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.
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
Given two models of a physical behavior, the “better” model has the highest correlation (or lowest error) to the data and the lowest number of degrees of freedom (#DOF) in terms of tunable parameters. This ratio CC/#DOF of correlation coefficient over DOF is routinely used in automated symbolic regression algorithms and for scoring of online programming contests. A balance between a good error metric and a low complexity score is often referred to as a Pareto frontier.
So for modeling ENSO, the challenge is to fit the quasi-periodic NINO34 time-series with a minimal number of tunable parameters. For a 140 year fitting interval (1880-1920), a naive Fourier series fit could easily take 50-100 sine waves of varying frequencies, amplitudes, and phase to match a low-pass filtered version of the data (any high-frequency components may take many more). However that is horribly complex model and obviously prone to over-fitting. Obviously we need to apply some physics to reduce the #DOF.
Since we know that ENSO is essentially a model of equatorial fluid dynamics in response to a tidal forcing, all that is needed is the gravitational potential along the equator. The paper by Na  has software for computing the orbital dynamics of the moon (i.e. lunar ephemerides) and a 1st-order approximation for tidal potential:
The software contains well over 100 sinusoidal terms (each consisting of amplitude, frequency, and phase) to internally model the lunar orbit precisely. Thus, that many DOF are removed, with a corresponding huge reduction in complexity score for any reasonable fit. So instead of a huge set of factors to manipulate (as with many detailed harmonic tidal analyses), what one is given is a range (r = R) and a declination ( ψ=delta) time-series. These are combined in a manner following the figure from Na shown above, essentially adjusting the amplitudes of R and delta while introducing an additional tangential or tractional projection of delta (sin instead of cos). The latter is important as described in NOAA’s tide producing forces page.
Although I roughly calibrated this earlier  via NASA’s HORIZONS ephemerides page (input parameters shown on the right), the Na software allows better flexibility in use. The two calculations essentially give identical outputs and independent verification that the numbers are as expected.
As this post is already getting too long, this is the result of doing a Laplace’s Tidal Equation fit (adding a few more DOF), demonstrating that the limited #DOF prevents over-fitting on a short training interval while cross-validating outside of this band.
This low complexity and high accuracy solution would win ANY competition, including the competition for best seasonal prediction with a measly prize of 15,000 Swiss francs . A good ENSO model is worth billions of $$ given the amount it will save in agricultural planning and its potential for mitigation of human suffering in predicting the timing of climate extremes.
 Na, S.-H. Chapter 19 – Prediction of Earth tide. in Basics of Computational Geophysics (eds. Samui, P., Dixon, B. & Tien Bui, D.) 351–372 (Elsevier, 2021). doi:10.1016/B978-0-12-820513-6.00022-9.
 Pukite, P.R. et al “Ephemeris calibration of Laplace’s tidal equation model for ENSO” AGU Fall Meeting, 2018. doi:10.1002/essoar.10500568.1
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.
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
This is what amounts to forecast meteorology — a seasoned weatherman will look at the current charts and try to interpret them based on patterns that have been associated with previous observations. They appear to have equal powers of memory recall, pattern matching, and the art of the bluff.
I don’t do that kind of stuff and don’t think I ever will.
If this comes out of a human mind, then that same information can be fed into a knowledgebase and either a backward or forward-chained inference engine could make similar assertions.
And that explains why I don’t do it — a machine should be able to do it better.
For the LTE formulation along the equator, the analytical solution reduces to g(f(t)), where g(x) is a periodic function. Without knowing what g(x) is, we can use the frequency-domain entropy or spectral entropy of the Fourier series mapping an estimated x=f(t) forcing amplitude to a measured climate index time series such as ENSO. The frequency-domain entropy is the sum or integral of this mapping of x to g(x) in reciprocal space applying the Shannon entropy –I(f).ln(I(f)) normalized over the I(f) frequency range, which is the power spectral (frequency) density of the mapping from the modeled forcing to the time-series waveform sample.
This measures the entropy or degree of disorder of the mapping. So to maximize the degree of order, we minimize this entropy value.
This calculated entropy is a single scalar metric that eliminates the need for evaluating various cyclic g(x) patterns to achieve the best fit. Instead, what it does is point to a highly-ordered spectrum (top panel in the above figure), of which the delta spikes can then be reverse engineered to deduce the primary frequency components arising from the the LTE modulation factor g(x).
The approach works particularly well once the spectral spikes begin to emerge from the background. In terms of a physical picture, what is actually emerging are the principle standing wave solutions for particular wavenumbers. One can see this in the LTE modulation spectrum below where there is a spike at a wavenumber at 1.5 and one at around 10 in panel A (isolating the sin spectrum and cosine spectrum separately instead of the quadrature of the two giving the spectral intensity). This is then reverse engineered as a fit to the actual LTE modulation g(x) in panel B. Panel D is the tidal forcing x=f(t) that minimized the Shannon entropy, thus creating the final fit g(f(t)) in panel C when the LTE modulation is applied to the forcing.
The approach does work, which is quite a boon to the efficiency of iterative fitting towards a solution, reducing the number of DOF involved in the calculation. Prior to this, a guess for the LTE modulation was required and the iterative fit would need to evolve towards the optimal modulation periods. In other words, either approach works, but the entropy approach may provide a quicker and more efficient path to discovering the underlying standing-wave order.
I will eventually add this to the LTE fitting software distro available on GitHub. This may also be applicable to other measures of entropy such as Tallis, Renyi, multi-scale, and perhaps Bispectral entropy, and will add those to the conventional Shannon entropy measure as needed.
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.
Our book Mathematical Geoenergy presents a number of novel approaches that each deserve a research paper on their own. Here is the list, ordered roughly by importance (IMHO):
Laplace’s Tidal Equation Analytic Solution. (Ch 11, 12) A solution of a Navier-Stokes variant along the equator. Laplace’s Tidal Equations are a simplified version of Navier-Stokes and the equatorial topology allows an exact closed-form analytic solution. This could classify for the Clay Institute Millenium Prize if the practical implications are considered, but it’s a lower-dimensional solution than a complete 3-D Navier-Stokes formulation requires.
Model of El Nino/Southern Oscillation (ENSO). (Ch 12) A tidally forced model of the equatorial Pacific’s thermocline sloshing (the ENSO dipole) which assumes a strong annual interaction. Not surprisingly this uses the Laplace’s Tidal Equation solution described above, otherwise the tidal pattern connection would have been discovered long ago.
Model of Quasi-Biennial Oscillation (QBO). (Ch 11) A model of the equatorial stratospheric winds which cycle by reversing direction ~28 months. This incorporates the idea of amplified cycling of the sun and moon nodal declination pattern on the atmosphere’s tidal response.
Origin of the Chandler Wobble. (Ch 13) An explanation for the ~433 day cycle of the Earth’s Chandler wobble. Finding this is a fairly obvious consequence of modeling the QBO.
The Oil Shock Model. (Ch 5) A data flow model of oil extraction and production which allows for perturbations. We are seeing this in action with the recession caused by oil supply perturbations due to the Corona Virus pandemic.
The Dispersive Discovery Model. (Ch 4) A probabilistic model of resource discovery which accounts for technological advancement and a finite search volume.
Ornstein-Uhlenbeck Diffusion Model (Ch 6) Applying Ornstein-Uhlenbeck diffusion to describe the decline and asymptotic limiting flow from volumes such as occur in fracked shale oil reservoirs.
The Reservoir Size Dispersive Aggregation Model. (Ch 4) A first-principles model that explains and describes the size distribution of oil reservoirs and fields around the world.
Origin of Tropical Instability Waves (TIW). (Ch 12) As the ENSO model was developed, a higher harmonic component was found which matches TIW
Characterization of Battery Charging and Discharging. (Ch 18) Simplified expressions for modeling Li-ion battery charging and discharging profiles by applying dispersion on the diffusion equation, which reflects the disorder within the ion matrix.
Anomalous Behavior in Dispersive Transport explained. (Ch 18) Photovoltaic (PV) material made from disordered and amorphous semiconductor material shows poor photoresponse characteristics. Solution to simple entropic dispersion relations or the more general Fokker-Planck leads to good agreement with the data over orders of magnitude in current and response times.
Framework for understanding Breakthrough Curves and Solute Transport in Porous Materials. (Ch 20) The same disordered Fokker-Planck construction explains the dispersive transport of solute in groundwater or liquids flowing in porous materials.
Wind Energy Analysis. (Ch 11) Universality of wind energy probability distribution by applying maximum entropy to the mean energy observed. Data from Canada and Germany. Found a universal BesselK distribution which improves on the conventional Rayleigh distribution.
Terrain Slope Distribution Analysis. (Ch 16) Explanation and derivation of the topographic slope distribution across the USA. This uses mean energy and maximum entropy principle.
Thermal Entropic Dispersion Analysis. (Ch 14) Solving the Fokker-Planck equation or Fourier’s Law for thermal diffusion in a disordered environment. A subtle effect but the result is a simplified expression not involving complex errf transcendental functions. Useful in ocean heat content (OHC) studies.
The Maximum Entropy Principle and the Entropic Dispersion Framework. (Ch 10) The generalized math framework applied to many models of disorder, natural or man-made. Explains the origin of the entroplet.
Solving the Reserve Growth “enigma”. (Ch 6) An application of dispersive discovery on a localized level which models the hyperbolic reserve growth characteristics observed.
Shocklets. (Ch 7) A kernel approach to characterizing production from individual oil fields.
Reserve Growth, Creaming Curve, and Size Distribution Linearization. (Ch 6) An obvious linearization of this family of curves, related to Hubbert Linearization but more useful since it stems from first principles.
The Hubbert Peak Logistic Curve explained. (Ch 7) The Logistic curve is trivially explained by dispersive discovery with exponential technology advancement.
Laplace Transform Analysis of Dispersive Discovery. (Ch 7) Dispersion curves are solved by looking up the Laplace transform of the spatial uncertainty profile.
Gompertz Decline Model. (Ch 7) Exponentially increasing extraction rates lead to steep production decline.
The Dynamics of Atmospheric CO2 buildup and Extrapolation. (Ch 9) Convolving a fat-tailed CO2 residence time impulse response function with a fossil-fuel emissions stimulus. This shows the long latency of CO2 buildup very straightforwardly.
Reliability Analysis and Understanding the “Bathtub Curve”. (Ch 19) Using a dispersion in failure rates to generate the characteristic bathtub curves of failure occurrences in parts and components.
The Overshoot Point (TOP) and the Oil Production Plateau. (Ch 8) How increases in extraction rate can maintain production levels.
Lake Size Distribution. (Ch 15) Analogous to explaining reservoir size distribution, uses similar arguments to derive the distribution of freshwater lake sizes. This provides a good feel for how often super-giant reservoirs and Great Lakes occur (by comparison).
The Quandary of Infinite Reserves due to Fat-Tail Statistics. (Ch 9) Demonstrated that even infinite reserves can lead to limited resource production in the face of maximum extraction constraints.
Oil Recovery Factor Model. (Ch 6) A model of oil recovery which takes into account reservoir size.
Network Transit Time Statistics. (Ch 21) Dispersion in TCP/IP transport rates leads to the measured fat-tails in round-trip time statistics on loaded networks.
Particle and Crystal Growth Statistics. (Ch 20) Detailed model of ice crystal size distribution in high-altitude cirrus clouds.
Rainfall Amount Dispersion. (Ch 15) Explanation of rainfall variation based on dispersion in rate of cloud build-up along with dispersion in critical size.
Earthquake Magnitude Distribution. (Ch 13) Distribution of earthquake magnitudes based on dispersion of energy buildup and critical threshold.
IceBox Earth Setpoint Calculation. (Ch 17) Simple model for determining the earth’s setpoint temperature extremes — current and low-CO2 icebox earth.
Global Temperature Multiple Linear Regression Model (Ch 17) The global surface temperature records show variability that is largely due to the GHG rise along with fluctuating changes due to ocean dipoles such as ENSO (via the SOI measure and also AAM) and sporadic volcanic eruptions impacting the atmospheric aerosol concentrations.
GPS Acquisition Time Analysis. (Ch 21) Engineering analysis of GPS cold-start acquisition times. Using Maximum Entropy in EMI clutter statistics.
1/f NoiseModel (Ch 21) Deriving a random noise spectrum from maximum entropy statistics.
Stochastic Aquatic Waves (Ch 12) Maximum Entropy Analysis of wave height distribution of surface gravity waves.
The Stochastic Model of Popcorn Popping. (Appx C) The novel explanation of why popcorn popping follows the same bell-shaped curve of the Hubbert Peak in oil production. Can use this to model epidemics, etc.
Dispersion Analysis of Human Transportation Statistics. (Appx C) Alternate take on the empirical distribution of travel times between geographical points. This uses a maximum entropy approximation to the mean speed and mean distance across all the data points.
A ridiculous paper on the uncertainty of climate models is under post-publication review at peerpub.com
What drives me more nuts is why everyone is trying to correct what a blithering idiot (P. Frank) is advancing instead of just solving the differential equations and modeling the climate variability. Does everyone think we will actually make any progress by correcting the poor sod’s freshman homework assignment?
Instead, let’s get going and finish off the tidal model of ENSO. That will do more than anything else to quash the endless discussion over how much natural climate variability is acceptable to be able to discern an AGW trend.
The modeled QBO cycle is directly related to the nodal (draconian) lunar cycle physically aliased against the annual cycle. The empirical cycle period is best estimated by tracking the peak acceleration of the QBO velocity time-series, as this acceleration (1st derivative of the velocity) shows a sharp peak. This value should asymptotically approach a 2.368 year period over the long term. Since the recent data from the main QBO repository provides an additional acceleration peak from the past month, now is as good a time as any to analyze the cumulative data.
The new data-point provides a longer period which compensated for some recent shorter periods, such that the cumulative mean lies right on the asymptotic line. The jitter observed is explainable in terms of the model, as acceleration peaks are more prone to align close to an annual impulse. But the accumulated mean period is still aligned to the draconic aliasing with this annual impulse. As more data points come in over the coming decades, the mean should vary less and less from the asymptotic value.
The fit to QBO using all the data save for the last available data point is shown below. Extrapolating beyond the greenarrow, we should see an uptick according to the red waveform.
Adding the recent data-point and the bluewaveform does follow the model.
There was a flurry of recent discussion on the QBO anomaly of 2016 (shown as a split peak above), which implied that perhaps the QBO would be permanently disrupted from it’s long-standing pattern. Instead, it may be a more plausible explanation that the QBO pattern was not simply wandering from it’s assumed perfectly cyclic path but instead is following a predictable but jittery track that is a combination of the (physically-aliased) annual impulse-synchronized Draconic cycle together with a sensitivity to variations in the draconic cycle itself. The latter calibration is shown below, based on NASA ephermeris.
This is the QBO spectral decomposition, showing signal strength centered on the fundamental aliased Draconic value, both for the data and the set by the model.
The main scientist, Prof. Richard Lindzen, behind the consensus QBO model has been recently introduced here as being “considered the most distinguished living climate scientist on the planet”. In his presentation criticizing AGW science , Lindzen claimed that the climate oscillates due to a steady uniform force, much like a violin oscillates when the steady force of a bow is drawn across its strings. An analogy perhaps better suited to reality is that the violin is being played like a drum. Resonance is more of a decoration to the beat itself.
 Professor Richard Lindzen slammed conventional global warming thinking warming as ‘nonsense’ in a lecture for the Global Warming Policy Foundation on Monday. ‘An implausible conjecture backed by false evidence and repeated incessantly … is used to promote the overturn of industrial civilization,’ he said in London. — GWPF
The Madden-Julian Oscillation (MJO) is a climate index that captures tropical variability at a finer resolution (i.e. intra-annual) than the (inter-annual) ENSO index over approximately the same geographic region. Since much of the MJO variability is observed as 30 to 60 day cycles (and these are traveling waves, not standing waves), providing MJO data as a monthly time-series will filter out the fast cycles. Still, it is interesting to analyze the monthly MJO data and compare/contrast that to ENSO. As a disclaimer, it is known that inter-annual variability of the MJO is partly linked to ENSO, but the following will clearly show that connection.
This is the fit of MJO (longitude index #1) using the ENSO model as a starting point (either the NINO34 or SOI works equally well).
The constituent temporal forcing factors for MJO and ENSO align precisely
This is not surprising because the monthly filtered MJO does show the same El Nino peaks at 1983, 1998, and 2016 as the ENSO time-series. The only difference is in the LTE spatial modulation applied during the fitting process, whereby the MJO has a stronger high-wavenumber factor than the ENSO time series.
This is the SOI fit over the same 1980+ interval as MJO, with an almost 0.6 correlation.