Question

Let’s stop ignoring the greenhouse effect and incorporate it into the previous problem in a very rough way. Assume the atmosphere is a single layer, a spherical shell around Earth, with an emissivity e=0.77 (chosen simply to give the right answer) at infrared wavelengths emitted by Earth and by the atmosphere. However, the atmosphere is transparent to the Sun’s radiation (that is, assume the radiation is at visible wavelengths with no infrared), so the Sun’s radiation reaches the surface. The greenhouse effect comes from the difference between the atmosphere’s transmission of visible light and its rather strong absorption of infrared. Note that the atmosphere’s radius is not significantly different from Earth’s, but since the atmosphere is a layer above Earth, it emits radiation both upward and downward, so it has twice Earth’s area. There are three radiative energy transfers in this problem: solar radiation absorbed by Earth’s surface; infrared radiation from the surface, which is absorbed by the atmosphere according to its emissivity; and infrared radiation from the atmosphere, half of which is absorbed by Earth and half of which goes out into space. Apply the method of the previous problem to get an equation for Earth’s surface and one for the atmosphere, and solve them for the two unknown temperatures, surface and atmosphere.

a) Derive equations for Earth's surface and atmosphere temperatures.
b) The problem is unsolvable without knowing the atmospheric pressure.
c) Solve for temperatures considering the Earth as a perfect blackbody.
d) The atmosphere's emissivity has no effect on the solution.

Answer 1

The problem involves incorporating the greenhouse effect into the previous problem. Equations can be derived and solved for Earth's surface and the atmosphere's temperatures. The atmospheric pressure is needed to solve the problem. The atmosphere's emissivity does have an effect on the solution.

Step-by-step explanation:

The problem at hand involves incorporating the greenhouse effect into the previous problem. In this scenario, the atmosphere is assumed to be a single layer spherical shell around Earth with an emissivity of 0.77 at infrared wavelengths. The atmosphere is transparent to the Sun's radiation, allowing it to reach the surface. The greenhouse effect is caused by the difference between the atmosphere's transmission of visible light and its absorption of infrared. To solve the problem, equations can be derived for Earth's surface and the atmosphere's temperatures, and then these equations can be solved for the two unknown temperatures.

a) To derive the equations, the energy transfers from solar radiation absorbed by the Earth's surface, infrared radiation from the surface absorbed by the atmosphere, and infrared radiation from the atmosphere need to be considered. Using the method from the previous problem, an equation can be obtained for both Earth's surface and the atmosphere's temperatures.

b) The problem cannot be solved without knowing the atmospheric pressure, as it affects the radiative energy transfers and thus the temperatures.

c) When considering the Earth as a perfect blackbody, the equations for Earth's surface and the atmosphere's temperatures can be solved for their respective values.

d) The atmosphere's emissivity does have an effect on the solution. It influences the absorption of infrared radiation from Earth's surface by the atmosphere and affects the temperatures of both Earth's surface and the atmosphere.