# Global temperature change computations using transient sensitivity, part two

In a recent post I made some crude estimates of projected global mean surface temperature (GMST) change to year 2090, based on the estimates of transient climate response/sensitivity (TCR/TCS) values taken from the AR5. The numbers I got were at the very low end of the AR5 90% confidence intervals and I couldn’t understand why. As mentioned there, my working assumption was that my numbers were most likely the indefensible ones. And that does indeed appear to be exactly the case, and fortunately the discrepancy between my values and the AR5’s was not too hard to spot. So, this post is to explain the mistake I made and provide a little associated discussion.

The very short version is that I forgot to account for the temperature time delay(s) in the earth system whenever an increasing RF is imposed over a number of years. Stupid mistake, but good learning experience. As a quick review, I got these median and mean GMST values, using the means and medians respectively of the 52 TCR values given in AR5 WG1 (45 are from Chapter 9, Tables 9.5 and 9.6). I applied the standard RF linear pro-rating to account for differences in delta RF from a doubling, the latter being how TCR is defined. For total anthro (TA) forcings, using the midpoints of the AR5 time intervals, I got these numbers:

```     Period   RCP dT.med dT.mean AR5.mid
1 1995-2090 rcp26  0.433   0.444     1.0
2 1995-2090 rcp45  1.145   1.176     1.8
3 1995-2090 rcp60  1.632   1.676     2.2
4 1995-2090 rcp85  2.726   2.799     3.7
```

Let’s call it ~2.75 deg C for the RCP 8.5 scenario, or almost a degree below the AR5 midpoint.

TCR is clearly defined–the GMST change expected from a 1% per year increase in RF CO2 (or an equivalent RF from all sources, CO2eq), once 2x RF CO2/CO2eq has been reached. A 1.0 percent increase always takes ~70 years to reach doubling [log(2, base=1.01)], so the time span–and hence the intensity–of the imposed RF change, is clearly defined. The longer it takes for a given increase in RF, the greater the T change by the end of the interval, relative to that at some future time beyond it (i.e., when equilibrium is reached). Due to ocean heating and large scale mixing, only some fraction of the RF-induced T effects can manifest in 70 years, which is why the Equilibrium Climate Senstivity (ECS) is always quite a bit higher than the TCR is. The AR5 reference time period midpoints (1995-2090, 95 years) give an interval just a little longer than the defined TCR 70 year period, not enough to matter hugely it would seem.

The lagged temperature increase in the system depends on several factors, but the AR4 (2007) best estimate was for an additional increase of 0.6 degrees C by year 2100. For the RCP 8.5 scenario, adding that value to 2.75 brings the estimate to 3.35, so now we’re definitely in the ball park just by that adjustment alone. But that 0.6 does not include any lagged response(s) that might originate in say the earlier decades of the 21st century but not be manifest until toward the end thereof. Trying to estimate that value requires a close look at the CMIP5 model outputs. I’ll look again in the AR5 for any such estimates (any help on the issue appreciated!), but it’s not hard at all to see that another 0.35 degrees, or more, could arise therefrom, giving a midpoint estimate of 3.7 degrees C, for the RCP 8.5.

The topics of time lags and slow feedbacks are very interesting ones IMO (in any field of science) and the fact that these are substantial in global T responses to rapid RF changes is of major importance. The fact that the mean/median TCR estimates declined between AR4 and AR5 is not necessarily a good thing, given that ECS estimates did not change much with them*. From a strict predictability perspective, forcing the system hard and fast is not a good thing, and you surely don’t want a big gap between TCR and ECS estimates, because it will tend to increase the difficulty of identifying and communicating cause and effect relationships. And since the scientific understanding on climate change and its attribution certainly affects policy, that’s not a good situation to be in.

* On reflection, one should also consider the fact that TCR estimates based on observational data have more confidence attached to them, than do ECS estimates from same, making the whole topic that much more intriguing.

Notes:
1. Edited several times for clarity.
2. The xkcd cartoon that spurred my original post is +/- unaffected by these considerations, i.e. still biased high, (but not by as much as I’d originally thought either). This is because that cartoon uses (presumably), the middle 20th century as its baseline point, and the lagged 0.6 degree C addition to the year 2100 T estimate, and whatever might be due to RF increases from 2000 to ~2030, are already +/- included in any estimate of T change from 1950 to 2100. Most GHG-induced warming is post-1950, when [CO2] was about 310 ppm.

## 10 thoughts on “Global temperature change computations using transient sensitivity, part two”

1. Hi Gim,
Just from a “design of experiments” perspective, it seems that if ECS, TCS, and the difference between the two are all unkowns, there is no way to calculate any of them from a single run of observational data. Wouldn’t model output just lead you in a circle back to the original parameterization? If I were confronted with this situation in my work, I would fill out a test matrix holding each variable constant, but then in my work, there is usually a system available for input-output testing.

• Yikes! Gim? All my typos are in the first five words because you cannot see the first five words on this WordPress comment form because they are covered up by “enter your comment here.”

• Demons at work Matt. I made so many typos and other mistakes on this I had to edit it ~10 times.

I’m still learning on the topic of the relationship between the two metrics (reading a lot right now). No source of estimates is without issues, that much seems clear. The observation-based estimates seem plagued by the recurring problem that the best quality data is the most recent…but also the shortest, and hence with great potential for confusion of signal and noise.

The only way that a systematic I-O analysis like you’re describing can be done for TCR/TCS, and especially for ECS, is via model output, as far as I can see. As far as I know, both TCR and ECS are emergent properties in most (all?) of the 45 AR5 models (30 GCMs, 15 EMICs), so there is no circular reasoning relating one to the other. My impression is that the less complex EMICs are the most efficient way to go about investigating this, because they subsume a lot of unnecessary spatio-temporal details inherent in GCMs.

• OK, thanks, now I understand what you are doing. GMST is certainly an emergent property of climate, I have to think some more about ECS.

• ECS is just expected delta GMST for a defined delta RF (3.7 Watts/m^2), after the forcing stabilizes, and the oceans and atmosphere have equilibrated. TCR is the same, but at end of 70 years (actually, mean of years 60 to 80), instead of at equilibrium. Therefore both are emergent, given the forcings and the model physics.

2. OK, reread both posts, getting closer…you are right of course. So at the beginning of a transient interval of 70 years, there is lagged heat “in the pipeline” (ocean, primarily) from earlier CO2 increases, but at the end there is also heat still in the pipeline due to the same lag. The heat still in the pipeline at the end would be ECR-TCR. The heat in the pipeline at the beginning would be the same, but would have to be adjusted both up and down. The up adjustment would be due to response decay as concentration of CO2 increases, and the down adjustment would be due to the fact that the lagging heat at the beginning was caused by less than a full doubling of concentration if you start at a year in the midst of anthropogenic releases such as 1995. But this confused me:

“But that 0.6 does not include any lagged response(s) that might originate in say the earlier decades of the 21st century but not be manifest until toward the end thereof.”

Does this refer to some of the difference between ECR and TCR due to the time interval being longer than 70 years (1995 to 2090)?

• Hang on Matt.

• To answer your question–yes. Since TCR is strictly defined as delta T for a 3.7 W/m^2 RF increase over ~70 years (which results from a 1% per year increase in CO2), delta T estimates over time periods > 70 yr must necessarily include some portion of the lagged (i.e. hidden, non-manifested) T increase that are not manifested at the 70 year point. The fraction of manifested T change has to go up as the time span increases, and there should also be definable relationships between the time span and the percentage manifested. For this reason, the idea that one can just linearly proportion the TCR for time periods not equal to 70 years, seems dubious to me. The rate of the forcing change needs to be factored in as well. I really wish the CMIP5 and/or AR5 had summarized these relationships better–they’re pretty darn important (and also real interesting, scientifically, at least to me).

I was uncertain of what you meant in your discussion of adjustments up and down though. In the TCR case, CO2 is increasing exponentially (1% per year), which gives an approximately linear increase in the RF with time. A constant CO2 increase (e.g. x ppm per year) will give a response decay. Note also that the RCP 8.5 forcing change over the 21st century is very close to a 1% per year in CO2 equivalent forcing.

3. Jim, I have a different take on your results. I’ve tried to indicate the softer spots in my argument.

You’ve used TCS (transient climate sensitivity) as the change in temperature divided by the change in forcing, multiplied by the forcing change associated with a doubling of CO2. That is, TCS=delta_T/delta_F*F_2xCO2. It’s plausible that the TCS so obtained will be similar to TCR for scenarios similar to the TCR scenario; that is, a linear increase of forcing over time. The RCP2.6 scenario, with non-monotonic forcing, doesn’t fit that mold, and one would expect the actual temperature change to exceed that computed from TCR and the applied forcing. [Alternately, the computed TCS will exceed TCR.] While this is not a quantitative argument, let’s put aside the RCP2.6 comparison for this reason. The RCP4.5 scenario, while it has a monotonically increasing forcing, has a negative acceleration in the latter half of the 21st century. So one might expect a similar effect, although muted by comparison with RCP2.6. Let me hand-wave away those two cases for now, and concentrate on RCP6.0 & RCP8.5, which have more-or-less linearly increasing forcing over the 21st century.

I looked at the multi-model mean temperature time series for these scenarios, from Climate Explorer. The series begins in 1861, so with a second hand-wave, I’ll consider this to be roughly the same as pre-industrial, ignoring away the small amount of forcing of that period. Now if we compare the changes from 1861-1880 to 2081-2100, we have:
Scenario delta_T(K) delta_F(Wm-2) TCS(K/doubling)
RCP6.0 2.85 5.21 1.88
RCP8.5 4.29 7.63 1.94
[I’ve used Forster et al.’s 3.44 Wm-2 as the forcing for CO2 doubling, notwithstanding your prior objections. That’s a third hand-wave, I guess.] Both TCSs are fairly close to the mean/median TCR of 1.8 K. Possible reasons for the difference include (a) the difference in conditions between TCR & TCS; and (b) different subset of models for the multi-model mean temperature vs. the 30 of Table 9.5 (or the 45 of Tables 9.5 & 9.6 combined).

OK, let me get back to your point. AR5 used a baseline period of 1986-2005, during which forcing averaged 1.65 Wm-2. Relative to that, the increase in forcing to the end-of-century (2081-2100) period is 3.56 & 5.98 Wm-2 resp. Using the above TCSs, that projects to an increase of 1.95 & 3.37 K resp. That’s short of the 2.2 & 3.7 K middle of the AR5 projections. The difference arises because the temperature gain to the baseline period (0.63 K) is too low compared to what one would expect from 1.65 Wm-2 and a TCS of 1.9. Which has been previously noted in e.g. Otto et al.

In brief, the models’ projected temperature rise from today is higher than the TCR would imply, because it includes making up for the “shortfall” to date. This “shortfall” is caused by the TCS to date (~1.3) being less than the model-mean TCR (1.8) or the TCS of the model-mean temperature series (1.9).

• Many thanks Harold, I really do appreciate you taking the time to look closely at the numbers. I think I got that, except for the last paragraph. A few disparate thoughts in response:

One is that the median TCR I get (combining the 45 AR5 models and 7 post-AR4 observational studies shown in the first post on the topic), is ~1.7, not 1.9. [Note that the obs. based estimates since AR4 differ substantially from the AR5 model-based estimates.] If I use Forster’s 3.44 W/m sq, instead of 3.71, for 2x CO2 delta RF, and a 1.7 TCR, I get (3.44/3.71) x 1.7 x (1.65/3.44) = 1.7 x (1.65/3.71) = 0.76, which is still > the 0.63 shortfall you mention, but not as much greater as if I used median TCR = 1.9, and 3.71 W/m sq for 2x CO2 RF.

Relatedly, did the AR5’s 45 models use Forster’s 3.4 W/m sq, or Gregory’s 3.7, to compute their TCR point estimates? And also, are those point estimates modal (i.e. max likelihood) or mean values over all ensemble runs?

To go from 1871 to 2090 (midpoint years), that’s 120/70 or 1.71 the length of the defined TCR period. I have to believe that will give a TCS value that is greater than just a linear scaling of TCR, because of accumulating, lagged T increases with time. But how much more I’m not sure, though I think I could make a rough guess if I crunched some more numbers. That would seem to easily get me to, or beyond the AR5 delta T midpoint estimates…