Thursday, January 14, 2016

Carbon Cycle: The Correlation Between Temperature and CO2

In our previous post we deduced from first principles a relationship between temperature and CO2 concentration in the atmosphere of our carbon cycle. In the appendices of this post, we show that our calculation is consistent with the van t'Hoff equation used by chemists to predict the concentration of CO2 above a water reservoir, and to measurements of the solubility of CO2 in water. In the main body of this post, we will see how our calculation compares to measurements of temperature and CO2 concentration in the Earth's atmosphere over the past half-million years.

In our carbon cycle equations, MR is the mass of carbon in the oceanic reservoir, MA is the mass of carbon in the atmosphere, kR is the fraction of oceanic reservoir CO2 molecules that will be emitted into the atmosphere each year, and kA is the fraction of atmospheric CO2 molecules that will be absorbed by the oceanic reservoir each year. The rate at which carbon is emitted by the oceanic reservoir is kRMR, and the rate at which it is absorbed by the oceanic reservoir is kAMA. At equilibrium, these two rates will be the same, so we have:

kRMR = kAMA      ⇒     MA = kRMR/kA      (Eq. 1)

In our previous post we concluded that kA is constant with temperature, while kR increases with temperature as e−2300/T. Our consideration of the Earth's carbon-14 inventory showed that MR = 77,000 Pg. In our natural, equilibrium carbon cycle, we found that MA = 650 Pg. Given that the carbon must reside somewhere, even if MA doubles or halves with temperature, MR will change by less than 1%. So we can assume, to the first approximation, that MR is constant with temperature. It is MA that varies with temperature, and it does so in proportion to kR.

MA = kRMR/kA ∝ e−2300/T      (Eq.2)

Aa two temperatures T1 and T2, the equilibrium values of MA, which we denote MA(T1) and MA(T2), are related by:

MA(T2) / MA(T1) = e−2300/T2 / e−2300/T1 = e2300(1/T1−1/T2)      (Eq. 3)

Our (Eq. 3) predicts a close, positive correlation between temperature and atmospheric CO2 concentration, and it gives us an estimate of the magnitude of the change in CO2 concentration with temperature. The graph below shows atmospheric CO2 concentration provided by Barnola et al. and global temperature relative to today provided by Petit et al. over the past 425,000 years as determined from the Vostok ice cores.


Figure: Absolute Atmospheric CO2 Concentration and Relative Temperature versus Time from Vostok ice cores. Click to enlarge. Local copies of data here and here.

We see close and sustained correlation between CO2 concentration and temperature, even as temperature varies by 12°C. If we assume today's average global temperature is 14°C = 287 K, the change from −9°C to +3°C relative to today is a swing from 278 K to 290 K. Our (Eq. 3) predicts an increase in the mass of carbon in the atmosphere by a factor of e2300(1/278−1/290) = 1.41. Because almost all carbon in the atmosphere is bound up in CO2 molecules, the concentration of CO2 in the atmosphere is proportional to the total mass of carbon in the atmosphere, so when the total mass increases by a factor of 1.41, the CO2 concentration should increase by a factor of 1.41 also. Looking at the graph, we see CO2 rising from 190 ppmv to 290 ppmv, which is a factor of 1.52. Given the many uncertainties in our calculations, and in the ice-core measurements themselves, we are well-satisfied with the agreement between our calculations and the magnitude of the CO2 concentration changes in the ice core measurements.

We conclude that the correlation between CO2 concentration and temperature in the Earth's atmosphere over the past half million years is due to the effect of temperature upon the exchange of CO2 between the atmosphere and the oceanic reservoir of the Earth's carbon cycle. As the temperature rises, the CO2 concentration rises, and when temperature falls, the CO2 concentration falls.

Appendix 1: The van t'Hoff equation for CO2 and water states that the concentration of CO2 above the water is proportional to e−2400/T. Our (Eq. 2) states that it is proportional to e−2300/T. We are well-satisfied with this agreement.

Appendix 2: Suppose we have pure CO2 gas at atmospheric pressure above a reservoir of water. No matter how much CO2 the water dissolves, we maintain the same pressure of CO2 above the water. In this arrangement, unlike the arrangement of our carbon cycle, the concentration of CO2 in the water can vary. Applying the same reasoning we presented in our previous post, the rate at which CO2 molecules are absorbed by the water remains constant with temperature. At equilibrium, the rate at which CO2 molecules are emitted by the water must equal the rate at which they are absorbed, which means the rate of emission must also remain constant. But our calculation states that the probability of any given CO2 molecule in the water being emitted in a certain interval of time must increase with temperature as e−2300/T. If the rate of emission is to remain constant, the concentration of CO2 in the water must decrease with temperature as e2300/T. Only then will the rate of CO2 emission by the water, which is the product of the probability and the concentration, remain constant with temperature. We examine the plot of CO2 solubility in water versus temperature here. As temperature increases from 10°C to 20°C, the solubility of CO2 in water drops from 2.5 g/kg to 1.25 g/kg, which is a factor of 0.50. Our calculation suggests that it should drop by a factor of 0.58. We are well-satisfied with this agreement.

Tuesday, January 5, 2016

Carbon Cycle: Effect of Temperature

Up to now, we assumed that the temperature of our carbon cycle remained constant. We never specified what the temperature of the ocean or the atmosphere was. Nor did we assume that the temperature of the ocean or the atmosphere was uniform. We merely assumed that the temperature in each part of the system remained constant. Today we estimate how the rate of transfer coefficients of our carbon cycle, kA and kR, will change with temperature. We recall that kA is the fraction of atmospheric CO2 molecules that will be absorbed by the oceanic reservoir each year, and kR is the fraction of oceanic reservoir CO2 molecules that will be emitted into the atmosphere each year.

Claim: To the first approximation, kA is independent of temperature.

Justification: Absorption of CO2 takes place at the surface of the ocean. The air pressure at sea-level is dictated by the weight of air per square meter pressing on the sea, which is roughly ten tonnes per square meter. When the atmosphere warms up, its mass does not change. The pressure at sea-level remains constant. But the air does expand. Because the air expands, the number of CO2 molecules per cubic meter decreases. Because the gas is warmer, each CO2 molecule is moving faster. Their average velocity is proportional to the square root of the absolute temperature. Suppose we increase the temperature from 14°C to 15°C. Absolute temperature increases from 287 K to 288 K. According to the gas law, the number of CO2 molecules per cubic meter at the ocean surface decreases by 0.35%. But their velocity increases by 0.18%, which means each CO2 molecule above the ocean surface collides with the ocean 0.18% more frequently than before. Combining these two effects, there will be a net 0.18% decrease in the number of opportunities for CO2 molecules to be absorbed. To the first approximation, this is no change at all. Let us also consider whether a faster-moving CO2 molecule, having collided with the ocean, is more or less likely to be absorbed. When CO2 dissolves in water, energy is released. The CO2 molecule does not need to supply any energy in order to be dissolved. A hotter CO2 molecule has more energy, but this energy is not required by dissolution, and will not make the reaction more likely. To the first approximation, therefore, a 1°C rise in temperature will have no significant effect upon kA.

Claim: To the first approximation, KR increases by a factor of 1.028 for each 1°C warming of the ocean surface.

Justification: Emission of CO2 takes place at the surface of the ocean. When the ocean warms by 1°C, its volume increases by 0.02%, which is insignificant. As we showed above, the warmer CO2 molecules will collide with the ocean surface 0.18% more often, but this, too, is insignificant. Let us consider whether a faster-moving CO2 molecule, having reached the surface of the ocean, is more or less likely to be emitted into the atmosphere. The emission of CO2 from solution requires heat. The enthalpy of dissolution for CO2 in water is −19.4 kJ/mole = −0.20 eV per CO2 molecule. Each CO2 molecule dissolving in water releases 0.20 eV of heat. Conversely, in order to be un-dissolved, a CO2 molecule needs to supply at least 0.20 eV from its own thermal energy. At 14°C, the average molecule has kinetic energy only 0.025 eV. But a tiny fraction will have, by random collisions, a much higher energy, as dictated by the Boltzmann distribution. In a given period of time, the number of molecules that attain energy E is proportional to eE/kT, where T is absolute temperature, k = 8.62×10−5 eV/K is the Boltzmann constant, and e = 2.72 is the exponential constant. The rate at which any reaction requiring thermal energy E takes place is proportional to eE/kT. Thus kR is proportional to eE/kT, which means there is some constant, A, with units of 1/yr, for which kR = A e−0.02/0.0000862T = Ae−2300/T. In our natural, equilibrium carbon cycle, kR = 0.00048 1/yr. Suppose the natural, equilibrium temperature of the ocean surface is 14°C = 287 K. We can obtain the value of A as follows.

kR @ 287K = Ae−2300/287 = A×0.00033 = 0.00048 1/yr ⇒ A = 1.45 1/yr.

We raise the temperature by 1°C to 15°C = 288 K and re-calculate the rate.

kR @ 288K = 1.45 e−2300/288 = 0.000493 1/yr.

The rate has increased by a factor of 0.000493/0.00048 = 1.028, which is 2.8%. If we use 10°C or 20°C as our initial ocean temperature, we get 2.9% and 2.7% rises respectively. To the first approximation, each 1°C increase in temperature increases kR by 2.8%, regardless of our guess at the initial temperature of the ocean surface.

We simulate the effect of a sudden 1°C warming of our carbon cycle in the following way. We begin with our natural, equilibrium atmosphere, described by our carbon cycle model. We increase kR by 2.8%. We set the fossil fuel emission, mF, to zero. We obtain the following plot of atmospheric CO2 concentration versus time.


Figure: Atmospheric CO2 Concentration Following a 1°C Step Increase in Temperature.

The increase in temperature causes an immediate 2.8% increase in the rate at which CO2 is emitted by the ocean, but no increase in the rate at which it is absorbed. The additional CO2 released by the ocean builds up in the atmosphere. As the concentration of CO2 in the atmosphere increases, the rate at which it is absorbed by the ocean increases, because there are more CO2 molecules available for absorption. When the atmospheric concentration has increased by 2.8%, the rate of absorption is once again equal to the rate of emission. The carbon cycle reaches its new equilibrium in roughly one hundred years.

In our next post, we will see how well our calculations agree with the many empirical observations of carbon dioxide in solution, and with the history of our atmosphere over the past half-million years.

Wednesday, December 23, 2015

Carbon Cycle: With Ten Petagrams per Year

Today we use our model of the Earth's carbon cycle to show us what will happen if we start adding ten petagrams of carbon to the atmosphere each year by burning fossil fuels. Ten petagrams is roughly the amount we emitted in 2015, so we are going to see how this addition would affect the concentration of CO2 in the atmosphere if it began suddenly, continued for thousands of years, and was the only phenomenon affecting change in the carbon cycle. In particular, our calculation assumes that the temperature of the atmosphere and the ocean remain constant, which may not be true if rising CO2 concentration enhances the greenhouse effect.

We begin with the carbon cycle in its natural, equilibrium state. The concentration of CO2 in the atmosphere is 300 ppm, which means it contains 650 Pg of carbon. The oceanic carbon reservoir, meanwhile, contains 77,000 Pg. Each year, the ocean emits 37 Pg into the atmosphere, and absorbs 37 Pg from the atmosphere. But now we start adding an extra 10 Pg/yr to the atmosphere by burning fossil fuels. In the the numerical equations that describe our carbon cycle, we set mF = 10 Pg/yr. You can download our carbon cycle spreadsheet here. In it, you will find the following plot, along with our calculations.


Figure: Atmospheric and Oceanic Carbon Mass versus Time, with Ten Petagrams per Year Fossil Fuel Emissions. The time scale is logarithmic. Note that 1 Eg = 1,000 Pg. Click to enlarge.

During the first ten years, the total mass of carbon entering the atmosphere each year is 47 Pg/yr. But the mass absorbed by the oceans remains 37 Pg/yr. The mass of carbon in the atmosphere increases by 10 Pg/yr.

After ten years, we have emitted 100 Pg by burning fossil fuels. The mass of carbon in the atmosphere has increased by 80 Pg to 730 Pg. The rate at which carbon is absorbed by the ocean has increased in proportion, because there are more CO2 molecules available to absorb. The ocean now absorbs 41 Pg/yr. With 37 Pg/yr emitted by the ocean, and 10 Pg/yr emitted by burning fossil fuels, the carbon mass of the atmosphere increases by 6 Pg/yr.

After a hundred years, we have emitted 1,000 Pg by burning fossil fuels. The mass of carbon in the atmosphere has increased by 180 Pg to 830 Pg. The rate at which atmospheric carbon enters the oceanic reservoir is 47 Pg/yr. The mass of the oceanic reservoir itself has increased by 820 kg to 77,820 kg, which is only 1%. The rate at which oceanic carbon emerges into the atmosphere is still close to 37 Pg/yr. The atmosphere is in equilibrium: carbon enters at 47 Pg/yr and leaves at 47 Pg/yr. Its carbon mass increases only by 0.1 Pg/yr.

After one thousand years, the mass of carbon in the oceanic reservoir has increased by 13% to 87,000 Pg. The rate at which the ocean emits carbon into the atmosphere has increased by 13% to 42 Pg/yr. The mass of carbon in the atmosphere has risen to 910 Pg, and every year 52 Pg of atmospheric carbon is absorbed by the oceanic reservoir. The atmosphere is still in equilibrium with the ocean: each year 52 Pg enters and 52 Pg leaves. Its carbon mass continues to increase by only 0.1 Pg/yr, while the carbon in the oceanic reservoir increases by 10 Pg/yr.

After six thousand years, the mass of carbon in the atmosphere has doubled, and the mass in the oceanic reservoir has almost doubled. The plot below shows how atmospheric CO2 concentration increases with time for the same scenario.


Figure: Atmospheric CO2 Concentration in Response to Emission of 10 Pg/yr of Carbon by Burning Fossil Fuels. Units are parts per million by volume. Click to enlarge.

Burning fossil fuels at the rate we are going today, it would take 100 years to raise the concentration of CO2 in our natural, equilibrium atmosphere from 300 ppmv to 400 ppmv, and 6,000 years to raise it from 300 ppmv to 600 ppmv.

Sunday, December 20, 2015

Carbon Cycle: Equations and Diagram

Suppose human beings start to add carbon to our natural, equilibrium atmosphere at a rate mF by burning fossil fuels. We will express mF in petagrams of carbon per year, or Pg/yr. Note that we have been working with carbon masses, not CO2 masses, but we can convert atmospheric carbon mass to atmospheric CO2 mass simply by multiplying by 3.7, which is the ratio of the molar mass of CO2 to the atomic mass of carbon-12. With the addition of mF, our carbon cycle now looks like the diagram below. The time t = 0 yr is the moment just before we start we start adding mF to the atmosphere.


Or, expressed as two differential equations, it looks like this:


We can solve these differential equations in the same way we already solved those of carbon-14 concentration. Examining the equations, we see that we can obtain the behavior of non-radioactive carbon by setting the decay constant, γ, to zero and inserting the the human emission of carbon in place of the cosmic ray creation of carbon-14. More convenient than the analytic solution for our purposes, however, is a numerical solution that we can implement in a spreadsheet and combine with the historical and projected values for mF as our study progresses.

In the numerical solution, we pick a time step small enough that changes in the masses and transfer rates are negligible during the step. Suppose our step is δt. Provided δt is small enough, we can assume, for example, that mA during each step is equal to kA times the value of MA at the beginning of the step. We do not have to account for the fact that MA may be changing during the step, because the time step is so small these changes will be negligible compared to MA. For our carbon cycle, it turns out that one year is always a small enough time step, and in long, slow developments, ten years is small enough.

The following equations are the numerical equivalents of our differential equations. They show how we calculate MA and MR at time tt using their values at time t.


The current rate at which humans emit carbon by burning fossil fuels is close to, 10 Pg/yr. In our next post, we will set mF to a constant 10 Pg/yr and calculate how our natural, equilibrium atmosphere responds over ten thousand years.

PS. If you find our carbon cycle drawing too confusing and drab, you can try this one drawn by my youngest son.

Thursday, December 17, 2015

Carbon Cycle: Rate of Transfer

Our study of the radioactive isotope carbon-14, which we began in Carbon-14: Origins and Reservoir, led us to the conclusion that roughly one in every eighteen atmospheric carbon atoms are absorbed by a vast oceanic reservoir each year. When the atmosphere and the reservoir are in equilibrium, the same amount of carbon flows out of the reservoir as into it, so that the mass of carbon in the atmosphere remains constant. But the oceanic reservoir contains over one hundred times as much carbon as the atmosphere.

We now embark upon a series of posts in which we calculate the effect of mankind's carbon emissions upon the atmospheric CO2 concentration. Our starting point for these calculations will be the natural, equilibrium atmosphere and oceans of the late nineteenth century. This atmosphere contains 300 ppmv CO2, which implies a total atmospheric carbon mass of 650 Pg (see here). The oceanic reservoir contains 77,000 Pg, and each year 37 Pg of carbon is exchanged between the atmosphere and the reservoir (see here).

As we showed in our previous post, the probability of an atmospheric carbon atom being transferred into the oceanic reservoir is independent of the number of carbon atoms in the atmosphere. The number of carbon atoms moving into the reservoir is equal to the total number of carbon atoms in the atmosphere divided by eighteen. If we have twice as many carbon atoms, the rate at which they move into the oceanic reservoir will double. If mA is the mass of carbon moving into the reservoir every year, and MA is the mass of carbon in the atmosphere, we must have:

mA = kAMA, where kA = 37 Pg/yr ÷ 650 Pg = 0.057 Pg/yr/Pg.

The probability of a carbon atom in the oceanic reservoir being released into the atmosphere is likewise independent of the number of carbon atoms in the reservoir. If mR is the mass of carbon leaving the reservoir every year and MR is the mass of carbon in the reservoir, we must have:

mR = kRMR, where kR = 37 Pg/yr ÷ 77,000 Pg = 0.00048 Pg/yr/Pg.

If we were to double suddenly the mass of carbon in the atmosphere, carbon would start to move into the oceanic reservoir at double the rate. The graph of atmospheric carbon dioxide concentration would look almost exactly like the graph of carbon-14 concentration after the nuclear bomb tests, which we present here. The movement of carbon between the atmosphere and the oceanic reservoir is governed by almost exactly the same equations as the movement of carbon-14. The only difference is that carbon-14 decays, while carbon-12 lasts forever.

UPDATE: The above calculations are consistent with Henry's Law of gasses dissolving in liquids. Henry's Law applies when the concentration of gas in the liquid has reached equilibrium at a particular temperature with the concentration of gas above the liquid. At equilibrium, the rate at which the liquid emits the gas is equal to the rate at which the liquid absorbs the gas. Henry's Law states that the equilibrium concentration of the gas in the liquid at a particular temperature is proportional to the partial pressure of the gas above the liquid. According to Boyle's Law, the partial pressure of a gas at a particular temperature is proportional to the number of gas molecules per unit volume. In our rate of transfer equations, the rate at which CO2 is absorbed by the ocean is proportional to the concentration of CO2 in the atmosphere, and the rate at which CO2 is emitted is proportional to the concentration in the liquid. If we double the concentration of CO2 in the atmosphere, our equations tell us that the rate of emission by the oceans will equal the rate of absorption only if the concentration of CO2 in the oceans also doubles, which is precisely what Henry's Law requires. Of course, we have not yet considered how changing the temperature of the atmosphere and ocean will affect the rates of transfer, but we will get to that later.

Wednesday, December 9, 2015

Carbon-14: Probability of Exchange

In the carbon cycle of our natural, equilibrium atmosphere, each carbon atom in the atmosphere has a certain probability each year of being absorbed by the reservoir. We call this the probability of exchange in the atmosphere. According to our calculations, 37 Pg of carbon is absorbed by the reservoir each year. Meanwhile, the total mass of carbon in the atmosphere is 650 Pg. To the first approximation, the probability of exchange in the atmosphere is 5.7% per year.

Likewise, a carbon atom in the reservoir has a probability of being released into the atmosphere each year. We have 37 Pg of carbon emerging from the reservoir each year, and the reservoir contains 77,000 Pg of carbon, so the probability of exchange for the reservoir is 0.048% per year.

Almost all carbon in the atmosphere is bound up in CO2. In our previous post we showed that the reservoir is the deep ocean. When a carbon atom enters the deep ocean, it does so as part of a CO2 molecule. The CO2 molecule arrives by chance at the ocean surface, and by further chance it dissolves into the salty water. The CO2 molecule turns into some kind of carbonate ion. This ion mixes down through the top thousand meters of water until it reaches the deep ocean. The carbon atom is now part of the reservoir. The probability of this happening each year is 5.7% for each and every CO2 molecule in the atmosphere.

Likewise, the probability of any carbon atom in the reservoir emerging into the atmosphere as part of a new CO2 molecule each year is 0.048%. The carbonate ion containing the carbon atom mixes up through the top one thousand meters of the ocean, arrives by chance at the surface, and by further chance emerges from the surface as an atmospheric CO2 molecule.

The exchange of carbon between the atmosphere and the ocean is a first-order chemical process. During the process, each carbon atom is acting alone. It does not require the cooperation of any catalyst to permit it to be dissolved in saltwater or released from saltwater. If we were to double the number of CO2 molecules in the atmosphere, so that six hundred out of every million air molecules were CO2 instead of only three hundred, the probability of any one of them being absorbed by the reservoir each year would remain the same.

If the reservoir were something more complex than the ocean, such as a forest, we would be unable to assert that the probability of exchange was unaffected by the number of CO2 molecules in the atmosphere. A forest needs water and sunlight to convert CO2 into sugar and oxygen. If we double the number of CO2 molecules in the atmosphere, we might find that CO2 molecules are lining up inside forest leaves waiting for enough water and sunlight to arrive before they are turned into plant matter. But our reservoir is the ocean, and entering and leaving it is a statistical process in which each carbon atom acts in isolation.

So far, we have assumed that the atmosphere and reservoir are staying at the same temperature. They could be at different temperatures, but they are neither warming nor cooling. But we note that the probability of exchange is strongly affected by temperature. We have only to look at the decreasing solubility of CO2 in water with temperature, as presented here, to see that this strong effect must exist.

For now, we assume our natural, equilibrium atmosphere, and its carbon reservoir, are neither cooling nor warming. The probability of exchange in the atmosphere remains constant at 5.7% per year, even if we halve or double the atmosphere's CO2 concentration. The probability of exchange in the reservoir remains constant at 0.048% per year, even if the mass of carbon in the reservoir halves or doubles. The probability of exchange is independent of concentration.

This concludes our series of posts on carbon-14. In our upcoming posts, we will apply what carbon-14 has taught us about the Earth's carbon cycle to predict how human CO2 emissions will affect the CO2 concentration of the atmosphere.

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 atmosphere, 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
Concentration
(ppt)
Relaxation
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.