Life is Like a Black Box of Chocolates

Guest Post by Willis Eschenbach

In my earlier post about climate models, “Zero Point Three Times The Forcing“, a commenter provided the breakthrough that allowed the analysis of the GISSE climate model as a black box. In a “black box” type of analysis, we know nothing but what goes into the box and what comes out. We don’t know what the black box is doing internally with the input that it has been given. Figure 1 shows the situation of a black box on a shelf in some laboratory.

Figure 1. The CCSM3 climate model seen as a black box, with only the inputs and outputs known.

A “black box” analysis may allow us to discover the “functional equivalent” of whatever might be going on inside the black box. In other words, we may be able to find a simple function that provides the same output as the black box. I thought it might be interesting if I explain how I went about doing this with the CCSM3 model.

First, I went and got the input variables. They are all in the form of “ncdf” files, a standard format that contains both data and metadata. I converted them to annual or monthly averages using the computer language “R”, and saved them as text files. I opened these in Excel, and collected them into one file. I have posted the data up here as an Excel spreadsheet.

Next, I needed the output. The simplest place to get it was the graphic located here. I digitized that data using a digitizing program (I use “GraphClick”, on a Mac computer).

My first procedure in this kind of exercise is to “normalize” or “standardize” the various datasets. This means to adjust each one so that the average is zero, and the standard deviation is one. I use the Excel function ‘STANDARDIZE” for this purpose. This allows me to see all of the data in a common size format. Figure 2 shows those results.

Figure 2. Standardized forcings used by the CCSM 3.0 climate model to hindcast the 20th century temperatures. Dark black line shows the temperature hindcast by the CCSM3 model.

Looking at that, I could see several things. First, the CO2 data has the same general shape as the sulfur, ozone, and methane (CH4) data. Next, the effects of the solar and volcano data were clearly visible in the temperature output signal. This led me to believe that the GHG data, along with the solar and the volcano data, would be enough to replicate the model’s temperature output.

And indeed, this proved to be the case. Using the Excel “Solver” function, I used the formula which (as mentioned above) had been developed through the analysis of the GISS model. This is:

 T(n+1) = T(n)+λ ∆F(n+1) * (1- exp( -1 / τ )) + ΔT(n) exp( -1 / τ )

OK, now lets render this equation in English. It looks complex, but it’s not.

T(n) is pronounced “T sub n”. It is the temperature “T” at time “n”. So T sub n plus one, written as T(n+1), is the temperature during the following time period. In this case we’re using years, so it would be the next year’s temperature.

F is the forcing, in watts per square metre. This is the total of all of the forcings under consideration. The same time convention is followed, so F(n) means the forcing “F” in time period “n”.

Delta, or “∆”, means “the change in”. So ∆T(n) is the change in temperature since the previous period, or T(n) minus the previous temperature T(n-1). ∆F(n), correspondingly, is the change in forcing since the previous time period.

Lambda, or “λ”, is the climate sensitivity. And finally tau, or “τ”, is the lag time constant. The time constant establishes the amount of the lag in the response of the system to forcing. And finally, “exp (x)” means the number 2.71828 to the power of x.

So in English, this means that the temperature next year, or T(n+1), is equal to the temperature this year T(n), plus the immediate temperature increase due to the change in forcing λ F(n+1) * (1-exp( -1 / τ )), plus the lag term ΔT(n) exp( -1 / τ ) from the previous forcing. This lag term is necessary because the effects of the changes in forcing are not instantaneous.

Figure 3 shows the final result of that calculation. I used only a subset of the forcings, which were the greenhouse gases (GHGs), the solar, and the volcanic inputs. The size of the others is quite small in terms of forcing potential, so I neglected them in the calculation.

 Figure 3. CCSM3 model functional equivalent equation, compared to actual CCSM3 output. The two are almost identical.

As with the GISSE model, we find that the CCSM3 model also slavishly follows the lagged input. The match once again is excellent, with a correlation of 0.995. The values for lambda and tau are also similar to those found during the GISSE investigation.

So what does all of this mean?

Well, the first thing it means is that, just as with the GISSE model, the output temperature of the CCSM3 model is functionally equivalent to a simple, one-line lagged linear transformation of the input forcings.

It also implies that, given that the GISSE and CCSM3 models function in the same way, it is very likely that we will find the same linear dependence of output on input in other climate models.

(Let me add in passing that the CCSM3 model does a very poor job of replicating the historical decline in temperatures from ~ 1945 to ~ 1975 … as did the GISSE model.)

Now, I suppose that if you think the temperature of the planet is simply a linear transformation of the input forcings plus some “natural variations”, those model results might seem reasonable, or at least theoretically sound.

Me, I find the idea of a linear connection between inputs and output in a complex, multiply interconnected, chaotic system like the climate to be a risible fantasy. It is not true of any other complex system that I know of. Why would climate be so simply and mechanistically predictable when other comparable systems are not?

This all highlights what I see as the basic misunderstanding of current climate science. The current climate paradigm, as exemplified by the models, is that the global temperature is a linear function of the forcings. I find this extremely unlikely, from both a theoretical and practical standpoint. This claim is the result of the bad mathematics that I have detailed in “The Cold Equations“. There, erroneous substitutions allow them to cancel everything out of the equation except forcing and temperature … which leads to the false claim that if forcing goes up, temperature must perforce follow in a linear, slavish manner.

As we can see from the failure of both the GISS and the CCSM3 models to replicate the post 1945 cooling, this claim of linearity between forcings and temperatures fails the real-world test as well as the test of common sense.

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TECHNICAL NOTES ON THE CONVERSION TO WATTS PER SQUARE METRE

Many of the forcings used by the CCSM3 model are given in units other than watts/square metre. Various conversions were used.

The CO2, CH4, NO2, CFC-11, and CFC-12 values were converted to w/m2 using the various formulas of Myhre as given in Table 3.

Solar forcing was converted to equivalent average forcing by dividing by 4.

The volcanic effect, which CCSM3 gives in total tonnes of mass ejected, has no standard conversion to W/m2. As a result we don’t know what volcanic forcing the CCSM3 model used. Accordingly, I first matched their data to the same W/m2 values as used by the GISSE model. I then adjusted the values iteratively to give the best fit, which resulted in the “Volcanic Adjustment” shown above in Figure 3.

[UPDATE] Steve McIntyre pointed out that I had not given the website for the forcing data. It is available here (registration required, a couple of gigabyte file).