where a is a parameter that represents the concentration of the absorbing material. The quantity ax is called the optical depth. The parameter a is the product of two terms. One is the concentration ρ of the absorber and the other is a characteristic of the absorbing material α, called its radiative efficiency.

The relationship is the one shown below:

The source of the nonlinearity may be thought of in terms of a saturation of the absorption capacity of the atmosphere in particular frequency bands. The concentration of greenhouse gases can make the atmosphere essentially opaque in a particular band. If the atmosphere absorbs 100 percent of the radiation in a band the absorption will not be increased when additional greenhouse gases are added. The atmosphere would then be said to be saturated in that particular frequency band. However full saturation may not occur; it is a matter of relative saturation.

Because of the nonlinear response a small increase in a greenhouse gas under conditions of low concentration can have more of an impact than a much larger increase under conditions of high concentration. In the diagram below the increase from A to B produces a much bigger impact on the proportion of radiation energy absorbed than the increase from C to D even though the magnitude of the increase from C to D is larger than the increase from A to B. In fact, from point C no increase in concentration no matter how large will produce as much of an impact as the increase from A to B.

The amount of energy entering the atmosphere can be represented as

L(t, θ)(1 − e^−ax)

where L is the energy intensity of the radiation, t is the time of the year including the time of day and θ is latitude.

If an increase absorber concentration of Δa produces an increase in energy absorbed of ΔE then the temperature increase ΔT is given by

ΔT = βmΔE

where β is the specific heat of the medium material and m is its mass density.

Increase in Temperature in Atmosphere
due to Increase in the Concentration of CO2

Let Δa be the increase in the concentration in CO2 everywhere. The increase in atmospheric temperature ΔT as a function of existing greenhouse gas concentration a is given by

ΔT = βmΔE = βm·e^−axΔa

The relationship between the increase in temperature ΔT due the increase in atmoapheric CO2 is as shown below

Overlap and Non-overlap

Each greenh ouse gas has a spectrum of radiation frequencies it will absorb and re-radiate. These are due to natural vibration modes of the molecules. If molecules only absorbed radiation of precisely those frequencies then very little interaction of molecules and radiation would take place because of the low probabilities of occurrences of radiation of exactly those frequencies. However there are factors which result in the absorption of radiation at frequencies near those in its spectrum, such as the Doppler effect resulting from the motion of the molecules. For more on this see Absorption Spectra.

The graph below is based upon absorption data for water vapor and carbon dioxide given in J.R. Houghton's The Physics of Atmospheres (Cambridge University Press, 1977). It shows some overlap in the absorption spectra of the two greenhouse gases. Until the absorption spectra of the two gases were measured accurately it was believed that carbon dioxide did not absorb any radiation that was not absorbed by water vapor. If there were complete overlap of the spectra there would be no significant role for the miniscule amount of carbon dioxide in the air to have a role in atmospheric warming. The non-overlapping spectral band for carbon dioxide was not discovered until about the early 1950's. For more on the history of the role of carbon dioxide in global warming see CO2 History.

The importance of this non-overlapping band in the carbon dioxide spectrum depends upon what portion of the thermal radiation occurs in that band. This will depend upon the surface and atmospheric temperatures. For more on this topic see Black Body Radiation.

Consider the case in which water vapor (H2O) and carbon dioxide (CO2) are the only greenhouse gases. Each absorbs radiation in two bands, one of which is a band where both absorb radiation, the overlap band. Human activities increase both CO2 and H2O. The three diagrams below depict the situation. The anthropogenic effects include an increase in both CO2 and H2O.

Carbon dioxide absorbs in a band in which water vapor does not, so this band is relatively unsaturated and the impact is relatively large.

The relationship between the increase in temperature ΔT due the increase in atmoapheric CO2 is as shown below

In the overlap band in which both absorb there is relative saturation. The change from C to D includes both the increase in H2O and CO2. The increase in CO2 has relatively little effect in this band.

The effect here of the anthropogenic increases is not much different than in the band in which only H2O absorbs.

The total energy absorbed depends upon how much of the energy of the thermal radiation is in the three bands. However it easy to envision the increase in global warming due to anthropogenic increases could be disproportionately from the increase in CO2 in the absorption bands that are exclusively for CO2. Water vapor and carbon dioxide are both greenhouse gas but carbon dioxide is a greenhouse gas with a difference.

Although the material above gives the proper perspective on greenhouse gases it leaves out the even more important factor of anthropogenic cloudiness on global temperatures. The anthropogenic changes in water vapor in the atmosphere are necessarily accompanied by changes in cloudiness. The changes in cloudiness on global temperatures are overwhelmingly more important than the changes in the atmospheric greenhouse gases. Almost everyone has experienced the effect of cloud cover on local temperatures. On clear nights the temperature falls drastically compared to what happens when there is a cloud cover. The effects of the greenhouse gases are the same on clear and cloudy nights so the difference comes entirely from the difference in cloudiness.

(To be continued.)