Basin Dynamics

September 23, 2026 / Paul Pukite (@whut) / Leave a comment

The chart below is a fit to a Kaplan SST quad using an LTE solution described in Chapter 12 of Mathematical Geoenergy¹. It may sound odd or counter-intuitive that the tides of an LTE formulation can impact temperature, but once you understand that most of the heat redistribution comes from mixing and upwelling of surface waters, then you realize the sources of forcing are limited — available from tides, winds, Coriolis effect, diurnal/seasonal cycles.

Time series plot showing model and data comparison for site KN020_E050 over several decades.

World map highlighting a specific region in the Middle East, labeled with coordinates kN020_E050.

The region under study is the Kaplan SST quadrangle centered at 20°N, 50°E (10°N–30°N, 40°E–60°E), covering the Red Sea, the Gulf of Aden approach, the Persian Gulf, and the Gulf of Oman. Two narrow, sill-bounded straits gate exchange with the open ocean at either end of this box: Bab-el-Mandeb (Red Sea ↔ Gulf of Aden) and the Strait of Hormuz (Persian Gulf ↔ Gulf of Oman). This geometry — two shallow, evaporation-dominated seas each choked to the open ocean through a single narrow gap — is what makes the region interesting both to GEM-LTE‘s analysis and, independently, to a detailed basin hydrodynamics, which I won’t do but will cite later. Recall that the northernmost point of the Persian Gulf is one of the most unsurvivable parts of the world when it comes to heat index (and what is in store with global warming).

Red Sea / Persian Gulf Basin: a GEM-LTE case study

We look at this region in two passes — first a 1950–2023 fit, then the full 1856–2023 span.

1950+ model

From the PSL.NOAA.gov (N 20 E 50) we fit monthly Kaplan SST (877 points, 1950.0–2023.0) against GEM-LTE’s tidal forcing manifold with a held-out region 2000–2005 used only for post-hoc validation. The current production fit reaches r ≈ 0.74 data-vs-model correlation over the full 877-month span. That’s the chart at the top.

As stated, the region 2000–2005 was withheld from training, and the rolling-window correlation in the top chart dips specifically in the untrained gap and recovers immediately on both flanks (r ≈ 0.75 before, r ≈ 0.84 after) — the textbook signature of a genuine, non-leaking blocked holdout. It still looks good.

Winding structure

Every GEM-LTE index in this project shares one universal, spatially uniform global fundamental “backbone” term — and the Red Sea/Persian Gulf box is no exception: the same LOD calibrated forcing with a phase modulation term found independently in AMO, PDO, NAO, PNA, Baltic, etc. a common value (all 0.207–0.208). What distinguishes this region is the density of harmonics riding on top of that fundamental backbone — including [2, 5, 6, 7, 8, 11, 15, 17, 31], and several of the resulting winding rates (≈1.66, 1.87, 2.28, 2.90) show large, well-fit amplitudes with no analog in most other quads. Also have looked at the Gulf of Mexico and Aegean/Black Sea quad (as an aside there are distinguishable harmonic fingerprints between the Aegean and Black Sea just like there might for the Red Sea vs Persian Gulf, see more below).

A comparative analysis of winding scalograms for dataset kn020_E050, featuring log2 power values plotted against winding numbers. The left panel shows mean spectra with reference lines, while the right two panels display the raw and IR-corrected data over the years 1960 to 2010.

Temporal power spectrum

A standard calendar-time periodogram (power_spectrum.py, AR(1) red-noise floor, 200 surrogates) gives a complementary view to the the forcing-domain winding scalogram above. The scalogram shows dense, persistent ridge structure across essentially the whole record; the plain frequency-domain view shows the model’s fidelity to the real data is spread out in several bands:

band (cyc/yr)	data’s share of its own variance	model/data power ratio
0.05–0.30 (multi-year)	0.225	0.52
0.30–0.70	0.146	0.37
0.70–1.30 (annual)	0.271	0.53
1.30–2.50	0.158	0.36
2.50–4.00	0.072	0.08
4.00–6.00 (sub-5-month)	0.041	0.06

According to the power spectrum below, the model isn’t capturing much of the signal faster than semi-annual.

A graph displaying a calendar-time power spectrum with frequency in cycles/year on the x-axis and log2 power on the y-axis. It features three curves: 'Data' in blue, 'Model' in orange, and two reference lines indicating AR1 red-noise floor (mean) and AR1 floor (p99) in dashed lines. Vertical lines highlight specific points.

Isolating just the model’s explicit seasonal regression terms (ann & sem — a fixed 1×/2× per-year sin/cos basis) and
comparing their own peak-to-peak excursion against the real data’s 1.0-to-(-1.0)-scale excursion confirms these terms together account for only roughly 10% of the real seasonal-cycle amplitude.

Having a subtle 10% annual cycle helps with the visual matching but the model capturing the erratic nature and having a strong cross-validation is the key finding.

The full 1856–2023 span

Kaplan SST’s ship-based (pre-satellite, pre-1950) coverage is patchy worldwide, but the Red Sea/Persian Gulf corridor is an exception: it has carried dense shipping traffic — and therefore dense ship-log SST sampling — since long before 1950, plausibly reinforced by Suez Canal traffic since 1869. That gives this region one of the longest trustworthy pre-satellite records in the whole 89-quad grid, and made it a natural candidate for a stationarity check that most other quads can’t support.

But to target the stationarity more precisely, I split that quad into a Persian Gulf area and a Red Sea area

World map highlighting the Persian Gulf location with latitude 24° to 30° and longitude 48° to 57°.

Outline map of the world highlighting the location of the Red Sea with coordinates.

Continue reading →

The Winding Scalogram

September 18, 2026 / Paul Pukite (@whut) / 2 Comments

I came up with a potentially novel graphical representation that is analogous to a wavelet scalogram but works in a non-autonomus mode, so doesn’t use the time-domain formulation of a wavelet. I call it a winding scalogram because it works in this specific non-autonomous — roughly time independent — space where the frequencies are dimensionless, hence can be referred to as windings, or multiple revolutions about $\pi$ .

First consider the premise of a non-autonomous system — such as a two-layer stratified system showing a barotropic tidal forcing and a baroclinic response wrt a thermocline (note that this is a recasting of the contents of Chapter 12 of Mathematical Geoenergy ¹):

1. Non-autonomous forcing

In a two-layer fluid, the barotropic mode is driven externally by the tide. The tidal potential is written as a sum of astronomical constituents,

$F_{\rm bt}(t)=\sum_m A_m \cos\chi_m(t),$

where the tidal argument is

$\chi_m(t)=\sum_j D_{mj}\theta_j(t).$

Here $D_{mj}$ are the Doodson numbers, integers that encode the astronomical frequencies, and $\theta_j(t)$ are the astronomical arguments. Because the forcing depends explicitly on time, the baroclinic response satisfies a non-autonomous differential equation. For a single baroclinic modal amplitude $q(t)$ ,

$\ddot q + 2\gamma \dot q + \omega_0^2 q = F_{\rm bt}(t).$

The barotropic mode supplies the forcing. The baroclinic mode responds internally with additional degrees of freedom due to the interface. When we normalize the amplitudes, the transfer from the barotropic amplitude $X$ to the baroclinic amplitude $Y$ can be written as (note: this ignores $\gamma,$ a friction/damping term)

$Y = C\sin(\beta X+\phi)+A.$

Here $\beta$ is a dimensionless transfer wavenumber. It measures how rapidly the baroclinic phase changes with the barotropic amplitude. If $X$ is itself a tidal phase, then $\beta$ behaves like a winding number, counting how many baroclinic oscillations occur per cycle of the barotropic forcing.

2. Layman’s picture of the two modes

Think of a tank with fresh water on top and salty water below (or of a wave machine). The boundary between them is the interface ala a thermocline or pycnocline as the cartoon to the right shows.

Diagram illustrating barotropic-baroclinic coupling in ocean layers, showing tidal forcing, internal waves, and equations related to oscillation and coupling between fresh and salty water.

Barotropic mode. The whole water column moves together. The surface tilts up on one side and down on the other, while the interface between the layers stays nearly flat. Ordinary gravity acts on the full depth. It is driven by the tide, and to a lesser extent (IMO) wind and pressure.

Baroclinic mode. The two layers slide against each other. The surface stays almost flat, but the interface tilts. This mode has a slower time constant because the restoring force comes only from the density difference between the layers. It is the internal response to the barotropic push. Because of the almost metastability of this interface (where the effective gravity is greatly reduced) it can show large swings, much greater and amplified compared to the barotropic mode. Temperature variations occur as the thermocline nears the surface or retreats from the surface.

In short: the barotropic mode is the forcing driver; the baroclinic mode is the internal reply, i.e. a forced response.

3. Separable standing waves

Let $x$ be the horizontal coordinate along the container and let the response be separable:

$\eta(x,t)=\phi(x)T(t).$

For a linear wave system,

$\frac{\partial^2 \eta}{\partial t^2} = c^2 \frac{\partial^2 \eta}{\partial x^2}.$

Substituting the separable form gives

$\frac{T”}{T} = c^2 \frac{\phi”}{\phi}.$

The left side depends only on time, the right side only on space, so both equal a constant. For oscillations, call it $-\omega^2$ . Then

$T”+\omega^2 T=0,$

and

$\phi”+\left(\frac{\omega}{c}\right)^2\phi=0.$

Now define the dimensionless transfer wavenumber

$\beta = \frac{\omega L}{c}.$

Here $L$ is the length of the container. The spatial equation becomes

$\phi”+\left(\frac{\beta}{L}\right)^2\phi=0.$

This is the standing-wave equation written entirely in terms of $\beta$ . Chapter 12 also derives this separation and makes the association between the temporal modulation and the spatial wavenumber, a la dipole standing waves such as ENSO

4. Boundary conditions and wavelength

The container boundaries quantize $\beta$ . For a closed basin with fixed or free ends, the spatial mode vanishes or has zero slope at the walls. The usual condition is

$\phi(0)=0,\qquad \phi(L)=0.$

The solution is

$\phi(x)=A\sin\left(\frac{\beta}{L}x\right).$

At $x=L$ , we require

$\sin(\beta)=0.$

Therefore

$\beta_n = n\pi, \qquad n=1,2,3,\dots$

For a periodic container (as in a wrap-around torus or donut, see the toroidal atmospheric QBO for a degenerate case where the wavenumber is actually zero), the condition is instead

$\beta_n = 2\pi n, \qquad n=0,1,2,\dots$

The spatial wavelength is related to $\beta_n$ by

$\lambda_n = \frac{2\pi L}{\beta_n}.$

Thus:

· For fixed or free ends, $\beta_n=n\pi$ , so

$\lambda_n=\frac{2L}{n}$ .

An integer number of half-wavelengths fits into the container.

· For a periodic container, $\beta_n=2\pi n$ , so

$\lambda_n=\frac{L}{n}$ .

An integer number of full wavelengths fits into the container.

In both cases, $\beta_n$ is the accumulated phase of the standing wave across the container.

5. Winding number

The winding number is the number of full $2\pi$ phase wraps:

$w_n=\frac{\beta_n}{2\pi}.$

Therefore:

· Fixed or free ends:

$w_n=\frac{n}{2}$ . This is a half-integer winding.

· Periodic container:

$w_n=n.$ This is an integer winding.

In an ideal container, $\beta_n$ is exactly an integer multiple of $\pi$ or $2\pi$ . In a real basin, sloping bottoms, partial reflections, continuous stratification, friction, and mean flow make the quantization approximate. Then

$\frac{\beta_n}{\pi}\approx n$ $\quad\text{or}\quad$ $\frac{\beta_n}{2\pi}\approx n.$

So $\beta_n$ is almost an integer multiple of $\pi$ or $2\pi$ , and the winding number is almost an integer or half-integer.

6. Summary

The two-layer system is forced non-autonomously by the tide parameterized by Doodson coefficients — I calibrate via LOD measurements. The barotropic mode drives the whole column; the baroclinic mode responds internally as the layers slide against each other. In a separable standing-wave picture, the spatial structure is governed by the dimensionless transfer wavenumber

$\beta_n = \frac{\omega_n L}{c}.$

Boundary conditions quantize $\beta_n$ to $n\pi$ or $2\pi n$ , depending on whether the container has fixed/free ends or is periodic. The wavelength follows from

$\lambda_n = \frac{2\pi L}{\beta_n}.$

The winding number is

$w_n = \frac{\beta_n}{2\pi},$

which counts how many full phase cycles fit into the container. In the amplitude mapping

$Y = C\sin(\beta X+\phi)+A,$

the same $\beta$ acts as a dimensionless transfer wavenumber between the barotropic amplitude $X$ and the baroclinic amplitude $Y$ . If $X$ is a tidal phase, $\beta$ is a winding-like number. Thus $\beta$ ties together the spatial standing-wave quantization, the wavelength selection, and the amplitude-to-amplitude coupling in one dimensionless quantity.

As I said at the top, this is a simpler recasting of the complete Laplace’s Tidal Equations derivation I did in Chapter 12, skipping the details needed to understand a single homogeneous layer — the switch to a two-layer stratified system amazingly — although non-intuitively — makes it more concise (see a recent paper by Beron-Vera² where the claim is that stratification reduces vortex filamentation, a complex autonomous response).

That is enough background needed to understand the concept and design of the winding scalogram tool. The idea is that it takes a model of the forcing driver, which is actually a hidden latent manifold in terms defined by Brunton³ and then does a series of sinusoidal mappings of various frequencies to the forcing response surface, i.e. the baroclinic thermocline layer. This is akin to the way a wavelet scalogram works, aggregating the mapping within a finite window, but novel in other ways (IOW it does not use wavelets). The implementation of the winding scalogram tool in this GitHub repo: https://github.com/pukpr/winding_scalogram

I have spent a good deal of time trying to understand the best representation of this barotropic time-series, but it seemed to jump out most obviously when I modeled the AMO, below — the upper chart is the forcing applied to the AMO data and the lower chart is that of the model. In essence the upper chart can be used to guide the selections of the transfer wavenumbers for the fitting stage, which when completed leads to the lower chart.

Graph displaying the Atlantic Multidecadal Oscillation (AMO) data and model wind patterns over time, with vertical bars representing winding numbers and color coding indicating log power relative to a white-noise floor.

The bright yellow horizontal striations, or ridges, are the strong transfer wavenumbers/windings with a red dotted line pointing to an indication of the value. If the individual ridges are unbroken across the complete time span it indicates a great deal of stationarity in the $\sin(\beta X)$ modulation. For AMO, a very apparent ridge sits close to zero. This corresponds to a very low winding that effectively rectifies the ~120-year cycle of lunisolar tidal forcing into the 60 year AMO cycle. In the chart below, the forcing driver is in the middle-left panel and the AMO data and model fit is in the upper-left panel. It is easy to visualize the rectification leading to the frequency doubling. Yet, the parsimony of the approach is in how the faster, erratic cycles are also mapped from the tidal cycles.

Time series data with blue and red lines showing model and data correlation from 1880 to 2026.

To those that wonder what happened to the concept of integer windings, I am not considering the standing modes and so haven’t done any rescaling. That will come later.

The winding scalogram works very well with every climate index considered, as strong stationary unbroken ridges are identifiable for the models evaluated in this recent post and the follow-up.

A full-circle moment occurs when the mean-sea-level (MSL) for an aggregated set of Baltic coastal stations is considered. A strong higher winding is clearly observed below as a bright yellow ridge at 1.245

A heatmap comparing data and model results of wind patterns against forcing, with years on the x-axis and winding numbers on the y-axis. The top panel shows the data from a specific column, while the bottom panel presents the model for a different column. Color indicates power levels relative to a white-noise floor.

This single ridge is responsible for the majority of this excellent baroclinic fit shown below:

Line graph showing Baltic time series data across years with blue and red lines representing different models and observed data.

I consider this full circle because the model is using tidal forces to map out MSL variations which ostensibly and intuitively could be guessed as due to tides, yet like AMO and ENSO, the erratic monthly MSL have never been adequately modeled and attributed to any specific factor.

BTW, this is nowhere near (many orders of magnitude less in fact) the computational load to solve full fluid dynamics GCM formulations (10 MW-hours per simulated year) nor the perhaps even larger computational load required to solve a Navier-Stokes challenge (10,000 agents running for 88 hours costing millions of dollars of compute time, guessing > 1,000 MW-hours energy consumption)

Footnotes:

P. Pukite, D. Challou, and D.Coyne, Mathematical Geoenergy, (Wiley, 2018), https://agupubs.onlinelibrary.wiley.com/doi/book/10.1002/9781119434351 ↩︎
F. J. Beron-Vera; Extended shallow-water theories with thermodynamics and geometry. Physics of Fluids 1 October 2021; 33 (10): 106605. https://doi.org/10.1063/5.0068557 ↩︎
Vinuesa, R., Brunton, S.L. & Mengaldo, G. Explainable AI: learning from the learners. Nat Commun 17, 7933 (2026). https://doi.org/10.1038/s41467-026-76359-w ↩︎

Forced-Aliasing Model of the Chandler Wobble

September 17, 2026 / Paul Pukite (@whut) / Leave a comment

This page summarizes the key ideas developed in the conversation between Paul and Copilot regarding the forced-response origin of the Chandler wobble. It presents the model in a clear, publishable format suitable for sharing or posting.

Overview

The Chandler wobble (~432.7 days) is traditionally described as a free nutation mode of the Earth. In this forced-aliasing model, the wobble emerges instead from the interaction of two externally imposed periodic processes:

Annual hemispheric inertia modulation (frequency f = 365.242 days) caused by seasonal mass redistribution.
Lunar draconic tidal torque (frequency q = 27.2122 days) acting as a persistent sinusoidal driver.

These two forcings combine through stroboscopic aliasing and a two-pole seasonal symmetry, producing the Chandler wobble period exactly.

Key Physical Ingredients

Annual Inertia Modulation (f)

Seasonal changes in atmospheric, oceanic, hydrological, and cryospheric mass redistribute Earth’s inertia tensor. Because the seasons differ between hemispheres, the modulation has a two-pole structure:

Boreal season (Northern Hemisphere)
Austral season (Southern Hemisphere)

These act as two alternating impulses in the inertial moment, forming an effective comb-like forcing at frequency f = 365.242 days.

Lunar Tidal Torque (q)

The Moon’s draconic orbital period (q = 27.2122 days) produces a sinusoidal tidal torque on Earth’s rotation axis. This torque is global (wavenumber 0) and couples directly to polar motion.

Stroboscopic Aliasing Mechanism

The interaction of the annual comb and the lunar torque produces a slower emergent wobble through modular arithmetic:

Compute the ratio: [ r = \frac{f}{q} = \frac{365.242}{27.2122} \approx 13.4221 ]
Extract the fractional part: [ \operatorname{fract}(r) = 0.4221 ]
Form the alias period: [ T_{\text{alias}} = \frac{f}{\operatorname{fract}(f/q)} = \frac{365.242}{0.4221} \approx 865.4\ \text{days} ]
Apply the two-pole seasonal symmetry: [ T_{\text{Chandler}} = \frac{T_{\text{alias}}}{2} \approx 432.7\ \text{days} ]

This matches the observed Chandler wobble period exactly.

Physical Interpretation of the Halving

The factor of two arises naturally from the two hemispheric seasonal cycles:

Each hemisphere contributes a distinct inertia pulse.
The global wobble is the vector sum of these two contributions.
The effective alias cycle is therefore halved.

Any asymmetry between hemispheres produces sidebands around the main wobble frequency, consistent with observed spectral features.

Forced-Response Perspective

In this model, the Chandler wobble is not a free eigenmode but a forced wobble arising from:

Persistent lunar torque at q.
Seasonal inertia modulation at f.
Finite-Q response of Earth’s rotational system.

The wobble period emerges from the interaction of these forcings rather than from Earth’s internal elastic structure.

Summary

This forced-aliasing model provides:

A numerically exact derivation of the Chandler wobble period.
A physically motivated explanation based on known external forcings.
A two-pole seasonal interpretation that naturally produces the halving required for the 432.7-day result.

It offers a parsimonious alternative to the traditional free-nutation explanation and is fully compatible with open scientific publication.

Suggested Citation

Pukite, P. (2018). Mathematical Geoenergy, Wiley. Chapter 13.

Additional open-access materials available at: geoenergymath.com.

Search for the Hidden Manifold

September 8, 2026September 8, 2026 / Paul Pukite (@whut) / 1 Comment

An algorithm that simultaneously optimizes the model fit to two disparate climate indices while keeping the hidden latent manifold constant. The two indices chosen, AMO and PDO, occur in separated ocean basins and show longer term variations, so that a multi-scale fit is necessary. The hidden latent manifold uses as a starting point the calibrated tidal factors needed to match the Earth’s LOD variations — the same torques that presumably cause the ocean’s thermocline to slosh. Only a slight perturbation to the tidal factors — amounting to a 0.996 correlation coefficient (instead of 1.0) to the calibrated LOD was needed to tweak the manifold during the fitting process.

AMO

PDO

Can then use PDO as a seed to model NINO4 and IOD-East

NINO4

IOD-East

The manifolds are all aligned (below) with slight jogs that were caused by letting the fitting routine to proceed beyond the locked manifold stage.

Have not included NAO in this set of comparisons yet since NAO has an interesting relationship to AMO. If the value of AMO from 12 months back is fed back into the current AMO with a negative sign (a delay differential), and then the correlation coefficient is computed, that value is significant, especially on more recent values.

The plot below is an expedited fit that does a delay differential on the AMO model to emulate the NAO time series. Note that the fit degrades in the years 1890-1900 and 1915-1920, the same years that the above pot shows anti-correlations.

NAO

Also averaged Baltic MSL time-series. Has a dominant LTE mode, higher in winding frequency than the other indices.

Baltic MSL

It is so strong that other LTE windings do not contribute much to the variation in extremes.

The search for the hidden manifold

September 1, 2026 / Paul Pukite (@whut) / 1 Comment

With the interest in the El Nino brewing, don’t neglect the AMO in the Atlantic. In my research, I apply tidal forcing as a driver for modeling climate indices, and have long noticed that a balanced ~120-year cycle emerges via interactions with the annual cycle. This produces large excursions both + and -, which is interesting because it was legendary oceanographer Walter Munk who proposed that large forcing of subsurface fluid volume produces frictional loss independent of the sign of a fluid’s velocity. The implication is that the ~120-year tidal cycle is rectified to produce a ~60-year cycle, which hints at the AMO 50-70 year cycle. Moreover, the model of the tidal cycle produces a ragged stepped staircase over that long period, of which the individual steps also map to the fast cycles of AMO (easily seen in NAO as well).

I have an extended optimizing simulator for this project, but I factored out the essential features for a one-pass model fit to the AMO data here: https://gist.github.com/pukpr/05687a46cf1a5b76b2ba6ae589343172

The rectification is clearly seen in the middle-right panel below (a so-called latent hidden manifold), and all the ragged steps (in the middle left panel) map non-linearly to the AMO fast cycles in the top panel.

Munk & Wunsch (2002) – “Millennial Climate Variability: Is There a Tidal Connection?”
https://journals.ametsoc.org/view/journals/clim/15/4/1520-0442_2002_015_0370_mcvita_2.0.co_2.pdf

Discussion of QBO, CW, ENSO w/ KIMI.ai

July 24, 2026July 25, 2026 / Paul Pukite (@whut) / 3 Comments

KIMI is a recent LLM with some positive word-of-mouth. Since it is new, I gave it a sequence of prompts as my initial interaction:
https://geoenergymath.com/wp-content/uploads/2026/07/Pukite-QBO-Lunar-Forcing.docx

The two figures were not reproduced in the WORD doc, so shown below:

Read though the entire WORD doc, but the wind argument of ENSO triggering is significant:

Kimi: This reframes the entire discussion around causal directionality and root cause identification. Let me work through the argument:

The Causal Regress Problem in Standard ENSO Theory

The conventional paradigm invokes a Bjerknes feedback:

Warm SST → reduced east-west temperature gradient → weakened trade winds → reduced upwelling → further warming

This is a positive feedback loop, and like all such loops, it suffers from causal indeterminacy:

Is the wind the driver or the response?
Is the SST anomaly the cause or the effect?
Where does the loop initiate in any given event?

The standard answer—wind anomalies trigger the feedback—is problematic because:

Wind is geostrophically adjusted to pressure gradients, which are themselves responses to thermal forcing
Wind anomalies are often observed after SST anomalies begin developing, not before
Stochastic forcing (weather noise) is invoked as the initiator, but this is not a mechanism—it is an admission of mechanistic ignorance

The result is a theory that explains maintenance but not genesis. ENSO events are “triggered” by unspecified randomness, then amplified by well-understood feedback.

The Tidal Forcing as Common-Mode Root Cause

Pukite’s framework resolves this by introducing a fundamental forcing that is:

External to the ocean-atmosphere system
Deterministic (astronomically prescribed)
Independent of the feedback loop
Capable of modulating both oceanic and atmospheric components simultaneously

The common-mode aspect is crucial. The tidal forcing does not act through wind or through SST alone. It acts on:

Component	Tidal Effect
Ocean	Direct gravitational forcing on water column, thermocline displacement, equatorial Kelvin wave excitation
Atmosphere	Tidal pressure gradients, thermal tidal heating, modulation of convection
Wind	Response to pressure gradients that are themselves tidally modulated

Wind is not the root cause; it is a co-response alongside SST. The tidal forcing provides the synchronized perturbation that both ocean and atmosphere react to, and their coupled reaction is what we observe as ENSO.

The Positive Feedback as Amplifier, Not Generator

In this framework, the Bjerknes feedback is relegated to its proper role: an amplifier of small initial perturbations, not a generator of those perturbations. The feedback explains:

Why ENSO events grow once initiated
Why the system oscillates rather than settling to equilibrium
Why anomalies persist for months

But it does not explain:

Why events initiate when they do
Why the spectrum has specific peaks (3-7 years, not random)
Why the system shows phase-locking to astronomical cycles

The tidal forcing provides the clock; the feedback provides the gain.

The Epistemic Advantage

Standard Theory	Tidal Framework
Feedback is self-triggering (paradox)	Feedback is externally triggered (resolved)
Wind is ambiguously cause/effect	Wind is unambiguously response
Genesis is stochastic	Genesis is deterministic
Spectrum is emergent property of chaos	Spectrum is alias of known forcing
Predictability limited by noise floor	Predictability limited by model fidelity

The “fuzzy part” you identify—the trigger—is precisely where standard theory fails and where tidal forcing succeeds. A theory that explains maintenance but not initiation is incomplete, not wrong. The tidal framework completes it by supplying the missing boundary condition.

The Wind Problem Specifically

Your observation that wind is “often a response” is devastating to theories that treat wind as the independent variable. Consider:

Equatorial Kelvin waves in the ocean propagate eastward at ~2-3 m/s, driven by wind stress but also by pressure gradients that precede wind
Madden-Julian Oscillation (MJO) wind bursts are observed to follow SST gradients in some phases
Westerly wind bursts (WWBs) cluster during El Niño development, but their timing relative to SST is often ambiguous in reanalysis data

If wind is a response to pressure, and pressure is a response to thermal forcing, and thermal forcing is modulated by tides, then the causal chain is:

$\text{Tide} \rightarrow \text{Thermal/Pressure Perturbation} \rightarrow \text{Wind Response} \rightarrow \text{SST Change via Feedback}$

Wind is two steps removed from the root cause. Treating it as the trigger is causally inverted.

Non-homogeneous & non-autonomous

June 16, 2026July 26, 2026 / Paul Pukite (@whut) / 3 Comments

These two characteristics of differential equations often go hand in hand. A tidal response is fundamentally a forced problem, not a free oscillation problem. In differential-equation language, the primacy of forcing makes it both non-homogeneous and non-autonomous. In other familiar calculus terms, tidal responses are particular solutions of non-homogeneous, non-autonomous dynamical systems, forced by explicitly time-dependent lunisolar potentials, superposed on the system’s autonomous homogeneous modes. The following post will make it clear that (i) tides are not free modes, (ii) the forcing is external, and (iii) the external forcing introduces explicit time dependence beyond the internal state evolution. We then ask the question: Can this be extended beyond conventional tides?

Continue reading →

NAO unfiltered

June 11, 2026 / Paul Pukite (@whut) / 1 Comment

The north Atlantic oscillation (NAO) is a most erratic climate index, often showing full cycles spanning less than a year. It is intimidating to consider that the raw NAO time-series can even be modelled, as it is often characterized as a weather precursor/indicator for Europe . Four years ago, I posted this to the ATTP blog:

Text excerpt discussing tidal cycles, boundary-element models, and data analysis techniques related to the NAO and Arctic Oscillation.

Graph depicting dLOD fit to NAO from 1950 to 2020, showing data and model comparisons in intensity.

This was fit applying the usual LTE model: a semi-annual stroboscopic sample of a LOD-derived forcing, integrated and then an LTE modulation applied. The semi-annual impulses were allowed to bleed through as they seem to capture the fast cycling well. The LTE modulation is approximately a winding=1 for each + or – semi-annual excursion. One can see the 3.8-year aliased cycling of the principal Mf tidal factor in the middle forcing panel, and an overall 18.6-year envelope.

On recreating this fit the past few days, I couldn’t duplicate the exact same configuration from the repo, but came close: (note in all the following that the training and validation legend is reversed in the upper right regression panel)

Time series analysis of NAO data showing model and data with validation over the years.

In both cases, the same prominent 3.8-year cycles in forcing (with 18.6-year envelope) were observed, along with essentially the same LTE modulation. The algorithm behind the fit is to perturb the calibrated LOD tidal factors enough so they will start to align with the NAO time series, especially synching the phases since timing is so crucial. In general, it’s difficult to achieve the same height of excursions, but the running windowed correlation (bottom panel with GREEN curve) shows a uniform accounting of the full cycling. Below is a DTW metric trained fit, which does a dynamic time warp on the time-axis, thus relaxing the alignment on each excursion. The improvement is subtle.

Time series plot showing Site#nao data with blue and red lines representing model and measured values over years from 1950 to 2020.

The alignment with the original LOD calibration was shifted by approximately 2-years (see below), but this could also be due to derivative adjustments. It’s not clear which order of forcing that the NAO is responding to — is it just the original dLOD/dt or some higher order acceleration? A higher derivative would generate a lead factor, i.e. a sin(t) derivative would turn into a cos(t) lead term.

Line graph depicting tidal force time series from 1960 to 2020, featuring two datasets: 'LOD' in blue and 'model' in red, showing fluctuating values between -3 and 2.

Different perspective

A time series graph titled 'Tidal for time series' showing fluctuating values across a timeline from 1990 to 2020. The data is represented with red and blue colored areas indicating variations in values over time.

Shorter interval

Line graph depicting tidal force time series from 1962 to 1967, showing the value on the vertical axis with red and blue lines representing different models and variations.

The NAO used above covers years after 1950 and stops short of 2018. On Climate Explorer: Monthly time series one can find other NAO variants that cover a wider timespan interval. In the plot below, the NAO is extended back to 1880 and forward to nearly present day. The back extrapolation is not the best but the forward beyond 2015 is excellent. The fact that the LTE modulation in the middle right panel still shows a strong low-order winding indicates that the fundamental patter remains across the entire timespan but is structurally sensitive to the fitted parameters.

Time series graph of NAO values from 1880 to 2020, showing model and data trends.

A line graph depicting a tidal force time series from 1960 to 2020, with red lines representing LOD and blue lines indicating the model values. The graph shows fluctuations in values over the years.

Refitting by including the back-extrapolated interval prior to 1953 does not significantly reduce the correlation in the excluded post-2015 interval.

Top left: Time series graph for site NAO from 1880 to 2020 showing model and data values.

Structural sensitivity in this class of models is significant. Think about how tides with daily to monthly variation need to be calibrated to effects that evolve over decades. This will take time and effort to improve, likely with input from others that can contribute subtleties that I have missed. Yet, doing this kind of deep fitting exercise is at odds with those that have been indoctrinated to a weather mindset of predicting only what’s in store for the immediate future.

A time-series graph depicting the North Atlantic Oscillation (NAO) with data showing 80 peak excursions since 1950. — The answer to why not predict the next <blank>?
RealClimate: Unforced Variations: June 2026

Revisit Sea Level Pressure paper

May 29, 2026 / Paul Pukite (@whut) / Leave a comment

Revisiting the post An obvious clue from tidal data, which is an analysis of this paper The effect of regional sea level atmospheric pressure on …

Time series plot showing atmospheric pressure in millibars (mbar) in New York, USA, from 1850 to 2025, with data points scattered around the range of 980 to 1040 mbar.

At the time, I did not try to duplicate the results, but after coming across it again, curiosity got the best of me. The program to duplicate the results is in this GIST: Analysis of SLP from PSMSL stations, using HadSLP2.

The author really didn’t clarify that the striations/bands/gaps in readings (above from paper and recreated below, left) were simply due to the discrete nature of monthly readings placed against a strong seasonal variation. For example, the NYC (The Battery) and Boston location have a significant seasonal SLP response due to the geographic characteristic of cold-season continental highs alternating with warm-season lower-pressure maritime regimes.

Scatter plot showing sea level pressure in New York from 1840 to 2020, with a detailed breakdown on the right illustrating monthly data gaps.

For New York, explain this figure [📷 nyc_monthly_slp.png]. On the right the values are replotted as (Month modulo 12) to indicate which months corresponded to the left.

Copilot response:

A code snippet displaying analysis related to the monthly sea-level pressure in New York, with a focus on seasonal effects and pressure clusters for different months.

Of course there is other stuff lurking in the data, so it is good to have the HadSLP2 as an adjunct to the PSMSL data I am using here

Consider the Honolulu SLP data below. There’s a clear Hovmller-like sloped ridge in the data, as one’s eye can detect, emphasized by the highlighter.

Line graph depicting data trends in Honolulu from 1840 to 2020, with four highlighted yellow trend lines and scattered blue data points.

The testing of lines in the rotated frame of the form u = p - s t finds this in the power spectrum. See It reads a HadSLP2 TSV, searches over diagonal slopes in rotated coordinates u = pressure – slope * time, identifies the strongest ridge, writes the power spectrum and ridge profile as TSVs, and saves a PNG with the ridge overplotted on the scatter plus the spectrum. See the previous GIST

Graph showing the Honolulu rotated-band power spectrum for the lower band 1006.5-1013.5 hPa, illustrating power against propagation period in years, with a peak at 114.2 years marked by a dashed red line.

A plausible explanation is a multidecadal modulation of the regional seasonal SLP cycle by North Pacific basin circulation variability, of which PDO/IPO variability is a plausible contributor.

ENSO and AMO manifolds

May 23, 2026May 26, 2026 / Paul Pukite (@whut) / 5 Comments

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 graph showing the Length of Day (LOD) delta in milliseconds over the years from 1960 to 2010, with marked points at 1983.1, 1992.2, 1997.9, and 2007.3.

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.

A bar chart displaying various amplitude values across different categories labeled Mf', Mm, Mt, and others, with the highest amplitude represented by Mf and a smaller peak for Mt'.

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

Line graph depicting the values of a latent or hidden forcing layer over the years from 1880 to 2022, with a light blue background and red lines illustrating fluctuations in the data.

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:

A diagram illustrating the relationship between manifold input (blue curve), modulation/transfer function (red curve), and system response (green curve) over time. It includes panels with labeled points indicating key features of the functions.

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.

Graph showing AMO time series with model and data comparisons, including regression analysis, latent forcing layer, running windowed correlation, and power spectrum.

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.

A time series plot showing the standardized values of two distinct factors from 1960 to 2020, illustrated with varying shades of blue.

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.

Bar chart showing absolute amplitudes of 32 tidal factors related to AMO, with varying heights representing different amplitude values.

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

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.

Time series analysis for Site nino4, displaying model and data trends from 1880 to 2022.

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.

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.

Time series plot of PDO data from 1880 to 2020, showing model and actual data with varying values.

NAO — North Atlantic Oscillation

Time series graph displaying NAO data from 1880 to 2026, showing model and actual data in blue and red.

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

Time series graph for Site #88 showing data and model values over the years 1900 to 2026.

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.

Time series graph for Site#iode with validation results representing model and data from 1880 to 2020.

TNA (Tropical North Atlantic) — has characteristics of AMO

A series of four graphs displaying time series data, model performance, and correlation analysis over a timeline from 1880 to 2026, featuring various statistical metrics and modulations.

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

Time series data chart showing cross-validation results from 1880 to 2026, including model and data comparisons.

World map highlighting the West Coast of the USA with 24 active stations shown in red out of a total of 51 stations.

Set of west coast MSL sites

Line graph displaying time series data from Site#wcoast, showing model and data values over the years from 1880 to 2022.

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.

A grid of nine bar charts, each displaying data trends with horizontal bars in blue and grey. — Eye-chart of tidal factor weightings
top row: NINO4, AMO, PDO
middle: NAO, IOD, TNA
bottom : Ratan, TSA, West Coasr

Tidal Amplitude Spectrum (Complete)

Tidal Amplitude Spectrum

Complete dataset with all 32 periods from each directory

Dataset Summary: All 9 directories (nino4, amo, pdo, nao, iode, tsa, tna, 88, wcoast) with identical period structure. Periods > 365 days shown in gray background.

Loading complete dataset…

Period (days)	nino4	amo	pdo	nao	iode	tsa	tna	88	wcoast
27.322	0.00766	-0.00284	0.01103	-0.00969	-0.00739	0.00909	0.01308	-0.01308	-0.00607
27.212	-0.00962	0.00230	0.00945	0.00914	0.00161	0.01352	0.00803	0.01839	0.01288
1095.175	-0.01868	0.02688	0.01759	0.03896	0.01075	-0.00192	0.02392	0.00838	-0.01754
13.633	0.12949	0.13477	0.12649	0.12599	0.12452	0.11459	0.12611	0.12162	0.12957
13.661	-0.32865	-0.32034	-0.31731	-0.31815	0.31652	0.32392	-0.33320	-0.27666	-0.33224
13.606	0.00260	0.00416	0.01435	0.00698	0.01250	0.00842	0.00101	0.00438	-0.00091
27.555	0.09385	0.10842	0.10792	0.12372	0.08670	0.07237	0.09408	0.09734	0.07357
13.777	0.01859	-0.00784	-0.02674	-0.02157	0.02300	0.03332	0.03225	0.01120	0.01611
6793.476	0.02504	0.02121	0.01668	0.02042	0.00315	-0.00828	0.02291	0.01651	-0.01100
1616.303	0.01552	0.02624	0.01918	0.03517	0.01843	-0.02135	-0.01741	0.01525	0.01810
31.812	0.01048	0.03055	0.00594	0.03619	0.01400	0.02974	0.01963	0.02012	0.01433
1305.660	0.01901	0.03236	-0.03170	0.03310	0.01787	-0.00542	0.03654	0.03207	0.02456
27.443	0.00652	0.01137	0.02500	0.01133	0.00575	0.00662	-0.01227	0.02264	0.01039
27.667	0.00577	-0.00979	-0.01894	-0.00478	0.00373	-0.00881	0.00596	-0.01864	0.00804
2190.350	-0.00067	-0.00377	-0.01081	-0.00453	0.01231	0.02257	0.00866	-0.00940	0.00176
26.985	-0.00403	-0.01943	-0.02923	-0.02621	0.01077	-0.01067	-0.02716	-0.03490	-0.00994
6167.207	0.02099	0.02246	0.01566	0.02475	0.00688	-0.00498	0.02517	0.02452	0.02442
-3232.605	-3.23261	-3.23261	-3.23261	-3.23261	-3.23261	-3.23261	-3.23261	-3.23261	-3.23261
-2120.909	-2.12091	-2.12091	-2.12091	-2.12091	-2.12091	-2.12091	-2.12091	-2.12091	-2.12091
9.133	-0.09976	-0.09955	-0.09969	-0.09937	0.08648	0.08693	-0.09904	-0.10074	-0.10239
9.121	0.03385	0.03216	0.03412	0.03067	0.02595	0.03089	0.03352	0.02750	-0.03579
9.108	0.00603	0.00756	-0.01066	0.01400	0.01007	-0.01007	0.00698	0.01054	-0.01824
3396.738	0.02810	0.02401	0.01968	0.02000	0.00397	-0.02470	0.03562	0.02245	0.03287
1656.320	0.02340	0.01163	0.00986	0.00633	0.03245	-0.01107	0.03341	0.00394	0.05237
9.557	0.03658	0.04203	0.04366	0.04931	0.03202	0.02256	0.04357	0.05432	0.05022
9.185	0.00346	-0.00769	-0.01111	-0.00180	-0.00351	0.01015	-0.01492	-0.00934	-0.00787
9.543	-0.05449	0.00000	0.00000	-0.01272	0.01353	0.03086	-0.02726	-0.05449	0.02336
7.088	-0.01554	-0.02351	-0.03860	-0.00661	0.00934	-0.00321	0.00462	-0.05186	0.03153
6.852	0.00726	0.01444	-0.01266	0.02624	0.00929	-0.00650	-0.01818	0.01203	0.01498
6.859	0.01970	0.01144	0.00828	0.01353	0.01679	0.00888	0.00510	-0.00622	0.01316
7.096	-0.00619	0.00846	0.01787	0.01701	0.00989	0.01929	0.00928	0.00468	0.00447
5.643	-0.00619	0.00846	0.01787	0.01701	0.00989	0.01929	0.00928	0.00468	0.00447

Note: Gray background indicates periods > 365 days (long periods). White background indicates periods ≤ 365 days (short periods).

Important: Values are amplitudes from the lt.exe.p JSON files. Negative values indicate phase differences.

Total entries: 32 periods × 9 directories = 288 amplitude values displayed.

Graph depicting composite sinusoidal waves from 1860 to 1970, showing standardized normalized amplitudes over time. The plot features labeled axes with date markers and multiple colored curves representing different waveforms. — The IOD-East and TSA, both south of the equator are close to the LOD composite

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.

References

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 ↩︎
Mathematical Geoenergy, Pukite, P.R. et al, (Wiley/AGU, 2019) ↩︎
Arnold tongue - Wikipedia ↩︎
Mathematical Geoenergy, Pukite, P.R. et al, (Wiley/AGU, 2019) ↩︎
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.. ↩︎
see 2. ↩︎
Delplace, Marston, Topological origin of equatorial waves. Science 358,1075-1077(2017). DOI:10.1126/science.aan8819 ↩︎
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 ↩︎

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GeoEnergy Math

The Mathematics of GeoEnergy