08.2c - Earth's Energy Balance

The fraction of radiation reflected (or scattered) depends on the albedo of the surface (this is related to the colour).

The Albedo (α) of a planet is defined as the ratio between the total scattered (reflected) radiation and the total incident radiation of that planet.

The albedo of a planet is affected by the following:

1. Season (cloud formations)
2. Latitude
3. Terrain (ocean has low albedo because it mainly absorbs and snow has high albedo because it mainly reflects)

The average albedo of the earth is assumed to be 0.3. Show that the average power reflected per m² by the earth's surface is 105 W m^-2.

General Resources:

Kognity Textbook: Chapter 8 (Use your ACS Login)
IB Physics Site: Topic 8 - Comprehensive notes
IB Physics Site: Topic 8 - More notes
Topic 8 Flashcards - Vocab Devo.

The Earth absorbs like a DISC (πr²) and reflects/radiates like a SPHERE (4πr²)

Is the Earth Warming?

Simple model of the Earth's energy balance

The temperature of the Earth can be calculated by applying the conservation of energy at the earth's surface. The diagram to the right shows this simple model.

At the Earth's surface some radiation from the Sun is reflected and the remainder is absorbed. The Earth heats up due to this absorbed radiation. The warm Earth then emits radiation back into space. If the energy is in balance then:

Power radiated by Earth = Power absorded by Earth

Power absorbed:

Power = IA (eq.1)

Some of the incoming radiation is reflected by the Earth's surface. The fraction reflected is equal to the albedo α, so the fraction absorbed is equal to (1 - α). So,

so: Power = (1-α)IA (Eq. 2)

The power absorbed is the same as the same that would be absorbed by a flat disk with the same radius as the Earth and the intensity I is the solar constant S (S =1360 W m^-2).

Area = πr^2 (area of circle or disc)

Power Absorbed is as shown to the right.

Power Radiated:

From the Boltzmann equation, the warm earth emits radiation from all of its surface i.e the surface area of a sphere, and the emissivity of Earth is assumed to be 1.

Surface Area of a SPHERE: A= 4πr²

Therefore, the power radiated is as shown (e = 1.0, therefore is not shown)

Setting the power radiated equal to the power absorbed (Eq. 8 / 8a)

Therefore, solving for T:

S =1360 W m^-2
α = 0.30
σ = 5.67×10⁻⁸ W⋅m⁻²⋅K⁻⁴

If the solar constant S is 1360 W m^-2 and the average albedo of the Earth is 0.30 then show that the Earth's temperature is -18.5 ºC.

This temperature is very cold and unrealistic because this simple model does not include the earth's atmosphere.

Eq 1

Eq. 2

Eq. 3 - S substituted for I and πr² for A

Eq 4 - Boltzmann's Equation

Eq. 6 - Surface area of a sphere

Eq. 7 Substituted Values

Eq. 8

Eq. 8a

Eq 9 - Rearrange Terms

Eq. 10 - Solve for T

A Better Model:

The model shown in the diagram includes the earth's atmosphere, which will also absorb some radiation and emit it towards earth or back into space. The conservation of energy can now be applied at two locations: the top of the atmosphere and at the earth's surface.

e_sun - emissivity of the atmosphere to radiation from the Sun

e_earth - emissivity of the atmosphere to radiation from the Earth

The atmosphere absorbs very little of the radiation from the Sun so the emissivity of the atmosphere to radiation from the Sun is very high and can be assumed to be approximately 0.80 (e_sun = 0.80).

The atmosphere absorbs a lot of the radiation from the Earth so the emissivity of the atmosphere to radiation from the Earth is very low and can be assumed to be approximately 0.10 (e_earth = 0.10).

Use the equation above to show that the temperature of the Earth will be 15 ºC when the atmosphere of the Earth is taken into account.

This model including the atmosphere, although better than the first model, is still very simple. The model can be made more complex by including for example the effects of convection currents and the effect of different layers within the ocean.

Sage Modeller Albedo Effect

Please:

Follow the link to this FILE.
DOWNLOAD the file.
From within the site: SAGE Modeller, open the file from your computer.
You can then save the file to your Google Drive.

All models are wrong, but some are useful.

- G. Box

Using the Model:

There are several connections that are not defined (dashed grey lines). Based on your predictions, add the relationships between the variables that are not defined.
Run the simulation with various initial settings to discern any trends in the model.

The atmosphere

The above model did not include the earth's atmosphere. The atmosphere contains mostly molecular gases; mainly nitrogen (N₂) and oxygen (O₂) along with noble gases and ozone (O₃). There are also a number of molecular gases with variable composition known as the greenhouse gases. The greenhouse gases to be considered are carbon dioxide (CO₂), methane (CH₄), nitrous oxide (N₂O) and water vapour (H₂O).

In molecules there are multiple ways in which energy can be stored. Energy can be absorbed by the transfer of energy when an electron moves to a higher energy level (as we studied in Topic 7). This is known as electronic energy. Energy can also be stored as rotational energy or vibrational energy.

1. Electronic energy - energy stored as potential energy when electrons are excited to higher energy levels.

2. Rotational energy - energy transferred to kinetic energy as a molecule rotates.

3. Vibrational energy - energy transferred to kinetic and potential energy as a molecule oscillates due to its bonding.

The way that electromagnetic radiation interacts with the gas molecules found in the earth's atmosphere can be investigated using the PhET simulation below.

Start with carbon monoxide gas and radiation from the microwave part of the electromagnetic spectrum.
Move the slider on the microwave emitter to the right so that microwave radiation passes through the carbon monoxide molecule.
Note if the carbon monoxide molecule absorbs any energy and try to identify whether the energy is stored as rotational or vibrational energy.
Then investigate if the molecule absorbs Infrared, Visible or Ultraviolet radiation in turn.
Once again note whether the energy is stored as rotational or vibrational energy.
Now replace the carbon monoxide molecule with nitrogen and expose it to microwave, infrared, visible and ultraviolet radiation as before.

Absorption graphs

The graphs below show the % absorption of radiation by three of the atmospheric gases: oxygen/ozone, carbon dioxide and water vapour. Use the results from the simulation to explain them.

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