This is an example of a global warming denier being, not just a crackpot, but dishonest. (Go ahead and sue me, Roy, I'd love to see how a working-class Pennsylvania jury would handle it.)

Dr. Spencer is a former NASA scientist with a Ph.D. in meteorology. He worked with satellite monitoring of Earth and atmosphere temperature conditions, *so he must know Beer's Law.* If you don't know what Beer's Law is, don't worry about it. I'll use it later to establish why his blog post is not just wrong, but dishonest.

The blog post in question is here:

Spencer's Dishonest Blog PostLook for the January 1st, 2009 entry titled "50 Years of CO2: Time for a Vision Test."

In it, Spencer gives us a series of charts to show how much carbon dioxide is in the atmosphere. On a chart with readings from 0 to 100%, we can't see the CO_{2} fraction at the bottom. On the next chart, 0 to 10%, we still can't see it. Etc. The ambient level of carbon dioxide is currently about 385 parts per million by volume (ppmv), which is a small fraction of the atmosphere, about 0.04%. Clearly, implies Spencer, this is too little to worry about.

Here's where the Beer's Law bit comes in. The Beer-Lambert-Bouguer law tells you how much a beam of light is diminished by passing through a transmitting medium:

I = I_{0} exp(k ρ dz) |
(1) |

Here,

- I is the intensity of the light received by the sensor
- I
_{0}is the original intensity of the beam - k is the coefficient of extinction
- ρ is the density of the medium
- dz is the distance convered

I is an "intensity," power per unit area per unit wavelength per unit solid angle, measured in the SI in W m^{-2} m^{-1} sr^{-1}. The units of k would be m^{2} kg^{-1}, ρ kg m^{-3}, and dz m.

Do you see a term for volume fraction in there? Or molar percentage? Neither do I.

Extinction can be by either of two means (assuming no phase change)--the substance the light is traveling through can absorb the light, or it can scatter it. For infrared light, like the thermal IR given off by the warm ground on Earth, scattering is very small and can be neglected, so k is an "absorption coefficient" rather than an "extinction coefficient."

How much carbon dioxide is above every square meter of the Earth? Not much, says Dr. S. Let's calculate how much is not much. We'll make the simplifying assumption that the IR from the Earth's surface only goes straight up, so we can modify Beer's Law to work with mere "flux density" in watts per square meter.

The mass of the atmosphere is estimated at about 5.136 x 10^{18} kg, and the Earth's surface area is about 5.1007 x 10^{14} m^{2}, so it turns out there are about 10,070 kilograms of air above every square meter of Earth. The volume fraction of CO_{2} is only 385 ppmv, as stated before, but we can't multiply a mass by a volume fraction and get the mass of CO_{2}. We have to multiply by the ratio of the molecular weight of CO_{2} to normally wet air as well. I've done the work for you; it's 44.01/28.92, so it turns out we have 5.9 kilograms of carbon dioxide above each and every square meter of the Earth's surface.

Now, from the tables of Houghton (2002) and some fancy mathematical footwork on my part, the absorption coefficient of carbon dioxide between the wavelengths of 14.3 and 16.0 microns is about 163 square meters per kilogram. The Earth, at a mean global annual surface temperature of 288.15 K and a surface emissivity of 0.95, radiates an average of 371.4 watts per square meter. Assuming what's called a Planckian distribution, about 7.85% of that falls between 14.3 and 16.0 microns in wavelength. That's about 29.2 watts per square meter.

How much of that could 5.9 kilograms of carbon dioxide in Earth's atmosphere absorb? Well, let's say it occupies a column 10 kilometers high (10,000 meters). Its density would then be about 0.00059 kilograms per cubic meter, the path length would of course be 10,000 meters, and the intensity getting through would be... about zero!

The infrared light in that range given off by the Earth is completely absorbed by the carbon dioxide!Now, that 5.9 kilograms per square meter used to be 4.26 kg m^{-2} before the industrial revolution. If you run through the example above with 4.26 instead of 5.9, you still get complete absorption. But let's not let Dr. S. invoke the old saturation argument against CO_{2} global warming; which is dealt with thoroughly here. Let's just note that that also specious argument would completely contradict Dr. S's argument that tiny amounts don't matter; it would be an argument that tiny amounts matter too much.

That tiny volume fraction of CO_{2} matters, and matters greatly.

The argument that "it's too small a fraction to make a difference" is prima facie stupid anyway. A 10-gram bullet can kill a 100-kilogram man, and that's a mass fraction of only 10 ppm. The lethal volume fraction of fluorine in air is 0.1 ppmv. That's one part in *ten million,* compared to carbon dioxide's one part in 2600. And that's why Dr. Roy Spencer is **WRONG.**

Page created: | 01/11/2009 |

Last modified: | 02/13/2011 |

Author: | BPL |