In Note 4 on the diagram, it is said that the longwave infrared radiation is emitted from the surface of the Earth due to the 168 W/m^2 of radiation from the sun that the surface has absorbed. In Note 5, it states that some of this radiation is absorbed by greenhouse gas molecules with half being emitted in the direction of space and half emitted back toward the Earth's surface. The infrared radiation from the greenhouse gas emission toward the surface is absorbed and the surface gains more heat. This additional heat causes more infrared radiation to be emitted, which again is absorbed by greenhouse gas molecules and half again is re-emitted toward the surface heating it still more.
This is very important in the greenhouse theory because the Earth's surface is desperately cold due to direct solar heating, since the black body temperature corresponding to a thermal power of 168 W/m^2 is only 233.3 K or -39.8 C. Fortunately, the theory claims half of the surface radiated infrared energy that is returned to the surface provides more heating of the surface, which radiates still more energy as it warms, but half of that is returned. In fact, the claim is made that a great deal of heat energy is added to the Earth's surface in this way. Let us examine the frequently cited Kiehl-Trenberth Energy Balance diagram to see how much.
This diagram comes in several versions and the incoming radiation from the sun absorbed by the surface differs somewhat in the various versions. In this diagram, it is 161 W/m^2, or a bit less than the 168 W/m^2 of the IPCC diagram above. The total energy added to the surface by greenhouse gas re-emission is said to be 333 W/m^2 here. This means that greenhouse gases are adding slightly more than two times the energy of the sun to warm the surface. They believe that a factor of two times the solar warming is due to the multiplier effect of greenhouse gas re-emission. Each time a greenhouse gas molecule absorbs ground radiation energy, it sends half of it back to the surface. The surface radiates all of that added energy and half of that half is returned to the surface by the gracious greenhouse gas molecule. The result is a geometric series of one-half raised to a power times the initial heat energy the surface had from the sun. It works like this:
Back Radiation = (161 W/m^2) (Sum from n=1 to infinity of terms 1/(2^n))
The greenhouse gas theory says that the sum of the geometric series is 2. Actually, the sum is 1. I suspect the error arose from the use of a general formula for geometric series which starts with the term for n=0 and one-half raised to power zero is one. The greenhouse gas theorists forgot to subtract this first term of one, which was not in the series they needed to sum. In any case, note that 2 times 161 W/m^2 is 322 W/m^2, or almost the 333 W/m^2 this diagram claims warms the surface with slightly more than twice the power of the sun itself.
There are many problems with the physics of these diagrams. This will not be a general discussion of all of these problems. I will note that it is interesting that the Kiehl-Trenberth diagram allows for some cooling of the surface by evaporation and by thermals or air convection currents, but these apparently become operative only after all of the direct solar warming radiation has already been doubled by the re-emission of infrared radiation back to the surface! One should note that no radiation is emitted by the nitrogen, oxygen, and argon that make up most of our atmosphere in this model. They are warm, as implied by the thermals, but they emit no radiation in their theory. [17 March 2013: Actually, oxygen molecules absorb and emit energy on the border of visible and infrared radiation (found in solar insolation, but not the Earth's emission spectrum) and nitrogen, oxygen, and argon all emit electromagnetic radiation in the very low energy microwave radiation spectrum. These represent very minor amounts of energy, however.]
Another interesting implication is that the absorbing greenhouse gas molecule always re-emits 100% of the infrared radiation it absorbs. It never transfers any energy to nitrogen, oxygen, or argon due to collisions, despite molecular collisions occurring at the rate of 6.9 billion per second at sea level. In reality, such collisions do occur before re-emission does and the absorbed energy often is transferred to nitrogen, oxygen, and argon molecules or atoms, which remain cooler than the Earth's surface under most conditions and times of day.
Another interesting implication is that the absorbing greenhouse gas molecule always re-emits 100% of the infrared radiation it absorbs. It never transfers any energy to nitrogen, oxygen, or argon due to collisions, despite molecular collisions occurring at the rate of 6.9 billion per second at sea level. In reality, such collisions do occur before re-emission does and the absorbed energy often is transferred to nitrogen, oxygen, and argon molecules or atoms, which remain cooler than the Earth's surface under most conditions and times of day.
As we know, heat is not transferred from cooler bodies to warmer bodies. The flow of heat is always from the warmer to the cooler body. This creates a problem since as one ascends in height in the lower atmosphere, called the troposphere, which extends to an altitude of about 15,000 meters, the temperature drops. Where the average surface temperature is 288 K, the temperature at 1000 meters altitude is 281.7 K, and at 5,000 meters it is 255.7 K, or about the same temperature the Earth appears to be as seen from space and assuming it is a black body radiator. Of course it is not a black body radiator, despite the fact that greenhouse gas theorists almost always assume it is. The everyday objects of our lives do emit infrared radiation, but with a reduced efficiency compared to a black body radiator. The Earth's surface is full of everyday objects and materials, none of which are black body radiators. Greenhouse gas theory says that the greatest increase in temperature due to added CO2 occurs at an altitude of 8,000 to 12,000 meters above the equator and the lower latitudes. At 10,000 meters, the gas temperature is 223.3 K. A greenhouse gas molecule at any of these altitudes is colder when at equilibrium with the surrounding gas molecules than the surface of the Earth at 288K. Even if we compare to the black body temperature corresponding to the 161 W/m^2 said to come straight from the sun, the molecules at 10,000 meters are colder than that surface would be and they could not heat it according to thermodynamics principles. Apparently, the greenhouse theorists believe the greenhouse gas molecule absorbs the infrared radiation it is able to absorb and that raises the temperature of the molecule to a higher temperature than the surrounding molecules in the atmosphere at that elevation and before the greenhouse gas molecule has a collision and transfers energy to those surrounding molecules, it re-emits infrared energy to the ground and warms it.
Let us now discuss the properties of a black body radiator. It is a very special cavity with a very thin skin. The volume within the cavity has a very special property. The energy density, or the energy per unit volume of space, is constant everywhere within it. That energy is determined by the photons of various frequencies that are emitted from the walls of the cavity as uniquely determined by the temperature of the walls of the cavity. One of the interesting properties of a black body cavity is that if the interior volume is doubled, then the emission of photons from the walls must also double, assuming the temperature remains the same. Let us examine the distribution of electromagnetic frequencies found in a black body at several temperatures of interest for the greenhouse gas theory of global warming. The diagram below shows the spectrum of frequencies or wavelengths for the sun and for some temperatures similar to that of the Earth.
The spectrum of wavelengths for the sun at 5525 K is shown at the left top of the diagram to have a peak in the visible light part of the electromagnetic spectrum, but also has a long tail in the infrared part of the spectrum. On the right top of the diagram, three spectra of emission energies are shown for black body radiators at 310, 260, and 210 K. These are the smooth lines of violet, blue, and black, respectively. The Earth's surface at about 288 K is closer to the spectrum for a body at 310 K. The blue area filled in within those black body spectra is the portion of longwave infrared radiation from the surface of the Earth which is not absorbed by the greenhouse gases. The wavelengths at which the greenhouse gases absorb radiation is shown for each gas below, as is the sum of their absorptions.
If the Earth's surface is at a certain temperature, then it too will have a black body-like emission spectrum. Now suppose that CO2 absorbs a particular wavelength of infrared radiation out of that spectrum and then re-emits that energy at that wavelength back to the Earth's surface. Can that photon absorbed by the surface raise the temperature of the surface? No. The reason it cannot raise the temperature of the surface is because to do so, the radiative spectrum has to move to the left in the diagram above. The shorter wavelengths on the left correspond to higher frequencies and to higher energies. For the surface to become warmer due to the absorption of the photon from a greenhouse gas, higher energy vibrational states must become occupied in the Earth's surface materials. A photon from a lower temperature emitter cannot warm the surface to a higher temperature because that lower energy photon cannot excite the necessary higher energy vibrational modes. That photon can slow down the cooling of surface at night, since its emission at night will cool the surface and the returned photon will be at a higher energy than the surface is by the time the photon returns. This is the equivalent of the process when we put hot coffee in a thermos, thereby slowing down its cooling rate. But, the returning photons from the reflective wall in the thermos never heat the coffee to a higher temperature than it was at when it was poured into the thermos.
Whether a photon is absorbed by a material or not is dependent upon the electronic and vibrational states in the material which can be excited and the energy of the photon. The fact that a photon is incident upon a material does not mean it will be absorbed. The greenhouse gas theorists recognize this when the material is nitrogen or oxygen molecules, but they assume the Earth's surface can absorb whatever strikes it, at least if it is a low energy or longwave infrared photon. But, just as visible light passes through window glass without absorption, this is not necessarily the case. The light photon is not absorbed in glass because glass has a wide energy band gap in which there are no occupied or unoccupied electronic states. Still higher energy ultraviolet energy may excite available unoccupied electronic states, which in turn will de-excite in time. Until they do, they can warm the glass. But visible light just passes through. The same is the case with some of the low energy, longwave infrared radiation returned from greenhouse gas molecule de-excitations. The Earth's surface will not accept them since the excitable vibrational states are already excited and vibrating assuming that its temperature has not dropped since the returned photon was emitted by the ground. There simply is no available energy state able to accept it. The fact that the greenhouse gas alarmists ignore this fact is a very egregious error.
For another approach to the problem of heating surfaces with black body radiation, let us return to the black body cavity. To make this simple to picture, let us assume it consists of two identical hemispheres which we have put together to make a sphere. We supply heat to the sphere wall to bring it to a temperature T. The power radiating from the black body walls is known to be equal to a constant times T to the fourth power. The interior is full of photons, which are incident upon the walls, which are also radiating photons. Let us image one of the hemispheres is on the left and one is on the right as we look at our black body radiator. When a photon from the right wall is incident upon the left wall, the temperature of the left wall does not go up. It remains constant. An incident photon did not cause a temperature rise. Of course, one may say that the incident fluxes of photons are equal to emitted fluxes, so it is expected the temperature will not increase. But, then if we have supposed that the greenhouse gas molecule that has absorbed a photon from the Earth's surface has not lost energy to a gas collision and has re-emitted the photon at the same energy back to the Earth's surface, then why would the Earth's surface temperature increase any more than the left hemisphere wall temperature would in our black body cavity? The answer is that it would not.
Perhaps you would object that there is something so special about the black body radiator that the Earth's surface cannot replicate it and will act differently. Indeed, there are differences, but it is the analogy to the black body radiator explicitly that is being used in the greenhouse gas theory. The differences of the Earth's surface from a black body radiator are important, but I cannot see how they help to allow one to claim that photons of an energy from the surface's black body radiation spectrum are able to drive the surface to higher temperature than it had when it first emitted a photon of the same energy.
Let us try still another thought experiment. Let us separate our two black body hemispheres ever so slightly so that we have two hemispheres facing each other and they are still at the same temperature. In the greenhouse gas theory, these two facing spheres would heat each other up to a higher temperature! The right sphere would emit a photon and it would be absorbed by the left sphere. The left sphere would then emit half of such photons back to the right sphere and half out into space from the outside surface of the hemisphere. But then half of those emitted from the inside surface would be incident upon the right hemisphere inside surface and be re-emitted toward the left hemisphere. Of course one might say the right hemisphere emits half inward and half outward, but note that in greenhouse gas theory no photons are emitted in the direction of the Earth's interior, despite the fact that they actually are. So assuming the right hemisphere acts like the Earth's surface and the left hemisphere acts like the greenhouse gas, we have set up the same "energy doubling" geometric series between our two hemispheres. The temperature of the right hemisphere must climb. Well, no, it does no such thing. The two hemispheres will remain at the same temperature just as if they were still together as a standard black body cavity. Well not quite. Actually, each will be a very slightly lower and equal temperature.
Let us now take our hemispheres and widely separate them. We will put the right one at a higher temperature than the left and then bring them into close proximity to almost make a spherical cavity. What will happen? The cooler left hemisphere will warm, but neither hemisphere will become warmer than the right hemisphere before they were brought together. The photons incident upon the warmer right hemisphere from the cooler left hemisphere cannot make the initially warmer hemisphere any warmer. Why would they when if we made the left hemisphere the same temperature as the warmer hemisphere, it could not warm that hemisphere. Indeed, the most ideal situation for warming one hemisphere would be to put the two hemispheres back together to make the true black body cavity, but with each hemisphere at a temperature T as we put them together, we would simply wind up with a black body cavity with a temperature of T.
In short, the greenhouse gas theory errs badly about the behavior of black body radiation and radiators. Yes, bodies above absolute zero temperature do radiate photons. But those photons are not always absorbed and they are able to raise the temperature of the absorbing body only if they are of a high enough energy to allow vibrational states to be excited which would be excited in the black body radiator at that higher temperature. The black body radiator requires that the energy levels of its spectrum be occupied and that they be filled to the prescribed extent given by the black body energy distribution for a given temperature. Having many photons incident at an energy level which is already filled, will not increase the temperature of the black body radiator material.
The consensus greenhouse gas theory hugely exaggerates the effect of greenhouse gases. This is not to say that there is no effect. As I have pointed out in a chapter of Slaying the Sky Dragon: Death of the Greenhouse Gas Theory (Stairway Press, Mount Vernon, Washington, 2011), greenhouse gases cool the Earth's surface during the daylight hours and retard its cooling in many cases at night. This is an important, though in no way catastrophic effect, and it has been observed to be happening as the CO2 concentrations have increased. But those effects of greenhouse gases do not violate the conservation of energy and do not require cooler or equal temperature bodies to warm a warmer or equal temperature body.
As I also pointed out in that chapter, the incident radiation from the sun is actually somewhat greater than that claimed in the diagrams above. The temperature of the Earth's surface given that higher energy flux or power density is also much higher than the equivalent black body radiator, since the Earth's surface has a much lower emissivity than does a black body. This means that it cools itself more slowly with radiation than a black body would. As a result, the Earth's surface is likely only about 9 C warmer than one would expect from the incident radiation from the sun. There are many other sources of energy that can account for this much smaller energy differential. Among these are the effects of gravity upon our atmosphere, the heat from the very hot interior of the Earth, the energy due to the tides in the ocean and land due to the moon and other bodies in our solar system, the energy from cosmic rays and dust, the energy from the solar wind, the energy from the interaction of the sun's magnetic field with that of the Earth, and the energy stored in the subsurface oceans and land. The urban heat island effect is also substantial in some areas, though relatively minor across the globe.