the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
A Novel Transformation of the Ice Sheet Stokes Equations and Some of its Properties and Applications
Abstract. A fullStokes model provides the most accurate but also the most expensive representation of ice sheet dynamics. The BlatterPattyn model is a widely used less expensive approximation that is valid for ice sheets characterized by a small aspect ratio. Here we introduce a novel transformation of the Stokes equations into a form that closely resembles the BlatterPattyn equations. The transformed exact Stokes equations only differ from the approximate BlatterPattyn equations by a few additional terms, while their variational formulations differ only by the presence of a single term in each horizontal direction (one term in 2D and two terms in 3D). Specifically, the variational formulations differ only by the absence (or the neglect) of the vertical velocity in the second invariant of the strain rate tensor in the BlatterPattyn model when compared to the Stokes case. Here we make use of the new transformation in two different ways. First, we consider incorporating the transformed equations into a code that can be very easily converted from a Stokes to a BlatterPattyn model, and viceversa, simply by switching these terms on or off. This may be generalized so that the Stokes model is switched on adaptively only where the BlatterPattyn model loses accuracy, hopefully retaining most of the accuracy of the Stokes model but at a lower cost. Second, the key role played by the vertical velocity in converting the transformed Stokes model into the BlatterPattyn model motivates new approximations that improve on the BlatterPattyn model, heretofore the best approximate ice sheet model. These applications require the use of a grid that enables the discrete continuity equation to be invertible for the vertical velocity in terms of the horizontal velocity components. Examples of such grids, such as the firstorder P1E0 grid and the secondorder P2E1 grid are given in both 2D and 3D. It should be noted, however, that the transformed Stokes model has the same type of gravity forcing as the BlatterPattyn model, i.e., determined by the slope of the ice sheet's upper surface, thereby forgoing some of the gridgenerality of the traditional formulation of the Stokes model. This is not a serious disadvantage, however, since in practice it has not impaired the widespread use of the BlatterPattyn model.
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RC1: 'Comment on egusphere20241052', Ed Bueler, 08 Jul 2024
Summary: This paper rewrites the standard glaciological (Glen law) Stokes model in a form which resembles a shallow approximation, the BlatterPattyn (BP) model. This expresses the saddlepoint structure of the Stokes problem in a form close to the unconstrainedoptimization form of the BP model. The stability and finite element (FE) analysis of the new form is addressed, and new mixed FE pairs for verticallyextruded meshes are propsed. Smallscale experiments are presented, and then prospective applications at larger scale are discussed. The resulting essentiallytheoretical paper is both frustrating and promising. The manuscript's current form is notably inefficient, with 1500 lines of text. The presentation is likely to be hard to read for those who have not already done battle with BP equations and related technical matters. Despite doing numerical experiments, the author provides no opensource code basis for further development by readers, a clear demerit in 2024. The manuscript avoids the functionspace understanding of the Stokes and BP problemsthis is the viewpoint from which these problems are known to be wellposed and by which they are solved by mainstream finite element librariesbut then it labors to build a fragmented substitute for this viewpoint. Despite these flaws, the paper illuminates important matters. It shows how the (transformed) Stokes equations are close to an "extended BlatterPattyn" (EBP) form, and thereby how the solvability conditions of the Stokes model work in practice over verticallyextruded meshes. The EBP model has similar numerical and stability issues as the Stokes problem, which is actually clarifying because the numerical and FE character of the standard BP and Stokes models otherwise appear very different. The infsup stability of the mixed Stokes problem is recognized here, when the mesh is extruded and when one simultaneously wants the EBP model to be solvable on the same mesh, as the requirement of unique solvability of the continuity (incompressibility) equation for the vertical velocity from the horizontal velocity. A necessary condition for this to work is that the number of vertical velocity and pressure unknowns must be exactly the same, or rather that a particular matrix in the blockwise form of the discrete equations must be invertible.
Recommendation: A manuscript which made the same points in half the length, and which provided open source code in a widelyused language, facilitating further development, would be an excellent paper. Of course it is not realistic to expect recoding at that level. However, significant revisions should be attempted. A muchshortened abstract is offered below, along with several other suggestions for trimming.
Specific Comments on Manuscriptlines 935: This long abstract could be halved without losing meaning, by removing the sales pitches and by other simple edits. However, changes are also needed to clearly identify the models (systems) under consideration. The following is a guess/suggestion for an abstract which meets these objectives. It has 191 words vs 371 in the original: """We introduce a novel transformation of the Stokes equations into a form that resembles the shallow BlatterPattyn (BP) equations. The two forms only differ by a few additional terms, and the variational formulations differ only by a single term in each horizontal direction, but the BP form also lacks the vertical velocity in the second invariant of the strain rate tensor. The transformed Stokes model has the same type of gravity forcing as the BP model, determined by the ice surface slope. An apparently intermediate "extended BlatterPattyn" (EBP) form is identified, which is actually the same as the standard BP model although it retains a pressure variable. The role played by the vertical velocity in the transformed Stokes and EBP forms, reflected in the blockwise structure of their discrete equations, motivates the construction of new finite element velocity/pressure pairs for verticallyextruded meshes. With these new pairs, examples of which are demonstrated in 2D and 3D, the discrete continuity equation can be uniquely and stably inverted for the vertical velocity. We describe how to incorporate the new forms into codes that adaptively switch between Stokes and BP models, where the latter would lose accuracy."""
line 41: "full" is unnecessary.
line 5272: The style of glaciology, used at unnecessary length in these lines, says some models are shallow and some are higher order. It is more accurate to say all are shallow, and to not claim some are "higherorder" because the order depends on which scaling argument is use.
line 99: "THE LOWER BOUNDARY OF an ice sheet ...". (A 3D ice sheet can't be divided the way the text says.)
lines 103105: This "vertical line of sight" phrase appears here and later. Surely one can just say: "We assume the glacier's geometry is described by an upper surface function z_s(x,y) and a lower surface function z_b(x,y)."
lines 105106: There is nothing about the rest of the paper, in my reading, that excludes the techniques being used for floating ice. (Put f_i=0 in equation (11)?) It is true that there must be sufficient dragsee the inequality in Schoof (2006)*somewhere at the base* so that the velocity field is unique, but the techniques apply across grounding lines.
lines 112126 Briefer notation is surely possible.
line 149: "positivedefinite" > "nonnegative"
line 178180: Whether or not the surface kinematical equations can be added "easily", the way this is said here is silly. The whole paper assumes fixed ice geometry.
lines 192195: I don't know what this means. "There are some stress boundary conditions and it is easier for the author to think about them in the variational formulation."? No need for this?
lines 197200: No need for this.
lines 204209: Is this option ever used later in the paper? (Line 233 suggests not.) If not, it can be removed and replaced with a simple declaration that the boundary conditions can be weakly imposed if desired.
lines 238252: This is a valuable observation, namely form (17) which shows ~P solves a trivialized problem. If this observation is original, then great. Otherwise cite it more clearly; did it appear in DPL 2010? (The nearby citations to DPL do not refer to this main idea as far as I can tell.)
Figure 2: This basic point is greatly appreciated: The deviation from hydrostatic is relatively small. However, in this and almost all figures, the fonts are too small! (Also these figures are bad on a monochrome printer, but I suppose that train has left ...)
line 282: I don't think (22) is actually used *here*.
around line 282: Warn the reader that "dummy variables" ("flag variables"?) are about to be used. As the text is written, they are finally explained on the next page.
lines 286 and onward: I find "modified" really unpleasant here. For (25) the tensor ~\tau_ij is actually modified; it is not equal to the original. But in (26) the tensor is merely rewritten; neither "modified" nor the tilde have the same meaning as they do in the equation above. Similarly (27) and (28) are not "modified" but merely rewritten, as far as I can tell. I therefore would not say "modified" or add a tilde; just write out the new form. Equality means equality.
line 325: "implies the use of" > "uses"
lines 327336: This is a rambling paragraph that can be shortened to something like "As noted earlier we require the upper and lower surfaces of the glacier to be functions of the horizontal coordinates x,y. That is, as expected in glacier modeling, overhangs are not permitted."
line 344348: Repetitive. Say *once* (earlier, presumably) that one could impose boundary conditions weakly, and that you won't do that.
line 360: Help the reader by referencing/comparing (23).
lines 361 and 404: Separate these into 2 displays. (Or better, just be more efficient. Use vector notation?)
lines 437439: This use of the continuity equation is completely mainstream in glaciology. It applies in all shallow theories including BP. (And the current manuscript illuminates it!) Please say this some other way.
lines 459460: Again, deriving FE discretizations from variational principles is the normal way to do business. Why "except"?
line 475: There is no reason to use capital "U" here, and it is a source of confusion because capital U is used shortly in subscripts with a different meaning.
line 495: "u, w, AND M_{UP}, M_{WP}"
Section 4.3: This section needs editing most. The main point of the entire paper is made in subsection 4.3.3, I believe. Roughlyspeaking the main point is that, for the transformed Stokes or EBP equations, the block M_{WP} must be invertible, thus square, when an extruded mesh with zaligned cells is used. This point is buried after laborious and repetitive text. The main point of the paper *does* require a blockwise presentation of the Newton step equations, so the text will necessarily be somewhat technical, but it doesn't have to bury the main idea. There would seem to be no reason not to start a section with (47) and (48); the notation here is obvious. In any case, this reader had to get 600 lines into the document before getting to the key lines (roughly starting at line 596), and only then have an "oh ... that is what he is trying to say ..." moment.
lines 596600: The main point of the paper, right? Which this reader appreciates! The blockwise form of the EBP model is therefore the central object of the paper, and could be put much earlier and more prominently.
lines 616618: I would not permit my undergrad linear algebra students to say what is said here. The necessary condition is that *M_{WP} must be nonsingular*, from which it *follows logically* that it must be square. The text literally says that nonsingularity is "in addition" to squareness, thereby asserting that square matrices are invertible! (Line 1521 is worse.) Equation (56) could instead say "M_{WP} is nonsingular"; one is allowed to put text in displayed LaTeX equations.
Section 5: I think the paper would be improved by removing this section. I understand that the transformed Stokes model is the same as the Stokes model, and the EBP model is the same as the BP model. So recapitulating the ISMIPHOM purpose, which is (I suppose) to examine how close BP results are to Stokes results, should not come out any differently here, and thus it is not worth doing. Of course it is true that different numerical approaches generate different results in detail. But what exactly should the reader know about this numerical comparison? Can this be summarized in a sentence or two?
lines 778780: For efficiency I assume that BP is first used everywhere, then some criteria is applied, and then Stokes is used where the criteria applies. But do you want to demonstrate that the Stokes calculation everywhere gives the nearly same criteriasatisfying region?
line 785: Is the "counterintuitive" aspect of this explained by noting that the effective viscosity is often actually largest in the top of the ice column, which implies the greatest longitudinal and bridging stress transmission up there? I often find that visualizing the effective viscosity, in these shearthinning flows, illuminates where stresses delocalize the problem.
line 811813: It is not the personal computer etc. which stops an analysis of the cost savings, but rather the lack of a performance model for the solver. This could be added, but it requires a bit of thinking.
Subsection 6.2 and Section 7: This seems like tedious overkill. If a reader gets the main points of the paper then they can probably imagine lagging the Newton iteration and/or dual grids and/or higher order. In any case, another 300 lines are burned before the summary. If these are important enough then they could be a separate paper? Otherwise most readers won't have the endurance; really I don't.
Section 8 (Summary): Too long.
Appendix AC: On and on.
Appendix D: The manipulations shown in (79) and (80) are again very close to the main novel point of the paper. I see no reason why they can't be written into a new and prominent form which makes subsubsection 4.3.3 into the central material.
line 1521: Again, please don't say that all square matrices are invertible. (Literally the text says "the solvability condition [n_u=n_p] implies the invertibility of M_{WP}". Just no.)
Citation: https://doi.org/10.5194/egusphere20241052RC1  AC1: 'Reply to RC1', John Dukowicz, 31 Aug 2024

RC2: 'Comment on egusphere20241052', Christian Schoof, 07 Aug 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere20241052/egusphere20241052RC2supplement.pdf
 AC2: 'Reply to RC2', John Dukowicz, 31 Aug 2024
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