calculation of atmospheric radiative forcing

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CALCULATION OF ATMOSPHERIC RADIATIVE FORCING (WARMING EFFECT) OF CARBON DIOXIDE AT ANY CONCENTRATION 1

H. Douglas Lightfoot1 and Orval A. Mamer2

Quebec’s Global Environmental and Climate Change Centre, Montreal, Canada Corresponding author: 8 Watterson, Baie D’Urfe, QC, H9X 3C2, Canada Email: [email protected] 2 The Goodman Cancer Research Centre, McGill University, 1160 Pine Ave. West, Montreal QC, Canada, H3A 1A3, Email: [email protected]

ABSTRACT The Beer-Lambert law does not apply strictly to the relationship between radiative forcing (RF) of CO2 and concentration in the atmosphere, i.e., ∆RF = 5.35ln(C/Co). It is an approximation because water vapour competes unevenly with CO2 over the IR absorption wavelength range. We propose a quadratic model as an improved approximation. It links concentration to RF thereby allowing RF calculation at any concentration, not just ∆RF. For example, at 378 ppmv of CO2, the level in 2005, it calculates RF = 8.67 W m-2, or approximately 2.7% of the total RF of all the greenhouse gases. A second and independent method based on worldwide hourly measurements of atmospheric temperature and relative humidity confirms this percentage. Each method shows that, on average, water vapour contributes approximately 96% of current greenhouse gas warming. Thus, the factors controlling the amount of water vapour in the air also control the earth’s temperature. Key words: radiative, forcing, logarithmic, quadratic, model, water vapour

1. INTRODUCTION The most important question relating to the contribution of the greenhouse gas (GHG) carbon dioxide (CO2) to global warming is “What is the warming effect, i.e., radiative forcing (RF) in W m-2, of CO2 at current levels, such as at 378 ppmv as in 2005? Why is this important? Because comparing this value with the back radiation of 324 W m-2 [1] of the total of all greenhouse gases tells us whether CO2 is a large or small cause of atmospheric warming over the 20th century. Because, as we show in section 2, there is considerable uncertainly about the RF of CO2 at current levels, the purpose of this paper is to derive a method of estimating the RF of CO2 at any level. In 1896, Arrhenius identified CO2 as a greenhouse gas and postulated the

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Energy & Environment · Vol. 25, No. 8, 2014

relationship between concentration and warming effect (radiative forcing, RF) was logarithmic. On this basis, the IPCC developed a simplified logarithmic equation, ∆RF = 5.35ln(C/Co), in the TAR for the range of 275 to 378 ppmv [2] based on the BeerLambert law. The Beer-Lambert law was derived for absorption of light by liquids. There are two parts to the Beer-Lambert law. The Beer law states absorption is proportional to the concentration of light absorbing material and the Lambert law states that absorption is proportional to the length of the light path. Much is known about the Beer-Lambert law and the absorption of light by liquids having a low concentration of an absorbing substance. The law says each layer of the same thickness of liquid absorbs the same percentage of the initial light. Thus, the absorbance of light by liquids is expressed as: A = - log10 (I/I0) When gases are involved in place of liquids, the relationship is expressed as: Á = ln (I/I0) Note: the conventions of logarithms to “base 10” and “base e” give different results for the same input data. For accuracy, the Beer-Lambert law depends upon pure and dilute solutions and monochromatic light. These conditions are not available for absorption of IR by CO2 as shown in Figure 1 [3]. Below approximately 5 µm wavelength there is negligible energy available [4]. Thus, essentially all of the available energy is absorbed in the range from 5µm to 30 µm. At various wavelengths over this range, water vapour competes with CO2 for absorption of IR as shown by the darker areas in Figure 1. The mix of absorption bands and the competition between CO2 and water vapour does not act like the single absorption band required for the correct application of the BeerLambert law.

Figure 1. Absorption of IR by CO2 is not uniform and, therefore, not logarithmic because of competition from water vapour over the absorbance wavelength range for IR from 5 to 30µm.

Calculation of atmospheric radiative forcing (warming effect) of carbon dioxide at any concentration

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The authors of the TAR, section 6.3.5, recognized the Beer-Lambert law as being only an approximation and suggested two other more complicated logarithmic approximations. Even though there is no theoretical basis for the Beer-Lambert formula, ∆RF = αln(C/Co), it has been accepted by the scientific community as a reasonable approximation. In this paper we propose an improved mathematical approximation that, like the Beer-Lambert law, has no theoretical basis. Nevertheless, it gives exactly the same ∆RF over the same range of 275 to 378 ppmv and represents the actual physical observations outside of the range. Importantly, the improved mathematical approximation allows calculation of RF at any concentration of CO2 and not just ∆RF. We accept the IPCC statement that there is a measurable increase in atmospheric temperature since 1850 of approximately 0.8 oC as given in the Intergovernmental Panel on Climate Change (IPCC) report Climate Change 2007: The Physical Science Basis (AR4) [1] that CO2 is a greenhouse gas and that its concentration in the atmosphere has increased since 1750. Part of this increase is probably due to human activity, and that the increase in CO2 concentration caused an increase in atmospheric temperature since 1750. For reference, note that the IPCC estimated the change in warming effect over the range of 275 to 378 ppmv as ∆RF = 5.22ln(C/Co) = 5.22ln(378/275) = 1.66 W m-2. The constant 5.22 is a slight change from 5.35 in the TAR [2], which gives ∆RF = 1.70 W m-2. The difference in constants makes little difference to our results. In this paper, parts per million by volume is “ppmv” and parts per million by mass is “ppm”. 2. UNCERTAINTY AROUND RF OF CO2 At present, there is no reliable method for determining the RF of CO2 at current levels. Even though there are peer reviewed papers that suggest such values, a search of AR4 shows none have been recognized or accepted by the IPCC. Work by Keihl & Trenberth [5] and Pearson & Palmer [6] suggest values of 32 and 218 Wm-2 respectively for RF while Schmidt et al. [7], publishing after the release of AR4, show several sources of data that convert to 22 to 43 W m-2. The authors of the latter underline the uncertainty by concluding with: “Given that the attribution (of CO2) is closer to 20% (31 W m-2) than 2% (3.1 W m-2), it might make more intuitive sense that changes in CO2 could be important for climate change” (text in parentheses added). Thus, there is considerable uncertainty in the RF of CO2 at current levels. It is evident that more accurate and precise numerical estimates are needed. 3. THE LOGARITHMIC MODEL FROM 275 TO 378 PPMV AR4 proposes a logarithmic model ∆RF = 5.22ln(C/Co) = 1.66 ± 10% W m-2 that is said to work within the CO2 concentration range of 275 to 378 ppmv. This model provides a unique curve between the two points that can only change if the constant 5.22 were to change. It is widely recognized that this relationship provides only ∆RF and its validity outside the range 275 to 378 ppmv is not established. The logarithmic model plotted in Figure 2 does not provide for absolute values of

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Figure 2. The generally accepted approximation for the relationship between ∆RF and CO2 concentration over an undefined range, which in this case is 275 to 378 ppmv. RF on the Y axis, and the actual RF at 378 ppmv, for example, is indeterminate. It does have wide support in the scientific community as an approximation because it is well defined and substantiated in TAR and AR4. It is, therefore, a firm starting point for considering mathematical models that may calibrate the Y-axis, which is the purpose of this paper. Plotted points within the 275 to 378 ppmv range are calculated with the equation proposed in TAR and added to define the curve over this range. 4. COMPARISON OF THE QUADRATIC AND LOGARITHMIC MODELS Figure 3 is the same as Figure 2 except the logarithmic model points are surrounded by the squares representing the points of the following quadratic model: RF = -0.00002480C2 + 0.03231C

(1)

This quadratic model was selected such that each logarithmic dot is exactly in the centre of each square over the range of 275 to 378 ppmv. There is no constant term because the quadratic was selected to start at zero concentration and, therefore, the constant is zero. Note that it is impossible to distinguish between the quadratic and the logarithmic models within the range of 275 to 378 ppmv. Each model provides the same ∆RF. The difference can only be determined by examining the end conditions as in Figure 4.


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Figure 3. A plot of the quadratic model (squares) accurately follows the logarithmic model points (dots) over the concentration range of 275 to 378 ppmv

Figure 4. The quadratic model starts at zero because RF = zero at zero CO2 concentration and provides a link to the RF scale. This link allows calculation of the actual RF rather than ∆RF between two concentrations.

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5. WHY THE QUADRATIC STARTS AT RF = ZERO PPMV In Figure 4, the quadratic is shown to start at zero. It is axiomatic that the RF of CO2 is zero at zero concentration. This is similar to the Beer-Lambert law where absorbance is zero at zero concentration. The zero is a defining point. It links the quadratic to the Y axis and RF. Thus, at 378 ppmv, RF = 8.67 W m-2. At some point above this, CO2 has absorbed all of the IR that it can absorb and it is impossible for additional CO2 to increase RF further. This is the concentration where the model plateaus. Because the quadratic equation is known, this concentration can be calculated exactly. Maximum RF = 10.5 W m-2 and occurs at 654.6 ppmv. Below 275 ppmv, the logarithmic model curves more than the quadratic model and crosses the concentration axis at approximately 70 ppmv. Above 378 ppmv, the curvature of the logarithmic model is less than that of the quadratic model, and continues to infinite RF. In both directions outside the range 275 to 378 ppmv, the quadratic is a better fit to what is known about the physical science than is the logarithmic model. 6. ARE THERE OTHER MODELS THAT ARE GOOD APPROXIMATIONS? It is possible there are other mathematical models that will start at zero concentration and plateau while passing through 275 and 378 ppmv and having the same curvature as the logarithmic model over the range. A case in point is that of a sine model of the form: RF = α(sine(Θ))

(2)

RF = 9.502(sine((C/654.6) x 90))

(3)

Specifically:

In Figure 5, this model starts at zero, coincides with the logarithmic curve over the range of 275 to 378 ppmv and plateaus at 654.6 ppmv. At 378 ppmv, RF = 7.48 W m-2, which is close to RF = 8.67 W m-2 of the quadratic. It is likely that any model that meets these restrictive conditions will give RF close to that of the quadratic model, and well within the usefulness of the approximation. We do not use the sine model because CO2 concentration has to be converted to equivalent degrees. The quadratic model is easier to use because the concentration values can be used directly. 7. INSTRUCTIVE PLOTS OF RADIATIVE EFFICIENCY (RE) In section 2.10.2 of AR4, the IPCC uses the term “radiative efficiency (RE)” to refer to the slope of the logarithmic model at a specific concentration of CO2 in the atmosphere. The units for RE and slope are W m-2 ppmv-1 (Watts per square metre per part per million by volume of CO2). For reference, the authors of Section 2.10.2 in AR4 used the formula ∆RF = 5.35ln(C/Co). In this paper we use the constant 5.22 as in ∆RF = 5.22ln(C/Co) because it gives ∆RF = 1.66 W m-2 as given in Figure SPM.2 of AR4.


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Figure 5. There are other models that also start at zero and link to the RF axis, such as this sine model. The RF of 7.48 W m-2 is close to 8.67 W m-2 of the quadratic because the conditions to be satisfied are the same for each model. By calculus, the slope, or RE, of a curve at any concentration point is given by the first derivative of the equation of the curve. For the logarithmic model, the first derivative is 5.22/C. This is the upper curve in Figure 6 and is asymptotic to both X and Y axes and intersects neither. A serious problem with the logarithmic model is there are no defined limits indicating where the model is valid and where it is not. The first derivative of the quadratic model, eqn (1), is the straight line: RE = -0.000496C + 0.03231

(4)

In Figure 6, the line of eqn (4) intersects the Y-axis at RE = 0.03231 W m-2 ppmv-1 where CO2 concentration equals zero. Where RE = 0, the derivative line intersects the X-axis at CO2 concentration of 654.6 ppmv. The intersections on both the X and Y axes are mathematically defined limits. In the range 275 to 378 ppmv, these two RE curves have the same slope. It is impossible to distinguish between them over the range 275 to 378 ppmv. Below this range, the quadratic model can be distinguished from the logarithmic model because it intersects the Y axis at zero concentration. Above the range, the quadratic model intersects the X axis at 654.6 ppmv. For interest, the RE plot of the sine model, RE = 0.02259cosine((C/654.6)*90)), is included in Figure 6 to show that it intersects both the X and Y axes. Above approximately 300 ppmv the sine model is essentially parallel to the quadratic model. All three models have the same slope over the range 275 to 378 ppmv.

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Figure 6. Plots of RE, which is the slope of the model line at a given point, show the logarithmic model has no defined limits. The quadratic and sine models have mathematically defined limits where the curves meet the vertical and horizontal axes. Note that all curves are parallel over the range of 275 to 378 ppmv and that above approximately 300 ppmv, the quadratic and sine models are parallel. 8. TOTAL BACK RADIATION OF ALL GHG Figure 7 is FAQ 1.1 Figure 1 from page 96 of AR4. It shows the radiation balance for the earth and that the back radiation of all of the greenhouse gases is 324 W m-2. This is the value used to calculate the RF of CO2 at 378 ppmv as (8.67/324)/100 = 2.7% back radiation of the total of all of the greenhouse gases. 9. OTHER GHGS, EXCEPT WATER VAPOUR AND CO2 All of the greenhouse gases, with the exception of water vapour, are well above their boiling points and act as ideal gases. Therefore, for the purposes of this study, methane and nitrous oxide obey the logarithmic model over the given ranges. Quadratic models of each are developed from the logarithmic models as for CO2. The general term for the logarithmic model is: ∆RF = αln(C/Co), and α = ∆RF/(ln(C/Co))

(5)

For methane over the concentration range of 700 to 1745 ppbv [2], ∆RF = 0.48 W m-2 and: α = (0.48)/(ln(1745/700)) = 0.525

(6)


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Figure 7. This is FAQ 1.1 Figure 1 from AR4 showing the radiation balance for the earth and the total back radiation of all of the greenhouse gases is 324 W m-2. Thus, the logarithmic model for methane is: ∆RF = 0.525ln(C/Co)

(7)

The quadratic model that follows the logarithm model over the range of 700 to 1745 ppbv is: RF = -0.0000002034C2 + 0.0009567C

(8)

From this quadratic model for methane, RF at 700 and 1745 ppbv is 0.57 and 1.05 W m-2 respectively. Similarly, for nitrous oxide, the logarithmic model is: ∆RF = 0.595ln(C/Co)

(9)

The quadratic model for nitrous oxide over the range of 240 to 314 ppbv is: RF = -0.0000008105C2 + 0.002611C

(10)

The RF at 240 and 314 ppbv is 0.58 and 0.74 W m-2 respectively. The values of 1745 ppbv for methane and 314 ppbv are from the TAR [2] and are for 1998. These concentrations were used in AR4.


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Table 1. This table shows the individual and total RF of all GHG except water vapour in 2005. Because the total is only 3.4% of the total RF of 324 W m-2, water vapour accounts for the remainder of approximately 96%. Carbon dioxide Methane Nitrous oxide Halocarbons Ozone Total

1750 7.01 0.57 0.58 0.0 -

˘ RF 1.66 0.48 0.16 0.34* 0.30*

2005 8.67 1.05 0.74 0.34 0.30 11.10

% of 324 2.7 0.3 0.2 0.1 0.1 3.4

In Table 1, the RF values in 2005 for carbon dioxide, methane and nitrous oxide are derived from the appropriate quadratic models. The RF for halocarbons is directly from Table SPM.2 of AR4 because there were none in 1750. There is a value of 0.25 W m-2 for ozone in 1750, which is very close to the level in 2005 and was ignored for purposes of this study. In Table 1, the total RF of all of the GHG except water vapour is approximately 11.10 W m-2, or 3.43% of the total RF of all GHGs of 324 W m-2. 10. THE IMPORTANT CONTRIBUTION OF WATER VAPOUR AS A GHG From Table 1, CO2 accounts for 2.7% of the global warming while all of the other gases account for approximately 0.7% for a total of approximately 3.4%. It becomes evident that, on average, water vapour accounts for approximately 96% of the current global warming. This is an important finding because it leads to the conclusion that the factors controlling the average level of water vapour in the atmosphere also control atmospheric temperature. 11. AN INDEPENDENT METHOD TO SHOW WATER VAPOUR ACCOUNTS FOR 96% OF THE WARMING BY GHG There is a relatively simple method that anyone can use to estimate the warming effect of CO2 at current concentration levels. The method uses the measured values of relative humidity (RH) and dry bulb (DB) temperature recorded hourly at many cities around the globe [8]. These measurements allow calculation of the weight of water vapour per kg of dry air using a psychrometric chart or program [9] and, therefore, the concentration of water vapour by weight, ppm. Psychrometric charts and computer programs are available on the internet. Calculation of the number of CO2 molecules from the ppm of water vapour is simple. For example, there are the same number of molecules of CO2 and water vapour for the same concentration measured by volume, ppmv. Because the molecular weight of CO2 (44 g/mole) is 1.52 times the number-average “molecular weight” of air (29 g/”mole”), 378 ppmv of CO2 in 2005 is equivalent to 575 ppm by mass. Similarly, 378 ppmv of H2O is equivalent to (18/29) x 378 = 235 ppm of H2O in terms of mass. Therefore, each 235 ppm of water vapour represents one molecule of water vapour.


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Figure 8. The ratio of water molecules to CO2 molecules for atmospheric temperatures from -10 to +30 oC and relative humidity from 0 to 100% for CO2 concentration of 378 ppmv in the atmosphere. Figure 8 is a chart of the number of molecules of water vapour per molecule of CO2 for relative humidity from zero to 100% and for atmospheric temperature from -10 oC to +30 oC. The number of CO2 molecules is based on CO2 concentration of 378 ppmv, the level in 2005 and that used by the IPCC in AR4. At 50% RH and 15 oC dry bulb temperature; there are approximately 23 molecules of water vapour for each CO2 molecule in the atmosphere. On a 25 oC summer day in Montreal with 70% RH, there are close to 60 molecules of water vapour for each CO2 molecule. 12. ATMOSPHERIC TEMPERATURE AND RH FOR WORLD CITIES The values of temperature and RH in Table 2 are from the Intellicast.com web site [8]. Intellicast makes available hourly readings of temperature and relative humidity for many cities around the globe. This is a sampling for Aug 15, 2010 between 19.00 and 20.00 GMT. Note the average RH of 64% and the average ratio of water molecules to CO2 molecules of 45 to 1. The values in Table 3 are for cities in South Africa close to Johannesburg. They are an example of a warm dry area. Note the average RH of 52% and the average ratio of water molecules to CO2 molecules of 21 to 1. For purposes of this study we assume an average world temperature of 15 oC and average RH is 50% [10]. This gives some allowance for the fact that at the poles, there is much less water vapour in the air. For reference, if RH = 60%, the ratio of water molecules to CO2 molecules would be approximately 27.4, rather than approximately 22.7.


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Table 2. From the Intellicast.com web site Aug 15, 2010, temperature at 20:00 GMT and RH at 19:00 GMT. The average RH is 64% and the average ratio of water molecules to CO2 molecules is 45 to 1. Bogata Cairo London Moscow New York Rio de Janiero Tokyo Melbourne Averages

Temp, oC 16 30 18 22 25 18 28 9 21

RH, % 77 52 64 50 60 56 79 76 64

H2O/CO2 38 60 36 36 52 31 82 23 45

Table 3. Data for the same date and time as for Table 2 for cities are in South Africa close to Johannesburg, an example of a warm dry area. The average RH is 52% and the average ratio of water molecules to CO2 molecules is 21 to 1. Antananarivo Capetown Gaborone Harare Johannesburg Johan-Maputo Lusaka Port Elizabeth Averages

Temp, oC 11 13 15 15 12 12 13 15 13

RH,% 100 94 17 39 22 22 51 72 52

H2O/CO2 35 38 8 18 8 8 20 33 21

Maurellis [11] showed in 2003 that because of its shape a water vapour molecule was more effective at warming than a CO2 molecule. He did not give any indication of how much more effective. At 50% RH, the ratio of warming effectiveness H2O/CO2 is approximately 1.6 as follows: (324 – 11.10)/(22.7)/8.67) = 1.6

(11)

Where: 324 W m-2 is the back radiation of all GHG, 11.10 and 8.67 W m-2 are from Table 1 and 22.7 is the average number of molecules of water vapour per molecule of CO2. If RH = 60%, the warming effectiveness ratio H2O/CO2 is approximately 1.32.


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13. THE OVERALL PICTURE As shown graphically in Figure 9, on average, each molecule of CO2 is surrounded by approximately 23 molecules of water vapour at ground level. This ratio is constant until the condensation temperature is reached at approximately 1500 metres altitude [12]. Condensation changes the water vapour into water droplets thereby reducing the ratio and releasing heat, some of which then radiates to space. If the warming effect of water molecules and CO2 molecules were the same, then the contribution of CO2 would be (1/22.7) = 4.4% of that of water vapour. But from the previous section, water molecules are 1.6 times more effective at warming than CO2 molecules. Using this value and the ratio of 22.7:1, the contribution of CO2 to warming of the atmosphere is approximately (1/22.7)/1.6 = 2.8% of that of water vapour. As water vapour is approximately 96% of the total RF of all of the GHG, the contribution of CO2 is approximately 4% less than this, i.e., 2.69%. If the average RH were 60%, the contribution of CO2 would be ((1/27.4)/1.32) x 0.96 = 2.65%. For practical purposes, these values are the same as the 2.7% obtained by the quadratic model.

Figure 9. The overall picture where, on average, each CO2 molecular is surrounded by approximately 23 molecules of water vapour from ground level until condensation occurs at approximately 1500 metres altitude. 14. DISCUSSION 14.1. AR5 In Figure SPM.5 of the AR5 Summary for Policymakers, the error bands for the GHG are doubled in width compared to those in AR4 for CO2 and N2O, and close to 9 times for halocarbons. Thus, the AR4 ∆RFs used in this study are well within the expanded

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error bands except for methane. In AR4 the ∆RF of methane was given separately, whereas in AR5 it is combined with some CO2 and water vapour. By measuring the length of the coloured bars, it appears the AR5 ∆RF of methane is at the edge of the AR4 range. Methane concentration increased from 1745 in 1998 to 1803 ppbv in 2011 [13]. RF increases from 1.05 to 1.064 Wm-2, or by 1.4%. The current RF of methane is approximately 1/8 that of CO2, and the small increase in methane has no measurable overall effect. 14.2. Doubling CO2 The total global warming of all of the greenhouse gases (GHG) of 324 W m-2 causes a temperature increase of 33 oC from -18 oC (255K), the earth’s temperature with no GHG, to the current average temperature of +15 oC (288K) [10]. By the quadratic model, ∆RF = 3.26 W m-2 for doubling of CO2 from 275 to 550 ppmv. On a pro rata basis, ∆RF = 3.26 W m-2 would generate an increase in atmospheric temperature of only 33 oC x (3.26/324) = 0.33 oC. An increase in the concentration of CO2 increases the temperature of the atmosphere, but the temperature does not “run away”. Any increase in temperature is balanced by an increase in radiation of heat to space. Radiation varies as the fourth power of the absolute temperature (K), i.e., radiation to space increases much faster than a linear temperature increase [6]. Thus, feedback, which theoretically exists, is small and controlled. 15. SUMMARY AND CONCLUSIONS There is no theoretical basis for the Beer-Lambert law as the mathematical relationship between RF of CO2 and concentration in the atmosphere. This is because water vapour and CO2 compete unevenly over the wavelength absorption range of IR. Thus, the Beer-Lambert law is only an approximation. The quadratic model gives the same ∆RF as the logarithmic model over the range 275 to 378 ppmv. Outside of this range, the quadratic model is a better fit to the origin point and to the plateau where CO2 has absorbed all possible back-emitted IR. It also provides an absolute value at 378 ppmv for RF of 8.67 W m-2, not just ∆RF. This is approximately 2.7% of the total RF of all of the GHG, including water vapour, contributing to global warming. An independent method estimating the world average ratio of water vapour molecules to CO2 molecules gives the same result, i.e., CO2 is responsible for approximately 2.7 % of the total RF of all of the GHG. Each of these two independent methods is sufficient by itself to prove CO2 is a much smaller contributor to global warming than has been previously thought. Another important finding by both methods is that, on average, water vapour accounts for approximately 96% of the current global warming. Therefore, the factors controlling the amount of water vapour in the atmosphere control atmospheric temperature.


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REFERENCES: [1] Solomon, S., Qin D., Manning M., Chen Z., Marquis M., K.B. Averyt K.B., Tignor M. and Miller H.L. (eds.). IPCC Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA. Available at: http://www.ipcc.ch/publications_and_data/publications_ipcc_fourth_assessment_report_ wg1_report_the_physical_science_basis.htm. See FAQ 1.1 Figure 1, page 96. [2]

Houghton J. T. and Ding Y. (Co-Chairs) of IPCC Working Group I, Climate Change 2001: Working Group I: The Scientific Basis, Intergovernmental Panel on Climate Change, 2001, Third Assessment Report (TAR), Cambridge University Press, Port Chester, NY, 2001. The currently accepted logarithmic model between RF and concentration of CO2 is eqn (1) from Section 6.3.5. Available at: http://www.grida.no/publications/other/ipcc_tar/?src=/climate/ipcc_tar/wg1/212.htm. The concentrations of CH4 and N2O are from Table 4.1.1. Available at: http://www.ipcc.ch/ipccreports/tar/wg1/130.htm#tab41a.

[3]

Lindsey R, NASA Earth Observatory, The Atmosphere’s Energy Budget, January 14, 2009. Available at: http://earthobservatory.nasa.gov/Features/EnergyBalance/printall.php

[4]

Hayden H C, A Primer on CO2 and Climate, Second Edition, Vales Lake Publishing, LLC, 2008. See pages 55 and 56.

[5]

Kiehl J T, Trenberth K E, Earth’s Annual Global Mean Energy Budget, Bulletin of the American Meteorological Society, Vol. 78, No. 2, February 1997, pp 197-208. Available at: http://www.geo.utexas.edu/courses/387h/PAPERS/kiehl.pdf

[6]

Pearson P N, Palmer M R, Atmospheric carbon dioxide concentrations over the past 60 million years, NATURE |VOL 406 | 17 AUGUST 2000, pp 695-699. See Figure 4, page 698, concentration approximately 350 ppmv (current level). Available at: http://lrg.elte.hu/oktatas/Elemek%20korforgasa%20PhD/NC/Pearson%20Palmer%202000%20Nature.pdf

[7]

Schmidt G et al., Attribution of the present-day total greenhouse effect, Journal of Geophysical Research, Vol. 115, D20106, doi:10.1029/2010JD014287, 2010. Available at: ftp://soest.hawaii.edu/coastal/Climate%20Articles/CO2%20role%20modern%20warmin g%202010.pdf

[8]

Intellicast.com, Current Relative Humidity maps of the world. Available at: http://www.intellicast.com/Global/Humidity.aspx. Intellicast.com, Current temperature maps of the world. Available at: http://www.intellicast.com/Global/Temperature/Current.aspx.

[9]

MegaWatSoft Psychrometric Calculator available at: http://www.megawatsoft.com/Default.aspx.

[10] UNEP (2005) Global surface temperature. United Nations Environment Programme, Vital Climate Graphics, Trend in global average surface temperature. Available at: http://www.grida.no/publications/vg/climate/page/3070.aspx.

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[11] Maurellis, A., (2003). The climatic effects of water vapour. Physics World (Print edition, May 2003). Available at: http://physicsworld.com/cws/article/print/17402. [12] The Engineering ToolBox. Air - Altitude, Density and Specific Volume. Available at: http://www.engineeringtoolbox.com/air-altitude-density-volume-d_195.html. [13] Climate Change 2013 The Physical Science Basis, Working Group I Contribution to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change, Summary for Policymakers, page 9, B.5 Carbon and Other Biogeochemical Cycles. Available at: http://www.ipcc.ch/report/ar5/wg1/docs/WGIAR5_SPM_brochure_en.pdf