Two modes to ENSO Variability

[mathjax]Since the last post in the SOIM series, where we identified a highly correlated behavior in tide gauge data to the ENSO SOI measure, it occurred to me that many of the earlier entries in this series seemed to go down a different path.  That path still involved Mathieu differential equations, but the parameters differed and the right-hand side (RHS) forcing function did not play as significant a role (described initially here with a couple of follow up posts).

Yet from the first of the tide gauge posts there was a clear indication that an additional distinct sloshing mode might apply, and that this could be related to the “phantom” Mathieu waveform we may have been chasing earlier.

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An ENSO Predictor Based on a Tide Gauge Data Model

[mathjax]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.

Fig 1:  The first step is to map a proxy (tide gauge data) to the SOI data

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.

$$ frac{d^2x(t)}{dt^2}+[a-2qcos(2omega t)]x(t)=F(t) $$

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

Fig 2: The second step is to model the tidal data in terms of a sloshing formulation. The biennial factor shows a phase reversal around 1953, switching from an even to odd year periodicity. The yellow highlighted area is one of the few regions that a correlation is clearly negative. Otherwise the fit models the behavioral details quite effectively.

This required an optimization of essentially three Mathieu factors, the a and q amplitudes, and the ω 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.

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 the If [ ] condition in the equation of 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.


[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.



Using Tidal Gauges to Estimate ENSO

[mathjax]— This is another post in the SOI Model project documented on this blog and at Azimuth.

A denier blogger (S.Goddard) recently created an interesting graphic:

Goddard’s description of the visual :

“Apparently they believe that water likes to pile up in mounds, and to help visualize their BS I created a 3D animation.”

This was evidently written with the intent to debunk something or other, but what the denier essentially did was help explain how ENSO works — which is a buildup of water in the western Pacific that eventually relaxes and sloshes back eastward, creating the erratic oscillations that are a characteristic foundation of the ENSO phenomena.

Seeing this visualization prompted me to consider whether any long-term tidal gauge records were available that we could possibly apply to modeling ENSO. In fact, one of the longest records (as provided by another denier blogger in Australia, J.Marohasy) is located in Sydney harbor, and available from the PMSL site.

The results are quite striking.

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Azimuth Project on El Ninos

A go-to place for ENSO and El Nino discussions is the Azimuth Project and its open source coding collaboration on predicting El Nino events.

The forum is for threaded discussions:

I have been hanging out there, as the collaborative environment is conducive to generating analysis ideas.

The most recent discussion thread I started concerns QBO and ENSO:

Another is on ENSO Proxy records, incorporating historical paleoclimate data from Michael Mann and others.

I will continue to write summaries of the progress on this blog.

The thing to remember when perusing the El Nino topics in the forum is that not everyone is taking the same approach. The main branches are:

  1. Evaluating and reproducing teleconnection approaches
  2. Looking at delay differential equations and the messy Lorenz chaotic formulations
  3. Data mining via machine learning concepts on the Earth’s volumetric satellite data
  4. Yours truly’s sloshing dynamics approach using Mathieu-type differential equations.

There is a significant difference between 2 and 4 in the analytical and computational complexity.  I think my approach is much more tractable and it may be on the verge of producing some predictive power by piggybacking on the more periodic QBO.

Anyone is free to join and contribute to the forum by registering for a login account. See the blog also and the main page Wiki.