The bare rock model

The temperature of the surface of the Earth is controlled by the ways that energy comes in from the Sun and shines back out to space as IR. The Sun shines a lot of light because the temperature at the visible surface of the Sun is high and therefore the energy flux I = sa T4 is a large number. Sunlight strikes the Earth and deposits some of its energy into the form of vibrations and other bouncings around of the molecules of the Earth. Neither the Earth nor the Sun is a perfect blackbody, but they are both almost blackbodies, as are most solids and liquids. (Gases are terrible blackbodies as we will learn in Chapter 4). The Earth radiates heat to space in the form of IR light. Earth light is of much lower frequency and of lower energy than sunlight.

We are going to construct a simple model of the temperature of the Earth. The word model is used quite a bit in scientific discussions, to mean a fairly wide variety of things. Sometimes the word is synonymous with "theory" or "idea," such as the Standard Model of Particle Physics. For doctors, a "model system" might be a mouse that has some disease that resembles a disease that human patients get. They can experiment on the mouse rather than experiment on people. In climate science, models are used in two different ways. One way is to make forecasts. For this purpose, a model should be as realistic as possible and should capture or include all of the processes that might be relevant in Nature. This is typically a mathematical model implemented on a computer, although there's a nifty physical model of San Francisco Bay you should check out if you're ever in Sausalito. Once such a model has been constructed, a climate scientist can perform what-if experiments on it that could never be done on the real world, to determine how sensitive the climate would be to changes in the brightness of the Sun or properties of the atmosphere, for example.

The simple model that we are going to construct here is not intended for making predictions, but is rather intended to be a toy system that we can learn from. The model will demonstrate how the greenhouse effect works by stripping away lots of other aspects of the real world that would certainly be important for predicting climate change in the next century or the weather next week, but make the climate system more complicated and therefore more difficult to understand. The model we are going to explore is called the layer model. Understanding the layer model will not equip us to make detailed forecasts of future climate, but one cannot understand the workings of the real climate system without first understanding the layer model.

The layer model makes a few assumptions. One is that the amount of energy coming into the planet from sunlight is equal to the amount of energy leaving the Earth as IR. The real world may be out of energy balance for a little while or over some region, but the layer model is always exactly in balance. We want to balance the energy budget by equating the outgoing energy flux Fout to the incoming energy flux Fin,

Fin = Fout

Let's begin with incoming sunlight. The intensity of incoming sunlight Iin at the average distance from the Sun to the Earth is about 1350 W/m2. We'll consider the Watts part of this quantity first, followed by the square meter part. If you've ever seen Venus glowing in the twilight sky you know that some of the incoming visible light shines back out again as visible light. Venus's brightness is not blackbody radiation; it is hot on Venus but not hot enough to shine white hot. This is reflected light. When light is reflected, its energy is not converted to vibrational energy of molecules in the Earth and then re-radiated according to the blackbody emission spectrum of the planet. It just bounces back out to space. For the purposes of the layer model, it is as if the energy had never arrived on Earth at all. The fraction of a planet's incoming visible light that is reflected back to space is called the planet's albedo and is given the symbol a (Greek letter alpha). Snow, ice, and clouds are very reflective and tend to increase a planet's albedo. The albedo of the bright Venus is high, 0.71, because of a thick layer of sulfuric acid clouds in the Venusian atmosphere, and is low, 0.15, for Mars because of lack of clouds on that planet. Earth's albedo of about 0.33 depends on cloudiness and sea ice cover, which might change with changing climate.

Incoming solar energy that is not reflected is assumed to be absorbed into vibra-tional energy of the molecules of the Earth. Using a present-day earthly albedo of 0.3, we can calculate that the intensity of sunlight that is absorbed by the Earth is 1350 W/m2 (1 - a) = 1000 W/m2.

What about the area, the square meters on the denominator? If we want to get the total incoming flux for the whole planet, in units of Watts instead of Watts per square meter, we need to multiply by a factor of area,

What area shall we use? Sun shines on half of the surface of the Earth at any one time, but the light is weak and wan on some parts of the Earth, during dawn or dusk or in high latitudes, but is much more intense near the Equator at noon. The difference in intensity is caused by the angle of the incoming sunlight, not because the sunlight,

Sunlight

Fig. 3.1 When sunlight hits the Earth, it all comes from the same direction. The Earth makes a circular shadow. Therefore, the Earth receives an influx of energy equal to the intensity of sunlight, multiplied by the area of the circle, n • r 2arth.

Sunlight

Fig. 3.1 When sunlight hits the Earth, it all comes from the same direction. The Earth makes a circular shadow. Therefore, the Earth receives an influx of energy equal to the intensity of sunlight, multiplied by the area of the circle, n • r 2arth.

measured head-on at the top of the atmosphere, is much different between low and high latitudes (Fig. 3.1). How then do we add up all the weak fluxes and the strong fluxes on the Earth to find the total amount of energy that the Earth is absorbing?

There's a nifty trick. Measure the size of the shadow. The area we are looking for is that of a circle, not a sphere. The area is

Putting these together, the total incoming flux of energy to a planet by solar radiation is

Our first construction of the layer model will have no atmosphere, only a bare rock sphere in space. A real bare rock in space, such as the Moon or Mercury, is incredibly hot on the bright side and cold in the dark. The differences are much more extreme than they are on Earth or Venus where heat is carried by fluid atmospheres. Nevertheless, we are trying to find a single value for the temperature of the Earth, to go along with a single value for each of the heat fluxes Fin and Fout. The real world is not all at the same temperature, but we're going to ignore that in the layer model. The heat fluxes Fin and Fout may not balance each other in the real world, either, but they do in the layer model.

The rate at which the Earth radiates energy to space is given by the Stefan-Boltzmann equation:

As we did for solar energy, here we are converting intensity I to total energy flux F by multiplying by an area A. What area is appropriate this time? Incoming sunlight is different from outgoing earthlight in that the former travels in one direction whereas the latter leaves Earth in all directions (Fig. 3.2). Therefore, the area over which the Earth radiates energy to space is simply the area of the sphere, which is given by

Sunlight

Earthlight

Fig. 3.2 When IR light leaves the Earth, it does so in all directions. The total rate of heat loss equals the intensity of the earthlight multiplied by the area of the surface of the sphere

Earthlight

Fig. 3.2 When IR light leaves the Earth, it does so in all directions. The total rate of heat loss equals the intensity of the earthlight multiplied by the area of the surface of the sphere

Therefore, the total outgoing energy flux from a planet by blackbody radiation is

Fout = 4n rearth^arth

The layer model assumes that the energy fluxes in and out balance each other (Fig. 3.3)

Fout = Fin which means that we can construct an equation from the "pieces" of Fout and Fin which looks like this:

Factors of n and rearth appear in common on both sides of the equation, which means that we can cancel them by dividing both sides of the equation by those factors. Also dividing by 4 on both sides gives units of Watts per area of the Earth's surface. We get

We know everything here except the Tearth. If we rearrange the equation to put what we know on the right-hand side and what we don't on the left, we get

What we have constructed is a relationship between a number of crucial climate quantities. Changes in solar intensity such as the sunspot cycle or the Maunder Minimum (Chapter 10) may affect Iin. We shall see in Chapter 4 that greenhouse gases

Table 3.1 The temperatures and albedos of the terrestrial planets. The intensity of sunlight differs with distance from the Sun

^solar

a

Tbare

^observed

T1 layer

(W/m2)

(%)

(K)

(K)

(K)

Venus

2600

71

240

700

285

Earth

1350

33

251

295

303

Mars

600

17

216

240

259

Sunlight Earthlight

Sunlight Earthlight

Fig. 3.3 An energy diagram for Earth with no atmosphere, just a bare rock in space.

are extremely selective about the wavelengths of light that they absorb and emit; in other words, they have complexities relating to their emissivity s values. The albedo of the planet is very sensitive to ice and cloud cover, both of which might change with changing climate.

If we calculate the temperature of the Earth, we get a value of 255 K or about — 15°C. This is too cold; the temperature range of Earth's climate is —80°C to about +55°C, but the average temperature, what we're calculating using the layer model, is closer to + 15°C than —15°C. Table 3.1 gives the values we need to do the same calculation for Venus and Mars, along with the results of the calculation and the observed average temperatures. In all three cases, our calculation has erred on the side oftoo cold.

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Responses

  • Bell
    How to solve the bare rock model equation?
    3 years ago

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