Critique of The Steel Greenhouse by Willis Eschenbach

Willis Eschenbach made a guest post entitled The Steel Greenhouse at Watts Up With That in November 2009 that reduces a critical aspect of the catastrophic man-made global warming hypothesis to a very simple model.  Some critics of catastrophic man-made global warming claim his model is incorrect and others embrace it.  In this post I will solve the same problem he does, but with fewer assumptions and I will not violate the energy density conservation rules of equilibrium electromagnetic fields given by Stefan’s Law in the simple limit of black body cavities and more generally given by electromagnetic field theory as Eschenbach does.  I will follow the mathematics from a non-equilibrium case to the radiative equilibrium case.

In one very important respect, Eschenbach produces a correct result, yet in another very important respect he buys into an error that causes a huge amplification of the effects of infrared-active or greenhouse gases when that concept of thermal radiation is applied to real climate issues.  If you have not read my prior post on thermal radiation physics which I reference below, you are a most unusually astute scientist if you really know and understand what Eschehbach’s widely shared error is.

I have previously discussed the fundamentals of black body thermal radiation and how it applies to real life materials in several postings.  The best single post to read to understand why it is improper to think about black body and thermal radiation generally as most scientists do is:

Solving the Parallel Plate Thermal Radiation Problem Correctly Provesthe Settled Science of Man-Made Global Warming Wrong

The Eschenbach model for his discussion of a fundamental aspect of the greenhouse gas warming effect is to imagine the Earth as a perfectly conducting sphere with black body emission closely surrounded by a perfectly conducting shell which also has surfaces that act as though they are black body absorbers and radiators.  Effectively, his model takes there to be only vacuum between the surface of the inner sphere and the surrounding shell and only vacuum and a T=0 K universe beyond the surrounding shell.  The only means for energy to flow in the system between the inner sphere and the outer shell is by thermal radiation, as it is also beyond the shell.  The very small correction for the different surface areas of the inner sphere and the outer shell will be ignored as Eschenbach did.  The geometric surface area correction is less than one part in a thousand.  This is not meant to be an accurate model of the Earth and its atmosphere.  It is a useful thought experiment.

Eschenbach posits that the inner sphere has its own source of heat which he sets at a thermal power density of 235 W/m² at the surface of the sphere.  Since this is the only source of heat, at equilibrium, the only very slightly larger shell around the sphere must radiate energy into space at a power of 235 W/m².  So far he is right.

He posits that the outer shell is a great conductor, so there is no temperature gradient in the shell between the inner and outer surfaces.  Now he applies standard issue knowledge of thermal radiation and says that if the two sided outer sphere is radiating power on the outside surface at 235 W/m², then it must be doing so also from the inner surface which has the same temperature, because the relationship between the power of radiation and the temperature is given by P = σT⁴, where P is the power per unit area, T is the temperature of the surface in Kelvin, and σ is a constant.  This relationship is the Stefan-Boltzmann law.  If the inner surface were radiating into a vacuum at T = 0K, this would be a correct application of the Stefan-Boltzmann Law.  This is not the case for the inner surface, though we will imagine that it is for the outer surface since space has an average temperature relatively close to absolute zero compared to an Earth surface temperature near 288 K.

Eschenbach goes on to observe that since the shell is radiating energy back to the inner sphere at 235 W/m² and the sphere surface already had a supply of power of 235 W/m², the sum of the two powers is now 470 W/m².  Putting a shell around the core sphere has doubled the radiating power of the core sphere.  This is the real greenhouse effect he says.  His solution is based on a flux of photons at 470 W/m² flowing outward from the sphere surface and a flux of photons at 235 W/m² flowing downward from the shell to the sphere surface.

Some people are bothered by the failure here to conserve energy, but not very many, because most people think it is only important to conserve energy at the sphere and at the shell.  Most people seem to examine this and say, well, the 470 W/m² radiating out from the inner sphere surface minus the 235 W/m² radiating into the inner sphere surface from the inner surface of the shell is still 235 W/m² which is supplied by the internal power supply of the core sphere.  QED, energy is conserved.  Never mind the fact that the energy of the photons issuing forth at the rate of 470 W/m² and the energy of the photons from the inside wall of the shell at 235 W/m² must add, not subtract, when we examine the energy density of the volume between the outer shell and the inner core sphere.  I will discuss this somewhat further on in this post, but the reference I gave above will be a much more thorough discussion of this critical issue.

Let us step back from this and talk a moment about black body cavity thermal radiation.  The principal characteristic of a black body cavity is that it is at thermal equilibrium and the energy density inside the cavity is everywhere the same and given by Stefan’s Law.  If the energy density is e, then e = a T⁴, where a is Stefan’s constant.  Within the cavity in equilibrium, there are just as many photons traveling in one direction as in its opposite direction.  If photons traveling in opposite directions had energies that cancelled one another out, then the energy density inside a black body cavity would be zero and would not be given by Stefan’s Law.

If you return to Jackson’s Classical Electrodynamics, you will also find that two oppositely directed electromagnetic plane waves will simply pass through one another and reappear as normal plane waves after their very brief interaction.  They most certainly do not sum up to zero energy.

Let us simplify the problem even more by just looking at two facing planes, one of which has a supply of power Q per unit surface area and only radiates that power from the surface facing the other plane which has two sides that can radiate power.  Imagine these to be a small section out of the Eschenbach inner core of a unit area of surface and of a unit area of outer shell.  This simplification of the model with its parameters for thermal radiation is shown below:

The power into the left plane representing a unit surface area of the inner core causes it to radiate power at a rate of P_S, when the power to the sphere is first turned on.  We will assume that the surrounding shell on the right of the drawing was at T=0 K when the power to the inner core was turned on.  Let us either assume that it has a finite heat capacity so that it has to warm up to its equilibrium temperature or we count on the finite speed of light to create a delay.  We are making this assumption so that we are not too quick to leap to false assumptions.  What is the general case before and when equilibrium is reached?  It is obvious that T_O will increase.  What will happen to T_S?

The power transferred from the inner core to the outer shell is P_S. The power radiated from the outer surface of the shell section will be P_O and that surface is in vacuum facing nothing but T = 0 K space.  For simplicity and in order to be strictly correct in applying the Stefan-Boltzmann Law, the space between the powered inner core and the spherical shell is in vacuum.  We have

Q = P_S  = σ T_S⁴ - σ T_O⁴

P_O = σ T_O⁴

At equilibrium, P_S = P_O, so

T_SE⁴ - T_OE⁴ = T_OE⁴, where the added E in the subscripts designates the equilibrium values.

Therefore, T_SE⁴ = 2 T_OE⁴ or T_SE = 1.189 T_OE and

P_S = σ ( 2 T_OE⁴ - T_OE⁴ ) = σ T_OE⁴.

But Q = P_S always, so when the shell was still at T=0 K, Q = P_S = σ T_SI⁴ , where T_SI was the initial temperature of the surface of the sphere when Q was first turned on and all the sphere surface saw as a T=0 K environment.  Consequently,

T_OE = T_SI

At equilibrium, the outward facing surface of the shell radiates energy at the same rate the initial core spherical surface did when it was surrounded by T=0 K.  The shell temperature has become what the initial core sphere surface temperature was.  Very importantly, the inner core surface temperature has increased to be

T_SE = 1.189 T_OE = 1.189 T_SI

Putting the shell around the inner core has sufficiently retarded its rate of cooling that with the same input power to the inner core, its temperature has increased by a factor of 1.189 or the one-quarter root of 2.  The reason for this is that the powered inner core is emitting energy from a surface of unit area 1, while the surrounding shell is retarding its emission with a surface of unit area 1 and emitting a power equal to the initial power emitted from the sphere from its outer surface of unit area 1.  In the similar problem with two planes both of which have two black body surfaces and one of which is supplied with power, the equilibrium condition has both planes at the T_SI temperature.  They create a black body cavity between them and the photon emission from the two facing inner surfaces is P = 0.  There is only P = σ T⁴ emission from the outward facing surfaces of each plane and the interior energy density is given by Stefan’s Law as

e = a T⁴

Let us return to Eschenbach’s post.  His inner sphere had a power of its own of 235 W/m² and the shell radiated 235 W/m² down upon the inner sphere, so he says the inner sphere surface radiates power away from its surface equal to the sum of the internal power and the radiated power from the surrounding shell, which is 470 W/m².  Applying the Stefan - Boltzmann Law:

P_S = 470 W/m² = σ T_SE⁴

T_SE = 301.74 K

In my case,

P_S = σ T_SI⁴ = 235 W/m²

T_SI = 253.73 K

T_SE = 1.189 T_SI = 301.68 K

So, both Eschenbach and my calculations yield the same, higher inner core surface temperature. 

Our important difference is that he supposes the vacuum between the inner core and the surrounding shell has a photon density corresponding to (470 + 235) W/m² = 705 W/m², while my photon density corresponds only to those emitted from the inner core surface and there are no photons emitted from the inner surface of the surrounding shell.  The reasons for this are given at length in my first reference above.  Consequently, the real photon density between the sphere and the shell is actually that corresponding to 235 W/m².  Eschenbach has multiplied the photon density by a factor of 3.

Why is the photon density critical when one more realistically addresses the catastrophic man-made global warming hypothesis?  One way one calculates the longwave infrared absorption warming attributed to greenhouse gases is with an experimentally measured absorption cross section for each frequency of photon energy for each greenhouse gas molecule such as water vapor and carbon dioxide.  One then multiplies the number of photons of each frequency times the value of the absorption cross section for that frequency to calculate the number of absorption events.  A factor of 3 exaggeration in the number of photons at each frequency is an important exaggeration of the greenhouse gas effect.

It is actually even worse than this when the proponents of the catastrophic man-made global warming hypothesis with a similar misconception set to work.  Let us look once again at the NASA Earth Energy Budget:

NASA has a surface radiation of 117% here and a back radiation of 100%.  This produces a corresponding photon density of 217%.  In reality, the photon density is 117% - 100% = 17%.  Consequently, NASA has amplified the photon density by a factor of 217% / 17% = 12.8.  This is the equivalent of amplifying the greenhouse gas effect by a factor of 12.8.

There are many who believe that the radiative forcing caused by a doubling of carbon dioxide in the atmosphere is 3.7 W/m².  Divide that radiative forcing value by 12.8 to account for the greatly exaggerated effect caused by an exaggeration of the number of photons that carbon dioxide can absorb and one gets a radiative forcing value of only 0.29 W/m².  This alone would make it much harder to experimentally document the warming effect of carbon dioxide and would explain why the global climate models have been exaggerating the effects of carbon dioxide so long and why it has been so hard for them to find that elusive hot spot in the upper troposphere in the tropics they predicted.

It has other important consequences as well.  Suddenly the cooling effects of carbon dioxide that are usually ignored with the claim that they are much smaller than the greenhouse gas warming effect are not so small in comparison.  These cooling effects include:

• The absorption of solar insolation in the atmosphere before it can reach the surface to warm the surface
• Carbon dioxide has a higher heat capacity than do nitrogen and oxygen molecules, so more carbon dioxide increases the heat energy carried upward by convection currents
• Because carbon dioxide radiates thermal energy from a warmer layer of air to a cooler layer of air above it and that energy is transported at the speed of light, albeit for a short distance in the troposphere, this is faster transport of energy than is the convection current that would otherwise transport this energy upward           

Even if each of these three cooling effects is smaller than the reduced greenhouse warming forcing effect for carbon dioxide of 0.29 W/m², the sum of the decrease on the net warming forcing effect may be quite significant.  What is more, these cooling effects probably do not saturate as quickly as the greenhouse warming effect does as one increases the concentration of carbon dioxide in the atmosphere from current levels.  Consequently, the small warming effect of 400 ppm of carbon dioxide may be reduced by further additions of carbon dioxide, if not now, then maybe as one adds more to 600 ppm of carbon dioxide in the atmosphere.  At this point, we do not know what happens as CO₂ is added in increments at higher concentrations than 400 ppm.

In addition, the diminished effect of carbon dioxide on warming should cause everyone to have more interest in understanding many natural causes or non-man-made causes of climate variability.  We have far too little knowledge of

• Solar irradiance variations
• Solar wind and the weakening solar magnetic field effects
• Cosmic ray seeding of clouds
• Other causes of cloud variations
• The condensation of water in dew and ground fog surface warming
• Precipitation effects on warming/cooling
• Evaporation of water as a function of temperature and humidity around the world
• Better understanding of the greenhouse effect of water vapor
• Ocean currents and cycles
• Effects caused by the weakening of the Earth's magnetic field
• Effects of aerosols
• Effects of dust
• Other effects not listed

Then there are other man-made effects, primarily man’s use of the land.

I believe that these other effects on climate will in some cases prove to be more important for our understanding of the climate and its changes than are the effects of additions to the carbon dioxide concentration in our atmosphere.  Carbon dioxide has a very small effect on the climate, especially so when one is concerned about the effect of additions to the present levels of carbon dioxide.

A Summary of Some of the Physics Errors of the NASA Earth Energy Budget

         I have previously discussed many errors in the physics of the NASA Earth Energy Budget which are critical to the argument backing the catastrophic man-made global warming hypothesis. These errors are essentially the same in the Earth Energy Budgets of the UN IPCC reports, though there are minor variations in the values of the heat transport powers in the Earth system consisting of its surface and its atmosphere. The NASA Earth Energy Budget is shown below, where the heat transport is denoted as a percentage of the average solar insolation at the top of the atmosphere:

Among these errors are:

•         The transport of heat in the atmosphere does not address the critical role in the temperature profile played by the action of gravity on air molecules. This is not an actual error in the Earth Energy Budget, but that budget does serve to misdirect attention toward a completely radiation and heat transport dominated view of the problem.

•         The 117% surface radiation from the Earth’s surface requires the Earth’s surface to directly interface to vacuum, with no atmosphere present. The Earth’s surface must be at 289.4 K, be a black body radiator with an emissivity of 1.00, and be surrounded only by space at very nearly 0 Kelvin (K). Note that 289.4 K is a higher temperature than that usually taken to be the Earth’s average surface temperature and that the Earth’s surface emissivity is usually said to be about 0.95. The lack of vacuum at the interface with the Earth’s surface is a serious problem because the surface oscillating dipoles that radiate infrared energy cannot provide that same kinetic energy that creates radiated energy to evaporating water or transfer it to air molecules colliding with the surface. Energy must be conserved. The higher temperature and emissivity used for the surface is a smaller error, but indicative of a cavalier attitude to the science.

•         The Conservation of Energy in a system in equilibrium does not allow the flow of energy into the Earth’s surface to exceed the rate at which energy enters the system. Energy enters this system at 100%, yet this NASA Earth Energy Budget claims it is incident upon the Earth’s surface at a rate of 7% reflected solar insolation plus 48% absorbed solar insolation plus 100% back radiation from the atmosphere for a total of 155%.

•         The atmosphere cannot possibly absorb as much radiation from the surface of the Earth as is claimed to be absorbed, because the atmosphere is not as absorbing as would be a black body absorber and a black body absorber would have to be at a lower temperature than any temperature in the Earth’s atmosphere to absorb as much radiation as the so-called settled science Earth Energy Budget claims is absorbed. This is because the power absorbed by a black body absorber at temperature T_A from a black body emitter at a temperature of T_E at equilibrium is P = σT_E⁴ - σT_A⁴. In the above schematic diagram, it is not possible for the surface to emit 1.17 P_SI, where P_SI is the solar radiation at the top of the atmosphere, and have (1.17 - 0.12) P_SI = 1.05 P_SI be absorbed by the atmosphere. See my discussion of this issue in A Critical Lesson from the NASA Earth Energy Budget.

•         In Solving the Parallel Plane Black Body Radiator Problem and Why the Consensus Science is Wrong, I proved that the consensus science method of applying the Stefan-Boltzmann Law of Thermal Radiation causes the essential characteristic energy density of a black body cavity in equilibrium to double relative the energy density given by Stefan’s Law. Stefan’s Law states that the electric field energy density in a black body cavity is e = aT⁴, where T is the temperature in Kelvin and a is Stefan’s constant. The correct energy density is maintained in the case of two parallel planes at temperatures T_W and T_C with T_W > T_C in the limit that T_C approaches T_W, if the radiation from the warmer plane toward the cooler plane is given by P_W = σT_W⁴ - σT_C⁴ and the radiation from the cooler plane toward the warmer plane is given by P_C = 0. The settled science thinks P_W = σT_W⁴ and P_C = σT_C⁴, which causes there to be many more photons with real energy between the planes than there really are and causes the doubling of the energy density known in Stefan’s Law. Applying this result to the NASA Earth Energy Budget one realizes that there is no equilibrium back radiation from the cooler atmosphere to the warmer surface, so the 100% back radiation is fictitious. Equally important, if the atmosphere were a black body, the radiation from the surface would also be much reduced to the extent that the atmosphere were absorbing some of it. Other critics have made the claim that cooler bodies do not radiate toward warmer bodies using a simple argument based on the Second Law of Thermodynamics, which by itself is not sufficient. However, coupling that law with a minimization of the total energy in the system, which provides the correct result to many a physics problem, does provide a pretty good argument for the same result that I worked out from electromagnetic field thermodynamics. Note that the elimination of back radiation eliminates a power incident upon the surface of 100% and therefore eliminates the violation of the Conservation of Energy at the Earth’s surface discussed in the third bullet above. There are serious consequences of using black body radiation theory in a manner that doubles the energy density of a black body cavity.

Further Discussing the Diminished Role of Radiation in the Lower Atmosphere

          Let us consider the equilibrium condition now at the Earth’s surface that the flow of energy into the surface per unit area must equal the flow of energy out of the surface per unit area. The power absorbed by the surface from solar insolation, P_ABS, according to the NASA Earth Energy Budget is 48%. We now know that the other input to the surface they claim from back radiation is zero in the equilibrium case in which the air cools with increasing altitude from the surface. This is not quite true on average for the real Earth system since there are occasions, commonly in the dawn hours and shortly afterwards, when the air temperature just above the surface is warmer than the surface. This is easily recognized as the cause of dew and ground fog. Consequently, I will allow that back radiation might be 1 or 2%, but the upcoming discussion will ignore this small effect.

          The flow of energy out of the Earth’s surface according to NASA is given by the sums of 5% power lost in convection, 25% power loss through evaporation, and the radiated power P_R. Consequently, we have

P_ABS = (0.48)(340 W/m²) = (0.05 + 0.25)(340 W/m²) + P_R

Solving for P_R, we get

 P_R = (0.18)(340 W/m²)

From the NASA Earth Energy Budget we know that radiation passing through the atmospheric window into space from the surface without any atmospheric absorption is a power, P_AW, of 12% of the top of the atmosphere solar insolation. The remaining power radiated from the Earth’s surface is absorbed by the atmosphere and converted into an upward power loss as convection, R_CC. Thus we have

P_R = P_AW + P_CC

(0.18)(340 W/m²) = (0.12)(340 W/m²) + P_CC

P_CC = (0.06)(340 W/m²)

          Consequently, if NASA has correctly measured the radiation emitted from the surface through the atmospheric window into space, the absorption of solar insolation by the surface, and the sum of the heat loss from the surface due to convection and water evaporation, then the fraction of the radiation from the surface which is absorbed in the atmosphere is only half that of the radiation from the surface that escapes into space without absorption in the atmosphere and it is one-third of the total radiation emitted by the surface. According to the NASA Earth Energy Budget the radiation emitted by the surface of 117% has all but 12% absorbed by the atmosphere, which means that water vapor and carbon dioxide and the various minor infrared-active gases, the greenhouse gases, are playing a huge role in absorbing a power of 105%. In the next to last bulleted item above, I showed that the atmosphere cannot possibly absorb so much infrared radiation from the surface. In reality, we see above that these gases only absorb 6% according to the NASA numbers after we eliminate those that are clearly wrong. The role of infrared-absorbing gases has thus been falsely magnified by a factor of

(105%) / (6%) = 17.5

In light of these observations, is it not interesting that so many are claiming that the science is settled and that there is a scientific consensus that mankind is faced with catastrophic global warming resulting from his generation of carbon dioxide and the use of fossil fuels?

          Given the errors in the science of climate change that I have pointed out here, one should wonder how accurate any of the NASA and the similar values used in the UN IPCC reports might be.

          There is another way in which the NASA Earth Energy Budget is quite misleading with respect to the atmospheric absorption of infrared radiation from the surface. In reality, in most of the world the main part of the surface radiation that is absorbed is absorbed within a very few meters of the surface and not far up into the atmosphere as the diagrams for energy budgets picture the absorption. There are some areas such as the polar regions and a few deserts where the distance for absorption is significant, but in most of the world the humidity is high enough that the absorption length is very short based on laboratory measurements of absorption cross sections or mean free path lengths. Surface radiation in the colder polar regions is substantially less than that from the warmer regions of the Earth, so the longer absorption lengths in those polar regions are also of less importance to the energy budget. That much smaller part of the absorption of surface radiation performed by carbon dioxide is also occurring very close to the surface, though it is a few times greater than the average distance for water vapor, but is also more uniform over the Earth since the concentrations of carbon dioxide in the atmosphere are more uniform.

          If the surface infrared emission is 18% and the atmosphere absorbs 6%, then the temperature a black body absorber in the atmosphere, T_A, would have to be at to absorb so much infrared radiation can be calculated from:

(0.06)(340 W/m²) = (0.18)(340 W/m²) - σT_A⁴

T_A = 163.8 K

This is a temperature lower than that found in the Earth’s atmosphere, so even a black body absorber cannot absorb such a large fraction of the infrared radiation emitted from the Earth’s surface as is implied by the NASA values in the Earth Energy Budget after we have eliminated the errors I pointed out in the bullets at the start of this post. The infrared-active gases can only absorb a fraction of what a black body absorber can, so they certainly cannot remove as large a fraction of the surface-emitted infrared as could a black body absorber.

          I expect the easiest power value for NASA to measure accurately is the 12% surface-emitted radiation through the atmospheric window into space. But, I expect that their measurements of the surface absorption of solar insolation, the loss of surface energy due to convection, and the loss of surface energy due to the evaporation of water are not very well-established numbers. Clearly, the fraction of the surface-emitted infrared energy absorbed by the atmosphere cannot be as high as one-third. NASA has probably substantially underestimated the sum of the heat loss of the surface by means of water evaporation and convection.

          Such is the sad state of the so-called settled science of man-made global warming and such is the foolishness of the scientific consensus on climate change, insofar as that exists.

Solving the Parallel Plate Thermal Radiation Problem Correctly Proves the Settled Science of Man-Made Global Warming Wrong

This is essentially a re-posting of an article of 2 November 2017 which has been edited with additions several times since the original version appeared.  The essential physics has not been changed, but I have tried to help the reader understand it more readily along with its implications for the fate of the catastrophic man-made global warming hypothesis.

I will present the Consensus, Settled Science solution to the parallel plane black body radiator problem and demonstrate that it is wrong.  I will show that it exaggerates the energy of electromagnetic radiation between the two planes by as much as a factor of two as their temperatures approach one another.  As a result, the calculations of the so-called Consensus, Settled Science dealing with thermal radiation very often result in violations of the Law of Conservation of Energy.

Their calculations of thermal radiation greatly exaggerate the density of the infrared photon radiation in the atmosphere and the extent of the absorption of infrared radiation by infrared-active molecules, commonly called greenhouse gases, such as water vapor and carbon dioxide.  Their theory of the transport of heat energy between the surface of the Earth and the atmosphere and through the atmosphere is very wrong.  It exaggerates the role of thermal radiation greatly and minimizes the role of the water evaporation and condensation cycle and the role of thermal convection.  It cannot be emphasized enough how harmful their mishandling of thermal radiation calculations is to their understanding of the critical issues pertaining to the Earth's climate and to man's role in changing the climate through the use of carbon-based (fossil) fuels. 

In the case in which the two black body parallel plane radiators have the same temperature, the volume between them becomes that of a black body radiator.  The fundamental characteristic of a black body radiator is the constant energy density in the cavity.  I will show that the so-called settled science treatment, which wrongly takes the primary characteristic of a black body radiator to be that the power of emission of radiant energy is given by the Stefan-Boltzmann Law, clearly violates the real principal characteristic of a black body cavity, namely that its constant energy density, e, is given by Stefan’s Law as

e = aT⁴,

where T is the temperature in Kelvin, a is Stefan’s Constant of 7.57 x 10^-16 J/m³K⁴ , and e is in Joules per cubic meter.

I have already demonstrated the failure of the settled science treatment of thermal radiation from black body radiators in the form of concentric spherical shells in a paper posted on 23 October 2017, entitled Thermal Radiation Basics and Their Violation by the Settled Science of the Catastrophic Man-Made Global Warming Hypothesis, but I want to post this solution with its simpler geometry so that the reason the consensus treatment is wrong will be even more apparent to thinking readers.

Numerous critics of the consensus science on catastrophic man-made global warming have argued that the Second Law of Thermodynamics claims that energy only flows from the warmer body to the colder body, but the consensus scientists have argued that thermodynamics only applies to the net flow of energy.  I have long argued that the reason that radiant energy only flows from the warmer to the cooler body is because the flow is controlled by an electromagnetic field and an energy gradient in that field.  I will offer that proof in this paper with some qualification on the energy flow from the cooler body to the warmer body within a tight constraint.  The Second Law of Thermodynamics is not invoked as the basis of the proof in this paper, but the minimum energy of a system consistent with the Second Law of Thermodynamics does turn out to be a consistent solution to the problem of thermal radiation, while the Consensus, Settled Science theory of thermal radiation does not minimize the system total energy, does not produce the correct energy density of a black body cavity, and is not consistent with the Conservation of Energy.  I have pointed out its failure to conserve energy in many prior posts. 

In a black body cavity, the electromagnetic radiation is in equilibrium with the walls of the cavity at a temperature T.  The energy density e is the mean value of 

½ E·D + ½ H·B,

where E is the impressed electric field, D is the displacement, which differs from E when the medium is polarized (i.e., has dipoles), H is the impressed magnetic field and B is the magnetic polarization of a medium.  If the cavity is under vacuum, then D = E and B = H in the cavity volume and |E| = |H|, so e equals |E|².  The mean value of the energy density of the electromagnetic field in the cavity depends on the temperature and is created by the oscillating dipoles and higher order electric poles in the cavity walls.  The energy density is independent of the volume of the cavity.  The radiation pressure on the cavity walls is proportional to the energy density.

This physics may be reviewed by the reader in an excellent textbook called Thermal Physics by Philip M. Morse, Professor of Physics at MIT, published in 1965 by W.A. Benjamin, Inc., New York.  Prof. Morse wrote it as a challenging text for seniors and first-year graduate students.  I was fortunate to use it in a Thermodynamics course at Brown University in my Junior year.  Alternatively, see my post The Greenhouse Gas Hypothesis and Thermal Radiation -- A Critical Review.

Inside a black body cavity radiator at a temperature T, the energy density, or the energy per unit volume of the vacuum in the cavity is constant in accordance with Stefan’s Law.  If one opens a small peephole in the wall of the black body cavity, the energy density just inside that peephole is the energy density of the black body cavity and that energy density is proportional to the square of the electric field magnitude there.  The Stefan-Boltzmann Law states that the flow rate of energy out of the peephole when the black body cavity is surrounded by vacuum and an environment at T = 0 K, is given as the power P per unit area of the peephole as

P = σT⁴

Note that P = (σ/a) e and that e in the T=0K sink is equal to zero.  A change of energy density in the vacuum volume immediately inside the peephole into the black body cavity as given by Stefan's Law to a value of zero in the T=0K outside environment causes a power of thermal radiation emission out of the peephole to the outside environment as given by the Stefan-Boltzmann Law.

Why is it that a surface which is not a peephole into a black body cavity might act like a black body radiator?  It has to be that the energy density very, very close to that surface has the characteristic of the energy density in a black body cavity radiator, namely that

e = aT⁴.

Any flow of energy out of the surface due to its temperature T must be caused by this electromagnetic field energy density at the surface generated by the vibrational motion of electric charges in the material of the surface.  Such flow of energy from the surface only occurs to regions with an energy density that is lower.  There is no flow of energy from the inside wall of a black body radiator because the energy density everywhere inside the cavity at equilibrium is equal.  P from the interior walls is everywhere zero.  A non-zero P is the result of a non-zero Δe.

In fact, while it is commonly claimed that photons inside the cavity are being 100% absorbed on the walls and an equal amount of radiant energy is emitted from the absorbing wall, the actual case is that the radiant energy incident upon the walls can be entirely reflected from the walls.  Planck had derived the frequency spectrum of a black body cavity from an assumption of complete reflection from the walls.

Here is the problem of the parallel plane black body radiators diagrammed, where T_C is the cooler temperature and T_H is the warmer temperature:

Let us first consider the case that each plane is alone and surrounded by an environment of space at T=0.  Each plane has a power input that causes the plane to have its given temperature.  Each plane radiates electromagnetic energy at a rate per unit surface area of 

P = σT⁴.

Consequently, if neither plane were in the presence of the other and each plane has a surface area of A on each side and P_CO , P_CI , P_HO , and P_HI are all radiation powers per unit area, we have

P_C = AP_CO + AP_CI = 2AσT_C⁴

And

P_H = AP_HO + AP_HI = 2AσT_H⁴ ,

since in equilibrium the power input is equal to the power output by radiation.

In the consensus viewpoint, shared by many physicists and by almost every climate scientist, the parallel plane black body radiators above are believed to emit photons from every surface of each plane even in the presence of the other plane with a power per unit area of

P_H = σT_H⁴ and P_C = σT_C⁴,

just as they would if they were not near one another and they only cast off photons into a sink at T = 0 K.  This viewpoint takes the emitted radiation as a primary property of the surfaces rather than an electromagnetic field with a known energy density as the primary property of the surfaces.

Thus, when these planes are in one another's presence this consensus viewpoint says that

P_C = AP_CO + AP_CI - AP_HI = 2AσT_C⁴ - AσT_H⁴

P_H = AP_HO + AP_HI - AP_CI = 2AσT_H⁴ - AσT_C⁴

Note that P_C becomes zero at T_C = 0.8409 T_H and below that temperature P_C is negative or a cooling power in addition to radiative cooling.  If T_H = 288K, then T_C = 242.2K, the effective radiative temperature of the cooler plane to space if the cooler plane is thought of as the atmosphere and the warmer plane is the surface of the Earth, both the atmosphere and the surface act like black body radiators, and the atmosphere receives only radiant energy from the surface.  This is in agreement with calculations I have presented in the past and is a result which I believe to be correct under the assumptions, even though in some critical respects this consensus viewpoint is wrong.

If these planes were isolated from one another and each plane faced only that T = 0 K vacuum, then one would have

e_H = aT_H⁴ and e_C = aT_C⁴,

because these are black body radiator surfaces.  P_HI , when the hotter plane is surrounded by T=0 environment, provides the photon flow near the emitting surface which causes the local energy density to be

e_H = aT_H⁴ in this case.

Unlike the case of concentric spherical shells, which I considered in Thermal Radiation Basics and Their Violation by the Settled Science of the Catastrophic Man-Made Global Warming Hypothesis, there is no divergence or convergence of the photons emitted from either surface.  The relationship of the radiative power P to the energy density due to that electromagnetic radiation is always e = (a/σ) P as one traverses the distance between the planes.  

Consequently, the energy density between the planes is

e = (a/σ) P_HI + (a/σ) P_CI = aT_H⁴ + aT_C⁴

anywhere between the two planes, because photons have energy no matter which direction they are traveling and they do not annihilate one another based on their direction of travel.  The total energy between the planes is that of the electric field or it is the sum of the energy of all the photons in the space between the planes.  The energy density e would then be the total energy of all the photons divided by the volume of vacuum between the planes.

Now, let us imagine that these planes are very close together and the ends are far away and nearly closed.  Let us have T_C → T_H, then

e → 2aT_H⁴,

but this space between the planes is now a black body cavity in the limit that T_C → T_H, and we know by Stefan’s Law that

e = aT_H⁴

in this case.  In addition, we have created a black body cavity radiator here and P for the walls inside the cavity is actually zero because the interior is in a state of equilibrium and constant energy density.  P is only P = σ T⁴ just outside the peephole facing an environment at T = 0 K.  In the above consensus viewpoint case, each plane surface is emitting real photons, but these cannot annihilate those photons of the opposite plane.  There are no negative energy photons.  These respective photon streams simply add to the total energy density.

If we did insist on claiming that oppositely directed photons had cancelling energies as is commonly done, then the energy density inside a black body cavity would be zero, not the finite energy given by Stefan's Law.  Given the thermal equilibrium in a black body cavity, there are just as many photons traveling in one direction as in the opposing direction, yet their respective energies must be added when determining the energy density.  Stefan's Law insists that all photons have positive energy.

The consensus treatment of black body thermal radiation doubles the energy density in a black body cavity, in clear violation of the principal characteristic of a black body cavity upon which their treatment must be based.  Their treatment greatly increases the energy density between the planes whenever T_C is anywhere near T_H, such as is the case of the temperatures in the lower troposphere compared to the Earth’s surface temperature.  Consequently, the sum of P_HI and P_CI must be much smaller than they are thought to be in the consensus treatment of this problem or in the similar concentric spherical shell problem.

Let us now examine the correct solution to this parallel plane black body radiators problem.  It is the electromagnetic field between the two planes that governs the flow of electromagnetic energy between the planes.  Or one can say it is the energy density at each plane surface that drives the exchange of energy between the planes due to the energy density gradient between the two planes.  The critical and driving parameter here is

Δe = e_H - e_C = aT_H⁴ – aT_C⁴ ,

where each black body radiator surface maintains its black body radiator requirement that the energy density at the surface is given by Stefan’s Law.

Electromagnetic energy flows from the high energy surface to the low energy surface, as is the case in energy flows generally.

P_HI = (σ/a) Δe = (σ/a) (aT_H⁴ – aT_C⁴ ) = σ T_H⁴ – σ T_C⁴

P_CI = 0,

which is consistent with experimental measurements of the rate of radiant heat flow between two black body radiators.  Note that as T_C approaches T_H, P_HI approaches zero as should be the case inside a black body cavity in thermal equilibrium.  There is no thermal emission from either of the black body cavity walls then, except with the absorption of a photon of equal energy.  Note also that when T_C = 0 K, P_HI is given by the Stefan-Boltzmann Law

P_HI = σ T_H⁴.

Let us recalculate P_H and P_C in this correct formulation of the problem:

P_C = AP_CO + AP_CI - AP_HI = AσT_C⁴ + 0 – [AσT_H⁴ - AσT_C⁴] = 2AσT_C⁴ - AσT_H⁴

P_H = AP_HO + AP_HI - AP_CI = AσT_H⁴ + [AσT_H⁴ - AσT_C⁴] - 0 = 2AσT_H⁴ - AσT_C⁴

And we see that the power inputs to each plane needed to maintain their respective temperatures as they cool themselves by thermal radiation are unchanged in this correct energy density or electromagnetic field centered viewpoint from the consensus viewpoint. Experimentally, the relationship between the power inputs to the thermal radiation emitting planes at given temperatures are exactly the same.  This fact causes the proponents of the consensus viewpoint to believe they are right, but they nonetheless violate the energy density requirements of electromagnetic fields and of black body radiation itself.

Because P_CI = 0, back radiation from a cooler atmosphere to the surface is also either zero or accompanied by an offsetting decrease in the emission from the surface and not 100% of the top of the atmosphere solar insolation as in the current NASA Earth Energy Budget.  Because P_HI = σ T_H⁴ – σ T_C⁴ , the Earth's surface does not radiate 117% of the top of the atmosphere insolation either.  If the net energy flow by radiation between the surface and the atmosphere is 17%, then any actual back radiation from the atmosphere would be offset by a reduction of radiation from the surface below the 17% level.  In other words, if there were a 2% back radiation from the atmosphere, then the surface radiation would be limited to 15% in order to preserve the proper energy density.  The NASA radiation flows are hugely exaggerated as are the similar Earth Energy Budgets presented in the UN IPCC reports.  See the NASA Earth Energy Budget below:

This is very important because reducing these two radiant energy flows of infrared photons reduces the effect of infrared-active gases, the so-called greenhouse gases, drastically.  Many fewer photons are actually available to be absorbed or emitted by greenhouse gases than they imagine.  This is a principal error that should cause the global climate computer models to greatly exaggerate the effects of the greenhouse gases, just as they have.

As I have pointed out in the past, one fatal consequence of the exaggeration of thermal radiation from the surface of the Earth is readily calculated from the fact that even if the atmosphere acted as a black body absorber, which it does not, it can only absorb the thermal radiation said to be emitted from the surface at an absurdly low atmospheric temperature.  Observe that the power of infrared radiation from the surface which is absorbed by the atmosphere P_SA is

P_SA = σ ( T_S⁴ – T_A⁴)

P_SA = (1.17 – 0.12) (340 W/m²)

The first equation is from the discussion above with T_S the surface temperature and T_A the temperature of the atmosphere.  The second is according to the NASA Earth Energy Budget above.  If one takes T_S to be 288K, then the temperature of the atmosphere required to absorb as much infrared energy as NASA claims is absorbed is 155.4K.  There is no such low temperature in the Earth’s atmosphere.  To find so low a temperature, one has to go far out into the solar system many times the radius of Earth’s orbit.  That being the case, any such thermal radiation absorbed by matter at such a low temperature is as much lost in the Earth Energy Budget as is the power equal to 12% of the solar insolation emitted from the Earth’s surface which NASA allows passes through the atmospheric window into space.  It should be apparent to the reader that the NASA Earth Energy Budget is nonsense.

It is not at all surprising that physics adheres to a minimum total energy in the system. The total flow of photons from the warmer and the colder bodies are constrained by the energy density requirement between the bodies.  The presence of two bodies in equilibrium at non-zero temperatures causes a reduction in the total photon generation in the system of the two bodies.  The normal radiation of each body compared to its being alone in a T=0K environment is reduced.  By means of the electromagnetic field between these two planes, the photon emission of the planes is coupled and affected by the presence of the other plane.  This is in no way surprising for an electromagnetic field problem.  One needs to remember that photons are creatures of electromagnetic fields.  Opposing streams of photons do not annihilate one another to cancel out energy, they simply add their energies.  Treating them as though one stream has a negative energy and the other a positive energy is just a means to throw the use of the Conservation of Energy out the window.  That is too critical a principle of physics to be tossed out the window.

Extending the Solution to Gray Body Thermal Radiators and Other Real Materials:

Many real materials do not behave like black body radiators of thermal radiation.  Those that do not radiate as black body materials would, radiate less than the black body radiator would.  Why would they radiate less?  This is because they do not create as high an electromagnetic field energy density at their surfaces as does a black body radiator.  From Stefan’s Law for a black body radiator, the energy density at the surface is

e = aT⁴

but for a gray body radiator the energy density at each wavelength λ is

e(λ) = εaT⁴.

This means the energy density at any given frequency is a constant fraction ɛ of that of a black body radiator, with 0 ≤ ɛ ≤ 1.

In general, a material may not behave like either a black body or a gray body radiator at a particular wavelength.  In that case,

e(λ) = ε(λ)aT⁴,

where the fraction of the black body output at wavelength λ is variable.

An isolated material surrounded by vacuum and a T=0 K environment then has a power per unit area output of

P(λ) = εσT⁴ for a gray body and ε is seen to be the emissivity, and

P(λ) = ε(λ)σT⁴, for a general material, such as carbon dioxide or water vapor, where the absorption and emission become variable fractions of that of a black body as a function of wavelength.

For our two parallel plates above, if both are gray bodies, then between the plates

Δe = e_H – e_C = ε_H a T_H⁴ – ε_C a T_C⁴

P_HI = (σ/a) Δe = ε_H σ T_H⁴ – ε_C σ T_C⁴

P_CI = 0.

Here we see that the emissivity which determines the electromagnetic field energy density at the surface is also playing the role of the absorptivity at the absorbing colder surface.  So of course, Kirchhoff’s Law of thermal radiation that the emissivity equals the absorptivity of a material in a steady state process applies.  There is really nothing at all to prove if one starts with the primary fact that the energy density is the fundamental driver of the thermal radiation of materials and we know its boundary conditions.

Update:  Clarified that any emission from a cooler body in equilibrium with a warmer body results in still less emission from the warmer body on 3 September 2018.