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relating back to the discussion:

Actually you gave me a fun idea, Graham! In a sandpile when the sand is at the critical angle of repose, as steep as possible, small landslides occur... and at least in theoretical models, these landslides are roughly scale-invariant: there are small ones and big ones and bigger ones, with the frequency of a landslide of size $x$ being $\propto x^{-p}$ for some power $p$. Under some conditions sand naturally organizes itself into dunes that are near the critical angle of repose: this is called self-organized criticality. The idea is that this system naturally has a second-order phase transition as some sort of attractor.

Maybe Pacific warm water that's just about ready to slosh back east is a bit like a sandpile at its critical angle of repose! If so, there might be a second-order phase transition here.

I feel this idea is a overly naive, but it might have some merit, or lead to some better ideas.

I was thinking about this idea and searched the forum to see if it came up before. The discussions of sloshing driving El Nino sound a lot like self organized criticality to me. That suggests one possible approach.

Bialek, Nemenman & co as well as Sejnowski & Saremi have papers on measuring criticality in complex natural signals, particularly images and neural data by treating the pixel/signal intensities as the order parameter. This approach could be applied to the various gridded data sets like the NOAA surface tempreature, pressure, humidity ... data sets. It sounds like the they should be in a near critical state most of the time, and El Nino's should correspond to departures from criticality.

I have seen a paper claiming that epilepsy attacks are departures from criticality I also think one that claims it for stock market crashes, but everything eventually get claimed to cause those.

Link strength sounds like a partial indicator of criticality. Looking for criticality on the full data could be more promising.

I found the following older paper:

- J S Andrade Jr, I Wainer, J M Filho, J E Moreira. Self-organized criticality in the El Nino southern oscillation Physica A: Statistical Mechanics and its Applications 215, 331--338 (1995). (The versions linked on Scholar are paywalled, but it is possible to find free versions on the internet.)

Following the citing and related papers on Scholar shows that scaling and criticality in climate is a lively cottage industry in its own right. One way this might be usable in El Nino prediction is to try and estimate the amout of energy built up in the system. This should primarily be a function of the water temperature differential across the the El Nino region and the sea level differential. If El Nino is really an SOC system the probability and likely size of the next event should be a function of the built up energy.

If El Nino is SOC, then there ought be mini/micro El Ninos happening on all space and time scales. These would be small backflows eastward from the warm pools. Is there a data set from which these would be detected easily? I know Paul has been looking into the details of the sloshing behavior. What data did you use to create your visualizations?

I should probably actually read the paper first :)

## Comments

Daniel, I don't have any visualizations of the sloshing behavior per se. I assume that whatever is there is a standing wave that has been stationary over time.

So in one dimension, I am assuming the standing wave is separable:

$ f(x,t) = g(x) \cdot h(t) $

All I am looking at is h(t).

The question on whether the ENSO time series are stationary is discussed here [1]

The current unresolved issue is if something changed at around 1981. I believe the behavior did change, best seen in a wavelet scalogram with the transition centered around 1200 months = 100 years after 1980.

The question is does the top scalogram look as if it has a tilt to it thus creating a continuous transition? Or is it 2 pieces as in the lower model?

There was a significant self-organized critical behavior that occurred in 1982 -- the massive El Chichon eruption. This was large enough that it did change the amount of solar insolation that the ocean received for several years.

If I were to do something with 2D sloshing visualization it would be something like this http://www.cpc.ncep.noaa.gov/products/analysis_monitoring/enso_update/wkxzteq.shtml

[1] Rosen, Ori, Sally Wood, and David S Stoffer. “AdaptSPEC: Adaptive Spectral Estimation for Nonstationary Time Series.” Journal of the American Statistical Association 107, no. 500 (2012): 1575–89.

`>"What data did you use to create your visualizations?" Daniel, I don't have any visualizations of the sloshing behavior per se. I assume that whatever is there is a standing wave that has been stationary over time. So in one dimension, I am assuming the standing wave is separable: $ f(x,t) = g(x) \cdot h(t) $ All I am looking at is h(t). The question on whether the ENSO time series are stationary is discussed [here [1]](http://www.stat.pitt.edu/stoffer/dss_files/adaptSPECjasa.pdf) >"The estimated posterior probabilities that the time series are stationary, that is,Pr(m=1|x) are 0.95, 0.93, and 0.99 for the SOI, Nĩno3.4, and DSLPA indices, respectively. These results confirm the findings of Rosen, Wood, and Stoffer (2009), Solow(2006), and Nicholls (2008). >One explanation for the difference between these findings and the earlier study of Trenberth and Hoar (1996) is that Trenberth and Hoar (1996) tested explicitly if there had been a change in frequency from 1981 onward. " The current unresolved issue is if something changed at around 1981. I believe the behavior did change, best seen in a wavelet scalogram with the transition centered around 1200 months = 100 years after 1980. ![wavelet](http://imageshack.com/a/img904/714/jkgkWf.gif) The question is does the top scalogram look as if it has a tilt to it thus creating a continuous transition? Or is it 2 pieces as in the lower model? There was a significant self-organized critical behavior that occurred in 1982 -- the massive El Chichon eruption. This was large enough that it did change the amount of solar insolation that the ocean received for several years. ![volc](http://imageshack.com/a/img537/1507/KO4WqM.gif) If I were to do something with 2D sloshing visualization it would be something like this <http://www.cpc.ncep.noaa.gov/products/analysis_monitoring/enso_update/wkxzteq.shtml> [1] Rosen, Ori, Sally Wood, and David S Stoffer. “AdaptSPEC: Adaptive Spectral Estimation for Nonstationary Time Series.” Journal of the American Statistical Association 107, no. 500 (2012): 1575–89.`

I thought you posted a 3D animation recently. You also had some tidal gauge plots.

`> Daniel, I don’t have any visualizations of the sloshing behavior per se I thought you posted a 3D animation recently. You also had some tidal gauge plots.`

These animated GIFs that John posted early on were inspirations for trying to understand what was going on:

http://johncarlosbaez.wordpress.com/2014/06/24/el-nino-project-part-2/

You look at that and compare to the dynamics of sloshing and can see similarities. Why has no one looked at the math of sloshing ?

Somebody made up this ridiculous fake plot "showing" how much the sea level rises around the Philippines that I pinched:

This is not 3D but is a numerical solution of the sloshing equation

This shows the surface and the subsurface computational mesh.

`These animated GIFs that John posted early on were inspirations for trying to understand what was going on: <http://johncarlosbaez.wordpress.com/2014/06/24/el-nino-project-part-2/> ![1](http://johncarlosbaez.files.wordpress.com/2014/05/t-dyn3.gif) ![2](http://johncarlosbaez.files.wordpress.com/2014/05/sst-wind-cur-eqt-20c1.gif) You look at that and compare to the dynamics of sloshing and can see similarities. Why has no one looked at the math of sloshing ? >I thought you posted a 3D animation recently. You also had some tdal gauge plots. Somebody made up this ridiculous fake plot "showing" how much the sea level rises around the Philippines that I pinched: ![goddard](http://imageshack.com/a/img540/6605/Qei3sz.gif) This is not 3D but is a numerical solution of the sloshing equation ![sloshing](http://imageshack.com/a/img823/6842/6fd5.gif) This shows the surface and the subsurface computational mesh.`