[mathjax]— 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.
The Sydney harbor tidal data extends back to 1886 and features two sets of data, which are easily sliced together to make it current.
The idea, related to delay differential equations, is to determine if it is at all possible to model at least part of ENSO (through the SOI) with data from a point in time in the tidal record with a compensated point from the past. This essentially models the effect of the current wave being cancelled partially by the reflection of a previous wave.
This is essentially suggesting that $$SOI = k f(t) – k f(t-Delta t)$$ where f(t) is the tidal record and k is a constant.
After some experimenting, a good fit is obtained when the current tidal data is set to 3 months ago, and the prior data is taken from 26 months in the past. To model the negative of ENSO, the 3-month old data is subtracted from the 26-month old data, Figure 1:
Fig 1: Model of ENSO uses Tidal Gauge readings from Sydney Harbor.
In Figure 2 below, I introduced a perturbation by adding a time series of a sloshing model with a period of approximately 6 years. This illustrates what kind of correction may be needed to capture additional details. Without this correction, the correlation coefficient drops from around 0.74 to 0.66 for the time interval shown in Figure 1 above. Although many of the peaks and valleys are capture with remarkable resolution, some are missed, such as the dip at around 1976-1977 and the peak at 1905.
Fig 2 : With a Mathieu-type correction. The correlation coefficient increases from 0.65 to 0.74 in the interval after 1953.
A delay differential equation is a perturbed differential equation, either of an ordinary linear differential equation or of a non-linear Mathieu equation. The latter are referred to as delayed Mathieu equations.
The fact that this does appear to work well leads to the question that if we can model the tidal gauge dynamics, then that can feed back into the ENSO model, along with a Mathieu-type sloshing model to fill in any gaps and improve the correlation.
So the question is whether the tidal gauge readings can be modeled directly. The data is charted in Figure 3 below, which features both collected gauge readings overlapped. There is a large yearly and bi-yearly component, that when filtered out reveals the intra-decadal signal that contributes to ENSO. There is also an overall trend, ostensibly due to global warming.
Fig 3: Unprocessed data from the tidal gauges in Sydney harbor
As a start, I tried to develop a forced Mathieu-like differential equation model of the tidal data, after filtering out the long-range fluctuating trend. It is important to retain the annual and bi-annual information, since the intent is to apply the quasi-two-year differencing algorithm afterward. Figure 4 below is a DiffEq with forcing functions at annual and bi-annual frequencies along with a slight Mathieu factor with a frequency of ~2.3 rads/year intended to help match the erratic waveform.
Fig 4: First pass at evaluating a Mathieu-type Differential Equation to the Sydney Harbor tidal gauge data. The upper curve is the low-pass filtered data, and the lower curve is the DiffEq. The highlighted area is a region of a good behavioral match.
Although not ready for full evaluation, the correlation coefficient of 0.77 indicates that this is likely at least qualitatively correct. Moreover, the region highlighted in yellow illustrates an interval that matches very well with the formulation.
This is very promising result for climate science research [1], and I have to thank the denier bloggers Goddard and Marohasy for the inadvertent #OwnGoal.
References
[1] Krauskopf, Bernd, and Jan Sieber. “Bifurcation Analysis of Delay-Induced Resonances of the El-Nino Southern Oscillation.” arXiv Preprint arXiv:1109.2818, 2011.
GeoEnergy Math 
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Hi Webby: I understand from a friend that you were trash talking me but in the same breath looking for help on ENSO, so I wandered over here to your blog to see what you were up to.
1, It’s not surprising that you can find an ENSO signal in Sydney sea level data. ENSO is the dominant mode of natural variability on the planet. You can find ENSO impacts in almost every dataset.
2, I assume you understand that the direct El Niño and La Niña processes, especially those relating to sea level, take place in the tropical Pacific and are focused on the equatorial Pacific. Yet you are attempting to use sea level data from a location (Sydney, Australia) that’s outside of the tropics and more than 3700 km (2300 miles) from the equator. Curious. So basically you’re studying the impacts of ENSO in Sydney sea levels, nothing more, nothing less.
3, As a result, the satellite-based sea level data from the western equatorial Pacific (that are directly impacted by ENSO) run in and out of synch with the Sydney Harbor tide gauge data, even over a short-term like 1993 to 2003:
4, Being so distant from the tropics suggests the variations in Sydney sea level data you’ve found are not caused by the direct effects of ENSO, that you’re seeing the secondary effects caused by changes in atmospheric circulation.
5, That you’ve inverted the SOI for your visual comparisons confirms that you are not seeing the direct impacts of the “ENSO sloshing” that takes place in the western tropical Pacific. Sea levels there fall during El Niños and rise during La Niñas.
6, That you had to lag data by 2 years also suggests you’re seeing aftereffects, not the direct effects. But we already know that ENSO impacts sea levels so I fail to see what we’re learning from your exercise.
7, You’ve “easily” spliced two tide gauge datasets together from Sydney Harbor. But you failed to show that they agree during the overlap period(s).
Last, I did see your “Pacific gyre garbage patch” comment at HotWhopper. Cute. In my series on the 2014/15 El Niño (an El Niño that still has yet to develop) I present ENSO basics. These are processes that have been known and documented for decades. I present data to help with the visualizations of those processes. Because WUWT has such a wide audience, I’ve helped thousands of people learn about basic ENSO processes this year. They read my posts, then go to—let’s say—the NOAA ENSO blog and the stories are the same in both places. And then you call my work garbage. That undermines your credibility, Webby, not mine.
You may not like my calling ENSO a chaotic, naturally occurring, sunlight-fueled, recharge-discharge oscillator. But I’ve documented that for—what?—5 years.
You may not like my presentation of the upward steps in sea surface temperatures in response to strong El Niño events. One reason you don’t like it is, you’ve never bothered to try to understand it. And the fact that you don’t like my data presentation does not make it wrong. In fact, last year, Trenberth began to confirm my interpretation was right by parroting it. But he conveniently overlooks the fact that he’s written in at least two papers that El Niños are fueled by sunlight, due to reductions in cloud cover during La Niñas. I’ve also documented and confirmed that with data and reanalyses numerous times in numerous ways.
Well, good luck with your research, Webby. As I noted above, you’ve got a few hurdles to deal with. And I’m glad you’re trying to learn about a miniscule portion of ENSO…the impacts of ENSO on Sydney sea levels.
I won’t be back to comment again.
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Lookie at what Wayman Tizzy said:
“I won’t be back to comment again.”
Nice. I got what I needed from his comments.
Tisdale and others should watch this Jerry X Mitrovica presentation where he advised scientists to “engage your skeptics”
Jerry also quoted FScott Fitzgerald, who said that “Conflict has a value beyond victory and defeat.”
That’s what #OwnGoals are all about.
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Just an aside, but many sceptics believe that there was a global, synchronous MWP as warm as or even warmer than the present. Irrespective of the validity of this belief, it’s interesting that the evidence suggests that a persistent La Nina-like state characterised the East Pacific during the period when there were episodes of regional warming in the NH (Diaz et al. 2011; Steinke et al. 2014).
La Nina. *Not* El Nino. Persistent El Nino states seem to be associated with the Little Ice Age. From the Steinke et al. abstract:
So millennial climate variability apparently demonstrates exactly the opposite relationship between ENSO and global temperature to that claimed by Bob Tisdale.
But conservation of energy comes first 🙂
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