There are numerous feedbacks in the climate system, and the feedbacks which affect global temperature are overwhelmingly positive. One example is water vapor feedback: increased temperature leads to greater absolute humidity in the atmosphere, and water vapor is a potent greenhouse gas, so this further increases warming. Another is ice albedo feedback: warmer conditions reduce snow and ice cover, which are highly reflective of incoming sunlight; when snow and ice are replaced with open land and ocean (which are far less reflective), earth’s albedo decreases, so the planet absorbs a greater fraction of incoming sunlight and again we experience further warming.
Suppose warming of amount 1 deg.C leads to a further warming of deg.C; we can call the feedback factor. Then we expect this additional warming of also to trigger the feedback mechanism, so we’ll see yet more warming, in the amount . This additional warming also triggers the feedback mechanism, causing yet more warming in the amount , and so on and so on, etc., etc. The total warming is therefore
Thus if the basic feedback factor is , then the total increase due to feedback will be different. We can call this the gain due to feedback, so
If there were no feedbacks in the climate, then we can estimate climate sensitivity with considerable precision: each increase in climate forcing of 1 W/m^2 (watt per square meter) would lead to temperature increase of between 0.3 and 0.31 deg.C. Another way to state climate sensitivity is that doubling CO2 concentration leads to an increase in climate forcing of about 4 W/m^2, so would lead to about 1.2 deg.C temperature increase. We can call this the no-feedback climate sensitivity to doubling CO2, or we can just call it . With feedbacks, this change will be multiplied by the gain factor , so the actual climate sensitivity to doubling CO2 will be .
A recent post on RealClimate discusses a new paper by Roe and Baker about why estimates of the probability distribution for climate sensitivity are so asymmetric; they have a long “tail” at the high end with small but not insignificant probabilities for very high values, but almost no probability of very low values. The fundamental reason is that we don’t really estimate the total feedback, i.e., the gain, , what we really estimate is the feedback factor .
Suppose we know the probability distribution for the feedback factor . This is a function such that the probability that lies in a small interval of width , centered on the value , is
What will be the probability distribution for the gain factor? It’s straightforward to compute that if we know the probability distribution for one variable (in this case, ), we can compute the probability distribution for another variable (in this case, ) which is simply a transform of the original variable:
where is the derivative of with respect to . Since we know that in this case , we get
There are two important consequences of this. One is that if the probability distribution for is symmetric, like for example the normal distribution, then the probability distribution for will not be. In fact it will have a long tail on the high end, with small but not insignificant probability of very high values, but almost no chance of very low values. Another is that if the probable accuracty of our estimated feedback factor is , then the probable accuracy of our estimated gain factor will be greatly magnified, approximately given by
An example may clarify. Suppose the probability distribution for is approximately the normal distribution, with mean value and standard deviation . Then the probability distribution for looks like this:
It follows that the probability distribution for climate sensitivity looks just like it (after all, is just a multiple of :
In this graph I’ve added dashed lines indicating the usual range of likely values quoted in the press, from about 2 to 5 deg.C per doubling CO2. There’s a striking difference between the tails of this distribution; the high end shows nontrivial likelihood extending even to very high values, while on the low end the probability quickly drops to near zero. There’s another difference too: the “spread” of the probability distribution, which is the basic uncertainty in our estimate of , is much greater than the spread in our distribution of . For this case, the uncertainty 0.14 in translates to uncertainty of approximately 1.6 in , and that translates to uncertainty of about 1.9 deg.C in climate sentivity .
It’s in the nature of things that methods of estimating feedback actually amount to estimating the feedback factor . The gain and sensitivity are then derived from this. It’s also in the nature of things that probability distributions which emerge from our estimates of are roughly normal. Hence the probability distributions for and will always have a bigger, longer tail at the high end, and much more uncertainty, than the distribution for . Even if we can estimate the feedback factor with reasonably good precision so is low, the precision in will be not so good.
Roe and Baker’s analysis has implications for policy choices, above and beyond its scientific interest. It has been suggested that it’s not really a good idea to set a “hard target” for limiting CO2 concentration (such as the sometimes-suggested 450 ppmv), because if sensitivity turns out to be substantially greater than 3 deg.C (an oft-quoted average value) then that goal won’t be good enough to avoid extremely dangerous climate change. Instead targets should be flexible, so that if sensitivity turns out to be higher than expected we can adjust the target to keep climate change in the less-dangerous range. However, we’re rapidly approaching the point at which it will be no longer realistic to aim for such targets. CO2 levels currently are about 382 ppmv, and rising at about 2.1 ppmv/yr, which if sustained will bring us to the 450 ppmv level right around the year 2040 (perhaps in my lifetime). And since CO2 emissions are increasing, and there are signs that the ability of oceans and biosphere to absorb part of the increased CO2 from human emissions, this would seem to be an optimistic forecast.
It also should be taken as a stern warning. It really is possible, even if unlikely, that climate sensitivity is on the high end of estimated ranges, or higher. If that’s true then we’re headed for real trouble. For me, this emphasizes the urgency of the problem. It’s not something to leave for the next generation, or even the next decade. We need strong action to limit tampering with the climate, and we need it now.