Numerical Modelling of Subglacial Erosion and Sediment Transport and its Application to th e North American Ice Sheet s over the Last Glacial
Cycle
by
@Al{'xan d reXlelan so n
A t.hes is submittedto theSchoolofGraduate St udiI's in partialfulfilhncnt oftlu:
requirem entsforthed{'gn '{' of
Masterof Science
Departmentof Phy sic sand Physic alOce an o gr aphy
Memori alUniversityofNewfound la nd July2012
St..John's Nowfouudlnu.l
Abstract
Present-day sediment dis tribut.ion offersapot ent.iall yst ro ng cons truiut011pas t in' sheetevolut ion. however, glacia lsyste m models(GS?lIs)cannotaddl'l'ssthiswhik- lackin gproper representa ti onsofsubglucialsod inu-nt productionandt.rnnsp ort. To- ward th isgoal.Ipresentacont.ine ut.alsubg lacialprocessmod elforsodinu-nt pro- duct.ion,cnt.rain mcnt,traus port.und dep ositi on.Themodel isdrivenh~'tlu-duta- calibra t ed?lIVN :3-0 GS;l.1and hy anewl ydevelop ed su bgla ciul hyd rol ogyCOIllJ)(}-- uent..Modelresultsarc compa redaguiustthepresent-daydist.ributiouofglacig pIlic sed iment overNorth Amer ica undgeologicall'sti Iltal<'SofLuurc ut.ido orosiou.Civon the mod el'sparam etri c scns iri vity, thecalculat ed erosiondepths fall withint.lu- gp- olog icalpst imat l's ofLauren ti de erosion,withthe excep t.ion ofHallpt'subrusion Inw tha tuudorestim a tos abrasion hy atleastallorderofmugn itu do.Ill.add it.io n.thr- mod el overest ima tessed ime ntcnt.ru iru ucntandt.huseng luciultranspo rt.whic hpit her sllggeststhat the la rge-scal e n-pn-scntat ionofl'l'geiat iolliut rusionis iundcquute or that thebasalhydrologymodul eunderestiruatr -swater pl'l'SSIll'l'S.
Acknowledgements
Isincerelythan kIllysu perv isors,Dr. L('\''IurasovawlDr. Trevor!3el!forgiv ing metheop port u n itytoworkon thisproject and foroffe ringthei rvnluab k-insight an dgu id ancethroughoutthe courseofIllyprogram.Specialtha nkstolilycol!eaglil's Robort !3riggsan d MarkKavanaghfo roffering.onsoverulocca sion.muchapp reciat ed practicalass istance,aswellastotherest of1.he:\IUl\' Gla cialSystemDv muuk-s grou p.
Trist anHauser.KeviuLcMo rzado« and Tuimuz!3aha d oryfortheircout.iuuedmoral sup por t. Fu n d ingfrom theNat uralScience'san dEngiw,pringH('smrchCouncilof Canada(l\'S EHC )andtheSchoolofGraduateStu diesisgra t efu l!ynekuowk-dgc d.
iii
Table of Contents
Abstract
Ack no w le dg ment.s
Tab leofContents
List of Tables
List of Figures
Listof Acrony ms
List of Sym bo ls
1 In t.r o d uct.io n an d Overview 1.1 Moti vations. 1.2 Objocti ves audapproach. 1.:1Structure
1.4 Co-aut horshipsta tement.
Numci-icalModollingofSu bglacialEr os io nan dSed ime nt Transport and itsAp plicat io nto theNort hAmerica n IceSheetsovert.heLa s t.
GlacialCycle
iv
iii vi
vii
xi
xii
xii i
2.1 Int rod uction. .).) l\lod eldescription.
2.2 . 1 Sensit ivityanalysis . ') ')') Xlodel umucrica. 2.2.3 Subglucialerosion.
2.2.:tl Procoss-ori ontod erosionlaws 2.2.:t2 Empiricalorosion Iaw 2.2.../ Eng lacin ltrans port.
2.2.4.1 Entra inment 2.2.,1.2 Verticalmixing. 2.2.5 Transpor t.bysu bglacial defonu ar.ion 2.2.G )CPdyn ami cs
2.2.7 131L~alhydrology. 2.:\ Results.
:2.3.1 Sedi menttransport.
:2.:3.1.1 Applicati ontoHudsonl3ay sed imcut. 2.:t1.2 Cont inentalsed iment dist.ribution.. 2.:\.1.:\ Engluciulvs,subglacia l transp ort. 2.3.2 Erosion
2.3.3 Sensiti vityanalysis . 2.-l Discussioll.
2.../.1 Ovcrcst.iu ia t.io nofsedimen t entrai n me nt 2.4.2 Com pa riso noferosio n la ws 2.4.:\ Sensi tiv ityana lysis. 2.5 Sum maryandconclusions
12 12 l:l I-I
2-1
:W
2!) :\1
.m
·11
·11
·1'
G2 G2
(ir;
3 Summa ry
;~.l Futurework .
Bib liogr aphy
vi
70
76
List of Tables
2.1 Sedimen tmodelparameters.Bracket ed symbolsreprese ntsth« ubh re- viat.ion susedin Figur e2.12.The rangecolum nshowslower. basp.and
uppervalues in order. ;\.1
»» Var iousresu ltsfromtheem pir icalandproct-ss-ori ontcderosion la ws . ;,7
vii
List of Figures
2.1 Initial sediment dist.ribut.i onat120 kyrDP,based011the mapof FullOil (199S) .A maximumofIi;//(/,t(base valueof ISm)ispl"<'scrill edover arms ofobse r ved till blank ets .Each colore dsq uarecorros po uds to a fu llgr id-cell..
')') Rati o of basal wate r pressur e andicc overb u rde n pressure atLC i\1.
Conto ursshowiccthicknessill metres(fro m ruull)md027.«alihrn t.io u setNSb t illTarnsovct ul.2(12)..
2.;) Time slicesshow ingcng lac ia landsu bglacia lsed ime ntth ickll('sSfrom 70to SOkyrDP.Pe rcen tu gc of tot al sed imentvolu me is show nlot cugla cial(E)andsubglacia l(S)sto rage.Are a ofinit.ial cover (uuiform 20 m)is out line d bydash edlinesillright.punt-Is.COIlt.OurSillk-It pan els show iccthickn essill meter s(fromrunID 1l1l1027 . cnlibrn r.ion set.Nfib t ill Tara sov otal.2(12).Label sill panel (a) showHudsonDay (H D), .Ia mes Day(.JI3),Manitoba (i\IA),Ontario (ON).North Dak ota (ND), Sou th Dak ota (SD) ,and Ohio(O H).Vnrinn co-ba scd color levels arc usedtoma xim izeinfonunt.ionill theI-ITrunge.
viii
I.J
;l:\
2.4 Time slices showingenglac ia lan dsu bg lac ia lsed inw n t. thickness1'1'01 11 40 to20 kyrnp.Per cen t a ge oftota lsed imentvolui ucis show n I'm engla cia l(E)andsu bg lac ia l(S) storago .Areaof initialcover(unifon u 20 m)isout lined bydash edlinesin rightpanel s. Con to ursin left panels show ice th ick ness in meters (from I'Im 10 nnlOn.ca lihrution setNbhtinTarnsovctul,2( 12).Lab elsin panel(a)show Hudsonnay (1113), .hu uesnay (.In). Muni toba(~I A).Ontario(ON) , Nort h Dakota (NO) , Sou t h Dak ot a(SO),and Ohio(O H).Vnri nuco-hasod color le\'('ls areused to ma ximizeinforma ti onin thel-rr range.
2.5 Predi ct edput.tern of present-day sed ime ntafter 120kyr oft.nuisp ort initi alizedwit hauniform thickne ssin Hudsonnay(ou tl ined bythi n bla ckda shedlines ) .Lab els sho w Hudsonnay(H n),.Imursnay(.ID).
Manit ob a(~ I A),Ontari o(ON). Nort h Da kot a(ND). Sou t h Dakot a (SD),and Ohio(O H).The ap prox ima teextentofthe carbo na tedisper- saltrainofNorthern Ontari ois ou t line d byshort,gree n/b lackdashes (Shilts,19S0 ;Hildes,20(ll) ,whe reas theappro ximat e sou rhornext.cu r ofobse rve domars is ou t line d bylon g, cyan/b la ck dush es (Pros t etal..
2( 00) .Vari an ce-ba sed colo r levelsareused toma ximi ze info ruu u.io n in theI-rr range.
2.G Integ rat edslid ingveloc ity(a bso lu teva lue)over the la stglacia lcyel('.
Varia nce - base dcalmle velsareusedtoma xim izeinfonn at.ionin thoI-a range .
2.7 Presen t-d aymodelled sed in u-n!thic k ness.Usingtheem p ir ica lerosio n la w an dthe initializati onfield of figure 2.1.Vari an ce-b as ed color leve ls areusedtoma ximizeinfortuut.ionintheI-arange.
ix
·ID
2.8 Cumulat ivesedimentdeformation over thelastglac ialcycle. Positive (negat ive) valuesreprese ntareaswhereaccumulatio u(dl'pld ion)of sedimentbydeformat ion occurre d. Varian ce-b ased colorlewis,\I"('used tomaxim izeinform a tioninthel-rrrange.
2.9 Cumula tiveerosionover thelastglacialcyclu predictedby t.ho ompiricn l erosionlaw.AVl'rageerosion depth=;).99ui.Dash edboxes out linet.lu- Dubawnt and DelfinIsla nd studyareas whereerosion dopt.h dataaI"<' availa ble.Vari ance-basedcolor levels areused tomaxim izeinfonu ution in thei-arange.
2.10Cumula t ive erosionoverthe last glac ialcyclepred icu «!byt.h« proC('SS- oriente derosion laws.Averageabrasion dep th=0.012in(Hallot.) and 1.58m(Boulton's).Averagequarr yingdepth=L)8ui.Dashedboxes out linetheDubawut and DaffinIslandstu dyareaswhore erosion dopt.h da taarcavaila ble.Variance-base dcolorlevels areuso.ltomnximi z«
inform a tion inthei-arange..
2.11Sensitivity analysisforthe englacin lsedi mcut t.hickucss atLC i\1. Each poin t representsamodelrunwit heither the lower(blue) orthonplH'1 (red)value of a given pa ram eter.The x-axisshowsthe sodimontruod cl pa rametersintheorder presen tedinTable2.1. TherunspresentedIWI"<' usee! the empiricalerosion law,whichexplainswhyCI: (nbbrev ia r.ion for0,71111')rep lacesall thepa nuuetors related to abrasionandquarry ing.
Baseenglacial thicklH'SS=1.33m.Some punuuetc rshaveno im pactand thus haveconfounded upperand lowerpoints.
2. 12Sens it iv ityana lys is for the eros ionlaws .Eachpointrepresentsamodel ru nwit heither thelower (b lue)ortheupper(red)valueofagiVl'u parameter.Resultsare at present-dayafte r modelruns of 120kyr.Th«
x-uxisshows thesed imentmodelpar am eter sintheorderpn'sent ('din Table2.1. Panel(d)wasobtainedwit hthe empir icalerosio n law.which exp lains whyCc(ab breviat ionforC,:",)replaces allthepartu uct.ors relatedtoabrasionand quarrying.Somepa ram etershave110im pact andthusha ve co nfoundedup per and lowerpoin ts..
2.13Time-seriesof eng lacia l(green),subg lacia l(re d),anderode d(blue, shown nega tive) volu meofsed ime nt over thelastglacialcycleand norma lizedagainsttheinit.iul sed ime ntvolume(clashedpin klin n).Th«
blackcurvesareobtaine dfro mthebaselinerun,whe reast.h«widt hof thecurvescorresponds totherangeproducedbytheupp er and lower valueofC;"'l"Inad d ition,icevolume(fro mrunII)mtl027. calibration setN5btillTarasov et al.2(12),normalizedagains tLG l\I volume.is
GO
showninlightblue forcom pa rison agains tcnglncialscdiuroiu\'OIU ll H' . GI
xi
List of Acronyms
BP GSM HB IRD JB
LGM LIS MA ND
OH ON SD
DeforePre-out GlacialSystem;'-!odel Hudson Day leeflaf tedDeb ri,;
.l.u uosDay LastGlacial Ma ximum Lau ren tid eIce Shed Mani tob a
North Dakota Ohio Out.ario Sout h Dakot a
xii
List of Symbols
Symbol Description Abrasion rate
Units
AI' A,
Effect.ivc area ofcontac t
Crosssectionalareaoftheabrad ingparticle Therm allyactiv at edGlen Howlawcoefliciout Basalmeltingrate
Sedimentcohesion Pa
C Englacial sediment conr-cntratio u by volume Basaldebr is concent.rution
C,~"il Crit icalenglacialsedimentconce utrnt.ionbyvolume Scalingfact orfor the empiricalerosion law
C':lIIo' Maximumenglacialsediment «onrcnt.rut.ionby volume
Quarry ing scalingfactor
Pa-I
D
ir
Do
E
Diffusioncoefficientfor vort.ical mixiug Diffusioncoefficientprefuctor
Newtoniansediment roferencodofonu ationrate Erosionrate
Erosion ratefor the empiricallaw
xiii
111:1:--;-1
Symbol Descript io n
J/::'<I Norma l bedmodificationfa ctor
'/;;':<1 Tangent ialbedmodifica tion factor
FN Norma lcontac tforce Gravitationalaccelerati on HI> Thickn essofthemodelledbasalice h; Sed ime ntsat ura tion thickn ess hj Wat cr filmth ickness h~"L1 Maxinuuniuitialsed iiucntthickn ess h""<1 Sedime nt thickn ess
1<"<1 Shiekliugfactor
Units
HV' Bedrock hardness Pa
h", Effective su bglacialwater thickn ess
1.'"1,, Abrasionwear coefficieut
1\" Appa rentcond uct ivityof the sedime nt Hydraulic cond uct ivityofthesediment kz Vertical gridcell idcnt.ifior
Entrainedarray dep th
Lf " " Latentheatof fusion of ice
I;"",r Cut-offeug lac ialarr aythick ness
11/1 ,,<1 Arraydepthmodifier
II:
Sed ime ntrheology exponentII; ,
Residualpressur e expo nent.forquarr yingxiv
Symbol Description
N; Nu mbe rof vert icalcells
Units
I~
Pi Pili PI'
r.
P", (2
(2.,
(2",
Efloctivcnormalpressure Iceoverb ur denpressure Pressur emolti ng coefficient Typicalvaluefor theresid ual pressur e Separation pressure
Basal waterpressur e Quarryingrate Sedimentdeform a t ionflux Horizon tal water flux Abradi ngparticlera dius Transit ion radius
I'a I'a
I'a I'a I'a
n;II"'''' Mean of the gra in-s ize distribution 5",
\~lIi,'
Basalwat er massbal an ce Time coordina te Horizont al icevelocity Maximum entrainme nt rate Verticalurixingcoefficient Norma l icevelocity Netcntraiumc nt/rlopositionrate Abrad ingparti cle 's absolu tevelocity Entrain me ntrate
Symbol Description
Slidingvelocity Typicalslidingvelocit y Hor izont alvelocitywithinthe sediment Verticaldistan cefromtheice/bedinterface
Un its
Attenuation fact orfor the z-de pendc nceof verticalmixing m I3edelevation
Sediment deformati ondepth Then na l resistivityofcuglacinl ma tcri ul Gridcellthickness
Effect iveiceviscosity I/.* Rock-rock frictioncoefficient
Newtoniansedimentreferenceviscosity zp* Sedi mentangle of internalfriction
IIIK\\,-1
Pas
Pas
Pi
P,,,
Density of icc Sediment density Wa tordensity Basalshearst ress Shearstresswith inthe sed iment
Pa Pa
(* I3asalroughness
xvi
Chapter 1
Introduction and Overview
1.1 Motivations
Asprimaryprod ucts of icc-bedinteraction s , glacialerosiona landdepositional land- forms offerapotentiall y st rongcons trainton pastice shed evolution.The gn' a lNort h Ameri can ice sheets,for inst ance,havebeenthe su bjectofextensive palHl'oglaciolog- icalrecons tructions basedonglacia lg<'omorphicdata(BonnandEvans,2(1 0).Mor«
precisely,past glacia lconditions,suchasicethickn ess,extent.and flowpat.t.orn,are inferre dfromsu bglacial bedf orms,tillcharacte ristics,andmoraineposition s,andan' temp orally cons t rained by geochro no logica l da ta.Th is typeof inversion modellingis hascrlOilgenet icclassificationsof sedime nt-land formassociationsandis ,t.horofor«.
dependentonfield interpret ations ofglac ia l landforms and dep osits.A furt.herchul-
!<lngeis introduce dhy the palimpsestnatureofglac ia l land scap es ,whichg<'n('rally limit sinterpret ati onstothelatePleistocen e.
Overthe course ofthelast fewdecad es ,numoricalglacialsyste m iuodcls(CSi\ls) have emergedas analterna tiveapproa chforreconst.ructi ng palaeoi<'('slll'ds.Their
adva ntageover inver sionmodels stemsfromtheir abilitytoquunt.itat.ivclvcOlllpa n ' their resultstovarioustype s of field obse rva tionandtoprovide spa tialand temporal coverage even in theabsenc e of constraintdata.In their cur rentsta ll'.CSi\1results arccoiuparablc wit hgeo physica lobse rvat ions.suchas rela tivesen-level dataand present-d ayrates ofsur face uplift. Theuse ofgoomorp h ic datais,however.limited to cons t ra in ingmarineandtcrrcs t.riallimitsbasl'd on icemarginal featu res,as wellas iceHowpa tternba sed on directi onalerosiona land dep ositi onallnudforms.!3('call sl' GSi\Isla ckproperrepr esent a tions ofsubglac ialerosion andsediment transp ortpro- Cl'SS('S,theycannot fu llypredictthe geomorp hir:andg{'ologicoutcomesofglac ia t ions.
Incorpora ting such basalprocess compo nentsinGSi\Isisthus a!H'Cl'SSary st ('P toward exploiting thebroad errange ofglacialgeomorphologicalobsorva t.ionsfor rous truiu!
purposes.
Thisthes is addressesthisresear chgapbydevelopin gacont.inc ntal-scnlosllbg lacinl pro cessmod elthat accounts forsed iment producti on ,eutruiumout.tran sport, .u id dep ositi on (he rea fte r referr edtoasthesedime n t '/II.odel).Th ismod elisapplie-d to theNort h Ameri canice shee tsover thelastglac inlcycle and predi ctst.hr-dist.ribut.i on ofglacigenicsedime ntand«umula r.iveerosion patterns.Theseresult s an 'compare d against thepresent-day sed imentCOVl'roverNort h America andgl'ologicnlost.iiuu tos of Lau rentide eros ion. Althou ghthepro cesses of sedi ment production.cut.ra iuru eu t., trans port and dep ositi onhavebeento somedegree ad dresso d th{'or eti cally.few at- tempts havebeenmad eto imp lem entt.hcrn inconti ne nta l-scale GSjlls.t.homost.nxr-nt of whic hwas thewor kofHildes(2001;2004) .Thisthesisist.hus a cont inuationof his attem pt todevelopa coupled basalprocess/ice sheet model.
Hildes(20(H: 200·])adoptedapro res s-orientod approuc hwhe n-dist.inct erosionaland trans po rtiuocha ui smswerephysicallymodelled based011oxist ingt.lu -orctical d('scr ip- t.ionsof these pro cesses.Thiscont.rastswit hthousr:ofompi ricnllawsmoreCOlI Il IlOIlI~' impleme nte d illGS~b(s('('Secti on2.2.3.2).Thisprocrss-ori ontr-dapp roac h wuxr-om- hin ed wit h alithologicaldescri ptionof thehorlton'I>I'('S(,1l1subg hH'ial pro c('ss('s011 thecoutinont ul-scnle .Thismeth odallowodthot.rujrx-torvofsp('ci ticlit hologir-, 101)(' predi ct ed.thustheextentandcom pos itionofuiod elk-dglacia l deposit»wer«COlll- pared againstthose ofobse rved dispe rsal trains. Althou ghIdonotincludesub-grid lithologicalinfonn at.ion ,I doado ptseveralofHildos app roach('s,inc-ludingt.lu-uscof Hallet'sab rns ionlaw(Hallet,1!l?Db)and Philip'slawforr('gel ati ollintrusion(Philip.
ID80 ). and thercprescntat.ionofver t icaland hori zon t ul('Ilglacialtrans port byvar ia- t.ionsilldebr is couccnt.ratio n .IIIad d it.ion.Ipresenta110\'(,1quarrvinglawbas('d011 theestima te dexten t.ofsu bg lac ia lcavities.implemen t Boulton'sannlvsis of abrasion (Bou lt on. ID?!l)forcom pa riso n with thatof Hallet.undad dasoft-boddoform utiou compo ne nt based 011themod elofJen son(199S).These asp('cts an'discuss('dilldotnil illSection2.2
1.2 Obj ective s and approach
Tlu-primary goalof thisthesisistodevelopaconr.im-ntul-sculesubglacia lerosionnud sed ime nttransport model cons istentwithg('ologicalostimat es ofcu mula ti voerosiou and wit hthe ohse rvodpatte rn ofglacigcllic deposits overNort hAmorir:n .IIIudd it.ion, Iaddress thefollowingkeyquesti ons :
•Call process-ori entederosion lawsbereplaced bysim p leciup iric ulrelut.ionship s be tween ero sion rateandsomeglaciolog icalvaria ble?
•Whenimplement edinGSllIs. doHallets andBou lto n'sabrasionlawsprodu cx- com parableerosion pa tt ern s?
•Canalar ge-scalequar ryi ng lawbased ontheextentof subglacial ca vit ips pr(}- ducerea list icerosion pa tt ern s'!
•Wh ich of su bglacia ldofonuati on and cngluciultruns port wast.hoIlIOSt ellic ipllt transpo rt mochanism of the NorthAu ieri cnnice sheets?
Toprop erl y ad dressthese questi on s . theresultsprese ntedin thist.lu-sis an' obta ined inthecontex t ofa sensitivity analysis.Asdiscu ssedinSoctiou 2.2.1 .this ap proach cons ist»in definingrealisticrangesforthe poorlyr-onstrn inod puuuu cte rsIlspdill thosed imentmodelaw l insys t.cmntically gene rati llg t.hcassoeialPdrangps ofiuo.k-l 0IltC01ll('S,Th is appro achis superior tousing a singh'set of panu notorvaluos b('('alls(' it allowstheparametricsensit ivityof themod eltobpox.u uino dawl t.lu-iruportauc-o of different pro cesses aw l vari ables tobeidentifi ed.Systcmat.i«sensiti\'ityallalysps.
aswellas atte m ptsto addresstheques ti on s out lined ubovo inth«contincn t ul-scnle co nt ext.havesofarbeenla ckingin thisline ofresearr-h.
Further more. whencond uct ingmodellingstudiesthat invol ve several diffprpnttopics.
such as the vari e ty ofsubglac ial processesiruplcmcntcd intlu-spd iIllpnt mod el.itis toin pt ing topresentthe entirerange of possibleresu ltsthatcan1)('gpn('ra l pd bytIl<' model . regard lessof theirrelevance ortheircouiput ihi litvwithphysicalobsorva tion s.
Suchanappro ach wouldyieldaver ydetail eddocum cnt.ati o noft.hemodol'»lx-huviou r.
but. would beatrisks of bein gapurelynumoricn!modelling('x prcisewith liruiu «]
scope ,Inthehopeofpres cu tiug a st udythat hasvalueto thei('(' sh('e tnl<l<lpllillg couuuu nity,Iattem pt to avoidthisapproa ch byprosoutingresu lts tha tcan. ing('II<'ral.
heCOllipared to availa bledata or to other models,aw l hy addn'ssingglac iologically releva nttop icsin mostofthe discu ssions .
1. 3 Structure
Th is thesisis writte ninmanuscrip t forma t (incontrasttotraditionalfonua t.).as such,its content ispresent ed inthestyleof journalnrtir-les aw lwill 1)('prcp a rodfor pu b licati on.Inthis case,the su pe rvisorycon nuittccagr eed thatasingleruanu script wasprefe rable giventhena tureawlscopeoftheresearchundert ak e n.Thisuuuruscript ispresentedinCha pter2and iswrittenasastand-aloneresearch pa pe r.itthus containssome backgroundmatorinltha tisrepeatedfromthisintrod uct or ychnpu -r.
Chaptcri!cons istsofsuuu uarycom me ntsanddescription sof pot en ti al futurorpsp,lITh directi onsresultingfromthisthesis.Thebibliographyserve» forthe who le t1l('sis.
1.4 Co -authorship st atement
Author shipforthe pap erpresentedinChapte r 2 isin thefollowing orde r:1\1r. Aloxan- dreMolan son(thesisauthor) ,Dr. TrevorBell(thes isco-su perv isor},and Dr. L(·\·
Tarasov(thesis supervisor).Dr.Bellis aPro fessor withtheDepurt.nu-ntofCpog- raphyof Memori al Univers ityofNewfound land.and Dr.Ta ras ov is anAssociat(·
Pro fessor withtheDepa rtmentof Physicsand Physical OceaIlOgraphy ofMom onnl Univers ityofNewfound land.
Dr. Tarasovdevelop edthein itia l idea and directi onof theproj ec t, afte rwh ich1\1r.
Melan sonconceptua lizedthedet ail edformandstru ctureofthe model presente-d hen'.
l\1r.Melanson subse q ue nt lywrotethe code for themodel and porforruodtheuumcrir-ul sim ulatious as wellthedata ana lysis.Relat edmodels wereprovided hyDr. Tarasov
(l\[UN3-DGSl\[,aswellasthecalihrat.odset ofGSl\[ pa rameters) and hyMr.Murk Kavanagh(basal hyd rologysolver),aMast er's stu de nt ofDr. Taras ov.Dr.Dellpro- videdsupe rvisoryguidu nco and keyknowledgepcrtaiuingtoglac ialgeOluorphology.
Theresear chpaperpresente dinChapte r 2 waswrit.runcuti rclybyl\l r.1\ldausou, wit hthe exceptionofSection2,2.6 tha t wasprovidedbyDr.Tarasov.Dr. Dellaud Dr.Tarasovthoro ughlyreviewed the manuscript andsuggestedminorrevisions,
Chapter 2
Numerical Modelling of Subglacial
Erosion and Sediment Transport
and its Application to the North
American Ice Sheets over the Last
Glacial Cycle
Abs tract
Present-d ay sed ime nt distributionoffersa potenr.inllv st ro ngcons rrnint011 pastice sheetcvolu t.io n.Tiowcve r,glaciu lsystc m11IOdels(GS i\ls)cannot.udd ress thiswhilelackingproperrcprcsc ntu ti ous ofsubg lacia lsed iment product.ion and trnnsport.Incorp ora ti ngthese cleme nts ill GSl\Isisalsorequiredill order to quantifytheimpactofchang ingscd hne utcover011glac ialcycle dyn ami cs .
Toward this goa l,Iprescntu subglacia l proc essmodel(hcrou ftcrrefcrrc d to ast hescd'irncn trn odcl)t ha t illcorp ora tes mec ha llisms forsed ime llt pro d uc t ioll, cnt.rninme nt.,transport,and depositi on .Bedrock eros ion is calc ulutcdeit he rb~' Hallet's or Boulton'sabrasionlaw ,andb~'a novelqunrryii u;lawpanunctnrized as afuncti onofsubglacialcavityext.cnr.,Theseprocoss-ori ont.cderoisouInws are compared agaiu st a simpleemp ir ica l relationshipIwt wl'l'll erosionrateand the workdon ebybasalst ress.Sedim ent ent.rninmont is rep rcscntedb~'Philip's lawforregelationintrusionundsoft- bod deformati onisincludedas a subglac ia l sed imenttra ns po rt mechani sm,therheology of whichisassum edto1)('wonkly lion-lin ea r.Themodelisdrivenbythedntn- calibrn tcdi\IUN:1-0 GSi\1and a newly develop ed subglac ial hydrologymodule.
The sed ime nt modelisappliedto thelast Nort hAmcri rnnglaciatio n and predi ct s sed ime nt thickn ess awlcumula tiveeros ion pnt.t.erus. Theseoutput ficldsuro com pa rod nguin stthepresent-day glacigellicsed ime llt dist.rihutionaud geologica lest imates of Laurr-nt.ideerosion.Givenplausibleparamet errall ges for the sed iment mod el(chose n a prioribased011availabl elitera tur eor011 heuri sti cargum ellt s),the ca lculatedero siolldepthsfall withillthe geological estima tes of La ure llt ide erns ioll,wit h t heexce p t ionof Ha llet'sa bras ion law t ha t undere stim at esabrasionby atleast an orderofm ag nitu do.III nrldit iou,most of rimsill the sensit ivityset produceunren.listicnllythick andcout.inuo us moraiuos alongthe eastern,sout he rnandwestern margin s of theNor t h Amcru-nuic(, co mplex ,wh ichsuggests thatthemodeloverestimates sedime ntonrra im uc nt
andthuscnglacinltra nsport.Thisis eit herexplainedb~'the misrep rr-a-ntat iou oflarge-scaleregolnt.ionintrusion orhyunderestimatedbasalwate r pressu res.
A realistic sediment dist ribu tionis onlyobtained whent.lieontraiuuu-ntrntr- is capped at theaveragebasalmeltingrate.whichsup;geststha t a renlistir:
balancebetween entrainmentand deposition is achie\'ed whenthe ourrauuu out and hasalmeltin g rnresareofthe sameorderofiuaguitudo.
2.1 Introduction
Asfirsthandevide nceof ice-bod inte ractions.glacialorosiounland dop osit.iouullund- forms offe rapotentially st ro ngconst ra int on pasticcslh'etevolu tio n.Althought.hoy are the prima rytoolsused in quulita t.ivcrcconst.ruct.iouof palPOi('('sh ppts (Glasspra lld Denne tt, 20(4), numeri calmod elling ofthe subg lacia l pn )("('SSPSthat(T('at ('dt.horn has beenhampere dhythela ck ofdirect observationsofthe ic('/l)('dint erfurc.However.
thead ventof new subglacial monitoring techn iques has increased the a(...pssibi lit.\,of couteuiporarygladprhods(BonnandEvans.2(10)aw l. in conjunction wit.hkovlab o- ratory expe riments(Ivorson.1990:Kamb.1991:Ive rson.!99;!:In 'rSOlIawlScu u uous.
1995:Tulaczyk ota!"2000;Bye rs ct a!"2(1 2).has allowedmodelsto hp fon uu lutod and validat ed againstthenewdat a,Thishasprog rossivoly ullowedgla cialsystpn l models(GS;\ls)toincorp o ra tequantit ati ve rcprescututionsofsubglacialerosioun l and trans portprocesses (e.g.Alley and ;\!a cAyeal. \99·1:Jonson otul.,199fl:Egh ohll r-t a!"2012),althoughonlyfew at toruptshav ebeenruudein t.hr- eont.inc utul-scuk- context(Tu lley,1995;Hildes, 20(H;Hild('seta!"200-1;.l.uuieson ot a!":2000. 2lllll).
lnd ccd,themajorityofcont.iuc utal-sc alcGSI\!sst ill lack sudIn-prr-sc-nt.atious and hen ce can notpredict the gpologicandgeomorphieout.conn-sofglaciations. This dclici encyprecludesthemfrom fully exp loiti ngthebro ad rangpofgpOlllorp hic data
10
forconst ra int purposes .In addition , a subg lac ia l processmodelfully cou pled with icedyn ami cs calcu la t ionsisnecessar yto prop erl y ad d ressthe impact ofsoft- hod deform ati on on icc shee t beha viour.This wou ld be su pe rior tothecurrentpra cti c(' of prescribingpot en ti al enha nce d flowareasbased on present -d ay sod imo ntCOW l'('.g.
Ta ra sov and Peltier,2(04) .
Towar dthese goals, I present a conti nenta l-sca lesubg lac ia l pro cessmodelthat ac- counts forsedim entproducti on , cntrui u me nt, transp ort , and depositi on .'I' homod el is ap plied to the Nort h Ameri canice sheets over the last glac ia lcycIPand pn-dicts the distribution ofglac ige nicsed ime ntandcumu la tiveeros ion patterns .This efl'or t is a cont inua t ionof theworkof Hildcs (20(1l:20( 4),whocom bine daproros s-oriout.od ap p roa chwit halitholo gicaldescriptionof thebedto rep rese nt these])J'oc ('sses on theconti ne nta l-sca le, Thismethod allowed thetraject or y ofspec ific lit.holouio»to bepredi cted ,thusthe exte ntandcom pos it ionof modelled glac ia l dep osit s was com- par ed agains ttha tofobse rved disper saltrains .Althou ghI donotincludesub-g r id lithologicalinformation ,I do ado ptsevera lof Hildcs ' approaches,includingt.house of Hall et's ab ras ion law (Ha llet, 1979b ) and Philip'slawfor regelati onintru sion (P h ilip, 19S0) , and therepr esent ati onof vertical and horizont al oug lac ia l tran sp or tbyvaria - t.iou sin debri s conce ntra t ion.Inaddi t ion , I presen ta nove]quarr yin glaw based011 theestima te dexte ntofsubg lac ia lcavit ies, implem entBoult.ou' s anulys i» ofab ras io n (Bo ulto n, 1979 )for com pa riso nwit htha tof Hall et , andadda soft- b od dr-torunuion co m po ne nt basedonthe model of.Icns on(1995). Thelatterisincluded toass('ssthe trans p ort pot enti al ofsu bg lac ia l deform ati onrelati ve to engl aciu l transp o rt. and to det ermine whe t her it has an impa ct on the eros ion pnt.teru.
11
Moreover.Itacklethe importan tquestion ofwhet herconti nenta l-scu losuhglacial erosion mod elsrequire theuse ofaprocess-orien ted ap proac h(where nhrnsiouand qua rryin g aremod elled separa telywit htheirownset ofcontrollingvariahlcs},or can he simplyreduced to ancmpirica llv- bascdscaling rela tionshipbet.weenerosionratt's andsomeglaciologica lvariables(e.g.ice thickness ,slidingveloc ity.etc.).Altho ugh both approac hes havebeen applied to thecont inental-scale(s('e Section:2.:2.3).t.lu-r«
hasboeuno attem ptto compareandevaluate them. Itliu»spr-cifiru llyaU('nlpt10fill thisrosearc hgap hyimplementin gboth approac hesin themode-landcOlnpari ng t.heiu systemat ically.By adopti ng this goal,Iimp licitly OpPOS('thenotionthatphysicall y- based approac hesshouldauto m.uically bedeemed superior.Rath er.Isuppo rt tha t thereisno apriorireasontoreject em pirical laws,especia llyin the larg('-scal(' coutcxt., where it isconcei va bletha t thesmall-scalevariab ilityofsubglaciul proc('sses a\'('rages- outandtherebyrevealsalargerpattern cont rolledhy theaverage valueof glaciological varia bles.
Most of theresults presen tedhere were obta ined in the contex tofa sousit.ivit.v anulysi«
(s('eSect ion :2.2.1).This approac hallowsthesod iuroutmod el' spa rnu u-tr icsellsit ivity to heexaminedandtheinfluence of diffcrontprocesses andvari nbk-s(e.g.outrn iunu-nt, sedimentdeformat.iou,basal wate r pressu re,ctc.)onthe predi ctedpnt.toru oferosiou anddepos ition tohe identi fied.This app roach,as wellasat.to rupt«to add n'ss the questionoutli nedabove inthe continental-sca lecontext. have sofar1)('('11lackingin thisline of research.
12
2.2 Model description
[ presentbelowatwo-component model thatqua nt ifiesprocesses ofsu bg lacia lsodi- uron tprod uction,entrain me nt, transportand dep osit ion .Thesubglaci alcorup ouont (2-D)acquir es sed ime ntfromerode d bedrock and frommelt ing of debris-ladeniCl"and tran sportsithorizont all yby soft-bed deforma ti on.Whcr« outraimueuts,ityieldstun- terialtotheetutlacialcompo nent(:\-D),whichallows vert icaland horizontal eng lacial tra nspo rtof debris,andsedime nt depositioninzonesofdominan tha salmelt ing.Tho mod elisdrivenbythe]\[U N3- DGSl\[(Tarasov ct al.2012;Sectio n2.2.G) ,that110W includesabasalhyd rol ogy solver(Ka vanagh 2012;Section2.2.7 ).Togl't.hpr.tll('y pro- vide key controlling variables,suchasicethickn ess,slidingveloc ityand basalwuter pressu re. Tobette r isolatet.he sedime ntmod elbohaviou rs,on ly passivo«oupliru;
betweenthe sed ime ntmodeland thesedriving com po nents is cUIT('nUyturnedOI l.
2.2.1 Sensit iv ityana lys is
Any attem pt tomode!sparsly-obscrvcdandsmall-scale pt'OCPSSl'S,such ast.hos«ill- elude din the sed imentmod el,ishoun d toinclude somelevelofidcalizntion and pa nuu ct.erizat.ionwit h resp ecttotheirlarge-scalerepresentations.Th isroq uirost.h«
uscofpoorly-oruncon strainedpar amet er sthat ca nhardlyloud toIirtu couclusion«
whenonly a single setof valuesisassignedtothem.Rath er.it isproforahlctodofiue aplausiblerangeforeachpa ram eter(inthis eas e,an upper,lowerand basl' value) andsystema tica llygenerate the associa ted rangeofmod elou tcomes.Tlu- pa rumot.ors towhichIassignarangeareherein referre d toassedimentmodel pann neirrsand art:ident.ifiod by anasterisk. Theirvaluosaredocumentedand explained in Table 2.1.ThemodelI'Imthat usest.hebase valueof each parameteris rd l'IT('dtoast.hc bas elinerun.Th is app roach is su pe riortousin g a sing leset ofpanunot.or valuo»\ll'-
1;1
cause itexplicitlyshows theboundsandtheHex ibilityofthemod el.awlidout.ifics theparameters (thus theprocesses )to which themod elismostsensiti ve. Fu rther- more,examining parametric sensitiv ityisan import an t. first stoptowarduucortu iutv quuu tifi cation.
2.2.2 Mod elnumer ic s
The sed ime nt.model's equa tionsarcdiscre tizcdbyfinitevolu mes011a stagg pl'l'dsp her- icalcoor d inategridspan ningNort hAme ricawit ha1°meridi onal and (UiOzonal resolutio n. Inaddition,theenglaoinlcom po nentis solved over
N ;
ver ti cal lavers expo nentiallyconce nt ra t ed ncart.heice-substrateinterface ,resu lti nginadom ain thickn essofHI,(Table2.1). Horizontal trans porttermsarct.cmporallvdiscr <'tizl'd by theexplicitEuler for ward scheme,wherea stheimplicit tri-diagonal nuu.rix algo- rith m is appliedto thosefor verticaltransport.Furthermore,thetransportHuxps at theinterfacesarc calculated front the upwind scheme.wh ilethosefor vr-rtical in- t.erf acesare comp utedfromthepower-law sche meforadvccti on-diflusionproble ms (Patan ka r,1980 ). The transport of materialbetweengridcellsandbet.weontnodl'l compo nentsconser ves totalsedi me nt volume.Themod elisrun from UOkyr1)('- for epresent(I3P)topresent-d aywithasynchro nous dynam ict.iruc-stoppiug;b('(\\'ppn thecomponents toenfor ceCFLconditions.Followin gthemethod ofHildos(20(Jl:200-1),themodelisiniti alized withpresent- day sed ime ntthickness (sc'c'Figur«2.1).
Based on the surficialgcology mapofCanada(Fu ltou.1995),Idefine a sub-grid sed ime ntcover factorto eachsur ficialca teg or y(c.g.1for t.illblankets.0.1fort.ill venee r}.Thenpscal ed,model-gridsed ime nt thicknessis thenohtaincd byaVl'raging the sedime ntcoverfacto rsand multiplyingby amaximum initial thickness ,II:"".,..As the sedime nt.dist ribution ofFigure2.1failst.ocap ture t.heint ra-catl 'gory variability of sedime ntthickness,itdocsnot. accuratelyrepresen t thethickness ofpn'sl'n l.-d ay
14
7G"N
55°N
sed i me nt. t.hickncss (m )
170"W 150aW 130aW llooW 90aW 700W 500W
Figur e2.1:Initi al sed iment distribution at 120 kyrI3P,based011the mapof Fulton (1995).A maximum ofh~",.r(basevalue of 15m)isprescrib ed overareasof ohservod till blank et s.Eac hcolore dsq uarecorros po ndstoafullgrid-c ell.
sed iment, butmerely servesas areali sticiniti aliza ti onmap.La stly,as tra ns port by proglacial strea ms isnegligibl e attheconti nenta lscale,itisnot includedin this study and Iapplyano-fluxboundar y cond it ionalong the icema rgin s.
2.2.3 Subglacialerosio n
Glacialabrasionand quarrying arecom mo nlyviewedasthedominant su bglac ia lcro- sion mechanisms(Drewry,1986;Glasserand Benn ett,2(04).Althoughnu-Itwutr»
erosion ha s been hyp oth esizedtoproduce significa nt local erosio n through cat as- trophi c ou t burs t flood events(Shaw,2(02) ,it isthough ttobenegligible rolu t.ivcto abrasionand qua rr ying over the cont ine nta l-scale(Drewry,1986:Iverson,2002:Co-
15
henet al.,2(06).With theirownset ofcontrollingvaria bles.abrasion andquarrying arcfavour edunder differentglaciologicalconditions.but have also I)('cnshownto1)(' controlledbylit hologicalpropertiesofthe bed(Glasser andBounot.f.,200·1;Kruhh cn- dam and Glasser,20 11).Thecom plexityofthe erosion pattern over North Amoricu is a consequenceof thesemultiple controlsonglac ialerosion,and it is still uuclru r wheth er this pattern ca nbe successfullyrepro d ucedby sim pleempiricalerosion laws (Hallet2011;Sect ion2.2.3.2),orrequiresa process-ori entedapproach.whor« nhrn- siouandquarryin g arcmod elled separately (Hildes2()(1l;Hildos ctal.200·1;Scxt.i o u 2.2.:3.1).Ra th erthanadopt ingonlyoneoft.hesoapproaches,Ifavoura compa rison bet ween themandim plement both in themodel.
2.2.3.1 Process-oriented er os io n laws
Quantitativemod els ofglacialabrasiontakethefon uofwear lawsfrom ma terial scienceada ptedtothe subglacialcontex t(Archard.1~)5:l;Hallet.1!l7!lb: Boulto n.
1!l7!l;Drewry .1986; Cuffey andAlley,1!l!l6;Hildes.2001;t«,200'1):
whereI.:,,/>,.isthe abrasionwear coefficient(0.2).HF*istheVickerhurducss of t.h«
bed rock,C/>(III-~)isthe size-depe nde nt basal deb risconcentration.FNt.hocontact force bet ween thebedrockand t.h«abrading part icle,Ristheparl.icloradius.and
II!,,,,,ist.he absolutevelocityatwhich theabra di ngpnrticlo isdragg('dalongth«Ix-d.
The oxpoucnt.ialfactorrepresentsshieldingofbed rockbydepositedsedinH'nt..whcr«
h.".,1andIi.:.."arc respectivelyt.hcsubglacialsedimentthicknessanda cha racte rist ic depthforthe shielding effect(hereafte r referred to asthe shielding factor;Hildes200 I;
Hildos etal.2004;Eghohnetal.2( 12). Notetha t. Equation(2.1)implicit.lvassumos thatthe abra ding par ti cle' shardn essisalways greate r than thatof t.h«bedrock.which
16
pot ent iallyleadsto overes tima tingtotalabrasion,as soft particles canno tphysically abra deaharderbedrock ,
As out lined below,cont rollingvariablesul'",.(f?)andF,v(R)dep endont.h«abra di ug part icle size. Rath erthanchoosinga sing lerepresentativ eclast size,10valuesof Ran' chose nunifor mly over thebase-H)logar it hmicscale,ranging from111111 to0.1 m(i.e.
clayto cobble).Theproductofvl'",. (R)andF,v(R)is weightedagainstt.hcgra in-s iz<' dist ri but ion,C/,(R),andsun uucdto obtainthetotalab rasionrate.The\\'eightsan' generated by anormally-distrib ute dparti cle size pro fileover theba s('-l0logarithm ic scale,andscaled wit hthebasal deb ris concent ra t ionby volu me,ascom pu te d fromthe model's cnglac ia ltransport compo nent(Section 2.2.4).This calculationaSSIllIH'Sthat thedebris concent ra tion by volume is equa ltotheareal debr is conce ntra ti on incontact wit hthe bed.Assu ming anormalparticle sizedist ri bu t ionis ahighly sim plified approac h,as thesedistribu t ion aretypicallybi-oreven poly-mod al(Huldorsou.19S1:
Bou lton,1978,1979;Drewry ,1986;Hubbard andShar p,1995: Cookd al..2( 11).
Thisspe cificdistri bu t ion was chose ntoincorporat e a widerangeof pa rt.iclc siz('swhile minimizing thenu mber of param et ersrequiredto generateit. Amore dctailodst.udv couldattempt todynamicall y gene rate thepart icleradiidistrib u tion;this wou ld, however,requiresta t ist ical datapertainin gtothe spocificsizeofdebr isprod uce d by abras ionandquarry ing.
\Vithacleardep end ence on bedrockhardn essthroughthe ab rasionm<'flici('nt, t.hc questionofwhethertheabrasion lawrequirestheincorpor ation ofde taik-dlit ho logical inform a tion arises.Suchan effort isnon-tri vialbecau sebed rock hurdu os» varieswith in amod el gridcell,promptin gtheuse ofanupscalingruleto obta ina singl<'grid-scal<' value.ThiswasdonebyHildes(200 1;2(0 4) withanarea-weight ingsc!H'nH'.Alt hough
17
it is likelyto affectthe spa t ia l pa tto rn ofabras ion. ItreatHv:as aunifonu and constant sed iment mod elpa ramet er, Asdiscussed illSection 2,:1,2,howe-ver.the ave rage abrasion depthpredi ct edbyHall ets abrasionlaw(seebelow)issimilar to thatofHildes(200 1;2004)
The exa ct formofFNand
v,,,,,,
differ accordi ngto thecontex tillwhichtheyan'ill- t.crprcted,i.c .Hall et'sor Boul ton'sabra sionla w,BOULTON'SMODE L
Bou lton(HJ74;1( 79) suggestedthattheforceexerted by theahrading pa rti cleOilthe bedrockisprop ort ion altotho effoctivc norm alpressure,p",such t.hutFN=A,.J~"
whereA"isthe effectiveareaofcontact..For a spherical purt.iclo of radiusRillcontuct witha waterfilmofthickn esshj,th is isgivenbyA"=7f(2Rhj- (hj)2)(Bye rs<'!ul., 20l2),Thefriction be t weentheclas t. and thebed ro ck causes the forme r to travel slower than the slidinj; ice (Bo u lto n,1(7 9),Byusing\VeertllI1Ul'Stheoryofbasal sliding,Boulto nostimatedthis reduced velocityas
wherev"is t.heslid ing velocity,ilr;is thetcm pcrat. ure- dc pc udc ntcodficiellt ill CI('ll's Howla w,II.'is the rock-ro ckfrict.ioncocffir-icn t.,A,.isthecross-scct.innu lan'aof the pa rt.icle,Pmis the pressu reuielt.ingcoefficient(7,42x10-1<1\ Pa-I) ,Lj " "isthela t.cnt hea t offusion (:l.:J4x1O-r,.J kg-I),andPiis theiceden si ty(910kgm-;\),
18
HALLET'S MODEL
13y recognizingthatabra ding particles arc genera llyem]H'dd{'d inpres su ri ze-dic{', Hallet(HJ79b;1981)sugges te d tha t stressconce ntrations requ iredfor clasts toincise thebedrockrema inconsta nt underchan ges of norm al dkctiv{'pressu re ,Hismod el thusassertstha teffectivecontactforcesare contro llod byiccflow towardt.hob<'d ratherthan by theeffective pressu re.The norma l drag exertedby t.he ic{' on t.lu:
embeddedclastsis expressedas(Watt, 1974):
(2.:1)
where,t;;;dis amodifi cat ion factoraccount.iug for the prosonco of t.ho b{'d(1.8: l3y{'rs etal. 2012),'Iis theeffect ive ice viscosity,Ris thecritica lrad iusat which t.h«
dragpercross-sectionalareareaches amaximum(Hallet.1979b)and isgi\'{'n by ('JIIPIII/Lfl/"fliZ,~ ) t Ii"is the norm alicc velocityandinprincipl e cOlllpris{'s t.lmx:
compo nents:thecom ponentoftheslidi ngvelocitynorm altothehod.t.hobasal melti ng rate,iJIII 1•U,andthenorui alcompo nent.oftheextensiouul icc velocity,Civcn thatthe relevan t vert ical length scaleis approximatelythe diamete r oftheclast s, theext ensionalcom po nent is showntobenegligibl e(I-lildes. 200 I). TIl('norm al slidi ngcom po nent.011theother hand,isdep end ent011thesmall-sca le(1-100Ill)bpd dip angle,c.g. positi ve onthestoOlSsideofbedprotru sions and n{'ga ti Vl' on tlu-ir Ice.Ideal treatme ntofthisterm wouldincorp ora te sta t ist icalinformatio non I)('ddip angles at therelevantscales.Lackin gthis stua ll-scalc rep resent a tion,Iassmuo that t.h«
roughnessof the bedis average d-o utoverthe mod elgrid-scale, yieldi ngI'"=ill/ I"'I'
Notethati)gravita tional buoyan cy ofthe abra ding part.iclois notiuclurlr-dinthe' contact forceexpressio n becau seit 's coutri butio n isnegligiblerolnt.ivcto t.h«viscous
I!)
dragforpar ticl es wit h radiuslessthanafewdocim etrc s(Drewry,lDSG:Iverson.2( 02).
andthat ii)the der iva ti on ofEquat ion(2.~l)assumesalinea r ice rheology.
Thepart iclevelocityover thebedis obta ined by assuming force equilibriu m bot.woon thetange ntia l dragimpart edbytheice011theabra dingpart icleand the frict ionwit h the bed,yield ing
wheref;:':<1is thetangentia l bedmodi ficationpar am et er(1.7:Ilildes 20(H) .
Alt hough lab ora tor y expe riments favourHallet's abras ion law(Iverson,IDDO:I3y('rs et.ul.,2( 12),questi onsremain whet her it is appro pria teincondi tio nsof highdebri s conce nt.rat.lon.whereclosepar ticl eintera ctionmightinduce adep elldenc('Oill'!r('ctiV(~
norm alpressur e (Hallet,ID7Db;Iverson,2(02).Thelack ofconclusivel'videlH'('I'm choosingtheappro priatelar ge-scaleinterpr eta t ivefram ework,as wellastill' under- estima te dabras ion resultin g fromHildcs'(200l:2004)atte m pt toimploui ontHullots law inacont inenta lGSf-.r(see Section2.:l.2) providesmoti vationfor iiupk-nu-nt.iru;
andcom pa ringHallet's and Boul to us abrasion laws. Note that in bot hiuipkuuout u- t.ions,abras ion isneut ralizedwhen i)the part icle velocity bl'conll'slll'ga U\"(\ii)till' icebecomes cold- base d,and iii)whenicc reachesHota tion.
:20
QUARRYING
Quarryingisregard ed asthedominantglacialerosion mechanism(BriuerandSwan- son,1998;Iver son,2002;Loso ctal.,200-1;Riihimuki,200G;Cohen ct.al..2000; KrabbendauiandGlasser,2( 11),yeta satisfying large-scal e rcprescntuti on ofitre- mainselusive.Th isisprim arilydue tothe com plexintera ction between glaciologica l.
lith ological,andhydrologicalcontrollingprocessesand to thepoorunders tandingof theirrelative cont rib ut iontolarge-scalequarryin grut.os.
Det ail ed theore t icalanalysisofstressfieldsin bothbasaliccandlx-drocksuggest thatthin,fastflowingiceopt imizesquarr yin g(Iverson,1991;Hallet,1( 90).Such cond itionsfavour theformation ofsubglacialcav itiesthatrod ucothe arm of ic(,-IH'd contactandconseque ntlyen hancethe stressconcent ra tiononcontact points .Hildos (20lll)evensuggests thatthe absenceofcavitiesneut.rul izosquarryingIH'cau seitpn'- dudesthe bed ro ckfromreachingtheminimumstress intensityrequ iredtopropagat e' crac ks .Theseana lysesassu methat snbcrit.iculcrac kgrowth undericeload rate-limits quarr ying,thusthatlith ology-sp ecificinform a ti onsuchasminimunstressintensity (estima ted fromthe crit ica lst ress inten sity)andcrackgrowthexponent(cont ro lst.hr:
relatio ns hip between crack propagation ratesandstressinten sities ) an'theprimar y lithological controlonquarrying.Hiklcs(20(H; 2(04)ada ptsthismod oIliugapproa ch to the continental-scaleby assu mingquarryingratesproporti on al to su hcrit.ir-nlcrack propagat ionrates,whicharecalcula tedhyupscaling't.h«aforementionedlithological par amet erstothe model grid.Alth oughtheiranalysispro ducesrealistic(hutslightly undorcstimatcd)quarryingrates,fieldevide ncesugg est thatth is roprosont.nrionof quarryingisincomp let e.
21
Indeed ,Gla sseretal. (1998),Duhnfort.hct.al.(2010),and KrabbondumandGlasser (20 11)showthatinter-and intra -li th ologyvariat ionsof preglaciallx-drockjoin tdon- sitycorrelate s withqua rrying efficiency.Furt.hcn uorc ,a recentst.udv(Hoove rctal., 2(12)interpret sthe strongcorre la tion betweentheorientationofqua rrk-dslll'faces aIHIthat of proglucinljoints (and the associated lackofcorrela t ionwithslid ingeli- roct ion)as evide nce fa lsifyin gthe assum pt ion thatsu bg lacially- ind ucedcrackgrowt h controlsquarrying.This assu mptionim pliesthatquarri ed slll'facessho uld align with principa lstresses impart edbyHowiugice.Rather,the authorssuggest thatt.h« bed sho uldbeidealized as"aseriesof blocks separa ted bydiscon tinu ouspn'glacial joints conta ining intact rock bridges".Theinflu cncoofsu hcri tk'ulcrackgrowt h wouldt.hen berestrictedtotherock bridges , and thusrdegatedto alesser rolethanin pre vious models ,
These stud iesshow that pa rameterstiedto subc ritic ulcrack propagati onan' not necessarilytheprimarylithological contro lon quarryingrut.es.there fore.usingthem inquarryingmod els isdifficult,to justif y, Rather.sub-g rid informationonh.xl rock join tdensity shouldalsobeincl udedand mighteven ])('moreinfluent ial.However.
aspointedoutinKrubbond am andGlasser(20 11),jointdensity canvarybyup to 2 ordersofmagnitudewit h inasing le lithology,renderingbedrockg('()!ogy iufo rm.u.ion imp ra cti calforincorporatingthis aspect inlar ge-sca lemod els.
Thela ck ofspa t ial informat ion011bedrockjoin tdensity forcesruetonssumcauuiform lit hological cont rolon quarr ying overthemod eldomain . Cons ideringthe hnpo rtuncc ofsu bg lacialcavities in thequarryin gprocess ,Iscale quarryingrat.oswit hthr-('sti- ma ted grid-sca leextent ofcavities. Thelatteris assu med propor t ional tot.h«rr-siduul pressure,i.e.the differ encebet weenbasal water pressure.PliO.aIHIsepa ra ti on pros-
sure,p."which is exp ress edas (Schweizer and Ikon .1992:Bonn aw lEvans.20 lO:
Cuffey and Paterson,20 10):
(2.fJ)
wherePiis the ice overb urde n pressur e,T/>isthebasalshearst ress(cu rrent.lvassu u u-d equul tothedrivingstress),and('istheha salrough ness (vert ica lvar iati onoflx -dro rk featuresovertheirchar act eristic length ) .
TIl('quarryingrateisthe ncalcu lut.odas:
(2.G)
whe reC,;,,,,,,is thequarryin g coefficient, ";,controlstheinflue nceoft.hc resid ua lprr-s- sure onquarrying,and
P ,.
isa typ icalvaluefor theresidual pressure(ISkPa l,cal- culute dfromPe=(J.:3;\IPa,(*=0.1,andT/>=lOOkPa(Cu lley and Pa t ors on.2( 10).Quarryingisneutra lized whe nP",<p...Theresu lt ingquarryin gla wisconsistout withtheacceptedviewof quarr y ing,i.e .prop orti on a ltobasal water pressureandslid- ing spe ed,and inver selyprop orti on al toicc overb urde n pressure(Class <'randBonuc t.t..
20(4).Note , however,tha tth isappro achisnoveland lacksfieldvalidation .
Troat.iug (' asased ime ntmodelpanunoter im plicitlyaSSUIIl('St.hatquarrvi ugra!<'s can beeflect.ivc ly est ima ted hyanaverageurcasu rcofbed rough ness.Thisis as('rio us sim plifica tion becausetopograph ic controlofquarryin g opera tes 011arango ofdifforout scales,as sugges t ed fromtherangeofrochemou toun ecsdimensions (13<'1111and Evaus . 2( 10).None the less,Iassumethatrealis t icqua rryingrates can heobtai nhy a vpragil1g
topographicvariationandtreat ingroughnessasauniformscd i u rcnt.modelpunuu ot.or (Tahle2. 1) .
Thereistheoretical(Iverson,1(01)and fieldevide nce(Cohenctnl.,200G)sugg<'sting thatfluctua ting ca vitywat erpressur ealsoenhancesquarryingrates.Thisisirre-levant at the continental-sca le,however,aspercolating sur facewaterisunlikelyton-ncht.hr:
bedand induce rapidcha ngesin hasnlwate rpressur e.\Vat prprr-sxurr:isthuslikely to va ry at amuchlowerfrequencythan thediuruulfluctu ationsobservr -dinalpine andoutlet glaciers(BonnandEvans.2( 10)
Although thereis gcnpwlugrcemcntonthedomin ance ofquarr yingrelntivctonhru- sion,quantita t iveevide nceof theirrela ti ve cont.ributionto totalglac ialerosionis spa rse. 13ymeasuringsuspendedsediment load and bedloudinout.lotst.rc-a msof alpineglaciers, Loso etal.(2004)and Riihimaki(2005)cstiuuu.odthatqua rryingis resp onsibleforrespe ctivelySO-OO%andGG%ofthetotalsed imentprodu ction. Quan- titati veana lysisofglacialerosionalland form morphology overarang<' ofbed rock propertie s (Kruhbend.u uandGlasser,2( 11),however,shows thatbedrock hardn ess andjointdensitydictatewhet her abrasion or quarryingis t.h« dom inan t erosional mechanism , even underuniformglacialconditions.Thus,forsoft rock withlowjoint density,land formsare created prcd ominuntl yhyabrasion.when -asquarr yingdom- inat esforhard rockwith highjoin tdensity.This suggests tha tseekingaunivorsnl quantificationoftherelat ive strengthof abrasionand quarr yingmight1)('anill-posed problem.
:2-1
2.2.3. 2 Empiricaler os io nlaw
Altho ug h process-ori ent ed erosion lawshavebeen appliedto courincntnl-scaloCS1\Is (Hildcs .2001:Hildcs et al.,2(04),thereha sbeennoattcui ptto evaluatewheth ersuch anapp roac hcaneffectively bereplacedby simplescaling relati on shipsbetwe en eros ion ratesandglaciologica lvariables,e.g.slidingvelocity,ice thicknessor drivin g stn 'ss (or somecombina t.ion). Suchempirica lerosionlawshave,however.[H'en ap plied inthecontextof alpineand tidewat er glac iers(Harbor.199:2;Herman aw l Braun. :2008;Kessler etal.,2008;Egholmet al.,2009, 2(12),to glacia l land scap e evolut.ion nuderice shee ts(.Jallliesonet al.,200G,2008,20 10;Wilson ot a!"2(12). and tolong- ter merosion depth est imutiou under the Laur entid elee Sheet(LIS;Hallot20II).
Thisapproac h hasthe ad vantageofrequ iring no theoreticaldescription of suial l-scalc su bg lac ial processes, wh ich minimizes therelia nce onassump t.ionsaw l idoaliza t.iou s . However,by estilllatingerosionfroma singleexpress ion,itisiiupl icitly assu modthat allglac ialeros ionalprocesses are controlled bythe salliegrid-scalevaria ble,t.huslosing theditlerout.iulrelati ve efficiencyofthepro cesses.
IfollowHallet(2011)and Pollard and Deconto(2009)and assumr:tha t glacia lerosion ratesscalewit hthe produ ct of basalshearst ress(assumedequaltothedriving st n'ss) andslidi ngveloc ity(herea fter referr edto asbasalpower ):
(2.7)
This pa rt.icular choiceofscaling variabl eis su ppo rte d byciupiricul ('videll('('(Hulk-t, 20 II)thatallowscstiiua t.iouofthecoefficient.0;,,,,1'(Table:2. 1). Siruilurlv to t.h«
process-ori en tedlaws ,no erosi on occur sunder cold-base d or float ing ic('.
2.2.4 Englacialtran sport
Obser vati ons oferra t ic lithologiesfarfromtheir sourcearea provideunequi vocal ov- idence forthe ability of ice sheets andglac iers totran sp ortdebris(Stru vors etul., 1992;Mah aney,1995;Larson ,200:3,2008;Beuu andEvans,2(10).Dobris-boariu g basalice in icesheetsandice st rea ms hasbeen observed toroach thickIH'ssesof to 20in(Drewry,1986;Cow and ?-leese,1096;Alley ctal.,10D?;Cook<'Ial.,2(11) . representi ng thepot enti alforsignificant deb ris storageand trausp ort.. Such basal layersare com pose dof severaltyp es of facies andarethustheprod uct of variouseu- t.ru iruucntaud mixingmechani sms.However , given the sma ll-scale nature of ('u glacial transport processes and theirlackofla rge-scaletheor etical representations.Iem ploy asimplifiedappro ach inwhich the widerangeof processesislimitedtojustafew.
luthisregar d,Iadaptthetran sportmodel ofHildes(2()()];2( 04).which consist s of representinghorizontal and vertic aldebris transport by variat.ious in thedebri s conce ntru tio n(by volume) ,C=C(;r,y, z):
ilft-
=-V'.ci;-iJ(~\;"t)
+\~"i.,
(2.8)whereVi=(I;(:r,y,z)isthehori zont alicevelocity. Thevert ical icevelocityis assuu u-d equal to the basalmelting rateand isincorpor atedillthenetcutru i u tuc ntZdcpositi ou rate,\~".t=\~,('lr,y)(see Equat ion2.12). \~"i.r=\~Ili.,.(:r,y)is the heuri sticall y defined vert ical mixing rate,also discussedin the sect ions holow,Note tha tCisuot allowed to exceedamaximum valueofC'~IlI.r;any excess concontrn tio n isrcdi sf.rihutcd to adjacentcells.
Thedesired outco meof this cnglaciultransport mod elisto reprodu cet.hnvcrt.ir-al pa ttern ofdebris couccntra t ion obser vedin basalice,rangiug from thedebrisrich
:2G
(:30-90%)stra t ified faciesloca tednear the bed,totheupper.disp ersed(orclot.tcd) faci('s with lowerdebri s conte nt« 10%) (Drewry ,19SG;Knight,1!J!J7;Walkrelul..
2000;CooketaI.,2011).
2.2.4.1 Entrainment
Alt houg h basalicc entra ins unconsolid a ted sediment by sovoraluux-hani sm s,I fol- low Eghohu's(2012)and Hildos'(2001;20()'[) first-ord er assumpt ionthat n'gplati on intru sioninto thesediment isthe domin an t one(dist inctfrom\Veer tll lHnn'g('lation acrossbasal obstacles,seefor exam ple Alley ct.al.1997). Thetheoret ical basisfor this process wasfirst established hyPhilip(19S0),who mod elled iceregelatin gdowninto anarrayofcylinde rs(represe nt ing porousbasal sodimcnt.). Labor at oryoxpt-ri tuouts (Iverson,1993;Iverson andScnuuons,19!J5)la ter confirmedtha tthe intru sionrnt r-,
fittedwellwit h therelat ionproposedhyPhilip(19S0):
where 1\"is theappare ntcond uct ivityof the sediment(ordr-rof 10-1',ur'Pa-Is-I, dep end ent onthesediment porosit y : seeIverson andSemmens19!J5, theirEquat.ions ? and3),andInis theeffect ive thickness of the entra inedsedi mentarray.Equa t ionp.!J) assumestha tthe pore-wa terpressureis equa ltothe grid-scu l« basal water pressu re providedhy the hyd rologymod el.
Inthecase whereu,is greate r(res pect ively,lower )thanthedep osition rate,I" will increase(decrease)untilit reaches anoquilibrium thickn essfor whichon t.raiiu ue n t bulau ccsdoposit.ion .Thedep end enceOIlI" apIH'a rs bocau sc iccmustn'g(>] al e around tho wholearrayofent rai ned ma terialin ordertointrudefurth erintobasal sodiuunt: thickerarrays thusprovid emoreresist an ce to int rus ion(Alloyct al.,1997).However ,
thisis onlythecaseif the array isclast-su ppor ted,that is,there isclose contac t bet weenembe d ded par ticles.Thisloads tocomplica tionswhen dolinim;t.hceffect i,'(' arraydep t h,I,,,hecausomixingprocessesdilutetheontraiucdarrayandena bleIur- t.horregclut.ion intrusion.I"mustthereforemeasu rethethickness of clast -sup port ed materi alrathertha nthe tot althicknessofent ra ine d deb ris.Asa t isfyingdofiuit.ion for thiswasprop osedbyHildos (200I)byint rod ucingan arraydepthmodifier:
I",,,,k:)=0.5{t anh[20(C (;;)-C,:ri/)]+I} (2.10)
Thiscflect.ivclyreprese ntsa smoothed unitstep function.andassigns lar geweightsto concent ra t ionaboveC,:ri/'andvice-ver sa.Theoffoctivearraydepthist.hon cak-ulntcd
I"=
L
Il/wd(kJf:,;;(kJk,
(2. 11)
wherek,isthe vert icalgridcell identifier, andf:,;;isthegridcell t.hickuoss.Thonet cntrai utucntj'dcpositio n rate,\!"d'cannowbedefinedas:
(2.12)
whe re the secondright-ha ndside termisthedepositi on rate,for whichitis ussu uu«!
tha tdepositedsedime nt instanta neouslyacquires apor osity9',andwhen'C(;;=0) isthedebri s conce ntra t ion byvolu meatthe icc/bedinterfu co.Conve rsely,fn 'shly ent rain eddebrisisassu medto have aconcent rat ionof 1-9'.
As shownin Figure2.2,thebasalhyd rologymod elpredi ctsonlyasmal lext ent. highl y pressurizedsu bglac ia lwat er ,which in turnyieldslarg(~areas ofunreali st.icout.raiumout
28
rates(up to[)orders of magnitudeabovethebasalmeltingrate),Thispromptedt.h«
add it ionof twoad hoe sed ime nt mod elparametersdesigm'dto wgu la t ('cut.rai m uc ut rates:v~,,,.,.imposesan upperbound on theentra iiuuc nt rate aIHI itsvulue-; an 'ch(JS('n bet ween theaverageand themaxinnnu of
i:
assum ing thatca p pingv,.at. thislovol will producemorerealisti cnet entrainment rat es,1~1(I."istheupperbound for the dfect ivearray dep th abo ve whichno ent ra inme nt is allowed. Thelnttorisjustifi ed from thethe oryofregela t ionent ra in me nt.which predi ct snocntru in mc ntafte r an eq u ilibri umarray thickn essisreach od.2.2.4 .2 Verticalmix ing
Fold ingof basalicehasbeeninferredfrom ice cores and obser vedin outlotand alpin«
glac iers(G ow and i\11'ese ,1996;Knight,1997;Waller ctaI.,2000;Cook<'Ial..2( 11).
andis argue dtobe theiuostefficientmixin gmechani sm (Alleyotal.,I(97),Gy assu m ing tha tfoldin g eventsoccur onarandombasiswheniceencoun tersbedrock obs ta cles,ver t ica l mixin gha sbeenideali zed as edd y-d iffusion(Alleyand i\lacAyml.
El9')). Based011sim plescaling argruucnt.,the diffusion cocllic icnt..D.is est illla lPdas throeorfourorde rsof magni tud elower tha n that forthonualdiffusion. \\,()adopt th isap proachandexp ress thever t ica l mixin gterm ofEq ua t ion(2.8)by :
\I"''',.-
- ~
D:::(D~)
D::: (2. 1:1)Bas ed on thedescriptionabove,it isheuristicall y conce ivub lotha tDsho u ldcOiTela t (·
wit h the slid ingspe ed,and that mixing sho uld bemor epronoun cednca rt.h«lxx].
kadingto the followin gparameteri zati on(fromHildos20ll!):
D=D*
(~)
exp (-;:)exp (__il" Cd)
u..., z* h;ct! (2.1.1)
where
ir
an d are scaling fa ctors.an di\is a typ irnlslid ingveloc ity(100IIIvr").The red uc ti onofdiffusionwith increa sing sed ime n t thickn ess asxu mox t.hatt.heroisno slid ingatthe ice-sedimentinterfa ce ,or thatitdoesnot.produceconsid oruhlcfolding.
2.2.5 Transport bysu bg laci a l deformation
Itis1I0Wgene ra llyaccep te d that su bg la ciulsedime nt deformsandenhan{'('sice dis- chargeun d er relative lylow bnsalshourstresses (Alley, 1991: Murray.199 7:Allov.
2000;Eva nsctal.,200G) .Itisalso sugg es t ed that thisprocesskadstolarg('-scat<' ice sheet instabilit ies (Cla rk,199-l;Clar kct al.,l!JD9).alth ou ght.hisno t.ionha s g(>I1- crate d debat e (Boult.onct aI.,20(H;Piot ro wskict a!..20(H,2( 02). Thefocusorthis pap erbeing glacia leros io n andsed imenttransport ,Iincludea soft-boddoformnt.io n compo ne nt notfor itsimpact on icedynamics (which wouldrequire2- wa y co u p ling betw een the i\IU N 3- D GSi\[andthesedi ment.mod el),hut. ratherforit.s s{'d inH>nt trans po rt pot enti al.In d eed,inferredandmo delledtransp ortdistuncohythisprocess rangesfro m tento seve ral hund red kilometres (Bou lto n.199Gb,a:Clar kandPollard.
1995:Larson,2003,200S).
Ca lc u la t io nsofthe deformationflux areba sedona stress -s train relati on sh ipIormass- inovem cut.in landslid es(Iver son,1985) . whichwasfirs t app lied to su bglacial ma!<'rial byJen sonetal.(1995), andlateriucorp ora tc d inseve ralglac ialsyste m model sby (Clar kct.aI.,199G;Licciardi ctaI.,1995 ; Cla rk andPoll ar d.[99S:Tarasovan d Peltier . 200-l: PollardandDecouto,20(9).
* c ...1!..L *
(iJU " "'I)
1/11.,T",."=C+[P"+(p" -p,p),!} :::]tanr.p+(2Do )",,-I1'0 ~
:10
where
T,,, ,"
ist.heshearstress with intho scd imeut.,assu me d equaltothe drivingstl"{'ss.y is thegravita r.iona lacceleru tio n(9.8mS-2),zisthe vertica lcoord ina te im'!'('asillg down wardfrom 0at. theice-sedimen t.iuturfa ce.p"andPu,areI"{'SIH'c!i \,p!ytho (k llsity oftho sed ime ntandofwater(2390 kgm-:\ and1000 kgm"),cis t.hosodiiuontcoho- sion an dgenera lly assum ed equa l tozero(Jensonet al.,1995;CuffevandPa terso n.2( 10),<.pis angleof intern alfricti on of the serlimont.,Dothe Newton ia nr<'l'pl"{'I)('('dp- for ma tio nrate(7.9 x10-7S-l;Pollardand Decon to20(0),II"thorlH'ologyPXIHlIH'llt.
and/1;'t.he Newtonia n referen ce viscosity. Theverti calvelocityprofile,o ..,,(: ).is oh- t.aincd hyintegratingEquati on(2.15).ASI'COlld intc gra t.ion,from:=0 tothohasp of thedcfonuinglayer(also called deforu mt.ioudep th).:= :",yields t.ho dofonunt.ion flux,0.,.Thethick nessofthe deforminglayeris calcu late d from:
z"=(iTt,1 -c-P,tan<.p' )/((p" -Pu,)ytan<.p') (2.IG)
whichis obtaine d from solvingII"",,(Z)=O.Modelled andobse rvedvalues fort.h«
deform ati ondep thran gefromafew ccnt.imctrcsto10 m(Alley .l!J!JI;JPllSOlIPt.al., 1!J!J5;Clarkand Pollard,1!J08;Alley,2000;Boultonct al.200!;Cu tfoyand Paterson. 2(10),
Thefollowingassum ptio nsare usedtoderi vet.hedeformin gflux cxprossio u:i)spd i- mont isthawed andsatu rated throughoutitsdepthwhoncverthoha St)isn)('lting.ii) thereisllO ver t ica lva ria t ion illwa ter pressur ewithin the sed imentand.iii)t.hc shea r stressimpa rtedbytheiceis indepe ude ntofz.Sed ime nt. t.hicku css ist.lu-nohtainod bysolving the conti nuityequat.ion:
(2. 17)
;\1
wherej;;istheerosionrateeit her fromtheprocess-ori entedor t.h«empirical erosion law.
As glacioH1lvialrewor kingof till doesnot transpo rtsignificantsedime nt.volumeover long,grid-scale distances,thisprocessisnotincludedinthis thesis.TIl<' calcu lat<'d sediment thicknessthusrepresents generalglac igenic deposit sand doesnotdist.iu- guishosbet weenprim aryandsecondary tills.
2.2 .6 Icedyn amics
The versionof theI\IUNGSI\I usedhereinclud es a3-Dt.licnuo-uior-haui rullvcou- pled sha llowicemodel,pa rametri zed climateforcing.asynch ro nously coupledsm- fa cedrainage solver,visco-elast icbedrockresponse.positivedegr ee-daysurfa romass- bal ance(withtempera t uredep end entdegree-d aymeltcoefficients) .anddotaik-dpa- nu ucteriznt ionsforbot.hmarineandlacust rineice calvingand iceshelves.Themod el hasbeen calibra tedagainst alarge set ofobserva tiona lgeophysica lcons t ruiuts.in- elud ingrela tivesea-leveldataand present-dayratesofsurfaceuplift. I\Iaril1<' limit andst rand-line datahave also beenusedtofurtherconstra inenscnih lomodelresul ts . Over successive calibrations theinitial Eeruiantopograp hyhasbeen adj usted tominimizediscrepancybetweenprosont-d aymod elled(i.e. afterafull glacialcycle)andobse rvedsolid-eart htopography.Alth ough a calihra te densemble ofdcglacia!chronologiesis availa ble,onlya sing le,highprobabilitysPlofGSI\1pa- ramctc rsisusedinth is study(nmID11Il1027fromcalibrations<'t NGb t inTarnsov PI al.2(12).Det ailsOilthe l\IUN3DGSMand its calibra t ionareprovidedin Tarnsov et al.(2012) and refer en cestherein.
2.2.7 Basalhydr ol ogy
Anew ad d itiontothe1\IUN3-D G81\[ is abasal hydrol ogymod ule(Ka vnna gh.:201:2).
Following the model of Flowers (Flowers,20(3),thebasalwate rpH'SSUH' is cal<'n lall'd fromanon -linearrelat ionship withtheeffecti ve subg lac ialwater thickn ess .hII':
( h"' )}
P",=Pi
hi
(:2.18)whereh;'isthe saturat ion thicknessof the sedim ent.Ideally,thiswouldbecak-u latod fromthe sed imentmod el;however, onlypassive couplingis currc ut.lyturuo.lOil[){'- t.wocn it andthehydrology solver.h;' isthus treatedas a sedime nt.mod elparanu-ter acting as a sim plecontrolonthe magnitu de ofwaterpressur e.
Theoffoctivewate r thicknessis calculated fromthe continuity equa t ion:
~
=- v ·
(2",+5", (:2.[~))where5",isthebasal massbalance(including aprescribed drainage10the nquifer).
Q",is thehorizon t al waterflux rcpruso ntinj;drainage thro ug hcavitynot.worksandis calculatcd fromDar cy 'slaw:
(:2.:20)
whew1\11'isthehyd rauli c cond uct iv ityofthe sed ime nt(par.u uet.orizodas afuncti on of h", )and[I",is thewater density.
The solveralsoinclu desa down-grad ient solver,adapted fromt.h«l\lUN:l-D C8l\['s surfacewater draiu agc com po nent, torepresenttu nne lforma tio n. Agrid-cellis
flagged as conta ining tunnelsif one of itsinterfa cefluxesrea ches a spec ificthresh old . Thewa teris the n re-di stributedbyfollowin gthe culcu luto dhydr auli chmdgrad i- ent..Figu re 2.2shows thebasal wate r pressurenormal ized aga inst t.heiceoverburden pressu re at La stClacialMaximum (LG r.I),
nonnalized waterpre s sur e
35"N--{~~~~~_I,
1700W 1500 W I30aW 1100W 900 W 700W 500W
Figure2,2: Ra ti o of basalwater pressure and ice overb urd en pressureatLGi\1.Con- tomsshow icet.hickn ossin metre s (from runlD nn1027,culibra t.ionsetNfihtin Ta rusov et al.2(12),
Table2.1: Sedime nt mod el para mete rs.Bracket ed symbo lsrepresentstheabbreviationsusedin Figur e2.12. Therange columnshowslower. base.and uppervalues inorder.
Sym bol De s cripti on Range
N; (Nz) Numberofverticallayersin [11,15,19]
the englacialcomponent
Justific ation
Corres pondsto amodel domainthickn ess of10.7.15.7.
and 21.4m,resp ectivel y,and doesnotaffect therelativ e gridspac ing.
C;rnp (Ce) Scaling consta nt bet ween [1.5,8,14] x10-11 FromHallet(2011).
basalpoweranderosionra t e Pa-I
HV *(HV) Vickerbedrockhardness [1.5, 4.7]x109Pa Rangefromhardn ess ofsediment a ry to met am orphic rocks(Hildes,2001).
1/*(mu) Rock-rock friction coeffi- [0.3,0.8.5,10]
cient
Bas ed onthecom pila t ionof Byerlcc (1978 ).whosug- gest 0.85fornormalpressur eabove5 :---IP a,but point s out tha t adependence onsurfaceroughnessatlower pressur es can inducevariat ionsin1/*.
Table2.1-Conti nued
Sym bo l Des cr ip t ion Ran ge Jus tifi cation
R;nean (R) l\lea nof the grain-size dis- [1O-~. 10-3• 10-2] Corres po nd to 2.3to -4.3on thed>-scaleand falls wit hin
hj(hf) tribution
\Vater film thickness
the observe d ran geof mean gra insize in basalicefacies (Boulton,1978,1979:Haldorsen.1981:Drewry ,1986:
HubbardandSha rp,1995: Cooket al.,2011).
[10-8,10-°.10-~]m From Drewry(1986):Hub ba rd andNienow(1997)and
B;(By) Temperature-dependent
Hallet(1979a).
[24,38,55]x10-25 From Cuffoyand Paterson(2010) Glen'sflowlaw coefficient s-lpa-3
C;lIa r(Cq) Qua rry ingcoefficient [10-11,10-10.10-9] Asafirst orderguess.I usethe same range astheabra -
Pa-1 sion factorkabr/HV'.
Ta ble 2.1-Continued
Sym bo l Descrip t io n
(*(zet) Basal roughness
Range
[0.01,0. 1,1]
Justifi cati on
Asrochesmou t onn eesaretheremnant s of quarr y- ing events,Iassume thattheyrepr esentthe relevan t scaleonwhich quarr yin g ope rates.Thebase value is thusobtained fromtheirobserve daverage dimen sions
n;,(np) Quarr yin g law exponent [0.3,0.5.1]
controllingtheinfluence of basal water pressure
(K rabbendamand Glasser,2011),italsocorres po ndsto thevalue usedbyWeertmanin his theo ret icalana lysis of basalsliding(Weertman,1957).
Theoreti calconside ra t ions(Cufiey andPaterson. 2010) suggest a valueless tha n1.asthepresenceofacrit ical water pressur elimitstheslidingvelocityto a globa lly- controlledvaluetha t sho uld neu tr ali zefur ther cavity growt hevenwit h increasin gwater pressur es.
