As requested by @richardbetts and @edhawkins here is a simple model to back up the arguments from the previous post – Stabilising Climate.
CO2 levels rise when the rate of change of the sources – S exceeds the rate of change of sinks – K. Without human emissions then S = K, when averaged over one year. However with ever increasing human emissions the situation becomes dynamic
If C is the yearly value of CO2, S the net sources of CO2 and K the net sinks, then at time t.
However it has been measured for at least the last 60 years that
Now let’s assume that the world manages to stabilise annual emissions at current rates of 34 Gtons CO2/year indefinitely. CO2 sinks currently absorb roughly half of that figure – 17 Gtons and have been increasing proportional to the increase in partial pressure of CO2 in the atmosphere – currently that of 400ppm. Stabilising emissions now results in a decreasing fractional uptake by carbon sinks as the partial pressure imbalance between the surface and atmosphere begins to fall. The simplest assumption is that the sink increase depends only on the partial pressure difference for a given year. Therefore if this pressure difference is reduced by half in one year then the next year it will be reduced by one quarter, then one eighth and so on. The same argument applies for the case that it takes longer to reduce pressure difference by a half.
Year 1: 50% Year 2: 25% Year 3: 12.5% Year 4: 6.25% etc. which is simply equal to the infinite sum
So in this simplest of models, CO2 levels in the atmosphere will taper off after just ~10 years to reach a new long term value equivalent to adding an additional one year of emissions 34 Gtons of CO2 to the atmosphere. The atmosphere currently contains 3.13 x 10^12 tons of CO2 so the net increase at equilibrium would in this simple model be just 1%. Therefore for the years following 2016 the resultant CO2 curve would look like the red curve below. If instead it takes say 4 years for the sinks to increase by then we get the blue curve. In this case it would take 30 years for CO2 levels to to stabilise and the increase would be 5 times larger.