Question

1) Given a planetary albedo of 30% and using a 1 atmospheric layer energy balance model for the earth, calculate what fraction of out-going long-wave radiation emitted by the earth is absorbed by the atmospheric layer in order to have the observed earth temperature of 288K. (Ignore convective and latent heat fluxes and assume that all non-reflected solar radiation is absorbed by the earth).

1a) Calculate the total rate of longwave radiation emitted by the atmospheric layer of the earth (using the above model and assume it's 5 km above the earth's surface) and compare that to the total global production of electricity.

1b) Greenhouse effect: assume that the absorption of long-wave radiation
by the atmosphere is increased by 10%. What is the temperature of
the earth now? What other key processes would modify this result?

1c) A long time ago, the earth's sun was 85% as strong as present, and
yet the earth was no colder than today. What long-wave absorption rate
would have been required by a 1 layer atmosphere to maintain present-day
temperature (288K)? Interpret your result (What could have changed the
absorption rate?,...).

Answer 1

The rocky, inner planets of our solar system vary in sizes, atmospheres, and temperatures. Mercury, the smallest and closest to the sun has no atmosphere and extremes of temperature that average to about that predicted by our simple black body model. Mars, the next largest and farthest from the sun, has a very tenuous atmosphere that is mostly CO₂ and an average temperature close to or just a bit above that predicted by the simple black body model. Venus is closest in size to Earth, but has an atmosphere that is much denser than Earth’s. Venus is continuously shrouded in clouds that make it impossible to observe its surface at visible wavelengths from outside the atmosphere and are responsible for the planet’s very high albedo. This table provides a comparison of the observed and predicted surface temperatures of the planets and the compositions of their atmospheres.

Credit: American Chemical Society

The table provides evidence that an atmosphere has a pronounced effect on the temperature at the planetary surface, causing it to be warmer than predicted by the simple black body model. Venus, with a thick atmosphere, has a surface temperature about 500 K above the prediction. The Earth, with a thinner atmosphere, has a mild 33 K warming. However, this warming of the Earth above the freezing point of water (273 K) has profound consequences, because life, as we know it, would not be possible on a planet where the water is permanently frozen—the snowball Earth instead of the “blue marble” shown in the figure.

The N₂/O₂/Ar composition of the Earth’s atmosphere is for dry air. There can be a significant percentage of water, ~4%, over warm tropical waters while the percentage is very low over frigid Arctic ice pack.

A common but misleading analogy to the atmospheric warming effect is a greenhouse. The clear glass or plastic walls and roof of a greenhouse allow the Sun’s radiant energy in the visible part of the spectrum to enter and warm the surfaces inside the enclosure. The warmed surfaces heat the air inside the greenhouse by conduction and convection. The warm air reaches the cool wall and roof, loses its extra energy, and recirculates to the warm surfaces, thus setting up a steady state of trapped warm air in which the surface of the soil and plants and the average air temperature are higher than the temperature outside.

The analogy of the atmosphere to a greenhouse is misleading. Planetary atmospheric warming depends mainly on how infrared radiation interacts with molecules in the atmosphere called greenhouse gases, not on trapped warm air. Although the simple black body model is not entirely adequate to explain the temperature of planets with an atmosphere, the basic idea that the energy absorbed by a planet has to equal the energy it emits remains valid even with the atmospheric warming effect. This figure shows how the energy budget is balanced for the Earth when the atmospheric warming effect due to greenhouse gases is included.

The greenhouse gets its name because plants (greenery) can grow inside it when the conditions outside are too cold for them.

The incoming solar energy, 341 W·m^-2, is what we calculated based on the Sun’s black body emission. The reflected radiation, 102 W·m^-2, shown at the far left accounts for the Earth’s albedo, 0.30. Of the remaining unreflected radiant energy in the near ultraviolet, visible, and short wavelength infrared, about 30% (78 W·m^-2 out of 239 W·m^-2) is absorbed by gases in the atmosphere (mainly O₃, O₂, H₂O, and CO₂) and warms the atmosphere. The remaining radiant energy, 161 W·m^-2, reaches the surface and warms it.

In the middle of the diagram, “thermals” represent air warmed by contact with the warm surface. The air expands and is buoyed up into the atmosphere where it delivers energy to the cooler surroundings at higher altitudes. Likewise, energy absorbed by water as it evaporates or is emitted as gas by plant transpiration is carried into the atmosphere as a gas, where it releases energy to the surroundings when it condenses to form clouds.

The major players in the energy balance are shown at the right of the diagram where the emission and absorption of energy as infrared radiation are represented. The surface radiation emission, 396 W·m^–2, is the energy flux calculated from the Stefan-Boltzmann equation for a black body at a temperature of 288 K, the observed average surface temperature of the Earth. A small fraction of this energy is lost directly to space, but the great majority is absorbed by gases and clouds in the atmosphere and re-emitted in all directions, including downward toward the surface, 333 W·m^–2 warming the surface at the far right of the diagram. The result is a greater warming of the surface than by the incoming solar radiation alone.

The rounded values at the top of the figure show total incoming solar radiation flux, 341 W·m^–2, equal to the total outgoing flux, the combination of the reflected short wavelength radiation, 102 W·m^–2. and the long wavelength radiation, 239 W·m^–2,

(incoming) 341 W·m^–2 = 102 W·m^–2 + 239 W·m^–2 (outgoing)

The outgoing infrared radiation flux, 239 W·m^–2, from the atmosphere, clouds, and a small amount from the surface, is essentially that calculated from the Stefan-Boltzmann equation for a black body at a temperature of 255 K, the predicted temperature of the Earth in the absence of an atmospheric warming effect.

Note, however, that the more precise values for the incoming and outgoing energy fluxes do not exactly balance.

(incoming) 341.3 W·m^–2 > 101.9 W·m^–2 + 238.5 W·m^–2 = 340.4 W·m^–2 (outgoing)

The Earth is warming and the only way this can happen is for the incoming energy to exceed the outgoing energy. This analysis indicates that the warming planet is retaining the equivalent of 0.9 W·m^–2 (very bottom of the figure). To find out more about the atmospheric warming effect at the molecular level, see How Atmospheric Warming Works, and to find out about the effects of adding more greenhouse gases to the atmosphere, see Greenhouse Gases.