One aspect of the 2020 pandemic is that everyone with any math acumen is becoming aware of contagion models such as the SIR compartmental model, where **S I R** stands for **S**usceptible, **I**nfectious, and **R**ecovered individuals. The Infectious part of the time progression within a population resembles a bell curve that peaks at a particular point indicating maximum contagiousness. The hope is that this either peaks quickly or that it doesn’t peak at too high a level.

In the modeling of oil depletion the equivalent **S I R** compartmental model corresponds to the stages of **S**equestered (in the ground), **I**dentified (i.e. discovered), and **R**ecovered (i.e. extracted). So the compartmental models of oil production and contagion growth (as it plays out during a pandemic) show intuitive parallels :

- Discovery of an oil reservoir is analogous to the start of infection — the leading indicator.

- Extraction from that reservoir to depletion is analogous to death — the lagging indicator.

- Keeping oil in the ground is the equivalent of recovering from the infection.

This has been progressing over the course of decades, with the global peak of the discovered oil occurring by the end of the 1960’s and on a downhill trajectory since then — a slow but relentless extraction drawdown with the majority of the citizenry barely being aware of this trend. The derivative of the recovery phase — the extraction — is shown below and is either at peak or near it.

This slow progression of global oil production is nowhere near as sudden as what we’re going through now as the full S I R coronavirus cycle completes in a matter of months. And this virus cycle may recur again, whereas the S I R version for oil will not — as oil does not regenerate.

Since the connection between the compartmental model of the oil shock model and the pandemic compartmental model is so striking, there are other ways to analogize between the two. Quarantining people from accessing the oil during the oil embargo of the 1970’s (see the figure above for the notch right before 1980) produces the same effect as quarantining during an epidemic (see the notch in the figure below).

When the quarantining policies are relaxed, the contagion will resume — the pattern is spike, suppress, resume. From our book, in the figure below, one can see how the “quarantining=embargo” suppression flattened the production curve below what it was likely trending toward, thus temporarily delaying the time to full depletion.

The analogy is that people are the contagion and will continue to extract oil until the equivalent herd immunity is reached — that is until when the finite oil resources are exhausted. Only an embargo shock prevented the consumption from proceeding too quickly, without which the supply would deplete much sooner.

Another analogy is in how dispersion works. When discovering oil, not every geographic location is exploited at the same time, leading to a spread in how quickly the reserves are exhausted. This is a stochastic element to the compartmental model, the figure below taken from Chapter 7 with the stochastic version of the logistic S-curve discussed recently at the Azimuth Project forum.

Similarly, with a global contagion not every spatially-separated population is infected at the same time, leading to an analogous dispersive effect in infected populations as in the figure below.

As stated earlier, the problem with contagions is that the **S I R** process can potentially repeat, as the populations are renewable. This recurrence is not possible with crude oil, where the supply is finite & non-renewable. While this is strictly true, the finiteness can be masked if there are hidden pockets of reserves, such as with the shale oil reserves of the Permian and Bakken formations. This can be seen in the USA production data shown below, where a spike recovery in production occurred starting in 2010 (the costly fracking of oil partly financed by government stimulation after the 2008-2009 market crash).

This is weakly analogous to a reemergence of a contagion. However, this only provides a temporary reprieve to global oil depletion as the shale oil is already proving to be near peak.

What does the global economy look forward to? Even when we eventually contain the pandemic, the inexorable decline of oil will continue. The pandemic itself will provide a transient suppressive shock to global oil production — via demand destruction of production this only temporarily delays the inevitability of continued oil depletion.

The book describes how to model this compartmental process in depth in Chapter 3 through 9.

Even though the idea of compartmental modeling is well-known in epidemiology, one can find little evidence as to its application to fossil fuel depletion. Google Scholar citations for “compartmental model” & “oil depletion”

The only hit shown refers to our work, and by expanding the keyword search, Google Scholar returns this:

Herrero, C., García-Olivares, A. and Pelegrí, J.L., 2014. Impact of anthropogenic CO2 on the next glacial cycle. Climatic change, 122(1-2), pp.283-298.

The way that Herroro *et al* apply a compartmental model is to model the compartments as oil, atmospheric CO2 from combustion of the oil, and sequestering of that CO2 in the ocean. We also model this compartmental flow in Chap 9.

As of April 1, there are many burgeoning collective efforts on optimizing pandemic contagion S I R models, as listed here (and I contributed a version of Hubbert Linearization for pandemic modeling on Medium.com). Perhaps eventually, the same level of attention will be paid to fossil fuel depletion compartmental models such as the oil shock model, as the impact on society is comparable.

]]>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! ).

Archibald, H. L. Relating the 4-year lemming ( Lemmus spp. and Dicrostonyx spp.) population cycle to a 3.8-year lunar cycle and ENSO. Can. J. Zool. 97, 1054–1063 (2019).

These are his main figures:

This 3.8 year cycle directly agrees with the ENSO model driven by the fortnightly tropical cycle (13.66 days) interacting with an annual cycle, which is indicated in the middle right pane in the figure below:

This image has been resized to fit in the page. Click to enlarge.

Below is an expanded view of the tidal forcing used in the climate index model in comparison to the peak years in lemming population. The tidal forcing meandering square wave aligns with the cyclic peak lemming populations. The 3.8 year cycle derives directly from the aliased beat cycle 1/(27-365.242/13.6608) = 3.794 years. The dotted lines are a guide to the eye

This image has been resized to fit in the page. Click to enlarge.

This does not validate the ENSO model but like the cyclic anchovy and sardine populations off the coast of Peru and Chile, it substantiates the view that climate variations impact wildlife populations. What the Lemming/Arctic Fox cycle does is show the ENSO-to-lunisolar-forcing connection more directly. In other words, the lemming population may be more sensitive to the precursor tidal forcing than to the resulting (and more erratic) ENSO cycles. Archibald discusses this aspect:

“One big question raised but not answered by this study: Is the ENSO cycle driven by or somehow connected with the 3.8-year lunar cycle? The correlation between predicted lemming/Arctic fox peak years and January-March SOI peak years (Table 1) suggests that ENSO might be driven by the 3.8-year lunar cycle. Zhang (2001) surmised, “There must exist some behind and deep mechanism of linking rodent outbreaks and ENSO together”. However, there is not much evidence for such a linkage. The only reference found to a relationship between a 3.8-year lunar cycle and atmospheric circulation is Wilson’s (2012) finding of a significant 3.78-year (+- 0.06) periodicity in the peak latitude anomaly of the subtropical high-pressure ridge over eastern Australia from 1860 to 2010. However, some worldwide climate oscillations contain a periodicity of ~3.8 years (selected examples in Table 4), and it is clear that these oscillations are interconnected. For example, Li and Lau (2012) examined the late winter teleconnection between the ENSO and the NAO and found that the average NAO index values in 19 El Niño winters and 20 La Niña winters were -0.35 and 0.51, respectively: positive NAO index values are more frequent during La Niña winters, and negative winter NAO index values are more frequent during El Niño events.”

Archibald (2019)

The solution to the Laplace Tidal Equations provide the direct linkage from 3.8 years to the observed ENSO pattern. The 3.8 year pattern remains embedded in this pattern along with the nonlinear superharmonics that are generated from the forcing input.

Will leave this recent citation here:

This may help explain why the LTE wave solution aligned precisely along the equator sustains a standing wave mode with characteristic Mach-Zehnder superharmonics. A Google Scholar search for “superharmonic cascade” only gives 4 hits.

]]>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 Dischargin**g.**(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 Noise****Model****(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.

(**) *In the blog comments, I will add follow-on discoveries that have been covered since the book was published.*

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

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

“Wavelet spectra of

Tang et al [2]f_{o}E_{s}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.”

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.

- 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
- 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).

The evidence points to a common tidal forcing for the cyclic behavior for the ocean indices. Even though the tidal forcing is allowed to vary slightly, the time-series inevitably matches the pattern as shown in** Fig.1**, with a cycle of 3.8 years.

From that forcing *f*(*t*), the Laplace’s Tidal Equation (LTE) response is straightforwardly an application of a *sin*(*A f*(*t*)) modulation to the appropriate time-series, as shown in **Fig 2** below.

The more gradual wavenumber modulation is the same for each (corresponding to the main ENSO dipole) but the IOD has a much stronger high wavenumber modulation, which is ~ 7× the fundamental.

The best-fit result after applying the appropriate LTE modulation in **Fig 2 **to **Fig 1** is respectively shown in **Fig 3** below.

Note what this result is saying : that in each case of a complex erratic cyclic index, a simple modulation can recreate the peaks and valleys remarkably well. This is difficult enough to do for a single index alone but to do it for both simultaneously is statistically impossible given the few degrees of freedom available for such a complex waveform — unless this is the actual standing wave fluid dynamics being modeled.

If this physics model holds up elsewhere, it points to the concept of a universal pattern that we can apply to modeling a standing wave dipole, which is the following data-flow for the LTE model (**Fig. 4**):

The following paper (thanks to Jim Stuttard @OctupusSnook) provides a way of mathematically thinking about data flows

Paulo Perrone, Notes on Category Theory with examples from basic mathematics (2020)

— from arXiv

In particular, this paper has a very good introduction for scientists and engineers interested in applying category theory to data-flow models. The gist of category theory is to organize mathematical formulations into common canonical patterns such that one can more easily understand what transformations can be applied. It may be coincidental that Perrone chooses just this rarely encountered construction, the LTE modulation highlighted in **Fig. 5** below, to introduce the *composite* data-flow formulation

The highlighted text is rare because the units of the inner and outer terms both have to be in radians. This is essentially the same LTE closed-form solution applied to transform **Fig 1** and **Fig 2** above into **Fig 3**, with only the sinusoidal LTE modulation differing for ENSO (*g1*) and IOD (*g2*) as shown below.

Where else this *sin*(*cos*(*x*)) formulation comes up in is in Mach-Zehnder modulation, where the physical data flow is described by a beam splitter, which mathematically transforms into a composite of a sinusoidally modulated inner phase term.

Follow the development of applied Category Theory at the Azimuth Project forum and we may be able to leverage ideas on how to formulate adjoint and topological transformations — these may be of help in automatically modeling such non-linear constructions.

p.s. Category is an overloaded term as this blog post falls into a Wave Energy category and also a Category Theory category as allowed by WordPress.

]]>The key to making an association between SOI and MJO is to analyze the daily measure of SOI and compare that to a high-resolution (pentad=5-day) time-series of MJO. Consider if the MJO traveling wave (see **Fig 1**) is kicked off by the ENSO equatorial disturbance, then an index such as SOI should lead MJO by ~20 days if MJO is traveling at 5 to 15 m/s and it has to fully propagate to be precisely measured.

Indeed, the SOI leads MJO by ~21 days according to this optimal overlay:

As an aside, one can now see why the high resolution of SOI is necessary. The issue with the monthly SOI readings is that if the MJO period is ~45 days, that’s above the Nyquist frequency, so it would be challenging to isolate these features via sampling the monthly time-series (I have tried and had little success).

**Fig 3** below shows that the correlation is ~0.47 at lead of 21 days and damps quickly with lead/lag shifts of a few days. The 2nd-order satellite wings are at +78 and -25, which would put the MJO period at around 57 to 46 days, which is a sanity check for the MJO typical range.

Revisiting the high resolution SOI model, but applying it to the MJO time-series, **Fig 4** below shows that a good correlation can be achieved by assuming a nearly identical tidal forcing and similar Laplace Tidal Equation (LTE) modulation parameters.

As a comparison, the high resolution SOI model is shown below. The daily SOI data goes back to 1991, so the MJO data provides the continuation of the time-series prior to that date (but wasn’t applied to the fitting procedure). The parameters and forcing for two models (MJO & SOI) are very similar, yet that should be expected from the known temporal correlation and short lag applied to MJO.

Now comes a novel signal processing algorithm based on the LTE model. We know from the Mach-Zehnder-like modulation of the LTE analytical result that the forcing level replaces the time parameter as the sinusoidal input. So instead of using time for the Fourier spectral analysis, if we use the forcing level then the wavenumber parameters should obviously be revealed in the amplitude spectra. We can then use these directly in the model as a means to quickly tune the fit. This is shown in **Fig 6** below for the SOI LTE modulation. The spectra to the left is related to the low frequency aspect of ENSO, while the spikes on the right correspond to the 40 to 50 day MJO cycles.

This also applies to the MJO time series with similarly isolated spikes shown in **Fig 7**. These are sharply delineated wavenumbers that drive the response via the forcing level .

Note that in the the Fourier domain of the actual temporal signal, the addition of a strong single LTE modulation will create a spread response in the spectrum as shown in **Fig 8**, creating in effect the broad 30 to 90 day spectral spread of MJO. This is just Mach-Zehnder in action, with the temporal => forcing transformation providing a handy adjoint conjugation. So it’s much easier to fit a model to data via **Fig. 6** or **Fig. 7** than by **Fig. 8**.

To elaborate further, the reason that these spikes show up so clearly is that they are driven by slight changes in the forcing level, which the Fourier series representation captures by grouping only level shifts, see **Fig 9**. It thus isolates the standing waves that can develop during the relatively flat tops of the annually-driven step response modulation to tidal forcing. The step change is likely the mechanism that provides a synchronization to the high wavenumber responses and thus long-term coherence between the model and data.

This only explains the math. According to most “just-so” explanations of the MJO behavior [2], a specific location provides the launching pad for the traveling wave to start.

and here’s video I made of a recurring flame soliton running along a groove in a burning oak log. This likely has some of the same math as the MJO, QBO, and Kelvin waves that race around the equator.

- Madden R. and P. Julian, 1971: Detection of a 40-50 day oscillation in the zonal wind in the tropical Pacific,
*J. Atmos. Sci*.,**28**, 702-708. - Wei, Y., Ren, H.-L., Mu, M. & Fu, J.-X. Nonlinear optimal moisture perturbations as excitation of primary MJO events in a hybrid coupled climate model.
*Climate Dynamics***54**, 675–699 (2020).

The new research by Professor Michael Mann in his peer-reviewed article called “Absence of internal multidecadal and interdecadal oscillations in climate model simulations” asks whether the decadal >10 year (and perhaps faster cycles) in the AMO and PDO behave as an internal property of the ocean or whether the cycles are externally forced. The quandary is that Mann does not deny that the ENSO behavior ** has** a strong internal oscillation, so what if anything makes ENSO special?

Like the AMO, the characteristic of PDO that distinguishes it from ENSO is in the longer decadal variation that it exhibits. In** Fig 1** below, we show a model fit to PDO that relies on essentially the same input tidal forcing that was applied to the ENSO model. What is most impressive about the fit is how naturally the interdecadal cyclic variation emerges.

The comparison between the ENSO and PDO tidal forcing is shown in** Fig 2** below. As with AMO, the clue to where the interdecadal cycle arises from is in the slight modulated curvature in the profile.

What is causing the curvature is the interference between two closely aliased primary tidal forcings. These are the fortnightly tropical cycle of 13.66 days and the monthly anomalistic cycle of 27.55 days as described here in LOD characterization and see** Fig A2** at the end of the AMO post.

The two primary forcings acting mutually are also observed in tidal tables, in what is known as the long-period 9-day tide labelled “Mt” . The period of this tide is ~9.133 days and so when applied to an annual impulse we get 365.242/9.133 = 39.99146. This number is very close to an integral value but not quite, so it will only reach a constructive interference against an annual impulse every 1/(40-39.99146) ~ 120 years.

Since the LTE model is nonlinear, the 120 year underlying cycle can readily transform into a 60 year harmonic. So — just as for the AMO — the ~60 year cycle emerges in the PDE time series with the appropriate LTE modulation as in **Fig 3** below.

Although this analysis is slightly different than what we presented at the 2018 AGU meeting, the same bottomline stands, in that the tidal forcing is nearly equivalent for the ENSO, PDO, and AMO climate indices as show in **Fig 4** below. This also applies to the IOD index, so that any long-term predictability may arise solely due to the ability to precisely refine this forcing along with the LTE modulation for the appropriate oceanic dipole.

The reality is that with the combination of AGW and the additive impact of multiple concurrently peaking oceanic dipoles, extremes in temperature can occur that may be larger than any time in modern times. Consider the possibility in **Fig 4 **of the IOD, ENSO, and SAM peaking simultaneously and the impact that has in Australia. And that is at least partly what is happening now.

TIW wavetrains are also observed in the equatorial Atlantic so would be considered alongside the AMO there as the high wavenumber and low wavenumber pairing.

According to the Laplace’s Tidal Equation (LTE) model that we applied to ocean thermocline dynamics, TIW wavetrains originate from the higher-order solutions to the LTE differential equation. As with any waveguide containing standing-wave behaviors, the spatio-temporal solution allows for multiple wavelength/frequency combinations — in the case of LTE, it’s solved resulting in a linear dispersion relation, *k ~ f*. So for the Pacific ENSO + TIW pairing, the ratio is ~15:1, with the TIW wavelength ~15 times shorter than the ENSO standing wave (see **Fig 1**) and the TIW LTE amplitude modulation also 15 times the ENSO LTE modulation. This faster modulation is shown in the inset in the upper panel in **Fig 2** below. The specific combination of slower and faster modulation provides the proper mix of harmonics to recreate the spikiness in the ENSO time-series.

As with any application of harmonics, the most important aspect is in modifying the shape (e.g. triangle, square, sawtooth) of the waveform and not in the overall period. Adding the higher-order modulation via the 15x TIW wavenumber is shown in **Fig 3 **below, with a clear enhancement of the sharpness in the overall fit by comparing the upper plot to the lower plot.

A recent paper titled “A simple theory for the modulation of tropical instability waves by ENSO and the annual cycle” [1] suggests a similar close relationship between ENSO and TIW. They refer to it as a simple model in that specific harmonics related to the ENSO cycle and annual cycle comprise the TIW wavetrain, which they empirically establish from an isolated TIW time-series. In contrast, in our model, the relation is determined by the nature of the LTE modulation. **Fig 4** below shows our unfitted TIW model (extracted as the 15x factor) alongside the model from the Boucherel paper.

This is unfitted in the sense that the principal higher wavenumber solution derives only from a best fit of the LTE model to the NINO34/SOI time-series data — essentially the higher harmonics contribution of **Fig 3**. This has the byproduct of fitting the observed TIW as in **Fig 4**, thus creating the faster surface temperature cycles.

What is not called out by the Boucherel paper is that the TIW may actually amplify ENSO, whereas they show it having an opposite polarity relationship. That is dependent on where they are measuring the amplitude of the standing wave as shown in **Fig 5** below. To investigate this further, it will be useful to have access to the complete set of data and reproduce their regression procedure to isolate the TIW component.

[1] Boucharel, J. and Jin, F.F., 2020. A simple theory for the modulation of tropical instability waves by ENSO and the annual cycle. *Tellus A: Dynamic Meteorology and Oceanography*, *72*(1), pp.1-14.

The new research on AMO by Professor Michael Mann appears to be meant to be somewhat provocative, which is OK as it spurred some discussion on Twitter. His peer-reviewed article is called “Absence of internal multidecadal and interdecadal oscillations in climate model simulations” and its takeaway is right in the title. Essentially, Mann *et al* are asking whether the ~60 year oscillation (and perhaps faster cycles) in the AMO behave as an internal property of the Atlantic ocean or whether the cycles are externally forced. ** Fig A1 **in the Appendix provides some background on the AMO time-series data.

The characteristic of AMO that distinguishes it from ENSO is the multidecadal variation of ~60 years that it exhibits. In** Fig 1** below, we show a model fit to AMO that relies on essentially the same input tidal forcing that was applied to the ENSO model. What is most impressive about the fit is how naturally the ~60 year cyclic variation emerges.

The comparison between the ENSO and AMO tidal forcing is shown in** Fig 2** below. The clue to where the 60 year cycle arises from is in the slight modulated curvature in the profile.

What is causing the curvature is the interference between two closely aliased primary tidal forcings. These are the fortnightly tropical cycle of 13.66 days and the monthly anomalistic cycle of 27.55 days as described here in LOD characterization and see** Fig A2** at the end of this post. These two, when interacting against a yearly annual impulse, produce a clear ~120 year repeat pattern as shown in **Fig 3**.

The two primary forcings acting mutually are also observed in tidal tables, in what is known as the long-period 9-day tide labelled “Mt” . From **Fig 4** below the period of this tide is precisely 9.132933 days and so when applied to an annual impulse we get 365.242/ 9.132933 = 39.99175. This will reach a constructive interference against an annual impulse every 1/(40-39.9917)=121.3 years.

This value is important to consider for understanding how a ~60 year cycle comes about from the model. Since the LTE model is nonlinear, the 120 year underlying cycle can readily transform into a 60 year harmonic. Or these longer periods may not be as obvious, as with the ENSO model. So it just so happens that the 60 year cycle emerges in the AMO time series with the appropriate LTE modulation as in **Fig 5** below (as it does with PDO).

The 60 year modulation also appears as an intra-spectral cross-correlation as described in the previous post .

Furthermore, scientists at NASA JPL and the Paris Observatory have long known about a 60 year link between LOD and climate variation see **Fig 6** below,.

The idea is that this multi-decadal period is integrated over time, creating a mutual interaction between the long-period forcing of the lunar + solar tides and the sloshing response of the ocean basins. The LOD or Universal Time (UT1) measure becomes a correlating measure of these forcing constituents. So the longer the period of potential constructive interference, such as with the 60 year near aliasing of the *Mt* constituent against a yearly impulse, the more that an inertial response can accumulate [4]. It may in fact be that the entirety of the LOD variations are due to the lunar + solar forcing and this is the unification between LOD and the climate dipole standing-wave behavior.

From [4]

- Cleverly, J.
*et al.*The importance of interacting climate modes on Australia’s contribution to global carbon cycle extremes.*Sci Rep***6**, 1–10 (2016). - Desai, S. D. & Wahr, J. M. Empirical ocean tide models estimated from TOPEX/POSEIDON altimetry.
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The context is looking for autocorrelations in the frequency domain of a time-series. Although not as common as performing autocorrelations in the time domain, it is equally as powerful.

The earlier idea was to look for harmonics in the periodicity of the ENSO signal, and the chart described in the post showed clear annual and higher harmonics in the time series. This was via a straightforward sliding autocorrelation in the power spectra.

As an additional technique, we can look for symmetric sidebands of the annual fundamental and harmonics frequencies by folding the spectra over about the annual frequency and performing a direct correlation calculation.

This correlation is painfully obvious and is well beyond statistically significant in demonstrating that an annual impulse signal is modulating another much more complex forcing signal (likely of tidal origin). This is actually a well-known process known as a double-sideband suppressed carrier modulation, used most commonly in facilitating broadcast transmissions. As shown in the equations below, the modulation acts to completely suppress the carrier (i.e. annual) frequency.

Read the previous post for more detail on the approach.

Ordinarily, the demodulation is straightforward via a standard mixing approach, as the carrier signal is a much higher frequency than the informational signal, but since annual and long-period tides are of roughly similar periods, the demodulation will only complicate the spectrum. This is not a big deal as we need to fit the peaks via the LTE formulation in any case.

This is a new and novel finding and not to be found anywhere in the ENSO research literature. Why it hasn’t been uncovered yet is a bit of a mystery, but the fact that the annual signal is completely suppressed may be a hint. It may be that we need to understand why the dog didn’t bark.

Gregory (Scotland Yard detective): “Is there any other point to which you would wish to draw my attention?”

— “

Holmes: “To the curious incident of the dog in the night-time.”

Gregory: “The dog did nothing in the night-time.”

Holmes: “That was the curious incident.”The Adventure of Silver Blaze” by Sir Arthur Conan Doyle

If that is not the case, and this has been published elsewhere, will update this post.

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