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 subsurface
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 "Earth’s rockequivalent topography as shape model" (Fig.1).
The map has been divided in 10° steps and all excess mass (assuming 2700kg/m^{3})
above 6351 sphere has been intergrated and summed into 648 (36x18) masspoints.
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 jpegnoise 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 0.52km in height, but not a continentsized 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  Perlongitude average elevation over 6355km and leastsquares fit of 4axis 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 perlongitude sums.
The best fit of quadrants to these perlongitude sums, using leastsquares fit, was
determined to be shifted from ordinary coordinates by 21°E, placing +Y axis onto Andes
mountains at 69°W, +X axis onto Africa at 21°E, Y axis east of Himalaya at 111°E,
and finally the X axis falls into the deepest part of Pacific at 159°W.
The irregularity of the shape can be summarized in a simple way, that Andes are counterbalanced
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 vector sum on the spinaxis 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...
Highfrequency component of LoD 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 highfrequency LoD Δω_{3} 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 highfrequency signal, if the force is integrated over the whole globe...?
Frequency analysis of LoD changes Δω_{3}: 
Frequency analysis of vertical component of tides: 




Tab.1  frequency analysis of tides and LoD Δω_{3} 
For calculating tides I used the former geoid rockequivalent height model, which causes some nonexactness, which can be estimated on Fig.3c comparing my results with IERS subtracting tides...
The Tides in Fig.3a, that also match the IERScalculated 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 Zcomponent of torque due to tide (that should influence LoD Δω_{3}) 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) compared with LoD Δω_{3} 
Fig.3b  Earth tides (vertical component) compared with LoD Δω_{3}  in detail 
Fig.3c  Earth tides (vertical component) compared with LoD Δω_{3}  in detail The difference calculated by IERS is in blue, the difference calculated by me is in red... (See fullwidth detailed version) 
The halfyear oscilation in Δω_{3} is by 2740 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 halfyearly cycle in Δω_{3}, 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 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 postnewtonian) 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 1e9.
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 landmass, 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 Δω_{3} 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 speedup and slowdown (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 Δω_{3}.
Fig.4a  Geoid torque vector components: X_{r21E} (blue), Y_{r21E} (green), Z (red). 
Fig.4b  Geoid torque sum, X_{r21E} (blue),Y_{r21E} (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, X_{r21E} (blue),Y_{r21E} (green),Z (red) components, compared with LoD changes Δω_{3} (purple) 
Fig.5b  same as Fig.5a, in detail... 
Fig.5c  Geoid torque damped sum, negative of Y_{r21E} component (black), with lowfrequency subtracted by DoG_{730} (difference from 2year gaussian), negative... 
Fig.5d  same as in Fig.5c in detail (See detailed fullwidth version) 
Fig.5e  same as in Fig.5c in more detail. Note, that highfrequency component is misaligned... 
Fig.5f  Geoid torque damped sum, negative of Y_{r21E} component (black), with middlefrequency highlighted by Gauss108,54, compared with IERS EOP C04 Δω_{3} with tidal variation removed (purple) with low frequency removed by 2year gaussian... 
Fig.5g  same as Fig.5f, in fullwidth detail... 
Although the halfyearly frequency and phase seems to match, in more detailed inspection the amplitude does not match, and the highfrequency 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:  

 
Tab.2  frequency analysis of Torque dampedsum 
Fig.6a  Z component of damped torque Sum (black) and it's derivative (violet) compared with LoD Δω_{3} variation (red) 
Fig.6b  same as in Fig.6a, in detail... (See detailed fullwidth version...) 
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 JupiterEarth meet period...
Fig.7a  Geoid torque due to planets only, vector components X_{r21E} (blue), Y_{r21E} (green)
and Z (red) Does the torque work in right angle from the influence? Then the +Y_{r21E} (69°W) axis assymetry will be due to an Africa assymetry at 21°E... 
Fig.7b  Geoid torque damped sum, due to planets only, vector components X_{r21E} (blue), Y_{r21E} (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 fullwidth version) 
Fig.7d  Geoid torque damped sum, X_{r21E} component (blue), Y_{r21E} (green),
separated highfrequency component by DoG_{730} difference from 2year gaussian,
which is shown as a lowfrequency component... 
Fig.7e  Geoid torque damped sum.
Y_{r21E} lowfrequency component (green),
Y_{r21E} highfrequency component extracted using DoG_{730} (aqua),
it's derivative (purple),
and their complex phase at bottom... (See detailed fullwidth version...) 
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 crossproduct.
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 nonspherical.")
The angular momentum of EMB (EarthMoon 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 27day 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 22year signed Halle Sunspot cycles, due to EarthJupiterVenus 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, darkgreen: 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 years  Description 
199.41 days  (199.39199.42)  0.55 year   half Jupiter meet period 
583.51 days  (583.35583.68)  1.6 year   Venus meet period 
291.88 days  (291.84291.92)  0.8 year   half Venus meet period 
132.96 days  (132.95132.97)  0.36 year   ? 
209.03 days  (209.00209.05)  0.57 year   ? 
1452.3 days  (1451.31454.3)  3.98 year   half EarthVenus 8year meet cycle? 
194.60 days  (194.58194.61)  0.53 year   ? 
5729.9 days  (5714.35745.6)  15.69 year   half Saturn orbit period ?? 
439.19 days  (439.10439.29)  1.20 year   beat between 365.25 and 199.41 
97.31 days  (97.3197.32)  0.27 year   ? 
116.77 days  (116.77116.78)  0.32 year   ? (similar to Venus apparent Solar day?) 
4315.1 days  (4306.34324.0)  11.81 year   Jupiter orbit period 
137.17 days  (137.16137.18)  0.38 year   ? 
The main frequency here is 199.41 days, half the EarthJupiter meet period.
Beat frequency between this freqency and Earth year 365.25 days, is:
f_{Beat} = f_{1}  f_{2}
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 sinewave, the wave looks like this, and it will be used to predict Chandler wobble:
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 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 earthorbit sinewave), and imaginary part is its daily derivative scaled (amount of planetary perturbation combined with earthorbit cosinewave?)...
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).
Daily data in eopc04 and longterm data in eopc01 (together available for download are versions, that I used...).
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 lineartrend average
(15year average for shorter eopc04 and 30year 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 (thinred line at chart top), that there is no
constant period, and the calmaverage is somewhere arround 400day (365 .. 439 day) range,
but there exist a lot of slowdowns (longer periods  peaks above) and some speedups
(shorter periods, troughs below).
Actual average of instantaneous frequency from EOPC data, when using ONLY values in
range 350450 days (or 300500 days) are:
Data  total days  Period Range  Filtered days  Average Period 
eopc04  19033  350450  8811  403.194 
eopc04  19033  300500  14078  410.126 
eopc01  53046  350450  10767  400.73 
eopc01  53046  300500  21306  403.017 
eopc04  19033  350500  12211  422.374 
eopc04  19033  300450  10678  390.400 
eopc04  19033  350550  13773  433.508 
eopc04  19033  300600  16444  428.594 
eopc04  19033  365450  7867  408.619 
In the calmmode (for ex. during 1995.51997.3),
the average period of Chandler wobble is arround 400 days, near the EarthJupiter meet time period,
or arround 439 days, beat frequency between 199.41 and 365.25 day periods...
Probably this 400day 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 noiseonly and perturbations,
so filtering a calmmode range 350450 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 6.2year 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 Earthorbit (yearly) period. For 435.5 day period leads to 6.2year beat, while 434.2 day period leads to 6.3year beat.
Beat frequency 431.25 .. 433.94 can be also reached as a beat between modified Geodetic precession and 365.25day Year, using formula Omega_{geodetic} = π/2 * Rs_{Sun} / 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 yearlyregular 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?
Here I will combine Chandler wobble with AM(EMB)+YearSine (purple serie in charts), with Geoid torque due to planets only (lightblue 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...
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 365,400,439 day periods)  
Chandler complex amplitude (always positive)  
Chandler r21E Phase (see description of quadrants above...) 


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 yearorbit sine, and imaginary part from it's derivative  amount of planetary perturbation to earth orbit and yearorbit cosine...?) (Fig.9b) 
Geoid torque due to planets only, lowfrequency part, X_{r21E}component (Fig.7a) 
Geoid torque due to planets only, lowfrequency part, Y_{r21E}component (Fig.7a) 
Lowfrequency component of damped sum of torque due to planets only, Y_{r21E} component, extracted by 2year gaussian (Fig.7e) 
Highfrequency component of damped sum of torque due to planets only, Y_{r21E} component, extracted by DoG_{730} (Difference of 2year Gaussian) (Fig.7e) 
Complex phase of highfrequency component of damped sum, Y_{r21E}, real part is running damped sum, imaginary part is it's derivative... (Fig.7e) 
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 EarthJupiter 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 yearsine, 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 outofphase,
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 (19622014) with analysis.
(large version)
Note the phase series at bottom. Green is Chandler wobble real data, purple is AM(EMB)+YearSine and aqua (light bluegreen) is Torque YdY complex phase. 
Fig.11b  Chandler wobble eopc01 r21E (18402008) 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, DoG_{730}) 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 speedsup to get to the nearest forcing phase backward (as in 2014,2015). 
Fig.12b  Same as Fig.12a, in fullwidth detail... 
Fig.12c  Chandler wobble comparing only with AM(EMB), marking 400day period slope in phasediagram at low chart, and showing various possible predictions until year 2020... 
Fig.12d  Chandler wobble eopc01 r21E (18402008) with phase, combined with AM(EMB)+YearSine and Geoid Torque YdY (damped sum, DoG_{730}) with their phases. It is possible to see, that the relation of phasesynchronization 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... 
Fig.13a  EOPC04  comparision of instantaneous period (frequency) between Chandler wobble (red) and AM(EMB)+YearSine (purple) ... 
Fig.13b  same as Fig.13a, fullwidth detail... 
Fig.13c  EOPC01  comparision of instantaneous period (frequency) fluctuations (top part) between Chandler wobble
and Torque damped sum YdY complex serie... (See detail fullwidth 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 spreadsheet 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...)
Highfrequency 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 middlefrequency 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 middlefrequency 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 19621963 may be due to inexact measurement, or other possible
influence?)
The Chandler wobble polar shift does not have the high frequency 13day component,
not even a little. It is very probably caused either by planetonly torque
to Earth (geoid) shape  but why planets only?, or by spinorbit 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 EarthJupiter 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 outofsync with
its forcing, it's amplitude is much smaller...
The wobble seems to align with EarthJupiter heliocentric syzigy on +Y (rotated,natural) axis,
and the frequency (period) anomalies are well insync 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 yearsine 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 gravityanomally
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 1020% ... If the forcing
was through atmosphere and not rocks and water, it would use another proportions...
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 lowfrequency of tidal variations but not the highfrequency, 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 counterbalancing mass in the core...
All calculations are done in SI units, if possible, using m as space distance, s as time unit and N as force, m/s^{2} 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 35°E 5°S weight 9.28e19 kg, 35°E 5°N 9.264e19 kg, 25°E 5°S 9.196e19 kg and least massive is 125°E 85°N, 9.11e17 kg.
Orbital angular momentum of EMB (E+M) rel. to Sun (during years 20052012)
is 2.7e40 kg m^{2}/s, and range (maxmin) 6.8e35 in same units, range (maxmin) of EMB rel. Ssb is 5e38 in same units.
Earth rel. Sun is 2.659e40 .. 2.662e40 and range (maxmin) 2.6e37 kg m^{2}/s, range (maxmin) of Earth rel. Ssb is 5.67e38 .
Average Orbital acceleration of Earth is 0.005931 m/s^{2}, yearly amplitude (fft) 9.512e5,
29.526 d amplitude 1.537e5, 182.426 d amp. 2.5e6, 14.252 d amp. 1.29e6, ... 1.6 year amplitude 5.586e7 ...
Average Ω_{EMB,Sun} = 1.991e7 ( 1.926e7 .. 2.059e7 ) ...
Spin angular momentum of Earth planet with ω = 7.2921150e5 rad/s, WGS84 ellipsoid 6378137 m assumed as sphere, mass 5.97219e24kg, yields momentum I = 2mr^{2}/5 = 9.718e37 kg m^{2} (in wiki/Polar_motion there is polar moment of inertia = 8.04e37 kg m^{2}, equatorial 8.014e37 ...) and L = I . ω = 7.087e33 kg m^{2}/s .
Chandler wobble  cca ± 0.2 mas = 5.55e5° = 9.696e7 rad (longterm 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 6.0923e6 rad  during 439 days = 3.7929e7s,
which is Δ L = 4.317e28 kg m^{2}/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 Y_{r21E} axis is +3.3097e16 rad/s ,
displacing from 0.035 arcsec at 1846 to +0.314 arcsec at 2008 (by +1.692e6 rad during 5.1122e9 s)
(steady displacement in +Y direction)
.
Long term drift on X_{r21E} axis was 1.45e17 rad/s during 19th century
and +3.713e17 rad/s during 20th century
displacing from 0.096 arcsec in 1846 to 0.1011 arcsec in 1900 (by 2.47e8 rad during 1.7e9 s)
and to 0.075 arcsec in 2008 (by +1.265e7 rad during 3.4e9 s)
(almost constant, 1020x smaller drift than in Y_{r21E}...)
.
This means average torque +2.35e18 N m on Y_{r21E} axis,
1.03e17 N m on X_{r21E} axis during 19th century and
+2.64e17 N m on X_{r21E} axis during 20th century ...
Average LoD Δω_{3} (from IERS EOP C04 values including tidal variation)
is cca 1.465 picorad/s = 1.5e12 rad/s (min 3.6756, max 0.9062),
which should be a base of Δ L = 7.1885e26 kg m^{2}/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 m^{2} 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 ...
X_{r21E} 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
Y_{r21E} 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 √(x_{r21E}^{2} + y_{r21E}^{2}) =
0.648e+21 ...
Calculated S,M,P* torque running undamped sum ( ∑ τ . t )
in Y_{r21E} direction sum 2.88e30 ,
average torque 5.63e20 N m ?
in X_{r21E} direction sum 1.62e30 ,
average torque 3.17e20 N m.
Average planetonly 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, ...
X_{r21E} 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,
...
Y_{r21E} 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,
...
Z_{r21E} 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?) 1.44287e6 N/kg, with fft amplitudes:
27.517 d amp=8.641e7, 13.66 d amp=3.18e7, 182.645 d amp=2.29e7, ... yearly amp=1.832e7 ...
If 3.18e7 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 = 2mr^{2}/3 = 1.3967e32 kg m^{2}, and L = I ω = 1.0185e28 kg m^{2}/s.
To produce Δ L of 1.2857e25 kg m^{2}, 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 Δω_{3} may not be caused by atmosphere expansion (possibly only by it's weight change but even that
is not enough), but how is weightchange 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 Δω_{3} 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 20000120 at 12:00 UTC, axis +X 0°E,+Y 90°W,+Z 90°N:
Reactive centrifugal acceleration (opposite to orbit acceleration) of Earth
is 0.57e2,0.16e3,0.21e2 = 0.6087e2 m/s^{2},
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 0.56867e2,0.1593e3,0.2093e2 = 0.60617e2 m/s^{2},
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 2.016e10,9.949e11,5.739e11 = 2.32e10 m/s^{2} ,
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 1e5 m/s^{2}.
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 1e10m/s^{2}...
Onto 10x10° model node centered at 35°E,5°N 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 0.5712e2,0.159e3,0.21e2 = 0.60888e2 m/s,
making onto that node force 5.29e17,1.476e16,1.948e17  = 5.641069e17 N.
Forces added are 0.915e14,0.64e14,0.68e14 = 1.308e14 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 24hour sums, here in appendix in FFT amplitudes it was divided...
Average coriolis force (∑_{i} 2M_{i} Ω x v_{i}) 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 nonconstant Ω_{EMB,Sun} = 1.991e7 ...
(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 m^{2}/s,
which is bigger than S,M,P* gravity?
Average euler force (∑_{i} M_{i} dΩ/dt x r_{i}) due to changing Ω_{Earth,Ssb}
on the minimum .. maximum rates dΩ 4.82e18 .. 5.73e18, 3.73e18 .. 5.51e18, 1.66e15 .. 1.66e15=1.628e14 rad/s^{2}
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 m^{2}/s .
Average centrifugal force (∑_{i} M_{i} Ω x (Ω x r_{i}) ) due to orbit,
which should mostly be canceled by gravity
(which is same at CoM_{Earth} 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 m^{2}/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 m^{2}/s .
For comparision (on another day):
Ω_{Earth,Sun} = 1.18e11,8.1977e8,1.8896e7=2.05977e7,
Ω x (Ω x r) = 6.16e21,3.37e22,1.463e22=3.727e22
Ω_{Emb,Sun} = 4.24e12,8.19e8,1.888982e7=2.059e7,
Ω x (Ω x r) = 6.23e21,3.411e22,1.48e22=3.77e22,
3.77e22 / 3.727e22 = 1.011
Ω_{EarthEmb} = 2.5e7,1.056e6,2.526e6=2.749e6,
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 5°E is on range
0.0029 N/kg at angle 85° to vertical at latitude 85°N to
0.0337 N/kg at angle 5° to vertical at latitude 5°N (the model has 10degree 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 34.5°E),
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 m^{2}/s
(pointing 140.29°E),
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 5.65e17 second per day during a century,
but rather it is the sum of last bit lost in FPU number rounding, being 1e18 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...?
2126.4.2015, 3.6. P.A.Semi