Assignment: Clouds in the Leaky Greenhouse Model#
Students completing this assignment will gain the following skills and concepts:
Continued practice working with the Jupyter notebook
Familiarity with the toy “leaky greenhouse” model
Conceptual understanding of the role of clouds in the planetary energy budget
This assignment requires some mathematics. You can present your work in this notebook, on hand-written paper, or a combination. Just make sure you communicate clearly which answers belong to which question.
For answers presented in the notebook, follow the usual procedures to ensure that your code is well commented and runs clearly without errors (see previous assignment instructions).
Consider the two-layer “leaky greenhouse” (or grey radiation) model from these lecture notes.
Here you will use this model to investigate the radiative effects of clouds.
Clouds simultaneously reflect shortwave radiation and absorb longwave radiation. These two effects often oppose each other in nature, and which one is stronger depends (among other things) on whether the clouds are low or high (i.e. in layer 0 or layer 1).
For this question we will suppose (as we did in the lecture notes) that there is no absorption of shortwave radiation in the atmosphere.
Suppose a cloud reflects a fraction \(\alpha_c\) of the shortwave beam incoming from above. \(\alpha_c\) is a number between 0 and 1. Provide a coherent argument (in words, sketches, and/or equations) for why the shortwave effects cloud should alway be a cooling on the surface. Is this cooling effect different if the cloud is low or high? Explain.
Because the liquid water droplets in a cloud are effective absorbers of longwave radiation, a cloud will increase the longwave absorptivity / emissivity of the layer in which it resides.
We can represent this in the two-layer atmosphere by letting the absorptivity of a cloudy layer be \(\epsilon + \epsilon_c\), where \(\epsilon_c\) is an additional absorptivity due to the cloud. Derive a formula (i.e. an algebraic expression) for the OLR in terms of the temperatures \(T_s, T_0, T_1\) and the emissivities \(\epsilon, \epsilon_c\) for two different cases:
a low cloud (the additional \(\epsilon_c\) is in layer 0)
a high cloud (the additional \(\epsilon_c\) is in layer 1)
Now use the tuned numerical values we used in class:
\(T_s = 288\) K
\(T_0 = 275\) K
\(T_1 = 230\) K
\(\epsilon = 0.586\)
and take \(\epsilon_c = 0.2\)
(a) Repeat the following for both a high cloud and a low cloud:
Calculate the difference in OLR due to the presence of the cloud, compared to the case with no cloud.
Does this represent a warming or cooling effect?
(b) Which one has a larger effect, the low cloud or the high cloud?
Based on your results in questions 1-3, which do you think is more likely to produce a net warming effect on the climate: a low cloud or a high cloud? Give an explanation in words.
How would your answer change if the atmosphere were isothermal, i.e. \(T_s = T_0 = T_1\)?
It is licensed for free and open consumption under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.
Development of these notes and the climlab software is partially supported by the National Science Foundation under award AGS-1455071 to Brian Rose. Any opinions, findings, conclusions or recommendations expressed here are mine and do not necessarily reflect the views of the National Science Foundation.