lördag 2 april 2011

A Simple Radiation Model

Inspired by Claes Johnsson I will here make an attempt to formulate a simple radiation model aimed at illustrating how the temperature gradient changes with an increase in absorption/emission parameter A. The equation reads

(A - A^2)U(x) - 1/2*A^2U''(x) = 0.

U(x) stands for the temperature. In this model we assume that the radiation is proportional to T and not T^4. The amount of radiation going directly into space is propotional to A - A^2, since the amount A^2 is absorbed by the atmosphere. The remaining amount of radiation is instead transported as diffusion. The physical solution to this equation is

U(x) = exp(-ax)

where a^2 = 2(1 - A)/A, which is a strictly decreasing function of A in the interval (0,1). Hence, the gradient flattens as the optical activity A increases, contrary to the greenhouse hypothesis. The situation with A = 1 corresponds to a fully opaque atmosphere. The strength of this model is that the A^2 term can be replaced by any term less than A and the main conclusion stands still. All other forms of heat transfer are neglected here which is the cause of the singularity at A = 0. Comments are welcome.

PS

The formula follows from some very simple assumptions which is most clearly seen in a discrete form, create a discrete mesh with index n, U(n) is the temperature at position n. The equation is now

-(A - A^2)U(n) + 1/2*A^2(U(n-1) - U(n)) - 1/2*A^2(U(n) - U(n+1)) = 0

The first term is the heat lost by direct radiation to space. The second term is the heat gained from the warmer lower position. The third term is the heat lost to the cooler upper position. The equation can be rewritten

(A - A^2)U(n) - 1/2*A^2(U(n+1) - 2U(n) + U(n-1)) = 0

And we can now identify the discrete second derivative. Excersize: Spot the "back-radiation" terms ;)