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 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:
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.
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.
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.
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 , and I have to thank the denier bloggers Goddard and Marohasy for the inadvertent #OwnGoal.
 Krauskopf, Bernd, and Jan Sieber. “Bifurcation Analysis of Delay-Induced Resonances of the El-Nino Southern Oscillation.” arXiv Preprint arXiv:1109.2818, 2011.