Numerical Simulations of the Ross Sea Tides

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 89, NO. C1, PAGES 607-615, JANUARY 20, 1984

DOUGLAS REED MACAYEAL1

GeophysicalFluid DynamicsProgram, Princeton University

Tidal currentsbelow the floating Ross Ice shelfare reconstructedby using a numericaltidal model. They are predominantlydiurnal, achievemaximum strengthin regionsnear where the ice shelf runs aground,and are significantlyenhanced by topographicRossby wave propagationalong the ice front. A comparisonwith observationsof the vertical motion of the ice shelf surfaceindicates that the model reproducesthe diurnal tidal characteristicswithin 20%. Similar agreementfor the relatively weak semidiurnal tideswas not obtained,and this calls attentionto possibleerrors of the open boundaryforcing obtained from global-oceantidal simulationsand to possibleerrors in mapping zones of ice shelf grounding.Air-sea contact below the ice shelf is eliminated by the thick ice cover. The dominant sub-ice-shelfcirculation may thus be tidally induced.A preliminaryassessment of sub-ice-shelfconditions basedon the numericaltidal simulationssuggests that (1) strongbarotropic circulation is driven along the ice front and (2) tidal fronts may form in the sub-ice-shelfcavity where the water column is thin and wherethe buoyancyinput is weak.

INTRODUCTION in Figure2), wherea bore hole was openedfor severaldays Tidal currents are the strongestobserved form of seawater [Cloughand Hansen, 1979; Jacobs and Haines, 1982]. motion in the cavity below the floatingice shelfin the south- Tidal currents can be difficult to reconstruct on the basis of ern Ross Sea [Williams and Robinson,1979, 1980; Jacobs and observedtidal amplitudeand phasealone because of complex Haines,1982]. The thickice platform shown in Figure1 elimi- basinshape and topography[Williams, 1976]; hence,the scar- nates air-sea contact; thus sub-ice-shelf ocean circulation and city of reliabletidal currentmeasurements presents a serious heat transport may be forcedprimarily by tidal currentsor by obstacleto the investigationof sub-ice-shelfoceanography. related processessuch as tidal current rectification and tidal Basintopography, for example,may exerta particularlyinflu- front formation [MacAyeal, 1983]. Given presentice flow pat- entialcontrol on the currentsbecause, in polarlatitudes, topo- terns,approximately 25% of the snow that accumulatesover graphic Rossbywaves can be excited by the diurnal tide. Antarctica flows through the Ross Ice Shelf and ablates into Thesewaves are commonlyobserved in the Arcticalong con- the Ross Sea by basal melting or by icebergcalving [Hughes, tinentalslopes and aboveisolated seabed bumps [Cartwright, 1975]. The Ross Sea tidal regime may thus provide a direct 1969;Huthnance, 1974, 1981; Cartwright et al., 1980;Thomson and influential li•k between the ocean and the earth's largest and Crawford,1982]. It is thereforereasonable to expectsuch ice mass..This paper presentsthe results of numerical tidal wavesalong sectionsof the ice front, along various seabed simulations undertaken to reconstruct the Ross Sea tidal cur- ridgesbelow the ice shelf,and along the continentalslope rents and to estimate their influence throughout the sub-ice- north of the Ross Sea. As a result of their strong currents, shelfcavity. thesewaves could induce significantstirring in the sub-ice- The Ross Ice Shelf is an integratedice massthat is flexible shelfenvironment. Current meter records, however, are gener- when deformed over large horizontal length scalessuch as ally requiredto detecttopographic Rossby waves [Cartwright, those imposedby tides in the water below. As a result of its 1969]; thus, to detect them below the ice shelf, the available slow horizontal movement,the ice shelf has provided a natu- observationsof tidal amplitudeand phasemust be coordi- ral platform upon which the tidal amplitude and phase have natedwith numericalsimulations capable of accuratelyrecon- been measured (the 10 observation stations are shown in structingthe tidal currents. Figure 2) [Williams and Robinson,1979, 1980; Williams, 1976, The numericaltidal simulationsconducted in this studyare 1979' Thiel, 1960; Thiel et al., 1960].These measurements intended to amplifythe existingobservations by calculating show that the diurnal tide is stronger than the semidiurnal the sub-ice-shelftidal currentsand by extendingthe mapsof tide and that the tidal amplitudesare largest in areas near tidal amplitudeand phaseacross regions not coveredby the data collection network. where the ice shelfruns aground. In contrastwith the tidal amplitudeand phase,tidal cur- MODEL EQUATIONSAND PROCEDURE rents and their effect on the sub-ice-shelf water column have not been measuredreliably becauseof the thick and impen- The governing equations for barotropic tidal motion em- etrable ice cover. The few available tidal current measure- ployedin this studyare [Nihoul, 1975,p. 51] ments come from north of the ice front (MCM and "current •(Du)/•t + V-(Duu)= -gDV(rl- •]e)-fDez x u- klulu meter" in Figure 2) [Heath, 1977; Gilmouret al., 1962; Jacobs and Haines, 1982] and from a singlesub-ice-shelf location (J9 + vDV2u (1) and

•Now at Departmentof the GeophysicalSciences, The University •rl/•t + V-(Du) = 0 (2) of Chicago. where u is the depth-averagedhorizontal velocity, r/ is the Copyright 1984 by the American Geophysical Union. departureof the sea surfaceor ice shelfbase from the level of Paper number 3C 1414. rest,D is the instantaneousdepth of the water layer (extending 0148-0227/84/003 C- 1414505.00 from the seabedto either the sea surfaceor the ice shelf base),

607 608 MACAYEAL' SIMULATION OF Ross SEA TIDES

100øW 110øW 120øW 150øWl 80 ø 150øE 120øE 110øE 100øE

80øS OOOE

30øS WestAntarctica .• • EastAntarctica / 11 • / "'.:•:•;:y-•,i•-,:... '- 10øE ...... 120øW , ..'1.-:'•'•' '•'-..••••'::•,•,•':J'•' :•...•?"•T ...... tarctic Mountains 20øE

ß.:.:-,:•. -•7.--:....: •.. '":":•:•:•f:f.:::':"::f:-'::''BCF::::':::-'...... "-: '. :.':•'•".."::ii?::: ...... •:..e. :•... :i:;•::.:::-•;i:::i:i::•::;:!:.:..:.::.::.i:::i::•:..:...... ,...... ======.'.... '- '.:---::L:-u:y•5:-:..',•.:.-.,• . :.'.:.:....:::;.';::•::;::;:::5::-::.:;;.::::..:;;•f:: ..:•.::•'::::...... :-:..::L::::.:•: ...... 130øW 130øE :..?-'.•'i:•.t•?.....:...e. •.9.....c...•.....i..• :..'•:"d?:•i.":':::.•:?!.:i':.:'':-'-•.'-!;• ...... ::::.-'•"':::i!i!:Y:":"?::?:.?'-•:sd':•:::::::::::::::::::::::::::::::::::::::i:..:- '....!•: ":!•:'..'"':i:::::::::11111;i!:i•;!•:i,•11:i::,:.; :.:-:ii':i::7:::.'•:•il;i;:!i'::ii::;•ii:.':':::;:...... '.. --:.:::.-'f.:.. •:•:-::.:-._ %-.•:: •:-'•. :•: L•.•?i:...•:% :.:%:,::.--,-... ß • ' """ '"• '"" '•' ""• '"--'••'••••••••••••'••,-•"• •.:c.•-•:-.t;.:. .• • ...... :-:• ::...--• •,, ... ß...... --:y-.:..- -.-:.-.::::.:.:: ...... •:...: ::.-:•-•-;:;.:..... - - ...... :'•&•'- '•..• .;' .z•-.'-.-.•' '"';:,,• ...... •,,,,,,,"• .....;...... ,•...... '.'"'.::":':'.; ":•...... '•-•'. '•c• '•'•:'•'•t";'%:.%:?•¾...... --' 140øE 140øW / .... :t•.- -...;-.•-:.:::: • ...... ;...... •...... -,.-' ...:...... : ..::::..:.::::•::•.'..•

,••• • o::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: o-•• • 500 ,ooo

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50øE 150øW / oo 70øS• RossSea • • ••/Ca•e__t) '• ,0os lO•

160øW 1 70øW W180øE 1-70øE 160øE Fig. 1. At themargins ofthe West Antarctic Ice Sheet, large floating ice shelves (indicated byshading) extend seaward from wherethe ice floatsfree of the seabed.The RossIce Shelf,shown above, occupies the southernhalf of the RossSea. It rangesin thicknessfrom 1100to 100m, coversan areaof 580,000km :, and flowstoward the openocean at ratesof up to 1200m/yr. Air-sea contact is prohibitedwithin the sub-ice-shelf cavity; hence, tidal currents and related tidally driven processesmay contribute to oceancirculation below the i•e shelfand associated basal-ice ablation. Although observations of the tidal amplitudesand phaseshave been made at 10 locationsdistributed across the ice shelfcovered portion of the RossSea (indicated above by dots)[Williams and Robinson,1979, 1980], this paperpresents the resultsof numericaltidal simulationsconducted to-investigate the sub-ice-shelftidal currents.This map is tracedfrom the polar-stereographic projectionof the AmericanGeographical Society of New York [ 1970]. lie is the equilibriumtidal elevationspecifying forcing by the significant influence within several kilometers of coasts sun and moon [Dietrich,1963, p. 443], g = 9.81 m/s2 is the [Hughes,1977; Holdsworth,1977; MacAyeal, 1983], its effect gravitationalacceleration, f-- 1.42 x 10-½ s-• is the Co- on tidal propagationthrough the centralpart of the basinis riolis parameterat the meanlatitude of the RossSea (78øS), k thoughtto be minor [Williamsand Robinson, 1981]. is the nondimensionalquadratic bottom-frictionparameter Boundaryconditions applied at coastsare u-e, = 0, where equalto 2.5 x 10-3 in ppenwater and 5.0 x 10-3 in iceshelf e, is the outwardpointing unit vectorthat is perpendicularto coveredwater [Rhmming and Kowalik,1980, p. 17], v = 100 the coastand •(u- e•)/•n + (2/Ax)uß e, = 0, wheree, is the unit m2/sis the eddyviscosity (selected arbitrarily to supressnu- vector tangent to the coast, •(u-et)/c•n is the gradient of the mericalnoise, but otherwiseof small importance),and e• is a longshoreflow in the direction of e., and fix is the finite unit vectorthat is perpendicularto the geoid. differencegrid point spacing(10 km). The first conditionlisted The elastic strength of the ice shelf and its inertia with above representsthe no cross-shoreflux condition, and the respectto vertical motion are disregardedin this study for secondcondition listed aboverepresents a computationally simplification.Although ice shelf tidal flexure may provide a efficientcompromise between the no-slip and free-slipcon- MACAYEAL'SIMULATION OFROSS SEA TIDES 609

85øS 85øS 180 ø

epth (rn)

80øS 80%

o 140OE 140øW

300

75øS 75ø: openboundary

O•'•'•Jo 0K m

160øW 180ø 160øE Fig.2. Thenumerical domain isbounded bythe mountainous coasts,the ice shelf grounding line,and the open boundary.Depthin the open portion ofthe Ross Sea isobtained fromHayes and Davey ['1974]. South ofthe ice front (indicatedbythe dotted linetransecting thebasin from the east tothe west), thewater depth isdefined asthe thickness of theseawater layer alone and is obtained from Greischar andBentley [1980].

Two additionalconditions applied at the openboundary ditionsassociated with the eddy viscosity term ir• equation (1). Thissecond condition produces no noticeableeffect on the areV .(Duu)=0 andvDV2u = O.These conditions represent model simulation. suppressionof!•e momentumflux convergence andthe vis- Tidalforcing is accomplishedby specifying the astronomic cous drag at theopen boundary where information required tocompute these terms must otherwise come from outside the tide-generatingpotential, r/e, and by specifying thesea-surface model domain. Outward wave radiationthrough the open elevation,r/,as a functionof time and position along the open boundaryshown in Figure2. Theopen boundary conditions boundary, such as that described by Blumberg and Kantha arethe strongest form of forcing and were obtained from Sch- [1982],was not accommodated explicitly in this study. As a widerski's[1980b, 1981a, b, c, d, e] global-oceansimulations of result,small-scale high-frequency waves introduced during the thediurnal and semidiurnal tides. Direct observations of tidal modelspin-up period were artificially reflected at theopen conditionsalong the open boundary do not exist; hence, Sch- boundary.These waves were damped during model spin-up by widerski'sdata, accurate to within0.05 m of amplitudeand raisingthe eddy viscosity. 15ø of phase [-Schwiderski, 1980a], represent the best available Thefinite difference grid representing theRoss Sea consists sourceof modelforcing data. Preliminary numerical simula- of a 110•.x•160 rectangular array of gridpoints with 10 km tionsencompassing minor alterations of the open boundary spacing. This grid spacing isapproximately thesame as the conditions,in an attemptto reproducebetter the observedhorizontal range of water parcel movement during the diurnal tidalfields, proved to be unsuccessfulandcomputationally tidal cycle and is thereforeadvantageous forsimulation of suchprocesses as tidal currentrectification [Zimmerman, expensive;therefore, further improvement ofthe open bound- 1981].Map magnificationfactors adjusting for thepolar aryforcing by a t[ial-and-errorprocess was not attempted. stereograp,hicprojection ofthe earth's surface were disregarded [KoW:h•ikand Untersteiner, 1978]. Topography of the TABLE 1. Tidal Constants modeldomain, displayed in Figure2, wasconstructed from Frequency, mapscompiled byHayes and Davey [1974], by Greischar and Symbol Name Species 10 -'• s- x Bentley[1980], and by Rose [1979]. The open boundary lies alongthe 600 m depth contour atthe porthern margin of the K1 declinationalluni-solar diurnal 0.72921 continentalshelf. The topographynear the openboundary O1 principallunar diurnal 0.67598 P1 j•rincipalsolar diurnal 0.72523 was smoothed,and somesections were straightenedto M2 principallunar semidiurnal 1.40519 suppressnumerical noise. S2 principalsolar semidiurnal 1.45444 The finite-differenceforms of equations(!) and(2), and the N2 ellipticallunar semidiurnal 1.37880 boundaryconditions used in thisstudy, are essentially the From Dietrich[1963, pp. 426-427]. same asthose used by L. Y. Oeyand G. L. Mellor(unpub- 610 MACAYEAL' SIMULATION OF Ross SEA TIDES

TABLE 2. Comparisonof the Simulated(S) and the Observed(O) Tidal Amplitude and Phaseat 10 RossSea LocationsShown in Figure (2)

Amplitude,10- 2 m

K1 O1 P1 M2 S2 N2

S O S O S O S O S O S O

Station LAV 41 34 32 25 14 11 9 3 6 5 4 McM 28 26 22 26 9 9 8 4 8 2 4 C13 35 30 27 34 11 10 7 3 7 4 3 O19 37 31 28 29 12 10 8 4 8 2 3 C36 45 37 35 32 15 12 8 3 5 6 4 RI 55 44 40 38 19 15 15 5 8 10 8 J9 50 37 36 37 16 12 4 7 3 8 3 C16 38 31 29 27 12 !0 6 3 6 2 2 BC 54 43 39 35 17 14 7 8 5 10 4 F9 57 41 39 40 19 14 16 8 16 11 8 •lS - 01/10 8.6 • 3.2 2.7 4.8 3.8 2.0

Phase*

K1 O1 P1 M2 S2 N2

S O S O S O S O S O S O

LAV 156 154 141 141 155 154 162 35 299 342 267 344 McM 229 196 204 186 228 196 147 6 307 268 264 234 C13 219 200 198 190 218 200 136 300 280 130 252 160 O19 224 208 202 196 222 208 128 340 271 190 241 185 C36 175 160 158 153 176 160 187 75 316 25 300 44 RI 162 160 145 140 162 160 209 130 333 26 322 5 J9 194 191 178 172 193 191 282 205 !12 106 32 78 C16 216 200 196 190 215 200 117 310 249 160 228 147 BC 193 186 176 174 192 186 290 213 100 112 39 94 F9 213 206 193 190 209 206 13 258 160 142 118 168 zlS-Ol/10 12 6 11 121 56 62

*Degrees with respectto the Greenwichmeridian. lishedmanuscript, 1982), so will not be describedhere. Simula- EVALUATION OF MODEL PERFORMANCE tions of each of the three diurnal and three semidiurnal tides The tidal simulationsare comparedwith the observations listedin Table 1 were begunfrom a stateof rest,run through obtained by Williams and Robinson[1979, 1980] in Table 2 five tidal cyclesto allow spin-up,and then Fourier analyzed and in Figures3 and 4. The diurnal tidal simulationsachieved on the sixth cycleto obtain the relevanttidal fields.Allowing the best agreementwith the observations.This is encouraging additional warm-up cyclesbefore analysisdid not alter the becausethe diurnal tides dominate the RossSea tidal regime resultsby more than 5%. and are, therefore,more likely to control the oceanographic conditionsbelow the ice shelf.Figure 3 indicatesthat the simulated diurnal amplitudesare approximately20% higher than ß i I

240

_ ß ß ß ß 220 ß ß

E 30 _ .•200 Legend c• 20 .e ß K1 18o ß 01 ß/z ß P1 160 / ' ?andP1 • Error ß ß 01

140 ObservedAmplitude (cm) i • i • Error Fig. 3. The simulatedand observedamplitude of the three diur- 140 1•0 1•0 ' 2•0 ' 210 ' 230 nal tidesat the 10 tidal measurementstations are comparedby plot- ObservedPhase (degrees) ting pointson a graphhaving the observedand the simulatedampli- tudes as coordinateaxes. Perfect agreement is achievedwhen points Fig. 4. The simulatedand observedphase of the three diurnal tides fall on a line extendingat a 45ø anglefrom the origin(the thin line). at the 10 tidal measurementstations are comparedin this diagram. MACAYEAL'SIMULATION OF ROSSSEA TIDES 611

85øS 180 ø 85øS simulatedsemidiurnal tidal fields along the Siple Coast. These effectsare expectedto be most pronouncedfor the semidiurnal tidesbecause the seasurface curvature is generallylarger than that of the diurnal tide and becausethe frequencyis relatively high. 80øI INTERPRETATION OF SIMULATED AMPLITUDE AND PHASE The characteristicsof the simulated amplitude and phase, shown for two representativetides K1 and M2 in Figures :5 and 6, respectively,may be explained in terms of Kelvin wave dynamics EGill, 1982, p. 378]. Both the diurnal and semidiurnal amphidromicpoints, for example,constitute destruc- 75øS tive interferenceproduced by Kelvin wave propagationabout the curved coastline of the Ross Sea. By using the basin- averageddepth of 400 m, the wavelengthsof diurnal and semidiurnal Kelvin wavesare :5400and 2700 km, respectively.The length of the curved Ross Sea coastline,from its easternpoint

160øW 180 ø 160øE on the ShiraseCoast to Cape Hallett, is approximately 3000 km. The Ross Sea will thus contain a singlewavelength of the Fig. 5. The simulatedK1 amplitudeis contoured(solid lines) in semidiurnal Kelvin wave. In this circumstance, the semi- centimeters,and the simulatedK1 phaseis contoured(broken lines) in degreesrelative to Greenwich. The other two simulated diurnal diurnal tide along the eastern and western sidesof the basin tides,O1 andP1, displayedpatterns similar to thoseabove. will be 180ø out of phase, and destructiveinterference will occur in the basin's center. Such an interferencepattern is consistentwith the central amphidromic point of M2 dis- the observeddiurnal amplitudes.The anticipateduncertainty played in Figure 6. of the open boundary forcing [Schwiderski,1980a], coupled The diurnal amphidromic point off Cape Hallett results with the tendencytoward tidal amplificationsouth of the open from interferencebetween a deep-oceantidal wave propagat- boundary (see Figure 5), may account for this disagreement. ing directly acrossthe mouth of the Ross Sea and the coastal Although correctionsof the boundary conditionscould have Kelvin wave that propagatesalong the interior coastlineof the been incorporated,a trial-and-error adjustmentwas rejected shallow basin. The 5400-km wavelengthof the interior Kelvin becauseof its potential expense. wave resultsin a 200ø phaselag betweenthe easternmargin of Disagreementbetween the simulated and observedsemi- the RossSea and Cape Hallett. Assumingthat the wavelength diurnal tidal fieldswas substantiallygreater than that between of the deep-oceantide is equal to the circumferenceof Antarc- the simulatedand observeddiurnal tides (Table 2). The simulated M2 tide shownin Figure 6, for example,consists of an

amphidromicregime occupying the ice shelf-coveredportion 85øS 180 ø 85øS of the basin. This pattern agreeswith the observedpattern [mapped in Figure 6 of Williams and Robinson,1980], with the exceptionof three discrepancies:the simulatedphase is shifted by approximately 180ø from the observedphase (see

Figure 6), the simulatedamphidromic point is shiftedapproxi- 80 o,, 80os mately 150 km toward the southeast,and the simulatedampli- tudedisplays greater amplification along the SipleCoast. 140oW' The phase error of the simulated semidiurnaltide may be attributed to the open boundary forcing. Williams and Robin- son [1980] noticed a similar discrepancybetween their data and global ocean tidal model simulationsavailable at the time of their study.This led them to constructtwo alternativemaps 75øS of the M2 phase contours(compare their Figures 6 and 9). One of these alternativesis qualitatively similar to that pre- sentedin Figure 6, but the other displaysa virtual amphidromic point [Schwiderski,1980a] residinginland from the Shi- M2 rase Coast. Mislocation of the simulated semidiurnal amphidromic 160øW 180 ø 160øE point and overamplificationalong the Siple Coast may result from errorsin the specifiedlocations of ice shelfgrounding or Fig. 6. The simulated M2 tidal amplitude is contoured (solid from the effectsof ice shelfflexure disregarded in equation(1). lines) in centimeters,and the simulated M2 phase is contoured (broken lines) in degreesrelative to Greenwich. The other two simu- Thomas and Bentley [1978] and K. C. Jezek (unpublished lated semidiurnaltides, S2 and N2, dislayedpatterns similar to those manuscript,1982), report severalareas of ice shelfgrounding above.A crossnorthwest of the central M2 amphidromicpoint indi- along the Siple Coast southeastof Crary Ice Rise. These cates one of the two alternative locations where Williams and Robin- groundedregions are not representedin the model domain; son [1980] have mapped an M2 amphidromicpoint from their data. thus, their influencecould be responsiblefor significantmodel For this alternative,the observed0 ø phase contour (not shown) ex- tendsfrom this crosstoward Minna Bluff (Figure 1). The other alter- error. The effectsof ice shelf rigidity and of the ice shelf's native map presented by Williams and Robinson[1980] exhibits a inertia during tidal rise and fall may additionally distort the virtual amphidromicpoint inland from the ShiraseCoast. 612 MACAYEAL:SIMULATION OF ROSSSEA TIDES

85øS 180ø 85øS ! SOUTH

80øS 150CLOCKWISE

40øE

75øS •5øS NORTH

Fig. 9. Jacobs [in Jacobsand Haines, 1982] obtained a 7-month current meter record from the position labeled "current meter" in K1 Figure 1. The observedK1 tidal current ellipsegiven by this record is smaller than that of the simulation. Both ellipses, however, have 160øW 180ø 160øE clockwisepolarization. Axis labels are in units of centimetersper second. Fig. 7. The imaginaryfigure traced by the tidal currentvector at a particularlocation during a completetidal cycle is calledthe tidal currentellipse. The simulatedK1 tidal currentellipses shown above Kelvin wave attains its maximum amplitudeat the coastand are drawnat everytenth grid point above.Shading denotes regions where the tidal current vectors rotate dockwise rather than counter- exhibitsexponential offshore decay, with an e folding decay clockwise.The K1 tidal currentsexhibit a strip of clockwiserotation scaleequal to the Rossbyradius of deformation,(gD/f2) •/2 just seawardof theice front resulting from topographic Rossby wave [Gill, 1982, p. 379]. In the vicinity of the Siple Coast, the propagationalong the ice front. Over most of thebasin, the K1 tidal offshoredecay scaledisplayed by the semidiurnaltide is ap- currentellipses are orientednorth-south. This feature accompanies a proximately220 km and is consistentwith the Rossbyradius geostrophicbalance between the tidal currentsand the seasurface of deformationdetermined from the local depth. The diurnal elevationgradient. tide is not as sensitiveto the shallow depth along the Siple Coastand, therefore, exhibits more gentle offshore decay. tica [Schwiderski,1981b], the deep-oceantide will exhibita 20ø phaselag betweenthe sametwo points.If the two waves SIMULATED TIDAL CURRENTS are in phaseat the easternmargin of the RossSea, then the The tidal currentellipse is the imaginaryfigure traced out two waveswill be 180ø out of phaseoff Cape Hallett, and by a tidal currentvector during a completetidal cycle.The destructive interference will result. Intensificationof the diurnal and semidiurnal amplitudes 4oW 3øW 2øW løW alongthe shallow Siple Coast may also be relatedto coastal Kelvin wave dynamics.In an oceanof constantdepth, a

80øS 80øS

140øE 140øW

4oW 3oW 2øW løW Fig. 10. The K1 tidal currentsat 0ø phase(180 ø phasemay be obtainedby reversingthe directionof the vectorspictured here) along the easternend of the ice front are unusuallystrong and display high crossice front shear.These featuresare manifestationsof topographic 160oW 180ø 160øE Rossbywave propagation along the discontinuityof depth found at [rig.8. Thesimulated M2 tidalcurrent ellipses south of theice the ice front. The falselatitude and longitudecoordinates indicated at front tendto be orientedalong the directionof the nearestcoastline the outsidemargins of this map representa grid coordinatesystem in andare very narrow. This is consistent with the dynamics of a coastal which the intersectionof the prime meridian and the equator is de- Kelvinwave propagating about the curvedcoastline of the basinas fined to be North Pole. This coordinate systemhas been adopted, by suggestedby the M2 amplitudeand phase distributions shown in agreement among investigators,for presentation of Ross Ice Shelf Figure6. Nearthe northern margin of theRoss Sea, the simulated field data. All directions referencedin the text, however, are spedfled M2 tidalcurrent ellipses indicate a balancebetween the acceleration relative to the true directions.The island in the top left of this figure is and the Coriolis force. Roosevelt Island. MACAYEAL:SIMULATION OF ROSS SEA TIDES 613

4ow 3øW 2øW low features are consistentwith the Kelvin wave dynamics dis- cussedpreviously in referenceto the M2 amplitude and phase patterns.Near the open boundary, the M2 ellipsesare nearly circular and are polarized counterclockwise.These features indicate a balance between the relative acceleration and the 10øS 10os Coriolis force.Although this balancecan be expectedfor the semidiurnaltide along the seawardmargin of the continental shelf [Munk et al., 1970], these simulated featuresmay also resultfrom numericalnoise generated at the open boundary. 11 o$ 11 øS DIURNAL PERIOD TOPOGRAPHIC ROSSBY WAVES Topographic Rossbywave excitation was exhibited by the diurnal simulationsalong some of the shallow seabedridges extendingnorthwest of the Siple Coast and, most notably, along the easternsegment of the ice front. The K1 flow along the ice front segment,shown in Figures 10 and 11, is unusually strongand displayshigh crossice front shearduring portions 4øW 3øW 2øW ]øW of the tidal cycle. Moreover, the diurnal tidal current ellipse Fig. 11. The K1 tidal currentsat 90ø phasealong the easternend polarization,shown in Figure 7, reversesacross the ice front of the ice front are nearlyin geostrophicbalance with the free surface and displays a strip of clockwisepolarization that transects slope(Figure 5). The strongcross ice front shearoccurring at 0ø phase the basin along the northern side of the ice front. There is (in Figure 10) northwestof RooseveltIsland has reduced,but another indirect observationalevidence in support of the strong cur- such feature now occursat the junction of the ice front with the Shirase Coast. rents along the ice front segmentshown in Figures 10 and 11. Truesdale and Kellogg [1979], for example, report a low diatom assemblageabundance in this area consistentwith polarization of this ellipseis either clockwiseor counterclock- strongwinnowing of the seabedsediments. wise. Figures 7 and 8 display the simulated tidal current el- Longuet-Higgins [1968], Rhines [1969], and Chapman lipsesof K1 and M2 at selectlocations throughout the Ross [1982], for example, have developed simple analytic treat- Sea. The shadedregions of Figures 7 and 8 are where clock- ments for topographicRossby wave propagation along a dis- wise polarization occurs.Figure 9 presentsa comparisonbe- continuity of depth. Becausethe ice front manifestsitself as a tween the simulatedand the observedK1 tidal-currentellipse cliff that blocks the upper 200 m of the water column, these at the ice front position labeled "current meter" in Figure 1. previous studiesmay be applied to explain the resultsof the Although the orientation and polarization of the simulated simulation. The equation relating frequency to wavelength and observedK1 tidal current ellipsesagree, the simulated derivedby Rhines[1969] definesthe maximum frequencysup- currentis strongerby approximately20%. porting topographicRossby wave propagation along a given The two most distinctive features of the simulated K1 tidal- ice front [Longuet-Higgins,1968]. If ro is this upper bound, current ellipsesare (1) the strip of anomalousclockwise polar- and h• and h2 > h• are the depthson the two sidesof the ice ization aligned with the ice front and (2) the north/south front, Rhines'[1969] dispersionrelation is written orientation of the semi-majoraxes. The anomalouspolariza- tion is related to a specialform of topographicRossby wave to/f= (h2/h,- 1)/(h2/h, + 1) (3) that propagateswestward along the depth discontinuityem- bodied by the ice front. This wave is discussedin the next The right-hand side of equation (3) is les.s than unity; there- section.The north/southorientation of the K1 ellipsesis relat- fore, ro must be less than the inertial frequencyf In polar ed to the balance of forcesgoverning the tidal currents.De- latitudes, the diurnal tides have a frequencyapproximately noting the westwardand southwardtidal current components half as large as f so will excite topographicRossby waves by u and v, respectively,the complexratio v/u indicatesthe whereverh2/h• > 2. The semidiurnaltides, in contrast,have a polarization, orientation, and ellipticity of the tidal current frequency nearly equal to f so will not excite topographic ellipse [Munk et al., 1970]. The simulationsindicate that the Rossbywaves along the ice front unlessh2/h• >>1. Although K1 currents exhibit the following approximate momentum ro doesnot dependon the wavelengthin Rhines'[1969] analy- balance: 8v/St + fu = 0 and 8u/St + gSrl/SX= 0, where 8rl/SX sis,the phasepropagation is in the direction having shallow is the gradient of r/in the westwarddirection. This approxi- water on the left (in the southernhemisphere). mate momentum balance indicates that the maximum south- The value of ro defining the maximum frequencyfor topo- ward flow is 90 ø in advance of the maximum free surface graphic Rossby wave propagation may be computed from elevationand that the southwardacceleration is driven by the equation(3) by usingthe observedratio h:/h• (Figure 2). This Coriolis force on the westward flow. The ratio v/u, in the value exceedsthe frequencyband of the diurnal tides along presentcircumstance, is -if/co, where cois the K1 frequency. two short segments'at the extremeeastern end of the ice front This ratio impliesthat (1) the semi-majoraxis of the K1 ellipse and at a positionnorthwest of RooseveltIsland. is twice as long as the semi-minoraxis (because f • 2to),(2) the The currents associatedwith topographic Rossby waves semi-majoraxis is oriented north-south,and (3) the polariza- have the following three characteristics[Rhines, 1969]' (1) the tion is counterclockwise.This last point explainswhy the strip current magnitudeis maximum at the ice front and decayson of clockwisepolarization transecting the basin along the ice either side with an e folding decay scaleequal to the wave- front is considered unusual. length(approximately 100 km in the simulations);(2) the long Over much of the Ross Sea, the M2 tidal currents are recti- ice front flow has oppositedirections on either side of the ice linear and are aligned parallel to the nearest coast. These front; and (3) the tidal-currentellipses are polarized clockwise 614 MACAYEAL'SIMULATION OFROSS SEA TIDES

g : / ,

t 200 400 600 800t Shir•eCo#t Ro• DISTANCEALOI• ICE FRONT (km) Fig.12. Thestrong tidal current shear and the tidal current ellipse polarization reversal at theice front indicate possibletopographic Rossby wave excitation bythe diurnal tides. The analysis ofRhines [1969] and Longuet-Higgins [1968]indicates that a maximumfrequency cutoff exists, above which such waves cease to propagate.This cutoff is a functionofthe ratio of the depths oneither side of the ice front (h2/hx) and is plotted above asa function ofposition along theice front by using observed depths. The frequencies ofthe diurnal tides fall below this curve along two segments ofthe icefront where the K1 tidal currents displayed inFigures (10) and (11)exhibit strong cross ice front shear. andcounterclockwise on the shallow and deep sides of theice respectively.The estimatedresidual circulation along the ice front, respectively.The simulateddiurnal tidal currentsare frontis largeand could lead to significantcross ice front heat consistentwith thesetraits along the two segmentsof the ice transport[MacAyeal, 1983]. frontwhere the depth ratio, h2/h•, exceeds 2. Thepolarization Tidal frontscomprise the boundariesbetween stratified re- reversalis displayedalong the entireice front, and thissug- gionsand areaswhere the watercolumn is verticallywell geststhat thesewaves influence the regionsbeyond the short mixedby the actionof the tidal currents[Simpson and icefront segments where their propagation is strictlyallowed. Pingree,1977]. A roughestimate of tidalfront positions below theRoss Ice Shelf may be accomplished bycomparing the rate RESPONSEOF THE SUB-ICE-SHELFCAVITY TO STRONGTIDAL at whichtidal currents dissipate energy by generatingsmall- CURRENTS scaleturbulence to the buoyancyflux maintainingstratifi- cation[Fearnhead, 1975]. Preliminary assessmentof tidal current rectification and Verticallywell-mixed conditions are expectedalong the tidal frontformation may be accomplishedby applyingthe SipleCoast where ice shelf shoaling reduces the watercolumn scalesof thesimulated tidal currents to simpleanalytic formu- thicknessbelow a criticalvalue and where the buoyancy influx las.Robinson's 1-1981-1 treatment of tidal rectification,for ex- is likely to be weak [MacAyeal,1983]. Hydrographic ample,provides an appropriateorder-of-magnitude estimate measurementsthrough the J9 bore hole [Foster,1983] are of thetime-independent barotropic circulation driven by tidal consistentwith the suspectedpresence of tidal frontsin the currents.His expressionsfor themagnitude of thiscirculation, southeasternsection of thesub-ice-shelf cavity. Moreover, hy- (u), alonga seabedridge and an ice front are (u)= 6.3 drographicsections along the ice front [Jacobs et al., 1979] x 10-3 (m-2 s-•) Ay Y •D/•y and(u) = 0.11(m s -•) AD/D, revealglacial meltwater emerging from the sub-ice-shelf cavity respectively.In theseexpressions, Y is the rangeof cross- at depthsin excessof 400 m. Correspondencebetween this isobathwater column displacement driven by tidal currents, depthand the large ice shelf draft along the Siple Coast sug- Ay is thewidth of theseabed ridge, •D/•y is thewater depth geststhat zones of basalmelting correlate with zones of strong gradient,and AD is the depthchange across the ice front. verticalmixing induced by tidalcurrents. Tidal currents may Usingparameter values representing the seabed ridges extend- thus providea dominantcatalyst for strongvertical heat ing from the SipleCoast and the ice front northwestof Roose- transfernecessary for basalmelting and associatedthermoha- veltIsland, the estimates of (u) are0.0025 m/s and 0.06 m/s, line circulation. MACAYEAL: SIMULATION OF ROSSSEA TIDES 615

Acknowledgments.I thank the facultyand staffof the Geophysical Munk, W., F. Snodgrass,and M. Wimbush, Tides off-shore:Transi- Fluid DynamicsProgram of PrincetonUniversity, and of the Oceans tion from California coastal to deep-seawaters, Geophys.Fluid and Ice Branchof Goddard SpaceFlight Center,for valuablescien- Dyn., 1, 161-235, 1970. tific assistance,preparation of figures,and computertime. In addition, Nihoul, J. C. J., Hydrodynamicmodels, in Modelling of Marine Sys- I thank the manyreviewers who helpedto improvethe qualityof this tems,Elsevier Oceanogr. Set., vol. 10, edited by J. C. J. Nihoul, pp. manuscript.This researchwas supportedby the NASA Student 41-67, Elsevier, New York, 1975. TrainingProgram grant NGT 031-001-800,by PrincetonUniversity, Ramming, H. G., and Z. 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