IceSheetModelingInthisexercise,wewilldosomeexperimentswithasimpleicesheetmodelbasedonaclassicpaperbyJohannesWeertman,from1976.Ourgoalsaretounderstandsomebasicthingsabouthowtheseicesheetsgrowandshrink,andhowtheycanrespondtosunlightvariationscausedbyorbitalchangesoftheEarthrelativetotheSun.

LargecontinentalicesheetssuchasGreenland(atleft)areimportantcomponentsoftheglobalclimatesystemthatplayacriticalroleinalteringtheplanetaryalbedo,whichisconnectedtoapotentpositivefeedbackmechanism,andalsoincontrollingthelevelofglobalsealevel.

TheirgrowthanddeclinehasbeenoneofthedominantfeaturesofthePleistoceneiceages,andtheircurrentdeclineisofgreatimportancetotherisingglobalsealevel.Thetimingoftheiceagesandinterveningwarmerperiodsarelargelycontrolledbyorbitalchanges,andoneofthegoalsofthismodelingexerciseistoseehowthisworks.

Itiscommontoassumethaticebehavesasadeformableplasticmaterial,whichmeansthatthereisacriticalshearstress,τ0,belowwhichnostrain(deformationorflow)willoccur,andabovewhich,thestrainislimitless.Stressisjustaforceactingonanarea,andshearstressisaforceappliedparalleltoasurfaceasopposedtoaforceappliedperpendiculartoasurface,whichiscalledanormalstress.Wetalkaboutstressesratherthanforces,sincestressesarewhatcancausematerialstodeform(whetherbyfloworbyfracture).Theshearstressatthebaseofapileoficeisafunctionofthesurfaceslope,thethickness,gravity,anddensity:

€

τ b = ρgh sinα where α is the slope angle (1)Thismeansthatwherethethicknessoftheiceisgreater,theslopecanbesmallerandstillachievethecriticalshearstress.Wheretheiceisthinner,youneedahigherslopetogetthecriticalshearstress.Consideringthattheheightorthicknessoftheicemusttaperto0attheedge,youcanseethattheslopeoftheglacierhastobegreatestrightattheedge(whichisillustratedinaschematicwayinthedrawingabove).Iftheslopeistoolow,thebasalshearstresswillnotmatchthecriticalshearstressτ0,butassnowpilesup,creatingmoreice,thethicknesswillincreaseuntilτ0isreached,atwhichpoint,flowwillbegin.Asflowbegins,theslopewilldecrease;thiscausesthebasalshearstresstodropbelowτ0andflowwillstop,butthensnowpilesupagainandτ0ismet.Theresultofthisisthattheglacierevolvestothepointwherethebasalshearstresshoversrightaroundthecriticalshearstressτ0andasteadystateconditionoccurs.Theresultofthisisthataglacierhasanequilibriumprofile,whichisdescribedbythefollowingequation:

How do continental ice sheets flow?

The ice piles up, creating a surface slope (!), which generates a basal shear stress (") that causes the ice to flow. As the ice piles up, the crust subsides to achieve isostatic equilibrium.

!="ghsin#

#

h

Todevelopourmodelofanicesheet,wehavetostartwithafewbasicsofhowiceformsandflows.Glacialicebeginsassnowfallthataccumulatesovertheyears.Asitgetsburiedundermoresnow,thesnowcrystalsundergoakindofmetamorphism,eventuallyturningintosolidice.Ice,asanaturallyoccurringpolycrystallinesolid,isreallyakindofrock,butunlikemostotherrocks,icecanactuallyflowatthesurfacewithoutmelting.Thissolid‐stateflowisquitefastrelativetoothergeologicprocesses,enablingglacierstobeverydynamicfeaturesofthesurface.

€

h(x) =2τ 0ρg

L − x( ) = λ L − x( )( )12 where λ =

2τ 0ρg

(2)

Here,histheheightorthicknessoftheiceatvaluesofx,whichisdistancealongthesurface;x=0isthecenteroftheicemassandListhedistancefromthecenteroftheicetotheedge.Theicesheetisconsideredtobeperfectlysymmetricalsoitlooksthesameinthe+xand–xregions.Weertmansaysthattypicalvaluesforλare8‐15.Ifyouintegratethisequation(2)fromx=‐Ltox=L,yougetthecross‐sectionalarea,andyoucanalsoflipthisaroundtogetthelengthfromthecross‐sectionalarea:

€

AX =43λ12L

32 and conversely, L =

34AX

2

λ

13

(3)

Hereiswhattheshapeoftheglacierlookslike,attwodifferenttimes,withdifferentcross‐sectionalareas:

Alsoshowninthisdiagramisthesnowline,whichseparatescolderareaswheresnowwillaccumulatetoformicefromwarmerregionswherethemeltingexceedssnowfallandtheglacierwillexperiencealossofice.Thesnowlineslopesgentlyuptotherighttowardsthewarmersideofthediagram.Wherethissnowlineintersectsthesurfaceoftheglacier(redcirclesabove),wedividetheglacierintoitsaccumulationzoneanditsmeltingzone.Thegroundingpositionofthesnowline(blackcircleabove)markstheplacewhereitintersectsanelevationofzero.Themodelstartswithaninitialglacierlength,andfromthat,wecancalculatetheprofileoftheglacieranditscross‐sectionalarea.Oncewehavetheprofile,wecanfindtheintersectionwiththesnowline,whichallowsustoseparatetheglacierintotheregionsabovethesnowlinewhereaccumulationcanoccurandbelowthesnowlinewheremeltingwilloccur.Wegetthesnowlinebysettingtheequationforthesnowlineequaltotheequationfortheshapeoftheicesurface,whichleadstoaquadraticequation.Oncewehavethesnowline,wecancalculatethechangeinthecrosssectionalareaasfollows:

€

dAX

dt= Lacvac + Labvab (4)

Here,Lacisthelengthoverwhichaccumulationoccurs,andLabisthelengthoverwhichmeltingorablationoccurs.Theselengthsaremultipliedbytheircorrespondingratesvacandvab(theablationrateisnegative)summedtogivethechangeincross‐sectional(AX)overagivenintervaloftime.Thebalanceofaccumulationandablation—thesignofequation4—thendeterminesiftheglacierwillshrinkorgrow;ineithercase,weassumethatitmaintainstheequilibriumprofile.Inthemodel,theaccumulationrate(vac)andablationrate(vab)arerelatedbyaparametercalledepsilon:

€

ε =vacvab

(5)

Ifwarmingoccurs,thegroundinglinemovestotheleft(‐xisconsideredtobetowardtheNorth),whereascoolingmovesittotheSouth(rightinthediagram).Basedonobservationsofthepresent,Weertmancalculatedthatthegroundingpositionofthesnowlinechangesby17.7kmforeveryW/m2ofmeansummerinsolationchange.Inthisway,wecanmakeaconnectionbetweentheorbitally‐drivenchangesinsummerinsolationtothemodelasawayofforcingtheglaciertogrowandshrink.Hereiswhatthemodellookslike:

Qtisthetime‐varyingsummerinsolation(=incomingsolarradiation)for55°NandQ0isthepresentdaysummerinsolationforthesameregion;dQisjustthedifferencebetweenQtandQ0anddxdQtellshowmuchthegroundinglinemovesgiventhe

changeininsolation(dQ).QtandQ0areconnectedtothemodelviaaswitchsothatwecandisablethemorenablethem.Theswitchallowsustodoaexperimentswithoutthecomplicationsoforbitalforcing.SPECMAPistheoxygenisotoperecordfromtheoceansthatgivesusasenseofthetimingandmagnitudeoficevolumechangesovertime;thisisjustsomethingwecanplottoseetheextenttowhichourlittleicesheetmodelmimicstheactualrecordoficegrowthandmelting.BothSPECMAPandQtgobackto300kyr.Timebeginsat‐300,000yearsandendsat0.Themodelalsoincludesaconvertercalledseedarea,whichcomesintoplaywhenthereisnoglacierandthegroundinglinemovesintothepositiverealm,indicatingcooling;thisjustallowstheglaciertogetgoingagain.ExperimentsTheseexperimentscaneitherbedonebyconstructingyourownmodelusingSTELLA,orbydownloadingapre‐madeversion,orbyworkingwithaversionthatrunsonline.Tobeginwith,makesurethattheMilankovitchorbitalvariationsofinsolationareturnedoffsotheydonotimpactthemodel.Experiment1:SteadyState?ResponseTime?Inthisfirstexperiment,let’sseewhathappenstotheglacier’slengthovertimewithsomereasonableinitialconditions.TimeSpecs:

Runfrom‐300,000to‐200,000years,withaDTof200andRunge‐Kutta4.ModelParameters:

accumulation_rate=1.2{m/yr}epsilon=.24{ratioofrates}initial_grounding_line=‐400{km}initial_length_km=400{kmstartinglength}lambda=14{icestrengthparameter}slope=0.002

BesurethattheMilankswitchisturnedoffforthisexperiment.a)Beforerunningthemodel,trytopredictwhatwillhappentotheicesheet.Willitfindasteadystate,orwillitjustshrinktonothingorwillitgrowindefinitely?Thenrunthemodelandexplainwhathappens.b)Now,changetheinitiallengthto3000km—averylargeicesheetinthiscase.Willitfindasteadystateagain?Willithavethesamesteadystatelengthasinthefirstcase?Atthestart,doyouthinkthattheaccumulationratetimesaccumulationareawillbegreaterthanorlessthantheablationratetimestheablationarea?c)Howquicklydoestheicesheetgetintoitssteadystate?Howfastcantheglaciergrowandshrink?Insystemsanalysis,thisiscalledtheresponsetime,andisoftendefinedasthetimeittakesasystemtoaccomplishabout2/3ofitschangetotheeventualsteadystate(soitreallyonlyappliestosystemsthattendtowardasteadystate).Inthecaseofourglacier,youcanfindthedifferenceinlengthbetweenthe

steadystatelengthandthestartinglength—thenfindthepointintimewhenabout2/3ofthischangehasbeenaccomplished;thatisyourresponsetime.Usethemodelset‐upandresultsfromthefirstexperiments(a&b)toestimatetheresponsetime,givingtheresultinkyr.Experiment2:CrossingtheThresholdtoRapidMeltingIntheaboveexperiment,welookedat2initiallengthsandfoundthatinbothcases,theglacierevolvedintoasteadystatelength,wheretheaccumulationareaaddedwasequaltotheablationarearemoved.Now,let’sexploreawiderrangeofinitiallengths,whichwillrevealaninterestingchange.Startwiththesamemodelset‐upasin1a,wheretheinitiallengthwas400km.BesurethattheMilankswitchisturnedoffforthisexperiment.a)Runthemodel,andyoushouldseetheglaciergrowtoalengthofabout2100kmandthenleveloff,havingreachedasteadystate.b)Now,decreasetheinitiallengthto300km.Whatlengthdoestheglacierendupat?c)Now,decreasetheinitiallengthto200km.Whatlengthdoestheglacierendupat?Describe,briefly,whathappenstotheglacierinthiscase.Notethatin(a),theglacierdeceleratesasitapproachesthesteadystate—itgetsthereverygradually.Howdoesthisdecelerationof(a)comparewiththebehaviorinthiscase?d)Nowincreasetheinitiallengthto210km.Whatlengthdoestheglacierendupat?Describe,briefly,whathappenstotheglacierinthiscase.e)Itshouldbecleartoyouthatthereisathresholdintheinitiallengththatseparatestwoverydifferentbehaviorsanddifferentoutcomes.Fiddlearoundwiththeinitiallengthuntilyoufindthethreshold(within1kmisfine).Experiment3:ChangingtheGroundingLineNowlet’sseewhathappensifwechangethepositionofthegroundingline,whichisshowngraphicallybelow,shiftedtotheleft(towardsmorenegativevalues):

Thisiskindoflikeimposingawarmingontheglacier.Startwiththesamemodelset‐upasin1a:TimeSpecs:

Runfrom‐300,000to‐200,000years,withaDTof200andRunge‐Kutta4.ModelParameters:

accumulation_rate=1.2{m/yr}epsilon=.24{ratioofrates}initial_grounding_line=‐400{km}initial_length_km=400{kmstartinglength}lambda=14{icestrengthparameter}slope=0.002

BesurethattheMilankswitchisturnedoffforthisexperiment.a)Runthismodeltoactasacontrol,takingnoteoftheendinglengthandthegeneralbehavior.Thenshiftthegroundinglineto‐500km.Makeapredictionaboutwhatwillhappen,thenrunthemodelanddescribehowthischangehasaffectedtheglacier.b)Nowshiftthegroundinglineto‐300kmandmakeapredictionabouthowthiswillaffecttheglacier,thenrunthemodelanddescribehowthischangehasaffectedtheglacier.Experiment4:ChangingtheIceStrength(λ)Nowwewillinvestigatetheaffectofchangingtheicestrengthparameter(λ),whichhasasitsmainvariablethecriticalshearstressforflowoftheice.Ifwelowerλ,thenweareeffectivelyloweringthecriticalshearstress,makingiteasierfortheicetoflow.Thiswouldmeanthatwithalesserthicknessand/orashallowerslope,theicewillflow.Tobegin,wewillusethestandardset‐upfromexperiment1a,whereλissettoavalueof14.Runthis“control”modelfirst,andtakenoteoftheendinglengthandmaximumthicknessoftheglacier.BesurethattheMilankswitchisturnedoffforthisexperiment.a)Changeλto12,thusmakingtheiceflowmoreeasily.Makeapredictionaboutwhatthiswilldototheglacierincomparisonwithourcontrol.Willtheglaciergrowtoagreaterorlesserlengthrelativetothecontrol?Willtheheightbelesserorgreater?

b)Runthemodelanddescribewhathappensandhowtheresultscomparewithyourpredictions.c)Nowchangeto10,andmakeaprediction.Thenrunthemodelanddescribewhathappensandattempttoexplainwhyithappens.Experiment5:RatioofAccumulationandAblation(ε)Howwillchangingtheratioofaccumulationandablation(melting)ratesaffectthegrowthoftheicesheet?Themodelparametercalledepsilon(ε)controlsthisratio.Wewillagainusethemodelset‐upfrom1aasourcontrol;hereεissetat0.24.Firstrunthismodeltorecallwhathappenstothelength.BesurethattheMilankswitchisturnedoffforthisexperiment.a)Nowchangeepsilonto0.28.Rememberthattheablationrateisequaltotheaccumulationratedividedbyepsilon.Whatwillchangingepsilontoalargervaluedototheablationrate—makeitgreaterorlesserthanthecontrol?Predicthowthischangewillaffecttheequilibriumlengthoftheglacier,andexplainyourreasoning.b)Then,runthemodelanddescribewhathappens,andexplainwhytheglacierrespondsthisway.c)Nowchangeepsilonto0.20.Howwillthischangetheablationraterelativetothecontrol,andhowwillthisaffectthegrowthoftheglacier?d)Runthemodelandthendescribewhathappens,andexplainwhytheglacierrespondsthisway.Experiment6:ChangingtheSlopeoftheSnowlineWhatwillhappeniftheslopeofthesnowlineincreases?Itisalreadysettoaverylowvalueof0.002.First,let’svisualizewhatthiswoulddototheglacier:

Youcanseethatitwillshortentheaccumulationlengthandincreasetheablationlength.So,whatwillthisdotothegrowthoftheglacier?

Asbefore,webeginwiththemodelset‐upfor1a,andthenrunthismodeltoremindourselvesofthecontrolcase.Takenoteofthebeginningablationlengthinthecontrol.a)Changetheslopeslightlyto0.0022.Firstmakeapredictionabouthowthiswillaffectthegrowthoftheglacierrelativetothecontrol.b)Runthemodelandexplainwhathappensandwhyithappens.Howdoesthebeginningablationlengthofthismodelcomparewiththecontrol?Howdidtheresultscomparewithyourprediction?c)Nowincreasetheslopeevenmoreto0.0024.Runthemodelanddescribewhathappensandwhy.Experiment7:OrbitalForcingNow,wewillconnecttheorbitalforcingtothemodelbyturningontheMilankswitch.Fortheweb‐basedversion,wewillnowshifttoadifferentmodelthatrunsforthefull300kyr(we’vejustbeenrunning100kyrsofar).Firstrestorealltheparameterstothewaytheywereforexperiment1a.Now,withtheMilankswitchturnedon,thechangingsummerinsolationduetoorbitalvariationswillforcethegroundinglinepositiontomovebackandforth.Higherinsolationpushesthegroundlinglinepositiontothenorth(towardmorenegativevalues,whileadecreaseininsolationmovesthegroundinglinetothesouth(morepositivevalues).Asyoushouldknowbynow,movingthegroundinglinepositionwillcausetheglaciertoadvanceandretreat.RunthemodelandplotthelengthinkmandQt(theorbitallycontrolledvariationinsummerinsolation),andstudytherelationshipbetweenthepeaksandtroughsinQtandthesizeoftheicesheet.a)Studytherelationshipbetweentheglacier’slengthandQt(theinsolationovertime).Aretheyperfectlyinsync,ordoesoneseemtolagtheother?b)WhatisthelagtimeinkyroftheicesheetrelativetoQt?c)Howconsistentisthislagtime?d)Whatistherangeofvariationinthelengthoftheglacierinkm?Forcomparison,theLaurentideicesheetexpandedandcontractedabout25°oflatitudefromit’scenterofmass(x=0)andthereare111kmperdegreeoflatitude.e)NowcomparetheicesheetlengthwiththeSPECMAPrecordofdel18O,whichispartlyameasureoficevolumeandpartlyameasureoftemperature—highervaluesrepresentmoreiceandcoldertemperatures.Howwelldotheyagree?Howsimilarordissimilararethetimesofthepeaksandtroughs?

f)Lookatthemostrecent10kyrofthemodel.HowisQtchangingduringthistime,andhowdoesthemodelglacierrespond?WhatdoesthissuggestmightbehappeningatthepresenttimeifwewerenotincreasingthegreenhouseeffectthroughelevatedCO2levels—enteringanothersmallglaciationorholdingsteadyormovingtoawarmerinterglacial?