Committed Near-Future Retreat of Smith, Pope, and Kohler Glaciers Inferredtransient by Model Calibration D

Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | over this pe- 1 − a 3 erent projection with re- ff 3 10 km wide), interconnected West ∼ , and B. Smith 3 4460 4459 grounded area. We contrast this prediction with one 1 − a , I. Joughin 2 2 , P. Heimbach 1 7–8 times smaller than that of Pine Island or Thwaites (Shepherd et, al. 2002 )– ∼ This discussion paper is/has beenPlease under refer review to for the the corresponding journal final The paper Cryosphere in (TC). TC if available. University of Edinburgh, School ofUniversity GeoSciences, of Edinburgh, Texas, UK Institute for Computational Engineering and Sciences/Institute for Applied Physics Laboratory, University of Washington, Seattle, USA Abstract A glacial flow model ofof Smith, Pope inverse and methods Kohler against Glaciers hasand time-varying, been velocities, annualy covering calibrated the resolved by period means observations 2002ibration” of to – produces 2011. ice an The height optimal inversion set –with of termed a time-mean, “transient time-evolving spatially state cal- varying that parameters accountsmodel together for dynamics. the Serving transient as nature an ofused, optimal observations initial with and condition, the no the additional estimateding forcing, state for line for 2011 predicting retreat is grounded overa ice the near-steady volume loss ensuing loss of 30 and grounded ground- years. ice The volume transiently of calibrated approximately model 21 km predicts spect to ice volume loss andlevels ungrounding. of Sensitivity unforced, studies i.e. suggest large committedreasonable near-future assumptions sea regarding level uncertainties contribution of from the these unknown ice parameters. streams under 1 Introduction Smith, Pope, and Kohler Glaciers, three narrow ( observations of the ice shelves and sub-shelf environments (e.g. Jenkins et al., 2010 ; focus is often placed upon these larger ice streams, with regard to both modeling and Antarctic ice streams, have exhibited substantialAs thinning these and ice speedup streams in are recent– years. smaller the than contribution neighboring of Thwaitesis Smith and Glacier Pine to Island total Glaciers Amundsen Embayment grounding-line flux Committed near-future retreat of Smith, Pope, and Kohler Glaciers inferredtransient by model calibration D. Goldberg The Cryosphere Discuss., 9, 4459–4498,www.the-cryosphere-discuss.net/9/4459/2015/ 2015 doi:10.5194/tcd-9-4459-2015 © Author(s) 2015. CC Attribution 3.0 License. 1 2 Geophysics, Austin, Texas, USA 3 Received: 25 July 2015 – Accepted: 5Correspondence August to: 2015 D. – N. Published: Goldberg 25 ([email protected]) August 2015 Published by Copernicus Publications on behalf of the European Geosciences Union. riod, as well as lossobtained of following 33 a km commonly usednot “snapshot” consider or time steady-state dependence inversion, andTransient which assumes calibration does all is observations shown to togrounding be achieve line contemporaneous. a retreat better histories, fit and with yields observations a of quantitatively thinning di and 5 15 25 20 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | sea level 1 − 0.06 mma ness) is found through ff ∼ (or 1 − a 3 21 km ∼ 4461 4462 ect. For instance, co-located gridded velocity and thickness data ff erent sensitivities. ff culties in estimating the state of an ice sheet at any given time. Unlike other ffi As the observational record grows, so does the availability of data for the same A number of studies have employed snapshot calibrations to make near-future pro- The problem of projecting ice sheet behavior is challenging, in part due to incom- Transient calibration of a model of an Antarctic ice stream with temporally-resolved stream, with the significanta benefit realistic of past producing trajectory.well Such initial developed an conditions in other approach, for areas which forecasting of“state we geophysics, from and e.g. parameter term in estimation” “transient oceanography (Wunsch where and calibration”,where it Heimbach , it is is2007), is known or known as reservoir as modeling “historyof matching” (Oliver such et al., a2008). calibration, Hereparameters we are applied present the the to results result Pope, ofoptimal Smith an fit and inversion to in a Kohler whichmodel 10 Glaciers. year a is Calibrated time time-evolving then series model model of produces integratedrapid an surface for grounding-line elevation an retreat and velocity continues additionalof observations. for 30 years. The grounded another In ice decade, the remains but transiently then near slows, calibrated constant while run, at loss requires interpolation for applicationfields, to potentially leading a to model transientallow whose the nonphysical model artifacts. discretization to An staggers adjust oft-usedmodel to these approach may these is inconsistencies then to before have conductingpotentially drifted experiments. di The to a state far from contemporaneous observations,geographic with areas at multipletemporal points resolution in for time. the It purpose is of sensible, constraining then, the to time-evolving make state use of of an this ice jection on a similar timescale. Inconsistencycan between have the a data similar and model e discretization components of the climate (Taylorto et al., the2012), present ice state,the sheet models as models cannot must the be be required “spunproperties initialized up” historic such from forcing observations, as fields whichwhich ice are are we thickness not will mostly and refer limited available. as velocity.which to Rather, A “snapshot” solves surface an calibration, widely-used inverse first methodology or introducedparameters optimal by is relating control(1992),MacAyeal one problem. to and In sliding to this stress technique, (and an possibly optimal ice-shelf set sti of Seroussi et al., 2014). These studiesof have these deepened ice our streams. understanding However, of the theis use behavior of potentially snapshot problematic: calibrations in anynonphysical ice temporal transients sheet projections which inconsistencies persist among for datasets decades, can which lead is not to ideal if the goal is projections of the behaviorforcing of scenarios Pine (Payne Island et and, al. 2004; ThwaitesJoughin Glaciers et in al., response2010, to2014; varying Favier et, al. 2014; to di a least-squares fit ofity the field. ice Time-dependence model’s is nonlinearstress not balance) considered, momentum is since non-inertial. balance the We to choosevelocity momentum the a and balance term geometry given (or “snapshot” at veloc- because rather a it applies single to instant, ice assumed to be the same for both datasets. plete understanding of physical processes (Vaughan and Arthern, 2007), but also due Tinto and, Bell 2011 ; Favierrates et have been, al. observed2014; nearJoughin theand et Smith Thwaites terminus, al. , evenShepherd ( ).2014 larger et However, thanretreat high that al., of thinning of the2002; Pine Smith Island McMillan grounding etthat line has al., the been2014). observed iceRignot ( Additionally,on et stream substantial al., Thwaites may2014), (Joughin suggesting be etdynamical, al. subject understanding2014 ). to As of the such thewhether same there it causes is will instability of continue a thought at this need similar to to retreat; rates. develop be and a underway if quantitative possible, to determine et al.(2014) used methodsbalance similar over the to Northeast thosetimetry. Greenland used No Ice future in projections Stream this were over made study a in to 6 their year infer study. period surface from mass laser al- plan-view data has not previously been carried out, though we point out that Larour contribution). We show that the predictedto high levels any of future ice changes loss in are relatively forcing, insensitive and to any systematic errors in our calibration. 5 5 15 20 10 15 25 20 10 25 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | as (3) β via gradient . Minimization i snap J the uncertainty of the ob- 2 ) with respect to the control i u ( are found by solving the adjoint η snap i J µ if the dependence of ice viscosity on is self-adjoint, i.e. the adjoint operator L 0 , is minimized with respect to unknown L is sometimes extended with an additional ∗ u : i . The adjoint method is popular for snapshot 4464 4463 snap µ enforces the stress balance equations exactly. L J i . µ λ (grid cell or node), and yield the gradient of i i µ (often referred to as a control variables), subject to the con- λ , (2) i with respect to L , (1) 0 i 2 J | µ ∗ i 2 of the linear sliding law , and observed velocity, 0 is the discretized form of the stress balance at node ) 1 are at location u i u N = β ∗ i i = X u 0. The misfit (or cost) function is expressed as − ( u ) 2 i η = β u | , ) − , which can in turn be used to carry out the minimization of cient u λ λ 1 and ( , N = i ffi i i X u u L u ( 2 snap = J L β , the linearization of the operator 0 = = L In (1992),MacAyeal the model considered is the Shallow Shelf Approximation (SSA) The Lagrange multipliers We proceed with detailing what we mean by “snapshot” vs. “transient” calibration 0 b snap above. Development of sophisticatedshelf glacial flow physics codes have andLarour led the et to consideration al., of the2004; ice- Joughinstress use et balances of al., (e.g. 2010; alternativeLeeMorlighemet or et al., et al., 2012). augmented al., 2015) control and2010; the spacesGoldberg use (e.g. and of higher-order , variables Sergienko 2011; Petra descent or quasi-Newton optimization methods. The of calibrations in glaciology duecan to be the solved fact by that thestrain same rates code is used ignored. to solve 2.2 Transient calibration When observations distributedmodel, in the “snapshot” time calibrationbration, are which can available consists be of together extended optimizing to with agreement what of a the we time-evolving model term with “transient” observational cali- data (Morland and, Shoemaker 1982; , MacAyeal 1989) and the control variable is “smoothing” term that penalizesMorlighem small-scale et variations al., in2010). the Theto control ice be parameters geometry known (e.g. exactly. (i.e. surface and bed elevation) is assumed J of the Lagrangian The coe τ is often used as the control variable servation. The constrained optimization problemone may by be introducing turned Lagrange multipliers into an unconstrained where J where (1992)MacAyeal applied such an optimalmodel control velocity, method, in which the misfit between straint that the velocityform satisfies the nonlinear stress balance, written in the generic of an ice flow(Sect.2). model, We and then describe show the howthe observational details ice data of sheet3), (Sect. the observations as calibrationprojection well are used as are used in the presented this in model study this in and these (Sect.4). results process Sect.5, Results to followed of plausible the by uncertainties calibration in an and the investigation parameter of estimates (Sect.6). the sensitivity2 of Model calibration 2.1 Snapshot calibration A widely used approachvariables, for using single-time a stress observations balance is model, to via the invert adjoint for or uncertain Lagrange multiplier control method. (or uncertain) variables 5 5 10 15 20 25 20 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | (5) repre- x ), and erentiation (AD; ) ff k ( cient information is x ( ffi , (4) F 2 = ∗ 2 ) 1) k ( i ) + s k k ( i ( s − x ) k η ( i s is the time index, and the asterisk ) s : k ( ki χ 1 snap N = are weights to impose relative importance J i X are equal to 1 if there is an observation at s 1 ) s T = ω ( ki k X 1) 4466 4465 χ s − k ( ω and now extends to one with time-evolving Lagrange x + u and 0 ) i 2 ω J u F

( ki ∗ ) 2 χ − k ( i ) ) k u k ( i ( i . x ects of vertical shearing are represented (Goldberg, 2011). − u ﬀ ) k ) can be carried out in a similar manner, by use of its gradi- ( i η trans k ( i J u µ

) , 0 otherwise. 1 trans u N = k ( ki J i X χ 1 1 T = N = k X i X 2 1 T = − k X u is ice surface elevation, the superscript ω s trans = J and time step i = In this framework, the control parameters may now be chosen to be time-dependent. The other data set is a series of annual surface digital elevation maps (DEMs) from In addition to these time-dependent data sets, we use the BEDMAP2 bed topography Minimization of 0 trans available to constrain the larger controltime-independent space. parameters In the are following, used. unless stated otherwise, 3 Observations The time-dependent observations of velocity andtwo surface recently elevation in generated Eq. data (4) sets. come OneSmith from contains Glacier InSAR-derived region, surface binned velocities annually of toet a the al., 500 m; 2009 gridMedley for et theCoverage al., years is 2006–20102014). not (Joughin Velocities spatially are uniform, available for but floating greater and in grounded later ice. years. 2001 to 2011 onavailable a seaward 1 of km the grid.stream 1996 Coverage ridges. grounding is Figure1 lineshows consistentation the (Rignot between geographic and et years, region thinning but al., of recorded data20012014), study by measurements, is along or the but not with on is transient the simply slowFurther data acceler- an inter- details sets. extrapolation of The backward this in 2001 data time set surface from are is later given not years. in(Fretwell from AppendixA. et al., 2013) andWe also the MEaSUREs use (450 the mturesArthern grid) in et data the region, al.(2006) set as (Rignot accumulation explained et dataset in al., to Appendix ). 2011 B1. estimate ice tempera- 4 Model and calibration setup The land icebach(2013). The model model’s used stressShelf balance in equations, is but this depth-integrated, the similarly e study to the is Shallow that described in Goldberg and Heim- indicates observational values. cell multipliers, i.e. J However, doing so is meaningful only if physically justified and if su J at multiple time levels, withtion both enforced the as nonlinear model stressfunction, equations. balance which This and should is ice be equivalent thickness compared evolu- to against the following constrained cost where the model equations are written in generic form where of observations. The Lagrangian cated, now requiring acontinuous-form adjoint time-dependent of adjoint the model model,models equations, (Tziperman as which and has, Thacker can been),1989 done or be by for means derived simplified of ocean via Algorithmic Di the ent with respect to the control vector. However, gradient calculation is more compli- sents the modelmodel state, and i.e. to evaluate the minimal set of variables needed to step forward the Griewank and, Walther 2008).et al., Used2005; extensively inWunsching and is ocean becoming Heimbach, modeling increasingly2013), Heimbach, (e.g. common2013; theHeimbach Larour (Heimbach use et and al., of Bugnion, 2014). AD; 2009 toolsGoldberg in and land ice model- 5 5 25 10 15 20 10 15 20 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | er- are ff y facing q y and x and q x . Where the domain j ects of firn on ice dynamics. ff as control parameters at y q 4468 4467 , as explained in more detail in Appendix .B2 In at the end of each model year, as are velocity τ and x q trans J ) for each rectangular cell boundary j τ and j σ can be defined along any horizontal boundary, floating or grounded. τ and σ and shear membrane stress σ is a control parameter. Our other control parameters, less common in glaciological Our results in Sect.5 are generated assuming time-invariant control parameters. In The observations used in our transient calibration are those described in Sect.3. 2 constraints in the yearsice and geometry locations from available. Forvelocities the are the 2002 used snapshot as DEM; constraints. calibrationmain As as we discussed excludes we use below, ice in shelves. do transient Weenable calibrations not carry the comparison, have out do- and snapshot 2002 the calibrationssient velocities, resulting in calibration. MEaSUREs parameters the Similar become same domain initial toβ to guesses other in ice our modelinversions, tran- arise calibrations, from the the basal natureexplained of sliding below. the parameter transient calibration and the data sets used,Sect. as 6.2 we allow for time-dependentresults. parameters, and consider the implications of the 4.1 Boundary stresses as control parameters Our transient surface observations onlyTime-resolved give annual values velocity inland observations offrom are the 2007 provided 1996 to for grounding 2010. the line. Includingof ice ice transient shelves ice-shelf shelves, in thickness but from our 2001–2011. only domain,constrained, Such then, as an would estimate Dotson require would and be estimation Crosson very shelves poorly were likely influenced by processes other The initial ice thicknessare in each applied model to run the is cost from function the 2001 DEM. Subsequent DEMs than thinning duringshear the margin 2001–2011 joining window, the such2014). shelves, We and as overcome ungrounding crevassing, this from weakeninglem problem pinning of by (see, points formulating the (Rignot e.g. an etgroundingGebbie open, al. line et boundary as estimation al.(2006) the prob- the for downstream grounding an line, boundary which oceanographic of wouldnow analogue), the otherwise be be with imposed domain part along (see the this ofmarsh, boundary. Fig.1). 1996 the The stress Stresses2006) action balance of at along solution, the membrane a must stress stress horizontal tensor (Hind- boundary has two components: normal membrane boundaries, respectively. These boundary fluxes enterequation, the model which through is the solved continuity viaa a finite-volume cell scheme, boundary. and Boundary areslow-moving fluxes treated ridges, as are or constant where not over boundary imposed stresses along are imposed. the Note internal that boundaries with the model, only used in transient calibration; forlead snapshot to calibration, high MEaSUREs velocities thinning do rates not boundary. in these regions, despite no-flow conditions at the upstream These boundary stresses aretially not known varying a (along priori,with and boundaries) two we control unknowns treat ( parameters them as to unknown be spa- estimated via calibration, 4.2 Boundary volume flux as aIn control parameter our transient calibrations,due the to ice the fluxupstream incomplete regions into poorly coverage the constrained, of leading domain tothis the anomalously must we high time-dependent be thinning. consider To estimated. velocities, address boundary This which fluxes is leaves the entiated using AD software.Fig.1 Weon solve the the land 500interpolated m ice to equations grid this in grid. of theHowever, This the the domain allows domain time-dependent for does shown resolution not velocity in include ofwe set, ice the explain shelf and below. relatively seaward We narrow all of do ice the not other 1996 streams. account grounding fields for line, the are as e borders a slow-moving ridge velocitiesapplied. are set to zero, and boundary stresses are not Grounding-line migration is implemented throughdescribed a in hydrostaticGoldberg floatation and condition. As ), Heimbach (2013 the model has been successfully di 5 5 20 25 10 15 20 25 10 15 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | , are τ and σ , which favor , s β ω and u ω , the control parameters in the snap- y q erent between the two simulations. In the ff and x 4470 4469 q cient to generate a better ice-stream state estimate. ffi set by a decrease in backstress along the grounding line ff 100 m higher than observed, a misfit that is larger than the ∼ 50 % or more of the observed velocity. The misfit is largest at the boundary ∼ erent to the one inferred from the snapshot calibration, with uniformly small misfits. This strengthening is o A noticeable feature of the transiently calibrated boundary stresses is that of “neg- For velocities in the snapshot-calibrated run, the misfit for 2010 – the last year in The grounding-line behavior too is very di Relative to the time integration with initial state and parameters obtained from the ff shot and transient calibrations haveexamine a how one-to-one the relationship. parameters Thusshot are it and adjusted is transient for interesting calibration, transient we to parts infer calibration. of an In area the both of glaciersters the very (Fig.4b). is weak snap- a The bed strengthening most in of(Fig.4c). the striking the fastest bed adjustment moving under of the basal trunks stress of Pope, parame- Smith and(Fig.4d). Kohler It Glaciers is possible thatthan one our combination snapshot of calibration boundary isvelocity stresses equifinal, and and bed elevation i.e. observations. parameters that to In there reproducerectly this is imposed estimate case the more our dynamic snapshot state calibrationthe of does the transient not system. observations cor- The is additional su information provided by pendix B2, boundary stressesstress, which are depends expressed on as bedan a assumed depth. fraction bed Negative that buttressing of is could unconfined too be shallow. membrane compensating for ative buttressing”, i.e. thewhat normal would be membrane felt stress withoutto in any observations, ice some is shelf. locations insensitive ThisHowever, to it is could could small-scale be larger also oscillations because than be the in due model, the to and boundary errors the in stress fit the field. bed topography data. As detailed in Ap- run the 2011 groundinging line line is has not retreated(2014) completely considerably. (digitized coincident The with and modeled the plottedSmith/Kohler 2011 observed grounding for ground- grounding region. comparison), line The particularlyevent, of cause theRignot in for ice et the this al. in discrepancy western5–10 years this is part (see region unclear; Sect. of does5.3). but the unground in in any our5.2 simulation, it is Adjustment simply of control delayed parameters by Aside from the boundary volume fluxes snapshot-calibrated run there is almost no retreat, while in the transiently calibrated surface elevation. time, and so only theerror final along years are the shown flow at transects this from level Fig.1c. of detail. Figure3 snapshotgives inversion, surface the transient calibrationspect gives to good surface agreement, elevation especially (Fig.2d).di with The re- 2011 surface elevation misfit field looks very impact of the thinning signal itself over the period of integration. The misfits grow with Figure3 (bottom row) shows the reductiontransects. in On transient surface Smith elevation and misfit along Kohler,substantial misfit the (Fig.2b). in 2010 The velocity relatively hasand low decreased, transient decrease though calibration it in can is velocity still be misfit explained between by snapshot our choices of 5 Results 5.1 Calibration results Our snapshot calibrationeas recovers except MEaSUREs the velocities margins of(Fig.4a). to the The high narrow control western parameters accuracy branch adjusted in of in Kohler all the entering snapshot ar- Dotson calibration, Shelf is up to with the slow-moving ridge, which mayby be the because the model no-flow is condition imposedthe not there accurate. grounding By line 2011, is modeled surface elevation within 20–30 km of then used in a transientthis (but non-calibrated) run run agrees from with 2002–2011.the The the top degree transient row to observations of which . Fig.3 is demonstrated in Fig.2a andwhich velocity c, observations and are available – is largest in Kohler and Smith glaciers, and 5 5 10 15 20 25 25 20 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | (6) usive time ff over the next three 1 where applicable) are − a y 3 q grounded ice volume loss , is BEDMAP2 bed elevation. 1 x − q R ey and, Paterson 2010). a ff 3 (Cu (and 1 τ 21 km − ∼ are ice and ocean densities, respectively, and 3 − σ 4471 4472 , β must be inferred from surface and bed data as fol- 1028kgm obs = h w ρ obs s i 0.06 mm sea level equivalent), while the snapshot calibrated and ρ i 3 ∼ ρ − − 25 % more. Thus there is a quantitative impact of the initial state, − w ∼ ρ , obs b R − 918kgm = obs i 150 years based on a nominal surface slope of 0.01, thickness of 1400 m, and max s ρ ∼ = = is surface elevation from the transient DEM set, and Both snapshot and transient calibrations predict continued contribution to sea level Spatial patterns of projected grounding-line position for the transiently calibrated inally, it is important to realize that these projections are unforced: the estimated obs obs obs held constant over thiswhich time unground. This period, is andcommitted the (Price no basis et submarine al., for melt2011). referring is to applied the projected to grounded any ice areas 6 loss as Uncertainties of estimated parameters 6.1 Uncertainty of sea level contributionThe projection projection of committeddecades grounded from volume 2011 loss onward ofmodel is 21 parameters. km The subject adjoint toable capabilities uncertainty bounds of due the on its modeleters, this allow implicit which uncertainty us dependence to can through on estimate calculation be reason- of integrated sensitivities against to parameter these field param- perturbations. For instance, run show significant retreat from 2011–2021 (Fig.6), followed by a slight slowdown rise. The transiently calibrated model projects model suggests from 2011 to 2041 ( and therefore of thethe type of region. calibration There used,retreat: is on in projected an the snapshot-calibrated sea even run, leveltransiently almost more contribution calibrated no pronounced from run ungrounding takes ungrounding impact place, isof while significant on the in (Fig.5b). transiently the projected Given calibrated the grounding simulation muchwe to line closer surface accept observations fit this in a simulationthe least-square as region. sense, a better estimate of the dynamic state of the glaciers in in retreat. In contrast,Grounding-line retreat thinning does rates not remain proceedKohler Glaciers, down high suggesting the throughout the deep retreat the troughspen predicted incised 30 by in yearRignot by the et integration. Smith near-term. al. (2014) and Welateral might stresses argue not from that hap- areas this of is shallowerstudies bed because suggest limit the grounding-line grounding troughs retreat. line are However, other can quite retreat be narrow, in and episodic Amundsen rather and2010; than Bellingshausen sustainedJamieson ice due et streams to al., retreat details2012). in of Thus the bed geometry we futurethroughout our (Joughin cannot (i.e. simulation. et discount beyond The al., imposed 2041), further masspected particularly fluxes to rapid at influence since the grounding the inland thinning line boundary results: rates are the not time remain ex- scale high (30 years) is less than the di parameters and boundary conditions scale for grounding line changesto to be propagate across thevelocity domain, scale in which the we upstream calculate regions of 100 ma s where 5.3 Projected ice loss and behavior The model state and parametersare estimated used via as either initial snapshot conditionsa or in transient two 30 year calibration 40 year prediction integrationsof out window cumulative to 2011–2041. loss 2041, of i.e. The extendingfloating volume results into shelves above or are floatation the (VAF) shown portion frompressure, of in 2001. a and grounded5a Fig. VAF thus column does is in that notobservational an would terms include be data, indicator supported thickness of by sea ocean lows: level contribution. To calculate VAF from the h b 5 5 25 20 10 15 15 20 25 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | (7) (the j σ anywhere β 10 years, is similarly small. = y t q and ). Then the perturbation to x y ), and it can be interpreted , q . (9) 2 x 2 ( 5 years to β ( ∗ P ∗ 2 ) = k δ t ( i ) s k ( i s − ) k ( σ i s erent experiments. The annual cost func- ff ) s is sign-definite, i.e. decreasing ( ki 4474 4473 ) are distinct for each cell face, and constitute χ β 1 we define (0) j N = i σ X k s ), the sensitivity of VAF loss to topography, is plotted ω R ( ∗ + (and δ 2 5). (8)

∗ (10) − 2 j ) is subject to a perturbation k t σ ( i ) ( 2 k u ( i β ect the ﬁt to observations – ice loss projections would change ) takes on the values u − (10) ﬀ j j y . We refer to this quantity as and ) σ d 2 k ( i σ x β + (5) j u d ) σ

t P ) ) u − ( 2 ki χ β ( 1 ∗ (10 N = δ i X is the model domain. (5) j D Z u σ D ω = by year. That is, for each year = = ) ) t If we assume an error of 100 % for each basal sliding parameter – an unlikely sce- The above estimation of uncertainty bounds is tentative. Our inverted parameters To facilitate the discussion we define an annual cost function, i.e. a breakdown of ( k VAF ( j trans In Fig.7c this value is plotted by year for di tions resulting from the snapshot and transient calibrations are plotted (although recall where in Fig.7b. Note that the influence of Isaac et al., 2014). Enablingture such research calculations goal. within our estimation framework is a6.2 fu- Time dependence of control parameters Our adjoint-based calibration framework allows forparameters the that estimation/adjustment vary of control not2007, only2013). in Justification space, for but doingparameters, also so e.g. in derives boundary time from the stresses (e.g. which physicalWunsch representing interpretation could and far-field change of Heimbach, stresses due these in totime the crevassing dependence or can ice ocean be shelves, melting. inferredvary from We the piecewise-linearly investigate observations. whether over In such predefined ourwith time framework, intervals parameters intervals of of 5 uniform years, length. and For over instance, the interval from normal stress at face σ as follows: assume the nario, as this would a riori uncertainties based onmethods observational that uncertainties infer may the be Hessian possible e.g. of through the cost function (Kalmikov and, Heimbach 2014; have no aa priori posteriori estimates uncertainties or or covariances.fidence uncertainties, Thus intervals we on and are ice unable loss our to based minimization provide on accurate observational does con- uncertainty. Estimation not of a provide poste- VAF loss that follows from this parameter perturbation is givenδ by increases ice loss,projected while 2041 lowering grounding line. the bed only increases ice loss upstreamby of at the most 57certainties. %. 100 % Other error parameters inprojection have the by lower boundary at influence, stress most parametersThe assuming 1 would full %. reasonable change range The un- the of influence icechange bed of the loss elevation projection input by errors fluxes at associatedwhile most our with 30 %. model the These is BEDMAP2 values nonlinearperturbations. are data – Of based set but on course, the would linear results thesethese sensitivities, are relatively fields low borne would sensitivities out we not byoverwhelmed anticipate experiments by vary that with its the independently finite projected uncertainty. – massa Thus loss level but our of value based conservative is committed uncertainty not sea on level analysis contribution suggests from the region. The parameters aFig.7 shows thesliding adjoint parameter sensitivity of transiently-calibrated VAF loss to the basal J additional parameters for themore system. calibration parameters Thus, are therameter involved. greater space, Considering the the the additional temporal increaseof information in resolution, the is size the calibration. meaningless of if the it pa- does not improveJ the fit 5 5 10 10 25 15 20 20 15 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 20 %). Given ∼ is allowed to vary linearly in time. In 2 β 4476 4475 is reduced, but the reduction is small ( ). Results from two additional calibrations are shown as trans J trans parameter is assumed time-invariant, but boundary stresses J 2 β entries in Eq.4). This requires understanding of measurement errors, η While transient calibration can potentially constrain time-varying behavior of poorly We emphasize that our results do not suggest negligible change in ice-shelf but- our calibration, we see thata allowing small for improvement time-varying of controlbuttressing fit, parameters (and and only bed strength) thus provides did we notbuttressing do change did from not decrease 2001–2011. over reject While this the it time,system is null it possible occurred is hypothesis that also long that possible before far-field thata some observations continued perturbation response began, to to and the thisdetails the of perturbation. 2001–2011 how More temporal retreat investigation observational isof is sampling poorly needed just is known regarding parameters. able the to constrain temporal structure known control parameters, careparameter must set be yields taken an thatformation improved the provided fit increase (relative with in to observations. dimension time-invariant of Otherwise, parameters) the the may be additional of in- limited use. For 7 Discussion We do not holdproducing our spatiotemporally snapshot resolved observations. calibrationSUREs For to velocities, this be which calibration the have weThis a best used choice was much possible MEa- made later in becauseresults time no the 2002 stamp demonstrate sense velocity than data that fo was the re- adata available. Nevertheless, ice sets snapshot our that geometry calibration might used. withical be inconsistent framework, non-contemporaneous with cannot data, each be or transient other expected calibration if to can used take reproduce at accountobservations, time-dependent face of value behavior, thereby time-varying in whereas data giving a in more orderbehavior. dynam- to confidence The better nonlinear in reproduce least-squares near-futuredata framework projections sets ensures of can that ice be mutuallyof properly sheet incompatible having weighted, to i.e. be interpolatedinceased simultaneously by care fullfilled the must exactly. model be Importantly,element dynamics, taken within (the to instead such provide a usefulpotential framework error systematic estimates biases, for and each representation observational errors. well. In the first,are the allowed to vary linearlyond, over the boundary 2001–2011 period stresses as are described above. constant In while the sec- that the snapshot calibration issient not misfit designed reflected nor intended by to explicitly reduce the tran- each case, the number ofdoubles. The degrees cost of function freedom which describe the time-variant control tressing or bed strength parametersbuttressing over provided the by Dotson 2001–2011 and period. Crossonobserved In ice fact, submarine shelves a over melt decrease this in period ratespinning is points (Pritchard likely, given (Rignot et et al., al. , treat,).2014 found2012) (The to and loss be of an lossmodel basal important of and stress mechanism therefore ice due by not toJoughin rumples implicitnamic grounding-line et in and nature re- al.(2014), inferred of is boundary the resolved subglacial stresses.) by hydrological Similarly,et system given our al., in the2013), the dy- Amundsen it region is2001–2011 (Schroeder conceivable period that as there well. were Itdue changes is to in possible the bed that shortness sliding the ofvarying strength small the vs. over reduction estimation time-mean the controls period, of does over the not whichmore cost influence rigour the the function is solution distinction is significantly. required between Additionally, in time- determiningtions whether regarding a the reduction level of of misfitparameters, temporal is and significant. data Ques- of resolution required appropriatetargets to criteria for constrain future to time-varying work. identify overfitting of such parameters, are the small reduction in misfitinformation with the added addition carries of little parameters, meaning. we conclude the additional 5 5 25 20 10 15 15 20 25 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | can be expected 1 − yr 3 ect the floatation condi- ff 21 km ∼ ects in order to achieve better agreement ff 4478 4477 ects of firn density in our model. Neither has our transient surface data ff Sensitivity calculations suggest that, under reasonable assumptions regarding pa- Extending the simulations beyond the 2001–2011 calibration period, both snapshot- As the catchment of Smith, Pope and Kohler Glaciers is relatively small, the potential We briefly consider potential reasons for the discrepancy between our modeled 2011 In addition to the control parameters discussed above (boundary stresses, upstream rameter uncertainties, a committed grounded ice loss of Still, future studies should account for firn e out any changes insea boundary level contribution. conditions That orbut of external with the significant forcing. transiently grounding Both calibrated line model show retreat and is a grounding nearly significant line-concentrated 20 thinning. % smaller, and transiently calibrated models are run in “predictive mode” from 2011 to 2041, with- with grounding line observations. 8 Conclusions Generalizing optimal control methods based onin steady-state adjoint glaciology models well-known to thoseform using model a calibration transient based on forwarda simultaneous and state adjoint nonlinear and model, parameter least-squares enables estimationsuch fit through us a of to transient per- a calibrationGlacier model for region to the based time-resolved on grounded2011. observations. velocity portion This and We transient of surface calibration perform the observations isregion Smith, compared covering based with the Pope on a years and instantaneous “snapshot”transient 2001– Kohler (and calibration calibration of assumed agrees the far contemporaneous) same tions, better observations. giving with increased The spatially confidence in and near-future temporally behavior resolved predicted observa- by the model. for sea level contribution iset not al., as2010, large2014). as that Nevertheless, of the Thwaites volume and loss Pine from Island (Joughin these glaciers is quite high tion (e.g. Griggs andcan Bamber, ),2011 explain the it is disagreementvations. reasonable between to Figure8 our ask modelledgives whether 2011grounding a these grounding lines, omissions detailed line as and comparison well obser- the between as BEDMAP2 the the data 2011 modeled via groundinggrounding and line line Eq. estimates, observed inferred (6). but it from There doesmodel. the not is explain Rather, 2011 slight the we DEM erroneously disagreement suggest grounded and regiondue between this in region the to our is latter buttressing anomalously two fromline, thick which the (and is not small therefore visible grounded) groundedgrounding in line the “island”Rignot agreement at et is al.)(2014 the not data. explicitly Smith Furthermore, accounted we Glacier for point grounding in out that our transient cost function. from the region, even inOur the sensitivity absence of analysis external doesof forcing not projected or ice climate-induced replace volume feedbacks. a loss,chain and comprehensive is a uncertainty more needed complete for quantification end-to-end transient uncertainty ice propagation model calibration. grounding line and that of countRignot for et the al.(2014). e Asbeen mentioned corrected in Sect.4, for we firn. do As not ac- the depth of the firn layer can a out these results maywhich depend is somewhat implicitly on imposed by ourand the prior we scaling stress of assumptions the cost of importance functionin their gradients of future (see choosing variability, transient Appendix conservative iceB3), and unbiased sheet prior calibrations. information fluxes, and sliding parameters),initial two (2001) others surface were elevation,considered initially and investigated: as adjustments adjustments potentially to to from important bed which for elevation. the These observational initial fieldsmeasurements, condition agreement, and is were as bed derived topography the is is(Durand an 2001 considered et backward-in-time, al. a DEM extrapolation2011; source ofMorlighem ofjustments et later uncertainty were al., not2011; for foundSun ice for et flow eithermillimeters, al., (the and2014). inversion the adjusted However, significant bed initial ad- on surface the onfit order the to of order observations. of meters), Thus and these their control inclusion did variables not were improve not the considered further. We point 5 5 15 10 25 20 25 20 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | . 1 − usion ff out of the bed is 2 ect flow of nearby − , and use its depth- ff B in Glen’s flow law, we fol- B 4480 4479 ness parameter ff erent year, and those measurements acquired within ff All heights are relative to the WGS84 ellipsoid. BEDMAP2 bed elevations are ad- Available data for the model include ICESat satellite altimetry data (Zwally et al. , (2013). Here we discuss in detail features specific to,B1 or developed for, this Temperature-dependent study. rheology For the temperature-dependent ice sti equation for temperature toupper steady surface state, temperature and with kinematiceterization velocity boundary of and conditionsWang geometry come and heldet from Hou(2009), al.(2006), fixed. the respectively. and The param- A from constantassumed. geothermal the From flux accumulation the of data steady-state 100 mWm set temperature of field weArthern calculate justed for this geoid. Appendix B: Model description A general overview of the ice flow model used is given in Goldberg and Heimbach sources is treated asfor a each collection point, of and pointson larger, with spatially which small, uniform the statistically biases data independentwell that errors were constrained are by collected. independent data, To we forwithin ensure use each that 1 only day km data all in for elevation-changethree points at estimates months that of are have least the a one reference repeatwith date measurement di an of initial 30 elevation Decemberlarger model, 2010. than We then three fit removed times the those the resulting standardcess data data deviation until points of set either whose all no model residuals further residuals,standard were repeating points deviation this were pro- (equal removed in toassumed an errors) the iteration, of or standard the until misfit deviation the reached of normalized unity. the residuals divided by their low the approach of Joughin et al. (2009) by stepping forward an advection-di average in all simulations, without adjustment. given their size, anddecades. our Furthermore, projection significant shows thinning no of indication the of it region slowing could in a the next few ice streams by changingintroduced surface in gradients. this The studyAntarctic methodology ice – of stream which – transient could has calibration benear-future applied not behavior. to To previously other do regions been this, ofobservations, applied better Antarctica for to availability both to better of grounded constrain a and spatially floating marine-basedeach and ice, observational temporally along element resolved with will credible be error essential. estimates for Appendix A: Generation of surface elevation fields The ice-sheet surface heighta used time-varying in surface the model modelsent to the is laser-altimetry surface as derived and a from photogrammetric referenceof surface, a data. corresponding elevation to We least-squares increments 30 repre- for fit December years 2010, of of and between an a 2002 irregular set and mesh. The 2012,while reference each surface defined each has for a elevation the mesh increment resolution nodes as has up a to a function around resolution 100 of m, to of time is the 2 km. found data The through pointsderivatives. an model’s The and iterative surface model measures minimization height fit of of is determinedroughness the its in of sum roughness part the of and by reference its the surface thethe numerical misfit and roughness weight the weights assigned of elevation-change to to increments; its the wedata give temporal selected errors expected of reference-surface around errors 0.06 m, due and to to random, give elevation-rate uncorrelated errors of around 0.03 myr Bridge program, (Krabill data2010; Blair derived and from Hofton , the2010), Worldview satellites, and for stereophotogrammetric 2011 and 2012. Each of these data 2012), and airborne scanning laser altimetry data supplied by NASA’s Operation Ice- It is likely, however, that spatialresulted correlation in in considerably data larger errors errors and in irregular some data places. distribution 5 5 20 15 10 25 10 15 20 25 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | (B3) , the τ ect of ff imposed are set by depend on n , and ectively, we · cf τ ff ` τ S = and directions. In this τ y σ . Rather, the bound- ff and and cf σ x = σ depend on the nonlocal solution direction (Fig.9b). We implement τ y and , the component normal to σ , (B1) σ within the ice sheet or ice shelf, the force x ` x u arises from hydrostatic pressure. We hence- as 4482 4481 v 2 n F · + + is surface elevation, and summation is over the y y ). S u v s θ and in a depth-integrated sense, is is ice basal elevation (Goldberg et, al. 2009). Inter- ( 4 )), then velocities and stresses within the ice would (e.g. Hindmarsh , 2006): , and τ s b θ ( ` z S y τ x v ), 2 v θ ) and ( + + θ σ ( x y cf (Fig.9a). Along a calving front, direction and some in the σ , the distance along the grounding line. If the stress balance u u σ ` τ 4 θ x γ , (B2) 2 b coincide with the grounding line. For a given solution to the stress s z i = will have a certain dependence along the grounding line, and in ν ` w ects will likely be limited to the vicinity of the grounding line.) In other τ ρ = ρ ff σ I ) , − ect of the ice shelf on grounded velocities (and thickness evolution) is h ρgH∂ cf 2 ff σ and σ , = H ) are the actual parameters. Notice that in this formulation is ocean density and is the restriction of the Cauchy stress tensor to the i n σ is vertical thickness, and , τ Tr( τ is the normal vector to b γ h F w as a set of parameters, with a separate value for each cell face. E γ τ H + H ρ n , σ τ + − ρg 2 h 0, σ − n γ σ (1 · ij = = In particular, let In some of our simulations, the boundary of the domain does not remain coincident Thus in our runs, the boundary of the computational domain is internal to the ice S and = = S index. Along an arbitrary horizontal line j cf cf S and where nally to the ice shelf and ice stream, however, balance, general will vary with were again solved, but only over the groundedbe part of the the same. domain, (Thisbalance with used is in mathematically this truenonhydrostatic study; e for while it the does depth-averaged not hydrostatic hold stress for the general Stokes balance, any bed depth at the cell faceFretwell according et to the al., topographic2013). data set (in this case BEDMAP2, with the grounding line, as there is grounding-line retreat. The grid cell faces along words, the e to Eq. (B2). imposed solely through body (and initially coincidesgrid, with this the boundary grounding is line). notin As a (i.e. continuous our normal line model to) but has aσ the a collection of rectangular cell faces, some directed ary condition is a forcingnot that as needs stresses to but be as estimated. an These excess parameters fraction are of expressed σ the unconstrained membrane stress. Thus the ice shelves on theby the grounded membrane ice. stress Within tensor the ice, horizontal stresses are described to be equal to the same ( implement a Neumann boundaryon condition; the albeit ice one thickness and that bed does depth, not as is depend the uniquely case for a calving cli B2 Boundary stresses Here, we describe in morefrom detail the how, domain in and our replaced experiments, with the a ice boundary shelves are condition omitted that represents the e τ where context, the stress balance solvedwritten by the ice model for depth-average velocity∂ can be where j acting on the line, per unit length forth refer to the two components of where H component parallel to local force balance: σ 5 5 10 25 15 20 20 10 15 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | ) y ( τ . The j and trans ) ∂q x ∂J ( σ were found for the in- ) are still representative 1 . Note than ) s − y ( ( a τ and a given flux parameter τ 2 i m σ erentiation software which TAF, is 4 and ff and ) σ y ( σ , boundary stresses along faces nor- ) x ( τ ), and ) x ( and σ 4483 4484 τ ) , x ) ( x ( σ σ γ a) were used for boundary stresses, inpt fluxes, and 1 erent control variables must be accounted for. For in- − ff to be several orders of magnitude larger than i trans ∂σ ectively impose the stresses on a portion of the shelf. However, ∂J ff The BEDMAP2 dataset is available as Supplement to the source cited. , and 10 Pa(m 1 momentum balance (and are therefore more relevant to flow predomi- − direction). a x 2 x direction (and likewise m x 4 10 × ey, K. and Paterson, W. S. B.: The Physics of Glaciers, 4th edn., Butterworth Heinemann, ff In Fig.4d we distinguish between using polarization of 4.3-cm wavelength microwaveD06107, emission,, doi:10.1029/2004JD005667 J. 2006. Geophys. 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100 (f) 80 60 x (km) 40

es, S., Elsevier, Oxford, UK, Norm of velocity change be- ﬃ (m) thinning cum. (c)

(b) 0 Crosson Crosson Shelf Ice Crosson Crosson Shelf Ice Dotson Shelf Ice Dotson Shelf Ice (b) 2011 2009 2010 2008 2007 2006 2005 2004 2003 2002 0 −10 −20 −30 −40 −50 −60 −70 −80 −90 Hövmoller plots of cumulative thinning

(e) 80 (d–f) 60 x (km) 4490 4489 40 20

0 Kohler Glacier Kohler

2007 2006 2005 2004 2003 2002 2011 2010 2009 2008

Smith Glacier Pope Glacier Pope cum. thinning (m) thinning cum. −40 −50 −60 −70 −80 −90 0 −10 −20 −30

60 (d) in descending order. Crosson Crosson Shelf Ice 40 Dotson Shelf Ice Cumulative surface thinning, 2001–2011 in the surface elevation dataset. (c) x (km) (a) 20 (c) Ice speed in the Pope/Smith/Kohler and Crosson/Dotson system. The white con-

0 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 tion and ice state estimates,Chap. in: 21,edited Ocean by: Circulation Siedler, and G.,553–579, Climate: Church, 2013. A J., 21st4465, Gould, Century4474 J., Perspective, and Gri Greenland Ice SheetData Altimetry Center, Data Boulder, Colorado,, (HDF5), USA, 2012. 10.5067/ICESAT/GLAS/DATA205 doi: NASA4479 DAAC at the National Snow and Ice Zwally, H. R., Schutz, H. R., Hancock, D., and Dimarzio, J.: GLAS/ICESat L2 Antarctic and along transects in The shaded region shows where data is available. Figure 1. (a) tour is the grounding line asof given the by transient BEDMAP2, and surface the elevationour magenta dataset. state contour The represents estimate the rectangular simulations limits boxboundary – shows ﬂuxes boundary the are stresses subdomain imposed are used along imposed the for light along blue the boundaries. black contour and tween 2006 and 20102006 within or the 2010. model domain, excluding the areas of no coverage in either Wunsch, C. and Heimbach, P.: Dynamically and kinematically consistent global ocean circula- 5

error (m) error error (m) error 80 60 40 20 0 80 60 40 20 0

100 100 50 50 dist (km) dist (km)

0 0

2010 2008 2006 2004 2002 2010 2008 2006 2004 2002

error (m) error error (m) error 80 60 40 20 0 80 60 40 20 0 (b) (d)

80 80 60 60 4492 4491 40 40 dist (km) dist (km) 20 20 , the green hatches give the 2011 grounding line posi-

(d) 0 0

2006 2004 2002 2010 2008 2010 2008 2006 2004 2002

error (m) error error (m) error 80 60 40 20 0 80 60 40 20 0 erent ﬂowlines. From left to right, panels correspond to ﬂowlines in

ﬀ 60 60 40 40 dist (km) dist (km) (a) (c) 20 20 erence between (top panels) modeled and observed velocities in 2010 (the last ﬀ

0 0 Di Comparison of transient misﬁt of modeled surface elevation between snapshot and 2010 2008 2006 2004 2002 2010 2008 2006 2004 2002 Fig.1c in descending order. Top rowcalibration. panels: snapshot calibration. Bottom row panels: transient transient calibration along di Figure 3. Figure 2. year available) and (bottom panels) modeledels: and snapshot observed calibration. surface Right elevation inmodeled panels: 2011. grounding transient Left lines calibration. pan- in The 2011. magenta In contours represent tion reported by Rignot et al.(2014) (digitized from the publication).

( σ γ 2 0 4 − β 0 (b) 2 (b) 030 2 0 2 0 2 total ungrounded area in domain 0 1 0 (b) The adjustment of 2 The pattern of the buttressing inferred (d) position. (b) Transient Calibration Data Snapshot Calibration y (c) (d) .

000 0 2

800 600 400 200

(a) 1200 1000 1400 −200 Ungrounding area Ungrounding

4493 4494 0 4 0 2 ers from that of Fig.2a and b). (a) ﬀ 030 2 0 2 0 2 of the “snapshot” calibration to MEaSURES velocities (Rignot et, al. 0 | 1 ∗ 0 2 u Snapshot Calibration Transient Calibration Data − which achieves the misﬁt in 2 u

| 000 0 β 2 800 600 400 200

1200 1000 (a) loss VAF (c) Sea level contribution from the region and Error corresponding to points on the boundary at the same calibration relative to that of thein snapshot calibration. the calibrations. Specifically, the profiles to the left of the figure show sliding parameter 2011) as in Eq. (1) (note colorscale di through 2041 based on snapshotDEM and data, transient BEDMAP2, calibrations and (solid Eq. curves) (6) and inferred (red from hatches). the Figure 5. (a) Figure 4. (a) Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | R (b) bed topography 2041 2021 (b) and 2 β 12

0 2 0 1 0 2 (c) 008 2 006 4496 4495 2 the sliding parameter Annual cost functions (cf. Eq.9) for various calibrated 4 00 2 (c) snapshot transient linear boundary stress linear friction parameter (a) 2 (a)

00 0 2031 2011 2

8000 6000 4000 2000

14000 12000 10000 J (k) Sensitivity of grounded volume (Volume above ﬂoatation, or VAF) loss from the do- Cumulative thinning since 2001 (shading) and grounding-line position (red contours) (see Sect. 6.1 above formodel explanation). runs. main over the 40 year integration to Figure 7. Figure 6. in 40 year runsuccessive from plots with transient green and calibration. brown The contours, respectively. 2021 and 2031 grounding lines are shown in Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | (b) σ τ −1460 τ σ −1480 τ σ −1500 x (km) 4498 4497 , velocities in the glacier would be the same in both −1520 σ ﬀ =0 τ H (a) H σ H τ Glacier H Grounding line Grounding −1540 Ice shelf Ice x σ −1560 τ=0 H H z σ H

−580 −600 −620 −640 −660

y y (km) y Glacier Calving cliff Calving Open ocean =0? τ Visualisation of boundary stresses. Top: schematic of depth-integrated normal x H , respectively. The blue contour is the modeled 2011 grounding line, and green 3 Schematic of representation of boundary stresses through parameters. Shaded cells A detailed comparison of modeled grounding lines, the grounding line implied by the − y (b) cases. represent computational domain, andwere white it cells included represent in the areaat domain. where each Separate an cell degrees ice of face. freedom shelf describe would normal be, and shear stress Figure 9. (a) and shear stress alongsame. a In vertical front the or caseice grounding shelf, of line. and the Bottom: stresses along planform icestresses the visualization shelf, were grounding of imposed the line along the depend the stress on calving balance this cli solution. must If be these grounding-line solved within the glacier and Figure 8. data used in the modelingarea study, and represents directly the observed portion grounding-line of position.with the The the domain red 2011 which shaded surface is DEM ungrounded1028 in and kgm 2011, BEDMAP2, inferred and from assuming ﬂoatation icehatches and give the ocean 2011 densities grounding ofis line 918 the position and computational from boundary,Rignotthe and etRignot the al.(2014). et thick The al.(2014) black thin data contour black does contour the not 1996 extend grounding to Pope line. Glacier. Note that