Sunday, April 21, 2013

How much will the Earth's temperature rise?

Climate sensitivity: how has the Earth responded in the past?

In debates about the change of temperature of the Earth that we can expect in the future, it is illuminating to take a backward glance at the past changes of the temperature of the Earth. How sensitive is the Earth to the buildup of greenhouse gases?

The climate sensitivity is the rise in temperature divided by the forcing (in W/m2). In the absence of any feedback, the climate sensitivity can be shown to be 1/(4 SIGMA T3), where SIGMA is the Stefan-Boltzmann constant, (SIGMA = 5.67 x 10-8 W/M2K4), and the temperature is in Kelvin.

Numerically this is a sensitivity of 0.3 K/(W/m2), meaning that if the increase in forcing is 1 W/m2, then the resulting increase in temperature is 0.3 C, in the absence of feedback, when the Earth has come to a new equilibrium with the higher greenhouse gases.

The doubling of CO2 from pre-industrial levels will produce a forcing of 4 W/m2, which implies, in the absence of any feedback, a temperature rise of

4 x 0.3 = 1.2 C (the no-feedback result).

While global climate models are necessary, it is also valuable to have a model-independent estimate of the climate response to increased CO2. What climate sensitivity did the Earth show during warming that ended the last ice age? This is the large climate change that is closest to us in time (about 10 K years ago), and therefore the most valuable. Earlier times are less relevant, because of continental drift. If you go back 200 million years ago, the continents were not even close to where they are today.

The change in temperature between the ice age and post-ice age was 5 C, and the change in solar forcing was 7.1 W/m2. This implies a climate sensitivity of 5/7.1 = 0.7 K /(W/m2). Multiplying this sensitivity by the forcing expected from a doubling of CO2 from pre-industrial levels, namely 4 W/m2, yielding a predicted temperature rise of

0.7 x 4 = 2.8 C (from the Earth’s climate record).

This is consistent with the IPCC prediction of the rise in temperature (in response to a doubling of atmospheric CO2), which is an increase in the range of 1.5 to 4.5 C. This is the change between the new higher equilibrium temperature and the past equilibrium temperature.

Notice that the paleoclimate data implies that the feedback is positive, at least on the time scales of centuries to millennia. (The feedback may well be positive on a much larger time scale of millions of years, but that is not the relevant time scale for human civilization).

Reference: Seinfeld and Pandis, Atmospheric Chemistry and Physics, (Wiley, New York, 1998).pp. 1102-1103. These authors cite a difference between glacial and interglacial periods of 5 C, and a forcing of 7.1 W/m2, and a radiative forcing for doubled CO2 of 4 W/m2.

This refutes some of the arguments made at WUWT here .


Ron Broberg said...

The post-industrial temp will rise 2.8C

With methane feedbacks, maybe as high as <a href=">4.8C</a>

This is based on a climate sensitivity of 3C and my estimates of fossil fuel use. If Ts is more like 2.5C, you can knock that down to more like 2.3-3.8C (est).

Anonymous said...

"Numerically this is a sensitivity of 0.3 K/(W/m2), meaning that if the increase in forcing is 1 W/m2, then the resulting increase in temperature is 1.2 C, in the absence of feedback, when the Earth has come to a new equilibrium with the higher greenhouse gases."

I think you meant 0.3 C from a 1 W/m2 increase in forcing.

Also, "While global climate models are very useful, it is valuable to have a model-independent estimate of the climate response to increased CO2" seems like a bit of a non-sequitur: you provide the no-feedback sensitivity in the previous sentence, and the paleo sensitivity in the subsequent sentence, so this "climate models are useful" thing is out of place. It would make more sense if you added a sentence giving the range of sensitivities given by models before this one. Nor do I think of models as the primary source of climate sensitivity estimates, except inasmuch as they are used in conjunction with historical data...


ps. But nice article: it is always good to get summaries like this.

Anonymous said...

Though... as I look more closely... I'd also rephrase "the change in solar forcing was 7.1 W/m2". Going to the IPCC Paleoclimate chapter I find their statement a little more clear: "Overall, the radiative perturbation for the changed greenhouse gas and aerosol concentrations and land surface was approximately –8 W m–2 for the LGM" - because there wasn't really a net global solar forcing change, there was just a redistribution due to changes in axial tilt, precession, and orbital eccentricity. And in general - sourcing numbers is good. I don't have my copy of Seinfeld & Pandis right next to me, or I'd check and see if your numbers all came from them (it would also be good to note that there is uncertainty in both the forcing and in the temperature change). Note, also, that it isn't always easy to distinguish forcings and feedbacks - in this case, all 8 W/m2 could be considered to be either forcing or feedback depending on the question asked...


ps. One last typo: you have an extra parenthesis in your definition of SIGMA.

James Annan said...

The LGM to pre-industrial temperature change was 4C, not 5C.

Source: me.

Lars Karlsson said...

I have long wondered what counts as 'forcing' and 'feedback' during the ice age transitions, because one should make the same categorisation when transferring CS calculations from these times to current conditions. Obvioulsy, change in albedo due to change in ice sheets is the most important factor, and I take it is treated as a forcing in Elis calculation ("change in solar forcing was 7.1 W/m2"). But when talking about present conditions, it is typically considerad as a feedback, right?

(When you make comparisons using GCMs, that distinction should not be important though).

Martin Vermeer said...

MMM, don't forget the ice-sheet albedo perturbation. Figure 6.5. I assume this is taken as part of the forcing here

Anonymous said...

Martin: Yes, the ice-sheet albedo is part of the 8 (it is the "land surface" in my quoted sentence).

On forcing vs. feedback: clearly, if we were to treat the ice sheets as a feedback (perhaps using only the CO2/CH4 change as a forcing), we would overestimate current climate sensitivity because there just isn't that much snow and ice left to lose. On the other hand, if we treat ice sheets solely as a forcing (which is a common approach) we might underestimate CS because there still is _some_ snow/ice albedo feedback.

And heck, by definition, current climate sensitivity does not take into account carbon cycle and methane feedbacks because, well, that could be complicated and confusing.


Paul S said...

Since James Annan has commented I'll point out a similar simple calculation appeared in the summary of the Annan and Hargreaves 2013 paper on LGM temperature reconstruction. They used their global temperature estimate of 4 +/- 0.8ºC and forcing range 6-11 W/m^2 to get median 1.7ºC, 95% 1.2-2.4ºC.

2.8ºC lies very much at the uppermost boundary when incorporating recent estimates for forcing and temperature change in this calculation.

However, the problem with this model (and it is a model) for finding sensitivity is that it assumes all forcings are alike, which they aren't. There's no reason to expect a 3W/m^2 ice sheet forcing applied only to NH high latitudes would change large scale temperatures with the same "efficacy" as 3W/m^2 forcing from WMGHGs, spatially biased towards the tropics. It's very likely that taking forcing efficacy into account would result in a lower value for forcing, possibly much lower.

Adopting a plausible efficacy-adjusted forcing range of 4-10 W/m^2 and using the 4 +/-0.8ºC LGM estimate gives a full sensitivity range of 1.2-4.4ºC, which agrees well with recent published estimates using GCM-data comparisons.

John said...

Thanks to MMM for catching some typos. In response to comments, I now cite the page numbers in Seinfeld and Pandis, who in turn are typically citing the IPCC (1995). The point about this estimate is that the range of predictions from the sophisticated computer models are consistent with the paleoclimate data. Deniers who claim that the feedback is very small, or negative, are inconsistent with the paleoclimate data.

Anonymous said...

Going to the original WTFUWT article, I'd argue that there are 3 critical flaws in the Happer pieces:

1) Prof. Happer has forgotten that other greenhouse gases exist, thinking that all the forcing has to come from CO2

2) he claims that Joe Romm thinks that 6 degrees will happen by 2050, despite linking to the article where Romm talks about how the 2050 was a mistake made by a journalist which Romm specifically corrected in his own article to be 2100).

3) Using a baseline CO2 concentration of today's 400 ppm, rather than the preindustrial 280 ppm. This is wrong twice: once because the original 6 degree estimate was based on preindustrial, and secondly because even if the projection was based on today's temperature, we aren't in equilibrium today, so you'd need to subtract out the "warming in the pipeline" (constant concentration commitment) before calculating how much additional GHG you'd need.


ps. Why do I ever bother to read WTFUWT? It is negative utility and information.

Anonymous said...

If you differentiate the Stefan-Boltzmann equation (which is where the "bare" sensitivity comes from), you get 4 sigma T^3

Sensitivity is 1/ (4 sigma T^3)

ie, T^3 instead of T^4


KAP said...

Further, most sources (i.e., IPCC TAR) agree that doubling of CO2 results in 3.7 W/m², not 4. So putting it all together (along with the corrections above) one might argue that climate sensitivity is not
(5/7.1)*4=2.8, but rather (4/8)*3.7=1.85. Still a lot larger than 1, though.

John said...

Dear Anonymous:

I stand corrected AGAIN (!!). Sensitivity (no feedback) is
1/(4 SIGMA T^3). This has been fixed in the essay.

Dear KAP
If you calculate a temperature rise of 1.85 C that is also consistent with the range of estimates by the IPCC, namely 1.5 to 4.5 C. My point is that the paleoclimate record gives an estimate of the Earth's sensitivity that supports (does not refute) the IPCC estimates. which are based on large-scale computer models. This in turn enhances confidence in the model.

Dear MMM: Happer makes a number of mistakes in talking with Watt.