180

8.2Energy balance of the earth

Figure 8.3 shows radiation from the sun reaching the earth. The sun is so far away that the rays arrive essentially parallel. The radiation that reaches the earth has to go through the area of a disc of radius R where R is the radius of the earth. The intensity of the incident radiation is called the solar constant S. The power going through the disc is then

P=S π R 2

Exam tip

The solar constant S is intensity. Intensity is power per unit area so P = SA.

Figure 8.3: The radiation reaching the earth must first go through a disc of area πR 2, where R is the radius of the earth.

181

Clearly, the earth’s surface receives radiation during the day when it faces the sun. But if we want to define a night and day average of the incident intensity, I av, we must divide the power through the disc (Sπ R 2) by the total surface area of the earth (4π R 2) to get

S π R 2 4 π R 2=S 4

Exam tip

You must be able to justify the factor of 4 in the average intensity received by earth

The albedo of the earth is α, and so a fraction α S 4 is reflected, leaving an average intensity reaching the earth surface of:

I av=(1−α)S 4

Because the average albedo of earth is about 0.30, this amounts to 0.7×1400 4=245 W m−2.

In other words, at any moment of the day or night, anywhere on earth, 1 m2 of the surface may be thought to receive 245 J of energy every second.

We are interested in the equilibrium temperature of the earth. This is the temperature attained by the earth when the energy input to the earth is equal to the energy output by the earth (Figure 8.4).

Figure 8.4: Energy diagram showing energy transfers in a model without an atmosphere. Note that the energy in equals the energy out.

The next worked example introduces a first glimpse of an energy balance equation an equation expressing the equality of incoming and outgoing intensities of radiation, leading to an estimate of the equilibrium temperature of the earth.

Worked example 8.2

Assume that the earth’s surface has a fixed temperature T and that it radiates as a black body. The average incoming solar radiation reaching the surface has intensity I av=(1−α)S 4=245 W m−2. (The albedo of 0.30 is provided by the atmosphere, but all other effects of the atmosphere are ignored.)

aWrite down an equation expressing the fact that the intensity received by the earth equals the intensity radiated by the earth into space (an energy balance equation).

bSolve the equation to calculate the equilibrium earth temperature.

cComment on your answer.

Answer

aThe average intensity reaching the surface is:

I av=(1−α)S 4=245 W m−2

The intensity radiated is:

I out =σ T 4

(Here we are assuming that the earth is a black body, so e = 1 and the surrounding space is taken to have a temperature of 0 K.)

Equating the incident and outgoing intensities we get:

245=σ T 4

bSolving the equation, we find:

T=245 5.67×10−8 4=256 K

This temperature is –17 °C.

182

cIt is perhaps surprising that this extremely simple model has given an answer that is not off by orders of magnitude! But a temperature of 256 K is 32 K lower than the earth’s average temperature of 288 K, and so obviously the model is too simplistic. One reason this model is too simple is precisely because we have not taken into account the fact that not all the power radiated by the earth actually escapes. Some of the power is absorbed by the gases in the atmosphere and is re-radiated in all directions including back down to the earth’s surface, causing further warming that we have neglected to take into account. In other words, this model neglects the greenhouse effect. This simple model also points to the general fact that increasing the albedo (more energy reflected) results in lower temperatures.

The model used in Worked example 2.2 assumes that the surface of the earth behaves as a black body (e = 1 and α = 0) but that the atmosphere does not (α = 0.30). Another drawback of the simple model is that it is essentially a zero-dimensional model. The earth is treated as a point without interactions between the surface and the atmosphere. (Latent heat flows, thermal energy flow in oceans through currents and thermal energy transfer between the surface and the atmosphere due to temperature differences between the two are all ignored.) Realistic models must take all these factors (and many others) into account and so are very complex.

The albedo of the earth varies which is why we use an average value. The variations depend on whether an area is covered by many or few clouds, latitude (high latitudes include more snow and ice coverage, α = 0.80), on whether one is over deserts (α = 0.40), forests (α = 0.14) or deep ocean water (α = 0.06) and so on.

Check yourself 2

In the simple example of worked example 8.2, would you expect the average temperature of the earth to increase or decrease if the albedo were to increase?

The greenhouse effect

The earth’s surface radiates as all bodies do. But the earth’s surface is at an average temperature of 288 K and using Wien’s law, we find that the peak wavelength at which this energy is radiated is an infrared wavelength. Unlike visible light wavelengths, which pass through the atmosphere mainly unobstructed, infrared radiation is strongly absorbed by various gases in the atmosphere, the so-called greenhouse gases gases in the atmosphere that are capable of absorbing infrared radiation. This radiation is in turn re-radiated by these gases in all directions. This means that some of this radiation is received by the earth’s surface again, causing additional warming (Figure 8.5).

Figure 8.5: A simplified energy transfer diagram to illustrate the greenhouse effect.

This is radiation that would be lost in space were it not for the greenhouse gases. Without this greenhouse effect the phenomenon in which re-radiated energy from the greenhouse gases returns to earth, warming the earth, the earth’s temperature would be 32 K lower than what it is now.

The greenhouse effect may be described as the warming of the earth caused by infrared radiation, emitted by the earth’s surface, which is absorbed by various gases in the earth’s atmosphere and is then partly re-radiated towards the surface. The gases primarily responsible for this absorption (the greenhouse gases) are water vapour, carbon dioxide, methane and nitrous oxide.

The greenhouse effect is a natural consequence of the presence of the atmosphere. There is, however, also the enhanced greenhouse effect the augmentation of the greenhouse effect due to human activities. This refers to additional warming due to increased quantities of the greenhouse gases in the atmosphere. The increases in the gas concentrations are due to human activity and are closely related to the burning of fossil fuels.

183

Greenhouse gas | Natural sources | Anthropogenic sources
H2O | evaporation of water from oceans, rivers and lakes | irrigation
CO2 | forest fires, volcanic eruptions, evaporation of water from oceans | burning fossil fuels in power plants, cars, airplanes, burning forests
CH4 | wetlands, oceans, lakes and rivers, termites | flooded rice fields, farm animals, processing of coal, natural gas and oil, burning biomass
N2O | forests, oceans, soil and grasslands | burning fossil fuels, manufacture of cement, fertilisers, deforestation (reduction of nitrogen fixation in plants)

Table 8.2: Sources of greenhouse gases.

Greenhouse gases in the atmosphere have natural as well as human-made (anthropogenic) origins (Table 8.2). Along with these sources of the greenhouse gases, we have ‘sinks’ as well, that is to say, mechanisms that reduce these concentrations. For example, carbon dioxide is absorbed by plants during photosynthesis; it is also dissolved in oceans.

Mechanism of photon absorption

As with atoms, the energy of molecules is discrete; molecules have energy levels as we will see in Chapter 21. The difference in energy between molecular energy levels is the energy of an infrared photon. This means that infrared photons travelling through greenhouse gases will be absorbed. The gas molecules that have absorbed the photons will now be excited to higher energy levels. But the molecules prefer to be in low-energy states, and so they will immediately make a transition to a lower-energy state by emitting the photons they absorbed. But these photons are not all emitted outwards into space. Some are emitted back towards the earth, thereby warming the earth’s surface (Figure 8.6).

Figure 8.6: Greenhouse gases absorb infrared (IR) photons and re-radiate them in all directions.

There is an alternative way to explain photon absorption by the greenhouse gases. This makes use of the concept of resonance, which we will discuss further in Chapter 15. All systems have their own natural frequency of oscillation. For example, a pendulum oscillates back and forth with a frequency that depends on the length of the string to which it is attached. Suppose we exert a force on the pendulum by pushing it at regular intervals of time. How will the pendulum respond to this external force? It turns out that the amplitude of the oscillations (how far out the pendulum swings to the sides) will be large only when the frequency with which the external force is applied is equal to the natural frequency of the pendulum. When this happens we have resonance. In the case of the greenhouse gases the natural frequency is the frequency of the photons the greenhouse gases can absorb. Resonance is achieved when the photons emitted by the earth’s surface have the same frequency as the natural frequency. In that case, photons will be absorbed.

184

Worked example 8.3

One consequence of warming of the earth is that more water will evaporate from the oceans. Predict whether this fact alone will tend to increase the temperature of the earth further or whether it will tend to reduce it.

Answer

Evaporating means that energy must be transferred to water to turn it into vapour, and so this energy will have to come from the surroundings, reducing the temperature. Further, there will be more cloud cover, so more solar radiation will be reflected back into space, further reducing temperatures. This is an example of negative feedback: the temperature increases for some reason, but the effect of this increase is a tendency of the temperature to decrease and not increase further. (There is, however, another factor of positive feedback that will tend to increase temperatures: evaporating water means that the carbon dioxide that was dissolved in the water will now return to the atmosphere!) To decide the overall effect, detailed calculations are necessary. (Negative feedback wins in this case.)

THEORY OF KNOWLEDGE

To reach reliable predictions about climate change and its consequences, very complex and time-consuming modelling is required. Even before models are constructed we need accurate and reliable data to be the input to the models. Models for climate behaviour are complex because of the very large number of parameters involved, the interdependence of these parameters on various kinds of feedback effects and the sensitivity of the equations on the initial values of the parameters. This makes predictions somewhat less reliable than we would like. This gives ammunition to some politicians not to accept the reality of climate change. Advances in computing power, the availability of more data and further testing and debate on the various models will improve our ability to predict climate change more accurately in the future, and this will hopefully force governments into action so as to limit the damage caused by climate change.

Links

Climate change is an instance where many scientific disciplines come into play. To understand how the greenhouse gases absorb infrared radiation we need to understand atomic physics (Chapter 21), to understand how they radiate we need to understand the Stefan–Boltzmann law (Chapters 7 and 8) and to understand how energy is exchanged between the surface and the atmosphere we need the methods of heat transfer (Chapter 7). We also need atmospheric physics to tell us how clouds are formed and how they move, and finally we need sophisticated computer code to model all the complex interactions taking place.

TEST YOUR UNDERSTANDING

6A surface absorbs 120 W m−2 of radiation and reflects 30 W m−2. What is the albedo of the surface?

7The following data are available for a climate model of a planet in equilibrium (Figure 8.7):

Incident intensity on planet | 340 W m−2
Incident intensity on surface | 160 W m−2
Intensity reflected from cloud | 75 W m−2
Intensity reflected from surface | 30 W m−2

Figure 8.7: For question 7.

185

aWhat is the albedo of the cloud?

bWhat is the albedo of the surface?

cWhat is the albedo of the planet?

dWhat is the intensity absorbed by the cloud?

eWhat is the total radiated intensity into space?

8The following data are available for a climate model of the earth in equilibrium (Figure 8.8):

Solar intensity | 180 W m−2
Intensity of greenhouse gases radiation | 320 W m−2
Convection currents | 85 W m−2

Figure 8.8: For question 8.

What is the intensity radiated by the surface?

9Construct an energy flow diagram for an energy-balanced model of a planet with the following data:

The whole planet is in equilibrium and so is the surface (S) by itself and the cloud and atmosphere system (CAS) by itself.

Incoming solar intensity | 100 W m−2
Incoming solar intensity absorbed by CAS | 23 W m−2
Intensity reflected by CAS | 23 W m−2
Intensity reflected by S | 7.0 W m−2
Intensity radiated by CAS into space | 58 W m−2
Intensity radiated by S | 116 W m−2
Convection currents and latent heat into CAS from S | 29 W m−2

Use your diagram to find the following intensities:

aSolar intensity absorbed by surface

bThe radiation emitted by the surface that escapes into space

cThe radiation emitted by the surface that gets absorbed by CAS

dThe radiation emitted by CAS back down to the surface

and

eCalculate the albedo of the surface.

10A researcher uses the following data for a simple climatic model of a planet without an atmosphere: incident solar radiation = 350 W m–2, absorbed solar radiation = 250 W m–2.

aMake an energy transfer diagram for these data.

bDetermine the average albedo for the planet that is to be used in the modelling.

cDetermine the intensity of the outgoing long-wave radiation.

dEstimate the temperature of the planet according to this model, assuming a constant planet temperature. Take the emissivity of the surface to be 1 minus the albedo in b.

11The present equilibrium temperature of the earth is 288 K. An increase in greenhouse gas concentrations in the atmosphere will lead to a new and higher equilibrium temperature of the earth. Carefully explain why. (Use the Stefan–Boltzmann law.)

12The distance d between the sun and the earth is increasing very slowly. This means the earth’s average temperature T will go down. The fraction of the power radiated by the sun that is received on earth is proportional to 1 d 2; the power radiated by the earth is proportional to T 4.

186

aDeduce the dependence of the temperature T of the earth on the distance d.

bHence estimate the expected drop in temperature if the distance increases by 1.0%. Take the average temperature of the earth to be 288 K.

cCan you think of a reason why the sun–earth distance might be increasing?

dThe distance is actually increasing at a rate of 1.5 cm per year. This distance is now 1.5 × 1011 m. After how many years will the distance increase by 1%?

13Assume that the planet Venus has a circular orbit around the sun with orbit radius 1.08 × 1011 m. The albedo of the planet is 0.75.

aShow that the solar constant at the position of Venus is about 2.6 kW m−2.

bEstimate the equilibrium temperature of Venus.

cThe surface temperature of Venus varies from 710 K to 760 K. What can we conclude about the atmosphere of Venus?

dMercury is closer to the sun than Venus but has a lower average surface temperature and a huge variation (700 K to 100 K) between day and night temperature. What can we conclude about the atmosphere of Mercury?

14A body radiates energy at a rate (power) P.

aDeduce that the intensity of this radiation at distance d from the body is given by:

I=P 4 π d 2

bState one assumption made in deriving this result.

15 aDefine the term albedo.

bState three factors that the albedo of earth depends on.

cState what is meant by the greenhouse effect.

dState the main greenhouse gases in the earth’s atmosphere, and for each give one natural and one human-made source.

16Outline the main ways in which the surface of the earth transfers thermal energy to the atmosphere and to space.

17 aCompare the albedo of a subtropical, warm, dry land with that of a tropical ocean.

bSuggest mechanisms through which the subtropical land and the tropical ocean transfer thermal energy to the atmosphere.

cIf the sea level were to increase, sea water would cover dry land. Suggest one change in the regional climate that might come about as a result.

18Evaporation is a method of thermal energy loss. Explain whether you would expect this method to be more significant for a tropical ocean or an arctic ocean.

19It is estimated that a change of albedo by 0.01 will result in a 1 °C temperature change. A large area of the earth consists of 60% water and 40% land. Calculate the expected change in temperature if melting ice causes a change in the proportion of the area covered by water from 60% to 70%. Take the albedo of dry land to be 0.30 and that of water to be 0.10.