the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 26 Mar 2024
Calving of Ross Ice Shelf from wave erosion and hydrostatic stresses
Abstract. Ice shelf calving constitutes roughly half of the total mass loss from the Antarctic ice sheet. Although much attention is paid to calving of giant tabular icebergs, these events are relatively rare. More frequent, smaller-scale calving events likely play an important role in the ice shelf frontal dynamics. Here, we investigate the role of bending stresses at the ice shelf front in driving calving on the scale 100 m – 1 km, perpendicular to the ice edge. We focus in particular on how buoyant underwater "feet" that protrude beyond the above-water ice cliff may cause tensile stresses at the base of the ice and ultimately lead to fracture. Indirect and anecdotal observations of such feet at the Ross Ice Shelf front suggest that this process may be widespread. We consider satellite observations, together with an elastic beam model and a parameterization of frontal wave erosion to estimate the size and frequency of such calving events. Our results suggest that foot-induced mass loss at Ross Ice Shelf may cause up to 25 % of the total frontal ablation. However, stresses induced through this process are likely not sufficient to initiate crevassing but rather act to propagate existing crevasses. In addition, the relatively strong ice thickness dependence of the frontal uplift suggests an important role for internal bending moments due to temperature gradients in the ice. The highly variable environment, irregularity of pre-existing crevasse spacing, and complex rheology of the ice continue to pose challenges in better constraining the drivers behind the observed deformations and resulting calving rates.
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RC1: 'Comment on egusphere-2024-571', Ravindra Duddu, 10 Jun 2024
This article investigates the role of elastic bending stresses at the ice shelf front in driving calving using the classical beam theory (Euler–Bernoulli equation) and compares it with satellite observations from ICESat and ICESat-2 elevation data. The novelty of this study lies in its focus on small scale calving arising from the footloose mechanism and parametrization of frontal wave erosion to estimate the size and frequency of calving mechanism. The article is well structured and easy to read, particularly, I liked how the discussion of model limitations was weaved in quite nicely with the model parametrizations.
However, the main limitation of the article is that it uses the classical elastic beam theory equation and its analytical solution. The authors do acknowledge that viscous processes may play a crucial role and cite the work of Mosbeux et al. (2020). The article almost feels like a report on the limitations of simple model parametrizations, serving to bring awareness to the community that “Further investigations are therefore warranted …” as remarked in the conclusion. While the article does not advance the state-of-the-art models or propose new parametrizations, I do think it would make a contribution to the literature. My detailed comments are listed below.
Detailed Comments:
- Line 25 – Two review papers (Benn et al., 2017, Alley et al., 2023) were cited. I think a recent article Bassis et al. (2024) discussing the stability of ice shelves and ice cliffs in changing climate would be a relevant reference. https://www.annualreviews.org/content/journals/10.1146/annurev-earth-040522-122817.
- In equation (1), why was the bending moment and axial compressive load at the ice shelf front due to the (linearly-varying) hydrostatic water pressure not considered. Instead, only a point load arising from increased buoyancy of the foot is considered. Is the moment neglected because there is no analytical solution like equation (2) for the case when moment is included. On line 137, it is remarked that the moment is neglected for simplicity and cited Mosbeux et al. (2020) that this will cause 15 – 20%. It would be great to see some more details and a quantitative analysis of the effect of the end moment and compressive pressure at the ice-ocean front in this paper, especially given the compressive pressure often tends to stabilize ice cliffs.
- Going from eq. (1) to eq. (2), please provide the expression used for the foot-related forcing term Q. It is not clear how equation (3) was obtained. Also, below equation (4) it is stated that x_RM does not depend on the size of the foot. Perhaps, these two things were clarified in Wagner et al. (2016), but would be good to add a clarification to this paper.
- Line 108 – Poisson’s ratio of 0.3 is used throughout this work, modelling ice as being compressible. Wouldn’t a Poisson ratio of 0.5 be more appropriate for the time-scales considered (years), as this would result in the ice being incompressible.
- Line 132 – Following Wagner et al. (2014) a simple yield stress based criterion is implemented for the calving event. I am afraid this is overly simplistic. In a recent article (Gao et al., 2023), we have proposed an advanced finite element based cohesive zone models for simulating nonlinear viscous and hydrofracture process in ice. I suggest the authors could perhaps acknowledge the need for some advanced process models for better representation and understanding of near-terminus small-scale calving events. https://ieeexplore.ieee.org/abstract/document/10271321
- Section 3.2 both describes the process creating the foot as “erosion” and “melting”. Please clarify whether this is driven by melting (thermal energy being advected to the ice-shelf to induce a phase-change) or erosion (wave motion mechanically removing material through impacts/abrasion).
- Line 162 – It is stated that equation (6) has not been validated comprehensively against real world conditions. Please clarify whether this is in the context of calved icebergs as done by Gladstone et al. (2001) with the parameters from Martin and Adcroft (2010) and England et al. (2020), or is their statement more about ice shelves. I think it is a missed opportunity that the authors have not made any advancement on this, but rather applied an empirical formula from the literature. At the least a discussion on how to improve the state-of-the-art in the context of wave erosion/melting would be useful. Also, is the use of the “mean over an ocean strip along Ross Ice Shelf” is well justified by the following statement that melt rate estimates are not sensitive to the choice of the ocean strip width. I found these sentences a bit confusing, so please rephrase and clarify.
- Line 170 – two dataset resolutions are specified in degrees and the other one for sea ice concentration is specified as 25 km. Please clarify what a 0.01 degree resolution means in terms of 25 km resolution, and various quantities were can regridded without interpolation.
- Line 204 – 207 – Several of the profiles having berm characteristics were excluded in Figure 4a showing Ross Ice Shelf elevation profiles. Please add a sentence on why these profiles exist and why the rampart-moat profile is not found everywhere. Does this correspond to locations where a small scale calving event occurred in the past leaving behind the berm shaped profile. Is it possible to look for the elevation data for berm profiles across time. Please clarify how much of the data-set was excluded by the criteria (presumably this is only a small portion of the dataset?) If not, please comment on how representative are the presented results if a significant portion of the reference data does not follow this mechanism?
- Just a comment - Banwell et al. (2019) assumed 1 GPa as well, which gave better match of vertical deflection of ice shelf due to flexure from surficial lakes. As remarked by the authors, I agree that the discrepancy between laboratory and field/observational values may be arising from viscous creep effects https://www.nature.com/articles/s41467-019-08522-5
- Line 224 – It is good that effect of firn layer is accounted by averaging the ice density to 850 kg/m^3. In Gao et al. (2023) our major finding was that deeper crevasses are possible in ice shelves because the firn layer increase the height of ice above the sea level. Such crevasses will of course change the effective ice thickness and the apparent flexural rigidity of the ice shelf, as remarked by the authors in lines 241 – 242. https://ieeexplore.ieee.org/abstract/document/10271321 Also, Line 245, I would think crevasse spacing also effect the flexural rigidity not just crevasse depth.
- Line 252 – Just curious if there is any physical interpretation for the quadratic scaling proposed for effective ice thickness h*
- I found Section 4.2 a little bit difficult to follow. Specifically, I did not follow how the authors determined L ~ 110 m, l_f = 0 – 32, and also how they judged l_f^max = 30 is consistent with the image of the calved iceberg in Fig. 1c. Also, from the histogram in Figure 6, the maximum foot size is about 30 m and is smaller than the theoretical critical foot size of l_f^max = 44 m for 50 kPa. The authors state “bending stress alone may not be sufficient to initiate crevasses” Are the authors referring to perhaps water pressure in basal crevasses or just nonlinear viscous deformation induced stresses. Please clarify.
- In Section 4.4, the authors state “… merely intended as back-of-envelope estimates.” Please add a statement to clarify why these estimates are still useful, perhaps from the point of view of global ice sheet model parametrizations. I also commend the authors for their honest statement that the agreement between the theoretical and observed calving rates is somewhat of a coincidence. See my comment below on the calculation of the average calving rate. Although it is good to acknowledge viscous process, the Deborah number calculation is defined for linear viscous material, whereas ice is nonlinear viscous with strain rate dependence. This means that the timescale of relaxation (or bending) can be heterogeneous depending on the local strain rate in an ice shelf or glacier. We address this in a paper under review in The Cryosphere https://egusphere.copernicus.org/preprints/2024/egusphere-2024-346/
- A simple fluid mechanics (mass balance) compatible calving rate law is implemented here. This calving rate law is typically defined by averaging over many calving events, thus more suited for glaciers or ice shelves that regularly calve ice bergs (see discussion in Section 6.1 in Bassis et al., 2024). However, in the case of the Ross ice shelf one calving event occurred between 2006 and 2007, which was averaged over span of 4 or 5 years to obtain the average calving rate of 200 – 300 m/yr. I find this a bit odd and almost seems like the model was forced to fit to the data. In my opinion, a calving rate law is not well suited to describe the footloose calving mechanism, however, large scale ice sheet models seem to use this simplistic calving laws. It would be great if the authors can add some more discussion on this in the paper. https://www.annualreviews.org/content/journals/10.1146/annurev-earth-040522-122817.
- Lines 328 and 329, I did not understand how theoretical and observation process times \tau were calculated as 0.5 and 5 years. Also, if theoretical estimate is 0.5 then should the Deborah number range from 0.3 – 40 instead of 0.3 – 10.
- Line 337 – the phrase “… maximum location of maximum stress …” is confusing. Can this be rephrased as – the location of global maximum of principal stress? By maximum stress are the authors referring to the maximum principal component and its global maximum?
Citation: https://doi.org/10.5194/egusphere-2024-571-RC1 -
AC2: 'Reply on RC1', Nicolas Sartore, 20 Aug 2024
We would like to thank the reviewers for their insightful and constructive feedback. As there was significant overlap between the two reviews we have chosen to present our responses in the form of one combined document, however, we will address each individual comment separately.
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RC2: 'Comment on egusphere-2024-571', Anonymous Referee #2, 18 Jun 2024
This paper makes the innovative proposal that the ‘footloose’ calving mechanism – well-established in Arctic contexts – might be an important process of mass loss from the Ross Ice Shelf. The paper is well structured and easy to follow, and basically proceeds in three stages of unequal length: 1. presentation of observational evidence for the ‘footloose’ process; 2. a simple model of the process is presented then used to predict calving rates, and 3. discussion of model performance and validation. Some of these are much more convincing than others.
Evidence for the process.
Evidence for the process is threefold: 1. A photograph of an iceberg provides striking evidence for the existence of an ice foot. 2. Good use is made of ICESat elevation data to show that rampart and moat profiles are common, providing convincing evidence of ‘bottom-out’ flexure of the ice front. 3. One example is given of a single calving event in a six-year ICESat record, which appears to have cut off the ice shelf at the point of maximum depression. It is not explicitly stated that this is the only example found in the record, but it seems reasonable to assume that it is. The evidence thus provides support for the idea that ice feet form at the margin of the Ross Ice Shelf, and that these cause buoyant flexure. The evidence also supports the idea that ‘footloose’ calving does remove ice from the Ross Ice Shelf, although the fact that this evidence is confined to a single set of ICESat profiles and one archive photograph indicates that such calving events are rather infrequent.
The model.
This is the least satisfactory part of the paper. As it stands, the model is highly simplified, and has the major omission that it neglects the effect of depth-varying back-pressure on the ice front. Uneven back-pressure is well known to cause ‘top out’ flexure of floating ice fronts, and thus opposes the ‘bottom out’ flexure due to the development of an ice foot. Since the two flexure processes oppose each other, it is essential to establish their relative importance as an ice foot grows, as this will determine the stress distribution in the ice and ultimately, the critical ice foot length required for calving. Because it neglects this process, the model is of little value in either an illustrative or predictive sense.
To model the footloose process properly would probably require the use of a finite-element model such as Elmer/Ice to explore the evolution of elastic stresses and viscous deformation during the growth of an ice foot. Such an approach would have the advantage of identifying the actual locations and magnitudes of stress concentrations, which would provide a much more secure foundation for calving predictions.
Validation.
In Section 4.4, there is some discussion about the differences between the results and the (limited) available data for validation. The text notes that there are ‘some differences’ but the differences are really quite major: calving frequency is evidently much less than the 2/yr suggested by the model, and the single example of calving in the ICESat data suggests that calving magnitudes may be typically much greater. Furthermore, the brief discussion of ice-front advection vs. ice velocity indicates that the calculated frontal wave erosion rate is too high. Taken together, these results indicate that the model does a poor job of describing the process and predicting calving rates. There is thus little justification for the claim that the footloose mechanism accounts for ‘up to 25%’ of frontal ablation of the shelf.
Having said this, the idea is an interesting one and I think the paper should be published in some form. I strongly suggest, however, that the authors make a big effort to improve the model. This will provide a more convincing picture of how the footloose process might actually work in an Antarctic context and will help identify the key factors that control calving losses. I recommend either one of two possible ways forward. Publication could be delayed until more work is done, especially with regard to developing a better model of the footloose process. Alternatively, the paper could be shorn of much of the modelling element to focus on the observational evidence that ice-foot development, bottom-out flexure, and footloose calving actually operates on the Ross Ice Shelf. In both cases, quantitative data on ice velocities and frontal position should be compiled to quantify cliff retreat rates. Whichever course is taken, I think that major revisions are required before publication.
Citation: https://doi.org/10.5194/egusphere-2024-571-RC2 -
AC1: 'Reply on RC2', Nicolas Sartore, 20 Aug 2024
We would like to thank the reviewers for their insightful and constructive feedback. As there was significant overlap between the two reviews we have chosen to present our responses in the form of one combined document, however, we will address each individual comment separately.
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AC1: 'Reply on RC2', Nicolas Sartore, 20 Aug 2024
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