in the assumptions of the models: i) Poisson ratio is assumed to be depth-invariant; ii) firn is assumed to be impermeable when evaluating the depth of meltwater-driven hydrofracture, neglecting the fact that meltwater will penetrate the porous firn layer instead of fracturing it. I suggest the authors reconsider the model assumption, or at least highlight the limitations, before it can receive further consideration.
Why do the authors neglect depth-variations in Poisson ratio? Shouldn’t Poisson ratio and Young’s modulus both strongly depend on the density? Will a depth-varying Poisson ratio (which is more realistic) significantly affect the results? Below attach some references on the depth variations in Poisson ratio [1, 2, 3]. One possible way to represent the depth varying mechanical properties could be developing empirical relationships between Poisson ratio/Young’s modulus, and the firn density.

The longitudinal stress was derived for compressible linear elasticity (Eqn.1 in the manuscript), why not viscous model? I think that a common approach, when looking at calving for example (e.g. Benn et al 2007, Ann [4]), is to calculate the background stresses from a viscous model (associated with long- term creep of the ice, and estimated perhaps from satellite-derived estimates of strain rate) instead of using an elastic model to calculate that background state. The authors might need some explanation justifying why they use linear elasticity to calculate the longitudinal stress.

Once the authors start to consider meltwater within the crevasse, it confuses me that the porous nature of firn is completely ignored. LEFM no longer holds for porous material and poromechanics [5] should be considered. Could the authors at least highlight the limitations of current results (Figure 5&7 in the main text)?

[1] Schlegel, R., Diez, A., Löwe, H., Mayer, C., Lambrecht, A., Freitag, J., ... & Eisen, O. (2019). Comparison of elastic moduli from seismic diving-wave and ice-core microstructure analysis in Antarctic polar firn. Annals of Glaciology, 60(79), 220-230.
[2] Smith, J. L. (1965). The elastic constants, strength and density of Greenland snow as determined from measurements of sonic wave velocity (Vol. 167). US Army Cold Regions Research & Engineering Laboratory.
[3] King, E. C., & Jarvis, E. P. (2007). Use of shear waves to measure Poisson's ratio in polar firn. Journal of Environmental and Engineering Geophysics, 12(1), 15-21.
[4] Benn, D. I., Cowton, T., Todd, J., & Luckman, A. (2017). Glacier calving in Greenland. Current Climate Change Reports, 3, 282-290.
[5] Coussy, O. (2004). Poromechanics. John Wiley & Sons.
Citation: https://doi.org/10.5194/egusphere-2024-660-RC1
- AC2: 'Reply on RC1', Emilio Martinez-Paneda, 04 Jun 2024
  
  Please see the attached document.
  
  Citation: https://doi.org/10.5194/egusphere-2024-660-AC2
RC2:
'Comment on egusphere-2024-660', Anonymous Referee #2, 03 May 2024

  The idea that it is worth considering how the properties of firn layers could affect the stresses that control surface crevasse opening is very compelling. However, this analysis makes the radical assumption that the stresses in an ice sheet or shelf are controlled solely by the elastic deformation of compressible ice. This is fine if one simply wants to go through a mathematical exercise, but the title, abstract and body of the paper imply that the results of this analysis applies to real ice sheets and shelves. I was particularly disturbed by the fact that the abstract does not make clear that this is an exercise based on ignoring viscous flow of ice. The fact that the Maxwell time of ice is on the order of days means that an ice sheet or shelf would have to have formed in less than a day for this analysis to be applicable.
   The major conclusion of the paper is that inclusion of low-density firn produces opposite effects for idealized ice sheets versus floating ice shelves. The abstract and a cursory reading of the paper makes this seem like a general conclusion. Upon closer reading it is clear that the ice shelf result is only for a particular region close to the edge of the shelf. The authors correctly note that assumption of perfect elasticity results in compression everywhere far from the shelf edge so that no surface crevasses should result for any assumed firn densities or Young’s Moduli! This is confusing because the paper only discusses analytic solutions for the stress field far from a shelf edge. To get surface crevassing on a compressible ice shelf with infinite viscosity requires bending stresses close to the edge of the shelf. The authors then use a finite element model to compute those stresses at a fixed position (250 m) from the shelf edge. At that position the predicted crevasse depth is increased by a decreasing firn density and Young’s Modulus. I assume that this is a robust result but it is hard to evaluate given the information in this paper. More importantly, the paper makes it seem that this is general result based on the analytical results derived in the paper, as is clear from the opening of the “Conclusions” section:
“In this paper, we derived analytical equations for the far field longitudinal stress including the effects of surface firn layers, described by depth-varying density and Young’s modulus profiles based on field data. These analytic expressions were used to perform fracture propagation studies on isolated air/water-filled surface crevasses in grounded glaciers and ice shelves …”
This certainly gave me the wrong idea when I first read the paper.
   The other major result of the paper is that low-density firn results in smaller crevasse depths for a grounded glacier compared to a uniform ice case. The authors note that this result contradicts the previous Linear Elastic Fracture Mechanics analysis of van der Veen (1998). I suspect that the difference with the previous study is caused by the assumption of purely elastic horizontal stresses which are less extensional at the ice sheet surface than the stresses assumed by van der Veen (1998). Thus, again I am not convinced that the results of the new analysis apply to the real world.
It is incumbent on these authors to make a case that the assumption of perfect elasticity gives insight into the opening of surface crevasses on real ice sheets and shelves.

Citation: https://doi.org/10.5194/egusphere-2024-660-RC2
- AC1: 'Reply on RC2', Emilio Martinez-Paneda, 04 Jun 2024
  
  Please see the attached document
  
  Citation: https://doi.org/10.5194/egusphere-2024-660-AC1

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

ED: Reconsider after major revisions (further review by editor and referees) (20 Jun 2024) by Josefin Ahlkrona

AR by Emilio Martinez-Paneda on behalf of the Authors (20 Jun 2024) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (06 Jul 2024) by Josefin Ahlkrona

RR by Anonymous Referee #2 (20 Jul 2024)

Suggestions for revision or reasons for rejection

I have two major problems with the paper.

(1) I do not trust the basic result that firn results in shallower crevasse penetration that is asserted by the authors. They claim that van der Veen go the opposite result because he only considered the effect of low-density firn on the vertical stress (σ_zz) and not on the difference between the horizontal and vertical stress (R_xx). It is pretty easy to show that low-density firn has a larger effect on the magnitude of σ_zz than on R_xx.

To do that I considered a simple analytic treatment of the effect of low-density firn on vertical and horizontal stresses. I assume the Weertman (1957) stress state (which the authors call the incompressible state). To make the problem super simple I assume a firn layer of thickness h_L has a density equal to half of the density of ice in the rest of the layer which has a density ρ_i. I start with the standard assumption that the vertical stress just equal to the weight of overburden so that below the firn layer the vertical stress magnitude is:
σ_zz=ρ_i g(H-z)-(ρ_i gh_L)/2

where, as assumed in the paper under review, that H is the ice layer thickness, z is the distance above the base of the layer and g is the acceleration of gravity. It is easy to show that the difference between the vertical and horizontal stress through the layer is:

R_xx=(ρ_i g)/2 (H-h_L+(h_L^2)/H)-(ρ_w g)/2 (h_W^2)/H

It makes sense to me that the magnitude of σ_zz is reduced by (ρ_i gh_L)/2 while the magnitude of R_xxdrops by only (ρ_i g)/2 (h_L-(h_L^2)/H). For crevasses the open only within the firn the reduction in vertical stress magnitude is even greater.
It is easy to calculate the depth of crevasse opening with the simple Nye (1955) assumption of no stress change on crevasse opening. Though that assumption is poor it allows for an analytic solution and in all cases I have seen the LEFM solution scales with the Nye solution. When you use the Nye assumption the crevasses are indeed deeper than for a case with uniform density.
This little exercise makes me think there is something wrong in the solution derived in this paper though I have not taken the time to go through their analysis.

(2) In the response to reviewers the discussion of the new section 5 “Non-linear Viscous Incompressible Rheology” is wrong in several fundamental ways. First, the authors claim that their earlier analysis of the problem with a Poisson’s ratio of 0.35 is based on a “common assumption if crevasse propagation occurs in a rapid brittle manner…” This completely missed to point, also made by reviewer 1, that the usual assumption is that the background stress is set by flow in the layer long before a crevasse forms. The layer is “commonly” assumed to have relaxed viscously (see Nye(1955), Weertman(1973), van der Veen (1998) and many others). Second, the deformation during crevasse opening is usually assumed to happen so fast that there is no viscous relaxation, but the opening magnitude certainly depends on the elastic constants (see Weertman (1980) for a clear discussing of this point) so the material cannot be treated as being incompressible. Finally, the paper still maintains that the purely elastic stress field for compressible cases are valid for fast crevasse propagation. That only works if an ice has no time to equilibrate through any viscous flow.

References
Nye, J. F. (1955). Comments on Dr. Loewe's letter and notes on crevasses. Journal of Glaciology, 2(17), 512–514. https://doi.org/10.3189/
s0022143000032652

Van der Veen, C. J. (1998). Fracture mechanics approach to penetration of bottom crevasses on glaciers. Cold Regions Science and Technology,
27(3), 213–223. https://doi.org/10.1016/s0165-232x(98)00006-8

Weertman, J. (1973). Can a water-filled crevasse reach the bottom surface of a glacier? In Symposium on the Hydrology of Glaciers, Cambridge,
7–13, September 1969 (pp. 139–145). International Association of Hydrologic Sciences.

Weertman, J. (1980). Bottom crevasses. Journal of Glaciology, 25. https://doi.org/10.3189/S0022143000010418

Hide

RR by Anonymous Referee #1 (25 Jul 2024)

ED: Publish subject to minor revisions (review by editor) (11 Sep 2024) by Josefin Ahlkrona

AR by Emilio Martinez-Paneda on behalf of the Authors (20 Sep 2024) Author's response Author's tracked changes Manuscript

ED: Publish as is (08 Oct 2024) by Josefin Ahlkrona

AR by Emilio Martinez-Paneda on behalf of the Authors (10 Oct 2024) Manuscript

Journal article(s) based on this preprint

03 Dec 2024

The influence of firn layer material properties on surface crevasse propagation in glaciers and ice shelves

Theo Clayton, Ravindra Duddu, Tim Hageman, and Emilio Martínez-Pañeda

The Cryosphere, 18, 5573–5593, https://doi.org/10.5194/tc-18-5573-2024,https://doi.org/10.5194/tc-18-5573-2024, 2024

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Theo Clayton, Ravindra Duddu, Tim Hageman, and Emilio Martinez-Paneda

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We develop and validate new analytical solutions that quantitatively consider how the properties of ice vary along the depth of ice shelves and can be readily used in existing ice sheet models. Depth-varying firn properties are found to have a profound impact on ice sheet fracture and calving events. Our results show that grounded glaciers are less vulnerable than previously anticipated while floating ice shelves are significantly more vulnerable to fracture and calving.


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