The Chandler Wobble Challenge

In the last post I mentioned I was trying to simplify the ENSO model. Right now the forcing is a mix of angular momentum variations related to Chandler wobble and lunisolar tidal pull. This is more complex than I would like to see, as there are a mix of potentially confounding factors. So what happens if the Chandler wobble is directly tied to the draconic/nodal cycles in the lunar tide? There is empirical evidence for this even though it is not outright acknowledged in the consensus geophysics literature. What you will find are many references to the long period nodal cycle of 18.6 years (example), which is clearly a lunar effect. If that is indeed the case, then the behavior of ENSO is purely lunisolar, as the Chandler wobble behavior is subsumed. That simplification would be significant in further behavioral modeling.

The figure below is my fit to the Chandler wobble, seemingly matching the aliased lunar draconic cycle rather precisely, taken from a previous blog post:

cw

The consensus is that it is impossible for the moon to induce a nutation in the earth's rotation to match the Chandler wobble. Yet, the seasonally reinforced draconic pull leads to an aliasing that is precisely the same value as the Chandler wobble period over the span of many years. Is this just coincidence or is there something that the geophysicists are missing?

It's kind of hard to believe that this would be overlooked, and I have avoided discussing the correlation out of deference to the research literature. Yet the simplification to the ENSO model that a uniform lunisolar forcing would result in shouldn't be dismissed. To quote Clinton: "What if it is the moon, stupid?"

In the current research literature, the Chandler wobble is described as an impulse response with a characteristic frequency determined by the earth's ellipticity.

https://en.wikipedia.org/wiki/Chandler_wobble "The existence of Earth's free nutation was predicted by Isaac Newton in Corollaries 20 to 22 of Proposition 66, Book 1 of the Philosophiæ Naturalis Principia Mathematica, and by Leonhard Euler in 1765 as part of his studies of the dynamics of rotating bodies. Based on the known ellipticity of the Earth, Euler predicted that it would have a period of 305 days. Several astronomers searched for motions with this period, but none was found. Chandler's contribution was to look for motions at any possible period; once the Chandler wobble was observed, the difference between its period and the one predicted by Euler was explained by Simon Newcomb as being caused by the non-rigidity of the Earth. The full explanation for the period also involves the fluid nature of the Earth's core and oceans .. "

There is a factor known as the Q-value which describes the resonant "quality" of the impulse response, classically defined as the solution to a 2nd-order DiffEq. The higher the Q, the longer the oscillating response. The following figure shows the impulse and response for a fairly low Q-value. It's thought that the Chandler wobble Q-value is very high, as it doesn't seem to damp quickly.

Impulse response

In contrast, ocean tides are not described as a characteristic frequency but instead as a transfer function and a "steady-state" response due to the forcing frequency. The forcing frequency is in fact carried through from the input stimulus to the output response. In other words, the tidal frequency matches the rhythm of the lunar (and solar) orbital frequency. There may be a transient associated with the natural response but this eventually transitions into the steady-state through the ocean's damping filter as shown below:

steady state

This behavior is well known in engineering and science circles and explains why the recorded music you listen to is not a resonant squeal but an amplified (and phase-delayed) replica of the input bits.

So why does the Chandler wobble appear close to 433 days instead of the 305 days that Euler predicted? If there was a resonance close to 305 days, any forcing frequency would be amplified in proportion to how close it was to 305 (or larger in Newcomb's non-rigid earth model). Therefore, why can't the aliased draconic lunar forcing cycle of 432.76 days be responsible for the widely accepted Chandler wobble of 433 days?

This is the biannual geometry giving the driving conditions, which I used in the QBO model.

geometry

And this is the strength of the draconic lunar pull at a sample of two times a year, computed according to the formula cos(2π/(13.6061/365.242)*t), where 13.6061 days is the lunar draconic fortnight or half the lunar draconic month.

aliased

We can count ~127 cycles in 150 years, which places it between 432 and 433 days, which is the Chandler wobble period.

Yet again Wikipedia explains it this way:

"While it has to be maintained by changes in the mass distribution or angular momentum of the Earth's outer core, atmosphere, oceans, or crust (from earthquakes), for a long time the actual source was unclear, since no available motions seemed to be coherent with what was driving the wobble. One promising theory for the source of the wobble was proposed in 2001 by Richard Gross at the Jet Propulsion Laboratory managed by the California Institute of Technology. He used angular momentum models of the atmosphere and the oceans in computer simulations to show that from 1985 to 1996, the Chandler wobble was excited by a combination of atmospheric and oceanic processes, with the dominant excitation mechanism being ocean‐bottom pressure fluctuations. Gross found that two-thirds of the "wobble" was caused by fluctuating pressure on the seabed, which, in turn, is caused by changes in the circulation of the oceans caused by variations in temperature, salinity, and wind. The remaining third is due to atmospheric fluctuations."

Like ENSO and QBO, there is actually no truly accepted model for the Chandler wobble behavior. The one I give here appears just as valid as any of the others. One can't definitely discount it because the lunar draconic period precisely matches the CW period. If it did't match then the hypothesis could be roundly rejected.

And the same goes for the QBO and ENSO models described on these pages. The aliased lunisolar models match the data nicely in each of those cases as well and so can't easily be rejected. That's why I have been hammering at these models for so long, as a unified theory of lunisolar geophysical forcing is so tantalizingly close -- one for the atmosphere (QBO), one for the ocean (ENSO), and one for the earth itself (Chandler wobble). These three will then unify with the generally accepted theory for ocean tides.

With that unification in mind, here is the math on the Chandler wobble. We start with the seasonally-modulated draconic lunar forcing. This has an envelope of a full-wave rectified signal as the moon and sun will show the greatest gravitational pull on the poles during the full northern and southern nodal excursions (i.e. the two solstices). This creates a full period of a 1/2 year.

Full wave

The effective lunisolar pull is the multiplication of that envelope with the complete cycle draconic month of 2π/ω0 =27.2122 days. Because the full-wave rectified signal will create a large number of harmonics, the convolution in the frequency domain of the draconic period with the biannually modulated signal generates spikes at intervals of :

2\omega_0,  2\omega_0-4\pi,  2\omega_0-8\pi, ... 2\omega_0-52\pi


According to the Fourier series expansion in the figure above, the intensity of the terms will decrease left to right as 1/n^2, that is with decreasing frequency. The last term shown correlates to the Chandler wobble period of 1.185 years = 432.77 days.

One would think this decrease in intensity is quite rapid, but because of the resonance condition of the Chandler wobble nutation, a compensating amplification occurs. Here is the frequency response curve of a 2nd-order resonant DiffEq, written in terms of an equivalent electrical RLC circuit.

q factor

So, if we choose values for RLC to give a resonance close to 433 days and with a high enough Q-value (>100), then the diminishing amplitude of the Fourier series is amplified by the peak of the nutation response. Note that it doesn't have to match exactly to the peak, but somewhere within the halfwidth, where Q = ω/Δω

response

So we see that the original fortnightly period of 13.606 days is retained, but what also emerges is the 13th harmonic of that signal located right at the Chandler wobble period.

That's how a resonance works in the presence of a driving signal. It's not the characteristic frequency that emerges, but the forcing harmonic closest to the resonance frequency. And that's how we get the value of 432.77 days for the Chandler wobble. It may not be entirely intuitive but that's the way that the math of the steady-state dynamics works out.

Alas, you won't find this explanation anywhere in the research literature, even though the value of the Chandler wobble has been known since 1891 ! Like the recent finding that the moon is proving a significant gravitational forcing in triggering earthquakes, the same could be asserted for the moon's direct influence on the Chandler wobble. Perhaps, it has taken this long because the data has become more abundant and more precise and it becomes harder to argue against the parsimonious correlation with a rather obvious and plausible physical explanation. For the Chandler wobble, nothing that complicated here at all, just a limited understanding on how seasonal physical aliasing can come about, and the admission that the forcing period carries through a resonance.

4 thoughts on “The Chandler Wobble Challenge

  1. If I understand correctly, with your ENSO model you are talking about the thermocline sloshing being one of the mechanisms of temperature change at the ocean surface.

    I'm now wondering how much is the ocean fluid mass relocating with that same tidal pull, being deep ocean mass shifting of water from east to west and back. Would that mass movement be enough to either excite or stimulate the Chandler wobble?

    How close is that lunar tidal mass movement of water to any of the specific harmonics in question?

    • Peter, I have strong evidence that ENSO dynamics is completely tidally driven. The Chandler wobble precisely aligns with the lunar draconic tide aliased with the seasonal forcing.

      • OK, thanks. I understand that to mean that the lunar gravitational effects directly on the earth are the major cause, and any centre of mass changes in the earth as a result of the oceans tidal movements would effectively be a very low order factor.

  2. Pingback: Tidal Model of ENSO | context/Earth

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