“If so, do you have an explanation why the diurnal tides do not move the thermocline, whereas tides with longer periods do?”
The character of ENSO is that it shifts by varying amounts on an annual basis. Like any thermocline interface, it reaches the greatest metastability at a specific time of the year. I’m not making anything up here — the frequency spectrum of ENSO (pick any index NINO4, NINO34, NINO3) shows a well-defined mirror symmetry about the value 0.5/yr. Given that Incontrovertible observation, something is mixing with the annual impulse — and the only plausible candidate is a tidal force. So the average force of the tides at this point is the important factor to consider. Given a very sharp annual impulse, the near daily tides alias against the monthly tides — that’s all part of mathematics of orbital cycles. So just pick the monthly tides as it’s convenient to deal with and is a more plausible match to a longer inertial push.
Sunspots are not a candidate here.
Some say wind is a candidate. Can’t be because wind actually lags the thermocline motion.
So the deal is, I can input the above as a prompt to ChatGPT and see what it responds with
I made one change to the script (multiplying each tidal factor by its frequency to indicate its inertial potential, see the ## comment)
At the end of the Gist, I placed a representative power spectrum for the actual NINO4 and NINO34 data sets showing where the spectral peaks match. They all match. More positions match if you consider a biennial modulation as well.
Now, you might be saying — yes, but this all ChatGPT and I am likely coercing the output. Nothing of the sort. Like I said, I did the original work years ago and it was formally published as Mathematical Geoenergy (Wiley, 2018). This was long before LLMs such as ChatGPT came into prominence. ChatGPT is simply recreating the logical explanation that I had previously published. It is simply applying known signal processing techniques that are generic across all scientific and engineering domains and presenting what one would expect to observe.
In this case, it carries none of the baggage of climate science in terms of “you can’t do that, because that’s not the way things are done here”. ChatGPT doesn’t care about that prior baggage — it does the analysis the way that the research literature is pointing and how the calculation is statistically done across domains when confronted with the premise of an annual impulse combined with a tidal modulation. And it nailed it in 2025, just as I nailed it in 2018.
What really is the color GREEN? What does it mean to different people? We actually have no idea. Now, consider The Dress on its 10 year anniversary. One can either argue with others whether it is BLUE and BLACK or WHITE and GOLD, or objectively look at it’s composition from a technical standpoint. Below is a cross-section of the dress image using the online image-picker application showing a palette from selected pixels in the image
A climate science breakthrough likely won’t be on some massive computation but on a novel formulation that exposes some fundamental pattern (perhaps discovered by deep mining during a machine learning exercise). Over 10 years ago, I wrote on a blog post on how one can extract the ENSO signal by doing simple signal processing on a sea-level height (SLH) tidal time-series — in this case, at Fort Denison located in Sydney harbor.
The formulation/trick is to take the difference between the SLH reading and that from 2 years (24 months) prior, described here
The rationale for this 24 month difference is likely related to the sloshing of the ocean triggered on an annual basis. I think this is a pattern that any ML exercise would find with very little effort. After all, it didn’t take me that long to find it. But the point is that the ML configuration has to be open and flexible enough to be able to search, generate, and test for the same formulation. IOW, it may not find it if the configuration, perhaps focused on computationally massive PDEs, is too narrow. That was my comment to a RC post on applying machine learning to climate science, see the following link and subsequent quote:
“ML-based parameterizations have to work well for thousands of years of simulations, and thus need to be very stable (no random glitches or periodic blow-ups) (harder than you might think). Bias corrections based on historical observations might not generalize correctly in the future.”
This same issue arises when using ML to simulate PDEs. The solution is to analytically calculate what the stability condition(s) is (are), then at each timestep to add some numerical diffusion that nudges the solution towards satisfying the stability condition(s). I imagine this same technique could be used for ML-based parametrizations.
“A key enabling assumption, sometimes called the manifold hypothesis [14], is that the data lie on or near a low-dimensional manifold; for physical systems with dissipation, such manifolds can often be rigorously shown to exist [15–18]. These manifolds enable a low-dimensional latent state representation, and hence, low-dimensional dynamical models. Linear manifold learning techniques, such as principal component analysis, cannot learn the nonlinear manifolds that represent most systems in nature. To do so, we require nonlinear methods, some of which are developed in [19–25] and reviewed in [26].“
Floryan, Daniel, and Michael D. Graham. “Data-driven discovery of intrinsic dynamics.” Nature Machine Intelligence 4.12 (2022): 1113-1120.
“Chaos is typically blamed for the lack of predictability beyond a forecasting time window, which is on the order of 10 days for weather forecasting. However, on the one hand, most turbulent and chaotic systems exhibit strong coherence in the flow, such as synoptic events in weather or coherent structures in turbulence. On the other hand, most physical model might have additional structural errors that limit their capacity to correctly forecast beyond a certain time horizon, independent of chaos.
We will show in this presentation that most chaotic and turbulent flows can be predicted on relatively long range, at least longer than expected with both physical models and standard deep learning, using a combination of a reduced order model (that captures the low-dimensional coherent structures in the flow) and generative AI to obtain the crisp results and details of the flow. We will conclude stating that current AI-based weather models might not have achieved a plateau in performance yet, especially at longer time scales, and that physics-based weather model still have room for improvements. Reduced order models might not be able to beat chaos but can lead to much longer-range prediction than currently expected.“
Reminded by a 20-year anniversary post at RealClimate.org, that I’ve been blogging for 20 years + 6 months on topics of fossil fuel depletion + climate change. The starting point was at a BlogSpot blog I created in May 2004, where the first post set the stage:
Click on the above to go to the complete archives (almost daily posts) until I transitioned to WordPress and what became the present blog. After 2011, my blogging pace slowed down considerably as I started to write in more in more technical terms. Eventually the most interesting and novel posts were filtered down to a set that would eventually become the contents of Mathematical Geoenergy : Discovery, Depletion, and Renewal, published in late 2018/early 2019 by Wiley with an AGU imprint.
The arc that my BlogSpot/WordPress blogging activity followed occupies somewhat of a mirror universe to that of RealClimate. I initially started out with an oil depletion focus and by incrementally understanding the massive inertia that our FF-dependent society had developed, it placed the climate science aspect into a different perspective and context. After realizing that CO2 did not like to sequester, it became obvious that not much could be done to mitigate the impact of gradually increasing GHG levels, and that it would evolve into a slow-moving train wreck. That’s part of the reason why I focused more on research into natural climate variability. In contrast, RealClimate (and all the other climate blogs) continued to concentrate on man-made climate change. At this point, my climate fluid dynamics understanding is at some alternate reality level, see the last post, still very interesting but lacking any critical acceptance (no debunking either, which keeps it alive and potentially valid).
The oil depletion aspect more-or-less spun off into the PeakOilBarrel.com blog [*] maintained by my co-author Dennis Coyne. That’s like watching a slow-moving train wreck as well, but Dennis does an excellent job of keeping the suspense up with all the details in the technical modeling. Most of the predictions regarding peak oil that we published in 2018 are panning out.
As a parting thought, the RealClimate hindsight post touched on how AI will impact information flow going forward. Having worked on AI knowledgebases for environmental modeling during the LLM-precursor stage circa 2010-2013, I can attest that it will only get better. At the time, we were under the impression that knowledge used for modeling should be semantically correct and unambiguous (with potentially a formal representation and organization, see figure below), and so developed approaches for that here and here (long report form).
As it turned out, lack of correctness is just a stage, and AI users/customers are satisfied to get close-enough for many tasks. Eventually, the LLM robots will gradually clean up the sources of knowledge and converge more to semantic correctness. Same will happen with climate models as machine learning by the big guns at Google, NVIDIA, and Huawei will eventually discover what we have found in this blog over the course of 20+ years.
Note: [*] In some ways the PeakOilBarrel.com blog is a continuation of the shuttered TheOilDrum.com blog, which closed shop in 2013 for mysterious reasons.
Trying out the Dante plug-in for GPT-4 LLM, using Mathematical GeoEnergy as a knowledge-base.
Try the prompt “What is the distribution of global variation of wind?”
The distribution of global variation of wind can be characterized by the entropic dispersion of wind speeds, which often follows a Rayleigh or Weibull distribution. This distribution shows a peak away from zero wind speeds and a rapid decline of frequency at higher velocities. Wind power distribution data from different regions, such as Ontario and northwest Germany, display the same damped exponential probability distribution function for the frequency of wind power derived from wind speed. This universal behavior in wind speed variability is influenced by the Earth’s climate and environment, which generate more states for the system to exist within, increasing the likelihood of variability.
Add “How does the BesselK distribution fit in to this?”
The BesselK distribution is another way to model the distribution of wind speeds and other natural phenomena. It is derived from the maximum entropy principle, which aims to find the most likely distribution given certain constraints. In the context of wind speed variability, the BesselK distribution can provide a good fit to the observed data, similar to the Rayleigh or Weibull distributions. The BesselK distribution has been used to model various natural processes, such as rainfall and wave heights, in addition to wind speeds. It is important to note that different distributions may be more suitable for specific regions or conditions, and the choice of the distribution should be based on the best fit to the observed data.
ENSO (El Niño-Southern Oscillation) is driven by lunar forcing through the interaction of lunar cycles with the Earth’s rotation rate and the seasonal cycle. The lunar cycles, such as the monthly and fortnightly cycles, are aliased due to their interaction with the annual cycle, resulting in ENSO cycles observed at the multiyear scale. The gravitational forcing from the lunar orbit and its interaction with the Sun’s yearly (lunar-perturbed) cycle can be applied to model ENSO effectively. The better the precise orbital path is obtained, the better resolved the fit.
The premise for the ENSO model is that a predictable lunar forcing leads to the erratic cycles of ENSO. The erratic nature of the cycles and any difficulty in matching these are directly the result of the physical aliasing that occurs from applying a low-frequency sampling (the yearly impulse) to a higher-frequency phenomenon (the monthly and fortnightly lunar cycles) which are incommensurate with the annual cycle.
How to solve LTE on the equatorial waveguide?
Start with Laplace’s tidal equations for a fluid sheet of average thickness D, the vertical tidal elevation ζ, and the horizontal velocity components u and v (in the latitude φ and longitude λ directions).
Simplify the equations by considering the small-angle approximation along the equator, where φ is close to zero.
Reduce the set of Laplace’s tidal equations to a single equation by applying the chain rule and making an adjoint connection between a change in latitudinal forcing with a temporal change.
Obtain a potentially highly nonlinear result by fixing φ(t) to a periodic function with a long-term mean of zero.
Derive the horizontal longitudinal acceleration of the fluid (v) by applying the solution to Laplace’s third tidal equation in simplified form.
Consider multiple solutions for the various standing waves, which will be used in fitting the model to the data.
Apply the known tidal forcing along with an annual harmonic (and subharmonic) forcing and observe if the result correlates or matches the ENSO time series. By following these steps, you can generate a concise solution that has all the characteristics of the measured ENSO. Keep in mind that this approach is a simplification of the full Navier-Stokes equations and may not capture all the complexities of the equatorial waveguide.
After this point, I provided ChatGPT prompts as italicized underlines after each sinewave icon , and ChatGPT provided responses. I executed the Julia code in a REPL window and saved the screenshotted plots and entered the image links below.
Model: GPT-4
Math equations for a spinning top in Julia
To simulate a spinning top in Julia, you can use the equations of motion derived from Euler’s equations for a rigid body. Here’s a simple example using the DifferentialEquations.jl package, which provides a convenient way to solve ordinary differential equations (ODEs) in Julia:
First, install the required packages by running:
using Pkg
Pkg.add("DifferentialEquations")
Pkg.add("Plots")
Now, you can write a script to simulate a spinning top:
using DifferentialEquations
using Plots
# Define the spinning top's properties
I1, I2, I3 = 0.1, 0.2, 0.3 # Moments of inertia along the principal axes (x, y, z)
Ω1, Ω2, Ω3 = 0.0, 10.0, 20.0 # Initial angular velocities (rad/s)
# Equations of motion for the spinning top (Euler's equations for a rigid body)
function spinning_top!(du, u, p, t)
Ω1, Ω2, Ω3 = u
du[1] = (I2 - I3)/I1 * Ω2 * Ω3
du[2] = (I3 - I1)/I2 * Ω1 * Ω3
du[3] = (I1 - I2)/I3 * Ω1 * Ω2
end
# Initial conditions
u0 = [Ω1, Ω2, Ω3]
tspan = (0.0, 10.0) # Time span for the simulation (in seconds)
# Solve the ODE
prob = ODEProblem(spinning_top!, u0, tspan)
sol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8)
# Plot the angular velocities
plot(sol, xlabel="Time (s)", ylabel="Angular velocity (rad/s)", title="Spinning Top", legend=:bottomright)
This code sets up and solves the ODEs that describe the motion of a spinning top with specified moments of inertia and initial angular velocities. The result is then plotted to show the evolution of the angular velocities over time.
Remember that this is a simple example that assumes the top is spinning in a vacuum, without any external forces or torques acting on it (such as gravity or air resistance). To model a more realistic spinning top, you will need to account for these factors in your equations of motion.
To the equations, add a slight periodic horizontal forcing impulse to the top of the spinning top
To add a slight periodic horizontal forcing impulse to the spinning top, you can modify the equations of motion accordingly. Let’s assume the forcing impulse acts along the x-axis and has the form F(t) = F0 * sin(ω * t), where F0 is the amplitude of the forcing impulse and ω is its angular frequency.
Here’s how you can update the spinning top simulation:
Modify the spinning_top! function to include the forcing impulse:
function spinning_top!(du, u, p, t)
Ω1, Ω2, Ω3 = u
F0, ω, I1, I2, I3 = p
# Forcing impulse along the x-axis
F = F0 * sin(ω * t)
du[1] = (I2 - I3)/I1 * Ω2 * Ω3 + F/I1
du[2] = (I3 - I1)/I2 * Ω1 * Ω3
du[3] = (I1 - I2)/I3 * Ω1 * Ω2
end
Update the initial conditions and ODE problem:
# Forcing impulse parameters
F0 = 0.01 # Amplitude of the forcing impulse
ω = 2π / 5 # Angular frequency of the forcing impulse (period = 5 seconds)
# Parameters to be passed to the spinning_top! function
p = [F0, ω, I1, I2, I3]
# Solve the ODE
prob = ODEProblem(spinning_top!, u0, tspan, p)
sol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8)
The updated spinning_top! function now includes a forcing impulse along the x-axis, which is passed as a parameter (F0, ω). The ODE problem is also updated to include these parameters.
By running the simulation with the updated code, you can observe the effects of the slight periodic horizontal forcing impulse on the spinning top.
Is there an analytical solution for small forcing perturrbations?
For small forcing perturbations, it may be possible to find an approximate analytical solution using perturbation methods, such as regular perturbation or multiple-scale perturbation techniques. These methods involve expanding the solution in a power series of a small parameter (in this case, the amplitude of the forcing perturbation) and then substituting the series into the equations of motion.
However, finding an exact analytical solution for the spinning top with small forcing perturbations is generally not feasible due to the nonlinear nature of the governing equations. The Euler equations for rigid body rotation are a set of coupled, nonlinear ordinary differential equations, and the addition of a forcing term makes them even more complex. In practice, numerical methods are often used to solve these equations, as demonstrated in the previous examples.
If you want to gain insights into the behavior of the system under small forcing perturbations, you can use linear stability analysis and perturbation methods to study the stability and dynamics of the system. These methods can help you find approximate analytical solutions that are valid for small perturbations and understand the qualitative behavior of the system. However, for a complete and accurate description of the spinning top’s motion with forcing, it is usually necessary to rely on numerical simulations.
Generate the equivalent Fourier series representation of the previous differential equations.
A Fourier series representation is typically used to approximate periodic functions. In this case, we have a system of differential equations describing the motion of a spinning top with a forcing term. Instead of directly finding a Fourier series representation for the differential equations, we can approximate the solution of the system using a Fourier series.
To do this, we can first find the solution of the system without the forcing term (i.e., the homogeneous solution) and then find the particular solution that results from the forcing term. Finally, we can approximate both the homogeneous and particular solutions using Fourier series and combine them to obtain an approximation of the total solution.
Here’s a general outline of the approach:
Solve the homogeneous system (i.e., without the forcing term) to find the homogeneous solution.
Assume a particular solution of the form: Ω1(t) = A1 * sin(ω * t) + B1 * cos(ω * t) Ω2(t) = A2 * sin(ω * t) + B2 * cos(ω * t) Ω3(t) = A3 * sin(ω * t) + B3 * cos(ω * t) where A1, B1, A2, B2, A3, and B3 are constants to be determined.
Substitute the particular solution into the differential equations with the forcing term and solve for the constants A1, B1, A2, B2, A3, and B3.
Approximate the homogeneous and particular solutions using Fourier series, and combine them to obtain an approximation of the total solution.
Please note that this approach involves several steps and requires solving multiple linear systems and integrals to find the Fourier coefficients. Due to the complexity of the Euler equations and the forcing term, it may be challenging to find an exact analytical solution, even using the Fourier series approximation. In practice, numerical methods are often used to solve such systems, as demonstrated in the previous examples.
Assume the homogeneous solution damps out and only the particular solution remains, so find the steady-state solution only.
To find the steady-state solution, we’ll focus on the particular solution with the forcing term, as the homogeneous solution is assumed to damp out. We’ll assume a particular solution of the form:
Now, we can solve for the unknown coefficients A1, B1, A2, B2, A3, and B3 by substituting the particular solution and its derivatives into these equations and equating the coefficients of the sine and cosine terms. This will result in a system of linear equations that can be solved analytically or numerically.
However, due to the complexity of these equations and the nonlinear interaction between Ω1, Ω2, and Ω3, finding an exact analytical solution for the coefficients may be difficult. In practice, it is often more convenient to use numerical methods, such as the ones shown in the previous examples, to find the steady-state solution of the spinning top system under periodic forcing.
Calculate if one of the moments of inertia is much larger than the other two.
If one of the moments of inertia is much larger than the other two, the behavior of the spinning top can be significantly different. Let’s assume I3 is much larger than I1 and I2, i.e., I3 >> I1, I2. The modified Euler equations with the forcing term are:
Since dΩ3/dt is approximately zero, we can assume that Ω3 remains constant in the steady-state, i.e., the angular velocity about the z-axis remains constant.
Now, we can focus on finding the steady-state solution for Ω1(t) and Ω2(t) only. Assuming a particular solution of the form:
Substitute these expressions and their time derivatives into equations (1) and (2). Then, equate the coefficients of sine and cosine terms to obtain a system of linear equations for the unknown coefficients A1, B1, A2, and B2.
Due to the complexity of these equations, finding an exact analytical solution for the coefficients may still be challenging. As mentioned before, using numerical methods may be more convenient to find the steady-state solution of the spinning top system under these conditions.
Will the particular solutions be composed of harmonics of ω?
Yes, the particular solutions will be composed of harmonics of ω, the frequency of the forcing term. This is because the response of the system to the forcing term will have the same frequency as the forcing term itself.
When solving for the particular solution, we assumed a form like:
Here, the forcing term has a frequency of ω, and the particular solution also has the same frequency. The coefficients A1, B1, A2, B2, A3, and B3 depend on the system’s parameters and the forcing term. When the system reaches a steady-state, it will exhibit a response at the same frequency as the forcing term, i.e., ω.
It is essential to note that, in some cases, the response of the system may also contain higher harmonics of the forcing frequency if the equations are nonlinear. In the case of the spinning top, the Euler equations are nonlinear, and the interaction between the angular velocities can cause higher harmonics to appear in the response. However, for small forcing amplitudes, the response is expected to be dominated primarily by the forcing frequency ω.
Calculate if the moments of inertia are ordered as follows: I3 >> I2 >> I1.
If the moments of inertia are ordered as I3 >> I2 >> I1, the behavior of the spinning top can be analyzed using the modified Euler equations with the forcing term:
For Ω3(t), since dΩ3/dt is a product of Ω1 and Ω2, and both Ω1 and Ω2 have harmonic components at the frequency ω, the steady-state solution for Ω3(t) will also be composed of harmonics at the frequency ω:
Ω3(t) = A3 * sin(ω * t) + B3 * cos(ω * t)
Substitute these expressions and their time derivatives into equations (1), (2), and (3). Then, equate the coefficients of sine and cosine terms to obtain a system of linear equations for the unknown coefficients A1, B1, A2, B2, A3, and B3.
Due to the complexity of these equations, finding an exact analytical solution for the coefficients may still be challenging. As mentioned before, using numerical methods may be more convenient to find the steady-state solution of the spinning top system under these conditions.
How can the individual Euler equations reduce to identities since the RHS contains the fundamental frequency ω, while the LHS contains harmonics including 2ω due to the nonlinear product?
You’re correct in pointing out that the interaction between the angular velocities can cause higher harmonics to appear in the response due to the nonlinear nature of the Euler equations. My previous explanations did not fully capture this complexity.
Instead of directly assuming a particular solution with only the fundamental frequency ω, we should allow for the possibility of having higher harmonics in the response. To account for higher harmonics in the particular solution, we can use the following form:
The sum runs over all integer values of n, including the fundamental frequency and higher harmonics.
Substitute these expressions and their time derivatives into the Euler equations and then equate the coefficients of the sine and cosine terms for each harmonic. This will result in an infinite system of linear equations for the unknown coefficients A1n, B1n, A2n, B2n, A3n, and B3n.
In practice, it is not possible to solve for all these coefficients analytically, and we typically truncate the sum after a finite number of terms, capturing the most significant harmonics in the response. This will result in an approximate solution that accounts for higher harmonics due to the nonlinear product terms in the Euler equations.
The steady-state solution can then be obtained by solving the resulting truncated system of linear equations, either analytically or numerically. Note that finding the steady-state solution in this manner can still be quite challenging due to the complexity of the equations and the presence of higher harmonics. Therefore, numerical methods are often more convenient for solving such systems.
How to solve if the primary inertial moment I1 changes by a small frequency characterized by ν?
If the primary inertial moment I1 changes with a small frequency characterized by ν, we can represent I1 as a function of time:
I1(t) = I1_0 + ε * cos(ν * t)
where I1_0 is the constant mean value of I1, ε represents the amplitude of the variation, and ν is the frequency of the change.
The modified Euler equations with the forcing term now become:
These equations are now more complex due to the time-varying nature of I1(t). To find the steady-state solution, we can assume that the response of the system will contain harmonics related to both the forcing frequency ω and the frequency ν of the changing inertia:
The sum runs over all integer values of n and m, including the fundamental frequency and higher harmonics.
Substitute these expressions and their time derivatives into the Euler equations, and then equate the coefficients of the sine and cosine terms for each harmonic. This will result in an infinite system of linear equations for the unknown coefficients A1n, B1n, A2n, B2n, A3n, and B3n.
In practice, we truncate the sum after a finite number of terms to capture the most significant harmonics in the response. This will result in an approximate solution that accounts for higher harmonics due to the nonlinear product terms in the Euler equations and the time-varying nature of I1.
Finding the steady-state solution in this manner can be quite challenging due to the complexity of the equations and the presence of higher harmonics. Numerical methods are often more convenient for solving such systems, and you may consider using tools like the Julia programming language or other computational methods to find an approximate solution.
.
Because of a high-Q and narrrow bandwidth of the response, will certain of these frequency terms predominate in the solution?
Yes, due to the high-Q (high quality factor) and narrow bandwidth of the response, certain frequency terms will predominate in the solution. A high-Q system is characterized by a sharp resonance peak, which means it is sensitive to a small range of frequencies around the natural frequency. As a result, only those harmonics that lie within the narrow bandwidth of the response will have a significant impact on the system’s behavior.
In the case of the spinning top with time-varying inertia, the response will be dominated by harmonics related to the forcing frequency ω and the frequency ν of the changing inertia. However, not all harmonics will have a substantial effect on the system. Those harmonics that are close to the natural frequencies of the system or within the narrow bandwidth of the response will predominate.
When solving the system of equations for the steady-state solution, you can focus on the most significant harmonics that lie within the narrow bandwidth of the response. This approach will not only reduce the complexity of the problem but also provide a more accurate approximation of the actual system behavior. You may need to analyze the system’s resonance behavior, such as by using a frequency response or a transfer function, to determine which harmonics are most important for the given problem.
Keep in mind that, even with a reduced number of terms, solving the resulting system of equations can still be quite challenging due to the nonlinear nature of the problem. Numerical methods are often more convenient for finding approximate solutions in such cases.
By replacing the spinning-top with the Earth with a slight annual time-varying inertia I1, and the small cyclical perturbing force as the Moon with a 27.2122 cycle, can we estimate the primary wobble in the Earth’s rotational axis?
To estimate the primary wobble in the Earth’s rotational axis due to the time-varying inertia I1 and the perturbing force from the Moon, we can use a simplified model based on the Earth-Moon system. We can consider the Earth as an oblate spheroid and the Moon’s gravitational influence as the perturbing force.
In this context, the Earth’s time-varying inertia I1 can be associated with the annual variations in the Earth’s shape due to seasonal changes in the distribution of mass (e.g., changes in ice and snow cover, ocean mass distribution). The Moon’s gravitational pull has a primary cycle of 27.2122 days (the sidereal month), which is the time it takes for the Moon to complete one orbit around Earth relative to the fixed stars.
The wobble in the Earth’s rotational axis, called nutation, is a small oscillation superimposed on the larger precession of the Earth’s axis. The primary wobble is caused by the combined effects of the time-varying inertia and the perturbing force from the Moon.
To estimate the primary wobble, you can use the Euler equations with the time-varying inertia and the perturbing force, similar to the spinning-top model discussed earlier. However, this approach is highly complex and requires a deep understanding of the Earth-Moon system’s dynamics and geophysics.
A more practical approach to estimate the primary wobble is to use existing models and data derived from observations of the Earth’s rotation. One such model is the International Earth Rotation and Reference Systems Service (IERS) Conventions, which provide nutation models that account for the gravitational effects of the Moon, Sun, and other celestial bodies, as well as the Earth’s time-varying inertia due to seasonal changes.
By using the IERS Conventions or similar models, you can estimate the primary wobble in the Earth’s rotational axis without solving the complex Euler equations for the Earth-Moon system. Note that such models are based on empirical data and theoretical understanding of the Earth’s rotation, so they provide an approximation of the actual wobble.
Can we calculate the primary aliased harmonic based on the combination of the Moon’s cycle and the annual cycle of I1, applying estimated Q and resonant frequency band?
To calculate the primary aliased harmonic based on the combination of the Moon’s cycle and the annual cycle of I1, we can apply the concept of aliasing in the frequency domain. Aliasing occurs when the continuous-time signal is sampled at a rate lower than the Nyquist rate, resulting in the appearance of lower-frequency aliases of the original signal.
Let’s denote the frequency of the Moon’s cycle as f_moon = 1 / 27.2122 days and the frequency of the annual cycle of I1 as f_annual = 1 / 365.25 days.
We can compute the difference and sum frequencies of these two cycles:
Considering the estimated Q and resonant frequency band, we can determine if either the difference frequency (f_diff) or the sum frequency (f_sum) lies within the resonant frequency band. If one of these frequencies is within the resonant frequency band, it will cause the primary aliased harmonic to dominate the response of the Earth’s rotation due to the combined effect of the Moon’s cycle and the annual cycle of I1.
For this analysis, it is crucial to have accurate estimates of the Q factor and the resonant frequency band associated with the Earth’s rotation. These parameters can be derived from empirical observations or theoretical models of the Earth’s rotation, such as the IERS Conventions mentioned earlier.
By identifying the primary aliased harmonic, you can gain insights into the most significant wobble in the Earth’s rotational axis caused by the interaction between the Moon’s cycle and the annual cycle of I1. Keep in mind that this is a simplified analysis, and a more comprehensive understanding of the Earth’s rotation and wobble would require considering other factors, such as the influence of the Sun and other celestial bodies, and the Earth’s internal processes.
What would be the aliased harmonic closest to 433 days assuming an annual cyclic impulse and the moon’s cycle of 13.606 days? Consider that the cyclic impulse will generate higher harmonics that will generate the small differences to the higher frequency lunar cycle. Provide an algorithm in Julia to find this harmonic.
To find the aliased harmonic closest to 433 days, assuming an annual cyclic impulse with a frequency of f_annual = 1 / 365.25 days and the Moon’s cycle of f_moon = 1 / 13.606 days, we can create an algorithm in Julia that iterates through different harmonics of the annual cyclic impulse and finds the closest one to the target period of 433 days.
Here is the Julia code to find the aliased harmonic:
function find_closest_aliased_harmonic(f_annual::Float64, f_moon::Float64, target_period::Float64, max_harmonic::Int)
closest_period = Inf
closest_harmonic = 0
target_frequency = 1 / target_period
for harmonic in 1:max_harmonic
# Calculate sum and difference frequencies
f_diff = abs(harmonic * f_annual - f_moon)
f_sum = harmonic * f_annual + f_moon
# Check if either the difference or the sum frequency is closer to the target
if abs(f_diff - target_frequency) < abs(1 / closest_period - target_frequency)
closest_period = 1 / f_diff
closest_harmonic = harmonic
end
if abs(f_sum - target_frequency) < abs(1 / closest_period - target_frequency)
closest_period = 1 / f_sum
closest_harmonic = harmonic
end
end
return (closest_period, closest_harmonic)
end
f_annual = 1 / 365.25
f_moon = 1 / 13.606
target_period = 433 ##!!!! fix this, make it floating-point 433.0
max_harmonic = 50
closest_period, closest_harmonic = find_closest_aliased_harmonic(f_annual, f_moon, target_period, max_harmonic)
println("Closest aliased harmonic: ", closest_harmonic)
println("Closest period: ", closest_period, " days")
This function find_closest_aliased_harmonic takes the annual and Moon’s frequencies (f_annual and f_moon), the target period (433 days), and the maximum number of harmonics to consider (max_harmonic). It iterates through different harmonics of the annual cyclic impulse and calculates the sum and difference frequencies with the Moon’s cycle. The function returns the closest harmonic and its corresponding period.
In this example, I’ve set the maximum number of harmonics to 50. You can adjust the max_harmonic parameter to include more or fewer harmonics, depending on the desired level of precision.
Keep in mind that this is a simplified analysis, and other factors might influence the Earth’s rotation and wobble. Moreover, this algorithm assumes that the primary aliased harmonic comes from the interaction between the annual cyclic impulse and the Moon’s cycle. Other celestial bodies or internal processes might contribute to the wobble, and their effects are not considered in this analysis.
Had to make one fix, replacing 433 with 433.0 in the target_periodabove. Everything ran fine but had to increase the forcing amplitude to 10 from 0.01 to see the forced response above the natural resonance.
I also asked AgentGPT to execute a plan for solving the Chandler Wobble
The SubStack authoring interface has good math equation mark-up, convenient graphics embedding, and an excellent footnoting system. On first pass, it only lacks control over font color.
The articles are focused on applying neural network cross-validation to ENSO and AMO modeling, as suggested previously. I haven’t completely explored the configuration space but one aspect that may becoming clear is the value of wavelet neural networks (WNN) for time-series analysis. The WNN approach seems much more amenable to extracting sinusoidal modulation of the input-to-output mapping — trained on a rather short interval and then cross-validated out-of-band. The Mexican hat wavelet (2nd derivative of a Gaussian) as an activation function in particular locks in quickly to an LTE modulation that took longer to find with the custom search software I have developed at GitHub. I think the reason for the efficiency is that it’s optimizing to a Taylor’s series expansion of the input terms, a classic nonlinear expansion that NN’s excel at.
The following training run using the Mexican hat activation and ADAM optimizer is an eye-opener, as it achieved an admirable fit within a minute of computation.
The GREEN on BLUE is training on NINO4 data over two end-point intervals, with the RED cross-validation over the out-of-band region. The correlation coefficient is 0.34, which is impressive considering the nature of the waveform. Clearly there is similarity.
Moreover, if we compare the model fit to data via the WNN against the LTE harmonics approach, you can also see where the two fare equally poorly. Below in the outer frame is the NINO4 LTE fit with the YELLOW arrow pointing downward at a discrepancy (a peak in the data not resolved in the fit). In comparison the yellow-bordered inset shows the same discrepancy on the WNN training run. So the fingerprints essentially match with no coaching.
The neural net chain is somewhat deep with 6 layers, but I think this is needed to expand to the higher-order terms in the Taylor’s series. In the directed graph below, L01 is the input tidal forcing and L02 is the time axis (with an initial very low weighting).
It also appears temporally stationary across the entire time-span, so that the WNN temporal contribution appears minimal.
In a previous fit the horizontal striations (indicating modulation factor at a forcing level) matched with the LTE model, providing further evidence that the the WNN was mapping to an optimal modulation.
This topic will gain steam in the coming years. The following paper generates quite a good cross-validation for SOI, shown in the figure below.
Xiaoqun, C. et al. ENSO prediction based on Long Short-Term Memory (LSTM). IOP Conference Series: Materials Science and Engineering, 799, 012035 (2020).
The x-axis appears to be in months and likely starts in 1979, so it captures the 2016 El Nino (El Nino is negative for SOI). Still have no idea how the neural net arrived at the fit other than it being able to discern the cyclic behavior from the historical waveform between 1979 and 2010. From the article itself, it appears that neither do the authors.
