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 …

Continue reading# ENSO

# The Indian Ocean Dipole

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 …

# 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:

Continue reading# 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.

Continue reading# Characterizing spatial standing waves of ENSO

In Chapter 12 of the book, we did not delve into the details of the spatial aspects of the LTE-based ENSO standing wave model to any great degree. We did include a concise derivation of the steps involved in creating separable equations, corresponding to solutions for the temporal and spatial parts of the standing wave dipole. However, only passing mention was made of the unique nature of the spatio-temporal coordination which emerges from the model — and which should be observed in the empirically observed behavior. We can do that now with the abundance of comprehensive data available.

Continue reading# Applying Wavelet Scalograms

Dennis suggested:

“I was thinking about ENSO model and the impulse function used to drive it. Could it be the wind shift from the QBO that is related to that impulse function.

5/19/2019

My recollection was that it was a biennial pulse, which timing wise might fit with QBO. “

They are somehow related but more than likely through a common-mode mechanism. Consider that QBO has elements of a semi-annual impulse, as the sun crosses the equator twice per year. The ENSO model has an impulse of once per year, with more recent evidence that it may not have to be biennial (i.e. alternating sign in consecutive years) as we described it in the book.

I had an evaluation *Mathematica *license for a few weeks so ran several wavelet scalograms on the data and models. **Figure 1** below is a comparison of ENSO to the model

# Synchronization, critical points, and relaxation oscillators

In Chapter 18 of the book, we discuss the behavior around critical points in the context of reliability, both at the small-scale in terms of component breakdown, and in the large-scale in the context of earthquake triggering which was introduced in Chapter 13. The connection is that things break at all scales, with the common mechanism of a varying rate of progression to the critical point:

As indicated in the figure caption, the failure rate is generally probabilistic but with known external forcings, there is the potential for a better deterministic prediction of the breakdown point, which is reviewed below:

# Forced Natural Responses to LTE Solution

In Chapter 12 of the book, we describe in detail the solution to Laplace’s Tidal Equations (LTE), which were introduced in Chapter 11. Like the solution to the linear wave equation, where there are even (cosine) and odd (sine) natural responses, there are also even and odd responses for nonlinear wave equations such as the Mathieu equation, where the natural response solutions are identified as MathieuC and MathieuS. So we find that in general the mix of even and odd solutions for any modeled problem is governed by the initial conditions of the behavior along with any continuing forcing. We will describe how that applies to the LTE system next:

# Implicit Interpolating Cross-Validation of ENSO

In Chapter 12 of the book, we describe the forcing mechanism behind the El Nino / Southern Oscillation (ENSO) behavior and here we continue to evaluate the rich dynamic behavior of the Southern Oscillation Index (SOI) — the pressure dipole measure of ENSO. In the following, we explore how the low-fidelity version of the SOI can reveal the high-frequency content via the solution to Laplace’s Tidal Equations.

# High Resolution Analysis of SOI

In Chapter 12 of the book, we have a description of the mechanism forcing the El Nino / Southern Oscillation (ENSO) behavior. ENSO shows a rich dynamic behavior, yet for the pressure dipole measure of ENSO, the Southern Oscillation Index (SOI), we find even greater richness in terms of it’s higher frequency components. Typically, SOI is presented with at least a 30-day moving average applied to the time-series to remove the higher-frequencies, but a daily time-series is also available for analysis dating back to 1991. The high-resolution analysis was not included in the book but we did present the topic at last December’s AGU meeting. What follows is an updated analysis …