by Bryce Johnson
An atmospheric radiation transport code, SpectralCalc™ (1) has been used to predict carbon dioxide’s contribution to earth heating. The code predicts that doubling atmospheric CO2 raises its temperature by only 0.22 degrees C. This temperature rise is very much lower than “conventional” predictions which indicate at least a 1 C rise for doubling (2). The code shows that increasing CO2 by a factor of eight would trigger a 0.86 C rise while an increase of 0.72 C would be manifest by only a 30 percent increase in water vapor.
Thirty percent is a typical daily variation in the amount of water vapor (3, 4). At the current rate of increase, 2000 years would be required to achieve an increase of a factor of eight in CO2. Another statement of the comparison would be that it takes carbon dioxide 2000 years to affect the temperature as much as water does in one day. It is doubtful that such a small increase as 0.22 C (the doubling effect) could trigger any “feedbacks” (effects caused by the heating which modify its result) which are considered by climate-change proponents to be the cause of most of the CO2 generated effects because they are claimed to be positive (increase the effect).
Conventional wisdom on global warming is contradicted by two additional findings. These are: 1) there is negligible contribution to earth warming at high attitudes; and 2) the dominant source of radiation to outer space from the atmosphere (which is its cooling mechanism) is from low, rather than high altitudes. These plus the altitude variation of ghg energy deposition from Figure 4 give ample evidence that ghg heating of the atmosphere is a low-altitude phenomenon.
The two known analyses (5, 6) have reported results comparable to this study use the same code or a similar method,
There is no reason to consider that carbon dioxide has anything but a negligible impact on earth warming.
Results and conclusions contained herein are dependent on the accuracy and adequacy of the SpectralCalc computer code. Requests for documentation of its accuracy and for a review of this study for appropriate use of the code from GATs, Inc., the owner of the code, have gone unanswered. The code is proprietary and unavailable for outside checking. The code has been used for a number of years and has had adequate opportunity for “debugging” and comparison with observations. The description provided for users indicates it produces valid and accurate results, but results of check runs have not been provided
The SpectralCalc™ Code and Greenhouse Effect
The greenhouse effect is the absorption by a molecule of a quantum (specific amount) of radiation energy and its immediate conversion to local heat energy. It occurs only in the infrared (IR) portion of the electromagnetic-wave spectrum (energy distribution). Its wavelength range is about 4 to 80 microns. It is produced by heated substances and when it is absorbed it produces heat.
The SpectralCalc code was designed for calculating radiation transmission and its interaction with matter. It is ideal for characterizing the greenhouse process and it has a built-in library of digitized greenhouse-gas and atmospheric data at successive concentric virtual spherical shells about the earth and a capability to accurately calculate transmission and absorption probabilities of specified radiation (uniquely characterized by its energy or its wavelength) between any two points in the atmosphere with properties altered at the boundaries of the individual shells traversed by the radiation. Its universality and accuracy provides a unique probe of major concerns of earth warming. Because it calculates only one path at a time an additional calculation is required to determine the radiation from a surface area. Microsoft’s Excel™ spreadsheet is used in this study for this calculation. .
“Saturation” effects are included in SpectralCalc. Saturation is the case where the infrared (IR) radiation has encountered enough IR absorbing molecules to remove all the energy of a specific set of wavelengths so that additional molecules of the absorber cannot increase absorption at these wavelengths. Saturation is a significant factor in limiting the possible ghg heat of a given absorber. It is exhibited by both H2O and CO2 and they are synergistic (the saturation of the two together is more than the sum of each of their individual values). It is likely that the low absorption values produced with this code are due to its detailed and accurate characterization of saturation and this observation is also expressed in Reference 6. Appendix A contains a description of the application of SpectralCalc in determining the fractional greenhouse gas (ghg) absorption of carbon dioxide.
The following approximations simplify the calculation without compromising the results.
- The only source of IR radiation (the exclusive source of greenhouse effect) is from the Earth. The sun’s fraction of IR radiation is too small to matter.
- Water and carbon dioxide can be considered the only ghg contributors to the atmosphere. Other ghg contributors are negligible compared to these.
- Clouds and precipitation can be ignored. This assumption assures a conservative result since liquid water and ice absorb more IR than water vapor and their absence augments the fraction attributable to carbon dioxide which is conservative for this analysis.
- Exclusion of feedback effects avoids complicating the analysis and, because of their controversial nature, adds confidence to the results.
The analysis is completed with two ratios:
- The ratio of carbon dioxide ghg atmospheric heating to that of water ghg heating, and
- The ratio of ghg heating of the atmosphere to total heating of the atmosphere.
The first ratio requires extensive use of the SpectralCalc code.
The second of these ratios is available from any number of earth-atmosphere energy balances (such as those shown in Figures 1, 2, and 3).
Determination of CO2 Contribution to Atmospheric Temperature
If the ratio of atmospheric absorption with added carbon dioxide to that without the added carbon dioxide (i.e., that with the current level of carbon dioxide) is designated as c and the ratio of ghg heat addition to total atmospheric heat addition as g, then the fraction of carbon dioxide ghg heating to total greenhouse heating is c/(1+c) and the fraction of carbon dioxide heat to total heat is gc/(1+c). Since radiation to outer
space is the only way the atmosphere can lose heat, the heat transfer equations for the total atmosphere with and without the added carbon dioxide heat are respectively written with the Stefan-Boltzmann equation as
(1) e s(Tw4 – To4) = H(1 + gc/(1+c))
(2) e s(Two4 – To4) = H
Where e, s are the atmospheric emissivity and Stefan-Boltzmann constant, respectively. Subscripts w, wo and o designate “with” and “without” added carbon dioxide, and “outer” space, respectively. H is the heat transfer rate without the additional CO2 . The T values are in units of absolute temperature, normally as degrees Kelvin. Division of Equation 1 by Equation 2 and approximating To as 0, reduces to
(3) Tw = Two(1 + gc/(1+c))1/4
The temperature of the atmosphere is generally considered to vary from 288K at the earth’s surface to 255K at the “top of the atmosphere” where most radiation to outer space is considered to be emitted. The definition of top of the atmosphere is arbitrary. There really is no unique “top” and radiation directly to outer space is emitted throughout the atmosphere. Little difference is seen in the added temperature to the atmosphere between using the “warmest” or the “coldest” of atmospheric temperatures as Two in the above equation. This means that the distribution of heat within the atmosphere has but a small effect on greenhouse heating. Figure 6 shows that the atmosphere is not preferentially cooled from high-altitude radiation. In fact, the opposite is true.
Atmospheric Heat Balance
Figure 1 is the atmosphere-and-earth heat balance produced by the International Panel on Climate Change (IPCC). This is useful as a point of comparison for the analysis presented here. It is the basis for the IPCC analysis of global warming, but has been subject to some valid criticism. The surface radiation emission shown as 396 w/m2 is very nearly the same as the prediction by the Stefan-Boltzmann equation for the accepted standard surface temperature of the earth at 288K which is 390 w/m2. The back radiation absorbed by the surface matches the so-called “downwelling” radiation produced by the heat of the atmosphere (6, 7). The other heat fluxes are approximately the same as those of depicted in the figures 2 and 3 diagrams, which depict only net heat flows for each type crossing the the atmospheric boundaries, as Figure 1 does for all but IR heat at the earth’s surface.
Figure 2. Atmospheric Energy Balance by U. S. Weather Service (9)
Figure 3. NASA Earth-Energy Budget (10)
When using the earth’s surface temperature as the starting point for IR calculation the back radiation is accounted for because of its effect on the surface temperature. The small imbalance of 0.9 w/m2 is not valid. It is an apparent attempt to illustrate a “warming earth” by the IPPC.
The net energies crossing the earth and outer atmosphere boundaries are similar for these balances, but the ratio of greenhouse absorption to that radiated directly to space is much larger for the NASA balance than for the other two. In order to avoid underestimating the effect of carbon dioxide the NASA balance is used for determining the g value of Equation 3.
The only inputs required for the analysis are the 288 K, for the earth’s surface temperature, the heat flux from the sun (341.3 w/m2 ) and the distribution of solar radiation entering the system among the various absorbers and reflectors. In calculating IR absorptions with varying amounts of atmospheric CO2 the SpectralCalc™ code accepts as input the fractions of all IR absorbers in the atmosphere as determined by those listed in the U.S. Standard Atmosphere which is included with SpectralCalc and controlled by the user with an input multiplying factor. These are used to assess the fractional IR absorption and transmission at a given limited wavelength interval between any two points in space as long as the code’s calculation capability is not exceeded. If it is exceeded the scope of the calculation can be reduced to match the code capability. For this analysis only water and carbon dioxide are used and the values between the earth’s surface to outer space are calculated at various angles. The output of the code is the probability of absorption and transmittance from a unit source at an emitting surface to a receiving surface. Source units are normally watts/m2 per steradian.
It was found that an average angle with the vertical was adequate for approximating the result from integrating over the unit hemisphere of emission from a flat surface. The approximation avoided many time-consuming SpectralCalc runs. It was found that 60 degrees produced the appropriate result. The accuracy of using a single angle is enhanced by the fact that the results are in terms of ratios which tends to cancel the effects of the error in the approximation.
Results were calculated for water vapor only and for carbon dioxide only in an otherwise standard atmosphere to illustrate their saturation synergism. They capture 16 percent less IR radiation acting together than they would if they could act independently. To determine the effect of added CO2, runs were made with the current level and twice, four times and eight times that level of CO2 with an otherwise standard-atmosphere. The fractional increase caused by each added ratio was then used in Equation 3 to determine the associated atmospheric temperature increase. The g value in Equation 3 is based on the NASA balance is 15/64 or nearly 25 percent. The NASA balance is used because it has the highest ratio of greenhouse heating to total heating of the atmosphere and thereby produces the greatest conservatism.
An additional calculation was made with a standard atmosphere and a 30 percent increase in water vapor concentration to demonstrate that the effect was much greater than that of doubling carbon dioxide.
Other determinations using the code were the altitude distribution of ghg heat deposited in the atmosphere, the back radiation and “out” or upward radiation from the heat of the atmosphere and a determination of the source distribution for the “out” or “upwelling” radiation. The altitude distribution shown in Figure 4 demonstrates the insignificance of high altitudes in greenhouse heating. The altitude distribution of ghg heat deposition simply sums the absorption between increments of altitude for the wavelength increments and plots them at the altitude of the midpoint of its increment
The back and out radiation results are shown in Figure 5, which is of interest but has no bearing on carbon dioxide’s contribution to atmospheric heating. Figure 6 shows the altitude distribution of the radiation sources that cool the atmosphere. The results indicated in Figure 5 are the exact opposite of the assumptions that have been used to date. It is fortunate that the distribution has only a secondary effect of carbon dioxide’s heating effect.
The results shown in Figures 5 and 6 require that a source magnitude be determined for the altitude points where upward and downward radiation is to be calculated. This magnitude is proportional to the product of moles per meter3 and radiance for the specific interval of wavelength used in the SpectralCalc code. The moles per meter3 are proportional to ratio of absolute pressure to absolute temperature according to the perfect gas law. These values for each altitude are from the U. S. Standard Atmosphere data library of SpectralCalc. The emittance value at the midpoint of the altitude increment is assumed to be the average for the increment The radiance of the wavelength increment is determined from the Planck spectrum of the blackbody radiation at the temperature of its altitude. The emittance past the altitude of the “top of the atmosphere” and that into the surface of the earth from the source at that altitude are calculated by SpectralCalc. Example calculations are in Appendix B.
Figure 5 values are from plotting upward and downward emittance values at each wave length increment and plotting against the wavelength at the midpoint of the increment. Figure 6 values are the sum of values of emittance at the top of the atmosphere (modeled at 30 km) from all the wavelength increments producing such emittance from the midpoint of a given altitude increment. These sums are then plotted at the midpoint of the altitude increment to produce Figure 6. The values so plotted are assumed to be the average emittance per km in the increment.
Table 1 summarizes the steps leading to determining the effect of added carbon dioxide. The number listed with the absorber is the factor applied to the standard-atmosphere values to determine its concentration. These values are extremely low compared to prior published results with the exception of References 5 and 6 which are comparable.
All three values lie within 0.2 C with each other for doubling atmospheric CO2. NASA ‘s latest estimate (2) is roughly eight times the result shown here. The commonality between references 5 and 6 and this study is the use of detailed tracking of the radiation at specific paths through the atmosphere with altitude dependent parameters made possible with large computer codes such as SpectralCalc.
These calculations apparently show much more saturation by molecules in the atmosphere than those of previous ones. Also it shows that carbon dioxide whose absorption spectrum is dominated by two large peaks is subject to more saturation than the relatively smooth shape of water absorption with lower peak values.
The comparison of water and carbon dioxide acting alone (the first two columns of Table 1) shows that water’s individual absorption is more than three times as effective as that of carbon dioxide. But this ratio does not represent that of the pure cross sections because the code varies the concentration with altitude.
Figure 4 shows the deposition of greenhouse heat as a function of altitude and indicates the negligibility of greenhouse effect at high altitudes.
Figure 5 is a spectrum of the upwelling and downwelling radiation from the atmosphere as calculated with SpectralCalc. The upwelling radiation was calculated as 80 percent greater than downwelling. Because it does not impact the goal of this study it has not been rechecked or reviewed. It is similar to the measured spectra of downwelling radiation. It has been included to illustrate the versatility of SpectralCalc.
Figure 5. Spectra of Upwelling and Downwelling Radiation
Figure 6. Magnitude vs. Altitude for Sources of Atmospheric Radiation Cooling
Figure 6 shows where the radiation comes from that cools the atmosphere. It definitely does not come from near the “top of the atmosphere,” as alleged by both sides of the global warming debate. It should be not be a surprise. The larger source at low elevations more than compensate for greater transmission values at high elevations. Figure 6 adds to the evidence that greenhouse heating is a low altitude phenomenon.
Appendix A SpectralCalc™ Application for GHG Modeling
The best illustration of SpectralCalc’s capability in ghg calculations is with the results of its use. The following figures are from the computations of absorption in atmosphere for the earth’s IR with different combinations of water vapor and CO2 concentrations and within the wavelength ranges indicated at the bottom of the figures. Absorption is plotted, but the transmittance is listed on the lower right. Figures A.1-1, A.1-2 and A.1-3 are at standard-atmosphere concentrations of water and carbon dioxide in the wavelength range of 16-to-18 microns. Fractional areas above these curves are the mean transmittance for that micron range and fractional areas below are the mean absorption. A.1-1 is for carbon dioxide only and the fact that the spectral lines are truncated at a probability of 1 indicates some saturation. The top graph on this figure is a logarithmic plot of the absorption of all molecules in the atmosphere that absorb in this range. They are color
Figure A.1 Absorption in Atmosphere of only Std. Atm. CO2
Figure A.2 is for water vapor by itself.
Figure A.2 Absorption in Atmosphere of only Std. Atm. H2O
Figure A.3. Absorption of Std. H2O + Std. CO2 in Atmosphere
These three figures illustrate both saturation and its synergism. Note that carbon dioxide by itself is capable of absorbing over 55 percent of incident radiation in this range. (Absorption is 1 – transmittance, as listed on the lower right of the graph). But when combined with water vapor it can contribute only 5 percent. This is because water almost completely saturates absorption in this micron range. Beyond 22 microns there is no more carbon dioxide absorption and water completely saturates the absorption to beyond 100 microns and is the main reason that water completely dominates ghg absorption.
Table A.1-1 is a summary of the SpectralCalc results that are used in Table 1 of the main report. In the row of GHGs, c is for carbon dioxide and h is for water. The numbers associated with these letters are the multipliers for the standard atmosphere values of these gases. The entries are the products of the radiance values for the wavelength range as determined from the blackbody sub-program of SpectralCalc and the absorption probability. They were calculated by the atmosphere paths sub-program which produced the three figures above.
The last row of the table provides the input values for the absorptions listed in the first row of Table 1 of the main report. Note that twenty four wavelength increments were required to span the total range of the earth’s IR at 288 K. A similar table was constructed for the transmittances.
Table A.1-1 Summary of SpectralCalc Calculations
Appendix B. Calculation of IR Transport Within the Atmosphere
Table B.1 is a summary of one of 24 such calculations that provide the inputs for Figures 5 and 6. For Figure 6 the sums of the up and the down transmissions are each plotted against the midpoint of the associated wavelength increment produce the spectra shown. For Figure 7 the up transmission values for each altitude increment are divided by the depth of the increment to obtain transmission per kilometer and this is plotted at the midpoint of the altitude increment. The up transmission is to 30 km which is well into the stratosphere. Each of these plots starts with a source term at the emission point.
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The author is indebted to Dr. Ed Berry and Mr. Neil Brown for advice and encouragement and to Dr. Berry for publishing a draft on his website for review.
About the Author
Bryce W. Johnson is a retired Professional Engineer in the State of California and has earned the following degrees: BS/ME, University of Idaho; MS/NE, North Carolina State University and PhD/ME, Stanford University. His career spanned 47 years research in nuclear power and nuclear weapons. The last 32 years were as a research scientist for Science Applications International Corporation (SAIC).