The Chandler Wobble and the SOIM

Seth Carlo Chandler Jr was an actuary who studied his namesake wobble for thirty years.

ENSO and the Southern Oscillation Index has confounded everyone with its unpredictability.

Could a connection exist between the ENSO and the Chandler Wobble ? [1]

Based on what I have been analyzing with respect to the ENSO data, I am leaning in that direction.  In the first post on the Southern Oscillation Index Model (SOIM), my initial analysis lead to a fundamental Mathieu frequency T of 6.3 years, a value of a = 2.83  and q = 2.72 :

I followed that up with additional checks and analogies to other physical phenomena :

That culminated with a trial fit of the SOI with a set of Mathieu parameters. Yet -- even though the fit was decent -- I was not satisfied with the result as it tended to overfit with respect to the adjustable parameters.  The ideal situation would limit the number of fundamental frequency terms.

Three observations lead me to a much simplified representation.

  • The main Mathieu frequency of 6.3 years seemed to vary over the historical record.
  • The pressure index of the SOI is essentially a differential measure, and so the derivative of the Mathieu function should be fit to the pressure, e.g. use MathieuCPrime and not MathieuC .
  • The connection between the original fit of 6.3 years and the Chandler Wobble beat frequency of 6.39 years (= 1/(1-365.25/433)), and the fact that this measure has been known to vary over the past 100+ years.

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓

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CSALT Volcanic Aerosols

The volcanic aerosol factor of the CSALT model is an example of a perfectly interlocking piece in the larger global surface temperature puzzle.  I thought I would present a more detailed description in response to the absolutely hapless recent volcano posts at the WUWT blog (here and here).  The usual deniers in the WUWTang Clan can't seem to get much right in their quest to intelligently spell out ABCD (Anything But Carbon Dioxide).

Fig 1 :  CSALT model using the GISS Stratospheric Aerosol forcing model.

The addition of the volcanic aerosol factor is no different than the other components of the CSALT model. Two flavors of volcanic aerosol forcings are provided. The standard forcing table is the GISS stratospheric aerosol optical thickness model maintained by Sato [1] and I use this table as is (see Figure 1).   The more experimental model that I generated is a sparse table that features only the volcanoes of Volcanic Explosivity Index (VEI) of 5 or higher.

The VEI scale is logarithmic so that a VEI of 6 contains 10 times as much ejected particulates by volume than a VEI of 5 (which recursively is 10 times as much as VEI=4, and so on). This means that by modeling VEI of 5 or 6 we should capture most of the particulates generated as discrete events.

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Decadal Temperature Variations and LOD

The two primary oscillating factors that we have identified in the CSALT model of global temperature are the Southern Oscillation Index (SOI) and the Length of Day (LOD).  The distinguishing factor in terms of impact  is that the SOI is characterized by intradecadal oscillations while the LOD fluctuates across decades [1].

If we can model the SOI deterministically, as demonstrated here, the hope is we may be able to model the LOD as well.  But first, we need to understand the significance of the LOD and its possible origin.

Fig 1:  As a premise for Length of Day (LOD) variations we consider that the rotational moment of inertia changes along the planetary surface.  If a band of water positioned along the equator shifts to higher latitude, the rotational moment of inertia decreases and the rotational velocity increases, thus shortening the length of day.

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SOIM and the Paul Trap

This post follows up on the idea of modeling the historical Southern Oscillation Index (SOI) record with details on how one can apply the SOIM to make accurate predictions.  Based on some some early encouraging success, I asserted that a more comprehensive model fitting would be possible.  That's what this follow-on post is about -- trying to verify that we can accomplish that "holy-grail" of prediction, the prediction of future El Nino / Southern Oscillation (ENSO) conditions.

To foreshadow what's to come, Figure 1 shows the comprehensive SOIM fit, which incorporates a grouping of optimally phased Mathieu functions (introduced in the previous post)

Fig. 1 : Fit of the full SOI historical record (in green) to the SOI Model (in blue).

This is a very promising result based on the premise of the last post. The principal additions to the simple model are (1) a multi-harmonic basis set of Mathieu functions and (2) a more constraining physical interpretation to the math.

What follows is the explanation and various verification checks, which include:

  1. Sensitivity of the model to parameter selection
  2. Comparison to fitting red noise (to show over-fitting is not an issue)
  3. Hindcasts and forecasts based on restricted training intervals
  4. Power spectrum of model and data

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The Southern Oscillation Index Model

A simple model of the Southern Oscillation Index (SOI) does not exist. I find it important to understand the origin of the SOI fluctuations, not only because it is an interesting scientific problem but for its potential predictive value -- in particular,  I could use a model to extrapolate the SOI factor needed by the CSALT model to make global surface temperature projections.  This has implications not only for long-term climate projections but for medium-term seasonal weather predictions, particularly in predicting the next El Nino.

The current thinking is that the index that characterizes the presence of El Nino and La Nina conditions (also known as ENSO)  is unpredictable enough to make any prediction beyond a  year or two pointless. That makes it a challenging problem, to say the least.

So although the SOI is defined as oscillatory (thus the name), these oscillations are not the typically sinusoidal, perfectly periodic waveforms that we are used to dealing with, but consist of uneven, sporadic pulses that remain virtually impossible to deconstruct. Yet, the problem may not be as intractable as we are lead to believe. The key to understanding the SOI is to decode the characteristics of the waveform itself shown below in Figure 1.

Fig 1 : The SOI is defined as the atmospheric pressure difference between Darwin, Australia and Tahiti in the south Pacific. Given that we have measurements that span over 130+ years, there may be a possibility that we can crack the code and decipher the fluctuating waveform. In the figure, the sloped dotted line gives us a clue to their nature.

The waveform is periodic alright but this is the periodicity that lies in the strange mathematical world of crystal lattices and warped coordinate systems.  Follow the math on the next page and the mystery is revealed.

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Relative strengths of the CSALT factors

For doing global surface temperature projections with the CSALT model, I find it critical to not over-fit if the training period is short. Over-fitting at short intervals can create oppositely compensating signs on factors, and these become sensitive to amplification when projected. The recommendation is then to rank the factors (or principal components) in order of their contributing strength to promoting a good fit via the correlation coefficient. See Fig. 1

Fig 1: Ranking of CSALT factors to generate best fit with fewest degrees of freedom.

With the original handful of CSALT factors, we can reach good correlation rather quickly. But after this point, the forcing factors from solar, lunar, and orbital become increasingly more subtle, providing progressively less thermal forcing as we run down the list of periods suggested by previous researchers. From the clear asymptotic trend, we would likely require several times as many factors to reach correlation coefficient levels arbitrarily close to 1.  Noise does not seem to be an issue as the vast majority of the temperature fluctuations appear to come from real forcing terms.  The noise residual in this case is at the 0.002 level or 0.2% of the measured signal.

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What missing heat?

As with the discussion about the "pause" in global surface temperatures, much consternation exists about the so-called "missing heat" in the earth's energy budget.

There are three pieces of the puzzle regarding this issue, which collectively have to fit together for us to be able to make sense about the net energy flow.

  1. The surface temperature time series, see the CSALT model
  2. Heat sinking via ocean heat content diffusion, see the OHC model
  3. The land and ocean surface temperatures have an interaction where they can exchange energy

We have an excellent start on the first two but the last requires a fresh analysis.   Understanding the exchange of energy is crucial to not getting twisted up in knots trying to explain any perceived deficit in heat accounting.

Consider Figure 1 below where the energy fluxes are shown at a level of detail appropriate for book-keeping.  On the left side, we have an energy balance of incoming flux perturbation (Li)  and outgoing flux (Lo) for the land area (we don't consider the existing balance as we assume that is already in a steady state).  On the right side we do the same for the sea or ocean area (Si and So).

The question is how do we proceed if we don't have direct knowledge of all the parameters.  The first guess is that they have to be inferred collectively. The complicating factor is that the sea both absorbs thermal energy (heat) into the bulk shown as the OHC arrow, and that some fraction of the latent and radiative heat emitted by the sea surface transfers over to the land.  There is little doubt that this occurs as the Pacific Ocean-originating El Nino events do impact the land, while regular seasonal monsoons work to redistribute enormous amounts as rain originating from the ocean and delivered to the land as moist latent energy. So the question mark in the figure indicates where we need to estimate this split.

Fig 1: Schematic of energy flow necessary to balance the budget.

Solving this problem would make an excellent homework assignment and perfect for a class in climate science.  Let's give it a try.


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Projection Training Intervals for CSALT Model

DC commented in the previous post that a training interval can be used to evaluate the feasibility of making projections of the CSALT model. His initial attempts hold great  promise as shown here.  One can see that the infamous "pause" or "hiatus" in global surface temperature is easily predicted using DC's training interval up to 1990.

Figure 1 below is my attempt at doing the projections with a more sophisticated version of the CSALT model.  The top chart is the model fit using all available data, and below that is a succession of projections with training intervals that end in 1990, 1980, 1970, 1960, and 1950.  Each successive chart uses fewer data points yet appears to hold fast to a credible projection, signifying the invariance of the model across the years.

Fig 1 : Set of training runs with temperature projection, the Green curve is the GISS data and Blue curve is the CSALT model.. The end of the training period is the upward pointing red arrow. The future temperature projections cover the interval spanned by the horizontal arrow. The correction term shown is for the WWII years where temperature readings were hot by a factor of 0.14 C.

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Reverse Forecasting via the CSALT model

DC of the Oil Peak Climate blog suggested that reverse forecasting to earlier dates using the CSALT model may be an interesting experiment.  Considering the growing sophistication of the model, I tend to agree.

The contributing factors to CSALT are a mix of empirical forcing terms and several periodic elements suggested by climate scientists with an interest in tidal and solar topics,  including Keeling from Scripps [1], R.Ray from NASA Goddard [2], Dickey from NASA JPL [3], and going back to Brier in 1968 [4]. These fall under the category of orbital influences discussed in a previous post.   Selecting the periods of the principle orbital most commonly cited, we get the staggered view of the individual contributions shown in Figure 2(see for an interactive version)

Figure 1 : Contributions of the various thermodynamic factors to the CSALT model of global average surface temperature. The "sme" term represents the Sun-Moon-Earth alignment harmonic of 9 years.

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