(see later posts here)

[mathjax]Seth Carlo Chandler Jr was an actuary who studied his namesake wobble for thirty years.

ENSO and the Southern Oscillation Index has confounded everyone with its unpredictability.

Could a connection exist between the ENSO and the Chandler Wobble ? [1]

Based on what I have been analyzing with respect to the ENSO data, I am leaning in that direction.  In the first post on the Southern Oscillation Index Model (SOIM), my initial analysis lead to a fundamental Mathieu frequency T of 6.3 years, a value of a = 2.83  and q = 2.72 :

• The Southern Oscillation Index Model

I followed that up with additional checks and analogies to other physical phenomena :

• SOIM and the Paul Trap

That culminated with a trial fit of the SOI with a set of Mathieu parameters. Yet — even though the fit was decent — I was not satisfied with the result as it tended to overfit with respect to the adjustable parameters.  The ideal situation would limit the number of fundamental frequency terms.

Three observations lead me to a much simplified representation.

• The main Mathieu frequency of 6.3 years seemed to vary over the historical record.
• The pressure index of the SOI is essentially a differential measure, and so the derivative of the Mathieu function should be fit to the pressure, e.g. use MathieuCPrime and not MathieuC .
• The connection between the original fit of 6.3 years and the Chandler Wobble beat frequency of 6.39 years (= 1/(1-365.25/433)), and the fact that this measure has been known to vary over the past 100+ years.

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓

The excuse for the overfitting I believe was due to the slight variation of the characteristic Mathieu frequencies over the 130+ year time span of the SOI. When fit over smaller intervals, the convergence turned out much better with far fewer frequency components. However, unless a pattern is found for such a variation in frequencies, it makes it difficult to use for projections. So if we can pinpoint the origin of the fundamental frequency and its variability, we can start  from there.

The first step I took was to use a brute force approach to deduce the characteristic Mathieu frequency.   We start with the Mathieu equation from the first post:

$$ frac{d^2Phi}{deta^2}+[a-2qcos(2eta)]Phi=0 $$

where

$$ eta = omega t $$

Then if we plot the ratio of the SOI index measurement and its second derivative, we can conceivably extract the varying periodic factor from the graphed profile:

$$ -frac{frac{d^2Phi}{deta^2}}{Phi} = a-2qcos(2eta) $$

If we make the assumption that dΦ/dt is the SOI index, that means we can use the first derivative of the SOI and its integral to find any periodic values. The tricky part is that the integral wanders away from zero, so I filtered out the low frequency modulation, leaving behind only the subdecadal fluctuation. Although a bit messy with singularities, since the denominator passes through zero as the index wanders through inflection points, the fundamental frequency is clearly discernible, see Figure 1:

The important point to note is that a sinusoidal component is clearly present, suggesting that this process does indeed reflect a nonlinear Mathieu equation, and that the main frequency definitely moves around, varying from double periods of 12.7 years paired with weaker indications of a 6.35 year period.

Now if we model the SOI data via piece-wise intervals over the range using only the low-frequency terms (44.1 years and 146.5 years) and the slightly varying 6.35 year fundamental frequency, we get Figure 2 :

Figure 3 is the differentiated form, which maps directly to the SOI. Though not close to perfect, enough correlation exists to strong SOI extremes (see 1983) to indicate that this is on the right track.

The agreement is good enough that it becomes tempting to add in a few extra fixed periods arising from fundamental tidal beat frequencies of 6 years, 8.85 years, 18.6, and 16.9 years, corresponding to various combinations of the draconic, sidereal, and anomalistic lunar months [2], see Figure 4.  In particular, Royer [3] has detected  the 16.9 year period in air temperatures.

Adding these cycles as Mathieu components, we get Figure 5.

At this point it is useful to understand how I came to apply the derivative of the Mathieu function, which is really the breakthrough in the analysis. Consider that the parametric plot of the result and it’s first derivative is very useful as a means to characterize a possible solution, as I showed in the initial post. Yet, I didn’t look closely enough at the phase relation, as one see the following figures.  First consider  Figure 6 and Figure 7 which are the index and the phase plot of the actual data.

Fig 6:  SOI data

Fig 7 :  Phase plot of SOI

This is interesting in that the “binocular” effect in the phase plot of Figure 7  matches better the first and second derivative of the Mathieu function (Figure 8), rather than the main and its first derivative (Figure 9) shown below.

Compare the above against the plain Mathieu phase plot in Figure 9.  The difference is primarily in how the lissajous figure traces out in the vertical versus horizontal directions.

The appendix is a rundown of some other Mathieu-like formulations that I looked at.  The last one is important in that it shows the phase plot of a red noise model, created using an Ornstein-Uhlenbeck process.  Some climate scientists propose that the variability in the Pacific ocean indices is close to red noise [7].  This is plausible , as one model of an O-U process is of a random walker oscillating back-and-forth within a potential well.

Yet, the evidence points to the even more plausible explanation of a limited set of periodic or quasi-periodic factors interacting with a nonlinear wave equation (i.e. Mathieu equation) to produce the fluctuating pattern in the SOI profile.

Discussion

This brings us to the Chandler wobble connection. As Gross suggested [1], the wobble itself is a result of a quasi-periodic oceanic disturbance, and so as a premise we somehow attribute it as a strong factor in the ENSO SOI evolution.

“Using calculations based on applying numerical models of the ocean to data obtained on the Chandler wobble between 1985-95, Dr Gross has determined the principles causes of the wobble. He found the wobble is due to fluctuating pressure on the bottom of the ocean, caused by temperature and salinity changes, and wind-driven changes in the circulation of the oceans.

Like a top wobbling as it spins – only much much slower – the Chandler wobble takes 433 days or 1.2 years to complete one cycle. The amplitude of the wobble amounts to about 6.5 metres at the North Pole. But like a top, eventually the wobble should stop. This is what has puzzled scientists, who have calculated that the Chandler wobble would have stopped in just 68 years, unless some force was acting to keep it going.

Dr Gross calculated that two thirds of the Chandler wobble is caused by changes in pressure at the ocean bottom, and the rest by fluctuations in atmospheric pressure. However, Dr Gross calculated that atmospheric winds and ocean currents had only a minor effect on the wobble. ” [link]

The pole tide caused by the Chandler wobble is only a few millimeters in height, so is not the cause of the effect but a side-effect of the deeper water activity.

The variations in the Chandler wobble have long been known and are best understood by a combination of a naturally damped sharp resonance (with a Q factor over 100) together with a semi-regular forcing factor which restarts the Chandler wobble at intervals of around 20 years, see Figure 10.

Before becoming too speculative, it is important to remember the strong  fundamental correlations between changes in angular momentum of the earth’s rotation and with oceanic and atmospheric fluctuations. Figure 11 below from Chao [6] shows how the LOD relates to ENSO, wind variations, and ocean tidal variations.

These really are pieces of an interlocking jigsaw puzzle and it is important to keep that in mind.

Zotov [4] has speculated that that the forcing is associated with the periodic 18.6 year tidal forcing (see figure to the right). This was based on spectral analysis of the Chandler Wobble data over different intervals.

Gibert et al [5] have also isolated the phase and frequency shifts over different intervals using wavelet analysis.

These approximately 20 year intervals map to the intervals that I use in the SOI Mathieu fit. This essentially provides a rationale for resyncing the fundamental 6.35 year frequency in the model fit along those same intervals.

The reason that this was not discovered earlier (in comparison to the Chandler Wobble measurements) was that the nonlinear wave equation obscures the periodic elements underling the SOI, overlaying what appears to be a random or “chaotic” waveform.   In fact, this is not chaotic at all, but an anharmonic waveform that is characterized by using the appropriate mathematical tools.

Hmmm, what can I find if I work on it for 30 more years? The mind reels.

Appendix

The following are plots of solutions to Mathieu-like equations solved using Mathematica. This demonstrates the variety of aperiodic waveforms that these non-linear equations can create.  Periods chosen reflected common tidal beat frequencies (e.g. 6 years, 4.45 years, 9 years).

d^2f(t)/dt^2 =-(pi/3.7)^2*2.81*(1+2*(cos(2*pi*t/9)+cos(2*pi*t/4.5))/2)*f(t)

Amplitude plot

Phase plot

d^2f (t)/dt^2 =-(pi/3.57)^2*2.81*(1+2*(cos(2*pi*t/6)+cos(2*pi*t/4.5))/2)*f(t)

Amplitude plot

Phase plot

d^2f(t)/dt^2 =-(pi/2.8)^2*2.81*(1+2*cos(2*pi*t/3)*cos(2*pi*t/4.5))*f(t)

Amplitude plot

Phase plot

d^2f (t)/dt^2 = -(pi/2.75)^2*2.81*(1+2*cos(2*pi*t/3)*(cos(2*pi*t/60)+cos(2*pi*t/7.3))/2)*f(t)

Amplitude plot

Phase plot

d^2f (t)/dt^2 = -(pi/3.15)^2*2.81*(1+2*cos(2*pi*t/3)*(cos(2*pi*t/60)+cos(2*pi*t/17))/2)*f(t)

Amplitude plot

Phase plot

d^2f(t)/dt^2 =-(pi/2.70)^2*2.81*(1+2*cos(2*pi*t/3)
 *(0.1*cos(2*pi*t/16.9)+0.9*cos(2*pi*t/7.3)))*f(t)

Amplitude plot

Phase plot

d^2f (t)/dt^2 =-(pi/2.65)^2*2.81*(1+2*cos(2*pi*t/3)*(cos(2*pi*t/16.9)+cos(2*pi*t/7.3))/2)*f(t)

Amplitude plot

Phase plot

d^2f(t)/dt^2 +df(t)/dt =-(pi/2.70)^2*2.81*(1+2*cos(2*pi*t/3)
 *(0.1*cos(2*pi*t/16.9)+0.9*cos(2*pi*t/7.3)))*f(t) + 25*sin(2*pi/60*t)

Red Noise model stepped over 1600 months, ~ 133 years.

RandomFunction[OrnsteinUhlenbeckProcess[0,0.001, 0.1], {0,1600, 1}]

Red noise

Red noise phase

 

References

[1] R. S. Gross, “The excitation of the Chandler wobble,” Geophysical Research Letters, vol. 27, no. 15, pp. 2329–2332, 2000.

[2] W. H. Berger, J. Pätzold, and G. Wefer, “A case for climate cycles: Orbit, Sun and Moon,” in Climate development and history of the North Atlantic realm, Springer, 2002, pp. 101–123.

[3] T. C. Royer, “High‐latitude oceanic variability associated with the 18.6‐year nodal tide,” Journal of Geophysical Research: Oceans (1978–2012), vol. 98, no. C3, pp. 4639–4644, 1993.

[4] L. Zotov, “Analysis of Chandler wobble excitation, reconstructed from observations of the polar motion of the Earth.”

[5] D. Gibert, M. Holschneider, and J. Le Mouël, “Wavelet analysis of the Chandler wobble,” Journal of Geophysical Research: Solid Earth (1978–2012), vol. 103, no. B11, pp. 27069–27089, 1998.

[6] B. Fong Chao, “Excitation of Earth’s polar motion by atmospheric angular momentum variations, 1980–1990,” Geophysical research letters, vol. 20, no. 3, pp. 253–256, 1993.

[7] D. L. Rudnick and R. E. Davis, “Red noise and regime shifts,” Deep Sea Research Part I: Oceanographic Research Papers, vol. 50, no. 6, pp. 691–699, 2003.