Chandler wobble

P.A.Semi, 2015/04..06

Geoid and natural coordinate system

Earth is not a sphere, and it is not a rotational ellipsoid. The surface is highly irregular.
The shape is decomposed to a regular sphere with radius 6351km, which does not contribute to any irregularity, and to other mass, consisting of equatorial bulge and continents and sub-surface gravitational anomalies.

I used the model from page Earth2012 - Spherical harmonic models of Earth's topography and potential provided by "Western Australian Geodesy Group" on "Curtin University", Bentley, named "Earths rock-equivalent topography as shape model" (Fig.1).
The map has been divided in 10 steps and all excess mass (assuming 2700kg/m3) above 6351 sphere has been intergrated and summed into 648 (36x18) mass-points.
Map has been read from JPEG image with 0.5km color steps (a sort of gridded format, with information not encoded in numbers but in colors), manually clearing meridian lines and their jpeg-noise using mouse painting software, since I could not use 4M sperical harmonic coefficients, considering it a computationally useless format...
This could introduce some error on the order 0.5-2km in height, but not a continent-sized error... The precision of my model thus created can be considered cca 10%...?

Fig.1 - Geoid model Fig.1b - Geoid model Fig.1c - color scale Fig.2 - Per-longitude average elevation over 6355km
and least-squares fit of 4-axis system.

For the purpose of considering spin differences (LoD and Chandler wobble), it is useful to divide the shape differences about the rotational axis, making per-longitude sums. The best fit of quadrants to these per-longitude sums, using least-squares fit, was determined to be shifted from ordinary coordinates by 21E, placing +Y axis onto Andes mountains at 69W, +X axis onto Africa at 21E, -Y axis east of Himalaya at 111E, and finally the -X axis falls into the deepest part of Pacific at 159W.
The irregularity of the shape can be summarized in a simple way, that Andes are counter-balanced by Himallaya and Eurasia, but Africa is not balanced by Pacific, thus Africa being the most assymetrical irregularity.
As the Earth rotates, some of the irregularity on left side at morning is compensated by it at right side at evening, but not exactly, since planet position changed in between (this difference being most important for Moon and for Venus when meeting it...).
(The geoid model (including my early torque calculation results) is available in JSON format zipped...)

Torque onto the geoid shape

Torque vector sum on the spin-axis has been evaluated, due to gravitational force of Sun,Moon and planets and centrifugal force due to Earth orbit arround Sun and EMB, using JPL ephemerides DE422. The spin centrifugal force and own Earth gravity was not considered, since it is assumed to be constant over time and not causing the irregularities.
Evaluating 24 steps per calendar day to average forces and torques from one side with the (almost) same on other side...

In order to remove linear trend and to simile to natural processes, the torques are summed with linear damping, using damping coefficient k = 0.988

See below Fig.4b and Tab.2 for comparing this with LoD changes, and rest of this document...

Tidal variation in LoD ( Δω3 ) differences

High-frequency component of LoD (Length of Day) change is caused very probably by atmospheric tides, whose difference is related to vertical ecliptic separation between Sun and Moon. Few or none other Moon phenomena are in this frequency 13.661 = 27.3/2 ...

Evaluating tides on Earth by Sun,Moon and planets shows very good synchronization with both amplitude and phase of high-frequency LoD Δω3 (difference of angular frequency of Earth spin) differences (Fig.3*)

The sum of horizontal component magnitude of tidal force is inverted, so that weaker tides mean faster rotation, stronger tides means slower rotation...
The sum of vertical component of tides (always inward, since more places have inward tidal force than places with outward tidal force) shows very similar relation...

Although the tides have been evaluated hourly, there is almost no high-frequency signal, if the force is integrated over the whole globe...?

Frequency analysis of LoD changes Δω3:
      Frequency analysis of vertical component of tides:
Frequency Amplitude (picorad/s)
362.695 d (0.993 y) 0.20845
181.201 d (0.496 y) 0.15175
13.661 d 0.1323
27.516 d 0.067
Frequency Amplitude (arbitrary units)
13.656 d 82.76
5230.3 d (14.32 y) 36.24
27.53 d 33.65
182.63 d 33.39
Tab.1 - frequency analysis of tides and LoD Δω3

For calculating tides I used the former geoid rock-equivalent height model, which causes some non-exactness, which can be estimated on Fig.3c comparing my results with IERS subtracting tides...

The Tides in Fig.3a, that also match the IERS-calculated tides, are simple scalar sum of absolute values of magnitudes of vertical part of the tidal vector, which does not make much physical sense? (Vector sum of symmetrical tides over a sphere should be a zero vector...?) The vertical part of the tidal vector can be responsible for lifting of the atmosphere or displacing something in the core or mantle...?
Also note, that the horizontal part of the tide vector has got a different frequency than this vertical part...
Also the Z-component of torque due to tide (that should influence LoD Δω3) has a different frequency, dominant 14.765 days (half of 29.53 days), whereas the vertical abs tide sum has a correct dominant frequency of 13.656 days (half of 27.312 days).

Fig.3a - Earth tides (vertical component, gray) compared with LoD Δω3 (Length of Day, difference of spin angular frequency, pink)
Fig.3b - Earth tides (vertical component) compared with LoD Δω3 - in detail
Fig.3c - Earth tides (vertical component) compared with LoD Δω3 - in detail
The difference calculated by IERS is in blue, the difference calculated by me is in red... (See full-width detailed version)


The half-year oscilation in Δω3 is by 27-40 days delayed after peaks of tidal forcing, usually arround Jul 22 and Jan 27, with varying amplitude but very common timing, is very probably caused by north and south summer expanding the atmosphere...? (North summer more intense, because more lands and more atmospheric heating...)

There is another possible explanation of half-yearly cycle in Δω3, that is very good synchronized with peaks on damped sum of X,Y r21E component of torques on Earth axis (Fig.5d), but the high frequency does not match...?

Real tides

Real tides differ from the simplification, based on Taylor expansion, used above.
The tide is a difference between gravitational attraction, that differs arround the Earth, higher below Moon and Sun and smaller on opposite points, and the reactive centrifugal force, that is same over all body of the planet.

The acceleration of the Earth is taken from JPL ephemerides. They also use 300 asteroids and PPN (parametrized post-newtonian) gravity. When evaluating these equations with 8 planets (all but Earth) and Moon, for the whole of Earth, the centrifugal force and gravity force on order 1e24N both cancel out and the residue is on the order of 1e15N, which estimates introduced error on the order of 1e-9.

These tides are not symmetric - the difference between antipode points is arround 1 .. 3%, which is bigger than the difference in radius from the planet center...
When the Earth rotates 1 day round, the X and Y components mostly cancel out, but the Z component, which is usually much weaker, is not canceled by Earth rotation.

Fig.31a - Tidal force vector sum parts: Blue X r21E , green Y r21E , red Z.
Fig.31b - same as Fig.31a in detail

It can be said, that the sum of tidal vectors over the globe points southward little east of Africa. Always southward, because north hemisphere has more land-mass, so the horizontal equatorward component of northern vectors is always bigger than the equatorward component of southern vectors...

The major frequency of X component is 13.658 days, the major frequency of Z component is 27.314 and 13.658 days, but major frequency of Y component is 14.762 days.

But the difference in LoD Δω3 should not be caused by a force, but by it's torque:

Fig.32a - sum of daily torque due to tidal force. Blue X r21E , green Y r21E , red Z.
Fig.32b - same as Fig.32a in detail.
(See below in section Numerical estimates).

Here we see, that speed-up and slow-down (Z component of the torque) is very small, and much bigger are the X and Y components of the torque (wobble).
Major frequency of X and Y components is 14.189 days, major frequency of Z component is 14.765 days (half of 29.53), and this is not seen in variation of LoD Δω3.

Geoid torque

When evaluating torques on Earth by Sun,Moon and planets, the vector trend looks like Fig.4a. The change ΔL is sum of that vector, here on Fig.4b.
Fig.4a - Geoid torque vector components: Xr21E (blue), Yr21E (green), Z (red).
Fig.4b - Geoid torque sum, Xr21E (blue),Yr21E (green),Z (red) components

Note the trend of undamped torque sum, as it will be later seen in the Chandler wobble linear trend...

When instead using damped sum with coefficient 0.988 (selected with some experimenting, to decrease the value within 1 year to 1.2%...), that probably has some natural explanation, the linear trend is removed and frequencies can be better analyzed...

Fig.5a - Geoid torque damped sum, Xr21E (blue),Yr21E (green),Z (red) components, compared with LoD changes Δω3 (purple)
Fig.5b - same as Fig.5a, in detail...
Fig.5c - Geoid torque damped sum, negative of Yr21E component (black), with low-frequency subtracted by DoG730 (difference from 2-year gaussian), negative...
Fig.5d - same as in Fig.5c in detail
(See detailed full-width version)
Fig.5e - same as in Fig.5c in more detail. Note, that high-frequency component is mis-aligned...
Fig.5f - Geoid torque damped sum, negative of Yr21E component (black), with middle-frequency highlighted by Gauss108,54, compared with IERS EOP C04 Δω3 with tidal variation removed (purple) with low frequency removed by 2-year gaussian...
Fig.5g - same as Fig.5f, in full-width detail...

Although the half-yearly frequency and phase seems to match, in more detailed inspection the amplitude does not match, and the high-frequency is not aligned... In even more detailed inspection, lot of high peaks do match...

If Y component of torque would influence LoD, it could be caused by horizontally displacing some mass in the mantle, moving it away from axis of rotation??

Torque sum damped, complex XY frequency: Torque sum damped, Z frequency:
Frequency Amplitude (arbitrary units)
182.134 d (0.4987 y) 2.108
363.215 d (0.9944 y) 1.961
14.189 d 0.7201
13.658 d 0.3307
27.555 d 0.3126
Frequency Amplitude (arbitrary units)
14.762 0.6361
365.214 0.4875
14.189 0.4193
9.613 0.1554
182.241 0.1328
Tab.2 - frequency analysis of Torque damped-sum

The Z component of torque sums, which would be expected to influence the LoD Δω3 variation, shows also different high-frequency, but also does not match the half-year cycle phase, but it's derivative does match the half-year cycle phase...
Fig.6a - Z component of damped torque Sum (black) and it's derivative (violet) compared with LoD Δω3 variation (red)
Fig.6b - same as in Fig.6a, in detail... (See detailed full-width version...)

Planets only

When evaluating torques due to planets only (cca 5% of the whole sum), excluding Sun,Moon and Reactive Centrifugal acceleration, it shows another trend, that will be little used later for comparing with Chandler wobble. I do not know, why Sun & Moon should be excluded?
The main cycle is 11.86 year Jupiter, and also shows cca 400 day Jupiter-Earth meet period...

Fig.7a - Geoid torque due to planets only, vector components Xr21E (blue), Yr21E (green) and Z (red)
Does the torque work in right angle from the influence? Then the +Yr21E (69W) axis assymetry will be due to an Africa assymetry at 21E...
Fig.7b - Geoid torque damped sum, due to planets only, vector components Xr21E (blue), Yr21E (green) and 100x Z (red,scaled)
Fig.7c - Geoid torque damped sum, Z component (red), it's derivative (violet) and their complex phase at bottom.
(See detailed full-width version)
Fig.7d - Geoid torque damped sum, Xr21E component (blue), Yr21E (green), separated high-frequency component by DoG730 difference from 2-year gaussian, which is shown as a low-frequency component...
Fig.7e - Geoid torque damped sum. Yr21E low-frequency component (green), Yr21E high-frequency component extracted using DoG730 (aqua), it's derivative (purple), and their complex phase at bottom...
(See detailed full-width version...)

Earth orbit

Angular momentum is calculated by:
      L = r x p = r x m v
where L is angular momentum vector, p is linear momentum vector (mass * velocity) relative to the center and r is position vector relative to the center, and x denotes cross-product.

During the planet orbit, Angular momentum is almost constant. When the planet is near the Sun, it orbits faster, when it is far, it orbits slower.

The sum of angular momentum of all planets, relative to system Barycenter (SSB) is a conserved property. This vector is a normal to the Invariable plane. (There reads a sentence in Wikipedia: "If all Solar System bodies were point masses, or were rigid bodies having spherically symmetric mass distributions, then an invariable plane defined on orbits alone would be truly invariable and would constitute an inertial frame of reference. But almost all are not, allowing the transfer of a very small amount of momenta from axial rotations to orbital revolutions due to tidal friction and to bodies being non-spherical.")

The angular momentum of EMB (Earth-Moon barycenter) with respect to center in Sun is much more constant, than the same with respect to SSB, which shows large variation due to Sun moving arround SSB.
The angular momentum of Earth body itself shows large 27-day variation, due to it's additional orbit arround EMB.
The elliptical orbit of EMB arround Sun is almost constant, with little variation (with amplitude on the order 10-6 of total AM) due to planetary perturbations on Earth orbit (Fig.8).
The changing modes of AM(EMB,Sun) differences well correspond with 22-year signed Halle Sunspot cycles, due to Earth-Jupiter-Venus repeating configurations...

Fig.8a - Orbital Angular momentum magnitude of Earth or Emb relative to Sun or Ssb.
blue: Earth rel. SSB, olive: EMB rel. SSB, light green: Earth rel Sun, dark-green: EMB rel. Sun ...
(EMB is above, because to it's mass is calculated Moon also...)
The variation of AM(E*) rel. SSB shows the signature of Sun orbiting the SSB.
AM(Earth) shows additionally a signature of Earth orbiting EMB (29.526 day frequency component).
AM(EMB) rel Sun is the most smooth (elliptical) orbit - arround the Sun, which further draws Earth arround the SSB with it's own movement arround SSB...
Fig.8b - EMB orbital angular momentum relative to Sun.
Fig.8c - same as Fig.8b in detail...

Frequency analysis of EMB AM variations, from 5000 year of JPL ephemerides, sorted by significance, also marking neighbour FFT slots to specify estimated precision:

Period(range)Period yearsDescription
199.41 days (199.39-199.42) 0.55 year - half Jupiter meet period
583.51 days (583.35-583.68) 1.6 year - Venus meet period
291.88 days (291.84-291.92) 0.8 year - half Venus meet period
132.96 days (132.95-132.97) 0.36 year - ?
209.03 days (209.00-209.05) 0.57 year - ?
1452.3 days (1451.3-1454.3) 3.98 year - half Earth-Venus 8-year meet cycle?
194.60 days (194.58-194.61) 0.53 year - ?
5729.9 days (5714.3-5745.6) 15.69 year - half Saturn orbit period ??
439.19 days (439.10-439.29) 1.20 year - beat between 365.25 and 199.41
97.31 days (97.31-97.32) 0.27 year - ?
116.77 days (116.77-116.78) 0.32 year - ? (similar to Venus apparent Solar day?)
4315.1 days (4306.3-4324.0) 11.81 year - Jupiter orbit period
137.17 days (137.16-137.18) 0.38 year - ?

The main frequency here is 199.41 days, half the Earth-Jupiter meet period.

Beat frequency between this freqency and Earth year 365.25 days, is:
fBeat = f1 - f2
1 / (365.25/199.41 - 1/1) *365.25 = 439.185374457308

This 439.2 day beat period is very near to stated Chandler wobble period...

When AM variation is combined with 365.25 day sine-wave, the wave looks like this, and it will be used to predict Chandler wobble:

Fig.9a - AM of EMB, combined with year-sine wave (0 at equinoxes, minimum at winter solstice, maximum as summer solstice)
Fig.9b - AM of EMB, combined with year-sine wave, decomposed:
YearSine wave (green) uses: sin(Ecl(Earth)-pi) -- position in ecliptic coordinates...
AM(EMB) wave (blue) uses: (VLength(VAngMoment(Emb,Sun)) -180.040575119159 - 3.35584265168733E-09*(JD-1028811.37805)) * (-435)
where VAngMoment() function calculates Vector angular momentum in units Mass/1024kg * Dist/AU * Speed km/s ...
Subtracted base value and linear trend, inverted and scaled to range -0.913 - 1.087 and uses 7-day Gaussian smoothing (needed, because source data was calculated in 7-day steps and re-filled linearly?)
Their sum is purple bold serie.
It's daily derivative * 50 is violet bold serie. Coefficient 50 selected to make their amplitude approximatelly same...
Combined into complex number, whose phase is displayed at bottom part... (What is it physically or mathematically? Value combined with it's own amplified derivative into a complex number...?! Why ??)

The YearSine wave could probably represent the Coriolis force differences due to changing inclination relative position of equator and ecliptic, the AM(EMB) wave could represent the Euler force due to changing orbital Ω , and putting them equivalent is probably an unjust simplification...? Next version of this text sometimes will make more precise computations of these forces.
By numerical estimates, these forces are on surface much larger, than difference between gravitation and reactive centrifugal force due to orbit (which mostly cancels gravitation of Sun, Moon and Planets)...
After making numerical estimates and evaluating these fictitious forces, they may not be causing this: the Euler force is very weak and also has a yearly signature, if the coriolis force (due to orbit) really existed, it would be unreasonably strong...

When converting this to a complex serie for determining phase and instantaneous frequency, the real part is this curve (earth orbital energy combined with earth-orbit sine-wave), and imaginary part is its daily derivative scaled (amount of planetary perturbation combined with earth-orbit cosine-wave?)...

When converted to SI units and multiplied by different constants, the units of real part will be J.s (joule second) and unit of imaginary part will be J (joule).

Chandler wobble

For analysis of Chandler wobble, 2 data-files are used:

Daily data in eopc04 and long-term data in eopc01 (together available for download are versions, that I used...).

Fig.10a EOPC04 Chandler wobble r21E, with complex phase and instantaneous period, 1962-2014.
The removed linear trends (15-year averages) are shown as bold dotted lines...
(Blue sinusoid - dXr21E, green sinusoid dYr21E, red complex amplitude, green "backslash" below is complex phase (with five lines marking "quarters" 0,π/2,π,3π/2 ), sharp red above is instantaneous period with three lines showing 365, 399 and 439 days)
(Scale of the charts is 93px / year)
Fig.10b - same as Fig.10a - in detail
Fig.10c EOPC01 Chandler wobble r21E, with complex phase and instantaneous period, 1846-2008.
Data before 1962 are highly chaotic, partly probably due to methods used to deduce them...
Data before 1900 are less chaotic then 1900-1962, probably due to yet different method to deduce them...?
Fig.10d EOPC01 Chandler wobble r21E instantaneous period detail...
The three lines mark periods 365, 399, 439 days
Fig.10e EOPC04 angular acceleration |dΩ| in range 0 .. 2.99e-17 rad/s2, with 27-day gaussian. (derived from X,Y components and delta LoD of Earth rotation parameters)
On calendar year 1983 measuring changed, and also a lot changed starting calendar year 1984, probably values between 1962 .. 1982 were smoothed by measurement method? Values in this range only little differ from 27-day gaussian.
Fig.10f EOPC01 angular acceleration |dΩ| in range 0 .. 1.27e-16 rad/s2, same method as Fig.10e . Trend highlighted by 54-day guassian smoothing.
It can be seen, that data before 1962 (before start of exact measurements) are rather chaotic. Rather than Earth behaviour changed with start of measurement it is probable, that earlier estimates are not exact, rather best what is possible to guess... Data between 1892 .. 1900 are different than data between 1846 .. 1892 and data between 1900 .. 1962 .

The difference between eopc01 and eopc04 data in instantaneous period at the time when they overlap are either due to sparse sampling in historical eopc01 data, and mainly due to a different window used to subtract linear-trend average (15-year average for shorter eopc04 and 30-year average for longer eopc01 serie...)

Since 1962, the Earth rotation is measured preciselly.
Former data (eopc01) uses some sort of interpolation, determined from various historical astronomic observations, and the data are thus very noisy... Data before 1900 are little less noisy than in range 1900 - 1962, which suggests, that before year 1900 different sources for interpolation have been used...? (The difference in behaviour just on year 1900 seems too much as an administrative artefact than a real change in physical conditions on century boundary...)

It can be noted from instantaneous period (thin-red line at chart top), that there is no constant period, and the calm-average is somewhere arround 400-day (365 .. 439 day) range, but there exist a lot of slow-downs (longer periods - peaks above) and some speed-ups (shorter periods, troughs below).
Actual average of instantaneous frequency from EOPC data, when using ONLY values in range 350-450 days (or 300-500 days) are:

Data total days Period Range Filtered days Average Period
eopc04 19033 350-450 8811 403.194
eopc04 19033 300-500 14078 410.126
eopc01 53046 350-450 10767 400.73
eopc01 53046 300-500 21306 403.017
eopc04 19033 350-500 12211 422.374
eopc04 19033 300-450 10678 390.400
eopc04 19033 350-550 13773 433.508
eopc04 19033 300-600 16444 428.594
eopc04 19033 365-450 7867 408.619

In the calm-mode (for ex. during 1995.5-1997.3), the average period of Chandler wobble is arround 400 days, near the Earth-Jupiter meet time period, or arround 439 days, beat frequency between 199.41 and 365.25 day periods...
Probably this 400-day value is an artifact of selecting a symmetric filter window?
When filtering assymetrically arround this period, the average is also moved away...
But when inspecting the instantaneous period chart (Fig.10d, Fig.10* ...), the values above cca 450 and below cca 365 are all noise-only and perturbations, so filtering a calm-mode range 350-450 can be justified...
Nevertheless, there cannot be stated a single frequency or period of chandler wobble, since the period varies a lot...

Approximatelly every 6.2 years there is a huge damping of wobble amplitude, connected with much larger period of the wave...
There is no known planetary phenomena arround the 6.2-year range, and a smaller amplitude implies intuitivelly a faster wave to conserve energy...?

6.2 year period is a beat frequency between 435 day period and 365.25 day Earth-orbit (yearly) period. For 435.5 day period leads to 6.2-year beat, while 434.2 day period leads to 6.3-year beat.

Beat frequency 431.25 .. 433.94 can be also reached as a beat between modified Geodetic precession and 365.25-day Year, using formula Omegageodetic = π/2 * RsSun / Dist(Earth,Sun), leads to T=2*π/ω=2305.9 day on start of May, beat with 365.25 day as 1/(1/t1)-(1/t2)) = 431.25 .. 433.94 with higher period on January and shorter period in July, differing with distance of Earth from Sun... Have not found a formula for magnitude of such precession, but there is nothing such yearly-regular as this in Chandler wobble...

6.2 year period is also 1/3 of 18.6 year Moon precession cycle, but how it could be 1/3 connected?

Chandler wobble prediction

Here I will combine Chandler wobble with AM(EMB)+YearSine (purple serie in charts), with Geoid torque due to planets only (light-blue bold serie and green dashed serie) and various planetary phenomena involving Earth,Jupiter and Venus (arrows and boxes), to explain most phenomena of Chandler wobble irregularities.

It has to be noted, that it is a "prediction" of past behaviour and how it could evolve, if no significant change in behaviour will occur...

Chart legend

Chandler dX r21E, linear trend
Chandler dX r21E, with linear trend removed
Chandler dY r21E, linear trend
Chandler dY r21E, with linear trend removed
Chandler instantaneous Period. (The 3 lines mark 365, 399, 439 day periods)
Chandler complex amplitude (always positive)
Chandler r21E Phase (see description of quadrants above...)
Earth-Jupiter heliocentric conjunction
Earth-jupiter heliocentric opposition
Earth-Jupiter heliocentric quadratures
Earth-Venus heliocentric conjunction
Earth-Venus heliocentric opposition
Venus-Jupiter geocentric conjunctions and oppositions
Mark boundaries of Ch.W. slow-downs or returns from them...?
Emb-Jup heliocentric angle at Earth-Venus syzigy (con,opo):
Marks Earth-Venus heliocentric con/opo,
also this is an outer envelope of 11y Sunspot cycle...
AM(EMB)+YearSine - one of prediction waves for Chandler wobble (Fig.9b)
AM(EMB)+YearSine - Phase
(complex wave formed by taking real part from AM(EMB)+YearSine - earth orbital energy combined with year-orbit sine, and imaginary part from it's derivative - amount of planetary perturbation to earth orbit and year-orbit cosine...?) (Fig.9b)
Geoid torque due to planets only, low-frequency part, Xr21E-component (Fig.7a)
Geoid torque due to planets only, low-frequency part, Yr21E-component (Fig.7a)
Low-frequency component of damped sum of torque due to planets only, Yr21E component, extracted by 2-year gaussian (Fig.7e)
High-frequency component of damped sum of torque due to planets only, Yr21E component, extracted by DoG730 (Difference of 2-year Gaussian) (Fig.7e)
Complex phase of high-frequency component of damped sum, Yr21E, real part is running damped sum, imaginary part is it's derivative... (Fig.7e)

Analysis charts

On Fig.11 is complex analysis chart for Chandler wobble, planetary forcing on Earth orbit and planetary torques on geoid shape...

All frequency (period) anomallies of Ch.W. peak near Earth-Jupiter syzigy, and all are surrounded very closely by peaks of AM(EMB)+YearSine wave frequency (periods)...

Chandler wobble phase is "predicted" by AM(EMB)+YearSine wave (purple serie, with phase at bottom) Earth Angular momentum combined with year-sine, or by Torque YdY complex serie (aqua serie, with phase at bottom).
It seems, that sometimes the synchronization is little better toward the AM(EMB)+YearSine serie, than to the Torque YdY complex wave...

Notice at bottom on phase diagrams - when Ch.W. phase is well aligned with AM(EMB) forcing phase, the wobble is "calm" in frequency and huge in amplitude.
But the forcing phase (either one) is highly irregular and when the wobble gets out-of-phase, more near to the next phase "run" of the forcing, it waits for it and attaches to the forcing again. During this "wait", the Chandler wobble period is highly prolonged and amplitude shallower (probably because it is out of phase with the forcing?)

See on Fig.12 extracted series simplified...

Fig.11a - Chandler wobble eopc04 r21E (1962-2014) with analysis. (large version)
Note the phase series at bottom. Green is Chandler wobble real data, purple is AM(EMB)+YearSine and aqua (light blue-green) is Torque YdY complex phase.
Fig.11b - Chandler wobble eopc01 r21E (1840-2008) with analysis (large version)
Fig.12a - Sample - Chandler wobble r21E (green Y, blue X, red amplitude) with phase (green), combined with AM(EMB)+YearSine with it's phase (purple) and Torque YdY complex (damped sum, DoG730) with it's phase (aqua).
The purple or aqua phase are pretended forcing, which are highly irregular. When the wobble (with best frequency of 400 days) gets too out of phase with the forcing (1985,1986,1987), it delays to get into the forcing phase again (1987,1988)...
Very seldom it happens, that it speeds-up to get to the nearest forcing phase backward (as in 2014,2015).
Fig.12b - Same as Fig.12a, in full-width detail...
Fig.12c - Chandler wobble comparing only with AM(EMB), marking 400-day period slope in phase-diagram at low chart, and showing various possible predictions until year 2020...
Fig.12d - Chandler wobble eopc01 r21E (1840-2008) with phase, combined with AM(EMB)+YearSine and Geoid Torque YdY (damped sum, DoG730) with their phases.
It is possible to see, that the relation of phase-synchronization is visible even in the early chaotic part of data...
Fig.12e - Chandler wobble eopc01 r21E - sample of damping region when attaching to the following phase of forcing wave...
Fig.12f - Chandler wobble eopc01 r21E - sample of even more fancy attaching to the following phase...
Fig.12g - Chandler wobble eopc01 r21E - another sample of phase attaching...
Comparing timing of period (frequency) fluctuations:
Fig.13a - EOPC04 - comparision of instantaneous period (frequency) between Chandler wobble (red) and AM(EMB)+YearSine (purple) ...
Fig.13b - same as Fig.13a, full-width detail...
Fig.13c - EOPC01 - comparision of instantaneous period (frequency) fluctuations (top part) between Chandler wobble and Torque damped sum YdY complex serie...
(See detail full-width version)
Fig.13d - same as Fig.13c, old region...

All wide charts use 93 px/year horizontal scale...

These last charts (Fig.10,Fig.11,Fig.12, 24Mb...) are available in XML format for program EphView...
(if you cannot process simple XML and when importing into some spread-sheet program like OpenOffice or Excel, copy 1 serie into clipboard and use " quote as separator and take 2nd column as date, 4th column as value, or use a similar logic with awk program...)


High-frequency component of LoD variation is caused by tides.
The middle frequency (0.5y) may be due to north and south summer atmospheric effects, or possibly by middle-frequency of torques on geoid shape, of which best synchronized seems surprisingly Y component of the torque vector. The derivative of Z component of torque vector, which would be expected to be related to angular velocity, is much more constant, much less matching the irregular amplitude of middle-frequency of LoD change.
The low frequency (18.5y?) of LoD change seems to be related again to the Moon 18.5 year precession cycle. (The only exception at start of record 1962-1963 may be due to inexact measurement, or other possible influence?)

The Chandler wobble polar shift does not have the high frequency 13-day component, not even a little. It is very probably caused either by planet-only torque to Earth (geoid) shape - but why planets only?, or by spin-orbit coupling, due to differences in Earth orbital energy (angular momentum, in J.s), which is itself irregular. Either of these two seem to be the cause of Chandler wobble irregularities.
When the wobble, which often maintains the Earth-Jupiter meet frequency of 400 days (199.4*2=398.8), gets out of phase with the orbital forcing, it either waits to get in sync, or seldom speeds up to get back to the nearest sync phase...
During this period/frequency irregularity, when the wobble is out-of-sync with its forcing, it's amplitude is much smaller...

The wobble seems to align with Earth-Jupiter heliocentric syzigy on +Y (rotated,natural) axis, and the frequency (period) anomalies are well in-sync with frequency (period) anomalies of AM(EMB)+YearSine wave or Geoid Torque +Y wave, possibly synchronized with other planetary phenomena.

No magnetic effects and probably no climate effects are needed to explain the Chandler wobble irregularities. (The year-sine wave part of the forcing is too constant to be caused by much more variable climate...?) Neither the small inner planet beats (as suggested by someone) are needed to explain this...


The work does not use false analytic simplifications of orbits or geoid shape to a rotational ellipsoid, rather uses numerical integration of gravity-anomally geoid shape (gridded to 36x18 points) and planet orbits (and acceleration) from numerically integrated JPL ephemerides...
The Geoid model, as simplified, may be inexact to within 10-20% ... If the forcing was through atmosphere and not rocks and water, it would use another proportions...

Discussion update

Since the continents are not symmetric arround the rotation axis, it is very probable, that there is an opposite assymetry deep in the core, below of that to what is sensitive the Earth2012 gravity model scanning.

Because the Chandler wobble shows low-frequency of tidal variations but not the high-frequency, and probably opposite to that, which would be expected on continents, and because tidal variation would be rather expected to make precession and nutation of external rotation axis and not the internal one (pole shift), it is rather a tidal variation to that counter-balancing mass in the core...

Appendix A - Numerical estimates

All calculations are done in SI units, if possible, using m as space distance, s as time unit and N as force, m/s2 as acceleration, rad angles, etc... Using these unless otherwise noted...

Geoid model uses sphere 6351km of mass 5.94817e24 kg, which does not contribute to torque, since it is assumed symmetrical. 648 (36*18) nodes spaced by 10 of summed surface irregularities weight 2.5425e22 kg, of those 648 nodes the most massive is 10 centered on 35E 5S weight 9.28e19 kg, 35E 5N 9.264e19 kg, 25E 5S 9.196e19 kg and least massive is 125E 85N, 9.11e17 kg.

Orbital angular momentum of EMB (E+M) rel. to Sun (during years 2005-2012) is 2.7e40 kg m2/s, and range (max-min) 6.8e35 in same units, range (max-min) of EMB rel. Ssb is 5e38 in same units.
Earth rel. Sun is 2.659e40 .. 2.662e40 and range (max-min) 2.6e37 kg m2/s, range (max-min) of Earth rel. Ssb is 5.67e38 .

Average Orbital acceleration of Earth is 0.005931 m/s2, yearly amplitude (fft) 9.512e-5, 29.526 d amplitude 1.537e-5, 182.426 d amp. 2.5e-6, 14.252 d amp. 1.29e-6, ... 1.6 year amplitude 5.586e-7 ...
Average ΩEMB,Sun = 1.991e-7 ( 1.926e-7 .. 2.059e-7 ) ...

Spin angular momentum of Earth planet with ω = 7.2921150e-5 rad/s, WGS84 ellipsoid 6378137 m assumed as sphere, mass 5.97219e24kg, yields momentum I = 2mr2/5 = 9.718e37 kg m2 (in wiki/Polar_motion there is polar moment of inertia = 8.04e37 kg m2, equatorial 8.014e37 ...) and L = I . ω = 7.087e33 kg m2/s .

Chandler wobble - cca 0.2 mas = 5.55e-5 = 9.696e-7 rad (long-term bias removed), with polar radius 6355km it makes 6.16 m wobble (in wiki/Chandler_wobble there is 9 m ...)
This is like to displace vector L 38.717m aside - by 6.0923e-6 rad - during 439 days = 3.7929e7s, which is Δ L = 4.317e28 kg m2/s , which requires torque 1.138e21 N m applied during 3.7929e7 s ...?
When calculating from angular acceleration τ = I dΩ/dt gives typical range |τ| = 2e20 .. 1.7e21 N m with average value 1.45e21 N m between 1962 and 2015.
Long term drift on Yr21E axis is +3.3097e-16 rad/s , displacing from -0.035 arcsec at 1846 to +0.314 arcsec at 2008 (by +1.692e-6 rad during 5.1122e9 s) (steady displacement in +Y direction) .
Long term drift on Xr21E axis was -1.45e-17 rad/s during 19th century and +3.713e-17 rad/s during 20th century displacing from -0.096 arcsec in 1846 to -0.1011 arcsec in 1900 (by -2.47e-8 rad during 1.7e9 s) and to -0.075 arcsec in 2008 (by +1.265e-7 rad during 3.4e9 s) (almost constant, 10-20x smaller drift than in Yr21E...) .
This means average torque +2.35e18 N m on Yr21E axis, -1.03e17 N m on Xr21E axis during 19th century and +2.64e17 N m on Xr21E axis during 20th century ...

Average LoD Δω3 (from IERS EOP C04 values including tidal variation) is cca -1.465 picorad/s = 1.5e-12 rad/s (min -3.6756, max 0.9062), which should be a base of Δ L = -7.1885e26 kg m2/s ... FFT amplitude of freq. 362.695 d (yearly) amp=0.2084, 181.2 d amp=0.1517, 13.661 d amp=0.1323, 27.516 d amp=0.067...
The 13.661 d wave of amp 0.1323 picorad/s represents Δ L = 1.2857e25 kg m2 during (13.661*86400) s, which means τ = 1.0893e20 N m ...

Average Torque magnitude on my geoid surface model (of weight 2.5e22kg) is 8.034e20 N m, fft amplitude of 14.189 d frequency is 2.395e20 , 27.554 d freq. 6.132e19 , 9.367 d freq. 3.722e19 , 13.66 d freq. 3.694e19 , 365.24 d freq. 2.806e19 ...
Xr21E component avg=-0.317e21 (max. pos. 0.67e21, max. neg. -1.588e21), fft amplitude of freq. 14.190 d amp=2.17e20, 9.367 d amp=5.49e19, 14.159 d amp=3.98e19, 13.66 d amp=2.48e19, yearly amp=1.9e19, 182.29 d amp=1.57e19, 27.554 d amp=1.48e19
Yr21E component avg=+0.565e21 (max. pos. 1.647e21, max. neg. -0.649e21) fft amplitude of freq. 14.19 d amp=2.08e20, 9.367 d amp=5.23e19, 13.66 d amp=4.56e19, 14.162 d amp=3.99e19, 182.3 d amp=2.65e19, 27.554 d amp=2.63e19, 16.997 Y amp=2.62e19
Z       component avg=-1.034e15 (max. pos. 0.723e20, max. neg. -0.707e20) fft amplitude of freq. 14.765 d amp=1.13e19, 14.191 d amp=8.48e18, 9.612 d amp=4.14e18, ...
Complex XY FFT: 14.191 d amp=1.31e20, 13.661 d amp=5.93e19 9.366 d amp=4.62e19 27.543 d amp=2.89e19, 182.37 d amp=2.76e19, 13.632 d amp=2.6e19, 17.187 Y amp=1.91e19, ...
Average √(xr21E2 + yr21E2) = 0.648e+21 ...
Calculated S,M,P* torque running undamped sum ( ∑ τ . t ) in Yr21E direction sum 2.88e30 , average torque 5.63e20 N m ? in Xr21E direction sum -1.62e30 , average torque -3.17e20 N m.

Average planet-only torque on my geoid surface model (of weight 2.5e22kg) is 1.405e20 N m, fft amplitude of 5.9 year frequency is 3.947e19 , 398.688 d (1.09y) freq. 2.10e19 , 1.33 year freq 1.86e19, ...
Xr21E component avg=-0.4562e17 (max. pos. 0.7704e20, max. neg. -0.829e20) fft amplitude of freq. 11.799 Y amp=3.28e19, 439.10 d (1.202y) amp=8.89e18, 2.66 Y=3.03e18, ...
Yr21E component avg=1.803e17 (max. pos. 0.3467e21, max. neg. -0.3164e21) fft amplitude of freq. 11.799 Y amp=1.36e20, 439.10 d (1.202y) amp=3.72e19, 2.656 Y amp=1.26e19, ...
Zr21E component avg=1.4955e15 (max. pos. 3.627e18, max. neg. 3.9845e18) fft amplitude of freq. 396.46 d amp=7.65e17 198.94d amp=3.38e17, 145.93d (0.4y) amp=1.78e17, 116.58d amp=1.63e17, ...
Complex XY FFT: 11.587 Y amp=7.29e19, 438.37 d (1.2y) amp=2.03e19, 975.2 d (2.67y) amp=1.14e19, 3.92y amp=1.12e19, ...

Average horizontal component of tidal force (or acceleration?) 1.44287e-6 N/kg, with fft amplitudes: 27.517 d amp=8.641e-7, 13.66 d amp=3.18e-7, 182.645 d amp=2.29e-7, ... yearly amp=1.832e-7 ...
If 3.18e-7 N/kg force (acceleration?) is applied to mass=X at distance 6378137 m from center to produce torque 1.09e20 N m, then mass X=5.374e19 kg (if applied ideally perpendicularly to rotation axis).
The weight of atmosphere is 5.15e18 kg, it's total moment of inertia I = 2mr2/3 = 1.3967e32 kg m2, and L = I ω = 1.0185e28 kg m2/s. To produce Δ L of 1.2857e25 kg m2, it would need to be all displaced by 227 km away during 13.6 days (impossible), or it's mass changed by 6.5011e15 kg, which is 1.26 ‰ (probably possible)... In wiki, there is: "an annual range due to water vapor of 1.2e15 or 1.5e15 kg" - which is 5x less than required...
It seems, that Δω3 may not be caused by atmosphere expansion (possibly only by it's weight change but even that is not enough), but how is weight-change of atmosphere related regularly to a tide...? (but it may be related to summer season, but again the weight range seems not enough?)
The tides also work through oceans and through the Earth crust, but there is very little yearly variation...?
Mass of all oceans (hydrosphere) is 1.4e21 kg, so the LoD Δω3 may sufficiently work through ocean tides, and surely by crustal tides...

Average tidal force (inertial+gravity forces) over globe per day is on the range: X 0.40e14 N .. 1.868e14 N (average 1.144e14 N), Y -0.047e14 N .. 0.66e14 N (average 0.26e14 N), Z -3.014e14 N .. -0.275e14 N (average -1.79e14 N). The torque due to this tidal force is on the range: X -1.61e21 N m .. 6.67e20 N m (average -3.28e20 N m), Y -6.5e20 N m .. 1.65e21 N m (average 5.63e20 N m), Z -6.88e19 N m .. 7.56e19 N m (average 2.51e18 N m).

The torque calculation decomposed, example day 2000-01-20 at 12:00 UTC, axis +X 0E,+Y 90W,+Z 90N:
Reactive centrifugal acceleration (opposite to orbit acceleration) of Earth is |-0.57e-2,-0.16e-3,0.21e-2| = 0.6087e-2 m/s2, making onto central sphere of Earth (CoM at R=0) with mass M=5.97257e24 kg force |-3.396e22,-9.5e20,1.25e22| = 3.62e22 N, calculated gravity acceleration by S,M,P* is |0.56867e-2,0.1593e-3,-0.2093e-2| = 0.60617e-2 m/s2, making onto that central sphere with mass M=5.97257e24 kg force |3.3964e22,9.5165e20,-1.25e22| = 3.62e22 N, causing no torque, because lever arm distance to center is 0. When these two forces are added together, the rest is |1.204e15,5.942e14,-3.43e14| = 1.386e15 N difference (almost canceled), that estimates error in acceleration calculation of |2.016e-10,9.949e-11,-5.739e-11| = 2.32e-10 m/s2 , caused by calculating Earth acceleration as a difference in velocity vector 0.1s apart multiplied by 10. Previously, when used 1 day apart divided by 86400, the difference was on the order 1e-5 m/s2. Using even shorter time interval for a derivative makes problems with numerical precission of FPU... Also the difference is caused, becaused I use Newtonian gravity equations, but take acceleration from ephemerides, for which JPL used PPN gravity, where the difference is somewhere arround 1e-10m/s2...
Onto 10x10 model node centered at 35E,5N with mass 9.26455e19kg with CoM at R=6.365e6 m, the orbital reactive centrifugal acceleration is same, making force |-5.291e17,-1.4825e16,1.9475e17| = 5.64399e17 N onto this mass, S,M,P* cause gravity acceleration of |0.5712e-2,0.159e-3,-0.21e-2| = 0.60888e-2 m/s, making onto that node force |5.29e17,1.476e16,-1.948e17 | = 5.641069e17 N. Forces added are |0.915e-14,-0.64e-14,-0.68e-14| = 1.308e-14 N.
Torque due to S,M,P* gravity force is |-7.167e23,1.30548e24,-1.848e24| = 2.3734e24 N m, torque due to orbital reactive centrifugal acceleration is |7.165e23,-1.305e24,1.847e24| = 2.3726e24 N m, and total torque is only |-2.12e20,4.04e20,-6.65e20| = 8.069e20 N m.
Total torque in 24 hours of this day makes a circulating vector: |-4.26e21,-2.033e22,2.023e20|=2.077e22 , |-9.634e21,-1.903e22,6.644e20|=2.134e22 , |-1.458e22,-1.645e22,8.431e20|=2.2e22 , |-1.873e22,-1.269e22,6.814e20|=2.263e22 , |-2.176e22,-7.946e21,2.19e20|=2.317e22 , |-2.34e22,-2.516e21,-4.161e20|=2.354e22 , |-2.347e22,3.218e21,-1.04e21|=2.371e22 , |-2.192e22,8.829e21,-1.464e21|=2.368e22 , |-1.885e22,1.388e22,-1.548e21|=2.346e22 , |-1.447e22,1.798e22,-1.238e21|=2.311e22 , |-9.124e21,2.081e22,-5.847e20|=2.273e22 , |-3.238e21,2.217e22,2.721e20|=2.241e22 , |2.732e21,2.199e22,1.137e21|=2.219e22 , |8.341e21,2.035e22,1.811e21|=2.206e22 , |1.321e22,1.742e22,2.14e21|=2.196e22 , |1.703e22,1.349e22,2.052e21|=2.182e22 , |1.964e22,8.862e21,1.573e21|=2.16e22 , |2.094e22,3.876e21,8.153e20|=2.131e22 , |2.094e22,-1.168e21,-4.881e19|=2.097e22 , |1.971e22,-6.002e21,-8.288e20|=2.062e22 , |1.736e22,-1.039e22,-1.361e21|=2.028e22 , |1.402e22,-1.415e22,-1.542e21|=1.998e22 , |9.869e21,-1.709e22,-1.355e21|=1.978e22 , |5.083e21,-1.907e22,-8.693e20|=1.975e22
and the sum of these 24 vectors is |-1.456e22,2.604e22,1.151e20|=2.984e22 , which means average torque |-6.068E20,1.085E21,4.795E18|=1.243E21 ...
Uncalibrated charts used 24-hour sums, here in appendix in FFT amplitudes it was divided...

The fictitious forces?

Average coriolis force (∑i 2Mi Ω x vi) on surface nodes due to movement of cca 0 .. 460m/s (due to Earth spin) in rotating frame of Earth orbiting relative to the SSB with average but non-constant ΩEMB,Sun = 1.991e-7 ... (it is a question, whether should be evaluated rel. to Sun or rel. to SSB?) is |6.39e16,1.53e16,-2.7e8|=6.57e16 N, causing an average torque |-5.88e22,1.05e23,-2.17e7|=1.20e23 kg m2/s, which is bigger than S,M,P* gravity?
Average euler force (∑i Mi dΩ/dt x ri) due to changing ΩEarth,Ssb on the minimum .. maximum rates dΩ |-4.82e-18 .. 5.73e-18, -3.73e-18 .. 5.51e-18, -1.66e-15 .. 1.66e-15|=1.628e-14 rad/s2 is in ranges |-9.09e11 .. 9.07e11, -3.98e12 .. 3.97e12, -1.35e10 .. 9.86e9| = 1.13e9 .. 4.08e12 N, causing a torque |-8.67e18 .. 8.33e18, -5.68e18 .. 5.07e18, -1.311e21 .. 1.305e21|=2.4e17 .. 1.31e21 avg 6.12e20 kg m2/s .
Average centrifugal force (∑i Mi Ω x (Ω x ri) ) due to orbit, which should mostly be canceled by gravity (which is same at CoMEarth as gravity force therein, but not necessarily at surface, the difference on surface between locally differing gravity force/acceleration and centrifugal force/acceleration due to globally same orbital acceleration, is called tides) is |-8.80e16 .. 7.93e16, -8.02e16,7.07e16|=2.66e16 .. 6.24e19 avg 3.82e19 N, causing a torque |-1.25e24 .. 1.34e24, -5.88e24 .. 5.49e24, -7.86e21 .. 7.26e21| = 8.54e20 .. 6.03e24 avg 3.68e24 kg m2/s .
Sum of centrifugal and gravity forces (a tidal force) is in ranges | 4.03e13 .. 1.87e14, -4.72e12 .. 6.61e13, -3.01e14 .. -2.75e13 | = 6.53e13 .. 3.42e14 avg 2.15e14, causing a torque | -1.61e21 .. 6.67e20, -6.51e20 .. 1.64e21, -6.87e19 .. 7.55e19 | = 9.31e18 .. 2.01e21 avg 8.069e20 kg m2/s .
For comparision (on another day):
ΩEarth,Sun = |-1.18e-11,-8.1977e-8,1.8896e-7|=2.05977e-7, Ω x (Ω x r) = |6.16e21,-3.37e22,-1.463e22|=3.727e22
ΩEmb,Sun = |-4.24e-12,-8.19e-8,1.888982e-7|=2.059e-7, Ω x (Ω x r) = |6.23e21,-3.411e22,-1.48e22|=3.77e22,      3.77e22 / 3.727e22 = 1.011
ΩEarth-Emb = |-2.5e-7,-1.056e-6,2.526e-6|=2.749e-6, Ω x (Ω x r) = |-8.47e19,-1.716e20,-8.0e19|=2.07e20      3.727e22 / 2.07e20 = 180.05

The fictitious forces due to Earth spin can probably be ignored, since local gravity is directed toward CoM making no torque, centrifugal acceleration out from rotation axis making almost no torque (and if any, then it will be constant), Coriolis force is none, since mountains do not move relative to Earth, and Euler force due to dΩ can probably be ignored and dM is not expected for mountains...

The centrifugal force due to Earth spin, that really is measurable in Earth rotating frame. The centrifugal acceleration on longitude 5E is on range 0.0029 N/kg at angle 85 to vertical at latitude 85N to 0.0337 N/kg at angle 5 to vertical at latitude 5N (the model has 10-degree stepping with node centers in middles) which is at most arround 0.35% (0.0035x) of local gravity. Multiplied by continent masses and summed over globe, gives a net force |1.08e19,7.43e18,0| = 1.313e19 N (pointing 34.5E), and since the force vector is not exactly toward CoM, it makes a net constant torque of |-1.84e25,1.53e25,-1.67e7|=2.399e25 kg m2/s (pointing 140.29E), which is surprisingly large...? The Z component of torque due to centrifugal force is very probably caused by rounding error in FPU. (it would slow down Earth rotation on a very small rate of 5.65e-17 second per day during a century, but rather it is the sum of last bit lost in FPU number rounding, being 1e-18 of the other components...). But the horizontal (eastward) component is 10000x larger than required for a Chandler wobble, which is strange...
This rather means, that the asymetry of continents is probably balanced by an asymetry deep in the Earth core, that is not measured in the surface gravity model Earth2012...?


PDF version (or another PDF version) of this article is available too, but the wide charts are of little use in PDF viewer (may require 1600% zoom...).
First version omits some charts in PDF print due to a bug in Firefox 19, the second version has yellow background due to a plugin installed in another version of Firefox...
Some more usable PDF output is planned...
The ZIP version of this HTML page is available too...



     21-26.4.2015, 3.6. P.A.Semi