the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
The effect of hydrology and crevasse wall contact on calving
- 1Department of Earth, Ocean and Atmospheric Sciences, University of British Columbia, BC, Canada
- 2Department of Mathematics, University of British Columbia, BC, Canada
- 1Department of Earth, Ocean and Atmospheric Sciences, University of British Columbia, BC, Canada
- 2Department of Mathematics, University of British Columbia, BC, Canada
Abstract. Calving is one of the main controls on the dynamics of marine ice sheets. We solve a quasi-static linear elastic fracture dynamics problem, forced by a viscous pre-stress describing the stress state in the ice prior to the introduction of a crack, to determine conditions under which an ice shelf can calve for a variety of different surface hydrologies. Extending previous work, we develop a boundary-element-based method for solving the problem, which enables us to ensure that the faces of crevasses are not spuriously allowed to penetrate into each other in the model. We find that a fixed water table below the ice surface can lead to two distinct styles of calving, one of which involves the abrupt unstable growth of a crack across a finite thickness of unbroken ice that is potentially history-dependent, while the other involves the continuous growth of the crack until the full ice thickness is cracked, which occurs at a critical combination of extensional stress, water level, and ice thickness. We give a relatively simple analytical calving law for the latter case. For a fixed water volume injected into a surface crack, we find that complete crack propagation almost invariably happens at realistic extensional stresses if the initial crack length exceeds a shallow threshold, but we also argue that this process is more likely to correspond to the formation of a localized, moulin-like slot that permits drainage, rather than a calving event. We also revisit the formation of basal cracks and find that, in the model, they invariably propagate across the full ice shelf at stresses that are readily generated near an ice shelf front. This indicates that a more sophisticated coupling of the present model (which has been used in a very similar form by several previous authors) needs modification to incorporate the effect of torques generated by buoyantly-modulated shelf flexure in the far field.
Maryam Zarrinderakht et al.
Status: final response (author comments only)
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RC1: 'Comment on tc-2022-37', Anonymous Referee #1, 08 May 2022
This article presents a linear elastic fracture mechanics (LEFM) approach to estimate the penetration depth of water-filled crevasses in an ice shelf. The key novelty is that the authors consider the introduction of crevasse generates elastic stress in the ice shelf, which is otherwise at equilibrium due to viscous stress. The proposed model is an improvement over the van der Veen (1998a,b) and Lai et al. (2020) in that it considers crack wall contact using the discontinuity boundary element method. With regard to water in crevasses, the paper considers both fixed water table and fixed water volume injected, which leads to different propagation conditions. The article is generally well written from Section 3 onwards, but Section 1 and 2 have a few typos and confusing sentences, which can be easily fixed. The conclusion of the paper is long and a bit hard to follow. Overall, I found the article is a good contribution and I recommend it for publication with minor revisions.
Detailed Comments:
- The introduction can be improved, as I found a few typos and grammatical errors. Also, it does not acknowledge a lot of prior work on this topic. For example, the article cites Lipovsky (2020) for numerical approaches for LEFM, but it was previously introduced in an article by Jimenez and Duddu (2018). https://www.cambridge.org/core/journals/journal-of-glaciology/article/on-the-evaluation-of-the-stress-intensity-factor-in-calving-models-using-linear-elastic-fracture-mechanics/0378315BDB37E88E37B1B07F6BC60426
- Replace the usage of the word “torque” with “moment”. In physics, the turning effect of a force is generally termed as torque, but in mechanics torque stands for torsional moment, whereas the seawater pressure on an ice shelf causes a bending moment.
- The model considered here is not a Maxwell model, as mentioned on page 3, line 75. In a Maxwell-type, the viscous stress must be equal to the elastic stress. The strains are additively split. I believe the assumption of this paper is a compressible Kelvin-type model. The introduction of the crack within an otherwise viscous ice shelf at equilibrium leads to elastic stress perturbations. These elastic stress vanish on the boundary far away from the crack. This is better clarified elsewhere in the paper, but not in the model description early on.
- Line 98, page 4, it is mentioned the stress field defined by (6) cannot be generated by an elastic rheology is not true. This stress field can be obtained with a nearly incompressible elastic rheology. It is really not the elastic or viscous nature but rather the incompressibility assumption that leads to this stress state. Please see Sun et al. (2021) Appendix A for the derivation of the elastic stress field, wherein if you plug in Poisson’s ratio of 0.5, you would recover the stress field defined by (6). https://www.sciencedirect.com/science/article/abs/pii/S2352431621000626
- Line 119 - 120, page 5, seems like a typo, there is no subscript on [v]^+_- and in equation (9) u should not be bold in [u]^+_-.
- (11) seems to have some wrong notation. The index j appears three times and this violates Einstein’s summation convention.
- Line 139, page 5, I do not understand why it is more natural to prescribe water volume. Isn’t it as poorly constrained as the water height in crevasses. Please explain how one would constrain water volume from observations.
- Line 159, page 6, please use text roman i for the subscript for ice density, so that it does not mix up with the subscript index i
- Line 205, page 8, The authors state that instead of solving a full dynamic crack problem, they can use the semi-analytical theory of Freund (1990). It is not clear to me why Freund’s approach is needed. Please explain why the simple stability criterion for steady state crack used in Lai et al. (2020) is not adequate for analysis.
- In Eq. (22), the quantity [t] comes out to be negative. Is that correct?
- In Eq. (32) you have the term (s - \eta – z) where z is has a dimension but \eta is nondimensionalized. Is there any typo there?
- I found the results section to be a bit hard to read. I felt like a lot of minor details were discussed which at time made me lose the big picture. I think the paper can be condensed a lot in this section.
- Throughout the paper, I found minor typographical errors that are a few too many, but I did not want to list them here. Please proofread the entire article before submitting the final version.
- In section 5, two calving laws were introduced one for basal crevasses and another for surface crevasses. A major critique is that unless these calving laws are incorporated in an ice sheet model and validated with observational data, we do not know if it is good. However, this maybe beyond the scope of this article.
- The conclusion of this paper is really long and I found it difficult to read. It will be good if it can be broken up into subsections to improve readability.
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RC2: 'Comment on tc-2022-37', Bradley Lipovsky, 18 May 2022
Dear Editorial Staff and Authors,
This manuscript by Zarrinderakht and co-authors was simply wonderful to read. It constitutes a significant advance in the field of glacier fracture mechanics and is obviously a stepping stone towards bigger things. I enthusiastically support the publication of this manuscript in the Cryosphere. I do have a few questions and comments that I hope will improve the quality of the manuscript. Many of these draw connections to my own work on glacier fracture mechanics, which isn't to suggest that my work be given any special pedestal, but is rather just to share how I think about some of the physics of these kind of problems. Please feel free to take or leave this work as you see fit.
All the best,
Brad
Questions and Commetns
Why assume the crack propagates slowly (i.e., equation 19)? We know very well that crevasses in ice shelves are seismogenic. See, for example, Aster et al., 2021, who interpreted the unique seismic characteristics of certain impulse ice shelf seismic observations to be caused by crevasses growth. In order for crevasses to generate seismic waves, they must propagate at inertial (or near-inertial) velocities. Furthermore, the full inertial treatment of crevasse growth maintains the form of Equation 18, it just changes the last multiplicative term on the right hand side. This situation was treated by Lipovsky (2018) which to my knowledge is the first-, and prior to the present manuscript the only, study to examine the dynamics of glacier fracture growth (albeit with horizontal propagation, although the authors will appreciate that the math is the same). If the crack does move suddenly then water compressibility may be important (also, see below). The equations necessary to treat compressible pressure gradient flow along hydraulic fractures were given by Lipovsky and Dunham (2015) with application to hydraulic fractures in glaciers.
I realized when reading the caption of Figure 3 (“…even where the crack is closed….” [sic]) that the authors assume hydrostatic pressure for what appear to be closed-off water blobs. Could the pressure be cryostatic? If so, this would provide additional reason to treat the compressibility of the water.
Section 2.1 Model description. Some readers might be interested to know that Lipovsky (2020) also used viscous pre-stresses in LEFM calculations. To my knowledge, this publication introduced these concepts in glaciological research. I gave a different physical explanation of the viscous pre-stresses but the form was mathematically identical to that used in the present manuscript. I do prefer the physical explanation given in the present manuscript, but I'm at least encouraged that the math is the same since I grappled with this for a while.
Experience hiking around glaciers with water-filled crevasses tells us that crevasses are often up to a meter wide (or more). It is unlikely that this meter of opening is due entirely to elastic stresses, as one can calculate that this would require enormous and unrealistic stresses. The explanation for the opening is instead that the ice surrounding the crevasse has deformed through flow. The crack would have non-zero width in the absence of the elastic tensions. In this case, not all crack closure would result in contact. It is therefore worth noting that —in at least some cases— negative crack opening (i.e., crack closure) does not result in contact, and instead simply results in the crack getting narrower but not having walls that touch. I'm curious now: can the numerical method in this manuscript handle nonzero initial crevasse widths?
If, on the other hand, the crevasse is assumed to be so narrow that the walls could touch, then fluid viscosity should become important [see again LD15]. Maybe these points are already acknowledged in line Line 150, where the reader is cautioned that more complexity in the fluid flow is warranted.
Section 4. Results
- The first sentence of this section seems to imply that the width of the domain is an important parameter in the problem. I don’t understand why this would be the case if Rxx is (conceptually at least) treated as a boundary condition at great distances (i.e., +/- infinity). Numerically, shouldn’t the simulations be run for a sufficiently large domain width so that the solutions do not depend on this parameter?
I am confused by the results in Figure 3. The model seems to be treating the case with constant water volume, but yet the water volume clearly changes from figure 3b1 to 3b2. I think this is supposed to mean that some water is stored at the surface. But if water is stored at the surface, then the appropriate tractions ought to be applied at the surface of the glacier. Instead, the surface of the glacier is taken to be traction free (Equation 3). It seems like a rigorous treatment of this situation must either include the surface load or else omit crevasse depths that are too shallow to hold the prescribed water volume.
Figure 4 and 5 are simply wonderful contributions to the literature on glacier fracture mechanics. Thank you for this.
Figure 6 / Line 455. See comment above about the surface load due to a lake. As I understand it, the model essentially has the water "coming from nowhere". Maybe the surface loading could resolve the paradox of stability at high prestress. There's an analytical SIF in Tada (2000) that you could compare to, see their section 8.9.
Discussion, particularly the "problem" on Line 676: I think this same issue was discussed by Rist et al (2002). Their solution was to introduce back stress from sidewall coupling. Maybe I'm wrong and they were solving a different problem, but either way I would appreciate a clarification.
References:
Lipovsky, Bradley P., and Eric M. Dunham. "Vibrational modes of hydraulic fractures: Inference of fracture geometry from resonant frequencies and attenuation." Journal of Geophysical Research: Solid Earth 120.2 (2015): 1080-1107.
Lipovsky, Bradley Paul. "Ice shelf rift propagation and the mechanics of waveâinduced fracture." Journal of Geophysical Research: Oceans 123.6 (2018): 4014-4033.
Lipovsky, Bradley Paul. "Ice shelf rift propagation: stability, three-dimensional effects, and the role of marginal weakening." The Cryosphere 14.5 (2020): 1673-1683.
Maryam Zarrinderakht et al.
Maryam Zarrinderakht et al.
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