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Saturation, Nonlinearity and Overlap
in the Radiative Efficiencies of Greenhouse Gases

If one pursues the question of how much of the greenhouse effect is due to each of the various greenhouse gases one finds a perplexing variety of answers in the literature. One source says that 95 percent of the greenhouse effect is due to water vapor, another 98 percent. These figures may be referring to the proportion, by weight or volume, of water vapor among the greenhouse gases of the atmosphere. Another source says that proportion water vapor is responsible for is between 36 and 70 percent. Water droplets in clouds account for another 10 to 15 percent so water as liquid or vapor accounts for between 46 and 85 percent of the greenhouse effect. The same source attributes 9 to 26 percent of the greenhouse effect to carbon dioxide (CO2).

The perplexingly wide range is explained by the source as being due to the nonlinearity of the response of the atmosphere to greenhouse gases and the overlap of the absorption spectra of the various greenhouse gases. The phenomenon is explained no further than evoking the terms nonlinearity and overlap. This material below is an attempt to clarify the situation.

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. This effect could be depicted as follows:

However the relationship may be more in the nature of the one below:

More on this later.

Overlap and Non-overlap

There could be a situation in which with some combination of greenhouse gases the atmosphere is saturated for one frequency band but not in another. The addition of a greenhouse gas which absorbs in both bands would then increase the absorption in the unsaturated band but not in the saturated band. This is the significance of the overlap of the absorption spectra of different greenhouse gases.

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 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.

Fundamental Analysis

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.

When there are more than one greenhouse gas the value of a is

a = Σ αiρi

where αi and ρi are the radiative efficiency and linear density of constituent i.

The product ax above is sometimes called the optical depth. The concentration could change over the path of the radiation. In such a case the optical depth D is

D = ∫0xa(z)dz

While the absorption is nonlinear the optical depth is linear in the concentrations and it is thus possible to specify what proportions of the optical depth for any band the various greenhouse gases are responsible for.

The Radiative Efficiencies of the Greenhouse Gases

The radiative efficiencies are available for some of the greenhouse gases.

Radiative Efficiencies of Greenhouse Gases
Given in the IPCC's Third Assessment Report
Concentration (1998)
in Atmosphere
H2ONot givenNot Given
CO2Not given365,000
Source: Tables 6.7 and 6.1, p. 388 and p. 358 of Climate Change 2001: The Third Assessment Report of the Intergovernmental Panel on Climate Change.

Climate Change 2001 gives a figure for the radiative efficiency of CO2 of 0.01548 W/m²/ppmv but emphasizes this figure is to be used only for the computation of global warming potentials. It is incredible how the scientific works on global warming can leave H2O entirely out of the picture. A diligent search of sources other than Climate Change 2001 reveals that the radiative efficiency of water vapor is fifty to sixty percent greater than that of CO2. For more on the role of water, in liquid and vapor forms, on the climate see Water.

It must be said that the radiative efficiency of a substance is not intrinsic to that substance. The line spectrum of a substance is intrinsic to the substance but the absorption by a substance depends not only upon its line spectrum. The line spectrum of a substance experiences broadening due to pressure and temperature effects. Also the amount of energy absorbed by a substance depends upon the distribution of energy in the incident radiation.

The diagrams shown below shows how the distribution of energy is dependent upon thee temperature of the radiating surface.

The climate modelers of course presume that the level of water vapor in the atmosphere is a function of the global temperature and is therefore a derived effect. Even if this were strictly true it would not hurt to have the data on water vapor displayed for comparison. However those modelers, despite the criticality of the assumption, never display a graph showing the data in which global humidity is a function of the level of global temperature. There is a crucial conceptual error involved in those models. It is one thing for water vapor concentration to be a function of global temperature, but an entirely different matter for water vapor concentration to be a function only of temperature. For more on this topic see Water Vapor.

Radiative Forcing and Global Warming Potential

The two concepts of radiative forcing and global warming potential (GWP) should not be confused with radiative efficiency. Radiative forcing is the change in the energy input to the Earth's climate system over some period of time due to some external change. It is measured in watts per square meter (W/m²). It is a useful concept and leads to the definition of the climate sensitivity parameter λ, i.e.,

λ = ΔTs/ΔF

where ΔTs is the change in the Earth's global mean surface temperature and ΔF is the radiative forcing.

Global warming potential is an interesting statistic for the greenhouse gases but it should not be confused with the radiative efficiency. The global warming potential for greenhouse gas combines its radiative efficiency with its average residence time in the atmosphere. There are some greenhouse gases which a miniscule components of the atmosphere but their introduction into the atmosphere is signficant because they may stay around for centuries contributing to global warming whereas others such as CO2 may reside for a relative short time before being absorbed by water. The global warming potential are measured over a specific period of time and in comparison to CO2. Some of the GWP's are shown below.

for 100 years
H2ONot Given
Source: Tables 6.7, p. 388 of Climate Change 2001: The Third Assessment Report of the Intergovernmental Panel on Climate Change.

The absence of a value for water vapor is perplexing. Even more perplexing are these question-and-answer presentations on U.S. Department of Energy websites:

Q. I am curious about the global warming potential of water vapor.
A. Water vapor is indeed a very potent "greenhouse" gas, in terms of its absorbing and re-radiating outgoing infrared radiation. It is commonly not mentioned as an important factor in global warming, because it is not clear that the atmospheric concentration (as compared with CO2, methane, etc.) is rising.

What is the global warming potential of water vapor? Are the anthropogenic water vapor emissions significant?
Water vapor is a very important part of the earth's natural greenhouse gas effect and the chemical species that exerts the largest heat trapping effect. Water has the biggest heat trapping effect because of its large concentration compared to carbon dioxide and other greenhouse gases. Water vapor is present in the atmosphere in concentrations of 3-4% whereas carbon dioxide is at 387 ppm or 0.0386%. Clouds absorb a portion of the energy incident sunlight and water vapor absorbs reflected heat as well.

The questions call for a numerical answer. Global Warming Potential (GWP) is a technical term widely used in the literature on global warming. The babbling answers suggests that the numerical value is not readily available even to people who are specialists in global warming. This ignoring of the technicalities of the role H2O in the climatology of the Earth is mind boggling.

The IGCC says that about half of the projected global temperature increase from a doubling of the concentration of CO2 is due to the water vapor feedback effect. All of the models have to have used a figure for radiative efficiency of water vapor in order to have projected a feedback effect yet nowhere does the IGCC literature give that crucial figure.

The Consequences of Nonlinearity for Global Warming

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.

Band Overlap and, More Importantly, the Band Non-overlap

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 not heavily saturated 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.

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 experience 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.)

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