JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 96, NO. C12, PAGES 22,117-22,124,DECEMBER 15, 1991
Retentionof GreenlandRunoff by Refreezing'
Implicationsfor ProjectedFuture Sea Level Change
W. TAD PPEFFER AND MARK F. MEmR
Institutefor Arcticand AlpineResearchand Departmentof GeologicalSciences,Universityof Colorado,Boulder
TISSA H. 1IJJANGASEKARE
Departmentof Civil, Environmental,and ArchitecturalEngineering,Universityof Colorado,Boulder
Meltingproduced
at the surfaceof subfreezing
permeable
tim doesnot freelypercolatebutinsteadrefreezes
near its point of origin, raisingthe tim temperature
and decreasing
the pore space.If sufficientmeltwateris
introduced,
thetim will warmto 0øC,andsubsequent
waterwill fill theremaining
porespacewithoutfreezing.
Only afterthe residualwatercontentof the tim is exceeded
will waterflow throughthetim and possiblyescape
asrunoff.This process
preventssummermelt on highArcticglaciersandice capsfrom escaping
in its entirety,
and it must be accountedfor in modelingglacierrunoff on the basisof surfaceenergybalance.A model is
presented
herewhichdescribes
in a simpleway the transientprocessof infiltration,refreezing,andmnoffin a
futurewarmingclimate.The modelis appliedto Greenland,for whichpredictions
of mnoff-inducedsealevel
risethatdonotconsider
therefreezing
process
couldbe asmuchas5.0 cmtoohighover150years.
INTRODUCWION
Globalsealevel changes
on the timescaleof 1 to 100yearsas
a consequence
of threemainprocesses:
densitychanges
in ocean
waterdrivenby watertemperature
changes;
changesin liquid
waterstorage
on land;andchanges
in waterstorage
asicein glaciers and ice sheets[Meier, 1990]. Changesin glaciervolumes
occuron time scalesof 100 to 10,000yearsby dynamicresponses
of ice massesto environmentalforcing [Johannesson
et al.,
1989], but on shortertime scalesof 1 to 100 years, volume
changesoccurin responseto changesin annualmassbalance
amplitudes[Meier, 1984] independent
of any dynamicadjustment. The time scaleof sealevel-glacierinteractionconsidered
here is the short-term(1 to 100 years)responseof sealevel to
changesin glaciermassbalance.
Changes
in glaciermassbalancecreatea net gainor lossof
waterfrom the worldoceanovera specificperiodof time accordingto whethergloballandicemassexperiences
a netlossor gain
overthe sameperiod.Individualglaciersandice sheetsaddto or
subtractfrom world oceanvolume on an annualbasis according
oceanasrunoff.Suchan approachwasusedin at leastonemodel
of future sea level change[EnvironmentalProtectionAgency,
1983]; the error and uncertaintydue to the neglectof refreezing
were notedin that analysisbut not quantified.
The refreezing phenomenonis pertinentprimarily in model
estimatesof glacierrunoff but alsobearson certainmeasurement
methodsfor glacier mass balance. Complete measurementsof
glacier mass balancein the accumulationarea (which measure
masschangesin a surfacelayercontainingseveralyears' accumulation) accountfor meltwaterinfiltrationandrefreezing,but mass
balance measurementsbased on surface elevation alone (e.g.,
satellitealtimetry)mustmake someassumptionaboutthe density
of thicknesschanges.The usualassumptionis that the densityis
constantat sometypical value. This works if conditionsare in a
steady state. Under warming climatic conditions, however,
increasing melt above the runoff limit could produce a
densificationof preexistingfirn and an accompanyingsurface
lowering which would be interpretedincorrectlyon the basisof
surface elevation measurements alone as a mass loss.
The purposeof thisreportis to quantifytheprocessof meltwa-
to whethertheyindividually
looseor gainmassannually.
Glaciers
ter refreezingin a simpleway, in termsof its effect on climatetypicallygainmassby accumulation
of snowandlosemassby driven changesin future sea level. In the analysisthat follows,
meltingof glacierice, butmasscanalsobe lostby meltingand we quantifyin a simple,approximateway, what errorsmight be
runoffof snowaccumulated
duringthepreviouswinter.
incurredin predictionsof futuresealevel changeif adjustments
of
glaciermassbalancein responseto climaticchangeare modeled
withoutconsideringthe effect of meltwaterrefreezing.Refreezing
differencebetween accumulationentering,and meltwaterplus
of surfacemelt occursto a significantdegreeonly in the highArcicebergcalvingleaving,theglaciersystem.
A number
of previous tic, soAntarctica(whereessentiallyno meltingoccurs)is not constudieshaveusedlocalenergyandmasstransfers
at a glaciersur- sidered,nor are the mountainglacier regionsoutsidethe Arctic
face to calculate local accumulation and melt. It is incorrect, how(where melting occursbut conditionsare too warm for extensive
ever,to equatetotalmeltwithtotalrunoff.Undercoldconditions, refreezing). Within the high Arctic, only Greenlandis conas are found in much of Greenlandand the circumpolarArctic,
sidered.This restrictionis madebecauseof the scarcityof infor-
Thenetannualocean-glacier
masstransferfor anypartor all of
the world's glaciersmust be determinedby calculatingthe
notall meltescapes
asrunoff,butinstead
refreezes
locallyin the marion on current mass balance and area-elevation distributions
surfacelayerof multiyearsnow(firn).Formulations
thatneglect of the other Arctic glaciersand ice caps.As a consequence,
this
refreezing
overestimate
theamount
of meltwater
discharged
to the analysiscan make claimsonly abouterrorsin the futurecontribution of Greenland
Copyfight1991by theAmericanGeophysical
Union.
Papernumber91JC02502.
0148-0227/91/91JC-02502505.00
to world sea level. Greenland
constitutes about
90% of the glacierizedarea of the high Arctic, however,and is
more influencedby meltwaterrefreezingthan is the remainderof
the Arctic owing to its higheraverageelevation.Accordingly,we
22,117
22,118
PFEFFER
ET AL.:RUNOFFREFREEZING
ANDSEALEVELCHANGE
expectthatthelimitedanalysis
hereaccounts
for something
better
where
than 90% of the effect of errors associated with meltwater refreez-
b(z,t) = C (z,t)-M(z,t)
h-<hr(t)
ingfromall glaciersources
in theworld.
b (z ,t ) = C (z ,t )
h
The modelingstrategyto be followedhereis to analyzethe
phenomena
of meltwaterinfiltration,refreezing,and runoffin The termf(calving) containsthe contributionof massflux across
snowandto adopttwomodelsof thehowthesephenomena
might the glacier boundarynot accountedfor by meltwaterrunoff or
respondto a futurechangingclimate.Thesetwo modelsbracket precipitation.This term is predominantlycalving in Greenland,
theactualprocess
(whichcannotbe modeleddirectlyat present), but for applicationto other settingsit could includeotherfluxes
so thatthe timingandmagnitude
of the actualtransient
behavior suchas meltingor freezingbeneathice shelvesor sublimationat
of glacierrunoffwill lie somewhere
in betweenthepredictions
of theglaciersurface.For thepresentpurpose,calvingis includedas
the two modelsdescribed.Simulationsof glaciermassbalanceare an unknownnonzeroterm,but its variabilitywill dropout of the
runusinga setof sixfutureclimatescenarios
asforcingfunctions. calculation, and need not be considered. The role of the runoff
Thesesix scenarios
bracketthe likely near-futureclimatechanges limit hr(t) is clear in the expressionfor the local massbalance
in theArctic.Ratherthanmakepredictions
of futuremassbalance b(z,t): below hr accumulationaddsand melting subtractsfrom
for all of Greenlandbasedon very incompleteinformation,this the massbalance,while abovehr, meltinghasno effect on mass
analysisis constructed
in sucha way thatthefinalproductis the balance because the melted mass is not removed from the local
errorin futuresealevelpredictions
only,stemingfromtheneglect area.
of theinfiltration,
refreezing,
andrunoffphenomena.
In thedis- Equation
(1) is an accurate
expression
for themassbalance
cussions
thatfollow,theterm"excess"
sealevelriseisemployedB (t) providedthatA (z), M (z,t), and C (z,t) are accurately
known. Present-dayratesof melt and accumulation
are,however,
well known only over limited areason a limited numberof glaciers, and are of coursevery poorly constrainedin the future.
Parameterizationshave been developedhere for applicationto
Greenland,to which we restrictthe following analysis;theseare
ROLE OF THE RUNOFF L•Mrr • GLACmR MAss BALANCE
drawnfrom data presentedby Benson[1962], principallyin FigThe time-varyingmassbalanceof a glacieror ice cap can be ures29, 30, 47, and 48. Thesedatahave been synthesized,
with
written in a simpleway if the following are known: the specific M (z ,to) (gramsper squarecentimeterper year) in theform
annualmelt M (z ,t ) and accumulation
C (z ,t ) (gramsper square
M (z ,to)= (0.14586)(2900-z)e
-{ø'ømø46)'
(2)
centimeterper year) as functionsof elevationand time, the areaelevation distributionA (z) (meters), and the elevation hr(t) for z in meters.This three-parameter
curvehasbeenfit to three
(meters)of the runoff limit as a functionof time. The runoff limit data: elevation of the dry snow line, magnitudeof melt at the
is definedas the elevationabovewhich surfacemelt percolates equilibriumline, and magnitudeof melt at the margin, all at a
locallyinto underlyingcold,permeablefirn and is refrozenin its representativelatitudeof 72øN. The accumulationrate C (z ,to) is
entirety.Below this elevation,the annualquantityof melt M is tabulatedas a linear interpolationof Figure29 of Benson,againat
large enoughto satisfy the requirementsof firn cold content 72øN. Theseare a very simplerepresentation
of continentalaver(latentheat of fusionrequiredto raiseinitial firn temperature
to ageratesof accumulation
andmelt in Greenland;in particular,it
0øC, including any superimposedice formation) and residual can be questionedwhetherdatafor 72øN (a mid-rangelatitudein
water content(sometimescalled minimum, or irreducible,water Greenland)is representative
in degreeor form, of the entireconcontent:the pore water immobilizedby capillaryforces)in the tinent. Similar functionshave been derived for the QueenElizaunderlyingpermeabletim, with somesurplusof meltwaterfree to beth Islands, Canada [Koerner, 1979], where annual melt and
percolatein the snowpack.Runoff occurswhen surpluswater accumulationare representedas functionsof elevationand latipercolatesdownslopeand ultimately out of the glacier system tude. Latitude dependenceis an importantparameterfor Green(eitherby a surfaceor englacialroute).Water that doesnot leave land also, but is poorly constrainedby data and has not been
the area where it was producedbut remains,either as residual includedhere for simplicityin this basic analysis.Values have
watercontentor as slushtrappedin an enclosedbasin,is refrozen been determinedfor mid-latitudeGreenlandand are regardedas
duringthe succeeding
winter. Surfacelakes,however,appearto representativein an averagesenseof the entire continent.The
remainpartlythawedthroughthe winterif theyare largeenough rates of accumulation and melt as functions of elevation, deter[Ohmuraet al., 1991]. The depthof permeablefirn to the tim-ice minedfrom thesedata, are shownin Figure 1.
transition
is denoted
hereasdr. Thepositionof therunofflimit
An initial (presentday) value for the runoff limit hr mustbe
within the schemeof glacierfacies[Benson,1962] correspondschosenbut is not simply availablefrom observationof facies
approximatelyto the position of the percolation/wettedzone boundaries.A reasonableassumption
is that the runoff limit lies
boundary(the actualpositionof therunofflimit will be seento be nearthe upperlimit of the wettedfacies.In the wettedfacies,all
a sourceof substantial
uncertainty).The runoff limit may be more snowbecomeswettedabovean impermeable
ice or superimposed
accuratelyregardedas a zone,or elevationinterval,acrosswhich ice horizon,while in thepercolationfacies,surfacemelt wetsonly
a varying fractionof the annummelt escapesas runoff, but we part of the annualincrementof accumulation,
leavingsomefracassumefor simplicitythat this zone will generallybe narrow tion of permeablefirn unwetted.The boundarybetweenthe wetenoughto be treatedas a singlealtitudefor the purposeof divid- ted and percolationfaciesis fairly narrowlyconstrained
in West
ing a glacierintoregimesof runoffandno runoff.
Greenland,accordingto Benson's[1962] data, at an elevation
If M (z ,t ), C (z ,t ), A (z), and hr(t) are known, the totalbalance near 1800 m. Alternatively,the runoff limit can be definedin
for a glacieror ice capto whichthesetermsapplycanbe written: termsof conditionswhichmustbe met by accumulation
andablah
tion ratesto allow runoff,andthe runofflimit elevationassigned
1
accordingto the elevationat which theseconditionsobtain. A
to refer to this error: it is the overestimation of future sea level
rise thatstemsfrom thefailureto considertheseprocesses.
B(t)=hIA(z)b(z,t)dz
+f(calving)
0
(1) simplebut robustcriterionis describedin the appendixwhich
PFEFFERET AL: RUNOFF REFREEZING AND SEA I_,E•L CHANGE
5OO
' ........
I .........
I .........
I .........
22,119
higher elevation. If the transientbehavior of the runoff limit is
neglected,future runoff and consequentsea level rise will be
overestimated.An alternativeresponseof the runofflimit is possibleif a futurewarmingclimatedevelopsin which accumulation
increasesmore significantlythan melt. In this case,the net effect
400
is an increase
in df overtime,anda corresponding
decrease
in
runoff limit elevation.
The criticalquestionin the predictionof a risingrunoff limit in
a warming climate is exactly how the runoff limit elevationhr
will respondto increasingmeltwaterinfiltrationand refreezing.
As the processis describedhere, the fundamentalrequirementfor
300
the establishment of runoff at some new elevation above the ini-
tial runofflimit is thatthetim-icetransition
depthd/ at thenew
200
elevationbe raisednear enoughto the surfacethat a significant
fractionof annualmelt remainsfor runoff after the requirements
of cold content and residual water content are satisfied. Limited
observations[Sharp, 1951; Benson,1962] indicatethat conditions
couldexist which would allow runoff to occuron Arctic glaciers
100
underpresent-day
circumstances
whendf is approximately
4m
or less. An initial firn-icetransitiondepthin excessof 4 m could
be raisedby a varietyof meansintermediatebetweentwo extreme
cases.At one extreme,if all the firn pore spaceis filled in with
ation
0
0
1000
2000
3000
4000
Elevation (m)
superimposed
icebetween
theinitialdepthd• andthedepth
where runoff is established•decadesor centuries might be
requiredto accumulate
the necessary
meltwaterat reasonable
futuremelt rates.Present-dayfirn depthsin Greenlandvary from
Fig.1.Elevation
distributions
ofannual
meltandaccumulation
synthesized
forGreenland.
Weighting
duetoarea-elevation
distribution
isnotshown
in roughly4 m [Benson,1962] at the presentrunoff limit elevation
to maximum depthsof 90 m at the highestelevations.The firn
thisfigure,
sooverall
melting
appears
greater
thanaccumulation.
depthdf is approximately
70 m in the highestlocations
where
meltingoccurstoday.This processdemandsthat all porespacein
the firn layerbe filled in with superimposed
ice. This is an unrealresults in the condition that the ratio of melt to accumulation,
istic requirementbut resultsin a calculationof fill-in time which
M/C, be approximately
equalto 0.7 at therunofflimit elevation.
is longerthan any otherprocessand as suchgivesan upperlimit
For the synthesizeddistributionsof melt and accumulation
on the time requiredto establishrunoff at somenew elevation.
describedabove,M/C = 0.7 placeshr at 1680 m underpresent
At the otherextreme,if near-surfaceice layers(suchas typidayconditions.
Thisistheinitialvalue
used
(hrø)inthefollowingcally
form in percolationzones)coalesceinto a singleimpermecalculations.
able ice horizon,runoff might be achievedalong a thin (10-50
cm) "perched"impermeable
horizon.In this case,the meltwater
FACTORSINFLUENCING INFIL••ON
AND THE ESTABLISHMENT
requirements
to fill in thefull thickness
of firndownto theinitial
depthdf wouldbe avoided,anda newrunofflimit mightbeestaUnder quasi-steady
stateconditions,surfacemelt, refreezing, blished
in onlya ,fewyears.Thisprocess
demands
a minimum
OF CONDITIONS FOR RUNOFF
andrunoffoccuron glaciersin predictablespatialpatternsdeter- amount of meltwater infiltration and refreezing to create an
horizonandprovidesa minimumtimefor theestabminedbysurfacemassandenergyfluxesandinternalinitialcon- impermeable
lishmentof runoff.Ice layers5 to 20 cm thick in otherwise
ditions
(firndepth,
temperature,
density,
hydraulic
conductivity).
Underchangingclimaticconditions,
ratesof meltM andaccumu- permeabletim arevery commonin Arcticlocationswherelimited
lationC (gramsper squarecentimeter
per year)will changein summermelt occurs[Wakahamaet al., 1976; Koerner, 1977; W.
directresponse
to climate,while therunofflimit elevationhr will T. Pfeffer, manuscriptin preparation,1991], but while often
traceableover long distances(kilometers),they are typically
changein responseto changesin boundaryconditionsC andM,
on themeterscale,andcouldnotform an impermebut will,lag in time behindthosechangesowing to the interaction discontinuous
betweeninfiltratingmelt andinitial andsubsequent
firn condi- able horizon over significanthorizontaldistanceS.While ice
tions. The demands of cold content and residual water content on
layerstypicalof thepercolation
zonewouldnotactasimperme-
melt M will be alteredby new ratesandtemperature
of accumula- ablehorizonsfor runoffin theirpresentlyobservedform, it is pos-
tion C and will also be alteredby a changingdepthdf to
siblethatseveralyearsof unusually
highratesof meltcouldfill in
gapsin a pre-existing
discontinuous
ice layerandform a continuimpermeable
ice,withd! increasing
withgreater
accumulation
anddecreasing
asgreaterquantities
of meltfill in firnwithsuper- ous,thick ice layer whichcouldtransmitrunoffover significant
'unposed
ice.In a futurewarming
climate
where
meltincreaseshorizontaldistances.Such unusuallythick ice layershave been
moresignificantly
thanaccumulation,
oneexpects
greaterquanti- observedin tim at Mer de GlaceAgassiz(ArcticCanada)followfiesof meltwaterto refreezein initiallycoldpermeabletim, consequentlydiminishingthe thicknessof tim layer andelevatingthe
ing unusuallywarmsummers(R. M. Koerner,personalcommunication,1990),but theirabilityto actasanimpermeable
horizonis
runoff limit. Total melt seasonrunoff (definedas total annualsur-
unknown.
facemeltover'theentireglacierbelowtherunofflimit elevation)
will increase,but not as much as total surface melt, for some frac-
tion of the melt goesinto establishingthe runoff limit at a new,
Accurate simulationsof the actual transientsituationdepend
upon detailed analysesof the phenomenaof grain-scale
infiltrationandrefreezing.Thesearein progress
[Illangaxekare
et
22,120
PFEF•R ET AL.:RUNOFFREFREEZINGAND SEALEVEL CHANGE
al., 1990;Pfeiferet al., 1990]with theobjectiveof modelingthe part of the year when melting is not occurring.This term is
large (kilometer)scalehydrologyof initiallycold snowfrom a roughlythe samein magnitudeas the warmingdueto refreezing
foundation
of grain-scale
porousmediaphysics,but withouthav- during the melt season,as can be seenfrom the fact that under
ing to treatthegrain-scale
phenomena
explicitlyin thelarge-scale steadystateconditionswinter coolingand summerwarmingbalvary in a steady-periodic
model.The analysispresented
hereis a simplified,approximate anceeachother, and firn temperatures
approach,
intended
to providea quantitative
estimate
of theeffect fashion around a constant mean.
The energybalancebetweenthe available
meltovert* years
of meltwaterrefreezingduringclimatechangein termsof future
and latentheat and water massrequirements
to fill in the firn of
sealevel response.
dfø-d•(i.e.,toestablish
runoff)
overthesame
time,can
Two possible
modelsareconsidered
whichbracketpossible thickness
actualoutcomesin termsof time requiredto establishhr at some be
new elevation,corresponding
to the two extremecasesdescribed
t*
above. Thesemodelspredictdifferentratesat which the runoff
limit will rise to a new elevation,but the final equilibriumvalue
hr is the samein eithercase. The physicsin questionis that
which influences the transient behavior of the runoff limit; the
solved for the time
c
= (•p/T/ - q•/ps,)(d/-dr)
<M>(1
+•)cfPsi
) _<C>(
•)c/Psi
+•Tc/)
pc/
Pc/
(3)
steadystateposition,asdiscussed
above,canbe estimated
much
moresimplyby theelevationat whichM/C = 0.7. Themodels Equation(3) givesthe time to establishrunoffat someelevationz
theinitialrunofflimithrø,by fillingin all underlying
cold
arestatedexplicitlyin thefollowingparagraphs,
andthespecific above
permeablefixnwith superimposed
ice. This quantitycan alsobe
formulations are described.
Maximum
Time Fill-in
calculatedrecursivelyusingthe explicitformsM (z ,t ) andC (z ,t )
ratherthanthe averages<M> and<C>; this is donein the model
calculations.When thisrelationshipis inverted,andthe elevation
z is varied as a parameter,the result is the future runoff limit
hr(t ) as a functionof time, as determinedby the chosenfuture
patternof climategiven by M (z ,t ), C (z ,t ), andby the elevation-
Model
All firnbetween
theinitialfirn-ice
transition
depth
dfø andthe
final depthd} mustbe filled in to the poredose-offdensityof
0.83 gcm-3 beforerunoffis established.
Intermediate
densitiesdependent
properties
of thetim:Tf (z), pf (z), anddf(z).
(e.g., 0.4-0.6) are observedin fully wettedzones of thewet snow
facieson glaciers,but theseare densitiesin the wet firn layer
Minimum Time Fill-in
Model
above the depth dr. Runoff in theseregionsoccursalong
All meltwateris trappedin a horizonnearthe surface(_<df ),
impermeableice surfaces,composedof glacierice or superimposedice. This model requiresthe most meltwater,and conse- formingan impermeable
horizon"perched"
abovefirn which
quentlythe greatestnumberof yearsof meltwaterinput, of any remains
permeable
evenaftertheestablishment
of runoff.
fill-in requirementfor some particularpattern of future melt
For modelingpurposes,
thenear-surface
impermeable
horizon
M(z,t) and accumulationC(z,t). The model is overconservative, is assumed
to requirea negligibleamountof meltwater
for its foris requiredto developwithno lag time
sincemeltwateris observedto movevery nonuniformlythrough marion,andconsequently
inhomogeneous
firn [Sharp, 1951; Marsh and Woo, 1984], and betweenthe onsetof increasedmeltingandthe movementof the
substantial
regionsof uninfiltratedtim maybe expectedto be iso- runoff limit to a higher positionwhere the ratio of melt to
is the equalto the originalvalueat the initial posilatedbelowthe final depthof the firn-icetransition
d}. This accumulation
simplification,
however,providesa maximumboundon the time tion of the runoff limit.
for runoff to be established.
To formulate the maximum fill-in time model, let a new runoff
limithr be established
at elevation
z overa periodof t* years,
duringwhich time dimate is warming.The patternof warmingis
represented
by time dependentboundaryconditionsM (z ,t ) and
MODELS OF FWFURE CLIMATE CHANGE AND MELTWATER
DISCHARGE
Ambachand Kuhn [1989] andAmbach[1989] haveanalyzedin
C (z,t ), whoseaverage
valuesovertheperiodt* are<34(z)> and detail how elevation distributionsof massbalancetermsrespond
<C (z)>. The totalmeltavailablefor bringingd! to somevalued• to changingclimate, and we follow the basicconceptsexpressed
sufficient
for runoffin t* yearsis then<34(z)>t* (gcm-2).The by thoseauthors.The modelsof futureclimatechangepresented
initialdepth
d/øoffirnischaracterized
by depth-averaged
values here are not claimed to be predictionsof
future eventsbut are
of temperature
T/, densityp/, andporosityq•/. Coldcontentand qualitativelyreasonableestimationswhich bracketin quantity
porefiR-inrequirements
forthecolumn
oftimbetween
d/øanddf mostprobablefuturechanges.Likewise,the massbalancemodel
by equation(1) is an idealizationwhichis notclaimed
are(c/L)pf Tf(a/ø-d})and•)/,ps,(af-a;),respectively,
where
c represented
andL are the heat capacityand latentheat of fusionfor ice, and
Psi is the final densityof superimposed
ice, takento be the pore
close-off
density
of0.83gcm-3.Thetemperature
T! istaken
here
to be the absolutevalue of the temperaturein degreesCelsius.In
addition to the requirementsof the initial column, cold content
and pore fill-in mustbe satisfiedin the subsequent
accumulation,
and heat lost over the courseof the year outsidethe melt season
must be replaced. These three quantifiesare, respectively,
(c/L)Tc•<C•>t*,(•)cfPsi/Pcf
)(<Cf>-<M>)t*, andQwt*/L
for theentiretimeintervalt*, wherethesubscript
cf denotes
valuesparticularto the subsequent
accumulation.
The annualheat
lossQw is the netheatflux crossingtheglaciersurfaceduringthe
to be a highly accuratepredictorof absolutevaluesof massbalance,but is usefulfor analyzingchangesin absolutevaluesdue to
climaticallyforced changesin the termsb (z ,t ) and h, (t). The
strategyused in the following calculationsis, for each of six
modelsof future climate, to compareB (t) calculatedunder two
assumptions:
B l(t ), where the existenceof the runoff limit h, is
neglectedaltogether,and all surfacemelt is allowedto escapeas
runoff; and B2(t), where the runoff limit is accountedfor as in
equation(1). The differencebetweenthesequantities,
h
1
B•(t
)-B2(t
)=-•!M(z,t)dz
(g
yr
-•)
(4)
PFEFFERET AL.: RUNOFFREFREEZINGAND SEA LEVEL CHANGE
TABLE
1. Parameters for Future Climate Scenarios
ClimateModel
AT,øC
Ac/c
VariationFromStandardModel
S
4
0.10
I
II
3
5
0.10
0.10
IøC cooler
IøC warmer
22,121
the mass balance model to the choice of boundaryconditions
M (z ,t ) andC (z ,t ). The parameters
of the six modelsare givenin
Table 1.
III
4
0.15
50% wetter
IV
4
0.5
50% drier
V
4
0.0
Figure2 showstheelevationof therunofflimit asa functionof
time followingthe onsetof climaticwarmingfor the 6 projected
future climates and minimum
nochange
in accumulation
from present
and maximum
fill-in time models of
runoff limit response.All future climatesproducea rise of the
runoff limit from the initial value of 1680 m during the first 150
years,rangingin magnitudefrom 138 m for climateI (+3øC,10%
increasein accumulation), maximum fill-in time model, to 340 m
is the rate of meltwatererroneouslycalculatedas dischargeto the for climate II (+5øC, 10% increasein accumulation),minimum
ocean through the neglect of the runoff limit. The integrated fill-in time model. Climate S (+4øC, 10% increasein accumuladifferenceover someperiodof time,
tion) forcesa rise in runoff limit of 190 m for the maximum fill-in
time model and 250 m for the minimum
t1
(B
•(t
)-B2(t
))dt(g)
fill-in time model. Varia-
tions in projected future temperatureof IøC evidently affect
changesin runoff limit elevationsmorethando 50% variationsin
projectedfuture accumulation(15% and 5% increasesfor climatesIIl and IV comparedwith 10% for climateS). ClimateV
(5)
is the totalerror in predicteddischargeto the oceanover the interval to to t• when the effect of meltwaterrefreezing above the
runofflimit is neglected.Thesequantitiesareprimarilydependent
upon hr(2,t ), and the climaticallyforced variationsof b(z,t )
within the elevation range covered by hr(2,t). Note that
f(calving), includedin equation(1), may still exist and vary, but
as long as it is not influencedby movementof the runoff limit it
will dropout at equation(4).
In all climate models used here, warming is simulatedby
translatingthe present-dayelevationdistributionof melt up in
elevationat a rate of 100 m for eachdegreeCelsiusof warming
(annual average increase),with the total warming assumedto
occurin a linearrise over 100 years.Accumulationis assumedto
changelinearly over the sameperiodof time, to someadditional
fractionof the initial accumulationat any given altitude.The total
climatechangeis thusparameterizedby the total annualaverage
temperature
increaseandtotal fractionalincreasein accumulation
over 100 years. Of the six models used, one ("standard")is
chosento be most directly comparableto projectedfuture climatesas discussed
by Ambachand Kuhn [1989]. The remaining
five are variationsaroundthe standardand test the sensitivityof
causes the runoff limit to rise somewhat faster than it would under
the influenceof the standardfuture climate (S), since the same
increasein temperatureis presentwithout the additionalannual
energydemandsof greateraccumulation.For themaximumfill-in
time model, in all casesthe runoff limit elevation continuesto rise
after the 100-yearperiod of climate forcingbecausefirn at the
higherelevationsof meltingis still respondingto the increased
meltwaterinput. Sinceno laggingmechanism
is includedin the
minimum fill-in time model, changesin runoff limit elevation
ceaseat 100 yearswhenthe drivingclimaticvariationscease.In
no caseis the calculationextendedpast 150 years,becauseother
dynamic glacier influencesmust be consideredat longer time
scalesandbecausethe simplecharacterof the assumedfutureclimatesdoesnotjustify very longprojections
in time.
The difference in integratedmass balance (equation (4))
betweenmassbalancemodelswhichincludeor neglectthe runoff
limit are shownfor the samefuture climatemodelsin Figure 3.
For eachclimatemodel,the plottedcurveshowsthe excessnegative massbalance(dischargeto the ocean)as a functionof time
followingtheonsetof climatewarming."Excess"heremeansthe
2100
Maximum
Fill-In
21001
....,....,....
Minimum
Time Model
Fill-In
Time Model
II
2000'
2000
v
II
1900
1900
1800
1800
1700
1700
III
b
a
....
1600
0
, ....
5O
, ....
100
1600
150
....
0
• ....
50
• ....
100
150
Time (yearssinceonsetof warming)
Fig.2. Elevationof theranofflimit asa functionof timefollowingtheonsetof climaticwarming.CurveslabeledI throughV andS
represent
differentproposed
futureclimatesaslistedin Table1. Maximumandminimumfill-in timemodelsaredescribed
in the
text.
22,122
PFEFFERET AL.: RUNOFFREFREEZINGAND SEA LEVEL CHANGE
TABLE 2. Runoff Error for Future Climate Scenarios
additionalincrementof runoff dischargewhich would be calculated to escapeto the oceanif the existenceof the runoff limit
were neglected.The relativedifferencesbetweenclimates$, I, II,
III, and IV seenin Figure 2 also appearin Figure 3: a cooler
future climate (I) producesa smallerexcessdischargethan the
Climate
Model
$
!
II
warmerfutureclimate(II), while slightlywetterand drier cli-
III
mates
(III andIV) produce
relatively
smaller
differences
fromthe
standard
futureclimate($). ClimateV, however,
showsa
IV
V
discharge
which
isdiminished
incomparison
to$:theopposite
Maximum
Fill-inTime Minimum
Fill-inTime
Average
4.3
4.0
3.7
3.5
5.o
4.4
4.1
4.5
4.2
3.9
4.2
3.8
4.3
4.0
responseto that seenfor climateII, despitethe fact that runoff Totalover150years;centimeters
sealevelequivalent.
limits for climatesII and V respondmore quickly (for eitherthe
minimum or maximum fill-in time model) than that for $. This
difference
is dueto twocompeting
influences
in thedeterminationimum fill-in models,however,the excessdischarge
will never
of runoff:greaterratesof meltor diminished
ratesof accumula-dropto zero,sincea newsteadystaterunofflimit will be estationforcetherunofflimitto respond
morequickly,thusdiminish- blishedfor the newclimate,andmelt occurring
abovethatlimit
ing morequicklythe areabetweentherunofflimit andtheupper will be incorrectlyassumed
to escapeto the oceanif the runoff
elevationlimit of surfacemelting,the regionfrom whichexcess limit is neglected.
discharge
originates.
At the sametime,.greater
ratesof melt
Integration
of thecurves
of Figure3 overthe150-year
time
increasethe total amountof melt occurringin thisregion. In the interval(equation(5)) givesthe accumulated
excesssealevelrise
caseof climateII, theintensityof melt abovehr(t) dominates
the dueto neglectof therunofflimit for the six modelsof futurecli-
calculation,
resulting
in an excessdischarge
greaterthanthatfor mate.The integrated
valuesare givenin Table2. As discussed
S. For climateV, therunofflimit responds
morequicklybecause earlier,the differencein massbalance(Bl(t)- B2(t)) is a good
of lower accumulation
ratesratherthangreatermelt rates,andthe measureof the effect of the runoff limit in massbalancecalculaexcessdischargeis diminishedby the dominanceof the more tionswithinthe assumptions
given,despitethe fact thatthe abso-
rapidlyshrinking
areaabovehr(t), ratherthanincreased
by lutevalues
ofthemass
balance
arenotclaimed
tobehighly
accugreatermeltratesaboveh•(t).
rate numbers.Excesssealevel rise due to neglectof the runoff
Excessdischargedue to neglectof the runoff limit increases limit, for Greenlandalone,is for all futureclimatesand for the
duringthe 100-yearperiodof warmingclimate,as surfacemelt maximumor theminimumfill-in timemodel,between3.5 and5.0
extendsto progressively
higherelevationsabovetheinitialrunoff cm over 150 years.While the excesssealevel rise is systematilimit elevation.A kink occursin the excessdischargecurvesat cally smallerfor calculationsbasedon the minimumfill-in time
100 yearsin responseto the assumedabruptterminationof the model(averageover six climatemodelsis 4.0 cm versus4.3 cm
changein climateat that time. After 100 years,excessdischarge for the maximummodel),the differencebetweenmaximumand
stillexistsbecause
meltis stilloccurring
abovetherunofflimitin minimumfill-in time calculations
is smallin comparison
to the
the new climateregime,but for the maximumfill-in time model, totalrunofferror.
the excessdiminishesover time after 100 yearsas the transient An importantaspectof theseresults,bestseenin Figure3, is
runoff limit rises toward its final, steadystate value. The thelargeinitialvalueofexcessdischarge.
Evenunderpresent-day
minimumfill-in time modelis alwaysin equilibriumwith the conditions,
beforeanyclimatechangeandwithoutconsideration
drivingclimaticconditions,
andno changeoccursafterclimatic of thetransient
response
of therunofflimit, an excess
discharge
conditions
stabilize
at100years.
Forboththeminimum
andmax- of approximately
5.8x 1013
kgyr-I existsif runoffis determined
0.0
0.0
Maximum
Fill-In
Minimum
Time Model
Fill-In
Time Model
-0.5
f
0
50
100
a
I
-1.0
N
-1.5
150
i
b
-2.0
....
0
, ....
50
' ....
100
150
Time (yearssinceonsetof warming)
Fig. 3. "Excess"
discharge
of runoffto the oceanas a consequence
of neglecting
refreezingabovethe runofflimit. The climate
modelsandfill-in timemodelsarethesameasthosein Figure2.
PFEFFER
ETAL.:RUNOFF
REFREEZING
ANDSEALEVELCHANGE
22,123
is acceptable
in theabsence
of morepreciseinforfromenergybalancewithoutconsideration
of therunofflimit. approximation
zone boundarycan be disThiserroralone
corresponds
toansealevelriseof0.160mmyr-•, mation. The percolation-wetted
tinguished
by
stratigraphy
and
has
already
beenmappedto a limor approximately
2.4cmover150years.
Thetotalrunofferrorin
neglecting
therunofflimitis 4.0 cm(fromtheminimum
model) itedextentthroughoutthearctic.
Finally,theaccuracy
of thisor anyprediction
of futureglacier
or 4.3 cm (fromthemaximummodel)over150years;2.4 cm of
conditions
is dependent
thosevaluescomesfromignoring
thepresence
of therunofflimit massbalancedrivenby climaticboundary
upon
knowledge
of
what
the
future
boundary
conditions
will be.
whereit liestoday,andthebalanceis theresultof ignoringthe
An efforthasbeenmadehereto investigate
a rangeof futureclifact that it can move in the future.
mate scenarios which bracket most reasonable actual future
DISCUSSION
We havepresented
a modelof surfacemelting,infiltration,
refreezing,
andrunoffona glacieror icecap.It hasbeenapplied
to theGreenland
ice sheet,with theobjectiveof discovering
what
events.Greatercertaintyof what futureclimatemay be will
translate
intocorrespondingly
morevaluablemodelpredictions.
APPENDIX:DEF'IN1TIONOF TI-IERUNOFFLIMIT ELEVATION
IN TERMSOFM (z ,t ) ANDC (z ,t )
errors
mightbeincurred
in theprediction
of futuresealevel(that
component
of sealevelchange
coming
fromchanges
in glacier The runoff limit elevation as defined here is that elevation at
runoff) as a consequence
of neglectingthe refreezing which the annual amount of surfacemelt generatedis just
phenomenon.
Neglect
of therefreezing
of surface
meltabovethe sufficientto satisfythe followingdemandsof heat and water to
elevationof the runoff limit in models of future massbalance is
allow motionof waterin the tim: (1) enoughwatermustrefreeze
firn to bringthe snowtemperature
to
shownto produceerrorsin projectedsea level rise of in the initially subfreezing
0øC (thisis theremovalof the "coldcontent"),and(2) afterthe
firn temperature
is raisedto the meltingpoint,additionalwater
mustbe addedto raisethe porewatercontentof the tim to the
minimumvalue("residual"
watercontent)to allowwatermotion
are considered. One model was chosen to maximize the influence
over capillaryforces.This requirement
was developed
first by
abovetherunofflimit, lessannual
of runofflag due to refreezing:all permeablefirn abovethe initial Colbeck[1976].At elevations
positionof the runoff limit mustbe filled in with superimposed melt will occur (assuminga monotonicallyinverserelation
approximately+4.3 cm (maximum fill-in time) or +4.0 cm
(minimumfill-in time) over 150 yearswhenappliedto Greenland
(averagesover the six climate models).Two simple modelsof
how refreezingmeltwaterchangesthe positionof the runofflimit
ice. Any other model would require less meltwater (and
correspondingly,
time) to establishtherunofflimit at a new elevation, and the lagging effect of meltwater infiltrationwould be
diminished. At the other extreme, the runoff limit is allowed to
between elevation and annual melt which in some situationsmay
not be the case),andconditionsallowingdownslope
percolation
of waterthroughthefirn will not develop.Belowthiselevation,
theseconditions
will be developed,and additionalmelt will be
respondinstantaneously
to new conditionsof melt and accumula- freeto percolatedownslope.
The subfreezing
permeablefirn into which meltwaterwill
tion. Any other model of runoff limit responsewould require
more time to establish the runoff limit at a new elevation.
The
penetrate
consists
of thecurrent
year'saccumulation
plusresidual
years.Thebottomof thefirnlayer(atdepthdf
minimumfill-in time model causesthe effect of excessdischarge firnfromprevious
ice. The total
to be reduced,but not by a substantialdegree:the averagesea below the surface)lies on relativelyimpermeable
level excessis 4.3 cm over 150 yearsfor the maximummodeland thickness
df of thefirnlayeris typically
4 to 10m in theareaof
4.0 cm for the minimum
model.
interest
but maybe of theorderof tensof metersin theareaof
Regardlessof the transientcharacterof the runoff limit under
changingclimatic conditions,a major error is introducedby
neglectingthe presenceof the runoff limit altogether:under
present-dayconditions,beforeconsideration
of climaticwarming
interestin a futurewarmingclimate.
limit. Even without any consideration
of the transientbehaviorof
the runoff limit, a substantialimprovementcan be madein mass
balancecalculationssimply by includingthe present-dayrunoff
limit elevationand leaving it fixed for the future. This raisesa
needfor observations:
the present-day
positionof the runofflimit
is poorly known for all glaciersand ice sheets,and is the first
parameterthat needsto be quantifiedin any model of this type.
This is not an easytask, for unlike the faciesdefinedby Benson
[Benson,1962; Williams et al., 1991], the runoff limit probably
cannotbe discoveredonly by an examinationof stratigraphybut
mustbe foundby somemeansof tracingwatermotionduringthe
meltseasonanddistinguishing
waterthatescapesinto theablation
zone from water that may percolate locally but ultimately
refreezes.A workable assumption,however, is to equate the
runoff limit with the percolation-wetted
zone boundary. This is
only an approximation
of the runofflimit position,but in termsof
residualvalue,sinceif thisconditionis goingto be met from one
A simpleconditioncanbe imposedat the runofflimit which
requiresno information
aboutfirn properties
beneaththecurrent
year'saccumulation.
Thiscondition
is thattheannualmeltat the
orresponse
of therunofflimit,approximately
5.8x 10•3kgyr-• of runofflimit, M (hr,t), mustbe at leastgreatenoughto satisfythe
of cold content,and completelysaturatethe snow
excessrunoff is erroneouslycalculatedas dischargeto the ocean requirement
alone.We requirehere
if thepresent-day
runofflimit is neglected.
This is roughlyhalf of porespace,for the annualaccumulation
ratherthanmerelyfilled to the
the total error incurredby neglectingthe presenceof the runoff that the snowporesbe saturated
error in the total area above and below the runoff limit, the
year to the next, eachyear's net accumulation
mustbecome
impermeable
at the endof themeltseason
to providea horizon
for runoffthefollowingyear. Thisrequirement
providesa necessary,butnotsufficient
condition
for runofffromthegiveneleva-
tion.If therequirement
isfoundtobemetat some
elevation
h*,
thenit is knownthattherunofflimit hr liesat or belowh*. The
conditionfor melt at hr is, then,thatthe annualamountof meltbe
equalto or greaterthanthisamount.
With M the annual melt and C the annual accumulation in the
current
year,therequirements
listedabovemaybewritten:
c
M>•-CT•.+(C-M)(
P•,c-Pc
)
(A1)
Pc
whereT! is theinitialfirntemperature
(in positive
degrees
Celsiusbelowfreezing),c andL aretheheatcapacityandlatentheat
of fusionfor ice,Pc is theinitialfirndensity,
andP•,cis thepore
22,124
PFEFFERET AL.: RUNOFFREFREEZI1NG
AND SEA LEVEL CHANGE
T. H., R. J.Walter,M. F. Meier,andW. T. Pfeffer,Modelclose-off
density
(0.83gcm-3).Thefirsttermontheright-handIllangasekare,
ing of meltwaterinfiltrationin subfreezing
snow,WaterResour.Res.,
sideis thecoldcontentof thecurrentyear'saccumulation,
andthe
26(5),1001-1012, 1990.
secondterm the is meltwaterrequiredto fill the porespace.The Johannesson,
T., C. F. Raymond,andE. D. Waddington,
Time scalefor
factor(C -M ) accounts
for thefactthattheamountof porespace
adjustments
of glaciersto changesin massbalance,J. of Glaciol.,
356(121),355-369,1989.
to be filled is diminishedby the masstakenawayfrom the accuKoemer,R. M., DevonIslandice cap:corestratigraphy
andpaleoclimate,
mulationC to providethemeltwater.
Science,196(4285),15-18, 1977.
Solvingfor theratio(M/C), we obtain
Koemer,R. M., Accumulation,ablation,andoxygenisotopevariationson
M> c
-
P•-P•)] I+(P•-P•
+(
(A2)
Substituting
typicalnumbers
for densityandtemperature
for Arctic surface
snowat thestartof themeltseason
(e.g.,Tf =-15øC
theQueenElizabethIslandsice caps,Canada,
J. of Glaciol.,22(86),2541, 1979.
Marsh,P., andM.-K. Woo, Wettingfrontadvanceand refreezingof meltwater within a snowcover, 1, Observationsin the Canadian Arctic,
WaterResour.Res.,20(12), 1853-1864,1984.
Meier, M. F., Contributionof smallglaciersto globalsealevel, Science,
226(4681),1418-1421,21 December1984.
andPc= 0.3gcm-3),M/C takesthevalue0.697.Thisnumber Meier,
M. F., Reducedrisein sealevel,Nature,343, 115-116,1990.
turnsoutto be quiteinsensitive
to reasonable
variations
in Tf and Ohmura,A., K. Steffan,H. Blatter,W. Greuell,M. Rotach,T. KonzelPc,andfor a widevarietyof firnconditions,
thenecessary
condi- mann,M. Latemser,A. Ouchi,andD. Steiger,Energyandmassbalance
tion for runoff canbe statedsimplyas
M
--=0.?
C
duringthemeltseason
attheequilibrium
linealtitude,
Paakitsoq,
Greenland ice sheet,ETH GreenlandExped.Prog. Rep. 1, Dept. of Geegr.,
SwissFed. Inst. of Technol.,Zurich, 1991.
Pfeffer,W. T., T. H. Illangasekare,
andM. F. Meier,Analysisandmodeling of meltwater
flow in dry snow,J. of Glaciol.,36(123),238-246,
1990.
Acknowledgments.
We thank C. S. Bensonand two anonymous Sharp,
R.P.,Features
of thetim onupperSeward
Glacier,
J. of Geol.,59,
reviewers
for helpfuldiscussions
andcomments.
Thisworkwassupported 599-621, 1951.
byDOEgrants
DE-FGO2-87ER60570
andDE-FG02-90ER61078.
Wakahama,G., D. Kuroiwa, T. Hasemi,andC. S. Benson,Field observationsandexperimental
andtheoretical
studies
on thesuperimposed
ice
zoneof McCallGlacier,Alaska,J. of Glaciol.,16(74),135-254,
1976.
Ambach,W., Effects of climaticperturbations
on the surface-ablation
regime of the Greenlandice sheet,west Greenland,J. Of Glaciol.,
35(121),311-316, 1989.
Ambach,W., and M. Kuhn, Altitudinalshift of the equilibriumline in
Greenland calculatedfrom heat balance characteristics,in Glacier Fluc-
tuationsand ClimaticChange,editedby J. Oerlemans,pp. 281 - 288,
Kluwer Academic, BostonMass., 1989.
Williams, R.S., D.K. Hall, and C.S. Benson,Analysisof glacierfacies
usingsatellite
techniques,
J. ofGlaciol.,37(125),120-128,1991.
T. H. Illangasekare,Departmentof Civil, Environmental,and
ArchitecturalEngineering,Universityof Colorado,Boulder,CO 80309.
M. F. Meier and W. T. Pfeifer, Institutefor Arctic and Alpine
Research
andDepartmentof GeologicalSciences,
Universityof Colorado,
Benson,C. S., Stratigraphic
studiesin the snowandtim of the Greenland Boulder, CO 80309.
ice sheet,SIPRERes.Rep. 70, U.S. Army ColdRegionsResearchand
Engineering
Laboratory,Hanover,N.H., July 1962.
Colbeck,S.C., An analysisof waterflow in dry snow,WaterResour.Res.,
12(3),523-527, 1976.
ProjectingFuture Sea Level Rise, EnvironmentalProtectionAgency
(EPA) Dec. No. 230-09-007, Washington,D.C., Oct., 1983.
(ReceivedJune3, 1991;
revisedSeptember
27, 1991;
acceptedSeptember
27, 1991.)