In my first climate-related blog I talked about why I was so angry that the world wasn’t taking action to prevent climate change – 10 years ago the wedges concept, introduced by Pacala and Socolow , gave a clear prediction of how our annual CO₂ emissions were increasing and identified ways to reduce emissions. Back then, the world was still thinking in terms of how CO₂ *emission rates* affect climate change. Since then we have realised that emission rates are relatively unimportant and that global cumulative CO₂ emissions are what we need to keep an eye on.
This is because CO₂ has a long residence time in the atmosphere – if we were to completely stop burning fossil fuels and emitting CO₂ today, it would take up to 35 thousand years for all of the CO₂ we emitted since the industrial revolution to be re-absorbed  (if you want to know more about that, there is a nice blog here). This means that our CO₂ emissions over decades, even centuries, are pretty much instantaneous on the carbon-cycle timescale.
The amount of global warming will be controlled by the amount of CO₂ hanging around in the atmosphere, and, because it hangs around in the atmosphere for so long, that is controlled by the total amount of CO₂ we have emitted, rather than how fast we are emitting it. In 2009, Allen et al  calculated the Earth will warm by ~2 °C for every 3.67 trillion tonnes of CO2 that we emit . If we want a good chance of avoiding global temperature rise of more than 2 °C, the most CO2 we can emit is 3.67 trillion tonnes – total! period! ever!.
Reducing emission rates just delays the 2 °C temperature rise – it doesn’t prevent it. Stabilising emission rates buys us time but doesn’t solve the problem. The only way to solve the problem is to completely stop emitting CO₂.
This next diagram, simplified from the IPCC 2014 report (Figure SPM.5, p9 ), is a nice illustration of how much we can expect the global temperature to rise because of the total amount of CO₂ we emit. The x-axis shows cumulative CO₂ emissions since 1870 in gigatonnes of CO₂. (GtCO₂) The y-axis shows the temperature increase compared to pre-industrial temperatures. There are different ways of modelling the climate response to carbon dioxide emissions, and that is why this graph is plotted as a grey band, rather than a single line – the band represents the range of expected global warming according to lots of different models and calculations. Those different models and calculations all agree pretty well.
According the graph, when we have emitted 3670 Gt CO₂ (3.67 trillion tonnes), we can expect the global temperature to have risen between ~1.4 and ~3.1 °C (black dotted line shows 3.67 trillion tonnes vs 2 ܄C).
The black ellipse labelled “Observed” plots our actual total CO₂ emissions between 1870 and 2005 against the observed warming (as an average of the years 2000 to 2009 compared to an average of the years 1861 to 1880). As you can see, it fits the predicted warming very well.
The red line shows the cumulative total CO₂ emitted as of today , December 23rd 2014 (2150 Gt CO₂) and that, no matter what happens, we are already committed to global warming of between 0.8 and 1.8 °C.
The other ellipses are predictions for the amount of CO₂ we will have emitted in the year 2100 for different CO₂ mitigation strategies, and the amount of global warming we would expect. Figure 3, below, is simplified from Figure SPM5 of the IPCC 2014 report and shows the corresponding CO₂ emission scenarios. The blue scenario (blue ellipse in Figure 2, blue band in Figure 3) represents the likely outcome if we use an aggressive strategy to cut CO₂ emissions and are actually able to completely stop emitting CO₂ and start creating negative emissions – actively taking CO₂ out of the atmosphere faster than nature. The yellow scenario represents a CO₂ reduction strategy that sees us managing to stabilise global CO₂ emission rates at their current levels (i.e. the emission rate doesn’t increase) and then decrease emission rates in about 30 years time. In this scenario, we would still be releasing around 50 Gt CO₂ every year, to start with, and that means we would have spent our CO₂ budget within about 30 years; by 2100 we would have emitted 4-5 trillion tonnes of CO₂ and would see global warming of 2.5 to 3 °C. The orange scenario shows what would happen if we slow the increase in emission rate and stabilise CO₂ emissions in about 3 years time – by 2100 we would have emitted ~ 6 trillion tonnes of CO₂ and would see global temperature rises of 3 – 4 °C. The red scenario represents “business as usual” – what we would expect to happen if we didn’t make any special effort to reduce CO₂ emissions. In this scenario, emission rates would continue to increase over time and we are looking at > 6.5 trillion tonnes of CO₂ emitted with temperature rises of > 4 °C by the year 2100.
In summary, if we want a good chance at avoiding a 2 °C global temperature rise, then we can’t emit more than 3.67 trillion tonnes of CO₂. We have already emitted more than half of that in just the last 145 years. Our global CO₂ emission rates are continuing to rise and if we don’t take action quickly, we will have reached that limit within 30 years.
Notes and references:
 Reference  talks about “the trillionth tonne” – why are you talking about a budget of 3.67 trillion tonnes?
CO₂ emissions can be described in terms of mass of CO₂, or mass of C (i.e. the mass of carbon in carbon dioxide): The element carbon has an atomic mass of 12 and oxygen has an atomic mass of 16. So carbon dioxide has a molecular mass of 44, which is 3.67 times greater than the mass of carbon. Allen’s paper  and the website http://trillionthtonne.org/ both discuss CO₂ emissions in terms of mass of carbon – emitting 1 trillion tonnes of carbon (as carbon dioxide) will produce 2 °C warming. To convert this to mass of CO₂ we just multiply by 3.67. The graphs I used in this blog post used emissions and emission rates for CO₂ rather than just C, so I used CO₂ emissions and emission rates to make things simpler. The 2150 Gt CO₂ I quoted as having been released between 1870 and today was calculated by looking up the real-time cumulative total C-emissions from http://trillionthtonne.org/ (586 Gt) and multiplying it by 3.67.