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.

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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.



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).

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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.

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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.

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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.


Proxy confirmation of SOIM

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.

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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.

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The SOIM Differential Equation

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 :)

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The SOIM: substantiating the Chandler Wobble and tidal connection to ENSO

(see later posts here)

My previous posts on modeling the Southern Oscillation Index as a periodically modulated wave equation -- in particular via the Mathieu equation -- are listed below:

  1. The Southern Oscillation Index Model
  2. SOIM and the Paul Trap
  3. 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.

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