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

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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

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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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