This document describes a model of the carbon cycle and its modification to account for human activities like fossil fuel combustion and deforestation. The model includes five reservoirs - atmosphere, terrestrial biosphere, ocean surface, deep ocean, and soil. Differential equations describe the natural flows between reservoirs. The modified model adds a fossil fuel reservoir and changes flows to represent human impacts. Running the model for 500 years shows increases in atmospheric and soil carbon and a corresponding rise in global temperature of around 3.5 degrees Celsius due primarily to increased CO2 levels.

1. Carbon Cycle and Global Warming
Yilun Tang Prianka Ball
Math 295-03 Final Project, Spring 2016
Abstract
Global warming is the phenomenon of gradual heating on Earth. Scientists have observed the increas-
ing average Earth temperatures since the Industrial Revolution. The impact of global warming includes
potential climate change such as extreme weathers that has been reported globally. Global warming
occurs when carbon dioxide and other greenhouse gases collect in the atmosphere and absorbed sunlight
and solar radiation that have reflected on the Earth’s surface. Investigations on the carbon cycle and
the correlations between greenhouse gases and temperature significantly affect future climate change. In
this project, we model the natural carbon cycle and include the factors of human activities in order to
investigate the effects of greenhouse gases on temperature.

2. 1 Introduction
Scientific evidence has showed that Earth’s climate system is changing. Global warming is used by scientists
to describe the gradual change in temperature due to the increase in concentrations of carbon dioxide and
other greenhouse gases in atmosphere. To understand how global warming is happening, it is important to
understand how the natural carbon cycle works and how changes in the carbon cycle have an impact on
earth’s temperature. Carbon dioxide increases temperature but other greenhouse gases like methane can
increase temperature as much as 21 times as that of CO2. In this paper, we would first understand how
the carbon cycle works and then proceed on to how changes in the carbon cycle due to human activities
is changing the earth’s temperature.Later on we would also see how methane is also changing the Earth’s
temperature. By predicting the future trend of greenhouse gas emissions by taking into account the current
trend of greenhouse gas, we can predict the future temperature increase. This could act as a warning to
decrease current emission and take the right action to keep temperature increase at a minimum.
2 Background Information
2.1 Carbon Cycle
Carbon is more familiar to us in the form of the gas Carbon Dioxide (CO2). Carbon (C) is a very common
element and is widely distributed in the planet. Carbon can be found combined with elements like calcium
and iron in the form rocks, dissolved in oceans and other water bodies and in all living things. Carbon
moves different forms among the four major environmental subsystems: lithosphere (ground and inside the
earth), atmosphere (air surrounding the earth), hydrosphere (lakes, rivers and oceans) and biosphere (all
living things). This movement of Carbon from one system to another is called the Carbon Cycle.
The picture above shows how Carbon moves from one place to another. CO2 from the atmosphere is
taken up by plants through photosynthesis into various other organic compounds. Plants and other animals
give out CO2 into the atmosphere through respiration. CO2 from the atmosphere also gets dissolves in
seawater and some goes back into the atmosphere from the solution. The atmosphere, terrestrial biosphere,
ocean surface, deep ocean, soil are all carbon reservoir where Carbon is stored in different forms. The transfer
of Carbon from one reservoir to another. The source is the origin carbon and the sink is the destination of
the carbon flow. The source and the sink are often one of the reservoirs.
2.2 Climate Change
Carbon Dioxide, Methane, Nitrous Oxide and other greenhouse gases can naturally trap heat in the atmo-
sphere and this effect is referred as the greenhouse effect by scientists. When carbon circulation is in its
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3. natural cycle without excessive human activities, the concentrations of these gases remained relatively stable
until the start of the Industrial Revolution. When human factors are included in carbon cycle, the increase
of heat-trapping emissions in the atmosphere increased.The emissions are created from burning coal, oil, and
gas to generate electricity and driving transportation vehicles.
Since then, greenhouse gas concentrations have risen 44%, increasing the Earth’s global temperature. Natu-
ral changes alone do not explain the temperature changes. The perturbation to the natural carbon cycle by
increasing human activities accounts for the changes.
The other direct consequence of global warming is an increase in both ocean evaporation into the atmosphere,
and the amount of water vapor the atmosphere can hold. High levels of water vapor in the atmosphere allow
weather conditions for heavier precipitation, which lead to intense rain and snow storms.
Some extreme weather events that have been recorded include flooding in Malaysia and India, heat and
droughts in New Zealand and Australia.
3 Model Design
3.1 Carbon Cycle Model
We have developed a simple model to describe the natural carbon cycle with five main reservoirs: atmosphere,
terrestrial biosphere, ocean surface, deep ocean and soil, represented by A(t), T(t), O(t), D(t) and S(t)
respectively.
Table 1: Major Reservoirs in Carbon Cycle
Reservoir Initial Amount of Carbon(Gt)
atmosphere A(t) 750
terrestrial biosphere T(t) 600
ocean surface O(t) 800
deep ocean D(t) 38,000
soil S(t) 1,500
We made a simplifying assumption that, for each process other than marine death, the transfer of carbon
between subsystems is proportional to the amount of carbon in the source. Marine materials sinking to the
deep ocean is assumed to be constant as a natural process, so the carbon flow of marine death is unchanged
instead of being proportional to the amount of carbon in ocean surface. The major fluxes in carbon cycle
are displayed in the table below for processes that are not disturbed by human activities.
Table 2: Major Fluxes in Carbon Cycle
Flux Rate(Gt C/yr) Source Sink
terrestrial photosynthesis 110 atmosphere terrestrial
marine photosynthesis 40 atmosphere ocean surface
terrestrial respiration 55 terrestrial biosphere atmosphere
marine respiration 40 ocean surface atmosphere
carbon dissolving 100 atmosphere ocean surface
evaporation 100 ocean surface atmosphere
upwelling 27 deep ocean ocean surface
downwelling 23 ocean surface deep ocean
marine death 4 ocean surface deep ocean
plant death 55 terrestrial biosphere soil
plant decay 55 soil atmosphere
By dividing the flux by the initial amount of carbon, we were able to calculate the proportionality constants
for all the process at the initial point. The rate of change for each reservoir, remains constant through
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4. time. Thus, the proportionality constants for all processes other than marine death at initial points can be
incorporated into our long term model.
Flux=rate of change * Quantity of Carbon
Processes related to atmosphere reservoir, A(t), can be classified in two categories. Carbon transfer dur-
ing terrestrial photosynthesis, marine photosynthesis and carbon dissolving processes is flowing out of the
atmosphere reservoir, while carbon transfer during terrestrial respiration, marine respiration, evaporation
and plant decay processes is flowing into the atmosphere reservoir. Systems of five differential equations
were established based on the input and output of reservoirs. Similarly we can think about other reservoirs.
Based on what went into the reservoirs and what went out of the reservoirs, we determined the minus and
plus signs in the differential equations. All of these helped us to generate the differential equations.
Figure 1: Carbon Atmosphere
Using all of the processes mentioned about, the differential equations of the five reservoirs are:
Atmosphere :
dA
dt
=
7
40
O +
11
300
S +
11
120
T −
1
3
A
Terrestrial Biosphere :
dT
dt
=
11
75
A −
11
60
T
Ocean Surface :
dO
dt
=
14
75
A +
27
38000
D −
63
800
O − 4
Deep Ocean :
dD
dt
= 4 +
23
800
O −
27
38000
D
Soil :
dS
dt
=
55
600
T −
55
1500
S
Using the differential equations, above, we used Runge-Kutta 4 method to estimate the amount of the carbon
in each of the reservoir.(See attached R code for details). By using Runge Kutta 4 method we will able to
generate 5 different graphs for each of the reservoirs. We will vary the number of time step and time to see
how the the amount of carbon will vary in each of the reservoir.
3.2 Modified Carbon Cycle Model
Climate change happens because of increase of carbon in the atmosphere due to human activities. Additional
human activities can be incorporated into the carbon cycle model mentioned in previous section. Fossil fuel
combustion and deforestation are two major factors in carbon circulation today.
In addition to the five reservoirs in general carbon cycle, fossil fuel reservoir (F(t)) is added to include fossil
fuel combustion. The initial values of carbon in fossil fuel reservoir and the flux is displayed in the table
below.
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5. Reservoir Initial Amount of Carbon(Gt)
fossil fuel deposit F(t) 4,000
When there is combustion, carbon leaves from the fossil fuel deposit and enters the atmosphere. That’s why
during combustion, fossil fuel reservoir is the source and atmosphere is the sink. During deforestation, as
trees will be cut down, carbon in terrestrial biosphere will decrease and carbon in atmosphere will increase
as there will be less trees to take in the carbon from the atmosphere using photosynthesis.The fluxes for
combustion and deforestation are 5 and 1.15 Gt C/year respectively.
Flux Rate(Gt C/yr) Source Sink
combustion 5 fossil fuel deposit atmosphere
deforestation 1.15 terrestrial biosphere atmosphere
Limited by the existing fossil fuel deposits, the rate of change of fossil fuel emissions has constrained growth
with a carrying capacity of 15Gt C/year and growth rate of 0.03/year. Thus differential equation for fossil
fuel emission is as follows.
Fossil Fuel Emission :
dE
dt
= 0.03E(1 −
E
15
)
Using the initial value of fossil fuel reservoir and flux 5 Gt C/year, we found the proportionality constant
0.00125. Fossil fuel is leaving the fossil fuel deposit through fossil fuel emission so 0.00125F is being sub-
tracted. Thus the differential equation becomes:
Fossil Fuel Deposit :
dF
dt
= −0.00125F
Due to deforestation, carbon is escaping the terrestrial biosphere. 1.15 Gt C/year(flux of deforestation) is
being divided with 600(initial amount of carbon in terrestrial biosphere) to find the proportionality constant.
Thus the new differential equation for terrestrial biosphere is:
Terrestrial Biosphere :
dT
dt
=
11
75
A −
11
60
T −
1.15
600
A
Fossil fuel combustion and deforestation increase the carbon in atmosphere. Therefore, the new differential
equation for atmosphere becomes:
Atmosphere :
dA
dt
=
7
40
O +
11
300
S +
11
120
T −
1
3
A + E + 0.00125F +
1.15
600
T
The increase in concentration of atmospheric carbon dioxide have an effect in on the average global tem-
perature. The new differential equation of carbon in atmosphere will be treated as mass of CO2 in the
atmosphere and we will use the following equation to find the change in carbon dioxide in ppm:
[CO2]in ppm = 350x(mass of CO2 in the atmosphere)/750
After finding the concentration of CO2, we will used the following equation to find the temperature change.
In this equation 350 in the current concentration of CO2 in atmosphere and 750 is the stabilization of CO2.
temperature change(C)over entire period = 0.01([CO2] − 350
Another form of greenhouse gas other than CO2 is methane.Methane is an important and powerful greenhouse
gas with the ability to absorb 21 times as much heat per molecule as CO2. Therefore, it can be treated
similarly as carbon, with more effect on the temperature. Methane is lighter than CO2 so mixes more readily
with the air. Methane concentration in 1978 was 1.52, and methane concentration increase by about 1% per
year until 1990.
Thus the methane concentration in the year 1990 is 1.52 ∗ (1 + 1%)1
2 = 1.713.
In 2011, the concentration was 1.818. We made the assumption that the rate of change for Methane con-
centration is constant through our simulations. By calculating the rate of change, we have included the
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6. stabilization of CH4 so the stabilization term is now zero. So we used to following equation to find the
change of temperature due to methane:
temperature change(C)over entire period = 21x(0.01([change ofCH4])
4 Results
4.1 Carbon Cycle
Using the differential equations mentioned in the previous section, we used RK4 for 200 years to see how
amount of carbon different reservoirs. Even after running the simulation for a long period of time, we can
observe that the amount of carbon in different reservoirs stays the same the initial amount of carbon in
reservoirs as shown in Table 1. There is no change as the carbon is moving in a circle between different
reservoirs.
0 50 100 150 200
010000200003000040000
Time (years)
AmountofCarboninReservoirs(Gt)
Terrestrial Biosphere
Ocean Surface
Atmosphere
Soil
Deep Ocean
Figure 2: Amount of carbon in different reservoirs
4.2 Modified Carbon Cycle
With the modified system of differential equations, we simulated the carbon concentration for 500 years
including fossil fuel combustion and deforestation. The figure below shows the amount of carbon in five
reservoirs: fossil fuel deposit, terrestrial biosphere, ocean surface, atmosphere and soil. The amount of
carbon in deep ocean is being shown in the figure next to it as the difference is too big.
The figure below shows that carbon in fossil fuel deposit has been decreasing from its initial value of 4000
Gt to around 2600 Gt in 500 years.As it approaches 500 years, the rate of change of carbon decreases over
time. The carbon in fossil fuel reservoir is decreasing due to the fossil fuel emission. But it does not go to
zero because of the constrained growth in fossil fuel emission.The carbon in soil increases from 1500 Gt to
5 of 12

7. around 2600 Gt. The rate of change of carbon in soil increases and then decreases over time. The carbon in
ocean surface and atmosphere also increases and becomes 1250 Gt in 500 years. The rate of change carbon
in these two reservoirs are less than that in soil. The carbon in terrestrial biosphere increases to 900 Gt
in 500 years. Apart from carbon in fossil fuel reservoir, all the reservoir has an increase in carbon due to
combustion and deforestation.
0 100 200 300 400 500
5001000150020002500300035004000
Time (years)
AmountofCarboninReservoirs(Gt)
Fossil Fuel
Terrestrial Biosphere
Ocean Surface
Atmosphere
Soil
Figure 3: Amount of carbon in different reservoirs
The figure below shows the amount of carbon in deep ocean. Carbon increases to 41000 Gt in 500 years.
The rate of change in carbon decreases for some time then the rate increases.
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8. 0 100 200 300 400 500
36000370003800039000400004100042000
Time (years)
AmountofCarboninReservoirs(Gt)
Deep Ocean
Figure 4: Amount of carbon in deep ocean reservoirs
With the current rate of change for deforestation and fossil fuel combustion, in 500 years the temperature
will rise for around 3.5 Celsius degrees due to carbon dioxide and methane. Even though Methane is 21
times stronger than, CO2, it is not contributing much to the change in temperature as the concentration of
methane is not that much. Methane makes the temperature increase 0.5 Celsius. Whereas CO2 makes the
temperature increase to around 2.8 Celsius in 500 years. The figure below shows the temperature change
due to CO2 and methane.
0 100 200 300 400 500
01234
time (years)
temperaturechangeduetothegreenhousegases(C)
Carbon Dioxide
Methane
Carbon Dioxide and Methane
Figure 5: Temperature change when methane is 21 times stronger than carbon dioxide over 500 years
7 of 12

9. We also varied the coefficient for methane effect on temperature. When methane is 2.1 times stronger
than carbon dioxide, with current concentrations, the influence of carbon dioxide still dominates in global
warming.
0 100 200 300 400 500
01234
time (years)
temperaturechangeduetothegreenhousegases(C)
Carbon Dioxide
Methane
Carbon Dioxide and Methane
Figure 6: Temperature change when methane is 2.1 times stronger than carbon dioxide over 500 years
When methane is 210 times stronger than carbon dioxide, with current concentrations, the overall influence
of methane is larger than that of carbon dioxide in global warming.
0 100 200 300 400 500
02468
time (years)
temperaturechangeduetothegreenhousegases(C)
Carbon Dioxide
Methane
Carbon Dioxide and Methane
Figure 7: Temperature change when methane is 210 times stronger than carbon dioxide over 500 years
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10. 5 Discussion
The carbon concentrations are constant for the five main reservoirs in natural carbon cycle without human
factors. Naturally, the carbon circulate around between reservoirs, so the carbon concentration does not
change in a complete carbon cycle.
Considering human activities such as fossil fuel combustion and deforestation, the fossil fuel carbon
concentration is declining over time. The carbon concentration in fossil fuel reservoir is decreasing due
to the fossil fuel emission. But it does not go to zero because of the constrained growth in fossil fuel emission.
The carbon concentration for all the other reservoirs are increasing because carbon dioxide is added to
the other reservoirs during the combustion and deforestation processes. We can conclude that in general,
human activities add carbon to the environment, leading to the rise of global temperature.
According to our simulation for temperature changes due to greenhouse gases over time, although the
methane has a much stronger effect on temperature than carbon dioxide does, the overall temperature due
to methane is still significantly lower than carbon dioxide. This is consistent with the fact that methane
has a much lower concentration in atmosphere, therefore carbon dioxide is still dominating in global warming.
However, it does not imply that control of methane emission is not important in the context of constraining
global warming. The influence of methane on global temperature is becoming more and more significant
according to our simulation. While we talk about carbon print and climate change, it is also time to start
considering how to decrease methane emission.
From another perspective, relatively large portion of fossil fuel deposit is being consumed during the carbon
emission process. Fossil fuels are fuels formed by natural processes such as anaerobic decomposition of
buried dead organisms, which takes millions of years to resume. While the demand for energy is still
increasing, fossil fuel supply is dropping, which lead our interest to alternative and renewable energy. Some
alternative sources of energy include nuclear, hydroelectric, solar, wind, and geothermal, but a lot of them
are still in experiments. Therefore, the search for promising alternative energy is crucial.
It should also be recognized that, human activities which are influencing emissions of greenhouse gases,
extend beyond fossil fuel combustion and deforestation. In the future we could present a bigger picture on
this issue, With more factors quantified and incorporated into our model.
A R Source Code
A.1 Code written to General Carbon Cycle
Question 1
deltaT = 0.01
simLength = 200
iA = 750
iT =600
iO =800
iD =38000
iS =1500
x = seq (0,200, deltaT)
estA = vector(length=length(x))
estA [1] = iA
estT = vector(length=length(x))
estT [1] = iT
estO = vector(length=length(x))
estO [1] = iO
estD = vector(length=length(x))
estD [1] = iD
estS = vector(length=length(x))
estS [1] = iS
APrime = function(o,s,t,a) {(7/40 * o) + (11/300 * s) + (11/120 * t) - (1/3 * a)}
9 of 12

11. TPrime = function(a,t){(11/75 * a) - (11/60 * t)}
OPrime = function(a,d,o){(14/75 * a) + (27/38000 * d) - (163/800 * o) - 4}
DPrime = function(o,d){4 +(23/800 * o) - (27/38000 * d)}
SPrime = function(t,s){(55/600 * t) - (55/1500 * s)}
for (i in 2: length(x)) {
estIndividualsMidA = estA[i -1] + APrime(estO[i-1], estS[i-1], estT[i-1], estA[i -1]) * deltaT
estIndividualsMidT = estT[i -1] + TPrime(estA[i-1], estT[i -1]) * deltaT
estIndividualsMidO = estO[i -1] + OPrime(estA[i-1], estD[i-1], estO[i -1]) * deltaT
estIndividualsMidD = estD[i -1] + DPrime(estO[i-1], estD[i -1]) * deltaT
estIndividualsMidS = estS[i -1] + SPrime(estT[i-1], estS[i -1]) * deltaT
estIndividualsMidA2 = estA[i-1] + APrime(estIndividualsMidO ,estIndividualsMidS ,estIndividualsMidT ,
estIndividualsMidA ) * deltaT/2
estIndividualsMidT2 = estT[i-1] + TPrime(estIndividualsMidA , estIndividualsMidT ) * deltaT/2
estIndividualsMidO2 = estO[i-1] + OPrime(estIndividualsMidA ,estIndividualsMidD , estIndividualsMidO ) *
deltaT/2
estIndividualsMidD2 = estD[i-1] + DPrime(estIndividualsMidO , estIndividualsMidD ) * deltaT/2
estIndividualsMidS2 = estS[i-1] + SPrime(estIndividualsMidT , estIndividualsMidS ) * deltaT/2
estIndividualsMidA3 = estA[i-1] + APrime(estIndividualsMidO2 ,estIndividualsMidS2 ,estIndividualsMidT2 ,
estIndividualsMidA2 ) * deltaT/2
estIndividualsMidT3 = estT[i-1] + TPrime(estIndividualsMidA2 , estIndividualsMidT2 ) * deltaT/2
estIndividualsMidO3 = estO[i-1] + OPrime(estIndividualsMidA2 ,estIndividualsMidD2 , estIndividualsMidO2 )
* deltaT/2
estIndividualsMidD3 = estD[i-1] + DPrime(estIndividualsMidO2 , estIndividualsMidD2 ) * deltaT/2
estIndividualsMidS3 = estS[i-1] + SPrime(estIndividualsMidT2 , estIndividualsMidS2 ) * deltaT/2
estIndividualsRightA = estA[i -1] + APrime(estIndividualsMidO3 ,estIndividualsMidS3 ,estIndividualsMidT3
, estIndividualsMidA3 ) * deltaT
estIndividualsRightT = estT[i -1] + TPrime(estIndividualsMidA3 , estIndividualsMidT3 )* deltaT
estIndividualsRightO = estO[i -1] + OPrime(estIndividualsMidA3 ,estIndividualsMidD3 , estIndividualsMidO3
)* deltaT
estIndividualsRightD = estD[i -1] + DPrime(estIndividualsMidO3 , estIndividualsMidD3 )* deltaT
estIndividualsRightS = estS[i -1] + SPrime(estIndividualsMidT3 , estIndividualsMidS3 )* deltaT
estA[i] = (1/6) * ( estIndividualsMidA + 2* estIndividualsMidA2 + 2* estIndividualsMidA3 +
estIndividualsRightA )
estT[i] = (1/6) * ( estIndividualsMidT + 2* estIndividualsMidT2 + 2* estIndividualsMidT3 +
estIndividualsRightT )
estO[i] = (1/6) * ( estIndividualsMidO + 2* estIndividualsMidO2 + 2* estIndividualsMidO3 +
estIndividualsRightO )
estD[i] = (1/6) * ( estIndividualsMidD + 2* estIndividualsMidD2 + 2* estIndividualsMidD3 +
estIndividualsRightD )
estS[i] = (1/6) * ( estIndividualsMidS + 2* estIndividualsMidS2 + 2* estIndividualsMidS3 +
estIndividualsRightS )
}
plot(x,estA ,type="l",col="red",xlab="Time (years)",ylab="Amount of Carbon in Reservoirs (Gt)",ylim=c
(300 ,2400))
lines(x,estT ,type="l",col="blue")
lines(x,estO ,type="l",col="limegreen")
lines(x,estD ,type="l",col="purple")
lines(x,estS ,type="l",col="orange")
legend (90 ,2400 , c(" Terrestrial Biosphere","Ocean Surface","Atmosphere","Soil"),lty=c(1,1,1,1), lwd=c
(1.5 ,1.5 ,1.5 ,1.5) ,col=c("blue","limegreen","red","orange"))
A.2 Code written to Modified Carbon Cycle with Combustion and Deforesta-
tion
deltaT = 0.05
simLength = 500
iA = 750
iT =600
iO =800
iD =38000
iS =1500
iF =4000
iE=5
x = seq (0,500, deltaT)
estA = vector(length=length(x))
estA [1] = iA
10 of 12

12. estT = vector(length=length(x))
estT [1] = iT
estO = vector(length=length(x))
estO [1] = iO
estD = vector(length=length(x))
estD [1] = iD
estS = vector(length=length(x))
estS [1] = iS
estF = vector(length=length(x))
estF [1] = iF
estE = vector(length=length(x))
estE [1] = iE
APrime = function(o,s,t,a,e,f) {(7/40 * o) + (11/300 * s) + (11/120 * t) - (1/3 * a)+e+0.00125*f
+(1.15/600)*t}
TPrime = function(a,t){(11/75 * a) - (11/60 * t) -(1.15/600)*t}
OPrime = function(a,d,o){(14/75 * a) + (27/38000 * d) - (163/800 * o) - 4}
DPrime = function(o,d){4 +(23/800 * o) - (27/38000 * d)}
SPrime = function(t,s){(55/600 * t) - (55/1500 * s)}
FPrime = function(f){ -0.00125*f}
EPrime = function(e){0.03*e*(1-e/15)}
for (i in 2: length(x)) {
estIndividualsMidA = estA[i -1] + APrime(estO[i-1], estS[i-1], estT[i-1], estA[i-1], estE[i-1], estF[i -1])
* deltaT
estIndividualsMidT = estT[i -1] + TPrime(estA[i-1], estT[i -1]) * deltaT
estIndividualsMidO = estO[i -1] + OPrime(estA[i-1], estD[i-1], estO[i -1]) * deltaT
estIndividualsMidD = estD[i -1] + DPrime(estO[i-1], estD[i -1]) * deltaT
estIndividualsMidS = estS[i -1] + SPrime(estT[i-1], estS[i -1]) * deltaT
estIndividualsMidF = estF[i -1] + FPrime(estF[i -1]) * deltaT
estIndividualsMidE = estE[i -1] + EPrime(estE[i -1]) * deltaT
estIndividualsMidA2 = estA[i-1] + APrime(estIndividualsMidO ,estIndividualsMidS ,estIndividualsMidT ,
estIndividualsMidA ,estIndividualsMidE , estIndividualsMidF ) * deltaT/2
estIndividualsMidT2 = estT[i-1] + TPrime(estIndividualsMidA , estIndividualsMidT ) * deltaT/2
estIndividualsMidO2 = estO[i-1] + OPrime(estIndividualsMidA ,estIndividualsMidD , estIndividualsMidO ) *
deltaT/2
estIndividualsMidD2 = estD[i-1] + DPrime(estIndividualsMidO , estIndividualsMidD ) * deltaT/2
estIndividualsMidS2 = estS[i-1] + SPrime(estIndividualsMidT , estIndividualsMidS ) * deltaT/2
estIndividualsMidF2 = estF[i-1] + FPrime( estIndividualsMidF ) * deltaT/2
estIndividualsMidE2 = estE[i-1] + EPrime( estIndividualsMidE ) * deltaT/2
estIndividualsMidA3 = estA[i-1] + APrime(estIndividualsMidO2 ,estIndividualsMidS2 ,estIndividualsMidT2 ,
estIndividualsMidA2 ,estIndividualsMidE2 , estIndividualsMidF2 ) * deltaT/2
estIndividualsMidT3 = estT[i-1] + TPrime(estIndividualsMidA2 , estIndividualsMidT2 ) * deltaT/2
estIndividualsMidO3 = estO[i-1] + OPrime(estIndividualsMidA2 ,estIndividualsMidD2 , estIndividualsMidO2 )
* deltaT/2
estIndividualsMidD3 = estD[i-1] + DPrime(estIndividualsMidO2 , estIndividualsMidD2 ) * deltaT/2
estIndividualsMidS3 = estS[i-1] + SPrime(estIndividualsMidT2 , estIndividualsMidS2 ) * deltaT/2
estIndividualsMidF3 = estF[i-1] + FPrime( estIndividualsMidF2 ) * deltaT/2
estIndividualsMidE3 = estE[i-1] + EPrime( estIndividualsMidE2 ) * deltaT/2
estIndividualsRightA = estA[i -1] + APrime(estIndividualsMidO3 ,estIndividualsMidS3 ,estIndividualsMidT3
,estIndividualsMidA3 ,estIndividualsMidE3 , estIndividualsMidF3 ) * deltaT
estIndividualsRightT = estT[i -1] + TPrime(estIndividualsMidA3 , estIndividualsMidT3 )* deltaT
estIndividualsRightO = estO[i -1] + OPrime(estIndividualsMidA3 ,estIndividualsMidD3 , estIndividualsMidO3
)* deltaT
estIndividualsRightD = estD[i -1] + DPrime(estIndividualsMidO3 , estIndividualsMidD3 )* deltaT
estIndividualsRightS = estS[i -1] + SPrime(estIndividualsMidT3 , estIndividualsMidS3 )* deltaT
estIndividualsRightF = estF[i -1] + FPrime( estIndividualsMidF3 )* deltaT
estIndividualsRightE = estE[i -1] + EPrime( estIndividualsMidE3 )* deltaT
estA[i] = (1/6) * ( estIndividualsMidA + 2* estIndividualsMidA2 + 2* estIndividualsMidA3 +
estIndividualsRightA )
estT[i] = (1/6) * ( estIndividualsMidT + 2* estIndividualsMidT2 + 2* estIndividualsMidT3 +
estIndividualsRightT )
estO[i] = (1/6) * ( estIndividualsMidO + 2* estIndividualsMidO2 + 2* estIndividualsMidO3 +
estIndividualsRightO )
estD[i] = (1/6) * ( estIndividualsMidD + 2* estIndividualsMidD2 + 2* estIndividualsMidD3 +
estIndividualsRightD )
estS[i] = (1/6) * ( estIndividualsMidS + 2* estIndividualsMidS2 + 2* estIndividualsMidS3 +
estIndividualsRightS )
estF[i] = (1/6) * ( estIndividualsMidF + 2* estIndividualsMidF2 + 2* estIndividualsMidF3 +
estIndividualsRightF )
estE[i] = (1/6) * ( estIndividualsMidE + 2* estIndividualsMidE2 + 2* estIndividualsMidE3 +
estIndividualsRightE )
}
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13. plot(x,estA ,type="l",lwd=3,col="red",xlab="Time (years)",ylab="Amount of Carbon in Reservoirs (Gt)",ylim=
c(500 ,4000))
lines(x,estT ,type="l",lwd=3,col="blue")
lines(x,estO ,type="l",lwd=3,col="limegreen")
lines(x,estS ,type="l",lwd=3,col="orange")
lines(x,estF ,type="l",lwd=3,col="purple")
legend (280 ,4200 , c("Fossil Fuel", " Terrestrial Biosphere","Ocean Surface","Atmosphere","Soil"),lty=c
(1,1,1,1,1,1), lwd=c(3,3,3,3,3,3),col=c("purple","blue","limegreen","red","orange"),cex = 0.73)
plot(x,estD ,type="l",lwd=3,col="yellow",xlab="Time (years)",ylab="Amount of Carbon in Reservoirs (Gt)",
ylim=c(36000 ,42000))
legend (270 ,37000 , c("Deep Ocean"),lty=c(1) , lwd=c(3),col=c("yellow"),cex = 0.73)
A.3 Code written to temperature change due to carbon dioxide and methane
massCO2=estA
concenCO2 =350*massCO2/750
Temp1 = 0.01*(concenCO2 -350)
plot(x,Temp1 ,type="l",xlab="time (years)", ylab=" temperature change due to the greenhouse gases (C)",col=
"red",ylim=c(0 ,4))
legend (0,4, c("Carbon Dioxide","Methane","Carbon Dioxide and Methane"),lty=c(1,1,1), lwd=c(3,3,3),col=c("
red","blue","purple"),cex = 0.73)
iCH4 =1.52
CH41 =1.52*(1.01^12)
CH41
CH42 =1.818
rate =(CH42 -CH41)/21
CH43=rate*x
Temp2 = 0.21*CH43
Temp=Temp1+Temp2
lines(x,Temp2 ,type="l",col="blue")
lines(x,Temp ,type="l",col="purple")
References
[1] Shiflet, Angela B., and George W. Shiflet. (2007), Introduction to Computational Science: Modeling
and Simulation for the Sciences. Princeton Press
[2] The Carbon Cycle.Ainsworth Energy (2016) https://urldefense.proofpoint.com/v2/url?u=
http-3A__ainsworthenergy.com_about-2Dus_environment_the-2Dcarbon-2Dcycle_&d=DQIFaQ&
c=WAopAOoYhwkYfEix65l8HkxJg6TNRdSFVjz7ONc5bdk&r=KsYz107P5nVgJzMtiXFQAQ068g2pIbjww_
BcQHLNBqU&m=hVytVnEqoSJ2m4nWrKIY7b0ER0g9cVocQ5RYDh17IyQ&s=0N8SK2Q9FLGt08zEq_
7ltaKV2Xfjv9oZsd7fUeIjp1I&e=.
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