Excitation of the Chandler wobble by the geophysical annual cycle

Excitation of the Chandler wobble by the geophysical annual cycle

Excitation of the Chandler wobble by the geophysical
annual cycle
W. Kosek
Space Research Centre, Polish Academy of Sciences, Bartycka 18A, 00-716 Warsaw, Poland.
Abstract. It was found that the change of the
Chandler oscillation amplitude is similar to the
change of the beat period of the Chandler and
annual oscillations and to the negative change
of the phase of the annual oscillation of the coupled atmospheric/ocean excitation. The beat period increases due to decrease/increase of the
phase/period of the annual oscillation, which
means that the annual oscillation period becomes
closer to the Chandler one. The exchange of the
atmospheric angular momentum and ocean angular momentum with each other and with the
solid earth at the frequency equal approximately
to 1 cycle per year is called in this paper the ’geophysical annual cycle’ which can be represented
by the broadband annual oscillation in the sum
of the atmospheric and oceanic angular momentum excitation functions. The phase variations
of this annual cycle are one of the most important source exciting the Chandler wobble.
1
Introduction
The excitation of the Chandler wobble (CW) has
been explained partially by many authors, who
have taken into account electromagnetic torques
acting on the core-mantle boundary, earthquakes
as well as the atmospheric and oceanic angular
momentum. Rochester and Smylie (1965) had
found that electromagnetic torques acting on the
core-mantle boundary play a negligible role in
excitation of the Chandler amplitude. O’Connel
and Dziewonski (1976) and then Mansinha et al.
(1979) concluded that large earthquakes was a
noticeable contribution to the excitation of the
CW, but their hypothesis was verified by Souriau
and Cazenave (1985) and Gross (1986) who concluded that this excitation was negligible. The
contribution of meteorological sources to the CW
excitation was estimated as about 11 to 19% by
Ooe (1978). Hameed and Currie (1989) found
the 14.7 month signal in the surface air pres-
sure which was identified as the ”atmospheric
pole tide”. Plag (1997) showed the 14-16-month
atmospheric pressure fluctuations are responsible for most of the oceanic pole tide attributed
to the CW. Furuya et al. (1996) and Aoyama
and Naito (2001) concluded that the atmospheric
wind and inverted barometer (IB) pressure variations maintain a major part of the observed
CW. Aoyama et al.(2003) have shown that the
quasi 14-month fluctuation of the atmospheric
wind of the European Center for Medium-range
Weather Forecast data plays and important role
in the CW excitation between 1980 and 1993
years. The ocean-atmosphere excitation compares substantially better with the observed polar motion excitation at the annual and Chandler frequencies than when only the atmosphere
is considered (Ponte et al. 1998). Celaya et
al.(1999) and then Brzeziński and Nastula (2002)
confirmed that some combination of atmospheric
and oceanic processes explains the observed CW
excitation. The most important mechanism exciting the CW was the ocean-bottom pressure
fluctuations (Gross 2000; Gross et al. 2003).
In this paper the idea of Jeffreys (1972): ”The
reason is that whatever produces a slow fluctuation of the annual motion is precisely what is
needed to maintain the free motion” has been
followed. Instead of looking for the ∼14 month
signal in the fluids (Furuya et al. 1996; Plag
1997; Aoyama and Naito 2001; Aoyama et al.
2003) only the variable annual oscillation of the
geophysical fluids is considered as one of the possible sources of the CW excitation. The annual
and Chandler oscillations are both treated as
stochastic processes, having variable phases and
amplitudes. Of course, we are aware of the fundamental difference between these two components. The annual wobble is a forced motion connected to the seasonal thermal cycle therefore its
phase can only fluctuate around its well-defined
expectation. In contrast, for the CW, which is a
free motion, the phase has not expected value.
Even the sudden reversal of phase is possible
for the observational evidence (see Vondrak and
Ron, this issue) and for the theoretical model
(see Brzeziński, this issue).
To detect variations of the amplitudes and
phases of the stochastic Chandler and annual
oscillations the least-squares (LS) method was
applied in finite time intervals sliding along the
whole time interval of the pole coordinates data.
The variations of the annual oscillation phase in
the pole coordinates data (Kosek et al. 2001,
2002) suggest that the phase of the annual oscillation in the geophysical fluids is also variable.
The variations of the phase and amplitude of this
annual oscillation in the geophysical fluids, which
is called in this paper as the geophysical annual
cycle, can be one of the most important source
exciting the CW due to leakage of the power from
the annual to Chandler frequency band in the
fluid excitation function.
2
oceanic (AAM+OAM) excitation functions were
computed and are shown in Figure 1. In this
analysis the AAM excitation functions are the
sum of wind and pressure modified by inverted
barometer correction terms and the OAM excitation functions are the sum of the mass and
motion terms. The MWT coherence as the running correlation coefficient between the wavelet
transform coefficients computed using the Morlet wavelet analyzing function (Schmitz-Hübsch
and Schuh 1999) shows the amplitude and phase
agreement as a function of time and frequency
in the geodetic and fluid excitation functions. It
can be noticed that after adding the OAM to the
AAM the coherence becomes greater in the vicinity of the Chandler and annual frequencies but
especially for oscillations with periods less than
1 year.
GE & AAM
500
300
Data
0.9
100
2E+003
3
Coherence between the geodetic and
atmospheric/oceanic excitation functions
The geodetic excitation (GE) functions were
computed from the C04 pole coordinates data using the time domain Wilson and Haubrich (1976)
deconvolution formula.
The time-frequency
Morlet Wavelet Transform (MWT) coherences
(Popiński and Kosek 2000; Popiński et al. 2002)
between the complex-valued GE and AAM as
well as GE and the sum of the atmospheric and
2E+003
2E+003
2E+003
2E+003
0.8
-200
period (days)
The following data sets were used in the analysis:
1) the x, y pole coordinates data from the IERS
C04 in 1962.0 - 2004.5 with 1 day sampling interval and the IERS C01 in 1846 - 2002 with 0.1 year
sampling interval in 1946-1889 years and 0.05
year sampling interval afterwards(IERS 2003).
In some cases the C04 data were extended into
the past before 1962 year by the smoothed and
interpolated with 1 days sampling interval C01
data. 2) the equatorial components of the effective atmospheric angular momentum (AAM)
reanalysis data in 1948.0-2004.0 from the U.S.
NCEP/NCAR, the top of the model is 10 hPa
(Barnes et al. 1983; Salstein et al. 1986, Kalnay
et al. 1996). 3) the equatorial components of
global oceanic angular momentum (OAM) mass
and motion terms from Jan 1980 to Mar 2002
with 1 day sampling interval (Gross et al. 2003).
2E+003
0.7
-400
-600
0.6
1970
1975
GE & AAM
1980
1985
1990
1995
GE & (AAM + OAM)
500
0.5
0.4
0.3
300
0.2
100
2E+003
2E+003
2E+003
2E+003
2E+003
2E+003
0.1
-200
-400
-600
1966 1970 1974 1978 1982 1986 1990 1994 1998
years
Fig. 1. The MWT spectro-temporal coherences between the complex-valued GE and AAM as well as
GE and AAM+OAM excitation functions.
Analysis of the phasor diagrams of the annual
excitation in the AAM and OAM excitation functions have shown that their sum has a better
agreement with the GE functions than the AAM
excitation function alone (Brzeziński et al. 2005;
Gross et al. 2003).
4
Amplitudes and phases of the Chandler
and annual oscillations
The Chandler and annual oscillation amplitudes
were computed together with their phases by the
LS method (Fig. 2). The time interval of the
complex-valued C04 pole coordinates data going
into the LS model which consists of the Chandler
circle, annual and semiannual ellipses (McCarthy
and Luzum 1991; Kosek et al. 2002) was equal
to 5 years. The amplitude and phase variations
of the Chandler oscillation are much smother
than of the annual one as it has been previously
noticed by Schuh et al. (2001), Kosek et al.
(2001, 2002) and Höpfner (2002, 2003). Next,
the Chandler amplitude was also computed as
the envelope of the Chandler oscillation filtered
from the extended into the past before 1962 year
complex-valued C04 pole coordinates data using
the Fourier transform band pass filter (FTBPF)
(Kosek 1995) with the optimum frequency bandwidth (Fig. 2). The optimum frequency bandwidth was chosen so that the variance of the
residuals after subtracting the filtered Chandler
and annual oscillations was a minimum. The
agreement between the amplitude variations of
the Chandler oscillation computed by the LS and
FTBPF methods is very good (Fig. 2).
amplitudes
arcsec
0.2
Ch
An
0.1
0.0
o
50
40
30
20
10
0
-10
1980
days
periods
440
420
400
380
360
340
Ch
An
o
C
phases
An
Ch
Ch - long period
1984
1988
1992
1996
4
2
0
-2
1980
Nino 1+2
1984
1988
Nino 3
1992
years
Nino 4
1996
2000
2000
years
Fig. 2. The envelope of the Chandler oscillations
filtered by the FTBPF (circles). The LS amplitudes
and phases of the Chandler (black line) and annual (x
- blue line. y - red line) oscillations, and the phase of
the Chandler oscillation after removing long period
variations by the LS method (dots).
The aim of this investigation is to find the excitation mechanism responsible for the variations
of the Chandler amplitude.
5
CW oscillation phase shows low frequency variations explained by variations of the Chandler
frequency (Vondrak 1985). The Chandler frequency variations were detected by many authors
(Carter 1981; Okubo 1982; Lenhart and Groten
1987; Vicente and Wilson 1997; De-Chun and
Yong-Hong 2004). To use eq. 2 for computation of the Chandler period variations from it’s
variable phase determined by the LS method the
longer period variations of the phase were removed (Fig. 2. dots) to keep the expected value
of the Chandler frequency constant. The long period variations of the Chandler phase were computed by the LS model fit to the last 50 years
of the Chandler phase data. This LS model consisted of oscillation with periods of 70 and 140
years.
The beat period of the Chandler and
annual oscillations
A change of a phase ∆ϕ(t) of an oscillation with
variable frequency is associated with an opposite
change of a period ∆T (t) according to:
2πt/Tm + ∆ϕ(t) = 2πt/(Tm + ∆T (t))
(1)
where Tm is the mean period of oscillation.
Assuming the mean values of the Chandler
and annual periods, equal to TCh = 434.0 days,
TAn = 365.2422 days, respectively their period
variations were computed (Fig. 3) from their
phase variations shown in Figure 2. The free
Fig. 3. The period variations of the Chandler (black
line) and annual (x - blue line. y - red line) oscillations computed from their LS phase variations shown
in Figure 2. The Niño 1+2, 3 and 4 indices.
It can be noticed that the Chandler period
is much smoother than the annual one and before the biggest El Niño events in 1982/83 and
1997/98 the period of the annual oscillation
had minimum values and increased during these
events (Fig. 3). It has been previously found
that the amplitude and phase/period variations
of the annual oscillation are correlated with the
biggest El Niño events (Kosek 2003, 2004; Kosek
et al. 2001, 2002). Next, the variable beat period Tb (t) of the Chandler and annual oscillations
were computed from the their period variations
by the formula:
1
1
1
=
−
(2)
Tb (t)
TAn + ∆TAn (t) TCh + ∆TCh (t)
where ∆TCh (t), ∆TAn (t) are the Chandler and
annual oscillation period variations about the
mean.
Most part of the beat period variations shown
in Figure 5 are caused by the phase (Fig. 2) or
period (Fig. 3) variations of the annual oscillation. The beat period variations can be also
computed from polar motion radius data given
by the formula:
p
R(t) = (x(t) − xm (t))2 + (y(t) − ym (t))2 (3)
where x(t), y(t) are the extended into the past
C04 pole coordinates data, and xm (t), ym (t) are
the mean pole coordinates data.
To obtain the radius the mean pole coordinates
data were computed by the Ormsby (1961) low
pass filter. The Ormsby filter parameters were
assumed to minimize the variance of the residual Chandler and annual oscillations in the mean
pole coordinates data (Kosek et al. 2004). The
most energetic oscillation of the radius shown in
Figure 4 has a period approximately equal to 67 years which is the beat period induced by the
superposition of the Chandler and annual oscillations.
arcsec
radius
0.3
0.2
0.1
0.0
o
phases
200
180
1962
1968
1974
1980
1986
1992
1998
2004
years
Fig. 4. The polar motion radius (black line) and the
phase variations of the 6-7 years oscillation computed
by the LS method in 12 (green line) and 13 (blue line)
years time intervals.
Next, the phase variations of the 6-7 years oscillation were computed by the LS method from
the radius data (Fig. 4). The LS model consists of oscillation with periods of 6.37, 20 and
40 years and it is fit to the 12 and 13 years of
the radius data. These phase variations together
with the mean beat period value estimated from
eq. 2 for ∆TCh (t) = ∆TAn (t) = 0 enable computation of the beat period variations by eq. 1.
The beat period variations computed from the
radius data are smoother than those computed
from the LS phases of the Chandler and annual
oscillation due to longer time span of averaging
in the first case (Fig. 5). The Chandler amplitude changes obtained as the first difference of
the Chandler amplitude variations computed by
the LS method are shown in Figure 5. The LS
model of the Chandler circle, annual and semiannual ellipses is fit to 4, 5 and 6 years of the
complex-valued C04 pole coordinates data.
Next, the phase variations of the annual oscillation were computed from the complex-valued
AAM+OAM excitation functions data by the LS
method (Fig. 5). The LS model consists of the
annual oscillation and it is fit to 3 and 4 years of
the AAM+OAM data.
years
beat period from the radius
6.6
6.4
6.2
years
8
7
6
5
4
beat period from the phases of Ch and An
mas/day
Chandler amplitude change
0.10
0.05
0.00
-0.05
-0.10
o
310
300
290
1980
AAM + OAM
1984
1988
1992
1996
2000
years
Fig. 5. The period variations of the 6-7 years oscillation computed from the LS phases in 12 (green
line) and 13 (blue line) years time intervals from the
radius data. The beat period computed from the LS
phases of the Chandler and annual oscillations in 4
(black line), 5 (blue line) and 6 (green line) year time
intervals. The Chandler amplitude change computed
by the LS method in 4 (black line) 5 (blue line) and
6 (green line) year time intervals. The LS phases of
the annual oscillation computed from the complexvalued AAM+OAM excitation functions in 3 (red
line) and 4 (black line) year time intervals.
6
The physical mechanism of the Chandler Wobble excitation
It can be noticed that after 1984 when the accuracy of polar motion data became better the
variations of the beat period estimated from the
radius data and from the LS phases of the Chandler and annual oscillations are similar (Fig. 5).
The beat period variations are also similar to the
Chandler amplitude change as well as to the negative change of the LS phase of the annual oscillation computed from the AAM+OAM excitation
functions (Fig. 5). The correlation coefficient between the beat period computed from the radius
data in 13 year time intervals and the Chandler
amplitude change computed by the LS method in
6 year time intervals is equal to 0.51. The correlation coefficient between variations of the beat
period of the Chandler and annual oscillations
computed from their LS phases in 6 year time in-
tervals and the Chandler amplitude change computed by the LS method in 6 year time intervals
is equal to 0.65. The correlation coefficient between the Chandler amplitude change computed
by the LS method in 4 year time intervals and
the phase variations of the annual oscillation in
the AAM+OAM excitation functions computed
by the LS method in 3 year time intervals is
equal to 0.52. All these correlation coefficient
were computed from 1984 to the end of available
data and are significant at 95% confidence level.
These results suggest that the Chandler amplitude change is correlated with the beat period
variations as well as with the phase variations
of the annual oscillation in the AAM+OAM excitation functions. The physical mechanism of
the Chandler excitation can be explained as follows (Kosek 2004): The variations of the phase
of the annual oscillation in the AAM+OAM excitation cause similar variations of phase of the
annual oscillation of polar motion. The decrease
of this phase or increase of the annual oscillation
period means that this period gets closer to the
Chandler one which causes the increase of the
beat period between the Chandler and annual
oscillation (Fig. 6). Thus, the Chandler amplitude change increases during decrease of the
phase of the annual oscillation in polar motion
and its AAM+OAM excitation.
TAn
TCh
Fig. 6. A graph showing that the increase/decrease
of the annual oscillation period/phase results in the
increase of the beat period of the Chandler and annual oscillations according to eqs. 1 and 2.
Conclusions
The amplitudes and phases of the Chandler oscillation are smoother than those of the annual
one. The change of the Chandler amplitude increases with the increase of the beat period of
the annual and Chandler oscillations and decrease of the phase of the annual oscillation in
the pole coordinates data and the coupled atmospheric/ocean excitation. The beat period increases because the annual period gets closer to
the Chandler one. Thus, one of the most possible source exciting the CW can be decrease of the
phase of the broadband annual oscillation in the
AAM+OAM excitation functions. The period of
the annual oscillation was a minimum before the
biggest 1982/83 and 1997/98 El Niño events and
increased during these events. Thus, the CW can
be excited during the biggest El Niño events.
Acknowledgments. This research was supported
by the Polish Ministry of Scientific Research and
Information Technology grant No. 5 T12E 039
24. The author thanks Aleksander Brzeziński for
his valuable comments and discussion.
References
Aoyama, Y., and Naito I., 2001, Atmospheric excitation of the Chandler wobble, 1983-1998. J.
Geophys. Res. Solid Earth, 106, 8941–8954.
Aoyama, Y., Naito I., Iwabuchu T., and Yamazaki
N., 2003, Atmospheric quasi-14 month fluctuation
and excitation of the Chandler wobble. Earth
Planets Space 55, e25–e28.
Barnes R.T.H., Hide R., White A.A., and Wilson C.A. 1983, Atmospheric Angular Momentum Fluctuations, length-of-day changes and polar motion, Proc. R. Soc. London, A387, 31–73.
Brzeziński A. and Nastula J., 2002, Oceanic excitation of the Chandler wobble. Advances in Space
Research, Vol. 30, No. 2, 195–200.
Brzeziński A., Nastula J. and KoÃlaczek B., 2005,
Oceanic excitation of polar motion from intraseasonal to decadal periods. in ”A Window on the
Future of Geodesy”, ed. F. Sanso, IAG Symposia,
Vol. 128, Springer Verlag, Berlin Heidelberg.
Carter W.E., 1981, Frequency modulation of the
Chandlerian component of polar motion., J. Geophys. Res., Vol. 86, No. B3, 1653–1658.
Celaya M.A., Wahr J.M. and Bryan F.O. 1999,
Climate-driven polar motion, J. Geophys. Res.,
104, 12813–12829.
De-Chun Liao and Yong-Hong Zhou, 2004, Chandler
Wobble Period and Q Derived by Wavelet Transform., Chin. J. Astron. Astrophys. Vol. 4 (2004),
No. 3, 247-257
Furuya M. Hamano Y. and Naito I., 1996, Quasiperiodic wind signal as a possible excitation of
Chandler wobble. J. Geophys. Res., 101, 25537–
25546.
Gross R.S., 1986, The influence of earthquakes on the
Chandler wobble during 1977-1983., Geophys. J.
R. astr. Soc. (1986) 85, 161–177.
Gross R.S., 2000, The excitation of the Chandler
wobble, Geophys. Res. Lett., Vol. 27, No 15,
Aug. 1, 2000, 2329–2332.
Gross, R. S., I. Fukumori, and D. Menemenlis, 2003.
Atmospheric and oceanic excitation of the Earth’s
wobbles during 1980-2000, J. Geophys. Res., 108
(B8), 2370, doi:10.1029/2002JB002143.
Hameed S. and Currie R.G., 1989, Simulation of the
14-month Chandler wobble in a global climate
model., Geophys. Res. Lett. Vol. 16, No. 3,
247–250.
Höpfner J., 2002, Parameter variability of the observed Chandler and annual wobbles based on
space-geodetic measurements. Zeitschrift fr Vermessungswesen, 6/2002, 127, Jahrgang, 397–408.
Höpfner J., 2003, Low-frequency variations, Chandler and annual wobbles of polar motion as observed over one century, Scientific Technical Report STR03/01, GeoforschungsZentrum Potsdam.
IERS 2003, The Earth Orientation Parameters,
http://hpiers.obspm.fr/eop-pc/.
Jeffreys H. 1972, The variation of latitude, P. Melchior and Yumi S. (eds.) J. Rotation of the Earth,
39-42.
Kalnay E. et al., 1996, The NCEP/NCAR 40-year
reanalysis project, Bull. Amer. Meteor. Soc., 77,
437–471.
Kosek W., 1995, Time Variable Band Pass Filter
Spectra of Real and Complex-Valued Polar Motion Series, Artificial Satellites, No 24, Vol. 30,
No 1, 27–43.
Kosek, W., McCarthy D.D. and Luzum B.J., 2001,
El Niño impact on polar motion prediction errors,
Studia geophysica et geodetica 45, 347–361.
Kosek W., McCarthy D.D. and Luzum B.J., 2002,
Variations of annual oscillation parameters, El
Niño and their influence on polar motion prediction errors, Proc. Journees 2001, Systemes de Reference Spatio-Temporels, 85–90.
Kosek W., 2003, Polar motion prediction by different methods in polar coordinate system. Proc.
Journees 2002, Systemes de Reference SpatioTemporels, 125–131.
Kosek W., McCarthy D.D., Johnson T.J., Kalarus
M., 2004, Comparison of polar motion prediction results supplied by the IERS Sub-bureau for
Rapid Service and Predictions and results of other
prediction methods. Proc. Journees 2003, Systemes de Reference Spatio-Temporels, 164–169.
Kosek W., 2004, Possible excitation of the Chandler
wobble by variable geophysical annual cycle., Artificial Satellites, Vol. 39, No 2., 135–145.
Lenhardt H. and Groten E., 1987, Chandler Wobble
Parameters from BIH and ILS data, Manuscripta
Geodetica 10, 296–305.
Mansinha L., Smylie D.E., and Chapman C.H., 1979,
Seismic excitation of the Chandler wobble revisited., Geophys. J. R. astr. Soc. (1979) 59, 1–17.
McCarthy D.D. and Luzum B.J., 1991, Prediction of
Earth Orientation, Bull. God., 65, 18–21.
O’Connell R.J. and Dziewonski A.M., 1976, Excitation of the Chandler wobble by large earthquakes.
Nature Vol. 262 July 22 1976. 259–262.
Okubo S., 1982, Is the Chandler period variable ?,
Geophys. J. R. Astr. Soc. 71, 629–646.
Ooe M., 1978, An optimal complex AR.MA model of
the Chandler wobble. Geophys. J. R. astr. Soc.
(1978) 53, 445–457.
Ormsby J.F.A., 1961, Design of Numerical Filters
with Application to Missile Data Processing, J.
Assoc. Compt. Mach., 8, 440–466.
Plag H.-P., 1997, Chandler wobble and pole tide in
relation to interannual atmosphere-ocean dynamics, in Tidal Phenomena Lecture Notes in Earth
Sciences, 66, H. Wilhelm, W. Zurn and H.-G.
Wenzel (eds.) 183–216.
Ponte R.M., Stammer D. and Marshall J., 1998,
Oceanic signals in observed motions of the Earth’s
pole of rotation., Nature, Vol. 391/29 January
1998, 476–479.
Popiński W. and Kosek W., 2000, Comparison of
various spectro-temporal coherence functions between polar motion and atmospheric excitation
functions, Artificial Satellites, Vol. 35, No. 4,
191–207.
Popiński W., Kosek W., Schuh H. and Schmidt M.,
2002, Comparison of the two wavelet transform
coherence and cross-covariance functions applied
on polar motion and atmospheric excitation, Studia geophysica et geodetica, 45, (2002), 455–468.
Rochester M.G. and Smylie D.E., 1965, Geomagnetic
Core-Mantle Coupling and the Chandler Wobble.,
Geophys. J.R. astr. Soc. (1965) 10, 289–315.
Salstein D.A., D.M. Kann, A.J. Miller, R.D. Rosen
1986, The Sub-bureau for Atmospheric Angular Momentum of the International Earth Rotation Service: A Meteorological Data Center with
Geodetic Applications, Bull. Amer. Meteor.
Soc., 74, 67–80.
Schuh H., Nagel S. and Seitz T., 2001, Linear Drift
and Periodic Variations Observed in Long Time
Series of Polar Motion. Journal of Geodesy, 74,
701–710.
Schmitz-Hübsch, H., Schuh, H., 1999, Seasonal and
Short-Period Fluctuations of Earth Rotation Investigated by Wavelet Analysis. Festschrift for
Erik W. Grafarend, Universitt Stuttgart, Techn.
Rep. Dep. of Geodesy and Geoinformatics, Report Nr. 1999.6-2, 421-431.
Souriau A. and Cazenave A., 1985, Reevaluation of
the Chandler wobble seismic excitation from recent data., Earth and Planetary Science Letters,
75, 410–416.
Vicente, R.O. and Wilson C.R.,1997, On the variability of the Chandler frequency, Journ. Geophys.
Res., Vol. 102, No B9, 20439–20445.
Vondrak J., 1985, Long period behaviour of polar
motion between 1900.0 and 1984.0, Annales Geophysicae, 3, 3, 351–356.
Wilson C.R. and Haubrich R.A. 1976, Meteorological
Excitation of the Earth’s Wobble, Geophys. J. R.
Astron. Soc. 46, 707–743.