## Tuesday, October 27, 2015

### Carbon-14: Establishing Equilibrium

We have so far assumed that one million years is sufficient time for the carbon-14 concentrations in our carbon cycle to reach their equilibrium values. That is to say: if we started with no carbon-14 in our carbon cycle, and cosmic rays added 7.5 kg/yr to the atmosphere, it would take less than one million years for the carbon-14 concentration to stabilize at 1.0 ppt in the atmosphere and 0.8 ppt in the reservoir. In our previous post, we obtained the following equations for atmospheric carbon-14 concentration, CA, and reservoir concentration, CR, starting from CA = CR = 0.0 ppt at time t = 0 yr. The equations assume units of ppt for concentration and years for time.

CA = 1.0 − 0.2 et/17 − 0.8 et/8200
CR = 0.8 + 0.002 et/17 − 0.802 et/8200

The figure below shows how the two concentrations increase with time. The scale is logarithmic, which allows us to see changes in the first year as well as in the final ninety thousand years.

After ten years, the atmospheric concentration has risen to 0.1 ppt. We have 65 kg of carbon-14 in the atmosphere, which is most of the 75 kg created by cosmic rays in ten years. The concentration in the reservoir remains close to zero (0.0002 ppt).

After one hundred years, the atmospheric concentration has settled upon a value of 0.2 ppt. The concentration in the reservoir remains close to zero (0.009 ppt). Cosmic rays have made 750 kg of carbon-14, but only 130 kg of this remains in the atmosphere. The exchange of 37 Pg/yr of carbon between the atmosphere and the reservoir is carrying 37 Pg/yr × 0.2 ppt = 7.4 kg/yr of carbon-14 from the atmosphere into the reservoir. This state of affairs continues through the first millennium: carbon-14 created by cosmic rays flows directly into the reservoir, with no significant amount of carbon-14 being carried back out again, because the concentration in the reservoir is still close to zero.

After a thousand years, the concentration in the reservoir has risen to 0.1 ppt. The reservoir is beginning to fill up. The concentration in the atmosphere is now 0.3 ppt. We note that 0.3 ppt is still 0.2 ppt higher than in the reservoir. The net flow of carbon-14 into the reservoir remains 7.4 kg/yr.

After ten thousand years, the reservoir concentration is close to 0.6 ppt. In the atmosphere it is close to 0.8 ppt. We still have a net flow of 7.4 kg/yr of carbon-14 into the reservoir each year, but now this 7.4 kg/yr is the difference between 28.7 kg/yr flowing into the reservoir and 21.3 kg/yr flowing out. At the same time, the growth in the reservoir concentration begins to slow. The rate at which carbon-14 is decaying in the reservoir is starting to be significant compared to the rate at which carbon-14 is being added to the reservoir. The reservoir now contains 45,000 kg of carbon-14, and it decays at 5.4 kg/yr.

After fifty thousand years, the concentrations have stabilized at 0.8 ppt in the reservoir and 1.0 ppt in the atmosphere. The difference is what is required to transport almost all carbon-14 created by cosmic rays into the reservoir. The reservoir now contains so much carbon-14 that the radioactive isotope decays into nitrogen at the same rate that carbon-14 is added from the atmosphere.

The time it takes to arrive at equilibrium is far shorter than one million years. Our model indicates that the concentration of carbon-14 in the reservoir will be set by the average carbon-14 creation rate over the past fifty thousand years. The concentration in the atmosphere, on the other hand, is the sum of the reservoir concentration and an increment that is proportional to the creation rate that has prevailed for the past fifty years. If cosmic rays were to stop suddenly, the carbon-14 concentration would drop to the reservoir concentration of 0.8 ppt within fifty years, and stay at 0.8 ppt for a thousand years before the drop in reservoir concentration became significant. Conversely, if cosmic rays were to double the rate of carbon-14 creation, the concentration in the atmosphere would rise to 1.2 ppt within fifty years, and stay at 1.2 ppt for almost a thousand years before the rise in reservoir concentration became significant.