Numerical Solution of Schwarzchild's Equation using the Modtran Infrared Light in the Atmosphere Website and a Spreadsheet

Atmospheric Physics textbooks emphasize the method of obtaining Outgoing Long Wavelength Radiation (OLR) by applying Schwarzchild’s Equation (S.E.), corrected using the diffusivity factor (D.F.) approximation. Integration over pi Sr for each of many successive small wavenumber increments is probably more complex to explain than the D.F. method. The usual value of the D.F. suggested in most textbooks is 5/3. One exception is contained in the text by Pierrehumbert, where it is pointed out that the optimum D.F. depends on the particular case studied. In Pierrehumbert's text, he uses a default value of two for the diffusivity factor. ¹

Recently, Zhao and Shi have reviewed this topic. They point out that there are two limits for the D.F. The smallest possible value is unity for paths with large optical depths. The largest possible value is 2 for paths with small optical depths. These authors obtain a function which yields the optimum value of the D.F. for any path as a function of the optical depth of that path. ² In this presentation I use various alternate formats by which I numerically obtain and check for accuracy the OLR obtained from the S.E. as corrected by the D.F. approximation. The paper by Zhao and Shi informs the choice of D.F. used in these computations. An U.S. Standard Atmosphere containing various single species of greenhouse gas is assumed. The only computational tool used is a spreadsheet.

1. Raymond T. Pierrehumbert, “Principles of Planetary Climate”, Cambridge University Press, 2010, p. 191.
2. Jian – Qi Zhao and Guang-Yu Shi, “An accurate approximation to the diffusivity factor”, Infrared Physics and Technology 56 (2013) 21-24