Global Warming

* Everything should be made as simple as possible, but not simpler – A. Einstein

Introduction

In this article I will develop a simple* model for understanding the complex phenomenon of global warming. By global warming I mean the rising trend in average surface temperature of the entire Earth that has occurred over the past century. I start by describing the fundamental principles driving global temperature, which are quite straightforward. Basically, the Earth receives energy from the sun, which warms the Earth, and radiates energy out into space, which cools the Earth. Radiation increases with higher temperature and so the temperature of the Earth is that temperature at which energy outgo equals energy income. How the Earth radiates energy depends on properties of its surface and atmosphere. Here is where the complexity comes in because many things affect the ability of the Earth to radiate energy and so affect the temperature. This article deals with the three major factors, greenhouse gases, solar effects and the effect of aerosols.

The model developed here is greatly simplified yet it contains enough elements to allow a nonexpert to gain an understanding of how greenhouse gases work to warm the climate and how clouds affect it. The model is used to describe an important recent development in climate science, the cosmic ray climate driver hypothesis. This theoretical mechanism has been cited by many global warming skeptics as an alternate explanation for warming besides greenhouse gases. This article shows that although the theory is useful for explaining early 20^th century warming, it does not explain recent warming.

Elementary climate dynamics

The ultimate driver of climate is the sun, or more specifically, the energy bestowed upon the Earth as sunshine. The sun produces a total power output of some 4 x 10²⁶ watts. At Earth’s orbital distance this power output works out to about 1370 watts per square meter of surface perpendicular to the sun’s rays. This energy flux is called the solar constant or S. The Earth presents a circular barrier to this power output of area πR², where R is the radius of the Earth. It follows that the amount of energy intercepted by the Earth is then πR²S. The Earth reflects about 30% of incident energy, the rest is absorbed. The fraction reflected is called the albedo or A. With this I can write the following expression for the rate of energy absorption by the Earth:

1. Energy absorption rate = πR² S (1-A)

The Earth also radiates energy into space. This radiation is in the form of infrared (IR) radiation, not visible light as is much of the sun’s energy. All bodies radiate energy as a function of temperature. The amount of radiation that is emitted by one square meter of surface at a given temperature is given by the Stefan-Boltzmann equation:

2. Radiated energy flux = ε σ T⁴

Here T is the absolute temperature in degrees Kelvin (K). The Kelvin temperature scale is simply the familiar Celsius temperature with 273.15 added to it. It measures absolute temperature; 0 K is absolute zero, the coldest temperature possible. The parameter σ is the Stefan-Boltzmann constant and is equal to 5.67x10⁻⁸ watt m^-2·K^-4. Finally, ε is the average emissivity of the earth. Emissivity is a property of the material emitting radiation that relates a material’s actual radiative properties to that of an ideal black body. A black body has an emissivity of one for all wavelengths and represents the maximum radiative potential. Real materials have values of ε between 0 and 1 as a function of wavelength. The ε in equation 2 refers to an average value over the relevant wavelengths and various surface types of the Earth.

Equation 2 refers to an energy flux, that is, the energy transfer rate per unit area (square meter). To determine the total radiation emitted by the Earth I multiply this flux by the surface area of the Earth, which can be approximated as the area of a sphere: 4 πR². Multiplying equation 2 by this area gives the rate that energy is radiated by the Earth:

3. Energy radiation rate = 4 πR² ε σ T⁴

Over time the Earth’s average temperature will adjust so that energy absorbed equals energy radiated. Setting equation 1 and 2 equal to each other and rearranging gives the following equation related Earth’s average temperature, ε, the solar constant, and albedo:

4. ε σ T⁴ = ¼ S (1-A) = I = 240 watts/m²

For a solar constant of 1370 watts/m²and albedo of 0.3, the right hand side of equation 4 works out to 240 watts/m². I will call this the quantity the incoming solar radiation or insolation (I) It refers to the average amount of solar energy absorbed per square meter of surface. The actual average surface temperature of the Earth is about 15º C (288 K). If this value is inserted into equation 4 and the result solved for ε, one obtains:

5. ε = I / (σ T⁴) = 240 / (5.67 x 10^-8 · 288⁴) = 0.615

Table 1. Emissivity of various terrains (typical values observed over the entire range of IR wavelengths)

Material	Emissivity	Reference
Ice and snow	0.94-0.99	1
Vegetation	0.94-0.99	1
Soil	0.75-0.99	1
Seawater	0.98-0.99	2
Grassland	0.95	3
Barren Land	0.93	3
Forest	0.96	3
Urban	0.95	3

This value is the apparent emissivity of the Earth as viewed from space. Table 1 lists emissivity for a variety of terrain types. Emissivities for many materials do not vary much with wavelength, making the use of an average value reasonable. As Table 1 shows, the emissivities from various types of terrain are not much different from each other either, making the use of a single average value to describe the Earth’s surface emissivity a good approximation. The emissivity of the Earth’s surface (ε_S) appears to be greater than 0.95. A value of 0.96 is used on some models³ and will be used here. With a 0.96 emissivity, equation 4 yields a temperature of 258 K or −15º C. An Earth surrounded by an IR-transparent atmosphere would be a frozen ice ball. The Earth is not frozen because the clouds and atmosphere are not transparent to IR radiation. Some of the radiation emitted from the Earth is absorbed by the clouds and atmosphere and re-radiated. This reduces the Earth’s apparent emissivity from the 0.96 value for the surface to an apparent value of 0.615.

Factors affecting the earth’s albedo and emissivity: the effect of clouds

Table 2 shows presents radiative properties of clouds. Clouds cover about 63% of the Earth’s surface on average.⁵ The albedo of the Earth is given by the weighted sum of surface average albedo (A_SURF) and the cloud average albedo (A_CLOUD): A = 0.3 = 0.37∙A_SURF + 0.63∙A_CLOUD. Overall radiation measurements indicate that in the absence of cloud, solar insolation would be 53.5 watt m^-2 greater.⁵ Thus, a cloudless Earth would see a solar insolation of 240 + 53.5 = 293.5 watt m^-2. From equation 4, this value for I is consistent with an albedo of 0.14. Thus, A_SURF = 0.14. It follows that A_CLOUD = 0.39 = (0.3 − 0.37∙0.14)/0.63. The 0.25 difference in albedo between clouds and the surface also affects climate. In the section above I showed that an Earth with an IR-transparent atmosphere would have a temperature of −15º C. This calculation ignored the effect of clouds on albedo. An Earth with an IR transparent atmosphere would not have any clouds; its albedo would be 0.14 rather than 0.3. With A = 0.14 and ε = 0.96, equation 4 gives T = 271 K or −2º C, not −15º C.

Table 2. Radiative properties of clouds

Type	Fraction coverage^5	Effective ε^4	Albedo^5
Low	26%	0.68	0.39
Medium	18%	0.55	0.39
High	19%	0.33	0.39
Surface	37%	0.96	0.14

Radiation absorbed by clouds is re-radiated out into space from a surface having a lower effective emissivity than the surface below it. This lower effective emissivity mostly reflects the lower temperature of clouds due to their elevation. Clouds thus act to reduce the overall emissivity of the Earth, warming it. They serve as an IR radiation shield. Those readers who have ever used a carport in cold climes know that despite the lack of enclosure, no frost collects on the car’s windows overnight. Frost is caused by radiation from the car cooling it below the air temperature, allowing frost to form. With the carport top blocking this radiation, frost is avoided. Clouds serve a similar function for the surface beneath them.

Table 2 shows the effective emissivities of clouds as a function of their elevation. These values are averages over plots of cloud emissivities as functions of wavelength obtained by Shippert et al.⁴ Low cloud is defined as clouds with a base height below 3 km and high clouds have base heights above 6 km. Medium-high cloud is in between. Higher clouds are more effective shields because they re-radiate at lower temperatures. Using data from Table 2 the average emissivity of Earth with clouds would be 0.37·0.96 (cloudless surface) + 0.26·0.68 (low cloud) + 0.18·0.55 (medium-high cloud) + 0.19·0.33 (high cloud) = 0.694 (apparent emissivity of Earth as viewed from space). With a 0.694 emissivity, equation 4 yields of temperature of 279 K or +6º C. With clouds in an IR-transparent atmosphere, the Earth would be 8 degrees warmer than without clouds, but it still would be colder than its actual temperature of +15º C.

Note that in this calculation cloud cover is treated as an opaque screen; no radiation from the surface is assumed to penetrate through the cloud layer, it is all either absorbed or reflected back to the surface. These opaque screens themselves radiate outward into space with a lower effective emissivity, reducing the Earth’s emissivity from the 0.96 value of the surface to the value given in Table 2. In actuality, thin cloud is not strictly opaque and the interaction of clouds with radiation is more complex than this shield model indicates. Nevertheless this approach gives a reasonably good semi-quantitative picture of the effect of clouds. For example, an Earth completely covered by high cloud would be warmer: with A = 0.39 and ε = 0.33, equation 4 gives a temperature of 325 K (52º C). In contrast, an Earth covered by low cloud would be cooler: with A = 0.39 and ε = 0.68, equation 4 gives a temperature of 271 K (−2º C). These results are consistent with more sophisticated models that also show warming from high cloud and cooling from low cloud. The exact amounts predicted are different, but one should not expect detailed accuracy from such a simple model.

Factors affecting the earth’s emissivity: the greenhouse effect

Certain gases in the atmosphere called greenhouse gases absorb and re-radiate IR radiation emitted from the surface, just like clouds. Unlike clouds, the greenhouse-gas containing atmosphere cannot be modeled as an opaque screen, but rather, as a translucent screen that partially shields the ground emissions. This effect serves to reduce the effective emissivity of the cloudless sky from the 0.96 of the surface to a new value (ε_T) that takes into account this partial shielding from greenhouse gases. The shielding provided by greenhouse gases is dependent of the wavelength of the IR radiation. Figure 1 shows a plot of the absorption of IR radiation by the cloudless atmosphere as a function of wave number. Wave number is the number of wavelengths per centimeter, so larger values means shorter wavelength. From this figure one can see effects of individual greenhouse gases: strong absorption by water vapor at 100-600 cm^-1 and 1400-1600 cm^-1, by ozone at 1060 cm^-1, and by carbon dioxide (CO₂ ) at 600-700 wavenumber.⁶

Figure 1. Difference between surface radiation and energy radiated into space, indicative of IR radiation absorbed by the atmosphere⁶

Table 3 shows a summary of four kinds of IR shields lying over the Earth’s surface: the three types of cloud and the translucent atmosphere. The simple model I am developing here holds that the total emissivity of the Earth (which we know to be 0.615) is simply the weighted sum of the effects of all these shields. The weights are the percent coverage of each type of screen. The effective emissivities for three of the four screens are known (Table 2) and are reproduced in Table 3. The contributions from these three sum to 0.3385. The rest of the emissivity comes from surface radiation through the cloudless, translucent atmosphere, which has an effective emissivity of ε_T. If I assume that ε_T is higher than that for all clouds, then it will only affect the radiation from the 37% of the surface not covered by clouds. A value for ε_T can be obtained by subtracting the cloud-provided shielding from the overall emissivity of 0.615 and dividing the difference by the cloudless fraction of the Earth’s surface: (0.615−0.3385)/0.37 = 0.747 for ε_T (see Table 3). Since 0.747 is greater than 0.68 the assumption that ε_T was higher than the emissivity for any type of cloud was valid.

Once again this translucent screen analogy to the IR-absorbing atmosphere is a simplification of a more complex situation. Absorption and re-radiation of IR radiation occurs throughout the atmosphere and interacts with clouds. A completely accurate assessment of this phenomenon requires very complex radiation balance models. Nevertheless, the results obtained with this simple model (see below) are quite sufficient to gain a reasonably good understanding of how CO₂ affects the climate.

Table 3. Effect of various IR shields on overall emissivity of the Earth

Type of Cover	%Coverage^5	Emissivity^4	Contribution
High Cloud	19%	0.33	0.0627
Med Cloud	18 %	0.55	0.0990
Low Cloud	26%	0.68	0.1768
Surface	37%	εT = 0.747	0.2764
Overall	100%	0.615	0.3385+0.37εT

The value of ε_T depends on the concentration of greenhouse gases in the atmosphere such as CO₂. Thus, adding CO₂ to the atmosphere affects ε_T, which affects the earth’s overall emissivity as shown by equation 6:

6. ε = 0.3385 + 0.37 ε_T

Changes in overall ε produced by rising CO₂ affect temperature according to equation 4.

The analysis of the translucent atmosphere is more complex than the effect of clouds, which were treated as opaque barriers with a fixed emissivity. The translucent atmosphere both absorbs and transmits (allows to pass through) radiation from the surface. This situation can be described by a radiation balance on the translucent atmosphere:

RADIATION INTO ATMOSPHERE = RADIATION OUT OF ATMOSPHERE

7. σ ε_S T_S⁴ = 2 σ ε_A T_A⁴ + (1− ε_A) σ ε_S T_S⁴

surface radiation into atm = atm. radiation into space & back to surface surface radiation transmitted through atm into space

Here ε_S is the emissivity of the surface (0.96) and ε_A is the intrinsic emissivity of the atmosphere (a function of greenhouse gas levels). T_S and T_A are the temperature (in Kelvin) of the surface and the atmosphere, respectively. The factor (1− ε_A) is called the transmission of radiation through an absorbing medium. The factor 2 occurs because the atmosphere radiates both outward into space and inward towards the surface. Equation 7 can be solved for T_A as follows:

8. T_A⁴ = ½ ε_S T_S⁴ → T_A = (½ ε_S)^¼ ∙ T_S = 0.83∙288 = 240 K

The total emissions from the translucent atmosphere into space are given by:

9. total emissions = σ ε_T T_S⁴ = σ ε_A T_A⁴ + (1− ε_A) σ ε_S T_S⁴

Note than the 2 is absent because only the radiation going into space from the atmosphere is considered here. Equation 8 can be substituted into equation 9 to obtain the result:

10. ε_T = ε_S (1 − ½ ε_A)

For the actual value of ε_T of 0.747 from Table 3 and ε_S = 0.96, equation 10 gives a value of 0.4438 for ε_A. The effect of IR absorbers on emissivity is described by Beer’s Law, which states that for a given wavelength, the logarithm of the transmission through a mixture is equal to the negative of the sum of the absorbance (A_i) of each IR-absorbing species present:

11. ln(1─ ε_A) = − ∑ A_i for each wavelength

The symbol ∑ means “the sum of, (in this case absorbances) for all i”. The absorbance of species i is given by A_i = a_i∙b∙C_i, where C_i is the concentration of species i, b is the length of light path through the i-containing medium and a_i is the molar absorptivity. The molar absorptivity is a function of wave number (η); that is, a_i should be written as a function, a_i(η). For the application here, b is dependent on the thickness of the atmosphere, which is a constant independent of C_i. Hence a_i(η) and b can be combined into a single parameter k_i(η) and equation 11 becomes:

12. ln(1─ ε_A) = − ∑ k_i(η)∙C_i → ε_A = 1 − exp[−∑ k_i(η)∙C_i] for each wavelength

In fact, an IR spectra like Figure 1 gives the value of k_i(η) implicitly as a function of wave number since the spectra is a measure of atmospheric absorbance for a fixed C_i. I will assume k_i(η) is a constant equal to its average value over all η. I will consider only two species, water vapor (H₂O) and CO₂. For these assumptions, equations 11 and 12 can be written for the entire spectrum of emitted radiation as follows:

13. ε_A = 1 − exp[−(A_H2O + A_CO2)] = 1 − exp(−k_H2OC_H2O − k_CO2∙C_CO2)

For ε_A = 0.4438, the sum of A_H2O and A_CO2 is equal to 0.5865. Based on the bands in Figure 1 it is clear that water vapor is responsible for most of the IR absorption by the atmosphere. Let us assume 79% of absorbance is due to water vapor, that is, A_H2O = 0.79·0.5477 = 0.4634. This leaves A_CO2 = 0.1232. The “base level” for CO₂ is set at 350 ppm (0.035%), that is, the assumed Earth temperature of 288 K occurs at C_CO2 = 0.035. From this I obtain k_CO2 = 0.123/0.035 = 3.52 and equation 13 becomes:

14. ε_A = 1− exp(−A_H2O − 3.52∙C_CO2)

To obtain a value for A_H2O for situations other than the base case I assume that changing global temperature does not affect average relative humidity, only absolute humidity. With this assumption the effect of temperature on the average concentration of water (C_H2O) of the atmosphere is governed by the Clausius-Clapeyron equation:

15. ln[C_H2O/²⁸⁸C_H2O] = ln[A_H2O/²⁸⁸A_H2O] = −ΔH^VAP/R∙[1/T − 1/288] → A_H2O = ²⁸⁸A_H2O∙exp(5294∙[1/288 − 1/T]) = 0.4634∙exp(18.382 − 5294/T)

Here ΔH^VAP is the heat of vaporization of water and R is the gas constant. The term ΔH^VAP/R has the value 5294 K. ²⁸⁸C_H2O is the concentration of water vapor at the standard temperature of 288 K. The ratio C_H2O/²⁸⁸C_H2O is equal to the ratio A_H2O/²⁸⁸A_H2O where ²⁸⁸A_H2O is the value of A_H2O at the standard temperature of 288K, which has already been determined as 0.4634. With these values A_H2O becomes a simple function of T as shown in equation 15. Substituting equation 15 into equation 14 gives:

16. ε_A = 1− exp(−0.4634∙exp(18.382 − 5294/T) − 3.52∙C_CO2)

The greenhouse model is now complete. For a given value of C_CO2, ε_A as a function of T is obtained from equation 16. The value of ε_A is used in equation 10 to obtain ε_T as a function of T. This value is used in equation 6 to obtain ε for the Earth as a function T. This value of ε then used to calculate temperature using equation 4. The value of temperature obtained is a function of T and must equal T. This will be true for a particular value of T which can be obtained using trial and error (the goal seek function in Microsoft Excel™ makes this easy to do). This procedure was used to calculate the change in temperature from 288K (ΔT) produced by changes in CO₂ level from 0.035% (Δ%CO2). The results are plotted in Figure 2.

Figure 2. Temperature change and forcing as a function of change in CO₂ level relative to 350 ppm

This simple model has a number of limitations. It ignores the effect of changing levels of water vapor on clouds. If changes in water vapor affect lower clouds preferentially, then increases in water vapor will exert a cooling effect that will offset at least some of the greenhouse warming effect of water vapor itself. On the other hand, if increases in water vapor affects high cloud more than lower cloud, then the cloud-mediated effect of water vapor could be greater than the effect calculated here.

The model does a reasonably good job of predicting the temperature increase for a doubling of CO₂ level from 280 ppm (the preindustrial value) to 560 ppm. The value obtained from this model is +1.7º C, which is pretty close to the +1.85º C corresponding value obtained from more sophisticated models. The model underestimates the effects of low levels of CO₂. For example, it predicts temperature would be 2.3º C cooler without any CO₂ in the atmosphere. More sophisticated models predict 3-4º C (= 9-12% of total 33º C greenhouse effect).⁷ On the other hand, the model overestimates the effects of higher CO₂ levels; quadrupling CO₂ levels from pre-industrial levels is predicted to increase temperature by 4.4º C as compared to 3.7º C from more sophisticated models.

It is important to note than this model made use of an adjustable parameter, sometimes called a fudge factor. This adjustable parameter was the fraction of the atmospheric absorbance under base conditions that was assumed due to water vapor. The value of 79% was selected in order to calibrate the model with respect to more sophisticated models. What this means is the greenhouse model presented here is representational, not predictive. One cannot use it to accurately predict the direct effect of rising CO₂ on temperature; the results obtained with it will not exactly match the results of the complex models. The purpose of this model is to provide an easy to understand (and manipulate) model that can represent the results of more complex models. The results obtained with it will not exactly equal the results of the complex models, but it will get the approximate magnitude and direction correct.

An example of the use of the simple model: converting Earth into Venus

As an example of how this model can be used, let us consider what would happen if greenhouse gases rose incessantly. Equation 10 shows ε_T varying between 0.96 for ε_A = 0 (transparent) to 0.48 for ε_A = 1.0 (opaque). Note that 0.48 is the effective emissivity of an opaque emitter like a cloud or carport at the atmospheric temperature (T_A) of 240 K. According to the simple model, with very high levels of greenhouse cases ε_A would become one and the translucent atmosphere would become an opaque screen like a cloud with emissivity of 0.48. The emissivities of low (ε = 0.68) and medium (ε = 0.55) cloud would be replaced by the more effective screen (ε = 0.48) provided by the IR-opaque atmosphere. This would reduce overall ε from 0.615 to 0.452, resulting in a warming to 38º C. Associated with higher temperatures would be higher levels of water vapor, which would mean more clouds. If this cloud showed up as low or medium cloud, the cooling effect of increased albedo would reduce the warming effect of the greenhouse gases. If increased cloud showed up as high cloud too, then both emissivity and albedo would be reduced, resulting in increasingly warmer temperatures all the way to the 52º C obtained earlier for 100% coverage by high cloud. This example shows that even in an extreme case, this simple model correctly predicts that a Venus-like “runaway greenhouse effect” in which the oceans boil away simply can not happen on Earth no matter how high greenhouse gas levels get. The temperatures predicted here are not predicted exactly, of course, but the qualitative result (they are below the boiling point) is valid.

As an exercise, let’s see what would happen to an Earth in Venus’s orbit. The solar constant for Venus is 2614 watts m^-2, much higher than Earth’s 1370 watts m⁻², reflecting its closer proximity to the sun. Let’s cover the Earth with maximum-cooling low cloud. What would be the temperature be according to the simple model? With A = 0.39, S = 2614, and ε = 0.68, equation 4 gives T = 319 K, or 46º C (115º F). Since Venus actually is completely covered by clouds, this calculation seems reasonable enough and suggests that a Venus with liquid water and life could exist, although would be a pretty hot place by Earth standards. This was the rationale for pre-1950’s science fiction stories that depicted a steamy hot swamp-like Venus.

But how realistic is it for all the cloud to be low-lying cloud? If here on Earth there is high cloud cover equal to 75% of low cloud cover, shouldn’t this be the case for a warmer Earth? A 100% low-cloud covered Earth would most likely have at least 75% high cloud coverage as well, which would lower ε from 0.68 to around 0.4. This would raise the temperature to 364 K (91º C), at which point water vapor would make up a substantial portion of the atmosphere. With such a wet atmosphere, cloud cover would likely be complete at low, medium and high elevations. With 100% high cloud cover, ε would be 0.33, and equation 4 gives the temperature as 382K (109º C). The oceans can now boil, filling the atmosphere with water vapor.

The mass of the oceans is more than 300 times that of the atmosphere. Boiling away the oceans would increase the mass of the atmosphere enormously. Let’s imagine a 100-fold increase, that is, surface atmospheric pressure rises to 100 times that on Earth (the actual surface atmospheric pressure on Venus is about 96 atmospheres so such large atmospheres can certainly exist on terrestrial-type planets). Such a thick atmosphere would expand the elevation at which clouds can form. On Earth, the 0.33 emissivity for high cloud provides a measure of the temperature of the atmosphere at which high cloud forms. This temperature is 218 K because (218/288)⁴ = 0.33. On Earth, atmospheric pressure in the troposphere (the thick portion of the atmosphere under 11 km elevation) roughly follows this relation:

17. P = P_S(T/T_S)^5.26

Here the subscript s refers to surface. I note that at the level of high cloud T/T_S = ε^¼. With this I can write:

18. P_HC = P_S∙ε^1.32 = 1.0∙0.33^1.32 = 0.23 atmospheres

Thus, high cloud is associated with a pressure of about 0.23 atmospheres. At pressures much below this, thick clouds apparently do not form, perhaps the air is too thin to keep ice crystals in suspension. If I assume that high cloud occurs where P_HC = 0.23, then I can use equation 18 to predict ε for high cloud for atmospheres thicker than the Earth’s. The relation is:

19. ε = (0.23/ P_S)^0.76 = 0.33 P_S^−0.76

For P_S = 100 atmospheres, ε becomes 0.01. For extremely thick clouds, it is probably more appropriate to use the albedo for Venus itself, which is covered by thick cloud. The albedo of Venus is 0.75.⁸ Using A = 0.75, ε = 0.01 and S = 2614 in equation 4 gives T = 733K (460º C)—hot enough to melt lead. The actual average temperature of Venus⁸ is 464º C, remarkably close to this calculation. A hot Earth in Venus’s orbit would gradually lose its water to photolysis by solar ultraviolet rays, producing hydrogen and oxygen. The former would escape and the latter would oxidize sulfur compounds released by volcanism to sulfuric acid. The heat would bake out CO₂ from surface carbonates, maintaining a high atmospheric pressure despite loss of water by photolysis. The end result would be a bone dry, extremely thick atmosphere composed mostly of CO₂ with the high temperatures maintained by thick high clouds composed of sulfuric acid, which is exactly what Venus is today. Thus, the simple model presented here can explain how an Earth in Venus’s orbit would become like Venus, while an Earth in Earth’s orbit will stay Earth-like.

More sophisticated models in which a more exact radiation balance is calculated for each wavelength of the spectrum can give more detailed predictions. I will use these results, which are expressed in terms of forcings (see next section) in my subsequent discussion of the greenhouse effect.

The concept of forcings

More physically realistic models do not provide a simple relation between greenhouse gases or cloudiness and effective emissivity like the simplistic model just presented. Rather than expressing the effects of greenhouse gases on ε in equation 4, the practice is to treat the effect of greenhouse gases as forcings (F), effective changes in insolation (I), the average amount of energy reaching the top of the atmosphere. These forcings produce a temperature effect ΔT. The effect of a forcing on temperature can be determined from equation 4 by placing I + F and T + ΔT in for S and T and then subtracting the unmodified equation 4 as follows:

20. ε σ (T + ΔT)⁴ = I + F = ε σ (T⁴ + 4T³ΔT + 6 T²ΔT² + 4T ΔT³+ ΔT⁴)

21. ε σ (T)⁴ = I

Subtracting equation 21 from 20 and dividing by T⁴ gives

22. ε σ 4 [(ΔT/T) + 6 (ΔT/T)² + 4T (ΔT/T)³+ (ΔT/T)⁴] = F / T⁴

If ΔT is small compared to T, the higher-power terms of (ΔT/T) on the left hand side of equation 19 will be so small that I can neglect them. In this case equation 22 becomes:

23. 4 ε σ ΔT/T = F / T⁴

Multiplying both sides by T⁴ and solving for ΔT gives

24. ΔT = F / (4εσT³) = λ F = 1/(4∙0.615∙5.67x10^-8∙288³) ∙ F = 0.3 F

The parameter λ depends on ε and T. For T = 288 K and ε = 0.615, its value is 0.3. A value for forcing can be obtained from the simple greenhouse model constructed earlier. The radiation balance in equation 7 had a term, 2σ∙ε_A∙T_A⁴, reflecting radiation from the atmosphere into space and back to the surface. The change in the half of this term going back to the surface is the greenhouse forcing. Thus I can write:

25. F = Δ(0.37∙σ∙ε_A∙T_A⁴) = 0.37∙5.67 x 10^-8∙240⁴∙Δε_A = 69.3∙ Δε_A = 69.3∙(ε_A − ^Bε_A) = 69.3 ε_A − 30.8

Here the Δ symbol refers to the change in back radiation (the forcing) produced by the factor (greenhouse gases) under consideration. The factor 0.37 is included because this forcing only applies to the 37% of the surface not covered by clouds. The parameter ^Bε_A refers to the base value of ε_A, for T = 288 and C_CO2 = 0.035%, which is equal to 0.4438, see the section above. A forcing is the “pure” effect of a variable, without including any feedback effects from increased water vapor. Thus, equation 14 with A_H2O held constant at 0.4634 should be used to calculate the value of ε_A in equation 25:

26. F = 69.3∙[1− exp(−0.4634 − 3.52∙C_CO2)] – 30.8 = 38.5 − 69.3∙exp(−0.4634 − 3.52∙C_CO2)] = 38.5 − 43.6∙exp(−3.52∙C_CO2)

The CO₂ forcing calculated by equation 26 is plotted in Figure 2. Values for λ can obtained by dividing the ΔT value in Figure 2 by F. Values around 0.45 are obtained.

Greenhouse gas forcings calculated from complex models are approximated by equations having a logarithmic form:

27. F = 5.35∙ln[C_CO2/^BC_CO2]

Here ^BC_CO2 is the base C_CO2 level of 0.035%. The values from equation 26 obtained for CO₂ levels between 280 and 560 ppm, which includes the entire range of interest, can be fit to an equation of the form given in equation 27 with K = 5.4. This close match is deliberate, as the water-CO₂ split of 79%:21% was chosen to make the forcing behavior accurately reflect the forcing results obtained from more sophisticated models.⁹ Unlike equation 26, forcings obtained from equation 27 with K = 5.35⁹ can be considered as accurate assessments of the direct effect of CO₂ on climate forcing. Of course, climate forcing isn’t the result I desire, temperature is. And to obtain temperature from a forcing, a value of λ is needed. As shown above, the simple model provides a value of 0.45 for λ when effects of water vapor were included. This value is different from the 0.3 value determined above. The 0.3 value for λ applies to effects of forcings without any considering any other effects. For example, a doubling of CO2 levels will produce a forcing of 3.7 = 5.35∙ln(2) watts/m² according to equation 27. The effect of CO₂ all by itself would be 0.3∙3.7 = 1.1º C. The simple model gives the same result if water vapor content is held constant. But with rising temperature caused by CO₂, there will be rising H₂O, which will exert its own effect on temperature. This secondary effect from H₂O is called a feedback effect. By ignoring cloud formation and assuming constant relative humidity to account for water-vapor greenhouse effects, the simple model gives a value of 0.45 for λ. More sophisticated models give different values depending on how they handle the greenhouse effects of water vapor. The consensus value for λ from these models is 0.5. For λ = 0.5 I can write

28. ΔT = 2.68∙ln[C_CO2/^BC_CO2]

Figure 3. Temperature relative to 1961-90 compared CO₂ forcing (data from refs 10-15)

Figure 3 shows a plot of equation 28 as “CO2 forcing”. Also shown is a plot of global temperature. Both plots are shown in terms of deviations from a 1961-1990 basis period. That is, the temperatures are temperatures relative to the average over these years and ^BC_CO2 is the average CO₂ level during these years. The CO₂ forcing shows a gradual rise that accelerates after 1970, which is roughly consistent with rising temperature. The extent of the rise is about the same, but the monotonic nature of the CO₂ forcing model fails to account for periods of flat and declining temperature in the temperature trend. Clearly, more is needed to explain the trends in global temperature since 1850 than just the greenhouse effect.

Sun-related factors

Another obvious effecter of climate is the sun. The sun exerts two types of effects. The first effect is through the solar constant S as described in equation 4. Solar insolation (I) is directly related to S:

29. I = ¼ S (1-A) = 0.175 S à F = ΔI = 0.175 ΔS

Figure 4 shows a plot of the solar constant over the past three decades. The solar constant is not actually constant. Rather, it shows a 1 watt/m² cyclical variation that is closely correlated with the 11-year sunspot cycle. Equation 29 shows that this cyclical fluctuation produces a cyclical forcing of 0.18 watt / m². Multiplication by λ = 0.5 gives a temperature effect of 0.09º C.

Figure 4. Total solar irradiance (solar constant) since 1976 (data from refs 16-18)

Figure 5 shows a plot of the global temperature since 1850. The grey circles and line shows the annual values for global temperature relative to the 1961-1990 average. The heavy black line shows the trend obtained from a moving 20-year linear regression analysis. The actual plot fluctuates around the trend value. The parallel black lines show the “track” within which the temperature remains approximately 82% of the time. That is, it is the value met or exceeded roughly twice every 11 years (top and bottom of each 11-year solar cycle). The distance between these lines is 0.24º C. It reflects the size of the short-term fluctuations in global temperature. Note that the monotonic nature of the CO₂ forcing means that CO₂ cannot explain these temperature fluctuations. The existence of them is proof of the operation of important factors other than CO₂ in global temperature change. An obvious question is whether some of these fluctuations could be explained by solar effects. At most, 0.09º C of this fluctuation can be caused by the cyclical fluctuations in solar constant shown in Figure 4. Apparently there is more involved in these fluctuations than can be explained by fluctuations in the solar constant.

Figure 5. Global temperature relative to 1961-1990 average (data from refs 11-15)

Another solar effect on climate: the cosmic ray hypothesis

Danish physicist Henrik Svensmark has proposed that the sun exerts an indirect effect on climate that amplifies the direct effect of solar irradiance (see Figure 6). Solar wind modulation by the sun’s magnetic field varies with the solar cycle (as measured by sunspot number). When solar activity is high (lots of sunspots) the solar magnetic field strength is high and vice versa. The solar magnetosphere shields the Earth from galactic cosmic rays that enter the solar system from distant sources. As a result, cosmic ray flux on Earth varies inversely with solar magnetosphere strength and with sunspots. Cosmic rays produce ionization in the lower atmosphere, which leads to the formulation of tiny particulates that serve as cloud condensation nuclei resulting in more cloudiness. Cloudiness affects climate as we have seen earlier.

Figure 6. Cosmic ray climate mechanism (figure from ref 19)

Figure 7 shows a plot of the solar cycle defined by sunspots and total solar irradiance (TSI). Also shown is the negative of cosmic ray flux (−CRF). The plots are shown as differences from their average/trend values divided by their standard deviations. This puts the disparate data on the same scale and allows a good comparison of the cycles shown by all three parameters. All three variables show closely aligned cycles. Temperature deviations from the trend shown in Figure 5 are also plotted. A three-year moving average is used to smooth the temperature data. An apparent correspondence can be seen between temperature and the solar cycle as defined by sunspots, TSI or –CSF. The temperature oscillation associated with the solar cycle shown in Figure 7 is 10 years long on average with 0.15º C average magnitude. The size of these oscillations is too large for TSI to be the sole cause of them. The cosmic ray mechanism provides another way for the sun to exert a cyclical effect on temperature.

Figure 7. Cycles in Temperature, TSI, Sunspots and −CRF (sunspot data from ref 20, CRF data from refs 21,22)

Figure 8 shows a plot of low cloud cover versus cosmic ray flux (CRF). A good correspondence is evident. The correlations in Figures 7 and 8 show that low cloud cover should be inversely proportional to TSI and sunspot number. Because low cloud cover exerts a cooling effect, CRF also exerts a cooling effect. This means than −CRF should exert a warming effect, which is what was observed in Figure 7. That is, the CRF-mediated cloud effect should add to the TSI cycle, effectively amplifying it. This second effect, in tandem with TSI, is a likely explanation for the cyclical temperature fluctuation aligned with the solar cycle.

Figure 8. Cosmic rays versus low cloud cover (from reference 23)

Measuring the size of the cosmic ray effect: estimation using Figure 8 and the simple model

Figure 8 indicates that the solar cycle shows an average change in low cloud cover over a cycle of about 1.6%, say from 25.2% to 26.8%, which implies a change of fraction cloudless from 37.8% to 36.2%. Equation 6 becomes:

30. ε = 0.3494 + 0.354 ε_T

Albedo changes by +0.004 = 1.6%∙(0.39-0.14), from which solar insolation decreases from 240 to 240∙(1-0.304)/(1-0.30) = 238.6 watts m^-2. This value is substituted for 240 in equation 4 to give

31. T = (238.6/εσ)^¼

The procedure is as before. Here C_CO2 = 0.035 and equation 16 gives ε_A as a function of temperature. Equation 10 gives ε_T from ε_A, from which T is calculated by equations 30 and 31. The trial and error solution is a 0.44º C decrease in temperature due to the increased cloud cover. Added to this is the effect of TSI, which I have previously calculated as 0.09º C to yield a total solar cycle effect of 0.53º C. We have seen that the actual size of cyclical fluctuations is 0.15º C, less than a third this value.

Measuring the size of the cosmic ray effect: empirical approach from past excursions in CRF

Another approach to measuring the size of the cosmic ray effect is to use historical “natural experiments” in which cosmic rays were known to have varied and temperatures with them. Once such natural experiment is shown in Figure 9 when a very long-term variation in cosmic rays was impressed upon the Earth’s climate by the solar system’s periodic passage through the galaxy’s spiral arms, which are rich in cosmic ray sources. The result of this movement is a 185 million year cycle in cosmic rays during which cosmic rays fluctuate by about 80% of their current magnitude. Associated with this cycle is a temperature cycle of average magnitude 2.8º C. This suggests a 0.035º C temperature effect per 1% change in cosmic ray flux.

Figure 9. Cosmic ray flux versus temperature over a very long time scale (fig from ref 24)

Using this sensitivity, the effect of CRF on temperature could in principle be calculated in terms of the deviation from its modern value. CRF data is only available beginning in 1951 (Figure 10). These data cannot be used directly with the 0.035º C/percent temperature effect of CRF obtained from Figure 9 because different measurements of CRF show different magnitude changes (see Figure 10). Data from the Haleakala-Huancayo detectors show about a 5% variation per solar cycle suggesting a 0.17º C temperature effect due to the cosmic ray variations of the solar cycle. On the other hand, the Climax detectors show a 16% variation suggesting a 0.56º C temperature effect. To these values the TSI-mediated effect of 0.09º C must be added to give estimates of 0.26º and 0.65º C. The theoretical value of 0.53º C falls in between these two values.

Figure 10. Cosmic Ray Flux from various detectors (data from refs. 20-21).

Using proxies to predict historical solar effect

When the data in Figure 10 are normalized by plotting them in terms of standard deviations, the CRF data from the different locations correspond very well. Indeed, Figure 7 showed CRF data from all three detectors expressed in a normalized fashion and averaged into a single composite profile. Figure 7 also showed the good correlation between CRF and sunspot number. It should be possible to extend CRF back before 1950 using this correlation. Figure 11 shows an extension of Figure 7 backwards in time. Taking the place of TSI is the total open solar magnetic flux (OSF) obtained from a paper by Lockwood.²⁵ This paper gives OSF in terms of a running 11-year average. To be consistent, sunspot number and CRF are presented as 11-year moving averages also. The data are scaled in the same way as in Figure 7, the values are the differences from the average value over 1951-2000 and divided by the standard deviation over the same period.

Figure 11. Trends in SSN²⁰, OSF²⁵, and –CRF^21,22 expressed in terms of standard deviations from 1950-2000 mean

Figure 11 shows that solar activity as measured by sunspots or OSF rose significantly during the first half of the 20^th century. Global warming skeptics often point out that solar activity is at record levels so temperatures should be. They are right, solar activity is very high today, but it has been very high for 50 years. The major rise in solar activity that resulted in the current high levels of solar activity occurred in the first half of the 20^th century, during which temperature rose as well (see Figure 5). The simultaneous rise in both temperature and solar activity strongly suggests that solar activity is responsible for the early 20^th century temperature rise (but not the more recent post-1970 temperature rise). I desire to calculate the magnitude of this solar effect to verify the hypothesis that solar factors produced the early 20^th century rise in global temperature.

Figure 11 shows that the size of the solar activity rise during the first half of the 20^th century is measured at 4.5-6 standard deviations, depending on the measure (sunspots vs. OSF) used. These standard deviations are for 11-year averages. The standard deviation of an average is the square root of N times smaller than that of the raw data, where N is the number of things being averaged. The standard deviations in Figure 11 are smaller than those in Figure 7 by a favor of 3.32, the square root of 11. The amplitude of an 11-year cycle of annual data points is 2.6 standard deviations. This follows because two points out of every 11 (the cycle maximum and minimum) define the cycle amplitude, while the other 9 (82%) fall closer to the trend than the cycle amplitude. On a normal bell curve, 82% of the area under the curve lies within ±1.3 standard deviations of the average (trend) value. Thus, a standard deviation in Figure 7 is 2.6 times smaller than the cycle amplitude. Since the standard deviations in Figure 11 are 3.32 times smaller than the standard deviations in Figure 7, they are also 8.62 (=3.32x2.6) times smaller than the cycle amplitude. Given a cycle size, one divides by 8.62 to obtain the temperature to standard deviation calibration needed to convert Figure 11 into a temperature effect.

So far I have four estimates for the size of the solar-driven temperature cycle, which range from 0.15º to 0.65º C. It is tempting to go with the 0.15º C value from Figure 7, since that is a “real observed effect”. Unfortunately if one looks as a continuation of Figure 7 back to 1900, one sees that over this time the neat correspondence between temperature and sunspots is no longer evident.

Figure 12. Temperature fluctuations (3 YMA) compared to the sunspot cycle 1900-1960