Prompt to Google AI: possible that NN are effective in creating realistic fluid dynamics because they are emulating winding numbers that naturally arise in waves?
More to come on this…
Nonlinear Fluid Dynamics and NN
Dynamics
Temporal and Spatial behavior of climate
Deep Seek of Wind + ENSO
This is a thread with DeepSeek, my prompts in italics bold. It was highly critical in ways that appeared very articulate, but it started to acquiesce when supplied with additional information. Not sure if this is due to a sliding or widening context window. In the past, one can try to argue about established physics with an LLM and it typically won’t budge.
The previous post was a discussion with the Gemini LLM and that seemed to be more agreeing and eventually convinced that a topological insulator approach seemed the best fit, which Deep Seek acknowledged as promising but did not use as a complete framing. Grade: Deep Seek – A for toughness, Gemini – B+ for diving into the data
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Wind drag and ENSO ocean dynamics
Is this realistic to think that the wind “drags the hot surface water from East Pacific and piles it up in the West Pacific” is that which causes the ENSO phenomenon? How much volume would this displace and is it even plausible? We know that can happen with a lake, yet that is a more confined basin.
This is an excellent and insightful question. You’ve zeroed in on a common oversimplification of ENSO (El Niño-Southern Oscillation) and are asking whether the “dragging and piling” mechanism is literally plausible across an entire ocean basin.
Let’s break it down.
The Short Answer
The idea that trade winds directly drag a significant volume of warm surface water from the East Pacific to the West Pacific, piling it up like water against one side of a bathtub, is an intuitive but physically incomplete picture of what causes ENSO. However, it captures a real, measurable effect — but the volume displaced is surprisingly small relative to the basin, and the “piling” is tiny compared to other forces. The real driver of ENSO is not just wind dragging water, but wind changing the slope of the thermocline (the boundary between warm surface water and cold deep water).
Let’s do the math and physics.
Continue readingGlobal Forcing, Local Modulation Revealed
Gemini — Conversation with Gemini
The two CSV files are for NINO4 (lte_results) and MSL at Warnemunde (lte_results11). The 1st column is a shared time base. The 3rd column is data, and 2nd column is a model that takes a nearly shared 4th column (a hidden latent forcing manifold) and applies non-autonomous sinusoidal modulations to the 4th column to arrive at the 2nd column. In each case, the model matches the data very well despite there being little correlation to the time-series. It is known that El Nino have far-reaching effects but little is know on the common-modes of expression. Deep think about how this can be further explicated.
The results from your provided data clearly confirm your premise. When we compare the raw Data (Column 3) for NINO4 and Warnemünde Mean Sea Level (MSL), there is virtually no correlation (Pearson r≈−0.0878). However, looking at the hidden latent forcing manifold (Column 4), the signals are nearly identical, showing a massive correlation of r≈0.9775.
This is a profound observation. It suggests that two geographically and dynamically distinct systems—equatorial Pacific sea surface temperatures and Baltic/North Sea coastal water levels—are actually synchronized to the same low-dimensional global “pacemaker,” despite their raw time-series appearing entirely decoupled.
To further explicate this finding, we can break the “deep think” into three primary dimensions: Physical Mechanisms, Mathematical/Dynamical Explication, and Future Analytical Pathways.
Continue readingForcing selection
The Perplexity response:
Continue readingGEM-LTE modeling
February 2026 Cross-Validation Experiments: GEM-LTE Mean Sea Level and Climate Index Modelling
Executive Summary
GEM-LTE/experiments/Feb2026 at main · pukpr/GEM-LTE
This directory contains results from a comprehensive cross-validation study applying the GEM-LTE (GeoEnergyMath Laplace’s Tidal Equation) model to 79 tide-gauge and climate-index time series spanning the 19th through early 21st centuries. The defining constraint of this study is a common holdout interval of 1940–1970: the model is trained exclusively on data outside this thirty-year window, and each subdirectory’s lte_results.csv and *site1940-1970.png chart record how well the trained model reproduces the withheld record.
The headline finding is that a single latent tidal manifold—constructed from the same set of lunisolar forcing components across all sites—achieves statistically significant predictive skill on the 1940–1970 interval for the great majority of the tested locations, with Pearson correlation coefficients (column 2 vs. column 3 of lte_results.csv) ranging from r ≈ 0.72 at the best-performing Baltic tide gauges to r ≈ 0.12 at the most challenging Atlantic stations. Because the manifold is common to every experiment while the LTE modulation parameters are fitted individually to each series, the cross-site pattern of validation performance is informative about which physical mechanisms link regional sea level (or climate variability) to the underlying lunisolar forcing—and about the geographic basin geometry that shapes each site’s characteristic amplitude response.
The GEM-LTE Model: A Common Latent Manifold with Variable LTE Modulation
read more below, and contribute here: Discussions · pukpr/GEM-LTE · Discussion #6
Continue readingForcing of NAO and MSL in the Baltic
Modeling LTE solutions from pukpr/GEM-LTE: GeoEnergyMath Laplaces Tidal Equation modeling and fitting software
Warnemunde Mean Sea Level (#11 from PSMSL.org). Dashed lines are cross-validation intervals.
North Atlantic Oscillation
NAO (top) and Warnemunde MSL (bottom) latent forcing layer
Hoyer Metric in LTE Model Fitting
Modern signal processing and system identification frequently require quantifying the sparseness or “peakiness” of vectors—such as power spectra. The Hoyer metric, introduced by Hoyer [2004], is a widely adopted measure for this purpose, especially in the context of nonnegative data (like spectra). This blog post explains the Hoyer metric’s role in fitting models in the context of LTE, its mathematical form, and provides references to its origins.
What Is the Hoyer Sparsity Metric?
Given a nonnegative vector (), the Hoyer sparsity is defined as:
Where:
- is the L1 norm (sum of absolute values).
- is the L2 norm (Euclidean norm).
- is the length of the vector.
The Hoyer metric ranges from 0 (completely distributed, e.g., flat spectrum) to 1 (maximally sparse, only one element is nonzero).
Why Use the Hoyer Metric in Fitting?
In signal processing and model fitting, especially where spectral features are important (e.g., EEG/MEG analysis, telecommunications, and fluid dynamics in the context of LTE), one often wants to compare not only overall power but the prominence of distinct peaks (spectral peaks) in data and models.
The function used in the LTE model, Hoyer_Spectral_Peak, calculates the Hoyer sparsity of a vector representing the spectrum of the observed data. When used in fitting, it serves to:
- Quantify Peakiness: Models producing spectra closer in “peakiness” to the data will better mirror the physiological or system constraints.
- Regularize Models: Enforcing a match in sparsity (not just in power) can avoid overfitting to distributed, non-specific solutions. It’s really a non-parametric modeling approach
- Assess Structure Beyond RMS or Mean: Hoyer metric captures distribution shape—crucial for systems with sparse or peaky energy distributions.
Hoyer Metric Formula in the Code
The provided Ada snippet implements the Hoyer sparsity for a vector of LTE manifold data points. Here’s the formula as used:
-- Hoyer_Spectral_Peak
--
function Hoyer_Spectral_Peak (Model, Data, Forcing : in Data_Pairs) return Long_Float is
Model_S : Data_Pairs := Model;
Data_S : Data_Pairs := Data;
L1, L2 : Long_Float := 0.0;
Len : Long_Float;
RMS : Long_Float;
Num, Den : Long_Float;
use Ada.Numerics.Long_Elementary_Functions;
begin
ME_Power_Spectrum
(Forcing => Forcing, Model => Model, Data => Data, Model_Spectrum => Model_S,
Data_Spectrum => Data_S, RMS => RMS);
Len := Long_Float(Data_S'Length);
for I in Data_S'First+1 .. Data_S'Last loop
L1 := L1 + Data_S(I).Value;
L2 := L2 + Data_S(I).Value * Data_S(I).Value;
end loop;
L2 := Sqrt(L2);
Num := Sqrt(Len) - L1/L2;
Den := Sqrt(Len) - 1.0;
return Num/Den;
end Hoyer_Spectral_Peak;
Where all (). This is exactly as described in Hoyer’s paper.
Example Usage
Suppose the observed spectrum is more “peaky” than the model spectrum. By matching the Hoyer metric (alongside other criteria), the fitting procedure encourages the model to concentrate energy into peaks, better capturing the phenomenon under study.
For the LTE study here, the idea is to non-parametrically apply the Hoyer metric to map the latent forcing manifold to the observed climate index time-series, using Hoyer to optimize during search. This assumes that sparser stronger standing wave resonances act as the favored response regime — as is observed with the sparse number of standing waves formed during ENSO cycles (a strong basin wide standing wave and faster tropical instability waves as described in Chapter 12 of Mathematical Geoenergy).
Using the LTE gui, the Hoyer metric is selected as H, and one can see that the lower right spectrum sharpens one or more spectral peaks corresponding to the Fourier series of the LTE modulation of the center right chart.
It’s non-parametric in the sense that the LTE modulation parameters are not specified, as they would need to be for the correlation coefficient metric that I ordinarily use. The index here (#11) is the Warnemunde MSL time-series.
Citation and References
The Hoyer sparsity metric was introduced in:
- Hoyer, P. O. (2004). “Non-negative matrix factorization with sparseness constraints.” Journal of Machine Learning Research, 5(Nov):1457–1469. [Link: JMLR Paper]
For further applications in neural data and spectral analysis, you may see usage such as:
- Bruns, A. (2004). “Fourier-, Hilbert- and wavelet-based signal analysis: Are they really different approaches?” Journal of Neuroscience Methods, 137(2):321-332.
Conclusion
The Hoyer metric is a robust, intuitive, and well-cited tool for quantifying sparsity in spectra or model parameters—encouraging interpretable, physiologically plausible solutions when fitting models to data. It seems to work better than similar metrics such as entropic complexity, see reference below, where I tried applying it in the same LTE problem solution domain.
Reference:
- Hoyer, P.O., “Non-negative matrix factorization with sparseness constraints,” JMLR, 5:1457–1469, 2004.\
- Pukite, P., & Bankes, S. (2011). Entropic Complexity Measured in Context Switching. In Applications of Digital Signal Processing. InTech. https://doi.org/10.5772/25520
Let me know if you’d like code snippets, visualization examples, or more advanced mathematical discussion!
The mystery of El Nino
Google Gemini makes predictions on the next El Nino when prompted — scroll to the end.
https://gemini.google.com/app/8d7c963c197e05c0
Continue readingThere has ALWAYS been stratification in the ocean via the primary thermocline. The intensity of an El Nino or La Nina is dependent on the “tilt” of the thermocline across the equatorial Pacific, like a see-saw or teeter-totter as the colder waters below the thermocline get closer to the surface or recede more to the depths.
The only mystery is to what provokes the motion. For a playground see-saw, it’s easy to understand as it depends on which side a kid decides to junp on the see-saw.
For the ocean, the explanation is less facile than that, explain.
Current status of research according to AI
explain the Pukite tidal theory
The Pukite Tidal Theory, primarily developed by researcher Paul Pukite, proposes that long-period tidal forcing is the underlying driver for several major geophysical and atmospheric cycles that have previously been considered erratic or unresolved. [1, 2]
The core of the theory is that small gravitational perturbations from the Moon and Sun, which are perfectly predictable, are “aliased” or modulated by seasonal cycles to create the complex behaviors seen in Earth’s systems. [3, 4]
Key Phenomena Addressed
Pukite applies this model to three main “unresolved mysteries” in geophysics:
- Quasi-Biennial Oscillation (QBO): A regular reversal of stratospheric winds. The theory argues that lunar nodal cycles, when combined with the annual solar cycle, create the roughly 28-month QBO period through a process called physical aliasing.
- El Niño Southern Oscillation (ENSO): An erratic oceanic temperature cycle. The model suggests ENSO is a “sloshing” response of the ocean to tractive gravitational forces, essentially treating it as a solution to Laplace’s Tidal Equations.
- Chandler Wobble: A small deviation in the Earth’s axis of rotation. The theory posits this is caused by an external lunar torque rather than internal Earth dynamics. [1, 2, 3, 5, 6, 7]
How the Model Works
- Lunar Gravitational Potential: The model starts by calculating the precise lunar gravitational potential over time.
- Seasonal Modulation: These fast lunar cycles are provoked by a seasonal (yearly) peak in energy. This “carrier” signal helps filter out short-term fluctuations and reveals long-term patterns.
- Physical Aliasing: Because these cycles are sampled or triggered by seasonal events (like solstices), the resulting data shows lower-frequency “aliased” harmonics that match the observed multi-year periods of ENSO and QBO. [4, 8, 9]
Scientific Reception
This theory is considered a novel and controversial alternative to standard geophysical models. While it offers high correlation with historical data, it has faced skepticism from mainstream physical oceanographers and meteorologists who argue that it may “shoehorn” complex fluid dynamics into a periodic framework. Much of the work is documented on the GeoEnergy Math blog and in discussion papers for the Earth System Dynamics (ESD) journal. [1, 4, 7, 10]
Would you like to see a more detailed breakdown of the mathematical aliasing used to explain the QBO’s 28-month cycle?
[1] https://esd.copernicus.org
[2] https://www.researchgate.net
[3] https://esd.copernicus.org
[5] https://acp.copernicus.org
[8] https://esd.copernicus.org
[10] https://esd.copernicus.org
This link may go stale, but here is the original response:
Hidden latent manifolds in fluid dynamics
The behavior of complex systems, particularly in fluid dynamics, is traditionally described by high-dimensional systems of equations like the Navier-Stokes equations. While providing practical applications as is, these models can obscure the underlying, simplified mechanisms at play. It is notable that ocean modeling already incorporates dimensionality reduction built in, such as through Laplace’s Tidal Equations (LTE), which is a reduced-order formulation of the Navier-Stokes equations. Furthermore, the topological containment of phenomena like ENSO and QBO within the equatorial toroid , and the ability to further reduce LTE in this confined topology as described in the context of our text Mathematical Geoenergy underscore the inherent low-dimensional nature of dominant geophysical processes. The concept of hidden latent manifolds posits that the true, observed dynamics of a system do not occupy the entire high-dimensional phase space, but rather evolve on a much lower-dimensional geometric structure—a manifold layer—where the system’s effective degrees of freedom reside. This may also help explain the seeming paradox of the inverse energy cascade, whereby order in fluid structures seems to maintain as the waves become progressively larger, as nonlinear interactions accumulate energy transferring from smaller scales.
Discovering these latent structures from noisy, observational data is the central challenge in state-of-the-art fluid dynamics. Enter the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm, pioneered by Brunton et al. . SINDy is an equation-discovery framework designed to identify a sparse set of nonlinear terms that describe the evolution of the system on this low-dimensional manifold. Instead of testing all possible combinations of basis functions, SINDy uses a penalized regression technique (like LASSO) to enforce sparsity, effectively winnowing down the possibilities to find the most parsimonious, yet physically meaningful, governing differential equations. The result is a simple, interpretable model that captures the essential physics—the fingerprint of the latent manifold. The SINDy concept is not that difficult an algorithm to apply as a decent Python library is available for use, and I have evaluated it as described here.
Applying this methodology to Earth system dynamics, particularly the seemingly noisy, erratic, and perhaps chaotic time series of sea-level variation and climate index variability, reveals profound simplicity beneath the complexity. The high-dimensional output of climate models or raw observations can be projected onto a model framework driven by remarkably few physical processes. Specifically, as shown in analysis targeting the structure of these time series, the dynamics can be cross-validated by the interaction of two fundamental drivers: a forced gravitational tide and an annual impulse.
The presence of the forced gravitational tide accounts for the regular, high-frequency, and predictable components of the dynamics. The annual impulse, meanwhile, serves as the seasonal forcing function, representing the integrated effect of large-scale thermal and atmospheric cycles that reset annually. The success of this sparse, two-component model—where the interaction of these two elements is sufficient to capture the observed dynamics—serves as the ultimate validation of the latent manifold concept. The gravitational tides with the integrated annual impulse are the discovered, low-dimensional degrees of freedom, and the ability of their coupled solution to successfully cross-validate to the observed, high-fidelity dynamics confirms that the complex, high-dimensional reality of sea-level and climate variability emerges from this simple, sparse, and interpretable set of latent governing principles. This provides a powerful, physics-constrained approach to prediction and understanding, moving beyond descriptive models toward true dynamical discovery.
An entire set of cross-validated models is available for evluation here: https://pukpr.github.io/examples/mlr/.
This is a mix of climate indices (the 1st 20) and numbered coastal sea-level stations obtained from https://psmsl.org/
https://pukpr.github.io/examples/map_index.html
- nino34 — NINO34 (PACIFIC)
- nino4 — NINO4 (PACIFIC)
- amo — AMO (ATLANTIC)
- ao — AO (ARCTIC)
- denison — Ft Denison (PACIFIC)
- iod — IOD (INDIAN)
- iodw — IOD West (INDIAN)
- iode — IOD East (INDIAN)
- nao — NAO (ATLANTIC)
- tna — TNA Tropical N. Atlantic (ATLANTIC)
- tsa — TSA Tropical S. Atlantic (ATLANTIC)
- qbo30 — QBO 30 Equatorial (WORLD)
- darwin — Darwin SOI (PACIFIC)
- emi — EMI ENSO Modoki Index (PACIFIC)
- ic3tsfc — ic3tsfc (Reconstruction) (PACIFIC)
- m6 — M6, Atlantic Nino (ATLANTIC)
- m4 — M4, N. Pacific Gyre Oscillation (PACIFIC)
- pdo — PDO (PACIFIC)
- nino3 — NINO3 (PACIFIC)
- nino12 — NINO12 (PACIFIC)
- 1 — BREST (FRANCE)
- 10 — SAN FRANCISCO (UNITED STATES)
- 11 — WARNEMUNDE 2 (GERMANY)
- 14 — HELSINKI (FINLAND)
- 41 — POTI (GEORGIA)
- 65 — SYDNEY, FORT DENISON (AUSTRALIA)
- 76 — AARHUS (DENMARK)
- 78 — STOCKHOLM (SWEDEN)
- 111 — FREMANTLE (AUSTRALIA)
- 127 — SEATTLE (UNITED STATES)
- 155 — HONOLULU (UNITED STATES)
- 161 — GALVESTON II, PIER 21, TX (UNITED STATES)
- 163 — BALBOA (PANAMA)
- 183 — PORTLAND (MAINE) (UNITED STATES)
- 196 — SYDNEY, FORT DENISON 2 (AUSTRALIA)
- 202 — NEWLYN (UNITED KINGDOM)
- 225 — KETCHIKAN (UNITED STATES)
- 229 — KEMI (FINLAND)
- 234 — CHARLESTON I (UNITED STATES)
- 245 — LOS ANGELES (UNITED STATES)
- 246 — PENSACOLA (UNITED STATES)
Crucially, this analysis does not use the SINDy algorithm, but a much more basic multiple linear regression (MLR) algorithm predecessor, which I anticipate being adapted to SINDy as the model is further refined. Part of the rationale for doing this is to maintain a deep understanding of the mathematics, as well as providing cross-checking and thus avoiding the perils of over-fitting, which is the bane of neural network models.
Also read this intro level on tidal modeling, which may form the fundamental foundation for the latent manifold: https://pukpr.github.io/examples/warne_intro.html. The coastal station at Wardemunde in Germany along the Baltic sea provided a long unbroken interval of sea-level readings which was used to calibrate the hidden latent manifold that in turn served as a starting point for all the other models. Not every model works as well as the majority — see Pensacola for a sea-level site and and IOD or TNA for climate indices, but these are equally valuable for understanding limitations (and providing a sanity check against an accidental degeneracy in the model fitting process) . The use of SINDy in the future will provide additional functionality such as regularization that will find an optimal common-mode latent layer,.