More to come on this ansatz …
preliminary model fits below the fold
Continue readingDynamics
Temporal and Spatial behavior of climate
MSet
Global Models of Sea Surface Temperature
These are a set of 6 EOFs that describe the global SST in terms of a set of orthogonal time-series — essentially non-overlapping, each having a cross-correlation of ~0.0 with the others, like a sine/cosine pair, but in both spatial and temporal dimensions.
Continue readingThe Power of Darwin
Using the LTE model on the NINO4 ENSO time-series, I excluded the interval between 1870-1875 for cross-validation. The fit below is from scratch on a dLOD-calibrated initial forcing, then allowing the values to vary slightly but the dLOD remains at 0.9999668 of the initial calibration.
The cross-validation doesn’t look good and actually is anti-correlated to a CC = -0.5 over that interval. But the region before 1875 appears suspiciously flat in any case, see the data on the KNMI site — it’s only a certainty that the El Nino peak at 1877-1878 is real and not estimated.
So, the next step is to take the fit to NINO4 and apply it to the Darwin (SOI) time-series, applying the same fitting interval from 1875-present. Immediately, it captures the 3 positive excursions in the interval 1870 to 1875. And then letting the fit proceed to a high CC, the result is shown below:
The resultant fit in the excluded cross-validation region reaches 0.58, thus reversing the anti-correlation and confirming that the NINO4 time-series data prior to 1875 is likely incorrect.
The fit data is at https://gist.github.com/pukpr/c26f1da00337e92dbb47671ca48af2cf?permalink_comment_id=4639783#gistcomment-4639783. The main modifications to tidal factors are in the very long periods — these values start small due to the dLOD calibration (the differential filters out low frequencies) but the 4.42 year amplitude is nearly tripled after the fit, the 8.85 & 9.3 year up by 50%. The 18.6 year is only up 14%, and the 3rd order 6 year and 15.9 year are down 63% and up 45% respectively. All the fortnightly and monthly values are stable. This is perhaps reasonable considering how much the LOD drifts over time at low frequency, and that the calibration is restricted to post-1962.
The Big 10 Climate Indices
The above diagram courtesy of Karnauskus
Overall I’m confident with the status of the published analysis of Laplace’s Tidal Equations in Mathematical Geoenergy, as I can model each of these climate indices with precisely the same (save one ***) tidal forcing, all calibrated by LOD. The following Threads allow interested people to contribute thoughts
- ENSO – https://www.threads.net/@paulpukite/post/CuWS8MFu8Jc
- AMO – https://www.threads.net/@paulpukite/post/Cuh4krjJTLN
- PDO – https://www.threads.net/@paulpukite/post/Cuu0VCypIi5
- QBO – https://www.threads.net/@paulpukite/post/CuiKQ5tsXCQ
- SOI (Darwin & Tahiti) – https://www.threads.net/@paulpukite/post/Cuu2IkBJh55 => MJO
- IOD (East & West) – https://www.threads.net/@paulpukite/post/Cuu9PYvJAG2
- PNA – https://www.threads.net/@paulpukite/post/CuvAVR7JN7R
- AO – https://www.threads.net/@paulpukite/post/CuvEz37JPFV
- SAM – https://www.threads.net/@paulpukite/post/CuvLZ2CMt1X
- NAO – https://www.threads.net/@paulpukite/post/CuvNnwns2la
(*** The odd-one out is QBO, which is a global longitudinally-invariant behavior, which means that only a couple of tidal factors are important.)
Is the utility of this LTE modeling a groundbreaking achievement? => https://www.threads.net/@paulpukite/post/CuvNnwns2la
Canonical Cross-Validation
The only hope for a non-controlled-experiment-verified model to gain acceptance is either by (1) showing repeated success in predictions, or, precluding that due to long cycle time (2) producing rock-solid cross-validation results. Why? Let ChatGPT-4 answer:
A Digital Twin of ENSO
The idea of a digital twin is relatively new in terms of coinage of terms, but the essential idea has been around for decades. In the past, a digital twin was referred to as a virtual simulation of a specific system, encoded via a programming language. In the case of a system that was previously built, the virtual simulation emulated all the behaviors and characteristics of that system, only operated on a computer, with any necessary interactive controls and displays provided on a console, either real or virtual. A widely known example of a VS is that of a flight simulator, which in historical terms was the industrial forerunner to today’s virtual reality. A virtual simulation could also be used during the design of the system, with the finished digital twin providing a blueprint for the actual synthesis of the end-product. This approach has also been practiced for decades, both in the electronics industry via logic synthesis of integrated circuits from a hardware description language and with physical products via 3D printing from CAD models.
Continue readingGist Evaluation
The Gist site on GitHub allows you to comment on posts very easily. For example, images of charts can be pasted in the discussion area. Also snippets of code can be added and updated, which is useful for neural net evaluation. The following is a link to an initial Gist area for evaluating LTE models.
LTE Modulation & Neural Nets via Taylor’s
Is there a connection between the modulation of Laplace’s Tidal Equation (LTE) solutions and the highly nonlinear fits of neural networks?
“Neural tensor networks have been widely used in a large number of natural language processing tasks such as conversational sentiment analysis, named entity recognition and knowledge base completion. However, the mathematical explanation of neural tensor networks remains a challenging problem, due to the bilinear term. According to Taylorโs theorem, a kth order differentiable function can be approximated by a kth order Taylor polynomial around a given point. Therefore, we provide a mathematical explanation of neural tensor networks and also reveal the inner link between them and feedforward neural networks from the perspective of Taylorโs theorem. In addition, we unify two forms of neural tensor networks into a single framework and present factorization methods to make the neural tensor networks parameter-efficient. Experimental results bring some valuable insights into neural tensor networks.”
https://w.sentic.net/new-perspective-for-neural-tensor-networks.pdf
The connection is via Taylor’s series expansion whereby neural nets try to resolve tight inflection points that occur naturally in the numerical flow of potentially turbulent fluid dynamics.
- Li, Wei, Luyao Zhu, and Erik Cambria. “Taylorโs theorem: A new perspective for neural tensor networks.” Knowledge-Based Systems 228 (2021): 107258.
- Zhao, H., Chen, Y., Sun, D., Hu, Y., Liang, K., Mao, Y., … & Shao, H. “TaylorNet: A Taylor-Driven Generic Neural Architecture”. submitted to ICLR 2023
Also wavelets can maneuver tight inflections.
“In this study, the applicability of physics informed neural networks using wavelets as an activation function is discussed to solve non-linear differential equations. One of the prominent equations arising in fluid dynamics namely Blasius viscous flow problem is solved. A linear coupled differential equation, a non-linear coupled differential equation, and partial differential equations are also solved in order to demonstrate the methodโs versatility. As the neural networkโs optimum design is important and is problem-specific, the influence of some of the key factors on the modelโs accuracy is also investigated. To confirm the approachโs efficacy, the outcomes of the suggested method were compared with those of the existing approaches. The suggested method was observed to be both efficient and accurate.”
https://www.nature.com/articles/s41598-023-29806-3
- Uddin, Z., Ganga, S., Asthana, R. et al. Wavelets based physics informed neural networks to solve non-linear differential equations. Sci Rep 13, 2882 (2023). https://doi.org/10.1038/s41598-023-29806-3
Given the fact that NN results are so difficult to reverse engineer to match a physical understanding could this be a missing link?
Continue readingLimits of Predictability?
A decade-old research article on modeling equatorial waves includes this introductory passage:
“Nonlinear aspects plays a major role in the understanding of fluid flows. The distinctive fact that in nonlinear problems cause and effect are not proportional opens up the possibility that a small variation in an input quantity causes a considerable change in the response of the system. Often this type of complication causes nonlinear problems to elude exact treatment. “
https://doi.org/10.1029/2012JC007879
From my experience if it is relatively easy to generate a fit to data via a nonlinear model then it also may be easy to diverge from the fit with a small structural perturbation, or to come up with an alternative fit with a different set of parameters. This makes it difficult to establish an iron-clad cross-validation.
This doesn’t mean we don’t keep trying. Applying the dLOD calibration approach to an applied forcing, we can model ENSO via the NINO34 climate index across the available data range (in YELLOW) in the figure below (parameters here)
The lower right box is a modulo-2ฯ reduction of the tidal forcing as an input to the sinusoidal LTE modulation, using the decline rate (per month) as the divisor. Why this works so well per month in contrast to per year (where an annual cycle would make sense) is not clear. It is also fascinating in that this is a form of amplitude aliasing analogous to the frequency aliasing that also applies a modulo-2ฯ folding reduction to the tidal periods less than the Nyquist monthly sampling criteria. There may be a time-amplitude duality or Lagrangian particle-relabeling in operation that has at its central core the trivial solutions of Navier-Stokes or Euler differential equations when all segments of forcing are flat or have a linear slope. Trivial in the sense that when a forcing is flat or has a 1st-order slope, the 2nd derivatives due to divergence in the differential equations vanish (quasi-static). This means that only the discontinuities, which occur concurrently with the annual ENSO predictability barrier, need to be treated carefully (the modulo-2ฯ folding could be a topological Berry phase jump?). Yet, if these transitions are enhanced by metastable interface instabilities as during thermocline turn-over then the differential equation conditions could be transiently relaxed via a vanishing density difference. Much happens during a turn-over, but it doesn’t last long, perhaps indicating a geometric phase. MV Berry also discusses phase changes in the context of amphidromic tidal singularities here.
Suffice to say that the topological properties of reduced dimension volumes and at interfaces remain mysterious. The main takeaway is that a working NINO34-fitted ENSO model is produced, and if not here then somewhere else a machine-learning algorithm will discover it.
The key next step is to apply the same tidal forcing to an AMO model, taking care not to change the tidal factors enough to produce a highly sensitive nonlinear response in the LTE model. So we retain an excluded interval from training (in YELLOW below) and only adjust the LTE parameters for the region surrounding this zone during the fitting process (parameters here).
The cross-validation agreement is breathtakingly good in the excluded (out-of-band) training interval. There is zero cross-correlation between the NINO34 and AMO time-series to begin with so that this is likely revealing the true emergent characteristics of a tidally forced mechanism.
As usual all the introductory work is covered in Mathematical Geoenergy
- https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch11 (wind)
- https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch12 (wave)
A community peer-review contributed to a recent QBO article is here and PDF here. The same question applies to QBO as ENSO or AMO: is it possible to predict future behavior? Is the QBO model less sensitive to input since the nonlinear aspect is weaker?
Added several weeks later: This monograph PDF available “Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects”. Ignoring higher-order time derivatives is key to solving LTE.
Note the cite to Billy Kessler
Gerstner waves
An exact solution for equatorially trapped waves
Nonlinear aspects plays a major role in the understanding of fluid flows. The distinctive fact that in nonlinear problems cause and effect are not proportional opens up the possibility that a small variation in an input quantity causes a considerable change in the response of the system. Often this type of complication causes nonlinear problems to elude exact treatment. A good illustration of this feature is the fact that there is only one known explicit exact solution of the (nonlinear) governing equations for periodic two-dimensional traveling gravity water waves. This solution was first found in a homogeneous fluid by Gerstner
Adrian Constantin, JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, C05029, doi:10.1029/2012JC007879, 2012
These are trochoidal waves
Even within the context of gravity waves explored in the references mentioned above, a vertical wall is not allowable. This drawback is of special relevance in a geophysical context since [cf. Fedorov and Brown, 2009] the Equator works like a natural boundary and equatorially trapped waves, eastward propagating and symmetric about the Equator, are known to exist. By the 1980s, the scientific community came to realize that these waves are one of the key factors in explaining the El Niรฑo phenomenon (see also the discussion in Cushman-Roisin and Beckers [2011]).
modulo-2ฯ and Berry phase