This could be considered as a continuation of the earlier series - Atmospheric Radiation and the “Greenhouse” Effect - but I’ve elected to start a new series.
It’s clear that many people have conceptual problems with the subject of what is, in technical terms called radiative transfer. That is, how radiation travels through the atmosphere and is affected by the atmosphere.
Radiation and Gas in a Box
First, let’s consider what happens as we shine an intense beam of infrared radiation at a narrow range of wavelengths (let’s say somewhere in the region of 15μm) through a box of CO2 gas at room temperature:
The red arrow on the left is the incident radiation. The graph indicates the spectrum. The spectrum on the right is made up of two main parts:
- the transmitted radiation – the incident radiation attenuated by the absorbing gas
- the emitted radiation due to the temperature and emissivity of the gas at these wavelengths
The yellow spectrum shows what we would measure from one of the sides. Note that the transmitted radiation that goes from left to right has no effect on this yellow spectrum (except in so far as absorption of the incident radiation affects the temperature of the gas).
If we increase the length of the box (left to right) – and keep the density the same – the transmitted radiation from the right side would decrease in intensity. If we reduce the length of the box (again, same density) the transmitted radiation from the right would increase in intensity.
But the emitted radiation from the top is only dependent on the temperature of the gas and its emission/absorption lines.
And the temperature of the gas is of course affected by the balance between absorbed and emitted radiation as well as any heat transfer from the surroundings via convection and conduction.
Hopefully, this is clear. If anyone thinks this simple picture is wrong, now is the time to make a comment. Confusion over this part means that you can’t make any progress in understanding atmospheric radiation.
Radiation in the Atmosphere
How does radiation travel through the atmosphere?
The idea shown here is a spectrum of radiation at different wavelengths incident on a “layer” of the atmosphere (see note 1). The atmosphere has lots of absorption lines of many different strengths. As a result the transmitted radiation making it out of the other side is some proportion of the incident radiation. The proportion varies with the wavelength.
The atmosphere also emits radiation, and the emission lines are the same as the absorption lines. More about that in Planck, Stefan-Boltzmann, Kirchhoff and LTE.
However, the emission depends on the temperature of the gas in the layer (as well as the absorption/emission lines). But the absorption depends on the intensity of incident radiation (as well as the absorption/emission lines), which in turn depends on the temperature of the source of the radiation.
So in almost every case, the sum of transmitted plus emitted radiation is not equal to the incident radiation. By the way, the spectrum at the top is just a raggedy freehand drawing to signify that the outgoing spectrum is not like the incoming spectrum. It’s not meant to be representative of actual intensity vs wavelength.
And – it’s a two way street. I only showed one half of the story in figure 1. The same physics affects downward radiation in exactly the same way.
Considering One Wavelength at a Time
To calculate the actual transmission of radiation through the layer we simply work out the transmissivity, tλ, of the layer at each wavelength, λ (tλ simply indicates that t will vary for each value of λ we consider). We do that by looking up values calculated by spectroscopic professionals. These values are per molecule, or per kg of particular molecules so we need to find out how much of each absorbing gas is present.
1. The incident radiation making it through the layer = Iλ x tλ – for example, it could be 90% making it through, or 20%.
2. The “new” radiation emitted from each side of the layer equals the “Planck blackbody function at the temperature of the layer and the wavelength of interest” x Emissivity of the gas at that wavelength.
In case people are interested this can be written as Bλ(T).ελ, where ελ = emissivity at that wavelength, and Bλ(T) is the “Planck function” at that temperature and wavelength. Well, emissivity = absorptivity (at the same wavelength) and absorptivity = 1-transmissivity, so the same equation can be written as Bλ(T).(1-tλ).
Perhaps (the Planck function showing up in an equation) this is where many blogs (Parady blogs?) get the idea, and promote and endorse the idea, that climate science depends on the assumption that the atmosphere emits as a blackbody. There are some cases where the atmospheric emission is not far from the “blackbody assumption” (e.g., in clouds), but that is due to reality not assumption. There is no “blackbody assumption for the atmosphere” in climate science. But there is a movement of people who believe it to be true.
Misleadingly, they like to be known as “skeptics”.
End of digression..
Doing the Calculation
So it’s not really that hard to understand how radiation travels through the atmosphere. It is difficult to calculate it, mostly due to having to read a million absorption lines, figure out the correct units, get a model of the atmosphere (temperature profile + concentration of different “greenhouse” gases at each height), write a finite element program and work out a solution.
One important point – it is not possible to do this calculation in your head. If you think you have done it in your head, even to a close approximation, please go back and read this section again, then look up all of the absorption lines.
Still convinced – post your answer in a comment here.
In the next article I’ll explain radiative-convective models and show some results from the atmospheric model I built in MATLAB which uses the HITRAN database. Now that we have a model which calculates realistic values for emission, absorption and transmission we can slice and dice the results any way we want.
Does water vapor mask out the effects of CO2? What proportion of radiation is transmitted through the atmospheric window? What is the average emission height to space?
Part Two - some early results from a model with absorption and emission from basic physics and the HITRAN database
Part Three – Average Height of Emission - the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions
Part Four – Water Vapor - results of surface (downward) radiation and upward radiation at TOA as water vapor is changed
Part Five – The Code - code can be downloaded, includes some notes on each release
Part Six – Technical on Line Shapes - absorption lines get thineer as we move up through the atmosphere..
Part Seven – CO2 increases - changes to TOA in flux and spectrum as CO2 concentration is increased
Part Eight – CO2 Under Pressure - how the line width reduces (as we go up through the atmosphere) and what impact that has on CO2 increases
Part Nine – Reaching Equilibrium - when we start from some arbitrary point, how the climate model brings us back to equilibrium (for that case), and how the energy moves through the system
Part Ten – “Back Radiation” - calculations and expectations for surface radiation as CO2 is increased
Part Eleven – Stratospheric Cooling - why the stratosphere is expected to cool as CO2 increases
Part Twelve – Heating Rates - heating rate (‘C/day) for various levels in the atmosphere – especially useful for comparisons with other models.
Note 1 – What is a layer of atmosphere? Isn’t the thickness of this layer somewhat arbitrary? What if we change the thickness? And doesn’t radiation go in all directions, not just up?
These are all good questions.
In typical physics terms the actual equation of “radiative transfer” is a differential equation, which expresses continual change. In practical terms, solving a differential equation in most real world cases requires a numerical solution which has finite thicknesses for each layer.
People trying to solve these kind of problems usually check what happens to the solution as they go for more of thinner layers vs less of thicker layers. There is a trade-off between accuracy and speed.
Radiation does go in all directions. The plane parallel assumption has very strong justification and – in simple terms – mathematically resolves to a vertical solution with a correction factor. You can see the plane parallel assumption and the derivation of the equations of radiative transfer in Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations.