2022 September 05

- Part 1: What is the greenhouse effect? An accessible, scientific introduction
- Appendix A: What is the atmosphere?
- Appendix B: Ozone
- Part 2: Physics of light and temperature
- Part 3: Temperature of the Earth without an atmosphere
- Part 4: A model of the greenhouse effect
- Part 5: Differences between model and reality

In this part we will use what we have learned about light and temperature to create a simple model of the Earth with its atmosphere, which we can solve to see what effect changing the atmosphere has on the temperature of the modeled Earth.

The model is not *quantitative*, meaning that it is not
accurate enough for the numerical results from the model to agree with
the true numbers, but it is *qualitative*, meaning that the
overall behavior of the model agrees with the overall behavior of the
Earth. Therefore by understanding how the model works we can improve our
understanding of the Earth system; in particular, the way the greenhouse
effect works in the model is the same way that the greenhouse effect
works on the Earth.

Consider the following figure of the Earth^{1}We
mean “the surface of the Earth” when we say “the Earth”, as the interior
of the Earth only very slowly exchanges heat with the surface, so it can
be ignored. and its atmosphere, with energy flowing
between them.

The arrows in the diagram represent energy flowing between the
different objects in the form of light; we ignore other forms of energy
transfer^{2}All energy exchanged with the Sun or with space
is in the form of light, but some of the energy exchanged between the
Earth and the atmosphere is in other forms. In particular, hot water
molecules that physically move from the surface into the air bring a
large amount of energy with them, called *latent heat*. Heat
conduction plays a lesser role. We also omit geothermal heating, which
is energy flowing from the interior of the Earth to the surface. This is
estimated to be 47 TW, or 0.092 watts per square meter..
On the left of the diagram is shortwave radiation, that is, visible
light. The variable is the rate at which energy from the Sun is being
absorbed by the surface of the Earth; we assume that none of this
shortwave radiation is absorbed by the atmosphere. We omit from the
diagram light from the Sun that strikes the Earth (or clouds in the
atmosphere) and is reflected to space; is only the portion that is absorbed. Recall from
part 3 that

On the right of the diagram is longwave radiation, that is, infrared
light. The variable is the rate at which radiation leaves
the top of the atmosphere to space; some of this is emitted by the
atmosphere, and the rest is emitted by the Earth and passed through the
atmosphere. The variable is the rate at which radiation
emitted by the atmosphere strikes the Earth. Finally, the variable is the rate at which radiation is emitted by the
Earth, which will either be absorbed by the atmosphere or transmitted to
space.^{3}We briefly remark on the arrows that are absent
from the diagram. The most interesting omission is the arrow from the
Sun to the atmosphere; we have already commented on that. A tiny
fraction of the light emitted to space goes on to strike the Sun or
other bodies, but we are uninterested in where exactly it goes once it
leaves the Earth. Space is filled with *cosmic microwave background
radiation*, so there should be arrows representing microwave light
from space to each of the other objects, but the amount is so tiny as to
be totally insignificant – only 1.6 GW of it reaches the Earth, or 3
microwatts per square meter. Finally, the Sun emits a tremendous amount
of light into space that does not strike the Earth, but we are not
interested in that.

As discussed in part 3 we are interested in the equilibrium of the system, that is, when the total amount of energy entering each object equals the energy leaving that object.

So far we have not made any assumptions about the physics of the atmosphere, so there is insufficient information to solve for the model’s equilibria. However before we go further let us look at what we can conclude so far.

Let be the average^{4}Whenever we say the “average” temperature of a
(spherical) object in the context of blackbody radiation, we mean the
fourth root of the arithmetic mean of the fourth power of the surface
temperature, weighted by surface area and emissivity. That is, we use
exactly the average that makes the Stefan-Boltzmann law work with the
result. For objects like the Earth, where the temperature does not vary
tremendously from one location to another, this average is close to the
ordinary arithmetic mean. For tidally-locked or slowly rotating objects
like Mercury or the Moon, the distinction can be very
important. temperature of the Earth. Then as before, we
know that

where is the radius of the Earth.

At this point in the calculations in part 3 we solved for the effective temperature that satisfied the equality . However, the introduction of the atmosphere changes this equation. Since the Earth is at equilibrium, the total flow of energy in and out of the surface is equal, giving the equality

where the additional term is downward heating from the atmosphere, which has the effect of increasing the temperature of the surface of the Earth, demonstrating the greenhouse effect. Our goal is to find the temperature of the Earth, which will require calculating and .

Since the atmosphere is at equilibrium, we also have

Combining this equation with gives us , which is to say that the total
amount of sunlight the Earth absorbs equals the total amount of light it
emits, much as we would expect^{5}In
fact, direct measurements of light going in and out of the Earth agree
with each other up to the accuracy with which they can be
measured..

To solve for the temperature of the Earth, we need the values of and , or equivalently we need to know what fraction of the energy emitted by the Earth is absorbed by the atmosphere or transmitted through it to space. We know that but that is not sufficient information by itself. To find these values we need some kind of additional assumption about the physics of the atmosphere.

The simplest assumption is the total absence of gases that interact with infrared light, so that the atmosphere has no greenhouse gas. In this case, all light from the Earth is transmitted directly to space, and the atmosphere neither absorbs nor emits any infrared light.

In this case, we have and . Then and we get exactly the situation of part 3, so that the temperature of the surface of the Earth equals the effective temperature: .

The next simplest assumption that can be made about the Earth’s atmosphere is to assume that it fully absorbs all upwards emissions from the Earth while maintaining a single uniform temperature, which is usually described by saying it has a “single layer”. Thus consists only of emissions from the atmosphere. As the atmosphere has a uniform temperature, it emits the same amount of energy upwards and downwards, so we get and

Combining with we get

so

This is 30 C or 86 F, somewhat above the true average temperature of the Earth.

Let us assume again the the atmosphere is a single layer of uniform temperature, but now suppose it does not have sufficient greenhouse gas to fully absorb all infrared emissions from the Earth. Let be the fraction of such emissions that are absorbed, so that the atmosphere absorbs and transmits . Since half of the atmosphere’s emissions are downwards (by the uniform temperature assumption again), we get , so that

and therefore .

When , we recover the situation of the model with no greenhouse gas, and get . When , we find again the result of the model with ample greenhouse gas, and get . As we vary the amount of greenhouse gas in the atmosphere from none to ample, varies between 0 and 1, and the temperature calculated from the model varies from to , increasing as greenhouse gas is added. This illustrates how adding greenhouse gases to the atmosphere causes the temperature to rise.

With the single-layer assumption we saw that the Earth’s temperature increased with additional greenhouse gases up to a fixed maximum. However, as more and more greenhouse gas is added to the atmosphere, the assumption that the atmosphere can maintain a uniform temperature would break down as the emissions from the Earth can’t penetrate past the very bottom of the atmosphere.

If we instead approximate the atmosphere as being composed of multiple layers, each fully absorbing all incident infrared radiation and individually have uniform temperatures, then each such successive layer of greenhouse gases causes the temperature to increase by a factor of . Equivalently, every fourth layer doubles the temperature.

Similarly, we could modify the model to accommodate anti-greenhouse
gases^{6}Hazy conditions can have an
anti-greenhouse-like effect, although the mechanism is not exactly the
same; for example, major volcanic eruptions cool the Earth for a few
years by putting sulfur aerosols in the stratosphere.
which absorb shortwave radiation and transmit longwave radiation. Each
such layer of anti-greenhouse gases decreases the temperature by a
factor of , countering the effects of one layer of
greenhouse gases.

We mean “the surface of the Earth” when we say “the Earth”, as the interior of the Earth only very slowly exchanges heat with the surface, so it can be ignored.↩︎

All energy exchanged with the Sun or with space is in the form of light, but some of the energy exchanged between the Earth and the atmosphere is in other forms. In particular, hot water molecules that physically move from the surface into the air bring a large amount of energy with them, called

*latent heat*. Heat conduction plays a lesser role. We also omit geothermal heating, which is energy flowing from the interior of the Earth to the surface. This is estimated to be 47 TW, or 0.092 watts per square meter.↩︎We briefly remark on the arrows that are absent from the diagram. The most interesting omission is the arrow from the Sun to the atmosphere; we have already commented on that. A tiny fraction of the light emitted to space goes on to strike the Sun or other bodies, but we are uninterested in where exactly it goes once it leaves the Earth. Space is filled with

*cosmic microwave background radiation*, so there should be arrows representing microwave light from space to each of the other objects, but the amount is so tiny as to be totally insignificant – only 1.6 GW of it reaches the Earth, or 3 microwatts per square meter. Finally, the Sun emits a tremendous amount of light into space that does not strike the Earth, but we are not interested in that.↩︎Whenever we say the “average” temperature of a (spherical) object in the context of blackbody radiation, we mean the fourth root of the arithmetic mean of the fourth power of the surface temperature, weighted by surface area and emissivity. That is, we use exactly the average that makes the Stefan-Boltzmann law work with the result. For objects like the Earth, where the temperature does not vary tremendously from one location to another, this average is close to the ordinary arithmetic mean. For tidally-locked or slowly rotating objects like Mercury or the Moon, the distinction can be very important.↩︎

In fact, direct measurements of light going in and out of the Earth agree with each other up to the accuracy with which they can be measured.↩︎

Hazy conditions can have an anti-greenhouse-like effect, although the mechanism is not exactly the same; for example, major volcanic eruptions cool the Earth for a few years by putting sulfur aerosols in the stratosphere.↩︎

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