Chandler wobble

Controlled Experiments

/ Paul Pukite (@whut) / 3 Comments

Sorry to have to point this out, but it’s not my fault that geophysicists and climatologists can’t perform controlled experiments to test out various hypotheses. It’s not their fault either. It’s all nature’s decision to make gravitational forces so weak and planetary objects so massive to prevent anyone from scaling the effect to laboratory size to enable a carefully controlled experiment. One can always create roughly-equivalent emulations, such as a magnetic field experiment (described in the previous blog post) and validate a hypothesized behavior as a controlled lab experiment. Yet, I suspect that this would not get sufficient buy-in, as it’s not considered the actual real thing.

And that’s the dilemma. By the same token that analog emulators will not be trusted by geophysicists and climatologists, so too scientists from other disciplines will remain skeptical of untestable claims made by earth scientists. If nothing definitive comes out of a thought experiment that can’t be reproduced by others in a lab, they remain suspicious, as per their education and training.

It should therefore work both ways. As featured in the previous blog post, the model of the Chandler wobble forced by lunar torque needs to be treated fairly — either clearly debunked or considered as an alternative to the hazy consensus. ChatGPT remains open about the model, not the least bit swayed by colleagues or tribal bias. As the value of the Chandler wobble predicted by the lunar nodal model (432.7 days) is so close to the cited value of 433 days, as a bottom-line it should be difficult to ignore.

There are other indicators in the observational data to further substantiate this, see Chandler Wobble Forcing. It also makes sense in the context of the annual wobble.

As it stands, the lack of an experiment means a more equal footing for the alternatives, as they are all under equal amounts of suspicion.

Same goes for QBO. No controlled experiment is possible to test out the consensus QBO models, despite the fact that the Plumb and McEwan experiment is claimed to do just that. Sorry, but that experiment is not even close to the topology of a rotating sphere with a radial gravitational force operating on a gas. It also never predicted the QBO period. In contrast, the value of the QBO predicted by the lunar nodal model (28.4 months) is also too close to the cited value of 28 to 29 months to ignore. This also makes sense in the context of the semi-annual oscillation (SAO) located above the QBO .

Both the Chandler wobble and the QBO have the symmetry of a global wavenumber=0 phenomena so therefore only nodal cycles allowed — both for lunar and solar.

Next to ENSO. As with LOD modeling, this is not wavenumber=0 symmetry, as it must correspond to the longitude of a specific region. No controlled experiment is possible to test out the currently accepted models, premised as being triggered by wind shifts (an iffy cause vs. effect in any case). The mean value of the ENSO predicted by the tidal LOD-caibrated model (3.80 years modulated by 18.6 years) is too close to the cited value of 3.8 years with ~200 years of paleo and direct measurement to ignore.

Encyclopedia of Paleoclimatology and Ancient Environments, 721–728.
doi:10.1007/978-1-4020-4411-3_172 

In BLUE below is the LOD-calibrated tidal forcing, with linear amplification

In BLUE again below is a non-linear modulation of the tidal forcing according to the Laplace’s Tidal Equation solution, and trained on an early historical interval. This is something that a neural network should be able to do, as it excels at fitting to non-linear mappings that have a simple (i.e. low complexity) encoding — in this case it may be able to construct a Taylor series expansion of a sinusoidal modulating function.

The neural network’s ability to accurately represent a behavior is explained as a simplicity bias — a confounding aspect of machine learning tools such as ChatGPT and neural networks. The YouTube video below explains the counter-intuitive notion of how a NN with a deep set of possibilities tends to find the simplest solution and doing this without over-fitting the final mapping.

So that deep neural networks are claimed to have a built-in Occam’s Razor propensity, finding the most parsimonious input-output mappings when applied to training data. This is spot on with what I am doing with the LTE mapping, but bypassing the NN with a nonlinear sinusoidal modulation optimally fit on training data by a random search function.

I am tempted to try a NN on the ENSO training set as an experiment and see what it finds.

April 2, 2023

“I am tempted to try a NN on the ENSO training set as an experiment and see what it finds.”

Does GPT understand geophysics?

/ Paul Pukite (@whut) / 12 Comments

Not sure what version of ChatGPT is being used on my browser at the present time, but I created a couple of GPT prompts (essentially queries) and compared and contrasted the results, and against expectations.

The first is to determine if ChatGPT could infer a wobble in a classical electromagnetic dynamics experiment. The setup is very simple and straightforward, illustrated by the schematic shown below:

As one can’t (as of yet) submit the schematic as a prompt to ChatGPT and illicit a response, we need to create a natural language prompt first:

Consider a levitating rotating magnetized sphere with north and south magnetic poles located at each endpoint of the axis of rotation. The rotation speed is 1 radian per second. An iron rod of the same length is placed in parallel with the axis of rotation, and wound with wire to form an electromagnet. At what frequency would the sphere wobble if the electromagnet was driven by a slight alternating current perturbation of 1 Herz, after the natural response subsided?

This is the response as of yesterday:

The motion of the levitating rotating magnetized sphere is governed by the interplay between the magnetic field of the sphere and the magnetic field of the iron rod, which is now an electromagnet due to the wire wound around it.

When the iron rod is magnetized by the alternating current, it generates a magnetic field that interacts with the magnetic field of the rotating sphere, causing a torque that can cause the sphere to wobble. The wobbling frequency can be determined by analyzing the equation of motion for the system.

Assuming the wobbling is small, we can linearize the equation of motion by taking the first-order Taylor series expansion. The resulting equation can be expressed as:

I * d^2θ/dt^2 + γ * dθ/dt + k * θ = M * B

where:

I is the moment of inertia of the sphere
θ is the angular displacement of the sphere from its equilibrium position
γ is the damping coefficient
k is the spring constant
M is the magnetic moment of the electromagnet
B is the magnetic field of the rotating sphere

The solution to this equation of motion is a damped harmonic oscillator, which can be expressed as:

θ(t) = A * exp(-γt/(2I)) * cos(ωt + φ)

where:

A is the amplitude of the wobbling motion
ω is the frequency of the wobbling motion
φ is the phase angle
The frequency of the wobbling motion can be found by substituting the solution into the equation of motion and solving for ω. After some algebraic manipulation, we get:

ω = sqrt((k/I) – (γ^2)/(4I^2)) – MB/(Isqrt((k/I) – (γ^2)/(4I^2)))

In this case, the sphere is rotating at a constant rate of 1 radian per second, so we can assume that the magnetic field of the sphere is also constant. Therefore, the only variable that changes due to the alternating current perturbation is the magnetic moment of the electromagnet, which oscillates at a frequency of 1 Hz.

Assuming that the natural response of the system has subsided, the wobbling motion will be dominated by the frequency of the alternating current perturbation. Therefore, the frequency of the wobbling motion will also be 1 Hz.

The response is to assert that the sphere will wobble at the same frequency as the applied forcing frequency, which is the intuitive response.

Next, change the prompt to create a similar behavior, but replacing the magnetic forces with gravitational forces.

Consider a rotating planet with massive extra weight at each pole. The rotation speed is 1 radian per Earth year. A small satellite moon orbits the planet with a significant declination. At what frequency would the planet wobble if the moon’s declination cycle was 250 days, after the natural response subsided?

This is a straightforward experiment to imagine, but difficult to perform in a lab. Submitting to ChatGPT today, this is the response:

The motion of the rotating planet with massive extra weight at each pole is governed by the conservation of angular momentum. The presence of the extra weight at the poles changes the distribution of mass of the planet, which in turn affects its moment of inertia.

Assuming that the extra weight is symmetric about the axis of rotation, we can approximate the planet as an oblate spheroid, which means that the moment of inertia of the planet is not constant but depends on the angle between the axis of rotation and the axis of symmetry of the spheroid. This angle is known as the obliquity of the planet and can change due to external perturbations such as the gravitational pull of a satellite moon.

The wobbling motion of the planet due to the gravitational pull of the satellite moon can be described by the following equation of motion:

I * d^2θ/dt^2 + γ * dθ/dt + k * θ = M * sin(ψ)

where:

I is the moment of inertia of the planet
θ is the angular displacement of the planet from its equilibrium position
γ is the damping coefficient
k is the spring constant
M is the mass of the satellite moon
ψ is the angle between the axis of rotation of the planet and the direction of the gravitational force exerted by the satellite moon

The solution to this equation of motion is a damped harmonic oscillator, which can be expressed as:

θ(t) = A * exp(-γt/(2I)) * cos(ωt + φ)

where:

A is the amplitude of the wobbling motion
ω is the frequency of the wobbling motion
φ is the phase angle

The frequency of the wobbling motion can be found by substituting the solution into the equation of motion and solving for ω. After some algebraic manipulation, we get:

ω = sqrt((k/I) – (γ^2)/(4I^2)) – Msin(ψ)/(Isqrt((k/I) – (γ^2)/(4I^2)))

In this case, the planet is rotating at a constant rate of 1 radian per Earth year, which corresponds to a period of approximately 2π × 365.25 days. The mass of the satellite moon and the gravitational force it exerts on the planet are also constant, so the only variable that changes due to the moon’s declination cycle is the angle ψ.

Assuming that the natural response of the system has subsided, the wobbling motion will be dominated by the frequency of the moon’s declination cycle, which is 1/250 Hz. Therefore, the frequency of the wobbling motion will also be 1/250 Hz.

Since the math of magnetic forcing and gravitational forcing follows the same inverse square law, the answers are similar as expected. ChatGPT gets the dimension wrong, using Hz instead of 1/(day), but other than that, this is also an intuitive result, as the forced response carries through after any natural response damps out.

Does GPT understand geophysics better than geophysicists?

I did this LLM AI experiment to find out whether ChatGPT would substantiate the model of the Chandler wobble that I had published in Mathematical Geoenergy (Wiley/AGU, 2018). The premise is that the nodal declination cycle of the Moon would force the Chandler wobble to a period of 433 days via a nonlinear interaction with the annual cycle. A detailed analysis of satellite peaks in the frequency spectrum strongly supports this model. Introducing the electromagnetic analog places it on an equal footing with a model that can be verified by a controlled lab experiment, see (A) vs (B) below.

Of course, if I had created a prompt to ChatGPT to simply inquire “What causes the Chandler wobble to cycle at 433 days?”, it would have essentially provided the Wikipedia answer, which is the hazy consensus culled from research since the wobble was discovered over 100 years ago.

Note that this “physics-free” answer provided by ChatGPT has nothing to do with the moon.

ADDED 3/26/2023

This blog post has a section on ChatGPT using MatPlotLib to create a visualization of orbiting planets.

https://andrewmayneblog.wordpress.com/2023/03/23/chatgpt-code-interpreter-magic/

Cross-validation

/ Paul Pukite (@whut) / 5 Comments

Cross-validation is essentially the ability to predict the characteristics of an unexplored region based on a model of an explored region. The explored region is often used as a training interval to test or validate model applicability on the unexplored interval. If some fraction of the expected characteristics appears in the unexplored region when the model is extrapolated to that interval, some degree of validation is granted to the model.

This is a powerful technique on its own as it is used frequently (and depended on) in machine learning models to eliminate poorly performing trials. But it gains even more importance when new data for validation will take years to collect. In particular, consider the arduous process of collecting fresh data for El Nino Southern Oscillation, which will take decades to generate sufficient statistical significance for validation.

So, what’s necessary in the short term is substantiation of a model’s potential validity. Nothing else will work as a substitute, as controlled experiments are not possible for domains as large as the Earth’s climate. Cross-validation remains the best bet.

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Nonlinear long-period tidal forcing with application to ENSO, QBO, and Chandler wobble

/ Paul Pukite (@whut) / 7 Comments
SupplementaryDocument-egu-2021Download
Model fitting process for ENSO

Back to EGU abstract and presentation

Addendum: After this presentation was submitted, a ground-breaking paper by a group at the University of Paris came on-line. Their paper, “On the Shoulders of Laplace” covers much the same ground as the EGU presentation linked above.

Their main thesis is that Pierre-Simon Laplace in 1799 correctly theorized that the wobble in the Earth’s rotation is due to the moon and sun, described in the treatise “Traité de Mécanique Céleste (Treatise of Celestial Mechanics)“.

Excerpts from the paper “On the shoulders of Laplace”

Moreover Lopes et al claim that this celestial gravitational forcing carries over to controlling cyclic climate indices, following Laplace’s mathematical formulation (now known as Laplace’s Tidal Equations) for describing oceanic tides.

Excerpt from the paper “On the shoulders of Laplace”

This view also aligns with the way we model climate indices such as ENSO and QBO via a solution to Laplace’s Tidal Equations, as described in the linked EGU presentation above.

Chandler Wobble Forcing

/ Paul Pukite (@whut) / 8 Comments

Amazing finding from Alex’s group at NASA JPL

What’s also predictable is that the JPL team probably have a better handle of what causes the wobble on Mars than we have on what causes the Chandler wobble (CW) here on Earth. Such is the case when comparing a fresh model against a stale model based on an early consensus that becomes hard to shake with the passage of time.

Of course, we have our own parsimoniously plausible model of the Earth’s Chandler wobble (described in Chapter 13), that only gets further substantiated over time.

The latest refinement to the geoenergy model is the isolation of Chandler wobble spectral peaks related to the asymmetry of the northern node lunar torque relative to the southern node lunar torque.

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ESD Ideas article for review

/ Paul Pukite (@whut) / 1 Comment

Get a Copernicus login and comment for peer-review

The simple idea is that tidal forces play a bigger role in geophysical behaviors than previously thought, and thus helping to explain phenomena that have frustrated scientists for decades.

The idea is simple but the non-linear math (see figure above for ENSO) requires cracking to discover the underlying patterns.

The rationale for the ESD Ideas section in the EGU Earth System Dynamics journal is to get discussion going on innovative and novel ideas. So even though this model is worked out comprehensively in Mathematical Geoenergy, it hasn’t gotten much publicity.

Stratospheric Sudden Warming

/ Paul Pukite (@whut) / 2 Comments

Chapter 11 of the book describes a model for the QBO of stratospheric equatorial winds. The stratified layers of the atmosphere reveal different dependencies on the external forcing depending on the altitude, see Fig 1.

Figure 1 : At high altitudes, only the sun’s annual cycle impacts the stratospheric as a semi-annual oscillation (SAO). Below that the addition of the lunar nodal cycle forces the QBO. The earth itself shows a clear wobble with the lunar cycle interacting with the annual.

Well above these layers are the mesosphere, thermosphere, and ionosphere. These are studied mainly in terms of space physics instead of climate but they do show tidal interactions with behaviors such as the equatorial electrojet [1].

The behaviors known as stratospheric sudden warmings (SSW) are perhaps a link between the lower atmospheric behaviors of equatorial QBO and/or polar vortex and the much higher atmospheric behavior comprising the electrojet. Papers such as [1,2] indicate that lunar tidal effects are showing up in the SSW and that is enhancing characteristics of the electrojet. See Fig 2.

Figure 2 : During SSW events, a strong modulation of period ~14.5 days emerges, close to the lunar fortnightly period as seen in these spectrograms. Taken from ref [2] and see quote below for more info.

“Wavelet spectra of foEs during two SSW events exhibit noticeable enhanced 14.5‐day modulation, which resembles the lunar semimonthly period. In addition, simultaneous wind measurements by meteor radar also show enhancement of 14.5‐day periodic oscillation after SSW onset.”

Tang et al [2]

So the SSW plays an important role in ionospheric variations, and the lunar tidal effects emerge as the higher atmospheric density of a SSW upwelling becomes more sensitive to lunar tidal forcing. That may be related to how the QBO also shows a dependence on lunar tidal forcing due to its higher density.

References

  1. Siddiqui, T. A. Relationship between lunar tidal enhancements in the equatorial electrojet and stratospheric wind anomalies during stratospheric sudden warmings. (2020). Originally presented at AGU 2018 Fall Meeting
  2. Tang, Q., Zhou, C., Liu, Y. & Chen, G. Response of Sporadic E Layer to Sudden Stratospheric Warming Events Observed at Low and Middle Latitude. Journal of Geophysical Research: Space Physics e2019JA027283 (2020).

Chandler Wobble according to Na

/ Paul Pukite (@whut) / 7 Comments

In Chapter 13 of the book, we have a description of the mechanism forcing the Chandler Wobble in the Earth’s rotation. As a counter to a recent GeoenergyMath post suggesting there is little consensus behind this mechanism, a recent paper by Na et al provides a foundation to understand how the lunar forcing works. 

Chandler wobble and free core nutation are two major modes of perturbation in the Earth rotation. Earth rotation status needs to be known for the coordinate conversion between celestial reference frame and terrestrial reference frame. Due mainly to the tidal torque exerted by the moon and the sun on the Earth’s equatorial bulge, the Earth undergoes precession and nutation.

Na, S.-H. et al. Chandler Wobble and Free Core Nutation: Theory and Features. Journal of Astronomy and Space Sciences 36, 11–20 (2019).
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Lunisolar Forcing of the Chandler Wobble

/ Paul Pukite (@whut) / 3 Comments

In Chapter 13 of the book, we have a description of the mechanism forcing the Chandler Wobble in the Earth’s rotation. Even though there is not yet a research consensus on the mechanism, the prescribed lunisolar forcing seemed plausible enough that we included a detailed analysis in the text.  Recently we have found a recent reference to a supporting argument to our conjecture, which is presented below …

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The Lunar Geophysical Connection

/ Paul Pukite (@whut) / 6 Comments

The conjecture out of NASA JPL is that the moon has an impact on the climate greater than is currently understood:

Claire Perigaud (Caltech/JPL)
and
Has this research gone anywhere?  Looks as if has gone to this spin-off.
According to the current consensus, variability in wind is what contributes to forcing for behaviors such as the El Nino/Southern Oscillation (ENSO).
OK, but what forces the wind? No one can answer that apart from saying wind variability is just a part of the dynamic climate system.  And so we are lead to believe that a wind burst will cause an ENSO and then the ENSO event will create a significant disruptive transient to the climate much larger than the original wind stimulus. And that’s all due to positive feedback of some sort.
I am only paraphrasing the current consensus.
A much more plausible and parsimonious explanation lies with external lunar forcing reinforced by seasonal cycles.

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