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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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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The MJO

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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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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The AMO

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 https://geoenergymath.com/2019/02/16/autocorrelation-of-enso-power-spectrum/.

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.

AO, PNA, & SAM Models

In Chapter 11, we developed a general formulation based on Laplace’s Tidal Equations (LTE) to aid in the analysis of standing wave climate models, focusing on the ENSO and QBO behaviors in the book.  As a means of cross-validating this formulation, it makes sense to test the LTE model against other climate indices. So far we have extended this to PDO, AMO, NAO, and IOD, and to complete the set, in this post we will evaluate the northern latitude indices comprised of the Arctic Oscillation/Northern Annular Mode (AO/NAM) and the Pacific North America (PNA) pattern, and the southern latitude index referred to as the Southern Annular Mode (SAM). We will first evaluate AO and PNA in comparison to its close relative NAO and then SAM …

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The Indian Ocean Dipole

In the book, we modeled the ENSO and QBO climate behaviors. Based on the approach described therein we have since extended this to the PDO, AMO, and NAO indices, with the IOD the focus of this post.

In Chapter 12, we concentrated on the Pacific ocean dipole referred to as ENSO (El Nino/Southern Oscillation).  A dipole that shares some of the characteristics of ENSO is the neighboring Indian Ocean Dipole and its gradient measure the Dipole Mode Index.

The IOD is important because it is correlated with India subcontinent monsoons. It also shows a correlation to ENSO, which is quite apparent by comparing specific peak positions, with a correlation coefficient of 0.2.  This post will describe the differences found via perturbing the ENSO model …

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North Atlantic Oscillation

In Chapter 12 of the book, we derived an ENSO standing wave model based on an analytical Laplace’s Tidal Equation formulation. The results of this were so promising that they were also applied successfully to two other similar oceanic dipoles, the Atlantic Multidecadal Oscillation (AMO) and the Pacific Decadal Oscillation (PDO), which were reported at last year’s American Geophysical Union (AGU) conference. For that presentation, an initial attempt was made to model the North Atlantic Oscillation (NAO), which is a more rapid cycle, consisting of up to two periods per year, in contrast to the El Nino peaks of the ENSO time-series which occur every 2 to 7 years. Those results were somewhat inconclusive, so are revisited in the following post:

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Teleconnections vs Common-mode mechanisms

The term teleconnection has long been defined as interactions between behaviors separated by geographical distances. Using Google Scholar, the first consistent use in a climate context was by De Geer in the 1920’s [1]. He astutely contrasted the term teleconnection with telecorrelation, with the implication being that the latter describes a situation where two behaviors are simply correlated through some common-mode mechanism — in the case that De Geer describes, the self-registration of the annual solar signal with respect to two geographically displaced sedimentation features.

As an alternate analogy, the hibernation of groundhogs and black bears isn’t due to some teleconnection between the two species but simply a correlation due to the onset of winter. The timing of cold weather is the common-mode mechanism that connects the two behaviors. This may seem obvious enough that the annual cycle should and often does serve as the null hypothesis for ascertaining correlations of climate data against behavioral models.

Yet, this distinction seems to have been lost over the years, as one will often find papers hypothesizing that one climate behavior is influencing another geographically distant behavior via a physical teleconnection (see e.g. [2]). This has become an increasingly trendy viewpoint since the GWPF advisor A.A. Tsonis added the term network to indicate that behaviors may contain linkages between multiple nodes, and that the seeming complexity of individual behavior is only discovered by decoding the individual teleconnections [3].

That’s acceptable as a theory, but in practice, it’s still important to consider the possible common-mode mechanisms that may be involved. In this post we will look at a possible common-mode mechanisms between the atmospheric behavior of QBO (see Chapter 11 in the book) and the oceanic behavior of ENSO (see Chapter 12). As reference [3] suggests, this may be a physical teleconnection, but the following analysis shows how a common-mode forcing may be much more likely.

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