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 C_A are on the left side of (1) and all terms in C_R 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 C_A, C_R, and time, t. All other parameters are constants that we have already calculated. We must eliminate C_R from the (3) so as to obtain an equation in C_A and t alone, which we can then solve. We eliminate C_R by multiplying (3) by the same factor that we observe on the left side of (4).



We do the same thing for C_R, arriving at a differential equation in only C_R 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 m_p^C14, γ, M_A, M_R, and m_e 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 C_A, which comes out as 1.0 ppt, which is what we expect, because we chose the value of m_e and M_R to make sure that the equilibrium concentration would be 1.0 ppt. Our equation for C_A is as follows, where concentration is in ppt and time is in years.

C_A = 1.0 − 0.2 e^−t/17 − 0.8 e^−t/8200.

At t = 0, we have C_A = 0 ppt, and when t→∞, C_A→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 C_R.



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 C_R is,

C_R = 0.8 + 0.002 e^−t/17 − 0.802 e^−t/8200.

At t = 0, we have C_R = 0 ppt, and dC_R/dt = k₁α+k₂β = 0.0 ppt/yr, and d²C_R/dt² = k₁α²+k₂β² > 0, all of which we expect, and also when t→∞, we have C_R→0.8 ppt, which is one of our starting assumptions.

We are pleased to have an analytic solution for C_A and C_R. A numerical solution to the differential equations turns out to be unstable for time steps greater than ten years. We want to plot C_A and C_R 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.

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.

CR (ppt)	me (Pg/yr of Carbon)	MR (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, C_R, carbon mass exchange rate in Pg/yr, m_e, and reservoir carbon mass in Pg, M_R. 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.

Carbon 14: Size of the Carbon Reservoir

Carbon-14 is produced at roughly 7.5 kg/yr in the atmosphere. Its atmospheric concentration is one part per trillion (1.0 ppt). Its half-life is 5700 years. These three observations imply the existence of the carbon cycle we described in our previous post. Our graphical representation of the carbon cycle uses the quantities listed in the table below. Together, these quantities provide a complete description of the cycle. Their numerical values apply to our natural, equilibrium atmosphere, which is based upon the Earth's atmosphere at the end of the nineteenth century. So far, we have values for all but two of these quantities: the mass of carbon in the reservoir, and the mass of carbon exchanged each year between the reservoir and the atmosphere.

Symbol	Quantity	Value
mpC14	production rate of carbon-14 in atmosphere	7.5 kg/yr
γ	decay rate of carbon-14	0.00012 kg/kg/yr
MA	mass of carbon in the atmosphere	650 Pg
CA	concentration of carbon-14 in atmosphere	1.0 ppt at equilibrium
MR	mass of carbon in the reservoir	unknown, Pg
me	exchange rate of CO2 between atmosphere and reservoir	unknown, Pg/yr
CR	concentration of carbon-14 in reservoir	0.8 ppt at equilibrium

Table: Carbon Cycle Quantities. We have 1 Pg = one petagram = 10¹² kg = 10¹⁵ g.
Another way to represent the carbon cycle is with two differential equations, as shown below, where d/dt represents the rate of change of a quantity with time, where t is time in years. Thus dC_A/dt is the rate of change of the concentration of carbon-14 in the atmosphere with time, in units of ppt/yr.


When our carbon cycle reaches equilibrium, the concentration of carbon-14 in the atmosphere and the reservoir settles to a constant value, so the d/dt terms are zero. This observation reduces our two differential equations to two simple algebraic equations, and we can solve for our two unknown quantities.


We have 37 Pg of carbon transported into the reservoir every year, including 37.0 kg of carbon-14 because the atmosphere's concentration of carbon-14 is 1.0 ppt. Another 37 Pg of carbon is emitted by the reservoir every year, including 29.6 kg of carbon-14 because the reservoir's concentration of carbon-14 is 0.8 ppt. The net flow of carbon-14 is 7.4 kg/yr into the reservoir (subtract 29.6 kg/yr from 37 kg/yr). The reservoir itself, meanwhile, contains 77,000 Pg of carbon, of which 62,000 kg is carbon-14. The reservoir's carbon-14 decays at 7.4 kg/yr (multiply 62,000 kg by 0.0012 kg/kg/yr ). As is required by the state of equilibrium, the net flow of carbon-14 into the reservoir each year is equal to the amount of carbon-14 that decays in the reservoir each year. We also have 0.1 kg of carbon-14 decaying in the atmosphere each year, so the total mass of carbon-14 that decays each year is 7.5 kg, which is equal to the amount that cosmic rays create, which is also required by the state of equilibrium.

Although our quantities deal only with masses of carbon, we note that all exchanges of carbon between the atmosphere and the reservoir take place in the form of exchanges of CO2. In particular, all carbon-14 in the atmosphere is bound up in CO2. We assumed earlier in our argument that the residence time of carbon carbon-14 in the atmosphere was much longer than the two months it takes for newly-created carbon-14 to react with oxygen to form CO2. We can now check the validity of this assumption: we have 650 Pg of carbon in the atmosphere, and 37 Pg of carbon flowing from the atmosphere into the reservoir each year. The residence time of a carbon-14 atom in the atmosphere is eighteen years (divide 650 Pg by 37 Pg/yr), which is much longer than two months, so our assumption turns out to be correct.

We conclude that our natural, equilibrium atmosphere exchanges 37 Pg of carbon each year with a reservoir that contains 77,000 Pg of carbon. Furthermore, because all carbon in the atmosphere is in the form of CO2, and all exchanges of carbon with the reservoir are in the form of CO2, the atmosphere exchanges 140 Pg of CO2 with the reservoir every year, and the reservoir itself contains 280,000 Pg CO2 (multiply by carbon mass by the molecular weight of CO2 and divide by the atomic weight of carbon). Meanwhile, the 650 Pg of atmospheric carbon is contained in 2,400 Pg of atmospheric CO2. That is to say: the carbon reservoir is a CO2 reservoir, and it contains more than a hundred times as much CO2 as the atmosphere.

Carbon-14: The Carbon Cycle

When a carbon-14 atom is created in the atmosphere by a cosmic ray, it quickly combines with oxygen to form carbon monoxide (CO). In a couple of months, this carbon monoxide combines with more oxygen to form radioactive carbon dioxide (CO2). As we showed previously, almost all carbon-14 created in the atmosphere ends up in a reservoir outside the atmosphere. But how long does the average carbon-14 atom spend in the atmosphere before it leaves? Our equilibrium atmosphere contains 650 kg of carbon-14, while 7.5 kg are created by cosmic rays every year. Almost one century's worth of carbon-14 production is stored in the atmosphere. It is most likely, therefore, that the averge carbon-14 atom spends several years in the atmosphere before it passes into the reservoir. Let us assume, for now, that the residence time of carbon-14 in the atmosphere is much longer than the two months it takes for carbon-14 to be bound up into a CO2 molecule. We will come back and check the validity of this assumption later. For now, we assume that all carbon-14 in the atmosphere is bound up in radioactive CO2.

The composition of the atmosphere is such that over 99.5% of its carbon is in the form of CO2. When we observe that one in a trillion carbon atoms is carbon-14, this is equivalent to saying that one in a trillion CO2 molecules is radioactive CO2. But radioactive CO2 is chemically identical to normal CO2. The extra two neutrons in its carbon nucleus have no effect upon its interactions with other molecules. When one molecule of radioactive CO2 leaves an atmosphere that contains 1 ppt (one part per trillion) of carbon-14, it does so in the company of one trillion normal CO2 molecules. When one carbon-14 atom leaves the atmosphere, it does so with one trillion other carbon atoms. The reservoir of carbon-14 that must exist outside the atmosphere must also be a much larger reservoir of normal carbon. The concentration of carbon-14 in this reservoir cannot be greater than in the atmosphere, because the atmosphere is where the carbon-14 is created. We already calculated that the reservoir contains 62 Mg of carbon-14, so it must also contain at least 62,000 Pg (sixty-two thousand Petagrams) of normal carbon (divide the mass of carbon-14 by the maximum possible concentration of carbon-14 in the reservoir). The reservoir contains one hundred times as much carbon as our equilibrium atmosphere.

In order for carbon-14 to leave the atmosphere, it must be carried by radioactive carbon dioxide, which in turn means that there must be a trillion times as much normal carbon dioxide leaving the atmosphere. But our equilibirum atmosphere is, by assumption, in in equilibrium. Its carbon dioxide content, and therefore its carbon content, is constant. If m_e kilograms of carbon leave the atmosphere every year and enter the reservoir, we must have the same m_e kilograms of carbon leaving the reservoir and entering the atmosphere every year. Thus the carbon content of the reservoir remains constant as well.

If the carbon-14 concentration of the reservoir were the same as the atmosphere's, we would have the same amount of carbon-14 leaving the atmosphere as returning, because the amount of normal carbon leaving is the same as the amount returning. Therefore, the concentration of carbon-14 in the reservoir must be lower than in the atmosphere. Let the concentration in the atmosphere be C_A and in the reservoir be C_R. The net flow of carbon-14 out of the atmosphere will be m_e(C_A−C_R), which we already calculated to be 7.4 kg/yr.

It remains for us to estimate the equilibrium concentration of carbon-14 in our carbon reservoir. It is well known that carbon is stored in vegetation and in the oceans. One kilogram of water at 14°C will hold around 0.5 g of carbon in the form of dissolved CO2. Given that the mass of the oean is roughly 1.4×10²¹ kg, the oceans have have the potential to store up to 700,000 Pg of carbon. The Earth's biomass, meanwhile, appears to contain only 2,000 Pg of carbon. So we will assume that the majority of the Earth's carbon reservoir is in the oceans. The concentration of carbon-14 in the deep oceans was measured by Bien et al. to be around 80% of the concentration in the atmosphere. So we will assume that the equilibrium concentration of carbon-14 in the Earth's carbon-14 reservoir is 0.8 ppt.

The diagram below illustrates what we have concluded so far about the exchange of carbon between the atmosphere and the reservoir. The masses of carbon in the atmosphere and the reservoir are M_A and M_R respectively. The decay rate of carbon-14 is γ.


The diagram assumes that the carbon exchange between the atmosphere and the reservoir has already reached equilibrium. But it does not assume that the carbon-14 concentration has reached equilibrium. We are going to obtain an analytic solution for the evolution of carbon-14 concentration from a starting point of no carbon-14 at all. So instead of pre-supposing that we have already reached carbon-14 equilibrium, the diagram states that, given infinite time, the carbon-14 concentration in the atmosphere and reservoir will eventually reach equilibrium at 1.0 and 0.8 ppt respectively.

What is expressed in the diagram we can write down in two differential equations with some boundary conditions. The two equations contain two unknown constants: the carbon exchange rate, m_e, and the size of the carbon reservoir, M_R. In our next post, we will use these two differential equations to deduce the values of m_e and M_R.

Carbon-14: Removal from the Atmosphere

We are going to estimate the mass of carbon-14 in the Earth's atmosphere prior to our burning significant quantities of fossil fuels, and prior to our detonating atomic bombs. We will use the Earth's atmosphere at the end of the nineteenth century as an approximation to the atmosphere in its natural, equilibrium state, undisturbed by human activity.

The concentration of carbon dioxide in our equilibrium atmosphere is 300 ppmv (parts per million by volume, as estimated for the turn of the nineteenth century here). The total mass of the atmosphere is 5.3×10¹⁸ kg. (Atmospheric pressure is generated by the weight of the atmosphere per square meter, so divide sea-level atmospheric pressure by gravitational acceleration and multiply by the surface area of the Earth to obtain atmospheric mass.) In accordance with the gas law, the density of CO2 is 1.5 times higher than the density of air (the molar mass of CO2 is 44 g, and of air is 29 g). Thus 300 ppmv of CO2 in the atmosphere is the same as 450 ppm (parts per million by mass). The mass of CO2 in our equilibrium atmosphere is 2.4×10¹⁵ kg (450 ppm of 5.3×10¹⁸ kg). The molar mass of CO2 is 44 g, and that of carbon is 12 g, so the mass of carbon in the atmosphere is 6.5×10¹⁴ kg (2.4×10¹⁵ kg × 12 g ÷ 44 g). We will use petagrams (Pg) to represent large masses, where 1 Pg = 10¹² kg = 10¹⁵ g. Our equilibrium atmosphere contains 650 Pg of carbon.

The concentration of carbon-14 in our equilibrium atmosphere is 1.0 ppt (parts per trillion by mass). Almost all carbon in the atmosphere is contained in CO2, so the mass of carbon-14 in our equilibrium atmosphere is 650 kg (650 Pg of CO2 × 1ppt). As we already showed, the equilibrium reservoir of carbon-14 on Earth is 62 Mg (7.5 kg/yr production by cosmic rays ÷ 0.00012 kg/kg/yr decay rate = 62,500 kg = 62 Mg). Of this reservoir, only 1% is to be found in the atmosphere. From now on, when we refer to the Earth's carbon-14 reservoir we will be referring to the 62 Mg that is not in the atmosphere.

The 650 kg of carbon-14 in our equilibrium atmosphere decays at 0.078 kg/yr (650 kg × 0.00012 kg/kg/yr) and is added to by cosmic ray production of 7.5 kg/yr. In order for the carbon-14 content of the atmosphere to remain constant, carbon-14 must pass out of the atmosphere at 7.4 kg/yr. Let us suppose, for the sake of argument, that this 7.4 kg/yr does not pass into the carbon-14 reservoir. In that case, the 7.4 kg/yr goes somewhere else, and a new reservoir starts to build up, while the existing reservoir decays, which would mean that our carbon-14 reservoir would not be in equilibrium, which contradicts our observation that the reservoir had millions of years to reach equilibrium before the nineteenth century. Thus 7.4 kg/yr of carbon-14 must pass directly from the atmosphere into the reservoir. It could be that the reservoir contains many sub-divisions communicating with one another in complex ways, but this does not alter the fact that 7.4 kg of carbon-14 is passing out of the atmosphere and into the reservoir every year.

The figure below illustrates the origin and fate of Carbon-14 on Earth. We use M for mass, and m for mass flow. We use subscript A for atmosphere, R for reservoir, D for decay, T for transfer, and P for production. Superscript C14 means carbon-14.


Where is the Earth's carbon-14 reservoir? How does it acquire 7.4 kg of carbon-14 from the atmosphere every year?

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 7×10¹⁵ 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.

Scientific Method and Anthropogenic Global Warming

Suppose we have ten sick people. Dr. Quack persuades them to take his Patent Medicine. Five of them die and five of them recover. Dr. Quack says, "My medicine saved five lives!" But Dr. Nay says, "Nonsense, your medicine killed five people." Both claims are 100% consistent with the facts, and yet they are contradictory. Consistency with the facts is a necessary quality for a scientific theory, but it is not sufficient. If we allow consistency with observation to be sufficient proof of a theory, we are practicing pseudoscience. The Food and Drug Administration (FDA) rejects both Dr. Quack and Dr. Nay's claims, saying the Patent Medicine has absolutely no effect upon the recovery of the patients until experiment has proved otherwise.

The FDA has adopted the null hypothesis. The null hypothesis is the foundation of scientific reasoning. We determine the null hypothesis with Occam's Razor, by which we cut off any unnecessary parts of our hypothesis until we arrive at the simplest possible explanation of our observations. And the simplest possible theory about the relationship between one thing and another is that there is no relationship at all. The only way to disprove the null hypothesis, according to scientific method, is with observations. Compelling arguments and sensible speculation are insufficient, nor is consensus among scientists, nor the authority of experts.

When it comes to the climate, our initial null hypothesis is that the climate does not vary at all from one year to the next, nor does its carbon dioxide concentration. We will have to disprove this with observations before we can begin to discuss how humans might cause climate change. So let us look at what observations are available to disprove this hypothesis. We observe valleys carved by glaciers in warm climates. We find fossils of tropical plants in cold places. We pull out ice cores and they suggest the Earth's temperature has varied dramatically in the past few hundred thousand years, along with its atmospheric carbon dioxide concentration (6-16°C and 200-300 ppm when measured with a 1000-year running average). These observations and many others disprove our null hypothesis. The climate does vary. It varies dramatically and naturally, whithout any human influence. This is our new null hypothesis. We call it the theory of natural variation.

We can use the infra-red absorption spectrum of carbon dioxide to argue that doubling the carbon dioxide in the atmosphere will warm the planet (by roughly 1.5°C if we ignore clouds, and 0.9°C if we account for clouds, according to our own simulations). Before we can hope to show that human carbon dioxide is affecting the climate, however, we must show that human carbon dioxide emissions affect the atmospheric carbon dioxide concentration, and on this subject our theory of natural variation states that human emissions have no effect. So we must disprove the theory of natural variation. Natural emission and absorption of carbon dioxide were already in equilibrium before man started burning fossil fuels. Our annual emissions are only 4% of the natural emissions (8 Pg/yr from burning fossil fules compared to 200 Pg/yr of natural emission). In the simplest chemical equilibriums, absorption is proportional to concentration, so our 4% increase in emission will, to the first approximation, cause atmospheric CO2 to increase by 4%, or 10 ppm. Such an increase is so slight that it's not clear how we could distinguish it from the larger natural variations.

Even if we could prove that the recent increase from 330-400 ppm atmospheric carbon dioxide was due to our burning of fossil fuels, we would still have to prove that increasing carbon dioxide concentration causes the world to warm up. We may have a compelling reason to suspect that is has this effect, but we cannot abandon the theory of natural variation until it has been disproved, and we need observations that contradict the theory to disprove it. We cannot disprove the theory with compelling arguments alone. Even if we accept that increasing carbon dioxide traps heat somewhere in the atmosphere, this does not neccessarily mean that the climate will, as a whole, warm up. It may seem obvious that lighting a fire in my fireplace will warm my house, but the reality is that my children's bedrooms get cold when I light a fire. That's because the thermostat is near the fireplace, so when I light the fire the radiators turn off. The climate is a complex system. It could be that it contains similar surprises, where we observe the opposite effect to the one we expected. In fact, looking at the ice cores, it appears that atmospheric carbon dioxide increases occurr one thousand years after increases in temperature, which could mean that carbon dioxide somehow stops the world from getting any warmer at the end of an ice age, by a process that we don't understand.

So far as we can tell, scientific method, when applied to our observations of the Earth's climate, arrives at the assumption that the climate varies dramatically and naturally, and that human carbon dioxide emissions have no effect upon it. We look forward to seeing this assumption disproved by observations of nature, but so far we have been disappointed.