GEOPHYSICAL RESEARCH LETTERS, VOL. 20, NO. 2, PAGES 253-256, FEBRUARY 5, 1993
EXCITATION
OF EARTH'S
POLAR
BY ATMOSPHERIC ANGULAR MOMENTUM
MOTION
VARIATIONS, 1980-1990
BenjaminFong Chao
Geodynamics
Branch,NASA GoddardSpaceFlight Center
Abstract. We compute the polar-motion excitation
function due to the atmospheric angular momentum
(AAM) for both IB (inverted-barometer)and non-IB
cases,as well as the excitationfunctionfrom geodetically
observedEarth orientationdata for the period 1980-1990.
The two are then comparedin studyingthe AAM contributionto the polar motionexcitation.The polar driftswith
periodslonger than --2 years have similar characteristics,
but the comparison is inconclusivebecause of data
(AAM). AAM variationhasbeen demonstratedto be the
dominant
cause for LOD
variations
shorter
than a few
years[e.g.,Hide and Dickey,1991].It hasalsobeen linked
to excitationof rapid polar motions[Eubankset al., 1988],
and as the major excitationsourcefor the progradeannual
wobble [e.g.,Chao and Au, 1991].Other possiblepolarmotion excitations include seismic movements, mass
transportin the hydrosphereand post-glacialrebound;but
convincinglinks for them have yet to be found.
uncertainties. For the seasonal wobble excitation, the
Formulation
agreementis poor exceptfor the progradeannualwobble,
indicatingthe influence of other geophysicalexcitations
than AAM. For the Chandler wobble excitation, a correlation
coefficient
of 0.53 for non-IB
The polar motion is a 2-dimensionalquantity,convenientlyexpressedin termsof the complexangulardistance
and 0.58 for IB are
found for 1986-1990. Togetherwith a coherencespectral
analysis,they clearlydemonstratea strongcontributionof
AAM
to the Chandler
wobble
and Data
excitation.
Introduction
The Earth's rotationvariesslightlywith time. It varies
both in its magnitude(i.e., length-of-day
or LOD) and in
the orientationof its axis.The latter variationwith respect
to the terrestrial reference frame is known as the polar
motion:The Earth's rotationaxistracesout in the vicinity
of the North Pole a quasi-circular,progradepath with a
varyingamplitudeof a few hundredmilliarcseconds
(1 mas
-- 3 cm). The polar motion consistsmainly of an annual
wobbleand the Chandlerwobblewith a period of about
14 months.In addition, the mean pole undergoesa slow
drift over the years.
The presentpaper dealswith the excitationof polar
motion. Without excitation any polar motion will
eventuallydissipateaway; and the very existenceof the
polarmotionimpliescontinualexcitation.The geophysical
problem is to identify the sourcesof the excitation.To
studyexcitation,one shouldbear in mind that the Earth's
polarmotionis a linear,oscillatoryresonantsystem[Munk
and MacDonaM, 1960].The observedpolar motionis the
temporal convolution of the excitation with the free
Chandleroscillation[cf. Chao, 1985].Thus one derivesthe
polar-motion excitation function by deconvolvingthe
observedpolar motion. This "observed"polar-motion
excitation function is then compared directly with
excitationfunctionscomputedfor geophysicalsources.
The specificgeophysicalexcitationtreated presentlyis
the variation of the atmospheric angular momentum
Thispaperisnotsubject
toU.S. copyright.
Published
in 1993bytheAmerican
Geophysical
Union.
Paper number 93GL00130
253
m -- m1 + i m2 in radians,wheresubscripts
1 and 2 refer
to thex (alongthe GreenwichMeridian)andy (alongthe
90øE longitude)coordinatesof the terrestrialframe. The
equationof motiongoverningthe excitationof the geodetically observedm is [Barneset al., 1983;Gross,1992]
rh =ira(m-g)
(1)
where • is the excitationfunctionand (o is the complex
Chandlerfrequency.The excitationdue to AAM variation
can be evaluatedby [Barneset al., 1983]
(2)
1.00
R4iPsin20eiXda
+rig(C-A)
1'43R3
f(ucos0
+iv)eiXda
dp
2g(C-A)
where R, g, and rl are respectivelythe Earth's mean
radius,mean surfacegravity,and mean angularspinrate;
C andA are the Earth's polar and equatorialmomentsof
inertia;P is the deviationof the surfaceair pressurefrom
the local mean, u and v are the eastward and northward
componentsof the wind velocity.The first integral over
the unit sphere(with surfaceelement da = sin0 dO d)•,
where 0 and Z are the co-latitudeand east longitude)
arisesfrom changesin the productsof inertia due to air
pressurechanges.The secondintegral(whichalsoincludes
the integration over the vertical height in terms of the
pressurep) accountsfor the relative angularmomentum
due to the winds.
In orderto evaluate(2) oneneedsglobalatmospheric
pressureand wind data. We usethe ECMWF (European
Centre for Medium-rangeWeatherForecasts)data seton
a 2 ø latitude by 2.5 ø longitudegrid at 12-hour intervals
spanning11 years: 1980-1990. For a more detailed descriptionof the data and the numericalintegrationscheme,
see Chao and Au [1991]. Two ideal casesare assumed
254
Chao:AtmosphericExcitationof Polar Motion
regardingthe ocean'sresponseto overlyingair pressure
loading:In the inverted-barometer(IB) casethe ocean
respondscompletelyisostatically,
whereasthe non-IB case
assumeszero yieldingof the oceanto air pressureloading.
During 1980-1990 ECMWF has had severalchangeovers and modifications
of its circulation
models
used in
the generationof the analysisdata. These changeslead to
systematicerrors in the form of discontinuitiesin the time
series.The severityof the discontinuitydependson the
parameter in question and the nature of the model
change.In particular,we found it necessaryto invoke an
empirical shift at May 1 of 1985 in order to bring two
sectionsof the AAM seriesto matchat that junction.
Furtherpre-processing
on the AAM •; wasperformed:
The time seriesis first low-passed
with a Butterworthfilter
of order 5, in both forward and reverse directions to
eliminateanyphasedistortion.The cutofffrequencyis 0.1
cycleper day,or about36 cpy(cyclesperyear).The series
is then decimatedby a factor of two into daily values.The
purposeof this procedureis to facilitatelater comparison
with the observeddata, processedas follows.
Equation (1) is the basis for the time-domain
deconvolution[cf. Wilsonand Haubrich [1976] used to
deducethe "observed"polar-motionexcitationfunction
wherewe adopt a nominalChandlerto corresponding
to
a progradeperiod of 435 solardaysand a Q value of 100
[WilsonandHaubrich,1976].For m we usethe "Space90"
data set obtainedby Grossand Steppe[1991] througha
Kalman filter combination of all the available space
geodetic measurementsof the Earth orientation. The
Space90values are providedat nominal daily intervals;
however,the actualtemporalresolutionis typically5 days
with a cutoff frequencyof about 36 cpy.
Computationand Results
The tasknowis to comparethe AAM •; series(from
equation2) withthe observed
•; series(fromequation1).
A plot of the AAM seriescanbe foundin ChaoandAu
[1991](seealsoFigure3).
(a) Polardrift:First,we extractthe long-period
variation
-28
-39
-?-9
-19
9
t9
?-9
39
x (mas)
Fig. 1.x-y path of polar drift, 1980-1990.The solidline is
for AAM (non-IB),thebrokenlinefor AAM (IB), andthe
heavycurvefor the observed.The arrowsgivethe senseof
the drift.
resonance
with the free Chandleroscillation)observedin
the polar motion.
It is immediatelyevidentfrom Figure 2 that for the
progradeannualexcitation
the excitation
powerof AAM,
both IB andnon-IB, agreeswell with that of the observed.
However, it has been shown [Chao and Au, 1991] that
there existsa substantial~40 ø phasedifferencebetween
them. Much poorer match is seen for other seasonal
excitations -- retrograde annual and semi-annual,
indicatingcontributions
of othergeophysical
processes
that
canexcitethe seasonalpolar motion.A detailedstudyhas
been reportedby Chao and Au [1991].The geophysical
budgetfor the seasonalpolar motionis far from "closed".
(c) Chandlerwobbleexcitation:Let us now removethe
seasonalsignals.We achievethat by first estimatingthese
signalsin a least-squares
senseandthen subtracting
them
from each of the series.The extractionis achievedagain
throughthe applicationof a 2-wayButterworthlow-pass
filter of order 5, at cutoff frequencyof 0.5 cpy. Zero
paddingon both endsof the seriesis invokedto reduce
end effectsduringfiltering.The resultantx-y "polardrift"
pathsduring1980-1990are shownin Figure1.
The observedpolar drift path doesnot follow either
of thosepredictedby AAM, IB or non-IB.However,no
final conclusioncan be drawn from this comparison
becausethe AAM-predicted paths have been contaminatedby the discontinuities
in the AAM seriesdueto the
ECMWF analysismodel changes(see above).Yet it is
interestingthat the drifts showsimilarcharacteristic
and
-49
magnitude.
(b) Seasonalwobble excitation:Next, we removeby
subtractingthe above polar drift (as "noise")from our
series.A periodogramis then computed:Figure 2 shows
Frequency(cycleper ),ear)
strongseasonal
(annualand semi-annual)
signalsin both
Fig.
2.
Periodogram
of
the polar motionexcitation(after
prograde(positivefrequency)and retrograde(negative
frequency)excitations.
The progradeannualexcitationis removalof the polar drift in Fig. 1) for (a) AAM (non1980-1990.
responsiblefor the large annual wobble (under near- IB), (b) AAM (IB), and (c) the observed,
Chao: AtmosphericExcitationof Polar Motion
from the time series. We can now concentrate on the
O.G
remainingpolar-motionexcitation.The latter, by
0.5
definition,is the Chandlerwobbleexcitationbecause,
when convolvedwith the free Chandleroscillation,they
give rise to the Chandlerwobble.Figure 3 showsthe x
components
gl; the y components
g2 are similar (not
shown).The seriesexhibit the characteristicsof a white
stochastic
process.Noticethe amplitudedifferencesin the
overall
a.
4
• (a)
non-
0.3
0.2
a. 4
fluctuation.
The
255
AAM-induced
and
the
observed
(•) m
Chandler
excitationare then subjectto two (complementary)
correlation
analyses:
Thetime-domain
complex
correlation
functionwhich givesan overall correlationestimateas a
-1008
-808
-688
-488
-288
0
200
400
600
800
1000
functionof relativetime lag, and the frequency-domain
Time Lag (days)
complexcoherencespectrumwhichprovidesa spectral
Fig.
4.
Magnitude
of
the
complexcorrelationvs. time lag
decomposition
of theoverallcorrelation[e.g.,Chao,1988].
of the observed Chandler
excitation
relative
to that
Figure4 showsthe correlationmagnitude
versusthe time
induced by (a) AAM (non-lB), and (b) AAM (lB),
lag of the observedrelative to the AAM series,for both
1986-1990.
IB and non-IB cases.The period selectedis 1986-1990,
duringwhichtheAAM seriesare relativelyuniformin tile
underlyingmodels(see above).The resultfor the entire meteorologicaldata for the period 1980-1990. We also
periodof 1980-1990is similar(but with somewhat
lower computedthe "observed"
polar motionexcitationfunction
correlationcoefficient,see below). Figure 5 showsthe
from geodeticallymeasuredEarth orientationdata for the
corresponding
squaredcoherencespectra.Here we used sameperiod.The two are thencompared
in studying
the
the multitaper techniqueof Thomson[1982] which AAM's contributionto the polar motion excitation.For
providesrobustandminimum-leakage
spectralestimates. the polar drift with periodslongerthan --2 years,the
Sevenorthogonal
taperswithtime-bandwidth
productof comparison
is inconclusive
because
of systematic
errorsin
4•r were adoptedin the analysis,so that the spectral theECMWF AAM seriesduring1980-1990,eventhough
resolution
is 8 elementary
frequency
bins.The degreeof the drifts show similar characteristicand magnitude
freedom is 7 and the corresponding
99% confidence (Figure1). For the progradeannualwobbleexcitation,
the
thresholdfor the squaredcoherence
is 0.54 [e.g.,Chao, agreement
is goodin the amplitudewith a --40ø phase
1988]. The geophysicalsignificanceof these resultsis
discrepancy,
whereasthe agreementfor the retrograde
discussed in the next section.
annualwobbleandthesemiannual
wobbles
isratherpoor
(see
Figure
2,
and
Chao
and
Au
[1991]),
indicating
the
Conclusions
influenceof other geophysical
excitations
than AAM.
Our main result is with respectto the Chandler
We havecomputed
the AAM-induced
polarmotion
excitation
forbothIB andnon-IBcases
using
theECMWF
excitation.
The comparison
is fruitfulin that it clearly
demonstratesa stronginfluenceof AAM on the Chandler
wobble(Figures
4 and5). Thecorrelation
at zerotimelag
is striking,reachinga correlationcoefficientof 0.53 for
non-lBand 0.58for IB during1986-1990(no significant
correlationcan be found at other time lags). The
-188
i
'
188
-100
8
'
a8
.._.•
bIB::
,I
(a)non-lB
]
8
'
8.8
. l•
'
(b
)m
]
8.6
8.4
8.2
-38
-100
-Z8
-18
8
18
28
38
I
1980
1982
1984
1986
1988
1990
Year
Fig. 3. gl, the x componentof the Chandlerwobble
excitationseriesfor (a) AAM (non-IB),(b) AAM (IB),
and (c) the observed,1980-1990.
Frequency(cycleper year)
Fig.5. Squared
coherence
spectrum
between
theobserved
Chandler
excitation
andthatinduced
by(a) AAM (nonIB), and (b) AAM (IB), 1980-1990.Compare
with the
99% confidence threshold of 0.54.
256
Chao: AtmosphericExcitationof Polar Motion
:Correlation
coefficient
for 1980-1990is somewhat
lower:
0.45for non-IBand0.52for IB, Presumably
degraded
by
References
R.T. H.,R.Hide,A.A. White,
andC.A. Wilson,
Systematic
errorsin theAAM series
in earlieryears.Of Barnes,
Atmospheric
angular
momentum
fluctuations,
lengthcourse,
observational
andprocessing
errorshavefurther
degraded the correlation estimates.The extent of the
of-day changes and polar motion, Proc. Roy. $oc.
Lond., A 387, 31-73, 1983.
degradation
isdifficultto assess,
buttheactualcorrelation
Chao,B.F:, On the excitationof the Earth'spolar motion,
is certainlyhigherthan the estimatesquotedabove.It
should be noted that the above correlation estimates
representaveragevaluesoverthe studyperiodbecausethe
"instantaneous"
correlationmaywell be time-variable
at
Geoœhys.
Res. Lett., 12, 526-529, 1985.
Chao, B. F., Correlation of interannuallength-of-day
variation with E1 Nifio/Southern Oscillation,
sub-yearlytime scales[Kuehneet al., 1992].
1972-1986, J. Geoœhys.
Res.,93, 7709-7715, 1988.
6f the•correlation
peak.Thehalfwidthofthepeakisonly
--10 days.This indicatesthat the correlatedenergy
96, 6577-6582, 1991.
Perhaps
•qually
siriking
fromFigure
4isthesharpnessChao,B. F., and A. Y. Au, Atmosphericexcitationof the
between
the AAM-induced
and the Observed Chandler
excitatiøn
is largelyconcentrated
in periodsshorterthan,
say,a month,supportinga previousfindingby Eubankset
at. [1988].
A considerableportion of the coherencespectrumin
Figure5 exceeds
the 99% confidence
threshold.(The high
coherence around zero frequency is spuriousas there
existsvirtuallyno energyafter the removalof the polar
driftas.'above.)Thephase.values
Cluster
around0ø (not
Shown),insteadOf randomlydistributing
in (-180ø,180ø]
as would
result
from
two uncorrelated
series. This also
Earth's annual wobble: 1980-1988,J. Geoœhys.
Res.,
EUbanks,
T. M., J. A. Steppe,J. O. Dickey,R. D. Rosen,
and D. A. Salstein,Causesof rapid motiønsof the
Earth's pole, Nature,334, 115-119,1988.
Gross, R. S., Correspondence
between theory and
observationsof polar motion, Geoœhys.
J. lnt., 109,
162-170, 1992.
Gross,R. S., and J. A. Steppe,A combination
of Earth
orientationdata: Space90,JPL Geod.Geophys.
Prep.,
1991.
Hide, R., and J. O. Dickey, Earth's variable rotation,
Science,253, 629-637, 1991.
vindicatesthe significanceof the correlation.
Finally, it is interesting to note that the IB effect
consistently
yields a higher correlationestimatethan the
non-IB case. The difference, over 10%, is significant
(barring possible systematic errors arising from
undertakingthe IB procedure).This can be enigmatic
because, as argued above, a major portion of the
correlationcomesfrom periodsshorterthan a month -an unfavorableconditionfor the IB effect to take place.
Similar situation has been found with respect to the
Kuehne,J., S. Johnson,and C. R. Wilson,Atmospheric
excitationof non-seasonal
polar motion,EO$ Trans.
Amer. Geophys.Union, 73, 126.
Munk, W. H., and G. J. F. MacDonald,The Rotationof
the Earth, CambridgeUniv. Press,New York, 1960.
Nerem, R. S., B. F. Chao, A. Y. Au, J. C. Chan, S. M.
Klosko,N. K. Pavlis,andR. G. Williamson,Temporal
variations of the Earth's gravitational field from
satellitelaser rangingto Lageos,in press,Geophys.
Earth's J2 variations caused by atmosphericmass
Thomson, D. J., Spectrum estimation and harmonic
analysis,Proc. IEEE, 70, 1055-1096, 1982.
Wilson, C. R., and R. A. Haubrich, Meteorological
excitation of the Earth's wobble, Geoœhys.
J. Roy.
redistribution [Nerem et al., 1992]. Incorporation of
realistic
ocean
models
should
be able
to resolve
this
enigma,leadingto a more accurateassessment
of the role
playedby the atmosphereand oceansin the excitationof
the Earth'srotational(and gravitational)variations.
Res. Lett., 1992.
Astron. $oc., 46, 707-743, 1976.
B. F. Chao, GeodynamicsBranch, NASA Goddard
SpaceFlight Center, Greenbelt,MD 20771,USA.
Acknowledgments. I am grateful to A. Y. Au for
preparing the AAM series, and to R. S. Gross for
providingthe Space90Earth orientationseries.
(ReceivedNovember24, 1992;
acceptedDecember 16, 1992.)