[mathjax]Thanks to KO, I have started subscribing to Robert Grumbine’s blog. The topic currently is on what provides the pacemaker for the Chandler wobble.

My own opinion, which I hinted at before (and Keith Pickering also pointed out), is the same general mechanism as for the model of the QBO — but twice as fast. This idea essentially relates that the modulation of the yearly solar orbit (365.242 days) with the draconic (or nodal) lunar month of 27.21222 days sets up the perfect cyclic forcing for the 433 +/- 1.1 days Chandler wobble period.

Think in terms of the maximum declination of (1) the moon with respect to the equator along with (2) the maximum declination of the sun with respect to the equator. For (1) this happens once every ½ of the 27.2122 day nodal cycle or 13.60611 days and for (2) this happens twice a year (once for the southern hemisphere summer and once for the northern hemisphere summer).

Calculating this out, the closest aliased value is $$2pi (365.242/13.606) – 52pi$$ = 5.303 rads/year, equivalent to a period of 432.77 days. That is close to the generally accepted value of 433 days for the wobble [1]. One can also produce this value graphically by sampling a sinusoid of period 13.60611 days every half a year (see below). That’s enough to reveal the solar-lunar declination synchronization pretty clearly. In 120 years, I count a little over 101 complete cycles, which is close to the 433 day period.

The Earth has a large inertia compared to the moon, so it is essentially picking up differential changes in gravitational mass forcing, which is enough to get the wobble in motion.

So that’s my simple explanation for what drives the Chandler wobble, yet this differs from Grumbine’s idea, and from many other theories, many of which refer to it as a free nutation stemming from a resonance [2]. I am closer to Grumbine, who thinks it is planetary-solar while I think it is luni-solar. Without getting into the plausibility of the gravitational dynamics, it’s just too simple and parsimonious to pass up without posting a comment. It is also in line with my general thinking that very few natural processes follow a natural resonance, and that external forcing should always be the initial hypothesis. That works for the QBO in particular; in that case, the primary forcing period is the full Draconic cycle, likely due to the stronger asymmetry in lithosphere response for the two hemispheres.

And besides the QBO, this potential mechanism likely holds more clues to the behavior behind the ENSO model. As I commented at Grumbine’s blog:

@whut said…The Chandler Wobble, QBO, LOD variation, ENSO, Angular Atmospheric Momentum, and oceanic and atmospheric tides all have varying degrees of connection, likely ultimately tied to luni-solar origins.

Keep plugging away at what you are doing because there are likely common origins to much of the behavior. If you can figure it all out, it will be useful in establishing the natural variability in climate, and thereafter the GCMs can then use this information in their models.

## Refs

[1] 435 days also happens to be a commonly cited number (see here), but I get the smaller 433 days when I look at the data myself . And Nastula and Gross recently came up with a value of 430.9 days from newly available satellite measurements.

[2] “On the maintenance of the Chandler wobble” Alejandro Jenkins http://arxiv.org/abs/1506.02810

This is what happens with a yearly sampling

This picks up the Chandler wobble envelope of ~6.45 years that the polar angular momentum varies over time.

This removes the yearly rotating reference frame.

Still unexplained are phase reversals around 1930

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Place a flipped version of the upper curve on the lower curve and you get this:

Both of the blue curves track the observational Chandler wobble rdot (in red) after 1930. Well known that something happens before 1930 that essentially phase inverts the signal.

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Of possible interest: The influence of the lunar nodal cycle on Arctic climate, Harald Yndestad

“

The Arctic Ocean is a substantial energy sink for the northern hemisphere. Fluctuations in its energy budget will have a major influence on the Arctic climate. The paper presents an analysis of the time-series for the polar position, the extent of Arctic ice, sea level at Hammerfest, Kola section sea temperature, Røst winter air temperature, and the NAO winter index as a way to identify a source of dominant cycles. The investigation uses wavelet transformation to identify the period and the phase in these Arctic time-series. System dynamics are identified by studying the phase relationship between the dominant cycles in all time-series. A harmonic spectrum from the 18.6-year lunar nodal cycle in the Arctic time-series has been identified. The cycles in this harmonic spectrum have a stationary period, but not stationary amplitude and phase. A sub-harmonic cycle of about 74 years may introduce a phase reversal of the 18.6-year cycle. The signal-to-noise ratio between the lunar nodal spectrum and other sources changes from 1.6 to 3.2. A lunar nodal cycle in all time-series indicates that there is a forced Arctic oscillating system controlled by the pull of gravity from the moon, a system that influences long-term fluctuations in the extent of Arctic ice. The phase relation between the identified cycles indicates a possible chain of events from lunar nodal gravity cycles, to long-term tides, polar motions, Arctic ice extent, the NAO winter index, weather, and climate.”A mathematical explanation of Yndestad’s conjecture regarding the sub-harmonics inducing a phase reversal can be found in Non-Linear Dynamics in Geosciences

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Thanks KO.

oh brother, that NAO data is very noisy. I am sure they tried to get rid of the yearly signal but it still is there, along with some frequencies with periods from 2.3 to 2.7 years.

Interesting take on sub-harmonics. Sub-harmonics of a fundamental frequency are definitely meta-stable with respect to phase reversals because nothing exists to show a preference for an initial up or down cycle.

As an example, consider a fundamental sine wave of period of 1 year. This has equal weighting above and below zero, with an average of zero. But then consider a sub-harmonic of a 2 year period. If this started in the up direction, after 1 year the average would be much greater than zero. That initial stimulus has to come from somewhere else, because the fundamental can’t show a preference as the average is the same (zero) from one 2 year period to the next.

By contrast, all conventional (i.e super) harmonics average to zero over the course of a single fundamental cycle. The preference on the initial slope direction is set by the shape of the composite Fourier series set.

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Place holder for comments I made in an Azimuth Project thread

The way to think about this is of a spinning top (the Earth) being tugged at by a force that is simultaneously rotating obliquely about that axis (the Moon). If the Moon was strictly rotating concentrically about the equator, no wobble would be observed. But since the Moon follows a path with varying levels of declination about the equator, one would naively imagine that this would induce a slight wobbling force to the Earth’s axis. That’s what they refer to as a nutation, i.e. a wobbling rotation. Yet this idea is largely dismissed in the literature, suggesting that the Chandler wobble period is a “free” rotation arising more from a resonance condition involving the density and other characteristics of the Earth.

I would take their word for it if the wobble period of 432.7 days calculated from a first-principle trig argument didn’t align so closely with the measured 433 days. Don’t you hate when that happens?

So is the wobble forced or is it free? Grumbine says the former but lays it on solar+planetary forcing, whereas I think the 433 day period points to the forcing as predominately solar+lunar. I posted a comment to Grumbine’s blog and will see how he responds.

Here is a histogram of citations of the Chandler wobble period from the year 2000 to now based on statistics from Google scholar. Search string “chandler wobble” “430 days”, etc.

The weighted average is 432.78 days.

If I found an annoying hum in some arbitrary electrical signal and wanted to rule out the AC power line voltage as the source, all I would have to do is show that the frequency of that hum was not 60 Hz. If the hum did happen to measure as 60 Hz, I still couldn’t prove it was due to the line, but some other method would be needed to verify that it wasn’t the line.

That’s what will need to be done to debunk the lunar-forced Chandler wobble model. And for that matter, the QBO model.

Very straightforward to do a simple fit using the Excel solver, assuming a yearly sinusoidal signal combined additively with a Chandler signal of varying period, amplitude and phase. For the figure below, the solver picked a frequency extremely close to the predicted aliased Draconic lunar signal (see the numbers in red).

The fit was based on a short training interval, and then extrapolated over the longer time series. I also included the aliased harmonics that are in the QBO signal, but those are at the 1% level of amplitude so do not seem to be significant.

Amazing that the agreement for this case is at the 5th decimal place in terms of precision! This fit says the Chandler wobble period is 432.74 days.

A spurious coincidence, or is the moon the pacemaker of the Chandler wobble, just as it is for the QBO?

The extent of the Chandler wobble is a change of ~9 meters in which the rotational axis intersects the Earth’s surface. Once this wobble is in motion, all it takes is a reinforcing pulse to keep it going. The kinetic energy in the wobble is built up over many years, and any dissipation via friction, etc must be minimal.

It’s not that puzzling that the declination cycle of the Moon with respect to the Earth’s axis was dismissed, as the period is only 27 days — but then if you consider a nonlinear mechanism involving an extra push from the 1/2-year seasonal solar declination cycle, the 433 day period emerges in the Moire pattern.

Doing a literature search on this possible mechanism, I found papers by N. Sidorenkov (an AGW denier) who tries to deduce the wobble period by putting together various lunar cycles (see here). But his number is not spot on, and not nearly as cogent an argument that I am putting forward.

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