# The Azimuth Project Blog - doubling CO2 part one: introduction and basics

This page is a blog article in progress, written by Frederik De Roo. To see discussions of this article while it was being written, go to the Azimuth Forum.

Superseded by Blog - Doubling CO2... then what?

# Doubling $CO_2$ … then what?

guest post by Frederik De Roo

The first prize in the Beauty of the Night Sky category of the 2012 Earth Sky Photo Contest was won by Jia Hao for this beautiful picture:

(I hope this picture attracted some readers.)

But if you want to spend a clear summer night out in the fields, watching the rotation of the constellations, the rising of the moon, and the passing of airplanes, warm clothing makes the night comfortable. Or do you like to feel as if you’re radiating all your body heat directly into deep space?…

It’s amazing to realize that in front of the stars there is another thin layer—the atmosphere—which absorbs part of our body radiation. And this atmosphere emits also radiation towards the earth, helping us to keep warm. Could it be possible to alter the absorption characteristics of the atmosphere and thereby influence the average temperature at the earth’s surface? Well, yes. For example, carbon dioxide is a so-called greenhouse gas, and it’s present in the atmosphere. If its concentration rises, the average temperature will rise too. Say we double the atmospheric carbon dioxide concentration. How much does the temperature change?

### One problem at a time

To understand how a change in atmospheric carbon dioxide concentration influences the global temperature of the earth (this relation is called climate sensitivity) we need to understand:

(1) How do greenhouse gases and electromagnetic radiation interact in the atmosphere, and what has the structure of the atmosphere to do with all this?

(2) If we concentrate on only one wavelength and investigate a pseudo-energy balance, what is the influence of a rise in carbon dioxide concentration? Here we will see that the response of global temperature is logarithmic. First, we will argue qualitatively which mechanisms lead to this logarithmic response. As soon as we understand how it works, we can take a closer look at the equations describing the response.

(3) If we look at the true energy balance, i.e. the energy balance considering all wavelengths together, what is the response of the global temperature? This should allow us to quantify the climate sensitivity—but this is the climate sensitivity in the absence of feedback effects!

(4) However, the earth is not simple. If the average temperature rises due some forcing, several feedbacks come into play. For example, in a hotter earth there will be more water vapor in the atmosphere, which influences the radiative forcing too. Or perhaps snow caps will melt, and the albedo will drop, changing the radiative forcing, etc.

The fourth part is the only part where there is uncertainty about the science. It is too complicated to talk about here. The first three parts are well-established physics and chemistry, understood theoretically and verified experimentally.

Of course, as soon as someone else writes “well-established” it starts to smell like an argument from authority, so the best thing to do is to check if it actually is! Unfortunately, “you only live one life, and make all your mistakes, and learn what not to do, and that’s the end of you.” So if you have other things to worry about, I would advise you to trust me that parts 1-3 are well-established science. If something is confusing or unclear, it’s due to my limited understanding and teaching skills.

### Matter and radiation: scattering, absorption and emission

So let us start with the first topic, the interaction of matter and radiation, and more specifically, the interaction of greenhouse gas molecules with infrared radiation. In the later blog posts we will introduce equations, but first, we need some qualitative understanding.

Maybe you have heard that light consists of light particles called ‘photons’. However, because we will consider a lot of light in deriving the climate sensitivity, it suffices to talk about light as waves. Only when we have to understand the absorption of light with a single atom or molecule, it may help to temporarily consider a photon instead.

Light can interact in different ways with matter:

• there is scattering, exemplified by a rainbow when sunlight is scattered by raindrops—though this makes a rainbow sound more simple than it actually is:
• Next there is reflection: think of a mirror. Yet, in principle, reflection and scattering are basically the same: light reflected by a mirror is simply the wave superposition (i.e. the sum of all the waves) scattered by the top layer of the reflective foil behind the glass.

• there is also absorption of light, a very relevant process right now as it allows you to read this text.

• Another relevant process while you’re reading this text – unless you would read from e-paper – is that light can be emitted by your computer screen, so we notice that there is also emission.

So that leaves scattering (including reflection as a special case), absorption and emission, as the available interactions between matter and radiation.

Now, let’s imagine we have a fluid enclosed in an aquarium, look at one side, and send light through the aquarium from the opposite side. The light that is transmitted through the fluid, is the light we started with, minus the light scattered inside the aquarium (and reflected towards many sides of the room—though some of the light can of course also be forward scattered towards our eye, and this part we should not subtract). There are physical laws for how this happens, depending on the material with which the light interacts, and we can calculate all this. (Though matters will complicate if we consider monochromatic and coherent light). On top of that, if the absorption in the aquarium would not be negligible, we also have to subtract the absorbed light from the transmitted light. And if, inside the material, there would be additional sources of light, we have to take these into account too.

But even for such an absorbing-emitting liquid inside an aquarium, it would still be possible to write up differential equations describing the outcome of all these three interactions: scattering, absorption, and emission. Of course, for determining the climate sensitivity we’re not interested in visible light passing through an aquarium, we’re interested in infrared light passing through the atmosphere.

To build some intuition for scattering and absorption, it’s fun to contemplate the colors of different liquids and gases—it’s good to do this while you’re having a drink. In the first place, does the color appear through preferential scattering of a certain wavelength, or rather through preferential absorption of the other colors? For example, when you’re deep underwater in a swimming pool, the color of the light is shifted toward the blue, as red is more likely to be absorbed by water. On the other hand, the blue color of the atmosphere arises because short wavelengths are scattered more. So both effects can play a role. And naturally, often also the thickness matters: imagine, instead of a glass of milk, wine or beer, a swimming pool full of milk, wine or beer. How would the color look from above? And how would the color be when you’re deep below the surface?

Unfortunately, we can only see a small part of the electromagnetic spectrum, which we call visible light: with violet light having the shortest wavelength and red the longest. But there is much more light than this: ultraviolet, infrared, microwaves, radio waves, etc. All this is electromagnetic radiation, and we will henceforth call it light, albeit invisible to us, and it follows the same rules as the visible light – with one remark, light of a different wavelength usually interacts with different types of matter, because the interaction between light and a specific type of matter can strongly depend on the wavelength.

### Black and white bodies

Because we can only see between violet and red, our intuition for what is ‘black’ and what is ‘white’ is biased, and in radiation physics this leads to problems when the so-called “black body” is introduced.

We would probably call an object white if it is bright over the whole visible part of the spectrum in a uniform manner because it reflects all visible wavelengths. We would probably call an object black if it appears dark because it absorbs visible light. But an object can also seem bright because it’s either a source of light in a dim environment, and in the dark even a very reflective object will appear dark. Putting these silly remarks aside, a first caveat is nevertheless that we can only look at the visible wavelengths for determining black and white from our common day sense, but when talking about radiation we have to consider all the relevant wavelengths—and at least infrared light is very important at the temperatures we’re accustomed too (we’ll see this in a moment).

A second caveat is the following: when an object is both a good absorber and a good emitter, should we call it white or black? So maybe it’s better to forget about our everyday appreciation of black and white altogether when we read words like ‘blackbody radiation’. In the context of radiation, a black body is a perfect absorber, which means that it absorbs any photon of any wavelength interacting with it. In reality, no object will be able to absorb any photon of any kind interacting with it. But we can still call an object black for a certain wavelength, which means that it perfectly absorbs these wavelengths. For example, snow seems white to us, but it’s pretty black in the infrared: it’s a good absorber of infrared radiation. Water molecules are a good absorber of infrared radiation, and so is carbon dioxide.

Let us image cloaking device for a certain object, that does not scatter any light, so we can obtain no information about the object: all incident light (of all wavelengths) is absorbed. Together with the cloaking device, the object would be black. But, since it absorbs all the incident radiation, the object will heat up and it will send out electromagnetic radiation according to a black body emission spectrum. So in fact there is no true cloaking possible and a black body is also a very good emitter.

Simplistically speaking, light and matter interact because moving charges can emit light, and moveable charges can absorb light (we’ll forget about the mechanism for scattering for a while). For infrared radiation what matters are the vibration modes of a molecule (electron transitions within the electron shell are important in the UV range) see e.g. Tim van Beek’s blog article A quantum of warmth ) and the more legs the molecule has, the better its absorption characteristics in the infrared because there are more vibrational modes.

Now, if an excited mode can relax by emitting a photon, reciprocity tells us we can also excite a relaxed mode by the absorption of a photon. This reciprocity was already explained by John Baez, but I would like to add to this for clarity that this not because the absorption coefficient equals the emission coefficient that absorption equals emission. UNGRAMMATICAL AND UNCLEAR If you point an infrared laser beam at an a glass of milk, it does not mean that the glass of milk will emit just as much laser light as it has absorbed. Usually there exist different channels of relaxation for the excited molecules: in other words the absorbed laser light will rise the temperature of the milk. But milk at a temperature will also emit radiation! Every object above the absolute zero emits radiation, how exactly is a more difficult problem, as it depends on the microscopic nature of the object (if there is no channel corresponding to a certain wavelength it’s impossible to emit light of that wavelength). But, for an ideal object like a black body, its internal composition should not matter (because it can absorb all possible wavelengths, it can also emit at all possible wavelengths) and then the emitted electromagnetic radiation depends only on the temperature. Such an ideal emitter we thus also call a blackbody. Of course, a blackbody will not always appear black: depending where its peak intensity of emission lies with respect to visible light it will have a certain colour.

More realistic objects are called greybodies, their emission spectrum is that of a perfect blackbody but multiplied by a wavelength-dependent emission coefficient. The existence of a typical black body radiation spectrum allows to assign a temperature (called brightness temperature) to an object whose radiation spectrum approximates blackbody radiation reasonably well.

Finally, below are the radiation spectra of sun and earth, approximated by black body radiation. As one can see, there is hardly any overlap, which justifies the nomenclature of near infrared radiation, which is infrared radiation emitted more strongly by the sun than by the earth.

### Greenhouse gases and infrared radiation

The vibration spectrum of carbon dioxide has many peaks in the infrared, so carbon dioxide is very effective in absorbing these wavelengths. Yet, if the earth were much cooler or much hotter than it is, this wouldn’t matter so much, because the fraction of the earth’s emission spectrum in these wavelengths would be small. But because the emission spectrum of the earth precisely has a peak among these wavelengths, carbon dioxide is an important greenhouse gas, especially because it is a trace gas present in the atmosphere. Nevertheless, a common objection is that a concentration of a few hundred ppm of something (such as a trace gas in the atmosphere) has to be negligible by definition. To answer that, I would like to borrow a sentence from Craig Bohren: “To those who snort that 340 ppm of anything must surely be of no consequence, I recommend 340 ppm of arsenic in their coffee.”

Let us illustrate the influence of greenhouse gases with two pictures from (add source). The first shows the black body spectrum of an object at (roughly) the freezing point of water, compared to the downward flux at mid-latitudes in winter. For most of the wavelengths, the downward flux almost equals that black body spectrum. What does this mean?

It means that the greenhouse gases which are listed (water, ozone, carbon dioxide) do a very good job in emitting (almost) the same amount of radiation back downwards to the earth, at least for certain wavelengths.

The second picture is more detailed and shows the (outgoing) radiation spectrum of the earth, with a set of curves depicting black body radiation. In this picture, the effect of the greenhouse gases is to diminish the outgoing radiation. This reduction is not effective for all wavelengths, see the atmospheric window where the outgoing radiation is emitted at a temperature of 320 K. Carbon dioxide severely suppresses the outgoing radiation around number micron, and the temperature of a black body emitting that radiation is only around 220 K, which means that this radiation effectively is radiation from higher up in the atmosphere (where it is colder) instead of from the earth’s surface.

### Outlook and references

That seems enough for now. Next time, we’ll concentrate on optical depth and the structure of the atmosphere. That should complete the prerequisites necessary to understand the mechanisms behind the response of temperatures to rising atmospheric carbon dioxide concentrations.

(Next part: Blog - the log forcing (part two))

• Tim van Beek Putting the earth in a box.

• Tim van Beek A quantum of warmth .

• John Baez, Mathematics of the environment (part 3).

• John Baez Mathematics of the environment (part 2) : more important when 1D radiation model discussed.

• Craig Bohren, Clouds in a Glass of Beer: Simple Experiments in Atmospheric Physics, Chapter 10: The greenhouse effect. This is a great introductory text to the physics behind the greenhouse effect. It covers some of the themes of the present blog article. In fact, my text has been influenced by that chapter, although I’ve tried to avoid copying, which probably made my text worse than if I had simply copied. The chapter is actually written in a mildly skeptical style, but that doesn’t matter, the physics remains the same.