Sunday, November 29, 2015

Carbon-14: The Reservoir Is the Ocean

Up to now, we have guessed that the carbon-14 reservoir in our carbon cycle is the deep ocean, where the concentration of carbon-14 is 80% of the concentration in the atmosphere. We are using 1.0 ppt (parts per trillion) as the concentration in the atmosphere, and 0.8 ppt as the concentration in the deep ocean, but we recognize that the absolute concentration in the atmosphere is hard to measure, and might be slightly higher than 1.0 ppt. But the concentration in the deep oceans is well-known to be 80% of that in the atmosphere. Today we argue that the reservoir of carbon-14 in our carbon cycle must be the deep ocean.

Our calculations so far go like this. Every year, cosmic rays create 7.5 kg of carbon-14 from atmospheric nitrogen. Carbon-14 decays back into nitrogen with a half-life of 5700 years, which means 0.012% of it decays into nitrogen every year. After fifty thousand years, the total mass of carbon-14 in the atmosphere will reach equilibrium. The total mass of carbon-14 will be 62,500 kg, because 62,500 kg multiplied by 0.012% is 7.5 kg per year, so the rate at which carbon-14 is created by cosmic rays is equal to the rate at which the Earth's reservoir of carbon-14 decays back into nitrogen.

The natural, equilibrium atmosphere of the early twentieth century contained 650 kg of carbon-14. The remainder of the Earth's 62,500 kg of carbon-14 is elsewhere, in the reservoir of our carbon cycle. Almost all carbon-14 in the atmosphere exists in CO2. By whatever means a carbon-14 atom moves in and out of the atmosphere, it does so in CO2 molecules. When one carbon-14 molecule moves into the reservoir, it does so in the company of 1÷1.0 ppt = 1.0 trillion CO2 molecules. If the reservoir is the deep ocean, where the concentration of carbon-14 is 0.8 ppt, we can further say that, whenever a carbon-14 atom re-enters the atmosphere from the ocean, it does so in the company of 1÷0.8 ppt = 1.25 trillion CO2 molecules.

We calculated that the reservoir, assuming it is the deep ocean, must contain 77,000 Pg of carbon, and that every year 37 Pg of carbon must be exchanged between the reservoir and the atmosphere, all of it moving with CO2 molecules. When 37 Pg of carbon moves from the atmosphere to the reservoir, it carries with it 37.0 kg of carbon-14. When 37 Pg of carbon moves from the reservoir to the ocean, it carries with it 29.6 kg of carbon-14. The difference is a net 7.4 kg of carbon-14 flowing into the reservoir every year. We add to this another 0.1 kg per year, which is the decay of the carbon-14 that remains in the atmosphere, and we arrive at a total of 7.5 kg of carbon-14 being removed from the atmosphere by exchange and decay, which gives us equilibrium with the 7.5 kg per year being created by cosmic rays.

With the above values of reservoir size and annual mass exchange, we obtained an analytic solution to the carbon-14 concentration in our natural, equilibrium atmosphere. We showed that this analytic solution was in near-perfect agreement with the relaxation-time of atmospheric carbon-14 concentration following the nuclear bomb tests of the 1950s and 1960s. Thus we are confident that our analytic solution is a reliable model for predicting the behavior of carbon-14, and therefore of all carbon, in the Earth's atmosphere.

Let us consider the possibility that the reservoir of our carbon cycle resides somewhere other than the deep ocean. The table below shows other candidates for the reservoir, their carbon-14 concentrations, and the relaxation-time of carbon-14 concentration that we would obtain if this candidate were indeed the carbon-14 reservoir. We refer to Arnold et al. for our normalized concentration values, in which the atmospheric concentration is taken to be 1.0 ppt.

Candidate Carbon-14
Time (yr)
Ocean, Below 1000 m0.8017
Ocean, Top 100 m0.963.5
Biosphere, Land1.000.0
Biosphere, Ocean0.963.5
Soil, Humus1.000.0

The above candidate reservoirs are the only ones known to us that exchange CO2 with the atmosphere. The relaxation-time of carbon-14 concentration after the nuclear bomb tests was roughly 15 years. None of the above candidates are even close to being consistent with the aftermath of the bomb tests, except for the deep ocean, which is in excellent agreement. We conclude that the reservoir of our carbon cycle is the deep ocean and only the deep ocean. The other candidate reservoirs do exchange CO2 with the atmosphere, but whatever effect they have upon the Earth's carbon cycle is dwarfed by the flow of carbon into and out of the deep ocean.

Thursday, November 19, 2015

Carbon-14: Absolute and Relative Concentration

Suppose we want to measure the carbon-14 concentration in a cubic meter of gas. If we can measure the rate at which the gas emits beta particles, we will know the number of carbon-14 decays occurring per second. Because 0.012% of carbon-14 atoms undergo beta decay every year, we can deduce the number of carbon-14 atoms in the gas from the decay rate. By some chemical procedure, we measure the amount of carbon in the gas, and so we can deduce the carbon-14 concentration.

To measure the rate at which the gas produces beta particles, we put the gas in a chamber. All around the chamber we arrange devices that detect beta particles. We will not be able to cover the entire surface area with beta detectors, but we can measure the fraction we cover. If we cover half the area, we can assume half the beta particles emerging from the gas will strike one of our detectors. The detectors may not detect every beta particle that strikes them. But we can perform experiments to measure their efficiency. We end up with a scaling factor by which we can multiply the number of beta particles we detect to obtain the number of beta particles that left the gas volume.

The beta particles have energy up to 156 keV, with a Fermi-Dirac distribution. The most energetic of them can penetrate 100 mm of air. But they are not certain to do so. The least energetic of them can penetrate only a few millimeters of gas. Only a small fraction of the beta particles emitted by carbon-14 decaying within our cubic meter of gas will ever make it to our detectors. But we can, with the help of the continuous slowing-down approximation, and the Fermi-Dirac distribution, estimate the fraction of beta particles that will emerge from the gas volume. And so we can obtain a scaling factor by which to multiply the number beta particles we detect to obtain the total number of beta particles emitted by the gas.

Having performed these calculations, we start to count beta particles. If the gas contains 300 ppmv of CO2 and 1.0 ppt of carbon-14, there will be of order 2.6×1011 carbon-14 atoms in the gas. Of these, 3.1×107 will decay each year, or 0.97 per second. Our chamber is 50% covered with detectors that are 50% efficient, and only 10% of our beta particles get out of the gas volume, so we expect to detect of order one beta decay every forty seconds.

Before we make our first measurement, we evacuate our chamber, to measure the background rate of beta particles. We find that our beta detectors are detecting one or two beta particles per second. These are due to radioactive isotopes in the chamber walls, cosmic ray showers, and electronic noise.

At this point, we realize that measuring carbon-14 in a gas is going to be hard. So we freeze the carbon dioxide out of our gas sample and put a pellet of solid CO2 into a much smaller chamber. We still have the same problems, but they are less severe. We can reduce our background beta-particle rate to one every ten seconds, while raising our carbon-14 beta rate to one per second. We will have to admit, however, that our measurement of carbon-14 concentration will be accurate to no better than ±20%. And indeed, the measurements of atmospheric carbon-14 concentration vary from 1.0 to 1.5 ppt.

But it is much easier to measure the relative concentration of carbon-14 in various samples of gas, water, or wood. We put one sample in our chamber and count beta particles, then another sample. If our first measurement is 20% too high, so will the second one, because the 20% error is a feature of the chamber, not the sample. The upshot of these practical considerations is that we can say that the deep oceans have carbon-14 concentration 80% of atmospheric, and be confident to ±2%, but we cannot say what the actual deep ocean concentration is to better than ±20%.

So far, we have assumed that the atmospheric concentration of carbon-14 is 1.0 ppt. In fact, it may be as high as 1.5 ppt. If it is 1.5 ppt, we have 980 kg of carbon-14 in our natural, equilibrium atmosphere, instead of the 650 kg we calculated assuming 1.0 ppt. This 980 kg is still negligible compared to the 62,500 kg of carbon-14 in the reservoir. And it is the relative concentration in the deep ocean that dictates the values of the mass exchange rate and reservoir size in our carbon cycle model, not the absolute concentration.

Thus, it may be that the atmosphere contains 1.5 ppt of carbon-14, not 1.0 ppt, but this has no significant effect upon our conclusions. We will continue to assume that our natural, equilibrium atmosphere contains 1.0 ppt, and quote concentrations in other potential reservoirs of carbon-14 as a fraction of the atmospheric concentration, just as the authors do in Arnold et a., where they use "relative specific activity corrected for fractionation".

Friday, November 6, 2015

Carbon-14: The Bomb Tests

Between the years 1945 and 1962, we detonated hundreds of atomic bombs in the atmosphere. Most of these, and certainly the largest, were detonated in the five years leading up to the Partial Test Ban Treaty in late 1963, which banned further atmospheric detonations. Atomic explosions produce carbon-14. By 1964, bomb tests had produced enough carbon-14 to double the atmospheric carbon-14 concentration. The graph below shows atmospheric carbon-14 concentration relative to its value before the bomb tests, as measured by various groups.

Figure: Atmospheric Concentration of Carbon-14 During and After the Bomb Tests. This graph is from an essay by Pettersson. The author combined measurements from several stations to produce the most complete graph we could find. For alternate plots, see here and here.

The relaxation of the atmospheric carbon-14 concentration from its peak follows an exponential decay with time constant roughly 15 years. After fifty years, the concentration is within a few percent of its value before the bomb tests. The carbon-14 did not linger in the atmosphere. It disappeared. Let's see if our model of the Earth's carbon cycle predicts this same relaxation or not.

In our model, carbon-14 concentrations are governed by two differential equations. We already solved these equations for a starting-point of zero in the atmosphere and the reservoir. Using the same procedure, we can solve the equations for a starting point of 2.0 ppt in the atmosphere and 0.8 ppt in the reservoir, which is the state the atmosphere was in at the time of the Partial Test Ban Treaty, when atmospheric tests were stopped. We obtain the following solutions for atmospheric concentration, CA, and reservoir concentration CR.

CA = 1.0 + 0.987 et/17 + 0.013 et/8200
CR = 0.8 + 0.0089 et/17 − 0.0089 et/8200

We note that this as t → ∞, we have CA → 1.0 ppt and CR → 0.8 ppt, as before. Whatever carbon-14 we add to the atmosphere, it will eventually decay, and we will be left once again with the concentrations generated by cosmic ray production. The above equations produce the graph of atmospheric and reservoir concentration shown below.

Figure: Carbon Cycle Model's Prediction of Atmospheric and Reservoir Carbon-14 Concentration After Sudden Doubling of Atmospheric Concentration.

Our simple, two-part, uniform-reservoir model of the carbon cycle predicts an exponential relaxation of carbon-14 concentration after the bomb tests, with a time constant of 17 years. Needless to say: agreement between the model and observation is superb.

Saturday, October 31, 2015

Carbon-14: Assessment of Our Carbon Cycle Model

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.

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.

Saturday, October 24, 2015

Carbon-14: Analytic Solution to Concentration Equations

We can describe the origin and fate of carbon-14 with a diagram or a pair of differential equations. One of the assumptions upon which we have based our reasoning so far is that one million years is more than enough for carbon-14 concentrations to reach equilibrium. Today we solve the two differential equations with a starting point of 0.0 ppt carbon-14 in the reservoir and the atmosphere. Our solution will tell us how long it takes for equilibrium concentrations to be established. For those of you who are not interested in following the derivation, we invite you to await our next post, in which we use the solution to plot graphs of carbon-14 concentration versus time.

We begin by referring to our equations (1) and (2) as shown here. We re-arrange the equations so that all terms in CA are on the left side of (1) and all terms in CR are on the left side of (2). In doing so, we treat d/dt as if it were just another factor, which may seem odd, but it's accurate.

The only variables in these two equations are CA, CR, and time, t. All other parameters are constants that we have already calculated. We must eliminate CR from the (3) so as to obtain an equation in CA and t alone, which we can then solve. We eliminate CR by multiplying (3) by the same factor that we observe on the left side of (4).

We do the same thing for CR, arriving at a differential equation in only CR and t.

At this point we pause to check the equations by considering how they behave as time approaches infinity, as shown here, and we find that they appear to behave correctly. Both equations have solutions of the same form: a constant plus two decay terms.

We insert values for mpC14, γ, MA, MR, and me and obtain values for all five constants in our solution. The coefficients α and β dictate how rapidly the concentration evolves with time. We have α = 0.0574 and β = 0.000122. We see that α is close to the fraction of the atmospheric carbon that is exchanged with the reservoir every year, while β is close to the decay rate of carbon-14. We have 1/α = 17 yr, which is the time constant of exchanges between the atmosphere and the reservoir, and 1/β = 8,200 yrs, which is the time constant of accumulation of carbon-14 in the reservoir. The weighting factors k1 and k2 are −0.2 and −0.8 respectively. Together, they add up to −1.0 ppt. The constant term is the equilibrium value of CA, which comes out as 1.0 ppt, which is what we expect, because we chose the value of me and MR to make sure that the equilibrium concentration would be 1.0 ppt. Our equation for CA is as follows, where concentration is in ppt and time is in years.

CA = 1.0 − 0.2 et/17 − 0.8 et/8200.

At t = 0, we have CA = 0 ppt, and when t→∞, CA→1.0 ppt, as we expect. The 17-yr decay term represents the flow of carbon-14 into the reservoir. The 8200-yr term represents the accumulation of carbon-14 in the reservoir. We obtain a similar solution for CR.

The coefficients α' and β' are the same as α and β. But the weighting factors are different, as is the constant term. Our equation for CR is,

CR = 0.8 + 0.002 et/17 − 0.802 et/8200.

At t = 0, we have CR = 0 ppt, and dCR/dt = k1α+k2β = 0.0 ppt/yr, and d2CR/dt2 = k1α2+k2β2 > 0, all of which we expect, and also when t→∞, we have CR→0.8 ppt, which is one of our starting assumptions.

We are pleased to have an analytic solution for CA and CR. A numerical solution to the differential equations turns out to be unstable for time steps greater than ten years. We want to plot CA and CR over a hundred thousand years. Ten thousand steps are cumbersome in a spreadsheet. Furthermore, the analytic solution us gives more insight into the way the parameters of the carbon cycle interact to govern its behavior.

Saturday, October 17, 2015

Carbon 14: Reservoir Concentration

Our study of carbon-14, which began with Carbon-14: Origins and Reservoir, reveals that the atmosphere is exchanging carbon dioxide with a reservoir that contains over a hundred times as much carbon dioxide as the atmosphere itself. This exchange must be taking place, because it is the only means by which carbon-14 can be transported out of the atmosphere. The concentration of carbon-14 in the reservoir must be less than in the atmosphere, or else the carbon dioxide exchange would be unable to generate a net flow of carbon-14 into the reservoir.

We guessed that the reservoir was the deep ocean, which does have an adequate capacity to hold the reservoir CO2 in solution. If the deep ocean is the reservoir, its carbon-14 concentration has been measured to be 0.8 ppt, which is indeed less than the atmosphere's 1.0 ppt. Assuming the concentration of the reservoir is 0.8 ppt, we conclude that the carbon dioxide exchange rate in our natural, equilibrium atmosphere is 140 Pg/yr of CO2 (or 37 Pg/yr of carbon), while the reservoir holds 280,000 Pg of CO2 (or 77,000 Pg of carbon). Because our equilibrium atmosphere itself holds 2,400 Pg of CO2 (or 650 Pg of carbon), we see that 5.6% of the CO2 in the atmosphere enters the reservoir every year and is replaced by a like amount emerging from the reservoir. Furthermore, the reservoir holds 120 times as much CO2 as the atmosphere.

In future posts, we will find that the vast size of the reservoir, combined with the ready exchange of carbon dioxide between it and the atmosphere, imply that continuing to burn fossil fuels at the rate we are today will take thousands of years to cause a doubling in the carbon dioxide concentration of the atmosphere. Before we proceed, however, let us consider how our estimate of the size of the reservoir and the magnitude of the exchange are affected by uncertainty in the one parameter that we have guessed at: the concentration of carbon-14 in the reservoir. This concentration must be less than 1.0 ppt, and we guessed it is 0.8 ppt. But perhaps it is 0.7 ppt or 0.9 ppt. Using the equations we derived last time, we re-calculate the carbon exchange rate and the reservoir size for these concentrations.

(Pg/yr of Carbon)
(Pg of Carbon)
0.7 24 88,000
0.8 37 77,000
0.9 74 69,000
Table: Effect of Reservoir Concentration. We have reservoir concentration of carbon-14 in ppt, CR, carbon mass exchange rate in Pg/yr, me, and reservoir carbon mass in Pg, MR. Multiply carbon masses by 44/12 to get CO2 masses.

In Arnold et al., the authors present measurements of carbon-14 concentration in various layers of the ocean, in vegetation, and in soil, relative to the carbon-14 concentration in the atmosphere. These vary from 80% to 96% of the atmospheric concentration, with the deep oceans having the lowest concentration. The concentration in our reservoir must lie somewhere in the range 0.8-1.0 ppt. If the concentration is higher than 0.8 ppt, the reservoir will be slightly smaller, but the exchange rate will be much higher. At 0.9 ppt, for example, the exchange rate will be double what it is for 0.8 ppt. We see that our estimate of the exchange rate, being based upon a reservoir concentration of 0.8 ppt, is a conservative one, while our estimate of the reservoir size is bound to be close. We can now proceed with confidence in our analysis, knowing that we are certain not to over-estimate the exchange of carbon dioxide between the atmosphere and the reservoir.