Demodulation and the SOIM

A few people have emailed me asking for a simplified explanation for the model of ENSO that I have been working on. I can appreciate this because presenting solutions to differential equations is not one of the most intuitive ways to convey the essential model.

I decided to go with a mathematical analogy that hopefully will appeal to a techie. Although analogies do not often work when presenting a scientific model, if it turns out that essentially the same mathematical formulation describes the foundation, then it hopefully will serve some use, as many people may already be familiar with the compared-to mathematical construct. So the idea is use the elements of the Southern Oscillation Index Model (SOIM), which includes (1) a basic wave equation, and (2) a forcing provided by stratospheric winds (i.e. the QBO) to convince you that you can understand a model that  effectively describes the seemingly erratic behavior of ENSO. Understanding this concept will be music to your ears if you are willing to listen and appreciate the concept.

Two modes to ENSO Variability

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.

An ENSO Predictor Based on a Tide Gauge Data Model

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.

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

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.

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.

Climate March Announcement

In honor of and timed for the People's Climate March taking place this Sunday, I will be announcing a breakthrough in making accurate El Nino predictions.  This is based on research accomplished on this blog and at the Azimuth Project forum.

March on to a non-fossil fuel and renewable energy future !

Using Tidal Gauges to Estimate ENSO

— 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.

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: http://azimuth.mathforge.org/

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.

SOIM fit to Unified ENSO Proxy

A previous post described the use of proxy records of ENSO to fit the Southern Oscillation Index Model (SOIM).  This model fit used one specific set of data that featured a disconnected record of coral measurements from the past 1000 years, see Cobb [1].

As the focus of this post, another set of data (the Unified ENSO Proxy set) is available as an ensemble record of various proxy measurements since 1650 -- giving an unbroken span of over 300 years to apply a SOIM fit [2].  This ensemble features 10 different sets, which includes the Cobb coral as a subset.

To fit over this long a time span is quite a challenge as it assumes that the time series is stationary over this interval. The data has a resolution of only one year, in comparison to the monthly data previously used, so it may not have the temporal detail as the other sets, yet still worthy of investigation. (an interactive version is available here).

Change of Tide in thought?

One of the long-standing theories in climate science has to do with the topic of atmospheric tides. Similar to the more-familiar concept of ocean tides, these can be measured as regular fluctuations in wind, temperature, density and pressure throughout the atmosphere, with the effect more pronounced at higher altitudes as the lower density requires less energy to create an impact. The on-going theory states that the 24-hour solar cycle is the greatest contributor to the periodic atmospheric tidal effect [1]. Note that the theory appears to have (at least partially) originated with the famed AGW skeptic Richard Lindzen.

Yet some recent research appears to be challenging these ideas of a principally solar influence, especially in regards to features that obviously don't match the periods expected of the solar cycle [2][3]. The counter theory is that of it being a lunar (gravitational) origin rather than a solar (thermal) origin. This is all based on the rich detail in the data available from length-of-day measurements [4] and the credible fits of the countervailing theory to that data.

If the impact of this effect is real, it likely has repercussions on how I look at the source of the Quasi-biennial Oscillation (QBO) forcing and of the El Nino Southern Oscillation (ENSO) forcing.

Correlation of time series

The Southern Oscillation embedded with the ENSO behavior is what is called a dipole [1], 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.

Sloshing Animation

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.