**Addendum – 27 May 2013**

*Just in case you don’t get as far as the comments, the answer, unsurprisingly, to the question I pose in the title is No. I did not properly understand what one of the terms in the equation actually was. For posterity, however, I will leave my post as I wrote it. I found it quite an interesting exercise and have certainly learned quite a lot about climate sensitivity while trying to work all this out. Nothing wrong with being wrong – IMO. *

As you may have noticed, I’ve become interested in the various climate sensitivities. The two that are typically considered are the transient climate response (TCR) and the equilibrium climate sensitivity (ECS). I explain these in the post linked to earlier, so won’t repeat it here.

There has been a recent paper by Otto et al. (2013) called Energy budget constraints on climate response. It’s received quite a lot of attention because it is predicting a lower TCR than many earlier studies. Not by a huge amount, but rather than a best estimate of 1.8^{o}C their study suggests that it is closer to 1.3^{o}C. This could be significant because this is an estimate for the change in temperature at the instant in time at which the CO_{2} concentrations have doubled (the actual definition is a little more complicated, but this is essentially what it represents). Hence, it is possible that the surface temperatures may rise more slowly than expected.

So, I’ve finally downloaded the paper and given it a quick read and am starting to wonder if they are simply wrong. What worries me though is that in my experience when you consider the results of a paper published by a number of high-profile scientists to be wrong, it’s much more likely that you are the one who is wrong. So, I more than happy for someone else to correct my thinking here and I’m writing this with an awareness that there is a good chance that I’m the one who is mistaken.

So, why do I think Otto et al. (2013) is wrong? I’m going to focus only on their calculation of the TCR. To estimate the TCR they use

where

*F*

_{2x}is the change in forcing due to a doubling of the CO

_{2}concentration. They use a value of 3.44 W m

^{-2}. The other terms are the change in global mean temperature, ΔT, and the change in radiative forcing, ΔF. They go on to say :

For ΔT, we use the HadCRUT4 ensemble data set of surface temperatures averaged globally and by decade. …. For ΔF, we use the multi-model average of the CMIP5 ensemble of climate simulations with emissions that follow a medium-to-low representative concentration pathway.

So, they use the HADCRUT4 data set to determine Δ*T*, the change in surface temperature. They then use the multi-model average of the CMIP5 ensemble of climate simulations to determine the change in radiative forcing, Δ*F*. I have no issue with their choice of Δ*T*, but their choice of Δ*F* seems wrong. They seem to have used the total change in radiative forcing. The TCR is the change in temperature due to a doubling of CO_{2}. The change in forcing due to a doubling of CO_{2} is estimated to be 3.44 W m^{-2}. Therefore, the TCR will equal Δ*T* when Δ*F* = *F*_{2x} = 3.44 W m^{-2}. Consequently, it seems that Δ*F* should only be the change in forcing due to CO_{2}, not the total change in radiative forcing. By using the total change in radiative forcing, they’re estimating the climate sensitivity due to a change in forcing of 3.44 W m^{-2} not the climate sensitivity due to a doubling of CO_{2}. In other words, Δ*F* could equal 3.44 W m^{-2} before the CO_{2} concentration has doubled and hence their assumptions will underestimate the TCR.

Firstly, it’s possible that I’m wrong or have mis-interpreted the Otto et al. (2013) paper. Also the change in forcing due to CO_{2} in recent times is quite close to the total change in forcing (depending on the time interval the change due to CO_{2} seems to be between 75% and 90% of the total change) so maybe their estimate is okay. One can, however, do a quick check. To estimate the change due to CO_{2} one can use

Δ*F*_{CO2} = 5.35 ln (*C*/*C*_{o}).

Since 1970, the CO_{2} concentration has increased from 325ppm to 400ppm. This gives Δ*F* = 1.11 W m^{-2}. The forcing used by Otto et al. (2013) for this period was Δ*F* = 1.21 +- 0.52 W m^{-2} and the temperature change was Δ*T* = 0.48 K. This gives a best estimate according to Otto et al. (2013) of 1.36 K while I would argue it should be 1.5 K (although I accept that I’ve ignored errors in my calculation).

So, I really would be keen if someone could let me know if I’m getting this horribly wrong. I also accept that the difference might be small, but given that people seem excited by a change in TCR of a few tenths of a degree this may still be significant. Again, I would be surprised if a group of established scientists could have made such a basic mistake, so I am assuming that their calculation is correct and I’m just misunderstanding what they’re doing or misunderstanding how the ECS and TCR are calculated.

Have you had a look at the supplementary info? http://www.nature.com/ngeo/journal/vaop/ncurrent/extref/ngeo1836-s1.pdf

Yes, and from their Figure S2 it seems that they’re using the total net radiative forcing, rather than that simply due to CO

_{2}. I think it should only be the contribution due to CO_{2}. However, I really could simply be wrong about this so am hoping someone can explain what it is that I don’t understand about all of this.Only skimmed through their work fast and got the impression their treatment of aerosols and sulfates is different from other, could be wrong about this.

Thanks for the comment. They do make claims about doing things that are different from others. What I’m confused about though is why one needs to consider aerosols and sulfates at all, if one is trying to determine the climate sensitivity due to CO

_{2}(the TCR and ECS). From my look at the equations, what you need are the change in forcing due to CO_{2}(ΔF) and the change in ocean heating rate due to CO_{2}(ΔQ). The measured ΔT, in a sense, contains the influence of all the other forcings, so you don’t need to include them explicitly. Having said that, I seem to be the only person suggesting this so I have to assume that I’m the one who’s mistaken and just don’t understand this particularly well.Climate sensitivity is not specific to CO2: any change in forcing of a given size should have about the same effect. So to me it seems like they have done the right thing. Anyhow, how would one separate the part of ΔT that is due to CO2 from the part that is due to other forcings?

But isn’t that the point. They’ve determined the climate sensitivity due to a change in forcing of 3.44 W m

^{2}, not the climate sensitivity due to a doubling of the concentration of CO_{2}. In the absence of feedbacks a doubling of CO_{2}produces a change in forcing of about 3.44 W m^{-2}and an increase in surface temperature of about 1^{o}C. One could assume therefore that almost any process that increases the forcing by 3.44 W m^{-2}will produce an increase in surface temperatures of about 1^{o}C. Therefore, it seems like their assumptions will, by default, lead to a climate sensitivity close to 1^{o}C. It seems, to me at least, for their approximation to be valid they should be extracting the contribution due to CO_{2}, otherwise they’re not – strictly speaking – determining the TCR and ECS as defined in terms of a doubling of CO_{2}. Again, I could be wrong about this, but I still haven’t had a good explanation for why their assumptions are appropriate.As far as separating the CO

_{2}contributions, you’d have to do it via modelling which is partly why I think the TCR and ECS can only really be determined by models and can only be approximated using the method in Otto et al. (2013).“Therefore, it seems like their assumptions will, by default, lead to a climate sensitivity close to 1oC.”

No, the point is that the climate sensitivity should include the feedbacks as well. That’s what makes it nontrivial to estimate. 1oC is just what you get as radiative equilibrium after adding 3.44 W m-2. But if you include the effect of e.g. water vapor (feedback), you’ll get more than 3.44 W m-2.

Yes, I realise that but consider the following. If you include all known forcings in Δ

Fthen it is possible that ΔFwill equal 3.44 W m^{-2}before CO_{2}has doubled. Therefore the estimate for the TCR will be the measured ΔTat the instant when ΔF= 3.44 W m^{-2}. However, CO_{2}concentrations have not yet have doubled. The ΔTat the instant in time at which CO_{2}has doubled will therefore be bigger than the TCR estimated from this approximation. According to my reading, the TCR is defined as the change in temperature after a doubling of CO_{2}(at a rate of 1% per year, but let’s ignore that for now) and not the change in temperature after an increase in forcing of 3.44 W m^{-2}. It therefore seems to me that the method followed by Otto et al. (2013) – and Nic Lewis in his earlier work – will underestimate the TCR unless – by chance – the forcing due to CO_{2}happens to equal the total net forcing.I should add that I do understand what you mean about the feedbacks. The total forcing that they’re using is still a model result and if the models were able to capture all the forcings then one would expect Δ

Tto be about 1^{o}C when ΔF= 3.44 W m^{-2}. That it doesn’t indicates that there are other feedbacks that are producing (for example) a larger ΔTthan would be expected based on the estimated forcings. However, I still think that what they’re estimating is the climate sensitivity to a change in known forcing of 3.44 W m^{-2}, not the sensitivity to a doubling of CO_{2}. To do this, I think they need to remove all the known forcings apart from that due to CO_{2}. Again, maybe I’m wrong.Okay, I may have just worked this out. I had assumed that

F_{2x}was the change in forcing due to CO_{2}only. I think that is not the case. It is the adjusted forcing and is, I think, the change in all known forcings when CO_{2}has doubled – CO_{2}contributing most but not all of the forcing associated withF_{2x}. If so, then the Otto et al. (2013) calculation is correct and the answer to the question posed in the title of my post is “no it isn’t just simply wrong”. My sense that it was much more likely that I was wrong than a large group a established scientists turned out to be a reasonable expectation. Maybe this is something that those who are wildly skeptical of climate science should consider now and again 🙂Yes, I think Lars was correct and you are too, with all known forcings and feedbacks. Have you read the Forster paper that was being discussed? Here is the link in case you missed it:

http://www.atmos.washington.edu/~mzelinka/Forster_etal13.pdf

Realclimate.org is always good to search. – real climate on sensitivity

Yes, indeed Lars was correct. I’d been focusing so much on the definition of the Δ

Fterm in the denominator that I hadn’t considered that my understanding ofF_{2x}was mistaken. As an aside, this is quite an interesting illustration (I think) of the typical scientific process. I quite commonly have trouble trying to work something out and spend days or weeks failing to get it right and then all of sudden realise that I was just ignoring something fairly basic and simple that I could have established very quickly if I’d just doubled check everything at the beginning.Yes, it was the Forster et al. paper that cleared this up for me. It’s when I saw the term “adjusted forcing” that I realised my definition of

F_{2x}might be wrong and when I realised that in fact their assumptions were correct – and that I was (as expected) the one who was wrong.Pingback: The capricious nature of weather and its effects upon us | Aden Baker