GEOPHYSICAL RESEARCH LETTERS, VOL. 27, NO. 15, PAGES 2329-2332, AUGUST 1, 2000
The excitation
Richard
of the Chandler
wobble
S. Gross
JetPropulsionLaboratory,CaliforniaInstituteof Technology,Pasadena
Abstract.
The Chandler
wobble
is an excited
resonance of the
Earth'srotationhaving a periodof about 14 months.Althoughit
has been under investigation for more than a century, its
excitation mechanismhas remained elusive. Here, the angular
momentumof the atmospherecomputedfrom the productsof a
numerical weather prediction analysis system and the angular
momentumof the oceanscomputedfrom a globaloceanicgeneral
circulation model driven by observedsurface winds and fluxes
are usedto showthat during 1985.0-1996.0 the Chandlerwobble
was excited by a combination of atmospheric and oceanic
processes,
with the dominantexcitationmechanismbeingoceanbottompressurefluctuations.
Introduction
Any irregularly shapedsolid body rotating about some axis
that is not aligned with its figure axis will wobble as it rotates.
For the Earth, this Eulerian free wobble is known as the Chandler
wobble
in honor of S.C.
Chandler
who first observed
it in 1891
when
the sum of the modeled
oceanic
current
and bottom
pressure excitation is added to the sum of the modeled
atmospheric wind and pressure excitation (the relative
contributionof the individual oceanicand atmosphericprocesses
to excitingthe Chandlerwobblewas not reported)øHowever, the
excitation of the Chandler wobble is a broad-bandprocessthat
occurswithin a band of frequenciescenteredat the Chandler
frequency.Investigatingthe excitationof the Chandler wobble
therefore requires computingthe excitation power that occurs
within the Chandlerband. Here, realisticatmosphericand oceanic
general circulation models are used to obtain time series of
modeledChandler wobble excitationfrom which the power in the
Chandler band is computedand comparedto that observedto
show that during 1985.0-1996.0 the Chandler wobble was
primarily excited by ocean-bottompressurefluctuations,with
atmosphericpressurefluctuationsbeing about half as effectiveø
Oceaniccurrentshad only a minor effect on the Chandlerwobble
during this time, while atmosphericwinds, being out-of-phase
with the pressureterms, actedto reducethe modeledChandler
excitationpowerto nearlythat observed.
[Chandler, 1891]. From observations of the Chandler wobble
taken since its discovery, its period and Q have been estimated
[Wilson and Vicente, 1990] to be 433.0 _+1.1 (1•) days and 179 Polar Motion Excitation
Functions
(with 1• boundsof 74 and 789), respectively,giving an estimated
Measurementsof the Earth's changingrotation are currently
e-folding amplitudedecay time of 68 years. Becausea damping
techniquesof satelliteand lunar laser
time of 68 yearsis shorton a geologicaltime-scale,the amplitude madeby the space-geodetic
of the Chandler wobble shouldquickly dampento zero unless ranging,very long baselineinterferometry,and global positioning
systeminterferometry[Lambeck,1980]. The Earth rotationseries
somemechanismor combinationof mechanismsare exciting it.
Sinceits discovery,many processeshave been evaluated,without used in this study is a combination of these space-geodetic
success,to determinewhetheror not they could be the excitation measurementsknown as SPACE97 [Gross, 1999] and consistsof
mechanism(s)of the Chandler wobble including atmospheric
processes[Wilson and Haubrich, 1976; Wahr, 1983; Furuya et
al., 1996, 1997], continental water storage [Chao et al., 1987;
Hinnov and Wilson, 1987; Kuehne and Wilson, 1991], coremantle interactions [Rochester and Smylie, 1965; Gire and Le
Mou•l, 1986; Hinderer et al., 1987; Jault and Le Mou•l, 1993],
andearthquakes[Souriauand Cazenave,1985; Gross, 1986].
Recently, Celaya et al. [1999], usingthe resultsof a coupled
atmosphere-ocean-ice-land climate model, concluded that
atmosphericprocessesalone, oceanic processesalone, or some
combinationof atmosphericand oceanicprocessesprobablyhave
enoughpowerto excite the Chandlerwobble.However, Celaya et
al. [1999] were not able to discriminatebetweenthesepossible
excitation processesbecause, due to the nature of the climate
model they used, they could not match time series of modeled
excitation
with
that
observed--their
conclusions
were
based
daily averagedvaluesof UniversalTime, polar motion, and their
rates of change spanning1976.7-1998.0. Strictly speaking,the
observedpolar motion parametersspecify the location of the
Celestial EphemerisPole (CEP) within the body-fixed terrestrial
reference frame and will be so interpreted here. However, for
periodslong comparedto a day, suchas for the Chandlerwobble,
and to sufficient accuracy,the observedpolar motion parameters
can be interpretedas specifyingthe location of the rotation pole
within the terrestrialreferenceframe [Gross, 1992].
Polar motion consistslargely of: (1) a forced annual wobble
having a nearly constantamplitudeof about 100 milliarcseconds
(mas), (2) the free Chandler wobble having a variable amplitude
ranging between about 100 to 200 mas, (3) quasi-periodic
variationson decadaltime scaleshaving amplitudesof about 30
mas known as the Markowitz wobble, (4) a linear trend having a
rate of about 3.5 mas/yr, and (5) smaller amplitude variations
occurringon all measurabletime scales.This rich polar motion
spectrumis causedby the rich variety of processesforcing polar
motion. In the absenceof external torques, the polar motion
solely upon statisticalanalysesof the observedChandler wobble
and the output of the climate model. Ponte and Stammer [1999]
have also recently investigated atmospheric and oceanic
excitation of the Chandler wobble by fitting a sinusoidat the parameters
Xpandypcanberelated
to theprocesses
forcingpolar
Chandler frequency to modeled and observed polar motion motion by linearizingthe Liouville equationwhich expressesthe
excitationseries,finding betteragreementwith the observations conservationof angularmomentumwithin a rotating,body-fixed
referenceframe [Lambeck, 1980]:
p(t) +
Copyright2000 by the AmericanGeophysicalUnion.
i
dp(t)
rYcw dt
= Z(t)
(1)
Papernumber2000GL011450.
where rYcwis the complex-valuedfrequencyof the Chandler
0094-8276/00/2000GL011450505.00
wobble
andthecomplex-valued
quantity
p--(Xp- i yp)specifies
2329
Since the oceanic current and ocean-bottom pressure
thex- andy-coordinates,
Xpandyprespectively,
of theCEPwith
excitation
functionsare given as 5-day-averagedvalues,5-dayXpbeingpositivetowards
theGreenwich
meridianandypbeing
averagedvalueswere alsoformedof the daily averagedobserved
excitation functionsand of the 6 hourly atmosphericwind and
pressureexcitationfunctions.In orderto reducespectralleakage
into the Chandlerfrequencybandof annualexcitationprocesses,
which can be written as [Wahr, 1982]:
a seasonal signal was removed from the 5-day-averaged
observed,atmospheric,
andoceanicexcitationfunctionsby leastsquaresfittingandremoving a mean,a trend,andperiodicterms
where C and A are the greatestand least, respectively,principal at the annual and semiannualfrequencies.This fit was done on
momentsof inertia of the Earth, the mean angularvelocityof the that subsetof the seriesspanning1985.0-1996.0in orderto fit an
Earth is 12, the complex-valuedquantity c(t)= c13(t) + i c23(t) integral number of annual and semiannualoscillations.All
analysiswill be doneon this 11-yearsubsetof the
representschanges in the two indicated elements of the Earth's subsequent
residual excitation series.
inertia tensor such as those due to atmosphericor oceanic mass
redistribution,andh(t) = hI (t) + i h2(t) represents
relativeangular Chandler
Wobble Excitation
momentum changes such as those due to changes in the
Figure 1 showspower spectraldensity (psd) estimatesof the
atmospheric winds or oceanic currents. The factor of 1.61
includesthe effect of core decouplingand the factor of 1.44 in the observed (black curve), atmospheric (red curve), and sum of
denominatoraccountsfor the yieldingof the solidEarth due to its atmosphericand oceanic(greencurve) excitationfunctionsfrom
which seasonalsignalshave been removed.A Hanning window
changingload.
Becausepolar motion is resonantat the Chandler frequency was applied to each seriesprior to forming the spectralestimates,
(Equation 1), investigationsof the excitation of the Chandler and the spectralestimatesare of the time seriesincluding noise.
wobble are usually conductedby frequency-domaincomparisons In agreementwith the conclusionof previousstudies[Wilsonand
of the observed polar motion excitation functions with those Haubrich, 1976; Wahr, 1983], it is seen that the sum of
computedfrom variousgeophysicalprocesses.
Here, the observed atmosphericwind and pressurefluctuations(red curve) doesnot
polar motion excitation functions are those determined from the have sufficient power to excite the Chandler wobble (the
SPACE97 polar motion valuesand ratesusing Equation 1 with Chandlerfrequencyof 0.8435 cycles/year(cpy) is indicatedby
the period and Q of the Chandlerwobble set to 433.0 days and the vertical dotted line). However, a good match to the observed
179, respectively [Wilson and Vicente, 1990]. The observed Chandler wobble excitation power is obtained upon adding the
excitation functions thus recoveredare then comparedto those excitation due to oceanic current and ocean-bottom pressure
causedby atmosphericwind and pressurechangesand by oceanic fluctuations to that due to atmospheric wind and pressure
current and ocean-bottom pressure changes. The atmospheric variations(greencurve).
Since the time serieswhose spectraare displayedin Figure 1
excitation functions used here are those computed from the
National Centersfor EnvironmentalPrediction(NCEP) / National consistof 800 5-day-averagedsamples,the frequencyresolution
Centerfor AtmosphericResearch(NCAR) reanalysisprojectand
positiveby conventiontowards90øW longitude.Equation 1 is the
expressionfor simple harmonicmotion in the complexplane with
the right-hand-sideZ(t) being the forcing, or excitation, function
1.61 h(t)+ 1.44
Z(t)= 12(C-A)
are available
from
the International
Earth
Rotation
(2)
Service
(IERS) SpecialBureaufor the Atmosphere[Salsteinet al., 1993].
The oceanicexcitationfunctionsusedhereare thosecomputedby
Ponte et al. [1998] and are available from the IERS Special
SPECTRA
OF POLAR MOTION
EXCITATION
SERIES
Bureau for the Oceans.
Ponte et al. [1998] computed the polar motion excitation
functionsdue to oceaniccurrentsand, separately,ocean-bottom
pressurechangesfrom the productsof a global oceanicgeneral
circulationmodel (OGCM) driven by 12-hourwind stressfields
and daily surface heat and fresh water flux fields from NCEP.
Atmospheric pressure was not used to force the OGCM. The
ocean-bottompressureexcitationterm was correctedby them for
the effects of volume changesdue to steric effects within the
BoussinesqOGCM by adding a uniform sea level layer of
fluctuatingthicknessto the seasurfaceheightfields producedby
the OGCM [Greatbatch,1994]. The resultingoceaniccurrentand
ocean-bottompressureexcitation functionsare 5-day-averaged
valuesspanningJanuary1985 to April 1996.
The available atmospheric wind and pressure excitation
functionscomputedfrom the NCEP/NCAR reanalysisprojectare
6-hour values spanning 1958.0 to the present. The pressure
excitationterm is availableundertwo different assumptions
for
the responseof the oceansto surfacepressurechanges:(1) the
inverted barometerassumptionwherein the oceansare assumed
to respond isostatically to the imposed surface pressure
variations,and (2) the rigid oceanassumptionwhereinthe oceans
are assumedto fully transmit without delay or attenuationthe
atmosphericpressurevariations to the ocean-bottom.Since at
periods long compared to a day the inverted barometer
assumptionshouldbe valid [e.g., Wunschand Stammer, 1997],
thepressuretermcomputedunderthisassumption
is usedhere.
o
•-
. , atmosphere
• atmosphere+ocean
I
I
I
-- 2.0
-- 1.00
0.0
1.0
2.0
frequency(cycles/year)
Figure 1. Powerspectraldensity(psd)estimatesin decibels(db)
computedfrom time seriesof polar motionexcitationfunctions
Z(t) spanning1985.0-1996.0of: (a) theobservedSPACE97polar
motion excitation function derived from space-geodeticEarth
rotationmeasurements(black curve), (b) the sum of the excitation
functions due to atmosphericw•nd and pressurechanges(red
curve) where the atmosphericpressureterm is that computed
assumingthe invertedbarometerapproximationis valid, and (c)
the sumof all atmospheric
andoceanicexcitationprocesses
being
studiedhere, namely,the sumof the excitationfunctionsdue to
atmosphericwinds, atmosphericpressure(inverted barometer),
oceanic currents, and ocean-bottom pressure (green curve). A
seasonalsignalhasbeenremovedfrom all seriesprior to spectral
estimationby least-squares
fitting and removinga mean,a trend,
andperiodictermsat the annualandsemiannualfrequencies.The
vertical dotted line indicatesthe Chandler frequency of 0.8435
cycles/year(cpy). The retrogradecomponentof polar motion
excitation is representedby negative frequencies,the prograde
componentby positivefrequencies.The Chandlerwobbleis a
strictlyprogradeoscillation.
Table 1. Chandlerband excitationpower
of these time series is 0.0913 cpy which is just sufficient to
resolve the Chandler frequency band. Near the Chandler
frequency, the spectral estimatesshown in Figure 1 are given at
frequenciesof 0.730 cpy, 0.822 cpy, and 0.913 cpy. Integrating
the power spectral density estimates of Figure 1 across the
Chandlerfrequencyband, taken here to rangebetween0.730 cpy
and 0.913 cpy, gives the power in the Chandler band shown in
Table 1 for the variousexcitationmechanismsbeing studiedhere.
Excitation
process
Power(mas
2)
Observed
4.87
Atmospheric
The observed
excitationpowerin thisbandis 4.87 mas2 with the
wind
0.32
pressure
(i.b.)
windpluspressure
(i.b.)
1.87
1.44
Oceanic
currents
sum of the power due to atmosphericwind, atmosphericpressure,
oceanic current, and ocean-bottom pressure excitation being
slightly more than this at 5.44 mas2. Ocean-bottompressure
fluctuationsare seen to be the single most important mechanism
exciting the Chandler wobble, containing about twice as much
power in the Chandler band as that due to atmosphericpressure
fluctuations.Oceanic currentand atmosphericwind variationsare
minor contributors to the Chandler wobble excitation, having
powerin the Chandlerbandof only 0.12 mas2 and0.32 mas2,
0.12
ocean-bottompressure
currentsplusocean-bottom
pressure
3.45
3.69
Atmosphericplusoceanic
wind pluscurrents
i.b. plusocean-bottom
pressure
0.67
6.28
Total of all atmosphericplus oceanic
5.44
i.b., inverted barometer
respectively.Even so, destructiveinterferencebetween the sum
of atmosphericwind and oceaniccurrentexcitationand that due
to the sum of atmosphericand ocean-bottompressureexcitation
atmospheric
wind, atmospheric
pressure,oceaniccurrents,and
ocean-bottompressureis coherent with greater than 99%
reducesthepowerin theChandlerbandfrom6.28 mas2 to 5.44
mas2.
Figure 2 shows the magnitude of the squared-coherence
confidence.Since the sum of atmosphericpressureand ocean-
bottompressure
is alsocoherentwith theobserved
excitationnear
between the observed excitation
functions
and those due to
the Chandlerfrequency,the additionof atmosphericwind and
atmosphericwind and pressurevariations(red curve), the sum of
oceaniccurrentexcitationreducesthe powerin the Chandlerband
ocean-bottom pressure and atmosphericpressure fluctuations
to nearlythatobserved,but doesnot affectthecoherence.
(blue curve), and the total sum of atmosphericwind, atmospheric
pressure,oceaniccurrents,and ocean-bottompressurevariations
(greencurve).The squared-coherence
estimateswere obtainedby
averaging over 5 frequency intervals and the 95% and 99%
confidencelimits on the magnitudeof the squared-coherence
estimatesare indicatedby the horizontaldashedlines. As can be
seen, near the Chandler frequency (indicated by the vertical
dotted line) atmosphericwind and pressureexcitation is not
coherent with the observed excitation, but that due to the sum of
COHERENCE
--
OF OBSERVED
otmos+ocean
pressure
atmosphere
Discussion and Summary
The resultsreportedherefor the observedexcitationpowerin
the Chandlerfrequency
band(4.87 mas2) wereobtainedusing
thosevaluesof 433.0 days and 179 for the period and Q of the
Chandler wobble, respectively,that were estimatedby Wilson
and Vicente [1990] from 86 years of optical astrametric polar
motion observations.Other recentestimatesfor the period and Q
AND MODELED
EXCITATION
/•
99%
- atmosphere
i
-3.0
-2.0
!
-1.0
I
I
I
0,0
1,0
2ø0
3.0
frequency(cycles/year)
Figure 2. The magnitude
of thesquared-coherence
betweentheobserved
polarmotionexcitationfunctions
spanning1985.0-1996.0
andthe excitationfunctionsdueto: (a) the sumof atmospheric
wind andpressurechanges(red curve)wherethe pressureterm is that
computedunderthe invertedbarometerapproximation,(b) the sumof atmospheric(invertedbarometer)and ocean-bottom
pressure
fluctuations
(bluecurve),and(c) thesumof all theatmospheric
andoceanicexcitationprocesses
beingstudiedhere,namely,thesumof
atmospheric
wind,atmospheric
pressure
(invertedbarometer),
oceaniccurrent,andocean-bottom
pressure
variations.
A seasonal
signal
hasbeenremovedfrom all seriesprior to coherenceestimationby least-squares
fitting and removinga mean,a trend,and periodic
termsat the annualandsemiannual
frequencies.
The verticaldottedline indicatestheChandlerfrequencyof 0.8435 cycles/year(cpy)
and the horizontaldashedlinesindicatethe 95% and99% confidencelevelsof the magnitudeof the squared-coherence.
of the Chandler wobble include those of Kuehne et al. [1996] References
who usedjust 8.6 years of modern space-geodeticpolar motion
observations
toobtain
aperiod
of439.5
_+2.1(16)days
andaQ Celaya,
M.A.,J•M.Wahr,
and
F.O.Bryan,
Climate-driven
polar
motion,
J. Geophys.Res.,104, 12813-12829, 1999.
of72(with16bounds
of30and500)andhence
anestimated
e- Chandler,
S.Co,Onthevariation
oflatitude,
I, Astron.
J.,11,59-61,
folding amplitude decay time of 28 years, and those of Furuya
1891.
and Chao [ 1996] who used just 10.8 years of modern polar Chao,B. F.,W. P.O'Connor,
A. T. C. Chang,
D. K. Hall,andJ.L. Foster,
motion
observations
toobtain
a period
of433.7+_1.8(16)days
and a Q of 49 (with 16 bounds of 35 and 100) and hence an
estimated
e-foldingamplitude
decaytimeof 19 years.The
Snow
loadeffect
ontheEarth's
rotation
andgravitational
field,
1979-1985,J. Geophys.Res.,92, 9415-9422, 1987.
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wobble,
Geophys.
J.Int.,127,693-702,
1996.
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windsignalasa
possibleexcitation of Chandler wobble, J. Geophys.Res.,101,
studysincethey are baseduponthe longestseriesof polar motion
25537-25546, 1996o
observations.In fact, of thesethree estimates,it is the only one
Furuya,M., Y. Hamano,and I. Naito, Importanceof wind for the
that is based upon polar motion observationsspanning a time
excitation of Chandler wobble as inferred from wobble domain
interval (86 years) that is greater than its estimatedChandler
analysis,J. Phys.Earth, 45, 177-188, 1997.
Gire, Co,and J-L Le MouEl, Flow in the fluid coreand Earth'srotation,in
wobblee-foldingamplitudedecaytime (68 years).
Since
the Chandler
wobble
is a resonance
in the Earth's
Earth Rotation: Solved and Unsolved Problems, edited by A.
Cazenave,pp. 241-258, D. Reidel,Dordrecht,Holland,1986.
rotation, the observed excitation power near the resonance Greatbatch,R. J., A note on the representation
of steric sea level in
frequency, that is, in the Chandler frequency band, will be
modelsthat conservevolume ratherthan mass,JøGeophys.Res., 99,
12767-12771, 1994.
sensitive to the assumed value for the period and Q of the
Chandler
wobble
(Equation
1). Forexample,
if instead
of using Gross,
R.S.,Theinfluence
ofearthquakes
ontheChandler
wobble
during
anassumed
value
of433.0
days
foritsperiod
and179foritsQ,
the period were to be kept unchangedat 433.0 days but the Q
were to be changed to its 16 lower bound of 74, then the
observedexcitationpower in the Chandlerfrequencybandwould
1977-1983,
Geophys.
J.Roy.
astr.
Soc.,
85,161-177,
1986.
Gross,R. S., Correspondence
betweentheoryand observations
of polar
motion,Geophysø
J. Int., 109, 162-170, 1992.
Gross, R. S., Combinations of Earth orientation measurements:
SPACE97, COMB97, and POLE97, J. Geodesy,in press,1999.
changefrom4.87 mas2 to 7.99 mas2. In fact,an assumed
period Hinderer,J., H. Legros,C. Gire, andJ-L Le MouE1,Geomagneticsecular
of 433.0 days and a Q of 138 yields an observedChandlerband
excitationpowerof 5.45 mas2,nearlymatchingthatcausedby the
sumof atmosphericwind, atmosphericpressure,oceaniccurrents,
andocean-bottom
pressure
excitation(5.44 mas2, seeTable 1).
Thus, the conclusion reached here that the Chandler wobble is
being excited by a combination of atmospheric and oceanic
processesis basedupon assumingthat the periodof the Chandler
wobble is nearly 433.0 days and that its Q is nearly 138. Given
the scatterin the abovecited estimatesfor the period and Q of the
Chandler wobble, and their rather large 16 uncertainties,more
accurateestimatesof the period and Q of the Chandlerwobble are
clearly required. Such improved estimates could perhaps be
obtained when series of realistic oceanic angular momentum
values become available that are as long as the 40-year-long
series of atmospheric angular momentum values currently
availablefrom the NCEP/NCAR reanalysisproject.This would
then allow the Chandlerwobble'speriodand Q to be determined
from 40 years of polar motion observationsand models of its
atmosphericandoceanicexcitationprocesses.
Numerousinvestigationshave been conductedduring the past
centuryin attemptsto elucidatethe excitationmechanismof the
Chandler wobble. Here it has been shown that during
1985.0-1996.0 the singlemostimportantmechanismexcitingthe
Chandler wobble has been ocean-bottompressurefluctuations,
which contribute about twice as much excitation power in the
Chandlerfrequencybandas do atmosphericpressurefluctuations.
Atmosphericwinds and oceaniccurrentshavebeen shownhere to
play only a minor role in exciting the Chandler wobble during
this time. The ability to elucidate the role of atmosphericand
ocean-bottompressure fluctuations in exciting the Chandler
wobble is a testamentto the fidelity of the atmosphericand
oceanicgeneralcirculationmodelsthat wereusedto computethe
atmosphericand oceanicangular momentumestimatesused in
this study. The wide distribution of these atmospheric and
oceanic angular momentum estimates by the IERS Special
Bureaus for the Atmosphereand Oceans enables the type of
interdisciplinaryresearchwhoseresultsare reportedhere.
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R. S. Gross,JetPropulsionLaboratory,Mail Stop238-332, 4800 Oak
Acknowledgments. The work describedin this paperwas performed GroveDrive, Pasadena,CA 91109 (e-mail: [email protected])
at the Jet Propulsion Laboratory, California Institute of Technology,
undercontractwith the National Aeronauticsand SpaceAdministration. (ReceivedFebruary1, 2000; revisedApril 14, 2000;
acceptedMay 12, 2000.)
Supportfor this work wasprovidedby NASA's Office of Earth Science.