# The Azimuth Project Blog - a quantum of warmth (Rev #11)

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 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 an introductory meteorology textbook:

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

Here is what atmosphere and the earth surface do to the incoming radiation from the sun:

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:

#### 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 conduction. The atoms or molecules stay where they are and transfer energy by interacting with their neighbours.

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

#### What is Latent Heat?

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.

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 have noticed that there is a lot of energy flowing downwards from the atmosphere to the surface. How can one understand that?

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

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.

If you are interested in the mathematical topic of 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 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

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.

#### Downward Longwave Radiation: No Infrared From the Sun

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

From the Planck density, we can determine that sun and earth, as black bodies, emit at 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 $H_2O$ and $CO_2$:

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.

#### 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 it can, and 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 $W m^{-2}$.

## 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.

category: blog