**Correction**

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 Δ*T* and the radiative forcing has changed by Δ*Q*. 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 π

**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, Δ*T*, together with estimates for the change in forcing over the same time interval, Δ*Q*, and the change in forcing due to a doubling of CO_{2}, Δ*Q*_{2x} (and I should make clear that these are adjusted forcings so include all the other forcings and feedbacks associated with a doubling of CO_{2}). The change in temperature at the instant when CO_{2} 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 CO_{2} you then get how much the temperature will change once CO_{2} 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 CO_{2} 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 Δ*Q*–*H* tells you, during the time interval considered, how much of the change in forcing was associated with Δ*T*. That then allows you to estimate how much the temperature must change by if the total forcing applied is Δ*Q*_{2x}. 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 Δ*Q* and Δ*Q*_{2x} are changes in radiative forcings. For example, Δ*Q* 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 Δ*Q* 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 Δ*Q*_{2x}) 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 10^{23}J 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 α*t* (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 α*T* = 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 Δ*Q*=1.21 Wm^{-2}, Δ*Q*_{2x}=3.44 Wm^{-2}, *H* = 0.35 Wm^{-2}, and Δ*T* = 0.48^{o}C. Using Otto et als. (2013) expression for the ECS, this gives ECS = 1.92^{o}C. If, however, we replace *H* with my estimate above of an effective change in forcing (*H*=0.72 Wm^{-2}) I get ECS = 3.4^{o}C.

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.

**Addendum**

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, Δ*Q* is the change in forcing during the time interval considered, and Δ*Q*_{r} is the *the forcing that gives an equilibrium warming of Δ T_{r}* (i.e., it is the forcing associated with the observed warming). The paper then says that (Δ

*Q*-Δ

*Q*

_{r}) 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 Δ

*T*

_{2x}(ECS) to determine the amount of uncommitted warming (Δ

*T*

_{e}– Δ

*T*

_{r}). 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.

I should add that I may be making corrections to this throughout the day as I do need to get some work done and may not have proofread this as carefully as I should have.

wotts, I think you got that one wrong. ΞQ is the change in radiative forcing between the 1860-1879 and the 1970-2010 period. So it is an average over the last 40 years, consistent with the average change in “OHC forcing” over that period of time. It is not the current forcing (which would be 2.0-2.2 W/m2)! The temperature change is calculated accordingly, with +0.48K being the change between 1860-1879 and 1970-2010. The rather low ECS is caused by the omission of non-linearities and the uncertainties in the forcing estimates, which, as you rightly say, aren’t constant over time and, moreover, affected by natural variability (such as volcanic eruptions) which complicates matters.

I did wonder if I’d got that wrong. As I said, I was just pondering this π

Maybe you can clarify something. If I understand things correctly, Δ

Q_{2x}is the total change in forcing for a doubling of CO_{2}(is this right, I guess). To get the TCR, you should then use the change in forcing for the time period considered, rather than the average (i.e., if you were to consider the full time period, the ΔQshould equal ΔQ_{2x}). However, to get the ECS you should use the average rather than the change in forcing. That would seem, to me at least, that the ΔQfor the TCR should be different to the ΔQfor the ECS. Unless, of course, I’m simply confused or wrong about the definition of ΔQ_{2x}.Maybe you could also clarify the issue with regards to Wrigley (2005). This seems to use the same basic method to estimate the uncommitted warming (using estimates for the ECS) so it seems that you shouldn’t then be able to use this to estimate the ECS unless you are making some kind of assumption about the uncommitted warming.

Karsten, I’m clearly still confused (which isn’t that surprising). I’ve been having another look at Otto et al. I think I now see where Δ

Qcomes from. It seems that they take an average of the forcing for the period 1860-1879 and then an average (for example) for the period 1970-2009. The difference between these is then ΔQ. This still seems to me to be an average change in forcing, rather than an average of the forcing (for the period 1970-2009). IfHis computed how I think it is (change in energy divided by area and time) this would still seem inconsistent. Having said that, I may well simply be confused.Actually, Karsten, I think I’m starting to get this. I spent the last couple of days thinking about this and still seem to have got it wrong. I think I’ve misunderstood what role is

His playing. I had thought it represented the rate at which energy had gone into to other parts of the climate system. If I’ve got it right now, it actually represents how much energy is still entering the rest of the climate system at the time when the radiative forcing has changed by ΔQand surface temperatures have changed by ΔT. In a sense it represents the lag and hence how much more surface temperatures need to rise by so as to reduce the energy imbalance (which is effectivelyH) to zero and hence reach equilibrium. I’ve written before about Otto et al. and got that one wrong and now I seem to have got this one wrong. Maybe one day I’ll actually understand it πΞQ2x is indeed the forcing for a doubling of CO2 (either effective = 3.44W/m2, or standard = 3.71W/m2). For TCR, everything works the same way as for ECS. Whether you take the difference between two average periods (say 1860-1879 and 1970-2009), or the difference between specified start and end dates (say 1880 and 2009, or 1970 and 2009) doesn’t make any difference as long as ΞQ and ΞT refer to the same time interval. TCR(1860-1879/1970-2009) would be 3.44×0.48/1.21=1.36K. TCR(1880-2009) would be (approximately) 3.44×0.85/2.0=1.46K. TCR(1970-2009) would be 3.44×0.55/1.2=1.57K. Looking at AR5, TCR(1980-2011) would be 3.44×0.45/0.95=1.63K. On average, TCR hovers around 1.5, which is best translated into ECS by assuming a TCR to ECS ratio of 0.55, according to the respective CMIP3/5 model average. So we end up at ECS numbers between 2.5 and 2.9K. In case of a specified start/end date in the ECS equation, the OHC forcing term (or system heat uptake for that matter) would be some value which represents the chosen start/end date best, e.g. a decadal average. Otto et al. did that by assuming some average H_start for the earlier period (1860-1879) and H_end for the last 40 years (1970-2009). H is the difference between the two.

Re Tom Wigley, I think he is assuming an ECS of 2.6K in his estimate, based on an earlier paper of him (reference 10). If I understand it correctly, ECS has to be known in order to estimate the committed warming. Btw, problem with this (seemingly straight-forward) approach is that it doesn’t take the Carbon uptake of the oceans once we stop emitting CO2 into account. As a result, the committed warming would be less, as demonstrated by Matthews and Weaver 2010. Instead, it’s the aerosols that will lead to some degree of committed warming (see Matthews and Zickfeld 2012 for more.

No worries, if I hadn’t discussed it with him for quite some time, I wouldn’t have understood all the details either. While I’m now quite confident that I got the important bits right, being a scientist, doubt always remains π

Thanks, Karsten. Indeed, I think I’ve finally worked this out. I had misunderstood the time intervals in Otto et al. (should probably read things more carefully, but there’s only so much time). I hadn’t quite appreciated that it was the average across the time interval relative to 1860-1879. So, yes, you’re quite right I had got this one wrong.

As far as Wrigley (2005) is concerned he was (at one stage in the paper) assuming an ECS of 2.6K and then using that to estimate the committed warming. I’m going to have to give the relationship between Wrigley (2005) and Otto et al. (2013) a little more thought as they do seem a little inconsistent (or have some hidden assumptions). I’ve been wrong before though π

I think I’ll just sit this one out.

π

I should have probably done the same π

Although I would venture to suggest that these calculations don’t take into account future perturbations of the carbon cycle forced by warming, eg permafrost melt and reduction of efficacy of ocean carbon sink. Both of which would increase ECS and presumably TCR although the effects become more pronounced with time.

We crossed! I’m going to put myself in moderation!

π

I think that’s probably right and is something I still haven’t quite got my head around with respect to these estimates. There was something I was going to add to the post, but I wanted to check it again before doing so. Only so many errors I can bring myself to make in a single post π

A bazillion blog posts should be starting with:

> I may […] be taking a risk writing this […]

What’s life without risk?