EXCITATION OF THE
CHANDLER WOBBLE BY
GEOPHYSICAL ANNUAL
CYCLE
Wiesław Kosek
Space Research Centre, Polish Academy of
Sciences, Warsaw, Poland
kosek@cbk.waw.pl
poster available:
http://www.cbk.waw.pl/~kosek
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 of the phase of the
annual oscillation, which means that the annual
oscillation period increases and 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 represents the ‘geophysical
annul cycle’ which can be expressed by the
annual oscillation in the sum of the atmospheric
and oceanic angular momentum excitation
functions. The phase variations of this annual
cycle are possibly responsible for the Chandler
wobble excitation.
DATA SETS
Pole coordinate data – x, y IERS EOPC04 in 1962.0–
2003.8 and EOPC01 in 1846-2002 (IERS 2003). The
geodetic excitation (GE) functions 1,  2 were computed
from the IERS EOPC04 pole coordinate data using the
time domain Wilson and Haubrich (1976) deconvolution
formula (Chandler period equal to 1/ Fc  435 days,
quality factor Qc  100 ). http://hpiers.obspm.fr/eop/pc/
Atmospheric angular momentum (AAM) excitation
functions -  AAM ,  AAM - the sum of the wind and
1
2
pressure modified by inverted barometric correction of
the equatorial components of the effective atmospheric
angular momentum (EAAM) reanalysis data in 1948.02003.8 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, Salstein and Rosen 1997, AER 2003),
http://ftp.aer.com/pub/collaborations/sba/
Oceanic angular momentum (OAM) excitation
functions  OAM ,  OAM - the sum of the mass and
1
2
motion terms of the equatorial components of global
oceanic angular momentum from Jan 1980 to Mar 2002
with 1 day sampling interval (Gross et al. 2003), Ocean
model: ECCO (based on MITgcm).
http://euler.jpl.nasa.gov/sbo/sbo_data.html
INTRODUCTION
Since discovery of the Chandler wobble there has been
an ongoing discussion about its excitation and dumping.
The Chandler wobble excitation has been explained
partially by many authors, who have taken into account
earthquakes, electromagnetic torques acting on the coremantle boundary as well as the AAM and OAM.
Recently, only the AAM and OAM are considered as a
possible source of the Chandler wobble excitation
because the electromagnetic torques acting on the coremantle boundary (Rochester and Smylie 1965), the
earthquakes and the fluid core (Souriau and Cazenave
1985; Gross 1986; Hameed and Currie 1989) play a
negligible role. The atmospheric wind and invertedbarometer (IB) pressure variations maintain a major part
of the Chandler wobble, however the wind signal
dominates over the IB pressure term in the vicinity of the
Chandler frequency (Furuya et al. 1996; Aoyama and
Naito 2001). Some combination of atmospheric and
oceanic processes probably have enough power to excite
the Chandler wobble (Celaya et al. 1999). The joint
ocean-atmosphere excitation compares substantially
better with the observed excitation at the annual and
Chandler frequencies than when only atmosphere is
considered (Ponte et al. 1998) and the importance of the
OAM and AAM to the excitation of the Chandler and
annual wobbles were found to be of the same order
(Ponte and Stammer 1999). The most important
mechanism exciting the Chandler wobble in 1985-1996
was ocean-bottom pressure fluctuations, which
contribute about twice as much excitation power as do
atmospheric pressure fluctuations (Gross 2000, Gross et
al. 2003). Brzeziński and Nastula (2002) concluded that
the coupled system atmosphere/ocean fully explains the
observed Chandler wobble in 1985-1996.
It is well known that the annual oscillation of polar
motion is mainly excited by the annual variations of the
equatorial components  ,  of the AAM and OAM
1 2
excitation functions. The inclusion of the OAM in
geophysical excitation leads to major improvements in
the agreement of the amplitude and phase of the annual
oscillation with observed one, shown on the phasor
diagrams, over what was obtained when considering
AAM signals alone (Ponte and Stammer 1999;
Brzeziński et al. 2003; Gross et al. 2003). Adding the
ocean excitation to the atmospheric significantly
improves the coherence and phase with that observed,
including near the Chandler frequency (Gross et al.
2003).
THE VARIATIONS OF THE
CHANDLER AMPLITUDE AND THE
BEAT PERIOD OF THE CHANDLER
AND ANNUAL OSCILLATIONS
To show the mechanism of the Chandler wobble
excitation it is necessary to explain the variations of the
Chandler amplitude (Fig. 1) and its change (Fig. 2)
(Kosek 2003). The phase variations of the Chandler
oscillation are smoother than of the annual one (Fig. 3).
A change of a phase  (t ) of an oscillation with
nonstationary frequency is associated with an opposite
change of a period T (t ) according to the formula:
2 t     (t ) 
2
t  mean  const
mean
Tmean
Tmean  T (t )
(1)
The period variations of the Chandler and annual
oscillations can be computed by eq. 1 from their phase
variations shown in Figure 3. Since the Chandler
oscillation is a free wobble oscillation and its phase is
not fixed in time a drift of the phase was subtracted by
the robust method (Priestley 1981) before the change of
the Chandler period was computed.
Next, the variable beat period of the Chandler and annual
oscillations (Fig. 6) was computed:
1 
1
1

(2)
Tbeat (t ) TAn  TAn (t ) TCh  TCh (t)
where TAn  365.2422 days and TCh  434.0 days are the
mean periods of the annual and Chandler oscillations,
and TAn (t), TCh (t) are corresponding variations of
their periods about the mean.
Since variations of the polar motion radius are dominated
by the 6-7 year oscillation, which is a beat oscillation of
the Chandler and annual terms, the variations of the beat
period were also computed from the radius data (Fig. 4):
2 
2



m
m
Rt   xt  xt    yt  yt  , t  1,2,...,n
(3)
where the mean pole coordinate data xtm, ytm were
computed by the Ormsby (1961) low pass filter.
The phase variations of the 6-7 year oscillation of polar
motion radius (Fig. 5) were computed by the LS method,
and next the period variations of this oscillation (Fig. 7)
were computed by eq. 1.
THE VARIATIONS OF THE
CHANDLER AMPLITUDE AND
PHASE OF THE ANNUAL
OSCILLATION
Next, the phase variations of the annual oscillation in the
sum of the atmospheric and oceanic excitation functions
were computed (Fig. 9). These LS phase variations are
similar to the negative variations of the change of the
Chandler amplitude (Fig. 8). Adding the ocean excitation
to the atmospheric significantly improves the coherence
between the atmospheric and the geodetic excitation
functions (Fig. 10). The correlation coefficients between
the variations of the change of the Chandler amplitude
(Fig. 8) and the beat period variations computed from the
polar motion radius in 13 year time intervals and
computed from the LS phase variations of the Chandler
and annual oscillations in 6 year time intervals are equal
to 0.510 and 0.654, respectively. The correlation
coefficients between the variations of the change of the
Chandler amplitude computed in 4 year time intervals
and the phase variations of the annual oscillation of
1AAM  OAM and 1AAM  OAM  i2AAM  OAM computed in 3
year time intervals are equal to -0.592 and -0.524,
respectively. All the correlation coefficients are
significant at 95% confidence level.
arcsec
Chandler amplitude
0.30
0.25
LS
0.20
0.15
FTBPF
0.10
0.05
0.00
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
Fig. 1. The amplitude variations of the Chandler circle
oscillation computed from the complex-valued pole
coordinate data using the Fourier Transform band pass
filter (FTBPF) (Kosek 1995) (blue) and the least-squares
(LS) (red) methods.
mas/day
0.2
0.1
change of the Chandler amplitude
FTBPF
LS
0.0
-0.1
-0.2
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
Fig. 2. The change of the Chandler amplitude computed
from the complex-valued pole coordinate data using the
FTBPF (blue) and the LS (red) methods.
o
120
100
80
60
40
20
0
-20
1980
An
Ch
1984
1988
1992
1996
2000
Fig. 3. The phase variations of the Chandler (red) and
annual (blue) oscillations computed by the LS method, in
which the LS model is fit to 5 year time intervals of
complex-valued pole coordinate data.
arcsec
0.4
0.3
0.2
0.1
0.0
1900
Radius
1920
1940
1960
1980
2000
years
Fig. 4. The polar motion radius.
o
280
LS phase of 6-7yr oscillation
260
240
220
200
1950
1960
1970
1980
1990
2000
Fig. 5. The phase variations of the 6-7 year oscillation
computed from polar motion radius by the LS method.
The LS model is fit to 12 (blue) and 13 (green) year time
intervals of the radius data shown in Figure 4.
years
8
7
6
5
4
1980
Beat period
1984
1988
1992
1996
2000
Fig. 6. The beat period of the Chandler and
annual oscillations computed from the LS phase
variations. The LS model is fit to 5 (blue) and 6
(green) year time intervals of pole coordinate
data.
years
6.8
6.6
6.4
6.2
1980
1984
1988
1992
1996
2000
Fig. 7. The period variations of the 6-7 yr
oscillation computed from the LS phase
variations. The LS model is fit to 12 (blue) and
13 (green) year time intervals of the radius data.
mas/day
change of the Chandler amplitude
0.10
0.05
0.00
-0.05
-0.10
1980
1984
1988
1992
1996
2000
Fig. 8. Daily change of the Chandler amplitude
computed by the LS method in which the LS
model of the Chandler circle and annual ellipse
is fit to 4 (black), 5 (blue) and 6 (green) year
time intervals of the pole coordinate data.
o
AAM + OAM (retrograde)
310
300
290
1980
4
3
1984
o
250
1988
1992
1996
2000
AAM + OAM
200
150
1980
4
3
1984
1988
1992
1996
2000
Fig. 9. The LS phase variations of the annual
oscillation computed in 3 (blue) and 4 (green)
year time intervals from the sums
(1AAM  OAM )  i( 2AAM  OAM ) and 1AAM  OAM of
the atmospheric and oceanic excitation
functions.
GE & AAM
500
300
0.9
100
2E+003
2E+003
2E+003
2E+003
2E+003
2E+003
0.8
-200
0.7
period (days)
-400
-600
0.6
1970
1975
1980
1985
1990
1995
GE & (AAM + OAM)
GE & AAM
500
0.5
0.4
0.3
300
0.2
100
2E+003
2E+003
2E+003
2E+003
2E+003
2E+003
1970
1975
1980
1985
1990
1995
0.1
-200
-400
-600
Fig. 10. The Morlet Wavelet Transform timefrequency coherence functions (Popiński et al.
2002) between the complex-valued geodetic
(GE)  1  i 2 and atmospheric 1AAM  i2AAM
excitation functions as well as the geodetic and
the sum of the atmospheric and oceanic
1AAM  OAM  i2AAM  OAM excitation functions.
CONCLUSIONS: THE CHANDLER
WOBBLE EXCITATION MECHANISM
The increase of the change of the Chandler
amplitude occurred during the increase of the
beat period of the annual and Chandler
oscillations and decrease of the phase of the
annual
oscillation
of
the
coupled
atmospheric/ocean excitation. The increase of
the beat period means that the period of the
annual oscillation increases and becomes closer
to the Chandler one. Thus, the change of the
Chandler amplitude increases during decrease of
the phase of the annual oscillation of polar
motion and of the sum of the atmospheric and
oceanic angular momentum excitation functions.
Thus, the Chandler wobble is possibly excited
during decrease of the phase of the annual
geophysical cycle.
Acknowlegements. This paper was supported by the
Polish Committee of Scientific Research, project No
8T12E 005 20 under the leadership of Dr. W. Kosek. The
author thanks Prof. A. Brzeziński his valuable comments
and discussions and Dr. W. Popiński the LS subroutines
of complex-valued time series.
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