They point out two limiting cases: 1. If the optical depth is quite large the limiting D.F.is one. If the optical depth is small the limiting D.F. corresponds to 60 degrees relative to vertical. So that 1/cos theta is 2. Consider 400 ppm CO2. If the wn band chosen is 2 wn through 2200 wn, and since the CO2 absorption is pretty much limited to the range 500 wn to 850 wn, even for a vertical path from ground to 70 km by Modtran6 the transmittance is over 0.9. So using 60 degrees is not bad. For a range near CO2 bending mode region – around 670 wn you can get transmittance of less than 0.1 and d.f is ~ 1. Zhao and Shi derive a complicated formula which gives the D.F. for a general case.

BTW Pierrehumbert uses 60 degrees for his default theta,with a discussion of merits of other angles. See p.821. Petty/Houghton use 54 degrees.

With Zhao and Shi one can taylor the d.f. to a particular case.

]]>Nevertheless, Planck’s Law is wrong: Emission at a single wavelength or across all wavelengths is usually less than predicted. Sometimes this happens for trivial reasons: 1) Absorption and emission don’t reach equilibrium before radiation leaves an object such as an optically thin layer of atmosphere (too few GHGs, some wavelengths absorb weakly if at all). 2) We have created a variety of devices (lasers, LED and fluorescent light) where a Boltzmann distribution, LTE, and even thermodynamically-defined temperature don’t exist. But this doesn’t explain why most dense materials, take water for example, don’t have unit emissivity.

I believe I first heard the suggestion from DeWitt that emissivity less than unity is caused by internal radiation being internally reflected or scattered at the surface. This attractive hypothesis provides an explanation for why emissivity equals absorptivity, but I haven’t located any references. SOD has an article on the emissivity of the ocean that says emissivity has an angular dependence, exactly what one might expect for a surface phenomena. However, many metals have thermal emissivities less than 0.1. Could that much reduction be due to internal reflection or scattering?

Looking for references, I finally did a search for “Rational for Kirchhoff’s Law absorptivity and emissivity” (rather than emissivity alone) and found that some solid state physicists and other materials scientists are actively pursuing these questions even today. And it suddenly dawned on me that metals, crystalline materials and other solids may not behave like quantized oscillators. That description may be reasonable for molecules with covalent bonds, but not solids with complicated phenomena such as phonons, conduction etc. The propagation of thermal infrared inside a solid is “conductivity”. The most disturbing reference I encountered is given below. (This may be a conference proceeding and not peer reviewed.)

“The Theory of Heat Radiation” Revisited: A Commentary on the Validity of Kirchhoff’s Law of Thermal Emission and Max Planck’s Claim of Universality

Pierre-Marie Robitaille and Stephen J. Crothers. http://www.rxiv.org/pdf/1502.0007v1.pdf

Affirming Kirchhoff’s Law of thermal emission, Max Planck conferred upon his own equation and its constants, h and k, universal significance. All arbitrary cavities were said to behave as blackbodies. They were thought to contain black, or normal radiation, which depended only upon temperature and frequency of observation, irrespective of the nature of the cavity walls. Today, laboratory blackbodies are specialized, heated devices whose interior walls are lined with highly absorptive surfaces, such as graphite, soot, or other sophisticated materials. Such evidence repeatedly calls into question Kirchhoff’s Law, as nothing in the laboratory is independent of the nature of the walls. By focusing on Max Planck’s classic text, “The Theory of Heat Radiation’, it can be demonstrated that the German physicist was unable to properly justify Kirchhoff’s Law. At every turn, he was confronted with the fact that materials possess frequency dependent reflectivity and absorptivity, but he often chose to sidestep these realities. He used polarized light to derive Kirchhoff’s Law, when it is well known that blackbody radiation is never polar- ized. Through the use of an element, dσ, at the bounding surface between two media, he reached the untenable position that arbitrary materials have the same reflective prop- erties. His Eq. 40 (ρ = ρ′ ), constituted a dismissal of experimental reality. It is evident that if one neglects reflection, then all cavities must be black. Unable to ensure that perfectly reflecting cavities can be filled with black radiation, Planck inserted a minute carbon particle, which he qualified as a “catalyst”. In fact, it was acting as a perfect absorber, fully able to provide, on its own, the radiation sought. In 1858, Balfour Stew- art had outlined that the proper treatment of cavity radiation must include reflection. Yet, Max Planck did not cite the Scottish scientist. He also did not correctly address real materials, especially metals, from which reflectors would be constructed. These shortcomings led to universality, an incorrect conclusion. Arbitrary cavities do not con- tain black radiation. Kirchhoff’s formulation is invalid. As a direct consequence, the constants h and k do not have fundamental meaning and along with “Planck length”, “Planck time”, “Planck mass”, and “Planck temperature”, lose the privileged position they once held in physics.

Fortunately for climate science, GHG’s are simple molecules with covalent bonds and no surface to reflect or scatter radiation or produce emissivity less than unity.

]]>I put in a reply to your comment, but it would not post. Maybe if I wait a day. If it is not up by then then perhaps we can figure out some other way to communicate with you.

]]>Please see Zhao,J.Q.and Shi, G.Y. (2013) “An Accurate Approximation to the diffusivity factor.” Infrared Physics and Techmology, 56, 21-24.

The optical depth of the chosen path is the parameter of choice. They first point out two limiting cases: (1) if the optical depth is quite large the limiting D.F. is unity (no correction.) (2) If the optical depth is small then the greatest possible D.F. is used, which is that corresponding to angle theta = 60 degrees.

Zhao and Chi derive a complicated equation for intermediate optica depths. The canonical choice, for instance in Petty and in Houghton, is the D.F. corresponding to angle ~ 54 degrees. But you can find situations where the 54 degrees is not near optimum. Thus, consider the situation where the band width is between 2 wn and 2200 wn Apply to – say – 400 ppm of CO2. Since it is a pretty good approximation to consider all the CO2 absorption to be between 500 wn and 850 wn, with all the other wn corresponding to the black body case with no GHG, you will find that – even for a path from ground to 70 km the transmittance is over 0.9, by Modtran6. Consider then vertical paths. The negative of Ln 1 is zero which is the optical depth for zero absorption. The negative of Ln 0.9 is ~ 0.1 for CO2 between 500 wn and 850 wn. But say you used a window in the vicinity of the CO2 bending mode resonance. You could easily get a transmittance of , say, 0.01 or less. negative Ln 0.01 is then ~ 4.6 and the best DF will be more in the direction of 1, i.e. no D.F. correction at all.

Furhermore, alone amongst my testbooks, Pierrehumbet considers 60 his default D.F. angle, not 54 degrees. See his page 191, where the relative merits of other choices of this angle are discussed. Apparently, Zhao and Shi have gone beyond Pierrehumbert text, as will happen with any textbook. Perhaps there will be a new text written someday that includes the Zhao – Chi formula.

As far as a Wikipedia article is concerned, I know nothing of how one adds something to Wikipedia. But the last time I looked, if one looks up the Wikipedia article on Schwarzchild himself, the only contribution mentioned was the “other” Schwarzchild’s equation, which is some kind of solution for a spherical black hole in general relativity. The fact that Schwarzchild’s equation of radiative transfer is crucial to our understanding of the Sun, and to all present day climate science is not considered to be of sufficient importance to include,apparently. A “low hanging fruit” contribution to Wikipedia might be to correct this glaring omission to the Wikepedia article on Shwarzchild himself.

]]>I used my article as a vehicle for discussing (perhaps resolving for some people) some controversies in terms of SE. So I have a section on what SE predicts about saturated wavelengths – no radiative forcing. And a section on SE and the GHE – and the importance of the lapse rate. And a section about its importance and use in climate science. Two stream approximation yielding OLR and DLR. Even the existence of a diffusivity factor shortcut. Linked the online MODTRAN and Spectracalc. How SE fits in with Planck’s Law/SB eqn, Beer’s Law, and more fundamental equations of radiation transfer.

As I’m sure you know from my struggles immediately above with W/sr/m2 vs W/m2 and numerous other times I’ve ventured beyond my competence and learned something new thanks to the generosity of readers, I may not have gotten everything exactly right. In fact, I’m sure I haven’t. I will appreciate any corrections and criticism whenever the article goes live. Or you can write me now at frankwhobbs and icloud.com . When the article goes live, I can add links to other articles and perhaps enlighten people who would never hear the term Schwarzschild’s Equation anywhere else.

Doug: I presume you have seen SOD’s own calculations about the diffusivity factor above. Wikipedia has a listing of radiation transfer codes, some of which may not use a diffusivity factor or have the option of not using a diffusivity factor. I don’t know if any of these programs are easy to use.

https://en.wikipedia.org/wiki/Atmospheric_radiative_transfer_codes

]]>I would like to look into this but I have been having problems logging into Science of Doom. I am presently working on a method of demonstrating a numerical solution to S.E. for a journal that publishes papers on geosciences education, as well as in the 2019 AMS meeting where I will show how one may use David Archer’s free online program “Modtran Infrared Light in the Atmosphere” to demonstrate a numerical solution to S.E. using only a spreadsheet as a tool. The problem I am having with S.E. is that I believe the “diffusivity constant” correction is much more difficult to apply than commonly believed and recent research by others indicates that my finding indeed is correct. If you use the standard D.F. of one over cos 54 degrees you will be lucky to get an answer comes close. One must use an appropriate Gaussian Quadrature – and those are difficult to find…perhaps even are trade secrets.

]]>Have you seen this?

http://www.barrettbellamyclimate.com/page47.htm

It seems pretty comprehensive to me.

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