# Wobble Perturbation

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:

1. First, install the required packages by running:
``````using Pkg
``````
1. 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 = (I2 - I3)/I1 * Ω2 * Ω3
du = (I3 - I1)/I2 * Ω1 * Ω3
du = (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:

1. 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 = (I2 - I3)/I1 * Ω2 * Ω3 + F/I1
du = (I3 - I1)/I2 * Ω1 * Ω3
du = (I1 - I2)/I3 * Ω1 * Ω2
end
``````
1. 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:

1. Solve the homogeneous system (i.e., without the forcing term) to find the homogeneous solution.
2. 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.
3. Substitute the particular solution into the differential equations with the forcing term and solve for the constants A1, B1, A2, B2, A3, and B3.
4. 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:

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)

Differentiating Ω1(t), Ω2(t), and Ω3(t) with respect to time, we get:

dΩ1/dt = ω * (A1 * cos(ω * t) – B1 * sin(ω * t))
dΩ2/dt = ω * (A2 * cos(ω * t) – B2 * sin(ω * t))
dΩ3/dt = ω * (A3 * cos(ω * t) – B3 * sin(ω * t))

Substitute these expressions into the modified Euler equations with the forcing term:

(1) (I2 – I3)/I1 * Ω2 * Ω3 + F/I1 = dΩ1/dt
(2) (I3 – I1)/I2 * Ω1 * Ω3 = dΩ2/dt
(3) (I1 – I2)/I3 * Ω1 * Ω2 = dΩ3/dt

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:

(1) (I2 – I3)/I1 * Ω2 * Ω3 + F/I1 = dΩ1/dt
(2) (I3 – I1)/I2 * Ω1 * Ω3 = dΩ2/dt
(3) (I1 – I2)/I3 * Ω1 * Ω2 = dΩ3/dt

Since I3 is much larger than I1 and I2, we can make the following approximations:

• The term (I2 – I3)/I1 ≈ -1 and (I3 – I1)/I2 ≈ 1, as I3 dominates both I1 and I2.
• The term (I1 – I2)/I3 becomes very small and can be neglected.

Now, the modified Euler equations can be simplified as:

(1) -Ω2 * Ω3 + F/I1 = dΩ1/dt
(2) Ω1 * Ω3 = dΩ2/dt
(3) 0 ≈ dΩ3/dt

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:

Ω1(t) = A1 * sin(ω * t) + B1 * cos(ω * t)
Ω2(t) = A2 * sin(ω * t) + B2 * cos(ω * t)

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:

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)

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:

(1) (I2 – I3)/I1 * Ω2 * Ω3 + F/I1 = dΩ1/dt
(2) (I3 – I1)/I2 * Ω1 * Ω3 = dΩ2/dt
(3) (I1 – I2)/I3 * Ω1 * Ω2 = dΩ3/dt

Since I3 is much larger than I1 and I2, we can approximate:

• The term (I2 – I3)/I1 ≈ -1, as I3 dominates both I1 and I2.
• The term (I3 – I1)/I2 ≈ 1, as I3 dominates I1.

Since I2 is much larger than I1, we can also approximate:

• The term (I1 – I2)/I3 ≈ -1, as I2 dominates I1.

Now, the modified Euler equations can be simplified as:

(1) -Ω2 * Ω3 + F/I1 = dΩ1/dt
(2) Ω1 * Ω3 = dΩ2/dt
(3) -Ω1 * Ω2 = dΩ3/dt

As before, we can focus on finding the steady-state solution for Ω1(t) and Ω2(t). We 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)

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:

Ω1(t) = ∑(A1n * cos(n * ω * t) + B1n * sin(n * ω * t))
Ω2(t) = ∑(A2n * cos(n * ω * t) + B2n * sin(n * ω * t))
Ω3(t) = ∑(A3n * cos(n * ω * t) + B3n * sin(n * ω * t))

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:

(1) (I2 – I3)/I1(t) * Ω2 * Ω3 + F/I1(t) = dΩ1/dt
(2) (I3 – I1(t))/I2 * Ω1 * Ω3 = dΩ2/dt
(3) (I1(t) – I2)/I3 * Ω1 * Ω2 = dΩ3/dt

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:

Ω1(t) = ∑(A1n * cos((n * ω + m * ν) * t) + B1n * sin((n * ω + m * ν) * t))
Ω2(t) = ∑(A2n * cos((n * ω + m * ν) * t) + B2n * sin((n * ω + m * ν) * t))
Ω3(t) = ∑(A3n * cos((n * ω + m * ν) * t) + B3n * sin((n * ω + m * ν) * t))

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:

f_diff = |f_moon – f_annual|
f_sum = f_moon + f_annual

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.

``````julia> closest_period, closest_harmonic = find_closest_aliased_harmonic(f_annual, f_moon, target_period, max_harmonic)
(432.36397250739594, 26)
julia> println("Closest aliased harmonic: ", closest_harmonic)
Closest aliased harmonic: 26
julia> println("Closest period: ", closest_period, " days")
Closest period: 432.36397250739594 days

``````

The bottom-line response !!!

Closest period: 432.36397250739594 days

Had to make one fix, replacing 433 with 433.0 in the `target_period` above. 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

# Atmospheric Science

I don’t immediately trust the research published by highly cited atmospheric scientists. By my count many of them seem more keen on presenting their personal views rather than advancing the field. Off the top of my head, Richard Lindzen, Murry Salby, Roy Spencer, Tim Dunkerton, Roger Pielke, Cliff Mass, Judith Curry are all highly cited but come across as political and/or religious zealots. One guy on the list, Dunkerton, is also a racist, who happened to make the Washington Post twice : “Physicist ousted from research post after sending offensive tweet to Hispanic meteorologist” and “Atmospheric scientist loses honor, membership over ethics violation“. Awful stuff and he hasn’t stopped spouting off on Twitter.

Granted that Dunkerton says dumb stuff on Twitter but his highly cited research is also off-base. That’s IMO only because recent papers by others in the field of atmospheric science do continue to cite his ideas as primary, if not authoritative. For example, from a recently published paper “The Gravity Wave Activity during Two Recent QBO Disruptions Revealed by U.S. High-Resolution Radiosonde Data”, citations 1 & 12 both refer to Dunkerton, and specifically to his belief that the QBO period is a property of the atmospheric medium itself

Straight-forward to debunk this Dunkerton theory since the length of the cycle directly above the QBO layer is semi-annual and thus not a property of the medium but of the semi-annual nodal forcing frequency. If we make the obvious connection to the other nodal forcing — that of the moon — then we find the QBO period is fixed to 28 months. I have been highlighting this connection to the authors of new QBO papers under community review, often with some subsequent feedback provided such as here: https://doi.org/10.5194/acp-2022-792-CC1 . Though not visible yet in the comments, I received some personal correspondence that showed that the authors under peer-review are taking the idea seriously and attempting to duplicate the calculations. They seem to be methodical in their approach, asking for clarification and further instructions where they couldn’t follow the formulation. They know about the GitHub software, so hopefully that will be of some help.

In contrast, Dunkerton also knows about my approach but responds in an inscrutable (if not condescending) way. Makes you wonder if scientists such as Dunkerton and Lindzen are bitter and taking out their frustrations via the media. Based on their doggedness, they may in fact be intentionally trying to impede progress in climate science by taking contrarian stances. In my experience, the top scientists in other research disciplines don’t act this way. YMMV

## UPDATE 3/17/2023

More activity related to my review comment https://doi.org/10.5194/acp-2022-792-CC1

As a review, this was after commenting earlier this year on a Copernicus open-science research article on atmospheric cycles and QBO (with proposed links to sunspots and ENSO) that was undergoing a review, making a suggestion to consider analyses I had presented and published 4 years ago and also prior to that.

Thought that was that and was happy to see that the authors indicated they would revise the manuscript and perhaps advance understanding. But then several days ago, the editor interceded and essentially demanded that the authors not cite my research work. Apparently, the authors were influenced by the editor’s instructions, as they immediately removed my cite and replaced it with a citation to a review article that the editor preferred. The discussion on the article was then closed with no way for me to rebut.

This was all after I spent several hours working with the primary author as they worked to replicate my analysis, sending emails back and forth several times. The editor claimed that my contribution was “a new idea that has not been published in a recognized journal and received peer review”. This is not the case as I said above: Google Scholar citations all ignored.

Also see this post I contributed to the Peak Oil Barrel blog : https://peakoilbarrel.com/predicting-stratospheric-winds/

# Limits 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-2π 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-2π 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-2π 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

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

# Myth: El Nino/La Nina transitions caused by wind

This 2-D heat map, from Jialin Lin’s research group at The Ohio State University, shows the eastward propagation of the ocean subsurface wave leading to switch from La Niña to El Niño.

The above is from an informative OSU press release from last year titled Solving climate’s toughest questions, one challenge at a time. The following quotes are from that page, bold emphasis mine.

Jialin Lin, associate professor of geography, has spent the last two decades tackling those challenges, and in the past two years, he’s had breakthroughs in answering two of forecasting’s most pernicious questions: predicting the shift between El Niño and La Niña and predicting which hurricanes will rapidly intensify.

Now, he’s turning his attention to creating more accurate models predicting global warming and its impacts, leading an international team of 40 climate experts to create a new book identifying the highest-priority research questions for the next 30-50 years.

Lin set out to create a model that could accurately identify ENSO shifts by testing — and subsequently ruling out — all the theories and possibilities earlier researchers had proposed. Then, Lin realized current models only considered surface temperatures, and he decided to dive deeper.

“After 20 years of research, I finally found that the shift was caused by an ocean wave 100 to 200 meters down in the deep ocean,” Lin said, whose research was published in a Nature journal. “The propagation of this wave from the western Pacific to the eastern Pacific generates the switch from La Niña to El Niño.”

The wave repeatedly appeared two years before an El Niño event developed, but Lin went one step further to explain what generated the wave and discovered it was caused by the moon’s tidal gravitational force.

“The tidal force is even easier to predict,” Lin said. “That will widen the possibility for an even longer lead of prediction. Now you can predict not only for two years before, but 10 years before.”

Essentially, the idea is that these subsurface waves can in no way be caused by surface wind as the latter only are observed later (likely as an after-effect of the sub-surface thermocline nearing the surface and thus modifying the atmospheric pressure gradient). This counters the long-standing belief that ENSO transitions occur as a result of prevailing wind shifts.

The other part of the article concerns correlating hurricane intensification is also interesting.

p.s. It’s all tides : Climatic Drivers of Extreme Sea Level Events Along the
Coastline of Western Australia

# “Wobbling” Moon trending on Twitter

This NASA press release has received mainstream news attention.

The 18.6 year nodal cycle will generate higher tides that will exaggerate sea-level rise due to climate change.

Yahoo news item:

https://news.yahoo.com/lunar-orbit-apos-wobble-apos-173042717.html

So this is more-or-less a known behavior, but hopefully it raises awareness to the other work relating lunar forcing to ENSO, QBO, and the Chandler wobble.

Cited paper

Thompson, P.R., Widlansky, M.J., Hamlington, B.D. et al. Rapid increases and extreme months in projections of United States high-tide flooding. Nat. Clim. Chang. 11, 584–590 (2021). https://doi.org/10.1038/s41558-021-01077-8

# Review: Modeling of ocean equatorial currents in the phase of El Niño and La Niña

https://www.sciencedirect.com/science/article/abs/pii/S037702652100018X#!

The equatorial zone acts as a waveguide. As highlights they list the following bullet-points, taking advantage that the Coriolis effect at the equator vanishes or cancels.

This is a critical assertion, since — as shown in Mathematical Geoenergy –the Chandler wobble (a nutational oscillation) is forced by tides, then transitively so is the El Nino. So when the authors state the consequence is of both nutation and a gravity influence, it is actually the gravity influence of the moon and sun (and slightly Jupiter) that is the root cause.

The article has several equations that claim analytical solutions, but the generated PDF format has apparently not rendered the markup correctly. Many “+” signs are missing from equations. I have seen this issue before when I have tried to generate PDF pages from a markup doc, and assume that is what is happening. Assume the hard-copy version is OK so may have to go to the library to retrieve it, or perhaps ask the authors for a hard-copy.

main author:

Sergey А. Arsen’yev

Dept. of Earth and Planetary Physics of Schmidt’s Institute of the Earth’s Physics, Russian Academy of Sciences, 10 Bolshaya Gruzinskaya, Moscow, 123995, Russia

# Filtering of Climate Data

One of the frustrating aspects of climatology as a science is in the cavalier treatment of data that is often shown, and in particular through the potential loss of information through filtering. A group of scientists at NASA JPL (Perigaud et al) and elsewhere have pointed out how constraining it is to remove what are considered errors (or nuisance parameters) in time-series by assuming that they relate to known tidal or seasonal factors and so can be safely filtered out and ignored. The problem is that this is only appropriate IF those factors relate to an independent process and don’t also cause non-linear interactions with the rest of the data. So if a model predicts both a linear component and non-linear component, it’s not helpful to eliminate portions of the data that can help distinguish the two.

As an example, this extends to the pre-mature filtering of annual data. If you dig enough you will find that NINO3.4 data is filtered to remove the annual data, and that the filtering is over-zealous in that it removes all annual harmonics as well. Worse yet, the weighting of these harmonics changes over time, which means that they are removing other parts of the spectrum not related to the annual signal. Found in an “ensostuff” subdirectory on the NOAA.gov site:

# Lemming/Fox Dynamics not Lotka-Volterra

Appendix E of the book contains information on compartmental models, of which resource depletion models, contagion growth models, drug delivery models, and population growth models belong to. One compartmental population growth model, that specified by the Lotka-Volterra-type predator-prey equations, can be manipulated to match a cyclic wildlife population in a fashion approximating that of observations. The cyclic variation is typically explained as a nonlinear resonance period arising from the competition between the predators and their prey. However, a more realistic model may take into account seasonal and climate variations that control populations directly. The following is a recent paper by wildlife ecologist H. L. Archibald who has long been working on the thesis that seasonal/tidal cycles play a role (one paper that he wrote on the topic dates back to 1977! ).

# Stratospheric Sudden Warming

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 .

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  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 

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).

# Lunisolar Forcing of the Chandler Wobble

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 …