(see later posts here)

[mathjax]My previous posts on modeling the Southern Oscillation Index as a periodically modulated wave equation — in particular via the Mathieu equation — are listed below:

The first post introduced the Mathieu equation and established a premise for mathematically modeling the historical SOI time-series of ENSO, the Southern Oscillation part of the El Nino/Southern Oscillation phenomenon. The second post was an initial evaluation of a multivariate fit, evaluated by exploring the parameter space. The third post was a bit of a breakthrough, which focused on a specific periodic process — the Chandler Wobble (CW) — which appeared to have a strong causal connection to the underlying SOI model.

This short post effectively substantiates the Chandler Wobble connection and provides nearly as strong support that other tidal beat periodicities force the modulation as well.

In the previous post, I used a piecewise aggregation of the Mathieu equation periodic modulation to fit the SOI time-series. The idea was that although the modulation was periodic (shown below as the* a – 2·q·cos* term), it could not maintain phase or a stable frequency over the course of 130 years worth of data.

$$ frac{d^2Phi}{dt^2} = – [a-2qcos(2t)]Phi $$

The Chandler Wobble is notorious for having this same variability, both in phase and frequency of the measured CW time series.

So the approach was to divide the SOI time series into 8 intervals, and fit each of these sections of data independently to a Mathieu formulation. As shown in **Figure 1**, the individual intervals (averaging roughly 17 years each in duration) stitch together fairly effectively.

When the fitted parameters for the cosine period, its phase, and the **a** and **q** parameters are collected and reconstructed in terms of an aggregated set of intervals, the periodically modulated waveform appears as the **blue curve** in** Figure 2**.

This is quite remarkable in terms of how well the reconstructed intervals match the pole radial excursions shown as the **red curve** — using numbers pulled from the XDOT and YDOT columns of the *JPL Kalman Earth Orientation Series: POLE99* data set — found at the FTP site below:

The variability in the Chandler Wobble data has been well known over the years. The CW profile has a period of over 6.5 years after 1925 and 6 years or below before 1925, with the transition region around 1925 undergoing a strong phase shift. Note that although this could also be a combination of two frequencies beating against each other across the entire interval (see this Eureqa fit that I generated), we use the variable CW model for this evaluation.

## Discussion

As Gross from JPL [1] first surmised, a causal connection between the dynamics of the deep ocean and the wobbly rotation of the Earth likely does exist. Others have further expanded on the relation between the Chandler Wobble and the dynamics of the Pacific ocean [3], linking it to tsunamis [2] and ENSO [3].

This set of posts continues along this path and is perhaps one of the first mathematical substantiations of the connection between the periodicity of the Chandler Wobble and what appears to be a chaotic SOI characterizing the ENSO. In fact, the chaotic nature of ENSO is likely no more chaotic than the variably periodic nature of the earth’s wobble, only obscured by the non-linear nature of the Mathieu formulation. This is very intriguing, as it holds great promise in being able to predict the dynamics of ENSO with greater certainty.

Consider that besides the Chandler Wobble, tidal periods of 18.6 years, 8.85 years, 6 years, and 16.9 years also exist, and if these get applied to as independent Mathieu periodicities, the fit improves markedly as shown in **Figure 3**. Importantly, these periods are maintained across the entire 1880-2013 year interval, as these lunar-derived periods do not vary over time, unlike the Chandler Wobble pole tide.

The model can only get better as we continue to iron out the wrinkles. As the SOI pressure dipole is not even considered the strongest teleconnection index [4], it makes sense to continue to evaluate other data sources. In fact, the same CW modulation parameters fit the Nino 3.4 Sea Surface Temperature (SST) time series nearly as well, see **Figure 4** below.

One of the next steps is to work out the Mathieu equation with a continuous varying periodic modulation, replacing the relatively coarse piecewise fit applied in this model. This should improve the model fit considerably, as is typically the case whenever a piecewise representation is replaced with a continuous model. The nonlinear mathematics of the Mathieu equation is as nasty as it gets, but the payback in doing this with more precision holds immense promise. With an El Nino in the offing for the coming months, the public’s interest in predicting the dynamics of ENSO is nearing a peak.

## References

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

[2] E. Sassorova and B. Levin, “Spatial, and Temporal Periodicity in the Pacific Tsunami Occurrence,” in *Submarine Landslides and Tsunamis*, Springer, 2003, pp. 43–50.

[3] D. Sonechkin and R. Brojewski, “ENSO: a quasiperiodic forced dynamical system,” International workshop on the low-frequency modulation of ENSO, Touluse, pp. 23–25, 2003.

[4] J. Kawale, S. Liess, A. Kumar, M. Steinbach, A. R. Ganguly, N. F. Samatova, F. H. Semazzi, P. K. Snyder, and V. Kumar, “Data Guided Discovery of Dynamic Climate Dipoles.,” presented at the CIDU, 2011, pp. 30–44. (link)

This is a more fine-grained fit, where I use 10-year intervals to match up the basis Mathieu functions. The periodic forcing perturbation is still aligned to the Chandler Wobble as seen in the lower graph.

I am becoming convinced that the Mathieu excitation is likely 80% of the ENSO amplitude.

If the Mathieu wavelets were toolbox-ready, a wavelet analysis would be interesting.

http://en.wikipedia.org/wiki/Mathieu_wavelet

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This paper is an excellent mathematical treatment of sloshing dynamics:

Frandsen, Jannette B. “Sloshing motions in excited tanks.” Journal of Computational Physics 196.1 (2004): 53-87.

Interesting that the vertical sinusoidal perturbation captures the Mathieu flavor while the horizontal sinuoidal perturbation generates a forcing to the equation. If only the vertical exists, the simple Mathieu equation drops out. If the horizontal is added, then the full equation-solving treatment needs to be applied, with additional beat frequencies the likely result.

It is possible that tidal forces are creating the vertical excitation while some variation of the Chandler Wobble in angular inertia is providing the horizontal.

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After reading the Frandsen paper and getting some extra motivation from the fact that

velocityis the measure being mapped to MathieuC and MathieuS and not MathieuCPrime and MathieuSPrime, I tried another fit.I did have it right the first time.

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