ENVST-UA0360 - MATH-UA0228
Earth Atmosphere and Ocean - Chapter 2 - Greenhouse model and Climate sensitivity
The homework focuses on energy balance and the greenhouse effect. Feel free to use some of the Jupyter Notebooks used in class, but make sure to explain how you got to your answers.
1. (10pt) The Voyager 1 probe was launched in 1977 and aims at exploring outer space. As it moves further and further away from the Sun, the amount of energy it receives from the Sun decreases as
where S0 = 1360W/m2
is the solar constant, rE(= 150e9m) is the distance between the Sun and the Earth, and r is the distance between the Sun and the probe. Assuming that the probe is a sphere of albedo 0.1, determine its temperature as a function of r.
2. (30pt) Consider that the Earth behaves as a leaky greenhouse as in section 2.3.2.
(a) Find its surface temperature assuming that the incoming solar radiation is S0 = 1368Wm−2, its albedo is α = 0.3, and the emissivity of the atmosphere is ϵ = 0.7.
(b) Consider that the increase in atmospheric carbon dioxide corre-sponds to an additional heating of 4Wm−2
. Find the change in surface temperature necessary to increase the infrared emission at the top of the atmosphere by 4Wm−2
if the emissivity is constant.
(c) Under the same condition as above, find the change in surface temperature if the emissivity increases by 0.01 per degree Kelvin.
3. (30pt) In the leaky greenhouse model discussed in class, emissivity is capped at 1. However, some planetary atmospheres, such as that of Venus, are extremely opaque to infrared radiation. To model this, we treat the atmosphere as multiple layers of absorbing greenhouse. We consider the greenhouse effect induced by a ’two-layer’ atmosphere like the one represented on Figure 2.9. of the textbook. The atmosphere is made of two fully absorbing layers, A and B - meaning that each layer absorbs all radiation coming from above and below it. (The lower layer (B) absorbs all the radiation coming from the top layer (A) and the surface. The upper layer (A) absorbs all the radiation from the lower layer.)
(a) Write the energy budget from both the atmospheric layer and the surface.
(b) Solve these equations to determine the temperature of each layer and for the surface, assuming a solar constant of 2600 Wm−2 and an albedo of 0.75.