CRed - The Community Carbon Reduction Project at UNC-Chapel Hill

Modeling the Carbon Cycle

Mathematical models can be used to predict the impact of a particular policy on atmospheric carbon. Models such as those used by the Intergovernmental Panel on Climate Change (IPCC) have been developed to simulate the emissions of greenhouse gases, their dynamics in the earth system, resulting concentrations in the atmosphere, and the effect on temperature and climate. The analyses performed on this web site use a simpler, reduced-scale model of the carbon cycle that is significantly easier to operate (and hence are useful in education and in policy analyses) and yet mimics the predictions of the more complex models within the range of uncertainty inherent in all such models. It was produced during an NSF sponsored project: Visualization Technology in Environmental Curricula, and the full model and associated materials can be obtained from that web site.

Scientific understanding of global warming begins with the study of how carbon dioxide moves throughout the environment - described by the carbon cycle shown in the upper left (green) portion of Figure 1. The cycle depicts the flow of carbon regardless of its chemical and physical form, since the model used in this analysis depicts total carbon content of a compartment. Carbon mass in the atmosphere can be converted to carbon dioxide mass in the atmosphere by multiplying the carbon mass by approximately 44/12 (the ratio of the mass of a carbon dioxide molecule to that of carbon atom).

Figure 1:  The compartments and processes of the reduced-scale model

While the cycle is more complex than shown in the figure above, the reduced scale model used here focuses on six primary compartments of the earth system because they dominate carbon fluxes. These are the five compartments in green- soil/litter, atmosphere, flora, mixing ocean, and deep earth- plus society in light purple and tan (specifically, the infrastructure). The flows of carbon between the compartments are represented in units of billions of metric tons of carbon per year, or BMT/year. The flow of carbon in and out of flora is controlled by land characteristics, shown in the figure in yellow.

The emission term from society to the atmosphere contains two components: the per capita emissions of carbon (carbon released to the atmosphere per person per year, with units of BMT/person-year) and the population size (units of "people"). These two quantities are specified separately for the developed and developing worlds, as they differ significantly. The product of these two components for the developed world yields the emission rate from society to the atmosphere for the developed world (in units of BMT/year), with a similar product for the developing world. The per capita emissions term may be broken down further into terms from the industrial, commercial/residential and transportation sectors, and into fuel types in these energy sectors. The per capita emission rate is a function of per capita energy needs, the efficiency at which these needs are met, and an emissions factor (mass of carbon released as carbon dioxide per unit fuel consumed). For a complete description of the model, and to download it for use, please see the Global Warming Module page.

Is the model valid?

In order to verify that the predictions of the model are reasonable, it is necessary to compare them with real world data or with the predictions of other accepted models. By comparing the projections of the reduced-scale model with those of the IPCC (Intergovernmental Panel on Climate Change, Working Group I. Climate Change 2001: The Scientific Basis), it has been established that the predictions of the reduced-scale model fall within an acceptable range of differences between the present model and the IPCC models, illustrated in Figure 2.

Figure 2: Comparison of reduced-scale model with IPCC predictions. The different IPCC predictions refer to different scenarios of society’s growth , although at the lower range of these predictions. The values of k refer to the fractional annual rate of increase in per capita energy consumption, currently between 0.01 and 0.02. .

What would a 60% reduction accomplish?

The 1997 Kyoto Protocol aims to stabilize carbon concentration at a level no higher than twice the level present in the atmosphere prior to the Industrial Revolution (the horizontal line in Figure 2). This means that the concentration of carbon in the atmosphere should not exceed 550 ppm, or approximately 1160 billion metric tons of carbon. At the current rate of increase, concentrations may exceed this level in less than seventy years (See Figure 3). The Energy White Paper states that a reduction in carbon dioxide emissions of 60% by 2050 in the developed world will be necessary to achieve stabilization at or below 550 ppm.

Figure 3: Predicted world atmospheric carbon levels under various scenarios

Figure 3 also shows the predicted effect of reducing carbon dioxide emissions by 60%, using the reduced-scale model employed in the present report. PCPgrowth refers to an annual increase in per capita energy consumption and carbon dioxide production following the initial 60% reduction. Note that most of these scenarios keep the carbon dioxide concentration from doubling before 2100.

How are the predictions made in the model?

The principle of conservation of mass is used in the model to produce the equations describing the rates of change of carbon in the compartments shown in green in Figure 1. The difference between the rate at which carbon is moving into a compartment and the rate at which it is moving out of the compartment must equal the rate at which carbon is accumulating in or disappearing from that compartment.

(1) Rate of change of carbon in a compartment =

rate of carbon moving into compartment – rate of carbon moving out of compartment

Rates into a compartment are from sources for that compartment, and rates out are to sinks for the compartment. Units of all rates of flow are BMT/year.

The modeling of transfer of carbon between compartments uses two concepts: transfer rates and transfer rate constants. Transfer rates describe the rate at which carbon is flowing from one compartment to the next at any moment. A process in which the transfer rates remain constant over time is a zeroth-order process. For example, the reduced-scale model is based on the assumption that the rates of transfer from the atmosphere to the flora and from the flora to the atmosphere do not depend on time, and are therefore zeroth-order terms.

Some rates of transfer between compartments change as the environmental system evolves. To account for this, transfer rates can be expressed as the mathematical product of two quantities: the amount of carbon in a source compartment at some point in time and a transfer rate constant. A transfer rate constant expresses the fraction of carbon in the source compartment moving to the sink compartment per unit time. For example, a transfer rate constant equal to 0.1 per year means that 10% of the contents of a compartment are transferred to another compartment each year. This type of process is known as a first-order transfer process, in which the rate of flow out of a compartment is proportional to the amount of carbon in that compartment at each moment in time. The mass balance equations generally have a mixture of zeroth-order and first-order terms.

A mass balance equation is a form of differential equation, because it describes the rate at which the amount of carbon in the compartment is changing with time. This rate of change for any compartment i can be written as dNi(t)/dt, where Ni(t) is the amount of carbon in a compartment (compartment i) at a moment in time, t. The equation describing a first-order transfer process can be written as follows:

(2) Rate of transfer of carbon from compartment i to compartment j at time t =

(Transfer rate constant from i to j) x (Amount of carbon in compartment i at time t)

The rate of transfer of carbon is shown by the symbol Rij(t), with the subscript i referring to the compartment from which the carbon is moving, and the subscript j referring to the compartment to which the carbon is moving. The transfer rate constant is shown by the symbol λij. Equation 2 can therefore be written as follows:

(3) Rij = λij x Ni(t)

The subscripts i and j are replaced with letters corresponding the compartments of the environmental system (A for atmosphere, F for flora, O for mixing ocean, and S for soil). For example, the amount of carbon in the atmosphere at any point in time, t, is written as NA(t). The transfer rate constant for the movement of carbon from the atmosphere to the mixing ocean is written as λAO. The rate at which carbon is transferred from the atmosphere into the mixing ocean at any moment in time is then described by the equation:

(4) RAO = λAO x NA(t)

The equations describing the rate of carbon transfer in each of the four main compartments of the environmental system may be written in the same way. All of the equations can be seen on the Global Warming Module web site.

Population growth is controlled by three variables: birth rate (BR), death or mortality rate (MR), and neonatal survival fraction (SF). The general equation is:

(5) Rate of change of the population = rate entering the population – rate leaving the population

The rate of people entering the population at any time can be found by multiplying the birth rate times the neonatal survival fraction (SF(t)); the rate leaving the population is simply the mortality rate. Again, the equations can be seen on the Global Warming Module web site. One of the clear lessons from global warming policy studies is the crucial role played by population growth. As a result, the model developed here allows for control of the population through a constant k. The numerical value of k is the fractional rate at which the population (in either the developed or developing world) is brought into equilibrium through some form of control on birth rates.

As previously mentioned, the rates of transfer from the atmosphere to the flora (RAF) and from the flora to the atmosphere (RFA) are described here by zeroth-order processes controlled by two factors: the total land area devoted to flora (A) and its net primary productivity (NPP):

(6) RAF – RFA = A x NPP

Net primary productivity differs for different land types. The total rate at which carbon is entering the flora from the atmosphere is equal to the sum of the rates for each land type: barren (subscript b), deciduous (subscript d), rainforest (subscript r), cultivated (subscript c), marsh (subscript m), and grassland (subscript g). As a result, the right hand side of Equation 6 may be written as the sum of the terms A x NPP for each of these land types. Again, the full equations may be seen on the Global Warming Module web site.

For the infrastructure, the source (emissions) term, RFF (where FF stands for fossil fuel) is modeled as:

(7) RFF(t) = PCP x POP(t)

where PCP is the per capita production or emissions of carbon and POP(t) is the population at the time for which the source term is being calculated. The model includes a factor, Kenergy, that allows for growth of PCP over time as societies become more energy intensive; it is equal numerically to the fractional growth in energy intensity (PCP) per year. It is the value of k in Figure 2. When this factor is used, the value of PCP at any moment is:

(8) PCP(t) = PCP x e^Kenergy t

PCP in turn equals:

(9) PCP = EEN x RF / EFF

where EEN is the per capita energy need (barrels of oil energy equivalent per person per year); RF is the release factor (BMT carbon released to the atmosphere per barrel oil energy equivalence); and EFF is the efficiency of energy generation, transmission and use combined. In the full model available through the Global Warming Module web site, separate values of PCP are calculated for each of the three energy sectors (industrial, residential/commercial and transportation) and six fuel types considered in the model, and then summed over all 18 terms weighted by the fraction (F) of energy in each sector supplied by a given fuel type; this is done for both the developed and developing worlds. In the simplified model presented here, average values for EEN, RF and EFF (averaged over the energy sectors and fuel types) are provided, allowing you to adjust overall values of PCP in Equation 9 while leaving the current mix of fuels (values of F) implicitly unaltered.

The model equations cannot be solved analytically due to their complexity. As a result, numerical methods are used here. The software STELLA allows for use of either Euler’s, Runge-Kutta 2 or Runge-Kutta 4 methods. Attention must be given to the time-step in these numerical solutions; it is recommended that it be set to 1 year or less to avoid problems with numerical errors. Numerical values for all parameters, including initial values for carbon content of the compartments, are shown in Table 1.

Table 1. Default values for model parameters.