I've seen estimates ranging from a few years to hundreds of years so there certainly doesn't seem to be a consensus about this. These estimates also seem to be passing comments rather than details on how they reached the estimate, as an example, an editorial by Richard Kerr in January’s Science magazine states:

Carbon dioxide will remain in the atmosphere for centuries, warming the world all the while, ...

There does not appear to be any references for this statement, although I can not be sure because I did not get the quote directly from the pay-walled editorial, but from a secondary source (WUWT).

Before I give my calculations on this here are some assumptions I am going to make:

1. CO2 levels in the atmosphere were relatively stable at 278ppm in pre-industrial times.
2. The mass of CO2 in the atmosphere at 278ppm is 2173Gt
3. 771Gt of CO2 are released into the atmosphere from natural sources every year.
4. The current level of natural CO2 released into the atmosphere is the same as in pre-industrial times.

The final assumption I am going to make is that CO2 removal from the atmosphere follows an exponential decay, in other words has a half-life. This isn't unusual as there are many things that have a half life. Some examples (mainly from wiki):
a) Radioactivity.
b) Heat transfer.
c) First order chemical reactions.
d) Electric charge in a capacitor.
e) Many drugs have a biological half-life in the bloodstream.
f) The water level in a leaky bucket.

In order to calculate the half-life we need to know what the level of CO2 is when the amount entering the atmosphere is the same as the amount leaving it (stable point) and we need to know the amount entering and leaving. According to conventional wisdom (the IPCC) this stable point is at 278ppm or 2173Gt. The IPCC also say that the amount of CO2 from natural source entering and leaving the atmosphere is 771Gt.

According to my bit of calculus, which I wont show because it is very difficult to show it in a comment, the formula for calculating the half life is:

h = S * ln(2) / E
where:
h = half-life
S = Stable amount of CO2 in the atmosphere
E = CO2 entering/exiting the atmosphere every year.
ln(2) = natural log of 2

Plugging in the values we get a half life of 1.95 years.

What this means in real terms is that if all the extra 110ppm (278+110=388) is due to human factors then if we stop producing CO2 it would only take 6 years for the level to drop below 300ppm.

Of course, the half-life might vary. For example, a change in ocean temperature will change the rate which it absorbs CO2. Remove a carbon sink from land and the half life will change. But still, it has a long way to go from the ~2 years I've calculated and the figure of several centuries given by Science magazine.

Does any body else have any thoughts on this?

Hi, Terry,

A couple of years back I did a little exercise based on the monthly CO2 PPM figures from the Mauna Loa Observatory. I was trying to derive a half-life based.

The starting position was this:
a. There's an annual ripple in the ever-rising PPM, from 312 PPM in 1958 to c392 in 2010.
b. Every year in Northern summer there's a decline, peak gradient between July and Aug.
c. At the start, in 1958, the peak decline rate was 0.572% per month (1.8 PPM on 312)
d. By 2009, this was an almost unchanged 0.561% per month (2.15 PPM on 385)

Now, here's a (slightly bonkers) thought experiment which I don't mean to be taken too literally: If the conditions which cause that summer decline were to be made permanent, and that 'great sucking sound' of CO2 leaving the atmosphere every August were to operate all year round, what would the graph do? (Locking the Earth's orbit in the August position would be a big ask of even the most enthusiastic geo-engineering project!) I derived an exponential decay curve. That curve is, of course, a simplification of the pesky Real World that (tut!) always makes things more complicated.

Anyway, I get a half life of 122 +/- 2 months.

My estimate is that in August CO2 is leaving the atmosphere at a net rate of 150GT per annum (that is twelve times 0.57% of 2173GT per month).

The simple formula I used was N(t) = N(o).e^kt, where
N(t) is the atmospheric concentration at some future month
N(o) is the starting point such as today's 390-odd PPM
e is the base of natural logarithms, 2.718
k is the constant derived above, i.e., 0.0057
t is the number of months from starting point.

Now, none of this yields a residence time for individual CO2 molecules - I wasn't interested in that. Neither does it say anything about where the graph is headed in future decades. Nor does it say anything about the sources of additions and subtractions. But it DOES show that the now-withdrawn statement by the Royal Society (or rather some numpty in their press office) that "Once our actions have raised concentrations of CO2 in the atmosphere, levels will remain elevated for more than a thousand years” is balderdash.

It's just occurred to me that "the sources of additions and subtractions" cuts to the heart of the matter - the assertation that manmade CO2 is upsetting nature's balance. Have you by any chance seen a CO2 equivalent to the gorgeous energy balance diagrams which so beautifully show what happens to that 1361 W/m2 coming in from the sun? Such a visualisation might be a good foundation for an attack on the CAGW mob's position, and focus minds on the (to my mind) reasonable conjecture of a 900-year lag between climate changes and ocean outgassing.

Hi Brian,

I think you have a typo in the equation, it should be:

N(t) = N(o)/e^kt

Otherwise N(t) would always be greater than N(o).

If h is the half life time then:

N(t) = N(o)/2 when t = h

Therefore:

N(o)/2 = N(o)/e^kh

Which resolves itself to:
h = ln(2)/k

To arrive at your figure of 122 you have assumed that the net rate of loss is between 1.8ppm in 312 to 2.15 in 385 or approximately 0.56 to 0.58%. This takes no account the CO2 being added to, and removed from, the atmosphere during August.
If you take into account how much is added per month from natural sources* (8.2ppm) then those percentages change to between 2.63 and 2.81% which gives a half life of 25/26 months. Of course, it is unrealistic to add the entire 8.2ppm (otherwise the ppm would not drop in August) and the effect of reducing the 8.2 would be to increase the half life.

* The 8.22 does not include human contributions. If human contributions are included then the half life will decrease.