I’m looking at side-band variants of the lunisolar orbital forcing because that’s where the data is empirically taking us. I had originally proposed solving Laplace’s Tidal Equations (LTE) using a novel analytical derivation published several years ago (see Mathematical Geoenergy, Wiley/AG, 2019). The takeaway from the math results — given that LTEs form the primitive basis of the GCM-specific shallow-water approximation to oceanic fluid dynamics — was that my solution involved a specific type of non-linear modulation or amplification of the input tidal. However, this isn’t the typical diurnal/semi-diurnal tidal forcing, but because of the slower inertial response of the ocean volume, the targeted tidal cycles are the longer period monthly and annual. Moreover, as very few climate scientists are proficient at signal processing and all the details of aliasing and side-bands, this is an aspect that has remained hidden (again thank Richard Lindzen for opening the book on tidal influences and then slamming it shut for decades).

So, considering the extreme tides at Bay of Fundy in the western Atlantic, I used the side-band draconic lunar cycle of 27.1036 day which explains the 4.53-year Bay of Fundy extreme repeat cycle when applied in conjunction with the perigee cycle of 27.554 days.

365.242 * (1/27.1036 – 1/27.5545) = 1/4.53

(see also section 3.3 on equinoctial tides in https://www.erudit.org/en/journals/ageo/2004-v40-n1-ageo_40_1/ageo40_1art01)

But all the side-bands when aliased against an annual impulse are also likely important.

365.242 * (1/27.1036 – 1/27.5545) mod 1 = 0.2205
365.242 * (1/27.1036 + 1/27.5545) mod 1 = 0.731

365.242 * (1/27.1036 – 2/27.5545) mod 1 = -0.0347
365.242 * (1/27.1036 + 2/27.5545) mod 1 = 0.9863

The 1st is again the 4.53-year period, the 2nd smaller, but the third and fourth relate to the 1st harmonic of the perigee, which occurs due to the pair of inline pulls corresponding to each of the solar and lunar eclipse alignments. The 3rd equates to nearly a 30-year cycle and the 4th a 70-year cycle. Note how close the latter is to the multidecadal AMO cycle.

Now, this isn’t just a selective situation where we can cherry-pick the periods that we want to analyze — they all have to work together when the declination and perigee terms are multiplied together creating all the various cross-harmonic terms. Same as with conventional tidal analysis. That means we will see fast cycles on the order of a year and all the other longer cycles simultaneously if we attempt this kind of fit. And that’s how straightforward it is to model all the short-term and long-term intricacies of the AMO time series. And PDO and ENSO if you go there, first PDO as a perturbation to AMO and then ENSO as a perturbation to PDO.

Tidal forcing is essentially grouping together the declination factors (the first 8 rows below, p=0) and the perigee factors (last 4 rows below, N=0). These groups of sinusoids are multiplied together after adding a constant to each, and then a differential form is calculated. This is basically a Taylor’s series expansion of a tractional force multiplied by the inverse square distance law of gravity, often used as a model for tidal forcing.

Tidal Factor	s	p	N	Tidal Freq	Amp	Phase
27.32165641	1	0	0	13.36822137	-0.000579248	0.113447614
27.21221597	1	0	1	13.42198487	-0.02642434	-13.38684873
27.10364878	1	0	2	13.47574838	0.544512224	5.106707386
13.6608282 (Mf)	2	0	0	26.73644274	-0.014754052	0.745325851
13.60610798	2	0	2	26.84396975	-0.201903555	15.78121594
13.63341319	2	0	1	26.79020625	-0.074501124	-32.72256985
13.55182439	2	0	4	26.95149676	0.23519421	7.460943623
13.57891194	2	0	3	26.89773325	-0.187981569	1.296986592
26.99594445	1	0	3	13.52951188	0.001252741	1.297603014
27.55461332 (Mm)	1	-1	0	13.25520147	3.346926596	2.836133748
13.77730666	2	-2	0	26.51040295	1.358356355	8.662766548
13.7188202	2	-1	0	26.62342285	0.749511902	34.92423684
27.09260548	1	1	0	13.48124127	-1.679928781	-2.18904

This table is also simplified by removing the annual harmonics, as an annual impulse is convolved with the result.

The center 100-year interval is used for training. Note that a 72-year cycle naturally arises from the aliased side-band described earlier. The cross-validation is nowhere near perfect but predictively suggests the massive heat spikes of 1878 and last year.

The approach works extremely well with minimal degrees of freedom to achieve cross-validation.

Across climate indices, minimal effort is required to further cross-validate. The PDO is the closest cousin to AMO as it also has a (subtler) decadal variation. In the chart below, the higher-end cross-validation is more impressive.

The LTE modulation is the strongest discriminator here, compare the lower right panels of the two charts. Despite being in different ocean basins, the tidal forcing terms do not differ by as much as would be anticipated by the variations in the two time-series. Compare line-by-line the amplitudes for PDO and AMO highlighted as rectangles.

The low DOF in the model enables robust cross-validation. Narrow down the training interval to 1910-1950 in the following chart.

with further over-fitting, the out-of-band test interval does not diverge as badly as would occur with models composed of many more DOF. (actually must look closely at the following overfit chart to see any difference to the previous — as an aside, if this was a neural net model, it would wildly diverge)

As PDO shows similarity to ENSO, we can perform a cross-validation to an index such as NINO3, shown below. Although it doesn’t capture the magnitude of the strong El Ninos, it impressively captures the timing of the peaks in the validation interval after the year 2000. It also arguably captures the timing of the twin massive heating spikes of 1878 and last year, which are likely remnants of the same 72-year interval shown more clearly in the AMO model. The LTE modulation is very high for ENSO as can be seen in the lower right panel.

In a RealClimate discussion, a commenter PO’27 has been constructively making lots of noise to explain how it can’t work, but that’s not enough — he will have to show how the excellent cross-validation results across all the climate indices — QBO, AMO, PDO, ENSO and the small scale Bay of Fundy results — can occur just by statistical chance. Good luck with that. As PO’27 was also asking about the physics, this is about as close as one can get — physics is about creating a unified view of plausible behavior that is both precise and parsimonious.

I can imagine that machine-learning will also discover the cross-validation eventually but the challenge there will be to reverse engineer the results. Fortunately, that’s something we don’t have to do here, as the analysis was physics-first.