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 pressure
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.

2E+003 2E+003 2E+003 2E+003 2E+003 2E+003
2E+003 2E+003 2E+003 2E+003 2E+003 2E+003
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 Data
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).
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
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.
period (days)
500
300
100
-200
-400
-600
500
300
100
-200
-400
GE & AAM
1970 1975 1980 1985 1990 1995
GE & AAM
GE & (AAM + OAM)
-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
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

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).
arcsec
0.2
0.1
amplitudes
0.0
o
50
40
phases
30
20
10
0
Ch - long period
-10
1980 1984 1988 1992 1996 2000
years
Ch
An
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 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/T m + ∆ϕ(t) = 2πt/(T m + ∆T (t)) (1)
where T m is the mean period of oscillation.
Assuming the mean values of the Chandler
and annual periods, equal to T Ch = 434.0 days,
T An = 365.2422 days, respectively their period
variations were computed (Fig. 3) from their
phase variations shown in Figure 2. The free
An
Ch
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.
days
440
periods
420
400
380
360
340
o
C
4
Nino 1+2 Nino 3 Nino 4
2
0
-2
1980 1984 1988 1992 1996 2000
years
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
T b (t) of the Chandler and annual oscillations
were computed from the their period variations
by the formula:
1
T b (t) = 1
T An + ∆T An (t) − 1
T Ch + ∆T Ch (t) (2)
where ∆T Ch (t), ∆T An (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
Ch
An

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:
R(t) = √ (x(t) − x m (t)) 2 + (y(t) − y m (t)) 2 (3)
where x(t), y(t) are the extended into the past
C04 pole coordinates data, and x m (t), y m (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 6-
7 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
200 phases
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 ∆T Ch (t) = ∆T An (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
6.6
beat period from the radius
6.4
6.2
years
8
beat period from the phases of Ch and An
7
6
5
4
mas/day
0.10
Chandler amplitude change
0.05
0.00
-0.05
-0.10
o
310 AAM + OAM
300
290
1980 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.
T An
T Ch
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.
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