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Dispersion, Nonlinearity, Saturation, and
NonOverlap of Absorption Spectra in the
Radiative Effects of Greenhouse Gases


The current public policy focus on carbon dioxide is a bit perplexing when ones looks at the concentrations of water vapor and carbon dioxide in the atmosphere. There is a reluctance to establish an overall concentration of water vapor in the atmoshere. That reluctance is justified because water vapor is not evenly distributed. There are humid areas and there are desert areas. The polar regions are as much deserts as the Sahara and the Gobi because of the dryness of their atmosphere. Carbon dioxide is widely and uniformly distributed. Consequently the increase in the concentration of carbon dioxide, although small compared to the global average of water vapor concentration, can have a significant effect in the desert regions. But that significant effect of increased carbon dioxide in desert regions does not mean it will have as significant an effect in the more humid regions.


In order to properly understand the greenhouse effect one must take into account the nonlinearity of the effect of increased concentration of greenhouse gases. One must also take into account that different greenhouse gases may have different spectra for the absorption of thermal radiation.

First consider the matter of the nonlinearity. According to the Beer-Lambert Law the proportion of radiation absorbed upon passing through a distance x of a medium is

1 − e−ax

where a is a parameter that reflects the concentration of the absorber and its radiative efficiency. 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 absorber 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.

Overlap and Non-overlap

Each greenhouse 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.

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.

Latitudinal Profiles

One of the earliest and consistent predictions of global warming theory is that the polar regions would increase in temperature to a far greater degree than the equatorial regions. This prediction is plausible for several reasons. First of all, the polar regions are subject to the ice-albedo feedback; i.e., as sea ice and snow fields melt the ground and open water absorb more of the Sun's radiation. Second, the air of the polar regions is dry, so dry that they are deserts as much as the Sahara is. Being dry the polar air has very little of the overwhelmingly most important greenhouse gas, water vapor. In moister regions carbon dioxide is a relatively small proportion of the greenhouse gases, but in the dry regions it is relatively more important. Thus if the concentration of carbon dioxide doubles there is relatively smaller effect in moister regions than in the dryer regions so the temperature effect of the increased carbon dioxide is greater in the dryer regions. But, if the atmosphere in the polar regions warms there will be more evaporation and thus a postive feedback from greenhouse effect of increased water vapor.

Some of the predictions from computer models are that there would be a five to one ratio of the increase in the temperatures in the polar region relative to the increase in the equatorial regions. All of this is plausible but how does the prediction compare with the facts. Theories and models fail generally not from what is included in them but from what is left out. Climate models are made up of components that derive from verified science such as thermodynamics and mechanics. Scientists from outside climatology look at these models and think they are valid because they consist only of components drawn from the hard sciences. The models can fail not because of what is there, but from what is not there. For example, the El Niño Southern Oscillation is not in the models. Neither is the Mid-Pacific Thermal Vent. Nor is the Multidecadal Pacific Oscillation. The real world is a much more complex system than what the models encompass. This is illustrated by the prediction of temperature change by latitude.

Robert C. Balling, Jr. gives a graph relevant to comparing a model prediction with the actual record. It is given in his article, "Observational Surface Temperature Records versus Model Prediction," which is published in Shattered Consensus: The True State of Global Warming (page 53). A fascimile of Balling's graph is given below.

Here we have the real world in all its complexity. Over the period 1970 to 2001 the Arctic region did have a greater temperature increase than the tropical region. It was not a five to one ratio however. The north polar region increased in temperature about 83 percent more than the tropic region. However in the south polar region there was no larger increase than the tropics, If anything the south polar region increased less in temperature than the tropics with Antarctica actually decreasing in temperature. If the increase in temperature in the north polar region is taken as a verification of the theory and global warming model then the record in the south polar regions is a denial of the validity of the theory and model. That is the real world in all its complexity.

Balling presents the estimates of temperature change by latitude from a model. He does not give the details of the model or models used for the estimates but he may be using results from models used by the Intergovernmental Panel on Climate Change. The model(s) generally overestimates the change in temperature but more so in the polar regions, particulary the Arctic. The model(s) predict that the temperature increase in the Arctic should be about four times what it is for the tropics. In the very high latitudes of the north polar region the temperature change is declining with latitude rather than increasing. It is not implausible that the actual temperature increase with latitude in the Northern Hemisphere is reflecting the effect of the band of urban-industrial civilization between 25°N and 70°N.

In the Southern Hemisphere the models predict an increase at 50°S that is about two and half times the actual increase. In Antarctica the model(s) predict a temperature increase which is about 30 percent larger than the temperature increase predicted for the tropics but the actual temperature change was negative.

The climate models have some successes but they are not hard science they are purported to be.


The small increase in the concentration of greenhouse gases due to the increase in CO2 can have a significan effect only where the concentration of H2O is very low; i.e., in deserts including the polar regions.

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