1. Earth’s Equilibrium TemperatureIf Earth had no atmosphere, and
therefore no greenhouse effect, the mean temperature of the planet would
be much colder. Using the principles discussed thus far, we can easily
estimate the magnitude of the greenhouse effect. To do so, we will
compute the equilibrium temperature for a planet having no atmosphere.
By comparing the computed and observed temperatures, we will see how the
atmosphere influences Earth’s temperature.First, assume the planet acts
as a blackbody with regard to longwave radiation, that the planetary
albedo is 30 percent, and that the solar constant is 1361 watts per
square meter (W/m2). If Earth were a flat disk perpendicular to incoming
radiation, each square meter would receive 1361 joules per second
(J/sec). But Earth is not a flat disk; it is a sphere, the surface area
of which is four times larger than that of a disk of the same radius.
Thus, the intensity of radiation averaged over the sphere is one-fourth
as large as for the imaginary disk. Because of this, each square meter
of Earth receives 1361/4, or 340 W/m2, (These 340 watts per square meter
are the 100 units of solar radiation discussed earlier in this chapter
in the section titled The Fate of Solar Radiation.) Given the planetary
albedo of 30 percent, it must be that 70 percent of this incoming
radiation is absorbed. In other words, total absorbed radiation is...The
planet must lose exactly as much energy as it gains, and the intensity
of radiation for a blackbody is determined by rearranging and applying
the Stefan-Boltzmann law. Recall that the Stefan-Boltzmann law for a
blackbody states that...We know, however, that the intensity of
radiation for the planet without an atmosphere must be 238.2 W/m2. We
then rearrange the equation to solve for T, rather than I, to
get...which can be reduced to...Using the values σ = 5.67 × 10−8 W/(m2
K4) and I = 238.2 W/m2, the equilibrium temperature works out to 254.6 K
(−18.6°C, −1.4°F). Thus, the mean temperature of Earth would be far
colder without an atmosphere.Note that our calculation is highly
simplified and somewhat questionable. For example, we used 30 percent
for the planetary albedo, but that value arises in part from the albedo
of the atmosphere, which our imaginary planet lacks. Should we therefore
have used present- day surface albedo in the computation? Perhaps, but
surface albedo on the real Earth is in part the result of temperature,
the very thing we are trying to compute! Ideally, we would treat albedo
as a variable, allowing it to respond to changes in temperature.We see
that even a beginning question about the global effect of greenhouse
gases raises complications that are not easy to address with a simple
model. Given this, it is not surprising that realistic computer models
of atmospheric behavior are enormously complex, requiring huge computer
resources. Nonetheless, computer models have become indispensable for
daily weather forecasting (see Chapter 13) and tell us much of what we
know about potential climatic changes due to human activity (see Chapter
16).Outline the general process of determining Earth’s equilibrium
temperature. Get solution
2. Earth’s Equilibrium TemperatureIf Earth had no atmosphere, and therefore no greenhouse effect, the mean temperature of the planet would be much colder. Using the principles discussed thus far, we can easily estimate the magnitude of the greenhouse effect. To do so, we will compute the equilibrium temperature for a planet having no atmosphere. By comparing the computed and observed temperatures, we will see how the atmosphere influences Earth’s temperature.First, assume the planet acts as a blackbody with regard to longwave radiation, that the planetary albedo is 30 percent, and that the solar constant is 1361 watts per square meter (W/m2). If Earth were a flat disk perpendicular to incoming radiation, each square meter would receive 1361 joules per second (J/sec). But Earth is not a flat disk; it is a sphere, the surface area of which is four times larger than that of a disk of the same radius. Thus, the intensity of radiation averaged over the sphere is one-fourth as large as for the imaginary disk. Because of this, each square meter of Earth receives 1361/4, or 340 W/m2, (These 340 watts per square meter are the 100 units of solar radiation discussed earlier in this chapter in the section titled The Fate of Solar Radiation.) Given the planetary albedo of 30 percent, it must be that 70 percent of this incoming radiation is absorbed. In other words, total absorbed radiation is...The planet must lose exactly as much energy as it gains, and the intensity of radiation for a blackbody is determined by rearranging and applying the Stefan-Boltzmann law. Recall that the Stefan-Boltzmann law for a blackbody states that...We know, however, that the intensity of radiation for the planet without an atmosphere must be 238.2 W/m2. We then rearrange the equation to solve for T, rather than I, to get...which can be reduced to...Using the values σ = 5.67 × 10−8 W/(m2 K4) and I = 238.2 W/m2, the equilibrium temperature works out to 254.6 K (−18.6°C, −1.4°F). Thus, the mean temperature of Earth would be far colder without an atmosphere.Note that our calculation is highly simplified and somewhat questionable. For example, we used 30 percent for the planetary albedo, but that value arises in part from the albedo of the atmosphere, which our imaginary planet lacks. Should we therefore have used present- day surface albedo in the computation? Perhaps, but surface albedo on the real Earth is in part the result of temperature, the very thing we are trying to compute! Ideally, we would treat albedo as a variable, allowing it to respond to changes in temperature.We see that even a beginning question about the global effect of greenhouse gases raises complications that are not easy to address with a simple model. Given this, it is not surprising that realistic computer models of atmospheric behavior are enormously complex, requiring huge computer resources. Nonetheless, computer models have become indispensable for daily weather forecasting (see Chapter 13) and tell us much of what we know about potential climatic changes due to human activity (see Chapter 16).How does Earth’s equilibrium temperature compare to its actual temperature near the surface? Why is there a discrepancy between the two temperatures? Get solution
2. Earth’s Equilibrium TemperatureIf Earth had no atmosphere, and therefore no greenhouse effect, the mean temperature of the planet would be much colder. Using the principles discussed thus far, we can easily estimate the magnitude of the greenhouse effect. To do so, we will compute the equilibrium temperature for a planet having no atmosphere. By comparing the computed and observed temperatures, we will see how the atmosphere influences Earth’s temperature.First, assume the planet acts as a blackbody with regard to longwave radiation, that the planetary albedo is 30 percent, and that the solar constant is 1361 watts per square meter (W/m2). If Earth were a flat disk perpendicular to incoming radiation, each square meter would receive 1361 joules per second (J/sec). But Earth is not a flat disk; it is a sphere, the surface area of which is four times larger than that of a disk of the same radius. Thus, the intensity of radiation averaged over the sphere is one-fourth as large as for the imaginary disk. Because of this, each square meter of Earth receives 1361/4, or 340 W/m2, (These 340 watts per square meter are the 100 units of solar radiation discussed earlier in this chapter in the section titled The Fate of Solar Radiation.) Given the planetary albedo of 30 percent, it must be that 70 percent of this incoming radiation is absorbed. In other words, total absorbed radiation is...The planet must lose exactly as much energy as it gains, and the intensity of radiation for a blackbody is determined by rearranging and applying the Stefan-Boltzmann law. Recall that the Stefan-Boltzmann law for a blackbody states that...We know, however, that the intensity of radiation for the planet without an atmosphere must be 238.2 W/m2. We then rearrange the equation to solve for T, rather than I, to get...which can be reduced to...Using the values σ = 5.67 × 10−8 W/(m2 K4) and I = 238.2 W/m2, the equilibrium temperature works out to 254.6 K (−18.6°C, −1.4°F). Thus, the mean temperature of Earth would be far colder without an atmosphere.Note that our calculation is highly simplified and somewhat questionable. For example, we used 30 percent for the planetary albedo, but that value arises in part from the albedo of the atmosphere, which our imaginary planet lacks. Should we therefore have used present- day surface albedo in the computation? Perhaps, but surface albedo on the real Earth is in part the result of temperature, the very thing we are trying to compute! Ideally, we would treat albedo as a variable, allowing it to respond to changes in temperature.We see that even a beginning question about the global effect of greenhouse gases raises complications that are not easy to address with a simple model. Given this, it is not surprising that realistic computer models of atmospheric behavior are enormously complex, requiring huge computer resources. Nonetheless, computer models have become indispensable for daily weather forecasting (see Chapter 13) and tell us much of what we know about potential climatic changes due to human activity (see Chapter 16).How does Earth’s equilibrium temperature compare to its actual temperature near the surface? Why is there a discrepancy between the two temperatures? Get solution
