The annual glaciohydrology cycle in the ablation zone of ...The annual glaciohydrology cycle in the...

The annual glaciohydrology cycle in the ablation zone of theGreenland ice sheet Part 2 Observed and modeled ice flow

William COLGAN12 Harihar RAJARAM3 Robert S ANDERSON45

Konrad STEFFEN12 H Jay ZWALLY6 Thomas PHILLIPS1 Waleed ABDALATI127

1Cooperative Institute for Research in Environmental Sciences University of Colorado Boulder CO USAE-mail williamcolgancoloradoedu

2Department of Geography University of Colorado Boulder CO USA3Department of Civil Environmental and Architectural Engineering University of Colorado

Boulder CO USA4Institute of Arctic and Alpine Research University of Colorado Boulder CO USA5Department of Geological Sciences University of Colorado Boulder CO USA

6Goddard Space Flight Center National Aeronautics and Space Administration Greenbelt MD USA7Headquarters National Aeronautics and Space Administration Washington DC USA

ABSTRACT Ice velocities observed in 200506 at three GPS stations along the Sermeq Avannarleqflowline West Greenland are used to characterize an observed annual velocity cycle We attempt toreproduce this annual ice velocity cycle using a 1-D ice-flow model with longitudinal stresses coupled toa 1-D hydrology model that governs an empirical basal sliding rule Seasonal basal sliding velocity isparameterized as a perturbation of prescribed winter sliding velocity that is proportional to the rate ofchange of glacier water storage The coupled model reproduces the broad features of the annual basalsliding cycle observed along this flowline namely a summer speed-up event followed by a fall slowdownevent We also evaluate the hypothesis that the observed annual velocity cycle is due to the annualcalving cycle at the terminus We demonstrate that the ice acceleration due to a catastrophic calvingevent takes an order of magnitude longer to reach CUETH (lsquoSwissrsquo) Camp (46 km upstream of theterminus) than is observed The seasonal acceleration observed at Swiss Camp is therefore unlikely to bethe result of velocity perturbations propagated upstream via longitudinal coupling Instead we interpretthis velocity cycle to reflect the local history of glacier water balance

1 INTRODUCTIONIce discharge from marine outlet glaciers is a function ofdeformational and basal sliding velocities It has beensuggested that relatively small increases in surface ablationmay result in disproportionately large increases in icedischarge via basal sliding (Zwally and others 2002Bartholomew and others 2010) Recent interferometricsynthetic aperture radar (InSAR) observations have con-firmed that an annual velocity cycle is spatially widespreadin the marginal ice of West Greenland This annual velocitycycle is most likely due to seasonal changes in basal slidingvelocity (Joughin and others 2008) Recently howeversome studies have speculated that projected increases insurface meltwater production will likely result in a netdecrease in basal sliding velocity due to a transition fromrelatively inefficient to efficient subglacial drainage (Schoof2010 Sundal and others 2011) This motivates the need toquantitatively address the physical relation between glacierhydrology and basal sliding velocity We therefore seek acomputationally efficient means of reproducing the ob-served spatial and temporal patterns of basal sliding so thatwe can ultimately explore the likely response of outletglacier dynamics to climate change scenarios Other obser-vations and models indicate that the ice discharge fromGreenlandrsquos marine-terminating glaciers is highly sensitiveto calving-front perturbations which are subsequentlypropagated upstream via longitudinal coupling (Hollandand others 2008 Joughin and others 2008 Nick andothers 2009) We also explore this alternative mechanism asa cause for the observed annual velocity cycle

The Sermeq Avannarleq ablation zone in West Greenlandexhibits an annual ice velocity cycle similar to that of analpine glacier The critical features of this cycle are asummer speed-up event followed by a fall slowdown event(Colgan and others 2011a) A qualitative comparison of thisannual ice velocity cycle to a modeled annual glacier waterstorage cycle suggests that enhanced (suppressed) basalsliding generally occurs during periods of positive (negative)rates of change of glacier water storage (Colgan and others2011a) This notion is consistent with alpine glacier studiesthat suggest that changes in basal sliding velocity are due tochanges in the rate of change of glacier water storage (dSdtor the difference between rates of glacier water input andoutput Kamb and others 1994 Anderson and others 2004Bartholomaus and others 2008 2011) In this conceptualmodel three general basal sliding states exist (1) when localmeltwater input exceeds the transmission ability of thesubglacial hydrologic system (ie dSdtgt0 or increasingglacier water storage) (2) when the transmission ability ofthe subglacial hydrologic system exceeds the local input ofmeltwater (ie dSdtlt0 or decreasing glacier water storage)and (3) when meltwater input and subglacial transmissionability are in approximate equilibrium In alpine settingspeak basal sliding velocity can be expected when dSdtreaches a maximum (although there is some evidence thatpeak sliding velocity exhibits a slight phase-lag behind peakdSdt values Bartholomaus and others 2008) Following thismaximum both dSdt and basal sliding velocity decreaseand dSdt becomes negative during the later part of the meltseason when efficient conduits can transmit more water than

Journal of Glaciology Vol 58 No 207 2012 doi 1031892012JoG11J081 51

httpsntrsnasagovsearchjspR=20140008938 2020-05-20T034045+0000Z

is delivered to the englacial and subglacial system throughsurface melt

Our goal is to reproduce the annual ice velocity cycleobserved in the Sermeq Avannarleq ablation zone withcoupled ice-flow and hydrology models In this paper wecouple a one-dimensional (1-D) (depth-integrated) ice-flowmodel to a 1-D (depth-integrated) glacier hydrology model(Colgan and others 2011a) via a semi-empirical sliding ruleOur goal is not to reproduce specific observed intra- orinterannual variations in ice velocity Rather we attempt toreproduce an annual glaciohydrology cycle in dynamicequilibrium that reproduces the critical features of theobserved cycle and may thus serve as a basis for future workinvestigating the influence of interannual variations insurface ablation on annual ice displacement The 530 kmSermeq Avannarleq flowline runs up-glacier from its

tidewater terminus (km0 at 69378N 50288W) to the mainice divide of the Greenland ice sheet (km530 at 71548N37818W Fig 1) This flowline lies within 2 km of threeGreenland Climate Network (GC-Net Steffen and Box2001) automatic weather stations (AWSs) JAR2 (km140 at69428N 50088W) JAR1 (km325 at 69508N 49708W)and CUETH (lsquoSwissrsquo) Camp (km460 at 69568N 49348W)(all positions reported for 2008) In a companion paper to thisstudy (Colgan and others 2011a) we suggested that in theSermeq Avannarleq ablation zone (1) englacial water tableelevation which may be taken as a proxy for glacier waterstorage oscillates around levels that are relatively close toflotation throughout the year and (2) observed periods ofenhanced (suppressed) basal sliding qualitatively correspondto modeled periods of increasing (decreasing) glacier waterstorage In this paper we propose a semi-empirical and site-specific sliding rule that relates variations in the modeled rateof change of glacier water storage to observed variations inbasal sliding velocity

2 METHODS

21 Observed annual ice surface velocity cycleWe characterize the annual ice surface velocity cycle atthree locations (JAR2 JAR1 and Swiss Camp) along theterminal 55 km of the Sermeq Avannarleq flowline usingdifferential GPS observations of 10 day mean ice surfacevelocity in 2005 and 2006 (Larson and others 2001 Zwallyand others 2002) This characterization provides a represen-tative annual ice velocity cycle against which the accuracyof the modeled annual ice velocity cycle can be assessed Atall sites these observations reveal that the ice moves atwinter velocity until the beginning of a summer speed-upevent in which ice velocities increase above winter velocity(Fig 2) The summer speed-up event is followed by a fallslowdown event in which ice velocities decrease belowwinter velocities Using the positive degree-days (PDDs)observed at each station as a proxy for melt intensity(Ohmura 2001 Steffen and Box 2001) the onset of thespeed-up approximately coincides with the onset of summer

Fig 1 The terminal 55 km of the Sermeq Avannarleq flowline overlaid on a panchromatic WorldView-1 image (acquired 15 July 2009) withdistance from the terminus indicated in km The GC-Net AWSs are denoted in red Inset shows the location of Sermeq Avannarleq in WestGreenland

Fig 2 Observed 10 day mean ice surface velocity (grey lines) andcumulative positive degree-days (PDD red lines) in 2005 and 2006at Swiss Camp (SC) JAR1 and JAR2 (where available) versus day ofyear Black lines denote the bi-Gaussian characterization of theannual ice surface velocity cycle at each station (Eqn (1))

Colgan and others Observed and modeled ice flow52

melt and reaches a maximum approximately halfwaythrough the melt season while the slowdown occurs afterthe cessation of melt

We approximate the annual ice surface velocity cycleusing two Gaussian curves These are overlaid on meanwinter velocity with one (positive) curve representing thesummer speed-up event and the other (negative) curverepresenting the fall slowdown event The use of a bi-Gaussian function allows the amplitude width and timing ofboth the summer and fall events to be parameterizedindependently Thus we characterize surface velocity usas a function of day of year j according to

us frac14 uw thorn ethumax uwTHORN exp j jmax

dmax

2

uw uw umineth THORNfrac12 exp j jmin

dmin

2eth1THORN

where uw is the mean winter velocity and umax and umin arethe summer maximum and fall minimum velocities respect-ively The remaining four parameters govern the timing andshape of the speed-up and slowdown curves jmax (jmin) is theday of year of summer maximum (fall minimum) velocitywhile dmax (dmin) is the duration of the summer (fall) velocityanomaly All parameters were prescribed manually by visualinspection (Table 1) For jmax and jmin we assess an estimateduncertainty equivalent to the temporal resolution of thevelocity data (10 days) This bi-Gaussian characterizationwas fitted to the aggregated 2005 and 2006 velocity data ateach station (Fig 2)

22 Ice-flow modelWe apply a longitudinally coupled 1-D (depth-integrated)ice-flow model to the Sermeq Avannarleq flowline Thismodel solves for the rate of change in ice thickness (Ht)according to mass conservation

Ht

frac14 b Qx

eth2THORN

where b is the annual mass balance Q is the ice dischargeper unit width and Qx is the horizontal divergence of icedischarge To generate dynamic equilibrium ice geometryand velocity fields the ice-flow model was subjected to a1000 year spin-up that was initialized with present-day icegeometry and a lsquocoolerrsquo climate with no hydrology cycle(described in Section 24) We characterize dynamic equi-librium as the transient solution of Eqn (2) that exhibits nosignificant changes in ice thickness (|Ht| lt 1mandash1) duringthe last 100 years of spin-up Alternative approaches wouldbe to produce (1) a fully transient non-equilibrium present-day snapshot of flowline ice geometry and velocity by spin-up under a prescribed climate scenario or (2) a steady-statesolution of flowline ice geometry and velocity underimposed spin-up conditions The former would certainlybe desirable for modeling future flowline evolution but issensitive to uncertainties in the prescribed climate forcingThe latter requires the implementation of boundary condi-tions at both ends of the flowline and steady-state calvingflux is not precisely known in this instance due touncertainty in flowline delineation Following the 1000 yearspin-up an annual basal sliding cycle is introduced via thecoupled 1-D (depth-integrated) hydrology model We use asemi-empirical three-phase sliding rule (described in Section23) to convert variations in the rate of change of glacier

water storage calculated by the hydrology model intovariations in basal sliding velocity

221 Annual balanceThe annual mass balance of a given ice column is the sum ofthe annual surface accumulation cs surface ablation asbasal accumulation cb and basal ablation ab

b frac14 cs thorn Fas thorn cb thorn ab eth3THORNwhere F is the hydrologic system entry fraction based on theratio of annual surface accumulation to annual surfaceablation (Pfeffer and others 1991 Colgan and others2011a) As F is the fraction of ablation assumed to enterthe glacier hydrology system and eventually flow out of theice sheet the quantity 1 F is the fraction of ablation thatrefreezes and does not leave the ice sheet This assumes thatpurely supraglacial transport to the margin is negligible atSermeq Avannarleq all runoff is expected to drain into theenglacial system via either crevasses or moulins (McGrathand others 2011) Annual surface accumulation is pre-scribed as the observed mean annual value over the period1991ndash2000 (Burgess and others 2010) Annual accumu-lation increases from 025m at the Sermeq Avannarleqterminus to a maximum (05m) at 100 km upstream anddecreases again to 025m at the main flow divide (530 kmupstream) In the ablation zone annual surface ablation asis taken to be a function of elevation based on previousobservations

as frac14 ethzs zelaTHORN aela eth4THORNwhere is the present-day ablation gradient with elevation(as zs taken as 000372 Fausto and others 2009) zs isthe ice surface elevation zela is the equilibrium-line altitude(ELA) and aela is the annual surface ablation at the ELA (takenas 04m) The observed regional ELA was 1125m over theperiod 1995ndash99 (Steffen and Box 2001) and 1250m overthe period 1996ndash2006 (Fausto and others 2009) Weprescribe an ELA of 1125m as it is more likely to beconsistent with the steady-state surface mass-balance forcingprior to the highly transient post-1990 period Annualsurface abalation is distributed over an annual cycle toyield surface ablation rate _as using a sine function torepresent the melt season solar insolation history (cf fig 4 inColgan and others 2011a)

In the ice-flow model we assume that annual basalaccumulation is negligible (cb 0mandash1) and that submarinebasal ablation is only significant beneath the floatingtidewater tongue (eg Rignot and others 2010) We usethe relative magnitudes of ice and englacial water pressures

Table 1 The value of each parameter in the bi-Gaussiancharacterization of the annual ice surface velocity cycle at JAR2JAR1 and Swiss Camp (Eqn (1))

JAR2 JAR1 Swiss Camp

uw (mandash1) 105 66 113umax (m andash1) 195 95 175umin (mandash1) 80 59 101jmax (days) 175 200 200jmin (days) 255 250 235dmax (days) 30 25 12dmin (days) 30 20 25

Colgan and others Observed and modeled ice flow 53

Pi and Pw respectively to determine which flowline nodesare grounded (Pi Pw) or floating (Pi lt Pw) in a given time-step During spin-up and dynamic equilibrium we prescribea constant submarine basal ablation rate of ab = 10mandash1 toall floating nodes This prescribed rate is less than thecontemporary submarine basal ablation rate at SermeqAvannarleq which is estimated to exceed 25mandash1 (Rignotand others 2010) The ice-flow model however does notreproduce a floating tongue when contemporary submarinebasal ablation rates are imposed for the duration of spin-upAs our intent is to reproduce the dynamic equilibrium icegeometry that precedes the current rapid transient state wedepart from the present-day submarine basal ablation rate

222 Ice dischargeWe include depth-averaged longitudinal coupling stress

0xx as a perturbation to the gravitational driving stress

derived from the shallow-ice approximation (Van der Veen1987 Marshall and others 2005)

frac14 igHzsx

thorn 2

xethH

0xxTHORN eth5THORN

where is total driving stress Depth-averaged longitudinalcoupling stress ( 0xx ) is calculated following the approachoutlined by Van der Veen (1987) This formulation deriveslongitudinal coupling stress by solving a cubic equationdescribing equilibrium forces independently at each nodebased on ice geometry and prescribed basal sliding velocityub

0 frac14 0xx3 2

zsx

Hx

zsx

thornH

2zsx2 1

2

thorn 0xx2

23Hx

32zsx

thorn 0xx 2 3zsx

Hx

thorn 32H2zsx2 2

zsx

2 16

( )

thorn 325Hx

14zsx

thorn 12A

ubx

eth6THORNThe depth-integrated longitudinally coupled ice velocity

due to deformation ud may be derived from the equationfor horizontal shear rate ud z (eg Marshall and

others 2005)

udz

frac14

2A igethzs zTHORN zsx

n1 igethzs zTHORN zs

xthorn 2

xH

0xx

eth7THORN

where zs x is ice surface slope along the flowline and n isan exponent of 3 in the empirical relation between stressand strain rate describing ice rheology (Glen 1955)Integrating Eqn (7) twice in the vertical and dividing by Hyields

ud frac14 2An thorn 2

igHzsx

n1

igHzsx

thorn 2

xH

0xx

H

eth8THORNWe calculate the flow-law parameter A as a function ofboth ice temperature T and thickness H using anArrhenius-type relation (Huybrechts and others 1991)

AethT HTHORN frac14 EethHTHORNAoethT THORN exp QeethT THORNRT

eth9THORN

where Ao is a coefficient that depends on ice temperature(taken as 547 1010 Pandash3 andash1 when T26315 K and114 10ndash5 Pandash3 andash1 when Tlt26315K) Qe is the creepactivation energy of ice (taken as 139 kJmolndash1 whenT26315K and 60 kJmolndash1 when Tlt26315K) R is theideal gas constant (8314 Jmolndash1 Kndash1) and T is the icetemperature At each flowline node we use the steady-stateice temperature at 90 depth derived from independentthermodynamic modeling of the flowline (personal commu-nication from T Phillips and others 2011) to calculate theflow-law parameter The majority of shear occurs at or belowthis depth Thus along-flowline variations in basal icetemperature result in along-flowline variations in the flow-law parameter

We enhance the flow-law parameter by a factor E toaccount for increased deformation due to the presence ofrelatively soft Wisconsin basal ice This enhancement factorlinearly transitions from its prescribed value where icethickness exceeds 650m to 1 (ie no enhancement) whereice thickness is lt550m At flowline nodes where ice isgt650m thick Wisconsin ice is expected to comprise asignificant portion of the basal ice (Huybrechts 1994) Weassume that the uncertainty associated with the calculatedvalues of A is small in comparison to uncertainty associatedwith the Wisconsin flow enhancement factor We evaluatedynamic equilibrium ice geometry and velocity fieldsfollowing spin-up with E ranging between 2 and 4 (Reeh1985 Paterson 1991) as in situ borehole deformationmeasurements beneath nearby Jakobshavn Isbraelig indicateEgt1 (Luthi and others 2002) Under the E=3 scenariocalculated flow-law parameter values range between8210ndash18 and 4310ndash16 Pandash3 andash1 For comparison therecommended unenhanced (E=1) flow-law parameter at273K is 21 10ndash16 Pandash3 andash1 (Paterson 1994 Fig 3)

Local ice discharge Q is obtained by multiplying thesum of basal sliding and depth-averaged deformationalvelocities by ice thickness

Q frac14 ub thorn udeth THORNH eth10THORNBasal sliding velocity is prescribed via a semi-empiricalsliding rule described in Section 23

Following the approach taken in previous ice-sheetflowline models (eg Van der Veen 1987 Parizek and

Fig 3 The flow-law parameter values A calculated according toEqn (9) using the Wisconsin enhancement factor values E (inferredby observed ice thickness) and modeled ice temperature values Talong the Sermeq Avannarleq flowline Recommended values forE=1 ice are shown for comparison (Paterson 1994)


Alley 2004) we neglect lateral velocities and stressesstemming from divergence and convergence While thisassumption is likely valid in interior regions of ice sheets itis less valid near the ice-sheet margin where substantialdivergence and convergence can occur We acknowledge inSection 3 that the failure to account for possible lateraleffects is a potential factor in the systematic overestimationof ice velocities at JAR1 and underestimation of icevelocities at the terminus Another inherent shortcoming ofany 1-D flowline model is the prescription of the ice-flowtrajectory For example if the flowline length has beenunderestimated in the accumulation zone modeled icedischarge across the equilibrium line will be underestimatedand the model will require a decrease in surface ablationrate to maintain the observed ice geometry In addition theice-flow trajectory is unlikely to have been constant throughtime Striations on the ice surface mapped by Thomsen andothers (1988) suggest that the flowline through JAR1 stationlikely terminated on land just north of Sermeq Avannarleq in1985 rather than flowing past JAR2 to the tidewaterterminus as shown in Figure 1 The recent acceleration ofJakobshavn Isbraelig immediately south of Sermeq Avannarleqhas caused substantial reorientation of ice flow throughoutthe Sermeq Avannarleq ablation zone (Colgan and others2011b) As the flowline used in this study was derived from a200506 InSAR ice surface velocity field (Joughin and others2010) obtained after the onset of the reorganization of iceflow (1997 Colgan and others 2011b) it does notaccurately reflect the long-term ablation zone flowlinetrajectory

23 Three-phase basal sliding ruleThe sliding rules employed in glacier models have improvedwith advances in the conceptualization of basal slidingInitial sliding rules prescribed basal sliding velocity asproportional to driving stress on the assumption that higherdriving stress results in greater till deformation (Weertman1957 Kamb 1970) Observations that subglacial water wascapable of enhancing ice velocities by lubricating andpressurizing the subglacial environment led to parameter-izations in which basal sliding velocity was taken asproportional to subglacial water pressure (Iken 1981 Ikenand others 1983) as well as sliding rules that included botheffective water pressure (Pi ndash Pw) and driving stress (Bind-schadler 1983 Iken and Bindschadler 1986) Recentmodels have utilized basal sliding rules that are Coulombfriction analogues whereby basal drag is parameterized totake on a maximum value at low sliding velocities anddecrease with decreasing effective pressure and increasingsliding velocity (Schoof 2005 Gagliardini and others 2007Pimentel and Flowers 2010) To have predictive value abasal sliding rule should ideally be capable of reproducingobserved sliding velocities from first principles of hydrologyand force balance with a minimum of free parameters Herewe describe an empirical and site-specific three-phase basalsliding rule that focuses on the hydrologic aspect of basalsliding We regulate the magnitude and sign of a perturb-ation to background basal sliding velocities using modeledrates of change of glacier water storage

The Swiss Camp GPS data indicate that ice velocity is onaverage 14 faster during the winter than in the midst ofthe slowest motion of the year during the fall slowdown(113 and 99mandash1 respectively Fig 2 Table 1) This fallvelocity minimum matches very well the ice surface velocity

predicted by internal deformation alone According to theshallow-ice approximation the ice surface velocity solelydue to deformation may be calculated as (Hooke 2005)

ud zseth THORN frac14 2An thorn 1

igzsx

nHnthorn1 eth11THORN

Taking ice thickness H as 950m ice surface slope zs xas 001 and A as 33310ndash16 Pandash3 andash1 (the flow-lawparameter for ice at 272K which is the pressure-meltingpoint beneath 1 km of ice enhanced by a Wisconsin factorof 3) yields an ice surface velocity of 99mandash1

We interpret this as suggesting that Swiss Campexperiences significant background basal sliding velocityduring the winter which is suppressed during the fallvelocity minimum Thus at Swiss Camp (and similarly atJAR1 and JAR2) the background basal sliding velocity maybe approximated as the difference between observed meanwinter (uw) and fall minimum (umin) velocities We linearlyinterpolate these basal sliding velocities along the flowlineto provide the background basal sliding boundary condi-tion used in spin-up of the ice-flow model (Fig 4) Theyear-round persistence of the englacial hydrologic systemin the Sermeq Avannarleq ablation zone provides amechanism capable of maintaining favorable basal slidingconditions year-round (ie available liquid water Cataniaand Neumann 2010) After spin-up we overlay an annualbasal sliding velocity cycle on the background basalsliding velocity

Following theoretical developments in alpine glacio-hydrology (Kamb and others 1994 Anderson and others2004 Bartholomaus and others 2008 2011) we propose asliding rule that depends on the sign of the rate of change ofglacier water storage to prescribe lsquospeed-uprsquo during periodsof increasing glacier water storage and lsquoslowdownrsquo duringperiods of decreasing glacier water storage We take rate ofchange of englacial water table elevation (or head het)as a surrogate for rate of change of glacier water storage(Colgan and others 2011a) We formulate a three-phasebasal sliding rule that imposes (1) background basal slidingvelocity during the winter when he t0 (2) enhancedbasal sliding during positive rates of change of glacier waterstorage (hetgt0) and (3) suppressed basal sliding duringnegative rates of change of glacier water storage (hetlt0)We accomplish this by conceptualizing ub as the sum of

Fig 4 Background basal sliding velocity ubo estimated at SwissCamp (SC) JAR1 and JAR2 as the difference between mean winter(uw) and fall minimum (umin) velocities (Table 1) A least-squareslinear interpolationextrapolation to the terminal 65 km of theflowline is also shown


background basal sliding velocity ubo and a perturbationub

ub frac14 ubo thornub limitub 0 eth12THORNThe alpine glaciohydrology literature suggests that we mayexpect basal sliding velocity to scale nonlinearly with therate of change of glacier water storage (ie ub (he t)

mwhere mgt1 Anderson and others 2004 Bartholomaus andothers 2008) We impose m=3 and express the perturb-ation to the background basal velocity as

ub frac14 ncethxTHORNkethxTHORN het

m sign he

t

if he

t

025md1

0 if het

lt025md1

(

eth13THORNwhere nc(x) is the number of subglacial conduits mndash1 in theacross-ice-flow direction (Colgan and others 2011a) andk(x) is a tunable site-specific sliding coefficient We imposean arbitrary threshold of 025mdndash1 (constant in time andspace) that rates of change of glacier water storage mustexceed in order to either enhance or suppress backgroundbasal sliding We tune the sliding coefficient values (whichrange between 025 and 075m13 d23) to reach aminimum in the vicinity of JAR1 based on the observationthat the annual velocity cycle at JAR1 is damped incomparison to JAR2 and Swiss Camp (Fig 5)

The number of conduits per meter in the across-ice-flowdirection reflects variations in the configuration of thesubglacial drainage system with distance upstream fromrelatively large widely spaced conduits near the terminus torelatively small closely spaced conduits near the equilibriumline These differences in subglacial hydrologic systemconfiguration can be expected to result in differing slidingresponses to a given rate of change of glacier water storageFor example the enhanced sliding due to a given meltwaterinput is expected to be greater for a subglacial networkcomprising lsquocavitiesrsquo than for a subglacial network compris-ing lsquochannelsrsquo (Schoof 2010) As the sign of the rate ofchange of glacier water storage he t changes betweenpositive and negative it effectively modulates the sign ofub

(ie specifying whether the perturbation is acting to enhanceor suppress background basal sliding velocity) By para-meterizing ub so that it goes to zero when |het| fallsbelow a critical threshold (taken as 025mdndash1) Eqn (13)provides the framework for three phases of basal sliding

24 Input datasets and boundary conditionsFollowing Colgan and others (2011a) the ice-flow model isinitialized with observed ice surface and bedrock topog-raphy (Bamber and others 2001 Scambos and Haran 2002Plummer and others 2008) During spin-up the observedsurface mass balance was adjusted by decreasing surfaceablation in order to reproduce the observed present-day icegeometry We evaluate dynamic equilibrium ice geometryand velocity fields following spin-ups by decreasing surfaceablation by a factor of 25ndash75 Previous Greenland icesheet modeling studies have implemented similar surfacemass-balance corrections during spin-up in order to achieveequilibrium present-day ice geometries (ie Huybrechts1994 Ritz and others 1997 Parizek and Alley 2004) Thisadjustment is typically justified by the notion that thepresent-day Greenland ice sheet geometry reflects colderlsquoglacialrsquo conditions (ie less surface ablation with no changein accumulation) Both the observed surface accumulation(Burgess and others 2010) and ablation (Fausto and others2009) datasets have been validated by in situ observationsincluding the GC-Net AWSs along the Sermeq Avannarleqflowline (Steffen and Box 2001 Fig 1)

The differential equations describing the evolution of icethickness (Eqns (2ndash10)) were discretized in space using first-order finite volume methods with grid spacing x=500mThe semi-discrete set of coupled ordinary differential equa-tions at the computational nodes was then solved usinglsquoode15srsquo the stiff differential equation solver in MATLABR2008b During the 1000 year spin-up the ice-flow modelwas solved with a time-step t of 10 years Following spin-up the ice-flow model was solved concurrently with thehydrology model with a 2 day time-step Post-spin-up basalsliding velocity was calculated according to the three-phasesliding rule described above We apply a second-type (zeroflux) Neumann boundary condition at the main ice-flowdivide as the upstream boundary condition (zs x=0 andQ=0 at x=530 km) When the glacier terminus reaches thedownstream boundary of the model domain (km0) theterminal node ice discharge (Qterm or calving flux) iscalculated as the difference between the ice discharge ofthe adjacent upstream node and the annual mass balance(negative in the ablation zone) of the terminal node(Qterm =Qtermndash1 + bx at x=0 km) When the glacier ter-minus does not reach the downstream boundary of themodel domain no calving flux is imposed as a dynamicequilibrium has been achieved in which ice inflow acrossthe grounding line is balanced by total ablation (both surfaceand submarine) in the floating ice tongue

The 1-D (depth-integrated) hydrology model tracksglacier water storage and discharge through time Glacierwater input is prescribed based on observed ablation rateswhereas glacier water output occurs through conduitdischarge Conduit discharge varies in response to thedynamic evolution of conduit radius When coupled theice-flow and hydrology models receive the same surfaceablation forcing at each time-step The ice-flow modelupdates ice geometry used by the hydrology model eachtime-step while the hydrology model updates the subglacial

Fig 5 Along-flowline distributions of the parameters used tocalculate basal sliding perturbation (Eqn (13)) (a) sliding rulecoefficient k (b) subglacial conduits mndash1 in the across-flowdirection nc (Colgan and others 2011a) and (c) daily profiles ofrate of change of glacier water storage het over an annual cycle(Colgan and others 2011a) Vertical dashed lines denote thelocations of JAR2 JAR1 and Swiss Camp (SC)


head values used by the basal sliding rule Specificparameterizations used in the hydrology model are shownin Table 2 (cf animation 1 in Colgan and others 2011a)

3 RESULTSThe GPS-observed ice velocities provide insight into therelative importance of basal sliding in the Sermeq Avannar-leq ablation zone (Fig 2) The net annual displacementsobserved at JAR2 JAR1 and Swiss Camp are about 114 69and 115m respectively The fall minimum velocities takento represent purely deformational velocities (ie no basalsliding) observed at each of these stations are about 80 59and 101mandash1 respectively Therefore basal sliding appearsto be responsible for about 30 14 and 12 of the annual netdisplacement respectively These observations suggest thatsummer speed-up events are responsible for about 13 4 and4m of net displacement at JAR2 JAR1 and Swiss Camprespectively equivalent to about 11 6 and 3 of the annualdisplacement These increases in net displacement due tosummer speed-up events are partially offset by decreases innet displacement due to fall slowdown events The fallslowdown events suppress annual displacement by about 41 and 2m at JAR2 JAR1 and Swiss Camp respectivelyequivalent to about 4 1 and 2 of the annual displacementThus at Sermeq Avannarleq year-round (or lsquobackgroundrsquo)basal sliding appears to contribute a larger fraction of annualnet displacement than seasonal basal sliding

We compared modeled dynamic equilibrium ice geom-etry and velocity fields of the ice-flow model with observedice surface elevation (Scambos and Haran 2002) andvelocity (Joughin and others 2008) following a 1000 yearspin-up We explored a range of Wisconsin enhancementfactors (2ndash4) and surface ablation perturbations (decreases of25ndash75 below contemporary rates) While this parameterspace produces a relatively narrow range of ice surfacegeometries that are similar to the observed profile itproduces a relatively wide range of ice velocities that candiffer substantially from the observed profile (Fig 6) A

comparison between modeled mean ice surface elevationand velocity versus observed mean ice surface elevation(688m) and velocity (116mandash1) along the terminal 50 km ofthe flowline suggests that observations are most accuratelyreproduced with a 50 decrease in surface ablation belowcontemporary rates and a Wisconsin enhancement factor ofE=3 (Table 3) This depression in contemporary surfaceablation may represent a combination of (1) an under-estimation of historical accumulation rate (2) the increase insurface ablation that has occurred since the termination ofthe last glaciation and (3) error in our delineation ofthe flowline

Under all spin-up scenarios the ice-flow model over-estimates ice velocities in the km25ndash35 portion of theflowline and underestimates ice velocities in the terminal5 km We speculate that both systematic errors are artifactsof the 1-D (flowline) character of the ice-flow model A 1-Dmodel is expected to overestimate ice discharge (and henceice velocity) in reaches of divergent ice flow Thus theoverestimation of ice velocities in the vicinity of JAR1 maysuggest that divergent ice flow is occurring in this regionSimilarly a 1-D model is expected to underestimate icevelocities in regions of convergent ice flow Failure toaccount for convergent flow at the terminus has likelycontributed to the severe underestimation of terminus ice

Table 2 Specific parameterization of the 1-D (depth-integrated)hydrology model (notation follows Colgan and others 2011a)

Variable Definition Value

Glacier hydrology length scale 20 km Bulk ice porosity 001Fr Firn meltwater retention fraction 05Qg Geothermal flux 57mWmndash2

f Conduit friction factor 005ntermc Conduit spacing at terminus 0005mndash1

r termmax Maximum conduit radius at terminus 2m

Fig 6 (a) Modeled ice surface elevation zs and (b) modeled ice surface velocity us over a range of Wisconsin enhancement factors E andfractional contemporary surface ablation values as Observed ice surface elevation (Scambos and Haran 2002) and velocity (Joughin andothers 2008) are shown for comparison The basal boundary conditions (BC) for ice surface elevation and velocity are observed bedrockelevation and prescribed background basal sliding velocity Vertical dashed lines denote the locations of JAR2 JAR1 and Swiss Camp (SC)


thickness Observations suggest the bedrock overdeepeningat the terminus contains grounded ice 600m thick(Plummer and others 2008) while the modeled terminus isfloating with ice thicknesses of 300m

Generally however the ice-flow model achieves adynamic equilibrium ice geometry and velocity field thatclosely matches observations throughout the Sermeq Avan-narleq ablation zone (ie terminal 60 km of the flowline)This suggests that the basal sliding boundary conditionimposed during the 1000 year spin-up (ie lsquobackgroundbasal slidingrsquo Fig 4) captures the essence of the basalsliding profile beneath the terminal 60 km of the flowlineWhen we allow the annual hydrologic cycle to operateproducing seasonal variations in basal sliding around thisbackground basal sliding profile the three-phase basalsliding rule produces a reasonable annual basal slidingvelocity cycle throughout the Sermeq Avannarleq ablationzone (Animation 1) While the absolute magnitude andseasonal timing of modeled maximum and minimumvelocities do not precisely match observations (discussed

in Section 41) the coupled model produces an annual basalsliding cycle that captures the essence of the observedannual velocity cycle namely (1) prescribed backgroundbasal sliding velocity during the winter (2) a summer speed-up event followed by (3) a fall slowdown event

4 DISCUSSIONWe developed a representation of basal sliding velocity thatdepends on the local rate of change of glacier water storageIncorporating this basal boundary condition in a 1-D ice-flow model captures the essence of the annual velocity cycleof the Sermeq Avannarleq flowline (ie summer speed-upand fall slowdown events) The mean annual glaciohydrol-ogy cycle reproduced by this coupled model may serve as abasis for future investigations into the transient response ofthe Sermeq Avannarleq flowline to predicted increases insurface meltwater production While we do not drive ourbasal sliding rule with absolute head values he but ratherwith the rate of change of head het we achieve a

Animation 1 Animation of the annual glaciohydrology cycle (a) Surface ablation rate as (b) Bedrock elevation (brown) transient icegeometry (black line) and transient englacial water table elevation (blue) (Colgan and others 2011a) (c) Basal sliding velocity calculatedfrom the semi-empirical three-phase sliding rule (Eqns (12) and (13)) (d) Bedrock elevation (brown) and transient ice geometry (blackline) with contour shading to denote ice surface velocity us (color bar saturates at 200mandash1) Vertical dashed lines denote the locationsof JAR2 JAR1 and Swiss Camp (SC) Model time is given in day of yearFull movie available at httpwwwigsocorghyperlink11J081_Animation1mov

Table 3 Discrepancy between modeled and observed mean ice surface elevation zs and mean ice surface velocity us along the terminal50 km of the flowline under various scenarios of fraction of contemporary surface ablation rate as and Wisconsin flow-law enhancementfactor E Discrepancies are expressed in both absolute and relative values

E 025 as 050 as 075 as

zs us zs us zs us

m mandash1 m mandash1 m mandash1

2 32 (5) ndash11 (ndash9) 2 (0) ndash27 (ndash24) ndash101 (ndash15) ndash42 (ndash36)3 48 (7) 17 (15) 30 (4) 4 (3) ndash11 (ndash2) ndash19 (ndash17)4 59 (9) 40 (34) 45 (7) 28 (24) 25 (4) 12 (11)


satisfactory annual basal sliding cycle (Fig 7) We interpretthis as suggesting that the het term contains importantinformation that modulates basal sliding This suggests thatsliding rules that are dependent on basal stress and absolutehead may overlook important variables that are correlatedwith het particularly those related to transient subglacialtransmissivity or higher-order features of subglacial hydro-logic geometry (eg the nc(x) lsquoconduit spacingrsquo parameterused in this study) Recent observations of hysteresis in theratio of inferred subglacial water storage to sliding velocityover the course of a melt season support this notion (Howatand others 2008a) Thus we suggest that an improvementtowards achieving a physically based sliding rule would beto combine an absolute headbasal stress rule that modulatesbackground (ie winter) sliding velocities (which areprescribed in this study) with a het-type parameterizationthat honors the empirical subtlety that sliding reaches amaximum when he t reaches a maximum (rather thanwhen he reaches a maximum)

41 Modeled speed-up and slowdown eventsThere are discrepancies in the absolute magnitude andtiming between modeled and observed velocity maxima andminima at all three stations (Fig 8) As our interest lies inproducing an annual velocity cycle using an annualhydrologic cycle we focus the following discussion primar-ily on temporal discrepancies We speculate that therelatively damped annual velocity cycle observed at JAR1is due to a relatively dampened annual hydrology cycleWhile the hydrology model forces subglacial water to flowover the bedrock high at km30 in reality subglacial waterlikely flows around this bedrock high in the y direction Thusthe hydrology model likely overestimates he t values alongthe km25ndash40 portion of the flowline for which the scalingparameter k(x) compensates In regard to the absolutemagnitude of the annual cycle we note that at any locationalong the flowline an infinite combination of the parametersk(x) andmmay be used to scale het to a desired value Wealso note that the modeled basal sliding velocity cyclesexhibit abrupt transitions between background sliding andspeed-up and slowdown events These abrupt transitions aredue to the imposition of a relatively high threshold of rate ofchange of glacier water storage to transition between the

three sliding phases (|he t| gt 025mdndash1) In reality thisthreshold likely varies in time and space

We take uncertainty in the timing discrepancy asequivalent to the temporal resolution of the velocity data(ie 10 days) Modeled maximum velocity precedesobserved maximum velocity by 2010 days at SwissCamp and 18 10 days at JAR1 (Fig 8) It is more difficultto assess the timing discrepancy of the summer speed-upevent at JAR2 where an apparently premature onset of themodeled slowdown event truncates the speed-up event butthe discrepancy appears to be lt10 10 days The relativelyearly onset of the modeled speed-up event can be attributedto a combination of (1) incorrect timing of the prescribedmeltwater input and (2) the assumption embedded in thehydrology model that meltwater produced at the surface isimmediately routed to the glacier interior where it raises theenglacial water table This second assumption assumes thatthere is no temporary supraglacial meltwater storage (egwithin a saturated snowpack or in ponded water) and thatsupra- and englacial travel times are negligible In realitytemporary supraglacial meltwater storage and travel timecan delay the initial meltwater pulse from reaching theenglacial water table for several weeks after the onset ofmelt (Fountain and Walder 1998 Flowers and Clarke 2002Jansson and others 2003) This lag would be expected to begreater near the equilibrium line (ie at Swiss Camp) than atthe terminus (ie at JAR2) In addition to temporarysupraglacial meltwater storage englacial transfer time canbe expected to range over two orders of magnitude as aconsequence of the morphology of the englacial hydrologicsystem (ie moulin- versus crevasse-type drainage Colganand others 2011b McGrath and others 2011) A moredetailed treatment of supra- and englacial meltwater routingin the hydrology model would likely reduce the discrep-ancies between modeled and observed velocity maxima

The discrepancy between modeled and observed velocityminima is more variable the model produces a significantlyearlier onset at JAR2 (6510 days) an extreme delay atSwiss Camp (11010 days) and a reasonable match atJAR1 (15 10 days Fig 8) We believe that this reflects thedifficulty in capturing the behavior of subglacial conduits inan along-flowline 1-D hydrologymodel The timing of the fallslowdown event depends on the timing of the negative (or

Fig 7Modeled timendashspace distribution of basal sliding velocity ubalong the terminal 60 km of the Sermeq Avannarleq flowlineVertical dashed lines indicate the locations of JAR2 JAR1 and SwissCamp (SC) Color bar saturates at 100mandash1

Fig 8 Modeled and observed basal sliding velocity versus day ofyear at Swiss Camp (SC) JAR1 and JAR2 Discrepancies in thetiming of the summer speed-up and fall slowdown events aredenoted with dashed lines


decreasing) glacier water storage (het) phase This phase isinitiated by the upstream propagation of a nickpoint inenglacial water table elevation that corresponds to theopening of efficient subglacial conduits that in turn lowerglacier water storage (Animation 1 eg Kessler andAnderson 2004) A two-dimensional (2-D) (xy) hydrologicalmodel which allows water to flow both parallel and perpen-dicular to the ice dynamic flowline would be inherentlymore realistic in propagating changes in englacial water tableelevation upstream from various outlets at the ice marginBecause the timing of the slowdown event is controlled bythe evolution of conduit sizes the fall slowdown event ismore dependent on an accurate representation of subglacialconduit dynamics than is the spring speed-up event Thespring speed-up event initiates as long as conduits havecollapsed to their minimum radii (ie lsquoclosedrsquo) which is arelatively simple geometry to capture given the lengthywinter period over which this geometry is achieved

42 Upstream limit of seasonal basal slidingThe observed dates of summer maximum (jmax) and fallminimum (jmin) velocity at JAR2 JAR1 and Swiss Campsuggest that the timing of the fall slowdown event is moresynchronous than the timing of the summer speed-up eventalong the flowline (Fig 9) GPS observations indicate thatthe summer speed-up event lsquopropagatesrsquo (or is transmitted)upstream at 12 kmdndash1 (with a 95 confidence interval of08ndash26 kmdndash1) while the fall slowdown event propagatesdownstream at 17 kmdndash1 (with a 95 confidence intervalof 10ndash52 kmdndash1) We interpret this as suggesting that thetiming of the speed-up event depends on a meteorologicalforcing (ie the onset of melt or a critical surface ablationthreshold) while the timing of the slowdown event isdependent on the development of efficient subglacialtransmission capacity (which is more likely to be synchro-nous over the flowline) We speculate that the theoreticalupstream limit to which the annual basal sliding cycle canpropagate (ie the distance inland of which the ice sheetdoes not experience an annual velocity cycle) is defined bythe upstream convergence of the dates of maximum andminimum velocity The 2005 and 2006 GPS data suggest

that this occurs at 695 km upstream on day of year 223This distance upstream corresponds to the 1350m iceelevation contour which is slightly above the regional ELAover the period 1996ndash2006 (1250m Fausto and others2009) The lower limit of this convergence positiondelineated by the 95 confidence envelope is 55 kmupstream GPS velocity observations at Up50 (km985 at69758N 48148W in 2008) which show no evidence of anannual velocity cycle (Colgan and others 2009) provide anupstream limit for the annual velocity cycle

43 Importance of longitudinal couplingAlthough in situ GPS data indicate that Swiss Camp exhibitsan annual velocity cycle these data cannot indicate whetherthis is due to (1) local meltwater-induced acceleration(Zwally and others 2002 Bartholomew and others 2010) or(2) an annual velocity cycle originating downstream of SwissCamp that is propagated upstream via longitudinal couplingThis downstream annual velocity cycle may be due to eitherlower-elevation meltwater-induced acceleration (Price andothers 2008) or the annual tidewater calving cycle (Howatand others 2008b Joughin and others 2008) A previousstudy has suggested that a 10ndash20 velocity increase at SwissCamp can be achieved by a roughly 100 velocity increaseinitiated 12 km downstream from Swiss Camp that ispropagated upstream through longitudinal coupling (Priceand others 2008) This previous study used a 2-D (cross-sectional) ice-flow model in which basal sliding was par-ameterized to occur through deformation of a fluid layerseveral meters thick underlying the ice This basal fluid layerwas assumed to have an effective viscosity of28104 Pa aFor comparison typical flow-law parameter values(ie A10ndash16 Pandash3 andash1) represent an effective viscosity of25106 Pa a when H=500m zs x=001 and n=3 Inaddition to a vertical structure that is conducive to transmit-ting longitudinal coupling the magnitude of the assumeddownstream seasonal velocity perturbation (ie 100 ordoubling) greatly exceeds that recorded by JAR1 station135 km downstream of Swiss Camp

In the present study the 1-D (depth-integrated) ice-flowmodel suggests that the absolute longitudinal coupling stress| 0xx| Eqn (6) is only significant (defined here as gt10 oftotal driving stress) along the terminal 6 km of the flowline(Fig 10) Upstream of the icefall at 6 km where observedice thickness decreases to lt300m the absolute longitudinal

Fig 9 Observed day of year of maximum (jmax red) and minimum(jmin blue) ice surface velocity at the GC-Net stations versusdistance upstream Vertical whiskers denote 10 days uncertaintyin jmax and jmin at each station Solid lines denote least-squareslinear fit Dashed lines denote 95 confidence bounds for thelinear fit The grey shading denotes the 95 confidence envelopefor the convergence of jmax and jmin The black dot denotes day 223and upstream distance km695

Fig 10 Total driving stress and depth-averaged longitudinalcoupling stress 0xx along the terminal 50 km of the flowline atdynamic equilibrium (ie post-spin-up) Dashed lines represent10 of total driving stress


coupling stresses seldom exceed 10 of total driving stressIn addition any perturbation to the tidewater tongue wouldalso experience rapid radial diffusion (in the x-y plane) withdistance inland This inference of minimal inland couplingstresses fits the theoretical notion that coupling stresses aretypically important only in the terminal few km of ice-sheetflowlines (where ice thickness becomes small) In theseterminal regions the magnitude of coupling stresses can beequivalent to or exceed driving stresses (Van der Veen1987) Where longitudinal coupling stresses are insignificantalong the Sermeq Avannarleq flowline the forces governingice flow can be assumed to be local in nature

The inability of a seasonal terminus perturbation (cfHowat and others 2008b Joughin and others 2008) topropagate upstream to produce the 55 seasonal velocityacceleration observed at Swiss Camp (from uw=113mandash1 toumax = 175mandash1) can also be demonstrated by a simplenumerical simulation In this fully transient simulation wespin-up the ice-flow model for 1000 years under the E=3and 50 of contemporary as scenario To isolate potentialchanges in ice dynamics we disable hydrological couplingand prescribe temporally constant background basal slidingalong the flowline At 1 year after spin-up we impose a first-type (specified head) Dirichlet boundary condition ofH=0m at km8 This boundary condition which instantlyremoves the terminal several km of the flowline is meant torepresent a catastrophic terminus perturbation Despiteimposing an unprecedented terminus perturbation that isan order of magnitude larger than the annual advanceretreatcycle longitudinal coupling stresses inland of JAR2 (km14)fail to exhibit any significant change in the 4 years followingthe perturbation (Animation 2) The ice acceleration ispropagated up to Swiss Camp (km46) not by longitudinalstresses but by changes in ice geometry (ie surface slopesteepening) over 275 years (Fig 11) The modeled

upstream propagation rate (17 kmandash1) is over an order ofmagnitude slower than the upstream propagation rate of thesummer speed-up event inferred from the GPS observations(12 kmdndash1) In addition upon reaching Swiss Camp themodeled velocity perturbation is only 5 of the modeleddynamic equilibrium winter velocity at Swiss Camp whichis an order of magnitude less than the observed 55seasonal acceleration Thus it is unlikely that terminusperturbations associated with the annual retreatadvancecycle (Howat and others 2008b Joughin and others 2008)are propagated upstream by longitudinal coupling (Price andothers 2008) to produce the seasonal velocity variationsobserved inland of JAR2 (km14) Instead we interpret the

Animation 2 Animation of the terminus perturbation simulation (a) Total driving stress and depth-averaged longitudinal coupling stress 0xx (b) Bedrock elevation (brown) dynamic equilibrium ice geometry at year 0 (grey line) and transient ice geometry (black line) withcontour shading to denote ice surface velocity us (color bar saturates at 200mandash1) The vertical dashed red line denotes the location of thefirst-type boundary condition imposed to represent a catastrophic terminus perturbation (c) Prescribed background basal sliding velocityVertical dashed lines denote the locations of JAR2 JAR1 and Swiss Camp (SC) Model time (in years) given as relative to the end of spin-upFull movie available at httpwwwigsocorghyperlink11J081_Animation2mov

Fig 11 Timendashspace distribution of ice surface velocity us in theterminus perturbation simulation (color bar saturates at 200mandash1)Labeled black contours represent the relative magnitude of thevelocity perturbation (relative to dynamic equilibrium velocity inyear 0) resulting from the catastrophic terminus perturbation event atyear 1 Vertical dashed lines denote the locations of JAR2 JAR1 andSwiss Camp (SC) Model time given as relative to the end of spin-up


annual velocity cycle at upstream sites to reflect localmismatches between glacier water inputs and outputs (ielocal melt-induced acceleration Zwally and others 2002Kessler and Anderson 2004)

5 SUMMARY REMARKSWe examined the annual glaciohydrology cycle in theablation zone of the Sermeq Avannarleq flowline Wecoupled a 1-D (depth-integrated) ice-flow model to apreviously published 1-D (depth-integrated) hydrologymodel via a semi-empirical and site-specific basal slidingrule This sliding rule imposes seasonal perturbations to thebackground basal sliding velocity that are dependent on rateof change of glacier water storage Following a 1000 yearspin-up the ice-flow model produces dynamic equilibriumice geometry and velocity fields that compare well withobservations After spin-up the coupled model reproducesthe broad features observed in the annual basal sliding cyclein the terminal 60 km of the flowline namely (1) prescribedbackground basal sliding during the winter (2) a summerspeed-up event and (3) a fall slowdown event and return towinter velocities While we have put forth a plausible rulethat connects sliding velocity to the state of the glacierhydrologic system there remains a significant challenge todevelop a sliding rule that is more firmly based upon firstprinciples (ie no free parameters and based on physicalmechanisms by which basal water produces sliding) Thisrequires both a proper characterization of all components ofthe glacier hydrologic system from the spatialndashtemporalpattern of melt generation to the complex evolution of en-and subglacial transport and storage and a physics-basedconnection between the state of the hydrologic system andthe basal sliding

GPS observations of ice surface velocity at three stationsduring 2005 and 2006 suggest the annual velocity cyclepropagates as far upstream as 70 km We examined therelative magnitude of driving and coupling stresses andperformed a simple simulation to assess the possible contri-bution of an upstream propagation of a terminus perturbationthrough longitudinal coupling to the annual velocity cycleWe find that an extreme terminus perturbation is unlikely toinfluence velocities upstream of JAR2 (km14) on the seasonaltimescale Thus we suggest the annual ice velocity cyclealong the majority of the flowline (km14ndash70) is insteadattributable to the evolution of the glaciohydrologic system inresponse to meltwater inputs Following previous alpinestudies we suggest this local acceleration is due to variationsin basal sliding that are governed by variations in the rate ofchange of glacier water storage due to local mismatchesbetween surface meltwater input and the ability of thesubglacial hydrologic system to transmit water

ACKNOWLEDGEMENTSThis work was supported by NASA Cryospheric ScienceProgram grants NNX08AT85G and NNX07AF15G to KSand US National Science Foundation (NSF) DDRI 0926911to WC WC thanks the Natural Sciences and EngineeringResearch Council (NSERC) of Canada for support througha Post-Graduate Scholarship and the Cooperative Institutefor Research in Environmental Sciences (CIRES) for supportthrough a Graduate Research Fellowship RSA acknow-ledges support through NSF grant EAR 0922126 We

thank Gwenn Flowers and an anonymous reviewer forthorough insights on the manuscript Helen Fricker was ourscientific editor

REFERENCESAnderson RS and 6 others (2004) Strong feedbacks between

hydrology and sliding of a small alpine glacier J GeophysRes 109(F3) F03005 (doi 1010292004JF000120)

Bamber JL Layberry RL and Gogineni SP (2001) A new icethickness and bed data set for the Greenland ice sheet 1Measurement data reduction and errors J Geophys Res106(D24) 33 773ndash33 780

Bartholomaus TC Anderson RS and Anderson SP (2008) Responseof glacier basal motion to transient water storage NatureGeosci 1(1) 33ndash37

Bartholomaus TC Anderson RS and Anderson SP (2011) Growthand collapse of the distributed subglacial hydrologic system ofKennicott Glacier Alaska and its effects on basal motionJ Glaciol 57(206) 985ndash1002

Bartholomew I Nienow P Mair D Hubbard A King MA and Sole A(2010) Seasonal evolution of subglacial drainage and accelera-tion in a Greenland outlet glacier Nature Geosci 3(6) 408ndash411

Bindschadler R (1983) The importance of pressurized subglacialwater in separation and sliding at the glacier bed J Glaciol29(101) 3ndash19

Burgess EW and 6 others (2010) A spatially calibrated model ofannual accumulation rate on the Greenland Ice Sheet (1958ndash2007) J Geophys Res 115(F2) F02004 (doi 1010292009JF001293)

Catania GA and Neumann TA (2010) Persistent englacial drainagefeatures in the Greenland Ice Sheet Geophys Res Lett 37(2)L02501 (doi 1010292009GL041108)

Colgan WT Phillips TP Anderson RS Zwally HJ Abdalati W andRajaram H (2009) Similarities in basal sliding between Green-land and Alpine glaciers [Abstr C23B-0499] Eos 90(52) FallMeet Suppl

Colgan W and 7 others (2011a) The annual glaciohydrology cyclein the ablation zone of the Greenland ice sheet Part 1Hydrology model J Glaciol 57(204) 697ndash709

Colgan W and 7 others (2011b) An increase in crevasse extentWest Greenland hydrologic implications Geophys Res Lett38(18) L18502 (doi 1010292011GL048491)

Fausto RS Ahlstroslashm AP Van As D Boslashggild CE and Johnsen SJ(2009) A new present-day temperature parameterization forGreenland J Glaciol 55(189) 95ndash105

Flowers GE and Clarke GKC (2002) A multicomponent coupledmodel of glacier hydrology 1 Theory and synthetic examplesJ Geophys Res 107(B11) 2287 (doi 1010292001JB001122)

Fountain AG and Walder JS (1998) Water flow through temperateglaciers Rev Geophys 36(3) 299ndash328

Gagliardini O Cohen D Raback P and Zwinger T (2007) Finite-element modeling of subglacial cavities and related friction lawJ Geophys Res 112(F2) F02027 (doi 1010292006JF000576)

Glen JW (1955) The creep of polycrystalline ice Proc R SocLondon Ser A 228(1175) 519ndash538

Holland DM Thomas RH de Young B Ribergaard MH and LyberthB (2008) Acceleration of Jakobshavn Isbraelig triggered by warmsubsurface ocean waters Nature Geosci 1(10) 659ndash664

Hooke RLeB (2005) Principles of glacier mechanics Secondedition Cambridge University Press Cambridge

Howat IM Tulaczyk S Waddington E and Bjornsson H (2008a)Dynamic controls on glacier basal motion inferred from surfaceice motion J Geophys Res 113(F3) F03015 (doi 1010292007JF000925)

Howat IM Joughin I Fahnestock M Smith BE and Scambos T(2008b) Synchronous retreat and acceleration of southeastGreenland outlet glaciers 2000ndash2006 ice dynamics and coup-ling to climate J Glaciol 54(187) 646ndash660


Huybrechts P (1994) The present evolution of the Greenland icesheet an assessment by modelling Global Planet Change9(1ndash2) 39ndash51

Huybrechts P Letreguilly A and Reeh N (1991) The Greenland icesheet and greenhouse warming Global Planet Change 3(4)399ndash412

Iken A (1981) The effect of the subglacial water pressure on thesliding velocity of a glacier in an idealized numerical modelJ Glaciol 27(97) 407ndash421

Iken A and Bindschadler RA (1986) Combined measurements ofsubglacial water pressure and surface velocity of Findelen-gletscher Switzerland conclusions about drainage system andsliding mechanism J Glaciol 32(110) 101ndash119

Iken A Rothlisberger H Flotron A and Haeberli W (1983) The upliftof Unteraargletscher at the beginning of the melt season ndash a con-sequence of water storage at the bed J Glaciol 29(101) 28ndash47

Jansson P Hock R and Schnieder T (2003) The concept of glacierstorage a review J Hydrol 282(1ndash4) 116ndash129

Joughin I Das SB King MA Smith BE Howat IM and Moon T(2008) Seasonal speedup along the western flank of theGreenland Ice Sheet Science 320(5877) 781ndash783

Joughin I Smith BE Howat IM Scambos T and Moon T (2010)Greenland flow variability from ice-sheet-wide velocity map-ping J Glaciol 56(197) 415ndash430

Kamb B (1970) Sliding motion of glaciers theory and observationRev Geophys Space Phys 8(4) 673ndash728

Kamb B Engelhardt H Fahnestock MA Humphrey N Meier M andStone D (1994) Mechanical and hydrologic basis for the rapidmotion of a large tidewater glacier 2 Interpretation J GeophysRes 99(B8) 15 231ndash15 244

Kessler MA and Anderson RS (2004) Testing a numerical glacialhydrological model using spring speed-up events and outburstfloods Geophys Res Lett 31(18) L18503 (doi 1010292004GL020622)

Larson KM Plumb J Zwally J and Abdalati W (2001) Analysis ofGPS data collected on the Greenland ice sheet Polar Geogr25(1) 22ndash40

Luthi M Funk M Iken A Gogineni S and Truffer M (2002)Mechanisms of fast flow in Jakobshavn Isbraelig West GreenlandPart III Measurements of ice deformation temperature andcross-borehole conductivity in boreholes to the bedrockJ Glaciol 48(162) 369ndash385

Marshall SJ Bjornsson H Flowers GE and Clarke GKC (2005)Simulation of Vatnajokull ice cap dynamics J Geophys Res110(F3) F03009 (doi 1010292004JF000262)

McGrath D Colgan W Steffen K Lauffenburger P and Balog J(2011) Assessing the summer water budget of a moulin basin inthe Sermeq Avannarleq ablation region Greenland ice sheetJ Glaciol 57(205) 954ndash964

Nick FM Vieli A Howat IM and Joughin I (2009) Large-scalechanges in Greenland outlet glacier dynamics triggered at theterminus Nature Geosci 2(2) 110ndash114

Ohmura A (2001) Physical basis for the temperature-based melt-index method J Appl Meteorol 40(4) 753ndash761

Parizek BR and Alley RB (2004) Implications of increased Green-land surface melt under global-warming scenarios ice-sheetsimulations Quat Sci Rev 23(9ndash10) 1013ndash1027

Paterson WSB (1991) Why ice-age ice is sometimes lsquosoftrsquo Cold RegSci Technol 20(1) 75ndash98

Paterson WSB (1994) The physics of glaciers Third edition ElsevierOxford

Pfeffer WT Meier MF and Illangasekare TH (1991) Retention ofGreenland runoff by refreezing implications for projected futuresea level change J Geophys Res 96(C12) 22 117ndash22 124

Pimentel S and Flowers GE (2010) A numerical study ofhydrologically driven glacier dynamics and subglacial floodingProc R Soc London Ser A 467(2126) 537ndash558

Plummer J Gogineni S Van der Veen C Leuschen C and Li J (2008)Ice thickness and bed map for Jakobshavn Isbraelig CReSIS TechRep 2008-1

Price SF Payne AJ Catania GA and Neumann TA (2008) Seasonalacceleration of inland ice via longitudinal coupling to marginalice J Glaciol 54(185) 213ndash219

Reeh N (1985) Was the Greenland ice sheet thinner in the LateWisconsinan than now Nature 317(6040) 797ndash799

Rignot E Koppes M and Velicogna I (2010) Rapid submarinemelting of the calving faces of West Greenland glaciers NatureGeosci 3(3) 141ndash218

Ritz C Fabre A and Letreguilly A (1997) Sensitivity of a Greenlandice sheet model to ice flow and ablation parameters con-sequences for the evolution through the last glacial cycleClimate Dyn 13(1) 11ndash23

Scambos TA and Haran T (2002) An image-enhanced DEM of theGreenland ice sheet Ann Glaciol 34 291ndash298

Schoof C (2005) The effect of cavitation on glacier sliding Proc RSoc London Ser A 461(2055) 609ndash627

Schoof C (2010) Ice-sheet acceleration driven by melt supplyvariability Nature 468(7325) 803ndash806

Steffen K and Box J (2001) Surface climatology of the Greenland icesheet Greenland Climate Network 1995ndash1999 J GeophysRes 106(D24) 33 951ndash33 964

Sundal AV Shepherd A Nienow P Hanna E Palmer S andHuybrechts P (2011) Melt-induced speed-up of Greenland icesheet offset by efficient subglacial drainage Nature 469(7331)521ndash524

Thomsen HH Thorning L and Braithwaite RJ (1988) Glacier-hydrological conditions on the inland ice north-east ofJacobshavnIlulissat West Greenland Rapp Groslashnl GeolUnders 138

Van der Veen CJ (1987) Longitudinal stresses and basal sliding acomparative study In Van der Veen CJ and Oerlemans J edsDynamics of the West Antarctic ice sheet D Reidel PublishingCo Dordrecht 223ndash248

Weertman J (1957) On the sliding of glaciers J Glaciol 3(21)33ndash38

Zwally HJ Abdalati W Herring T Larson K Saba J and Steffen K(2002) Surface melt-induced acceleration of Greenland ice-sheet flow Science 297(5579) 218ndash222

APPENDIX

Variable notation

Ablation gradienti Density of ice (kgmndash3) Total driving stress (Pa) 0xx Longitudinal stress (Pa)A Flow-law parameter (Pandash3 andash1)E Wisconsin enhancement factorF Englacial hydrology system entry fractionH Ice thickness (m)K Sliding rule coefficient (m13 d23)N Glen law exponentPi Ice pressure (Pa)Pw Water pressure (Pa)Q Ice discharge per unit width (m2 andash1)Qe Creep activation energy of ice (kJmolndash1)R Ideal gas constant (Jmolndash1 Kndash1)T Ice temperature (K)ab Annual basal ablation (m)as Annual surface ablation (m)_as Surface ablation rate (m andash1)b Annual mass balance (m)


cb Annual basal accumulation (m)cs Annual surface accumulation (m)dmax Duration of summer speed-up event (days)dmin Duration of fall slowdown event (days)g Gravitational acceleration (m sndash2)he Englacial water table elevation (or head) (m)j Day of year (days)jmax Day of year of summer maximum velocity (days)jmin Day of year of fall minimum velocity (days)m Sliding rule exponentnc Subglacial conduits per meter (mndash1)t Given time (years)ub Basal sliding velocity (m andash1)

ubo Background basal sliding velocity (m andash1)ud Deformational velocity (m andash1)umax Ice surface summer maximum velocity (m andash1)umin Ice surface fall minimum velocity (m andash1)us Ice surface velocity (m andash1)uw Ice surface winter velocity (m andash1)x Given distance upstream (m)z Given elevation (m)zb Bedrock elevation (m)zs Ice surface elevation (m)

All heads and elevations are in reference to sea level asdatum

MS received 6 May 2011 and accepted in revised form 22 September 2011


is delivered to the englacial and subglacial system throughsurface melt

Our goal is to reproduce the annual ice velocity cycleobserved in the Sermeq Avannarleq ablation zone withcoupled ice-flow and hydrology models In this paper wecouple a one-dimensional (1-D) (depth-integrated) ice-flowmodel to a 1-D (depth-integrated) glacier hydrology model(Colgan and others 2011a) via a semi-empirical sliding ruleOur goal is not to reproduce specific observed intra- orinterannual variations in ice velocity Rather we attempt toreproduce an annual glaciohydrology cycle in dynamicequilibrium that reproduces the critical features of theobserved cycle and may thus serve as a basis for future workinvestigating the influence of interannual variations insurface ablation on annual ice displacement The 530 kmSermeq Avannarleq flowline runs up-glacier from its

tidewater terminus (km0 at 69378N 50288W) to the mainice divide of the Greenland ice sheet (km530 at 71548N37818W Fig 1) This flowline lies within 2 km of threeGreenland Climate Network (GC-Net Steffen and Box2001) automatic weather stations (AWSs) JAR2 (km140 at69428N 50088W) JAR1 (km325 at 69508N 49708W)and CUETH (lsquoSwissrsquo) Camp (km460 at 69568N 49348W)(all positions reported for 2008) In a companion paper to thisstudy (Colgan and others 2011a) we suggested that in theSermeq Avannarleq ablation zone (1) englacial water tableelevation which may be taken as a proxy for glacier waterstorage oscillates around levels that are relatively close toflotation throughout the year and (2) observed periods ofenhanced (suppressed) basal sliding qualitatively correspondto modeled periods of increasing (decreasing) glacier waterstorage In this paper we propose a semi-empirical and site-specific sliding rule that relates variations in the modeled rateof change of glacier water storage to observed variations inbasal sliding velocity

2 METHODS

21 Observed annual ice surface velocity cycleWe characterize the annual ice surface velocity cycle atthree locations (JAR2 JAR1 and Swiss Camp) along theterminal 55 km of the Sermeq Avannarleq flowline usingdifferential GPS observations of 10 day mean ice surfacevelocity in 2005 and 2006 (Larson and others 2001 Zwallyand others 2002) This characterization provides a represen-tative annual ice velocity cycle against which the accuracyof the modeled annual ice velocity cycle can be assessed Atall sites these observations reveal that the ice moves atwinter velocity until the beginning of a summer speed-upevent in which ice velocities increase above winter velocity(Fig 2) The summer speed-up event is followed by a fallslowdown event in which ice velocities decrease belowwinter velocities Using the positive degree-days (PDDs)observed at each station as a proxy for melt intensity(Ohmura 2001 Steffen and Box 2001) the onset of thespeed-up approximately coincides with the onset of summer

Fig 1 The terminal 55 km of the Sermeq Avannarleq flowline overlaid on a panchromatic WorldView-1 image (acquired 15 July 2009) withdistance from the terminus indicated in km The GC-Net AWSs are denoted in red Inset shows the location of Sermeq Avannarleq in WestGreenland

Fig 2 Observed 10 day mean ice surface velocity (grey lines) andcumulative positive degree-days (PDD red lines) in 2005 and 2006at Swiss Camp (SC) JAR1 and JAR2 (where available) versus day ofyear Black lines denote the bi-Gaussian characterization of theannual ice surface velocity cycle at each station (Eqn (1))





dmax

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APPENDIX

Variable notation











dmax

2


dmin

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others 2005)

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APPENDIX

Variable notation












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APPENDIX

Variable notation













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APPENDIX

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APPENDIX

Variable notation





























































































APPENDIX

Variable notation








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