first principles derivation of CO₂ radiative forcing

"Real knowledge is built from the ground up. If someone is using a lot of fancy words and a lot of big concepts, they probably don't know what they are talking about. True understanding is about algorithms, it's about understanding how things connect to each other. So it's the mark of a charlatan to try and explain simple things in complicated ways and it's the mark of a genius to explain complicated things in simple ways." - Naval Ravikant, Feb. 27, 2017

The following five laws of nature will be used to build up an understanding of the atmosphere.

1. Beer-Lambert Law

This 167-year-old law of physics states that the transmittance of light through a medium drops exponentially depending on the mass of absorbing material along the path of the beam of light.

2. Newton's Law of Gravity

This 332-year-old law describes the force with which every particle attracts every other particle. This law describes the mechanism that holds the atmosphere on earth. For calculations on Earth's surface, the mass of Earth (m₁), our distance to Earth's center (r) and the gravitational constant (G) are often combined into g

3. Ideal Gas Law

This 185-year-old law relates the pressure, temperature, volume and moles of a gas via the ideal gas constant "R."

4. Planck's law

This 119-year-old law of physics precisely describes the quantity of radiation (light, infrared, microwaves...) emitted by a black body at each wavelength given the temperature of the black body.

5. Kirchhoff's Law

This 159-year-old law of physics states: if a gas is a good absorber at some wavelength, it must also be a good emitter at that same wavelength. Were it not so, it would be possible to take a mirrored chamber and place inside columns of warm and hot gas, both strong IR emitters (ε≈1.0) but if the warm gas were a weak IR absorber (𝘢₁≈0.1) while the hot gas were a strong absorber (𝘢₂≈1.0) the resulting net energy flow would be from warm to hot gas, violating 2nd law of thermodynamics.

Equation (19) is extremely useful

This equation describes the atmospheric transmittance (transparancy) from any altitude to top-of-atmosphere.

This equation tells you how far you can see vertically (in IR) through the "fog" of CO₂ in the atmosphere.

This equation tells a satellite how far it can peer down into the atmosphere at a particular wavelength λ.

This equation tells you the effective height of IR thermal emission from the atmosphere at wavelength λ.

Equation (19) and its first derivative are plotted with atmospheric density. Across a range of mass absorption coefficients kₐ, notice there is an elevation where dt(z)/dz is maximized. This is the effective elevation of emission to space.

  • Much below this elevation, transmittance ≈ 0 and all light or IR at the wavelength λ is absorbed

  • Much above this elevation, there are too few molecules emitting due to low atmospheric density.




What does a satellite see?

Transmittance equation and AIRS satellite views at 10.5-9.8 μm IR.

A IR-sensitive satellite viewing Earth at λ=10.5 μm will observe surface-emitted IR passing through a transparent atmosphere. A small wavelength shift to λ=9.8 μm enters a band where the ozone layer absorbs surface-emitted IR. That doesn't mean there's no IR left to detect; it means the detected IR is emitted by the cooler atmosphere.

  • Increasing concentration of absorbing gas molecules renders the atmosphere opaque to infrared light at particular wavelengths corresponding to the absorption line(s) of the gas molecule.

  • The atmosphere's density decreases exponentially with altitude such that eventually infrared light escapes, even though there are still strong absorbers present (they are just too spread out to stop the infrared light anymore).

  • This leads to the concept of an "effective emission elevation" identified where the transmittance of the atmosphere t(z) is changing rapidly from 0 to 1 (or where the first derivative dt(z)/dz is maximized).

From Grant Petty's "A First Course in Atmospheric Radiation" from Nimbus-4 IRIS data from Goddard EOS Distributed Active Archive Center, instrument team leader Dr. Rudolf A. Hanel

Another example that demonstrates the concept of effective elevation of emission is a pair of IR scans over the Pacific: one viewing clear sky, the other from above a thunderstorm cloud. CO₂ is so strongly absorbing at 15 μm that the detected infrared is identical, despite large differences throughout the rest of the spectrum. This shows that for both A and B views, the effective elevation of emission at 15 μm is above the clouds.

Unfortunately the IRIS satellite only functioned for 8 months. The hyperspectral infrared measurements gathered were useful for checking the accuracy of computerized line-by-line radiative transfer models (LBLRTMs). We rely on these models to predict how effectively the Earth can cool itself by emitting longwave radiation. But there haven't been comparison studies between changing outgoing infrared over time as predicted by models and what's measured by satellites.