# Contents

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

## The Case of the Missing 33 Kelvin

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.

## 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.

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.

Tim van Beek: I would like to relate to math from now and then, but would also like to warn those readers with less interest in mathematics, or with less background knowledge, that they may or should skip such sections. Any suggestions how to do this?

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.

## Explaining the 33 K Gap: IR-Backradiation

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

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.

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

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

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 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.

category: blog