Climate Physics and Chemistry - Problem Set 1 - Solutions - Edubirdie

University:
Massachusetts Institute of Technology
Course:
12.842 | Climate Physics And Chemistry
Academic year:

2023
Views:

196

Pages:

2
Author:

Nathalie Berg

Climate Physics Problem Set 1 Solutions Generalize the simple radiative equilibrium problem described in class to an arbitrary number of layers (n), as shown below. Solve this system to find the temperature of each layer as a function of Te. Solution: First define and index i corresponding to the layer number, counting downward. (So i=1 is the top layer, i=2 is the next layer down, etc.) Then the radiative balance of each interior layer i is 2σ Ti 4 = σ Ti+14 + σ Ti−14 . This can be written instead σ Ti+14 − σ Ti 4 = σ Ti 4 − σ Ti−14 . This implies that ∆T 4 = a, where a is a constant. Integrating this difference equation gives Ti 4 = ai + b, (1) where b is another constant. Now energy balance at the top of the atmosphere gives T1 = Te , while solution of the problem with just two layers gives 1 T2 = 2 4 Te This implies that in equation (1), a = Te 4 and b = 0 . So the solution (1) may be written 1 Ti = i 4Te (2) Now the surface has to be treated as a special case, since all solar radiation is absorbed there. The surface energy balance is σ Ts 4 = σ Te 4 + σ Tn 4 . Using the solution (2) for Tn in the above gives 1 Ts = ( n +1) 4 Te . This together with (2) constitutes the solution to the problem.

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