If we don't have enough evidence that the forcing of ENSO is due to lunisolar cycles, this piece provides another independent validating analysis. What we will show is how well the forcing used in a model fit to an ENSO time series — that when isolated — agrees precisely with the forcing that generates the slight deviations in the earth's rotational speed, i.e. the earth's angular momentum. The latter as measured via precise measurements of the earth's length of day (LOD). The implication is that the gravitational forcing that causes slight variations in the earth's rotation speed will also cause the sloshing in the Pacific ocean's thermocline, leading to the cyclic ENSO behavior.
Much of tidal analysis has been performed by Fourier analysis, whereby one can straightforwardly deduce the frequency components arising from the various lunar and solar orbital factors. In a perfectly linear world with only two ideal sinusoidal cycles, we would see the Fourier amplitude spectra of Figure 1.
This is an ENSO fit that only has knowledge of data prior to 1980. The data is 80% NINO34 and 20% SOI, with the latter providing finer structure.The lower fit includes a slight variation of the Draconic month according to this NASA page. It doesn't seem to do much.
A previous fit used values of A=0.785, M=0.3, K=-0.15 to 1920. This used A=0.866, M=0.207, K=-0.16.
The conjecture out of NASA JPL is that the moon has an impact on the climate greater than is currently understood:
11) Biweekly Oceanic and Atmospheric Tropical Instability Waves: are they forced? coupled? or resonant with the Moon?Claire Perigaud (Caltech/JPL)
07/05/2011 Dr. Claire Perigaud JPL Earth-Moon-Sun alignments influencing El Niños and water/air mass momentum
How do we determine confidence that we are not fitting to noise for the ENSO model ? One way to do this is to compare the data against another model; in this case, a model that provides an instrumentally independent measure. One can judge data quality by comparing an index such as NINO34 against SOI, which are instrumentally independent measures (one based on temperature and one on atmospheric pressure).
If you look at a sliding correlation coefficient of these two indices along the complete interval, you will see certain years that are poorly correlated (see RED line below). Impressively, these are the same years that give poor agreement against the ENSO model (see BLUE line below). What this tells us is the poorly correlated years are ones with poor signal-to-noise ratio. But more importantly, it also indicates that the model is primarily fitting to the real ENSO signal (especially the peak values) and the noisy parts (closer to zero crossings or neutral ENSO conditions) are likely not contributing to the fit. And this is not a situation where the model will fit SOI better than NINO34 -- because it doesn't.
The tracking of SOI correlating to NINO34 matches that of Model to NINO34 across the range with the exception of some excursions during the 1950's, where SOI fit NINO34 better that the model fit NINO34. The average correlation coefficient of SOI to NINO34 across the entire range is 0.75 while the model against NINO34 is less but depending on the parameterization always above 0.6.
As a result of this finding, I started to use a modification of a correlation coefficient called a weighted correlation coefficient, whereby the third parameter set is a density function that remains near 1 when the signal-to-noise (SNR) ratio is high and closer to zero where the SNR is closer to zero. This allows the fit to concentrate on the intervals of strong SNR, thus reducing the possibility of over-fitting against noise.
Or is it really all noise? (Added: 5/17/2017)
As I derived earlier, the solution to Laplace's tidal equations at the equator for a behavior such as QBO leads to a sin(k sin(f(t))) modulated time-series, where the inner sinusoid is essentially the forcing. This particular formulation (referred to as the sin-sin envelope) has interesting properties. For one, it has an amplitude limiting property due to the fact that a sinuosoid can't exceed an amplitude of unity. Besides this excursion-limiting behavior, this formulation can also show amplitude folding at the positive and negative extremes. In other words, if the amplitude is too large, the outer sin modulation starts to shrink the excursion, instead of just limiting it. So if there is a massive amplitude, what happens is that the folding will occur multiple times within the peak interval, thus resulting in a rapid up and down oscillation. This potentially can have the appearance of noise as the oscillations are so rapid that (1) they may blur the data record or (2) may be unsustainable and lead to some form of wave-breaking. I am not sure if the latter is related to folding of geological strata.
So the question is: can this happen for ENSO? I have been feeding the solution to the delayed differential Mathieu equation as a forcing to the sin-sin envelope and find that it works effectively to match the "noisy" regions identified above. In the figure below, the diamonds represent intervals with the poorest correlation between NINO34 and SOI and perhaps the noisiest in terms of SOI. In particular, the regions labelled 1 and 6 indicate rapid cyclic excursions.
By comparison, the model fit to ENSO shows the rapid oscillations near many of the same regions. In particular look at intervals indicated by diamonds 1 and 6 below, as well as the interval just before 1950.
Now, consider that these just happen to be the same regions that the ENSO model shows excessive amplitude folding. The pattern isn't 100% but also doesn't appear to be coincidental, nor is it biased or forced (as the fitting procedure has no idea that these are considered the noisy intervals). So the suggestion is that these are points in time that could have developed into massive El Nino or La Nina, but didn't because the forcing amplitude became folded. Thus they could not grow and instead the strong lunar gravitational forcing went into rapid oscillations which dissipated that energy. In fact, it's really the rate of change in the kinetic energy that scales with forcing, and the rapid oscillations identify that change. Connecting back to the theory, that's what the sin-sin envelope describes — its essentially a solution to a Hamiltonian that conserves the energy of the system. From the Sturm-Liouville equation that Laplace's tidal equations reduce to, this answer is analytically precise and provided in closed-form.
The caveat to this idea of course is that no one else in climate science is even close to considering such a sin-sin formulation. Consider this:
What I did was use the modern instrumental record of ENSO — the NINO34 data set — as a training interval, and then tested across the historical coral proxy record — the UEP data set.
The correlation coefficient in the out-of-band region of 1650 to 1880 is excellent, considering that only two RHS lunar periods (draconic and anomalistic month) are used for forcing. As a matter of fact, trying to get any kind of agreement with the UEP using an arbitrary set of sine waves is problematic as the time-series appears nearly chaotic and thus requires may Fourier components to fit. With the ENSO model in place, the alignment with the data is automatic. It predicts the strong El Nino in 1877-1878 and then nearly everything before that.
From a previous post, we were exploring possible solutions to the Mathieu equation given a pulsed stimulus. This is a more straightforward decomposition of the differential equation using a spreadsheet.
The Mathieu equation:
can be approximated as a difference equation, where the second derivative f''(t) is ~ (f(t)-2f(t-dt)+f(t-2dt))/dt. But perhaps what we really want is a difference to the previous year and determine if that is enough to reinforce the biennial modulation that we are seeing in the ENSO behavior.
Setting up a spreadsheet with a lag term and a 1-year-prior feedback term, we apply both the biennial impulse-modulated lunar forcing stimulus and a yearly-modulated Mathieu term.
I was surprised by how remarkable the approximate fit was in the recent post, but this more canonical analysis is even more telling. The number of degrees of freedom in the dozen lunar amplitude terms apparently has no impact on over-fitting, even on the shortest interval in the third chart. There is noise in the ENSO data no doubt, but that noise seems to be secondary considering how the fit seems to mostly capture the real signal. The first two charts are complementary in that regard — the fit is arguably better in each of the training intervals yet the test interval results aren't really that much different from the direct fit looking at it by eye.
Just like in ocean tidal analysis, the strongest tidal cycles dominate; in this case the Draconic and Anomalistic monthly, the Draconic and Anomalistic fortnightly, and a Draconic monthly+Anomalistic fortnightly cross term are the strongest (described here). Even though there is much room for weighting these factors differently on orthogonal intervals, the Excel Solver fit hones in on nearly the same weighted set on each interval. As I said in a previous post, the number of degrees of freedom apparently do not lead to over-fitting issues.
One other feature of this fit was an application of a sin() function applied to the result. This is derived from the Sturm-Liouville solution to Laplace's tidal equation used in the QBO analysis — which works effectively to normalize the model to the data, since the correlation coefficient optimizing metric does not scale the result automatically.
Pondering for a moment, perhaps the calculus is not so different to work out after all:A bottom-line finding is that there is really not much complexity to this unique tidal formulation model of ENSO. But because of the uniqueness of the seasonal modulation that we apply, it just doesn't look like the more-or-less regular cycles contributing to sea-level height tidal data. Essentially similar algorithms are applied to find the right weighting of tidal factors, but whereas the SLH tidal data shows up in daily readings, the ENSO data is year-to-year.
Further, the algorithm does not take more than a minute or two to finish fitting the model to the data. Below is a time lapse of one such trial. Although this isn't an optimal fit, one can see how the training interval solver adjustment (in the shaded region) pulls the rest of the modeled time-series into alignment with the out-of-band test interval data.
This is remarkable. Using the spreadsheet linked in the last post, the figure below is a model of ENSO derived completely by a training fit over the interval 1900 to 1920, using the Nino3.4 data series and applying the precisely phased Draconic and Anomalistic long-period tidal cycles.
Not much more to say. There is a major disturbance starting in the mid-1980's, but that is known from a Takens embedding analysis described in the first paper in this post.