Nic Lewis has another post on Watts Up With That (WUWT) called Updated climate sensitivity estimates using aerosol-adjusted forcings and various ocean heat uptake estimates. Basically he tries various different changes in radiative forcing and ocean heat uptake rates to determine various values for the Transient Climate Response (TCR) and the Equilibrium Climate Sensitivity (ECS) and gets some *surprising results*.

The table below shows one of his sets of results. Firstly, his results don’t seem that surprising. The 5-95% range for the ECS (1 – 4^{o}C) is very similar to other studies. I’m also not sure if his best guess value for the ECS is the mode or the median. The 3 numbers one might quote when describing a distribution are the mode, the median or the mean. These are skewed distributions (as can be seen from the fact that the best guess is closer to the lower side of the range than the higher) and so the mode will be smaller than the median, which will be smaller than the mean. If he is quoting the mode, then there is a greater than 50% chance that the actual value will be bigger than the “best estimate”.

In a sense I found this post quite interesting in that I learned quite a lot about how to get the TCR and the ECS. The TCR is fairly straightforward. It is given by

TCR = *F*_{2x} Δ*T*/Δ*F*,

where *F*_{2x} is the change in forcing resulting from a doubling of the CO_{2} concentration, Δ*T* and Δ*F* are the change in temperature and forcing over the time interval considered.

The ECS is similar but also includes the heating rate of the oceans, Δ*Q*. It is given by

ECS = *F*_{2x} Δ*T*/(Δ*F* – Δ*Q*),

where the terms are the same as for the TCR. This makes sense. The TCR tells you the actual change in temperature at the instant at which the CO_{2} has doubled. The ECS includes that some of the energy is going into the ocean and hence that the ECS will be higher than the TCR and that there will be lag (i.e., ECS will be reached after TCR, obviously).

Basically, all that Nic Lewis has done is consider various estimates for the forcings and for the ocean heating rate. I thought I would quickly do a calculation of my own. Below is a figure showing the ocean heat content from Levitus et al. (2012). The total change in ocean heat content since 1955 is 2.5 x 10^{23}J. The oceans cover 70% of the surface of the Earth, so this gives a heating rate of Δ*Q* = 0.4 W m^{-2}.

The figure below shows the radiative forcing from the HADCRUT4 dataset. This seems to indicate that the change in radiative forcing since 1955 was about Δ*F* = 1.35 W m^{-2}.

The next figure shows the change in global surface temperature which indicates that the change in surface temperature since 1955 is Δ*T* = 0.5 K.

*F*

_{2x}are 3.7 W m

^{-2}and so now we have everything we need to calculate the TCR and ECS. What I get are

TCR =

*F*

_{2x}Δ

*T*/Δ

*F*= 1.37

^{o}C,

ECS =

*F*

_{2x}Δ

*T*/(Δ

*F*-Δ

*Q*) = 1.94

^{o}C.

These are very simple calculations with no errors estimates and everything determined by eye, but they seem reasonable. A TCR close to 1.5

^{o}C and an ECS close to 2

^{o}C. I guess, I’m not sure why what Nic Lewis has determined is all that surprising. The numbers might be a little lower than some other estimates, but not by much and the 5-95% ranges are quite consistent. I’m also not quite sure what the point of this post was. Maybe an attempt to write something moderately positive about a WUWT post.

**Addendum**

Maybe someone who knows more than me can clarify something. I was assuming that Δ*F* was the change in total radiative forcing. However, looking at the form of the TCR makes me think that it should be only the change due to CO_{2}. We want TCR = Δ*T* when the CO_{2} concentration has doubled and hence require (at that point) that Δ*F* = *F*_{2x}. If so, CO_{2} levels have increased from *C*_{o} = 315 ppm in 1955 to *C*=400 ppm today. The equation for the change in forcing due to CO_{2} changes is

Δ*F* = 5.35 ln(*C*/*C*_{o}),

which gives (for the period 1955-2013)

Δ*F* = 1.28 W m^{-2}.

The next issue is how does one determine Δ*Q* as this should now be the change in ocean heating due to increased CO_{2} only. Well, if the total change in forcing is 1.75 W m^{-2} and that due to CO_{2} is 1.27 W m^{-2}, then we can assume that 73% of the change in ocean heating rate was due to CO_{2}. Given that the total change is 0.4 W m^{-2}, that due to CO_{2} would be Δ*Q*_{CO2} = 0.29 W m^{-2}.

If I use this to recalculate the TCR and ECS I get,

TCR = *F*_{2x} Δ*T*/Δ*F* = 1.46^{o}C,

ECS = *F*_{2x} Δ*T*/(Δ*F*-Δ*Q*) = 1.9^{o}C.

So the TCR is higher than before, but the ECS is about the same. Firstly, am I right about this? Secondly, was Nic Lewis using the correct values for Δ*F* and Δ*Q*?

Sorry, I can’t answer your questions but I have another. I was wondering whether these values from Nic Lewis and the Otto paper include all the feedbacks expected? For instance, I’ve read that carbon will be released from peatlands, soils and bacteria but it’s not clear that this carbon is counted in those estimates.

I’m not sure, but I was actually wondering if this is relevant. In a sense the

F_{2x}is a model estimate (i.e., how much will the forcing due to CO_{2}increase by if CO_{2}is doubled). That makes me think that ΔFshould only be the change in CO_{2}forcing for the time interval considered (which also has to be based on a model). The ΔTis therefore the measured change in temperature and, in a sense, contains contributions due to all the other forcings. If the other forcings (feedbacks I guess) are large then ΔTwill be large and the TCR or ECS will be big. If the other forcings are small, then the TCR and ECS will be small. I’m not sure if this is correct, but it makes sense to me (which doesn’t necessarily make it correct).The thread’s getting long, but if you go to James Annan’s blog with this, hopefully Paul S and Karsten will still respond.

Thanks, I’ve given that a try.

Secondly, was Nic Lewis using the correct values for ΔF and ΔQ?Sorry for rather dodging this last night, but too late and too tired.

The simple answer is “probably”. The uncertainties in OHC reconstructions are large and increase with every decade one goes back in time. The uncertainties in negative aerosol forcing estimates likewise. The consequence at present is that the available data yield low estimates for ECS/TCR, as you know. The short time-scales involved do not help. The poorly constrained effects of natural variability further reduce the confidence that can be vested in the results.

Unscrupulous trumpeting of the lowest-possible estimates that can be wrung from the data by the usual suspects – ably cheer-led by NL – does not help.

Thanks for replying again. I assume you mean that they are using the correct values for Δ

Fand ΔQ. I suspect you’re correct, but I’m still clearly confused. The equation for the TCR isTCR =

F_{2x}ΔT/ΔF,which means that the TCR = Δ

Twhen ΔF=F_{2x}. If Nic Lewis (and Otto et al.) are using the total forcing rather than that simply due to CO_{2}then it seems as though the TCR will equal ΔTbefore CO_{2}has doubled (given that the CO_{2}forcing seems to be lower than the total forcing). This suggests that what they’re determining is the climate response to a change in forcing ofF_{2x}rather than the change due to a doubling of CO_{2}.Having said that, I do find it hard to believe that so many have got this wrong, so I’m more inclined to believe that I’m mistaken and don’t really understand this very well.

You are right and I’m as puzzled as you are now:

3.44 x 1.3/3.44 = 1.3

I’m not sure about this though. CO2 forcing is greater than the net of forcings.

Well, one possible explanation is that they’re essentially the same. If you consider the IPCC forcings (which are based on the time period 1750 – today) then the mean of the CO

_{2}forcing seems about the same as the total net forcing, but the errors are different. However, if you consider more recent time intervals, this doesn’t seem to be necessarily true. For example, I calculate the CO_{2}forcing for the period 1970 – 2013 to be 1.11 W m^{-2}while the total net radiative forcing is 1.21 W m^{-2}(with some errors, which I’ll ignore here). Not much of a difference admittedly, but enough to increase the TCR by 0.1 – 0.2^{o}C compared to that determined by Otto et al. (2013).More puzzled. As I understand it, CO2 forcing is higher than total net forcing – eg GISS forcings.

I think the figure that you link to shows GHG forcings (upper curve) and net forcing (lower curve). CO

_{2}makes up about 60% of the GHG forcing and so the CO_{2}forcing is therefore similar to the net forcing, rather than quite a bit bigger than the net forcing. This could be what they’re assuming, but I haven’t found any discussion by Nic Lewis or in the Otto et al. (2013) paper that actually says that they’re assuming that the CO_{2}is essentially the same as the net forcing.Okay, I think I’ve worked this out and it was me who was wrong (if my new understanding is correct). I had thought that

F_{2x}was the change in forcing due only to a doubling of CO_{2}. I think this is not quite correct. It is the adjusted forcing due to a doubling of CO_{2}. In other words what is the change in known forcings when CO_{2}has doubled, not what is the change due to CO_{2}alone. If so, then the assumptions by Nic Lewis and Otto et al. are fine. The numerator has the change in total forcing after a doubling of CO_{2}and the denominator has the change in forcing ΔFassociated with a change in temperature ΔT.Point taken about the proportion of total GHG forcing attributable to CO2. More importantly, thanks for the update on the oddly puzzling equation. I think you are probably correct, not least because it makes the problem go away in a logical manner.

I don’t think I’ve been at all helpful here, so my apologies. I have to admit that I don’t normally pay much attention to instrumentally-derived estimates for S for the reasons I’ve already set out. This has at least made me read the SI and think, which can only be a good thing 😉

No need to apologise. I think I’ve been complicating and confusing things quite easily all by myself. I have found this all quite interesting and have also learned quite a lot about climate sensitivity and how it’s calculated.

I agree with you about instrumentally derived estimates for the sensitivity. As you mention in your earlier comment there are issues with the short timescales and the variability. Also, the TCR and ECS are defined in such a way (for example the TCR being due to CO

_{2}rising at 1% per year) that make the estimates used here not as precise as a full model. I think that these estimates are probably fine in some sense, but I wouldn’t regard the low values that they get as particularly robust.