Thursday, September 24, 2015

Carbon-14: Origins and Reservoir

This is the first of a series of posts in which we use our knowledge of carbon-14 concentrations to arrive at firm conclusions about the way in which carbon dioxide (CO2) cycles between the atmosphere and the oceans. The implications of the atmosphere's carbon-14 concentration were studied thoroughly and objectively prior to 1960, in papers such as Arnold et al. But these authors did not have available to them the results of the nuclear bomb tests of the 1960s, so their conclusions could not be as firm as ours. The same implications have been studied more recently in work such as Mearns and Pettersson, but these authors did not attend to the rate of production of carbon-14 by cosmic rays, and so did not appreciate the necessary size of the global CO2 reservoir. Modern models of the CO2 cycle are presented in papers such as Archer et al., but these models are contradicted by carbon-14 observations, so they cannot be correct.

Carbon-14 has been produced in our atmosphere by cosmic rays for billions of years. A cosmic ray is an energetic particle arriving from space. Most are protons. Some have energy 1×1020 eV. (The Large Hadronic Collider, for comparison, produces protons with energy 7×1013 eV.) Cosmic rays collide with atmospheric nuclei and produce showers of photons and particles. Among the particles produced are neutrons, and these neutrons can react with nitrogen-14 nuclei to produce carbon-14.

A nitrogen-14 nucleus has seven protons and seven neutrons. During its reaction with a neutron, it ejects a proton but retains the neutron. The result is a nucleus with six protons and eight neutrons, which is carbon-14. The carbon-14 nucleus is unstable. Eventually, one of its neutrons will emit an electron and turn into a proton. The nucleus is once again nitrogen-14. The electron shoots out of the nucleus with energy up to 156 keV. It is called a beta particle, and the decay of carbon-14 is called a beta decay. The decay happens at random, but the probability that any given carbon-14 nucleus will decay each year is 0.012%. If we have one kilogram of carbon-14, there will be only half a kilogram left after 5700 years.

The electrons emitted by carbon-14 decay have sufficient energy to penetrate 50 mm of air. With care, we can measure the concentration of carbon-14 in a sample of air, or in a sample of wood, cloth, or animal tissue, by counting the electrons it produces, and weighing its carbon content. We find that one in a trillion carbon atoms in the atmosphere is a carbon-14 atom. The rest is carbon-12, with one part in a thousand carbon-13.

Almost all the carbon-14 in our atmosphere ends up in CO2 molecules. One in every trillion atmospheric CO2 molecules contains carbon-14. The rate at which cosmic rays produce carbon-14 is of order two atoms per square centimeter of the Earth's surface per second (see Lingenfelter for measurement 2.5 atoms/cm2/s and Kovaltsov et al. for 1.7 atoms/cm2/s). The creation rate varies as the Earth moves through the galaxy, and with cycles of solar activity, but to the best of our knowledge, the creation rate has been constant to within ±25% over the past ten million years.

Because we know carbon-14's rate of decay and its rate of production, which has been stable for at least a million years, we can calculate the equilibrium quantity of carbon-14 on our planet. Cosmic rays produce 2 atoms/cm2/s of carbon-14, so they produce 7.5 kg of carbon-14 every year. (Multiply 2 by the Earth's surface area in square centimeters, the number of seconds in a year, the atomic weight of carbon-14, and divide by Avogadro's number to get the number of grams produced per year.) In the past million years, cosmic rays produced 7.5 million kilograms of carbon-14. But each carbon-14 nucleus has a 0.012% chance of decaying each year, so only a small fraction of this 7.5 million kilograms still exists. Suppose 75,000 kg remained. In the coming year 9.0 kg would decay (0.012% of 75,000 kg) and only 7.5 kg would be created. The Earth's reservoir of carbon-14 would be decreasing at 1.5 kg/yr. Suppose only 50,000 kg remained. In the coming year, only 6.0 kg would decay (0.012% of 50,000 kg) and 7.5 kg would be created. The Earth's reservoir would be increasing at 1.5 kg/yr. The equilibrium size for Earth's carbon-14 reservoir is 62,000 kg (7.5 kg ÷ 0.012%). At this size, the rate at which carbon-14 in the reservoir decays is equal to the rate at which new carbon-14 is added to the reservoir by cosmic rays.

Historically, carbon-14 atoms have been produced exclusively by cosmic rays. But in the 1960s, nuclear bomb tests doubled the concentration of carbon-14 in the atmosphere. Since then, the concentration has relaxed to its historical value. For ethical and practical reasons, it is hard to perform experiments upon the Earth's atmosphere and climate. But the doubling of the carbon-14 concentration by bomb tests amounts to a gigantic experiment upon the atmosphere, and this experiment turns out to be profoundly revealing when it comes to estimating the effect of human CO2 emissions upon the climate.


  1. Unfortunately carbon-14 from bomb tests tells you almost nothing about the effect of human CO2 emissions upon the climate. This is because the decay of bomb carbon-14 can only tell you about the residence time of carbon in the atmosphere (the average time a particular molecule of CO2 remains in the atmosphere before being transferred to another reservoir) but not the adjustment time (the time taken for atmospheric CO2 to respond to changes in the sources and sinks - essentially the characteristic timescale of the decay of atmospheric CO2 should we cease all anthropogenic emissions today). The residence time for CO2 is not equal to the adjustment time because of the vast exchange fluxes that constantly exchange CO2 between reservoirs, but this is a straight swap, so it doesn't change atmospheric concentrations. 14C has a short residence time, but mostly because it is just being exchanged with 12C and 13C from the oceans and terrestrial biota. This is a somewhat counterintuitive idea that is often misunderstood, which is why I wrote a paper about it,

    Gavin C. Cawley, On the atmospheric residence time of anthropogenically sourced carbon dioxide, Energy & Fuels, volume 25, number 11, pages 5503–5513, September 2011.

    which you can find here:

    preprint here:

    I wrote a blog article about it for SkS, which you can find here:

    The residence time is not controversial, and estimates range from about 4 years to about 20. The adjustment time is much longer 50-200 years for the initial phase, but the decay has a long tail.

  2. Gavin, thank you for your comment. I'm trying to understand what you are saying, but I'm having difficulty. Which of the following facts do you contest?

    (1) Cosmic rays make roughly 8 kg carbon-14 per year in the upper atmosphere.
    (2) One in eight thousand carbon-14 atoms decays each year.
    (3) For equilibrium, there must be 64,000 kg carbon-14 on Earth.
    (4) There are 800 Pg of carbon in the atmosphere with carbon-14 at 1 part per trillioin.
    (5) Therefore, there are 800 kg of carbon-14 in the atmosphere.
    (6) There must exist a reservoir of carbon-14 containing the remaining 63,200 kg. Each year there is a net flow of 8 kg of carbon-14 into this reservoir.
    (7) Carbon-14 does the same thing as carbon-12 or carbon-13. It is chemically identical.

    From these facts, we learn a great deal about the carbon cycle of the atmosphere. In particular, that it will take six thousand years to double the CO2 concentration of the atmosphere emitting 10 Pg/yr of carbon as we are now.

    If you can counter any part of the above argument, I'd like to hear it. I have looked at your blog post and other similar discussions of carbon-14, and you do not discuss the production rate of carbon-14. Nor do you show calculate the effect of temperature upon CO2 concentration in the atmosphere, which is easy to do once you have calculated the rate of exchange between the atmosphere and the ocean.

    So I think you will find that there is something to learn from considering the carbon-14 production rate, and I would be delighted to have someone with your keen eye point out any errors in my calculations. Yours, Kevan

  3. I don't think I particularly disagree with any of them, however I haven't checked the details, I am happy to take your word for it, for the moment.

    The point is that the bomb carbon-14 data can only reliably tell you about the residence time of carbon in the atmosphere, but not the adjustment time and it is the adjustment time that is relevant to anthropogenic influence on atmospheric carbon dioxide. This is because carbon-14 acts as a tracer and allows us to estimate the residence time more or less directly. To make an argument about the adjustment time, you would need a model that dealt with the full complexity of the carbon cycle (such as the stratification and circulation of the ocean), at which point you would end up with a similar model to that used by Archer et al.

    This is a complicated issue, that is frequently misunderstood, so the first stage is to reach agreement on the difference between residence time and adjustment time.

  4. Dear Gavin, I'm not sure what you mean by "adjustment time". The residence time most definitely allows you to calculate the anthropogenic effect upon the atmospheric concentration, both in the short term and in the long term. The bomb test curve is something that our carbon-cycle model predicts correctly. Do you agree that doubling the number of CO2 molecules in the atmosphere will double the rate at which they are absorbed by the ocean? Yours, Kevan

  5. Dear Gavin, I'm not sure what you mean by "adjustment time". Then we have a problem as understanding the difference between residence and adjustment time is very important. This also suggests you have not read my paper yet as it explains the difference between residence and adjustment times (I also defined it in my initial comment). Please read my paper, especially the quote from the first IPCC report that states it is the adjustment time that is relevant (page 10 of the preprint). There is no point in discussing any other question until we agree on the residence/adjustment time distinction as it is critical to understanding our impact on the carbon cycle.

  6. Dear Gavin,

    Forgive me for being vague, and thank you for your continued attention. When I say "I'm not sure what you mean by adjustment time..." that's my polite way of saying, "I have studied what you mean by adjustment time, and I still cannot understand how you fail to see that the adjustment time can be deduced from the residence time."

    My best guess is that you have not allowed yourself to use Henry's Law, in which doubling the partial pressure of CO2 in the atmosphere will double the uptake of the ocean. The argument I present here begins with those seven observations and works out their logical consequences, then predicts the rise from 300 ppmv to 400 ppmv as a result of our 10 Pg/yr emissions, predicts the aftermath of the bomb tests, and predicts the response of CO2 concentration to temperature over the past 400k years. You will find the math here, if you are interested.

    If you don't want to wade through those equations, please look at your own model and ask yourself: how does 8 kg of carbon-14 leave the atmosphere every year?

    Yours, Kevan

  7. O.K. It is probably best to be direct in discussing science on blogs.

    "I have studied what you mean by adjustment time, and I still cannot understand how you fail to see that the adjustment time can be deduced from the residence time."

    sure, provided you have a valid model that encapsulates your assumptions, and the deduction is only as valid as the model.

    "My best guess is that you have not allowed yourself"

    It is probably best to avoid assuming what someone means and just ask them. None of us are mind readers and by assuming you know what someone is saying is a good recipe for not being able to correct that assumption if you are wrong as you end up interpreting what they say through the lens of that assumption.

    Henry's law only governs the immediate absorption of CO2 from the atmosphere into the thin surface layer. If you base a model on that assumption alone, you will draw incorrect conclusions. To a first approximation, uptake is indeed proportional to the difference between atmospheric and surface water concentrations. However the surface water is only a small proportion of the ocean as a whole, which doesn't act as a homogenous reservoir.

    You could take my model and swap anthropogenic emissions for carbon-14 in the model and run it again, and at equilibrium you would indeed find that the creation rate matches the uptake (and decay) rate. However, as I said in my paper, such simple models are good for illustrating the basic concepts of the carbon cycle, but they are of little use in producing realistic quantative conclusions.

    you write:

    "Modern models of the CO2 cycle are presented in papers such as Archer et al., but these models are contradicted by carbon-14 observations, so they cannot be correct."

    have you identified the specific flaw in the models used by Archer et al.?

  8. Dear Gavin,

    It does not matter that the reservoir is not homogenous. Just draw the envelope around the reservoir and make the interface the surface of the ocean. Whatever mixing goes on within the reservoir is now internal, and will also take place according to diffusion gradients, all of which are proportional to concentration. So you double the concentration in the surface layer, you double the rate of diffusion into the lower layers, and so on. You can model the reservoir as one body, two bodies, three bodies, or a hundred bodies, the results are pretty much the same.

    If your model can show 8 kg of carbon-14 moving out of the atmosphere into a reservoir of 63,200 kg of carbon-14 that must exist on Earth, then it's the same as my model. There is only one model that fits the data.

    You say the carbon cycle is "complex". Why? To +-10% we can model it with a two-reservoir system and a second-order differential equation. As systems go, it's simple. The model I have presented, which is the same as the one in Arnold et al, but with different units, makes three unambiguous predictions that are all accurate.

    Archer et al's models all ignore the fact that the rate of absorption into the vast carbon reservoir in the ocean is proportional to the concentration in the atmosphere. Their representation of the ocean model is wrong because it cannot account for the movement of carbon-14 every year from the atmosphere into the deep ocean, which has been going on for millions of years. There are roughly 64,000 kg of carbon-14 in the ocean, where the concentration is 0.8 ppt, so there is around 80,000 Pg of carbon in the ocean, and the atmosphere is freely exchanging CO2 with the deep ocean. The Archer et al models assume such a free exchange does not take place, but they are wrong, because the exchange must take place, or else carbon-14 would not reach the deep ocean fast enough to keep the inventory in the atmosphere down to only 100 years worth of production. So: yes, I have reviewed them, and they are all wrong.

    Yours, Kevan

  9. PS. Maybe you are concerned about calculating the transfer rate of CO2 from the atmosphere to the ocean from first principles, without any empirical observations to guide us. We have

    CO2 Pg/yr Transferred
    = Some Value Deduced from First Principles.

    Well, that would be hard, because you could get the constant wrong by an order of magnitude easily, depending upon how you implement all the layers in the ocean in your model.

    But we are in the happy position where we don't have to deduce the constant from first principles. The production rate of carbon-14 tells us the constant. It's roughly 40 Pg/yr for atmospheric concentration 300 ppmv.

    Now we wonder what the equilibrium rate of transfer will be at some new CO2 concentration, and by dimensional analysis or a any number of other thought experiments, the equilibrium rate will be proportional to the atmospheric CO2 concentration so long as the deep ocean concentration remains constant.

    It could be that by "adjustment time" you mean how long it takes for the rate of absorption to reach its new equilibrium value. In a two-reservoir system, this time constant is equal to the residence time. In a more complex system, this time constant could be 20% different from the residence time, but that's about it. Given that the residence time is 17 years, and it has taken 70 years for our carbon emissions to go from near 0 to 10 Pg/yr, any error in the adjustment time has no effect upon my prediction that it will take 6000 years to double atmospheric concentration at the rate we are going now.

    Does that make sense?

    Yours, Kevan

  10. Re: On the atmospheric residence time of anthropogenically sourced carbon dioxide.

    Looking at your paper, I see that your "residence time" arises from the rapid exchange between the top of the ocean and with the biomass, so it's only a few years. In my two-reservoir model, the residence time is much longer, 17 years, because the ocean is lumped together with everything that is not the atmosphere. Thus, the residence time is the adjustment time.

    You say "However, this objection does not apply to the mass balance arguments, as the net environmental flux, Fi − Fe, is not calculated from uncertain estimates of Fi and Fe, but from the difference between dC/dt and Fa" That's a nice idea. In the case of the change in Fe (absorption of CO2 into the ocean) with concentration in the atmosphere, C, you introduce a second parameter to make Fe linear but not proportional to C. At this point, you have enough free parameters to fit any near-linear change in CO2 concentration over the past century. But your method will work provided that the "adjustment time" is significantly greater than the time over which we added the CO2. If, however, the adjustment time is decades, you are simply fitting a line to the rising equilibrium concentration of CO2 in the atmosphere, and suggesting that the new equilibrium is far higher, but we have not got there yet.

    I see no physical basis for your non-proportional absorption rate from the atmosphere. Maybe there is an effect of ocean acidification, but my calculation suggests that's not going to be more than 10%. To the first approximation, the absorption rate will be proportional.

    Your model does not derive its exchange rates from the carbon-14 creation rate, nor do I see any mention of the creation rate in your paper, nor any explicit statement that you have modeled the descend of 8 kg/yr of carbon-14 into the deep ocean. These would be additional constraints. They give you the absolute rate of exchange with the deep ocean, and there is no need to work with the derivative of CO2 concentration as observed in the last sixty years. Instead, you can simply predict that increase, as I have done, and get it right to within 10%.

    Having said all that: thank you for the link to your paper, and thank you for your attention.

    Yours, Kevan

  11. "It does not matter that the reservoir is not homogenous."

    Kevan, I'm sorry but the last forty years of research into the carbon cycle says otherwise. I think what you are doing on this blog (working things out from first principles) is great, but it needs to be done with some humility and it is a good idea to do what I did, which is to collaborate with some carbon cycle specialists who can point out some of your misunderstandings. The thing that I really gained from doing this was becoming aware of the things I didn't know I didn't know. We can't deduce the actions of the entire carbon cycle by just looking at 14C, and if we could, don't you think the carbon cycle specialists would have found that out by now? Perhaps if they do things differently, there is a good reason for it, and you should find out what that reason is before dismissing their work.

    Regarding "I have studied what you mean by adjustment time, and I still cannot understand how you fail to see that the adjustment time can be deduced from the residence time", I gave that some thought overnight and my answer is now "no". The residence time depends on the mass of the reservoir and the magnitude of the flux out of it. The adjustment time also depends on the natural fluxes into the atmosphere, and they cannot be deduced from the residence time so they are based on the assumptions of the model and are not constrained by the 14C observations as far as I can see.

  12. Incidentally, the sort of carbon cycle models that were used in the research summarized in the first IPCC WG1 report do model 12C, 13C and 14C and are calibrated using observations including those for 14C in the different reservoirs and cosmic ray and bomb 14C (e.g. Enting and Pearman 1987, Tellus (1987), 39B, 459-476, e.g. Figure 7). I'd be rather surprised if Archer et al produce results that are seriously inconsistent with those of Enting and Pearman. Note that Enting and Pearman use an ocean model with six layers.

    An interesting question remains about the residence time estimated from bomb 14C, there have been many papers published on this and the estimates range from about 4 to about 20 years. This suggests that perhaps just estimating it from the decay in the obvious manner may be naive and there may be some complicating factor. I'm looking into that as your posts have stimulated my curiosity on that question.

  13. Dear Gavin,

    Please show me with a calculation that a multi-layer reservoir produces a significantly different result from a two-reservoir system. My two-reservoir system fits the carbon-14 data and predicts the ice-core correlation, bomb test aftermath, and recent CO2 rise. And yet you don't like it. Why?

    By all means show me a calculation or an observation of nature that contradicts my model.

    Your own paper is based upon an assumption that violates the laws of molecular chemistry: you assume that absorption of CO2 into the ocean is not proportional to CO2 concentration, even though you are assuming constant temperature and ocean concentration. The probability of a CO2 molecule being absorbed is independent of the number of gaseous CO2 molecules in the atmosphere. Molecules act independently. What do you imagine is happening with your linear rather than proportional model? Are the CO2 atoms talking to one another and agreeing not to go into the water?

    Furthermore, you take the derivative of CO2 measurements, which amplifies any non-linear error in the measurement or any deviation from the simple behavior of the system, and then fit a straight line to what's left, so that your fundamental parameter is the second derivative of the C measurement. Your 74 year residence time has no physical significance whatsoever: it is a function of the errors in your measurement of C and your assumptions of constant ocean concentration and temperature.

    So: your model is wrong because it is based upon faulty assumptions and abuse of data. I don't even need to compare it to the data. It can't be right. But even if I were to try to compare it to the data, I'd have trouble, because you don't confirm any of the absolute exchange rates. So its' not clear your model makes any statement about the world that we could confirm or deny. It's not falsifiable.

    I don't like to take apart other people's work. So please don't push me to it. Just stick to presenting calculations that take apart my work.

    Yours, Kevan

  14. "I don't like to take apart other people's work. So please don't push me to it. Just stick to presenting calculations that take apart my work."

    Sorry, if you are going to be hostile and confontational about this then there is little point in continuing. Part of scientific method is that you need to be welcoming of criticism, especially as I tried to state it as reasonably as I could. I would say however that if you read my paper, I make it very clear that my model is only useful to illustrate a few basic points about the carbon cycle and that if you want to make quantitive conclusions about the carbon cycle then you need to use the sort of carbon cycle model that carbon cycle researchers do. Perhaps you should also ask yourself why you are citicising my model for being unrealistic when I have already pointed out the (even larger) flaws in it already. Please do yourself a favour and understand why current carbon cycle models are constructed the way they are (and note that they are already calibrated against 14C observations).

  15. Alternatively, write up your model as a paper and submit it to a journal where carbon cycle modelling is commonly published. I for one would be happy to read it there, but I am no longer willing to try and discuss it with you here.

  16. Dear Gavin,

    Am I being hostile and confrontational. Rats. I am sorry. That's no way to treat a guest at my site. I am not used to the culture in this field of atmospheric science, so what you think is rude is me being respectful, and what you think is respectful, I think is rude.

    In this case, if I'm going to tell someone their paper is wrong, I should do it fast, with a sword, showing that I know the recipient of my critique is tough enough to take it.

    Of course, it did not help that I was in a rush to get to a dentist's appointment.

    From your side: please be assured that it is extraordinarily rude in my field to argue from personal authority or authority of the published authors as you are doing. We don't do that. It's considered an admission of incompetence. We listen to the new proposal and then point out the experimental evidence that contradicts the proposal. If we can't contradict the proposal with evidence, and assuming the proposal actually makes correct predictions, we are intrigued and impressed.

    What you are doing is the opposite of polite in my field: you are accepting that my model fits the data, but saying that's not enough, I have to first show respect for other models, and I should read more papers, and so on, and that because I am ignorant of other people's models, that means I don't know what I'm talking about. In my field, that's bombastic and arrogant, no matter how sugar-coated it sounds. But I understand it's the norm in this field, so I am trying to be patient.

    How about this: let's stick to calculations in the future.

    As to your paper, I apologize if the tone of my critique was rude within the culture of this field, and will try to do better next time, but your paper is fundamentally flawed, and I stand by that assertion. Is there a polite way to say that? I don't know.

    Yours, Kevan