Timescales of outletglacier flow with negligible basal friction: Theory, observations and modeling
 ^{1}Potsdam Institute for Climate Impact Research (PIK), Potsdam, Germany
 ^{2}Department of Geography and Environmental Sciences, Northumbria University, Newcastle, UK
 ^{3}Institute of Physics, University of Potsdam, Potsdam, Germany
 ^{1}Potsdam Institute for Climate Impact Research (PIK), Potsdam, Germany
 ^{2}Department of Geography and Environmental Sciences, Northumbria University, Newcastle, UK
 ^{3}Institute of Physics, University of Potsdam, Potsdam, Germany
Abstract. The timescales of the flow and retreat of Greenland's and Antarctica's outlet glaciers and their potential instabilities are arguably the largest uncertainty in future sealevel projections. Here we derive a scaling relation that allows the comparison of the timescales of observed complex ice flow fields with geometric similarity. The scaling relation is derived under the assumption of fast, laterally confined, geometrically similar outletglacier flow over a slippery bed, i.e., with negligible basal friction. According to the relation, the time scaling of the outlet flow is determined by the product of the inverse of 1) the fourth power of the widthtolength ratio of its confinement, 2) the third power of the confinement depth and 3) the temperaturedependent ice softness. For the outflow at the grounding line of streams with negligible basal friction this means that the volume flux is proportional to the ice temperature and the bed depth, but goes with the fourth power of the gradient of the bed and with the fifth power of the width of the stream. We show that the theoretically derived scaling relation is supported by the observed velocity scaling of outlet glaciers across Greenland as well as by idealized numerical simulations of marine icesheet instabilities (MISIs) as found in Antarctica. Assuming a proportionality between the flow itself and its spatial derivative, we combine the scaling relation with a statistical analysis of the topography of 13 MISIprone Antarctic outlets. Under these assumptions the timescales in response to a potential destabilization are fastest for Thwaites Glacier in West Antarctica and Mellor, Ninnis and Cook Glaciers in East Antarctica; between 16 and 67 times faster than for Pine Island Glacier. While the applicability of our results is limited by several strong assumptions, the utilization and potential further development of the presented scaling approach may help to constrain timescale estimates of outlet glacierflow, augmenting the commonly exploited and comparatively computationally expensive approach of numerical modeling.
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Johannes Feldmann and Anders Levermann
Status: final response (author comments only)

RC1: 'Comment on tc2022141', Anonymous Referee #1, 14 Sep 2022
Review of Timescales of outletglacier flow with negligible basal friction: Theory, observations and modeling.
This paper uses the similarity principle applied to the SSA neglecting the basal friction with the goal to infer the time scale of marine ice stream retreats in Greenland and Antarctica. The paper considers idealized setup along with specifically chosen glaciers for Greenland and Antarctica and shows relatively good agreements with the theory.
I found the paper really well written and organized. I would like to thank the authors to have spent the time on both aspects. I enjoyed reading this manuscript. While working with 3D complex models is becoming more of a norm, it is refreshing to see that simple mathematical arguments (derived from equations used in the 3D models) remain useful to understand or give a sense on dynamical principals. While the manuscript is very mathematically driven, the authors carefully left the details in the appendix while leaving the basic understandings of the method in the main text. Doing so results in a relatively short manuscript with little dilution from the mathematical details and well suited for a quick read.
I would fully support the publication of this publication after minor revision. Main and minor comments are following.
Main Comments
 In comparing your similarity principle to Greenland glaciers, you set aside a couple of the glaciers and argued that physical properties of the glaciers break the similarity assumption. Did you check that it was simply the case that these glaciers behaved with respect to a similarity principal for which the main balance in the momentum equation is between the basal friction and the driving stress?
 Since you choose to neglect the basal friction term in your similarity discretization, why did you choose a Weertmantype sliding law as opposed to a Coulombderived sliding law (Schoof (2005), Leguy et al. (2014), Tsai et al. (2015) ) in your idealized setup? A Coulomb sliding law has the advantage to go to zero at the groundline line by design. Please discuss this point in your manuscript.
 How can you compare your Antarctic theoretical computation with actual retreat rates? Could you compare them using the data from Rignot et al. 2019? You have found an elegant way in obtaining a retreat rate with a simple method, it would be super useful to show that the order of magnitude matches current observation in a way.
 While I understand that textbooks are a rich resource of information, using them for citations should be a last resort and when doing so, please mention the chapter and section or page number (Cuffey and Paterson 2010) is a rather large volume! You have used book references very often while papers would have been more appropriate. I will try to address a few of them bellow.
Figures
Figure 1, panel C: Please describe the meaning of each color shade. Also, are you using different values of A in order to obtain each profile starting from the most advanced profile? (I thought this was not very clear in the text or caption.)
Figure 4, panel B: I don’t understand what the ice softness indication “A” is doing on the figure. Please clarify or remove it.
Figure 7: please give a reference value for the blue and red semicircle for your reference glacier (PIG).
Minor comments:
Page 4, line 93: replace the citation “Greeve and Blatter, 2009” by Glen (1955).
Page 4, line107: The citations to reference the “Observations and laboratory studies” are wrong. Schoof 2007 and Haseloff 2015 use the value in their modeling effort. Instead, cite Duval (2013). Also, n cannot be observed directly as it is a parameter in a law which best fit data. Instead, I would replace “Observations and laboratory studies” by simply “Laboratory studies”.
Page 4 line 114: see remark above regarding Cuffey and Paterson.
Page 5 line 120: you can also cite Leguy (2015, chapter 7.1.5) who derived a relation for buttressing that is inversely proportional to the width of the bed. (Note that I do not expect people to read the dissertation of an author, I just happen to know his work having collaborated with him.)
Page 6, line 167: what did you modify in the MISMIP+ setup. Please add the relation you use in your bed topography derivation (in the appendix).
Page 8, line 216: Please remove the part of the sentence “might have been destabilized by recent oceanic warming.” It is highly speculative hence unnecessary here unless you can support the claim.
Page 10, line 278279: for the reason you mention here, why not running the model using a Coulomb friction law?
Page 10, line 279: Replace the citation by (Schoof (2005), Martin et al. (2011), Leguy et al. (2014))
Page 1112, code and data availability: The code, data and simulation setup should be available prior to the publication of the manuscript. Make sure that it is the case and replace the wording “will be” by “is”. At this point I could access your data.
Page 12, line 358: citation, see remark above.
Page 12, line 365: I don’t see the necessity in introducing the scalings Sx and Sy. Why not directly using L and W?
Page 12, line 366: Please, introduce the scaling x*=x/L and y*=y/W for clarity.
Page 13, line 381: Please define the O symbol as the Landau notation for nonmathematicians. And technically speaking, the leading order in Eq.(A3) is O(1/R) (if we account for the square root).
Page 14, line 399: citations, replace with Glen (1955) and Duval (2013).
Page 14, line 412: You argue that the velocity shape of the Rink Isbrae glacier near the terminus is ground to discard this glacier from your analysis. Why then keep Upernavik Isstrom N which exhibit a similar pattern?
Page 15, line 442: Please give the uniform rate of SMB you used in your simulations.
Page 15, line 444: replace citation with Glen (1955)
Page 15, line 449: Add chapter to citation.
References:
Rignot, E., Mouginot, J., Scheuchl, B., Van Den Broeke, M., Van Wessem, M. J., & Morlighem, M. (2019). Four decades of Antarctic Ice Sheet mass balance from 1979–2017. Proceedings of the National Academy of Sciences, 116(4), 10951103.
Glen, J. W. (1955). The creep of polycrystalline ice. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 228(1175), 519538.
Duval, P. (2013). Creep behavior of ice in polar ice sheets. In The Science of Solar System Ices (pp. 227251). Springer, New York, NY.
Leguy, Gunter. The effect of a basalfriction parameterization on groundingline dynamics in icesheet models. New Mexico Institute of Mining and Technology, 2015.
Schoof, C. (2005). The effect of cavitation on glacier sliding. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461(2055), 609627.
Martin, M. A., Winkelmann, R., Haseloff, M., Albrecht, T., Bueler, E., Khroulev, C., and Levermann, A.: The Potsdam Parallel Ice Sheet Model (PISMPIK) – Part 2: Dynamic equilibrium simulation of the Antarctic ice sheet, The Cryosphere, 5, 727–740, doi:10.5194/tc57272011, 2011. 6
Leguy, G. R., AsayDavis, X. S., & Lipscomb, W. H. (2014). Parameterization of basal friction near grounding lines in a onedimensional ice sheet model. The Cryosphere, 8(4), 12391259.
Tsai, V. C., Stewart, A. L., Thompson, A. F.: Marine icesheet profiles and stability under Coulomb basal conditions, J. Glaciol., 61, 205–215, doi:10.3189/2015joG14j221, 2015. 28, 76, 77

AC1: 'Reply on RC1', Johannes Feldmann, 21 Nov 2022
We would like to thank Referee#1 for their willingness to review our manuscript and for the helpful comments and suggestions. We are glad for the referee’s positive assessment of our study and are happy to hear that they would fully support the publication in TC after minor revisions. We will be glad to address the points raised by the referee in a revised version of the manuscript.
Sincerely,
Johannes Feldmann

RC2: 'Comment on tc2022141', Camilla Schelpe, 21 Nov 2022
In this study, the authors derive a scaling relation which provides a simple prediction for the characteristic timescales of outlet glaciers in Greenland and the Antarctic. They assume the flow can be described by the SSA and neglect basal friction in the momentum conservation equations. Based on the width of these outlet glaciers being an order of magnitude less than the length of the flow, they determine the leading order terms in the momentum equations and show that the driving stress is balanced by the lateral shear stresses for these geometries. This leads to a dimensionless relationship which compares ice flows with similar properties. It is a clean and simple formulation which, by focussing on similitude, abstracts many of the complex interactions which govern the ice flow. The resulting scaling relation only requires the geometrical properties of the outlet glaciers (depth, width and length) together with ice softness, to determine the characteristic flow timescale. The derived relationship does not make absolute predictions, but instead makes predictions relative to other glaciers with similar properties.
The authors then go on to test this relationship thoroughly. First, they compare the predictions against the timescale inferred from velocity measurements of various Greenland outlet glaciers which exhibit similar topographic properties. These experimental results are promising. Second, they compare the predictions against the retreat timescale from idealised numerical simulations performed within the PISM ice sheet model. This comparison requires the assumption that the flow timescale and retreat timescale are proportional. In this idealised simulation that does indeed appear to hold true, and the authors find excellent agreement to their scaling relation. Finally, they use the derived scaling relationship to make predictions of the relative retreat timescale for a number of Antarctic outlet glaciers with a retrograde bed that may be prone to MISItype retreat.
I enjoyed reading the paper. The manuscript was well laid out, with the complexity of the mathematical derivations, and technical details of data extraction and experimental setup nicely compartmentalised into separate appendices. The results are well explained, with helpful interpretations of the intuition behind a number of the mathematical results. I also appreciated the comparison in Sec 5 to their earlier work (Levermann and Feldmann, 2019) which considered a 1D flowline with basal shear stresses included, but lateral drag neglected. Taken together, the conclusions from these two studies can be considered to give a range of predictions under the differing assumptions.
I have some comments and questions about the study, but I’m hoping these can be addressed through adding a bit more discussion to the manuscript rather than requiring any major changes to the results. I would fully recommend publication with these additions.
Specific Comments:
 This study neglects the contribution of basal shear stresses, which the authors justify through reference to various papers (L81L83) which infer a low basal friction coefficient for the rapidly sliding ice stream outlets of Antarctica and Greenland. Since a low basal friction coefficient, if combined with rapidly flowing ice, doesn’t necessarily translate to negligible basal shear stress, it would be good if the authors discussed this decision a bit more in the manuscript. Maybe they could discuss the expected dominance of lateral drag in deep, narrow confinement channels. And/or the results from the idealised Antarctic simulation which includes basal friction, could be included as a posthoc justification?
 It was only after reading the paper fully that I understood the distinction between the flow timescale which is inferred from the surface velocity of the ice stream and used in the derivation of the scaling relation; and the retreat timescale for the speed of grounding line retreat, which is that simulated and predicted for the Antarctic outlet glaciers. I think it would be helpful for the reader if that distinction was emphasised in the introductory section of the text, and added as a fourth point in the potential limitations listed in Sec 5.
 Related to the above point, the excellent agreement between the timescale for grounding line retreat in the idealised Antarctic simulations, and that predicted theoretically by the flow timescale, is perhaps surprising. It suggests a mathematical relationship which holds true in this idealised setup. The authors allude to this on L156: “Grounding line retreat depends on the divergence of groundingline discharge, i.e., on the divergence of the flow speed at the grounding line. If we were to seek a relation fo the groundingline retreat , we could make the assumption that the retreat speed of an outlet glacier if proportional to its flow speed.” Could the authors include the mathematical reasoning for this? This would also help explain under what conditions the assumption that “retreat timescale = flow timescale” is correct and thus how those conditions are being met for the retrograde slope in the idealised simulations. Related to this I think the commentary on L11. and in Fig 1 caption that, "the flow velocity and its spatial derivaative are proportional" may be misleading. If I have worked out this relationship correctly, I believe it should be that ∂h/∂t ∝ h × ∂u/∂x ? Not u ∝ ∂u/∂x ?
 In the experimental testing of the predicted scaling relation to the grounded Greenland outlet glaciers, the authors take the average over 60km0km upstream of the grounding line for the estimated width, and 60km20km for the estimated velocity. I understand the authors’ comment that the velocity is being cut off to avoid pollution from the iceocean interactions for the last 20km, but the velocities generally seem to increase in the last 20km coinciding with the width narrowing. Therefore, would it not make more sense to also exclude the last 20km from the width estimation so that you are comparing likewithlike? How sensitive is the fit of the data to these choices? The good fit of the scaling relation to the Greenland outlet glaciers lends confidence to the similitude approach being valid across real word glaciers, i,e. that glaciers exhibit enough similarities that this simple scaling can be applied across them. It therefore seems important to make sure the conclusions are robust, and not sensitive to these slightly arbitrary choices.
 For the plots in Figures 3, 6 and S4, would it make sense to use a log plot so that the predicted scaling relationship gives a straight line (with the gradient equal to the exponent in the scaling law)? Deviation from the expected behaviour would then perhaps be easier to see by eye. It would also be helpful to plot the OLS estimate from the data and compare the gradients.
Technical Corrections:
 There are a number of places in the text which refer to the scaling relation being linear in the ice temperature: L8, L129, L253. However, my understanding is that the linear relationship being referred to is to the ice softness A. The ice softness is temperature dependent, but not linearly dependent on temperature, I believe?
 L465L468. I found this description of extracting the length scale confusing. Could it be rewritten? Is it just a justification for setting L=1/S in the scaling equations, or is something else going on here?
 Is there a reason for picking p1 to p1+20km as the distance over which to estimate W? Naively I think I would have expected the estimate to be the average over p1 to p2.
 Clarify that these multiple combinations are coming from the multiple flow lines for each outlet glacier. (Unless I have misunderstood, in which case even more clarification needed!)
 Could you elaborate on why you have chosen the 17^{th} and 83^{rd} percentiles, rather than using the 5^{th}95^{th} percentile range?
 Delete “respectively”.
 This reads as if the uncertainty range in Table 3 reflects the breakdown in the similitude requirement. My understanding was that the uncertainty range still assumes geometric similarity and that the scaling relation holds; instead it reflects uncertainty in the appropriate average value to take for the different geometrical quantities due to topographic variation in the outlet. Are those two things the same?
 On its own this explanation of running a separate set of experiments with reduced C is a bit confusing. I would make it clear that the first set of simulations had a nonnegligible C, but in both cases the scaling relation held, which suggests the conclusions in this paper are unaffected by ignoring the basal friction in the derivation.
 Should this be 0.07 not 0.7?

AC2: 'Reply on RC2', Johannes Feldmann, 21 Nov 2022
We are grateful for the willingness of Camilla Schelpe to review our manuscript and for her constructive remarks, questions and suggestions. We are pleased by the positive assessment of our manuscript and are delighted to read that the Referee recommends the publication of our study subject to minor revisions. We are happy to address the Referee's comments in a revised version of the manuscript.
Sincerely,
Johannes Feldmann
Johannes Feldmann and Anders Levermann
Johannes Feldmann and Anders Levermann
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