the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 19 Dec 2024
Brief communication: velocities and thinning rates for Halfar’s analytical solution to the Shallow Ice Approximation
Abstract. Analytical solutions to approximations of the Stokes equations are invaluable tools for verifying numerical ice-sheet models. Halfar (1981) derived a time-dependent solution to the Shallow Ice Approximation (SIA). Here, I derive the associated ice velocity vector field, and the resulting thinning rates, which may serve as additional checks for numerical ice-sheet models.
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RC1: 'Comment on egusphere-2024-3610', Logan Mann, 04 Mar 2025
This is an excellent contribution to the existing literature on analytic methods for ice sheet and glacier modeling. I was initially skeptical that this calculation is entirely novel. However, after some literature review, this is, to the best of my knowledge, an entirely novel contribution. This is likely to be particularly useful for benchmarking implementations of mass conservation in ice sheet models. While time-dependent SIA models are increasingly rare in ice sheet modeling, they are still widely used for mountain glacier modeling. The derivation is rigorous, detailed, and mathematically well presented. I have only a few notes on places where some more background information could be provided and some more clarity could be added to the derivation.
13. While numerical benchmarks are the most straightforward application of this work, I think this section could be strengthened with an appeal to the intellectual value of exact solutions. They are a uniquely insightful and complete way to understand a problem.
19. While the solution in Schoof (2006; JFM) only computes a quasi-static solution for the SSA velocity field, steady state analytic solutions for the velocity field and surface profile of SSA do exist. The first analytic solution to SSA was computed by Gunnar Böðvarsson (1955; Jökull), long before the formal application of free-film models to glaciology by Morland, Macayeal, etc. This relatively obscure article developed an exact, steady state solution for the flow field and surface profile of a grounded ice sheet (albeit with a very particular longitudinal stress distribution). A bewildering aspect of this solution is that it is not entirely clear how he derived it, beyond a remarkably effective educated guess, and it may represent a previously unknown solution to a class of 2nd-order nonlinear ODEs. Often misinterpreted as an approximate solution, Bueler (2014; JGLAC) showed that it is in fact an exact solution, and demonstrates how to couple it to an ice shelf solution from Van der Veen (1983; 2013), for an exact, steady state solution for a marine ice sheet, albeit a very particular one. The Böðvarsson solution is an important part of glaciological history and worth mentioning. Also, performing a similar calculation, to calculate the vertical velocity component of Böðvarsson’s solution, may be an interesting avenue for future work.
25. “… has never been checked against analytical solutions…” My interpretation of this is that the vertical velocity has not been analytically calculated from the Halfar solution. It may be too sweeping a claim to say that the vertical ice velocity has not been checked against analytical solutions at all. I’d rephrase to say that numerical simulations have not been checked against the vertical velocity component of the Halfar solution.
38.“defining the extensional strain rates at the ice margin presents additional problems, as the ice thickness and velocity there are discontinuous, and the margin generally does not coincide exactly with any grid point.”Shapero and de Diego (2024; JGLAC) partially addresses this with the 'dual-form’ of SSA, essentially inverting the constitutive equation in the variational form of SSA, which renders a mathematically consistent description of the ice margin. This might be worth mentioning, but it is not directly applicable to the SIA model, so I leave it at author's discretion.
90. Is the Q term a flux term? It might be worth explicitly saying that. In general, if some of the variable definitions that you've introduced have a straightforward physical interpretation, calling that out directly could make it easier for readers follow the derivation and develop their own intuition for it.
183. It might be worth elaborating on the ice divide problem as a known problem with the SIA model itself, rather than a unique aspect of your analysis for a reader to be concerned about.
Figure 1. A comparison to numerical results would be helpful for readers, but I leave this to the author's discretion.
-Logan E. Mann
Citation: https://doi.org/10.5194/egusphere-2024-3610-RC1 -
RC2: 'Comment on egusphere-2024-3610', Ian Hewitt, 21 Mar 2025
This paper presents various formulas leading to analytical expressions for the velocity components in a particular similarity solution of the Shallow Ice Approximation, referred to as Halfar’s solution. It is argued that such expressions may be useful for verifying the velocity field produced by numerical methods for more complicated ice-sheet models.
Whilst I agree such solutions are useful for these comparisons, I guess I don’t really feel there’s enough here to warrant a publication. It is basically an algebra exercise, which takes a little while to work through but is not hard, and I’m not sure we learn much from it. If it is to be published, I think the presentation could be improved. I also wonder whether The Cryosphere is the best venue for this. I feel a better venue might be a journal focussed on numerical methods, such as Geoscientific Model Development.
The presentation seems to me rather laboured and not particularly helpful to the reader. There are more steps of algebraic manipulation than are needed, and far more variables/notations introduced than are necessary. For example, the whole solution is axisymmetric, so it would be neater to work in cylindrical polar coordinates. If introducing shorthand notation to abbreviate an expression, it would be better to introduce it before you write the expression, rather than first writing out a long version, then saying let’s introduce a shorthand, and then writing the expression again. The choice of notation is also unfortunate in some cases; Q, for example, is so commonly used to mean a flux, that using it for the quantity it’s used for here is quite confusing, and there seems to be no logic in which quantities depend on what: f_1 and f_2 are functions of time whereas f_3 is a function of space. When introducing new variables to represent quantities that are obviously negative, it might be more intuitive to define them the other way around (e.g. why include the minus sign in Q in (20) and in the constant c in (25)?)
In fact, while reviewing this I found the notation sufficiently confusing that I decided to just go through the calculation myself, so I will try to attach what I think could be a neater way to present it (I can't promise there are no mistakes).
Other comments on the text:
Line 25 - I would be quite suspicious whether this statement is really true. I expect checks of the vertical velocity have been done, even if not reported. (I have only worked with depth-integrated models myself, for which these analytical solutions are used to *find* the vertical velocity rather than to check it, but I’ve often done similar checks that I wouldn’t necessarily report the details of in a publication).
Line 30 - I’m a bit confused by this statement. The continuity equation (or an alternative for non-constant density) is required for any solution of the Stokes equations. In depth-integrated models it’s often not discussed explicitly because it’s often only used in its depth-integrated form (which provides the evolution equation of the ice thickness).
Equation (4) - I think the ‘5’ in the definition of Gamma here ought to be (n+2) - (which is of course 5 for n=3, but you haven’t specialised to n=3 anywhere else).
Equation (37) - I don’t think you’ve actually defined V_1 and V_2 anywhere (though they are relatively easy to guess).
Line 186 - I believe there are known to be singularities at a contact line even for the full Stokes equations - the ‘moving contact line paradox’ - for which work-arounds can include appealing to some amount of slip at the base, or to some sort of molecular scale physics that is not accounted for by the continuum approximation. (e.g. Snoeijer & Andreotti 2013 Moving Contact Lines: Scales, Regimes and Dynamical Transitions, Ann. Rev. Fluid Mechanics. https://doi.org/10.1146/annurev-fluid-011212-140734).
Status: closed
-
RC1: 'Comment on egusphere-2024-3610', Logan Mann, 04 Mar 2025
This is an excellent contribution to the existing literature on analytic methods for ice sheet and glacier modeling. I was initially skeptical that this calculation is entirely novel. However, after some literature review, this is, to the best of my knowledge, an entirely novel contribution. This is likely to be particularly useful for benchmarking implementations of mass conservation in ice sheet models. While time-dependent SIA models are increasingly rare in ice sheet modeling, they are still widely used for mountain glacier modeling. The derivation is rigorous, detailed, and mathematically well presented. I have only a few notes on places where some more background information could be provided and some more clarity could be added to the derivation.
13. While numerical benchmarks are the most straightforward application of this work, I think this section could be strengthened with an appeal to the intellectual value of exact solutions. They are a uniquely insightful and complete way to understand a problem.
19. While the solution in Schoof (2006; JFM) only computes a quasi-static solution for the SSA velocity field, steady state analytic solutions for the velocity field and surface profile of SSA do exist. The first analytic solution to SSA was computed by Gunnar Böðvarsson (1955; Jökull), long before the formal application of free-film models to glaciology by Morland, Macayeal, etc. This relatively obscure article developed an exact, steady state solution for the flow field and surface profile of a grounded ice sheet (albeit with a very particular longitudinal stress distribution). A bewildering aspect of this solution is that it is not entirely clear how he derived it, beyond a remarkably effective educated guess, and it may represent a previously unknown solution to a class of 2nd-order nonlinear ODEs. Often misinterpreted as an approximate solution, Bueler (2014; JGLAC) showed that it is in fact an exact solution, and demonstrates how to couple it to an ice shelf solution from Van der Veen (1983; 2013), for an exact, steady state solution for a marine ice sheet, albeit a very particular one. The Böðvarsson solution is an important part of glaciological history and worth mentioning. Also, performing a similar calculation, to calculate the vertical velocity component of Böðvarsson’s solution, may be an interesting avenue for future work.
25. “… has never been checked against analytical solutions…” My interpretation of this is that the vertical velocity has not been analytically calculated from the Halfar solution. It may be too sweeping a claim to say that the vertical ice velocity has not been checked against analytical solutions at all. I’d rephrase to say that numerical simulations have not been checked against the vertical velocity component of the Halfar solution.
38.“defining the extensional strain rates at the ice margin presents additional problems, as the ice thickness and velocity there are discontinuous, and the margin generally does not coincide exactly with any grid point.”Shapero and de Diego (2024; JGLAC) partially addresses this with the 'dual-form’ of SSA, essentially inverting the constitutive equation in the variational form of SSA, which renders a mathematically consistent description of the ice margin. This might be worth mentioning, but it is not directly applicable to the SIA model, so I leave it at author's discretion.
90. Is the Q term a flux term? It might be worth explicitly saying that. In general, if some of the variable definitions that you've introduced have a straightforward physical interpretation, calling that out directly could make it easier for readers follow the derivation and develop their own intuition for it.
183. It might be worth elaborating on the ice divide problem as a known problem with the SIA model itself, rather than a unique aspect of your analysis for a reader to be concerned about.
Figure 1. A comparison to numerical results would be helpful for readers, but I leave this to the author's discretion.
-Logan E. Mann
Citation: https://doi.org/10.5194/egusphere-2024-3610-RC1 -
RC2: 'Comment on egusphere-2024-3610', Ian Hewitt, 21 Mar 2025
This paper presents various formulas leading to analytical expressions for the velocity components in a particular similarity solution of the Shallow Ice Approximation, referred to as Halfar’s solution. It is argued that such expressions may be useful for verifying the velocity field produced by numerical methods for more complicated ice-sheet models.
Whilst I agree such solutions are useful for these comparisons, I guess I don’t really feel there’s enough here to warrant a publication. It is basically an algebra exercise, which takes a little while to work through but is not hard, and I’m not sure we learn much from it. If it is to be published, I think the presentation could be improved. I also wonder whether The Cryosphere is the best venue for this. I feel a better venue might be a journal focussed on numerical methods, such as Geoscientific Model Development.
The presentation seems to me rather laboured and not particularly helpful to the reader. There are more steps of algebraic manipulation than are needed, and far more variables/notations introduced than are necessary. For example, the whole solution is axisymmetric, so it would be neater to work in cylindrical polar coordinates. If introducing shorthand notation to abbreviate an expression, it would be better to introduce it before you write the expression, rather than first writing out a long version, then saying let’s introduce a shorthand, and then writing the expression again. The choice of notation is also unfortunate in some cases; Q, for example, is so commonly used to mean a flux, that using it for the quantity it’s used for here is quite confusing, and there seems to be no logic in which quantities depend on what: f_1 and f_2 are functions of time whereas f_3 is a function of space. When introducing new variables to represent quantities that are obviously negative, it might be more intuitive to define them the other way around (e.g. why include the minus sign in Q in (20) and in the constant c in (25)?)
In fact, while reviewing this I found the notation sufficiently confusing that I decided to just go through the calculation myself, so I will try to attach what I think could be a neater way to present it (I can't promise there are no mistakes).
Other comments on the text:
Line 25 - I would be quite suspicious whether this statement is really true. I expect checks of the vertical velocity have been done, even if not reported. (I have only worked with depth-integrated models myself, for which these analytical solutions are used to *find* the vertical velocity rather than to check it, but I’ve often done similar checks that I wouldn’t necessarily report the details of in a publication).
Line 30 - I’m a bit confused by this statement. The continuity equation (or an alternative for non-constant density) is required for any solution of the Stokes equations. In depth-integrated models it’s often not discussed explicitly because it’s often only used in its depth-integrated form (which provides the evolution equation of the ice thickness).
Equation (4) - I think the ‘5’ in the definition of Gamma here ought to be (n+2) - (which is of course 5 for n=3, but you haven’t specialised to n=3 anywhere else).
Equation (37) - I don’t think you’ve actually defined V_1 and V_2 anywhere (though they are relatively easy to guess).
Line 186 - I believe there are known to be singularities at a contact line even for the full Stokes equations - the ‘moving contact line paradox’ - for which work-arounds can include appealing to some amount of slip at the base, or to some sort of molecular scale physics that is not accounted for by the continuum approximation. (e.g. Snoeijer & Andreotti 2013 Moving Contact Lines: Scales, Regimes and Dynamical Transitions, Ann. Rev. Fluid Mechanics. https://doi.org/10.1146/annurev-fluid-011212-140734).
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