# The Azimuth Project Blog - the color of night (Rev #9)

## Or: how big is the "greenhouse effect" really?

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

When we talked about putting the Earth in a box, we saw that there is a gap of about 33 kelvin between the temperature of a black body in Earth’s orbit with an albedo of 0.3, and the estimated average surface temperature on Earth. An effect that explains this gap would need to

1) have a steady and continuous influence over thousands of years,

2) have a global impact,

3) be rather strong, because heating the planet Earth by 33 kelvin on the average needs a lot of energy.

Last time, in a quantum of warmth, we refined our zero dimensional energy balance model that treats the Earth as an ideal black body, and separated the system into a black body surface and a box containing the atmosphere.

With the help of quantum mechanics we saw that:

• Earth emits mainly far infrared radiation, while the radiation from the sun is mostly in the near infrared, visible and ultraviolett range.

• Only very special components of the atmosphere react to infrared radiation. Not the main components $O_2$ and $N_2$, but minor components with more than two atoms in a molecule, like $H_2 O$, $O_3$ and $CO_2$. These gases react to infrared radiation: They absorb and re-emit a part of Earth’s emission back to the surface.

• This downward longwave radiation (DLR) leads to an increased incoming energy flux from the viewpoint of the surface.

This is an effect that certainly matches points 1 and 2: It is both continuous and global. But how strong is it? What do we need to know in order to calculate it? And is it measurable?

#### Survival in a combat zone

There has been a lively - sometimes hostile - debate about the “greenhouse effect” which is the popular name for the increase of incoming energy flux caused by infrared active atmospheric components, so maybe you think that the heading above refers to that.

But I have different point in mind: Maybe you heard about guiding systems for missiles that chase “heat”? Do not worry if you have not. Knowlegeable people working for the armed forces of the USA know about this, and know that an important aspect of the design of aircrafts is to reduce infrared emission. Let’s see what they wrote about this back in 1982:

The engine hot metal and airframe surface emissions exhibit spectral IR continuum characteristics which are dependent on the temperature and emissivity-area of the radiating surface. These IR sources radiate in a relatively broad wavelength interval with a spectral shape in accordance with Planck’s Law (i.e., with a blackbody spectral shape). The surface- reflected IR radiation will also appear as a continuum based on the equivalent blackbody temperature of the incident radiation (e.g., the sun has a spectral shape characteristic of a 5527°C blackbody). Both the direct (specular) as well as the diffuse (Lambertian) reflected IR radiation components, which are a function of the surface texture and the relative orientation of the surface to the source, must be included. The remaining IR source, engine plume emission, is a composite primarily of C02 and H20 molecular emission spectra. The spectral strength and linewidth of these emissions are dependent on the temperature and concentration of the hot gaseous species in the plume which are a function of the aircraft altitude, flight speed, and power setting.

This is an excerpt from page 15 of

• Military Handbook SURVIVABILITY ENHANCEMENT, AIRCRAFT CONVENTIONAL WEAPON THREATS, DESIGN AND EVALUATION GUIDELINES, MIL-HDBK-268(AS), 5 August 1982

You may notice that the authors point out the difference of a continuous black body radiation and the molecular emission spectra of $CO_2$ and $H_2 O$. The reason for this, as mentioned last time in a quantum of warmth, is that according to quantum mechanics molecules can emit and absorb radiation at specific energies, i.e. wavelengths, only. For this reason it is possible to distinguish far infrared radiation that is emitted by the surface of the Earth (more or less continuous spectrum) from the radiation that is emitted by the atmosphere (more or less discrete spectrum).

Last time I told you that only certain molecules like $CO_2$ and $H_2O$ are infrared active. The authors seem to agree with me. But why is that? Since last time we had some discussions about wether there is a simple explanation for this, I would like to try to provide one. When we try to understand the interaction of atoms and molecules with light, the most important concept that we need to understand it that of a electric dipole moment

#### Why is the dipole moment important?

Let us switch for a moment to classical electrostatic theory. If you place a negative electric point charge at the origin of our coordinate system and a positive point charge at the point $\vec{x}$, I can tell you the electric dipole moment is a vector $\vec{p}$ and that:

$\vec{p} = \vec{x}$

For a more general situation, let us assume that there is a charge density $\rho$ contained in some sphere $S$ around the origin, then I tell you that the electric dipole moment $\vec{p}$ is again a vector that can be calculated via

$\vec{p} = \int \vec{x} \rho(\vec{x}) d \vec{x}$

Okay, you say, so it is simple to calculate it, but what is its significance? Let’s say that we would like to know how a test charge flying by the sphere $S$ is influenced by the charge distribution in $S$. If we assume that our charge density $\rho$ is constant in time, then all we need to calculate is the electric potential $\Phi$. In spherical coordinates and far from the sphere $S$, this potential will fall of like $1/r$ or faster, so we may assume that there is a series expansion of the form

$\Phi(r, \phi, \theta) = \sum_{n = 1}^{\infty} f(\phi, \theta) \frac{1}{r^n}$

When our test charge is far away from the sphere $S$, only the first few terms in this expansion will be important to it.

We still need to choose an orthonormal basis for the coordinates $\phi$ and $\theta$, that is an orthonormal basis of functions on the sphere. If we choose the spherical harmonics $Y_{l m}(\phi, \theta)$, with proper normalization we get what is called the multipole expansion of the electric potential:

$\Phi(r, \phi, \theta) = \frac{1}{4 \pi \epsilon_0} \sum_{l = 0}^{\infty} \sum_{m = -l}^{l} \frac{4 \pi}{2 l +1} q_{l m} \frac{Y_{l m}(\phi, \theta)}{r^{l+1}}$

$\epsilon_0$ is the electric constant

The $l = 0$ term is called the monopole term, it is proportional to the electric charge $q$ contained in the sphere $S$. So the first term in the expansion tells us if a charge flying by $S$ will feel a net attractive or repulsive force.

The terms for $l = 1$ form the vector $\vec{p}$, the dipole moment. The next terms in the series $Q_ij$ form the quadrupol tensor. So, for the expansion of the potential we get

$\Phi(r, \phi, \theta) = \frac{1}{4 \pi \epsilon_0} (\frac{q}{r} + \frac{\vec{p} \cdot \vec{x}}{r^3} + \frac{1}{2} \sum Q_{ij} \frac{x_i x_j}{r^5} + \cdot \cdot \cdot)$

For atoms and molecules the net charge $q$ is zero, so the next relevant term in the series expansion of their electric potential is the dipole moment. This is the reason why it is important to know if an atom or molecule has states with a nonzero dipole moment: Because this fact will in a certain sense dominate the interactions with other electromagnetic phenomena.

If you are interested in more information about multipole expansions in classical electrodynamics, you can find all sort of information in this classical textbook:

• John David Jackson: Classical Electrodynamics (Wiley; 3 edition (August 10, 1998))

In quantum mechanics the position coordinate $\vec{x}$ is promoted to the position operator; as a consequence the dipole moment is promoted to an operator, too.

#### Molecular Emission Spectra or: Only Greenhouse Gases are infrared active? Really?

For atoms and molecules interacting with light, there are certain selection rules. A strict selection rule in quantum mechanics rules out certain state transitions that would violate a conservation law. But for atoms and molecules there are also heuristic selection rules that rule out state transitions that are far less likely than others. For state transitions induces by the interaction with light, a heuristic transition rule is

Transitions need to change the dipole moment by one.

This selection rule is heuristic: Transitions that change the dipole moment are far more likely than transitions that change the electric quadrupole moment only, for example. But: If an atom or molecule does not have any diploe transitions, then you will maybe still see spectral lines corresponding to quadrupol transitions. But they will be very weak.

Tim van Beek: Compare black body radiation to the emission spectrum of CO2 and H2O.

Another important point is of course the part

The spectral strength and linewidth of these emissions are dependent on the temperature and concentration of the hot gaseous species...

Of course the temperature, pressure and concentration of atmospheric components are not constant throughout the whole atmosphere. We should keep that in mind for later, when we take a closer look at the theory of atmospheric radiation.

But back to the continuous versus discrete spectrum part:

Since we can distinguish surface radiation and radiation from specific gases, we can

• point some measurement device to the sky, to measure what goes down, not what goes up and

• check that the spectrum we measure is the characteristic molecular spectrum of $CO_2$, $H_20$ etc.

and be fairly sure that we have indeed measured the part of the radiation that was re-emitted from the atmosphere to the surface.

What would be a good place and time on Earth to do this?

#### Measuring DLR

What is the place with the least water wapor, the clearest night sky, on Earth?

Tim van Beek: Insert measurement results from the antarctic region.

• MICHAEL S. TOWN, P. WALDEN, STEPHEN G. WARREN: Spectral and Broadband Longwave Downwelling Radiative Fluxes, Cloud Radiative Forcing, and Fractional Cloud Cover over the South Pole here

Also:

• Dan Lubin, David Cutchin, William Conant, Hartmut Grassl, Ulrich Schmid, Werner Biselli: Spectral Longwave Emission in the Tropics: FTIR Measurement at the Sea Surface and Comparison with Fast Radiation Codes, online here.

Also:

• “Measurements of the radiative surface forcing of climate”, online here.

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

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

AlsÜ

• Baseline Surface Radiation Network (BSRN) here.

Also have a look here.

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

There is also the HITRAN database: You can look up radiative properties of different molecules there. HITRAN was founded by the US air force. Why? I don’t know, but I guess that they needed the data for air craft design. Look out for the interview with Dr. Laurence Rothman for some background information.

category: blog, climate