the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 09 Jan 2023
The evolution of isolated cavities and hydraulic connection at the glacier bed. Part 1: steady states and friction laws
Christian Schoof
Abstract. Models of subglacial drainage and of cavity formation generally assume that the glacier bed is pervasively hydraulically connected. A growing body of field observations indicates that this assumption is frequently violated in practice. In this paper, I use an extension of existing models of steady state cavitation to study the formation of hydraulically isolated, uncavitated low-pressure regions of the bed, which would become flooded if they had access to the subglacial drainage system. I also study their natural counterpart, hydraulically isolated cavities that would drain if they had access to the subglacial drainage system. I show that connections to the drainage system are made at two different sets of critical effective pressure, a lower one at which uncavitated low-pressure regions connect to the drainage system, and a higher one at which isolated cavities do the same. I also show that the extent of cavitation, determined by the history of connections made at the bed, has a dominant effect on basal drag while remaining outside the realm of previously employed basal friction laws: Changes in basal effective pressure alone may have a minor effect on basal drag until a connection between a cavity and an uncavitated low-pressure region of the bed is made, at which point a drastic and irreversible drop in drag occurs. These results point to the need to expand basal friction and drainage models to include a description of basal connectivity.
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hydraulically connected(how easily water can flow along the glacier bed) plays a central role in determining how fast ice can slide.
Christian Schoof
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2022-1380', Anonymous Referee #1, 08 Feb 2023
General comments
Schoof’s manuscript (Part 1 of two related manuscripts) reports a theoretical study which extends the modelling of subglacial cavitation in two dimensions assuming constant ice viscosity and steady state, by exploring the possibility that some areas of the bed lack hydrological connection to the background subglacial drainage system. Calculations show how areas on the lee sides of basal bumps that aren’t connected to background drainage retain ice--bed contact with low normal stresses --- instead of forming new cavities --- when background effective pressure decreases; this occurs unless such areas get flooded as a (connected) cavity located further upglacier expands over the bump to reach those areas. This kind of flooding is shown to be irreversible in that subsequent increase in background effective pressure causes the ice to recontact the bump top, sealing off the flooded area as an isolated cavity, which cannot be eliminated afterwards: the state of an uncavitated lee side cannot be restored. Besides exploring the conditions behind these states and processes (inc. the effective pressures at state transitions) for a variety of bed geometry, Schoof examines their implications for bed friction, finding that the form of the basal sliding law depends on the historical evolution of the cavity configuration and can become multi-valued.
The study is new and interesting, and the mathematical modelling work substantial and rigorous, yielding insights into complex behaviour that is relevant for understanding basal processes and that has not been examined before. Although the modelling approach has limitations, these are clarified. I find the numerical experiments to be very well designed in illustrating key aspects of the system behaviour, and the manuscript well written with an effective structure and clear and detailed explanations.
The work reported in this Part 1 is also focussed and self-contained; it warrants a standalone manuscript. It makes sense to report the viscoelastic modelling in a separate paper (Part 2). Overall I feel that the work makes a worthwhile contribution to the field and think that it is of interest to the readership of The Cryosphere. I am happy to recommend it for publication after suitable revision.
My observations and queries in the ‘specific comments’ below aim at making the model setup easier to understand and making the study more complete. I think that improvement of the manuscript towards Specific Comment 1 (see below) is essential, and improvement towards Specific Comment 2 highly desirable (I leave it to the author to decide how much to investigate). The other specific comments should not be difficult to address. Also I note numerous typographical corrections and small edits/suggestions. They are listed at the end.
Specific Comments
1. In the model setup and introduction, the explanation of ‘access’ (to ambient drainage) and ‘permeable patch’ is unclear. The current combination of text and Figure 1 makes these ideas confusing and harder to understand than necessary, through to page 4 (past Figure 1). Only quite far in Section 2 do I know for sure from the mathematical descriptions how drainage connections are set up in your model and what they mean. You should stabilise the terminology and clarify /elaborate on the meaning of various terms and phrase ideas more carefully. Here are some of the key issues:
p2, lines 26-28: “Access to ambient drainage system is defined through a permeable bed patch P on which effective pressure N is prescribed; elsewhere, effective pressure is defined through… connectedness to patch P, or through…”. I found this passage to be cryptic. The indirect phrasing “is defined through” (used twice) causes vagueness. What N refers to isn’t clear, except it is an effective pressure. What “access” means isn’t clear, nor how its meaning differs from “connection/connectedness”. In “permeable bed patch P”, I find the word “permeable” to be distractive; I guess it is used to relate P to the context of the permeable part of the wider/ambient subglacial system, but currently I sense that possibly the permeability of P will be modelled or quantified (but it isn’t). Consider using direct phrasing such as “P locates the connection, where N is fixed at…”.
Across p2 to p4, P also switches between “patch”, “location”, “portion”, “access”. For example, “the location P of ambient drainage system access” on p3 (line 7) is difficult to understand at that stage of the manuscript. Overall the descriptions of this topic towards the end of Section 1 don’t communicate the picture/setup well as a foundation for Section 2. The mix of terms also conflicts with Fig. 1, which uses what looks like a vertical line to portray P (yet the caption refers to ‘portion’).
Fig 1 and its caption: In specifying part P of the bed interface, I think you have at least lateral connection in mind, as described mid-page on p5. Up to page 4, the concept of a vertical connection doesn’t seem to be an emphasis. The vertical line therefore confused me: it doesn’t indicate a horizontal extent, and I didn’t understand how it portrayed “patch” or “portion” as it is so thin. The caption says “beige… permeable portions (plural)” but this refers to figures throughout the paper and doesn’t specifically explain the current figure. I think you need to rework the caption, adding text to (i) clarify what P means and how the figure portrays it (lines 2-3 currently do not do a good job), and (ii) say explicitly that Cavity 1 is connected because it overlaps with P (/is intercepted by P), explaining also why Cavity 2 is unconnected. [Explanation (ii) is given in the text later, above equation (8), but I think you need it also in the figure caption.]
2. Choice of x_P:
Although you give motivation for placing P around “where cavities first form [in the fully permeable case]” (p5, line 20) --- close to the middle/steepest point on lee sides, this choice is restrictive. What if the connection occurs elsewhere on a lee side, away from that location, or on a stoss side? The analysis would be more complete if the former (lee-side) case at least is discussed properly, preferably aided by a suitable experiment to show the resulting behaviour.
You choose P to be narrow. What happens if it is wide/wider zone (even if you continue to assume only one such zone in (0, a))? Brief discussion of this scenario would be useful.
3. As reported clearly in the manuscript, there can be different solutions (cavity configurations) for the same N* depending on evolution history (in the quasi-steady sense), depending on whether an unconnected lee zone has yet been flooded --- e.g. at N* slightly above N*_disconnect in Fig. 5a (and within the corresponding sequence in Fig. 4). Does the solution method include or require a specific device or algorithm that tracks an aspect of the history to reach the different solutions, e.g. some iteration that uses the last configuration as initial guess? I might have missed it in the descriptions of p5 and the appendix, or perhaps it isn’t necessary and I misunderstand the solution method in (A3), or relevant details exist in the referenced literature. Clarification about this in the manuscript would be welcome.
4. N*_drown: On p11, N*_drown is explained clearly, but the meaning of “drown” there (describing separation of ice and bed everywhere across the domain and violation of force balance) seems different from what “drown” describes elsewhere (coverage over a bed protrusion by expanding cavity). Consider using another symbol rather than N*_drown?
5. Some passages consider or refer to bed protrusions or bumps as “less/more prominent” or “smaller/larger (or taller)”, e.g. on p17, also elsewhere. Although the literature on this subject may use these adjectives following an agreed convention, or specialists will always understand them, I think you should define their meaning early on. Probably you are comparing maximum elevations, not amplitude (so a “little” short bump superposed onto the peak of a large long bump is the largest/tallest in your analysis), but I am not in fact sure; I may have misinterpreted the meaning.
Technical corrections: typos, minor suggestions, etc.
The meaning of “process-scale”/“process-scale model” (several times in section 1) is unclear.
p2, line 6, unfinished sentence
p3, line 10, “robustness… obtained to changes”. Consider rephrasing this to make it easier to understand.
Fig 1: locate x = 0
Fig 1 caption: h(x) is cavity roof “elevation” (instead of height)? Line 3, v and y instead of w and z in the partial derivative.
p4, line 1, missing full stop
p5, line 25, “while”. Consider using “whereas” or “in contrast” to bring out the contrast against Part 2 better.
p6, line 9, “dimensionless combination of effective pressure” doesn’t describe eqn. (11)
Fig 3a: At first I read Fig 3a as showing the spatial distribution of N*. The horizontal axis needs to be labelled as b_j*, c_j*, as it isn’t x* (although the scale is the same). This issue extends to Fig. 5a to 5b and Fig. 9a to 9d, which are also missing the right axis labels.
Fig 3 caption, “bed shape b*(x*) against x*”. Change “shape” to “elevation”. Also in several other figures.
Fig 4 caption, line 1: should b*(c*) be b*(x*)? This arises also in the captions of Figures 2 and 7.
p8, line 3, “disappear at N*”. Is something missing here?
p8, line 6, is the colon before the equal sign a typo?
p9, line 26: “panels (a) and (e) … correspond to the same effective pressure”. Figure 4 caption gives two different values, 4.01 and 4.02.
p9, line 33: “bed an only”
Fig 6: since you plot two lines, the figure is clearer if you label the y-axis as “N2*, N1*” and add a label next to each line
Fig 7 caption, line 3: please check whether a minus sign is missing from the second sigma_nn
Fig 8, h_0 missing from the y-axis label
p14, line 18: trimple --- triple
Fig 9 caption: lines 8 to 10 have hiccups and duplication in various places. For example, “Panel (c). Panel (e): s abruptly”. And “Panel (c)” on line 9 had been introduced on line 6, and I would expect panel (e) (not d) to be covered last.
Fig 11: (i) in the y-axis label, should N2* be N1*? (ii) Although the gentle slope of the “flat portion“ of the blue line is described in the caption and the text, this feature would be much clearer if you set the axes’ aspect ratio at 1:1 (natural as both axes are effective pressures). Consider the same for Figure 6.
p17, line 16: “… depend on the geometry of the bed, and on the parts of the bed…”. (i) Clearer if you add “depend” after “and” or remove the comma. (ii) Clearer if you reword “parts”, because the intended meaning is the location/whereabout of the parts, not some undefined properties of parts.
p17, lines 23-26: I don’t understand how the implied possibility reported in the final sentence follows logically from the previous passage. Is something explanation missing?
p19, line 34: the complication or the definition is being stressed. This sentence is easier to read if written as “The definition of a friction law for an impermeable bed has a second complication that deserves to be stressed”
p20, line 23-24: A further limitation that should be mentioned is the limited range of locations x_P chosen in the experiments (on steepest points of lee sides). (Otherwise I think your design of putting x_P on different bumps on the topography works fantastically in illustrating the range of behaviour.)
p21, first line of A1, spurious comma after 54
p21, line 23, “whree”
Eqn. (A3): the integrand on the left should be a sum instead of difference. Capital C in h^prime_c, and b'' should be b'.
p23, line 15, hiccup after G(infty)
p24, line 2, "!98"
p24, line 5, to help readers, it is useful to (re)qualify V_j as cavity water volumes here. Again, capital C in h^prime_c ?
p24 last line, full stop before "To"
Citation: https://doi.org/10.5194/egusphere-2022-1380-RC1 - AC1: 'Reply on RC1', Christian Schoof, 23 May 2023
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RC2: 'Comment on egusphere-2022-1380', Anonymous Referee #2, 10 Feb 2023
In this paper, the author presents a novel model based on Fowler, 1987 and Schoof, 2005 for steady state subglacial cavitation over bedrocks that are hydraulically permeable in some regions, and impermeable in others. I find this work to be very well written and its scientific/mathematical content to be of excellent quality. Therefore, I recommend its publication after the minor considerations/corrections detailed below are considered.
General comments:- In this work, once a cavity becomes hydraulically isolated, the model enforces that the cavity have constant volume until it becomes hydraulically connected once again. Under these conditions, the water pressure of the cavity is an unknown to compute. Something that came to mind is the possibility of hydraulically isolated cavities that develop an air-filled gap in between the water and the ice. I can understand the reasoning behind the constant volume condition when, e.g., the far-field ice speed decreases, since the cavity would want to close but it cannot compress the water in it. However, upon an increase in this far-field speed, I find it logical that, at a certain point, when the water pressure becomes sufficiently low (perhaps 0?), a partially air-filled cavity starts to develop. In other words, it is reasonable to assume that a hydraulically isolated cavity is still pneumatically connected, since the opposite sounds unrealistic for most glacier conditions. In the context of your model, allowing the formation of air-filled cavities might mean enforcing pneumatic connectedness when a low water pressure is reached, and therefore allowing the volume to increase beyond the water volume. I mention this point because it would mean that the unbounded increase in basal stress that you see for the single cavity in Figure 8 would perhaps no longer hold. If this comment is sensible, I think it should be commented in the discussion.
Specific comments:p3, line 18: Consider starting sentence with a non-mathematical symbol.
p3, line 23: "the normal component of velocity vanishes" > "the normal component of the velocity vanishes".
p 5, line 14: In Stubblefield et al. (2021), the rate of change of the volume is enforced, not the total cavity volume. Mathematically, this is carried out by setting the integral of the normal velocity at the lower boundary equal to this prescribed rate of change. The Lagrange multiplier associated to this constraint is the water pressure.
p5, line 16: As you write, I also find the specification of a permeable point x_P along the bed awkward, from a physical point of view. It seems to me that the fact that you choose it to be the location where cavities first form is effectively equivalent to choosing that a particular cavity, once it forms, is connected to the external drainage system. I think it would be helpful to the reader to clarify the intention behind choosing x_P to be this point.
p 6, line 29: "by plotting cavity end point position" > "by plotting the cavity end point positions"
p 8, figure 4: I think it would be helpful to the reader to include a subtle line in the plots for the normal stress that indicates the values of N*. It would make it much easier to notice the areas of low pressure and the difference in water pressure between isolated and connected cavities. Same for figure 7.
p 9, line 3: "see also fig 3" > "see also figure 3"
p 9, line 9: "decreased." > "decreased"
p9, line 33: "we start with an uncavitated bed an only lower N*", this sentence does not make sense.
p10, Figure 5: Close brackets in line 3 of caption.
p11, line 6: "cavity cavity" > "cavity"
p13, line 5: "by arrow" > "by the arrow"
p13, line 12: Consider starting sentence with a non-mathematical symbol.
p14, line 19: "trimple" > "triple"
p16, line 7: "is" > "are"
p16, line 8: "again be unbounded" > "again unbounded"
p17, line 30: close brackets.
p17, line 32: "it easier" > "it is easier"
p18, line 20: "constraint" > "constraints"
p18, line 33: "If drainage system access" > "If the drainage system access"
p18, line 34: "effective pressure" > "the effective pressure"
p19, line 18: "bed," > "bed."
Citation: https://doi.org/10.5194/egusphere-2022-1380-RC2 - AC2: 'Reply on RC2', Christian Schoof, 23 May 2023
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2022-1380', Anonymous Referee #1, 08 Feb 2023
General comments
Schoof’s manuscript (Part 1 of two related manuscripts) reports a theoretical study which extends the modelling of subglacial cavitation in two dimensions assuming constant ice viscosity and steady state, by exploring the possibility that some areas of the bed lack hydrological connection to the background subglacial drainage system. Calculations show how areas on the lee sides of basal bumps that aren’t connected to background drainage retain ice--bed contact with low normal stresses --- instead of forming new cavities --- when background effective pressure decreases; this occurs unless such areas get flooded as a (connected) cavity located further upglacier expands over the bump to reach those areas. This kind of flooding is shown to be irreversible in that subsequent increase in background effective pressure causes the ice to recontact the bump top, sealing off the flooded area as an isolated cavity, which cannot be eliminated afterwards: the state of an uncavitated lee side cannot be restored. Besides exploring the conditions behind these states and processes (inc. the effective pressures at state transitions) for a variety of bed geometry, Schoof examines their implications for bed friction, finding that the form of the basal sliding law depends on the historical evolution of the cavity configuration and can become multi-valued.
The study is new and interesting, and the mathematical modelling work substantial and rigorous, yielding insights into complex behaviour that is relevant for understanding basal processes and that has not been examined before. Although the modelling approach has limitations, these are clarified. I find the numerical experiments to be very well designed in illustrating key aspects of the system behaviour, and the manuscript well written with an effective structure and clear and detailed explanations.
The work reported in this Part 1 is also focussed and self-contained; it warrants a standalone manuscript. It makes sense to report the viscoelastic modelling in a separate paper (Part 2). Overall I feel that the work makes a worthwhile contribution to the field and think that it is of interest to the readership of The Cryosphere. I am happy to recommend it for publication after suitable revision.
My observations and queries in the ‘specific comments’ below aim at making the model setup easier to understand and making the study more complete. I think that improvement of the manuscript towards Specific Comment 1 (see below) is essential, and improvement towards Specific Comment 2 highly desirable (I leave it to the author to decide how much to investigate). The other specific comments should not be difficult to address. Also I note numerous typographical corrections and small edits/suggestions. They are listed at the end.
Specific Comments
1. In the model setup and introduction, the explanation of ‘access’ (to ambient drainage) and ‘permeable patch’ is unclear. The current combination of text and Figure 1 makes these ideas confusing and harder to understand than necessary, through to page 4 (past Figure 1). Only quite far in Section 2 do I know for sure from the mathematical descriptions how drainage connections are set up in your model and what they mean. You should stabilise the terminology and clarify /elaborate on the meaning of various terms and phrase ideas more carefully. Here are some of the key issues:
p2, lines 26-28: “Access to ambient drainage system is defined through a permeable bed patch P on which effective pressure N is prescribed; elsewhere, effective pressure is defined through… connectedness to patch P, or through…”. I found this passage to be cryptic. The indirect phrasing “is defined through” (used twice) causes vagueness. What N refers to isn’t clear, except it is an effective pressure. What “access” means isn’t clear, nor how its meaning differs from “connection/connectedness”. In “permeable bed patch P”, I find the word “permeable” to be distractive; I guess it is used to relate P to the context of the permeable part of the wider/ambient subglacial system, but currently I sense that possibly the permeability of P will be modelled or quantified (but it isn’t). Consider using direct phrasing such as “P locates the connection, where N is fixed at…”.
Across p2 to p4, P also switches between “patch”, “location”, “portion”, “access”. For example, “the location P of ambient drainage system access” on p3 (line 7) is difficult to understand at that stage of the manuscript. Overall the descriptions of this topic towards the end of Section 1 don’t communicate the picture/setup well as a foundation for Section 2. The mix of terms also conflicts with Fig. 1, which uses what looks like a vertical line to portray P (yet the caption refers to ‘portion’).
Fig 1 and its caption: In specifying part P of the bed interface, I think you have at least lateral connection in mind, as described mid-page on p5. Up to page 4, the concept of a vertical connection doesn’t seem to be an emphasis. The vertical line therefore confused me: it doesn’t indicate a horizontal extent, and I didn’t understand how it portrayed “patch” or “portion” as it is so thin. The caption says “beige… permeable portions (plural)” but this refers to figures throughout the paper and doesn’t specifically explain the current figure. I think you need to rework the caption, adding text to (i) clarify what P means and how the figure portrays it (lines 2-3 currently do not do a good job), and (ii) say explicitly that Cavity 1 is connected because it overlaps with P (/is intercepted by P), explaining also why Cavity 2 is unconnected. [Explanation (ii) is given in the text later, above equation (8), but I think you need it also in the figure caption.]
2. Choice of x_P:
Although you give motivation for placing P around “where cavities first form [in the fully permeable case]” (p5, line 20) --- close to the middle/steepest point on lee sides, this choice is restrictive. What if the connection occurs elsewhere on a lee side, away from that location, or on a stoss side? The analysis would be more complete if the former (lee-side) case at least is discussed properly, preferably aided by a suitable experiment to show the resulting behaviour.
You choose P to be narrow. What happens if it is wide/wider zone (even if you continue to assume only one such zone in (0, a))? Brief discussion of this scenario would be useful.
3. As reported clearly in the manuscript, there can be different solutions (cavity configurations) for the same N* depending on evolution history (in the quasi-steady sense), depending on whether an unconnected lee zone has yet been flooded --- e.g. at N* slightly above N*_disconnect in Fig. 5a (and within the corresponding sequence in Fig. 4). Does the solution method include or require a specific device or algorithm that tracks an aspect of the history to reach the different solutions, e.g. some iteration that uses the last configuration as initial guess? I might have missed it in the descriptions of p5 and the appendix, or perhaps it isn’t necessary and I misunderstand the solution method in (A3), or relevant details exist in the referenced literature. Clarification about this in the manuscript would be welcome.
4. N*_drown: On p11, N*_drown is explained clearly, but the meaning of “drown” there (describing separation of ice and bed everywhere across the domain and violation of force balance) seems different from what “drown” describes elsewhere (coverage over a bed protrusion by expanding cavity). Consider using another symbol rather than N*_drown?
5. Some passages consider or refer to bed protrusions or bumps as “less/more prominent” or “smaller/larger (or taller)”, e.g. on p17, also elsewhere. Although the literature on this subject may use these adjectives following an agreed convention, or specialists will always understand them, I think you should define their meaning early on. Probably you are comparing maximum elevations, not amplitude (so a “little” short bump superposed onto the peak of a large long bump is the largest/tallest in your analysis), but I am not in fact sure; I may have misinterpreted the meaning.
Technical corrections: typos, minor suggestions, etc.
The meaning of “process-scale”/“process-scale model” (several times in section 1) is unclear.
p2, line 6, unfinished sentence
p3, line 10, “robustness… obtained to changes”. Consider rephrasing this to make it easier to understand.
Fig 1: locate x = 0
Fig 1 caption: h(x) is cavity roof “elevation” (instead of height)? Line 3, v and y instead of w and z in the partial derivative.
p4, line 1, missing full stop
p5, line 25, “while”. Consider using “whereas” or “in contrast” to bring out the contrast against Part 2 better.
p6, line 9, “dimensionless combination of effective pressure” doesn’t describe eqn. (11)
Fig 3a: At first I read Fig 3a as showing the spatial distribution of N*. The horizontal axis needs to be labelled as b_j*, c_j*, as it isn’t x* (although the scale is the same). This issue extends to Fig. 5a to 5b and Fig. 9a to 9d, which are also missing the right axis labels.
Fig 3 caption, “bed shape b*(x*) against x*”. Change “shape” to “elevation”. Also in several other figures.
Fig 4 caption, line 1: should b*(c*) be b*(x*)? This arises also in the captions of Figures 2 and 7.
p8, line 3, “disappear at N*”. Is something missing here?
p8, line 6, is the colon before the equal sign a typo?
p9, line 26: “panels (a) and (e) … correspond to the same effective pressure”. Figure 4 caption gives two different values, 4.01 and 4.02.
p9, line 33: “bed an only”
Fig 6: since you plot two lines, the figure is clearer if you label the y-axis as “N2*, N1*” and add a label next to each line
Fig 7 caption, line 3: please check whether a minus sign is missing from the second sigma_nn
Fig 8, h_0 missing from the y-axis label
p14, line 18: trimple --- triple
Fig 9 caption: lines 8 to 10 have hiccups and duplication in various places. For example, “Panel (c). Panel (e): s abruptly”. And “Panel (c)” on line 9 had been introduced on line 6, and I would expect panel (e) (not d) to be covered last.
Fig 11: (i) in the y-axis label, should N2* be N1*? (ii) Although the gentle slope of the “flat portion“ of the blue line is described in the caption and the text, this feature would be much clearer if you set the axes’ aspect ratio at 1:1 (natural as both axes are effective pressures). Consider the same for Figure 6.
p17, line 16: “… depend on the geometry of the bed, and on the parts of the bed…”. (i) Clearer if you add “depend” after “and” or remove the comma. (ii) Clearer if you reword “parts”, because the intended meaning is the location/whereabout of the parts, not some undefined properties of parts.
p17, lines 23-26: I don’t understand how the implied possibility reported in the final sentence follows logically from the previous passage. Is something explanation missing?
p19, line 34: the complication or the definition is being stressed. This sentence is easier to read if written as “The definition of a friction law for an impermeable bed has a second complication that deserves to be stressed”
p20, line 23-24: A further limitation that should be mentioned is the limited range of locations x_P chosen in the experiments (on steepest points of lee sides). (Otherwise I think your design of putting x_P on different bumps on the topography works fantastically in illustrating the range of behaviour.)
p21, first line of A1, spurious comma after 54
p21, line 23, “whree”
Eqn. (A3): the integrand on the left should be a sum instead of difference. Capital C in h^prime_c, and b'' should be b'.
p23, line 15, hiccup after G(infty)
p24, line 2, "!98"
p24, line 5, to help readers, it is useful to (re)qualify V_j as cavity water volumes here. Again, capital C in h^prime_c ?
p24 last line, full stop before "To"
Citation: https://doi.org/10.5194/egusphere-2022-1380-RC1 - AC1: 'Reply on RC1', Christian Schoof, 23 May 2023
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RC2: 'Comment on egusphere-2022-1380', Anonymous Referee #2, 10 Feb 2023
In this paper, the author presents a novel model based on Fowler, 1987 and Schoof, 2005 for steady state subglacial cavitation over bedrocks that are hydraulically permeable in some regions, and impermeable in others. I find this work to be very well written and its scientific/mathematical content to be of excellent quality. Therefore, I recommend its publication after the minor considerations/corrections detailed below are considered.
General comments:- In this work, once a cavity becomes hydraulically isolated, the model enforces that the cavity have constant volume until it becomes hydraulically connected once again. Under these conditions, the water pressure of the cavity is an unknown to compute. Something that came to mind is the possibility of hydraulically isolated cavities that develop an air-filled gap in between the water and the ice. I can understand the reasoning behind the constant volume condition when, e.g., the far-field ice speed decreases, since the cavity would want to close but it cannot compress the water in it. However, upon an increase in this far-field speed, I find it logical that, at a certain point, when the water pressure becomes sufficiently low (perhaps 0?), a partially air-filled cavity starts to develop. In other words, it is reasonable to assume that a hydraulically isolated cavity is still pneumatically connected, since the opposite sounds unrealistic for most glacier conditions. In the context of your model, allowing the formation of air-filled cavities might mean enforcing pneumatic connectedness when a low water pressure is reached, and therefore allowing the volume to increase beyond the water volume. I mention this point because it would mean that the unbounded increase in basal stress that you see for the single cavity in Figure 8 would perhaps no longer hold. If this comment is sensible, I think it should be commented in the discussion.
Specific comments:p3, line 18: Consider starting sentence with a non-mathematical symbol.
p3, line 23: "the normal component of velocity vanishes" > "the normal component of the velocity vanishes".
p 5, line 14: In Stubblefield et al. (2021), the rate of change of the volume is enforced, not the total cavity volume. Mathematically, this is carried out by setting the integral of the normal velocity at the lower boundary equal to this prescribed rate of change. The Lagrange multiplier associated to this constraint is the water pressure.
p5, line 16: As you write, I also find the specification of a permeable point x_P along the bed awkward, from a physical point of view. It seems to me that the fact that you choose it to be the location where cavities first form is effectively equivalent to choosing that a particular cavity, once it forms, is connected to the external drainage system. I think it would be helpful to the reader to clarify the intention behind choosing x_P to be this point.
p 6, line 29: "by plotting cavity end point position" > "by plotting the cavity end point positions"
p 8, figure 4: I think it would be helpful to the reader to include a subtle line in the plots for the normal stress that indicates the values of N*. It would make it much easier to notice the areas of low pressure and the difference in water pressure between isolated and connected cavities. Same for figure 7.
p 9, line 3: "see also fig 3" > "see also figure 3"
p 9, line 9: "decreased." > "decreased"
p9, line 33: "we start with an uncavitated bed an only lower N*", this sentence does not make sense.
p10, Figure 5: Close brackets in line 3 of caption.
p11, line 6: "cavity cavity" > "cavity"
p13, line 5: "by arrow" > "by the arrow"
p13, line 12: Consider starting sentence with a non-mathematical symbol.
p14, line 19: "trimple" > "triple"
p16, line 7: "is" > "are"
p16, line 8: "again be unbounded" > "again unbounded"
p17, line 30: close brackets.
p17, line 32: "it easier" > "it is easier"
p18, line 20: "constraint" > "constraints"
p18, line 33: "If drainage system access" > "If the drainage system access"
p18, line 34: "effective pressure" > "the effective pressure"
p19, line 18: "bed," > "bed."
Citation: https://doi.org/10.5194/egusphere-2022-1380-RC2 - AC2: 'Reply on RC2', Christian Schoof, 23 May 2023
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hydraulically connected(how easily water can flow along the glacier bed) plays a central role in determining how fast ice can slide.
Christian Schoof
Christian Schoof
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