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, M_R is the mass of carbon in the oceanic reservoir, M_A is the mass of carbon in the atmosphere, k_R is the fraction of oceanic reservoir CO2 molecules that will be emitted into the atmosphere each year, and k_A 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 k_RM_R, and the rate at which it is absorbed by the oceanic reservoir is k_AM_A. At equilibrium, these two rates will be the same, so we have:

k_RM_R = k_AM_A      ⇒     M_A = k_RM_R/k_A      (Eq. 1)

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

M_A = k_RM_R/k_A ∝ e^−2300/T      (Eq.2)

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

M_A(T₂) / M_A(T₁) = e^−2300/T₂ / e^−2300/T₁ = e^{2300(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 e^{2300(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 e^2300/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.

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, k_A and k_R, will change with temperature. We recall that k_A is the fraction of atmospheric CO2 molecules that will be absorbed by the oceanic reservoir each year, and k_R is the fraction of oceanic reservoir CO2 molecules that will be emitted into the atmosphere each year.

Claim: To the first approximation, k_A 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 k_A.

Claim: To the first approximation, K_R 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. (Compare to 0.42 eV latent heat of vaporization and −0.062 eV latent heat of fusion for an H20 molecule, and 0.26 eV latent heat of sublimation for a 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 e^−E/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 e^−E/kT. Thus k_R is proportional to e^−E/kT, which means there is some constant, A, with units of 1/yr, for which k_R = A e^{−0.2/0.0000862T} = Ae^−2300/T. In our natural, equilibrium carbon cycle, k_R = 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.

k_R @ 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.

k_R @ 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 k_R 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 k_R by 2.8%. We set the fossil fuel emission, m_F, 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.