The greenhouse effect is key concept in physics that has enabled the development of life on Earth but, in its enhanced form, also threatens our civilisation and the libes of many people and animals. We will learn now the greenhouse effect helps to maintain life on Earth, and how human activities can enhance the effect. We will also look at atmosphere can be modelled as a system to quantify the Earth-atmosphere energy balance.
Conservation of energy is a fundamental concept in physics. In the last topic we looked at the concept of internal energy and how the temperature of an object is a measurement of the average kinetic energy of the molecules in the object. Considering the Earth as an object, the Sun acts as an energy input to the Earth via the heat transfer method of radiation. As a hot object the Earth is also outputting energy into space through radiation. If the system is in equilibrium, input equals output then the internal energy and the temperature of the Earth will be unchanged. If, however, the input exceeds the output then the internal energy, and therefore the temperature, of the Earth will be increasing. Increasing the temperature will also increase the output (by the Stefan-Boltzmann law) but the new equilibrium will be achieved at a higher temperature.
The Earth has a hot core, kept hot by the radioactive decay of heavy elements, but little of that heat makes its way to the surface. The sun's energy output (its apparent brightness as we learned in the previous topic) is the only significant energy input to the surface of the Earth. The apparent brightness of the sun at the distance of the Earth is called the solar constant, symbol S and with units of watts per metre squared. The value of this constant is determined by luminosity of the sun and the distance of the Earth from the Sun using the equation from the previous topic, b = L / 4.𝛑.d².
L☉, the solar luminosity, is about 3.828×10^26 W. Meanwhile d is the distance between Sun and Earth, the radius of the Earth's orbit, which is a quantity known as one astronomical unit (AU). One AU is defined as 149 597 870 700 m. Using these figures, the solar constant is about 1 361 kW / m². This value varies slightly throughout the year as the Earth's orbit is not exactly circular. It also varies by about 0.1% on an 11-year cycle due to variations in the output of the sun, something known as the solar cycle.
The solar constant, the watts per metre squared of energy at the radius of the Earth's orbit and applies to a surface at right-angles to the Sun-Earth line. To determine the average amount of energy incident on the Earth we need to consider the geometry of the Earth. The total amount of energy that hits the Earth is determined by the cross-sectional area of the Earth (π r²). That energy is distributed over the entire surface area of the Earth (approximately a sphere, so 4π r²). Since π r² / 4π r² equals 1/4, the average power incident at the top of the Earth's atmosphere per metre squared is 1/4.S or about 340 watts. Again, this is measured at the top of the Earth's atmosphere and is an average. During the equinoxes (in March and September) when Sun shines directly down on the Earth's equator, the equatorial areas of the Earth receive the full solar constant of 1361 watts per metre squared, while the poles get amost no energy. This explains why the Earth's equatorial regions are the hottest while the poles are cold. the varying tilt of the Earth throughout the year, most extreme at the solstices in June and December, is the cause of the Earth's seasons.
The solar constant, S, as described above, describes the power incident per square metre at the top of the Earth's atmosphere. Since most of the solar radiation is in the visible part of the spectrum, and the Earth's atmosphere is transparent to visible light, most of the incident radiation can pass through the Earth's atmosphere. However, during the passage through the atmosphere and on reaching the surface some of the radiation is reflected or scattered away into space. The percentage of the incident radiation that is scattered is determined by the albedo of the surface, a property of the surface hit by the radiation. The definition of albedo is:
albedo = total scattered power / total incident power
As a ratio, albedo has no units.
The albedo of the Earth varies daily and is dependent on weather and location. Fresh snow has an albedo of up to 0.8, clouds between 0.4 and 0.8. Most land is between 0.08 and 0.28 and water, most of the Earth's surface, has an albedo of about 0.08. The average albedo of the Earth is about 0.3. This means that, on average, about 30% of the incoming solar radiation is scattered back into space, reducing the total amount of radiation input into the Earth-atmosphere system.
The Sun's luminosity determines the apparent brightness of the sun and hence the solar constant. An assumption we can make for the Sun is that it is an almost perfect emitter of radiation. Perfect emitters are known as black-body objects. Perfect emitters of radiation are also perfect absorbers of radiation and a perfect absorber of radiation looks black unless it is hot enough to visibly glow, hence the term black-body. However, since the Earth has a temperature about absolute zero it is also an emitter of radiation. This is the only output of energy from the Earth system. Stars are a reasonably good approximation to black-body emitters, but planets are not. The ratio between power radiated by a perfect black-body and the power radiated by an imperfect, real-life, emitter is known as the emissivity of the object. As a ratio, emissivity doesn't have a unit. As an equation, the formula for emissivity is:
emissivity = power emitted per unit area / σ.T⁴
where power per unit area is in units of watts per metre squared, T is temperature measured in kelvin and sigma (σ) is the Stefan-Boltzmann constant, 5.67×10^-8 W/m²/K⁴.
Emissivity is important in this topic because the only way the Earth can emit energy is through radiation and the emissivity of the Earth is a major factor in the amount of energy it can emit per unit time.
It is easy to confuse emissivity and albedo. Recall that emissivity is related to the emission of radiation, while albedo is related to the absorption and reflection of radiation.
In an equilibrium system the total input energy must equal the total output energy. How can we use the information given above to determine the average temperature of the Earth?
Input energy: The solar constant, S, described above has an average value of 340 watts per metre squared. However, the average albedo of the Earth is approximately 0.3. If 30% of the incident radiation is scattered then the actual energy absorbed by the Earth is (340 × 0.7 =) = 238 W/m².
Output energy: The Earth is emitting energy, and the amount of energy emitted is determined by the Stefan-Boltzmann law: L = σAT⁴. Rearranging, the output per square metre of the Earth is L/A = σT⁴.
Balancing the energies: Since we are describing an equilibrium, the input must equal the output. Therefore, 238 W/m² = σT⁴. Rearranging to find the temperature T we get T = the fourth root of 238 / σ. Since σ is 5.67×10^-8 W/m²/K⁴ , this gives a result of about 254 K or -18 ºC. This is a surprising result because the Earth isn't that cold on average in reality. The emissivity of the surface of the Earth, on average, is about 0.95 which would make the equilibrium temperature about 257 K or -15 ºC. Warmer but still not close to the real value of about 288 K or 15 ºC.
The solution to this paradox is that not all of the Earth's emitted radiation vanishes into space. Instead some of that emitted radiation is absorbed by the Earth's atmosphere and then, partially re-emitted back down toward the surface of the Earth, keeping it warmer than expected. This is the greenhouse effect.
The actual average surface temperature of the Earth is about 288 K. Using this value and Wien's displacement law we can calculate the maximum or peak wavelength of this emission. That works out to be about 1×10^-5 m, which is well into the infrared. We previous claimed that the Earth's atmosphere is transparent to visible light, after all we can see the sun and the stars, but that is not true in the infrared. Certain gases in the Earth's atmosphere, known as greenhouse gases, are significantly opaque to infrared radiation. This absorbed energy warms the atmosphere. The warmer atmosphere reradiates energy in all directions. Some of that energy is emitted back down to the ground, keeping it warmer than it would be otherwise.
Significant greenhouse gases in the Earth's atmosphere include methane (CH4), water vapour (H20), carbon dioxide (CO2) and nitrous oxide (N2O). These gases are good at absorbing infrared radiation due to two phenomena. All molecules have energy levels (equivalent to atomic energy levels you will meet in topic E: Nuclear and quantum physics), but are less precisely defined. For the greenhouse gases these molecular energy levels coincide with sections of the infrared part of the EM spectrum. The other physical process that leads to absorption of infrared range energies is resonance. Greenhouse gases have structures that can vary, such as a double bond with one atom and a single bond with another, and the structure of the gases can vary, or resonate, between the two forms. This absorbs energy at certain wavelengths which, for greenhouse gases, corresponds to EM radiation in the infrared region.
Simulation: A PhET simulation of the Greenhouse Effect.
The greenhouse effect is a natural consequence of the presence of greenhouse gases in the atmosphere, all of which are present due to natural effects of the Earth's biological and geological cycles. This is overwhelmingly a good thing. Without greenhouse gases the average temperature of the Earth would be as low as -18 to -15 ºC as described above. Far too cold for human or indeed almost any life to survive.
However, the effects of human industrial and agricultural actions (particularly the burning of fossil fuels and the release of previously trapped carbon as carbon dioxide) have led to increasing amounts of greenhouse gases in the atmosphere. This has lead to the enhanced greenhouse effect and a corresponding increase in temperature. If the level of greenhouse gases reaches a new equilibrium then the global average temperature will also reach a new equilibrium, but as long as the amount of greenhouse gases increases then so will global average temperatures.
Energy production statistics for the UK and the USA. Climate impact maps for electrcity production.
Image credits:
Banner Sourav Mohapatra, CC BY 2.0 <https://creativecommons.org/licenses/by/2.0>, via Wikimedia Commons