Bulletin ofGlaciologicalResearch22（2005）7579

二：■；⑥JapaneseSocietyofSnowandIce

Numericalsimulationofglaciaトva＝eycross−SeCtionevolution
HakimeSEDDIKl．shinSUGIYAMA2andRenjiNARUSEl

lInstituteofLowTemperatureScience，HokkaidoUniversity，SapporoO600819Japan
2LaboratoryofHydraulics，Hydrology，andGlaciology，ETH，ZtirichCH−8092Switzerland（Received October4，2004；Revised manuscriptreceived October27，2004）

Abstract
A numericalmodelfor glacial−Valley cross−SeCtion evolution has been developed．The modelallows the simulation of the development of U−Shaped valleys by coupling aniceflow modelin atransverse section with an erosion model．The core of the cross−SeCtion development modelis theCalculationofthetwo−dimensionalflowspeedfieldinatransversecross−SeCtionconsideringthelateraldragfromglaciersidewallsandthebasal−StreSSdependentslidingspeed．Assumingthattheglacial
erosionis aquadratic functionoftheslidingspeed，the modelshowsrapidevolutionofaV−Shaped

profi1eintoarecognizableglacialformwithslidingvelocitiesrangingfrom3ma1to8mal．

nalflow pattern computation andits coupling withthe subglacialerosion model．The developed valley
evolution modelwas tested byinvestigatingtheinfl−
uence of thelateralshear stress component on the1．lntroduction
The U−Shaped valleyis one of the most well−known products of alplne glaciation，yet relatively
littleisknown howiceflowandglacialerosioninter−actinthedevelopmentofthischaracteristicformand

formationofaU−Shaped valley．

piricalstudieshaveconfirmedthegeneralnotionthat
manyglaciatedvalleyshaveapproximatelyparabolic

2J．ダわ抑〝70deJ

dependingonicevelocityandeffectivepressurehave

direction is

Whyglacialerosionshouldproducesuchaform．Em−

2．Methods
TheproblemtobesoIvedisthe velocityfieldina（U−Shaped）crosssections（Graf，1970；Doornkampand
transverse cross−SeCtion of a glacier with a uniform
King，1971；Girard，1976；Aniya and Welch，1981）．To geometryalongtheglacier．TheCartesiancoordinate
investigate the formation processes of U−Shaped val−
SyStemis taken with the x−aXis along the glacier，y1eys，iceflowattheglacierbedwasstudied（Johnson， acrosstheglacier，andzperpendiculartothex−yplane1970）and quantitativeabrasion and plucking models（Fig．1）．Momentum balance of theicein the x−
been proposed（Boulton，1974；Hallet，1979；Robertsand Rood，1984）．Moreover，COuPling oficeflow anderosion models allowed better understanding of theevolution of cross−SeCtional（Harbor，1990，1992，1995）
andlongitudinalprofiles（MacGregoretal．，2000）．

With his glacialcross−SeCtion evolution modeland by assuming a quadratic function of the sliding

Velocityforanerosionlaw，Harborsuccessfullysimu−

1ated a proper erosion pattern（centralminimumin
thebasalslidingvelocityatthevaueycenter）forthe

賢＋豊＝βg石（1）∂S

WhereTxyandTrzaretheshearstresses，Pisthedensity

Ofice，glS the acceleration of gravity，and Sis theSurfaceelevation．Glentiverelation，SOthat

＝恥，

∂〟

∂z

sfiowlawisusedasaconstitu−＝2ダちz．

（2）

U−Shapedchanneldevelopment．However，Harborpro− The termFisthefluidity，defined as

Videdlittleinformation on the computation of thebasalshear stress，Whichis required for successful
erosion modeling．Particularly the stress conditions
required for the development of a U−Shaped valley
have not been described．

This paper presents the details of two−dimensio−

ダ＝A（かて呂F・（3）Here，Teistheeffectivestress，andtheratefactorAand
theflow−1awexponentnarematerialparameters．We

usedthecommonvaluesofn＝3andA＝214MPa3at

BulletinofGlaciologlCalResearch76

bedrock surfaceis calculated asg＝C〟言，（7）

where Cis an erosion constant equaltolO4a ml（Harbor，1992；MacGregoretal．，2000）．Equation（7）isusedherebecauseitrepresentsthegeneralformoftheabrasionlawproposedbyHallet（1979）．
ヱ、一？．l （J／／り・（−〜・（−／J〟MJ川M（か／As theinitialglacier and valley geometries，Weprescribed a V−Shaped cross section with maximum

ice thickness of480m，Surface width of1200m，and
downglacierslopeof40（Fig．2a）．TosoIveEquation（1）for theflow speed within this cross section，a tWO−
dimensiona135×35 finite−difference grid was em−

Ployed．Asetoffinite−differenceequationswassoIved
Fig．1．Initialglacier shape and the coordinatesystem

usedin the simulation．

With the LU factorization method assuming that thefluidity was constant and that the sliding speed wasZerO．Then，the computed velocityfield was used toSOlve Equation（3）for thefluidity with a Newton−

maticalsingularity caused by aninfinite viscosity

Raphsonschemesothatthenewvaluesofthefluidity
WereuSedinthenextiterationstep．Thevelocityfield
WaS also used to compute the stressfield withEqua−

COnSistsofvanishingsheartraction，SO thatmodeltoinvestigatethetemporalevolutionofglacial（Paterson，1994）．T（）WaSintroducedtoavoidthemathe−tion（2）tointroduce sliding speedin the next step
whenstressesapproachzero（Blatter，1995）andavalue
OfTo＝0．5KPahasbeenchosensmallenoughsothatit usingEquation（5）．Thecomputationwasiterated un−tilthevelocityfieldconvergeswithin2×104mal．
doesnotaffectthecomputationalresult．
The flow model was coupled with the erosion
Atthefreesurface（z＝S），theboundarycondition
Ts＝れ∫丁て，，＋乃1・ち、1＋乃zTtz＝0．

Valleys．For thefirst timestep，theaboveprocedures（4）

At the basalsurface，the form of the boundary
COnditions depends on thelocalconditions that are

Were uSed to calculate aflow pattern for theinitial

V−Shapedvalley．Calculatedbasalslidingspeedswere
Substitutedin Equation（7）tocomputethepatternof
COnSidered．lnthepresentworkbasalslidinglSintro− erosion rate across the profile．Then themodelcalcu−

duced by relating the sliding speed ub tO the shear

lates new coordinates for the glacier and the valley

1968，1979），

Theicesurfaceelevationwascalculatedinawaythat
thetotalicemasswaskeptconstantduringthesimu−

StreSSaCtingonthebedT／，（Weertman，1964；Lliboutry， CrOSS−SeCtion，Whichwereusedforthenexttimestep．
〟ゎ＝Cポ，（5）

mass was allowed toincrease during the simulation，
due to theparticularhigherosionratesatthevalley

withTゎ＝乃ユて−、1＋乃zちz，

1ation．However，for the simple model，the totalice（6）where c＝50m a1（MPa）1，is the sliding coef汽cientwhichisconstantacrosstheglacierbed，and hasbeen
Chosentoobtainvaluesforslidingvelocitythatallow

forglacialerosion．Weassumen′＝1．AtthemarglnS，
the月owspeedonthesurfaceisconstrainedtobezero．
2．2．EγOSgO乃〝70（ブeJCenter（Fig．2c，d）．Theseprocedureswererepeatedfor

aglVennumberoftimesteps．

3．Results and discussion
ThepresentstudyaimedtoshowtheimportanceOf thelateraldrag（Try）includedin Equation（1）andEquation（6），reSpeCtively．Therefore to highlightits
influence on the formation of U−Shaped valleys，WeGlacialerosion consists of severalprocesses，in−cludingabrasion，Plucking，Subglacialfluvialerosion， first ran a simple model，Which neglects thelateraldrag．Omittingthefirstterm，integrationofEquationand chemicaldissolution by subglacialwater．The（1）gives thebasalshearstressas
highly complex natureofthesemechanisms haspre−

cluded the developmentofphysically completemod−
els forthese processes，and only afew emplricalandtheoreticalanalyses have providedindication of theprlmary COntrOIs on rates of glacialerosion．In thesimulation described here，erOSion rate normaltotheTゎ＝βg好・（8）

WhereHis theice thickness．As shownin Figure2b，

the pattern of the sliding velocity for theinitialV−Seddik etal，

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400600800 1000

1 0

ValtoycrosむSOC血n（m）

400

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800 1000

Valloy￠r08ふ￡OCdon（m）

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皇倉10竜＞
晋⊇
S

200

400

600

800

Va川oyc10SS−S創出on（m）

1000 1200

Fig．2．（a）（c）InitialV−Shaped valleyanderoded va11eyobtainedwiththesimplified model，Velocitycontours
in m al．（b）（d）Corresponding cross−glacier variation of sliding velocities（fullline）and erosion rates（dottedline）．

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﹈
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400
600
800
Va川8yCrOSS−SOCぬn（m）

200

1000 1200

400

600

800

VallQyCrO創トSOC的n（m）

1000 1

Fig・3・（a）InitialV−Shaped va11ey cross−SeCtion，Velocity contoursin m a L．（b）Corresponding cross−glacier variation
Ofslidingvelocities（fullline）anderosionrates（dottedline）．

Shapedprofileisrepresentedbyarapidincreaseinthe

Sliding speed toward the center of the valley and adecreasetowardthemargins．Becauseofthenatureof

the erosionlaw，erOSion values follow the same pat−tern（Figs．2b and2d）．This pattern of erosion wasapplied to theinitialV−Shaped valley，and Figure2c

Showstheglacialvalleyerodedduringthesimulation

for50ka．Asobserved，thevalleyshaperemainssirni，
1ar to a V−Shaped profi1e，Characterized by a deep

Channelatthevalleycenter，Withoutthedevelopment
OfaUTShapedprofi1e．
The results of the simplified modelwere com−Bulletin ofGlaciologlCalResearch∴ヾ87︵Td∈︶
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．

200

400

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Valloycross−SO￠don（m）

1000 1200∧J200400

800

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Valloycro∬−8∝加n（m）

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ValloyαOSかSOCtbn（m）

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200400600800

Valloycros＄−SOCポon（m）

1000 1

Fig．4．（a）（c）（e）Modelresultsforasimulation ofglacial−Valley development．Glacial−Valley cross−SeCtion atthreedifferent time−StepS，Velocity contoursin m al．（b）（d）（f）Corresponding cross−glacier variation of sliding
velocities（fullline）anderosion rates（dottedline）．

drag．Figures 3a and b show theinitialV−Shaped

representedinFigure3b．Thisispreciselytheerosion
pattern required to convert a V−Shaped valley to a

imaatthecenterofthecrossprofi1e．Theincreaseof
drag associated with the constricted form at thecen−tion．InclusionofbothshearstresscomponentsTryand
TrzWaSneCeSSarytOObtaintheslidingvelocitypatternpared with thoseof the modelwhichincludeslateral

Valley，andthecomputedslidingvelocityanderosion
rate across the valley．Characteristic features of thesliding velocity and the erosion rate are increases

U−Shapedchannel，Whichdonotappearinthesimpli−
fied model（Fig．2b）．This comparison shows clearly
theimportanceofthedragfromthesidewalls（Txy）for
towardtheinterioroftheglacier，but withlocalmin− the modeling of glacialerosionin a va11ey−CrOSS SeC−

ter of the V−Shaped valley reduces velocities there，

resultingin a centralminimum for the erosion as

required forU−Shaped valleyformation．Figures4a，C

andeshowtheevolutionofacross−SeCtionvalleyforSeddik etal．

7夕

threedifferenttime−StepS．Themodelpredictstheevo−
1utionoftheV−Shaped pronleintoarecognizablegla−

Geography，Universit6Louis−Pasteur for his helpful

to8ma1（Figs．4b，d，f）andforthelasttime−Step，theReferences

cialformwithslidingvelocitiesranglngfrom3ma1

model could simulate the formation of a deep andcomments andinterestin this work．

Anya，M．and Welch，R．（1981）：Morphologicalanalyses ofglacialva11eys and estimates of sediment thickness ontimescaleissensitivelydependentonthevalueofthe
theva11eyfloor：VictoriaValleysystem，Antarctica．The
erosion constantin the erosionlaw，the modelsug−
Antarctic Record，No．71，7695．
Blatter，H．（1995）：Velocity and stressfieldsin grounded gla−
geststhataglacialvalleycanbedevelopedafter50ka
Ciers：a Simple algorithm forincluding deviatoric stress
Or during a single glaciation under the condition ofgradients．］．Glaciol．，41（138），333−344．
realistic sliding speed．This observation agrees withBoulton，G．S．（1974）：Processesandpatternsofglacialerosio11．
previous results from Harbor（1992）．As the valleyis
Coates D．R．，GlacialGeomorphology．Binghamton，New−
York，State University ofNew−York，4187，
progressively transformedinto a U−Shaped form，theDoornkamp，］．C．and King，C．A．M．（1971）：NumericalAnaly−
Centralminimain velocity and erosion tend to be
Sisin Geomorphology．London，England，Edward Ar−
removedanddisappearatthelasttimeTSteP（Fig．4f）．
nold，608p．
Girard，W．W．（1976）：Size，Shape and symmetry ofthe cross−profilesofglacialvalleys Ph．D．thesis．IowaCity，Iowa，4．Conclusion
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ThepresentstudydescribedthedevelopmentofaCrOSS−SeCtion．ArticandAlpineResearch，2，303−312．
Hallet，B．（1979）：A theorethicalmodelofglacialabrasion．］．
two−dimensionalmodel，Which has been successfully
Glaciolリ23（89），3950．
used forthe simulationofglacial−Valleyevolution．A
Harbor，）．M，（1990）：Numericalmodelingofthedevelopment
realisticU−Shapedva11eywasobtainedandtheimpor−
Of glacial−Valley cross sections［Ph．D dissert］．Seattle，tance oflateraldragln the development of glacial−
Washington，University ofWashington，219p．
Harbor，］．M．（1992）：Numericalmodelingofthedevelopment
Valleyswasshownbythenumericalexperiments．To
Of U−Shaped va11eys by glacialerosion．GeologlCalSoci−improvethemodelingofglacialvalleyformation，itis
etyofAmericaBulletin，104，13641375．
important to know accurate values oficeflow andHarbor，J．M．（1995）：Development of glacial−Valley cross−erosion parameters．Comparison of modeling results
SeCtionsunderconditionsofspatiallyvariableresistancetoerosion．Geomorphology，14（2），99107，
Withfield data of valley formation willprovide anJohnson，A．（1970）：PhysicalProcessesin Geology，San FraIl−
OPpOrtunitytoconstrain thoseparameters．
Cisco，California，Freeman，Cooper＆Co，577p．
Lliboutry，L．（1968）：Generaltheory of subglacialcavitation
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Acknowledgments
Lliboutry，L．（1979）：Localfrictionlawsforglaciers：aCriticalreviewand newopenings．J．Glaciol．，23（89），67−95．
We wouldlike to acknowledge Martin Ltlthifor
MacGregor，K．R．，Anderson，R．S．，Anderson，S．P，and Wa（
his valuable advices and comments that helped todington，ED．（2000）：Numericalsimulations of glacial−
Valleylongitudinalprofileevolution．Geology，28，No．11，
improve the manuscript substantially．We are also

Well−defined U−Shaped valley（Fig．4e）．Although theVery gratefulto Ralf Greve of theInstitute of Low
Temperature Science，Hokkaido University for hisSuppOrtin theiceflow modeldevelopment and hisguidanceforthemathematicalwork．WethankHeinz
Blatter of the Institute of Atmospheric and Climate

Science，ETH forsolidguidanceonglaciermodeling．

We also thankJean−Luc Mercier of the Faculty of1031−1034．

Paterson，W．S．B．（1994）：ThePhysicsofGlaciers．Tarrytown，
NewYork，PergamonandEIsevierScience，480p．
Roberts，M．C．and Rood，K．M．（1984）：The role ofice contril）−
utingareainthemorphologyoftransversefjords，Briト
ishColumbia．Geografiska Annaler，66A，381393．
Weertman，J．（1964）：Thetheoryofglaciersliding．］．Glaciolリ5（39），287−303．