Annals of Glaciology 34 2002 # InternationalGlaciological Society

Anew tide model forthe Antarctic ice shelves and seas

Laurie Padman,1 Helen A. Fricker,2 Richard Coleman,3, 4 Susan Howard,1 Lana Erofeeva5 1Earth &Space Research,1910 FairviewAve.E.,Suite102 ,Seattle,WA98102-3620,U.S.A. 2Institute of Geophysics and Planetary Physics,Scripps Institution of Oceanography,University of California SanDiego , LaJolla,CA92093-0225 ,U.S.A. 3Antarctic CRCand School of Geography and Environmental Studies,University ofTasmania,Box 252-80,Hobart,Tasmania 7001,Australia 4CSIROMarine Research, Box 1538,Hobart,Tasmania 7001,Australia 5College of Oceanic &Atmospheric Sciences,Oregon StateUniversity ,Ocean Admin. Bldg.104,Corvallis,OR97331-5504,U.S.A.

ABSTRACT.Wedescribe anewtide modelfor the seas surroundingAntarctica, includ- ingthe oceancavities under the floatingice shelves. Themodel uses dataassimilation to improveits fit toavailable data. T ypicalpeak-to-peak tide rangeson ice shelves are1^2 m butcan exceed 3 mforthe Filchner^Ronneand Larsen Ice Shelvesin the WeddellSea. Springtidal ranges are about twice these values.M odelperformance is judgedrelative to the 5^10cmaccuracythat isneededto fully utilize ice-shelf heightdata that willbe collected¹ with the Geoscience Laser Altimeter System,scheduledto be launched on the Ice, Cloudand land Elevation Satellite in late 2002. Themodel does not yet achieve this levelof accuracyexcept very near the fewhigh-quality tidal records that havebeen assimilated into the model.Some improvement in predictive skill is expectedfrom increased sophistication of modelphysics, but we also require better definitionof ice-shelf groundinglines andmore accuratewater-column thickness datain shelf seas andunder the ice shelves.Long-duration tide measurements (bottom pressure gaugeor globalpositioning system) incritical data- sparse areas,particularly near and on the Filchner^Ronneand R oss Ice Shelvesand Pine IslandBay ,arerequired to improve the performanceof the data-assimilationmodel.

INTRODUCTION (ICESat).GLASwill providemeasurements ofice-shelf sur- faceheight at 5cm accuracyduring its expected3 year ¹ Most ofthe mass loss fromthe Antarcticcontinent takes mission lifetime. On this time-scale, however,the height placefrom the floatingice shelves, viaiceberg calving from changesmay be small compared with the high-frequency their outermargins and basal melting beneath them (Ja- variabilityof tide height:perhaps tens ofcentimeters per year cobsand others, 1992).Thesurface heightof an ice shelf var- comparedwith the order1 mstandarddeviation of the tides. ies intime withocean tides, atmosphericpressure, ocean Tides,with periods of 0.5 and 1day,areundersampled by ¹ ¹ andice density,snowloading, firn compaction, ablation or satellite repeat periods.I CESat,for example, will have a accretionof ice atthe ocean/ice interface,and ice dynamics. 183day repeat intervalfor most ofits planned3 yearmission, Theseprocesses actover a widevariety of time-scales: from after 3months ofan 8 dayrepeat cycle.Orbits canbe ¹ hoursto decades and longer .Themain cause of short-time- designedto allow removal of tides froma longrecord of satel- scales: (less thanseasonal) height variability will generally lite altimeter data(Parke and others, 1987).Atthe time of beocean tides, withpredicted peak-to-peaktide-induced writing,however,TOPEX/Poseidon(T /P)is the onlysatellite displacements being 43mundersome shelves. Thelargest that is designedfor this purpose(Smith, 1999;Smith and tides occurin the WeddellSea: a gravimeterrecord from others, 2000),andit onlyprovides coverage to latitude nearthe groundingzone of the RutfordIce Stream inthe 66.2³S. Therefore,the most effectivemethod for predicting ¹ southern RonneIce Shelfshows peak-to-peak tidal changes tides forremoval from satellite dataover Antarcticice shelves ofabout6 m(Doake,1992).Toaccuratelymonitor long-term is throughnumerical modeling . trends inice-shelf surface height,the tide componentmust Inthis paperwe describe recent progress inmodeling beremovedfrom the heightmeasurement, such that allice- tides inthe SouthernOcean. Wetreat tides asnoise that must shelf heightsare referred toa ``tide-free’’datum. beremoved from satellite datacollected over ice shelves,so Thevertical displacement ofanice shelf canbe meas- wefocus here onpredictionof tidal height rather thantidal uredto high accuracy using global positioning system currents.Wenote,however ,that tides contributedirectly to (GPS) receivers placedon the ice (e.g.Kingand others, the dynamicsand thermodynamics of ice shelves (see, e.g. 2000),andinterferometry fromtime-separated satellite syn- MacAyeal,1 984;Makinson and Nicholls, 1 999)andplay a thetic aperture radar(SAR) images(e.g .Rignotand majorrole in setting the oceanicand sea-ice conditionsnorth others, 2000).Satellitealtimetry is the best approachto ofthe ice shelves (Robertsonand others, 1998;P admanand monitoringchanges in ice-shelf heightover time-scales of Kottmeier,2000).Thedistribution of tidal currents rather months toseveral years. A significantnew dataset on ice- thanheight variability is the most significantfactor affecting shelf surface heightwill be obtained from the Geoscience these processes; therefore accurateprediction of currents is Laser Altimeter System (GLAS),scheduledfor launch in alsoan important goal of our tidal studies, butis not late2002 on the Ice, Cloudand land Elevation Satellite discussed here. 247 Padman and others:Tide modelforAntarctic ice shelves and seas

TIDE-MODEL DESCRIPTION aremodeled on a gridwith node spacing of 1/4³61/12³(about 10kmspacingnear the Antarcticcoast at 70³S) ,usingthe ¹ Severalmodels of tides inthe SouthernOcean already exist, depthgrid described inthe Appendix.The ` `prior’’solution includingthe globalFinite-Element Simulation,version 95 .2 (i.e. the dynamicallybased ` `first estimate’’)is alinearized (FES95.2:le Provostand others, 1997)and regional models, formof the CATSmodel, version 00 .10(CATS00.10).Four such asthose forthe Ross Sea(MacA yeal,1 984)andthe diurnal (O1, K1, P1, Q1),foursemi-diurnal (M 2, S2, K2, N2) WeddellSea (Smithson andothers, 1996;R obertsonand andtwo long-period (Mm, Mf) constituents aremodeled. others,1998;Makinson and Nicholls, 1999).TheCircum-Ant- arctic TidalSimulation (CA TS)(Padman and Kottmeier , 2000)coversthe entire SouthernOcean south of 56³ S. Table 1.Tidal records used in model for assimilation or ¹ While the predictiveskill of recent versionsof CA TSis better validation thanthat of previous models, comparisons of predictionswith avarietyof datasets indicatethat significantfurther improve- ment isstill required. No. Name Location Type Record Comments length Errors inpresent-day tide models arise fromthree sources: errors inforcing, primarily due to errors inopen- days boundaryspecifications; simplifications in model physics; anderrors inthe water-columnthickness grid.Simplifica- 1 Faraday 65.25³S, 64.27³W CTG 365 2 Forster 70.77³S, 11.87³W BPR 307.25NU 1 tionsof the physicsare necessary toreduce computation 3 Rothera 67.57³S, 68.12³W CTG 365 time fora modelrun. Some simplifications (e.g .parameter- 4 Signy 60.70³S, 45.60³W BPR 384.96 izationsof bottomfriction and lateral mixing) apply to all 5 PTC 4.2.1 77.12³S, 49.05³W BPR 4.2NU :shortrecord 2 tide models.When ice shelves areincluded, additional terms 6 PTC 4.2.2 74.43³S, 39.40³W BPR 30NU 7 PTC 4.2.3 74.38³S, 37.65³W BPR 180 arerequired toaccount for some ofthe inelasticbehavior of 8 PTC 4.2.5 70.43³S, 8.30³W BPR 324 the ice. Inour model, the shelves aretreated aspassiveele- 9 PTC 4.2.6 71.05³S, 11.75³W BPR 367 ments freelyfloating on a perturbed ocean` `free’’surface, 10 PTC 4.2.7 72.88³S, 19.62³W BPR 316 andtheir onlyeffects areto reduce the waterdepth and to 11PTC 4.2.1960.05³S, 47.08³W BPR 408 12PTC 4.2.2073. 13³S, 72.53³W BPR 357RonneEntrance providean additional frictional surface (the ice/oceaninter- 13PTC 4.2.246 1.47³S, 61.28³W BPR 320 face)for dissipation of tidalenergy (MacA yeal,1984).Water- 14PTC 4.2.2760.85³ S, 54.72³W BPR 378 columnthickness ( d)isthe same aswaterdepth for the open 15PTC 4.2.3059. 73³S, 55.50³W BPR 377 ocean,and is the verticaldistance between the ice baseand 16PTC 4.2.3362. 13³S, 60.68³W BPR 358 17PTC 4.2.3470. 13³S, 2.65³W BPR 653 the seabedunder the ice shelves (see fig.1inSmithson and 18 PTC 4.1.3 60.02³S, 132.12³E BPR 21NU:shortrecord others (1996)fora diagramof tide-model geometry).Errors 19 Mawson 67.60³S, 62.87³E BPR 365 in d x; y arelargest under the ice shelves. Indeed,even the 20 Davis 68.45³S, 77.97³E BPR 365 … † precise locationof the groundingline for sections ofsome ice 21 Casey 66.28³S, 110.53³E BPR 365 22 RIS Base 82.53³S, 166.00³W Grav.44NU :validation shelves is still notknown, although progress is nowbeing 23 RIS C13 79.25³S,1 70.37³E Grav.34NU :validation madethrough satellite mapping(see, e.g.,Grayand others 24 RIS C16 81.19³S, 170.50³E Grav.45NU :validation 2002;F ricker andothers, inpress) .Thewater-depth grid is 25 RIS C36 79.75³S, 169.05³W Grav.34NU :validation slowlybeing improved (see Appendix)using additional 26 RIS F9 84.25³S,1 71.33³W Grav.58NU :validation 27 RISJ9 82.37³S, 168.63³W Grav.30NU :validation datafrom cruises andre-analysesof existingice-penetrating 28 RIS LAS 78.20³S, 162.27³W Grav.32 radardata, but more depthdata are needed. 29RIS McMurdo 77.85³S, 166.66³E BPR 365 Inthe longterm, improvements intide modelingwill 30 RIS O19 79.53³S, 163.36³E Grav.39NU :validation result froman increase inmodel sophistication combined 31 RIS RI 80.19³S, 161.56³W Grav.36NU :validation 32AIS BeaverLake 70.80³ S, 68.18³E BPR 39 witha more accuratewater-depth grid. In the shorter term, 33AIS HWD-200069. 71³S, 73.58³E GPS 31 however,weadopt a data-assimilation(or ` `inverse’’) 34R OPEX C-159.87³S, 30.10³W BPR 445 approach.An inverse model is aformalizedhybrid of a 35R OPEX C-273 .69³S, 34.61³W BPR 393 purelyempirical model (in which tidal constituents are 36R OPEX M-276.58³ S, 32.01³W BPR 377 37Rutford G.L. 78.55³S, 82.97³W Grav./tilt. 4/43 determined frommeasurements) anda dynamicalmodel in whichpredictions are based on solutions to the equationsof Notes:Faraday,Forster,Rotheraand Signy data were obtained from the fluidmotion and known forcing (R obertsonand others, Proudman OceanographicLaboratory (ACCLAIM dataset).PTC 1998).Dynamicalmodels, including CA TS,are called ` `for- stationsare from the Pelagic Tidal Constantspublication (Smithson, ward’’models,because they are run by time-stepping the 1992,and amendments) .Mawson,Davis and Casey data are from the SouthernOcean Database at Australia ’sNationalTidal Facility(NTF) . modelequations and analyzing the time series after the RossIce Shelf (RIS)stations are as reportedby Williams andRobinson modelhas reached equilibrium. One canthink of assimila- (1980).The AIS BeaverLake tide-gauge data were analyzed by the NTF , tionas usingdata to objectively ` `nudge’’aforwardmodel such andthe hot-water drill site(AIS HWD-2000)was collected by static asCA TStowards satisfactory agreement with the data,or GPS during January^February2000 as partof theAustralian Antarctic Sciencefield program.R OPEX stationsare unpu blished data,provided usinga physicalmodel to provide a dynamicallybased byK.W .Nichollsand M. J.Smithson. Rutfordgrounding line (G.L. ) scheme forinterpolating and extrapolating tide valuesfrom dataare based on 4 daysof gravimeterdata calibrated against tidal ana- sparse measurement sites ontoa uniformgrid. The details of lysisof 43 days of tiltmeterdata (Doake,1 992;Robinson and others,1 996). the data-assimilationmethod that wefollow are described in Datatypes are coastal tide gauges (CTG) ,bottompressure recorders (BPR),gravimeterson the ice shelf (Grav .),orGPS. Egbert (1997).Here, webriefly describe the assimilated data- 1 NU,notused. Forsterdata disagree significantly with theforward model set andthe relevantfeatures ofthe finalmodel, which we call anddata from Kapp Norwegia (PTC 4.2.6)and so are not used. the Circum-Antarctic DataAssimilation tides model,version 2 PTC4.2.2,a 30day record from the shelf break of the southern W eddell 00.10(CADA00.10). Sea,disagrees significantly with PTC4.2.3,a muchlonger nearby record, InCAD A00.10,tides forthe entire oceansouth of 58³ S andso is notused. 248 Padman and others:Tide model forAntarctic ice shelves and seas

Theseconstituents arethe same asthose inthe globaltides inthe T/Pdatadomain (north of 66.2³S) .Locationsof all ¹ modelTPX O5.1(personalcommunication from G .D. representers andthe eightRoss Ice Shelfgravimeter records Egbert, 2000),whichwe use formodel boundary conditions. areshown in Figure1 .Notethat the datasouth of the T/P TheCA TS00.10modelis drivenby time-stepping TPXO5.1- datadomain account for 510% ofthe totalnumber of derivedsea surface heightalong the northernopen bound- representers. Theirinfluence on the finalmodel is greater aryat58³ S, and by the astronomicaltide-generating poten- thanthis, however,becausethe model^datamisfit isscaled tialwithin the modeldomain. Sea ice isincludedneither in bya priorerror covariancemap, which in turn dependson this forwardmodel nor in the data-assimilationversion (see the amplitudeof the tides inthe priormodel solution. Since followingparagraph) .InCA TS00.10,the equationsare solved the largesttidal amplitudes are found in the southernand forall constituents concurrently,andnon-linearities can enter western WeddellSea and Siple Coast section ofthe Ross intothe finaltidal solutions through the quadraticbottom fric- Sea(see Fig.2),datafrom these regionshave a significant tionand momentum advection.One potentiallyimportant influenceon the finalinverse solution. N evertheless, most of outcomeof non-linearity in tide modelsis the establishment of the inversemodel improvement relative to the forwardmodel meanflows, even though the basicforcing is entirelyperiodic is dueto the tightconstraints imposedby the 270T /P-based (Makinsonand Nicholls, 1 999).Suchflows are output by the representer sites northof 66.2³ S. ¹ forwardmodel, but are lost inthe assimilationmodel because Asameasure ofthe efficacyof data assimilation, inT able2 it is basedon linearized equations to make the inversecompu- wepresent the root-mean-square(rms) error forthe four tationstractable . majorconstituents, M 2, S2, K1 and O1.Theseerror estimates Dataassimilation was performed usingthe Oregon arecalculated in the time domain,i.e. errors ofboth ampli- State UniversityTidal Inversion Software (OTIS: http:/ / tude andphase are included in the calculations.W euse two www.oce.orst.edu/po/research/tide/index.html),whichuses datasets, the 25non- T/Psites used inthe assimilation,and the the ``representer’’approachto data assimilation (Egbert eightgravimeter stations onthe Ross Ice Shelf.N otethat andothers, 1994;Egbert andBennett, 1996;Egbert, 1997).A since the former datasetis used inthe assimilation,improve- totalof 37 tide-gauge stations liein our model domain ments fromthe forwardto the inversesolution are expected. (Table1 ),includingcoastal sea-level gauges and bottom Incontrast, the gravimetersites providean independent pressure recorders, andgravimeter measurements onthe checkof the valueof data assimilation as adynamicalmeans Ross andRonne I ceShelves.T welveof these records were ofdata interpolation/ extrapolation.With the exceptionof M 2 excluded,4 becauseof uncertainties intheir quality,and8 onthe Ross Ice Shelf,assimilation significantly improves the gravimeterstations onthe Ross Ice Shelfwhich are used to fit betweenthe modeland measurements. Thefit ofthe provideindependent validation of model performance in inversesolution to the 25assimilated sites, rms errors of2^4 this region.Representers werelocated at each of the remain- cmper constituent,is constrainedby the choiceof our expected ing25 sites, andan additional270 representers werelocated error foreach height m easurement andour choice of a valuefor

Fig.1.Representer and data locations for the CADA00.10 model.The T/Psatellite radar altimetry measurements are all north of 66.2³S (indicated by the dashed line).The ¹ 270 representer locations within the T/Pcoverage area are shown assmall dots.Asterisks indicate locations of non-T/P data records (Table1)that are also used asrepresentersites in the assimilation. Solidsquares on the Ross Ice Shelf show the locations of eight gravimeter records that are used in validat- Fig.2.Rmstide height ( ¼±)for the entire circum-Antarctic ing the tide models but are not used in the assimilation. Solid seas to 60³S.The thick white line isthe SCAR1993 ice-shelf black contours indicate the1000and 3000misobaths,and the edge.The black line is the 1000 mwater-depth contour as a gray contour represents the ice fronts. guide tothe location of the continental slope. 249 Padman and others:Tide modelforAntarctic ice shelves and seas the OTISparameter , ¼",whichessentially measures the rela- i.e.the heightwhen all 1 0explicitlymodeled constituents are tiverole of dynamics and assimilated data in constraining the inphase. Wedonot show maps of this value,but the distribu- finalsolution. Overconstraining the inversemodel to height tion of ± x; y =¼ x; y is narrow,witha meanvalue of max… † ±… † dataresults inheight fields that are` `bumpy’’,andunrealistic 2.4.The maximum peak-to-peak tidal range is 2± , or ¹ ¹ max velocityfields anddynamic residual errors (the ``error’’for- 4^5 ¼ .Slightlyhigher values are possible when the many ¹ ± cingthat is requiredto nudge the forwardmodel solution to minortidal constituents areincluded in the summationin the inversesolution) .Our choiceof ¼",whichis chosensepa- Equation( 2)(see, e.g.,le Provostand others, 1997). ratelyfor each species (semi-diurnal anddiurnal) ,is presently Inthe followingsubsections wesummarize the distribu- basedon a combinationof subjectivetolerance for height field tion of ¼± forthree sectors ofthe SouthernOcean. Color ver- variability,andinformal comparisons of model predictions sions ofthe figuresin this papercan be foundat our web site, withother data types such ascurrent-meter dataand satellite http://www.esr.org/antarctic.html. interferometry. Thecomparison of forward and inverse model skills in Weddelland Bellingshausen Seas Table2 indicatesthat further improvements arerequired underthe Ross Ice Shelf.In a studyof R oss Seatides that will Modelvalues of ¼± exceed150cm intheW eddellSea under sec- bereported elsewhere, weshow that assimilatingthe eight gravimeterrecords fromthe ice shelf significantlyimproves modelperformance as judged by model comparisons with SARinterferometry datafrom the SipleCoast (personal communicationfrom I .Joughin,200 1). Thefull model grids for both CA TS00.10andCAD A00.10 canbe obtained on CD-R OM(inM atlab 2 form) fromthe first author.

MODEL RESULTS

Wecharacterizetidal-height variability by the standard deviationof the modeledtidal-height fields summed over alltidal constituents. Thisvalue, ¼±,isgivenas afunction of position x; y by … †

n 10 ˆ ¼ x; y h2 x; y ; 1 ±… † ˆ vn…† … † u n 1 uXˆ t where hn is the heightamplitude of the nth tidalcoefficient. Thetypical magnitude of the tidalrange (low to high tide) is 2¼ .Wealsodefine a maximumheight, ± , given by ¹ ± max n 10 ˆ ±max hn ; 2 ˆ n 1 … † Xˆ

Table 2.Comparison of the rms error (incm) between the modeled and measured tide heights for the four major tidal

constituents,M 2, S2, O1and K1

Fig.3.Close-up ofrmstideheights ( ¼±)for three sectors.Aster- M2 S2 O1 K1 isks indicate locations of non-satellite tide-height records listed cm cm cm cm inTable 1.(a)W eddell and Bellingshausen Seas.The highest values of ¼± in the southern Filchner^Ronne Ice Shelfare 25assimilated sites 4180cm south of the Henry and Korff Ice Rises (HKIR). CATS00.10 3.94. 75.13.4 The locations of tidalmeasurements at Ronne Entrance CADA001.0 2.42.4 3.9 2.0 8RISgravimeter sites (R.E.)and the Rutford grounding line (RutfordG.L.)are CATS00.10 3.67 .98.95. 7 indicated. (b)Ross Seato Pine Island Bay (PIB).The highest CADA00.10 3.94.5 5.4 4. 0 values of ¼± in the eastern Ross Ice Shelfare 480cm along the SipleCoast.The locations of tidalmeasur ements at McMurdo Notes:The comparisonis performedfor each site in thetime domain, i.e. Soundand the LittleAmerica Station(LAS) are indicated. (c) bothamplitude and phase errors between modeled and measured con- East Antarcticsector includingtheAIS.Thehighest valuesof ¼± stituent valuesare used. The twodatasets for which meanrms statistics in the southern AISar e 70 cm.The locationsof tidalmeasure- ¹ arepresented are: the 25 non- T/Pdatarecords that are included in the ments at Beaver Lake (B.L.)on the western side of theAIS,the assimilation;and the 8 gravimeterrecords on theRoss Ice Shelf (RIS) thatwere not assimilated. Results are presented for the forward model, GPSmeasur ements at the hot-water-drillingsite (HWD)and CATS00.10,and the inverse solution, CADA00. 10. the Mawson, Davis and Casey stations are indicated. 250 Padman and others:Tide model forAntarctic ice shelves and seas tionsof the Ronne^Filchnerand Larsen Ice Shelves(Fig .3a). SipleCoast instead of being a separatefeature as indicated Thespring tide peak-to-peakrange exceeds 7 minthe chan- inpresent coastlinecharts (Grayand others, 2002). nelsouth of the Henryand Korff Ice Rises inthe southern Tidalrecords forthe Ross Ice Shelfare mainly gravi- RonneI ce Shelf.This prediction is consistent withgravi- meter time series obtainedin the 1970s(Williams and meter records fromnear the groundinglines ofthe Doake Robinson,1 980).With the exceptionof the Little America Ice Rumples andRutford Ice Stream, andsite S902(Smith, Station(LAS) point,these gravimeterdata are not included 1991;Doake,1 992;Robinson and others, 1996).Themajor inthe data-assimilationruns discussed herein.The typical semi-diurnal constituents, M 2 and S2,dominatetide heights amplitudesfor the twomost energetic diurnalconstituents, exceptnear their amphidromicpoints close to the center of K1 and O1,areabout 30^40 cm. Therms error forthe unas- the RonneIce Shelffront. similated gravimeterpoints (T able2 )isabout4 and5 cm, Our modeledtide heightsin this regionare acceptable respectively,forK 1 and O1,i.e.1 0^20%of the true signal. formost present purposes,even without assimilation: in dif- Most ofthis error arises throughphase errors: modeled ferentialSAR interferograms (DSIs) fromthe frontof the diurnalamplitudes are within 2^3 cm ofmeasured values. Ronneand Filchner I ceShelves(Rignot and others, 2000), Theweaker semi-diurnal constituents arevery poorly repre- the typicalerror betweenthe differentialheight signal in sented inthe model,even with assimilation of LAS and the DSIs andthe same signalsynthesized fromthe non- McMurdoSound tide records. Bothamplitude and phase assimilativeCA TSmodelwas about 1 0cm. Tests inwhich errors arelarge. F orone example, the S 2 amplitudeand phase weincreased the benthicfriction coefficient under the ice atF9 (in the southeastern cornerof the Ross Ice Shelf)is 11cm shelves, asasimple means toparameterize additionaltidal- and142³from data, but 28 cm and187³from CAD A00.10.The energysinks, suggest that higherenergy losses underthe ice rms error resultingfrom this misfit is 15cm, withmaximum ¹ shelf further improvethe fit ofthe CATSmodelto the ice- values of 22cm. Thatis, the error inS alonecan exceed our ¹ 2 frontDSIs, althoughat the cost ofpoorerfits toopen-ocean nominalrequirement of 10cm fortidal prediction accuracy . ¹ data.Thebest fit wasfound for an under-ice dragcoefficient The S2 constituent isdifficultto predict becausesome ofthe S 2 of C 0.015,which is 5times greaterthan is typically signalin data records is associatedwith ocean response to D ˆ ¹ used forbenthic friction in ocean tide models.Smithson atmosphericradiational tides, andso is notmodeled correctly andothers (1996)cameto a similar conclusion:a higher bythe shallow-waterequations. value of CD underice shelves isrequired toreproduce some Threemain factors contribute to the difficultyof modeling data,particularly near the groundingline, but also at the tides inthe Ross Sea.First, T/Paltimetry ofthe open-ocean expenseof degrading agreement at open-water sites. While surfaceis notavailable south of 66.2³S ,more than1 000km ¹ some ofthe additionalenergy loss canbe attributed toturbu- northof the Ross Ice Shelffront. The T /Pdataprovide the lence generationat the ice-shelf base,other energy sinks are most rigorousconstraint on the inversemodel solution neededto explain this largevalue of CD.Possibilities include becauseof the goodspatial coverage of the datain the north- inelasticflexure in the groundingzone, non-linear transfers ern partof our model domain. Second, d x; y underthe Ross … † toother frequencies, andgeneration of baroclinic tides. Ice Shelfis poorlyknown, and may also have changed signifi- Identifyingand parameterizing these additionalenergy sinks cantlysince the surveysin the 1970s(see Grayand others, underice shelves is apriorityfor future studies. 2002).Third,no high-quality modern tidal records exist for Smallervalues of ¼ ( 50cm) occurin the Bellings- the Ross Sea.The quality of the gravimeterrecords is ± ¹ hausenSea. Three tidal stations, Faraday,Rotheraand unknown;hence, some ofthe differencebetween the model RonneEntrance (T able1 ),areavailable to test forward andthe datamay be due to data errors rather thanmodel modelperformance in this region,and to constrain the quality.(However,aswenoted previously ,assimilationof all inverse solution.In early versions of CA TS,tides atR onne the gravimetersites doesimprove model comparisons with Entrance( 73.13³S, 72. 53³W) werevery poorly represented SARinterferometry estimates oftidal displacement. ) whenthe modelwas run with the ETOPO-5global bathy- The value of ¼ inPine Island Bay ( 105³ W) is 50^60 ± ¹ ¹ metry grid,in which the cavityunder the GeorgeVI Ice cm. PineIsland and Thwaites Glaciers aredynamic ice Shelfwas 20 m thick.The forward model performed much streams drainingthe West Antarcticmarine ice sheet, and better,however,whenthe gridof d x; y wasadjusted to arebelieved to be presently outof dynamic equilibrium … † agreewith values based on seismic soundingtransects across (Rignot,1998;Wingham and others, 1998).Tidemodeling is the GeorgeVI Ice Shelf(Maslanyj, 1 987).Thesetransects importantfor this region,both to assess the possiblerole of showed that d exceeds 600m ina troughthat runs underthe tides inthe loss ofshelf ice andto remove tide-height vari- entire lengthof this ice shelf, providinga conduitfor tidal abilityfor satellite sensingof trends inice-shelf thickness. energyflux around the eastern side ofAlexanderIsland that Thereare, however ,notide dataand very few bathymetry wasessentially closed with the ETOPO-5grid. datain this region,so even our data-assimilation model is poorlyconstrained at this time. RossSea to PineIsland Bay EastAntarctica

Underthe Ross IceShelf, ¼ is 60cm, exceptfor a small Alongmost ofthe coastof East Antarctica, ¼ is 40^55 cm ± ¹ ± ¹ regionof higher ¼± (upto 1 40cm) alongthe southernSiple (Fig.2).However,underthe AmeryIce Shelf(AIS) at the Coastwhere Ice Streams AandC andWhillansI ce Stream southernend of Prydz Bay , ¼ increases to 65cm nearthe ± ¹ enter the Ross embayment(Fig. 3b) .Tidesare predomi- groundingline (Fig .3c).Tidesin this sector aremixed diurnal^ nantlydiurnal except along the SipleCoast. However ,the semi-diurnal.Tidalcurrents underthe ice shelf aresmall, typi- distributionof d inthis areais poorlyknown, and even the cally 55 cm s^1.With the exceptionof the AIS,much ofthe grounding-linelocation is uncertain:recent satellite data East Antarcticcoastline lies closeto the southernlimit ofT/P suggestthat the CraryI ce Rise is actuallyconnected to the altimetry coverage,unlike the southernportions of the major 251 Padman and others:Tide modelforAntarctic ice shelves and seas embayments oftheWeddelland R ossSeas.Tidal-heightpre- dictionsin this sector aretherefore relativelyreliable in our model,because of the assimilationof T/Pdata.As expected, dueto the T/Pconstraints inCAD A00.10,our model predic- tions for ¼± atMawson, Davis and Casey stations areclose tothe measured values.F orthe AIS,errors inour forward model(CA TS00.10),asjudgedby time-series comparisons betweenshort GPSrecords andmodelpredictions, increase further south.These data are not, however ,ofsufficient durationto be assimilated inCAD A00.10,and so can only beused formodel validation studies. When tides atBeaver Lake(a tidal lake to the west ofthe ice shelf) andthe hot- waterdrilling site (HWD-2000)near the northwestern cornerof the AISare assimilated, the rms errors between modelpredictions and the GPSdata further southare slightly reduced.M uchof the error may,however,bedue to the Fig.4.Example of satellite aliasing problem.The thin gray line exclusionof non-tidal sources ofheight variability ,andso is the predicted hourly tide height for the point 71.5³S ,70³E on wouldnot be removed even if the tidalmodel were perfect. theAIS.Thethickblackline isthesametidaltimeseriessampled at the GLASsampling frequency during the 8day repeat,Verifi- cation Phaseofthe ICESatmission.This phase willonlylast for ANEXAMPLE OFTIDE-MODEL APPLICATION 3months,but we show 1year of prediction to demonstrate the TOSATELLITEDATA rangeofpossibleoutcomesforan aliased time seriesthat isshorter than the aliasing periods of majorconstituents. Satelliteorbit repeat periods( ¢T)arelong compared with tidaltime-scales, andso the satellite dataundersample tidal variability.Tidalenergy appears in the satellite recordat an signalwas interpreted asbeing non-tidal, the apparentheight ``aliasingfrequency’ ’that fallsbetween 0 and1 /(2 ¢T). If the change (¢h)over3 months ofdata collection could be as aliasingfrequencies forthe majorconstituents areseparable largeas 60cm. If the GLAStime series wascorrected witha duringa mission life,then tidalinformation can be retrieved tidalmodel, ¢h wouldbe smaller butwould still containthe fromthe satellite records. Thisis true forthe T/Psatellite, aliasedsignal of the tide-model error.Thus,without an whoseorbit was designed specifically for tidal retrieval accuratetide model,tidal-height variability seen inthe (Parkeand others, 1987).Forother satellites, however,this is GLASdata may be misinterpreted aslong-time-scale,non- notthe case.Smith (1999)andSmith andothers (2000)com- tidal,geophysical variability . paredaliasing periods for GEOSA T,the EuropeanR emote- Anotherpotential source oferror overice shelves that sensingSatellite (ERS) andT /P,andfound that several maybe significant in satellite altimeter datais the inverse problemsarise. First, aconstituent maybe effectively barometereffect (IBE).Theisostatic response ofthe ocean ``frozen’’;that is, its aliasperiod is longcompared with the surfaceto changing air pressure ( P )is adepressionof satellite’smission life,or eveninfinite. In this case,regardless atm 1cm foreach 1 mbarincrease in P (Gill, 1982,p.337).A ofhowlarge the constituent amplitudeis, the recordedvari- ¹ atm pressure anomalyof 30m barassociated with a typical abilityof that constituent is negligibleduring a mission. Sec- ¹ polarlow therefore leadsto a 30cm increase insea level. ond,two or more constituents maybe aliasedto nearly the ¹ Thereis some evidencefrom differential SAR interferometry same frequency,andso cannot be separated from each other . that the IBEcan be an important component of the residual Third,constituents maybe aliased to other frequencies at height-changesignal after tides havebeen removed (Rignot whichnon-tidal energy may be present. Themost significant andothers, 2000). source ofnon-tidal energy is the annualresponse tocycles in oceanicor atmospheric conditions. The diurnal constituents K1 and P1 inthe 35day repeat phasesof the ERS missions are bothaliased to 1yearperiods and so cannot be separated CONCLUSIONS ¹ fromeach other or from non-tidal, annual variability . Todemonstrate the aliasingproblem, we show a 1year Wehavepresented anewtide modelfor the Antarcticice time series ofpredicted tide heightsfor the point7 1.5³S, shelves andseas southof 58³ S. Data assimilation was used 70³E onthe AIS,and the same time series sampledby the toconstrain the modelto better agreementwith measure- GLAS 8dayrepeat ``VerificationPhase’ ’(Fig.4).Weshow ments oftide height.T ypicalpeak-to-peak tidal ranges under ¹ afullyear of predictions, even though the VerificationPhase most ice shelves fringingAntarctica are 1^2m. Thiscan ¹ is onlyplanned to be 3 months long,todemonstrate the range increase to2^4 m atspring tides, andoccasionally exceeds ofpossible outcomes from a short mission.Theapparent quasi- 6m, notablyunder the southof the Filchner^RonneI ce Shelf annualcycle in the GLAStime series arises becausetwo of the (FRIS) inthe southernWeddellSea. Tides provide the major fourlargest semi-diurnal tides, S 2 and K2,havealiasing peri- short-periodsignal that willbe detected bysatellite-based ods of 1year.Anotherenergetic constituent, the diurnalK , techniques such asSAR and laser altimetry.Theseinstru- ¹ 1 hasan aliasing period of about 2 years.During the first ments candetect heightvariability of 5cm orsmaller ,yet ¹ 100days, GLAS sees significantshort-period variability wecannot provide tidal predictions at accuracies better than associatedwith the tide. From day1 00today 200 the trend 10^20cm forseveral critical segments ofthe Antarcticice- inheight is downward,until it recovers inthe last halfof the shelf area,notably the southernFRIS and the eastern side of yearbut with reduced short-period variability .If this entire the Ross Ice Shelf.Thus, the qualityof tide modelsis pre- 252 Padman and others:Tide model forAntarctic ice shelves and seas sently oneof the factorspreventing the fullexploitation of 99006to GLAS T eamMember J.-B.Minster.Workon the these datasets. AmeryI ce Shelfwas supported by grants to R.C. from the Forwardtide models basedon the shallow-waterequa- AustralianR esearch Counciland the AntarcticScience tionscan be improved by increasing the sophisticationof Grant (ASG) scheme. Tidalsea-level analyses for Mawson, the modelphysics and the accuracyof the water-column Davisand Casey stations andBeaver Lake were supplied by thickness grid.In an attempt toaddress the former require- the NationalTidalF acility,TheFlinders U niversityof South ment, weare currently investigatingadding parameteriza- Australia,copyright reserved. Wethankthe reviewers for tionsfor tidal energy loss tobaroclinic tide generation, their valuablecomments onthe manuscript. followingJ ayneand St. Laurent( 2001),andchanging the bottomdrag coefficient to improve model fit tovarious data- REFERENCES sets. Wehavefound that increasingthe parameterizedtidal energydissipation under ice shelves improvesmodel skill Brancolini,G. and12 others .1995.Descriptivetext for the seismic strati- alongthe Ronneand Filchner ice fronts whencompared to graphicatlas of theRoss Sea, Antarctica. In Cooper,A. K., P.F.Barker displacements observedin DSIs (Rignotand others, 2000). andG. Brancolini, eds.Geology andseismic stratigraphyof theAntarcticmargin 1 . Washington,DC, AmericanGeophysical U nion,27 1^286.(Antarctic However,the additionalenergy loss impliedby this param- ResearchSeries 68. ) eterizationis much largerthan one would expect from extra Doake,C. S. M. 1992.Gravimetric tidal measurement on Filchner Ronne turbulenceproduction at the ice-shelf base.This suggests Ice Shelf. In Oerter, H., ed.Filchner^Ronne Ice ShelfProgramme. Report No.6 that there is atleast onemore majorphysical mechanism (1992).Bremerhaven,Alfred WegenerInstitute for Polar and Marine Research,34^39. forenergy loss underice shelves. Possibleenergy sinks not Egbert,G. D .1997.Tidal datainversion: interpolation and inference. Progress accountedfor in ourpresent forwardmodel include inelastic inOceanography , 40, 53^80. flexurein the groundingzone, baroclinic tide generation, Egbert,G. D .andA. F.Bennett.1996.Data assimilation methods for ocean and/ortransfer oftidal energy to other harmonics and terms tides. In Malonette-Rizzoli,P ., ed.M odernapproaches to dataassimilation in oceanmodeling .Amsterdam,Elsevier ,147^179. throughnon-linear interactions, especially in the shallow Egbert,G .D.,A. F.Bennettand M .G.G.Foreman.1994.TOPEX/POSEIDON waternear the groundingzones. tidesestimated using a globalinverse model. J.Geophys. Res. , 99(C12), Thealternative approach to improving tide modeling, 24,821^24,852. usingan inverse technique such asdata assimilation, Fricker,H.A. and 8 others.Inpress. Redefinitionof the grounding zone of AmeryIce Shelf, EastAntarctica. J.Geophys. Res. requires that weincrease the number oftidal records for Gill,A. E.1982. Atmosphere^oceandynamics . SanDiego, Academic Press. (Inter- assimilation.Satellite altimetry aside,most newdata will nationalGeophysics Series 30 .) consist ofbottompressure sensors mooredin the oceannear Gray, L. and 6 others.2002.RADARSA Tinterferometryfor Antarctic the ice fronts, orstatic GPSon ice shelves controlledwith grounding-zonemapping. Ann.Glaciol. , 34 (seepaper in this volume). Greischar,L. andC. R.Bentley .1980.Isostatic equilibrium groundingline nearbyGPS on bedrock (e.g .Kingand others, 2000).Inthis betweenthe W estAntarctic ice sheet and the Ross Ice Shelf. Nature, case,care should be taken to obtain the ice-shelf records 283(5748),651^654. well away (410km) fromthe groundingline and shear Jacobs,S. S., H.H. Hellmer ,C. S. M. Doake,A. Jenkinsand R. M. Frolich. margins,since the tide models presently donot include the 1992.Melting of ice shelves and the mass balanceof Antarctica. J. Glaciol., 38(130),375^387. glacialrheology needed to model tide-forced displacements Jayne,S. R.and L. C. St. Laurent. 2001.Parameterizingtidal dissipation withinthe flexuralboundary layer .Models canpredict overrough topography . Geophys.Res. Lett. , 28(5),811^814. phase and relative amplitudeinformation closer tothe Johnson,M .R.and A. M.Smith.1997.Seabedtopography under thesouthern groundingline but will overestimate absoluteamplitude andwestern Ronne Ice Shelf, derivedfrom seismic surveys. Antarct.Sci. , 9(2),201^208. (Riedel andothers, 1999).Ideally,the newmeasurements King,M., L. Nguyen,R. Colemanand P .J.Morgan.2000. Strategies for wouldbe oflong duration (e.g .1year),allowinggood reso- highprecision processing of GPS measurementswith applicationto the lutionof the majortidal constituents. However,shorter AmeryIce Shelf, EastAntarctica. GPS Solutions , 4(1), 2^12. records arestill useful:records of30 days or longercan be leProvost, C., F.Lyard,J .M. Molines,M. L. Gencoand F .Rabilloud.1997. Ahydrodynamicocean tide model improved by assimilating a satellite analyzedfor the majortidal constituents, andshorter altimeterderived dataset. J.Geophys.Res. , 103(C3),5513^5529. records canbe used formodel validation. Toourknowledge, MacAyeal,D .R.1984.N umericalsimulations of the Ross Sea tides. J. Geophys. the onlycurrent planfor significant tidal data collection is Res., 89(C1),607^615. forthe AmeryIce Shelf.Other areas,particularly the out- Makinson,K. and K.W .Nicholls.1 999.Modeling tidal currents beneath Filchner^RonneIce Shelf andon the adjacent continental shelf: their lets ofthe majorW est Antarcticice streams alongthe Siple effecton mixingand transport. J.Geophys.Res. , 104(C6),13,449^13,465. Coastand Pine Island Bay ,require newtide measurements. Maslanyj,M .P.1987.Seismicbedrock depth measurementsand the origin of One additionalsource ofice-shelf heightvariability that GeorgeVISound, AntarcticP eninsula. Br.Antarct.Surv .Bull. 75, 51^65. isnotaccounted for in our tide modelis the IBE,which, as NationalOceanic and A tmosphericAdministration (NOAA).1988. ETOPO- 5bathymetry/topographydata. Digital relief of thesurface of theEarth. Boulder, wediscussed above,causes achangeof 1cmper 1mbar ¹ CO,U.S. Departmentof Commerce. National Oceanic and A tmospheric changein atmospheric pressure. Sincetypical pressure Administration. NationalGeophysical Data Center .(DataAnnounce- anomaliesfor polar low-pressure systems are 30 mbar, an ment88-M GG-02.) ¹ IBEresponse of 30cm changein sea level is possible.This NationalOceanic and A tmosphericAdministration (NOAA).1992. GEODAS ¹ CD-ROMworldwide marinegeophysical data. Boulder,CO,U.S. Department must beconsidered when attempting to ` `de-tide’’satellite- ofCommerce. N ationalOceanic and A tmosphericAdministration. derivedmeasurements ofice-shelf heights. NationalGeophysical Data Center .(DataAnnouncement 9288-M GG-02.) Nicholls,K.W .,L.Padmanand A. Jenkins.1 998.First physicaloceanograph y results fromtheR OPEX98cruise to thesouthernW eddellSea. In Oerter, H., comp.Filchner^RonneIce ShelfP rogramme (FRISP).Report No.12( 1998) . Bremer- ACKNOWLEDGEMENTS haven,Alfred WegenerInstitute for P olarand M arineResearch, 51^58. Padman,L. andCh. Kottmeier.2000.High-frequency ice motion and Theocean tidalmodeling was supported by grants to Earth & divergencein theWeddellSea. J.Geophys. Res. , 105(19),3379^3400. SpaceR esearch fromthe U.S.N ationalScience F oundation Parke,M. E., R.H. Stewart, D .L. Farlessand D .E. Cartwright. 1987.On thechoice of orbits foran altimetric satellite to study ocean circulation Office ofP olarPrograms (O PP-9896041)andN ASA(N AG5- and tides. J.Geophys. Res. , 92(C11),11,693^11,707. 7790).Fundingfor H .A.F.wasprovided by N ASAgrant N AS5- Riedel,B., U.Nixdorf,M. Heinert,A. Eckstallerand C. Mayer.1999.The 253 Padman and others:Tide modelforAntarctic ice shelves and seas

responseof the Ekstro « misen(Antarctica) grounding zone to tidal forcing . APPENDIX Ann.Glaciol. , 29, 239^242. Rignot,E. J.1998.F astrecession of a WestAntarctic glacier . Science, THECATS/CADADEPTH GRID 281(5376),549^551. Rignot,E., L. Padman,D .R.MacA yealand M .Schmeltz.2000. Observa- Our 1/4³61/12³model grid of d x; y is basedon E TOPO-5 tionof ocean tides below the Filchner andRonne Ice Shelves, Antarctica, … † usingsynthetic aperture radar interferometry: comparison with tide (NOAA,1988).Smith andSandwell ( 1997)havedeveloped a modelpredictions. J.Geophys.Res. , 105(C8),19,615^19,630. high-resolutiondepth grid for regions north of 72³ S, using Robertson,R., L. Padmanand G. D. Egbert. 1998.Tides in theWeddellSea. gravityanomalies obtained from satellite altimeter heights InJacobs,S. S. andR. F .Weiss, eds.Ocean, ice andatmosphere :interactionsat tointerpolate water depth between ship-track depth sound- theAntarctic continental margin .Washington,DC, American Geophysical Union,34 1^369.(Antarctic Research Series 75 .) ings.However ,wehave not yet developed a satisfactory Robinson,A.V .,C. S. M. Doake,K.W .Nicholls,K. Makinson and D .G. scheme formerging this gridwith our grid for regions south Vaughan.1 996. ESAMCA tidalcorrection report. Cambridge,British Ant- of72³ S, and so have not yet implemented this improvement. arcticSurvey . Instead,we have updated several areas of specific interest ScientificCommittee on Antarctic Research (SCAR) .1993. Antarcticdigital database.A topographicdatabase for usewith PC ARC/INFO,ArcView 2 and usinglocal depth grids. F orthe openW eddellSea, we use a ArcCAD2.CD-ROMversion 1.0,September1993 edition. Cambridge,Scott griddeveloped by R obertsonand others (1998),whichhas PolarResearch Institute, British AntarcticSurvey and the W orld Con- beenupdated with depth data for the southwestern Weddell servationMonitoring Centre. Seaacquired during the 1998R onnePolynya Experiment Smith, A. J.E. 1999.Application of satellitealtimetry to global ocean tidal modeling.(Ph.D. thesis,Delft University of Technology.DelftInstitute (ROPEX-98)(Nicholls andothers, 1998).Forthe openR oss forEarth-Oriented SpaceResearch. ) Sea,we use agridgenerated from data acquired from the Smith, A. J.E., B.A. C. Ambrosius andK. F .Wakker.2000.Ocean tides U.S.N ationalGeophysical Data Center (NOAA,1992).The fromT/P,ERS-1,andGEOSA Taltimetry. J. Geod., 74(5),399^413. resultant mapof water depth for the open-watersection of Smith, A.M. 1991.The useof tiltmeters to study the dynamics of Antarctic ice-shelfgrounding lines. J. Glaciol., 37(125),51^58. the Ross Seais similar tothat presented byBrancolini and Smith, W.H.F .andD .T.Sandwell. 1997.Globalsea floor topography from others (1995).Forthe AmundsenSea, including Pine Island satellitealtimetry and ship depth soundings. Science, 277(5334),1956^1962. Bay,weuse ETOPO-5,butreplace shallow shelf depths in Smithson, M. J.1 992. Pelagictidal constants 3. DelMar ,CA, International ETOPO-5with a more typicaldepth of 400 m. Unionof Geodesy and Geophysics, International Association for the PhysicalSciences of theOceans. (IAPSO PublicationScientifique 35. ) Water-columnthickness forthe oceancavities under the Smithson, M.J.,A.V .Robinsonand R. A. Flather.1996.Ocean tides under majorice shelves wasobtained from various sources. theFilchner^Ronne Ice Shelf, Antarctica. Ann.Glaciol. , 23, 217^225. Gridded water-columnthickness datafor the Filchner^ Williams, M. J.M., R.C. Warnerand W.F.Budd.1998.The effectsof ocean RonneIce Shelfwere obtained from J ohnsonand Smith warmingon melting and ocean circulation under theAmery Ice Shelf, EastAntarctica. Ann.Glaciol. , 27,75^80. (1997),andthose forthe Ross Ice Shelfwere provided by D . Williams, R.T.andE. S. Robinson.1 980.The oceantide in thesouthern Holland(personal communication, 1 999)basedon the Ross Sea. J.Geophys. Res. , 85(C11),6689^6696. measurements reported byGreischar andBentley ( 1980). Wingham,D .J.,A. L. Ridout,R. Scharroo, R. J. Arthern andC. K.Shum. TheR oss Ice Shelfcavity geometry was updated using the 1998.Antarctic elevation change 1992to 1996. Science, 282(5388),456^458. 1993SCAR coastline(SCAR, 1993).Depths forthe George VI Ice Shelfwere obtained from Maslanyj ( 1987).Forthe AIS, d x; y wasobtained from Williams andothers (1998), … † andthe ice shelf wasextended further southbased on results ofhydrostaticcalculations (F ricker andothers, inpress) .For allother ice shelves, d x; y wastaken from ETO PO-5. … †

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