Cosmic rays turn roughly 7.5 kg of atmospheric nitrogen into carbon-14 every year. Carbon-14 decays at 0.00012 kg/kg/yr. There must be 62,500 kg of carbon-14 on Earth, because 62,000 kg decays at 7.5 kg/yr. At the end of the nineteenth century, the atmosphere contained 300 ppmv CO2, or 650 Pg of carbon, of which 1 ppt was carbon-14, or 650 kg. Roughly 0.1 kg/yr of this 650 kg decayed each year, but the remaining 7.4 kg/yr had to leave the atmosphere somehow, entering some kind of carbon-14 reservoir.
This reservoir would have to contain the remaining 62,000 kg of carbon-14 we know must exist on Earth. Given that carbon-14 is chemically identical to normal carbon, the passage of one carbon-14 atom into the reservoir implies without any ambiguity that one trillion normal carbon atoms enter at the same time. Because carbon-14 is created in the atmosphere, the concentration in the reservoir could not be greater than 1.0 ppt, which means the reservoir would have to contain at least 62,000 Pg of carbon in order for it to hold 62,000 kg of carbon-14.
These observations are all unassailable facts of nineteenth century Earth, and indeed of Earth today. All of the quantities we have listed remain the same to within 10%, with the exception of the CO2 concentration, which has risen by 30%. We still cannot escape the conclusion that there exists a carbon reservoir on Earth of at least 60,000 Pg that is in ready communication with the atmosphere.
Having advanced the above argument in detail, we made two assumptions about the carbon cycle in order to develop a simple model of its behavior. We assumed that the carbon-14 concentration in the reservoir was uniform, and we assumed that this concentration was 0.8 ppt, which is the concentration in the deep oceans. With these assumptions we obtained two differential equations describing the carbon-14 concentration in the Earth's carbon cycle. The equations contained two unknown quantities: the mass exchange rate between the atmosphere and the reservoir, and the total size of the reservoir. In order to make the model fit our observed atmospheric concentration and our assumed reservoir concentration, we determined that the mass exchange rate had to be 37 Pg/yr and the reservoir had to contain 77,000 Pg of carbon.
Our one-reservoir, uniform-concentration model is already fully constrained by our observations of carbon-14 and the atmosphere. No adjustment to any of its parameters is possible without compromising its accuracy. The model predicts how the carbon-14 concentrations will respond to any change we care to imagine. In our previous post, we presented this plot of how the model predicts the concentrations will develop from a starting point of zero.
But how are we to know if these predictions are correct? What we have done so far is akin to plotting two observations on a graph, drawing a straight line through both, and declaring the straight line to be the place where all future observations will lie. It is true that there is only one straight line that we can draw through two points, but it is also true that we can draw a straight line through any two points. It is only when we have three or more points lying on the same straight line that a straight-line model becomes convincing. In the case of our model, we can always pick a value of mass exchange and reservoir size to make it fit our observations. If we are to have more confidence in our model, we must test it against other, independent observations of carbon-14 concentration. Until then, our model is merely a credible hypothesis: worth thinking about, but not conclusive.
This inconclusive position was the one Arnold et al. found themselves in when they published their study of the carbon cycle in 1956. They did the same analysis we have done, and they studied two-reservoir and three-reservoir models as well. The more complex models produced almost identical predictions to the one-reservoir model, so they concluded that the one-reservoir model was good enough. But they were unable to test the one-reservoir model with an independent observation of carbon-14 behavior in the atmosphere. No such observations existed at the time.
Now, sixty years later, we have the observations we need to test our carbon cycle model. In the 1960s, we inadvertently performed an experiment on the entire atmosphere, in which we doubled the mass of carbon-14 it contained. Our model makes a clear and unambiguous prediction of what will happen after such an event. If this prediction is inaccurate, our model must wrong. But if the prediction is accurate, the model must be reliable, because the chance of the model being accurate by chance is close to zero.