Willis Eschenbach has a new post on Watts Up With That (WUWT) called Model climate sensitivity calculated directly from model results. HotWhopper has already covered this in a new post, but I thought I would add my two cents worth anyway.

Willis claims that he wants to treat a climate model as a “black box” in which all you need to know are the inputs (forcings) and the outputs (global surface temperatures). His model has only two parameters, the lag time, τ, and the climate sensitivity, λ. The problem, according to Willis, is that he has never had the data that was necessary to show that complex climate models are “functionally equivalent” to his simple approach. Recently, however, Willis discovered the data he needed in a paper by Forster et al. (2013). Figure 2 of this paper shows a multi-model mean of the computed climate forcing. The climate forcing, Δ*F*, is just what Willis needs to carry out his analysis, so he reads the values of the climate forcing from this Figure using a digitiser.

Now that Willis has what he needs, he can run his calculation and produces the figure below. Looks quite impressive, doesn’t it?

So, the equation that Willis uses is in the figure above, but I include it below for clarity. The data that Willis obtained from Forster et al. (2013) is the Δ

*F*term, and he then varies the lag time, τ, and the climate sensitivity, λ, until he gets a best fit for the data. In the above figure, the fit is to the multi-model mean surface temperature, but he also repeats this by fitting to the GISS data. I’m not quite sure what the last term is. I assume Δ

*T*(

*n*) is simply

*T*(

*n*) –

*T*(

*n*-1). If so, it seems a little odd that the temperature at time

*n*+1 somehow depends on the difference between the temperature at

*n*and

*n*– 1.

So, what does Willis conclude.

1. The models themselves show a much lower climate sensitivity (1.2°C to 1.6°C per doubling of CO2) than the canonical values given by the IPCC (2°C to 4.5°C/doubling).

Well, I think that the climate sensitivity he gets here is the Transient Climate Response (TCR), not the Equilibrium Climate Sensitivity (ECS). I believe the IPCC current best value for the TCR is around 1.8^{o}, not between 2 and 4.5^{o}C and so, here, Willis is not comparing the same quantities. It also seems that what he gets is quite consistent with IPCC values. I can’t see how his model can give the ECS because he is trying to fit to “known” temperatures, not trying to determine the equilibrium temperature.

2. The time constant tau, representing the lag time in the models, is fairly short, on the order of three years or so.

Well the lag time in Willis’s calculation is of the order of 3 years, but what does this mean? He seems to be suggesting that this is the lag time to reach equilibrium, which it clearly is not. It also, as far as I can tell, doesn’t have any real physical significance. It is simply a number that his model needs so that the surface temperatures that his model determines match the data he is trying to fit.

3. Despite the models’ unbelievable complexity, with hundreds of thousands of lines of code, the global temperature outputs of the models are functionally equivalent to a simple lagged linear transformation of the inputs.

This, I think, is Willis’s biggest misconception. The climate forcings Δ*F* are not inputs to the climate models, they are essentially the outputs. Some, such as solar forcing, may be an input, but the greenhouse gas forcings are essentially what these climate models are trying to determine. That’s why they’re hundreds of lines long. The only reason Willis’s calculation doesn’t need hundreds of lines of code is because he’s taken the output from the climate models (Δ*F*) and used it as input to his model.

4. This analysis does NOT include the heat which is going into the ocean. In part this is because we only have information for the last 50 years or so, so anything earlier would just be a guess. More importantly, the amount of energy going into the ocean has averaged only about 0.25 W/m2 over the last fifty years. It is fairly constant on a decadal basis, slowly rising from zero in 1950 to about half a watt/m2 today. So leaving it out makes little practical difference, and putting it in would require us to make up data for the pre-1950 period. Finally, the analysis does very, very well without it …

The heat going into the oceans may not be explicitly included in Willis’s calculation, but it is included implicitly. He’s determining the parameters in his model by fitting to either measured surface temperatures (GISS) or to values determined by other climate models. The reason these surface temperatures are what they are is because energy has gone into the oceans. If less energy had gone into the oceans, the surface temperatures would be different and his model parameters would be different.

5. These results are the sensitivity of the models with respect to their own outputs, not the sensitivity of the real earth. It is their internal sensitivity.

Well, I’m not sure what he’s saying here so maybe he’s right about whatever he’s trying to say here.

So, essentially Willis Eschenbach takes climate forcing outputs from detailed climate models (which he assumes are actually inputs) uses these forcings in a simple model of his own to get a lag time and a climate sensivity. He confuses the TCR (which is what I believe he is calculating) with the ECS (which is what he compares his results to), he gets a small lag time (which he thinks is significant, but I think is irrelevant), and then claims that his model doesn’t even need to consider the ocean heat content (which it doesn’t only because its already incorporated into the data he is trying to fit). Sadly, rather than illustrating that one can do climate modelling with a really simple – two parameter – calculation, Willis has really just illustrated that he doesn’t actually know what he is talking about. To be fair, maybe I don’t know what I’m talking about either, so feel free to correct anything that I’ve got wrong.

**Addendum**

The comment on the WUWT post about the units in Willis’s equations made me wonder if his errors aren’t even more severe. Climate sensitivity, λ, is typically defined through

Δ*T* = λ Δ*F*,

so is the change in temperature for a 1 W m^{-2} change in radiative forcing. In the absence of feedback a doubling of CO_{2} produces a 3.7 W m^{-2} increase in radiative forcing and produces a 1K increase in temperature. In such a case λ = 0.27 K/(W m^{-2}).

Therefore, as far as I can see, to get the climate sensitivity from Willis’s equations you need to solve

λ Δ *F* = λ’ Δ *F*/τ + Δ*T*(*n*) exp^{-1/τ},

where λ’ is the term that, I think, Willis is claiming is the climate sensitivity. If I ignore the second term on the right-hand side (because I don’t know what Δ*T*(*n*) is), I can then approximate λ as λ’/τ. Depending on which of his values I use, I get λ between 0.46 and 0.55 K/(W m^{-2}). This converts to a climate sensitivity of between 1.7 and 2^{o}C, which is higher than claimed by Willis but also ignores a second term (whose values I don’t know) that I think would make it larger still. I’ve, however, now become slightly confused as to whether this is the TCR or the ECS. I think it is still the TCR (as he is not solving for the equilibrium temperature, but fitting to temperature data) but I could be wrong.

**Addendum – 23/05/2013**

I was thinking a little more about Willis’s choice of “black box” function and why he had a second exponential term. I downloaded the data he used and tried simply using

*T*(*n*+1) = *T*(*n*) + λΔ*F*.

Given that the total forcing is 1.67 W m^{-2} and the net change in temperature is 0.88^{o}C, an estimate for λ would be about 0.55 K/(W m^{-2}). I tried a few values close to this and ended up using λ = 0.525 K/(W m^{-2}) which gives the graph below. The red line is the multi-model mean that Willis used for his fit, and the blue line is my new fit using the above function. Clearly it doesn’t do quite as well as Willis’s function, but the correlation is still 0.9 and the TCR would be 1.94^{o}C.

I would think that forcings (e.g. GHGs from fossil fuel emmissions, solar irradiance) are mainly provided as inputs, although e.g. GHGs due to changes in the carbon cycle could be considered as output (or internal).

Se e.g. here</.

I did wonder that, but I’m not sure. Figure 2 of the paper that Willis uses refers to them as the “computed net forcings”. The paper actually says

I take this to mean that a big part of climate modelling is computing the forcings. It’s possible that one set of codes compute the forcings which are then used as inputs to another set of codes that calculate the surface temperatures, but that’s essentially the same. The forcings still need to be computed and clearly Willis is incapable of doing this.

In the comments to the WUWT post, someone (Nick Stokes) has pointed out something I should have noticed. Climate sensitivity has units of oC/(W m-2). The term λ that Willis calls his climate sensitivity appears to have units oC/(W m-2)/yr, so has the wrong units.

Okay, so I actually downloaded Willis’s Excel spreadsheet and it does seem as though he has calculated the climate sensitivity, λ, correctly in the sense that what he presents is actually λ/τ corrected according to 3.7 W m-2 produces 1oC of warming. So, he just hasn’t explained himself properly. I’m still confused, though, about the role of the second term in his equation as that would seem to complicate things somewhat.