Box 2.1 and Chapters 4 and 6 show that emission of carbon dioxide is closely correlated with energy use, thus connecting the data from Figures 2.1 and 2.4. Chapter 6 shows the strong connection between atmospheric concentrations of carbon dioxide and climatic change, thus correlating the findings in Figures 2.3 and 2.1, regardless of how skeptical one might be of the findings in Figure 2.3. We will introduce model building using a simple model that demonstrates that the increase in the atmospheric carbon dioxide concentration has a large anthropogenic (man- made) component, resulting in climate changes. State- of- the- art modeling of future climatic changes is described in Chapter 8.
We start with a simple description of the two key chemical processes that living systems use to acquire energy to function: photosynthesis and respiration.
In Chapter 4, I will show that the emission and sequestration of carbon dioxide are the two main components of the carbon cycle. The carbon cycle is the dominant pump that provides energy to the biosphere, which contains all living organisms on Earth. The energy is provided either directly through the photosynthetic process or indirectly through the digestion of products of the photosynthetic process through respiration reactions. Th e chemical reactions that summarize these two processes are shown in Box 2.1. Th e photosynthetic reaction sequestrates carbon dioxide to produce sugar and, in the process, stores the solar energy in the form of chemical bonds richer in energy than those of carbon dioxide and water, which are the starting materials for the process. The respiration reaction uses high-energy chemicals, such as sugars, which in turn produce low-energy products in the form of carbon dioxide and water. The excess energy is available for use in biochemical reactions that require additional energy.
15
Box 2.1
The chemical reactions that occur between atoms and molecules (see Box 1.1) are often described in terms of chemical equations. Here I describe three such reactions: the photosynthetic reaction, the digestion of sugar in the respiration process, and the reaction between methane and oxygen during the combustion of fossil fuels.
Reactions require that the same number of atoms of each element are on the left side (reactants) of the reaction as on the right side (products).The numerical subscripts indicate the number of atoms in a molecule and the numerical coefficients indicate the number of molecules in the reaction. in equation 2.1, 6CO2 means six molecules of carbon dioxide with a molecular formula of CO2 (which is one carbon atom bonded to two oxygen atoms).
A schematic representation of the photosynthetic reaction is given byO sun
6CO + 6H→ CHO + 6O. [2.1]
2261262
C6H12O6 represents one of the simplest sugars: glucose.
A schematic representation of respiration is given in the following:
CHO + 6O→ 6CO + 6HO + biologically stored energy. [2.2]
61262 22
Burning of Fossil Fuel
Last, a schematic representation of the combustion of fossil fuel is
CH + 2O→ CO + 2HO + energy. [2.3]
42 22
CH4, methane, is the main constituent in natural gas.
As I will discuss in Chapter 4, plants perform a significant fraction of the terrestrial chemical sequestration of carbon dioxide. In the case of an imbalance between sequestration and emission, the difference will change the atmospheric concentration of carbon dioxide. The relationship is not straightforward because it depends on the equilibration of carbon dioxide between the atmosphere and Earth’s oceans. The timing and extent of this equilibration process will be discussed in Chapter 4. Mankind contributes to this balance by burning fossil fuels in chemical reactions similar to the methane reaction shown in Box 2.1 and through the destruction of forests and other changes in land use resulting from population growth.
Fossil fuels are remnants of trees and other plants that died as early as 600 million years ago. Fossilization occurs when dead plants are prevented from decomposing back to carbon dioxide and water due to lack of oxygen. Instead they are buried in the ground where they are subjected to temperatures and pressures high enough to induce chemical changes that remove the oxygen from the sugar derivatives and other biomolecules. In this way, plants are converted to the hydrocarbons that constitute fossil fuels ranging from oil, gasoline, and natural gas, to coal, which is mostly carbon. In essence, this process stores the photosynthetically converted solar energy for millions of years for us to use as a relatively inexpensive and convenient energy source. In terms of the carbon dioxide balance, it provides us with the “opportunity” to emit long-sequestrated carbon dioxide.
In 2005, total world energy consumption was 4.6 × 1017 Btu (15 TW; see Box 2.2). About 85% of the energy sources used were carbonated fossil fuels. Because all this fuel originated from carbon dioxide sequestrated a long time ago, its burning constitutes a net emission without the compensating sequestration.
Figure 2.1 shows air sample concentrations collected continuously from 1958 at Mauna Loa on the island of Hawaii.1 Samples of air were collected four times a day, and the concentration of carbon dioxide is measured by infrared absorption spectroscopy, using the same properties of carbon dioxide that makes this material a greenhouse gas (GHG). The Mauna Loa site is considered one of the most favorable locations for measuring undisturbed air because possible local influences of vegetation or human activities on atmospheric carbon dioxide concentrations are minimal and any influences from volcanic vents can be excluded from the records. The measurements show an increase of around 17% from 1958 until 2004. Although Mauna Loa is the oldest site in which such measurements are made, the monitoring has expanded to different sites at diff erent latitudes.
Box 2.2
A common reference point throughout the book will be energy use per year. Energy per unit time is power.Thus
power × time = energy. The kilowatt (kW) is a unit of power, whereas kilowatt- hours (kW·h) is a unit of energy that can be directly converted to other energy units such as Btus (Appendix 1):
1 Btu = 2.93 × 10– 4 kW·h. When we specify the 2005 world energy consumption as 4.6 × 1017 Btu, we are actually specifying power and can easily convert it to the more usual power units: 2005 energy use = 4.6 × 1017 Btu/year = 1.3 × 1014kW·h/year =
1.5 × 1013 W = 15 terawatts (TW).
For very large numbers we will often use prefix multipliers:
Giga (G) = 109 Tera (T) = 1012 Peta (P) = 1015
An example from data taken in Antarctica is shown in Figure 2.2. Th e pattern of a monotonic increase in the carbon dioxide concentration is found at every site. However, the two figures also show something else. Superimposed on the steady increase at Mauna Loa, and almost completely absent from the South Pole measurements, are very regular oscillations. These oscillations directly represent the yearly cycle of the diff erence between the source and the sink of carbon dioxide at Mauna Loa. The full complexities of the carbon cycle as it equilibrates between land, air, and ocean will be discussed in Chapter 4. Here we deal only with the size of human contributions as measured against global natural phenomena. Th e yearly cycles in Figure 2.1 were attributed to the large fraction of trees in the northern hemisphere that shed their leaves in the fall and regrow them in the spring. The result is that less carbon dioxide is captured during the winter than in summer.
Figure 2.1. Carbon dioxide concentrations as observed at Mauna Loa
Figure 2.2. Carbon dioxide concentrations as observed at the South Pole Source: Keeling and W hort (2004).1
In Chapter 5, I discuss why an increase in carbon dioxide and other GHGs in the atmosphere is a cause for alarm when considering the climate of Earth. Figure 2.3 shows global temperature data for the last 130 years.
The long-term mean temperature data for Earth were calculated by processing data from thousands of worldwide observation sites on land and sea for the entire period of record of the data. Earth’s long-term mean temperatures were calculated by interpolating over uninhabited deserts, inaccessible Antarctic mountains, and so on, taking into account factors such as the decrease in temperature with elevation. By adding the long- term monthly mean temperature for Earth to each anomaly value, one can create a time series that approximates the temperature of Earth and how it has been changing through time.
The data start around 1880 because the mercury thermometer was invented by Gabriel Fahrenheit in 1714, and therefore the second part of the 19th century was the earliest that one could gather reliable direct temperature measurements over a wide enough geographic distribution to calculate average global temperature.
Figure 2.3 does not appear to be a very alarming curve. In fact, when it was presented to students and faculty (on separate occasions) the response was a big yawn. The “noise,” due to short-term fluctuations, is large and the overall trend in the deviations is small compared to everyday experience. Members of the faculty even suggested that I not include it in an introductory course that includes discussion of global warming for fear that the data would confuse inexperienced minds. These data are even less impressive when viewed in the context of historic global temperature fl uctuations that will be discussed in the next chapter. Th ere, we will see that the data for the carbon dioxide increase clearly demonstrate a dominant anthropogenic contribution, whereas at present, the temperature anomaly data do not. However, in Chapter 5 I will discuss the correlation between anthropogenic atmospheric changes and temperature changes with the conclusion that the cause- effect relationship between the two is supported by very solid science developed out of controlled experiments and works very well, not only on Earth, but also throughout the universe.
Figure 2.3. Deviations of mean annual global temperature from a 1951– 1980 average of 14°C Source: NASA Goddard Institute.2
W hy is carbon dioxide so special? W hy can’t we regulate carbon dioxide emission the same way the international community regulates the use of chlorofluorocarbons found to be destructive to the stratospheric ozone layer? The answer again is in Box 2.1. Extraction of energy from fossil fuels results in the emission of carbon dioxide. Thus banning emission of carbon dioxide means banning the use of fossil fuels as an energy source (not counting the accompanying respiration processes). Finding “substitutes” for carbon dioxide means finding alternative energy resources. The prospects for using alternative energy will be discussed in Chapter 11.
The reason the continuing use of fossil fuels is such a pressing political issue is best summarized in Figure 2.4, which shows the relationship of GDP per person with the energy consumption per person for various countries in the world. The data for this figure were taken from the US Central Intelligence Agency website, which includes a detailed collection of data for all the world’s countries. Similar data can be found on the World Bank website and those of various national energy agencies. Almost all the data are supplied by the individual countries.
The GDP per capita is a measure of the economic development of a country. The ratio of energy use to GDP is called energ y intensity. In Chapter 13, when I discuss attempts to reach a global consensus on global warming, the energy intensity issue will play a dominant role in describing the official US view as to how to approach this issue. In Chapter 12, I will discuss in greater depth the relations between energy use and economic indicators. Because GDP is measured in monetary units, we will examine the problems associated with comparing GDP between countries with different monetary and economic systems.
Readers from different countries might also ask questions specific to data about their countries: why, for example, is Singapore where it is in Figure 2.4? Because the data for the figure were taken from a single source, the data for a particular country might simply be wrong.
However, for the purpose of our discussion, there are three aspects of Figure 2.4 very robust with respect to different definitions of economic wellness or where a particular country ends up in the graph. If we look at the overall trend, then GDP per person increases linearly (in a straight line) with the energy use per person until about $25,000 per person and then becomes saturated; further energy use does not result in a GDP increase. There is also a great deal of scatt er in the graph. If we look at constant energy use per person, say 0.2 billion Btu per person, and trace a line parallel to the vertical axis, we will find differing information for the following countries: Russia has a GDP per capita of $4500; South Korea and Taiwan have a GDP per capita of about $13,000; European countries such as Germany, England, Italy, and France are clustered around $20,000 to $22,000; and Japan lands around $24,000. That represents more than a factor of six in the difference in energy intensity. In other words, Japan is more than six times more effi cient in using energy to produce economic output as compared to Russia. If we now take a complementary trace and put our finger at a GDP per capita of $22,000 and trace a line parallel to the horizontal axis, then we will find Italy with 0.13 billion Btu per capita; Germany, England, and France with 0.17 billion Btu per capita; the Netherlands with 0.24 billion Btu per capita; Singapore with
Figure 2.4.
GDP as a function of energy use for various countries Source: US Central Intelligence Agency.30.24 billion Btu per capita; and Canada with 0.41 billion Btu per capita. Again, we see more than a factor of three in the difference in energy intensity. In Chapter 12, we will investigate the history of this correlation and try to determine the factors that affect a country ’s energy effi ciency and the possible reasons for the saturation eff ect.
The saturation point will allow us to estimate the consequences, in terms of energy and carbon dioxide emission, of the developing countries that strive to reach the economic development of the developed countries. The graph also helps to explain the conflict between economically developed and developing countries in terms of the global solution to the issue of anthropogenic release of carbon dioxide. If economic development is driven by low-cost energy sources, then it is difficult for developing countries to accept the argument that the use of such sources should be banned at the point where the developed countries are no longer benefiting from their availability.
In Chapter 11, I will explore the issue of alternative energy sources not based on stored fossil fuels and thus not contributing to the emission of carbon dioxide. We will find that, at present, all these sources are considerably more expensive than traditional fossil fuels. The net result is that a large-scale shift to such sources will require a shift of resources toward energy production.
I will get back to this issue in Chapters 10 and 11, where I will try to estimate the total amount of fossil fuels still available and calculate the amount of carbon dioxide that will be emitted as a consequence of burning this fuel and the affect of this on the atmospheric composition and the world’s climate. I will describe the problems that we encounter in estimating how long these resources will be available and how to define their exhaustion point.
I will try to develop the argument that a satisfactory solution is that by the time oil and natural gas are exhausted, we should be in position to replace them with noncarbon sources.
In 1888, Henri Louis Le Chatelier, a French industrial chemist, made the observation that systems in equilibrium respond to stress by restoring equilibrium through minimizing the sources of the stress. This observation is known as Le Chatelier’s principle. Over the years, the principle has withstood countless challenges, and it now constitutes one of the most important building blocks of modern science. An argument can be made that nature, on its own, is able to compensate for the anthropogenic stresses that we impose through our growing use of fossil fuels, and thus we really need not bother with the search for potentially painful solutions. However, Le Chatelier’s principle does not specif y the method or rate of restoration. One possible mechanism is creating conditions that will eliminate the source of the disturbance: the human race. Thus we obviously cannot leave the job of maintaining equilibrium between nature and man to this principle alone.
Will the “ business as usual” scenario result in the end of the world (Armageddon)? Most probably not. In Chapter 8, I will explore the state- of- the- art mechanisms used to try to predict the future, which are based on modeling. Most models of climatic consequences, and the resulting rise in sea level, span about 100 years. The projected average global temperature increases are between 1°C and 3°C. The projected average global sea level rise is between 20 and 90 cm. Most of the rise in sea level originates from the thermal expansion of liquid water and not from the melting of the ice caps. Ice cap melting is not contributing much because most of the ice is in Antarctica, and because one of the most important consequences of global warming is increase in precipitation, the increase in snow precipitation in Antarctica is predicted to be larger than the melting, and as a result Antarctica acts as a water sink and not a source.
The projected temperature increases and consequent sea- level changes will certainly cause local dislocations and changes, some of them already visible. These early signs will be discussed in Chapter 14. These changes do not predict a global Armageddon. W hy do the projections stop at 100 years? This time period is hardly beyond our definition of “now.” One reason is simply technical: our ability to make projections based on modeling involves assumptions and simplifications that introduce a great deal of uncertainty. These uncertainties increase dramatically with an increase in the length of time that we try to project. If we take the worst-case scenario, and the average temperature of a significant portion of Antarctica rises above 0°C (the melting point of ice), melting the Antarctic ice cap, the average global sea- level increase is projected to be around 80 m (about 260 ft .). This will flood the living habitat of 90% of the world’s population.
In Chapter 10, I will explore another reason for stopping the projections at 100 years. Over this period, there is high probability that we will run out of recoverable fossil fuels. Th e projections here are also uncertain but most likely true for low-cost, convenient sources such as oil and natural gas. Coal is a bit more problematic, and a big uncertainty rests on the vast reserves of fossil resources such as shell oil, sand tar, and in particular, deep ocean deposits of methane hydrates. Yet Chapter 4 will make it clear that the anthropogenic contributions of carbon dioxide to the atmosphere are very small compared to the natural flux of the equilibrium between the atmosphere, the oceans, and land biota. Small, climate- induced changes to these balances can have large effects, but the triggering of such changes can be man induced or “natural.” Chapter 7 tries to explore the meaning of “natural.” It is now agreed, although not necessarily fully understood, that the major past climatic changes originated from astronomical variations in the sun-Earth energy balance. These variations can take the form of changes in orbital motion that affect distribution and in insolation (solar radiation that intercepts Earth). Some of the variations were shown to be periodic on a scale in which climatic records are available and some are not. In this sense, even if we do all the “right things” and ensure the maintenance of a “natural” atmospheric equilibrium, we do not necessarily ensure ourselves against catastrophic changes. We only ensure ourselves against catastrophic changes on a human time scale. The rest is not up to us.
W hat will follow is a “ back-of-the-envelope,” simplistic model designed to achieve two objectives: first, to make the case that the issues are real and, second, to provide us with more technical leverage when we read about what other people and “scientists” are doing. Th e model will require us to work with numbers from real databases. The models are interactive in a sense that the results strongly depend on the data that we choose to use and the details of the simplifications that we apply. We can choose to skip it or we can choose to play with it, but in both cases we assume control.
This simple model will estimate the amount of carbon dioxide presently sequestrated. We want the model to be transparent and based on scientific principles. Th is requires that the model be as simple as possible. These requirements will cost us in terms of the accuracy of the estimates that this model can derive.
Before we embark on such an estimate, it is useful to address the degree of accuracy we should try to achieve. Box 2.3 shows that for the purpose of this book, accuracy of an order of magnitude (within a factor of 10) is suffi cient. Attempts at higher accuracy will inevitably cost us in transparency. Stated differently, political decisions that our grandchildren will need to make should be regarded as our own problems.
Figure 2.5 shows a photograph of the landscape that we are modeling. It is a “typical” landscape of the Amazon rain forest. Rain forests such as this are usually described in terms of the layers of different greenery, starting from the tallest emergent layer with trees that can grow as high as 60 m (200 ft.), followed by the canopy, which is the primary layer of the rain forest that captures about 80% of the sunlight. Trees in the canopy can rise to 45 m. The next layer is the understory, which receives only a small fraction of the light and where plants can rarely grow above 3.5 m. The last layer is the forest floor, which receives almost no light and allows very few plants to grow.
Such a complex landscape is definitely difficult to model. I will try to model the Amazon rain forest by using a much simpler picture. I will describe the forest as an ensemble of identical trees, equally spaced from each other. Box 2.4 provides the details. I am estimating each tree to be 20 m high and 0.5 m wide, and I assume a distance of 10 m between trees. I assume that the trees are only bark without leaves and branches. I further assume that the cross sections of the trees are square.
Figure 2.5. A “typical” view of the Amazon rain forest Source: Gutro (2004).4
Box 2.3
What time periods are we concerned about? Because the main objective of this book is to collect sufficient information to provide us with tools to make choices and enforce them in the political process, the relevant time period is important.
In an arbitrary way, let us define the time span of cohabitation of three generations as “now.” In my case, three generations include (using 2005 as a reference year) me; my grandchildren Justin, Samantha, and Jack; and my son, their father. On average, my grandchildren’s life expectancy is about 75 years.
Suppose that the average yearly increase of our economic activity, as measured by GDP, is about 3%. If we assume a constant increase by the same 3%, then the increase will apply to a larger and larger base. This is an example of exponential growth. A useful parameter to characterize such growth is the doubling time. A good approximation of the relation between doubling time and percentage growth is given by
70Doubling t ime = Percentag e of g rowth .
The doubling time for constant 3% growth is 70/3 ≈ 23 years. Let us designate present GDP as 1 and try to determine the GDP in coming years.The following table shows the increase.
Table 2.1
Simple example of GDP growth with an annual growth of 3%
| GDP | Time (years) |
|---|---|
| 1 | Present |
| 2 | 23 |
| 4 | 46 |
| 8 | 69 |
Based on this rate of growth, within the time span that we have designated as “now,” we will get approximately a factor of 10 increase in GDP, as well as the good and not- so- good things that come with an increase in economic activity.We will see throughout this book that there are many uncertainties based on the estimated future increases in economic activity, energy use, and emission of GHGs. Any pretension of “exact” numbers is misleading. Thus an estimate within a factor of 10 should provide us with the tools to make informed decisions.
Among all the assumptions made, the one that caught the eye of students and faculty to whom the model was presented is the assumption of square trees. Everybody knows that trees are not square, so the model must be wrong. The approximation involved in treating the cross sections of the trees as square is analyzed in Box 2.5. Actually this approximation is unnecessary— calculations that involve circular cross sections are as easy as the ones that involve square cross sections. However, it is an eye- catching approximation and is amenable to exact analysis. Now, if any one of us has problems with any of the other assumptions made, it is a good exercise to change the assumptions and study the model’s sensitivity to them and try to determine if the conclusions we draw from the model are sensitive to the assumptions we make. If we can accomplish that, then it means our first-principle analysis works. If we cannot, then we try again.
Box 2.4
Assume that an Indian teak tree is 20 m high, 0.5 m in diameter, and has a density = 0.7 g/cm3. Thus weight of the tree = volume × density, volume = base × height = (assume square base) (50 cm)2 × 2000 cm = 5 × 106 cm3 , weight = (5 × 106 cm3) × (0.7 g/cm3) = 3.5 × 106 g = 3.5 tons. We take the tree to be made of structural cellulose, a derivative of sugar: sugar ≡ cellulose ≡ (HCOH)n .
The subscript n indicates that cellulose is a polymer of repeating structural units that consists of HCOH, similar to a simple sugar like glucose. The length of individual polymer chains may vary, but the relative ratio of hydrogen (H), carbon (C), and oxygen
(O) remain approximately constant.Thus
fraction of carbon = 12/(12 + 2 + 16) = 0.4 = 40% (See Box 2.1 and Box 1.1),
amount of carbon in each tree = 3.5 × 0.4 = 1.4 tons = 1.4 × 106 g.
Thus you can calculate the amount of carbon in the rain forest. Assume that the space between trees is 10 m:
area = 2.7 × 106 miles2 = 7 × 106 km2 ,
number of trees in the forest = area/(area/tree) = 7 × 1012 m2/100 m2 = 7 × 1010 trees (70 billion),
total amount of carbon in the rain forest: = number of trees × amount of carbon/tree
= 7 × 1010 × 1.4 tons/tree
= 1 × 1011 tons of carbon.
Thus the carbon density is 1 × 1011 tons/7 × 106 km2 = 1.4 × 104 tons of carbon/km2 .
National forest inventory and direct satellite measurements are quoted at 8 × 103 tons/km2, which is a difference less than a factor of 2.5 (Whereas it almost looks like we have adjusted the parameters to arrive at such an agreement, this is wrong.)
Burning of glucose (equation 2.2) releases 684 Cal. Glucose has 6 carbon atoms.The amount of released energy per carbon atom is 114 Cal/carbon atom:
amount of energy released by complete burning of the forest: = total carbon × energy/g C
= (1 × 1011 tons) × (106 g/ ton) × (114 Cal/12 g of C)
= 1 × 1018 Cal =3.8 × 1018 Btu (1 Btu = 0.252 Cal).
Box 2.5
The model for the energy stored in the Amazon rain forest was presented to a group of faculty members at Brooklyn College as part of a working group that tried to introduce quantitative reasoning across the curriculum.The issue that caught the attention of many of the participants and took about half of the discussion time was the assumption that the base of the trees is square. Let us examine this assumption. Figure 2.6 shows a circle inscribed in a square.
Figure 2.6. Circle in a square
The side of the square and the diameter of the circle are equal. Let us designate them with the letter d.The radius of the circle is half its diameter; let us designate it with the letter r.Thus d = 2 × r.
The area of the square is d2 = (2 × r)2 = 4r2 .
The area of the circle is πr2, where π = 3.14 to a very good approximation.
The ratio of the area of the square to that of the circle is 4r2/πr2 = 1.27.
The error in representing the bases of the trees as squares instead of circles amounts to 27%.This is a very small error compared to the other approximations in the model.
The Amazon rain forest constitutes approximately 60% of all the area of tropical forest on Earth. The carbon stocked in tropical forests constitutes approximately 50% of carbon stocked in vegetation of all kinds on Earth. The final result in Box 2.4 leads us to 3.8 × 1018/(0.6 × 0.5) = 1.2 × 1019 Btu of energy stored in all vegetation on Earth, which occurs mostly in forests.
The anthropogenic energy consumption of carbonated fossil fuels in 2005 was 3.9 × 1017 Btu. This is 3.2% of all the energy stored in vegetation on Earth. The sequestration in the forests takes place over the lifetime of the trees in the forests (and other vegetation), whereas the release of carbon from the fossil energy use is annual. Because the area of the forests is decreasing and the anthropogenic carbon dioxide emission is increasing, the release will outpace sequestration within the boundaries of our definition of “now ” in Box 2.3. As will be discussed in Chapter 4, the carbon balance is more complicated than the one depicted here, but it will point to the same conclusions.
Can we support or refute the conclusions from this simple model based on direct, independent measurement? It turns out that we can.
Figure 2.7 is an enlargement of Figure 2.1 over a 1-year period. The shaded area represents sequestration of carbon dioxide (yearly reduction from the maximum level for that year). Calculating the shaded area can tell us the amount of yearly sequestration that diffuses to that particular location.
Inspection of Figure 2.7 shows that the shaded area covers approx imately half the yearly rectangle between the minimum and max imum levels for the year. The size and frequenc y of these oscillations seem to be more or less constant. Figure 2.8 approximates
Figure 2.7. Enlargement of 1 year in Figure 2.1 Source: Keeling and W hort (2004).1
Figure 2.8. Simplified depiction of the oscillation in Figure 2.7 Source: Keeling and Whort (2004).1
this distribution for a geometry that is much easier to calculate. The yearly sequestration that occurs in broad- leaf trees takes place only in the downward pointing rectangles, during the summer. One of the rectangles is shaded for emphasis. From Figure 2.4, the height is approximately 5.3 parts per million volume (ppmv) and the width is half a year. Calculating this area is very simple, and one should be able to directly compare these measurements with our model calculations in Box 2.4. However, there is a problem. Figure 2.1 provides data for air concentration, whereas what we need is the absolute amount of carbon dioxide to compare with our previous model. As we will see shortly, the units of ppmv express the ratio of carbon dioxide molecules to air molecules, so to extract the absolute amount of carbon dioxide molecules sequestrated per year from Figure 2.1, we need to estimate the number of molecules (or moles, see Box 1.1) of air. To do that we need a little bit of atmospheric science. Box 2.6 gives us a rough estimate of the number of molecules in the atmosphere
Box 2.6
Our objective here is to create a simple model that will allow us to convert the data in Figure 2.1 into the total amount of carbon dioxide in the atmosphere and thus enable us to compare the direct atmospheric concentration of carbon dioxide with the amount stored in vegetation calculated in Box 2.3 and the amount released annually from combustion of fossil fuels.
Earth’s atmosphere is an extremely thin sheet of air extending from the surface of Earth to the edge of space. Gravity holds the atmosphere to Earth’s surface. The sun heats the atmosphere through two main mechanisms: direct heating of Earth followed by the transfer of surface heat to higher altitudes through movement of air masses, and through directly heating the air molecules through absorption of the ultraviolet solar radiation by ozone in the stratosphere. Measurements of atmospheric temperature as a function of height above the surface of Earth, primarily conducted by the US National Aeronautics and Space Administration (NASA), produce the results shown in Figure 2.9.
Figure 2.9. Variation of temperature with height in the troposphere and the low stratosphere
Source: NASA (2009).6
First, the temperature decreases sharply as the height increases, and then it stabilizes around – 55°C. Around an altitude of 25 km, the temperature starts to rise again due to the ozone layer’s direct absorption of solar energy in the stratosphere. The low region in which the temperature decreases is the troposphere, and the region above it is the stratosphere. Based on these data, the transition between the troposphere and the stratosphere is around 11 km.
Variation of temperature with height in the troposphere
| Height (km) | Temperature (°C) | Pressure (Atm) | n (1019 moles) |
|---|---|---|---|
| 0 | 15.0 | 1.00 | 2.15 |
| 1 | 8.5 | 0.89 | 1.96 |
| 2 | 2.0 | 0.78 | 1.76 |
| 3 | –4.5 | 0.69 | 1.59 |
| 4 | –10.9 | 0.61 | 1.44 |
| 5 | –17.4 | 0.53 | 1.28 |
| 6 | –23.9 | 0.47 | 1.17 |
| 7 | –30.4 | 0.41 | 1.05 |
| 8 | –36.9 | 0.35 | 0.92 |
| 9 | –43.4 | 0.30 | 0.81 |
| 10 | – 49.9 | 0.26 | 0.73 |
| 11 | – 56.4 | 0.22 | 0.63 |
Source: NASA (2009).6
The data for temperature and pressure as a function of height for the troposphere are given in tabular form in the first three columns in Table 2.2. We should now convert these data to the number of molecules (or moles). This conversion is performed through a law sometimes referred to as the ideal gas law, which gives the relationship between the volume, pressure, temperature, and the number of moles of the gas in the volume for a dilute gas.This relationship is given in equation 2.4:
PV = nR(T + 273). [2.4]
In this equation, P is the pressure in units of atmospheres,V is the volume in units of m3, and n is the number of moles (remember Box 1.1). T is the temperature in degrees Celsius. We add the 273 to convert the Celsius temperature into a different scale, but this is another issue that is explained in chapter 5. R is a constant given by 8.23 × 10– 5 (or 0.0000823) Atm·m3/mole·deg. With these units, the units on the left side of the equation will be the same as the units on the right side of the equation, as it should always be.
Because the temperature and pressure change with height, we will model the atmosphere as thin layers of 1 km depth surrounding Earth. A schematic presentation of these layers is shown in Figure 2.10.
Figure 2.10. A schematic representation of the 1 km layers of the troposphere surrounding Earth
The approximate volume of each of these layers is given by the surface area of the sphere multiplied by 1000 m (= 1 km).The area of a sphere is given by 4πr2, where r is the radius of the sphere.The radius of Earth is 6380 km or 6.38 × 106 m. So the area of Earth will be 4 × 3.14 × (6.38 × 106)2 = 5.1 × 1014 m2, and the volume of the first layer will be 5.1 × 1014 × 1000 = 5.1 × 1017 m3. Because 1 km is very small compared to the radius of Earth, we can treat all the layers as having the same volume. Each row in Table 2.1 now represents one of these layers, in which we assume that the pressure and temperature remain constant. We apply equation 2.4 to each layer to calculate the number of moles.The results of these calculations are shown in the last column. We will sum up the numbers in the last column and get 1.5 × 1020 moles of air in the troposphere.
We now go back to Figure 2.2. One ppmv will be equal to this number divided by a million or 1 ppmv = 1.5 × 1020 × 1 × 10– 6 = 1.5 × 1014 moles of carbon dioxide. From Figure 2.2 the height of the oscillations is approximately 5.3 ppmv, and because the sequestration takes only half a year, the number of moles of carbon dioxide sequestrated is 5.3 × 1.5 × 1014/2 = 4 × 1014 moles of carbon dioxide per year.
From Box 2.6, the total number of moles of carbon (which is the same as the number of moles of carbon dioxide) sequestrated in the atmosphere by broad- leaf trees in the northern hemisphere, as observed at the Mauna Loa site, is 4 × 1014. Each mole of carbon weighs 12 g (see Box 1.1). So the weight of carbon sequestrated every year by these trees is 4 × 1014 × 12/106 = 5 × 109 (5 billion) tons of carbon.
Again, the amount of carbon released through burning of fossil fuel in 2005 (again ignoring the 15% nonfossil fuel energy sources) is approximately 27 × 109 (27 billion) tons. Th e two numbers are well within the same order of magnitude.
