SOIM fit to Unified ENSO Proxy

A previous post described the use of proxy records of ENSO to fit the Southern Oscillation Index Model (SOIM).  This model fit used one specific set of data that featured a disconnected record of coral measurements from the past 1000 years, see Cobb [1].

As the focus of this post, another set of data (the Unified ENSO Proxy set) is available as an ensemble record of various proxy measurements since 1650 — giving an unbroken span of over 300 years to apply a SOIM fit [2].  This ensemble features 10 different sets, which includes the Cobb coral as a subset.

To fit over this long a time span is quite a challenge as it assumes that the time series is stationary over this interval. The data has a resolution of only one year, in comparison to the monthly data previously used, so it may not have the temporal detail as the other sets, yet still worthy of investigation. (an interactive version is available here).

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The SOIM: substantiating the Chandler Wobble and tidal connection to ENSO

(see later posts here)

[mathjax]My previous posts on modeling the Southern Oscillation Index as a periodically modulated wave equation — in particular via the Mathieu equation — are listed below:

  1. The Southern Oscillation Index Model
  2. SOIM and the Paul Trap
  3. 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.

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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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The Southern Oscillation Index Model

(Note dated 4/18/2019 — an update from the future. This was the first in a long series of posts trying to understand ENSO, which finally culminated in a publication described here on Book Info
(also see later posts on this blog here))

[mathjax]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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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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Tidal component to CSALT

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 [1], whose work has been referenced by skeptic Clive Best  here [2] .  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[1], 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.

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Temperature Induced CO2 Release Adds to the Problem

As a variable amount of CO2 gets released by decadal global temperature changes, it makes sense that any excess amount would have to follow the same behavior as excess CO2 due to fossil fuel emissions.

From a previous post (Sensitivity of Global Temperature), I was able to detect the differential CO2 sensitivity to global temperature variations. The correlation of temperature anomaly against d[CO2] is very strong with zero lag and a ratio of about 1 PPM change in CO2 per degree temperature change detected per month.

Now, this does not seem like much of a problem, as naively a 1 degree change over a long time span should only add one PPM during the interval. However, two special considerations are involved here. First, the measure being detected is a differential rate of CO2 production and we all know that sustained rates can accumulate into a significant quantities of a substance over time. Secondly, the atmospheric CO2 has a significant adjustment time and the excess isn’t immediately reincorporated into sequestering sites. To check this, consider that a slow linear rate of 0.01 degree change per year when accumulated over 100 years will lead to a 50 PPM accumulation, if the excess CO2 is not removed from the system. This is a simple integration where f(T(t)) is the integration function :
$$ [CO2] = f_{co_2}(T(t)) = \int^{100}_0 0.01 t\, dt = \frac{1}{2} 0.01 * 100^2 = 50 $$
The sanity check on this is if you consider that a temperature anomaly of 1 degree change held over 100 years would release 100 PPM into the atmosphere. This is simply a result of Henry’s Law applied to the ocean. The ocean has a large heat capacity and so will continue outgassing CO2 at a constant partial-pressure rate as long as the temperature has not reached the new thermal equilibrium. (The CO2 doesn’t want to stay in an opened Coke can, and it really doesn’t want to stay there when it gets warmed up)

So, if we try the impulse response we derived earlier (Derivation of MaxEnt Diffusion) to this problem, with a characteristic time that matches the IPCC model for Bern CC/TAR, standard:

As another sanity check, the convolution of this with a slow 1 degree change over the course of 100 years will lead to at least a 23 PPM CO2 increase.

Again, this occurs because we are far from any kind of equilibrium, with the ocean releasing the CO2 and the atmosphere retaining what has been released. The slow diffusion into the deep sequestering stores is just too gradual while the biotic carbon cycle is doing just that, cycling the carbon back and forth.

So now we are ready to redo the model of CO2 response to fossil-fuel emissions (Fat-Tail Impulse Response of CO2) with the extra positive feedback term due to temperature changes. This is not too hard as we just need to get temperature data that goes back far enough (the HADCRUT3 series goes back to 1850). So when we do the full combined convolution, we add in the integrated CO2 rate term f(T), which adds in the correction as the earth warms.

$$ [CO2] = FF(t) \otimes R(t) + f_{co_2}(T(t)) \otimes R(t) $$

When we compute the full convolution, the result looks like the following curve (baseline 290 PPM):

The extra CO2 addition is almost 20 PPM just as what we had predicted from the sanity check. The other interesting data feature is that it nearly recreates the cusp around the year 1940.  The previous response curve did not pick that up because it is entirely caused by the positive-feedback warming during that time period. The effect is not strong but discernible.

We will continue to watch how this plays out. What is worth looking into is the catastrophic increase of CO2 that will occur as long as the temperature stays elevated and the oceans haven’t equilibrated yet.