[mathjax]The enduring and existential problem with modeling of climate is that we never have a controlled experiment to evaluate our scientific theories against. We can interpret the model against recent instrumental data, but this is often not good enough for the skeptics that claim that it is 5-parameter elephant fitting.

So what is often done is to search for other data, such as selecting from what is available from historical proxy records. This can provide extra dimensions of the sample space for verifying results that were essentially trained and fit to recent data only.

For the SOI data, we have modern day instrumental data that goes back to about 1866.  However, impressive historical proxy results have been unearthed by Cobb through an analysis of coral oxygen levels.  After calibration of recent coral growth to modern equatorial sea-surface temperature (SST) records, the correlation is expected to sustain back through history.  This makes it an adequate proxy representation for the Southern Oscillation Index (SOI) that we have been using to understand and potentially predict ENSO dynamics.

The verification experiment is to take several sets of coral measurements and determine if the same general Mathieu-equation fit that was used to model the SOI data could be applied universally.  The answer is yes, the SOIM essentially uses similar parameters for the 12th, 14th, and 17th century ENSO proxy data.

To get to the bottomline quickly, the general SOIM equation we use is as follows:

$$ frac{d^2x(t)}{dt^2}+[a-qcos(omega t)]x(t)=c_1 sin(w_1 t+psi_1)+c_2 sin(w_2 t+psi_2)+c_3 sin(w_3 t+psi_3) $$

The LHS is the nonlinear Mathieu wave response, which has two amplitude functions, a and q, and the nonlinear modulation frequency, ω.

The RHS has three sinusoids, with frequencies, w₁, w₂, and w₃, representing periodic forcing functions that likely arise from the same lunar tidal influences as govern the quasi-biennial oscillations (QBO).

The following figures show the original SOI Mathieu equation fit (Figure 1) along with coral proxy measurements from centuries centered around the years 1300 AD (Figure 2), 1600 AD (Figure 3), and 1100 AD (Figure 4).  The Palmyra coral data were pulled from this NOAA FTP site and were filtered to roughly match the figures that Cobb provided. The delta value of measured oxygen isotope (¹⁸O) scales to the level of warming observed (up is warmer climates or a more negative SOI)

Fig 1 :  The SOIM fit to modern day Southern Oscillation Index (SOI) records

Fig 2 : The SOIM fit to 14th century proxy coral measurements

Fig 3 : The SOIm fit to 17th century coral proxy records

Fig 4 : The SOIM fit to 12th century coral proxy records.

This is a mix of various length proxy records, with more than reasonable fits considering the complexity of mapping a nonlinear equation to an erratically oscillating data set.

And the remarkable feat is that the key parameters governing the oscillatory profile remain roughly the same over the different data sets (see Figure 5 below), including the original SOI data fit!

Fig 5: Parameters applied over the different data sets

What this analysis substantiates is that the underlying dynamics of ENSO maps to the same nonlinear differential equation regardless of the time frame, making it likely a highly stationary process.  The ENSO sloshing behavior is thus not a purely chaotic process, but again likely the result of a periodic external force acting on a nonlinear system which destroys the chaos and results in a quasi-periodic regime [2].   So the Mathieu equation describing the sloshing of water in an excited volume [3] appears to remain a highly viable theory and one that needs to be pursued — both for predictive potential and for continued understanding of natural variability in the Earth’s climate with respect to the global warming trend.

 References

1. Cobb, Kim M, Christopher D Charles, Hai Cheng, and R Lawrence Edwards. “El Nino/Southern Oscillation and Tropical Pacific Climate during the Last Millennium.” Nature 424, no. 6946 (2003): 271–76.

2. Osipov, Grigory V, Jürgen Kurths, and Changsong Zhou. Synchronization in Oscillatory Networks. Springer, 2007.

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