Week1-Globalwarmingandthecarboncycle (1)

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Global warming and the carbon cycleWeeks 1–2

The inuence of human activity on the Earth’s climate has become anissue of major concern, involving not just science but also politics,economics and international relations. In this project, you will look at akey scientic aspect of this issue – the accumulation of carbon dioxide inthe atmosphere before adopting the role of one of three types of ’specialist’in the subject. You will then work with fellow students to prepare andwrite a scientic report.

During the rst two weeks of the project you will get a taste of all threespecialist areas. You will then discuss your ndings and results with therest of your team in your rst online meeting. During the meeting, andbased on your experiences over these rst two weeks, you will decide whoin your team takes which role: the carbon specialists , the laboratory specialists and the modelling specialists .

At the end of the eight-week project period, each member of the team willcontribute a report about your project. A suggested list of sections for thisreport is shown at the end of this document. Each team member will writetheir own abstract, introduction and general background and conclusions.The body of the report is to be written together by your team as adocument shared by the whole team.

In the sections of this guide that follow, you will cover some importantbackground knowledge about global warming and the carbon cycle. Youmay have come across some of this material before in your studies but it isre-presented here in the context of this team project. If you choose to be

the carbon specialist in your project team then you will delve much deeperinto the science behind global warming and the carbon cycle over thecoming weeks than if you become the laboratory or modelling specialist.

Global warming refers to the increasing temperature of the Earth’s You may have come across thisbackground to global warmingin S104.

atmosphere. This issue is likely to remain a major issue for the foreseeablefuture. Understanding this topic is far from easy – the Earth’s climate isextremely complicated. Furthermore, if a person’s only exposure to thescience of global warming has been through the general media (newsstories, etc.) then it may be difficult for them to get a view of what isreally going on.

Arguments about whether the Earth’s atmosphere-ocean system really is oris not warming up, and the causes of any effect, are now on many politicaland economic agendas. Some groups deny the existence of global warming,or at least any relationship with human activity on the planet, while othergroups are accused of scaremongering. There is no doubt that globalwarming has become a fascinating topic regardless of the science behind it.

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Rumours of catastrophe

1 Rumours of catastrophe

No doubt you have seen newspaper or magazine articles, web pages,television programmes or news reports about the issues surrounding globalwarming. Understandably, many of these stories focus on the potentialeffects of a rise in the Earth’s temperature. Figure 1 shows a few suchheadlines which could give you the impression that the world is facingsome sort of global catastrophe.

Sections 1 and 2 are adaptedfrom S104. We will be buildingon these topics over the next

8 weeks during your project.

Figure 1 Cuttings from newspaper stories focusing on some of theextreme consequences of global warming.

The overall concept of higher temperatures caused by global warming, andthe idea that global warming could have many uncomfortableconsequences, is already embedded in popular culture. Indeed, wheneveran unusual weather event occurs, people wonder whether it is aconsequence of global warming: be it a particularly powerful storm, or alarge number of hurricanes in one particular year, or a headline such as‘hottest July on record’.

However, care must be taken not to jump to conclusions. For those livingon the plains of the Midwest, tornados are not uncommon. Whereas mostUK residents would be extremely surprised to see a tornado coming theirway. In fact, very small tornados are not uncommon in the UK. Usually

they are weak and relatively benign; few will do any damage to buildingsfor example. However, if you wait long enough, the more powerful’extreme event’ will happen, just as if you roll ve dice repeatedly for longenough, eventually you will get ve sixes.

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A balance of energy ows

Nevertheless, some unusual weather events may well be linked to an overallglobal effect. The point is that it is hard to untangle the extreme eventfrom the overall slow, gradual underlying changes. To get to the bottom of long-term changes usually means looking at the average behaviour overlong periods of time.

2 A balance of energy ows

The Earth is warmed mainly externally by the Sun. There is also a steadyinternal contribution from radioactive elements, such as uranium, whichhave been part of the Earth since its formation. The radioactive decay of these elements is accompanied by the release of energy, warming the Earthslightly, but this contribution is small compared to that of the Sun, andstays practically constant over thousands of years.

The Sun emits electromagnetic radiation, some of which is incident on theEarth. The Earth directly reects some of this radiation and absorbs therest. The Earth also emits radiation by virtue of being warm. The meantemperature of the Earth is largely determined by the balance of theseenergy ows. There are many possible inuences that can affect the

Figure 2 The Sun is ourmajor source of heat.

balance including: the output of the Sun, the shape of the Earth’s orbit,the angle of the Earth’s tilt, the reective nature of the Earth’s surface,and so on. Some of these factors have inuenced the Earth’s climate in thepast, leading to ice ages with global average temperatures that were 5 ◦ Ccolder than today, and to warm periods that were 10 ◦ C hotter than today.

The Global Mean Surface Temperature (GMST) depends on the rate at

which the Earth’s surface gains energy, and the rate at which it losesenergy.

The Sun is the ultimate source of most of the energy arriving at theEarth’s surface. The next largest is the energy that ows out from theinterior of the Earth, but that rate of energy ow is 2 000 times less thanthe rate at which the surface receives solar energy. From now on the heatfrom the interior will be neglected. The Earth’s surface loses energy byvarious means. For now they will be grouped together to give one overallrate of energy loss, as shown in the highly simplied model in Figure 3.

Figure 3 Rates of energy gain and loss by the whole of the Earth’ssurface, represented by arrows to and from the Earth’s surface.

The downward-pointing arrow in Figure 3 represents the rate at which thewhole of the Earth’s surface gains energy, and the upward-pointing arrow

that originates at the Earth’s surface represents the rate at which thewhole of the Earth’s surface loses energy. If the rates were equal then theGMST would be constant. If the rates were not equal, the GMST wouldchange.

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If the rate of energy loss exceeded the rate of energy gain, what wouldhappen to the GMST?

In fact, the rates are not exactly equal every second and, through the dayand the year, there are moments when the gain slightly exceeds the loss,and other moments when the loss slightly exceeds the gain. However, overa period of a few years, the gains and losses balance out. This is why,when the GMST is averaged over a few years, the average is very nearlythe same as over the previous few, or the following few, years.

To explore the relationship between the GMST and the rates of energygain and loss in more detail, it helps to begin by examining a simpleanalogy that shares the key features of the system illustrated in Figure 3.This analogy will help you understand how the system works in this sense,you are constructing a model of the real system.

2.1 Models in scienceInsight into the behaviour of complex natural systems can often beobtained by constructing a model. In this context, a model is just asimplied description of something in the real world. Scientic models aidunderstanding by focusing only on some important aspects of the system.

For example, Figure 4a shows an aerial image of the Walton Hall area inMilton Keynes, UK – home to The Open University’s main campus. If youwanted to direct someone to a particular office, you might draw a simplesketch as in Figure 4b. The sketch is more usable than a photograph, and

gives the person trying to nd the office enough information to do so.However, Figure 4b is not much like reality it is a model of the realsituation. The important components of reality (the roads, theroundabouts, where to turn left and right and where to park) arerepresented in the model, and that is all that is needed.

Figure 4 (a) An aerial photograph of The Open University main campusat Milton Keynes. (b) A simple sketch showing directions, whichrepresents a model of reality.

Looking back at Figure 3, you should now realise that this treatment of the balance of energy gains and losses is itself a model of the real worldsystem. The entire Earth’s surface is being represented as a singlecomponent, with a single gain and loss. Clearly, this is not like reality, but

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it is a model of reality that represents the main components in anunderstandable way. Models that link a familiar, simple situation with amore complicated phenomenon often aid our understanding. For example,the motion of a simple pendulum illustrates the concept of ”periodicity”that can be also observed in the tides; reservoirs of water, pumps andpipework provide a conceptual model of electric circuits; snooker providesa macroscopic model for collisions between subatomic particles.

2.2 Modelling the behaviour of the GMST using aleaky tank analogy

The aim here is to model the behaviour of the Earth’s GMST, i.e. developa simple description of how the GMST relies on the rates of transfer of energy to and from the Earth’s surface. To do this, it is useful to consideran analogy of the way energy transfers to and from the Earth’s surface.The analogy we will use is a leaky tank into which water is pouring.

Figure 5 shows a tank of water with a tap feeding water in at a steady rate,and a vertical slot cut in the side of the tank, letting water out. The rateat which water is fed into the tank represents the rate of energy gain bythe Earth’s surface; the rate at which water leaks out of the slot representsthe rate of energy loss from the Earth’s surface. The level of water in thetank represents the GMST: the higher the level, the higher the GMST.

In the leaky tank shown in Figure 5, the rate at which water ows out of the slot depends on how much water is in the tank at a given moment.The greater the depth of water in the tank, the greater the rate of waterloss from the slot.

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Figure 5 The leaky tank. (a) Initially the tank is empty, and the water is

owing in at a certain steady rate. The water level rises until the leak rateequals the rate of input, whereupon the level becomes steady. (b) Theinput rate is increased, and the water rises to a new steady level that ishigher. (c) The input rate is decreased, and the water falls to a newsteady level that is lower.

Figure 5a shows a sequence that starts with the tank empty, and the waterowing in at a steady rate. Initially, the leak rate is less than the rate of input, so the water level rises. As the water level rises, there is a greaterlength of slot to let the water out, so the leak rate increases, and itcontinues to increase until the leak rate equals the rate of water input. Atthis point, the water level stops rising, and it stays at the level it hasreached.The water level is now in a steady state, i.e. it is not changing. Of course,water is pouring into and out of the tank, so this is a dynamic steady staterather than a static steady state. The crucial condition for the dynamicsteady state is that the input and output rates are equal. This equality of rates can be expressed as:

input rate = output rate (1)

The graph in Figure 6a shows how the water level in the tank changes withtime for the scenario in Figure 5a. You can see that in the rst 10 secondsafter the tap is turned on, the water level rises from zero to about 17 mm.In the time interval 10 to 20 seconds after the tap is turned on, it risesfrom 17 to about 30 mm, i.e. a further 13 mm in the next 10 seconds.

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Figure 6 showing how the water level rises with time for scenarios (a),

(b) and (c) in Figure 5.

How many millimetres does the water level rise in the time interval 20to 30 seconds after the tap is turned on?

Thus, as the water level rises, the rate at which the level changes decreases(i.e. the rate of change in level ‘slows down ’). In the graph this isapparent from how the curve bends over. You can see that, ultimately, thecurve attens out and stays at the same water level the steady-state level.At this constant level, the leak rate equals the rate of input.

Figure 5a starts with the analogy of a cold Earth (the empty tank) and theSun having just been turned on (the tap). The Earth’s surface gains solarenergy (the water) and so the GMST (the water level) rises. As it does so(and this is a crucial point) the rate of energy loss from the Earth’s surfaceincreases. This must be happening in reality since if it did not the Earthwould never reach a steady mean temperature.

So, using the leaky tank as an analogy for the way energy transfers to andfrom the Earth’s surface, a model has been developed for the behaviour of the Earth’s GMST and how it relates to the energy gains and losses of thesurface. That is, given a consistent rate of energy supplied to the Earth’ssurface, a steady state will develop where the GMST remains at a constantvalue. However, if the rate of energy supplied to the Earth’s surfacechanges, a new steady state will develop, characterised by a differentGMST value.

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So by modelling the behaviour of the system in this way, you have gainedan insight to why the Earth has remained at (approximately) the sametemperature for a huge period of time (it has been in a steady state). Theslight variations in GMST that have occurred in the past are a result of slightly different steady states being reached after changes in the rates of gain and loss of energy to the surface.

The leaky tank itself can be modelled with mathematics and themathematical model will in turn be adaptable as a model of the GMST.As we cannot do an experiment of the carbon cycle directly, some of yourteam will continue to think about the leaking tank in more detail. In laterweeks, the laboratory specialists will run a series of experiments on leakingtanks.

A taste of being an experimental specialist (Allow 1 h)Before you start, read through all the instructions in this box carefully and do a risk assessment to make sure you can work safely.

Find an old container that you no longer need (a thin ice-cream tubor a disposable cup or a take-away container for example).

Although the example given above is for a box with a slit, here youshould concentrate on a simpler system: a box with a single hole .

Make a small hole in the container near the base to create a leakingtank of your own. Remember that you can always make the holelarger but you cannot easily make it smaller.

In a sink or bath, turn the tap on to the ‘slowest’ steady ow you canset. Measure the ow rate of water. You can do this with a measuring

jug (or any known volume) by timing how long it takes for say onelitre to ow from the tap.

With the tap still running at the same rate, place your leaking tankin the water ow and time how long it takes to get to a steady depthof water in the container; at the steady state the water level will stoprising. You may have to try a few different ow rates from the tap toget a conveniently measurable depth.

Repeat the experiment with different ow rates and measure theheight of the water above the hole at the steady state and the time ittakes to reach the steady state.

Plot a graph of your results – what can you deduce from your data?

Over the next few weeks, the laboratory specialist will continue theseexperiments with the aim of producing data similar to that shown inFigure 6, but rst we need to delve more into the mathematics of theproblem.

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Numerical solutions

3 Numerical solutions

At the end of the previous section we considered an experiment to modelthe real world. We can also model the real world mathematically usingnumerical methods . The method described here will give you a glimpseinto a branch of mathematics called numerical analysis, which specialisesin solving problems numerically on a computer.

We have just been considering the analogy to the Earth’s GMST using aleaking tank. In your experiment, you have considered a tank with a holenear the base, Figure 7. Figure 7 A diagram of atank with a hole. The hole is

at a height zhole with theheight of the surface of thewater at any time given byzwater . The hole has an areaa and water with density ρ

ows from the hole with aspeed v. The tank base hasan area A that is uniform forall values of z.

Water is virtually incompressible, the only energy stored in the tank is thegravitational potential energy. This drives the outow, for which thekinetic energy density must equal the potential energy density:

ρg (zwater − zhole ) = 12

ρv2 (2)

(Notes: energy density, (J m − 3 in SI units) is equivalent to pressure (Nm − 2 in SI units), as a joule is the same as a newton metre.

The rate of ow of water out of the hole is given by the area of the holemultiplied by the speed of the water (this gives the volume of waterowing per second). If we let a be the area of the hole in the tank, thenthe rate of the ow of water out of the tank Q (volume per second) is givenby a × v and can be written as:

Q = a 2g (zwater − zhole ) (3)If the hole is at the bottom of the tank, then zwater − zhole = h where h isthe height of water in the tank giving:

Q = a 2gh (4)Finally, if the area of the tank base is given by A (so that V = Ah ) thenwe can rewrite this equation as:

Q = a 2gV A (5)In your experiment you have had water owing into the tank from the tapat a constant rate. If we call this constant rate of water owing into the

tank S then we can write an equation for the variation in the volume of water in the tank with time:dV dt

= S − CV 1

2 where C = a 2gA (6)When calculating quantities with formulas like this it is prudent to convertall quantities to SI units before inserting values.

Although this equation can be solved to nd an exact solution, we can alsouse numerical analysis. More information about setting up differentialequations can be found in the resource document Setting up differential equations .

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Numerical solutions

A taste of being a modelling specialist (Allow 2 h)Using the resource document Solving differential equations by computer: basic techniques as a guide, implement the Euler methodto create a mathematical model of the leaking tank, as described byEquation 6, in either Maxima programming language or in aspreadsheet.

Insert your own parameters into the constant C from yourexperiment.

Does your model match your experimental results?

What factors might affect how well your model matches yourreal-world experimental results?

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The greenhouse effect

4 The greenhouse effect

In the following sections you will be introduced to some backgroundinformation that will lead to the development of the so-called ASD model that the modelling specialists in your project team will bedeveloping. The information provided will give you an idea of thetype of research that will be required by the carbon specialists in yourteam to help to direct the modelling and experimental work of otherteam members.

Human civilization has developed in the period since the last ice age, soour way of life is adapted to temperatures that are fairly cool in historicterms. In the absence of human intervention, the Earth’s climate might beexpected to be fairly stable over a few thousands of years, but the effects

of industrialisation may have produced more rapid changes, and thesechanges are likely to grow in the future.

The electromagnetic radiation emitted by the Sun covers a broad spectrumthat peaks near the visible range. Some of this radiation is reected (byclouds or the oceans, for example), some drives chemical reactions such asphotosynthesis, but a large fraction is converted into heat energy, warmingour planet.

Any warm object emits a spectrum of electromagnetic radiation. In theideal case, this spectrum follows a characteristic pattern, known as ablack-body spectrum (or a Planck spectrum). The peak of this spectrumoccurs at a frequency that is proportional to the absolute surfacetemperature of the object.The Sun has a surface temperature of 5 800 K, and its peak output is nearthe visible range. The Earth is much cooler, with a surface temperature of about 290 K, so the peak of its emission spectrum is at a frequency that isroughly 20 times smaller – rmly in the infrared range.

Molecules in the Earth’s atmosphere, such as water, methane and carbon

Figure 8 In 1861, JohnTyndall discovered that watervapour and carbon dioxideare greenhouse gases, whosepresence in the atmospherewarms the Earth signicantly.

dioxide, do not absorb visible radiation very efficiently, but they are strongabsorbers of infrared radiation, which excites their molecular vibrationsand rotations. Consequently, electromagnetic radiation incident from theSun passes down to ground level quite easily, but the infrared energy

re-radiated by the Earth is intercepted, and some of the energy nds itsway back to the ground. Gases that are strong absorbers of infraredradiation are called greenhouse gases although in reality glass-boxgreenhouses are not a perfect analogy.

From a human perspective, the presence of greenhouse gases is a goodthing. Without it, the surface of the Earth would be about 30 ◦ C colder –a very harsh environment for life. The early interest was on ice ages, andwhether they could be triggered by reductions in greenhouse gases. Butthis is not the current situation. Over the last two or three hundred yearsthere has been a dramatic increase in the amount of carbon dioxide in theatmosphere. Increases in greenhouse gases might be expected to lead to

increases in surface temperature, and this is the current concern.There are several greenhouse gases, so why focus on carbon dioxide? Onereason is that human activity has led to an extremely rapid increase in thisgas. To understand why, we must go far back into history.

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The greenhouse effect

During photosynthesis, a plant takes in carbon dioxide from theatmosphere and converts it into other carbon-containing molecules thatbecome incorporated in the body of the plant. Normally, when the plantdies, the carbon-containing molecules recombine with oxygen in the air,producing gaseous carbon dioxide. So carbon from the atmosphere istemporarily borrowed by the plant, and then handed back again. However,if the plant dies in a swamp, it may not be easy for oxygen to reach it, anddecomposition is hindered. Under these circumstances, the carbon can belocked up, ultimately forming deposits of fossil fuels such as coal or oil,which accumulate over geological spans of time.

In about 1780, mankind discovered how to drive machines using the energyreleased by burning fossil fuels. At rst, the machines were in factories,but they spread to modes of transport such as cars and planes. Heatingand the generation of electricity also relied heavily on burning coal or gas.All of this activity released carbon dioxide into the atmosphere.

Figure 9 Burning fossil fuelsreleases carbon dioxide intothe atmosphere.

Many natural factors affect the amount of carbon dioxide in the

atmosphere. For example, erupting volcanos and breathing animals emitcarbon dioxide, while oceans and photosynthesising plants absorb it.Under normal circumstances, a rough balance is maintained and theamount of carbon dioxide in the atmosphere remains more or less constantover long periods of time. However, the burning of fossil fuels hasdisturbed this balance. The slow capture of carbon in fossil fuels, whichtook hundreds of millions of years to accumulate, is being released into theatmosphere in just a few hundred years. Other industrial processes, suchas cement production, add further amounts of carbon dioxide to theatmosphere. This is a leap into the unknown of epic proportions.

The rate at which industrial processes have released carbon dioxide can be

estimated, and the data are shown in red in Figure 10. An exponentialcurve tting these data is shown in blue.

Figure 10 An estimation of the total rate of emission of carbon dioxideinto the atmosphere due to industrial processes (such as the burning of fossil fuels and cement production). The zero of time is taken to be 1750,the start of the industrial revolution [1 Pg = 10 12 kg].

The data on which this graph is based can be found at:

cdiac.esd.ornl.gov/ndps/ndp030.html .

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Three awed arguments

5 Three awed arguments

By the rst half of the twentieth century, it was known that the burning of fossil fuels was releasing huge amounts of carbon dioxide into theatmosphere. Yet many people assumed that the global climate would notchange by much. Three main arguments were advanced to support thisview.

1. Some scientists assumed that the Earth would be self-regulating, sothat processes that increase the temperature would lead to otherprocesses that reduce the temperature. Any temperature change wouldthen be relatively modest.

2. The amount of carbon dissolved in the oceans is about fty timesgreater than that in the atmosphere. We might suppose that theadditional carbon dioxide released by the burning of fossil fuels willend up mostly in the oceans. If only a small fraction remains in the

atmosphere, this may be insufficient to cause much warming.3. Some scientists claimed that, even if the amount of carbon dioxide inthe atmosphere does increase, this will make little difference to globaltemperatures at ground level. It was thought that existing levels of carbon dioxide and water vapour are sufficient to absorb nearly all theinfrared radiation emitted at the Earth’s surface. And so, theargument ran, further additions would make no difference.

Unfortunately, all these arguments are seriously awed.

The rst argument is merely a statement of faith, rather than a scienticargument. So far as we know, the hypothesis that the Earth isself-regulating cannot be derived from basic physical principles, and thereis no compelling reason to believe it. It is true that some mechanismsprovide negative feedback, inhibiting a rise in atmospheric carbon dioxide.For example, an increase in atmospheric carbon dioxide, and theaccompanying rise in temperature, both increase the efficiency of photosynthesis. The resulting increase in plant growth removes some of thecarbon dioxide from the atmosphere, limiting the accumulation of CO 2 inthe atmosphere.

However, other mechanisms provide positive feedback, reinforcing a rise inatmospheric carbon dioxide. For example, as the amount of carbon dioxidedissolved in seawater increases, and as the temperature increases, theability of seawater to take up more carbon dioxide becomes restricted.Also, as the temperature rises, more water will evaporate from the oceans,and methane might be released from frozen ground in Siberia. Since bothwater vapour and methane are greenhouse gases, temperatures could bedriven even higher. To say that the Earth is self-regulating may thereforebe wishful thinking.

For practical purposes, the second argument is incomplete because it lacksany discussion of timescales. To give an analogy, after torrential rain weknow that the most of the rainwater will drain away, nding its way intorivers and oceans. However, this does not mean that oods are impossible,nor that they will last for a short time. A similar situation applies tocarbon dioxide. We know that it is being pumped into the atmosphere at avery fast rate. If there is a bottleneck in the process of transferring carbondioxide from the atmosphere into the ocean, it will accumulate in theatmosphere instead, and might stay there for ages.

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Measurements of carbon dioxide

Finally, the third argument turns out to be too simplistic. In the 1950sand 60s Gilbert Plass and Fritz M¨ oller realised that that the ow of energythrough the atmosphere has to be modelled layer by layer, with the mostcrucial layer being high up, where the Earth nally emits infraredradiation into space. The addition of extra carbon dioxide moves this nalemission layer outwards, where it is cooler. Because the nal layer iscooler, it does not radiate so well, and more heat is trapped around theEarth; this eventually feeds back into lower levels, and warms the Earth’ssurface. The detailed modelling of these energy ows lies beyond the scopeof this project. The only signicant point to remember is that increases inatmospheric carbon dioxide are expected to produce increases in globaltemperatures at ground level.

6 Measurements of carbon dioxide

In 1938, the engineer Guy Callendar collated historic measurements of atmospheric carbon dioxide and found a pattern of increasing levels overtime. By 1958, he had extended his analysis, and conrmed the increasewith greater condence (Callendar, 1958).

Around this time, improvements in infrared technology allowedmeasurements of atmospheric carbon dioxide concentration to be madewith unprecedented precision. Starting in 1958, and continuing for manyyears, David Keeling measured the concentration of carbon dioxide atMauna Loa in Hawaii (Keeling, 1970). His results, known as the Keelingcurve , show an inexorable rise with small seasonal variations. Figure 11shows the raw data in red, with an exponential t in blue.

Figure 11 The Keeling curve showing carbon dioxide concentration inparts per million (ppm). The underlying trend is an exponential increase.Superimposed on this is a seasonal variation related to the annual growthand decay of plants in the Northern Hemisphere.

Full information about the Keeling curve is available from the web sites

scrippsco2.ucsd.edu/keelingcurve.ucsd.edu .

The second site compares modern values with measurements of aircomposition in ice cores going back 800 000 years. Before the industrialrevolution the typical concentration of carbon dioxide was 225 parts permillion (ppm), with uctuations of ± 50 ppm occurring slowly overtimescales of order 50 000 years. Since the Industrial Revolution (around1750), there have been much larger increases (around 130 ppm) in only 250

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Units of measurement for carbon dioxide

years. Clearly, carbon dioxide is entering the atmosphere at a faster ratethan it is being removed.

Tracking particular isotopes of carbon gives further evidence on the originof the surplus carbon dioxide. The isotope C 14 is continually created whencosmic rays collide with molecules in the upper atmosphere. It is

radioactive, and typically survives for about 6 000 years. It is thereforepermanently present in the atmosphere, but has vanished from fossil fuels,which were laid down millions of years ago. If the carbon dioxide fromfossil fuel burning is making a signicant contribution to the totalatmospheric carbon dioxide, recent tree rings should show a decit in C 14 ,compared to tree rings formed before the industrial revolution. This decithas been measured.

Two further pieces of evidence indicate that the surplus carbon dioxide islinked to human activity rather than natural processes such as volcaniceruptions.

1. The concentration of carbon dioxide is greater in the NorthernHemisphere than in the Southern Hemisphere; this is readilyunderstood because anthropogenic emissions occur mainly in theindustrialised countries north of the Equator.

2. For the last 50 years, anthropogenic carbon dioxide emissions haveincreased almost exponentially; this is reected in the measured carbondioxide concentration in the atmosphere, which has also increasedexponentially.

7 Units of measurement for carbon dioxide

The amount of carbon dioxide in the atmosphere is sometimes expressed inparts per million (ppm). If the amount of carbon dioxide is n ppm, thereare n carbon dioxide molecules in every million molecules of dry air (i.e.air with all water vapour removed).

This way of measuring carbon dioxide is good enough for the atmosphere.However, carbon dioxide is absorbed by seawater, by plants duringphotosynthesis, and by rocks during weathering. In all these processes,chemical reactions take place that convert carbon dioxide into othercarbon-containing compounds. In these reactions, the carbon dioxidemolecules are destroyed, but the number of carbon atoms remains xed.Rather than tracking the carbon dioxide, it makes sense to ‘follow thecarbon’ in these reactions.

For this reason, the amount of carbon dioxide in the atmosphere is oftenexpressed in terms of carbon mass equivalents . This is the total mass of thecarbon atoms in the carbon dioxide molecules. A convenient unit is1 petagram carbon mass equivalent, written as 1 PgC, which correspondsto 1015 g = 10 12 kg of carbon atoms. In the atmosphere, the carbon atomsare in the form of carbon dioxide molecules, but carbon mass equivalentscan be used to quantify all stores of carbon, no matter what the carbon iscombined with.

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Modelling the ow of carbon

Conversion rates1ppm CO 2 in dry air ≡ 2.11PgC

1PgC ≡ 0.47ppm CO 2 in dry air .

Because a carbon dioxide molecule has a mass that is 44 .01/ 12.01(= 3 .66)greater than the mass of a carbon atom, the total mass of carbon dioxide inthe atmosphere is found by multiplying its carbon mass equivalent by 3 .66.In pre-industrial times, the total amount of carbon in the atmosphere was

280 ppm CO 2 of dry air ≡ 591 PgC ≡ 2160 Pg of CO2

By 2014, the amount of carbon dioxide has grown to

400 ppm CO 2 of dry air.

How many petagrams of carbon dioxide does this represent?

400PgC of dry air ≡ 2.11 × 400Pg CO 2

≡ 3.66 × 2.11 × 400 Pg CO 2

= 309 Pg CO 2

8 Modelling the ow of carbon

To understand why carbon dioxide accumulates in the atmosphere, and itsability to persist there for a long time, we need a quantitative model. It isuseful to imagine a series of reservoirs for carbon, representing theatmosphere, the oceans, the biosphere and sedimentary deposits. In theatmosphere, the carbon is in the form of carbon dioxide, but in otherreservoirs it may be in other forms such as carbonates, cellulose (biomass)or fossil fuels. Over long periods of time, carbon atoms move through thesereservoirs in the so-called carbon cycle . In this cycle, some reservoirsrelease carbon dioxide into the atmosphere, and some reservoirs absorb it.

Climate modelers have developed complex models involving many differentreservoirs. An example is shown in Figure 12, but even more elaborateversions exist such as that shown in the fth report of theIntergovernmental Panel on Climate Change, (IPPC, 2013 Figure 6.1).

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Figure 12 A model of the carbon cycle with reservoirs for theatmosphere, fossil fuels, vegetation, marine biota, surface ocean,intermediate and deep ocean and surface sediment, taken from Sarmientoand Gruber (2002, p.31).

In Figure 12, reservoir sizes (also called carbon stocks) are given in PgCand shown inside the boxes. Rates of ow are given in PgCyr − 1 andshown on the arrows. Black gures refer to the situation that existed justbefore the Industrial Revolution (around 1750). Red gures in boxesrepresent the cumulative changes to pre-industrial carbon stocks that hadoccurred by 1989, and red gures on arrows represent additional uxes(above pre-industrial values) estimated for 1989.

The models have become very complicated. This is as it should be – theEarth is a complex system, and the implications are so important forhumanity that every avenue must be explored to achieve as much certaintyas possible. The latest report (2014) on the physical science basis of climate change was compiled by 250 experts, is around 1500 pages long,and cites 9 000 scientic papers (IPPC, 2013). There is clearly a danger of getting bogged down in the detail, and this is something you must guardagainst when writing your project report.

The scope of this project is strictly limited. The modelling specialist(s) inyour project team will develop simplied models of the carbon cycle, basedon two or three reservoirs. These models are not as reliable as those

developed by experts working with supercomputers, but this is notimportant here. Simple models may not convince climate-change sceptics,but they have the merit of exploring the main features of the problem inthe most direct way.

We will use a model with just three carbon reservoirs, in addition to thefossil fuel source. The reservoirs are the atmosphere , the surface oceanand the intermediate and deep ocean (referred to simply as the deepocean from now on). The model may be called the ASD model from theinitial letters of the reservoirs. Compared with Figure 12, you can see thatthree reservoirs are being ignored:• marine biota (such as algae, plankton and sh)• surface sediment• vegetation, soil and detritus.

The rst two omissions are not very signicant. The marine biota carbonreservoir is small, with relatively small net carbon ows, so it cannot play

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a major role in the whole cycle. We will not include this reservoir, but wemust therefore add its net ow from the surface to the deep oceans. Thesurface sediment carbon reservoir at the bottom of the ocean also involvesa tiny carbon ow, which does not recirculate on any reasonable timescale.We therefore ignore it altogether.

The decision to ignore the carbon reservoir associated with vegetation, soiland detritus is more drastic. Complex factors come into play. For example,cutting down and burning trees releases carbon dioxide into theatmosphere and reduces the capacity of a forest to absorb more carbondioxide. On the other hand, increased carbon dioxide concentrationsproduce greater rates of photosynthesis, increasing the capacity of plantsto absorb more carbon dioxide.

It is not clear how things will evolve in the future. However, one of theconclusions of the Intergovernmental Panel of Climate Change (2014) isthat the net balance of all terrestrial ecosystems, those affected by land usechange and those not, has been close to neutral since 1750. By ignoring

the vegetation carbon reservoir, we are in effect assuming that thisneutrality will continue.

The division of the ocean into surface and deep regions may seem arbitrary,but is a real feature (Craig, 1958). The surface and deep regions areseparated by a thin layer called the thermocline , where the temperaturedrops more rapidly than in the layers above or below it (Figure 13). Thesurface ocean is well-mixed by waves and convection. The deep oceanis less well mixed because the narrow temperature range means thatconvection is weak. The thermocline is typically a few hundred metresbelow the surface. As a global average, the surface ocean contains about2% of the total volume of the seawater, and the deep ocean about 98%.

Figure 13 Measurements of temperature versus depth inthe Atlantic ocean, showing athermocline at 100–200 m.

It is essential for carbon to pass through the surface layer to reach thedeep ocean, where it is ultimately stored over long timescales. However,the chemistry in the surface ocean is such that it acts as a bottleneck –anthropogenic carbon dioxide is being emitted so rapidly that it cannot getthrough the surface layer quickly enough, and accumulates in theatmosphere. The role of the surface ocean as a bottleneck makes itimportant to treat the two regions separately. For simplicity, we assumethat the surface and deep oceans can both be treated as homogeneous.

A diagrammatic representation of the ASD model is shown in Figure 14.The atmosphere, surface and deep ocean reservoirs are indicated by boxesmarked a , s and d, and the carbon mass equivalents for these reservoirs areM a , M s and M d . The reserve of fossil fuels is shown by a circle, marked f .This acts as a source of carbon dioxide released into the atmosphere. Weassume that there are ample reserves, so that fossil fuels are not about torun out, but we do not need to know exactly how much remains.

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Figure 14 The ASD three-box model of the carbon cycle.

Carbon ows between the reservoirs are shown by arrows in Figure 14, andthe rates of carbon transfer from one reservoir to another are denoted bysymbols such as Ras . It is important to note the convention used here.

Convention for subscripts

The symbol R ij represents the rate of ow of carbon equivalents to See the notes on Setting updifferential equations for furtherdiscussion of this ‘back-to-front’convention.

reservoir i from reservoir j .

The units of R ij are carbon equivalents per unit time (e.g. PgC yr− 1 ).

In addition to the various ows between reservoirs, carbon enters theatmosphere from the burning of fossil fuels. The rate of ow of carbonequivalents into the atmosphere from this source is denoted by F (t). Thisis a function of time that has been increasing almost exponentially for thelast 250 years, as shown earlier in Figure 10.

To complete the model, we must make an assumption about the rate of ow of carbon between the reservoirs. The following assumption is a goodstarting point.

Flow rates of carbon between reservoirs

We assume that the rate of ow of carbon from one reservoir to The notes on Setting updifferential equations made asimilar assumption for migratinghuman populations.

another is proportional to the amount of carbon in the initial reservoir. So, if we are considering the ow of carbon to reservoir i from reservoir j , the ow rate is proportional to M j , and is given by

R ij = kij M j , (7)

where kij is called the rate constant for ows to reservoir i fromreservoir j . This constant depends on the nature of the reservoirs, butis independent of the amounts of carbon in them.

To give a specic example, the rate of ow of carbon to the atmosphere from the surface ocean is given by

Ras = kas M s , (8)

where kas is the rate constant for ows to the atmosphere from the surfaceocean.

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Preparing for your rst meeting

We consider a small time interval δt, during which the carbon massequivalent in the atmosphere changes by δM a . Following the methoddescribed in the notes on Setting up differential equations , and referring toFigure 14, we see that

δM a = Ras δt − R sa δt + F (t)δt,

where the contributions on the right-hand side have plus or minus signsaccording to whether they increase or decrease M a .

Using the linear proportionality specied in Equation 7, this can bewritten as

δM a = kas M s δt − ksa M a δt + F (t)δt.

Dividing by δt, and taking the limit as δt tends to zero, we get thefollowing differential equation

dM adt

= kas M s − ksa M a + F (t). (9)

for the rate of change of M a , the carbon mass equivalent of carbon dioxidein the atmosphere.

Setting the foundations for the ASD modelConstruct equations for the rates of change of M s and M d using thesame approach as that leading to Equation (9).

The system of three rst-order differential equations for M a , M s and M d .The system contains various rate constants and the function F (t), which isassumed to be known. By solving the system of differential equations, for

an appropriate set of initial conditions, you can nd out how M a , M s andM d depend on time. You can then compare your results with historicaldata, and make predictions about the future.

9 Preparing for your rst meeting

At the end of the second week on the project you will need to have anonline meeting with the rest of your team. During this meeting you willdecide, as a team, who will take on each role. As far as possible, try to

split the members of your team evenly between the roles of carbon specialist , laboratory specialist and modelling specialist . It will be helpful if your chosen project manager is also one of the modelling specialists.

Before the meeting, have a look at the short brieng notes for eachtype of specialist and think about the tasks that you have been doingfor the past two weeks:

What have you enjoyed most?What have you found most difficult?Which role will be the most fullling for you?

These questions will help guide you towards your role in the team.

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ReferencesBolin, B. and Eriksson, E. (1959) ’Changes in the carbon dioxide contentof the atmosphere and sea due to fossil fuel combustion’, in Bolin, B. (ed)The Atmosphere and the Sea in Motion , New York, Rockefeller InstitutePress, pp. 130–42.

Callendar, G.S. (1958) ’On the amount of carbon dioxide in theatmosphere’, Tellus , vol. 10, no. 2, pp.243–8.

Craig, H. (1958) ’A critical evaluation of the radiocarbon techniques fordetermining mixing rates in the ocean and atmosphere’, paper presented atthe Second UN International Conference for the Peaceful Use of Atomic Energy , Geneva, 1–13 September.

IPPC (2013) Climate Change 2013: The Physical Science Basis [Online].Working Group 1 contribution to the Fifth Assessment Report of theIntergovernmental Panel on Climate Change (AR5), Cambridge UniversityPress. Available at http://www.ipcc.ch/report/ar5/wg1/ (Accessed 9

March 2015).Keeling, C.D. (1970) ’Is carbon dioxide from fossil fuel changing man’senvironment?’ Proceedings of the American Philosophical Society , vol. 114,no. 1, pp. 10–17.

Möller, F. (1963). On the inuence of changes in the CO 2 concentration inair on the radiation balance of the Earth’s surface and on the climate’,Journal of Geophysical Research , vol. 68, no. 13, pp. 3877–86 [Online].DOI. www.dx.doi.org/10.1029/JZ068i013p03877 (Accessed 9 March 2015).

Sarmiento, J.L. and Gruber, N. (2002) Sinks for anthropogenic carbon,Physics Today vol. 55, no. 8, p.30. [Online].

www.dx.doi.org/10.1063/1.1510279 (Accessed 9 March 2015).

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Appendix: Project report plan

Figure 15 The structure of the project report. Each team member willwrite an abstract and a general introduction as Question 1 (Q1) of theEMA. The team should prepare the remainder of the introduction and thesections on the method results and extension as a shared document thatteam members each submit for Q2 of the EMA. The nal conclusion willform Q3 of the EMA and is to be done individually. The references, whichwill be marked as part of Q2, should combine elements from the individualintroduction (Q1)and the shared body of the report (Q2).

Documents

Week1-Globalwarmingandthecarboncycle (1)