Mechanical Models

Guest Post by Willis Eschenbach

[NOTE the update at the end of the post.] I’ve continued my peregrinations following the spoor of the global climate model data cited in my last post. This was data from 19 global climate models. There are two parts to the data, the inputs and the outputs. The inputs to the models are the annual forcings (the change in downwelling radiation at the top of the atmosphere) for the period 1860 to 2100. The outputs of the models are the temperature hindcasts/forecasts for the same period, 1860 to 2100. Figure 1 shows an overview of the two datasets (model forcings and modeled temperatures) the nineteen models, for the historical period 1860-2000.

Figure 1. Forcing (red lines, W/m2) and modeled temperatures (blue lines, °C) from 19 global climate models for the period 1860-2000. Light vertical lines show the timing of the major volcanic eruptions. The value shown in upper part of each panel is the decadal trend in the temperatures.  For comparison, the trend in the HadCRUT observational dataset is 0.04°C/decade, while the models range from 0.01 to 0.1°C/decade, a tenfold variation. The value in the lower part of each panel is the decadal trend in forcing. Click any graphic to enlarge.

The most surprising thing to me about this is the wide disparity in the amount, trend, and overall shape of the different forcings. Even the effects of the volcanic eruptions (sharp downwards excursions in the forcings [red line]), which I expected to be similar between the models, have large variations between the models. Look at the rightmost eruption in each panel, Pinatubo in 1991. The GFDL-ESM2M model shows a very large volcanic effect from Pinatubo, over 3 W/m2. Compare that to the effect of Pinatubo in the ACCESS1-0 model, only about 1 W/m2.

And the shapes of the forcings are all over the map. GISS-E2-R increases almost monotonically except for the volcanoes. On the other hand, the MIROC-ESM and HadGEM2-ES forcings have a big hump in the middle. (Note also how the temperatures from those models have a big hump in the middle as well.) Some historical forcings have little annual variability, while others are all over the map. Each model is using its own personal forcing, presumably chosen because it produces the best results …

Next, as you can see from even a superficial examination of the data, the output of the models is quite similar to the input. How similar? Well, as I’ve shown before, the input of the models (forcings) can be transformed into an accurate emulation of the output (temperature hindcasts/forecasts) through the use of a one-line iterative model.

Now, the current climate paradigm is that over time, the changes in global surface air temperature evolve as a linear function of the changes in global top-of-atmosphere forcing.  The canonical equation expressing this relationship is:

∆T = lambda * ∆F           [Equation 1]

In this equation, “∆T” is the change in temperature from the previous year. It can also be written as T[n] – T[n-1], where n is the time of the observation. Similarly, “∆F” is the change in forcing from the previous year, which can be written as F[n] – F[n-1]. Finally, lambda is the transient climate response (°C / W/m^2). Because I don’t have their modeled ocean heat storage data, lambda does not represent the equilibrium climate sensitivity. Instead, lambda in all of my calculations represents the transient climate response, or TCR.

The way that I am modeling the models is to use a simple lagging of the effects of Equation 1. The equation used is:

∆T = lambda * ∆F * ( 1-e^( -1/tau )) + ( T[n-1] – T[n-2] ) * e^(-1/tau)  [Eqn. 2]

In Equation 2, T is temperature (°C), n is time (years), ∆T is T[n] – T[n-1], lambda is the sensitivity (°C / W/m^2), ∆F is the change in forcing F[n] – F[n-1] (W/m2), and tau is the time constant (years) for the lag in the system.

So … what does that all say? Well, it says two things.

First, it says that the world is slow to warm up and cool down. So when you have a sudden change in forcing, for example from a volcano, the temperature changes more slowly. The amount of lag in the system (in years) is given by the time constant tau.

Next, just as in Equation 1, Equation 2 scales the input by the transient climate response lambda.

So what Equation 2 does is to lag and scale the forcings. It lags them by tau, the time constant and it scales them by lambda, the transient climate response (TCR).

In this dataset, the TCR ranges from 0.36 to 0.88 depending on the model. It is the expected change in the temperature (in degrees C) from a 1 W/m2 change in forcing. The transient climate response (TCR) is the rapid response of the climate to a change in forcing. It does not include the amount of energy which has gone into the ocean. As a result, the equilibrium climate sensitivity (ECS) will always be larger than the TCR. The observations in the Otto study indicate that over the last 50 years, ECS has remained stable at about 30% larger than the TCR (lambda). I have used that estimate in Figure 2 below. See my comment here for a discussion of the derivation of this relationship between ECS and TCR.

Using the two free parameters lambda and tau to lag and scale the input, I fit the above equation to each model in turn. I used the full length (1860-2100) of the same dataset shown in Figure 1, the RCP 4.5 scenario. Note that the same equation is applied to the different forcings in all instances, and only the two parameters are varied. The results are shown in Figure 2.

Figure 2. Temperatures (hindcast & forecast) from 19 models for the period 1860 to 2100 (light blue), and emulations using the simple lagged model shown in Equation 1 (dark blue). The value for “tau” is the time constant for the lag in the system. The ECS is the equilibrium climate sensitivity (in degrees C) to a doubling of CO2 (“2xCO2”). Following the work of Otto, the ECS is estimated in all cases as being 30% larger than “lambda”, which is the transient climate response (TCR). See the end note regarding units. Click to enlarge.

In all cases, the use of Equation 2 on the model forcings and temperatures results in a very accurate, faithful match to the model temperature output. Note that the worst r^2 of the group is 0.94, and the median r^2 is 0.99. In other words, no matter what each of the models is actually doing internally, functionally they are all just lagging and resizing the inputs.

Other than the accuracy and fidelity of the emulation of every single one of the model outputs, there are some issues I want to discuss. One is the meaning of this type of “black box” analysis. Another are the implications of the fact that all

of these modeled temperatures are so accurately represented by this simplistic formula. And finally, I’ll talk about the elusive “equilibrium climate sensitivity”.

Black Box Analyses

A “black box” analysis is an attempt to determine what is going on inside a “black box”, such as a climate model. In Figure 3, I repeat a drawing I did for an earlier discussion of these issues. I see that it used an earlier version of the CCSM model than the one used in the new data above, which is CCSM4.

Figure 3. My depiction of the global climate model CCSM3 as a black box, where only the inputs and outputs are known.

In a “black box” analysis, all that we know are the inputs (forcings) and the outputs (global average surface air temperatures). We don’t know what’s inside the box. The game is to figure out what a set of possible rules might be that would reliably transform the given input (forcings) into the output (temperatures). Figure 2 demonstrates that functionally, the output temperatures of every one of the climate models shown above in Figure 2 can be accurately and faithfully emulated by simply lagging and scaling the input forcings.

Note that a black box analysis is much like the historical development of the calculations for the location of the planets. The same conditions applied to that situation, in that no one knew the rules governing the movements of the planets. The first successful solution to that black box problem utilized an intricate method called “epicycles”. It worked fine, in that it was able to predict the planetary locations, but it was hugely complex. It was replaced by a sun-centered method of calculation that gave the same results but was much simpler.

I bring that up to highlight the fact that in a “black box” puzzle as shown in Figure 3, you want to find not just a solution, but the simplest solution you can find. Equation 2 certainly qualifies as simple, it is a one-line equation.

Finally, be clear that I am not saying that the models are actually scaling and lagging the forcings. A black box analysis just finds the simplest equation that can transform the input into the output, but that equation says nothing about what actually might be going on inside the black box. Instead, the equation functions the same as whatever might be going on inside the box—given a set of inputs, the equation gives the same outputs as the black box. Thus we can say that they are functionally very similar.

Implications

The finding that functionally all the climate models do is to merely lag and rescale the inputs has some interesting implications. The first one that comes to mind is that regarding the models, as the forcings go, so goes the temperature. If the forcings have a hump in the middle, the hindcast temperatures will have a hump in the middle. That’s why I titled this post “Mechanical Models”. They are mechanistic slaves to the forcings.

Another implication of the mechanical nature of the models is that the models are working “properly”. By that, I mean that the programmers of the models firmly believe that Equation 1 rules the evolution of global temperatures … and the models reflect that exactly, as Figure 2 shows. The models are obeying Equation 1 slavishly, which means they have successfully implemented the ideas of the programmers.

Climate Sensitivity

Finally, to the question of the elusive “climate sensitivity”. Me, I hold that in a system such as the climate which contains emergent thermostatic mechanisms, the concept of “climate sensitivity” has no real meaning. In part this is because the climate sensitivity varies depending on the temperature. In part this is because the temperature regulation is done by emergent, local phenomena.

However, the models are built around the hypothesis that the change in temperature is a linear function of forcing. To remind folks, the canonical equation, the equation around which the models are built, is Equation 1 above, ∆T = lambda ∆F, where ∆T is the change in temperature (°C), lambda is the sensitivity (°C per W/m2), and ∆F is the change in forcing (W/m2)

In Equation 1, lambda is the climate sensitivity. If the ∆F calculations include the ocean heat gains and losses, then lambda is the equilibrium climate sensitivity or ECS. If (as in my calculations above) ∆F does not include the ocean heat gains and losses, then lambda is the short-term climate sensitivity, called the “transient climate response” or TCR.

Now, an oddity that I had noted in my prior investigations was that the transient climate response lambda was closely related to the trend ratio, which is the ratio of the trend of the temperature to the trend of the forcing associated with each model run. I speculated at that time (based on only the few models for which I had data back then) that lambda would be equal to the trend ratio. With access now to the nineteen models shown above, I can give a more nuanced view of the situation. As Figure 4 shows, it turns out to be slightly different from what I speculated.

Figure 4. Transient climate response “lambda” compared to the trend ratio (temperature trend / forcing trend) for the 19 models shown in the above figures. Red line shows where lambda equals the trend ratio. Blue line is the linear fit of the actual data. The equation of the blue line is lambda = trend ratio * 1.03 – 0.05 °C / W/m-2.

Figure 4 shows that if we know the input and output of a given climate model, we can closely estimate the transient climate response lambda of the model. The internal workings of the various models don’t seem to matter—in all cases, lambda turns out to be about equal to the trend ratio.

The final curiosity occurs because all of the models need to emulate the historical temperature trend 1860-2000. Not that they do it at all well, as Figure 1 shows. But since they all have different forcings, and they are at least attempting to emulate the historical record, that means that at least to a first order, the difference in the reported climate sensitivities of the models is the result of their differing choices of forcings.

Conclusions? Well, the most obvious conclusion is that the models are simply incapable of a main task they have been asked to do. This is the determination of the climate sensitivity. All of these models do a passable job of emulating the historical temperatures, but since they use different forcings they have very different sensitivities, and there is no way to pick between them.

Another conclusion is that the sensitivity lambda of a given model is well estimated by the trend ratio of the temperatures and forcings. This means that if your model is trying to replicate the historical trend, the only variable is the trend of the forcings. This means that the sensitivity lambda is a function of your particular idiosyncratic choice of forcings.

Are there more conclusions? Sure … but I’ve worked on this dang post long enough. I’m just going to publish it as it is. Comments, suggestions, and expansions welcome.

Best regards to everyone,

w.

A NOTE ON THE UNITS

The “climate sensitivity” is commonly expressed in two different units. One is the c

hange in temperature (in °C) corresponding to a 1 W/m2 change in forcing. The second is the change in temperature corresponding to a 3.7 W/m2 change in forcing. Since 3.7 W/m2 is the amount of additional forcing expected from a change in CO2, this is referred to as the climate sensitivity (in degrees C) to a doubling of CO2. This is often abbreviated as “°C / 2xCO2

DATA AND CODE: As usual, my R code is a snarl, but for what it’s worth it’s here, and the data is in an Excel spreadsheet here.

[UPDATE]. From the comments:

Nick Stokes says:
December 2, 2013 at 2:47 am

In fact, the close association with the “canonical equation” is not surprising. F et al say:

“The FT06 method makes use of a global linearized energy budget approach where the top of atmosphere (TOA) change in energy imbalance (N) is split between a climate forcing component (F) and a component associated with climate feedbacks that is proportional to globally averaged surface temperature change (ΔT), such that:
N = F – α ΔT (1)
where α is the climate feedback parameter in units of W m-2 K-1 and is the reciprocal of the climate sensitivity parameter.”

IOW, they have used that equation to derive the adjusted forcings. It’s not surprising that if you use the thus calculated AFs to back derive the temperatures, you’ll get a good correspondence.

Dang, I hadn’t realized that they had done that. I was under the incorrect impression that they’d used the TOA imbalance as the forcing … always more to learn.

So we have a couple of choices here.

The first choice is that Forster et al have accurately calculated the forcings.

If that is the case, then the models are merely mechanistic, as I’ve said. And as Nick said, in that case it’s not surprising that the forcings and the temperatures are intimately linked. And if that is the case, all of my conclusions above still stand.

The second choice is that Forster et al have NOT accurately calculated the forcings.

In that case, we have no idea what is happening, because we don’t know what the forcings are that resulted in the modeled temperatures.