It turns out that the Darwin location of the Southern Oscillation Index (SOI) dipole is brilliantly easy to behaviorally model on it’s own.
The input forcing is calibrated to the differential length-of-day (LOD) with a correlation coefficient of 0.9997, and only a few terms are required to capture the standing-wave modes corresponding to the ENSO dipole.
As a bonus, the couple of years outside of the training interval are extrapolated from the model. This shouldn’t be hard for climate scientists, …. or is it still too difficult?
If that isn’t enough to discriminate between the two, the power spectra of the LTE mapping to model and to data is shown below. This identifies a couple of the lower frequency modulations as strong peaks and a few weaker higher harmonic peaks that sharpen the model’s detail. This shows that the data’s behavior possesses a high amount of order not apparent in the time series.
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Why isn’t the Tahiti time-series included since that would provide additional signal discrimination via a differential measurement as one should be the complement of the other? It should accentuate the signal and remove noise (and any common-mode behavior) if the Darwin and Tahiti are perfect anti-nodes for all standing-wave modes. However, it appears that only the main ENSO standing-wave mode is balanced in all modes.
In that case, the Darwin set alone works well. Mastodon
Experimenting with linking to slide presentations instead of a trad blog post. The PDF linked below is an eye-opener as the NINO34 fit is the most parsimonious ever, at the expense of a higher LTE modulation (explained here). The cross-validation involves far fewer tidal factors than dealt with earlier, the two factors used (Mf and Mm tidal factors) rivaling the one factor used in QBO (described here).
Sea-level height has several scales. At the daily scale it represents the well-known lunisolar tidal cycle. At a multi-decadal, long-term scale it represents behaviors such as global warming. In between these two scales is what often appears to be noisy fluctuations to the untrained eye. Yet it’s fairly well-accepted  that much of this fluctuation is due to the side-effects of alternating La Nina and El Nino cycles (aka ENSO, the El Nino Southern Oscillation), as represented by measures such as NINO34 and SOI.
To see how startingly aligned this mapping is, consider the SLH readings from Ft. Denison in Sydney Harbor. The interval from 1980 to 2012 is shown below, along with a fit used recently to model ENSO.
As cross-validation, this fit is extrapolated backwards to show how it matches the historic SOI cycles
Much of the fine structure aligns well, indicating that intrinsically the dynamics behind sea-level-height at this scale are due to ENSO changes, associated with the inverted barometer effect. The SOI is essentially the pressure differential between Darwin and Tahiti, so the prevailing atmospheric pressure occurring during varying ENSO conditions follows the rising or lowering Sydney Harbor sea-level in a synchronized fashion. The change is 1 cm for a 1 mBar change in pressure, so that with the SOI extremes showing 14 mBar variation at the Darwin location, this accounts for a 14 cm change in sea-level, roughly matching that shown in the first chart. Note that being a differential measurement, SOI does not suffer from long-term secular changes in trend.
Yet, the unsaid implication in all this is that not only are the daily variations in SLH due to lunar and solar cyclic tidal forces, but so are these monthly to decadal variations. The longstanding impediment is that oceanographers have not been able to solve Laplace’s Tidal Equations that reflected the non-linear character of the ocean’s response to the long-period lunisolar forcing. Once that’s been analytically demonstrated, we can observe that both SLH and ENSO share essentially identical lunisolar forcing (see chart below), arising from that same common-mode linked mechanism.
 F. Zou, R. Tenzer, H. S. Fok, G. Meng and Q. Zhao, “The Sea-Level Changes in Hong Kong From Tide-Gauge Records and Remote Sensing Observations Over the Last Seven Decades,” in IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 14, pp. 6777-6791, 2021, doi: 10.1109/JSTARS.2021.3087263.
For the tidal forcing that contributes to length-of-day (LOD) variations , only a few factors contribute to a plurality of the variation. These are indicated below by the highlighted circles, where the V0/g amplitude is greatest. The first is the nodal 18.6 year cycle, indicated by the N’ = 1 Doodson argument. The second is the 27.55 day “Mm” anomalistic cycle which is a combination of the perigean 8.85 year cycle (p = -1 Doodson argument) mixed with the 27.32 day tropical cycle (s=1 Doodson argument). The third and strongest is twice the tropical cycle (therefore s=2) nicknamed “Mf”.
These three factors also combine as the primary input forcing to the ENSO model. Yet, even though they are strongest, the combinatorial factors derived from multiplying these main harmonics are vital for generating a quality fit (both for dLOD and even more so for ENSO). What I have done in the past was apply the recommended mix of first- and second-order factors that appear in the dLOD spectra for the ENSO forcing.
Yet there is another approach that makes no assumption of the strongest 2nd-order factors. In this case, one simply expands the primary factors as a combinatorial expansion of cross-terms to the 4th level — this then generates a broad mix of monthly, fortnightly, 9-day, and weekly harmonic cycles. A nested algorithm to generate the 35 constituent terms is :
Counter := 1;
for J in Constituents'Range loop
for K in Constituents'First .. J loop
for L in Constituents'First .. K loop
for M in Constituents'First .. L loop
Tf := Tf + Coefficients (Counter) * Fundamental(J) *
Fundamental(K) * Fundamental(L) * Fundamental (M);
Counter := Counter + 1;
This algorithm requires the three fundamental terms plus one unity term to capture most of the cross-terms shown in Table 3 above (The annual cross-terms are automatic as those are generated by the model’s annual impulse). This transforms into a coefficients array that can be included in the LTE search software.
What is missing from the list are the evection terms corresponding to 31.812 (Msm) and 27.093 day cycles. They are retrograde to the prograde 27.55 day anomalistic cycle, so would need an additional 8.848 year perigee cycle bring the count from 3 fundamental terms to 4.
The difference between adding an extra level of harmonics, bringing the combinatorial total from 35 to 126, is not very apparent when looking at the time series (below), as it simply adds shape to the main fortnightly tropical cycle.
Yet it has a significant effect on the ENSO fit, approaching a CC of 0.95 (see inset at right for the scatter correlation). Note that the forcing frequency spectra in the middle right inset still shows a predominately tropical fortnightly peak at 0.26/yr and 0.74/yr.
These extra harmonics also helps in matching to the much more busy SOI time-series. Click on the chart below to inspect how the higher-K wavenumbers may be the origin of what is thought to be noise in the SOI measurements.
Is this a case of overfitting? Try the following cross-validation on orthogonal intervals, and note how tight the model matches the data to the training intervals, without degrading too much on the outer validation region.
I will likely add this combinatorial expansion approach to the LTE fitting software on GitHub soon, but thought to checkpoint the interim progress on the blog. In the end the likely modeling mix will be a combination of the geophysical calibration to the known dLOD response together with a refined collection of these 2nd-order combinatorial tidal constituents. The rationale for why certain terms are important will eventually become more clear as well.
Ray, R.D. and Erofeeva, S.Y., 2014. Long‐period tidal variations in the length of day. Journal of Geophysical Research: Solid Earth, 119(2), pp.1498-1509.
The red data points are the spectral values used in the ENSO model fit.
The top panel below is the LTE modulated tidal forcing fitted against the ENSO time series. The lower panel below is the tidal forcing model over a short interval overlaid on the dLOD/dt data.
That’s all there is to it — it’s all geophysical fluid dynamics. Essentially the same tidal forcing impacts both the rotating solid earth and the equatorial ocean, but the ocean shows a lagged nonlinear response as described in Chapter 12 of the book. In contrast, the solid earth shows an apparently direct linear inertial response. Bottom line is that if one doesn’t know how to do the proper GFD, one will never be able to fit ENSO to a known forcing.
In Chapter 12 of the book, we presented a math model for the equatorial Pacific ocean dipole known as ENSO (El Nino /Southern Oscillation). We argued that the higher wavenumber (×15 of the fundamental) characteristic of ENSO was related to the behavior known as Tropical Instability Waves (TIW). Taken together, the fundamental and TIW components provide enough detail to model ENSO at the monthly level. However if we drill deeper, especially with respect to the finer granularity SOI measure of ENSO, there are rather obvious cyclic factors in the 30 to 90 day range that can add even further detail. The remarkable aspect is that these appear to be related to the behavior known as the Madden-Julian Oscillation (MJO), identified originally as a 40-50 day oscillation in zonal wind .
In Chapter 12 of the book, we describe the forcing mechanism behind the El Nino / Southern Oscillation (ENSO) behavior and here we continue to evaluate the rich dynamic behavior of the Southern Oscillation Index (SOI) — the pressure dipole measure of ENSO. In the following, we explore how the low-fidelity version of the SOI can reveal the high-frequency content via the solution to Laplace’s Tidal Equations.
In Chapter 12 of the book, we have a description of the mechanism forcing the El Nino / Southern Oscillation (ENSO) behavior. ENSO shows a rich dynamic behavior, yet for the pressure dipole measure of ENSO, the Southern Oscillation Index (SOI), we find even greater richness in terms of it’s higher frequency components. Typically, SOI is presented with at least a 30-day moving average applied to the time-series to remove the higher-frequencies, but a daily time-series is also available for analysis dating back to 1991. The high-resolution analysis was not included in the book but we did present the topic at last December’s AGU meeting. What follows is an updated analysis …