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This page is a blog article in progress, written by Tim van Beek.

The Case of the Missing 33 Kelvin Continued

Last time, when we talked about putting the Earth in a box, we saw that a simple back-of-the-envelope calculation about the energy balance and the resulting average temperature of the earth is surprisingly close to the real world. But there is a gap: The temperature predicted by a zero dimensional energy balance mode is about 33 Kelvin lower than the estimated average surface temperature on earth.

In such a situation, as theoretical physicsists, we congratulate ourselves on a successful first approximation, , and look out for the next most important effect that we need to include in our model.

Most of you will of course heard about the effect that climate scientists talk about, which is often - but confusingly - called “greenhouse effect”, or “back radiation”. The term that is most accurate is downward longwave radiation (DLR), however, so I would like to use that instead. But first we’ll have to peek into our simple model’s box and figure out what is going on in there in more detail.

Peeking into the Box and Splitting up: Surface and Atmosphere

To get a better approximation, instead of treating the whole earth as a black body, we’ll have to split up the system into earth itself, and its atmosphere. The For solid the surface of the earth consists of a lot of different materials with different radiation properties, so that it is still a good approximation to say that it is a black body.

The atmosphere is more complicated. In a next approximation step, I’d I would like to pretend that the atmosphere is a body of its own, hovering above the surface of the earth, as a separate system. So we will ignore that there are several different layers in the atmosphere doing different things, including interactions with the surface. Okay, we are not going to ignore the interaction with the surface completely, as you will see.

Since one can quickly get lost in details when discussing the atmosphere, I’m going to cheat and look up the overall average effects in an introductory meteorology textbook:

  • C.Donald Ahrens: Meteorology Today, 9th edition, Books/Cole 2009.

Here is what atmosphere and the Earth’s earth surface do to the incoming radiation from the sun: sun (from page 48):

Of 100 units of inbound solar energy flux 30 are reflected or scattered back to space without a contribution to the energy balance of the Earth. This corresponds to an overall average albedo of 0.3 of the Earth.

The next graphic shows the most important processes of heat and mass transport caused by the remaining 70 units of energy flux, with their overall average effect: effect (from page 49):

Maybe you have some questions about this graphic; I certainly do.

Conduction and Convection?

Introductory classes for partial differential equations sometimes start with the one dimensional heat equation: This equation describes the temperature distribution of a rod of metal that is heated on one end. The kind of heat transfer ocurring here is called conductionconduction. The atoms or molecules stay where they are and transfer energy by interacting with their neighbours.

Now, Heat heat transfer by conduction isnegligible for gases ; why is it there in the graphic? The answer is that conduction is still important for border boundary layers. So, we don’t forget border boundary layer interaction completely, as I promised.

What is Latent Heat?

Latent There heat is energy input that does not result in a temperature label increase, “latent or heat” energy on output the that left does part not result in a temperature decrease. This can happen, for example, when there is a phase change of a component of the system: atmosphere: When fluid water at 0°C freezes, it turns into ice at 0°C while losing energy to its environment.Latent heat is energy input that does not result in a temperature increase, or energy output that does not result in a temperature decrease. This can happen, for example, when there is a phase change of a component of the system: When fluid water at 0°C freezes, it turns into ice at 0°C while losing energy to its environment. But the temperature of the whole system stays at 0°C.

The picture above shows a forest with water vapor (invisible), fluid (dispersed in the air) and snow. As the sun sets, parts of the water vapor will eventually turn into ice, releasing energy to the environment. During the phase changes there will be energy loss without a temperature decrease of the water.

Downward Longwave Radiation: The Atmosphere is not a Black Body

You’ll Last have time noticed we pretended that there is a lot of energy flowing downwards from the atmosphere Earth to as the a surface. whole How behaves can like one a understand black that? body.

The But reason you for may this notice is that the atmosphere is transparent for some frequencies and opaque for others. or, to be more precise, thetransmissivity of the atmosphere depends on the wavelength of the radiation.

The a) atmosphere consists of a couple lot of gases sunlight only; passes through the atmosphere and we reaches know the from surface quantum and mechanics that a gas consisting of, say,O 2O_2 molecules, or CO 2CO_2 molecules, has very different properties with regard to photon absorption and emission as a black body. In fact, this is one of the reasons for the invention of quantum mechanics in the first place.

If b)there you is are a interested lot in the mathematical topic of operator energy theory, flowing you’ll downwards know from the definition atmosphere of to the surface in form of infrared radiation.spectrum of an operator. If you haven’t looked into quantum mechanics, however, you’ll be surprised to hear that the spectrum of an operator is actually related to the emission and absorption spectrum molecules: Simplifying somewhat, an eigenvalue of the Hamiltonian operator that describes a molecule corresponds to one line in the emission spectrum of a gas consisting of such molecules.

Degrees of Freedom

This shows that the atmosphere does not act like a black body at all. Instead, it has a nonzero transmittance, which means that not <i>all</a> incoming radiation is absorbed.

The reason for this is that the atmosphere consists of a couple of different gases, like 0 2,N 20_2, N_2 and CO 2CO_2. These molecules can absorb and emit radiation at certain frequencies only. This observation lead to the development of quantum mechanics, which can be used to calculate the characteristic emission spectrum for every molecule.

Molecules and Degrees of Freedom

When a photon hits a molecule, the molecule can absorb the photon either by

  • an electron climbing the stairs to a higher energy level,

  • stronger vibration or

  • stronger rotation.

To get a first impression of the energy levels involved in these three processes, let’s have a look at this graphic:

This is taken from the book

  • Sune Svanberg: “Atomic and Molecular Spectroscopy”, Springer, 4th edition

Wikipedia The has energy a funny animated picture with the different vibrational modes on the page Y-axis is measured inIR spectroscopyelectron volt . : This is the energy that an electric charge takes up when it travels across an electric potential difference of one volt.

It is possible to use the characteristic absorption and emission properties of molecules of different chemical species to analyze the chemical composition of an unkown probe of gases (and other materials, too). These methods are usually called names involving the word spectroscopy, like, for example, infrared spectroscopy for methods that examine what happens to infrared radiation when you send it to your probe.

By the way, Wikipedia has a funny animated picture with the different vibrational modes of a molecule on the page infrared spectroscopy.

Downward Longwave Radiation: No Infrared From the Sun

Here The is reason why so much of the radiation of the sun passes through, but a nice lot overview of infrared radiation is reflected back, involves the spectrum specific property of electromagnetic certain radiation: components of the atmosphere.

But first things first: Here is a nice overview of the spectrum of electromagnetic radiation:

From the Planck density, distribution, we can determine that sun and earth, as black bodies, emit radiation mostly at very different wavelenghts:

This graphic is sometimes called “twin peak graph”.

Only some components of the atmosphere emit and absorb radiation in the infrared part, the part where earth’s spectrum is. These are called - somewhat misleading - greenhouse gases. Two prominent ones are H 2OH_2O and CO 2CO_2:

The “atmospheric window” at 8 to 12μm is quite transparent, which means that this radiation passes from the surface to the atmosphere without much ado. Therefore, this window is used by satellites to estimate the surface temperature.

It is not a coincidence that molecules with different species like CO 2CO_2 react to infrared radiation, while those with one kind of atom like O 2O_2 do not. The deeper reason is that molecules with different species may have a dipole moment.

Downward Longwave Radiation: Where does it come from?

When you measure some infrared radiation coming from the sky, there are two indications why this radiation has to come from “greenhouse gases” in the atmosphere:

  • It is infrared. Sunlight doesn’t have much radiation in the infrared regime.

  • It is not black body radiation, but is concentrated at the characteristic wavelengths of well known gas components of the atmosphere. Infrared radiation from the surface is to a good approximation black body radiation.

Devices to measure the infrared radiation of the planetary surface are called pyrgeometer, for pyr = fire and geo = earth.

Can a Cold Body Warm a Warmer Body?

Downward longwave radiation warms the surface, but: The atmosphere is colder than the surface, so how can radiation from the colder atmosphere result in a warmer surface temperature? Doesn’t that violate the second law of thermodynamics?

The answer is: Yes No it can, does and not. no it doesn’t. It turns out that others have already taken pains to explain this on the blogosphere, so I’d like to point you there instead to try to do a better job here:

Tim van Beek: The following is just a random collection of material right now!

Tim van Beek: I would like to add radiation measurements, maybe some can be found here:

Also have a look here.

Just to have a number, the flux of DLR (downwards longwave radiation) is about 300 Wm 2W m^{-2}.

From Zero to One Dimension

The zero dimensional model has a homogeneous inbound energy flux and an averaged albedo. In the next step to refine our model, we could insert a dependency of both the radiation and the albedo of latitude. This results in a one dimensional energy balance model.

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