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.

# Geophysical Energy

# Chandler Wobble according to Na

In Chapter 13 of the book, we have a description of the mechanism forcing the Chandler Wobble in the Earth’s rotation. As a counter to a recent GeoenergyMath post suggesting there is little consensus behind this mechanism, a recent paper by Na *et al* provides a foundation to understand how the lunar forcing works.

Continue readingChandler wobble and free core nutation are two major modes of perturbation in the Earth rotation. Earth rotation status needs to be known for the coordinate conversion between celestial reference frame and terrestrial reference frame. Due mainly to the tidal torque exerted by the moon and the sun on the Earth’s equatorial bulge, the Earth undergoes precession and nutation.

Na, S.-H.et al.Chandler Wobble and Free Core Nutation: Theory and Features.Journal of Astronomy and Space Sciences36, 11–20 (2019).

# 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:

# Lunisolar Forcing of the Chandler Wobble

In Chapter 13 of the book, we have a description of the mechanism forcing the Chandler Wobble in the Earth’s rotation. Even though there is not yet a research consensus on the mechanism, the prescribed lunisolar forcing seemed plausible enough that we included a detailed analysis in the text. Recently we have found a recent reference to a supporting argument to our conjecture, which is presented below …

# Length of Day

Continuing with Chapter 13 of the book, we also analyze the factors that perturb the Earth’s rotation rate and therefore the observed *length-of-day* (LOD). Researchers continue to update the characterization of the LOD, such as here, which we can compare against a straightforward model based on the accepted lunisolar gravitational forcing factors …

# The Chandler Wobble

In Chapter 13 of the book, we have a description of the mechanism causing the Chandler wobble of the Earth’s polar axis. Last fall, NASA updated a description of their understanding of the axis drift [1] described here, which we can place in context with the model …