(see later posts here)
My previous posts on modeling the Southern Oscillation Index as a periodically modulated wave equation -- in particular via the Mathieu equation -- are listed below:
- The Southern Oscillation Index Model
- SOIM and the Paul Trap
- The Chandler Wobble and the SOIM
The first post introduced the Mathieu equation and established a premise for mathematically modeling the historical SOI time-series of ENSO, the Southern Oscillation part of the El Nino/Southern Oscillation phenomenon. The second post was an initial evaluation of a multivariate fit, evaluated by exploring the parameter space. The third post was a bit of a breakthrough, which focused on a specific periodic process -- the Chandler Wobble (CW) -- which appeared to have a strong causal connection to the underlying SOI model.
This short post effectively substantiates the Chandler Wobble connection and provides nearly as strong support that other tidal beat periodicities force the modulation as well.
(see later posts here)
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 ? 
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.
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
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.
As we look at attribution of global warming to various physical mechanisms, one of the puzzling observations we can make is that many researchers place too much emphasis on a single cause. This is especially true of the research from those that have skeptical views of GHG-caused warming. For instance, Scafetta is convinced that the orbital forces are the key, and may also prove to be the cause of any long-term trends we are seeing -- yet he makes a concerted effort to downplay the effects of the CO2 control knob, giving the CO2 TCR a very low value. That is OK if he is truly being skeptical but not so good if he wants to retain objectivity.
From a previous post, we added Scafetta's orbital cyclic parameters to the CSALT model. These include orbital parameters that are lunar as well as solar and planetary. If we look at the periods that control lunar tides -- the 18.613 year period and the 8.848 period -- CSALT generates an amplitude and phase that lines up remarkably well with the diurnal tidal analysis of R.Ray at NASA Goddard , whose work has been referenced by skeptic Clive Best here  . See Figure 1 below:
Fig 1: The top panel shows the CSALT extracted 18.6-year diurnal tidal period amplitude (right axis) along with the temperature phasing. The left axis shows the yearly averaged actual tidal amplitude from R.Ray, which is completely in-phase with the temperature factor. The middle panel shows a higher resolution look at the tidal amplitudes over a shorter time interval. Both the 18.6 year and a faint 8.85/2 year extracted temperature signal are in phase and of comparable relative amplitudes as the data. The bottom panel shows the semidiurnal amplitude with a 8.85/2 temperature signal which has a different sign than the diurnal signal.
Let's start with a telling quote :
James Annan: "There’s so little interesting stuff going on in climate science these days."
... well of course, if we don't have anything worthwhile to say. But then we also find this:
Peter Ván: "The basic mystery in thermodynamics is the universality. The validity of thermodynamic equations and theories regularly exceed the expectations." 
The CSALT model of the global temperature anomaly has no right to work as well as it does. After all, it solves no dynamical behavior and requires little information with regards to the complexity of the earth's surface. Yet, it still captures all the useful detail in the historical temperature record, leaving behind a residual close to being in the white noise regime.
Fig. 1 : Residual noise of the CSALT model is flat and close to white noise
Cowtan and Way's hybrid correction to the HadCRUT global temperature series  has provoked expected interest by auditor Steve McIntyre. This is always welcome, because as with the majority of of these nosy irritants, the more that they try to find something wrong with a well-reasoned comprehensive analysis, the more that they lay out a cookie trail for us to follow. So guys like McIntyre make our job easier because what they try to expose backfires on them and it just gives climate scientists further substantiation of their own models -- not exactly what McIntyre had in mind.
This case is no different, starting with McIntyre's figure below:
Fig 1: Delta between CW Hybrid (basis 1961-1990) and HadCRUT4. From McIntyre http://climateaudit.org/2013/11/18/cotwan-and-way-2013/. Note the divergence in recent years.
Caldeira and Myrhvold  attempted the obvious by compiling various global warming models to try to extract simple thermal behavioral patterns, see their online paper.
They succeeded in my opinion.
What is most important about their work is that they placed diffusional models of warming, as pioneered by James Hansen , on an equal footing with first-order kinetics model. The first-order models use damped exponentials and are favored by analysts that want to keep the math simple. Caldeira and Myrhvold understand this and provide mixed exponential models that will piece-wise map to the diffusional responses normally seen in the numerical simulations.