The limits of carbon dioxide’s influence on world temperature

by Bryce Johnson


Proponents of anthropogenic global warming (AGW) hold that mankind causes the increase in atmospheric carbon dioxide (CO2) by burning fossil fuels and that without urgent measures to reduce its increase it will cause a runaway temperature increase within a few decades that will have unacceptably severe consequences to mankind. This article addresses the limits of the ability of fossil-fuel consumption to increase atmospheric CO2 level and on its associated temperature increase.  Primary findings are summarized in the following paragraph:

Consumption of all the world’s fossil fuels at any credible rate will not achieve a doubling of the current CO2 level and will increase world temperature by barely one-half degree Celsius. The achievable level of atmospheric CO2 is proportional to its release rate because its dilution by exchange with the land and ocean takes time.  An instantaneous release is the worst case and if the entire world’s CO2 from fossil fuels were so released (impossible, of course) it represents an extreme upper bound to the CO2 level. And even that would achieve an increase of less than 3 degrees Celsius. Most significant, however, is the strong evidence that feedback from increased CO2 is negative.

It is ironic that the only substance on which all life on earth depends should be so demonized and that analyses such as this one should be required to dispel the misconception of its great potential for harm.  It imparts a small contribution to overall greenhouse effect and greenhouse effect is a small contribution to the overall heat to the atmosphere. Also, its effect per unit CO2 addition decreases with each addition (as shown in Figure 6) so it is physically difficult to achieve any significant effect.

Limits on Possible CO2 Level

Figure 1 is a world carbon balance derived by WoodsHoleResearchCenter (1).  There are variations among researchers in this area over what the rates of carbon exchange and magnitude of carbon in the regions should be, but the variations are not large enough to invalidate the results from using the Figure 1 values. Other balances show the exchange between the shallow ocean and the deep ocean to be approximately the same as that between the shallow ocean and the atmosphere which means that the ocean can be modeled as a single unit containing 40,800 petagrams of carbon (a petagram is one billion metric tons).

In the figure the quantity in the atmosphere and its increase in carbon from use of fossil fuels are listed at 820 petagrams and 7.7 petagrams per year, respectively. In Figure 2 the rate of carbon insertion into the atmosphere between the years 1800 and 2007 are depicted and indicate a 2007 value of 9 petagrams of carbon, an increase over that indicated in Figure 1. Figure 3 shows the increase in CO2 in the atmosphere since1959 (in ppm).

 Figure 1 (from Reference 1)

The 820 petagrams is 0.0164 percent of the total weight of the atmosphere (2), which converts to 394 ppm CO2 closely matching the current Mauna Loa measurement of 393 ppm (Figure 3)  Figure 1 shows an atmospheric CO2 exchange rate with the land and ocean of 200 petagrams per year.  The time to replenish is the ratio of its content to the exchange rate, or 820/200 = 4.1 years.  This is the average retention time for a molecule of atmospheric CO2.

Four coupled simultaneous differential equations govern the time-dependent concentration in each of the four regions (atmosphere, land, ocean and fossil fuel reserves). These are displayed and solved by numerical means in Appendix A using the assumption that the rate of expulsion of CO2 from the atmosphere, land and ocean is proportional to the total concentration in each

The availability of data on carbon emission in Figure 2 (2) lags that of Figure 3 (ppm concentration) and the latest emission graph (petagrams emitted per year) is through 2007, but emission data through 2010 is now available. Carbon emission for 2010 was 10 petagrams (3) which conforms to the increasing insertion rate shown in Figure 2.

Figure 2. Atmospheric Carbon Insertion Rate by Year from 1800 to 2007 (2).

                                   (1 billions metric tons = 1 petagram)

Figure 3. Atmospheric Carbon Dioxide (source as indicated)

Figure 3 indicates the change in atmospheric CO2 concentration since 1959.


The Modtran computer code (4) developed by the Air Force and considered the standard for computing radiation transport in the atmosphere is used to solve the increase in IR energy absorption as a function of increase in CO2 concentration. Modtran is described in Appendix B.

This analysis assumes that Modtran uses sufficiently accurate calculation and atmospheric parameters to produce reliable results. The analysis includes feedback to the extent that results are compared with very different moisture conditions in the atmosphere, and the basis for claimed feedback is the additional atmospheric moisture caused by increased atmospheric CO2.

The elevated temperatures claimed by AGW proponents depend on positive feedback. Water vapor’s greenhouse effect frequently increases more in a day than CO2 can increase in decades.  Thus, if positive feedback were possible the world cataclysm that AGW proponents predict would have occurred centuries ago due to water alone.  And nature’s feedbacks are predominantly negative.  Both history and physics are aligned against the assumption of positive feedback.

Figure 4 shows the Modtran computed IR flux in watts per square meter across the atmosphere in both the upward and downward direction.  Upward flow stabilizes and downward flow disappears beyond about 30 km altitude because IR interactions are effectively zero due to the extremely low density of the atmosphere. Where the slope of the deposition is downward in the direction of the radiation, IR is being absorbed faster than it is being generated (by the heated molecules) and the opposite is true for an upward slope. 

The “thermals” and “latent heat” shown in Figure 5 entering from the earth’s surface are not IR.  That absorbed directly from sun into the atmosphere is radiation, but very little in the IR range. Its frequency degrades in the atmosphere to the IR energy range where it is absorbed as heat.  These non-IR components, 17, 80 and 78 watts/m2, respectively sum to 175 and must be added to the IR input value in order to derive a heat balance.

Figure 5. Energy Balance by IPCC (5)

Figure 5 values are similar to other heat balances, such as that by the weather service (6) and NASA (7).  But only the IPCC balance of Figure 5 indicates a net heat absorption into the earth, whose existence is highly controversial. The IR exiting the atmosphere to outer space is the only heat loss mechanism of the earth-atmosphere system.  Modtran is used to calculate input and output IR energy for all the other levels of CO2 of concern which are indicated in Table 1.

Figure 6 illustrates the baseline case of 400 ppm, the approximate current level of atmospheric CO2.

The heat rate to the atmosphere, H, for each CO2 level is determined with the following calculation:

H = IRin +175 – IRout                                                                     

Where H is the heat retained in the atmosphere at the particular CO2 level and 175 is the non-IR heat input, This equation forms the basis for calculating temperature rise. To maintain at least a short-term steady state the heat added must be removed and the only mechanism whereby the earth-atmosphere system can remove heat is by radiation to outer space, according to the Maxwell-Boltzmann equation for radiative heat transfer:

H =  caTa4 – coTo4

Where subscripts a and o refer to atmosphere and outer space, respectively;

H is heat rate into the atmosphere,

T is temperature in degrees Kelvin.

Ta is the temperature within the atmosphere which would match the aggregate of all radiative transfers to outer space when used in the above equation. In this analysis it is assumed to be the maximum (at the earth’s surface) because that produces the maximum (most conservative) value.

c is a constant (Maxwell-Boltmann constant times emissivity).

Outer space temperature is a factor of 100 lower than the characteristic atmospheric temperature, so that its 4th power is a factor of nearly a billion smaller and the second term on the RHS can be ignored without compromising accuracy. Subscripts b and g refer to conditions before and after the CO2 addition, respectively.  Dividing the Hg equation by that for Hb, the constant term cancels out and this simplified equation results:

Tg =  Ta*(Hg/Hb)1/4                                                (1)

Temperature rise due to added CO2 is the difference between Tg and Ta.


Figure 6 is a plot of the temperature rise in the atmosphere as a function of CO2 content and Table 1 summarizes the inputs and results of the computation.

A long period for burning the fossil fuel assures a relatively small maximum CO2 level because of the extra time available for mixing with land and ocean.  Under the scenario of maintaining the Figure-1 level of 10 petagrams per year to the atmosphere, the complete depletion of the fossil-fuel carbon occurs at 1000 years.

These scenarios depend on fossil fuels ending abruptly with no diminishing over time. Such scenarios are unrealistic but they ensure that maximum atmospheric CO2 content is not underestimated. The value of 0.5-percent-per-year increase of the fossil input rate to the atmosphere matches the current rate at which CO2 level is increasing in the atmosphere according to Figure 3.  A 1.5- percent-per-year value is the maximum increase indicated for carbon insertion of Figure 2.

These small temperature numbers of Table 1 are not likely to be underestimates.  It does not appear conceivable that either a half degree C temperature rise or a doubling of current CO2 level could be achieved.  Even if all that CO2 could be dumped in instantaneously, less than a 3 oC temperature rise would be seen. The maximum CO2 level and associated temperature rise is proportional to the rate of CO2 input.

In Figure 6, 400 ppm is assumed to be the current CO2 level in the atmosphere and all increases are from the temperature at that level.  Figure 6 shows the declining effect of increased CO2, but the most significant result is the dramatic decline in temperature rise caused by increasing atmospheric moisture. The Modtran code description, Appendix B, indicates that the middle curve best represents world average conditions.  Top-curve values are those in Table 1 because these are the most conservative.

Figures 7 through 10 show the time-dependent atmospheric CO2 content (as a ratio to the current content) and the resulting temperature rise above the current value as a result of the atmospheric carbon insertion rates indicated in Table 1.


The results show that increased carbon dioxide from the burning of fossil fuels causes an inconsequential impact on world temperature. The most significant conclusion, however, is that increased moisture, the presumed cause of feedback, causes a decrease in the warming effect of CO2 (negative feedback).


  1. “Understanding The Global Carbon Cycle,” Richard Houghton, WoodsHoleResearchCenter, 2007
  2. Fossil Fuel – Wikipedia,  Global Fossil Carbon Emission by Fuel Type from 1800 through 2007.
  3. CO2 Now.Org, CO2 Now | Home, Global Carbon Budget 2010
  4. Modtran Infrared Atmospheric Radiation Code,
  5. Trenberth, Kevin E., et al “Earth’s Global Energy Budget,” BAMS, March, 2009. This is by the Intergovernmental Panel on Climate Change (IPCC)
  6. http//
  7. budget


The author is indebted to Calvin M. Wolff for his in-depth critique of the draft of this document and for sharing his insight regarding the global warming issue.

The Author

Bryce Johnson is a retired professional nuclear engineer in the State of California.  His career spanned 45 years work in nuclear power and nuclear weapons research.  His education includes:  BS(ME), University of Idaho, MS(NE) North CarolinaStateUniversity and PhD(ME), StanfordUniversity.

Appendix A.  Numerical Integration

Four coupled simultaneous differential equations for the time-dependent concentration in each of the four regions of Figure 1 (atmosphere, land, ocean and fossil fuels—coal-oil-gas) are written as follows:

A’ (t) = fa(t) + la(t) + ca(t) –al(t) –ac(t)

L’(t) = la(t) +al(t)

C’(t) = ac(t) –ca(t)

F’(t) = -fa(t)

Where ‘ indicates differentiation with respect to time, t.  As indicated, all terms are functions of time, t.  A, L, C, and F are the CO2 quantities in the atmosphere, land, ocean and fossil fuels, respectively.  The time dependent transfer from atmosphere to ocean is ac; ca from ocean to atmosphere, etc.

The goal of these solutions is to determine the maximum CO2 concentration in the atmosphere for CO2 addition scenarios indicated in Table 1.  No guidance could be found in the literature on the transfer rates or concentration as functions of time. For this analysis it is assumed that the transfer rate out of a region is proportional to its concentration. (The only justification for this assumption is in its logic).

Therefore la(0)*L(t) = 100, and L(0) is 2000, so la = 0.05L(t).  Similarly al = ac = 0.122*A(t) and ca = 0.002*c(T).

All these fractional rates are per year. The first 10 years of the excel integration of the case of a 0.005 fractional increase in the fossil rate is listed below and the associated entries for each year n+1 in terms of the entries for year n are as follows:  For year 0, the current values of the carbon content are listed.

An+1 = An(1- 0.244)  + 0.05Ln  + 0.002Cn  + 10(1- 0.005n)

Ln+1   = Ln(1-0.05)  + 0.122An

Cn+1  = Cc(1-0.002) + 0.122An

Fn+1    =Fn  – 0.005*10(n+1)







The integration continues beyond when the value of F becomes 0 to determine the rate of temperature decline thereafter.  Maximum value of A occurs when F becomes zero. This time depends, of course, on the assumption of the rate of fossil fuel consumption.


Appendix B.  (Provided by Calvin M. Wolff)


MODTRAN (MODerate resolution atmospheric TRANsmission) is a computer program designed to model atmospheric propagation of electromagnetic radiation for the 100-50,000 cm-1 (0.2 to 100 um) spectral range.

The most recently released version of the code, MODTRAN5, provides a spectral resolution of 0.2 cm-1 using its 0.1 cm-1 band model algorithm.

Some aspects of MODTRAN are patented by Spectral Sciences Inc. and the US Air Force, who have shared development responsibility for the code and related radiation transfer science collaboratively since 1987. The acronym MODTRAN was registered as a trademark of the US Government, represented by the US Air Force, in 2008.

All MODTRAN code development and maintenance is currently performed by Spectral Sciences while the Air Force handles code validation and verification. Software sublicenses are issued by Spectral Sciences Inc., while single-user licenses are administered through Spectral Sciences’ distributor, Ontar Corporation.

MODTRAN5 is written entirely in FORTRAN. It is operated using a formatted input file. Third parties, including Ontar, have developed graphical user interfaces to MODTRAN in order to facilitate user interaction and ease of use.

MODTRAN is accessible to the public at

When you access the url above, a menu will appear, as follows:

Modtran IR in the Atmosphere

Where Iout is the infrared heat radiated outward from the earth at 70 km altitude.

On the graph, the smooth lines represent perfect blackbody radiation at the temperatures cited in the legend on the graph.  The red, jagged line is the earth’s actual infrared emission outward at 70 km altitude.  The horizontal axis is in units of wavenumber, proportional to frequency and inversely proportional to wavelength. To convert wavenumbers to wavelength in microns, simply divide the wavenumber value into 10,000; i.e., 10,000 wavenumbers corresponds to a wavelength of 1 micron. The visible spectrum is from 0.8 to 0.4 microns.

Please note that the result is for the tropical latitudes, no clouds or rain, with the instrument or observer looking down to the earth.

To run simulations for the average earth, set “Locality” to “1976 U. S. Standard Atmosphere” and change “No clouds or rain” to “NOAA Cirrus Model (LOWTRAN 6 Model)”.

When you simulate at these conditions, you will see that the ground temperature changes from 299.7K to 288.2K, corresponding to the 15C that is usually taken as earth’s average surface temperature. The radiation emitted from earth, Io is 242.782 w/m^2.

To compare the heat loss from earth at various CO2 levels, use the 1976… and NOAA…. settings, leave all the rest the same, and set the CO2 ppm to 390, which is closer to the current amount.  Record Io (watts/m^2) for that simulation. Then change increase the CO2 amount to whatever you choose. Doubling atmospheric CO2 would be 2 x 390 ppm = 780 ppm. When you double CO2 (780 ppm), you will see that the new Io (heat lost to atmosphere) drops to  240.336.  Therefore, using Modtran, the heat loss from the earth by doubling CO2 is 242.782 – 240.336 = 2.446 w/m^2, which is the greenhouse effect of doubling CO2 (an estimate).

To study (estimate) the effects of changes in other atmospheric constituents (CH4, Ozone & water vapor) at any given, constant CO2 ppm, do as follows:  multiply the default quantities (17 ppm CH4, 28 ppb O3, water vapor scale) by the 1 + the amount you want to change them. If you want a 20% increase, multiply by 1.2.

Modtran gives a good, but not the best estimate of radiative heat loss. Other programs, like SpectralCalcTM  should give more accurate estimates. Remembering we are averaging over the entire earth over an entire year.

5 thoughts on “The limits of carbon dioxide’s influence on world temperature”

  1. While the paper has significance in terms of the release of CO2 and its effect on CO2 concentration in the atmosphere , there are some missing bits of information.
    Where is the creditable experiment showing that the "greenhouse gas effect exists"?
    It is also important to keep our terms straight= Where is the proof that the "Greenhouse effect" can occur in the atmosphere? Professor Robert W. Wood in 1909 showed that the "Greenhouse effect" is confined space heating. How can there be "confined space heating " in the atmosphere? Dr Nasif Nahle has shown that the work of Professor Wood is valid( available on this web-site) Also see the work of Slaying the Sky Dragon – Death of the Greenhouse Gas Theory by
    Tim Ball (Author), Claes Johnson (Author), Martin Hertzberg (Author), Joseph A. Olson (Author), Alan Siddons (Author), Charles Anderson (Author), Hans Schreuder (Author), John O'Sullivan (Author)

    At G3 of this web-site section 10 is an experiment that demonstrates that the “greenhouse gas effect” does not occur. Shortly there will be a revised draft of this experiment – no new data just more readable. Anyone willing to do the experiment can get their own balloons from look for clear Deco Bubble Mylar balloons- 24” are available. Do the experiment – doing science experiments is fun. Other sources for the balloons and other materials can be found on the internet.

  2. Very interesting work! It'll take me a while to digest all of this, but, on first scan it looks like a good piece of work…. thanks!

  3. To comment 1:
    My only goals of the paper were to use "standard" tools of the trade to establish reasonable limits on the content and temperature increase of CO2. The different temperature levels achieved by different CO2 contents as shown by the Modtran calculations, I feel, does establish that greenhouse effects do exist.

    Bryce Johnson

  4. @1 Berthold,
    The global warming problem is played on more than one court. You are playing on Court A where the players are challenging the whole idea of a greenhouse effect. This is an interesting and important area in its own.

    Bryce is playing on Court B where the IPCC boys are engaged. He is using their own basic hypothesis that models like MODTRAN describe the heat transfer in the atmosphere. His paper above shows their calculations are wrong.

    Playing in Court B is a more direct way to show the IPCC is wrong than to play in Court A where the IPCC will not even acknowledge the argument is valid. I appreciate your arguments in Court A but they should not be used to conclude the debate in Court B is invalid or not needed.

  5. Dr.Ed,
    Very interesting calculations. Pretty much what one would expect. If you say increased CO2 will increase water vapor in the atmosphere, then one would assume you would get cloud formation until the water precipitates. A recent paper pointed out that water vapor in the atmosphere has not increased as the IPCC said it would. More clouds would equal cooler so any feedback would be negative. Logically speaking.

    I just did a quick check of correlation between temperatures from UAH since 2002 against CO2 measurements from Mauna Loa. The correlation Coefficient came out to +0.03. That is about as close to zero as data can get. No correlation at all.

    Why does the IPCC have any credibility at all?

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