The Hidden Harmony of ENSO

With this analysis, I wanted to demonstrate the underlying order of the most concise SOI Model. This model characterizes the salient fit parameters:

  1. Two slightly offset forcing sinusoids which match the average QBO forcing cycle
  2. A forcing sinusoid that maps to the frequency of the Chandler wobble beat
  3. A Mathieu modulation perturbing the 2nd-order DiffEq with a periodicity of about 8 years

This set of four parameters was used to model both modern day records corresponding to the atmospheric pressure data describing the Southern Oscillation Index, as well as to proxy records of historical coral data. The parameters seem to match closely over widely separated time intervals (see Figure 5 in the latter link).

Figure 1 is the modern-day SOI record, suitably filtered to show the multi-year excursions.

Fig 1: SOI Data. The waveform is erratic, to say the least.

It is amazing that this erratic a waveform can be modeled by a limited set of parameters that actually make some physical sense, but that is nature for you and the idea behind “sloppy modeling” — models that use just a few parameters to accurately describe a behavior. The simplified model strongly suggests that there is a hidden harmony acting to drive ENSO.

To get to the results, Figure 2 is the Mathematica model of the second-order Mathieu model describing hydrodynamic sloshing of the equatorial Pacific waters:

Fig 2: Simple model of SOI. The yellow regions show the excursions from a perfect alignment to the data.  A couple of extra factors corresponding to multi-decadal variations (ramp and dilate) as well as an 8.85 year residual possibly related to tides were added to slightly improve the fit. The forcings were also refactored as residuals to address the possibility of a linear effect. None of these latter factors impact the aligned locations of the El Nino and La Nina peaks, which are primarily driven by the DiffEq solution.

So even though the modeled waveform is erratic, the forcing factors are anything but ! (see Figure 3 below).  The Chandler wobble waveform (cw) appears weak but it is closer to the characteristic frequency for sloshing of ~4.3 years so has a greater resonant impact than the ~2.3 year average QBO period.

Fig 3: Forcing factors — QBO and Mathieu — and Mathieu modulation

This fit was accomplished primarily manually via adjusting the sinusoidal coefficients to achieve the highest correlation coefficient.

As substantiation, the Eureqa machine learning tool (differential solution guide here) finds the same general set of parameters for a higher complexity solution as shown in Figure 4.

Fig 4:  Eureqa will find a DiffEq fit with forcing sinusoids close to the QBO and Chandler wobble frequency along with a Mathieu modulation.

Figure 5 is one selected Eureqa fit:

Fig 5: A Eureqa-style DiffEq fit  to the data.  This fit uses the second-derivative data, D(soi,Time,2), implicitly to estimate the other factors by minimizing the error between the LHS and RHS of a differential equation. The Mathieu modulation is detected as well!

Figure 6 is the QBO factor that Eureqa discovers:

Fig. 6: The Eureqa detected QBO modulation featuring 4 sinusoids.  For the concise fit we use a less-complex combination of two sinusoids.

The less complex Eureqa solutions feature the solutions shown in Figure 3.

What is interesting is that the periods for QBO variability, when aliased, fold into the synodic lunar cycle in which the actual time between lunations may range from about 29.18 to about 29.93 days. In Figure 3, one QBO frequency corresponds to 29.3 days and one to about 29.4 days.  The average synodic cycle is 29.53 days, and 13 sideral months alias the same as 12 synodic months.  See this post on possible impacts that the lunar force may have on QBO.

It may be true as well that there is some essential harmony behind the Chandler wobble, QBO, and lunar/tidal effects which are manifestations of the same geophysical process relating to periodic changes in angular momentum.


— see index to other SOI posts

3 thoughts on “The Hidden Harmony of ENSO

  1. I’m having a hard time replicating the filter you use on the NCAR SOI data. I’ve got the data and I’ve tried both running means and LOESS smooth, which can both give “pretty close” but not exact replication to what you’ve posted.

    I’m sure you’ve mentioned your filter in some previous post somewhere, but if you can point me to where it would be helpful.


  2. Keith,
    Recently I have been combining the Darwin and Tahiti data directly to get SOI.

    In Mathematica, I will do this

    tahiti=Import["tahiti.txt", "table"];
    darwin=Import["darwin.txt", "table"];
    R = MeanFilter[darwin[[All,2]],7] - MeanFilter[tahiti[[All,2]],7] ;  

    which is a 7 month filter. It ends up looking very much like the NINO3.4 curve.

    All that is needed is a filter to remove the fast month-to-month changes, which is what the 7 month filter accomplishes.


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