JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. B9, PAGES 20,151-20,163, SEPTEMBER 10, 1996

Diurnal/semidiurnal polar motion excited by oceanic tidal B.F. Chao,1 R. D. Ray,2 J.M. Gipson,3 G. D. Egbert,4 and C. Ma I

Abstract. The axialcomponent of the oceanictidal angularmomentum (OTAM) hasbeen demonstratedto be responsiblefor mostof thediurnal and semidiurnal variations in 's rotationalrate. In thispaper we studythe equatorial components of OTAM andtheir correspondingeffects on the orientationof Earth'srotational axis, or polarmotion. Three oceantide models derived from TOPEX/Poseidonsatellite altimetry are employed to predict thepolar motion excited by eightmajor diurnal/semidiurnal (Q1, O1, P1, K1, N2, M2, $2, K2).The predictions are compared with geodetic measurements of polar motion from both long-termobservations and during the intensivecampaign Cont94. The progradediurnal and progradeand retrograde semidiurnal periods are treated, whereas the retrograde diurnal polar motionis not treated(because it cannotbe observeddirectly and uniquely.) The comparison showsgenerally good agreement, with discrepanciestypically within 10-30micro-arc- secondsfor thelargest tides. The eightfides collectively explain nearly 60% of the total variancein subdailypolar motion during Cont94. This establishesthe dominant role of OTAM in excitingthe diurnal/semidiurnalpolar motion and paves the way for detailed studiesof short-periodnontidal polar motion. The presentaccuracy, however, is inadequate to shedlight on the progradediurnal polar libration.

Introduction excitationvia the conservationof angularmomentum by the ocean tides. Two mechanisms are involved in this excitation Earth'srotation varies slightly with time dueto geophysical process [e.g., Munk and MacDonald, 1960]: (1) tidal processesthat involve mass movement on or withinthe Earth. deformationin the form of tidal heightvariation that changes We say that these geophysicalprocesses "excite" Earth Earth'sinertia tensor (referred to as the "mass"term); and (2) rotationalvariations, which manifestthemselves as changesin tidal motionrepresented by tidal currentthat evokesangular boththe rotational speed and the rotational axis orientation.The momentumrelative to the solid Earth as a whole (referred to as scalarvariation in the rotationalspeed is usuallyexpressed in the "motion"term). Connectedby the conservationof mass, termsof thetime derivativeof the universal time UT1 (or often theyare collectively called the oceanic tidal angularmomentum the length-of-dayif the variationin questionis longerthan a (OTAM). In the presenceof OTAM the conservationof the day). The vectorialvariation in the rotationalaxis orientation total angular momentumof the solid Earth-oceansystem relativeto the terrestrialreference frame is known as the polar dictateschanges in the solidEarth' s rotation. motion. Thispaper compares the polar motion observed by thespace One importantexcitation source is the luni-solartides, as geodetic techniquesof very-long-baselineinterferometry they influenceEarth's rotation in a variety of ways [e.g., (VLBI) and (SLR), with model Lambeck, 1980]. The one under considerationhere is the predictionsof diurnaland semidiurnal OTAM derivedfrom the TOPEX/Poseidonaltimetry data [e.g., Fu et al., 1994; Le Provostet al., 1995]. At any givenperiodicity (such as tidal) the polar motioncan be decomposedinto the progradeand 1LaboratoryforTerrestrial Physics, NASA Goddard Space Flight Center,Greenbelt, Maryland. retrogradecomponents (or "wobbles") [Munk and Mac- 2HughesSTX, NASA Goddard Space Flight Center, Greenbelt, Donald, 1960]. Here we treatboth componentsat the diurnal Maryland. and semidiurnalperiods, except the retrogradediurnal polar 3NVI, Inc.,NASA Goddard Space Flight Center, Greenbelt, motion for which, for reasonsdescribed below, only the Maryland. modelpredictions will be presented.In addition,results for 4Collegeof Oceanicand Atmospheric Sciences, Oregon State UT1 thatare updated fxom Chao et al. [1995]will be presented University,Corvallis. for comparison.

Copyright1996 by theAmerican Geophysical Union. Polar Motion Excitation Dynamics

Papernumber 96JB01649. Both kinematic and dynamic complicationsset the polar 0148- 0227/96/96JB-0164959.00 motionproblem apart from the UT1 problem.Kinematically,

20,151 20,152 CHAOET AL.:DIURNAL/SEM•IURNAL POLAR MOTION

UT1 is a scalarquantity and all observersagree on its value in a right-handedterrestrial coordinate system. Its physical independentof the referenceframe. The rotationalaxis excitationconsists of (1) the massterm: the complex-valued orientation,however, is a referenceframe-dependent vectorial quantityc whosereal andimaginary parts are the variationin quantity:While methods that are sensitive to absoluterotation thexz andyz componentsof theEarth's inertia tensor due to the (suchas those based on gravimetryor gyroscopicprinciples) massredistribution; (2) the motionterm: the complex-valued can potentiallymake direct measurement of the "true"polar relativeangular momentum h whosereal andimaginary parts motion[Chao, 1991], presentprecise measurements of Earth are simply the x and y componentsof the relative angular rotation have been solely obtained from space geodetic momentum.Here ctoo andh collectivelyrepresent the oceanic techniqueswhich measurethe relativerotation. The space tidal angularmomentum (OTAM). When normalizedagainst geodetictechniques do not determinethe polar motion A too as expressedin equation (1), they representthe uniquely;instead, they observe the orientation of thecelestial (dimensionless)excitation for the polarmotion. ephemerispole [e.g., Eubanks,1993; Soverset al., 1993; The first term in (1) arises from the Herring and Dong, 1994]. As a result,the retrogradepolar resonance. The numerical factors are the transfer functions motion at the diurnaltidal periodsis observationallyaliased [Munkand MacDonald, 1960] accountingfor Earth'selastic with andhence inseparable from thecorresponding . yielding effect and the decouplingof the fluid core from the Theseterms are relativelylarge as theyinclude the nutations mantle. The second term arises from the FCN resonance. driven by the externaltidal torquesand the polar motion Except in the retrogradediurnal band near the resonance,its resultingfrom the largesolid Earth fides. transfer functions are much smaller than their Chandler Dynamically,in contrastto the simplesystem function for counterpart(especially for the motionterm), so thatthe FCN UT1 excitation,the polar motion systemfunction can be term generallycontributes much less to the polar motion characterizedby a linearoscillatory system with two distinct excitationthan the Chandlerterm, and possibleerrors in the natural frequenciesof resonance:the progradeChandler FCN transferfunctions become numerically unimportant. wobblewith a periodof about434 days,and the retrograde free The mass term c in OTAM arisesfrom the tidal height, core (FCN) of the fluid core manifesting as a whereas the motion term h arises from the tidal current: retrogradenearly diurnal free wobble with frequency -(1+1/430) cycleper siderealday (cpsd)in the terrestrial frame [e.g., Herring et al., 1986; Merriam, 1994]. In the co Aa>o •cw - (•

Ocean Tide Convention and Models {DO 6.17x10'4 h(6o) + 0.0734 c(60)60oß (1) Atoo Atoo fit)fcn - O0 To evaluatec(t) and h(t) of equation(2) requiresglobal modelsof oceantidal height and current fields. As is traditional, whereA is theEarth's equatorial moment of inertia,o) o is the thesemodels are decomposedinto a setof discretefrequency meanangular rate of rotation,tOcw is the positiveChandler components,or constituents,thus enabling evaluation of c(to) angularfrequency corresponding to prograde434.45 sidereal andh(to) in (1). This paperconsiders in detailthe eight major dayswith a Q of 170 adoptedfrom Wilsonand Vicente[1990] diurnaland semidiurnal tidal constituents listed in Table1 (Q•, (butsee Furuya and Chao[1996]), and (.Ofca is the negative 01, P•, K1 , N2 , 342, S2 , andKa). Additional smaller fides are FCN angularfrequency at -1.00232 cpsd,corresponding to availablefrom at leastone of our analyses,and they may, in any retrograde23.88 hours. event, be roughlyinferred throughadmittance relationships The complex-valuedpole locationp is givenin radians;its [Munkand Cartwright,1966]. realpart is thex componentalong the Greenwichmeridian, and This sectiondiscusses three global (or near-global)tide itsimaginary part is they componentalong the 90øE longitude, modelsand presents their impliedestimates of polar motion CHAO ET AL.: DIURNAL/SEM[DIURNAL POLAR MOTION 20,153

Table 1. BasicProperties of theEight Major Diurnaland Semidiurnal Tides Studied in thisPaper and the Free Core Nutation(FCN)

Tide Period, Alias Period,* DoodsonArgument** Potential,***

(Type) hours days •: s h p X cm

Diurnal

Q• (L. elliptic) 26.868 9.13 1 -2 0 1 -90 ø 5.02 O• (L. princ.) 25.819 13.66 1 -1 0 0 -90 ø 26.22 P• (S. princ.) 24.066 182.62 1 1 -2 0 -90 ø 12.20 K• (S/L declin.) 23.935 ,• 1 1 0 0 +90 ø 36.87

FCN '23.88 -430 -

Semidiurnal

N2 (L. empti½) 12.658 9.13 2 -1 0 1 0 ø 12.10 M 2 (L. princ.) 12.421 13.66 2 0 0 0 0 ø 63.19 S2 (S.princ.) 12.000 182.62 2 2 -2 0 0 ø 29.40 K 2 (S/L declin.) 11.967 o• 2 2 0 0 0 ø 7.99 Tide typesL: lunar;S: solar. * As observedin inertialspace. ** Coeff•zientslisted for thefollowing angles: •:, mean lunar Greenwich time; s, meanlongitude of Moon;h, meanlongitude of Sun; p, meanlongitude of lunarperigee. Note thatif t is meansolar time, then '• = t- s + h. *** From Cartwrightand Tayler [ 1971].

parameters.We begin, however, with a brief review of some lagsproperly reflect the ocean'sphysically similar response to tidal conventions;such a discussionappears warranted in so forcingat nearbyfrequencies; see, for example,our Tables5 interdisciplinarya topic even thoughit will be well known to and6 wherephases vary smoothlyacross both the diurnaland experts. semidiurnal bands. Not shownin Table 1, but evidentin expansionssuch as that Tidal Phase Conventions of Cartwright and Tayler [1971], are the many additional smallerspectral lines comprising each tidal constituent.For the Along with the Kelvin-Darwin nomenclaturefor tidal eightconstituents listed, these lines occur significantly in only constituents,Table 1 liststhe tidal (cosine)arguments for the lunar tides; they differ in the fifth Doodson number (not eight fides. These argumentsimply use of standardphase shown), denotingdependence on the longitudeof the lunar conventionslong used in oceanography[Doodson, 1921]. nodeand thus differing in frequencyfrom the fundamentalline Unlikesome other conventions, the Doodson arguments, when by one or two cyclesin 18.6 years.These nodal modulations tabulatedin numericalorder, automatically classify the fides by areparticularly important for 0•, K•, andK 2. Obviously, they species, group, and constituent,without overlap and in should be accountedfor in any tidal analysisbased on short ascendingorder of frequency. (Recall that the classical time spans and in any tidal prediction. They have been definitionsof a tidalspecies, group, and constituent are a setof rigorouslyaccounted for in the oceanmodels discussed below. tidal spectral lines whose first, first two, and first three In VLBI and SLR analysis,authors have been inconsistent: Doodsonnumbers, respectively, are identical[Doodson, 1921; some have and somehave not allowed for them; fortunately, Munk and Cartwright,1966]. Different species, groups, and many of thesespace geodetic data setsare approachingtwo constituentsdiffer in frequency by at least 1 cycle/d, 1 decadesduration, so the neglectmay not be so crucial. cycle/month,and 1 cycle/yr,respectively.) The additional90 ø Onemethod of allowingfor nodalmodulations is to adoptthe increments in the diurnal band stem from Doodson's use of "nodalfactor" F and U, which vary periodicallywith the lunar cosine for all terms of (n+m) even but sine for (n+m) odd, node [Doodsonand Warburg, 1941] and are thussensibly where m is speciesnumber (1 for diurnal,2 for semidiurnal) constantover periodsof a few months.Any tidal constituent and n is the sphericalharmonic degree of the tidal potential withamplitude A andphase lag G can thenbe expressedas [A (n=2 for all tideslisted here). Finally, K• is 180ø offset, F(t) cos(lS'/(t)+ Z + U(t) - G)], where15 is a 4-vectorof becauseits amplitude in theharmonic development of the tidal Doodsonnumbers and l(t) is a corresponding4-vector of the potential[Doodson, 1921; Cartwrightand Tayler, 1971] is of astronomical variables in Table 1. This standard use of F and opposite sign relative to the other three diurnals. Direct U assumesa constanttidal responsefor all lines within a allowancefor this in thetidal argumentensures that tidal phase constituent,which is sufficientlyaccurate for our purposes, 20,154 CHAO ET AL.: DIURNAL/SEMIDIURNALPOLAR MOTION saveone exception: retrograde diurnal lines near (.0fcn, where factthat satellite altimeters are sensitiveonly to the geocentric theresonance response in (1) violatesthis constant-admittance tide:the sum of theocean tide, the body tide, andthe radial load assumption.The aboveexpression is followedin our tidal tide.For the latter two, purelyelastic models were adopted. predictionsof Figure 6 below, which does not include The tidal heights of model B were estimatedwith the retrogradediurnal components. responsemethod [Munk and Cartwright, 1966], which Modelpredictions of tidal variationsin polarmotion follow determinesthe shapeof the (complex)admittance across the fromequations (1) and(2). The tOcwand tOfc n values we adopt diurnal and semidiurnal bands rather than the (complex) for (1) follow Gross [1993] in orderto facilitatecomparison. amplitudesof individual tidal constituents.The response Our phaseconvention for progradeand retrogrademotions methodtherefore yields essentially all major and minor fides followsMunk and MacDonald [ 1960, equation(6.7.5)], and within each band, although we use here only the eight is consistentwith Gross [1993]. The complexpolar motion, constituents of Table 1. However, rather than quasi- correspondingto p(to) of (1), is writtenas independentanalyses of binneddata as in modelA (and as in theGeosat investigation ofCartwright and Ray [ 1990]that did p(t) = A•,exp[ io•,+i{(t) ] + Ar exp[i•-i{(t) ] usethe response method), model B expandseach response weight as a series of specialprecomputed oceanic normal where {[}(t)is the tidal argumentas given in Table 1, and modes.This requiresa large-scaleglobal inversion (of 5300 estimatedcoefficients in thepresent case), but it resultsin more subscripts p and r indicate prograde and retrograde, physicallyrealistic smoothing of the tidal maps,dependent on respectively.Tables 6 and7 will be in termsof the amplitudes the• bathymetrythrough the normal modes. The additional A• Ar andphases ai,, e•. atmosphericforcing at the S2 frequencywas accountedfor accordingto Cartwrightand Ray [1994]. Ocean Tide Models FollowingCartwright et fl. [ 1992], we estimatebarotropic tidal currentsfor modelsA and B throughhydrodynamical The threeocean models adopted here are basedon analysis relationshipswith the heightgradients. The methodrequires of the TOPEX/Poseidon(henceforth T/P) satellitealtimetry solvingthe following depth-averaged momentum equations (a measurements.T/P, launched in 1992 into a circular orbit of modifiedform of Laplace'stidal equations)for u andv at each 66 ø inclinationand 1336 km altitude,has generated a multiyear locationat which currentvelocities are required: time series of open-ocean height measurementswhich, consideringtheir near-globalcoverage, are of unprecedented 0t U -fv = -(g/a)csc0 0z(• - •o - T) (3a) accuracy,approximately 5 cmrms or less[Fu et al., 1994]. The mission benefits from several improvementsin altimetric technology, including use of a two-channel altimeter to 0t V'!' f u = (g/a)0o(t• - t•o - T) (3b) compensatefor ionosphericpath delay and, mostlynotably, tracking systemsthat have yieldedradial orbit ephemerides where g is the mean surfacegravitational acceleration, f = accurateto approximately3 cm rms [Marshall et al., 1995]. A 2tOoCOS0is the Coriolis parameter, •o is the"equilibrium" tide number of tidal analysesof T/P data have been carried out givenby (1 +k2-h2)/g times the primary tidal potential, with the (severalare describedin the T/P specialissue of Journal of Lovenumbers ka andh 2 allowingfor theelastic deformation of GeophysicalResearch, 99 (C12), 1994) [seealso Le Provost the body tide. The term T accountsfor loadingand self- et al., 1995;Andersenet al., 1995]. The threemodels adopted attractionof the tide and is here computedthrough a high- here, althoughrelying on essentiallythe samealtimetric data degreespherical harmonic expansion (to degree180) of the sets, employ different methodologiesand are therefore heightfield • [Hendershott,1972]. complementary.The modelsare those by Schramaand Ray F_xluations(3) contain no advection terms, no viscosityterms, [1994] (version9405, hereinafterreferred to as modelA), Ray in fact,no dissipativeterms whatsoever. They areappropriate et fl. [1994b](version 941230, hereinafter referred to as model strictlyto thedeep ocean. This restriction is acceptablebecause B), and Egbert et fl. [1994] (versionTPXO.2, hereinafter even though shelf current velocitiesexceed deep-ocean referredto as modelC). Briefly,both models A andB consist velocitiesby 1 or 2 ordersof magnitude,the small mass of tidalheights derived through strictly empirical analyses of (shallowdepth) associated with these currents implies that their thealfirne•c data,with tidal currentssubsequently deduced by angularmomentum is of little importance(see below). Of dynamicalrelationships; model C resultsfrom a formof data course,in purelynmx•cal hydrodynamicalmodels of thetides, assimilationthat yieldssimultaneously global fields of tidal carefulhandling of dissipativemechanisms in shallowseas is heightsand currents. Other models for whichthe tidal currents crucial;here thesemechanisms are implicit in our adopted are not explicitly providedcannot be used for OTAM (measured)height fields. calculation. The right-handsides of (3) are knownonce • is known. For model A, Schrama and Ray [1994] applied simple Velocitycomponents u, v arethen easily found, excepting at the harmonicfits to datacaptured in smallbins coveting the ocean criticallatitudes sin'l(o•tOo), where f= toand the equations surface,the bin sizebeing dictated by considerationsof tidal are indeterminate,in fact, are essentiallysingular (because of aliasing.The four major constituents M 2, S2, K1 andO 1 were physicalapproximations and measurement errors in thefight- determinedin the form of correctionsto the Schwiderski [ 1980] sideterms). The critical latitudesfor diurnaltides lie between model. As with all three models, allowancewas made for the 26.45ø (for Q1) and 30 ø (forK1); for semidiurnal fides they lie CHAO ET AL.: DIURNAL/SEMIDIURNAL POLAR MOTION 20,155 between70.94 o (forN 2) and 90 ø (forK2). As Cartwrightet al. accomplishthis. For now,we mustrely on model differencesas [1992] discuss,if u and v are decomposedinto rotarycyclonic thebest guide to errors.In fact,given the difficultiesof properly and anticycloniccomponents, it is only the anticyclonic definingrealistic covariances, including in inverse-theoretical componentsthat are indeterminateat the critical latitudes approaches,reliance on differencesof independentmodels is (stemmingphysically from the possibleexistence of inertial not necessarilya hindrance. currentsof identicalfrequency); the cycloniccomponents may Nonetheless,the height fields of thesemodels have beenwell be evaluated as usual. Test calculations show that the studied.They have been compared among themselves and with anticyclonicvelocity estimates become unstable within about othermodels, with in situtide gauge and oceanbottom pressure 5 ø of the critical latitudes. We have elected simply to measurements and with satellite altimeter measurements interpolatethe currentsacross the 10ø gap. Lesselementary [Andersenet al., 1995;C. K. Shumet al., Accuracyassessment approaches,invoking higher-order spatial gradients and Taylor of recent ocean tide models, submitted to Journal of seriesexpansions, might overcome the criticallatitude problem GeophysicalResearch, 1996]. The comparisonssuggest that more accurately(G. W. Platzman,personal communication the modelA heightsmay be slightlymore accurate than those (1994), who points out an earlier discussionby Brillouin of modelsB and C, but the differencesare small, and all three [1932]).We havenot pursued this in any depthpartly owingto modelsare considerably more accurate than any pre-T/P global the difficulties inherent in higher derivatives of measured model. (hencenoisy) height fields. To our minds,assimilation methods No detailed study of the current velocity fields has been hold morepromise, and model C to whichwe now turn is an undertaken. Yet we are confident that the model A currents are example. less accuratethan thoseof modelB or C, particularlyin the Model C is derived using a generalized inverse or diurnalband, because model A heightshave more pronounced assimilationapproach with a shallow-waterhydrodynamic high-wavenumbernoise (see the map of Schrarnaand Ray model [Bennett,1992; Egbertet al., •.994]. Tidal heightand [1994]),which is compoundedbythe requiredspatial gradients current fields are foundby minimizJ,•gthe misfit to both sea of equation(3). The normal-modeand representer expansions level elevations observed by F/P and the classical of modelsB and C ensurethat their heightsare relatively hydrodynamicalequations of Laplace,including the equationof smooth.In Table2 we showa comparisonof the modelcurrent continuity and appropriate boundary conditions.For the velocitiesto reciprocalacoustic measurements in the Pacific dynamicalconstraints, linearized shallow water dynamics are OceanbyDushaw et al. [1995] andto a currentmeter mooring assumed,essentially as in (3), butwith an additionaldissipation of Luther et al. [1991]. (The quoted uncertaintiesin the termincorporated as a linearparameterization of bottomdrag. acousticand mooring measurements are between0.3 and 0.5 A simplescalar correction (similar to that of Accad and mms4.) Although this single comparison cannot replace a Pekeris[1978]) is usedto accountfor oceanself attractionand detailedglobal study, it doesshow that the model currentsare load tides. The model domain extends farther north and south sensiblyrealistic. A comparisonbetween model C and tidal thanthe T/P coverage(see below), and all shallowcontinental constantsestimated using acoustic tomography data from the shelfareas connected to the openocean were included,to the AMODE experimentin the Atlantic Ocean [Dushawet al., extent possiblewith the roughly2/3 degreegrid. Details, 1994] showsa similar level of agreement.There is also a includingthe relativeweighting of dynamicsand data, and the suggestionthat diurnal tides may be less accurate than computationalapproach, are described by Egbertet al. [ 1994]. semidiurnals. Unlike the methods of models A and B, the inverse Models A and B originally extendedacross only those methodologyof model C providesa formal mechanismfor latitudesoverflown byT/P: 66øS to 66øN. Model C, by relying computingerror covariancesand hence for placingstandard on hydrodynamicmodeling, covered a slightlylarger range: errorson Earth rotationparameters. Work is in progressto 80øSto 70øN. The threemodels are here supplementedin the

Table 2. EastwardBarotropic Current Velocity at 40.6øN, 163.0øW

Diurnal Tides Semidiurnal Tides

Q, o, P, K, N2 M e $• K2 Acoustic* 1, 64 ø 4, 101o 2, 132ø 8, 128ø 1,191 o 13, 223 ø 5,272 ø 1,268 ø

Currentmeter** 1,110 ø 5, 122ø 3, 141ø 7, 135ø 2, 216ø 13, 218 ø 5,280 ø 1,280 ø

ModelA -- 7, 149 ø -- 9, 126ø -- 13, 226 ø 8, 272 ø --

Model B 1,104 ø 6, 116ø 4, 135ø 11,135 ø 2, 225 ø 12, 216 ø 8, 265 ø 2, 263 ø

Model C 1, 95 ø 5, 110ø 2, 129ø 7, 131ø 2, 193ø 14, 219 ø 6, 271ø 2, 275ø Amplitudesinmms -1, Greenwich phase lags in degrees. Model A fromSchrama and Ray [1994], model B fromRay et al. [ 1994b], andmodel C fromEgbert et al. [ 1994]. * FromDushaw et al. [1995],representing an averageacross a 745 km path. ** From Luther et al. [ 1991]. 20,156 CHAO ET AL.: DIURNAL/SEMIDIURNAL POLARMOTION

polar regionswith tidal heights adoptedfrom Schwiderski Data Comparison Results [1980] modelfor model A, from the strictlyhydrodynamic Polar Motion Data model of Le Provost et al. [1994] for model B, and with tidal heightsand currents from the hydrodynamicmodel of Kowalik As in workby Chaoetal. [1995] (hereinafterreferred to as and Proshutinsky[1994] for modelC (majorfides only). The CRE95), twotypes of polarmotion data will be employedin reasonfor threedifferent supplemental models is merelydue to comparisonswith tide models:"long-term" and "intensive". convenienceand model availability during the courseof the The primary long-term data are based on multiyear work. In any event,the polar regions,being small and located observationsfrom various VLBI networks,typically with 24- in extremehigh latitudes,are found to contributelittle to the hour observingsessions separated by gapsof a few days. polar motion excitation.A detailed treatmentof the Arctic Independentalgorithms have been applied to theseraw data by Ocean,including the handling of massconservation as it affects variousresearch groups [Sovers etal., 1993; Gipsonetal., tidal UT1, is givenby Ray et al. [1996]. 1993;Herring and Dong, 1994; Gipson, 1996], resulting in In Table 3 we tabulatethe completeOTAM for all three spectralestimates of thediurnal/semidiurnal tidal parameters. componentsfor theeight fides of Table1, brokendown to the Similarsolutions from long-term SLR measurements to satellite massterm (heigh0 and motion term (current) contributions. For LAGEOSare provided by R. Eanes(personal communication, spacelimitations we showonly the results from model C. In a 1996), which representupdated values from Watkinsand latersection we listthe vector totals of the polarmotion excited Eanes[1994]. These estimates for theeight major diurnal and by theseOTAM for all models.For variationsin UT1 the semidiurnaltides are given in Tables5-7. motion tenm from tidal currents dominate the mass terms from As for intensiveobservations of Earth rotation,there have tidalheights [Baader et al., 1983;Ray et al., 1994a].Table 3 been four VLBI specialcampaigns, each 1-2 weekslong: shows similar behavior for semidiurnal OTAM. The situation ERDE(October 15-31, 1989), Search92 (July 31 to August9, reverses for diurnal OTAM in x and y directionswhere the 1992),Cont94 (January 12-26, 1994), and Cont95 (August 23- massterms are generally, although not always,larger. 28, 1995).Sampling intervals as shortas 1 hourare achieved. Before leavingthe topic of modelresults, we alsouse the Asin CRE95,the present paper will focuson Cont94 [Gipson OTAM integralsof modelC to investigatethe influenceof etal., 1994], becausethe formalerrors during this intensive shallowseas and to justifythe neglect of shallow-seacurrents campaignwere a factorof roughly3 betterthan ERDE and a in modelsA and B. Table 4 showsthe x and y OTAM factor of 2 betterthan Search92. The formalerrors during componentsfor the M 2 tideof modelC, withand without the Cont95 are onlyslightly worse than those of Cont94,but the shallow seas.As expected,the currentsin shallow areas timespan is shorter and the data processing is stillpreliminary contributean insignificantamount to OTAM. Suchis notthe atthis writing. For the present purpose that focuses on subdaily casefor the tidal heights,however. Tidal heightsin shallow variations,each day during Cont94 is analyzedindependently; watercan be quite large, and they contribute appreciably to the the agreementof the seriesacross the daily boundariesserves inertia tensor.Unfortunately, fides in theseregions are also assome indication of the goodnessof the estimation. ,.,,'"t,,',-,,,•, vm'5'ingrapidly with d,o,,,-,•,•,;ø*... and axe not well UT1 Variation: An Update mappedwith altimetry. Shallow-water heights and deep-water currentsare undoubtedlythe primarysources of error in our First, we shall updatethe major resultsof CRE95 with tide-predictedpolar motion. respect to the diurnal/semidiurnaltidal UT1 variationswith

Table3. OceanicTidal AngularMomentum in All ThreeComponents Broken down to theMass (or Height)and Motion (or Curren0 TermsAccording to Model C

Diurnal Tides Semidiurnal Tides

Q1 Ol P1 K1 N2 M2 S2 K2

X

Height 1.1,342 ø 4.4, 334 ø 1.4, 316 ø 4.5,312 ø 1.4, 349 ø 5.3, 12ø 1.4, 50 ø 0.3, 53 ø

Current 0.8,305 ø 3.4,311 ø 1.5,293 ø 5.1,291 ø 1.5,247 ø 10.7,257 ø 5.8,293 ø 1.5,300 ø

Y

Height 2.5, 218 o 11.3,222ø 4.3,222 ø 13.5,222 ø 0.4, 260 ø 4.3,307 ø 3.3, 5 o 0.9, 2 ø

Current 0.8, 209 ø 4.6, 207 ø 2.4, 192ø 7.8, 188 ø 2.8, 160ø 18.5, 167ø 10.4, 199 ø 2.7, 199ø

z

Height 0.8, 136o 2.3, 159o 0.4, 0 ø 1.5, 26 o 0.8, 69 o 6.3, 82 o 2.8, 127 o 0.9, 126 o

Current 1.3, 106 ø 5.8, 119 ø 2.3, 125ø 7.6, 125ø 3.0, 326 ø 15.1,318 ø 7.3,344 ø 2.0, 336 ø Amplitudesin 1024 kg m 2 s'l; Greenwich phase lags in degrees. CHAO ET AL.: DIURNAL/SEMIDIURNAL POLAR MOTION 20,157

Table4. OceanicTidal Angular Momentum for M•. According to Model C

MassTerms (Tidal HeighO Motion Terms (Tidal Curten0

x y x y Global(minus Arctic Ocean) 5.76, 12.3o 4.23,306.4ø 10.36, 257.7 ø 18.31,165.3 Noshallows (cutoff is 200 m) 5.36•18.3 o 3.43•305.4 ø 10.39•257.7 ø 18.31,165.3 Amplitudesin1024 kg m 2 s'l; Greenwich phase lags in degrees. furtherfindings within the present context. For comparison,two CRE95 basedon work by Gipsonet al. [1993], in thatmany pre-T/P tide modelswill also be listedbut withoutmuch moresmall tidal termsare includedin the datareduction using discussion.One is basedon the empiricallyconstrained model theresponse method, and the nodal modulation is allowedfor of Schwiderski [1980]; its currents were calculated by in 01,K1, M2 and K 2. Cartwrightet al. [1992] (seealso Ray et al. [1994a])in the Figure 1 examinesthe VLBI and SLR estimatesof UT1 same manner as for models A and B. The other is the comparedwith T/P tide model predictionsfor the best theoreticalmodel by Broscheet al. [1989]followed by Seiler determinedtide M 2. It tellsa morecomplicated story than [1991],whose values were later reinterpreted by Gross[ 1993]. presentedin Figure2 of CRE95.The latter,which is identical Table 5 augmentsTable 1 of CRE95 with two (yet to Figure 1 herebut withoutthe two SLR estimates,showed unpublished)independent sets of estimatesfor UT1 excitation thatthe M2 spinlibration [Chao et al., 1991] can largely obtainedfrom long-termSLR: that by R. J. Eanes(updated explainthe departureof tide modelpredictions from VLBI from Watkinsand Eanes [1994], see above),and by Padis estimates.However, now with the two SLR estimateslying on [1994].The VLBI estimatesaccording to Gipson[1996] also the other side of the model predictions,the discrepancies representsan improvedset of valuesover thosequoted in betweentidal predictions and SLR estimatesobviously cannot

Table 5. TheUT1 Amplitudesand Greenwich Phase Lags for theEight Diurnal and Semidiurnal Tides in Table1

Diurnal Tides Semidiurnal Tides

Q1 O1 P1 K1 N2 M2 S• K

Observations

VLBI

Soverset al. 6.6, 37 ø 21.4, 39 ø 7.2, 27 ø 15.5, 13ø 3.0, 221 ø 18.2, 235 ø 5.2, 266 ø 2.8, 251 ø

Herring/Dong 5.3, 36ø 23.6, 47ø 7.1, 34ø 18.9,20 ø 3.2, 240ø 17.9,233 ø 8.6, 269ø 3.8, 282ø Gipson 5.6, 31o 22.2, 37ø 5.8, 25ø 18.6,29 ø 3.7, 239ø 18.6,236 o 8.0, 264ø 2.9, 283o

SLR

Eanes 7.0, 29 ø 22.6, 400 5.8, 28 ø 18.2, 24 ø 3.5,247 ø 17.0, 251 o 9.6, 281 o 2.7, 283 ø

Pavlis 5.0, 23 ø 25.2, 52 ø 6.6, 250 22.7, 36 ø 4.6, 234 ø 18.8, 261 o 4.2, 240 ø 3.3,225 ø

Predictions

Brosche/Seiler* -- 35.2, 22 ø 7.1, 59 ø 18.7, 52 ø 7.5,231 o 35.3, 230 ø 18.1,260 ø --

Schwiderski** 4.3, 18ø 20.4, 36 ø 5.9, 19ø 21.2, 27 ø 3.6, 240 ø 18.7, 244 ø 7.7, 256 ø 1.9, 260 ø

TOPEX/Poseid

ModelA -- 20.5, 29 ø -- 22.3,250 -- 19.4, 244 ø 7.7, 262 ø

Model B 4.8, 32 ø 23.2, 39 ø 8.3, 39 ø 24.2, 38 ø 3.8, 250 ø 17.6, 25 1o 7.7, 261 o 2.1,260 ø

Model C 5.6, 26o 20.1, 37 ø 5.9, 29 ø 19.7, 26 o 4.1,248 ø 17.7, 246 o 7.6, 267 ø 2.0, 259 ø , • , Amplitudesin micro-seconds(Its); phase lags in degrees(see text for phaseconvention). The observationsinclude those derived from long-termVI_BI measurements andthose from SLR. References: Sovers et al. [ 1993];Herring and Dong [ 1994];Gipson[1996]; Eanes [personalcommunication, 1996]; Padis [1994]; Broscheet al. [1989], Seller[1991];.Schwiderski [1980]. The TOPEX/Poseidon predictionsare computed according toT/P oceantide models; model A fromSchratna and Ray [1994],model B fromRay eta/. [1994b], andmodel C fromEgbert et al. [ 1994]. *As reinterpretedby Gross[ 1993]. **Tidal currentscomputed by Cartwrightet al. [ 1992]. 20,158 CHAO ET AL.: DIURNAL/SEMIDIURNAL POLAR MOTION

Table 6. Sameas Table 5, but for the ProgradePolar Motion

DiurnalTides Semidiurnal Tides

Q1 O1

Observations

VLBI

Sovers et al. 49, 54 ø 132, 54 ø 69, 92 ø 134, 51ø 23, 125 ø 22, 57 ø 21, 73 ø 32, 160 ø

Herring& Dong 35, 72 ø 199, 63 ø 60, 54 ø 152, 61 o 17, 135 ø 58, 91 o 12, 99 ø 39, 173 ø

Gipson 33, 81 ø 148, 74 ø 51, 60 ø 166, 63 ø 16, 108 ø 62, 110 ø 14, 89 ø 15, 104 ø

SLR

Fanes 31, 81 ø 121, 71 o 57, 65 ø 178, 66 ø 24, 144 ø 72, 117 ø 31,110 ø 8.8, 115ø

Predictions

Brosche/Seiler 181, 78 ø 55, 67 ø 172, 68 ø 22, 126 ø 105, 31, 68 ø

Schwiderski 29, 72 ø 142, 71 ø 54, 55 ø 179, 58 ø 22, 115ø 81,104 ø 24, 76 ø 6, 80 ø

TOPEX/Poseidon

Model A 135, 76 o -- 160, 65 ø -- 72, 111 o 23, 82 ø

Model B 22, 82 ø 114, 67 ø 49, 57 ø 150, 57 ø 15, 127 ø 74, 120 ø 28, 86 ø 7, 90 ø

Model C 27, 77 ø 142, 70 ø 56, 63 ø 171, 63 ø 17, 135ø 75, 116 ø 29, 86 ø 7, 96 ø Amplitudein micro-arcseconds(gas). be explainedby spinlibration. In fact, the two setsof estimates inadequatetidal loading corrections at groundstation sites. The (VLBI and SLR), thoughagreeing relatively well within resolution of this matter awaits further research. themselves,disagree by more than two standarddeviations. Figure6a replotsthe predictedUT1 curveof T/P modelsB •his impliesthe presence of significantsystematic errors in at and C, superimposedon the hourly Cont94 data from the leastone set of observations,and possibly both, due possibly to NASA R&D networkas in Figure 1 of CRE95, exceptthat

Table 7. Sameas Table 6, but for 'theRetrograde Polar Motion

Diurnal Tides Semidiurnal Tides

Q1 Ol P1 K1

Observations

VLBI*

Sovers et al. 37, 267 ø 265, 273 ø 174, 303 ø 62, 286 ø

Herring& Dong 48, 282 ø 265, 272 ø 120, 304 ø 31,328 ø

Gipson 46, 269 ø 257, 272 ø 128, 303 ø 18, 346 ø

SLR*

Fanes 28, 253 ø 253, 267 ø 124, 297 ø 33, 301 o

Predictions

Brosche/Seiler 83, 201 ø 881,153 ø 8865, 148 ø 40, 249 ø 303, 240 ø 232, 287 ø

Schwiderski 39, 304 ø 142, 305 ø 804, 139 ø 9400, 135 ø 35, 267 ø 251,278 ø 135,303 ø 38, 306ø

TOPEX/Poseidon

Model A -- 167, 305 o _ 9510, 129 o 245, 271ø 121,298 ø

Model B 68, 321 o 200, 314 ø 838, 138 ø 9700, 135 ø 46, 267 ø 262, 274 ø 140, 303 ø 40, 301 o

Model C 45, 307 ø 132, 297 ø 797, 136 ø 9670, 133 ø 45, 269 ø 263, 271ø 131,298 ø 34, 301 o *The observedretrograde diurnal values are absentfor the reasonsgiven in the text. CHAO ET AL.' DIURNAL/SEMIDIURNAL POLAR MOTION 20,159

I I I 50

Libration

0 - .__

•. -50

P -100

o ._o •-•5o

'• -200 VLBI O

-15 - -25O SLRI VLBI Ocean models I -300 I -20 • I I I i i i i i i i i -20 -15 -10 -5 0 5 50 - 100 -50 0 50 100 150 200 In-phase(rnicro-arcsecond) AUT1 In-phase (I,I.s) Figure 1. The phasordiagram for UT1 variationat theM 2 Figure 2. The phasordiagram for the retrogradeM 2 polar period:Three VLBI- andtwo SLR-observedvalues (with motion accordingto Table 7: Three VLBI- and one SLR- formalerror circles), and four tide-model predictions (three T/P observedvalues are givenas thin linesproperly labeled. The models,as well as the Schwiderskimodel which will not be three T/P tide model predictionsare shown as the heavy shownin Figures2-5) are givenaccording to Table5. The arrows. Consult Table 7 for individual identifications. correspondingcontribution of theEarth spin libration is also plotted. here the Cont94 data are preprocessedin a procedure 3. Theagreement between T/P fide-modelpredictions and somewhatimproved from those in CRE95,and the NEOS observationsis generally fairly good, especially for tideswith networkdata used in CRE95 in extendingthe seriesand filling relativelylarge amplitude. The best determined component is the1-day gap are not included here. The processing adopted retrogradeM2; the agreementis withinseveral percent, Or heregives slightly better agreement with the tidal models than typically10 I. tas. The next largest component, prograde K•, also presentedin CRE95. seesagreement within about 15%, or some 20 I.tas.Overall, the bestagreement is found between VLBI estimatesby Gipson Polar Motion: "Long-Term" Comparison [1996] andT/P modelC predictions. 4. Theagreement ispoorer for smaller tides. This is alsotrue Tables6 and7 listthe prograde and retrograde polar motion, amongthe VLBI estimates;Sovers et al. [1993]differ with respectively,forthe eight diurnal/semidiurnal tideconstituents. Others for the prograde M2 andretrograde K2 byup to a factor Figures2-5 plotthe phasor diagrams for someof thebetter of 3, andless severely for theretrograde S2 andprograde K•. determinedtides with polar motion amplitudes exceeding 100 Thereason may be related to theinternal inconsistency among micro-arcseconds(Bas), or about3 mmin poleposition. On Soverset al.'s estimatesfor these fides obtainedfrom different eachdiagram, three T/P tide modelpredictions (the heavy VLBI networks. arrows,consult Tables 6 and7 for individualidentification), 5. Thereare often systematic differences between VLBI and threeVLBI estimates,and one SLR estimateare shown(and so SLR estimates.The mostnotable case is the progradeK 2, labeled).The separationbetween estimates indicates the wheretwo of thethree VLBI estimateshave anomalouslylarge uncertaintiesassociated with eachtype of estimate. amplitudes(judging from the small tidal potential for this fide Several facts are noted as follows: • in Table 1) and diversephases. Gipson's [1996] VLBI 1. Thelarge amplitudes in the model-predicted retrograde estimate for this tide is much less anomalous,while the SLR diurnalcomponents due to FCN resonanceare listed here only estimateagrees much better with model predictions. for reference.As discussedearlier, no direct observationsfor 6. The Schwiderskimodel (with Cartwrightet al. [ 1992] themare available. The largestamplitudes depend sensitively currents)agrees well with the T/P modelsin theirpolar motion onthe actualFCN periodand Q. predictions,whereas the Brosche/Seiler model often predicts a 2. In general,the three T/P tidemodels agree well in their muchlarger effect. This is also found to be the case for UT1 by polar motionpredictions, typically within 5-10 •tas in Ray et al. [1994a].The accuracyof Schwiderskimodel is an•litudeand 10 ø in phase.This is consistent with findings in somewhatsurprising, given that the T/P elevationmodels are CRE95 for UT1. significantlymore accurate (Shum et al.,submitted manuscript, 20,160 CHAO ET AL.: DIURNAL/SEMIDIURNAL POLAR MOTION

I 180 2O

160

140 -2O

120 o -40 o

o 100 • -6o 30 .o 80 v E -80

SLR • -lOO 60

40 o -12o

20 -140 Libration

-160 I i I i i i i -50 0 50 1 O0 -50 0 50 100 150 In-phase(micro-arcsecond) In-phase (micro-arcsecond)

Figure3. San• as Figure 2 but for retrograde S2 polar motion. Figure 5. San• asFigure 4 butfor progradeO1 polar motion.

1996).The reason for thispresumably lies in certainfortuitous B and C (modelA is not plottedbecause it has fewer tide cancellationof the errorswhen the modelis integratedto form constituents). the Earth rotationparameters. The matchbetween model predictions and observations is apparentlyquite good, although not as good as the top panel for Polar Motion: "Intensive" Comparison UT1. This matchcan be quantifiedas follows.The total varianceof the Cont94polar motionseries (in its complex Figures6b and 6c presentthe hourlydata of the x andy form)is found to be 215 in 'units of 103Ilas 2, while the noise componentsofpolar motion during Cont94 intensive campaign. variancegiven the formal error [Gipson et al., 1994]is about Superimposedonthem are the predicted curves of T/P models 26.As expected, we find the total variance greatly reduced, to ! !0, aftersubtraction of thetides (them•e!ves of variance122 accordingtomodel C). Therefore,assuming uncorrelated noise, approximately1-(110-26)/(215-26) or nearly60% of the 160 SL subdailypolar motion power can be explained by theeight tides (comparedto thecorresponding -90% forUT1). Thison one 140 LBI handestablishes the dominantrole of fidesin excitingsubdaily polarmotion, and on the other hand indicates the existence of •-120 importantnontidal contributions. Besides observational noises o • 100 andmodeling errors, the majority of thediscrepancies seen in Figure6 canpresumably be attributedto thefollowing effects ,?.o80 not consideredin the oceantide models:(1) the presenceof

._ E v othergeophysical sources such as subdailyatmospheric and .e 60 oceanicangular momentum variations, daily andhalf-daily atmosphericthermal "fides," Earth librations (see below), and (, 40 0 possiblyearthquaY; (2) contributions from the response ofthe

2o internalinhomogeneities of the solidEarth to diurnaland Libration semidiurnaltidal forcing; (3) theneglect of smallertides in the tide modelspresented here [cf. Seilerand Wlinsch,1995]. I Closerexamination of Figure6 showsthat the tide models -2O I , almostalways underpredict the diurnaland especiallythe -50 o 50 100 5O In-phase(micro-arcsecond) semidiurnalpeaks. This is truefor both UT1 (seeCRE95) and polarmotion. Similar phenomenon is found with respect to Figure4. Sameas Figure 2 butfor progradeK 1polar motion Cont95data as well [Gipsonet al., 1996].The initial study by accordingto Table6. The correspondingcontribution of the Ma et al. [1995] has indicatedthe importanceof the Earth polar librationis alsoplotted. Consult Table 6 for atmosphericangular momentum and thermal tides; but, clearly, individual identifications. morecomprehensive investigations will beneeded. CHAO ET AL.: DIURNAL/SEMIDIURNAL POLAR MOTION 20,161

i i i i i i i ...... 50 -- --

500 - -

0- -

-500 - -

500 -

0 -

-500

Diurnal retrograde removed I I I I I I I I I I I I 13 14 15 16 17 18 19 20 21 22 23 24 25 January1994

Figure6. ComparisonofEarth rotation variations during Cont94 Campaign. The hourly observations with standarderrors are those made by VLBI (theNASA R&D network);the solid lines are predictions according toT/P tide Models B andC (seetext). (a) UT1 variationin unitsof Its;(b) x and(c) y componentsof the polar motion(in unitsof Bas)without the retrograde diurnal components.

Polar Libration accuraciesare inadequate for the libration to "close"the budget evenfor the largesttidal polar motions. Besidesthe semidiurnalspin librations (discussed above), the luni-solartidal torquesexerted on Earth's equatorial Summary ellipticitygive rise to diurnalpolar librations in theprograde direction [Chao et al., 1991]. The observedpolar motion Theoceanic tidal angular momentum (OTAM), consisting of shouldcontain the tide-generatedpolar motion and the polar the massand motion terms,excites Earth rotationvariation at librations,the largest amplitude of which(for K1) is probably all tidalperiods. CRE95 established that the axial component nomore than 15 gasdepending on the interiorproperties of the of OTAM is responsiblefor asmuch as 90% of thesubdaily Earth. Thus, if sufficientlyaccurate, the observedand tide- (primarilydiurnal and semidiurnal) power of theUT1 variation. generatedpolar motioncan be differencedto reveal the Hereupdated examples are presented inFigures 1 and6a. The librationcontribution. One hopesthat this will lead to useful mainpurpose of thepresent paper, however, is to studythe constraintson Earth's interior properties,particularly the equatorialcomponents of OTAM andtheir corresponding equatorialellipticity of thecore-mantle boundary [Chao et al., effectson the polarmotion. 1991;Herring and Dong, 1994;CRE95]. The equatorialcomponents of OTAM for eightmajor Accordingtoformulae given by Chao et al. [ 1991] underthe diurnal/semidiurnalfides (Q1, O1, P1, K1, N2,342, S2, K2) are presentphase convention, the predicted polar libration is (14.6 computedaccording to threeocean tide models derived from gas,150 ø) forK 1 and(11.3 gas, 150 ø) for Ot asplotted in TOPEX/Poseidonsatellite altimetry. Agreeing well among Figures4 and 5, respectively.Unfortunately, the departures themselves,these predictions are then compared with geodetic withinestimates from each technique, and hence the estimation measurementsof polar motion made by VLBI andSLR long- uncertainties,are generallylarger than the libration.Thus, term observations as well as VLBI intensive measurements similarlyfor UT1, the presentmeasurement and modeling duringthe Cont94campaign. The progradediurnal and 20,162 CHAO ET AL.' DIURNAL/SEMIDIURNAL POLAR MOTION progradeand retrograde semidiurnal polar motionsare treated; Doodson,A. T., The harmonicdevelopment of the tide-generating the retrogradediurnal polar motion is not treatedbecause it potential,Proc. R. Soc.London A, 100, 305-329,1921. (Reprinted cannotbe observeddirectly and independentlyof the nutations in Int. Hydrog.Rev., 31, 11-35, 1954.) by geodeticmeans. Doodson,A. T., andH. D. Warburg,Admiralty Manual of Tides,His The comparisonshows generally good agreement,with Majesty's StationOff., London,1941. Dushaw, B. D., P. F. Worcester,B. D. Cornuelle,and B. M. Howe, discrepanciestypically within 10-30 Basfor the largesttides Barotropicand baroclinicfides of the westernNorth Atlantic, Eos andprogressively poorer for smallertides. Examples are given Trans.AGU, 75 (44), Fall Meet. Suppl.,307, 1994. in Figures2-5. Overall, the bestagreement is foundbetween Dushaw,B. D., B. D. Cornuelle,P. F. Worcester,B. M. Howe, and D. VLBI estimatesby Gipson[1996] andtide modelpredictions S. Luther, Barotropicand baroclinicfides in the centralNorth accordingto T/P modelof Egbertet al. [ 1994].During the 2 Pacific Oceandetermined from long-rangereciprocal acoustic weeksof Cont94, the eight tidescollectively explain nearly transmissions,J. Phys. Oceanogr., 25, 631-647, 1995. 60% of the totalvariance in sub-dailypolar motion, as shown Egbert, G. D., A. F. Bennett, and M. G. G. Foreman, in Figures 6b and 6c. This on one hand establishesthe TOPEX/Poseidonfides estimated using a globalinverse model, J. dominantrole of OTAM in excitingthe diurnal/semidiurnal Geophys.Res., 99, 24,821-24,852, 1994. Eubanks, T. M., Variations in the orientation of the Earth, in polarmotion and on the otherhand paves the way for detailed Contributionsof Space to Geodynamics:Earth studiesof nontidalpolar motion. The excitationsources for the Dynamics,Geodyn. Ser., vol. 24, editedby D. E. Smithand D. L. latter include subdaily atmosphericand oceanic angular Turcott,pp. 1-54,AGU, Washington,D.C., 1993. momentumvariations, the daily and half-daily atmospheric Fu,L. L., E. J. Christensen,C. A. Yamatone,M. Lefebvre,Y. Menard, thermal"tides", Earth librations, earthquakes, and the response M. Dorrer, and P. Escudier,TOPEX/Poseidon mission overview, of the internal inhomogeneitiesof the solid Earth to J. Geophys.Res., 99, 24,369-24,381, 1994. diurnal/semidiurnaltidal forcing.A full, quantitativeresolution Fumya,M., andB. F. Chao,Estimation of periodand Q of the awaitsfurther data and investigations. Chandlerwobble, Geophys. J. Int., in press,1996. Gipson,J. M., VLBI determinationof neglectedtidal termsin Acknowledgments. We thank E. Pavlis and R. Eanes for the high-frequencyEarth orientationvariation, J. Geophys. unpublishedSLR results.This work is supportedby the NASA Research,in press,1996. GeophysicsProgram. Gipson,J. M., B. F. Chao,and C. Ma, RapidEOP variationsmeasured by VLBI, Eos Trans.AGU, 74 (16), SpringMeet. Suppl.,103, References 1993. Gipson,J. M., C. Ma, T. M. Eubanks,and A. P. Freedman,Diurnal Accad,Y., and C. L. Pekeris,Solution ofthe tidal equations for the M 2 andsubdiurnal EOP variationsduring CON'I94, Eos Trans.AGU, and S2 tidesin the worldoceans from a knowledgeof the tidal 75 (16), SpringMeet. Suppl.,I I l, 1994. potentialalone, Philos. Trans. R. Soc. 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Tayler,New computationsof the tide- Lainbeck,K., TheEarth's Variable Rotation, Cambridge Univ. Press, genexafingpotential, Geophys. J. R. Astron.Soc., 23, 45-74, 1971. New York, 1980. Cartwright,D. E., R. D. Ray,and B. V. Sanchez,A computerprogram Le Provost,C., M. L. Genco,F. Lyard, P. Vincent,and P. Canceil, for predictingoceanic tidal currents,NASA Tech• Memo., TM- Spectroscopyof the world ocean fides from a finite element 104578, 1992. hydrodynamicmodel, J. Geophys.Res., 99, 24,777-24,797, 1994. Chao, B. F., As the world turns,II, EOS Trans. AGU, 72, 550-551, Le Provost,C., A. F. Bennett,and D. E. Cartwright,Ocean tides for 1991. and from TOPEX/Poseidon, Science,267, 639-642, 1995. Chao,B. E, D. N. Dong,H. S. Liu, andT. A. Herring,Libration in the Luther, D. S., J. H. Filloux,and A.D. Chave,Low frequency Earth'srotation, Geophys. Res. Lett., 18, 2007-2010, 1991. motionallyinduced electromagnetic fields in the ocean,2, Electric Chao,B. E, R. D. Ray,and G. D. 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