JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. B9, PAGES 20,151-20,163, SEPTEMBER 10, 1996
Diurnal/semidiurnal polar motion excited by oceanic
tidal angular momentum
B.F.Chao,
1 R.D.Ray,
2 J.M. Gipson,
3 G.D. Egbert,
4 andC.MaI
Abstract. The axialcomponent
of theoceanictidalangularmomentum(OTAM) hasbeen
demonstrated
to be responsible
for mostof thediurnalandsemidiurnal
variationsin Earth's
rotationalrate.In thispaperwe studytheequatorialcomponents
of OTAM andtheir
corresponding
effectson theorientationof Earth'srotationalaxis,or polarmotion.Three
oceantidemodelsderivedfrom TOPEX/Poseidon
satellitealtimetryareemployedto predict
thepolarmotion
excited
byeightmajordiurnal/semidiurnal
tides(Q1,O1,P1,K1,N2,M2,$2,
K2).Thepredictions
arecompared
withgeodetic
measurements
of polarmotionfromboth
long-termobservations
andduringthe intensivecampaignCont94.The progradediurnaland
prograde
andretrograde
semidiurnal
periodsaretreated,whereastheretrograde
diurnalpolar
motionis not treated(becauseit cannotbe observeddirectlyanduniquely.)The comparison
showsgenerallygoodagreement,
with discrepancies
typicallywithin 10-30micro-arcseconds
for thelargesttides.The eightfidescollectivelyexplainnearly60% of thetotal
variancein subdailypolarmotionduringCont94.Thisestablishes
thedominantrole of
OTAM in excitingthediurnal/semidiurnal
polarmotionandpavesthe way for detailed
studiesof short-period
nontidalpolarmotion.The presentaccuracy,however,is inadequate
to shedlight on theprogradediurnalpolarlibration.
Introduction
excitationvia the conservationof angularmomentumby the
ocean tides. Two mechanisms are involved in this excitation
Earth'srotationvariesslightlywith timedueto geophysical
processes
thatinvolvemassmovementon or withintheEarth.
We say that these geophysicalprocesses"excite" Earth
rotationalvariations,which manifestthemselvesas changesin
boththerotationalspeedandtherotationalaxis orientation.The
scalarvariationin the rotationalspeedis usuallyexpressedin
process[e.g., Munk and MacDonald, 1960]: (1) tidal
deformationin the form of tidal heightvariationthat changes
Earth'sinertiatensor(referredto as the "mass"term); and(2)
tidal motionrepresented
by tidal currentthatevokesangular
momentumrelativeto the solid Earth as a whole (referred to as
the "motion"term). Connectedby the conservationof mass,
theyarecollectively
calledtheoceanictidalangularmomentum
of the
the length-of-dayif the variationin questionis longerthana (OTAM). In the presenceof OTAM the conservation
day). The vectorialvariationin therotationalaxisorientation total angularmomentumof the solid Earth-oceansystem
in thesolidEarth's rotation.
relativeto the terrestrialreferenceframe is known asthe polar dictateschanges
termsof thetime derivativeof the universaltime UT1 (or often
Thispapercompares
thepolarmotionobserved
by thespace
geodetictechniquesof very-long-baselineinterferometry
(VLBI) and satellite laser ranging (SLR), with model
Lambeck, 1980]. The one under considerationhere is the predictions
ofdiurnalandsemidiurnal
OTAM derivedfromthe
TOPEX/Poseidon
altimetrydata [e.g., Fu et al., 1994; Le
Provostet al., 1995]. At anygivenperiodicity(suchastidal)
the polar motioncan be decomposed
into the progradeand
1Laboratory
forTerrestrial
Physics,
NASAGoddard
Space
Flight
retrograde
components
(or
"wobbles")
[Munk and MacCenter,Greenbelt,Maryland.
motion.
One importantexcitationsourceis the luni-solartides,as
they influenceEarth's rotation in a variety of ways [e.g.,
2Hughes
STX,NASAGoddard
Space
FlightCenter,
Greenbelt,Donald, 1960]. Here we treatboth componentsat the diurnal
Maryland.
and semidiurnalperiods,exceptthe retrogradediurnalpolar
3NVI, Inc.,NASAGoddard
SpaceFlightCenter,Greenbelt,motion for which, for reasonsdescribedbelow, only the tide
Maryland.
modelpredictionswill be presented.In addition,resultsfor
4College
of Oceanic
andAtmospheric
Sciences,
OregonState UT1 thatareupdated
fxomChaoetal. [1995]will be presented
University,Corvallis.
for comparison.
Copyright1996by theAmericanGeophysical
Union.
Polar Motion Excitation Dynamics
Papernumber96JB01649.
0148- 0227/96/96JB-0164959.00
Both kinematicand dynamiccomplicationsset the polar
motionproblemapartfrom theUT1 problem.Kinematically,
20,151
20,152
CHAOETAL.:DIURNAL/SEM•IURNAL
POLARMOTION
UT1 is a scalarquantityandall observers
agreeon its value
independentof the referenceframe. The rotationalaxis
orientation,
however,is a referenceframe-dependent
vectorial
quantity:
Whilemethods
thataresensitive
to absolute
rotation
(suchasthosebasedon gravimetryor gyroscopic
principles)
canpotentially
makedirectmeasurement
of the "true"polar
motion[Chao, 1991], presentprecisemeasurements
of Earth
rotation have been solely obtainedfrom space geodetic
techniques
which measurethe relativerotation.The space
geodetictechniquesdo not determinethe polar motion
uniquely;
instead,theyobservetheorientation
of thecelestial
ephemerispole [e.g.,Eubanks,1993; Soverset al., 1993;
Herring and Dong, 1994]. As a result,the retrograde
polar
motion at the diurnaltidal periodsis observationally
aliased
withandhenceinseparable
fromthecorresponding
nutations.
Thesetermsare relativelylargeastheyincludethe nutations
driven by the externaltidal torquesand the polar motion
resultingfromthelargesolidEarthfides.
Dynamically,in contrastto the simplesystemfunctionfor
UT1 excitation,the polar motion systemfunctioncan be
characterized
by a linearoscillatory
systemwith two distinct
natural frequenciesof resonance:the progradeChandler
wobblewitha periodof about434 days,andtheretrograde
free
core nutation (FCN) of the fluid core manifestingas a
in a right-handedterrestrialcoordinatesystem.Its physical
excitationconsistsof (1) the massterm:the complex-valued
quantityc whosereal andimaginarypartsare the variationin
thexz andyz components
of theEarth'sinertiatensordueto the
massredistribution;(2) the motionterm:the complex-valued
relativeangularmomentumh whosereal andimaginaryparts
are simplythe x and y componentsof the relative angular
momentum.
Here ctoo andh collectivelyrepresent
theoceanic
tidal angularmomentum(OTAM). Whennormalizedagainst
Atoo as expressedin equation(1), they representthe
(dimensionless)
excitationfor thepolarmotion.
The first term in (1) arises from the Chandler wobble
resonance.
The numerical
factors are the transfer functions
[MunkandMacDonald, 1960]accounting
for Earth'selastic
yielding effectand the decouplingof the fluid core from the
mantle. The second term arises from the FCN
resonance.
Exceptin the retrogradediurnalbandnearthe resonance,its
transfer
functions
are much
smaller
than their Chandler
counterpart(especiallyfor the motionterm),so thattheFCN
term generallycontributesmuch less to the polar motion
excitationthan the Chandlerterm, andpossibleerrorsin the
FCN transferfunctionsbecomenumericallyunimportant.
The mass term c in OTAM arisesfrom the tidal height,
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
(2a)
frame[e.g.,Herring et al., 1986; Merriam, 1994]. In the
retrogradediurnalband the near-resonance
effect is very
sensitiveto the FCN period and Q [e.g., Wahr, 1981]
(especially
forcomponents
closest
to FCN periodsuchasK1),
sothatdetailed
knowledge
abouttheOTAM excitationof polar
motionandnutation
canin principleprovideusefulconstraints Inthese
equations,
a isEarth's
mean
radius;
p= 1035kgm'3is
c<t)=
-a4
pf f ocean
;(•,t)sinOcosOet•'d•
h<t)
=-aSpf
f o•ean
[u<f•,t)cos0
+iv<fLt)lH(f•)ea'df•<
onnutation
modelsandthedetermination
of FCN periodand Q. the mean densityof seawater;f• is an abbreviationfor the
For the abovetwo reasons,we shallnot treatthe retrograde
diurnal
polarmotionspersein thispaperbutwill onlypresent
thetide-heightandcurrentmodelpredictions
in thiscase.
Theequation
ofmotionforpolarmotioncanbederivedfrom
the Liouvilleequationbasedon theconservation
of angular
momentum [Munk and MacDonald, 1960]. Written in the
frequency
(to)domain,it is [SasaoandWahr,1981]
p(6o) =
1.124
h(to)+ 0.772 c(tø)
tøo
Aa>o
Aa>o
(DO
Atoo
Atoo
northwardspeed,respectively,of thetidalcurrentassumedto
be barotropicandhenceuniformoverthe watercolumnwith
depth H. Thesetidal variablesare providedby tide models
describedin the next section.Thus our studyof polar motion
procxeds
by simpleapplicationof theprincipleof conservation
of angularmomentum;the natureof the torquesthatgive rise
to this motion need not concern us here.
•cw - (•
{DO
6.17x
10
'4h(6o)
+ 0.0734 c(60)60o
ß
elementfor the integral over the oceans.The tidal height
relative to the seabedis (; u and v are the eastwardand
Ocean Tide Convention
and Models
(1)
fit)fc
n - O0
To evaluatec(t) and h(t) of equation(2) requiresglobal
modelsof oceantidalheightandcurrentfields.As is traditional,
whereA is theEarth'sequatorial
momentof inertia,o)oisthe thesemodelsaredecomposed
into a setof discretefrequency
components,
or
constituents,
thus
enablingevaluation
of c(to)
meanangularrate of rotation,tOcw
is the positiveChandler
angularfrequencycorresponding
to prograde434.45 sidereal andh(to) in (1). This paperconsidersin detailtheeightmajor
dayswitha Q of 170 adoptedfrom Wilsonand Vicente[1990] diurnal
andsemidiurnal
tidalconstituents
listedin Table1 (Q•,
(butseeFuruyaandChao[1996]),and(.Ofca
is thenegative 01, P•, K1, N2, 342, S2 , andKa).Additional
smallerfidesare
FCN angularfrequencyat -1.00232 cpsd,corresponding
to
retrograde23.88 hours.
The complex-valued
pole locationp is givenin radians;its
realpartis thex component
alongthe Greenwichmeridian,and
itsimaginary
partisthey component
alongthe90øElongitude,
availablefromat leastoneof our analyses,andtheymay, in any
event, be roughlyinferred throughadmittancerelationships
[Munkand Cartwright,1966].
This sectiondiscussesthree global (or near-global)tide
modelsandpresentstheir impliedestimatesof polar motion
CHAO ET AL.: DIURNAL/SEM[DIURNAL POLAR MOTION
20,153
Table 1. BasicProperties
of theEightMajor DiurnalandSemidiurnalTidesStudiedin thisPaperandtheFreeCore
Nutation(FCN)
Tide
Period,
AliasPeriod,*
(Type)
hours
days
DoodsonArgument**
•:
s
h
Potential,***
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/Ldeclin.)
23.935
,•
1
1
0
0
+90 ø
36.87
'23.88
-430
N2 (L. empti½)
12.658
9.13
2
-1
0
1
0ø
12.10
M 2 (L. princ.)
12.421
13.66
2
0
0
0
0ø
63.19
S2 (S.princ.)
12.000
182.62
2
2
-2
0
0ø
29.40
K2 (S/L declin.)
11.967
o•
2
2
0
0
0ø
7.99
FCN
-
Semidiurnal
TidetypesL: lunar;S: solar.
* As observed
in inertialspace.
** Coeff•zients
listedfor thefollowingangles:
•:,meanlunarGreenwich
time;s, meanlongitudeof Moon;h, meanlongitude
of Sun;
p, meanlongitudeof lunarperigee.Notethatif t is meansolartime,then'• = t- s + h.
*** From Cartwrightand Tayler [ 1971].
parameters.We begin, however, with a brief review of some lagsproperlyreflectthe ocean'sphysicallysimilarresponse
to
see,for example,ourTables5
tidal conventions;
sucha discussionappearswarrantedin so forcingat nearbyfrequencies;
interdisciplinarya topic even thoughit will be well known to and6 wherephasesvary smoothlyacrossboththe diurnaland
experts.
semidiurnal
bands.
Not shownin Table 1, but evidentin expansionssuchasthat
of Cartwright and Tayler [1971], are the many additional
smallerspectral
linescomprisingeachtidal constituent.
For the
listed,theselinesoccursignificantlyin only
Along with the Kelvin-Darwin nomenclaturefor tidal eightconstituents
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), denotingdependenceon the longitudeof the lunar
conventionslong used in oceanography
[Doodson, 1921]. nodeandthusdifferingin frequencyfrom the fundamentalline
Unlikesomeotherconventions,
theDoodsonarguments,
when by oneor two cyclesin 18.6 years.Thesenodalmodulations
tabulated
in numerical
order,automatically
classifythefidesby areparticularly
important
for 0•, K•, andK2.Obviously,
they
species, group, and constituent,without overlap and in shouldbe accountedfor in any tidal analysisbasedon short
ascendingorder of frequency. (Recall that the classical time spans and in any tidal prediction. They have been
accounted
for in the oceanmodelsdiscussed
below.
definitions
of a tidalspecies,group,andconstituentare a setof rigorously
tidal spectral lines whose first, first two, and first three In VLBI and SLR analysis,authorshave been inconsistent:
Doodsonnumbers,
respectively,
areidentical[Doodson,1921; some have and somehave not allowed for them; fortunately,
Munkand Cartwright,1966].Differentspecies,
groups,and many of thesespacegeodeticdata setsare approachingtwo
constituentsdiffer in frequencyby at least 1 cycle/d, 1 decadesduration,so the neglectmay not be socrucial.
Onemethodof allowingfornodalmodulations
is to adoptthe
cycle/month,
and1 cycle/yr,respectively.)
The additional90ø
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) constantoverperiodsof a few months.Any tidal constituent
A andphaselag G canthenbe expressed
as [A
and n is the sphericalharmonicdegreeof the tidal potential withamplitude
Tidal
Phase Conventions
(n=2 for all tideslistedhere).Finally,K• is 180ø offset, F(t) cos(lS'/(t)+ Z + U(t) - G)], where15is a 4-vectorof
because
itsamplitudein theharmonicdevelopmentof thetidal
potential
[Doodson,1921; Cartwrightand Tayler, 1971] is of
opposite sign relative to the other three diurnals. Direct
allowance
forthisin thetidal argumentensuresthattidal phase
Doodsonnumbersandl(t) is a corresponding
4-vectorof the
astronomical variables in Table 1. This standard use of F and
U assumesa constanttidal responsefor all lines within a
constituent,which is sufficientlyaccuratefor our purposes,
20,154
CHAO ET AL.: DIURNAL/SEMIDIURNALPOLARMOTION
saveoneexception:
retrograde
diurnallinesnear(.0fcn,
where factthatsatellitealtimetersare sensitiveonly to the geocentric
theresonance
response
in (1) violatesthisconstant-admittance tide:thesumof theoceantide,thebodytide, andtheradialload
assumption.The aboveexpressionis followedin our tidal tide.For the lattertwo, purelyelasticmodelswere adopted.
The tidal heights of model B were estimatedwith the
predictionsof Figure 6 below, which does not include
response
method [Munk and Cartwright, 1966], which
retrogradediurnalcomponents.
Modelpredictions
of tidalvariationsin polarmotionfollow determinesthe shapeof the (complex)admittanceacrossthe
fromequations
(1) and(2). The tOcw
andtOfc
nvalueswe adopt diurnal and semidiurnal bands rather than the (complex)
for (1) follow Gross [1993] in orderto facilitatecomparison. amplitudesof individual tidal constituents.The response
Our phaseconventionfor progradeand retrogrademotions methodthereforeyieldsessentiallyall 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 quasiindependent
analysesof binneddataasin modelA (andasin
corresponding
to p(to)of (1), is writtenas
theGeosat
investigation
of CartwrightandRay [ 1990]thatdid
usetherespons
e method),modelB expands
eachresponse
p(t) = A•,exp[io•,+i{(t)
] + Arexp[i•-i{(t) ]
weight as a seriesof specialprecomputedoceanicnormal
modes.This requiresa large-scaleglobalinversion(of 5300
where{[}(t)is the tidal argumentas given in Table 1, and
estimated
coefficients
in thepresentcase),but it resultsin more
subscripts p and r indicate prograde and retrograde,
physicallyrealisticsmoothingof the tidalmaps,dependenton
respectively.
Tables6 and7 will be in termsof theamplitudes
the•
bathymetry
throughthenormalmodes.The additional
A• Ar andphases
ai,,e•.
atmospheric
forcingat the S2 frequency
was accounted
for
according
to
Cartwright
and
Ray
[1994].
Ocean Tide Models
FollowingCartwrightet fl. [ 1992],we estimatebarotropic
tidal currentsfor modelsA and B throughhydrodynamical
The threeoceanmodelsadoptedherearebasedon analysis
relationships
with the heightgradients.The methodrequires
of the TOPEX/Poseidon(henceforth
T/P) satellitealtimetry
solving
the
following
depth-averaged
momentumequations
(a
measurements.T/P, launched in 1992 into a circular orbit of
modifiedformof Laplace'stidal equations)for u andv at each
66ø inclination
and1336km altitude,
hasgenerated
a multiyear
time series of open-oceanheight measurementswhich,
consideringtheir near-globalcoverage,are of unprecedented
accuracy,
approximately
5 cmrmsor less[Fu et al., 1994]. The
mission benefits from several improvementsin altimetric
technology, including use of a two-channel altimeter to
compensatefor ionosphericpath delay and, mostlynotably,
tracking systemsthat have yieldedradial orbit ephemerides
accurate
to approximately
3 cm rms [Marshallet al., 1995]. A
number of tidal analysesof T/P data have been carried out
(severalare describedin the T/P specialissueof Journal of
Geophysical
Research,99 (C12), 1994) [seealsoLe Provost
et al., 1995;Andersenet al., 1995]. The threemodelsadopted
here, althoughrelyingon essentially
the samealtimetricdata
sets, employ different methodologiesand are therefore
locationat whichcurrentvelocitiesarerequired:
0t U-fv = -(g/a)csc00z(• - •o - T)
(3a)
0t V'!'f u = (g/a)0o(t•- t•o- T)
(3b)
where g is the mean surfacegravitationalacceleration,
f =
2tOoCOS0
istheCoriolisparameter,
•o isthe"equilibrium"
tide
givenby(1+k2-h2)/g
timestheprimarytidalpotential,
withthe
Lovenumbers
ka andh2 allowingfortheelasticdeformation
of
the body tide. The term T accountsfor loadingand self-
attractionof the tide and is here computedthrougha highdegreesphericalharmonicexpansion
(to degree180) of the
heightfield• [Hendershott,
1972].
F_xluations
(3) contain
noadvection
terms,no viscosityterms,
complementary.
Themodelsarethoseby SchramaandRay
[1994](version9405,hereinafterreferredto asmodelA), Ray in fact,no dissipative
termswhatsoever.
They areappropriate
tothedeepocean.Thisrestrictionis acceptable
because
et fl. [1994b](version941230,hereinafterreferredto asmodel strictly
B), andEgbert et fl. [1994] (versionTPXO.2, hereinafter even though shelf current velocitiesexceed deep-ocean
referredto asmodelC). Briefly,bothmodelsA andB consist velocitiesby 1 or 2 ordersof magnitude,the small mass
depth)associated
withthesecurrentsimpliesthattheir
of tidalheightsderivedthroughstrictlyempiricalanalyses
of (shallow
is of little importance
(seebelow).Of
thealfirne•c data,with tidal currentssubsequently
deducedby angularmomentum
inpurelynmx•cal hydrodynamical
modelsof thetides,
dynamicalrelationships;
modelC resultsfroma formof data course,
in shallowseasis
assimilation
that yieldssimultaneously
globalfieldsof tidal carefulhandlingof dissipativemechanisms
heights
andcurrents.
Othermodelsfor whichthetidalcurrents crucial;here thesemechanismsare implicit in our adopted
are not explicitlyprovidedcannotbe used for OTAM (measured)heightfields.
calculation.
The right-handsidesof (3) are knownonce• is known.
u, v aretheneasilyfound,excepting
at the
For model A, Schrama and Ray [1994] applied simple Velocitycomponents
latitudes
sin'l(o•tOo),
where
f= toandtheequations
harmonic
fitsto datacapturedin smallbinscovetingthe ocean critical
in fact, are essentially
singular(becauseof
surface,the bin sizebeingdictatedby considerations
of tidal are indeterminate,
andmeasurement
errorsin thefightaliasing.
Thefourmajorconstituents
M2,S2,K1 andO1 were physicalapproximations
determinedin the form of correctionsto the Schwiderski[ 1980]
model. As with all three models,allowancewas made for the
sideterms).The criticallatitudesfor diurnaltideslie between
26.45ø(forQ1)and30ø(forK1);forsemidiurnal
fidestheylie
CHAO ET AL.: DIURNAL/SEMIDIURNAL
POLAR MOTION
20,155
between
70.94o(forN2)and90ø (forK2).As Cartwrightet al.
accomplish
this.For now,we mustrely on modeldifferencesas
thebestguideto errors.In fact,giventhedifficultiesof properly
definingrealisticcovariances,includingin inverse-theoretical
approaches,relianceon differencesof independentmodelsis
notnecessarily
a hindrance.
Nonetheless,
theheightfieldsof thesemodelshave beenwell
be evaluated as usual. Test calculations show that the
studied.
Theyhavebeencomparedamongthemselvesand with
anticyclonicvelocityestimatesbecomeunstablewithin about othermodels,within situtidegaugeand oceanbottompressure
5 ø of the critical latitudes. We have elected simply to measurements and with satellite altimeter measurements
interpolatethe currentsacrossthe 10ø gap. Lesselementary [Andersen
etal., 1995;C. K. Shumet al., Accuracyassessment
approaches,
invokinghigher-order
spatialgradientsandTaylor of recent ocean tide models, submitted to Journal of
seriesexpansions,
mightovercomethe criticallatitudeproblem GeophysicalResearch,1996]. The comparisons
suggestthat
more accurately(G. W. Platzman,personalcommunication themodelA heightsmay be slightlymoreaccuratethanthose
(1994), who points out an earlier discussionby Brillouin of modelsB and C, but the differencesare small, and all three
[1932]).We havenotpursuedthisin any depthpartlyowingto modelsareconsiderably
moreaccuratethananypre-T/P global
the difficulties inherent in higher derivatives of measured model.
(hencenoisy)heightfields.To ourminds,assimilationmethods
No detailed studyof the current velocity fields has been
hold morepromise,andmodelC to whichwe now turnis an undertaken. Yet we are confident that the model A currents are
example.
less accuratethanthoseof modelB or C, particularlyin the
Model C is derived using a generalized inverse or diurnalband,becausemodelA heightshavemorepronounced
assimilationapproachwith a shallow-waterhydrodynamic high-wavenumber
noise(see the map of Schrarnaand Ray
model [Bennett,1992;Egbertet al., •.994]. Tidal heightand [1994]),whichis compounded
bythe requiredspatialgradients
current fields are foundby minimizJ,•gthe misfit to both sea of equation(3). The normal-modeandrepresenter
expansions
level elevations observed by F/P and the classical of modelsB and C ensurethat their heightsare relatively
hydrodynamical
equations
of Laplace,includingtheequationof smooth.
In Table2 we showa comparisonof themodelcurrent
continuity and appropriateboundaryconditions.For the velocitiesto reciprocalacousticmeasurements
in the Pacific
dynamicalconstraints,
linearizedshallowwaterdynamicsare OceanbyDushawet al. [1995] andto a currentmetermooring
assumed,
essentially
asin (3), butwith an additionaldissipation of Luther et al. [1991]. (The quoted uncertaintiesin the
termincorporated
asa linearparameterization
of bottomdrag. acousticandmooringmeasurements
are between0.3 and 0.5
A simplescalarcorrection(similarto that of Accad and mms4.)Although
thissinglecomparison
cannot
replace
a
Pekeris[1978]) is usedto accountfor oceanself attractionand detailedglobalstudy,it doesshowthatthe modelcurrentsare
load tides. The model domain extends farther north and south
sensiblyrealistic.A comparisonbetweenmodelC and tidal
thantheT/P coverage(seebelow),andall shallowcontinental constantsestimatedusingacoustictomographydata from the
shelfareasconnectedto the openoceanwere included,to the AMODE experimentin the AtlanticOcean [Dushawet al.,
extent possiblewith the roughly2/3 degreegrid. Details, 1994] showsa similar level of agreement.There is also a
includingtherelativeweightingof dynamicsanddata,andthe suggestionthat diurnal tides may be less accurate than
[1992] discuss,if u andv aredecomposed
into rotarycyclonic
and anticycloniccomponents,it is only the anticyclonic
componentsthat are indeterminateat the critical latitudes
(stemmingphysicallyfrom the possibleexistenceof inertial
currents
of identicalfrequency);the cycloniccomponents
may
computational
approach,
aredescribed
by Egbertet al. [ 1994].
Unlike
semidiurnals.
the methods of models A and B, the inverse
Models A and B originally extendedacrossonly those
overflown
byT/P: 66øS to 66øN. Model C, by relying
methodologyof model C providesa formal mechanismfor latitudes
computingerror covariances
andhencefor placingstandard on hydrodynamicmodeling,covereda slightlylarger range:
in the
errorson Earth rotationparameters.Work is in progressto 80øSto 70øN. The threemodelsarehere supplemented
Table 2. EastwardBarotropic
CurrentVelocityat 40.6øN, 163.0øW
Diurnal Tides
Semidiurnal Tides
Q,
o,
P,
K,
N2
Me
$•
K2
Acoustic*
1, 64 ø
4, 101o
2, 132ø
8, 128ø
1,191 o
13, 223ø
5,272 ø
1,268 ø
Currentmeter**
1,110 ø
5, 122ø
3, 141ø
7, 135ø
2, 216ø
13, 218ø
5,280 ø
1,280 ø
ModelA
--
7, 149ø
--
9, 126ø
--
13, 226 ø
8, 272 ø
--
Model B
1,104 ø
6, 116ø
4, 135ø
11,135 ø
2, 225ø
12, 216ø
8, 265ø
2, 263ø
ModelC
1, 95ø
5, 110ø
2, 129ø
7, 131ø
2, 193ø
14, 219ø
6, 271ø
2, 275ø
Amplitudes
inmms-1,Greenwich
phase
lagsindegrees.
Model
A fromSchrama
andRay[1994],
model
B fromRayetal.[1994b],
andmodelC fromEgbertet al. [ 1994].
* FromDushawet al. [1995],representing
anaverageacrossa 745 km path.
** From Luther et al. [ 1991].
20,156
CHAO ET AL.: DIURNAL/SEMIDIURNAL POLARMOTION
polar regionswith tidal heightsadoptedfrom Schwiderski
[1980] modelfor modelA, from the strictlyhydrodynamic
Data ComparisonResults
Polar Motion
model of Le Provost et al. [1994] for model B, and with tidal
Data
heightsandcurrentsfrom thehydrodynamicmodelof Kowalik
Asin workby Chaoetal. [1995] (hereinafterreferredto as
and Proshutinsky
[1994]for modelC (majorfidesonly).The
CRE95),twotypesofpolarmotiondatawill beemployed
in
reasonfor threedifferentsupplemental
modelsis merelydueto
convenienceand model availability during the courseof the
work. In any event,the polarregions,beingsmallandlocated
in extremehigh latitudes,are foundto contributelittle to the
polar motion excitation.A detailed treatmentof the Arctic
Ocean,including
thehandlingof massconservation
asit affects
tidalUT1, is givenby Ray et al. [1996].
In Table 3 we tabulatethe completeOTAM for all three
comparisons
with tide models:"long-term"and "intensive".
The primary long-term data are based on multiyear
observations
fromvariousVLBI networks,
typicallywith24hour observing
sessions
separated
by gapsof a few days.
Independent
algorithms
havebeenappliedto theserawdataby
variousresearchgroups[Soversetal., 1993; Gipsonetal.,
1993;HerringandDong,1994;Gipson,1996],resulting
in
spectralestimates
of thediurnal/semidiurnal
tidalparameters.
fromlong-term
SLRmeasurements
to satellite
components
for theeightfidesof Table1, brokendownto the Similarsolutions
byR. Eanes(personal
communication,
massterm(heigh0andmotionterm(current)contributions.
For LAGEOSareprovided
space
limitations
we showonlytheresults
frommodelC. In a 1996), which representupdatedvaluesfrom Watkinsand
for theeightmajordiurnaland
latersectionwe listthevectortotalsof the polarmotionexcited Eanes[1994].Theseestimates
by theseOTAM for all models.For variations
in UT1 the semidiurnaltidesaregivenin Tables5-7.
As for intensiveobservationsof Earth rotation,therehave
motion tenm from tidal currents dominate the mass terms from
been four VLBI specialcampaigns,
each 1-2 weekslong:
shows similar behavior for semidiurnal OTAM. The situation ERDE(October
15-31,1989),Search92
(July31 to August9,
(January
12-26,1994),andCont95(August23reversesfor diurnal OTAM in x and y directionswhere the 1992),Cont94
28, 1995).Samplingintervalsasshortas 1 hourareachieved.
masstermsaregenerally,althoughnotalways,larger.
paperwill focusonCont94[Gipson
Beforeleavingthe topicof modelresults,we alsousethe AsinCRE95,thepresent
theformalerrorsduringthisintensive
OTAM integralsof modelC to investigate
the influenceof etal., 1994],because
shallowseasandto justifytheneglectof shallow-sea
currents campaignwere a factorof roughly3 betterthanERDE anda
in modelsA and B. Table 4 showsthe x and y OTAM factorof 2 betterthanSearch92.The formalerrorsduring
components
fortheM2 tideof modelC, withandwithoutthe Cont95are onlyslightlyworsethanthoseof Cont94,but the
andthedataprocessing
is stillpreliminary
shallow seas.As expected,the currentsin shallowareas timespanisshorter
purpose
thatfocuses
onsubdaily
contribute
aninsignificant
amountto OTAM. Suchis notthe atthiswriting.Forthepresent
variations,
each
day
during
Cont94
is
analyzed
independently;
casefor the tidal heights,however.Tidal heightsin shallow
of theseriesacrossthedailyboundaries
serves
watercanbequitelarge,andtheycontribute
appreciably
to the theagreement
of the estimation.
inertia tensor.Unfortunately,fidesin theseregionsare also assomeindicationof the goodness
and axe not well
,.,,'"t,,',-,,,•, vm'5'ing
rapidlywith d,o,,,-,•,•,
;ø*...
UT1 Variation: An Update
mapped
withaltimetry.Shallow-water
heightsanddeep-water
currentsare undoubtedly
the primarysourcesof error in our
First, we shall updatethe major resultsof CRE95 with
tidalheights[Baaderet al., 1983;Rayet al., 1994a].Table3
respect to the diurnal/semidiurnaltidal UT1 variationswith
tide-predicted
polarmotion.
Table3. OceanicTidalAngularMomentumin All ThreeComponents
Brokendownto theMass(or Height)andMotion
(or Curren0TermsAccordingto Model C
Diurnal Tides
Semidiurnal Tides
Q1
Ol
P1
K1
N2
M2
S2
K2
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
X
Y
Height
2.5, 218 o
11.3,222ø
4.3,222 ø
13.5,222 ø
0.4, 260ø
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, 188ø
2.8, 160ø
18.5, 167ø
10.4, 199ø
2.7, 199ø
Height
0.8, 136o
2.3, 159o
0.4, 0 ø
1.5, 26o
0.8, 69 o
6.3, 82 o
2.8, 127 o
0.9, 126o
Current
1.3, 106ø
5.8, 119ø
2.3, 125ø
7.6, 125ø
3.0, 326 ø
15.1,318 ø
7.3,344 ø
2.0, 336 ø
z
Amplitudes
in 1024
kgm2s'l;Greenwich
phase
lagsindegrees.
CHAO ET AL.: DIURNAL/SEMIDIURNAL
POLAR MOTION
20,157
Table4. Oceanic
TidalAngular
Momentum
forM•.According
toModelC
Motion Terms (Tidal Curten0
MassTerms(Tidal HeighO
x
y
x
y
Global(minusArcticOcean)
5.76, 12.3o
4.23,306.4ø
10.36, 257.7 ø
18.31,165.3
Noshallows
(cutoff
is200m)
5.36•18.3o
3.43•305.4ø
10.39•257.7ø
18.31,165.3
Amplitudes
in1024
kgm2s'l;Greenwich
phase
lags
indegrees.
furtherfindings
withinthepresent
context.For comparison,
two CRE95 basedon workby Gipsonet al. [1993], in thatmany
pre-T/P tide modelswill also be listedbut withoutmuch moresmalltidal termsare includedin the datareductionusing
discussion.
One is basedon theempiricallyconstrained
model theresponse
method,andthenodalmodulation
is allowedfor
of Schwiderski [1980]; its currents were calculated by
in 01,K1,M2andK2.
Figure 1 examinesthe VLBI and SLR estimatesof UT1
comparedwith T/P tide model predictionsfor the best
theoretical
modelby Broscheet al. [1989]followedby Seiler determined
tideM2. It tellsa morecomplicated
storythan
[1991],whosevalueswerelaterreinterpreted
by Gross[ 1993]. presented
in Figure2 of CRE95.Thelatter,whichis identical
showed
Table 5 augmentsTable 1 of CRE95 with two (yet to Figure 1 herebut withoutthe two SLR estimates,
unpublished)
independent
setsof estimates
forUT1 excitation thattheM2 spinlibration[Chaoet al., 1991]canlargely
obtainedfrom long-termSLR: thatby R. J. Eanes(updated explainthe departureof tide modelpredictions
from VLBI
However,now with the two SLR estimateslying on
from Watkinsand Eanes [1994], seeabove),and by Padis estimates.
[1994].The VLBI estimatesaccordingto Gipson[1996] also the other side of the model predictions,the discrepancies
tidalpredictions
andSLR estimates
obviouslycannot
represents
an improvedset of valuesover thosequotedin between
Cartwrightet al. [1992] (seealsoRay et al. [1994a])in the
same manner as for models A and B. The other is the
Table 5. TheUT1 Amplitudes
andGreenwich
PhaseLagsfor theEightDiurnalandSemidiurnal
Tidesin Table1
Diurnal Tides
Q1
O1
P1
Semidiurnal Tides
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
18.9,20ø
3.2, 240ø
17.9,233ø
8.6, 269ø
3.8, 282ø
Gipson
5.6, 31o
22.2,37ø
5.8, 25ø
18.6,29ø
3.7, 239ø
18.6,236o
8.0, 264ø
2.9, 283o
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
SLR
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
--
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
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
Schwiderski**
TOPEX/Poseid
,
•
,
Amplitudes
inmicro-seconds
(Its);phaselagsin degrees
(seetextforphaseconvention).
The observations
includethosederivedfrom
long-term
VI_BImeasurements
andthose
fromSLR.References:
Sovers
etal. [ 1993];HerringandDong[ 1994];Gipson[1996];
Eanes
[personal
communication,
1996];Padis [1994];Broscheet al. [1989], Seller[1991];.Schwiderski
[1980]. The TOPEX/Poseidon
predictions
arecomputed
according
toT/P ocean
tidemodels;
modelA fromSchratnaandRay [1994],modelB fromRayeta/. [1994b],
andmodelC fromEgbertet al. [ 1994].
*As reinterpreted
by Gross[ 1993].
**Tidal currentscomputed
by Cartwrightet al. [ 1992].
20,158
CHAO ET AL.: DIURNAL/SEMIDIURNAL
POLAR MOTION
Table 6. SameasTable5, but for theProgradePolarMotion
DiurnalTides
Q1
Semidiurnal
Tides
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, 160ø
Herring& Dong
35, 72 ø
199, 63 ø
60, 54 ø
152, 61 o
17, 135 ø
58, 91 o
12, 99 ø
39, 173ø
Gipson
33, 81 ø
148, 74 ø
51, 60 ø
166, 63 ø
16, 108ø
62, 110 ø
14, 89 ø
15, 104ø
31, 81 ø
121, 71 o
57, 65 ø
178, 66 ø
24, 144ø
72, 117ø
31,110 ø
8.8, 115ø
SLR
Fanes
Predictions
Brosche/Seiler
Schwiderski
29, 72 ø
181, 78 ø
55, 67 ø
172, 68 ø
22, 126 ø
105,
31, 68 ø
142, 71 ø
54, 55 ø
179, 58 ø
22, 115ø
81,104 ø
24, 76 ø
135, 76 o
--
160, 65 ø
--
72, 111 o
23, 82 ø
6, 80 ø
TOPEX/Poseidon
Model A
Model B
22, 82 ø
114, 67 ø
49, 57 ø
150, 57 ø
15, 127ø
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, 116ø
29, 86 ø
7, 96 ø
Amplitudein micro-arcseconds
(gas).
be explained
by spinlibration.In fact, the two setsof estimates inadequate
tidalloadingcorrections
at groundstationsites.The
(VLBI and SLR), thoughagreeingrelativelywell within resolution of this matter awaits further research.
themselves,
disagreeby more than two standarddeviations.
Figure6a replotsthe predictedUT1 curveof T/P modelsB
•his impliesthepresence
of significantsystematic
errorsin at and C, superimposed
on the hourly Cont94 data from the
leastonesetofobservations,
andpossiblyboth,duepossiblyto NASA R&D networkas in Figure 1 of CRE95, exceptthat
Table 7. SameasTable6, but for 'theRetrogradePolarMotion
Diurnal Tides
Q1
Ol
Semidiurnal
P1
Tides
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
28, 253 ø
253, 267 ø
124, 297 ø
33, 301 o
SLR*
Fanes
Predictions
Brosche/Seiler
83, 201 ø
881,153 ø
8865, 148 ø
40, 249 ø
303, 240 ø
232, 287 ø
39, 304 ø
142, 305 ø
804, 139 ø
9400, 135 ø
35, 267 ø
251,278 ø
135,303 ø
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
Schwiderski
38, 306ø
TOPEX/Poseidon
*The observedretrogradediurnalvaluesareabsentfor thereasonsgivenin thetext.
CHAO ET AL.' DIURNAL/SEMIDIURNAL POLAR MOTION
I
I
20,159
I
50
Libration
.__
0 -
•. -50
P -100
o
._o
•-•5o
'• -200
VLBI
-15
O
-25O
SLRI
I
Ocean models
-300
-20
-20
•
I
I
I
-15
-10
-5
0
I
i
5
AUT1 In-phase (I,I.s)
VLBI
50
i
- 100
i
-50
i
i
i
i
i
0
50
100
150
200
In-phase(rnicro-arcsecond)
M 2 polar
Figure 1. ThephasordiagramforUT1 variationat theM2 Figure 2. The phasordiagramfor the retrograde
motion
according
to
Table
7:
Three
VLBIand
one
SLRperiod:ThreeVLBI- andtwo SLR-observed
values(with
formalerrorcircles),
andfourtide-model
predictions
(threeT/P observedvaluesare givenasthin linesproperlylabeled.The
models,as well as the Schwiderskimodelwhichwill not be
shownin Figures2-5) are givenaccording
to Table5. The
corresponding
contribution
of theEarthspinlibrationis also
three T/P tide model predictionsare shown as the heavy
arrows. Consult Table 7 for individual identifications.
plotted.
3. Theagreement
between
T/P fide-model
predictions
and
observations
is
generally
fairly
good,
especially
for
tides
with
somewhat
improved
fromthosein CRE95,andtheNEOS
largeamplitude.
Thebestdetermined
component
is
networkdatausedin CRE95 in extendingtheseriesandfilling relatively
retrograde
M2;
the
agreement
is
within
several
percent,
Or
the1-daygaparenotincluded
here.Theprocessing
adopted
10I.tas.Thenextlargest
component,
prograde
K•, also
heregives
slightly
betteragreement
withthetidalmodels
than typically
sees
agreement
withinabout
15%,orsome20 I.tas.Overall,the
presented
in CRE95.
bestagreement
isfoundbetween
VLBI estimates
byGipson
here the Cont94 data are preprocessed
in a procedure
Polar Motion: "Long-Term" Comparison
[1996] andT/P modelC predictions.
4. Theagreement
ispoorer
forsmallertides.Thisis alsotrue
Soverset al. [1993]differwith
Tables6 and7 listtheprograde
andretrograde
polarmotion, amongtheVLBI estimates;
fortheprograde
M2 andretrograde
K2byuptoa factor
respectively,
fortheeightdiurnal/semidiurnal
tideconstituents.Others
fortheretrograde
S2 andprograde
K•.
Figures2-5 plotthephasordiagrams
for someof thebetter of 3, andlessseverely
mayberelatedto theinternalinconsistency
among
determined
tideswithpolarmotionamplitudes
exceeding
100 Thereason
micro-arcseconds
(Bas),or about3 mmin poleposition.
On Soverset al.'s estimatesfor thesefides obtainedfrom different
eachdiagram,threeT/P tidemodelpredictions
(theheavy VLBI networks.
differences
between
VLBI and
arrows,consultTables6 and7 for individualidentification), 5. Thereareoftensystematic
The mostnotablecaseis the progradeK2,
threeVLBI estimates,
andoneSLR estimateareshown(andso SLR estimates.
labeled).The separation
betweenestimatesindicatesthe wheretwo of thethreeVLBI estimateshave anomalouslylarge
amplitudes
(judging
fromthesmalltidalpotential
forthisfide
uncertainties
associated
with eachtype of estimate.
Several facts are noted as follows:
•
in Table 1) and diversephases.Gipson's [1996] VLBI
1.Thelargeamplitudes
in themodel-predicted
retrogradeestimatefor this tide is much less anomalous,while the SLR
agrees
muchbetterwithmodelpredictions.
diurnal
components
duetoFCN resonance
arelistedhereonly estimate
6. The Schwiderskimodel(with Cartwrightet al. [ 1992]
for reference.As discussedearlier, no direct observationsfor
agrees
wellwiththeT/P models
in theirpolarmotion
themareavailable.Thelargestamplitudes
dependsensitively currents)
predictions,
whereas
theBrosche/Seiler
modeloftenpredicts
a
ontheactualFCN periodandQ.
larger
effect.
ThisisalsofoundtobethecaseforUT1 by
2. In general,
thethreeT/P tidemodelsagreewellin their much
of Schwiderski
modelis
polar motionpredictions,
typicallywithin 5-10 •tas in Rayet al. [1994a].The accuracy
surprising,
giventhattheT/P elevation
models
are
an•litude
and10ø in phase.
Thisisconsistent
withfindings
in somewhat
CRE95 for UT1.
significantly
moreaccurate
(Shumetal.,submitted
manuscript,
20,160
CHAO ET AL.: DIURNAL/SEMIDIURNAL POLAR MOTION
180
I
2O
160
140
-2O
o
120
-40
o
o
•
100
-6o
30
.o
v
E
80
-80
SLR
60
• -lOO
40
o -12o
20
-140
Libration
-160
I
i
-50
0
i
i
50
100
150
-50
I
i
i
0
50
1 O0
In-phase (micro-arcsecond)
In-phase(micro-arcsecond)
Figure
3.San•asFigure
2 butforretrograde
S2polar
motion.Figure5. San• asFigure4 butfor progradeO1polarmotion.
1996).Thereasonfor thispresumablyliesin certainfortuitous B andC (modelA is not plottedbecauseit hasfewer tide
cancellation
of the errorswhenthe modelis integratedto form constituents).
The matchbetweenmodelpredictionsandobservations
is
the Earth rotationparameters.
apparently
quitegood,
although
notasgoodasthetoppanelfor
UT1. This matchcan be quantifiedas follows.The total
varianceof the Cont94polar motionseries(in its complex
Polar Motion: "Intensive" Comparison
tobe215in'units
of103Ilas2,whilethenoise
Figures6b and 6c presentthe hourlydataof the x andy form)isfound
giventheformalerror[Gipsonet al., 1994]is about
components
ofpolarmotionduringCont94intensivecampaign. variance
wefindthetotalvariance
greatly
reduced,
to
Superimposed
onthemarethepredicted
curvesof T/P models 26.Asexpected,
! !0, aftersubtraction
of thetides(them•e!ves
of variance122
according
tomodelC).Therefore,
assuming
uncorrelated
noise,
approximately
1-(110-26)/(215-26)
or nearly60% of the
subdaily
polarmotion
powercanbeexplained
by theeighttides
(compared
to thecorresponding
-90% forUT1).Thisonone
SL
160
140
handestablishes
thedominantrole of fidesin excitingsubdaily
LBI
polarmotion,
andontheotherhandindicates
theexistence
of
important
nontidal
contributions.
Besides
observational
noises
andmodeling
errors,themajorityof thediscrepancies
seenin
Figure6 canpresumably
beattributed
to thefollowing
effects
•-120
o
• 100
,?.o80
not considered
in the oceantide models:(1) thepresenceof
E
othergeophysical
sources
suchassubdaily
atmospheric
and
oceanicangularmomentum
variations,
dailyandhalf-daily
atmospheric
thermal"fides,"Earthlibrations
(seebelow),and
possibly
earthquaY;
(2)contributions
fromtheresponse
ofthe
internalinhomogeneities
of the solidEarthto diurnaland
semidiurnal
tidalforcing;(3) theneglectof smallertidesin the
tide modelspresented
here[cf. Seilerand Wlinsch,1995].
._
v
.e 60
(,
40
0
2o
Libration
Closerexamination
of Figure6 showsthat the tide models
almostalwaysunderpredict
the diurnalandespecially
the
I
-2O
-50
I
o
,
50
100
In-phase(micro-arcsecond)
5O
semidiurnal
peaks.
ThisistrueforbothUT1 (seeCRE95)and
polarmotion.
Similarphenomenon
is foundwithrespect
to
Figure4. SameasFigure2 butfor prograde
K1polarmotion Cont95dataaswell [Gipsonet al., 1996].Theinitialstudyby
according
to Table6. The corresponding
contribution
of the Ma et al. [1995] has indicatedthe importanceof the
angular
momentum
andthermal
tides;
but,clearly,
Earth polar librationis alsoplotted.ConsultTable 6 for atmospheric
individual identifications.
morecomprehensive
investigations
will beneeded.
CHAO ET AL.: DIURNAL/SEMIDIURNAL
i
i
i
i
i
i
i
POLAR MOTION
20,161
......
50 --
--
500 -
-
0-
-
-500 -
-
500
-
0
-
-500
Diurnal retrograde removed
13
I
I
I
I
I
I
I
I
I
I
I
I
14
15
16
17
18
19
20
21
22
23
24
25
January1994
Figure6. Comparison
ofEarthrotation
variations
during
Cont94
Campaign.
Thehourly
observations
with
standard
errors
arethosemadebyVLBI (theNASAR&D network);
thesolidlinesarepredictions
according
toT/PtideModels
B andC (seetext).(a)UT1 variation
in unitsof Its;(b)x and(c)y components
of thepolar
motion(in unitsof Bas)withouttheretrograde
diurnalcomponents.
Polar Libration
accuracies
areinadequate
forthelibrationto "close"thebudget
evenfor the largesttidalpolarmotions.
Besidesthe semidiurnal
spinlibrations(discussed
above),
the luni-solartidal torquesexertedon Earth's equatorial Summary
ellipticitygiveriseto diurnalpolarlibrations
in theprograde
direction [Chao et al., 1991]. The observedpolar motion
Theoceanic
tidalangular
momentum
(OTAM),consisting
of
shouldcontainthetide-generated
polarmotionandthepolar the massand motionterms,excitesEarth rotationvariationat
librations,
thelargestamplitude
of which(forK1)isprobably
nomorethan15 gasdepending
ontheinteriorproperties
of the
Earth. Thus,if sufficientlyaccurate,the observedand tidegenerated
polar motioncan be differencedto reveal the
all tidalperiods.
CRE95established
thattheaxialcomponent
of OTAM is responsible
for asmuchas90% of thesubdaily
(primarily
diurnal
andsemidiurnal)
poweroftheUT1variation.
Hereupdated
examples
arepresented
inFigures
1 and6a.The
librationcontribution.
One hopesthatthiswill leadto useful mainpurpose
of thepresent
paper,however,
is to studythe
constraintson Earth's interior properties,particularlythe equatorial
components
of OTAM andtheircorresponding
equatorial
ellipticity
of thecore-mantle
boundary
[Chaoet al., effectson thepolarmotion.
1991;HerringandDong, 1994;CRE95].
The equatorial
components
of OTAM for eightmajor
According
toformulae
givenbyChaoet al. [ 1991] underthe diurnal/semidiurnal
fides(Q1,O1,P1,K1,N2,342,S2,K2)are
present
phase
convention,
thepredicted
polarlibration
is(14.6 computed
according
to threeoceantidemodels
derived
from
gas,150ø) forK1 and(11.3gas,150ø) for Ot asplottedin TOPEX/Poseidonsatellitealtimetry.Agreeingwell among
these
predictions
arethencompared
withgeodetic
Figures4 and5, respectively.
Unfortunately,
the departures themselves,
ofpolarmotionmadeby VLBI andSLRlongwithinestimates
fromeachtechnique,
andhencetheestimation measurements
uncertainties,
are generallylarger than the libration.Thus, term observations as well as VLBI intensive measurements
The prograde
diurnaland
similarlyfor UT1, the presentmeasurement
and modeling duringthe Cont94campaign.
20,162
CHAO ET AL.' DIURNAL/SEMIDIURNAL
progradeandretrogradesemidiurnalpolarmotionsare treated;
the retrogradediurnal polar motion is not treatedbecauseit
cannotbe observeddirectlyandindependently
of the nutations
by geodeticmeans.
The comparisonshows generally good agreement,with
discrepancies
typicallywithin 10-30 Basfor the largesttides
andprogressively
poorerfor smallertides.Examplesaregiven
in Figures2-5. Overall,the bestagreementis foundbetween
VLBI estimates
by Gipson[1996] andtidemodelpredictions
POLAR MOTION
Doodson,A. T., The harmonicdevelopment
of the tide-generating
potential,
Proc.R.Soc.LondonA, 100,305-329,1921.(Reprinted
in Int. Hydrog.Rev.,31, 11-35, 1954.)
Doodson,
A. T., andH. D. Warburg,AdmiraltyManual of Tides,His
Majesty's StationOff., London,1941.
Dushaw, B. D., P. F. Worcester,B. D. Cornuelle,and B. M. Howe,
Barotropicand baroclinicfidesof thewesternNorthAtlantic,Eos
Trans.AGU, 75 (44), Fall Meet. Suppl.,307, 1994.
Dushaw,B. D., B. D. Cornuelle,P. F. Worcester,B. M. Howe, and D.
S. Luther, Barotropicand baroclinicfidesin the centralNorth
according
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Pacific Oceandeterminedfrom long-rangereciprocalacoustic
transmissions,
J. Phys.Oceanogr.,25, 631-647, 1995.
weeksof Cont94,the eight tidescollectivelyexplainnearly
60% of thetotalvariancein sub-dailypolarmotion,asshown Egbert, G. D., A. F. Bennett, and M. G. G. Foreman,
TOPEX/Poseidon
fidesestimated
usinga globalinversemodel,J.
in Figures 6b and 6c. This on one hand establishesthe
dominantrole of OTAM in excitingthe diurnal/semidiurnal Geophys.Res.,99, 24,821-24,852, 1994.
Eubanks, T. M., Variations in the orientationof the Earth, in
polarmotionandon the otherhandpavestheway for detailed
Contributionsof Space Geodesy to Geodynamics:Earth
studies
of nontidalpolarmotion.The excitationsourcesfor the
Dynamics,Geodyn.Ser.,vol. 24, editedby D. E. SmithandD. 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",Earthlibrations,earthquakes,
andtheresponse M. Dorrer, and P. Escudier,TOPEX/Poseidonmissionoverview,
of the internal inhomogeneitiesof the solid Earth to
J. Geophys.
Res.,99, 24,369-24,381, 1994.
diurnal/semidiurnal
tidalforcing.A full, quantitativeresolution Fumya,M., andB. F. Chao,Estimationof periodandQ of the
awaitsfurtherdata and investigations.
Chandlerwobble,Geophys.J. Int., in press,1996.
Gipson,J. M., VLBI determination
of neglectedtidaltermsin
Acknowledgments. We thank E. Pavlis and R. Eanesfor the
unpublished
SLR results.This work is supportedby the NASA
Geophysics
Program.
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