To AGW Doubters, Skeptics, "Deniers", and Anyone Interested in the Science Behind Global Warming

Almost all of what follows come from Wikipedia, but as it agrees with my own knowledge of chemistry and physics from many sources over the years, it makes a good if sometimes hard to follow summary of the science behind anthropogenic CO2 enhanced global warming.  It is theory however, so how much warming it has caused in the Earth's atmosphere over the last ~ 150 years, and the climatological consequences of that is left up to scientific debate.
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We start with Svante Arrhenius, in the latter 19'th century:

Greenhouse effect

Arrhenius developed a theory to explain the ice ages, and in 1896 he was the first scientist to attempt to calculate how changes in the levels of carbon dioxide in the atmosphere could alter the surface temperature through the greenhouse effect.^[8] He was influenced by the work of others, including Joseph Fourier and John Tyndall. Arrhenius used the infrared observations of the moon by Frank Washington Very and Samuel Pierpont Langley at the Allegheny Observatory in Pittsburgh to calculate the absorption of infrared radiation by atmospheric CO₂ and water vapour. Using 'Stefan's law' (better known as the Stefan-Boltzmann law), he formulated his greenhouse law. In its original form, Arrhenius' greenhouse law reads as follows:

if the quantity of carbonic acid [CO₂] increases in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression.

The following equivalent formulation of Arrhenius' greenhouse law is still used today:^[9]

ΔF = α Ln(/)

Here is carbon dioxide (CO₂) concentration measured in parts per million by volume (ppmv); denotes a baseline or unperturbed concentration of CO₂, and ΔF is the radiative forcing, measured in watts per square meter. The constant alpha (α) has been assigned a value between five and seven.^[9]

Arrhenius at the first Solvay conference on chemistry in 1922 in Brussels.

 

Based on information from his colleague Arvid Högbom (sv), Arrhenius was the first person to predict that emissions of carbon dioxide from the burning of fossil fuels and other combustion processes were large enough to cause global warming. In his calculation Arrhenius included the feedback from changes in water vapor as well as latitudinal effects, but he omitted clouds, convection of heat upward in the atmosphere, and other essential factors. His work is currently seen less as an accurate prediction of global warming than as the first demonstration that it should be taken as a serious possibility.

Arrhenius' absorption values for CO₂ and his conclusions met criticism by Knut Ångström in 1900, who published the first modern infrared spectrum of CO₂ with two absorption bands, and published experimental results that seemed to show that absorption of infrared radiation by the gas in the atmosphere was already "saturated" so that adding more could make no difference. Arrhenius replied strongly in 1901 (Annalen der Physik), dismissing the critique altogether. He touched the subject briefly in a technical book titled Lehrbuch der kosmischen Physik (1903). He later wrote Världarnas utveckling (1906) (German: Das Werden der Welten [1907], English: Worlds in the Making [1908]) directed at a general audience, where he suggested that the human emission of CO₂ would be strong enough to prevent the world from entering a new ice age, and that a warmer earth would be needed to feed the rapidly increasing population:

"To a certain extent the temperature of the earth's surface, as we shall presently see, is conditioned by the properties of the atmosphere surrounding it, and particularly by the permeability of the latter for the rays of heat." (p46)

"That the atmospheric envelopes limit the heat losses from the planets had been suggested about 1800 by the great French physicist Fourier. His ideas were further developed afterwards by Pouillet and Tyndall. Their theory has been styled the hot-house theory, because they thought that the atmosphere acted after the manner of the glass panes of hot-houses." (p51)

 

"If the quantity of carbonic acid [CO₂] in the air should sink to one-half its present percentage, the temperature would fall by about 4°; a diminution to one-quarter would reduce the temperature by 8°. On the other hand, any doubling of the percentage of carbon dioxide in the air would raise the temperature of the earth's surface by 4°; and if the carbon dioxide were increased fourfold, the temperature would rise by 8°." (p53)

 

"Although the sea, by absorbing carbonic acid, acts as a regulator of huge capacity, which takes up about five-sixths of the produced carbonic acid, we yet recognize that the slight percentage of carbonic acid in the atmosphere may by the advances of industry be changed to a noticeable degree in the course of a few centuries." (p54)

 

"Since, now, warm ages have alternated with glacial periods, even after man appeared on the earth, we have to ask ourselves: Is it probable that we shall in the coming geological ages be visited by a new ice period that will drive us from our temperate countries into the hotter climates of Africa? There does not appear to be much ground for such an apprehension. The enormous combustion of coal by our industrial establishments suffices to increase the percentage of carbon dioxide in the air to a perceptible degree." (p61)

 

"We often hear lamentations that the coal stored up in the earth is wasted by the present generation without any thought of the future, and we are terrified by the awful destruction of life and property which has followed the volcanic eruptions of our days. We may find a kind of consolation in the consideration that here, as in every other case, there is good mixed with the evil. By the influence of the increasing percentage of carbonic acid in the atmosphere, we may hope to enjoy ages with more equable and better climates, especially as regards the colder regions of the earth, ages when the earth will bring forth much more abundant crops than at present, for the benefit of rapidly propagating mankind." (p63)

Arrhenius clearly believed that a warmer world would be a positive change. His ideas remained in circulation, but until about 1960 most scientists doubted that global warming would occur (believing the oceans would absorb CO₂ faster than humanity emitted the gas).^{[citation needed]} Most scientists also dismissed the greenhouse effect as implausible for the cause of ice ages, as Milutin Milankovitch had presented a mechanism using orbital changes of the earth (Milankovitch cycles).^{[citation needed]}
Nowadays, the accepted explanation is that orbital forcing sets the timing for ice ages with CO₂ acting as an essential amplifying feedback.

Arrhenius estimated that halving of CO₂ would decrease temperatures by 4–5 °C (Celsius) and a doubling of CO₂ would cause a temperature rise of 5–6 °C.^[10] In his 1906 publication, Arrhenius adjusted the value downwards to 1.6 °C (including water vapor feedback: 2.1 °C). Recent (2014) estimates from IPCC say this value (the Climate sensitivity) is likely to be between 1.5 and 4.5 °C. Arrhenius expected CO₂ levels to rise at a rate given by emissions in his time. Since then, industrial carbon dioxide levels have risen at a much faster rate: Arrhenius expected CO₂ doubling to take about 3000 years; it is now estimated in most scenarios to take about a century.
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And now onto the 20'th century, and the quantum-mechanical explanation of why certain gasses exhibit the greenhouse effect, i.e., molecular vibrations and infrared radiation absorption:

Molecular vibration

Molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion. The frequency of the periodic motion is known as a vibration frequency, and the typical frequencies of molecular vibrations range from less than 1012 to approximately 1014 Hz.

In general, a molecule with N atoms has 3N – 6 normal modes of vibration, but a linear molecule has 3N – 5 such modes, as rotation about its molecular axis cannot be observed.^[1] A diatomic molecule has one normal mode of vibration. The normal modes of vibration of polyatomic molecules are independent of each other but each normal mode will involve simultaneous vibrations of different parts of the molecule such as different chemical bonds.

A molecular vibration is excited when the molecule absorbs a quantum of energy, E, corresponding to the vibration's frequency, ν, according to the relation E = hν (where h is Planck's constant). A fundamental vibration is excited when one such quantum of energy is absorbed by the molecule in its ground state. When two quanta are absorbed the first overtone is excited, and so on to higher overtones.

To a first approximation, the motion in a normal vibration can be described as a kind of simple harmonic motion. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. In reality, vibrations are anharmonic and the first overtone has a frequency that is slightly lower than twice that of the fundamental. Excitation of the higher overtones involves progressively less and less additional energy and eventually leads to dissociation of the molecule, as the potential energy of the molecule is more like a Morse potential.

The vibrational states of a molecule can be probed in a variety of ways. The most direct way is through infrared spectroscopy, as vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum. Raman spectroscopy, which typically uses visible light, can also be used to measure vibration frequencies directly. The two techniques are complementary and comparison between the two can provide useful structural information such as in the case of the rule of mutual exclusion for centrosymmetric molecules.

Vibrational excitation can occur in conjunction with electronic excitation (vibronic transition), giving vibrational fine structure to electronic transitions, particularly with molecules in the gas state.

In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave equation, the energy states for each normal coordinate are given by

,

where n is a quantum number that can take values of 0, 1, 2 ... In molecular spectroscopy where several types of molecular energy are studied and several quantum numbers are used, this vibrational quantum number is often designated as v.^[7][8]

The difference in energy when n (or v) changes by 1 is therefore equal to , the product of the Planck constant and the vibration frequency derived using classical mechanics. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency (in the harmonic oscillator approximation).

See quantum harmonic oscillator for graphs of the first 5 wave functions, which allow certain selection rules to be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one,

but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because vibrations are anharmonic. Another consequence of anharmonicity is that transitions such as between states n=2 and n=1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to a hot band.

Intensities

In an infrared spectrum the intensity of an absorption band is proportional to the derivative of the molecular dipole moment with respect to the normal coordinate.^[9] The intensity of Raman bands depends on polarizability.

Symmetrical stretching	Asymmetrical stretching	Scissoring (Bending)
Rocking	Wagging	Twisting