6.3 The carbon cycle

$CO_{2}(t)$ is calculated as a function of the concentration at the previous time step by a mass budget equation:

$$CO_{2}(t)=CO_{2}(t-dt)+F(t)cdotfrac{CO_{2}^{act}}{M_{CO_{2}}^{act}}cdot dt$$

where $CO_{2}(t)$ is the $CO_{2}$ concentration in ppm and $F(t)$ is the $CO_{2}$ flux towards the atmosphere in GtC/year. Note that $CO_{2}$ fluxes are expressed in GtC/year of Carbone. To convert these fluxes in Gt of $CO_{2}$ per year, you need to multiply the fluxes by 44/12. The factor $frac{CO_{2}^{act}}{M_{CO_{2}}^{act}}$ allows us to convert a $CO_{2}$ mass in Gt ($10^{9}$t) into a concentration in ppm: $M_{CO_{2}}^{act}$ is the $CO_{2}$ mass in the present-day atmosphere (750 Gt) and $CO_{2}^{act}$ is the present-day $CO_{2}$ concentration (405 ppm).

The $CO_{2}$ flux, $F(t)$, is the sum of several contributions:

• anthropogenic emissions;
• emissions associated with volcanism and oceanic ridge activity, $F_{volc}$. By default, $F_{volc}$=0.0083 GtC/year;
• Biological storage, i. e. the storage of organic matter in fossil form (oil, coal);
• continental alteration;
• $CO_{2}$ exchanges between the atmosphere and the ocean;
• absorption of some of the emissions by the ocean and vegetation.

Anthropogenic and volcanic emissions are assumed to be constant throughout the simulation.

6.3.1 Biological storage and continental alteration

We assume that the $CO_{2}$ fluxes leaving the atmosphere by biological storage and continental alteration are proportional to the $CO_{2}(t)$ concentration, by analogy with chemical reactions in which $CO_{2}$ is the reagent:

$$F_{conso}(t)=-scdot CO_{2}(t)$$

where $s$ is the $CO_{2}$ consumption rate in GtC/ppm/year.

The user chooses the consumption rate of $CO_{2}$ by biological storage $s_{bio}$ and by continental alteration $s_{alt}$. When the Earth is completely frozen (snowball), these consumption rates are canceled regardless of the choice of the user: in fact, freezing does not allow the consumption of $CO_{2}$ by these processes, which allows the exit of the snowball.

By default, $s_{alt}$ is such that continental alteration balances volcanism for long time scales: $s_{alt}^{ref}=frac{F_{volc}}{CO_{2}^{ref}}$. $s_{bio}$is null by default, because the current biological storage can be neglected. At Carboniferous, $s_{bio}$=-0.0014 GtC/ppm/year, according to the $CO_{2}$ fluxes reconstructed at that time ([Berner, 2003]).

6.3.2 $CO_{2}$ solubility in the ocean

In nature, the $CO_{2}$ solubility in the ocean depends on the temperature. As a result, an increase in temperature leads to $CO_{2}$ degassing into the atmosphere whereas a decrease in temperature leads to $CO_{2}$ pumping into the ocean. This phenomenon acts on time scales of a few thousand years, and probably played a role in $CO_{2}$ variations observed during glacial-interglacial oscillations (section 3.2.3).

In the model, this is represented by a flux $F_{oce}$, in GtC/year, written as:

$$F_{oce}=frac{1}{tau_{oce}}cdotfrac{CO_{2}^{act}}{M_{CO_{2}}^{act}}cdotleft(CO_{2}^{eq}(T)-CO_{2}(t)right)$$

where $CO_{2}^{eq}(T)$ is the atmospheric $CO_{2}$ concentration in equilibrium with the ocean at temperature $T$ and $tau_{oce}$ is the relaxation time scale of the $CO_{2}$ concentration towards this equilibrium.

$CO_{2}^{eq}(T)$ is parameterized according to the temperature so that (1) a cooling of 10°C (e.g. interglacial-glacial cooling) leads to a reduction of $CO_{2}$ concentration down to 180 ppm (2) the model simulates a 1°C increase for a 90 ppm increase from the pre-industrial period to present-day. This function is empirical rather than physical.

6.3.3 Absorption of part of $CO_{2}$ emissions by the ocean and the vegetation

The aim is to represent in a simple way that the superficial ocean and the vegetation absorb some of the $CO_{2}$ emissions: it is estimated, for example, that at present 35% of the current anthropogenic emissions are absorbed by the vegetation and 20% by the ocean. This plays a role especially at small time scales. In the model, we multiplies the $CO_{2}$ fluxes by $1-puit_{bio}-puit_{oce}$, where $puit_{bio}$=35% and $puit_{oce}$=20%.