Vertical shear is recognized today as a key component of the stress balance of ice shelves. However, the first ice shelf models were built on the neglect of vertical shear. Partly due to its historical treatment, it remains common to discuss vertical shear as though it were still considered negligible in ice shelf models. Here, we offer a historical perspective on the changing treatment of vertical shear over time, and we emphasize the term\u27s non-negligibility in current ice shelf modeling. We illustrate our discussion in the simplest context of an analytic, isothermal, shallow-ice-shelf model

Bartholomaus, Timothy C.

Enderlin, Ellyn M.

Miele, Chris

English

Boise State University - ScholarWorks

Boise State University ScholarWorks Geosciences Faculty Publications and Presentations Department of Geosciences 7-12-2023 Brief Communication: Is Vertical Shear in an Ice Shelf (Still) Negligible? Chris Miele University of Idaho Timothy C. Bartholomaus University of Idaho Ellyn M. Enderlin Boise State University The Cryosphere, 17, 2701–2704, 2023https://doi.org/10.5194/tc-17-2701-2023© Author(s) 2023. This work is distributed underthe Creative Commons Attribution 4.0 License.Brief communication: Is vertical shear in an ice shelf(still) negligible?Chris Miele1, Timothy C. Bartholomaus1, and Ellyn M. Enderlin21Department of Earth and Spatial Sciences, University of Idaho, Moscow, ID, USA2Department of Geosciences, Boise State University, Boise, ID, USACorrespondence: Chris Miele (mielebzu@uw.edu)Received: 22 November 2022 – Discussion started: 9 January 2023Revised: 26 May 2023 – Accepted: 12 June 2023 – Published: 12 July 2023Abstract. Vertical shear is recognized today as a key com-ponent of the stress balance of ice shelves. However, the firstice shelf models were built on the neglect of vertical shear.Partly due to its historical treatment, it remains common todiscuss vertical shear as though it were still considered negli-gible in ice shelf models. Here, we offer a historical perspec-tive on the changing treatment of vertical shear over time,and we emphasize the term’s non-negligibility in current iceshelf modeling. We illustrate our discussion in the simplestcontext of an analytic, isothermal, shallow-ice-shelf model.1 IntroductionAnalytic models of floating ice shelves date back to at least1957, when Johannes Weertman derived expressions for thetension and velocity gradients within a uniform-thickness iceshelf (Weertman, 1957). Weertman found that, for a shelfwith uniform surface elevation h= hT ; uniform density ρ;and no lateral flow, the depth-averaged longitudinal devia-toric tension, τxx , could be calculated viaτxx =14ρghT . (1)Using a depth-averaged constitutive relation, expressionsof this form permit the calculation of strain rates and ve-locities. Nearly 2 decades later, Thomas (1973) set out togeneralize Weertman’s expression for shelves of nonuniformthickness. Using the same underlying assumptions as Weert-man but imposing no restrictions on the surface elevationh= h(x), Thomas obtained an expression nearly identical toWeertman’s, wherein the depth-averaged deviatoric tensionisτxx =14ρgh. (2)By Thomas’ analysis, Weertman’s solution is valid regardlessof how h varies along a shelf. Thomas’ expression remainsthe generally accepted description of a nonuniform-thicknessshelf in longitudinal extension, and it is routinely cited or in-dependently derived in the literature (Sanderson, 1979; Cuf-fey and Paterson, 2010; Gudmundsson, 2013; Oerlemans,2021; Millstein et al., 2022). However, though Eq. (2) haspersisted, the formulation of this model has quietly under-gone a conceptual shift over the decades. This conceptualshift relates to the role of vertical shear in ice shelves – atopic which is sometimes incompletely communicated today,and on which we seek to provide clarification.2 The conceptual evolution of Thomas’ modelIn originally deriving the nonuniform-thickness model ofEq. (2), Thomas’ “sole restriction [was] that of zero shearstresses in vertical planes.” The neglect of vertical shearstress, τxz, was universal in the formulation of ice shelfmodels at the time (Weertman, 1957; Thomas, 1973; Robin,1975; Sanderson, 1979). However, it was understood bysome authors to be theoretically suspect. In a seminal papertitled “Is Vertical Shear in an Ice Shelf Negligible?”, Sander-son and Doake (1979) argued that vertical shear is funda-mentally linked with the thickness gradient of an ice shelfand that, strictly speaking, vertical shear “cannot be preciselyzero” except in the case of uniform thickness. This observa-tion did not challenge the practical utility of Eq. (2) (Sander-son and Doake, 1979, found vertical shear to be small enoughPublished by Copernicus Publications on behalf of the European Geosciences Union.2702 C. Miele et al.: Is vertical shear in an ice shelf (still) negligible?that its neglect was, in fact, justified, answering their own tit-ular question in the affirmative), but it highlighted a relation-ship that had been missed in Thomas’ analysis.The formulation of Thomas’ model evolved with the de-velopment of the shallow-shelf approximation (SSA) (Mor-land, 1987; MacAyeal, 1989; Schoof and Hindmarsh, 2010).The SSA, besides empowering a leap forward in computa-tional glaciology, was accompanied by two key theoreticaladvances in ice shelf modeling: how the neglect of terms isjustified and which terms are neglected.The SSA is built on the fundamental assumption that thethickness-to-length aspect ratio, ε, of an ice shelf is small(this is the “shallowness” of the SSA). With ε 1, largerpowers of the aspect ratio attain smaller values. In contempo-rary terminology, an “nth-order approximation” is obtainedby neglecting any term appearing as a coefficient of εn+1 af-ter nondimensionalization (i.e., setting those terms to zero inthe approximate model, with the understanding that they arenot exactly zero in real life). This dimensional-analysis ap-proach to excluding terms adds quantitative rigour to the ap-proach of Sanderson and Doake (1979), for whom negligibil-ity was more qualitatively assessed. As we sketch out in thenext section and contrary to the postulates of the first ice shelfmodelers, dimensional analysis does not lead to the whole-sale neglect of vertical shear from shallow-shelf models. Us-ing the SSA as a starting point to derive an analytic modelfor a longitudinally extending ice shelf, Thomas’ Eq. (2)results, albeit without the assumption of vanishing verticalshear. This is the modern approach to deriving Eq. (2).However, the inclusion of vertical shear stress in thepresent-day interpretation of Thomas’ model may somewhatclash with intuition, not least because the vertical shear termdoes not actually appear anywhere in Eq. (2). Even in liter-ature postdating the development of the SSA, it is commonto encounter language which, to a novice glaciologist, mightseem to imply that vertical shear is still discarded entirelyfrom shallow-ice-shelf models.1 This potential misstep is theprimary motivation for the present paper. In the sections be-low, we briefly illustrate (a) that dimensional analysis does1For example, in constructing the ice shelf model of Pattynand Decleir (1995), “the [vertical] shear stress term in [the x-momentum equation] is omitted.” Bueler and Brown (2009) statethat “D(v)13,D(v)31,D(v)23,D(v)32 [i.e., the vertical shear strainrates] are all negligible in the SSA.” Cuffey and Paterson (2010)specify that, to construct a nonuniform-thickness ice shelf model,the “assumption must be made that the slope at the bottom sur-face of the shelf is small so that the stress τxz will be negligible.”Larour et al. (2012) introduce the SSA as obtained by “assumingthat vertical shear is negligible” and then specify that ε̇xz = ε̇yz= 0.In an ice shelf model intercomparison, Pattyn et al. (2013) write,“A further approximation, known as the shallow-shelf approxima-tion (SSA), is obtained by neglecting vertical shear.” Bondzio et al.(2016) describe the SSA as an approximation which “neglects allvertical shearing but includes membrane stresses”, and Rückampet al. (2019) affirm that “the SSA neglects vertical shearing.”Figure 1. An ice shelf cross-section alongside a visual descriptionof several geometric parameters commonly used to describe iceshelf dynamics. H , h, and b represent the thickness, surface ele-vation, and basal elevation of the shelf. z= 0 is the waterline, andx = xT is the terminus. People on terminus for scale.not entail the neglect of vertical shear from the stress balanceof even a zero-order shallow-ice-shelf model and (b) that, de-spite the term’s absence in Eq. (2), the vertical shear stress ofan ice shelf can be directly calculated. For compactness, wepresent this discussion in the simplest context of an isother-mal, unconfined ice shelf in one horizontal dimension (as de-picted in Fig. 1).3 Pre-construction: which terms are neglected fromshallow-ice-shelf models by dimensional analysis?The typical balance of momentum for a 2D ice shelf cross-section, in x and z, can be expressed as∂∂xτxx +∂∂zτxz =∂∂xP, (3a)∂∂xτxz+∂∂zτzz =∂∂zP, (3b)where each τij term is a deviatoric stress and P is pressure.Both historically and today, simplification of these equationshas typically been carried out by neglecting the “bridgingterm” ( ∂∂xτxz of Eq. 3b). Possibly in an effort to maintaininternal consistency, the pioneering authors discussed abovetended to additionally neglect all other appearances of τxz,including the second term of Eq. (3a).In contrast to their approach and with asterisks denotingappropriately scaled parameters, Eq. (3a)–(3b) can be nondi-mensionalized as in Weis et al. (1999) (see also MacAyeal,1989, or Schoof and Hindmarsh, 2010) to become∂∂x∗τ ∗xx +∂∂z∗τ ∗xz =∂∂x∗P ∗, (4a)ε2∂∂x∗τ ∗xz+∂∂z∗τ ∗zz =∂∂z∗P ∗. (4b)The Cryosphere, 17, 2701–2704, 2023 https://doi.org/10.5194/tc-17-2701-2023C. Miele et al.: Is vertical shear in an ice shelf (still) negligible? 2703By this approach, in the zeroth- and first-order approxima-tions, the bridging term in Eq. (4b) will be neglected as a co-efficient of ε2 (i.e., omitted for the purpose of further simpli-fication), in agreement with Weertman, Thomas, and others.However, the vertical shear term in Eq. (4a) is retained, asit represents a coefficient of ε0. Thus, dimensional analysisprovides an internally consistent means of neglecting ∂∂xτxzwhile retaining ∂∂zτxz.4 Constructing the simplest shallow-shelf modelFollowing the workflow presented in Sect. 5.2 of Greve andBlatter (2009), the neglect of ∂∂xτxz in Eq. (4b) yields themodified x-momentum equation shown below (where wehave omitted asterisks for readability).2∂∂xτxx +∂∂zτxz = ρg∂∂xh (5)Although the most general shallow-shelf models are con-structed by depth-integrating equations of the above form(resulting in, for example, Eq. 6.55 of Greve and Blat-ter, 2009), we can provide an even simpler solution for anisothermal shelf, for which viscosity is depth invariant, and,consequently, τxx is depth invariant and equal to its depth-averaged value, τxx . It can be verified, by direct substitution(see also Schoof and Hindmarsh, 2010), that Eq. (5) is solvedby the systemτxx =14ρgh, (6a)τxz =12ρgz∂∂xh, (6b)where Eq. (6a) is simply the isothermal case of Thomas’Eq. (2), and Eq. (6b) is the (nonzero!) vertical shear stresswhich must accompany that solution. In fact, if we had setτxz to 0, Eq. (6a) would not actually solve Eq. (5), unlesswe additionally assumed that ∂∂xh= 0. This observation suc-cinctly illustrates the work of Sanderson and Doake (1979):τxz = 0 only to the extent that a shelf has uniform thickness.5 Concluding remarksIn discussions of shallow-ice-shelf models, it is fairly com-mon to hear vertical shear spoken of as “zero”, “neglected”,or otherwise unimportant. While this certainly was an ap-proximation made by early ice shelf modelers, this languageis at odds with current modeling practice. Indeed, as firstshown by Sanderson and Doake (1979) in “Is Vertical Shearin an Ice Shelf Negligible?”, nonzero vertical shear stressis a fundamental requirement of a nonuniform-thickness iceshelf. With dimensional analysis now enabling modelers tomore rigorously define negligibility, the present-day answerto their question is a resounding “no”.Data availability. No data sets were used in this article.Author contributions. CM devised the project and wrote themanuscript, with guidance from TCB and EME. TCB and EMEboth reviewed and edited the manuscript and acquired the financialsupport necessary for the project. TCB directly supervised CM.Competing interests. The contact author has declared that none ofthe authors has any competing interests.Disclaimer. Publisher’s note: Copernicus Publications remainsneutral with regard to jurisdictional claims in published maps andinstitutional affiliations.Acknowledgements. This work was made possible by grantnos. 1716865 and 1933105 from the US National Science Foun-dation and grant no. 80NSSC18K1477 from NASA.Financial support. This research has been supported by the Na-tional Science Foundation (grant nos. 1716865 and 1933105)and the National Aeronautics and Space Administration (grantno. 80NSSC18K1477).Review statement. This paper was edited by Reinhard Drews andreviewed by four anonymous referees.ReferencesBondzio, J. H., Seroussi, H., Morlighem, M., Kleiner, T., Rück-amp, M., Humbert, A., and Larour, E. 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Brief Communication: Is Vertical Shear in an Ice Shelf (Still) Negligible?

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Brief Communication: Is Vertical Shear in an Ice Shelf (Still) Negligible?

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