I knew I was taking a bit of a risk writing this (as I pointed out at the beginning anyway) as there was a reasonable chance that I misunderstood something fundamental. Unsurprisingly, this appears to be true. I may even be taking a risk writing this correction, as this may also end up being wrong. Anyway, my issue with the ECS calculation is probably incorrect. I had misunderstood the role of the term H. I had seen it as some kind of average of the rate at which the rest of the system had gained energy over the time interval. This is not right (I think). It is meant to represent the rate at which the rest of the system is still gaining energy when surface temperatures have changed by ΔT and the radiative forcing has changed by ΔQ. So the basic way it’s been implemented seems quite correct (as one might expect).
It still does seem, though, that it is still sensitive (as Karsten has pointed out in the comments) to short term variations. Anyway, for posterity, I’ll leave the post as I wrote it. Just bear in mind that some of it is almost certainly wrong (no surprise I hear some of you say). I’ve certainly learned something if noone else has 🙂
Original post starts here
I should start by saying that this post could be complete nonsense and is really just something I’ve been pondering for the last few days. It could also end up being quite convoluted and confused, so apologies in advance. One of the talking points of the recent report (SPM) from the Intergovernmental Panel on Climate Change (IPCC), is that they included a range for the Equilibrium Climate Sensitivity (ECS), but not a best estimate. Some see this as maybe significant. Personally, I don’t think it is.
I believe that the reason why they have not produced a best estimate is that the three primary methods for estimating the ECS cannot be reconciled to give a single best estimate. The three methods used are essentially paleo-climatology, modelling, and constraints using recent observations. The one that is apparently most divergent is the method that relies on recent observations.
One of the methods for estimating climate sensitivities (see, for example, Otto et al. 2013) is to use recent changes in surface temperature, ΔT, together with estimates for the change in forcing over the same time interval, ΔQ, and the change in forcing due to a doubling of CO2, ΔQ2x (and I should make clear that these are adjusted forcings so include all the other forcings and feedbacks associated with a doubling of CO2). The change in temperature at the instant when CO2 has doubled is known as the Transient Climate Response (TCR) and can be estimated using
This makes quite a lot of sense to me. You measure how much the temperature has changed during some time interval. You then estimate the change in forcing over the same time interval. That then tells you how the temperature has changed due to that change in forcing. If you know how much the forcing will change due to a double of CO2 you then get how much the temperature will change once CO2 has doubled. There are some issues, though. Surface temperatures don’t rise smoothly, so this method is sensitive to short-term variations in surface temperature. Therefore, ideally, one should consider a time period that’s long enough to be confident that the change in temperature is a fair representation of the long-term trend. Another issue is that other factors (solar variability, aerosols) can also influence the surface temperature on short timescales and so some of the temperature change (if the timescale is short enough) could be due to factors other than changes in atmospheric CO2 concentrations.
So, how does one then get an estimate for the ECS? To do that one needs an estimate for what fraction of the change in forcing is associated with changing surface temperatures. In other words, you need an estimate for the rate at which the rest of the system gains energy, H. Most of the excess energy entering the system goes into the oceans and so one method is to use the change in ocean heat content to estimate to estimate H. Having done this, the ECS is then given by
I can kind of see how the above equation works. The term ΔQ–H tells you, during the time interval considered, how much of the change in forcing was associated with ΔT. That then allows you to estimate how much the temperature must change by if the total forcing applied is ΔQ2x. It does, however, seem remarkably simple for something so complicated. It also suffers from all the same issues as the TCS if the timescale used is too short.
There is, however, possibly a more fundamental problem with this estimate for the ECS. Admittedly, I’m no expert at this and so suggesting that a group of, presumably, very bright people have got something fundamentally wrong may be a little presumptious. I’m sure Karsten, Tom, Victor, or Dana can put me straight if so. As far as I understand it, the terms ΔQ and ΔQ2x are changes in radiative forcings. For example, ΔQ tells you how much greater the radiative forcing is today than it was at the beginning of the time interval considered. The term, H, on the other hand seems to be an average. As far as I understand it, this term is computed by calculating how much the energy has increased in the other parts of the climate system (mainly oceans) and then dividing by the surface area of the Earth and the time. That seems, to me at least, to suggest that ΔQ and H are not quite consistent.
To make it consistent, you would need to either calculate the average change in forcing (which you’d also need to apply to ΔQ2x) or calculate an effective change in forcing for the energy entering the rest of the climate system. If we consider the ocean heat content, then it’s increased by 2.5 x 1023J since 1970. That means it’s associated with an average rate of increase of 0.4 Wm-2 over that time interval. That appears to be consistent with the numbers used by Otto et al. (2013) for the total system heat uptake.
If I want to estimate an effective change in forcing for the ocean heat content, I could assume that it would have the form αt (i.e., I assume that the change in forcing associated with the increase in ocean heat content changes linearly with time). I then need to integrate this term (multiplied by surface area) over the time interval considered – with the condition that it matches the total change in ocean heat content – to get an effective change in forcing.
In the above, A is the surface area of the Earth, E is the change in ocean heat content, and T is the time interval considered. If I plug the various numbers in I get α = 5.3 x 10-10. The effective change in forcing is then αT = 0.7 Wm-2. This is quite a bit bigger than the value used by Otto et al. (2013).
If we then consider the values used by Otto et al. for the period 1970-2009, it was ΔQ=1.21 Wm-2, ΔQ2x=3.44 Wm-2, H = 0.35 Wm-2, and ΔT = 0.48oC. Using Otto et als. (2013) expression for the ECS, this gives ECS = 1.92oC. If, however, we replace H with my estimate above of an effective change in forcing (H=0.72 Wm-2) I get ECS = 3.4oC.
So, to summarise, it would seem that combining a change in forcing with an average forcing (to determine the ECS) is not quite consistent. Having said that, maybe this is well known and it has been done properly in the studies and I just haven’t realised that. Maybe also, I just don’t know what I’m talking about and don’t appreciate the subtleties associated with estimating the ECS using recent observations. If I am right, however, it’s no great surprise that the ECS based on recent observations is lower than that from other studies since the approximations used essentially guarantee that the result will be a lower limit.
While writing this I came across a paper called The Climate Change Commitment (Wrigley 2005). It has a similar form for an expression associated with the ECS
In the above, ΔQ is the change in forcing during the time interval considered, and ΔQr is the the forcing that gives an equilibrium warming of ΔTr (i.e., it is the forcing associated with the observed warming). The paper then says that (ΔQ-ΔQr) is approximately the heat flux into the ocean (which it estimates as 0.7 Wm-2). What’s interesting about this paper, though, is that it uses estimates of ΔT2x (ECS) to determine the amount of uncommitted warming (ΔTe – ΔTr). So, this paper appears to be using estimates for the ECS to determine how much warming we have already locked in at some instant in time. This would seem to produce another inconsistency. How can you use the same method to determine the uncommitted warming (using the estimates of the ECS) and also use it to estimate the ECS? It would seem that you can’t estimate the ECS using this method unless you know the uncommitted warming locked in at the instant in time at which the estimate is made.
I did a bit more, which I won’t include here as this is a bit long already, which makes me think that using this method to estimate the ECS has an implicit assumption that the uncommitted warming at the instant when the estimate is made is effectively the TCR of the, as yet unused, forcing. Again, this would seem to imply that the estimate is a lower limit rather than something that should be regarded as robust. However, as I’ve already said, maybe I don’t know what I’m talking about so feel free to correct me through the comments.