Diamonds - and CO2 - are forever II (wonkish and long)

I have been thinking a lot about David Archer's models for the long-timescale chemical reactions of CO₂ and the its radiative forcing integrated over its entire lifetime. In particular, I was interested in the dynamics described in this figure reproduced from his book, The Long Thaw.

 The caption says a great deal: "Model simulation of atmospheric CO₂ concentration for 40,000 years following after a large CO₂ release from combustion of fossil fuels. Different fractions of the released gas recover on different timescales."

In the original Archer model[1-4], an instantaneous pulse of CO₂ is injected into the atmosphere at time t₀ and is chemically removed through interactions with the oceans, silicate rocks, and igneous rocks; each of these reactions is modeled with first-order kinetics but with different lifetimes.

This model can be used to address two questions:
(1) What is the ratio of the total accumulated energy due to global warming to the energy of combustion?

ΔF = α ln(C/C_o) [see, for example, NOAA AGGI]

[The time required for this energy to equal the heat of coumbustion combustion [OG: Maybe someday I will start believing that spell checker] for a given fossil fuel was discussed in Onymous Guy's earlier post, Diamonds - and CO2 - are forever]
  
(2) What is the effect of the logarithmic nature of the forcing function on the amount of total accumulated energy due to global warming added to the earth's energy budget, in particular, does it level off?

Here is the mathematical development:
(Apologies ahead of time, but these are not text, but images from a Mathematica notebook obtained via Grab. I am open to suggestions on improvements in capturing these sorts of images.)

Let's take a look at this pulse function when atmospheric CO₂ is instantly doubled:

The different decay processes are a little more visible in a plot logarithmic in time.

This plot covers a million years; clearly, if this model is at all accurate, one must conclude that although most CO₂ is removed from the atmosphere within a few hundred years, a significant fraction persists in the atmosphere for times that are long on the geological time scale.

Additional Questions
What about the lifetime integrated radiative forcings as the magnitude of the initial CO₂ pulse is increased? What effect will dominate, the very long lifetime of 25% of atmospheric CO₂, or the plateau due to the logarithmic increase in forcing?

Let's compare the exact (equation (9) ) and approximate (linearized, as in equation (12) ) lifetime integrated radiative forcings:

As you can see, there is little difference between the exact and linearized forcings, suggesting that the effect of the logartihmic function is negated by the fraction of atmospheric CO₂ which persists on the geological time scale.

The ratio of the approximate forcing to the exact term is shown in the next graph.

We observe that radiative forcing is very nearly linear in terms of the magnitude of the initial pulse of CO₂; furthermore, the logarithmic term reduces radiative forcing by less than 6% relative to the linear term even up to a pulse above ambient pressure up to 800 ppmv CO₂, a net amount nearly three times the amount in the atmosphere today.

The orange curve in the above figure has a slight curvature, so we investigated the effect of linear, quadratic, and cubic fits to a sequence of data separated by 10 ppmv.

Here is a graphical comparison of all three models with the exact result.

It is difficult to distinguish the exact result from the quadratic model except for points well away from the center of the fitting region.

A Final Remark
The integrated "pulse function" g(t) (equation (11) above) provides an effective lifetime for atmospheric CO₂ of 17303.3 yrs - as far as radiative forcing is concerned. This allows one to explicitly compare the total energy added to the atmosphere from the enthalpy of combustion of a fossil fuel with the accumulated energy due to global warming arising from the CO₂ added to the atmosphere through combustion. The following table summarizes, by fuel type, the ratio of lifetime-accumulated radiative forcing to the enthalpy of combustion; this ratio is dimensionless.

Conclusions
(1) As one can see from the above table, the effects of global warming are ten thousand times larger than that of combustion, and twice that for coal and coal products such as coke.
(2) The effects of global warming over the lifetime of CO₂ are very nearly linear in the amount of CO₂ added to the atmosphere.

The effects of carbon combustion are nearly eternal, as David Archer put it. And the addition of carbon dioxide to the atmosphere is an unforgiving and relentless burden, with no cushion at all. The more carbon that is burnt, the worse off we will be.

I would call this a very strong argument for decarbonization of the economy.

On Models
The Archer model is just that and the conclusions discussed above should not be oversold; I think this quote from his 2005 work is quite relevant:

However, the 300 year simplification misses the

immense longevity of the tail on the CO₂ lifetime, and

hence its interaction with major ice sheets, ocean methane

clathrate deposits, and future glacial/interglacial cycles. One

could sensibly argue that public discussion should focus on

a time frame within which we live our lives, rather than

concern ourselves with climate impacts tens of thousands of

years in the future. On the other hand, the 10 kyr lifetime of

nuclear waste seems quite relevant to public perception

of nuclear energy decisions today. A better approximation

of the lifetime of fossil fuel CO₂ for public discussion might

be ‘‘300 years, plus 25% that lasts forever.’’

References
[1] Multiple timescales for neutralization of fossil fuel CO₂
Archer, David;  Kheshgi, Haroon;  Maier-Reimer, Ernst (1997),
Geophysical Research Letters,vol. 24 (4) p. 405-408, doi:10.1029/97GL00168.

[2] Archer, D. (2005), Fate of fossil fuel CO₂  in geologic time, J. Geophys. Res., 110, C09S05, doi:10.1029/2004JC002625.

[3] Archer, D. The Long Thaw: How Humans Are Changing the Next 100,000 Years of Earth's Climate (Princeton Univ. Press, 2008).

[4] Inman, M. (2008), Carbon is Forever, Nature Climate Change, 2008(12), doi:10.1038/climate.2008.122.

Supplement

Onymous Guy supposedly features "The odd ejecta from around the world". In that spirit, here is the lurid cover to "Diamonds are Forever", a slogan that Onymous Guy hopes surely galls The Sage of Corbett. And here are two other lurid covers, because Kurt Vonnegut, Jr. - who almost certainly would have had no patience with the denialisti - used this device in his novels. The Sirens of Titan remains one of my favorite novels. This novel also introduced me to that Vonnegut character, Winston Niles Rumfoord. Venus On The Half Shell acquired a life of its own outside of Vonnegut's novels, although it was supposedly one of Kilgore Trout's novels, and was actually written by Phillip Jose Farmer, another author I admired.

From Wikipedia

Trout, who has supposedly written over 117 novels and over 2000 short stories, is usually described as an unappreciated science fiction writer whose works are used only as filler material in pornographic magazines.

Hence the lurid covers.