Dewitt: Thanks for your reply. You are, of course, completely correct in saying that I should have cited the integrated form of the Schwartzschild eqn to describe emission from a slab of atmosphere. Mentioning a 1,000,000X increase in GHG while considering the differential form of the equation is pretty misleading (a problem I partially acknowledged). However, your integrated emission term has a factor of 1-e^(-x). This term increases fairly linearly with x between 0 and 1. My analysis remains correct when x (optical thickness?) is <1.

I did a little more looking into slab models, such as the one at Barret Bellamy’s page and models in Taylor's Elementary Climate Physics. In both cases, they quietly mention an "optically thick" slab of atmosphere. This puts them in the region where the 1-e^(-x) term is 1 and doesn't vary with x. Unfortunately, neither appears to mention that the Earth's atmosphere is not optically thick at many wavelengths. (I remember reading once that the average was 0.7, which may be related to the value you derived above from MODTRAN.)

You said above that: "Single slab, gray atmospheres are crude toy models. They’re useful for demonstrating principles, but can’t be considered quantitative." The principle these models often demonstrate is that emission doesn't depend on GHG mixing ratio. Is this a principle climate scientists should be spreading without a full explanation of its limitations, especially with regard to the earth?

In the past, this problem was particularly acute when I tried to "picture" radiative forcing at the tropopause. This altitude is above roughly 80% of the atmosphere and essentially all of the water vapor, so the atmosphere is optically thin at most infrared wavelengths. If one imagines an optically thick slab atmosphere at this key altitude, one gets the completely misleading idea that emission (ie radiative cooling) doesn't increase as GHGs increase.

]]>True, of course. The math to describe the radiative output of the different objects, and, if you could take measurements of such hypothetical objects, they would be identical and you could not distinguish between them. I thought of the other types, but decided that it was more likely that new students would be instructed to consider only the surface and the temperature as the only factors that mattered.

]]>The alternative to the hollow shell is a superconducting solid with infinite heat capacity. Or you put a source of energy at a constant rate inside the shell.

]]>One can imagine a grad student teaching undergrads an introduction to Planck, Stefan-Boltzmann, and black bodies. In a real world scenario, as a body lost energy through radiation, its temperature would cool. This would lead to too much complication to explain to a student new to the subject; the energy emitted would have to be calculated over the range of temperatures as the body cooled and the surface temperature would also be dependent on the conductive properties of the body. So, for the sake of getting the primary lesson across, the grad student describes the black body as a hollow shell with no volume and a constant temperature. This creates two disjoints between the model and a real world body: a real body is does not emit uniformly under a Planck curve, and a real body of course has volume and internal conduction. In the world of thousands of undergrad students, some will get confused as to the important difference, regarding the absorptivity/emissivity, and the inconsequential one regarding the instruction for the purpose of getting the answer on the test to treat the black body as a hollow shell with constant temperature.

]]>I just tried a little experiment with MODTRAN. Looking up from the surface, I calculated atmospheric emission at different surface temperature offsets (-5 to +5 C) changing nothing else (1976 US standard atmosphere, clear sky). The intensity was a lousy fit to the S-B equation whether one allowed the emissivity or the exponent to vary. The key is that Teff ( I = σTeff^4) for the atmosphere turns out to be proportional to Tsurf, but the slope isn’t one. The OLS fit has a slope of 0.7395 with an R^2 = 1.

Single slab, gray atmospheres are crude toy models. They’re useful for demonstrating principles, but can’t be considered quantitative.

]]>You might also want to check out Barret Bellamy’s page on the Schwarzchild equation:

http://www.barrettbellamyclimate.com/page45.htm

He goes into a lot more detail.

]]>If the density of GHGs in a slab of atmosphere (at a given temperature) increases by 10X, the emission term grows by 10X.

I think the differential form of the Schwarzchild equation is causing confusion. A more appropriate form is:

I = [I0(exp(-κρz))]Tabsorption + {B[1 – exp(-κρ’z)]}Temission

What’s important is the value of exp(-κρz) for absorption and (1-exp(-κρz)) for emission. As the value of κρz, the optical depth τ, increases, the fraction transmitted approaches 0 and emission approaches {B}Temission. Increasing ρ by 1E06 does not increase emission by the same factor unless the value of κρz was vanishingly small to start with.

Let’s see if I got the Greek letters right.

]]>Dewitt: If I understand correctly, the HITRAN database is a tool for integrating the Schwartzschild equation along a path, for example from the surface of the earth to space. In contrast, blackbody (e=1) and graybody (e<1) radiation tell us how much radiation is emitted by a surface. In both cases, we can consider the simpler problem of emission at a single wavelength, while recognizing that the complete solution is obtained by integrating over all wavelengths. I'm complaining about the problems that develop when the equations for black/graybody radiation are applied to the surface of a slab of atmosphere.

A slab model of the atmosphere says that emission from the surface of the slab depends on emissivity and temperature. However, the Schwartzschild equation makes different predictions about emission:

dI = -Ikrdz + B(T)krdz

where r is the density of GHG and k is the absorption/emission coefficient at a given wavelength. If the density of GHGs in a slab of atmosphere (at a given temperature) increases by 10X, the emission term grows by 10X. If increased by 1,000,000X (producing an atmosphere about like Venus), the emission term will increase by 1,000,000X. When such dramatic changes in GHG content are made, the photons emitted from the surface of a slab of atmosphere will be coming from different depths (and many may be absorbed before they get to the surface). But slab models don't have the flexibility to handle changes in GHGs with an emissivity term – especially when we think of that term as being a constant, as it is for solid and liquid surfaces. Schemes proposing that we can calculate the flux from a slab of the earth's atmosphere using W = eoT^4 (and without integrating along a path, as HITRAN presumably does) seem grossly unrealistic and are the source of some of the confusion SOD presented in the post.

In summary, slab models are simply a trick that allow alarmists focus attention on the absorption term of Schwartzschild's Equation while pretending that emission is fixed by temperature and emissivity (W = eoT^4). In these models, increased GHGs warm by absorption, but do not increase radiative cooling. Perhaps SOD will someday do a post on "Are Slab Models of the Atmosphere an Alarmist Trick?" (Or perhaps his less scrupulous competitors will.)

If a slab of atmosphere is thick enough, then the problem simplifies to blackbody radiation, as shown below. However, optical density of the earth's atmosphere varies greatly with wavelength, so the following mathematics doesn't apply to our atmosphere. If radiation passes far enough through a homogeneous atmosphere, eventually an equilibrium will be reached where emission along the path is exactly balanced by absorption:

dI/dz = -Ikr +B(T)kr = 0

I = B(T) except when k or r = 0

In the limiting case of an optically thick slab of atmosphere, emission will be blackbody, B(T), at absorbing wavelengths. At non-absorbing wavelengths, emission will be due to whatever radiation is coming from behind the slab (the surface of a planet or empty space, both of which emit blackbody radiation at different temperatures than the atmosphere). As the optical thickness decreases, more and more of the emission will originate from behind the slab. In NO case, is radiation a simple fraction (e) of what would be expected from oT^4 – it's always a mixture of radiation from behind diminished by absorption and supplemented by emission.

]]>That should be Kirchhoff, not Kirckhoff.

]]>How does one map the concept of emissivity (a number between zero and one) onto the concept of GHGs (which can take on any value from zero to infinity)?

By calculating absorptivity at any given wavelength and then integrating over the spectral range.. Absorptivity is also bounded between zero and 1. By Kirckhoff’s Law, absorptivity must equal emissivity when local thermal equilibrium applies. The HITRAN database contains the information about the spectral lines to allow calculation of absorptivity for any temperature, pressure and mixing ratio of a given ghg. The atmospheric profiles of temperature, total pressure and partial pressure of atmospheric components with altitude can be measured or estimated. Emission is then product of the emissivity at a given wavelength and the Planck function for that wavelength and the local temperature integrated over the spectral range.

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