The Southern Oscillation embedded with the ENSO behavior is what is called a dipole , or in other vernacular, a standing wave. Whenever the atmospheric pressure at Tahiti is high, the pressure at Darwin is low, and vice-versa. Of course the standing wave is not perfect and far from being a classic sine wave.
To characterize the quality of the dipole, we can use a measure such as a correlation coefficient applied to the two time series. Flipping the sign of Tahiti and applying a correlation coefficient to SOI, we get Figure 1 below:
Fig 1 : Anti-correlation between Tahiti and Darwin. The sign of Tahiti is reversed to see better the correlation. The correlation coefficient is calculated to be 0.55 or 55/100.
Note that this correlation coefficient is "only" 0.55 when comparing the two time-series, yet the two sets of data are clearly aligned. What this tells us is that other factors, such as noise in the measurements, can easily drop correlated waveforms well below unity.
This is what we have to keep in mind when evaluating correlations of data with models as we can see in the following examples.
The models of ENSO for SOI and proxy records apply sloshing dynamics to describe the quasi-periodic behavior. see J. B. Frandsen, “Sloshing motions in excited tanks,” Journal of Computational Physics, vol. 196, no. 1, pp. 53–87, 2004.
The following GIF animations are supplementary material from S. S. Kolukula and P. Chellapandi, “Finite Element Simulation of Dynamic Stability of plane free-surface of a liquid under vertical excitation.”
Detuning Effect.gif shows the animation of sloshing fluid for the fourth test case, with frequency ratio Ω3 = 0.5 and forcing amplitudeV = 0.2: test case 4 as shown in Figure 4. This case corresponds to instability in the second sloshing mode lying in the first instability region. Figure 8(b) shows the free-surface elevation and Figure 9 shows the moving mesh generated in this case.
Dynamic Instability.gif shows the animation of sloshing fluid for the second test case which lies in the unstable region, with frequency ratio Ω1 = 0.5 and forcing amplitude kV =0.3: test case 2 as shown in Figure 4. Figure 6 shows the free-surface elevation and Figure 7 shows the moving mesh generated in this case.
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.
After having some very good success at modeling the ENSO via the Southern Oscillation Index Model (SOIM) but discovering a few loose ends, this post provides some puzzle pieces that may ultimately determine the source and synchronization of the ENSO forcing.
The significant finding with regards to the SOIM was that the input forcing had a period very close to the fundamental frequency of the Quasi-Biennial Oscillation (QBO) of stratospheric winds. Since measurements began in 1953, the fundamental period of the QBO has varied about a period close to 28 months. And this value is close to what the SOIM uses as a forcing input -- fit with a few Fourier series terms culled from the long-term QBO time series. But the open question was whether a mutual connection exists between the SOI and the QBO, or whether perhaps they share a common forcing input, external to both the ocean and stratosphere.
Is there such a thing as too simple a model? Take a look at this fit to ENSO
Fig 1: The SOI fit explained in this post. The time axis is in months from 1880.
My original approach to reconstructing the SOI time series using basis Mathieu functions is useful but not the only way to do the modeling. I believe that I took that as far as I could go and so decided to bite the bullet and solve the SOIM (Southern Oscillation Index Model) differential equations numerically. What pops out is a gloriously simple climate model that fits the data remarkably well -- and BTW, one of those models deemed not to exist by the climate science deniers.
What follows is the description of the model and how it connects to the previously described Chandler Wobble along with a crucial link to the quasi-biennial oscillation (QBO). Some type of forcing is responsible for providing the necessary energy to get the ENSO sloshing behavior going, and these are as good of candidates as any.
Read on to see how it can't possible be done
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 Southern Oscillation Index Model
- SOIM and the Paul Trap
- The Chandler Wobble and the SOIM
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.
Seth Carlo Chandler Jr was an actuary who studied his namesake wobble for thirty years.
ENSO and the Southern Oscillation Index has confounded everyone with its unpredictability.
Could a connection exist between the ENSO and the Chandler Wobble ? 
Based on what I have been analyzing with respect to the ENSO data, I am leaning in that direction. In the first post on the Southern Oscillation Index Model (SOIM), my initial analysis lead to a fundamental Mathieu frequency T of 6.3 years, a value of a = 2.83 and q = 2.72 :
I followed that up with additional checks and analogies to other physical phenomena :
That culminated with a trial fit of the SOI with a set of Mathieu parameters. Yet -- even though the fit was decent -- I was not satisfied with the result as it tended to overfit with respect to the adjustable parameters. The ideal situation would limit the number of fundamental frequency terms.
Three observations lead me to a much simplified representation.
- The main Mathieu frequency of 6.3 years seemed to vary over the historical record.
- The pressure index of the SOI is essentially a differential measure, and so the derivative of the Mathieu function should be fit to the pressure, e.g. use MathieuCPrime and not MathieuC .
- The connection between the original fit of 6.3 years and the Chandler Wobble beat frequency of 6.39 years (= 1/(1-365.25/433)), and the fact that this measure has been known to vary over the past 100+ years.
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
The volcanic aerosol factor of the CSALT model is an example of a perfectly interlocking piece in the larger global surface temperature puzzle. I thought I would present a more detailed description in response to the absolutely hapless recent volcano posts at the WUWT blog (here and here). The usual deniers in the WUWTang Clan can't seem to get much right in their quest to intelligently spell out ABCD (Anything But Carbon Dioxide).
Fig 1 : CSALT model using the GISS Stratospheric Aerosol forcing model.
The addition of the volcanic aerosol factor is no different than the other components of the CSALT model. Two flavors of volcanic aerosol forcings are provided. The standard forcing table is the GISS stratospheric aerosol optical thickness model maintained by Sato  and I use this table as is (see Figure 1). The more experimental model that I generated is a sparse table that features only the volcanoes of Volcanic Explosivity Index (VEI) of 5 or higher.
The VEI scale is logarithmic so that a VEI of 6 contains 10 times as much ejected particulates by volume than a VEI of 5 (which recursively is 10 times as much as VEI=4, and so on). This means that by modeling VEI of 5 or 6 we should capture most of the particulates generated as discrete events.
The two primary oscillating factors that we have identified in the CSALT model of global temperature are the Southern Oscillation Index (SOI) and the Length of Day (LOD). The distinguishing factor in terms of impact is that the SOI is characterized by intradecadal oscillations while the LOD fluctuates across decades .
If we can model the SOI deterministically, as demonstrated here, the hope is we may be able to model the LOD as well. But first, we need to understand the significance of the LOD and its possible origin.
Fig 1: As a premise for Length of Day (LOD) variations we consider that the rotational moment of inertia changes along the planetary surface. If a band of water positioned along the equator shifts to higher latitude, the rotational moment of inertia decreases and the rotational velocity increases, thus shortening the length of day.
This post follows up on the idea of modeling the historical Southern Oscillation Index (SOI) record with details on how one can apply the SOIM to make accurate predictions. Based on some some early encouraging success, I asserted that a more comprehensive model fitting would be possible. That's what this follow-on post is about -- trying to verify that we can accomplish that "holy-grail" of prediction, the prediction of future El Nino / Southern Oscillation (ENSO) conditions.
To foreshadow what's to come, Figure 1 shows the comprehensive SOIM fit, which incorporates a grouping of optimally phased Mathieu functions (introduced in the previous post)
Fig. 1 : Fit of the full SOI historical record (in green) to the SOI Model (in blue).
This is a very promising result based on the premise of the last post. The principal additions to the simple model are (1) a multi-harmonic basis set of Mathieu functions and (2) a more constraining physical interpretation to the math.
What follows is the explanation and various verification checks, which include:
- Sensitivity of the model to parameter selection
- Comparison to fitting red noise (to show over-fitting is not an issue)
- Hindcasts and forecasts based on restricted training intervals
- Power spectrum of model and data