GLOWING BRIGHTER: The Greenhouse Effect

(pages 75-79)

“What is the use of a house if you haven’t got a tolerable planet to put it on?” Henry David Thoreau {6/4}

Of the planets of the solar system, Mars, Venus and Earth could be three sisters. Earth and Venus especially are almost exactly the same size and quite similar in composition, while Mars isn’t too different. But cold, red Mars or hot, white Venus would have been intolerable for Thoreau’s house, with only blue-green Earth just right. Why?
 
There is an easy answer—planets closer to the sun are hotter. But, Venus is especially bright in our night sky because almost all the sunlight reaching Venus is reflected away without warming the planet. A naïve calculation including this reflection would make the surface of Venus almost as cold as Mars and frozen solid, not hot enough to melt lead as documented by space probes. {6/5} Let’s take a short look at the temperature of your dinner, to see why Venus is so warm and why fossil-fuel burning will turn up the Earth’s thermostat.
 
Touch the heating element—the “burner”—of an electric stove or toaster that has been turned off for a long time, and you’ll find that the burner is the same temperature as the rest of the room. Now, turn on the stove. You are using energy to push electrons through the burner, and the resistance to their motion generates heat. Initially, that heat primarily goes into warming the burner. But, as the burner gets hotter, it begins to dump more of that extra heat into its surroundings, warming them through some combination of conduction, convection and radiation, as described next.
 
If you were silly enough now to touch the burner, you would very quickly understand how it got its name. The rapidly vibrating atoms of the burner would collide with the atoms of your skin, speeding them up and perhaps knocking them out of your skin altogether as heat is conducted into your finger to make it sizzle and smoke. If you are wiser and keep your finger away, the air right next to the burner will be heated instead, expanding and rising to warm its surroundings by convection. And, as you watch the burner, you will see that it is producing light, first a faint deep red glow, then brighter and yellower as the burner warms. This glow is called electromagnetic radiation, or just radiation for short. {6/6}
 
Eventually, if you keep supplying electricity at the same rate, the temperature of the burner will stabilize, when the burner transfers energy to the air or your dinner at the same rate that electrical energy is supplied. If you then turn the stove to a higher setting, supplying more electric energy, the burner will warm before stabilizing at a new, hotter temperature, glowing more brightly and bluer in color. The burner rather quickly reaches the temperature that is needed to send out energy at the same rate that it is received.
 
To lose heat by conduction, the burner must touch something, and to lose heat by convection, whatever the burner touches must move away. But, the radiation that you see as a red glow doesn’t require touching anything. All things that are warmer than absolute zero are always radiating energy, glowing. Warmer things glow more brightly, and have more of their glow at shorter wavelengths with higher energy, moving from beyond the infrared range where we humans can’t see, into red, orange, yellow and up to or past blue into the ultraviolet and beyond. {6/7}
 
A planet cannot lose or gain enough heat by conduction or convection to be important, because space is simply too empty—there is virtually no stuff out there to conduct or convect. So, the temperature of a planet is controlled by radiation.
 
To start, we can assume that the incoming energy and outgoing energy for a planet are equal. {6/8} The incoming energy is virtually all sunlight—the stars just aren’t close enough to matter, and the radioactive decay within the planets supplies too little energy to be important (we’ll come back to this later, because the planet’s heat can help power humanity). Venus gets more incoming energy than Earth, and Mars less, because of their distance from the sun. {6/9} Some of the sunshine reaching a planet is reflected from clouds or snow or other things and bounces right back to space without warming anything. Venus, with its thick cloud layer, reflects about 80% of the sunlight reaching it, so we say it has an albedo of 80% (“alb” means “white”, as in “albino”, so albedo is the whiteness). Earth—with dark oceans balanced out in part by puffy white clouds and gleaming snow—reflects about 30%, and the almost-cloudless Mars is darker yet with an albedo of only about 22%. The rest of the sunshine is absorbed to warm each planet, and must be radiated back to space by the glow of the planet. {6/10}
 
From the basic physics of radiation, doubling the absolute temperature of something (the number of degrees above absolute zero) causes the rate at which that thing radiates energy to go up sixteen-fold. Stated differently, a 1% increase in the absolute temperature of a thing causes a 4% increase in the energy it radiates. {6/11} Many physicists have commented on the remarkable ways in which the universe is favorable for our existence—tweak the fundamental laws or constants a little bit, and we’re not here. Because of radiation physics, it takes a huge amount of energy to raise the temperature much, so we don’t fry when the sun comes up in the morning, or freeze when the sun sets in the evening, and as a consequence, we’re here.
 
Anyway, if you figure out how much sunshine is arriving at the top of the atmosphere of the red, white and blue-green planets, subtract off the reflected part, distribute the rest around the planet, and ask what temperature is needed to radiate that much energy back to space, you get some rather surprising answers: Mars is cold at -78oF (-61oC), as expected, but Earth comes in at 0oF (-17oC), and Venus at a remarkable -63oF (-53oC). So much sunshine is reflected from Earth, and even more from Venus, that they should join Mars in being frozen. But, they don’t. The surface of Mars, beneath its thin air, is 11oF (6oC) warmer than this simple calculation would indicate. Earth, with its thicker atmosphere, also is warmed more, by 59oF (-33oC). And Venus, shrouded in its thick atmosphere, has a surface that could melt lead, 855oF (475oC) warmer than indicated by this simple calculation for radiative equilibrium.
 
The calculations done here are good, and based on good physics. So, our very simple model must have omitted something. We know this something as the “greenhouse effect”. {6/12}
 
Start by looking at Venus with a telescope, and you will see sunlight reflected from the tops of clouds, not the rocky surface. If you could put on infrared goggles, and see the longer wavelengths being emitted by Venus, you’d still be looking at the tops of clouds—“seeing” the surface of Venus through the thick clouds is very difficult. {6/13} To send out as much energy as it receives, Venus must radiate at -63oF (-53oC), but that is the temperature at the top of the clouds, not at the surface or in the center of the planet, because radiation cannot escape directly to space from those places.
 
Any mountain climber on Earth knows that a high peak is colder than the base camp, for fairly simple physical reasons. {6/14} Venus has a really “thick” atmosphere that weighs about 90 times as much as Earth’s atmosphere (the surface pressure is a bit over 1300 pounds per square inch or about 9 million Pascals), and that is primarily made of CO2. As we’ll see in a moment, the heat-trapping “greenhouse” effect of that CO2 is what makes Venus’ surface hot enough to melt lead.
 
On the other end, Mars has a tiny greenhouse effect. Look at Mars through either your regular telescope or your infrared goggles and you will still see the surface, or close to it, except when it is fuzzed out by an occasional dust storm. The radiation you see from Mars is coming from closer to the surface than on Earth or Venus. Mars’ atmosphere is mostly CO2 as on Venus, but the pressure at the surface is only 0.6% of that on Earth—not much! The greenhouse effect on Mars is still 11oF (6oC), so the thin atmosphere does a little, but not a lot.
 
Now, for Earth. Look down from a spaceship or satellite, and at times your eyes can see beautiful blue oceans, green forests, and dazzling white ice caps, but sometimes you look down on the tops of clouds or on smoke and haze from human-caused or natural fires. The perky television weather person sometimes shows you how the Earth looks in a particular infrared wavelength that is blocked by water vapor. You see the surface in dry places, but this view is blocked by water vapor in other places, and you see the little bit of radiation coming from cold places high in the atmosphere above most of the water vapor, highlighting the fascinating swirls of wet storms. At certain other wavelengths, you could see the effects of CO2, methane, ozone, or other gases, and even see the plume of CO2 coming off cities. {6/15}
 
This greenhouse interaction of the atmosphere with radiation is observed fact. We have had satellites looking down for decades, and we use their views to help forecast weather, monitor pollutants, and in other ways. The atmosphere does absorb most of the energy emitted from the Earth in certain wavelengths. {6/16} Physical understanding based on laboratory measurements and quantum-mechanical calculations does allow very accurate calculation of the radiation measured on Earth and in space. Satellites have seen the drop in radiation leaving Earth in certain wavelengths as greenhouse gases have risen, and that drop is very accurately explained by the physically calculated effects of the measured rise in greenhouse-gas concentrations. {6/3} Although the blogosphere remains muddied by counter-claims, I know of no credible scientific objection to the warming effect of rising concentrations of greenhouse gases.
 
Footnotes:
6/3 Harries, J.E., H.E. Brindley, P.J. Sagoo and R.J. Bantges, 2001, Increases in greenhouse forcing inferred from the outgoing longwave radiation spectra of the Earth in 1970 and 1997, Nature 410, 355-357. Also see Griggs, J.A. and J.E. Harries, 2007, Comparison of spectrally resolved outgoing longwave radiation over the tropical Pacific between 1970 and 2003 using IRIS, IMG, and AIRS, Journal of Climate 20, 3982-4001.
 
6/4 Thoreau, H.D., May 20, 1860, Letter to Harrison Blake, p. 360, in The Writings of Henry David Thoreau. VI. Familiar Letters, Edited by F.B. Sanborn, Houghton Mifflin and Company, Boston, 1906.
 
6/5 See Venus, in Spencer Weart’s The Discovery of Global Warming, American Institute of Physics, http://www.aip.org/history/climate/Venus.htm, accessed May 31, 2010.
 
6/6 The radiation from an atomic bomb includes both electromagnetic radiation and rapidly moving particles; the radiation from your burner is just the electromagnetic part, unless you have a very strange burner! We’ll return to radiation when we discuss nuclear power later in the book.
 
6/7 In case it bothers you that shorter wavelengths have higher energy, try this: get a friend to hold the other end of a jump rope or other short rope, then move your hand up and down to make waves. If you move your hand slowly, all of the rope will go up and then down, for a very long wave. Move your hand faster, and you can get two wiggles between you and your friend, with the center of the rope hardly moving while the part of the rope closer to you than the center moves one way and the rope farther away moves the other. Move still faster, and you can get more waves along the length—and you’ll find your hand getting tired quickly. You must be more energetic to get the shorter waves.
If the idea of everything always radiating bothers you, note that bodies at the same temperature simply trade energy with no net effect; bodies at different temperatures drift towards a common temperature as the warmer body radiates more to the colder body than vice-versa.
 
6/8 Recall how much warming occurs over a few hours from the cold just before dawn to the heat of mid-afternoon. Then, consider this back-of-the-envelope calculation. To raise the temperature of 1 kg of dry air at sea level at constant pressure requires approximately 103 J/K, where I’m using “K” for “degrees Kelvin”, which is the same as “degrees Celsius” in this usage. The weight of the atmosphere sitting over 1 square meter of the Earth at sea level is about 104 kg/m2. Thus, to warm the air over 1 square meter by 1 degree Kelvin takes about 107 J, giving 107 J/m2/K. The incoming sunlight supplies about 240 W/m2, and multiplying by the 3.16x107 s in a year yields 7.6x109 J/m2/yr. Finally, dividing 7.6x109 J/m2/yr by 107 J/m2/K yields 760 K/yr, or multiplying by 1.8 to convert to oF, 1400oF/yr. That’s a lot of warming. In turn, we can have high confidence that outgoing energy is very close to incoming energy, because the atmosphere is certainly not warming by 1400 degrees F per year!
However, because the Earth is warming from the human-caused increase in CO2, as described in the next chapters, incoming energy is slightly greater than outgoing now. Decadally averaged at the turn of the millennium, the human-caused rise in greenhouse gases was causing the Earth to receive ~0.75 W/m2 more from the sun than was going back to space (see Hansen, J., L. Nazarenko, R. Ruedy, M. Sato, J. Willis, A. Del Genio, D. Koch, A. Lacis, K. Lo, S. Menon, T. Novakov, J. Perlwitz, G. Russell, G.A. Schmidt and N. Tausnev, 2005, Earth’s energy imbalance: Confirmation and implications, Science 308, 1431-1435). Only ~0.04 W/m2 of this was going into warming the air, with the great majority going into the ocean. This recent energy imbalance is actually quite large; the imbalance needed to melt the great ice sheets of the ice age over 10,000 years was only ~0.1 W/m2, following the calculations of Hansen et al. in the reference just above. Overall, while incoming and outgoing energy are not perfectly equal, and the planet warms when incoming is larger but cools when outgoing is larger, the two are not hugely different.
 
6/9 If you were lying on the sun deck of your spaceship in orbit right next to Mars, being twice as far from the sun as at Venus would mean that you would take about four times as long to get a sunburn. Only the tiniest bit of the sunlight reaching the orbit of Venus is blocked by Venus or Earth or dust or comets before reaching Mars. But, the sunlight is spreading in all directions. Think of the sun suspended at the center of a balloon. Initially the balloon is blown up just big enough to touch the orbit of Venus, with the sunlight illuminating the inside of the balloon. Next, inflate the balloon, making it stretch in all directions until it touches the orbit of Mars. The distance from the sun to the surface of the balloon is r, the energy leaving the sun is S, and the surface area illuminated is 4πr2; the brightness of the sunshine is then S/4πr2. This is the energy available at the top of the atmosphere of a planet. This amounts to about 592 W/m2 at Mars, 1370 W/m2 at Earth, and 2460 W/m2 at Venus. Remember that you personally internally generate about 100 W by burning food, and 1 m2 is just over 3 feet by 3 feet, or a small table top, so sunlight is equal to the energy from 6 of you on a table top at the orbit of Mars, 14 at Earth, and 25 at Venus.
 
6/10 Take the incoming radiation at the top of the atmosphere, divide by 4 because the energy must be spread around the spherical planet, and multiply by (1-A), where A is the reflected part or albedo so 1-A is the absorbed part, and you express albedo as a decimal so that an albedo of 30% means A=0.3. The energy to warm the surface is then 115 W/m2 for Mars, 240 W/m2 for Earth, and 132 W/m2 for Venus. This is about the internal energy of one person per tabletop on Mars and Venus, and 2 on Earth. So much of the sun’s energy is reflected from Venus that it has much less energy available to heat its surface than Earth does!
 
6/11 The rate at which energy is radiated away is equal to sT4, where s=5.67x10-8 W/m2/K4 is the Stefan-Boltzmann constant and T is the absolute temperature in degrees Kelvin, indicated K. Setting this equal to the absorbed energy at the surface, assuming incoming equals outgoing energy, and solving for T gives the equilibrium radiative temperature.
In the main text, I stated that a two-fold increase in absolute temperature gives a sixteen-fold increase in emitted radiation, so let’s do that first. If you let the energy radiated be e, then at some temperature T1, the emitted energy e1=sT14. At temperature T2=2T1, the energy radiated is e2=sT24=s(2T1)4=16sT14=16e1. Thus, doubling the absolute temperature increases the outgoing radiation 16-fold.
The text also states that a 1% change in absolute temperature causes a 4% change in outgoing radiation. This isn’t absolutely exact; more properly, an infinitesmal relative change in temperature causes a four-times-larger relative change in outgoing radiation, but 1% is small enough for this to be a good approximation. To check, you could just plug in a value for temperature—say, 255oK for equilibrium radiation for the modern Earth—and calculate the outgoing radiation using the Stefan-Boltzmann law given above, finding 240.16 W/m2. Repeat with a temperature 1% higher (257.55oK), and you’ll find 249.92 W/m2, an increase of ((249.92-240.16)/240.16)*100% = (9.76/240.16)*100% = 4.06%, pretty close to 4%.
If you want to do it right, note that the percentage increase is either dT/T for temperature or de/e for energy, multiplied by 100%, with dT indicating a small change in T, and de a small change in e. If you happen to know calculus, you can take the derivative of the Stefan-Boltzmann law with respect to temperature, de/dT=4sT3. Next note that sT3= sT4/T=e/T from the Stefan-Boltzmann law, so de/dT=4sT3=4e/T. Rearrange, and you have de/e=4dT/T, which says that a small relative change in absolute temperature gives a four-times bigger relative change in energy emitted, or a 1% change in absolute temperature gives about a 4% change in emitted energy.
If you don’t know calculus, you might write the Stefan-Boltzmann law for some temperature as e=sT4, and for a slightly higher temperature as e+de =s(T+dT)4, with de and dT again indicating small changes to e and T, respectively. If you expand (T+dT)4, you end up with (T+dT)4=T4+4T3dT+(additional terms that have higher powers of dT). But, if dT is small, (dT)2 is much smaller, and (dT)3 and (dT)4 are truly tiny, so those “additional terms” just don’t matter much. Thus, to good approximation, e+de=s(T4 +4T3dT). If you subtract the original Stefan-Boltzmann law from this, subtracting e from the left and the equivalent sT4 from the right, you are left with de=4sT3dT. If you then divide by e on the left and by the equivalent sT4 on the right, you get de/e=4dT/T, the same as we got using the calculus, and again showing that small relative changes in absolute temperature cause four-times larger relative changes in radiated energy. Whether you do it by examples, or calculus, or expansion and approximation, you get the same answer—we are fortunate that radiation physics keeps temperature changes small.
 
6/12 Some of our friends in meteorology HATE it when the warming effect of gases in the atmosphere is called the greenhouse effect. The glass in a greenhouse does affect radiation in the same way as “greenhouse” gases, letting visible light enter but blocking the exit of infrared radiation (Silverstein, S.D., 1976, Effect of infrared transparency on the heat transfer through windows: A clarification of the greenhouse effect, Science 193, 229-231). But, this gives only part of the warming in the greenhouse. The greenhouse is warmed primarily because the glass stops convection—when the sun heats the ground, which heats the air near the ground, outside a greenhouse the hot air rises in a convection cell that takes the heat away, but inside the greenhouse this motion is blocked by the glass. Unfortunately, use of the term “greenhouse effect” for the effects of certain gases on radiation is so widely known that that we couldn’t change the terminology if we wanted to, and we haven’t found a really good alternative. So, if you happen to be concerned about this, sorry, but that’s life, unless you can come up with a better idea and convince a lot of people that your idea is better.
 
6/13 Very long wavelength radar waves can get through, but Venus doesn’t send out much energy at those wavelengths.
 
6/14 Take some volume of air, inside an imaginary balloon. Warm the air a little so that the balloon floats upward. As it rises, there is less air above it, so the air pressure that is squeezing it drops, so it expands. But, the expansion requires that it push some surrounding air out of the way. The energy to do this work comes from the internal energy—the heat—in the initial air volume. So, provided that there are not other major sources of energy, the air cools as you go upward in a convection cell. Regions that are frequently affected by convection will follow this rule, even if at a given moment a convection cell is not active.
 
6/15 European Space Agency, SCIAMACHY satellite instrument, http://www.esa.int/esaEO/SEMZHVM5NDF_index_0.html, accessed October 12, 2009.
 
6/16 These same wavelengths are absorbed in energy coming down from the sun. But, at these wavelengths, the Earth supplies more energy to the atmosphere than the sun does. The total energy coming down from the sun equals the total energy going out from the Earth, but the sun sends most of its energy in the shorter, visible wavelengths whereas the Earth sends most of its energy in the longer, infrared wavelengths that interact with greenhouse gases.