Some Climate Science for Nuclear Scientists Submitted to Nuclear News. Rejected on the grounds that it is not specifically about nuclear. (Never mind the editorials in NN about how nuclear will save the world from “climate change.”) Howard “Cork” Hayden, Prof. Emeritus of Physics, UConn corkhayden@comcast.net January 21, 2022
ABSTRACT Everybody in the nuclear industry is familiar with the erroneous conclusion that since power plants emit radiation it is dangerous. The problem is one of scale, or course. Exposure to radiation from a power plant is minuscule, but non-zero, and that gets mathematically challenged people up in arms. In a similar vein, it is true that adding CO2 to the atmosphere causes some global warming. Again, the problem is one of scale. Herein, we will discuss the relevant unassailable physical laws and delve into the numbers.
INTRODUCTION If we add some heat to a fixed amount of water in a well-insulated container, we can easily calculate the temperature rise. If we add some heat to (say) a conference room (with no people present) that has excellent insulated walls as well as tables, pitchers of water, chairs, draperies, slide projectors, and so forth, we can calculate the temperature rise if we account for all masses and specific heats. Now imagine adding a certain heat flux (in watts per square meter) all over the earth. The temperature rise of a square meter of puddle will be considerably different than the temperature rise of a square meter of ocean, a square meter of snow, a square meter of leaf, a square meter of sand or a square meter of rock. To do a proper evaluation, it would be necessary to know the specific heats and masses of everything, as well as the amounts of ice that could melt and the amount of water that could evaporate. Of course, if that heat flux impinges on something thin that has low thermal conductivity, the fraction of heat radiated away will be higher than the fraction that penetrates through the object. Get out your supercomputer! It is thus a fool’s errand to try to answer that question. If we turn the question around, however, we get unambiguous answers on a slide rule: If the temperature of the surface of the earth rises (say) 1ºC, how much more heat does it radiate? This simple question is the key to evaluating the results of climate models.
STEP 1: PLANETARY HEAT BALANCE Absent any significant source of energy within a planet, at equilibrium the heat the planet absorbs from the sun will equal the heat radiated to space. If the solar flux at orbit is Isun, the flux averaged over the spherical surface is Isun/4. Every planet reflects some sunlight with its own albedo (reflectivity) α, so the absorbed sunlight is
. It follows that
(1) Before going on to the next step, note that Equation 1 shows that the infrared emission to space at equilibrium depends on exactly two variables: the solar intensity at orbit and the albedo of the planet. The 1
amount of sunlight absorbed by the planet depends only on the intensity of sunlight and how much is reflected into space. Alternatively, if the radiation to space increases it can only be because the solar intensity increases or the albedo decreases. At our orbit, the intensity of sunlight is presently about 1,366 W/m2. Estimates of its variability for the last few hundred years are in the 3-10 W/m2 range. The albedo of the earth has long been measured by measuring earthshine off the moon, and the measurement is confirmed by recent satellite measurements at 30%. By contrast, the solar intensity at Venus is 2,601 W/m2, and its albedo is 75%. Interestingly Venus radiates about 162 W/m2 to space, whereas the earth radiates almost 50% more at 239 W/m2. Equation 1 is universally applicable, not only to planets in our solar system but to all other planets (without internal heat sources) orbiting other suns. The equation has a name: Planetary Heat Balance.
STEP 2: SURFACE RADIATION Returning to our answerable question, we can calculate the radiation from a body at absolute temperature T from the Stefan-Boltzmann radiation law. Applied to the surface of planet, the law says, for emissivity ε (2) In Equation 2, σ is the familiar Stefan-Boltzmann radiation constant, 5.67 × 10–8 W/(m2K). The emissivity of the earth varies a bit from place to place, and is, on average, about 94%, but many climate scientists take it to be 100% as a close-enough approximation. We will do the same. Again, let us compare our planet with Venus. According to the IPCC, the surface of the earth, at an average temperature of 289 K (16ºC), radiates 398 W/m2 of IR. Venus, with about 92 bar of atmospheric pressure, has an average surface temperature of 737 K (464ºC), and therefore radiates 16,730 W/m2.
STEP 3: THE GREENHOUSE EFFECT Let us look at two pairs of numbers mentioned above. The earth’s surface radiates 398 W/m2, yet the radiation to space is 239 W/m2. The surface of Venus radiates 2,601 W/m2, yet Venus radiates 162 W/m2 to space. Respectively, the differences between surface radiation and radiation to space are 159 W/m2 and 2,439 W/m2, and can be accounted for only by the differences in the atmospheres. The term greenhouse effect has historically been used as a compound noun to refer to a phenomenon. Only after 30 years of reports has the IPCC used the term greenhouse effect in its Sixth Assessment Report (AR6 [1]) to mean the difference between the surface radiation and the radiation to space, and assigned it a symbol, G. The phenomena by which the atmosphere reduces the IR are many and complex, but the result G is calculable by simple subtraction. (3) The simplicity of Equation 3 hides the spectral dependence because the I-values are integrals over the IR spectrum. In spectral graphs, it is important to understand that the important matter is the area, rather than the y-value. In Figure 1, the area under the smooth black curve is the total IR emission from the surface (398 W/m2, according to IPCC.) The area under the jagged black curve is the total emission to space (239 W/m2, IPCC). The area between the smooth curve and the jagged black curve is the greenhouse effect G, and is (IPCC numbers) 398 – 239 = 159 (W/m2).
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Figure 1: The spectrum of IR to space, from van Wijngaarden and Happer [2], calculated for no CO 2 (green), present CO2 (400 ppm, black), and doubled CO2 (red). The smooth black line represents the radiation from the surface of the earth. The “frequency” is the number of wavelengths per centimeter (cm–1). The ordinate is in W/m 2 *cm, so that a vertical stripe is in units of W/m2. The first measurements of this spectral data were made over Guam in 1970 with the Nimbus satellite.
Curiously, the IPCC has long used a different symbol to represent changes in the greenhouse effect, calling it “radiative forcing ΔF,” with the Δ clearly representing a change. In the current terminology, ΔF should be represented as ΔG or dG. Notice in Figure 1 that there is very little difference between the jagged black line representing CO2 at 400 ppm and the jagged red line representing doubled CO2 at 800 ppm. The net blockage of IR in the CO2 band (black line) is about 30 W/m2, and the increase to 800 ppm is about 3 W/m2. However, to maintain consistency with IPCC’s numbers, we will use their 3.71 W/m2 as the “radiative forcing.” Either amount is very small compared to the 159 W/m2 greenhouse effect, representing changes of 1.9% and 2.3% respectively. We can represent the IR situation in Figure 1 pictorially as shown in Figure 2.
Figure 2: The relationship of Equation 3, shown graphically.
An Aside The quantity G is simply calculated by subtraction, but the detailed calculation from first principles is extremely involved. Van Wijngaarden and Happer [2] calculated the curves in Figure 1 from over 300,000 spectral lines of five greenhouse gases, taking account of altitude, as both temperature and pressure affect the line widths and collision frequency. Collisions can de-activate excited CO2 molecules that would otherwise radiate, and they can activate CO2 molecules, causing them to radiate. The value of G is not the amount of IR absorbed by the atmosphere, but rather the net amount of IR absorption. Note that adding CO2 to the atmosphere increases both the absorption of IR, but also the emission of IR to outer space at high altitude.
STEP 4: DOING THE ARITHMETIC We can construct a final, very useful equation by combining Equations 1 and 3: 3
Climate Constraint Equation
(4)
Equation 4 shows a relationship between four variables: the surface temperature, the solar intensity, the albedo of the planet, and the greenhouse effect. It applies at equilibrium to any planet with a surface, with any atmosphere whatsoever, and no internal heat source, that is in orbit around any sun. It is exceptionally general and can be derived on the back of a matchbook. Since we are mostly interested in changes, let us find the differential, noting that the greenhouse effect G = GCO2 + Gother: (5) To save the reader some calculation, let us fill in known numbers in Equation 5: (6) The IPCC has long attempted to assess the equilibrium climate sensitivity (ECS), defined as the equilibrium temperature rise due to the doubling of CO2 concentration, and has generally settled on a best estimate of 3ºC rise, with the current “very likely” range of from 2ºC to 5ºC [1]. Using 3ºC in Equation 6, we get 16.4 W/m2 for the first term on the right side. The IPCC has long used ΔF = 5.35ln(C/C0) (W/m2) = 3.71 W/m2 for CO2 doubling; that value is dGCO2 in Eq. 6. If their prognostications are correct, then somehow the 12.7 (= 16.4 – 3.71) W/m2 imbalance must be made up by some combination of an increase in dGother and a decrease in albedo, as IPCC considers the solar intensity to remain constant. The reader may well wonder how a “radiative forcing” of 3.71 W/m2 can cause the surface to increase its radiation by 16.4 W/m2. Unfortunately, the IPCC has neither asked nor answered the question. Let’s press the issue a bit further. Al Gore and his followers tell us that the correlation between CO2 concentration for the last several glacial/interglacial cycles is caused by variations in CO2 concentration, which has varied less than a factor of 2. The temperature variation has been around 10ºC, which means that the variation in surface IR has been over 50 W/m2. How, pray tell, can this be caused by less than 3.7 W/m2 of radiative forcing from CO2?
THE POSITIVE FEEDBACK SCENARIO We discussed the Fool’s Errand of asking what the temperature rise of the surface would be if a certain heat flux were added. If we assume that the sun remains constant and that the albedo does not change, then an increase in greenhouse effect must be matched exactly by an increase in surface radiation. The 3.71 W/m2 from CO2 doubling, by itself, will raise the surface temperature by 0.68ºC. How, then, can IPCC assert that it is very likely (their terminology) that the surface temperature will rise between 2ºC and 5ºC? The putative answer lies in positive feedback. Warming from CO2 (not CO2 itself!) causes more evaporation and melts ice and snow, resulting in lower albedo (but without doing the proper accounting for the energy). As all engineers who have worked with feedback systems know, positive feedback leads to infinities. Heat begets more heat which begets more heat, and so forth. The feedback factor is about +4.4 (= 16.4/3.71). One unit of heat begets 4.4 units of heat, which in turn begets 4.42 units of heat, and so on. This exponential increase in heating cold be caused by any source of heat whatsoever, and would surely have done so ages ago.
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THE LESSON FOR THE NUCLEAR INDUSTRY The Climate Constraint Equation (Eq. 4) cannot be used for predicting future climate; however, it provides a constraint on models. It cannot tell what will happen, but it does tell what cannot happen. Any model that results in an unbalanced equation is wrong, pure and simple. The burden of proof lies with the modeler to show that the model results in a balanced Equation 4. Think of the cries of the anti-nuclear crowd that a fraction a millirad of radiation dose is dangerous. Now think of the decades-long daily drumbeat telling us that everything bad can be attributed to “climate change” and that you are responsible for destroying the planet. There are literally thousands of such claims and all of them are false. In time, the weather will change, and the “climate change” scare will be replaced with screams of certainty that an ice age is upon us (and we are responsible), just as it was in the 1970s. When that time comes, and it will, the proponents of nuclear power who got on the “climate change” bandwagon to promote nuclear power at the expense of fossil-fuel power will be beset with claims that “YOU LIED TO US!” The nuclear industry is in no need of such opposition. Nuclear power plants are the safest energy source of electricity on the planet, and they run around the clock. It is high time for the nuclear industry to drop the feeble “climate crisis” argument. The merits of nuclear far outweigh those of other sources, especially solar, wind and (non-existent) “other” renewables for generating electricity. Please bear in mind that every source of energy, every means of transporting energy, and many uses are regularly opposed by groups of people self-identified as environmentalists. Such groups oppose coal, nuclear, oil, natural gas, hydro, wind, solar, and geothermal. Groups oppose trucks, trains, pipelines, and power lines. Some tell you to turn down your thermostat, drive less, stay home, and turn down the lights. In other words, there is a war against energy. The nuclear industry should not join that war. Remember this: by itself, a doubled atmospheric concentration of CO2, can increase the surface temperature by only 0.68ºC. That is not a strong case for any changes whatsoever in human behavior.
REFERENCES [1] “AR6 Climate Change 2021: The Physical Science Basis,” https://www.ipcc.ch/report/ar6 /wg1/#FullReport [2] W. A. van Wijngaarden1 and W. Happer, “Dependence of Earth's Thermal Radiation on Five Most Abundant Greenhouse Gases,” https//arXiv:2006.03098v1 4 June 2020. 1/21/22
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