This topic will gain steam in the coming years. The following paper generates quite a good cross-validation for SOI, shown in the figure below.

Xiaoqun, C. et al. ENSO prediction based on Long Short-Term Memory (LSTM). IOP Conference Series: Materials Science and Engineering, 799, 012035 (2020).

The x-axis appears to be in months and likely starts in 1979, so it captures the 2016 El Nino (El Nino is negative for SOI). Still have no idea how the neural net arrived at the fit other than it being able to discern the cyclic behavior from the historical waveform between 1979 and 2010. From the article itself, it appears that neither do the authors.

For the tidal forcing that contributes to length-of-day (LOD) variations [1], only a few factors contribute to a plurality of the variation. These are indicated below by the highlighted circles, where the V_{0}/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;
end loop;
end loop;
end loop;
end loop;

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.

References

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.

In Chapter 12 of the book we model — via LTE — the canonical El Nino Southern Oscillation (ENSO) behavior, fitting to closely-correlated indices such as NINO3.4 and SOI. Another El Nino index was identified circa 2007 that is not closely correlated to the well-known ENSO indices. This index, referred to as El Nino Modoki, appears to have more of a Pacific Ocean centered dipole shape with a bulge flanked by two wing lobes, cycling again as an erratic standing-wave.

If in fact Modoki differs from the conventional ENSO only by a different standing-wave wavenumber configuration, then it should be straightforward to model as an LTE variation of ENSO. The figure below is the model fitted to the El Nino Modoki Index (EMI) (data from JAMSTEC). The cross-validation is included as values post-1940 were used in the training with values prior to this used as a validation test.

The LTE modulation has a higher fundamental wavenumber component than ENSO (plus a weaker factor closer to a zero wavenumber, i.e. some limited LTE modulation as is found with the QBO model).

The input tidal forcing is close to that used for ENSO but appears to lead it by one year. The same strength ordering of tidal factors occurs, but with the next higher harmonic (7-day) of the tropical fortnightly 13.66 day tide slightly higher for EMI than ENSO.

The model fit is essentially a perturbation of ENSO so did not take long to optimize based on the Laplace’s Tidal Equation modeling software. I was provoked to run the optimization after finding a paper yesterday on using machine learning to model El Nino Modoki [1].

It’s clear that what needs to be done is a complete spatio-temporal model fit across the equatorial Pacific, which will be amazing as it will account for the complete mix of spatial standing-wave modes. Maybe in a couple of years the climate science establishment will catch up.

An intriguing yet under-reported finding concerning climate dipole cycles is the symmetry in power spectra observed. This was covered in a post on auto-correlations. The way that this symmetry reveals itself is easily explained by a mirror-folding about one-half some selected carrier frequency, as shown in Fig. 1 below.

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 discuss tropical instability waves (TIW) of the equatorial Pacific as the higher wavenumber (and higher frequency) companion to the lower wavenumber ENSO (El Nino /Southern Oscillation) behavior. Sutherland et al have already published several papers this year that appear to add some valuable insight to the mathematical underpinnings to the fluid-mechanical relationship.

“It is estimated that globally 1 TW of power is transferred from the lunisolar tides to internal tides[1]. The action of the barotropic tide over bottom topography can generate vertically propagating beams near the source. While some fraction of that energy is dissipated in the near field (as observed, for example, near the Hawaiian Ridge [2]), most of the energy becomes manifest as low-mode internal tides in the far field where they may then propagate thousands of kilometers from the source [3]. An outstanding question asks how the energy from these waves ultimately cascades from large to small scale where it may be dissipated, thus closing this branch of the oceanic energy budget. Several possibilities have been explored, including dissipation when the internal tide interacts with rough bottom topography, with the continental slopes and shelves, and with mean flows and eddies (for a recent review, see MacKinnon et al. [4]). It has also been suggested that, away from topography and background flows, internal modes may be dissipated due to nonlinear wave-wave interactions including the case of triadic resonant instability (hereafter TRI), in which a pair of “sibling” waves grow out of the background noise field through resonant interactions with the “parent” wave”

One of the frustrating aspects of climatology as a science is in the cavalier treatment of data that is often shown, and in particular through the potential loss of information through filtering. A group of scientists at NASA JPL (Perigaud et al) and elsewhere have pointed out how constraining it is to remove what are considered errors (or nuisance parameters) in time-series by assuming that they relate to known tidal or seasonal factors and so can be safely filtered out and ignored. The problem is that this is only appropriate IF those factors relate to an independent process and don’t also cause non-linear interactions with the rest of the data. So if a model predicts both a linear component and non-linear component, it’s not helpful to eliminate portions of the data that can help distinguish the two.

As an example, this extends to the pre-mature filtering of annual data. If you dig enough you will find that NINO3.4 data is filtered to remove the annual data, and that the filtering is over-zealous in that it removes all annual harmonics as well. Worse yet, the weighting of these harmonics changes over time, which means that they are removing other parts of the spectrum not related to the annual signal. Found in an “ensostuff” subdirectory on the NOAA.gov site:

Appendix E of the book contains information on compartmental models, of which resource depletion models, contagion growth models, drug delivery models, and population growth models belong to.

One compartmental population growth model, that specified by the Lotka-Volterra-type predator-prey equations, can be manipulated to match a cyclic wildlife population in a fashion approximating that of observations. The cyclic variation is typically explained as a nonlinear resonance period arising from the competition between the predators and their prey. However, a more realistic model may take into account seasonal and climate variations that control populations directly. The following is a recent paper by wildlife ecologist H. L. Archibald who has long been working on the thesis that seasonal/tidal cycles play a role (one paper that he wrote on the topic dates back to 1977! ).

The Indian Ocean Dipole (IOD) and the El Nino Southern Oscillation (ENSO) are the primary natural climate variability drivers impacting Australia. Contrast that to AGW as the man-made driver. These two categories of natural and man-made causes form the basis of the bushfire attribution discussion, yet the naturally occurring dipoles are not well understood. Chapter 12 of the book describes a model for ENSO; and even though IOD has similarities to ENSO in terms of its dynamics (a CC of around 0.3) the fractional impact of the two indices is ultimately responsible for whether a temperature extreme will occur in a region such as Australia (not to mention other indices such as MJO and SAM).

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 [1].