Warming Document

By Charles Dollins 12/30/22

The earth is heated by the sun. That heat is radiated back into space through black body radiation as described by Planck. A black body radiator at about 58 degrees F radiates infrared radiation with wavelengths near 15 microns. 15 micron radiation can easily be absorbed by CO₂ molecules and then released by the molecules in random directions. One such direction is back towards earth where that energy can be absorbed by the earth, hence, global warming. It can be said that the 15 micron radiation diffuses through the CO₂ molecules into space or back to earth. One method to describe diffusion mathematically is Fick’s second law. Is the diffusion of infrared photons through CO₂ molecules Fickian? To be Fickian the photon motion must be Brownian, i. e., jump randomly, jump with a more or less fixed distance and with a more or less fixed jump frequency. Scientists assume that the diffusion of infrared radiation in the atmosphere has a fixed jump distance under fixed conditions. To measure this they fill a glass tube with CO₂ at a known concentration, N, where N is the number of CO₂ molecules per unit volume. They then shine a beam of photons with fixed frequency or wavelength down the tube. The distance down the tube where 50 percent of the radiation has been absorbed is considered the jump distance for photons of the energy and for CO₂ molecules of that concentration. They then define the capture cross section, σ, of CO₂ for photons of that energy as

where λ is the length at which 50% of the photons have been absorbed. In equation (1) N and λ are known and σ is defined. The capture cross section is assumed to be independent of temperature and concentration N. Capture cross sections at many photon frequencies can be found at http://vpl.astro.washington.edu/spectra/co2pnnlimages.htm . For our purposes equation (1) is better written as

One thing to note here is that if you double the CO₂ concentration you cut the jump distance in half.

Another Fickian requirement is a known jump frequency. In low concentrations of CO₂ the molecules holds the photon for several seconds but in the real world environment things are much different. At room temperatures and pressures “air molecules” are traveling at about twice the speed of sound and with concentrations of about 10²⁵ molecules per cubic meter. They collide with each other and with the CO₂ molecules about 10⁹ times per second. These collisions are quite violent. The molecules have kinetic energy of about 30 eV where as the absorbed 15μ photon has energy of about 0.29 eV. It is, therefore, assumed that the photons will be knocked out of the CO₂ molecules after about 10^-9 seconds resulting in a jump frequency of 10⁹ jumps per second. In the future we will find that the jump frequency is not very important because the time component of Fick’s second law is zero. It is zero because the photons moving at the speed of light are always in a steady state with their environment. I use the phrase air molecules realizing that the atmosphere consists primarily of N₂, O₂, and Ar. These gases have similar atomic masses and kinetic energy so bare with me as I will use the term air molecule frequently.

Fick’s laws are laws not theories. They are laws based and derived from the three principles discussed above: known jump distance, known jump frequency, and random direction when jumping. Fick’s second law is

where t is the time, x is the direction normal to the earth’s surface and y and z are the axes parallel to the earth’s surface, D is the diffusion coefficient, and C is the concentration of CO₂ molecules occupied by a photon (in an excited state). In our steady state problem and with jumping parallel to the earth’s surface having no effect Equation(3) becomes

Equation(4) looks simple but it is not. The problem is that the diffusion coefficient is dependent on the altitude increasing as the altitude rises. Fick defined the diffusion coefficient as follows:

where ν is the jump frequency and λ is the jump distance determined from Equation(2). If D were constant then the concentration, C, as a function of altitude would be straight determined by the boundary conditions applied to Equation (4). The complexity of D is caused by the complexity of the atmosphere.

The atmosphere is frequently divided into three parts: troposphere, stratosphere, and mesosphere. The troposphere goes from the earth’s surface to about 10km. In the troposphere the temperature drops with altitude. The stratosphere goes from the troposphere to about 50 km and the temperature rises with altitude. The temperature rise is caused by the latent heat released by the production of ozone. The mesosphere goes from the stratosphere to about 86 km where outer space starts and where it is assumed here the concentration of infrared rays captured by CO₂ goes to zero. These temperature variations make it difficult to calculate the air density, N of equation(2), versus altitude. For simplicity yet realizing that something has be done a recommended formula for the air density versus altitude is

where N is in units of 1/m³ and x in the altitude in meters. This means that with altitude the jump distance increases and the diffusion coefficient, D, increases.

The countervailing effect is that with altitude the molecule collision rate also drops and with this drop jump frequency drops and the diffusion rate decreases.

I do not have access to a technical library but have searched using Google scholar and have not found any references to articles using Fick’s law to determine the diffusion of radiation from the earth. If you know of any please let me know at cdollins@gmail.com. Since the subject seems to be very important someone should be doing such calculations.

A little about my education. I have a PhD in Material Science and Engineering from the University of Illinois. While a graduate student I taught a course in solid state physics to advanced undergraduates and graduate students in the college of engineering at Illinois. In my work life I developed models to predict fast neutron damage in nuclear reactor materials.