Earlier this year, I decided to see how far I could get in characterizing the El Nino / Southern Oscillation through a simple model, which I referred to as the Southern Oscillation Index Model, or SOIM for short (*of course pronounced with a Brooklyn accent*). At the time, I was coming off a research project where the task was to come up with simple environmental models, or what are coined as context models, and consequently simple patterns were on my mind.

So early on I began working from the premise that a simple nonlinear effect was responsible for the erratic oscillations of the ENSO. The main candidate, considering that the ENSO index of SOI was clearly an oscillating time-series, was the Mathieu equation formulation. This is well known as a generator of highly erratic yet oscillating waveforms. Only later did I find out that the Mathieu equation was directly used in modeling sloshing volumes of liquids [1][2] -- which makes eminent sense as the term "sloshing" is often used to describe the ENSO phenomena as it applies to the equatorial Pacific Ocean (see here for an example).

Over the course of the year I have had intermittent success in modeling ENSO with a Mathieu formulation for sloshing, but was not completely satisfied, largely due to the overt complexity of the result.

However, in the last week I was motivated to look at a measure that was closer to the concept of sloshing, namely that of sea surface height. The SOI is an atmospheric pressure measure so has a more tenuous connection to the vertical movement of water that is involved in sloshing. Based on the fact that tidal gauge data was available for Sydney harbor (Fort Denison here) and that this was a long unbroken record spanning the same interval as the SOI records, I did an initial analysis posted here.

The main result was that the tidal gauge data could be mapped to the SOI data through a simple transformation and so could be used as a proxy for the ENSO behavior. The excellent correlation after a delay differential of 24 months is applied is shown in **Figure 1** below.

That was the first part of the exercise, as we still need to be able to quantify the tidal sea surface height oscillations in terms of a Mathieu type of model. Only then can we make predictions on future ENSO behavior.

As it turns out the model appears to greatly simplify, as the forcing, *F(t)*, for the right hand side (RHS) of the Mathieu formulation consists of annual, biannual (twice a year), and biennial (once every two years) factors.

The last biennial factor, though not well known outside of narrow climate science circles [3], is critical to the model's success.

Although the Mathieu differential equation is simple, the solution requires numerical computation. I (along with members of the Azimuth Project) like to use Mathematica as a solver.

The complete solution over a 85-year span is shown in **Figure 2** below

This required an optimization of essentially three Mathieu factors, the ** a** and

**amplitudes, and the**

*q***modulation (along with its phase). These are all fixed and constitute the LHS of the differential equation. The RHS of the differential equation essentially comprises the amplitudes of the annual, biannual, and biennial sinusoids, along with phase angles to synchronize to the time of the year. And as with any 2nd-order differential equation, the initial conditions for y(t) and y'(t) are provided.**

*ω*As I began the computation with a training interval starting from 1953 (aligning with the advent of QBO records), I was able to use the years prior to that for a validation. As it turns out, the year 1953 marked a change in the biennial phase, switching from odd-to-even years (or vice versa depending on how it is defined). Thus the validation step only required a one-year delay in the biennial forcing (see thecondition in the equation of

` If [ ] `

**Figure 2**).

The third step is to project the model formulation into the future. Or further back into the past using ENSO proxies. The Azimuth folks including Dara and company are helping with this, along with two go-to guys at the U of MN who shall remain nameless at the present time, but they know who they are.

Ultimately, since the model fitting of the tide data works as well as it does, with the peaks and values of the sloshing waters effectively identified at the correct dates in the time series, it should be straightforward to transform this to an ENSO index such as SOI and then extrapolate to the future. The only unknown is when the metastable biennial factor will switch odd/even year parity. There is some indication that this happened shortly after the year 2000, as I stopped the time series at this point. It is best to apply the initial conditions y and y' at this transition to avoid a discontinuity in slope, and since we already applied the initial conditions at the year 1953, this analysis will have to wait.

The previous entries in this series are best observed by walking backwards from this post, and by visiting the Azimuth Forum. Science is messy and nonlinear as practiced, but the results are often amazing. We will see how this turns out.

## References

[1] Faltinsen, Odd Magnus, and Alexander N Timokha. *Sloshing*. Cambridge University Press, 2009.

[2] Frandsen, Jannette B. “Sloshing Motions in Excited Tanks.” Journal of Computational Physics 196, no. 1 (2004): 53–87.

[3] Kim, Kwang-Yul, James J O’Brien, and Albert I Barcilon. “The Principal Physical Modes of Variability over the Tropical Pacific.” Earth Interactions 7, no. 3 (2003): 1–32.

# Keep it Lit

Good luck to the People's Climate Marchers. I read Bill McKibben's book Long Distance several years ago, and realize that persistence and endurance pays off. I also realize that there are no leaders in the movement, and that we all have to pull together to get off of fossil fuel. If we each do our share, the outcome will tend more toward the good than to the bad.

"...along with two go-to guys at the U of MN..."

America's large research universities are really quite impressive. We've taken them for granted (sic) and if some had their way they'd all be turned into vo-techs.

Grade A stuff. I'm unesnutioqably in your debt.

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I found your research fascinating, and from my humble stance as a curious layman, I may have an interesting perspective to offer you.

During my investigations of fifty years of rainfall records for my town in Southern California, I was struck by the pronounced periodicity observable in the graph, a distinctly sawtooth pattern.

In this part of the country we do not get much rain, and so any temporary increase in precipitation stands out in the totals for that year. I calculated that for any down year, there was an 80% chance that the next one would be higher. The third year had the same chance of being lower than the second, and so on.

Counting up the number of peaks, I divided into the count of years and I obtained a figure of 2.2 yrs.

This is a very crude result, but it may partially explain the phase shift in 1953 by precession. Personally, I think the actual figure of 2.2 may be somewhat lower since we have only one phase shift since that time.

Let us carry this idea a bit further. To use a term from acoustics, we may have to cycles of slightly different frequency "beating" against each other. On analogy with two audible tones, the result would be two additional frequencies, the sum and difference, which in this case, would be .2 and 4.2 yrs. I have no idea what this might mean or if it is relevant to your investigations.

I hope these comments prove useful.

TCG

Thanks TCG. El Nino definitely has an impact on Cali rainfall patterns. The biennial pattern that you found is fascinating and certainly should be considered in the mix. Same thing with yearly salmon catches, which show the same even/odd year pattern. High one year and then down the next.

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