Ce qu'on voit et ce qu'on ne voit pas

Just floating a back-of-envelope calculation about ¹²C:¹³C ratios in different parts of what I am going to call the 'dynamic' carbon cycle: the bit where carbon is moving around a lot, leaving out both the slow deposition of fossil fuels and carbonates and the geological or anthropogenic processes that get them out of the ground again.

I've taken some numbers from one of many pictures of the carbon cycle available on Our Friendthe Interwebz, and shown them to scale the way the ancient Greeks would have, geometrically. The area of the arrows and the squares are to the same scale, showing the amount of carbon in each place and the amount moving from one place to another in the course of a year. There are three populations of carbon in this simplified picture: Biomass (plants and soil), the atmosphere, and the oceans. And the greatest of these is the ocean.

Note that we basically have a fast equilibrium (the atmosphere and biomass equilibrating rapidly, on the rough order of a decade to exchange half the carbon in the biomass) and a slow equilibrium (the atmosphere and the ocean doing the same on the order of a few hundred years).

Before moving on, I just want to say that the overall ratio of ¹²C:¹³C in this whole system should remain pretty much the same for periods of geological time that are long compared the time scales in which lots of interesting things can happen. Good evidence for this is that the amount of variation in carbonates laid down since the beginning of the Pliocene is not very much at all: less than +/- 1 ‰ (Ghosh & Brand 2003).

Now, the relative amount of ¹²C:¹³C in each box will be governed by two things:

(1) The relative amounts of ¹²C:¹³C in the box(es) it is in equilibrium with; and,

(2) Any isotopic selectivity in the transitions between boxes.

Now, the isotopic selectivity due to the more rapid diffusion of  ⁴⁴CO₂ over ⁴⁵CO₂ through eensy-weensy membranes in plants is very well understood: and this generates a clear difference in the ¹²C:¹³C ratio between biomass and the atmosphere, despite the rapid interchange between them.

But, if there was no life on Earth, the atmosphere would still be enriched in ¹²C:¹³C relative to the ocean. Because there is also a solid basis for isotopic selectivity in the ocean:atmosphere equilibrium. Most of the carbon in the ocean is present as hydrogen carbonate ions in the deep ocean. To get to the surface, this material has to run a gauntlet of a layer of warm water hundreds of metres thick where calcium carbonate is stable. I haven't been able to find any decent data on isotopic dependence of hydrogen carbonate ion diffusion rates - just some molecular dynamics simulations that didn't find a significant difference* - but a priori, if you have a column of fluid some hundreds of metres high, surely the bottom of the column is going to be enriched in the component with a molar mass of 62 rather than 61.

Now it seems to be the because of the fast interchange between the biomass and the atmosphere, the relative distribution of carbon isotopes between the ocean and the total (biomass + atmosphere) component has to be governed by the slow equilibrium. And thus the main driver of the isotopic ratio in the atmosphere has to be the slow equilibrium. Yes? If you see a problem with this, let me know.

So: looking back at those boxes. And going back to that number from Ghosh & Brand 2003 and some other numbers. Taking -6 ¹³C ‰ for the pre-industrial recent atmosphere and -25 ¹³C ‰ for terrestrial biomass (Epstein 1969) gives an overall value of -20 ¹³C ‰ for the (biomass + atmosphere) component. Now, if we have a ¹³C ‰ of 0 ± 0.8 for oceanic carbonates since the Pliocene, this means that the error we should reasonably associate with this component should be of order ± 10 ¹³C ‰.

This may be a conservative overestimate: but looking at the size of the boxes where the carbon is sitting, the error in the small boxes has to be bigger than the error in the small boxes. And there is one more small problem, which is that while the ¹²C:¹³C selectivity in plants is well-understood and not likely to have changed much at all since the little fellers evolved their carbon fixation pathways, the ¹²C:¹³C selectivity of ocean:atmosphere transport ought to be dependent on the distribution of the layer of warm water, which will rely on the vicissitudes of global climate and over a longer period of time how the continents are shifting about to change global ocean circulation.

So, my point is: relying on ¹²C:¹³C ratios alone to tell us details about the source of a particular carbon-containing deposit at some remote period of time is probably not a good idea, because there are likely to be big systematic errors which we know not what of.

Perhaps that's not a very big point. 

* And only just found a significant difference for CO₂, where we know one exists experimentally.