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A famous mathematician wrote me, saying:

Hi John, Usually I’m asking you a math question but I’ve heard you sent a couple years on climate issues so perhaps I can ask a question in that subject as well. There is a number that seems very relevant to the climate discussion which I have never heard: The expected residence time or half-life of a molecule of CO_2 once released into the atmosphere.

I read statements like “most of the worlds fossil fuel reserves must never be burned”. If one takes “never” to mean “not in the next 500 years” then this statements seems right if the half-life is long, say >100 years. On the other hand if the half life is short, say <10 years, then it seems a question of limiting production of CO_2 to the absorption rate. I realize this is not quite as simple as a single number because , for example, CO_2 is absorbed in building coral reefs and if climate change destroys these reefs it will lower the absorption rate. On the other hand hire CO_2 could plausibly expand forests increasing the absorption rate. But setting these finer points aside, do you know the current ½-life of CO_2 in the atmosphere?

I replied:

Great to hear from you! The question of the half-life of CO2 is a bit complicated for two reasons.

First, an individual molecule of CO2 will leave the atmosphere and go into the ocean or plant material fairly soon: perhaps about 3-5 on average. But these molecules then come back into the atmosphere fairly rapidly too, since plants respire - they emit CO2 when they metabolize, just like us - and oceans both absorb CO2 and emit it.

So, the relevant quantity is not the mean time an individual molecule of CO2 stays in the atmosphere before it is first absorbed. A more relevant quantity is this: if we emitted a pulse of CO2 into the atmosphere, say a gigatonne, what fraction of this gigatonne would remain in the atmosphere as a function of time?

Second, this function does not decay exponentially over time, so the concept of half-life is not an adequate way to summarize what happens. Different processes occur at radically different time scales - quick processes on land, slower absorption by seawater, even slower absorption by deeper layers of the ocean, even slower conversion into limestone.

It is rather complicated to estimate this function, but here's what some experts think:

Very roughly: once carbon dioxide is put into the atmosphere, about 50% of it will stay there for decades. About 30% of it will stay there for centuries. And about 20% will stay there for thousands of years

This is the sense in which "most of the worlds fossil fuel reserves must never be burned".

The other aspect, of course, involves how much carbon there is down there, available to be burned. For now I'll just say there's a lot.

Here is some more information on the CO2 lifetime issue, from the "Azimuth Project":

http://www.azimuthproject.org/azimuth/show/Carbon+is+forever

It's also worth noting that the "impulse response" described in this graph (mathematically a Green's function) is good only in a linear approximation. As you noted, it's possible, even likely, that we are heating up the Earth enough to push certain quantities into nonlinear regimes - "tipping points" - that may affect the rates at which carbon is processed. These tipping point issues are quite hard to predict, though hundreds of scientists are trying.

Best, jb

## Comments

John, The Green's function for CO2 sequestering is a diffusional impulse response function. And a diffusional impulse response is definitely not characterized by a first-order damped exponential. Diffusion invariably generates fat-tail responses.

The paper ("Atmospheric Lifetime of Fossil Fuel Carbon Dioxide") cited on the Azimuth wiki mentions this

Incidentally, there is one trillion $$$ industry that depends on a fat-tail property to manufacture their products. That is the semiconductor fab industry and how they control the diffusion of dopants into silicon wafers. If you draw a schematic of how dopants are diffused from the gas phase into a heated solid, it will look exactly like the flow of gaseous CO2 into the oceans. And the math of the dopant diffusion is necessarily characterized to a high degree because the manufacturing process demands it. So it is exceedingly well understood.

I have explained this analogy of dopant diffusion and CO2 sequestering diffusion many times, but all I get is some blank stares in response. Of course, when I worked at IBM Research and did mathematical modeling of diffusion, I didn't have that problem :) They seemed to appreciate the math. Perhaps if we explained this analogy to every engineer working in Silicon Valley today, we could make AGW believers out of every last one of them :)

The fundamental mathematical physics never changes, its only the application that changes.

`John, The Green's function for CO2 sequestering is a diffusional impulse response function. And a diffusional impulse response is definitely not characterized by a first-order damped exponential. Diffusion invariably generates fat-tail responses. The paper ("Atmospheric Lifetime of Fossil Fuel Carbon Dioxide") cited on the Azimuth wiki mentions this > "Common measures of the atmospheric lifetime of CO2, including the e-folding time scale, disregard the long tail. Its neglect in the calculation of global warming potentials leads many to underestimate the longevity of anthropogenic global warming." Incidentally, there is one trillion $$$ industry that depends on a fat-tail property to manufacture their products. That is the semiconductor fab industry and how they control the diffusion of dopants into silicon wafers. If you draw a schematic of how dopants are diffused from the gas phase into a heated solid, it will look exactly like the flow of gaseous CO2 into the oceans. And the math of the dopant diffusion is necessarily characterized to a high degree because the manufacturing process demands it. So it is exceedingly well understood. I have explained this analogy of dopant diffusion and CO2 sequestering diffusion many times, but all I get is some blank stares in response. Of course, when I worked at IBM Research and did mathematical modeling of diffusion, I didn't have that problem :) They seemed to appreciate the math. Perhaps if we explained this analogy to every engineer working in Silicon Valley today, we could make AGW believers out of every last one of them :) The fundamental mathematical physics never changes, its only the application that changes.`

WebHubTel wrote:

Yes, that's basically what I said. I didn't assert that this

invariablyhappens. It's bound to happen when there are processes occurring over a wide range of time scales, and that's pretty common.`WebHubTel wrote: > John, The Green's function for CO2 sequestering is a diffusional impulse response function. And a diffusional impulse response is definitely not characterized by a first-order damped exponential. Diffusion invariably generates fat-tail responses. Yes, that's basically what I said. I didn't assert that this _invariably_ happens. It's bound to happen when there are processes occurring over a wide range of time scales, and that's pretty common.`