by Bryce Johnson (Revised, January 31, 2011)
The SpectralCalc™ Code
In a recent publication (1) Richard J. Petschauer utilized the SpectralCalc™ computer code (2) to demonstrate that carbon dioxide is but a “bit player” in earth warming by greenhouse gases (ghg’s). The code is extremely detailed and sophisticated, containing all the needed atmospheric and IR absorption parameters needed for an accurate description of “greenhouse” heating throughout the entire atmosphere. Figure 1 is an example of the detail available.
The colored plot at the top with logarithmic values indicates where in that particular range the atmospheric ghg components are active. The colors of the lines correspond to the color of the printed name of the molecule on the right. Absorption is plotted, but the value printed out on the bottom right is for transmittance (1 – absorption). The detail of the code limits it to a few microns of range and 20 such increments were utilized to cover the region where infrared absorption occurs.
A verification of Petschauer’s results using a different approach is reported in this document. The study is limited to the effect of carbon dioxide on heating of the earth.
The power and versatility of the SpectralCalc code offers insights into the subject of greenhouse heating that enhance the general knowledge of the subject. The major findings resulting from the application of SpectralCalc are as follows.
Petschauer’s results of the insignificance of CO2 in global warming are verified and reinforced (results here indicate that the effect of CO2 is even less than that predicted by Petschauer).
Most greenhouse activity occurs in the troposphere. Almost nothing happens at higher altitudes.
The code permits an altitude distribution of the deposition of ghg heat
Numerically, the total concentration of the CO2 in the atmosphere can be increased by a factor of 8 without exceeding a 1-degree C rise in temperature. At the past century’s rate of increased atmospheric CO2, this will require 2000 years.
The CO2 contribution to earth warming is low enough to call into question the entire issue of feedback.
The calculations are based only on water and carbon dioxide. The other greenhouse gases in the atmosphere are not sufficient contributors to warrant the effort to include them. Water plus carbon dioxide then are considered to be the total greenhouse contribution. The greenhouse contribution to total atmospheric heat is assumed to be ten percent. This is derived from the values indicated in Figures 2, 3 and 4 which are heat balances from the IPCC, the National Weather Service and from NASA. Figures 2 and 3 are for the earth-atmosphere and 4 is for the atmosphere, only. The results are 0.096, 0.10 and 0.085, respectively. Ten percent is then an appropriately conservative value.
The only contribution of greenhouse heat to the atmosphere considered is that generated at the earth’s surface due to its average temperature of 288K (396 watts/m2 by Figure 2 and 390 watts/m2 by SpectralCalc). The infrared fraction of the solar energy impinging on the earth is considered negligible compared to this value. The 355 watts/m2 back radiation value shown in Figure 2 returns some of its heat to the atmosphere by augmenting the temperature of the earth’s surface. There is some controversy about its contribution to global warming (5) but this analysis determines earth radiation by its known temperature eliminating the need for its consideration.
The internally generated IR radiation of the atmosphere, of which back radiation is a part, is a product of its own heat and no substance can spontaneously increase its own temperature.
The analysis is dependent on two ratios: that of CO2 ghg effect to H2O effect and that of greenhouse-generated heat to total atmospheric heat. The ratio calculation is much less prone to error than calculation of absolute heat rates. By varying these ratios valuable temperature insights are afforded. Feedback is not addressed in this analysis and it is not considered an appropriate concern.
In Figure 5 below are shown the angular increments used for numerical integration. Four sectors are used between 0 and 90-degrees and these are rotated around the vertical to give the total solid angle for that sector. Two 30-degree sectors and two 15-degree sectors are used as shown. Near 90 degrees the angle-dependent absorption increases rapidly due both to the longer path required to reach the 20km altitude (the maximum altitude of the integration) and the increased density at paths nearer the earth which is why only a 15-degree increment is used there. Rotating the system 360 degrees around the vertical of the unit hemisphere generates the total solid angle associated with the increment dq about q, which is
sin q * 2p*dq. (1)
The total flow through it from an isotropic unit source on the surface in the q direction is
cos q*sin q*2p*dq (2)
Where cos q projects the flat surface of the earth onto an area perpendicular to q direction.
Equation 2 is exactly integrable to p*sin2 q. Evaluation of this expression over the four increments of solid angle produces these values:
0-30 degrees: 3.141593
30-60 degrees: 2.299806
60-75 degrees: 0.627693
75-90 degrees: 0.214084
These values can be described as the total solid angle subtended by the increment times the average cosine over the associated angle. To obtain total greenhouse gas absorption in that solid angle this value is multiplied times the fraction of IR energy absorbed (as determined by SpectralCalc) in the angular increment times the earth’s IR emission per unit steradian in the IR wavelength interval used. The average of this value in the angular increment is determined by the average of the values on each of its boundaries.
The sum over the angular increments is the total for that wave-length interval and the sum of all the wave length intervals is the earth total for that particular combination of water and carbon-dioxide percentages. So 100 separate SpectralCalc determinations are required for each C02-H2O combination. Eight such combinations were used. Table 1 is a summary of the calculations required for absorption in the13-15 micron increment of earth radiation with 0.0125 and 0.004 as the volume fractions for CO2 and H2O, respectively. The maximum-angle that can be calculated by SpectralCalc without earth interference is 84.9 degrees and it is substituted for the 90-degree angle in Table 1.
Table 1 Sample Calculation for Absorbed Greenhouse Energy
Figure 6 is a plot of the SpectralCalc calculation that produced the transmission probability for the 60-degree angle in Table 1.Figure 7 is the plot of the earth’s blackbody spectrum between 13 and 15 microns. The total emitted energy for this plot is 12.8508 watts/m2/sr
Converting Atmospheric Heat Addition to Temperature Increase
By the first law of thermodynamics, any body, or substance, at steady temperature has to lose as much heat as it gains. Therefore, we simply equate the heat input of the atmosphere to its heat loss and utilize the only mechanism available to the atmosphere for heat loss, and that is by radiation to outer space. The atmosphere has only two boundaries, the earth and outer space, and it cannot lose heat to the earth because the earth’s surface is warmer than the entire atmosphere. The only mechanism for transfer to outer space is by radiation, which by the Stefan Boltzmann Law is proportional to T4, with T in absolute degrees, as in degrees Kelvin. Since some radiation from the earth’s surface goes directly to outer space, we can assume radiation at all levels in the atmosphere with their associated temperatures also transmits directly to outer space. It is impractical to try to determine all of these and it is totally unnecessary. It can
be assumed that some temperature between the earth’s surface temperature and that at the atmosphere’s outer limits is an effective temperature for calculating its heat rejection, and that the outer space temperature is 0 degrees Kelvin (it’s actually about 3 degrees, but the
Figure 7. Blackbody Earth Radiation Emittance Between 13 and 15 Microns
added convenience is well worth the slight inaccuracy. With these assumptions the relation between temperature and heat addition is:
Th/T = (1 + Ha)0.25 …………………………… (3)
Where Th is the new temperature after the heat Ha is added to the atmosphere whose average or effective temperature was T before the addition. Using the highest value of atmospheric temperature for T, 288 K at the earth’s surface, produces the highest (most conservative) temperature increase. Some have suggested that the temperature at the “top of the atmosphere,” 255 K, is appropriate (6). CO2 contribution is low enough that the difference in this choice is insignificant.
Temperature effect of CO2 addition
Table 2 shows the results of this calculation and compares them with those of Petschauer (1)
Table 2. Variation of Heat and Temperature Increase due to Atmosphere with C02
This study values are roughly half those of Petschauer. Petschauer included back radiation as a source of heat to the atmosphere(1) and this is sufficient to explain the difference. As explained under ‘Assumptions’ above, back radiation is a product of heat in the atmosphere and, therefore, cannot be a source of it. But both these results and those of Petschauer predict a much lower temperature rise than has been previously reported.
Using half the water density (but still considering greenhouse heating to be ten percent of total atmospheric heating) increases the temperature effect of added CO2, which is to be expected. However, the increase is small. This would indicate that CO2 saturation does not depend on overlapping water absorption at its absorption bands in the IR spectrum to push it to its limit.
SpectralCalc code predicts IR energy emissions and subsequent atmospheric energy absorption all the way out to 200 microns. But most analysts consider the limit of IR wavelengths to be 70 microns. Using this cutoff instead of 200 microns would increase the calculated effect of CO2 because only water absorbs out in the long wave lengths. If for the 0.0125 percent water vapor case, 70 microns had been the cutoff, doubling CO2 would have increased the temperature rise from 0.156 degrees C to 0.181 degrees C; quadrupling would have increased the temperature rise from 0.36 to 0.43 C and octupling would have changed it from 0.65 these 0.72. These are noticeable but not very significant.
Two additional studies were performed with the code. These used a single angle to represent the total angles shown in Figure 5 (60 degrees) to map the altitude deposition of IR energy emerging from the earth and to estimate the exchange of IR radiation between the altitude locations of the earth. To have used the same angular integration indicated in Figure 5 would have been untenable because of the vast number of calculations with SpectralCalc that would have been required. The results are of academic interest only since they don’t impact greenhouse warming and they are approximations. Table 3 illustrates the fate of the earth’s IR upon interaction with the atmosphere.
“Back radiation” is defined as that radiation which returns to the earth as a result of IR capture in the incident IR radiation to the atmosphere, mainly from the earth. But since such radiation is isotropic it sends radiation out to space also. This analysis accounts for both directions. Table 3 illustrates the calculation for the 13-15mm wave-length range.
Table 3. Calculating Secondary Radiation From Earth IR
The fourth column of Table 3 shows the altitude distribution of absorption from Earth IR. That plotted in Figure 8 is for the sum of all four increments summarized in Table 4.
Table 4. Secondary Radiation for Wave-Length Increments
The secondary radiation was also computed from the IR of the ranges of 20-22, 13-1510-12 and 4.4-5.4mm. The first interaction from the earth produces a fraction, f, of the incident energy absorbed, the second produces ff, the third fff, etc, and the sum of all such successive encounters is
S f n from n = 1 to infinity. (3)
As long as f < 1, the series sums to the unique finite value of f/(1-f), which is the source of the values for the fourth column of Table 4. The reason the absorption in the up direction is greater than that in the down direction is that most primary absorption occurs at very low altitudes (Figure 8) and the down directed radiation therefrom soon encounters an abrupt end to its source of air molecules. Using the same f value for successive interactions is approximate because the distribution of successive radiant energies emitted cannot duplicate those of the previous ones. But the f values are small enough that the infinite series can be reasonably approximated by the first term, so the error in this approximation is tolerable.
The purpose of this exercise is to demonstrate how quickly the absorbed IR radiation is converted to heat. For the wave-length increments calculated, over 90 percent of the energy is converted on the first interaction. And this first interaction from the earth’s IR is the only one contributing to greenhouse heat of the atmosphere because the subsequent ones are from that fraction of the earth’s IR that has already heated the atmosphere.
Figure 8. Altitude Deposition of Earth’s IR.
Additional calculation with SpectralCalc determined IR exchange between different locations in the atmosphere. These are of academic interest, only, since this exchange alters only the spatial distribution of atmospheric heat, it does not impact its overall increase or decrease. The main difference noted between that exchange and IR radiation from the earth was the decreased saturation effect within the atmosphere. This is because the exchange internal to the atmosphere involves receipt of a large fraction of IR impingement from nearby molecules where the IR radiation has not traversed sufficient moles to be significantly impacted by saturation. This is true for earth IR for only those molecules close to the earth.
Significance of Results
The adequacy and accuracy of these results depends entirely on the SpectralCalc™ code (2). The author has neither access to the inner workings of the code nor the technical background to be able to assess its validity and accuracy. It has been accessible to the public for a number of years. So it has had ample user interface and opportunity for “bug” elimination. The Internet providers of the code have been supplied with an early draft of this document with the request for a review for proper use of their code. No response has been received as yet. They are also being provided with this final version.
The code instructions for use include a caveat that it should not be used to prove or disprove global warming. That is not the use that has been made in this document. This document is limited to determining the maximum role that carbon dioxide can play in global warming. The code documentation would indicate that it is adequate for this purpose.
The study results depart substantially from previous predictions of the effect of carbon dioxide. They are very much lower, and only about half of Petschauer’s predictions (1) which were, themselves, significantly lower than previous predictions. These results are sufficiently low that, if they are accepted, all justification for continued effort at carbon dioxide reduction is eliminated.
At less than 0.2 degrees C for doubling today’s content, they are surely lower than even the daily variation in the effect of water vapor and likely well below the accuracy with which atmospheric temperature can determined. At today’s rates, 2000 years of increase will still leave the temperature increase less than one degree centigrade. Today’s research and argument regarding climate change center on feedback mechanisms and whether they are positive or negative. But with such a low trigger temperature, how can any feedback be generated, either positive or negative?
- Petschauer, Richard J., “Carbon Heat Trapping: Merely a Bit Player in Global Warming,” 2008. (posted on Climate Clash)
- SpectralCal.com of GATS, inc.
- Trenberth, Kevin E. et al “Earth’s Global Energy Budget,” BAMS, March 2009
- IPCC Third Assessment Report, Climate Change 2001
- Roy H. Spencer, PhD, “Help! Back Radiation has Invaded my Backyard!” Home/Blog, August, 2010
- The Climate System, EESC, Spring, 2007
Availability of Calculations
All calculations with the results of SpectralCalc, were performed and recorded with Microsoft Excel Spreadsheets. These are too voluminous to include with the report, but can be made available by email to anyone requesting them from firstname.lastname@example.org.
About the Author
Bryce W. Johnson is a native of Idaho with a BS Degree in Mechanical Engineering from the University of Idaho, an MS Degree in Nuclear Engineering from North Carolina State University and a PhD in Mechanical/Nuclear Engineering from Stanford University. He is a registered professional nuclear engineer (retired) in the State of California. Following two years in the Air Force his active career spanned 48 years of research in nuclear technology, the last 32 of which were as a Senior Staff Scientist at Science Applications International Corporation. He is a lifetime member of the American Nuclear Society. He has been actively investigating climate change for over two years as a private citizen.