f3 fse fc0 sc0 ls0 ws1e">angular momentum are neither in the same physical
dimension nor on the same order of magnitude, and their
normalized values are not necessarily negligibly small or
physically insignificant in comparison to the normalized
differences of the axial and equatorial moments of inertia;
the latter comprises the fundamental Chandler frequency
constituents. On the other hand, if motion and mass
redistribution can excite the Chandler amplitude, there is
no reason to assume that they are too small to affect the
Chandler frequency. It is hence inappropriate to neglect
them before they are physically identified indeed too
small to contribute to the Chandler frequency. Take into
account of above considerations and let the only mathe-
matical perturbation be
= (mx, my, 1 + mz), where (mx,
my, mz) are dimensionless small quantities [8], the lineari-
zation of the Liouville equation in the absence of external
torques becomes [10,12],
2
()
2
xyxxyxy z
x
xy
xx
yzxz y
xz
zz
xx
xyxy z
y y
x
yy
xy y
x
xx
yy
yzxzyz x
zz
yy
yzxz yyz
xz
xx
zz
xz yz
II I
mmm
II
IIh
Imm
II
IIh
mm
II
IIIh
mm
II
IIh I
Imm
II
II



 
 
 
 
 
 



y y
y
xx
y
z
Ih
m
I
II
mm
I
m
I






,
z

yz
yz z
zz
hI
mm m
II

(11)
where zx
y
x
I
I
I

and zy
y
x
II
I


,,
are not inde-
pendent components but the constituents of the funda-
mental Chandler frequency arising from matter distribu-
tion in a slightly triaxial and axially near-symmetrical
Earth [12], and
x
yz
is the excitation function
in Equation (10). The coefficients of left-side terms in
Equation (11) provide frequencies to the Chandler wob-
ble, thus not only moments of inertia but also products of
inertia and motion will contribute to the Chandler fre-
quency [12]. Equation (11) is three dimensional, and its
x- and y-components can no longer be mapped into a
complex plan mathematically as that in the Munk and
MacDonald scheme [8]. The solution of Equation (11)
[10,12] gives a slow damping Chandler wobble of
multiple frequency-splits as well as secular polar drift,

consistent with the observation that the Chandler wobble
hardly changes except exhibiting a “beat” phenomenon
of resonant coupled oscillations [8,12,15,31,32]. This
confirms that single frequency is not the intrinsic
property of the Chandler wobble, but is only for the free
rotation of a biaxial or slightly triaxial rigid Earth under
an assumed initial condition of a slight misalignment be-
tween the rotation and major principal axes. The ob-
servation of apparent single Chandler frequency is be-
cause the length of data analyzed is shorter than the
resonance cycle and in a time span within the modulation
envelope of the oscillations [15].
The solution of Equation (11) [10,12] is very compli-
cated. In the solution [12], the wobble frequency consists
of a natural frequency plus or minus three small feedback
frequency series that are equivalent to adding of small
springs and dashpots in series with the main oscillator.
Yet, the natural frequency can further be separated into a
fundamental frequency attributing to the Earth’s slight
triaxiality just like that of a rigid Earth, and also three
small feedback frequency series that are equivalent to
adding of small springs and dashpots in parallel with the
main oscillator. Such a feedback mechanism causes the
multiple splits of the Chandler frequency. The small
feedback frequency series are due respectively to instan-
taneous inertia, relative angular momentum, and inertia
variation arising from the same motion and mass redis-
tribution that excite the Chandler amplitude [16-18].
However, physical details of the frequency excitation are
not yet identified; the Liouville equation and its solution
can be further simplified if the orders of magnitude of
some of the terms are physically identified to be negligi-
bly small.
8. Rotation Instability
The solution of Equation (11) [10,12] gives an exponen-
tially damping Chandler wobble together with an ex-
ponentially increasing secular polar drift, suggesting the
Earth’s rotation is unstable. Secular polar drift represents
the Earth’s attempt to eliminate its products of inertia, so
it is always associated with the Chandler wobble that is
also involved with the products of inertia [14,22]; they
together constitute polar motion for the Earth to seek
rotation stability. The damping relaxation time for a
Chandler wobble of multiple frequency-splits is on the
order of 104 to 106 years [12], and available observation
[15] indicates the Chandler wobble has yet to reach a
complete multiple-resonance cycle. In a multilayered,
deformable, energy-generating and dissipative Earth,
once the rotation, major principal, and instantaneous
figure axes are separated from each other, they will no
longer be able to revert back to their original position of
alignment in the Earth again, while rheological equatorial
bulge will migrate with secular polar drift accordingly
[8]; so it is an unstable rotation. In an Earth in unstable
Copyright © 2012 SciRes. IJG
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938
rotation, secular internal torques [10,13,22], particularly
those due to the gyroscopic effect or gyricity from
rotation to motion [13,22,27], dominate secular global
geodynamics and also cause free nutation. The Earth will
reach a stable rotation via self-deformation and quadru-
polar adjustment according to the law of conservation of
angular momentum and the three-finger rule of the
right-handed system, until its rotation, major principal,
and instantaneous figure axes are all completely realig-
ned with each other to arrive at the minimum energy
configuration of the system [14]. Then, (mx, my, mz) = 0,
, and (x, y, z) = (a, b, c).

,0

9. Chandler and Markowitz Wobbles,
Secular Polar Drift
Assuming the Chandler wobble to be a complex time
series, Okubo [33] tests the variability of the Chandler
amplitude, and concludes that it is an artifact depending
on the analysis methods. Here we further examine this
particular problem via direct analysis of observation, to
see whether the multiple splits of the Chandler frequency
are artifacts. The data used for this analysis is the 105-
year POLE2004 series [15,34], which is, as error-
analysis below will show, reliable and useful for the
study of the continuation of polar motion. Complex Liou-
ville equation [8] predicts only a non-damping single-
frequency Chandler wobble with no secular polar drift, as
if the Earth were rigid after an amplitude excitation. It is
therefore not appropriate to map real observations into a
complex plan for the analysis of the frequency excitation
it does not cover; our analysis reflects Equation (11). The
analysis tool is a simplest radix-2 FFT; no further
assumptions or parameters are added to generate artifacts.
All artifacts in an FFT are due to the finite truncation of
input, and among them only the Gibbs phenomenon
smears the whole spectrum; the others are uncom-
pensated spectral leakages at local frequencies that will
neither contaminate the signals nor transfer to the time
domain to induce amplitude modulation. So after the
Gibbs phenomenon is removed, no other artifacts will
smear the spectra or induce amplitude modulation, while
digital filtering is exact, not approximation like conven-
tional filters.
We start from Figure 1, the power density spectrum of
the POLE2004 series from 0.0 to 1.1 cycle/year, includ-
ing secular polar drift, Markowitz, Chandler, and annual
wobbles as well as background noises. There is no Gibbs
phenomenon in a power density spectrum, so the baseline
tilting method [15,35] that removes the Gibbs pheno-
menon but will introduce a near-DC component into the
spectrum, is not applied. What in Figure 1 are thus either
input signals/noises or uncompensated leakages due to
finite truncation of the input. For the study of the Chan-
dler wobble, we remove the annual wobble from polar
motion, it then gives a waveform as that in Figure 4 and
a power density spectrum in Figure 5.
Pan [15] observes that the bandwidth of the Chandler
wobble is a constant 0.79 - 0.875 cycle/year regardless of
data length, data quality, time span, and time sampling
rate; whereas, that of the annual wobble varies from 0.99 -
1.01 cycle/year for the 105-year (1900-2005) POLE-
2004 series at 30.4375-day intervals to a broader 0.975 -
1.025 cycle/year for the more modern 42-year (1962-
2005) COMB2004 series at daily intervals. A broad band-
width consists of more than a single discrete frequency
even it has only a single peak [15]. The Chandler spec-
trum splits within its bandwidth with the increase of time
span regardless of time sampling rate, but the annual
wobble only shifts its frequency content. This exhibits
that the annual wobble varies timely in response to
seasonal fluctuations of the atmosphere; whereas, splits
of Chandler frequency reflect the amplitude modulation
cycles within the time span [15]. The Gibbs phenomenon
is already removed, and spectral leakages will not
contaminate the wobble frequencies, so the splits within
the constant bandwidth all belong to the Chandler com-
ponents that will induce amplitude modulation in the
time span. To further look into it, we isolate the Chandler
bandwidth; its power density spectrum is shown in
Figure 6, and waveform in Figure 7 (also [15]). A com-
parison of Figures 4 and 7, we can see that their dif-
ference is only that Figure 4 still contains all the remain-
ders of polar motion, secular polar drift, the Markowitz
wobbles, and background noises except the annual
wobble, while Figure 7 consists of only what is within
the Chandler bandwidth. Figures 4 and 7 reflect each
other; both exhibit the resonant oscillations or amplitude
modulation cycles that a single-frequency wobble cannot
have. However, there is a general belief that if the Chan-
dler frequency is to split, it is a single split [31,32,36,37].
So one may still suspect the side-splits of the Chandler
frequency are artifacts. To test this, we remove the side-
splits from the Chandler frequency. This is equivalent to
spectral leakages are totally compensated as if the input
length were infinite, which hence will not affect the
signals. Figure 8 is the power density spectrum of the
main-split and Figure 9 is its waveform, which displays
a typical single coupled oscillation obviously different
from the multiple amplitude modulations as that in Fig-
ures 4 and 7. A comparison of Figures 7 and 9, we can
easily conclude that the side-splits cannot be artifacts but
belong to the Chandler components, for artifacts or
spectral leakages are not able to add to the main-split to
induce the multiple amplitude modulations beyond a
single coupled oscillationas that shown in Figure 7.
For further confirmation, it is also of interest to see
how the side-splits alone wil behave. Figure 10 is the
l
Copyright © 2012 SciRes. IJG
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939
1900 1910192019301940 195019601970 19801990 2000
-4
-2
0
2
4
6x 10
5
Time (year)
X-Com pon en t (mu as)
Polar motion with annual wobble removed: January 20, 1900 - J anuary 20, 2005
1900 1910192019301940 195019601970 19801990 2000
-4
-2
0
2
4
6x 10
5
Time (year)
Y-Com p one nt (mu as)
Figure 4. Polar motion of the POLE2004 series (Gross, 2005) with annual wobble removed, span January 20, 1900 to January
20, 2005 at 30.4375-day intervals.
00.20.4 0.6 0.8 1
0
1
2
3
4
5
6
7
8
9
10 x 10
10
Frequency ( cycle/year)
Power Density Spe ctra (mua s**2/cpy)
Polar motion wi th annual wobble r emoved: January 20, 1900 - Januar
y 20, 2005
x-component
y- component
Figure 5. The power density spectrum of polar motion of the POLE2004 series (Gross, 2005) with annual wobble removed,
span January 20, 1900 to January 20, 2005 at 30.4375-day intervals.
C. PAN
940
00.2 0.4 0.60.8 1
0
1
2
3
4
5
6
7
8
9
10 x 10
10
Frequency ( c yc le/year)
Power Density Spectra (muas ** 2/cpy )
The C handler wobble only : January 20, 1900 - January 20, 200
5
x-com ponent
y-component
Figure 6. The power density spectrum of the Chandler wobble from the POLE2004 series (Gross, 2005), span January 20,
1900 to January 20, 2005 at 30.4375-day intervals.
1900 19101920 19301940195019601970 1980 1990 2000
-4
-2
0
2
4
6x 10
5
Time (year)
X-Component (mu as)
The C handler wobble only: J anuary 20, 1900 - January 20, 2005
1900 1910192019301940 1950 19601970 198019902000
-4
-2
0
2
4
6x 10
5
Time (year)
Y-Component (mu as)
Figure 7. The Chandler wobble from the POLE2004 series (Gross, 2005), span January 20, 1900 to January 20, 2005 at
30.4375-day intervals.
Copyright © 2012 SciRes. IJG
C. PAN 941
00.20.4 0.60.8 1
0
1
2
3
4
5
6
7
8
9
10 x 10
10
Frequenc y ( cycle/yea r)
Power Density S pectra (muas**2/c py)
The C handler wobble mai n-split only: January 20, 1900 - J anuary 20
, 2005
x-component
y- component
Figure 8. The power density spectrum of the main-split of the Chandler wobble from the POLE2004 series (Gross, 2005),
span January 20, 1900 to January 20, 2005 at 30.4375-day intervals.
1900 1910 1920 19301940 1950 1960 19701980 19902000
-4
-2
0
2
4
6x 10
5
Time (y ear)
X-Com ponent (mu as)
The C handler wobble main-split only: Januar y 20, 1900 - January 20, 2005
1900 1910 1920 19301940 1950 1960 19701980 19902000
-4
-2
0
2
4
6x 10
5
Time (y ear)
Y-Compon ent (mu as)
Figure 9. The waveform of the main-split of the Chandler wobble from the POLE2004 series (Gross, 2005), span January 20,
1900 to January 20, 2005 at 30.4375-day intervals.
Copyright © 2012 SciRes. IJG
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Copyright © 2012 SciRes. IJG
942
0.6 0.650.70.75 0.8 0.850.9 0.95 11.051.1
0
0.5
1
1.5
2
2.5
3
3.5
4x 10
10
Frequency ( cyc le/year)
Power Density Spectra (muas**2/cpy)
The C handler side-splits: January 20, 1900 - J anuary 20, 2005
x-component
y- component
Figure 10. The power density spectrum of the side-splits of the Chandler wobble from the POLE2004 series (Gross, 2005),
span January 20, 1900 to January 20, 2005 at 30.4375-day intervals.
power density spectrum of the side-splits and Figure 11
is its waveform, which further exhibit that the side-splits
are not leakages but components of the resonant oscilla-
tions of the Chandler wobble that are missed in Figures 8
and 9. Note in Figure 10 there are two non-zero uncom-
pensated leakage peaks within the original bandwidth of
the main-split, but which will not contaminate the side-
splits or induce amplitude modulations in the time domain.
In Figure 11, the magnitude of the amplitude modulation
is slowly decreasing, which may reflect the imbalances
of the splits at each side of the removed main-split. We
now remove both the Chandler and annual wobbles
wholly to see how the remainders of polar motion will
behave; Figure 12 is the power density spectrum and
Figure 13 is the waveform. The remainders in Figure 12
are secular polar drift, the Markowitz wobbles, back-
ground noises, as well as uncompensated leakages. How-
ever, as what is shown in Figure 13, none of the re-
mainders will generate resonant oscillations or amplitude
modulations. This further exhibits that artifacts or
uncompensated spectral leakages have nothing to do with
the multiple splits of the Chandler frequency.
In Figure 1 or 4 we can also find that near the zero-
frequency of the power density spectrum, there exist
three conspicuous and one minor spectral peaks in the
x-component. The y-component is dominated by secular
polar drift, but there are yet two peaks that can be
identified corresponding to those in the x-component.
Figure 14 plots the enlarged part of the spectrum from 0.0
to 0.3 cycle/year, and Table 1 lists the measurements of
those low-frequency spectral peaks. Because of the do-
mination of secular polar drift in this near-DC frequency
range, only two peaks, respectively at 0.029 cycle/year
(34.48 years) and at 0.047 cycle/year (21.28 years), can
be commonly identified from both the x- and y- com-
ponents, which are close to the Markowitz wobble. Gross
[18] reports the Markowitz wobble has a period of 24
years and an amplitude of 30 mas. The wobbles in this
frequency range are within or close to the bandwidth of
secular polar drift; their measurements are therefore
corrupted by it and are also heavily dependent on spectral
resolution. The corruption of the wobbles by secular
polar drift is another reason that to map polar motion into
a complex plan may be misleading. However, what listed
in Table 1 are yet apparent and cannot be taken too
seriously. Longer observation is needed for more detailed
study of these long-period wobbles. It is not yet certain
whether Equation (11) will predict such long-period
wobbles. If it does not, then they are not free wobbles.
Finally, we can also make a glance at secular polar
drift. Based on Figure 14, we pick 0.000 - 0.012 cycle/
year as its bandwidth. Then, its power density spectrum
is shown in Figure 15, and waveform is plotted together
with the original polar motion observation in Figure 16.
C. PAN 943
1900 19101920 19301940 195019601970 1980 1990 2000
-4
-2
0
2
4
6x 10
5
Time (y ear)
X-Compon ent (mu as)
The Chandler side-splits: January 20, 1900 - January 20, 2005
1900 19101920 19301940 195019601970 1980 1990 2000
-4
-2
0
2
4
6x 10
5
Time (y ear)
Y-Compon ent (mu as)
Figure 11. The waveform of the side-splits of the Chandler wobble from the POLE2004 series (Gross, 2005), span January 20,
1900 to January 20, 2005 at 30.4375-day intervals.
00.20.4 0.6 0.8 1
0
0.5
1
1.5
2
2.5
3
3.5
4x 10
10
Frequency ( cycle/year)
Power Density Spectra (muas**2/c py)
The Chandler and annual wobbles removed: J anuary 20, 1900 - J anuar
y 20, 2005
x-com p onent
y- component
Figure 12. The power density spectrum of the remainders of polar motion from the POLE2004 series (Gross, 2005) with
Chandler and annual wobbles removed, span January 20, 1900 to January 20, 2005 at 30.4375-day intervals.
Copyright © 2012 SciRes. IJG
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944
1900 191019201930 19401950 1960 1970 198019902000
-4
-2
0
2
4
6x 10
5
Time (ye a r)
X-Com pon en t (mu as)
The Chandler and annual wobbles removed: J anuar y 20, 1900 - J anuar y 20, 2005
1900 191019201930 19401950 1960 1970 198019902000
-4
-2
0
2
4
6x 10
5
Time (ye a r)
Y-Com ponent (mu as)
Figure 13. Remainders of polar motion from the POLE2004 series (Gross, 2005) with Chandler and annual wobbles removed,
span January 20, 1900 to January 20, 2005 at 30.4375-day intervals.
00.05 0.10.15 0.2 0.25 0.3
0
1
2
3
4
5
6
7
8x 10
10
Frequency ( c yc le/year)
Power Density Spectra (muas**2/cpy)
Polar Motion: January 20, 1900 - J anuary 20, 2005
x-component
y- component
Figure 14. The power density spectrum of polar motion of lower frequencies from the POLE2004 series (Gross, 2005), span
January 20, 1900 to January 20, 2005 at 30.4375-day intervals.
Copyright © 2012 SciRes. IJG
C. PAN 945
Table 1. The long-period (markowitz) wobbles.
Frequency (cycle/year) 0.006 0.018 0.029 0.047
Period (year) 166.67 55.56 34.48 21.28
x-amplitude (μas) 4.1 × 105 2.5 × 105 2.0 × 105 0.7 × 105
y-amplitude (μas) 4.4 × 105 2.7 × 105
00.050.1 0.15 0.2 0.25 0.3 0.35 0.4
0
1
2
3
4
5
6
7
8
9x 10
11
Frequency ( c ycle/year)
Power Density Spectra (muas**2/cpy)
Sec ular Polar Dri ft: J anuary 20, 1900 - J anuary 20, 2005
x-component
y -component
Figure 15. The power density spectrum of secular polar drift from the POLE2004 series (Gross, 2005), span January 20, 1900
to January 20, 2005 at 30.4375-day intervals.
1900 1910 1920 1930 1940 1950 1960 1970 19801990 2000
-4
-2
0
2
4
6x 10
5
Time (y ear)
X-Compon ent (muas)
Polar Motion and Secular Polar Dr ift: J anuar y 20, 1900 - J anuary
20, 2005
1900 1910 1920 1930 1940 1950 1960 1970 19801990 2000
-4
-2
0
2
4
6x 10
5
Time (y ear)
Y-Compon ent (muas)
Polar motion
Secular polar drift
Figure 16. Secular polar drift and polar motion from the POLE2004 series (Gross, 2005), span January 20, 1900 to January
0, 2005 at 30.4375-day intervals. 2
Copyright © 2012 SciRes. IJG
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Copyright © 2012 SciRes. IJG
946
10. Error Analysis of the ILS Data
The observation examined above includes the less reli-
able ILS data [15,34]; the high noise level in the ILS data,
particularly those recorded during the 1920-1945 War
period, may introduce errors into the analysis and thus
lead to misinterpretation. However, Pan [15] observes
that the noises in the data are mostly random and inco-
herent between x- and y-components, and the incohe-
rency is higher in higher frequencies. As exhibited by the
observation analysis above, such incoherent random
noises will not affect the periodic signals in the Chandler
frequency range much, for they are incapable of periodi-
cally feeding enough energy back to split the Chandler
frequency [12,38], while noises with periods less than a
month are already eliminated by the monthly sampling of
the data. If the noises could ever affect the wobble fre-
quencies, they would separate the x- and y-components
of the wobbles incoherently rather than nearly identical
to each other as what is observed in Figure 1 (and Fig-
ure 18). On the other hand, since complex Liouville equa-
tion predicts only a non-damping single-frequency wob-
ble of constant amplitude and no secular polar drift,
mapping the x- and y-components of observation, par-
ticularly those contain incoherent background noises,
into a complex plan is misleading. In order to further
clarify the problem, we will do an error analysis of the
ILS data, particularly those of 1920-1945 War years, in
three directions:
1) Time domain: Figure 17 plots the original polar
motion data of the POLE2004 series, span 20 January
1900 to 20 January 2005, at 30.4375-day intervals, in-
cluding the ILS data [34]. As shown, with the presence
of the annual wobble, the amplitude modulation in 1920-
1945 is slightly lower but not exceptionally low. How-
ever, the incoherency between the x- and y-components
is conspicuous, as is also exhibited by the amplitude
spectra of the data in Figure 18, which indeed reflect the
War disturbances. Figure 18 shows the incoherency gets
worse at higher frequencies, but yet hardly gets into the
bandwidths of the Chandler and annual wobbles. Now
we remove the annual wobble from the data, as that in
Figure 4, then the much lower amplitude modulation and
the incoherency between the x- and y-components in
1920-1945 become more conspicuous, which lead to a
belief that the split of the Chandler frequency is caused
by the “phase ambiguity” associated with the exception-
ally low amplitude in 1920-1945. However, here we need
to note that, as is already mentioned above, the x- and
y-components of the data are not mapped to a complex
plan but each treated alone and then plotted together. The
amplitude spectra are all zero phase, so there is not
“phase ambiguity” but incoherent noises as that shown in
Figure 18. Yet, the multiple splits of the Chandler fre-
quency are still there intact, not disappeared with “phase
1900 1910 19201930 194019501960 1970 1980 1990
6x 10
5
2000
-4
-2
0
2
4
Time (ye a r)
X-Com pon ent (mu as)
Polar Motion Observation: January 20, 1900 - January 20, 2005
1900 1910 19201930 194019501960 1970 1980 1990
6x 10
5
2000
-4
-2
0
2
4
Time (ye a r)
Y-Com pon ent (mu as)
Figure 17. Polar motion observation from the POLE2004 series, span 20 January 1900 to 20 January 2005, at 30.4375-day
intervals (Gross, 2005).
C. PAN 947
Figure 18. Amplitude spectra of polar motion from the POLE2004 series; Gibbs phenomenon removed.
ambiguity”. There are also questions: Why the annual
wobble seems less affected by the Wars but the inco-
herency around it becomes worse? Why the amplitude
modulation during 1914-1918 WWI period was not as
low as the years afterward? These questions will become
clear below. Figure 7 is the waveform of the Chandler
wobble extracted from its exact bandwidth in Figure 18,
which exhibits clearly not a single frequency motion but
multiple resonant oscillations with lowest amplitude at
1927. The incoherency between the x- and y-components
that is conspicuous in Figures 4, 17 and 18 is disap-
peared in Figure 7. The incoherent noises are thus sepa-
rable and removable, as exhibited by a comparison of
Figure 18 to Figure 6, since the noises do not contami-
nate the wobble frequencies. On the other hand, Figure 9
is the waveform of the main-split of the Chandler spec-
trum, which exhibits a single coupled oscillation with its
lowest amplitude at 1932, no longer at 1927 as that in
Figure 7, but there were no major wars in 1927-1932. If
the low amplitude modulation in 1920-1945 was indeed
caused by the Wars, then there should be only one lowest
point in the period, more likely closer to 1939-1945 or
even 1914-1918. The envelope of amplitude modulation
as that shown in Figures 7 and 9 will then be interrupted
at the same lowest point, and there will also be no shift of
the amplitude modulation cycles corresponding to the
Chandler frequency splits as that shown in Figures 7, 9
and 11. The amplitude modulation in Figure 9 is a typi-
cal single coupled oscillation, no longer reflects the excep-
tionally low amplitude in 1920-1945 as that in Figure 4.
The shift of amplitude modulation from Figures 7 to 9 is
thus due to the removal of the three side-splits from the
Chandler spectra, not because of Wars. More importantly,
as is also exhibited in Figure 9, a resonant coupled os-
cillation cannot have an open end; it must be cyclic. Only
one low amplitude end is not able to physically split the
natural frequency of the Chandler wobble; it needs an
energy feedback mechanism to achieve it [38].
2) Frequency domain: As is already demonstrated
above, the bandwidth of the Chandler wobble is a con-
stant regardless of data length, data quality, time span,
and time sampling rate, while that of the annual wobble
shifts its frequency content. The incoherent noises intro-
duced during the 1920-1945 War period are separable
from the constant Chandler bandwidth and removable, so
the Wars have not affected the frequency content of the
Chandler spectrum. A broad bandwidth contains more
than a single discrete frequency; the splits of the Chan-
dler spectrum within its bandwidth are expected all to be
the Chandler components. Fourier theory says a periodic
waveform can always be decomposed into a series of
harmonics each having its individual amplitude and fre-
quency. So the multiple amplitude modulations of the
Chandler wobble, as demonstrated above, are due to the
multiple splits of its bandwidth, and not a single fre-
quency motion with time-varying amplitude¸ which is
only apparent.
3) Synthetic simulation: Following the synthetic simu-
lation of 5-component Chandler wobble and annual
wobble [15], the upper plot in Figure 19 is a simulation
Copyright © 2012 SciRes. IJG
C. PAN
948
of the 105-year polar motion, while the lower plot is the
same simulation but with the annual wobble removed. In
the plots magnitude and time span are not exact but rela-
tive. By comparing these two plots respectively with
Figures 4 and 17, we can find the Chandler amplitude
modulation in certain time span can become conspicu-
ously lower without War interruptions. On the other hand,
Figure 20 shows the same simulations but free of noises.
By comparing the lower plot of Figure 20 with Figures
7 and 9, we can see after the annual wobble is removed,
the Chandler amplitude modulation can become excep-
tionally low in certain time span, and the envelope of a
resonant coupled oscillation cannot have an open end but
cyclic.
From above error analysis, it can be concluded that
what the War disturbances during 1920-1945 introduced
5
010 20 30 40 50 60 70 80 90 100
-5
0
Time (yea r)
Original Synthetic
010 20 30 40 50 60 7080 90 100
-5
0
5
Time (yea r)
Annual W obble Removed
Figure 19. Synthetic simulation of the 105-year polar motion of the POLE2004 series. Upper plot is the original simulation;
lower plot is with the annual w obble removed. Magnitude and time span are not exactly simulated.
5
010 20 304050 6070 80 90 100
-5
0
Time (year)
Original Synthetic
010 203040 50 60 7080 90 100
-5
0
5
Time (year)
Annual W obble Removed
Figure 20. Synthetic simulation of the 105-year polar motion of the POLE2004 series free of noises. Upper plot is the original
imulation; lower plot is with the annual wobble re moved. Magnitude and time span are not exactly simulated. s
Copyright © 2012 SciRes. IJG
C. PAN 949
into the ILS data are mainly incoherent noises, which are
separable from polar motion and removable because they
are either random or in much higher frequencies. The
lower amplitude modulation around the time span 1920-
1945 is not due to the Wars but is barely a coincidence.
The ILS observations are hence still reliable and useful
data for the study of the continuation of polar motion,
and the multiple splits of the Chandler frequency so ob-
served are real.
11. Conclusion
The rotation of a biaxial or slightly triaxial rigid body is
not able to depict the complexity of the rotation of a
multilayered, deformable, energy-generating and dissipa-
tive Earth that allows motion and mass redistribution.
Munk and MacDonald’s linearization of the Liouville
equation to represent the rotation of a non-rigid Earth has
oversimplified polar excitation physics, which ends up
equivalent to a rigid Earth with polar excitation superim-
posed on independent of rotation. It hence still gives a
non-damping single-frequency Chandler wobble of con-
stant amplitude with no secular polar drift. The problems
encountered in the Munk and MacDonald scheme are
reviewed, analyzed, and improved according to funda-
mental physical laws. The terrestrial reference frame is
most crucial, and the selection of a both theoretically and
observationally practical reference frame for the study of
the rotation of the physical Earth is a most difficult
problem. It should be physically located in the Earth,
unique, consistent with observation, and always asso-
ciated with polar motion; i.e., the reference frame should
be able to express the Liouville equation as the gene-
ralized equation of motion for the rotation of the physical
Earth. Physical angular momentum perturbation appears
as a relative angular momentum arising from motion and
mass redistribution about the same terrestrial frame
rotating with the Earth relative to an inertial frame fixed
in space as the whole system does, and cannot bypass the
Earth’s rotation and directly about an inertial frame.
Motion and mass redistribution in a rotating Earth is not
the same as that in an inertial Earth or flat Earth. At polar
excitation, the direction of the Earth’s rotation axis in
space does not change besides nutation and precession
around the invariant angular momentum axis, while the
principal axes shift responding to mass redistribution.
The rotation of the Earth at polar excitation is unstable,
and the Earth becomes slightly triaxial and axially near-
symmetrical even it was originally biaxial. Two physi-
cally distinct inertia changes will appear simultaneously
to superimpose each other at polar excitation; one is due
to mass redistribution, and the other arises from the axial
near-symmetry of the perturbed Earth. During polar
motion, the instantaneous figure axis or mean excitation
axis around which the rotation axis physically wobbles is
not a principal axis. The Earth’s principal axes are to be
geodetically determined, also is the Earth’s axial near-
symmetry angle pair
,
. The Chandler wobble
possesses not only a single frequency but multiple splits
and is slow-damping, which exhibits the “beat” pheno-
menon of resonant coupled oscillations. Secular polar
drift is after the products of inertia and is always asso-
ciated with the Chandler wobble; the two together con-
stitute polar motion for the Earth to seek polar stability.
The conventional belief that the axis around which the
rotation axis wobbles is the major principal axis is not
true in a non-rigid Earth.The Earth will return to its stable
rotation via self-deformation and quadrupolar adjustment
according to the law of conservation of angular momen-
tum and the three-finger rule of the right-handed system,
until its rotation, major principal, and instantaneous
figure axes are all completely realigned with each other
to arrive at the minimum energy configuration of the
system. Multiple splits of the Chandler frequency are
further confirmed by directan alysis of observation;
Markowitz wobbles are also observed. Error analysis of
the ILS data demonstrates that the incoherent noises from
War disturbances in 1920-1945 are separable from polar
motion and removable, so the ILS data are still reliable
and useful for the study of the continuation of polar
motion. The rotation of the physical Earth must follow
fundamental physical laws; legitimate mathematics may
not necessarily represent true physics.
12. Acknowledgements
Thanks are due to Richard Gross and Ben Chao for their
supply of the up-to-date polar motion observation and
help in the analysis, which further confirmed the multiple
splits of the Chandler frequency, the observation of the
Markowitz wobbles, and also enabled error analysis of
the ILS data. I am grateful to Steven Dickman’s reading
of the manuscript. High appreciation is due to the Edi-
tor-in-Chief and the four anonymous reviewers for their
detailed and in-depth comments and suggestions, which
greatly helped the rewriting of the manuscript. I am also
grateful to Lanbo Liu and one of the anonymous review-
ers of my past papers, who pointed out the need to syn-
thesize an overall review and systematic examination of
the linearization of the Liouville equation.
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