QBO

QBO Metrics

/ Paul Pukite (@whut) / Leave a comment

In addition to the standard correlation coefficient (CC) and RMS error, non-standard metrics that have beneficial cross-validation properties include dynamic time warp (DTW), complexity invariant-distance (CID) see [2], and a CID-modified DTW. The link above describes my implementation of the DTW metric but I have yet to describe the CID metric. It’s essentially the CC multiplied by a factor that empirically adjusts the embedded summed distance between data points (i.e. the stretched length) of the time-series so that the signature or look of two time-series visually match in complexity.

   CID = CC * min(Length(Model, Data))/ max(Length(Model, Data))

The authors of the CID suggest that it’s a metric based on “an invariance that the community seems to have missed”.

And a CID-modified DTW is thus:

CID = DTW * min(Length(Model, Data))/ max(Length(Model, Data))

I have tried this on the QBO model with good cross-validation results featuring up to-data data from https://www.atmohub.kit.edu/data/qbo.dat

These have similar tidal factor compositions and differ mainly in the LTE modulation and phase delay. As discussed earlier, any anomalies in the QBO behavior are likely the outcome of an erratic periodicity caused by incommensurate annual and draconic cycles and exaggerated by LTE.

from https://gist.github.com/pukpr/e562138af3a9da937a3fb6955685c98f

REFERENCES

[1] Batista, Gustavo EAPA, et al. “CID: an efficient complexity-invariant distance for time series.” Data Mining and Knowledge Discovery 28 (2014): 634-669.R
https://link.springer.com/article/10.1007/s10618-013-0312-3

Lunar Torque Controls All

/ Paul Pukite (@whut) / 13 Comments

Mathematical Geoenergy

The truly massive scale in the motion of fluids and solids on Earth arises from orbital interactions with our spinning planet. The most obvious of these, such as the daily and seasonal cycles, are taken for granted. Others, such as ocean tides, have more complicated mechanisms than the ordinary person realizes (e.g. ask someone to explain why there are 2 tidal cycles per day). There are also less well-known motions, such as the variation in the Earth’s rotation rate of nominally 360° per day, which is called the delta in Length of Day (LOD), and in the slight annual wobble in the Earth’s rotation axis. Nevertheless, each one of these is technically well-characterized and models of the motion include a quantitative mapping to the orbital cycles of the Sun, Moon, and Earth. This is represented in the directed graph below, where the BLUE ovals indicate behaviors that are fundamentally understood and modeled via tables of orbital factors.

The cyan background represents behaviors that have a longitudinal dependence
(rendered by GraphViz)

However, those ovals highlighted in GRAY are nowhere near being well-understood in spite of being at least empirically well-characterized via years of measurements. Further, what is (IMO) astonishing is the lack of research interest in modeling these massive behaviors as a result of the same orbital mechanisms as that which causes tides, seasons, and the variations in LOD. In fact, everything tagged in the chart is essentially a behavior relating to an inertial response to something. That something, as reported in the Earth sciences literature, is only vaguely described — and never as a tidal or tidal/annual interaction.

I don’t see how it’s possible to overlook such an obvious causal connection. Why would the forcing that causes a massive behavior such as tides suddenly stop having a connection to other related inertial behaviors? The answers I find in the research literature are essentially that “someone looked in the past and found no correlation” [1].

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Order overrides chaos

/ Paul Pukite (@whut) / 5 Comments

Dimensionality reduction of chaos by feedbacks and periodic forcing is a source of natural climate change, by P. Salmon, Climate Dynamics (2024)

Bottom line is that a forcing will tend to reduce chaos by creating a pattern to follow, thus the terminology of “forced response”. This has implications for climate prediction. The first few sentences of the abstract set the stage:

The role of chaos in the climate system has been dismissed as high dimensional turbulence and noise, with minimal impact on long-term climate change. However theory and experiment show that chaotic systems can be reduced or “controlled” from high to low dimensionality by periodic forcings and internal feedbacks. High dimensional chaos is somewhat featureless. Conversely low dimensional borderline chaos generates pattern such as oscillation, and is more widespread in climate than is generally recognised. Thus, oceanic oscillations such as the Pacific Decadal and Atlantic Multidecadal Oscillations are generated by dimensionality reduction under the effect of known feedbacks. Annual periodic forcing entrains the El Niño Southern Oscillation.

In Chapters 11 and 12 in Pukite, P., Coyne, D., & Challou, D. (2019). Mathematical Geoenergy. John Wiley & Sons, I cited forcing as a chaos reducer:

“It is well known that a periodic forcing can reduce the erratic fluctuations and uncertainty of a near‐chaotic response function (Osipov et al., 2007; Wang, Yang, Zhou, 2013).“

But that’s just a motivator. Tides are the key, acting primarily on the subsurface thermocline. Salmon’s figure comparing the AMO to Barents sea subsurface temperature is substantiating in terms of linking two separated regions by something more than a nebulous “teleconnection”.

Likely every ocean index has a common-mode mechanism. The tidal forcing by itself is close to providing an external synchronizing source, but requires what I refer to as a LTE modulation to zero in on the exact forced response. Read the previous blog post to get a feel how this works:

Common forcing for ocean indices

As Salmon notes, it’s known at some level that an annual/seasonal impulse is entraining or synchronizing ENSO, and also likely PDO and AMO. The top guns at NASA JPL point out that the main lunisolar terms are at monthly, 206 day, annual, 3 year, and 6 year periods, and this is what is used to model the forcing, see the following two charts

Now note how the middle panel in each of the following modeled climate indices does not change markedly. The most challenging aspect is the inherent structural sensitivity of the manifold1 mapping involved in LTE modulation. As the Darwin fit shows, the cross-validation is better than it may appear, as the out-of-band interval does not take much of a nudge to become synchronized with the data. Note also that the multidecadal nature of an index such as AMO may be ephemeral — the yellow cross-validation band does show valleys in what appears to be a longer multidecadal trend, capturing the long-period variations in the tides when modulated by an annual impulse – biennial in this case.

Model config repo: https://gist.github.com/pukpr/3a3566b601a54da2724df9c29159ce16?permalink_comment_id=5108154#gistcomment-5108154

1 The term manifold has an interesting etymology. From the phonetics, it is close to pronounced as “many fold”, which is precisely what’s happening here — the LTE modulation can fold over the forcing input many times in proportion to the mode of the standing wave produced. So that a higher standing wave will have “many folds” in contrast to the lowest standing wave model. At the limit, the QBO with an ostensibly wavenumber=0 mode will have no folds and will be to first-order a pass-through linear amplification of the forcing, but with likely higher modes mixed in to give the time-series some character.

Fundy-mental (continued)

/ Paul Pukite (@whut) / 1 Comment

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

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Bay of Fundy subbands

/ Paul Pukite (@whut) / 1 Comment

With the recent total solar eclipse, it revived lots of thought of Earth’s ecliptic plane. In terms of forcing, having the Moon temporarily in the ecliptic plane and also blocking the sun is not only a rare and (to some people) an exciting event, it’s also an extreme regime wrt to the Earth as the combined reinforcement is maximized.

In fact this is not just any tidal forcing — rather it’s in the class of tidal forcing that has been overlooked over time in preference to the conventional diurnal tides. As many of those that tracked the eclipse as it traced a path from Texas to Nova Scotia, they may have noted that the moon covers lots of ground in a day. But that’s mainly because of the earth’s rotation. To remove that rotation and isolate the mean orbital path is tricky.  And that’s the time-span duration where long-period tidal effects and inertial motion can build up and show extremes in sea-level change. Consider the 4.53 year extreme tidal cycle observed at the Bay of Fundy (see Desplanque et al) located in Nova Scotia. This is predicted if the long-period lunar perigee anomaly (27.554 days and the 8.85 absidal precessional return cycle) amplifies the long period lunar ecliptic nodal cycle, as every 9.3 years the lunar path intersects the ecliptic plane, one ascending and the other descending as the moon’s gravitational pull directly aligns with the sun’s.  The predicted frequencies are 1/8.85 ± 2/18.6 = 1/4.53 & 1/182, the latter identified by Keeling in 2000.  The other oft-mentioned tidal extreme is at 18.6 years, which is identified as the other long period extreme at the Bay of Fundy by Desplanque, and that was also identified by NASA as an extreme nuisance tide via a press release and a spate of “Moon wobble” news articles 3 years ago.

What I find troubling is that I can’t find a scholarly citation where the 4.53 year extreme tidal cycle is explained in this way. It’s only reported as an empirical observation by Desplanque in several articles studying the Bay of Fundy tides. 

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Proof for allowed modes of an ideal QBO

/ Paul Pukite (@whut) / 6 Comments

In formal mathematical terms of geometry/topology/homotopy/homology, let’s try proving that a wavenumber=0 cycle of east/west direction inside an equatorial toroidal-shaped waveguide, can only be forced by the Z-component of a (x,y,z) vector where x,y lies in the equatorial plane.

To address this question, let’s dissect the components involved and prove within the constraints of geometry, topology, homotopy, and homology, focusing on valid mathematical principles.

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Are the QBO disruptions anomalous?

/ Paul Pukite (@whut) / 10 Comments

Based on the previous post on applying Dynamic Time Warping as a metric for LTE modeling of oceanic indices, it makes sense to apply the metric to the QBO model of atmospheric winds. A characteristic of QBO data is the sharp transitions of wind reversals. As described previously, DTW allows a fit to adjust the alignment between model and data without incurring a potential over-fitting penalty that a conventional correlation coefficient will often lead to.

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Unified Model of Earth Dynamics

/ Paul Pukite (@whut) / Leave a comment

Lorenz turned out to be a chaotic dead-end in understanding Earth dynamics. Instead we need a new unified model of solid liquid dynamics focusing on symmetries of the rotating earth, applying equations of solid bodies & fluid dynamics. See Mathematical Geoenergy (Wiley, 2018).

Should have made this diagram long ago: here’s the ChatGPT4 prompt with the diagramming plugin.

Graph

Ocean Tides and dLOD have always been well-understood, largely because the mapping to lunar+solar cycles is so obvious. And the latter is getting better all the time — consider recent hi-res LOD measurements with a ring laser interferometer, pulling in diurnal tidal cycles with much better temporal resolution.

That’s the first stage of unification (yellow boxes above) — next do the other boxes (CW, QBO, ENSO, AMO, PDO, etc) as described in the book and on this blog, while calibrating to tides and LOD, and that becomes a cross-validated unified model.

Annotated 10/11/2023

ontological classification according to wavenumber kx, ky, kz and fluid/solid.

Added so would not lose it — highlighted tidal factor is non-standard

Geophysically Informed Machine Learning for Improving Rapid Estimation and Short-Term Prediction of Earth Orientation Parameters

Heuristics

/ Paul Pukite (@whut) / Leave a comment

In the book Mathematical GeoEnergy, I mention the word heuristic or heuristics 82 times. In scientific research, it’s an important signpost because it identifies where a physical understanding is lacking, which is the point I was trying to make in a recent online discussion.

In the preface, I specifically cite the sunspot cycle as a heuristic, which I commented on:

“For example, the 11 -year sunspot cycle is considered a heuristic but not the annual or daily cycles, which are trivially explained. Tried to get ChatGPT to agree with me:
https://chat.openai.com/share/2706730c-2767-4060-b65e-08549b538d0e“

chatGPT is correct.

whats your problem

I responded with two examples of heuristics that can conceivably be replaced by plausible and parsimonious physics.

Chandler wobble.

Heuristic: The wobble is approximately 433 days, thought to be excited by fluctuations of mass on Earth achieving a natural resonance condition.
Physics: The wobble of precisely 433 days is a frequency side-band of the lunar draconic cycle interacting with the annual cycle. Not resonant just as the annual wobble is not resonant, and due to a forced angular momentum response.

QBO.

Heuristic: The oscillation is approximately 2+ years, thus the name “quasi-biennial”, thought to be excited by atmospheric waves.
Physics: The oscillation of 2.37 years period follows from the semi-annual oscillation at higher altitudes locked to the longer period by the lunar tidal forcing at lower altitudes of the stratosphere. The cycle is commensurate with simultaneous nodal crossings of both the moon and sun across the ecliptic plane.

In both these cases, the heuristic could be discarded and the description of the behaviors updated. Another example may be Milankovitch cycles, which replaces the heuristic of glacial cycles with a plausible physical mechanism that matches the observations. Whether Chandler, QBO, or Milankovitch will stand the test of time is another question, but these new models aren’t considered heuristics because they predict precise values for the observations. Discovering replacements for heuristics are rare nowadays as most of the heuristics (such as ocean tidal cycles) were resolved long ago. Circadian rhythms were a heuristic replaced by an encoded mechanism 20-30 years ago. Explaining predator-prey population cycles are at the heuristic stage still, IMO and may not get resolved as humans modify the environment.

The suggestion is that an unresolved heuristic is always a good candidate for a thesis topic.

NEW

Discussing relatively recent discoveries that replaced a heuristic, one that also came up is the Quantum Hall Effect. This is a subtle one because although the effect was experimentally discovered by von Klitzing, a Japanese team did roughly predict it a few years earlier, but added a caveat that they did not believe their own calculations! They also did not predict the quantization was in exact integer multiples. So that remained a heuristic for only a short time until it was physically explained (I remember giving a talk on this derivation for my solid-state physics class, which the instructor was very happy about). But then the experimental discovery of a fractional Quantum Hall Effect occurred, which apparently is still a heuristic because a strong consensus has yet to emerge on the physics behind it. Over the years, there have been at least 3 Nobel Prizes shared among 7 physicists to topological QHE research.

I bring this up because there’s a recent Quanta Magazine article dated July 18 titled “How Quantum Physicists Explained Earth’s Oscillating Weather Patterns” which describes how the QHE math can be conceivably applied to making predictions for equatorial patterns, and thus removing at least some of the heuristic nature. Geoffrey Vallis, who is an expert on geophysical fluid dynamics, is quoted in the article saying that the new result is a significant advance that will provide a “foundational understanding” of Earth’s fluid systems. The intriguing aspect is that this did not require the periodic order of a lattice — quoting from the article:

“I was surprised to see that topology could be defined in fluid systems without periodic order,” said Anton Souslov, a theoretical physicist at the University of Bath

Curry has a tweet on this, a reply tweet here because I am blocked

The Big 10 Climate Indices

/ Paul Pukite (@whut) / 2 Comments

The above diagram courtesy of Karnauskus

These correspond to the geographically defined climate indices

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

  1. ENSO – https://www.threads.net/@paulpukite/post/CuWS8MFu8Jc
  2. AMO – https://www.threads.net/@paulpukite/post/Cuh4krjJTLN
  3. PDO – https://www.threads.net/@paulpukite/post/Cuu0VCypIi5
  4. QBO – https://www.threads.net/@paulpukite/post/CuiKQ5tsXCQ
  5. SOI (Darwin & Tahiti) – https://www.threads.net/@paulpukite/post/Cuu2IkBJh55 => MJO
  6. IOD (East & West) – https://www.threads.net/@paulpukite/post/Cuu9PYvJAG2
  7. PNA – https://www.threads.net/@paulpukite/post/CuvAVR7JN7R
  8. AO – https://www.threads.net/@paulpukite/post/CuvEz37JPFV
  9. SAM – https://www.threads.net/@paulpukite/post/CuvLZ2CMt1X
  10. 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