Many processes in climate and elsewhere have a rate of change linearly proportional to the displacement from equilibrium conditions. This is the linear relaxation model. Many reactions are governed by this or at least can be approximate to this for small excursions. This has many advantages computationally and much of climate science is based on this kind of approximated relationship. The solution of this equation leads to the exponential decays so commonly found natural and chemical processes.

The linear relaxation model of an impulse change leads to an exponential decay. Convolution of the input variable with the appropriate exponential decay “impulse response” provides the net system response to this input variation. This is a standard method in engineering and science and is not controversial.

The annual emission data can be convolved with such an impulse response function to give the accumulation time series.

One property of such a model is that, for a constant rate of increase, the output is the same constant rate of increase delayed by a time lag equal to the time constant of the decay function.

If the CDIAC estimation of global CO2 emissions is processed with a 14 year delay function. The result can be seen to be quite close to the original lagged by 14 years, demonstrating this effect where the rise is roughly linear. The figure of 14 is an example only, this relationship holds for any value.

Approximating the partial pressure difference in uatm as the same number of ppmv and noting the lag of the linear response, the time delay corresponding to the pressure difference is :

p.press difference (ppmv) / rate of change (ppmv/year) = number of years lag

Taking the 7 microatmosphere “average” difference between ocean surface partial pressure and atmospheric partial pressure of CO2 [1] and attributing the current annual increase of 2ppm entirely to emissions, leads to a time constant of 3.5 years for a linear relaxation model.

This does not agree with anyone’s estimation of the reaction response time.

The value of 14 years, estimated from decay of C14 after the end of airborne nuclear testing, would imply a rate of increase of 0.5 ppmv per annum resulting from the absorption of emitted CO2. The remaining 1.5 ppmv/a must therefore be due to out-gassing.

Unless, the true decay function is nearer to 3.5 years than 14.

An explanation of why we cannot use the variation of CO2 and temperature across the last deglaciation ( as is often done ) is given here:

https://climategrog.wordpress.com/?attachment_id=412

Three very thorough papers looking at the detailed kinetic processes are provided by Gösta Pettersson. [2]

They conclude (paper 3) that recent out-gassing is component is 40% higher than residual emissions, compared to 3:1 ration provided by this simplistic analysis.

[1] http://www.pmel.noaa.gov/pubs/outstand/feel2331/exchange.shtml

The average pCO2 of the global ocean is about 7 µatm lower than the atmosphere, which is the primary driving force for uptake by the ocean

[2]

http://www.false-alarm.net/wp-content/uploads/2013/06/paper1.pdf

http://www.false-alarm.net/wp-content/uploads/2013/06/paper2.pdf

http://www.false-alarm.net/wp-content/uploads/2013/06/paper3.pdf

[UPDATE]

Pettersson’s paper have now been revised. The updated versions are :

http://www.false-alarm.net/wp-content/uploads/2014/05/paper4.pdf

http://www.false-alarm.net/wp-content/uploads/2014/05/paper5.pdf