The Oil Shock Model and Compartmental Models

Chapter 5 of the book describes a model of the production of oil based on discoveries followed by a sequence of lags relating to decisions made and physical constraints governing the flow of that oil. As it turns out, this so-named Oil Shock Model is mathematically similar to the compartmental models used to model contagion growth in epidemiology, pharmaceutical/drug deliver systems, and other applications as demonstrated in Appendix E of the book.

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 Susceptible, Infectious, and Recovered 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.

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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.

undefinedOne 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! ).

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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 [1].

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

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

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

Tang et al [2]

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


  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).

Australia Bushfire Causes

The Indian Ocean Dipole (IOD) and the El Nino Southern Oscillation (ENSO) are the primary natural climate variability drivers impacting Australia. Contrast that to AGW as the man-made driver. These two categories of natural and man-made causes form the basis of the bushfire attribution discussion, yet the naturally occurring dipoles are not well understood. Chapter 12 of the book describes a model for ENSO; and even though IOD has similarities to ENSO in terms of its dynamics (a CC of around 0.3) the fractional impact of the two indices is ultimately responsible for whether a temperature extreme will occur in a region such as Australia (not to mention other indices such as MJO and SAM).  

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In Chapter 12 of the book, we presented a math model for the equatorial Pacific ocean dipole known as ENSO (El Nino /Southern Oscillation).  We argued that the higher wavenumber (×15 of the fundamental) characteristic of ENSO was related to the behavior known as Tropical Instability Waves (TIW). Taken together, the fundamental and TIW components provide enough detail to model ENSO at the monthly level. However if we drill deeper, especially with respect to the finer granularity SOI measure of ENSO, there are rather obvious cyclic factors in the 30 to 90 day range that can add even further detail. The remarkable aspect is that these appear to be related to the behavior known as the Madden-Julian Oscillation (MJO), identified originally as a 40-50 day oscillation in zonal wind [1].

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In Chapter 12 of the book, the math model behind the equatorial Pacific ocean dipole known as the ENSO (El Nino /Southern Oscillation) was presented.  Largely distinct to that, the climate index referred to as the Pacific Decadal Oscillation (PDO) occurs in the northern Pacific. As with modeling the AMO, understanding the dynamics of the PDO helps cross-validate the LTE theory for dipoles such as ENSO, as reported at the 2018 Fall Meeting of the AGU (poster). Again, if we can apply an identical forcing for PDO as for AMO and ENSO, then we can further cross-validate the LTE model. So by reusing that same forcing for an independent climate index such as PDO, we essentially remove a large number of degrees of freedom from the model and thus defend against claims of over-fitting.

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Tropical Instability Waves

In Chapter 12 of the book, we present the hypothesis that tropical instability waves (TIW) of the equatorial Pacific are the higher wavenumber (and higher frequency) companion to the lower wavenumber ENSO (El Nino /Southern Oscillation) behavior. See Fig 1 below.

Figure 1 : Tropical Instability Waves along the equator have about a ~15x higher wavenumber than the ENSO wave.

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.

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In Chapter 12 of the book, we focused on modeling the standing-wave behavior of the Pacific ocean dipole referred to as ENSO (El Nino /Southern Oscillation).  Because it has been in climate news recently, it makes sense to give equal time to the Atlantic ocean equivalent to ENSO referred to as the Atlantic Multidecadal Oscillation (AMO). The original rationale for modeling AMO was to determine if it would help cross-validate the LTE theory for equatorial climate dipoles such as ENSO; this was reported at the 2018 Fall Meeting of the AGU (poster). The approach was similar to that applied for other dipoles such as the IOD (which is also in the news recently with respect to Australia bush fires and in how multiple dipoles can amplify climate extremes [1]) — and so if we can apply an identical forcing for AMO as for ENSO then we can further cross-validate the LTE model. So by reusing that same forcing for an independent climate index such as AMO, we essentially remove a large number of degrees of freedom from the model and thus defend against claims of over-fitting.

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Autocorrelation in Power Spectra, continued

This is a short comment pointing to an addition to a previous post

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.

Lower x-axis is the lower sideband interval (blue) and
upper x-axis is the symmetric upper sideband interval (red) shown in reverse

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.