Basic geophysics to start.

AMO is measured in the north Atlantic, and influenced by an annual cycle — at a latitude that is inclined more to the Sun in the summer (peak declination is at summer solstice) than winter.

ENSO resides on the equator, subject to the topological constraints of that boundary condition. It therefore gets influenced by a northern hemisphere cycle and a southern hemisphere cycle. This turns into a semi-annual cycle.

Mechanical torques to the Earth’s rotation are measured by deviations in the Earth’s length-of-day (LOD) — see the time-series below¹. There is a clear annual and semi-annual cycle apparent as evidenced in the top panel, and also a gradual multi-decadal variation. Much stronger underlying this variation is a steady lunar tidal cycling, see bottom panel, where it is most easily revealed by taking the time-derivative of LOD — a real torque, or instantaneous acceleration. This decomposes to fundamental Mf, Mm, and Mf ‘ tidal cycles, with the Mf and Mm interfering to create an 8.848y perigean cycle, and the Mf and Mf ‘ interfering to create an 18.6 year nodal cycle. These beat envelopes can clearly be seen in the lower panel, along with occasional disturbances related to El Nino events (e.g. very strong 1983, 1988, 1992, very strong 1998, 2008)

A geophysical ansatz cooperatively linking LOD changes to climate cycles such as El Nino (ENSO), lies in the annual and semi-annual impulses that likely reinforce the instantaneous tidal torque that occurs at that time². The premise is that torque over an impulse duration leads to an incremental level shift in LOD and that generates an internal (i.e. hidden) latent forcing manifold for the ocean’s fluid dynamics. This is particularly sensitive along the subsurface thermocline, where effective gravity is reduced.

Consider the northern Atlantic first. The forcing manifold is generated via a convolution (i.e. essentially integrated) of the annual impulses with the tidal torque at that instance. The strongest constituent tidal factor Mf shown below generates a ~3.8 year cycle over time as it alternates between reinforcing or canceling in sign. The value of 3.8 is determined via modulo arithmetic, 365.242/13.66 mod 1 ~ 3.8. Similarly, the Mf ‘ and Mm lead to ~4.8 and 3.9 year cycles.

These are the strongest cycles by amplitude, but due to a fortuitous commensurate alignment with the annual signal, the Mt tidal has a significant impact on the shape of the manifold. The fact that 40 of the 9.133 Mt tidal cycles fit almost precisely into a year means that constructive interferences gradually accumulate over a 60-year period and then change sign and decrement over the next 60-year interval. This rides on top of the faster 3.8, 3.9, and 4.8 year cycles creating an erratic staircase as shown below. There is a behavior known in fluid dynamics called a devil’s staircase which likely has a meaningful relationship to this form.³

But this is just the manifold, a forcing that can be considered as almost a phase envelope — we are not yet seeing the oceanic basin’s response to his forcing. That’s why considering it a phase makes intuitive sense, as the response may simply be a sine wave acting on this phase, i.e. A sin (k*phase)+ B cos(k*phase) where k is a constant. This is where the fluid dynamics mathematics of Laplace’s Tidal Equations (LTE) and LTE modulation fits in, as described in detail in Chapter 12 of Mathematical Geoenergy². That text provides a non-intuitive grounding to what until now has a first-order physics explanation.

To get a feel for what this — in reality a non-linear response — involves computationally, consider the modulation/transfer function shown below:

That’s what it looks like with a k-modulation over the entire phase envelope, as it essentially doubles the frequency, changing the 120-year cycle to a 60-year cycle — not coincidentally the same period as the multidecadal period of the AMO.

Yet that 60-year cycle is only a single feature of the AMO, which is also characterized by wildly erratic fluctuations in value that almost obscure the multidecadal envelope. What actually works better as a model is if the shorter Mf steps on the staircase resolve to a complete single-period sinusoidal response, what in mathematical parlance is referred to a winding number of 1. The effective model thus becomes $\sin(k \phi + \epsilon \sin(k \phi+\phi_1) +\phi_2)$, where $\phi$ = phase, shown below for k = 1.55 and $\epsilon$ = 0.5. The phase slippage due to Mt causes the response to wander about zero over several decades.

The mapping to LOD remains largely intact through this as the 18.6 year and 8.85 year envelope is still clear. Note that a model of LOD is required to extrapolate before 1962, when the first LOD precision measurements were available.

The monthly and fortnightly remained near the same, but longer term tidal factors greater than 1 year in period had to be included (see bars in gray below), ostensibly to accommodate the drift in LOD estimated over the ~150 year time span of AMO.

The promise of the LOD-calibrated mathematical modeling further explains how the AMO itself may feed back into the LOD itself, as many have noted the multi-decadal variation in LOD resembles that of AMO⁵

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The next step is to evaluate NINO34 (i.e. ENSO), lying along the equator. The distinction here is that the annual impulse, used for AMO, must be converted to a semi-annual impulse (one positive [+] excursion alternating with one negative [-] excursion). Note that the semi-annual nature destroys the constructive interference of the Mt ordinal, which created the 120-year staircase. Instead, we have a strong ~3.8yr up and down devil’s staircase manifold. So, we can evaluate the following chart — middle left shows the estimated manifold and middle right shows the sin(k) LTE modulation applied to achieve the top panel left model fit in red.

The semi-annual forcing is doing as predicted — it breaks the Mt-driven secular 120-year build-up and replaces it with a bounded, alternating step manifold that behaves like an ENSO-scale oscillator rather than an AMO staircase.

• Top left: the red model tracks the broad phase/envelope of the blue NINO4 series well.
• Top right: the scatter is clearly elongated along a positive slope, so the fit is not accidental. Some of the validation points are outside the regression set.
• Middle left: this is the key result. The latent forcing is no longer a undulating staircase; it sits on recurring discrete bands, mostly between about -1.6 and -0.3, with occasional jumps toward 0 to +0.3. That is exactly the signature of the alternating +/- semi-annual impulse: an up/down “devil’s staircase” rather than constructive accumulation.
• Middle right: the red sin(k) modulation overlays the dominant blue bands fairly well, especially on the main latent levels, so the modulation is using the tightened manifold. It occasionally underrepresents rare extremes.
• Bottom left: the 50-month running correlation is usually high (~0.6–0.9) but drops down in the cross-validation interval. This could mean that the response is intermittently organized rather than uniformly phase-locked.
• Bottom right: the PSD match is strongest at the low-order peaks; model and data line up well at the main maxima, while the data keeps more high-frequency power than the model. So the semi-annual latent structure captures the core resonant bands, but not all of ENSO’s fast variance.

Bottom line: this figure supports the idea that for equatorial ENSO/NINO4, the correct latent driver is a semi-annual alternating pulse, yielding a compact, oscillatory ~3.8-year staircase manifold instead of the long constructive Mt staircase used for AMO/NAO. The manifold looks physically coherent and the modulation is plausible, but the weak validation window says the current mapping is still less robust and more regime-dependent than the AMO case.

An amazing concordance is that the k = 1.55 and $\epsilon$ = 0.5 are essentially the same for ENSO as for AMO, indicating this is likely a common-mode temporal response. It’s possible that these are related to Arnold tongue resonances⁶ In terms of plausibility and parsimony of these preliminary results, note how modest the Arnold winding is on the middle right panel (winding=2 suggests one winding for northern hemisphere and one for southern hemisphere) which may be related to the topological time reversal symmetry rules of the equatorial region⁷. If the equatorial latent manifold were showing many wraps, that would look more like a flexible fitting device; a winding of about 2 is close to the minimal nontrivial topology you would expect for an equatorial interface problem.

The reason this is plausible is that the equator is special: the Coriolis sign flips across it, so north and south contributions should enter with opposite handedness rather than accumulate into the same long constructive winding. In that setting, a two-sheet / two-turn organization — one branch associated with the northern side, one with the southern side — is a natural first-order picture. That the interpretation of the middle-right panel: the modulation is not over-twisted; it is just wrapped enough to separate the main latent bands and recover the top-left fit. That is also consistent with the Delplace/Marston style topological view of the equator as an interface where symmetry strongly constrains admissible structure.

On parsimony, this is good news. The model already uses a semi-annual sign-alternating impulse, which by itself
suppresses the Mt constructive staircase and forces a bounded oscillatory manifold. Once that choice is made, a
small winding number is the simplest way to map that latent staircase into ENSO-like oscillation. So, the topology is
doing real work without needing a large number of wraps, a high-order phase map, or a visually baroque modulation.

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So, what would one expect for PDO? Since it inhabits the northern Pacific, one would expect an annual impulse. Borrowing the parameters from AMO, it fits the pattern cleanly with a sharply delineated LTE modulation. Note that even though PDO is considered to have some of the character of ENSO, the fact that the k = 1.55 and $\epsilon$ = 0.5 parameters are again the same, indicates the common-mode behavior of these climate indices.

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NAO — North Atlantic Oscillation

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It does also work for coastal mean sea level (MSL) tidal stations : Ratan, Sweden

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IOD East (Indian Ocean Dipole) — Letting it free fit drove the Mt amplitude to a lower value. This indicated that the 120-year cycle was weaker, so adjusted this by adding a partial semi-annual component of -2/3 the amplitude of the annual impulse. The Indian Ocean straddles the equator but Asia to the north really clips off that lobe.

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TNA (Tropical North Atlantic) — has characteristics of AMO

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TSA (Tropical South Atlantic). Is this more like ENSO?

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Set of west coast MSL sites

All the tidal factors were allowed to vary as that was the easiest way to optimize and escape local minimum, but the distribution of weightings remained roughly the same in the 9 cases fitted above. Since the k and $\epsilon$ values also stayed even tighter, it’s possible that the cyclic fingerprint of each index is a combination of slightly different tidal factor contributions and the balance between annual and semi-annual impulses for that geospatial location. In fact, it might turn out that a more efficient fitting process is to start from a Bayesian-average tidal factor configuration instead of from the LOD calibration. This would reinforce the idea that this is truly a commo-mode behavior.

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Is this the deeper physics?

Delplace, Marston, and Venaille showed that equatorial Kelvin and Yanai waves arise as topologically protected edge modes, associated with a bulk Chern number of 2 for the rotating shallow‑water Poincaré spectrum in (k,ω) space. Their result is an abstract existence theorem: it guarantees robust equatorial waves but does not specify how they are forced or parameterized in time for prediction. In contrast, the LTE manifold used here selects an equatorial standing mode consistent with that topology and embeds it in a time‑domain, lunisolar‑forced framework, with explicit annual impulses and nonlinear modulation fitted directly to ENSO, AMO, and tide‑gauge records. In this sense, the LTE formulation provides a practical parameterization that connects the topological structure of equatorial waves to applied, data‑driven prediction in physical time.

From a dynamical‑systems perspective, the LTE manifold treats ENSO and related indices as the response of a phase‑locked forced oscillator, living in a low‑dimensional latent space and driven by a small set of quasi‑periodic forcings (lunisolar tides plus an annual impulse). In the language of nonlinear dynamics, this is an explicitly parameterized instance of mode locking on a torus (Arnold tongues, Devil’s staircase, Farey‑ordered p:q plateaus), while in the language of topological fluids it corresponds to driving a protected equatorial edge mode (Kelvin/Yanai‑like) selected by the bulk Chern structure of the rotating shallow‑water system. In modern ML terms, the construction is a physics‑informed analogue of SINDy/KAN latent‑manifold models ⁸: a shared, low‑dimensional latent driver is specified a priori, and simple nonlinear mappings (amplitude, phase, sinusoidal folding) are fitted to map that latent trajectory into many observed time series, providing an interpretable bridge between abstract topological theory and data‑driven prediction.

There is enough here for ML to extend, but the proviso is that the detailed LOD forcing must be applied -- I don't think it will work unless enough of the constituent tidal factors (ranked strongest to weakest) are included. The complexity of ENSO or AMO is a result of a Mach-Zehnder-like encryption of an already multi-constituent cycle - that's essentially impossible to decode without a valid manifold key.

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References

1. hpiers.obspm.fr/eop-pc/products/combined/C04.php?date=3&eop=3&year1=1962&month1=1&day1=1&year2=2012&month2=12&day2=31&SUBMIT=Submit+Search ↩︎
2. Mathematical Geoenergy, Pukite, P.R. et al, (Wiley/AGU, 2019) ↩︎
3. Arnold tongue - Wikipedia ↩︎
4. Mathematical Geoenergy, Pukite, P.R. et al, (Wiley/AGU, 2019) ↩︎
5. Marcus, S. L., 2016: Does an Intrinsic Source Generate a Shared Low-Frequency Signature in Earth’s Climate and Rotation Rate?. Earth Interact., 20, 1–14, https://doi.org/10.1175/EI-D-15-0014.1.. ↩︎
6. see 2. ↩︎
7. Delplace, Marston, Topological origin of equatorial waves. Science 358,1075-1077(2017). DOI:10.1126/science.aan8819 ↩︎
8. SINDy-KANs: Sparse identification of non-linear dynamics through Kolmogorov-Arnold networks
   AA Howard, N Zolman, B Jacob, SL Brunton, P Stinis
   arXiv preprint arXiv:2603.18548, 2026•arxiv.org ↩︎