IceSheetModelingInthisexercise,wewilldosomeexperimentswithasimpleicesheetmodelbasedonaclassicpaperbyJohannesWeertman,from1976.Ourgoalsaretounderstandsomebasicthingsabouthowtheseicesheetsgrowandshrink,andhowtheycanrespondtosunlightvariationsrelatedtoorbitalchangesoftheEarthrelativetotheSun.

LargecontinentalicesheetssuchasGreenlandareimportantcomponentsoftheglobalclimatesystemthatplayacriticalroleinalteringtheplanetaryalbedo,whichisconnectedtoapotentpositivefeedbackmechanism,andalsoincontrollingthelevelofglobalsealevel.

Itiscommontoassumethaticebehavesasadeformableplasticmaterial,theremeansthatthereisacriticalshearstressτ0belowwhichnostrain(deformationorflow)willoccur,andabovewhich,thestrainislimitless.Stressisjustaforceacting

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#

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h

Todevelopourmodelofanicesheet,wehavetostartwithafewbasicsofhowiceformsandflows.Glacialicebeginsassnowfallthataccumulatesovertheyears.Asitgetsburiedundermoresnow,thesnowcrystalsundergoakindofmetamorphism,eventuallyturningintonearlysolidice.Ice,asanaturallyoccurringpolycrystallinesolid,isreallyakindofrock.Butunlikemostotherrocks,icecanactuallyflowatthesurfacewithoutmelting.Thissolid‐stateflowisquitefastrelativetoothergeologicprocesses,enablingglacierstobeverydynamicfeaturesofthesurface.

TheirgrowthanddeclinehasbeenoneofthedominantfeaturesofthePleistoceneiceages,andtheircurrentdeclineisofgreatimportancetotherisingglobalsealevel.Thetimingoftheiceagesandinterveningwarmerperiodsarelargelycontrolledbyorbitalchanges,andoneofthegoalsofthismodelingexerciseistoseehowthisworks.

onanarea,andshearstressisaforceappliedparalleltoasurfaceasopposedtoaforceappliedperpendiculartoasurface,whichiscalledanormalstress.Wetalkaboutstressesratherthanforces,sincestressesarewhatcancausematerialstodeform(whetherbyfloworbyfracture).Theshearstressatthebaseofapileoficeisafunctionofthesurfaceslopetimestheheighttimesgravitytimesdensity:

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τ b = ρgh sinα where α is the slope angle (1)Thismeansthatiftheheightoftheiceisgreater,theslopecanbesmallerandstillachievethecriticalshearstress.Wheretheiceisthinner,youneedahigherslopetogetthecriticalshearstress.Consideringthattheheightorthicknessoftheicemusttaperto0attheedge,youcanseethattheslopeoftheglacierhastobegreatestrightattheedge(whichisillustratedinaschematicwayinthedrawingabove).Iftheslopeistoolow,thebasalshearstresswillnotmatchthecriticalshearstressτ0,butassnowpilesup,creatingmoreice,theslopewillincreaseuntilτ0isreached,atwhichpoint,flowwillbegin.Asflowbegins,theslopewilldecrease;thiscausesthebasalshearstresstodropbelowτ0andflowwillstop,butthensnowpilesupagainandτ0ismet.Theresultofthisisthattheglacierevolvestothepointwherethebasalshearstresshoversrightaroundthecriticalshearstressτ0andasteadystateconditionoccurs.Theresultofthisisthataglacierhasanequilibriumprofile,whichisdescribedbythefollowingequation:

€

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;dQisjustthedifferencebetweenQtandQ0anddxdQtellshowmuchthegroundinglinemovesgiventhechangeininsolation(dQ).QtandQ0areconnectedtothemodelviaaswitchsothatwecandisablethemorenablethem.Theswitchallowsustodoaexperimentswithoutthecomplicationsoforbitalforcing.SPECMAPistheoxygenisotoperecordfromtheoceansthatgivesusasenseofthetimingandmagnitudeoficevolumechangesovertime;thisisjustsomethingwecanplottoseetheextenttowhichourlittleicesheetmodelmimicstheactualrecordoficegrowthandmelting.BothSPECMAPandQtgobackto300kyr.Timebeginsat‐300,000yearsandendsat0.Themodelalsoincludesaconvertercalledseedarea,whichcomesintoplaywhenthereisnoglacierandthegroundinglinemovesintothepositiverealm,indicatingcooling;thisjustallowstheglaciertogetgoingagain.