Greenhouse Effect Simulation

About this experiment

Without its atmosphere the Earth would be a frozen ball at about -18 °C. Sunlight passes easily through the air and warms the ground, but the warm ground gives off infrared radiation, and gases such as water vapour and carbon dioxide absorb much of it and send part back down. This greenhouse effect keeps the average surface near 15 °C. Follow the energy flows, add carbon dioxide, change how much sunlight is reflected, and watch the surface warm towards a new balance.

Learning goals

Describe the flows of solar and infrared radiation between the Sun, the atmosphere and the surface
Explain how greenhouse gases warm the surface without creating energy
Explain radiative balance and why a planet warms until outgoing radiation matches absorbed sunlight
Explain the role of albedo and of the water vapour feedback
Interpret the warming expected for a doubling of carbon dioxide

How to use it

1Switch off the atmosphere. What is the surface temperature? Switch it back on. How much warmer is the Earth with an atmosphere?
2Look at the arrows at balance. Check that the energy arriving at the surface equals the energy leaving it.
3Choose Doubled (560 ppm). How much warmer does the surface become, and how long does it take?
4Switch off the water vapour feedback and repeat. How much of the warming is caused by the feedback?
5Raise the albedo to 0.4, as if more of the Earth were covered in ice or cloud. What happens?
6Set carbon dioxide to today's level. Compare the warming with the 1.2 °C observed since about 1850.

The model

\frac{( 1 - α ) S}{4} = ε σ T_{a}^{4} + (1 - ε) σ T_{s}^{4}

Balance at the top of the atmosphere: absorbed sunlight equals outgoing infrared

T_{s} = (\frac{2}{2 - ε})^{1/4} T_{e}, T_{e} = (\frac{( 1 - α ) S}{4 σ})^{1/4} \approx 255 K

Surface temperature of the one-layer model

Δ F = 5.35 ln \frac{C}{C _{0}} W/m^{2}

Radiative forcing of carbon dioxide

C \frac{d T _{s}}{d t} = absorbed - outgoing

The ocean stores heat, so warming takes time

Assumptions

A single atmospheric layer absorbs a share ε of the infrared from the surface and radiates the same share at its own temperature, up and down. It lets sunlight straight through.
The layer absorbs 77.9 % of the infrared with pre-industrial CO₂, which gives 15 °C. CO₂ changes this share so that it produces the standard forcing.
The water vapour feedback adds 1 % more infrared absorption for each kelvin of warming, giving about 3 °C for doubled CO₂, close to the central estimate of climate science. Clouds and ice are only represented by the albedo slider.
The heat capacity is that of about 100 m of ocean. The deep ocean, which delays warming for centuries, is left out.

Try this in class

Q1Use the Stefan-Boltzmann law to show that the Earth without an atmosphere would be about 255 K.
Q2Why does a greenhouse gas warm the surface but cool the upper atmosphere?
Q3Explain why the water vapour feedback amplifies warming caused by carbon dioxide.
Q4Why does the temperature keep rising for several years after the carbon dioxide stops increasing?
Q5Melting sea ice lowers the albedo. Is this a positive or a negative feedback? Explain.

Frequently asked questions

Does the greenhouse effect break the conservation of energy?

No. The surface receives sunlight plus the back radiation from the atmosphere, so it must be warmer to send all of that away again. At the top of the atmosphere the Earth still sends out exactly as much energy as it absorbs.

Why is the effect of carbon dioxide logarithmic?

The centre of the main CO₂ absorption band is already almost opaque, so adding more gas mainly widens the band at its edges. Each doubling adds roughly the same forcing, about 3.7 W/m².

How realistic is a one-layer model?

It gets the size of the natural greenhouse effect and the warming for doubled CO₂ about right, which makes it a good teaching model. Real climate models resolve many layers, clouds, oceans and ice, and give a likely range of 2.5 to 4 °C for doubled CO₂.