I wanted to mention a Thomas Stocker paper called The Closing Door of Climate Targets. I came across it thanks to Paul Price, on Twitter. It’s an attempt to illustrate what we would need to do in terms of emission reductions to achieve certain targets, and what targets become unachievable if we wait too long. It’s quite a nice little calculation since it is quite simple, but one should always be a little careful given the uncertainties in climate sensitivity and carbon cycle feedbacks.
The basic idea is to assume that our emssions grow at some rate until a time and then are reduced at a rate . One can therefore express the emissions as
where is the emissions at time , is the emissions today (about 9.3 GtC/yr), is taken in Stocker (2013) to be 1.8%/year, and is the rate at which emissions are reduced. The next step is to determine the cumulative emissions, because the amount of warming depends almost linearly on cumulative emissions. The cumulative emissions are
which can be solved to give
where is the cumulative emissions today (530 GtC).
The equilibrium warming due to cumulative emissions is somewhere between and oC per 1000 GtC (Equilibrium Climate Senstivities of between 1.7oC and 4.8o – if I’ve done the conversion correctly. I also think this includes some carbon cycle feedback uncertainty). To get the equilibrium warming you then use
You can also estimate an effective minimum temperature limit by assuming that the fastest emission reduction rate would fix the cumulative emissions at the level when . I think there is a slight subtlety that the paper doesn’t quite get; if emissions were halted completely, then atmospheric concentrations would drop and warming would largely cease; warming to equilibrium requires reducing emissions to the level where atmospheric concentrations are fixed. This, however, requires such a low level of emissions, that it is probably a reasonable representation of the minimum level of warming. The result is
So, now you have a set of fairly basic equations that you can use to determine relationships between rates of emission reduction, start year of emission reduction, and climate sensitivity. For example, the figure on the right shows the contours of peak warming plotted against start year and rate of emission reduction, assuming oC per 1000 GtC, and %. Even now, to keep warming to 2oC would require reductions of > 3% per year. Beyond 2030, it becomes virtually impossible.
You can also consider the rate of emission reduction that would be required to meet a particular target. This is shown in the left hand panel of the figure below. For example, waiting till 2040 before starting to reduce emissions, would require annual reductions of 10% to keep warming below 2oC. It also shows what isn’t possible. For example, beyond 2050, 2oC is no longer possible. The right panel shows the minimum achievable target, for different values, and for different rates of emission increase .
There are of course a lot of parameters that one can play with. You can try different climate sensitivities, , and different rates of emission increase, . If you want to consider transient, rather than equilibrium, response then use a range for of between 1oC and 2.1oC per 1000 GtC. Most of the numbers used here are probably reasonable middle values, so one should probably think of the results here as giving us about a 50% chance of keeping warming below some level. The main point, though, is that the longer we wait, the more extreme the emission reductions will need to be if we realise that we should be aiming to keep warming below some preferred level, and the more likely it becomes that some targets will effectively be no longer possible.