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The Case of the Missing 33 Kelvin Continued

Last time, when we talked about putting earth into 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 some gap, because the temperature predicted by a zero dimensional energy balance model is 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 solid 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 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 a introductory meteorology textbook: It is

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

Here is what the atmosphere does to the incoming radiation from the sun:

The next graphic shows the most important processes of heat and mass transport, with their overall average effect:

But the atmosphere consists of a couple of gases only, and we know from quantum mechanics that a gas consisting of, say, $O_2$ molecules, or $CO_2$ 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.

math technobabble:

If you are interested in operator theory, you’ll know the definition of the spectrum of an operator. If you haven’t looked into quantum mechanics, however, you’ll be surprised to hear that “spectrum of an operator” is actally 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.

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 has a funny animated picture with the different vibrational modes on the page IR spectroscopy.

Explaining the 33 K Gap: IR-Backradiation

• Roy Spencer, blogpost: Yes, Virginia, Cooler Objects Can Make Warmer Objects Even Warmer Still

Here is a nice overview of the spectrum of electromagnetic radiation:

BTW, if you doubt that a colder black body can emit low energy photons that are then absorbed by a hotter black body, increasing its energy in the process, you may ponder the question how a microwave oven works.

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

Only some components of the atmosphere emit and absorb radiation in the IR part, the part where earth’s spectrum is. These are called - somewhat misleading - “greenhouse gases”. Two prominent ones are $H_2O$ and $CO_2$:

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

• Military Handbook MIL-HDBK-268(AS)

• Caltech Infrared Spectroscopy

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

From Zero to One Dimension

The zero dimensional model has a homogenous 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.