This is a very simple explanation of the greenhouse effect with a suitably simple diagram. An explanation of the warming effect of greenhouse gases follows with a simple model. Then follows a slice of history about the origins of the greenhouse effect and the part played in it by John Tyndall.

The Greenhouse Effect

The Earth's atmosphere is moderately trans­parent in the visible part of the spectrum and the majority of the solar radiation can pass through the atmosphere without being absorbed and is absorbed by the surface which is thus warmed. On the other hand, minor atmospheric constituents, the greenhouse gases, of which water vapor is the most important, ab­sorb strongly in the infrared region which is where the Earth's surface emits. The atmosphere is largely opaque to the terrestrial infrared heat radiation.

What happens when the atmosphere absorbs radiation emitted from the surface of the planet? The atmosphere cannot steadily accumulate energy or it would become hotter and hotter. Instead, it emits radiation at the same rate as it absorbs when it has become hot enough to establish radiative equilibrium. The radiation is reemitted in all directions, and a substantial part of it is intercepted and absorbed by the surface. So the surface of the planet is heated not only by direct sunlight but also by infra­red radiation emitted by the atmosphere. For this reason the surface of a planet must radiate away more energy than it receives directly from the Sun, and the surface can have a temperature that exceeds the effective temperature of the planet. This is the greenhouse effect in simple outline. The details are far from simple and references to papers are available on subsequent pages.

 
That the atmosphere is warmed by the presence of greenhouse gases can be understood in terms of an analogy based upon tanks with variable overflows. The height of the overflow represents the effectiveness of the greenhouse gases as they hinder the escape to space of terrestrial radiation. One fundamental point is the when any particular system has come to equilibrium, there must be radiative equilibrium between the incoming solar radiation and that escaping to space. Otherwise the system would either be warming up or cooling down. The diagram below compares two tanks with one overflow higher than the other.
 
 
 

 

 In the left-hand tank there is a particular concentration of CO2 that assists in the positioning of the overflow. With a constant flow of sunlight into the tank, at equilibrium when the overflow is equal to the incoming sunlight, there is a flow to space of infrared radiation from the planet of equal intensity. Solar energy entering = IR1. This equilibrium occurs when the tank is sufficiently full of energy to cause overflow and the energy in the tank at that stage determines the temperature of the system.

In the right-hand tank the concentration of CO2 has been increased, placing the overflow at a higher level. This has the consequence of making the thermal reservoir, the heated system, warmer. Note that at radiative equilibrium the above equation still stands;

Solar energy entering = IR2 = IR1

There is more energy contained within the system which is then warmer.

The tank analogy can be extended to the extreme case where there are no greenhouse gases present in the atmosphere so that the Earth's surface can radiate energy directly to space without hindrance. This is shown in the next diagram.

 

 

 

 

Here the 'overflow' is place at the bottom of the tank so that any solar radiation entering has an unhindered escape. Again radiative equilibrium is attained:

Solar radiation entering = IR0 = IR1

In this case the thermal reservoir is empty and the planet would be very cold with a temperature equal to its emission temperature of 255 K.

The Bohren & Clothiaux simple radiative model

A simple radiative model demonstrating the main features of greenhouse warming is described in Fundamentals of Atmospheric Radiation, Bohren & Clothiaux, Wiley-VCH, 2006. A graphic of their model is shown in the diagram below showing a surface and an atmospheric slab. The atmosphere is completely transparent to incoming solar radiation; intensity S. The atmosphere absorbs a fraction of the terrestrial radiation, the remainder being transmitted directly to space. The temperature of the atmosphere is a uniform TA and the temperature of the surface is a uniform TS. Some approximations are required. The absorptivity (1 - τ) of the atmosphere slab is α, as averaged over the spectral range of terrestrial radiation [τ is the transmisivity or transmission of the slab, 100τ is its percentage transmission]. Using the ideal version of Kirchhoff's law the absorptivity can be equated with the emissivity of the atmospheric slab; α = ε. The law applies to individual frequencies at equilibrium, but this use is for the range of frequencies emitted by the surface.

 

 

The relationships between τ, σ and ε are: τ = 1 - α, α = ε, and τ = 1 - ε.

For radiative balance, the equation for space is:

S = (1 ‒ ε)σTS4 + εσTA4

The equation for the atmosphere slab is:

εσTS4 = 2εσTA4

Eliminating TA gives:

S = (1 ‒ ε)σTS4 + ½εσTS4

Rearranging gives:

TS4 = 2S/σ(2 ‒ ε)

The downward flux density is given by:

F↓ = σTA4 = εS/(2 ‒ ε)

This result indicates that the downward flux density for an atmosphere slab that absorbs terrestrial radiation completely [ε = α = 1] is given by S and if the slab is transparent to the radiation [ε = α = 0] F↓ = 0. Downward emission requires the emissivity of the atmosphere slab to be > 0. An absorbing atmospheric slab must emit radiation downwards (as well as upwards).

The total radiation absorbed by the surface is given by:

S + F↓ = S + εS/(2 ‒ ε) = 2S/(2 ‒ ε)

For the two extreme values of ε, this gives the total radiation absorbed by the surface as S if there is no downward flux density and 2S if the downward flux operates fully.

For a situation nearer to that on Earth the radiant flux density absorbed by the surface would be between the two limiting values of S and 2S. Taking the value of S to be the 235 W m-2 observed to be the mean emission flux density from the Earth to space, the surface temperature for the lower limit would be (235/σ)1/4 = 253.7 K.

That for the higher limit would be (2 × 235/σ)1/4 = 301.7 K

This shows that with an atmosphere that absorbs a fraction of the radiation emitted by the Earth's surface the surface temperature would be higher than its emission-to-space temperature of 253.7 K, but lower than 301.7 K. The generally accepted global mean surface temperature is 288.2 K, but that is with all factors operating including the very important non-radiative transfers from the surface to the atmosphere: water evaporation and thermal transfers that have cooling effects. The model assumes that all the absorbed solar radiation is absorbed by the surface. For the Earth, the 235 W m-2 absorbed from solar radiation is distributed between the stratosphere, troposphere and surface as 10 W m-2, 57 W m-2 and 168 W m-2 respectively.

The conclusion from the simple model is that there cannot be a greenhouse effect unless the atmosphere absorbs terrestrial radiation and that it operates by emitting a considerable downward flux density to be absorbed by the surface. The atmosphere impedes the passage of energy from the surface to space and warms the surface, but all the energy for that process originates in the Sun.

If ε > 0 greenhouse warming is inevitable!

 

 

 

The Greenhouse Effect & John Tyndall; a bit of history

John Tyndall (1820-1893) conducted experiments at the Royal Institution on the radiative properties of various gases. He discovered the vast differences in the abilities of 'perfectly colorless and invisible gases and vapours' to absorb and transmit radiant heat. The 'elementary gases,' oxygen, nitrogen, and hydrogen, were almost transparent to radiant heat, while more complex molecules, even in very small quantities, absorb much more strongly than the atmosphere itself.

 

He identified the importance of atmospheric trace constituents as efficient absorbers of long wave radiation and as important factors in climatic control. Specifically, he established beyond a doubt that the radiative properties of water vapour and carbon dioxide were of importance in explaining meteorological phenomena such as the formation of dew, the energy of the solar spectrum, and possibly the variation of climates over geological time. 

 

He concluded that 'The solar heat possesses the power of crossing an atmosphere; but, when the heat is absorbed by the planet, it is so changed in quality that the rays emanating from the planet cannot get with the same freedom back into space. Thus the atmosphere admits of the entrance of the solar heat, but checks its exit; and the result is a tendency to accumulate heat at the surface of the planet.' 

 

He recognized that water vapour, among the constituents of the atmosphere, was the strongest absorber of radiant heat and was the most important gas controlling the Earth's surface temperature. He proposed the analogy that 'The aqueous vapour constitutes a local dam,

by which the temperature at the earth's surface is deepened; the dam, however, finally

overflows, and we give to space all that we receive from the sun.' He recognized the importance of radiative equilibrium in which the Earth receives energy from the sun and emits the same amount of energy to space so that the system has a long-term balance and a steady, but within limits a very variable climate. 

He gave credit to his predecessors Saussure, Fourier, and Pouillet, among others, for the intuition that "the rays from the sun and fixed stars could reach the earth through the atmosphere more easily than the rays emanating from the earth could get back into space." The experimental verification of this phenomenon was done by Tyndall. 

Tyndall's dam analogy can be exemplified by what happened to the Colorado River. Its flow rate is around 620 cubic metres per second. The building of the Hoover Dam caused the formation of Lake Mead which contains a maximum of 35.2 cubic kilometers of water. That works out at 1.8 years of flow. An equivalent calculation for the atmosphere reveals a thermal reservoir that would be filled by about four months of absorbed sunlight. Additional CO2 would add to this reservoir, just as making the Hoover Dam higher would increase the size of Lake Mead. If the dam should burst, the water flow would be unchanged, but the 'thermal reservoir' would be empty. Likewise, a world without greenhouse gases would be very cold.

The troposphere has a negative lapse rate because it contains radiatively active gases and they are the cause of global warming

Without any radiatively active content the troposphere would be isothermal; i.e., the lapse rate would be zero. That a column of gas without any radiatively active content would be isothermal has been a topic of argument and is regarded as a paradox by some authors, unbelieved by others.

In the absence of any greenhouse gases the emission level of the Earth would be its surface and the troposphere would adopt the same temperature as that of the surface. The mean black body temperature of the Earth’s surface would be ~255 K to achieve radiative balance with space, some 33 K cooler than it is observed to be.

Treating the troposphere as an isolated system; one which does not permit the transfer of mass or radiation from or to the system from its surroundings, there would be no reason why it should not be isothermal. The gravitation of the Earth would cause a pressure gradient. Intermolecular forces and random collisions would allow the Maxwell-Boltzmann distribution of molecular speeds to be established at all levels, but the system will be isothermal.

The reason for this conclusion arises from the 2nd law of thermodynamics which indicates that heat cannot spontaneously flow from a colder body to a hotter one. It can and does flow spontaneously from a hotter body to a colder one, because there is a positive change in the entropy of the system.

The classical thermodynamic treatment of entropy change (ΔS) allows the conclusion to be reached. If, say for simplicity, two halves of the isolated gaseous system did have different temperatures, T1 and T2, such that T2 > T1 and if an amount of heat represented by ΔQ were to transfer from the hotter part to the colder part, the accompanying entropy change would be given by:

ΔS = -ΔQ/T2 + ΔQ/T1

The value of ΔS would be positive and the system would adjust itself until T2 = T1 when ΔS would be zero, the value corresponding to the state of equilibrium, synonymous with its being isothermal.

More mathematical treatments of the problem are discussed and explained in a paper by Coombs & Hale [A paradox concerning the temperature distribution of a gas in a gravitational field, Am. J. Phys., 53, 272, (1985)] and in the book by Chapman & Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd Edition, Cambridge Mathematical Library, 1970, re-issued as a paperback in 1990, reprinted in 1995.

In the absence of greenhouse gases, if the isothermal column was allowed to receive energy from below (i.e., from the surface), but energy was not permitted to escape at the top, the column would still be isothermal depending upon the temperature of the surface.

Additionally, if energy was permitted also to escape from the top of the column a temperature gradient would be established only with the presence of radiatively active greenhouse gases which could emit radiation to space. Such emission would cause negative lapse rate to be established. The greenhouse gases would also restrict the rate of passage of energy up the column and cause the surface to be warmer than in their absence.

Conclusion: The presence of greenhouse gases in the Earth’s troposphere causes the Earth’s surface to be warmer than in their absence and causes the troposphere to have a negative lapse rate.