Use of a new finite element continuity model to ... - Wiley Online Library

JOURNAL

OF GEOPHYSICAL

RESEARCH,

VOL.

92, NO.

B9, PAGES

8941-8949, AUGUST

10, 1987

Use of a New Finite Element Continuity Model to Study the TransientBehavior of Ice StreamC and Causesof Its PresentLow Velocity J. L. FASTOOK Inst•?ute/brQuaternaryStud•?s,Um•;ers•?y ofMaybe,Orono, Maybe The finite elementtechniqueis usedto solvethe continuity equationfor two modelingexperiments which investigatethe responseof Ice StreamC to changesin the boundaryconditions.By comparisonof the resultsof theseexperimentsit ispossibleto delineatethe mechanismresponsiblefor the anomalously low observedvelocity.Theseexperimentsare ( 1) the suddencaptureof a major portion of Ice StreamC's catchmentareaby neighboringIce StreamB and (2) the suddenremovalof a slidingbed conditionalong a major portion of Ice StreamC. The resultsof theseexperimentsfavorthe secondscenariooverthe first asbeing responsiblefor the presentstateof Ice StreamC and suggestthat the slidingconditionon Ice StreamC disappearedapproximately2000 yearsago. INTRODUCTION

the grounding line, all as a function of time. Details are Ice Stream C on the 8iple Coast of Antarctica presents presentedin the next section. The experimentsassumepresent-dayboundaryconditions somewhatof a p to glaciologists. While to a great extent it (i.e., currentaccumulation,catchmentarea,bedrocktopography displaysthe characteristics of a major ice stream (crevassed surface,concaveprofile, low surfaceslope,etc.), it lacksone [Drewry, 1983]) and modelsa flow band which is in balance with its catchment area. A reasonable estimate of the accumumajorcharacteristic of ice streams,velocity.While Ice Stream lation rate over the catchment area of Ice Stream C leads to a C's neighborsare all movingroughly 1 km/yr, a value that talliesroughlywith estimates oftheirbalancevelocities [S/•abtaœebalancevelocity of between400 and 600 m/yr. A fit to the eta/., 1986], Ice StreamC ismovinglessthan 50 m/yr, a value presentsurfaceis obtained by assuminga region of basal that in no way reflectsthe largecatchmentareathat is drained decouplingthat extendsnearly 1000 km into the interior of WestAntarctica.This regionof decoupling,which couldalso by this ice stream. be modeledby assumingsofterice at the bed where sliding Further, in the ice shelf thickness data from the RISP project [BentleyandJezek,1981], one notesthat Ice StreamC occurs,presumablyis due to somedeformablematerialat the eta/. [ 1986]on Ice displaysa pronouncedicetongue,similarin character,if not in bed similarto that observedby B/ankensht• Stream B. This configuration is used as the initial condition magnitude to those displayedby its neighboring active ice for both of the time-dependent experiments, described briefly streams.Indeed, the ice tongueassociated with Ice StreamC extendsasa recognizablefeaturealmostt•vothirds of the way here and more extensivelyin the resultssection. The first experimentconsists of deprivingIce StreamC of a to the ice front. Near the groundingline of Ice StreamC this major portion of its upstream catchment area,aswould be the tongue is lesspronounced,and the ridge of this tongue is a case if Ice Stream B were to expand its own catchment area. muchnarrowerandelongatedfeaturethen the ridgesassociated with currentlyvigorousice streams.This is an indicationthat Reducing the catchment area reducesthe balance velocity, andthishasbeensuggested asa possiblemechanismexplaining Ice Stream C was once an active ice stream similar to Ice Ice Stream C's low outlet velocity. The time-dependentresponse StreamsA and B, but that at somepoint in the pastitsvelocity of the system shows the velocity in the downstreamregion decreaseddramatically.As this remnant ice tongue is carried alongby the generalmotionof the RossIce Shelf,the ridge is decreasinggraduallyto the new equilibrium value dictatedby elongatedand smoothed.Velocity estimateson the RossIce the newsmallercatchmentarea.At no point in the simulation Shelf along the trajectoryfollowedby ice dischargedby Ice dothe velocitiesin the downstreamregiondrop to the observed Stream C range from about 500 m/yr in the region of the remnantice tonguedownto a fewtensof metersper year near the presentgroundingline. Assumingan averagevelocity of 250 m/yr and allowingthat the remnantridge startsabout500 km from the presentgroundingline, a roughestimateof when Ice StreamC becameinactivewould be about2000 yearsago. With this in mind, two modeling experiments can be performedusing a time-dependentfinite element flow band

values.

The secondexperimentconsists of removingthe deformable bed which is necessaryto obtain the low surfaceslopesand high velocitiescharacteristicof an ice stream. The velocity decreases dramaticallyto a valuefar belowthe balancevelocity for the unchangedcatchmentarea. One must note of course that once Ice Stream C returnsto equilibrium, its discharge velocitywill againattainvaluesin agreementwith its balance velocity.With the new bed conditions(the deformablelayer model. This model differs from the traditional stress-strain finite element model in that it utilizes a column average now removed)the ice streamwill need a much greatersurface estimateof the velocity field to obtain ice thicknessesand slope to drive the higher velocities. These higher surface will onlycomeaboutasthe icestreamthickens.Generally velocitiesalong the flow band from the continuity equation. slopes low accumulation ratesdictatethat this will be a slowprocess Output of the modelincludessurfaceelevationprofiles,velocity and that during this processwe can expect to seevery low distributions, basalstressdistributions, and the massflux across dischargevelocitiesalongIce StreamC. Both of theseexperimentsare performedusing Heaviside Copyright 1987 by the American GeophysiscalUnion. stepfunctionsfor the changingboundary conditions.Use of stepfunctionsin the changingboundary conditionstendsto Paper number 6B6112. 0148-0227/87/006B-6112505.00 exaggeratethe initial responseof the systemsincethe systemis 8941

8942

FASTOOK: USE OF A FINITE

ELEMENT CONTINUITY

MOOEL

immediatelyfar out of balance.Suddenchangesin boundary wherethe constantof proportionalityobtainedby combining conditionscan also lead to spuriousspikesin the first few equations(2), (3), (4), and (5) is given by time-dependentvelocityor basalstressprofiles,sincetheseare m derived quantitiesinvolving numerical derivatives.A more gentle ramp or sinusoidalvariation of boundaryconditions wouldallowa certainamountof relaxationduringthe transition periodwhichwouldlessen the impactofthe changingboundary n conditions.Programmingconsiderations aswell asuncertainties aboutthe durationofthe transitionperioddictatedthe useof a + (n+ 2• } (7) step function.

kx - -w{f[Pg

THE FINITE-ELEMENT

FORMULATION

The finite element method is par ticularly powerful in solvingellipti nonhomogeneous differentialequationssuchas the continuityequations[Becker eta/., 1981]. Most continuity equationscanbe formulatedin sucha way that the time rateof changeof the statevariableisrelatedto the spatialvariationof a flux like (i.e., conserved)variable and to sourcesor sinksof this flux like variable.

The state variable

can assume several

one then obtainsthe constitutiveequationfor the flow band model in terms of the particular choiceof flow and sliding laws.Different choicesof flow and/or slidingrelationswould yield differentformsforkoxbut would not materiallyaffectthe followingdiscussion. Indeed one of the majorstrengthsof the finite elementmethodisthe easewith which onecanvary the form of the constitutiveequation. These approximations,of course,do not apply in the more complexstressfield near an actualgroundingline. However, theseapproximationsare sufficientlyaccuratefor the present modelandindeedin theabsence ofsignificant creepdeformation are actually quite accurate. The continuity equationbecomes

differentinterpretations,dependingon the type of problem, but in every casethis statevariable shouldbe relatableto the flux by someconstitutiveequationwhere the flux is assumed to be proportionalto the spatialderivativeof the statevariable. Of interestto glaciologists isthe flow bandmodelof iceflow [LangdonandRaymond,1978;Fastook,1985, 1986] where the statevariablecorresponds to the icesurfaceelevation(relatable T to theicethickness throughknowledge ofthebedconfiguration), the flux like variable correspondsto the actual flux of ice throughthe flow band, and the constitutiveequationis some A solutionof this equationexpressedin its "strong"form is form of a flow or slidinglaw relatingice flux to surfaceslope. oftenvery difficult,especiallywhendealingwith the irregular Sources andsinksof fluxcorrespond to accumulation or ablation boundary conditionsso often encounteredglaciology.This can be overcomeby castingthe problem in its "weak" or at the surfaceaswell asmelting or freezingat the bed. variational principleformulation.If a solutionto equation(8) The continuity equationfor this situationcan be written isto be obtainedin someregion0 to L then the appropriately weightedintegralof equation(8) will alsoprovidea solution. •Sh de =a___w (1) This "weak" formulationis given by 6t dx

d -kx - w -

vdx - fLwa- ••]vdx (9) f•d[_kx dh

The flux at a point x is given by

0

(2)

The velocitycanbe written asa combinationof slidingand internal deformation(flow) proportionalto the fractionof the bed which

is considered

to be melted:

- fua + ( -f) uv

(3)

wherev isan appropriateweightingfunction.Due to equation (9)'s"average"natureit ismoreamenableto solutionwith the irregular data and boundary conditions encountered in glaciology.Any solutionof equation (9) will automatically satisfyequation (8). In addition, it can be shownthat in the averagesensethis type of relationis satisfiedfor any arbitrary finite interval

wherethe slidingvelocity[ Weertman, 1964]isgivenby m

(4)

in the domain

0 to L.

The finiteelementapproximation utilizesthisfactby breaking thisintegralinto finitesteps(elements)which aresmallenough sothat kx, w, and a can be approximatedasconstantwithin that interval. Equation (9) on one suchinterval becomes

and the flowvelocity[Glen,1955]is givenby

- kx ! h vdx- wa

$X,.

(n+2) •xx

vdx- w •

X/

6t

hvdx

X/

( 10)

the prime indicatesdifferentiationwith respectto x. (5) where Integratingthe left-handsideof equation(10) by parts

Expressingthe flux in termsof the surfaceslope, dh

a-

(6)

-kxXh'v'dxkxh'v =waXl' vdxw•dX l' (11) l' IX•

FASTOOK: USE OF A FINITE ELEMENT CONTINUITY MODEL

8943

one obtains the symmetric variational formulation (/}, the (16) unknown variable, and v, the weighting function, occur • •x{•} symmetricallywith respectto their derivatives}.This symmetry =[--1 1 ]/Lss{/}}--[B]{/}} will be important in the formulation of the stiffnessand capacitancematricesto be discussed later. Note alsothat the With derivativesexpressedin thisfashionequation( 11) can secondterm on the left-hand sideinvolvessimply the fluxes now be written as {equation{6}) into and out of the interval in question.This feature of the symmetricformulation is very important in simplifyingthe specificationof boundaryconditions.In this model, boundary conditions are a specified thickness at downstreamend of the flow band and a specifiedinput flux at the upper end. In the caseof a flow band extendingfrom the groundingline to the dome,the specifieddownstreamthickness is the flotationthicknessof ice for the appropriatebed depth, and the specifiedupstreaminput flux is zero. a While kx, w, and a are assumedto be constant over the (17) interval Xi