# Better explanation of Radiative Transfer in Atmosphere especially with Saturation

1. Jan 20, 2010

### stevecarson

Re: the absorption & re-emission of long-wave radiation in the earth's atmosphere.

I thought it all made sense until I was trying to work out how to explain it better to others!

So the Beer-Lambert equation describes the absorption at a given wavelength as exp(-k.z) where k is a constant (capture cross-section x number of molecules), z is vertical height through the atmosphere.

Simple stuff. But each layer absorbs energy, warms up and re-radiates. The radiation is equal to the absorption under a simple model so that the layer doesn't warm up. All seems fine. Of course in practice convection takes over but we haven't got to that yet.

Then I started to think about saturation, or high absorption. Take CO2 at 15um. Actually I'm not sure what k is for CO2 at 15um (anyone know?), haven't tried to look it up yet.

Case A
======
Let's suppose for sake of argument it's 0.01 m-1 close to the earth's surface.
From Beer-Lambert, only considering absorption..
At 10m the amount of 15um radiation is down to 90%
And at 100m the amount is down to 37%

Case B
======
Then suppose a 10x increase in CO2, just for sake of argument, and now k is 0.1m-1.
At 10m the amount of 15um radiation is down to 37%
And at 100m it's zero (0.005%)

Big difference.. or is it?

What difference does this really make? Because in case A, the atmosphere heats up due to the energy it absorbs and re-emits slightly cooler (if no convection) radiation. And in case B, the same thing happens. In case A there is less absorption, but less re-radiation. In case B more absorption but also more re-radiation.

So actually you haven't reduced the 15um radiation down at all. Or have you?

How different are case A and case B?
What does saturation really mean if, as a layer absorbs some CO2, it just heats up and re-emits?
Doesn't that mean that you can't really saturate a wavelength?

I hope someone can understand my new-found confusion and help me see what is really going on. Suddenly nothing makes much sense.

2. Jan 20, 2010

### Barakn

First I'd like to point out that you've modeled the atmosphere as if it was a constant density from top to bottom, whereas a better approximation of the density profile is e^(-z/h), where h is what's commonly called the scale height (for Earth, ~ 8 km). The intensity will not fall off as quickly with height as in your example. In regards to saturation, this is part a fallacious argument by opponents of global warming theory. If a particular wavelength is saturated then the average photon of that wavelength will be absorbed by the time it reaches the "top" of the atmosphere. But suppose we doubled the concentration of CO2 so that wavelength is "doubly saturated"? This means the average photon at that wavelength will be absorbed at an even lower height in the atmosphere, and thus will be emitted or transferred to other molecules at a lower height in the atmosphere, and thus more likely to make it back to the ground. Thus increasing CO2 levels will cause more of a greenhouse effect even at wavelengths that were already saturated.

3. Jan 20, 2010

### stevecarson

But, for example, Prof. F.W. Taylor of Oxford University in Elementary Climate Physics describes a band model and comments about the "strong condition":

He goes on to derive from that particular band model the modification to the absorption equations as a result. So I don't think the saturation concept is something which is from people who don't understand climate physics.

But since I started to think about it I can't get my head around what this really is in a qualititive sense. In that sense I am agreeing with you, but the "saturation concept" seems mainstream.