Do Additional Greenhouse Gases Warm or Cool the Earth?

If the Earth’s atmosphere had no infrared-active gases, commonly and confusingly called greenhouse gases, at all, the Earth would be colder on average.  The Earth’s surface would absorb more of the sun’s insolation, since water vapor would not be present to absorb the incoming energy from the sun and there would be no clouds.  Some of the heat absorbed by the surface would still be transferred to the nitrogen, oxygen, and argon molecules or atoms striking the Earth’s surface.  The remaining energy would be radiated from the surface.  Virtually all the energy radiated from the Earth’s surface would travel at the speed of light through the atmosphere into space and be almost instantaneously lost.  The day to night temperature changes would be much more dramatic than they are now.  The greenhouse gases benefit us greatly by moderating the day to night temperature changes.  At sufficiently low concentrations, each infrared gas with a non-overlapping absorption frequency with respect to other infrared-active gases already present, will slow down the rate of cooling at the surface and in the troposphere.  This allows the surface and the troposphere to be warmer than they would be were the infrared-active gas not present.

This is how the idea of a greenhouse effect comes about.  This paper Is not disputing that infrared active gases allow the Earth’s surface to be warmer than it would be if they were not in the atmosphere.  The question being examined is whether the further addition of an infrared-active gas will warm or cool the Earth’s surface and its lower atmosphere, the troposphere, when its atmospheric concentration is increased.

How does a low concentration of an infrared-active gas significantly slow down the cooling rate of the surface and the troposphere?  Suppose this gas molecule absorbs the longwave thermal radiation emitted from the Earth’s surface in the lower troposphere and enters an excited vibrational state.  If that absorbed energy were simply immediately re-emitted and carried off the previously absorbed energy at the speed of light, the absorption event would have no significant effect on the temperature of the surface.  The key fact here is that the excited molecule has billions of collisions per second with the 2500 times as plentiful non-infrared-active molecules of nitrogen and oxygen and atoms of argon.  That absorbed infrared energy is converted into kinetic energy passed to the molecules that collided with the excited molecule long before the lifetime of the excited molecule for re-emission of the absorbed energy by radiation.  What is the overwhelmingly dominant means of energy transport through the troposphere at this point?  It is the convection transport from warmer to cooler portions of the atmosphere, which is generally upward and to the higher latitude regions of the Earth.  The speed of that transport of energy is about 8 orders of magnitude slower than the speed of light.  So that first act of longwave thermal radiation absorption from the Earth’s surface is of immense importance, but after that initial conversion of infrared radiation from the surface into kinetic energy shared by all the molecules (mostly nitrogen and oxygen) and atoms (argon mostly) of our troposphere, the role of any further radiation from infrared-active molecules is a faster means of cooling than is the convection cooling mechanism.

Let us examine why this is true.  If the mean free path for absorption of a given wavelength of the longwave thermal radiation from the surface is short enough that there is an absorption event by an infrared molecule in the lower troposphere, subsequent absorptions of any emitted thermal radiation by molecules at that wavelength at higher altitudes will prevent that radiation from escaping into space.  This does slow down the cooling of the lower atmosphere in this very limited context.  This is what the standard view of greenhouse gases focusses on.  The problem is that the alternative to that infrared-active gas emitting thermal radiation to a higher altitude is its being much, much more slowly transported to higher altitude by convection.  Adding more of that infrared-active gas to the atmosphere results in moving energy upward through the troposphere in steps at the speed of light instead of having more of it moving upward in the slow convection currents.  The implication here is that a very low concentration of an infrared gas in the atmosphere will produce a warmer Earth surface and troposphere, but subsequent additions simply speed up the transport of energy to higher altitudes.  Then the added infrared emitting gas molecules in the upper troposphere and the stratosphere radiate thermal energy at a faster rate directly into space.  For an infrared-active molecule, the initial effect of adding it to the atmosphere is likely to be warming effect, but its warming effect rapidly passes through a maximum and then further additions start bringing down the temperature at the surface and in the troposphere.

Let us pause and get a better understanding of how carbon dioxide absorbs longwave radiation from the surface of the Earth.  An infrared-active molecule has an absorption spectrum over a range of wavelengths with absorption probabilities varying with wavelength over orders of magnitude.  The absorption probability at a wavelength is usually given in terms of an effective cross section, as though the size of the molecule were different for the absorption event at each wavelength.  Here, from a figure in Prof. Howard “Cork” Hayden’s The Energy Advocate, February 2020 (Vol.24, No.7) is the absorption spectrum near the main 15 µm (micrometer) absorption line for CO₂:

As Prof. Hayden explains, if the concentration of carbon dioxide in the atmosphere were only 40 ppm by volume (ppmv) or a bit less than one-tenth the present concentration, any radiation at the wavelengths above the red line in the figure at 1 x 10^-22 m² cross-section would be absorbed within a travel distance of 10 meters.  At that same very low concentration of CO₂, the absorption distance for radiation in the weaker absorption peaks above the lowest red line is less than 100 meters.  The troposphere in the U.S. Standard Atmosphere is 11,000 meters in altitude.  All parts of the absorption spectrum above the lower red line for as low a concentration of carbon dioxide as 40 ppmv are already absorbed many times traveling through the troposphere from the surface to the upper troposphere.  At 400 ppmv, carbon dioxide will add absorption events for the first time in the lower cross-section parts of the spectrum, but the additional first time absorptions are decreasing rapidly as more CO₂ is added to the atmosphere, while the rapid transport effects of carbon dioxide are moving more and more energy to space more quickly than convection would at an increasing rate as CO₂ is added.

I am going to develop a simple model for how the infrared-active gases warm the Earth, according to those who believe in catastrophic or even moderate man-made global warming.  This is not a model that I believe is correct.  This is intended as an exercise in determining a very generous upper limit on the size of the greenhouse gas warming claims based on additions of such gases to the atmosphere above those of the present and then showing that those claims are false in the context I have set up above.

I will start with a two layer atmosphere in which radiation from the Earth’s surface, is absorbed entirely in the lower layer L1, which radiates half that energy upward to atmospheric layer L2 where it is absorbed and half back down to the surface where all of it is absorbed.  All of the surface absorbed half from L1 is re-emitted upward and is re-absorbed in L1.  Layer L2 emits half the energy it absorbed from L1 directly to space, where it is permanently lost, and half back to L1 where all of it is absorbed.  Each layer L1 and L2 always emits half of its absorbed radiation energy upward and half downward.  The surface always emits its absorbed energy back to L1, where it is absorbed.  I believe this isotropic emission idea is incorrect, but we are going to do this exercise because it will give us insight and because it is rather a fun task to work out.  Note also that I am entirely ignoring convection as an energy transport mechanism for the sake of this argument.

The results are that space receives a series of emission energy originating with an emission of one unit of energy from the Earth’s surface, Sp, whose sum is

So, all of the unit of energy emitted from the surface is eventually emitted into space.  In fact, the first 5 emissions to space already total 0.7627 of the total energy of 1.  In this crazy model we have assumed all the energy emitted from the surface is absorbed in L1, despite the fact that the atmospheric window actually allows about 70% of all surface radiated longwave energy to pass through the atmosphere unabsorbed and directly into space.  So Sp really equals 0.3 and the first five emissions to space really equal 0.2288.  In this simple model with no thermalization of the absorbing molecule and no significant half-life before it re-emits energy as radiation, sending more than 3/4ths of the surface emitted energy into space takes about the time it would take radiation traveling at the speed of light, 3 x 10⁸ m/s, to travel through the troposphere about 11,000 m high five times.  That time is 3.7 x 10^-5 s.  In comparison, it takes hours for the alternative heat dissipation process of convection currents to raise surface energy to the top of the troposphere where it can be directly emitted to space as radiation.

Of course, the catastrophic man-made global warming argument does not emphasize the speed with which energy is emitted to space by a radiation-centric model.  They emphasize the added time radiation energy spends near the surface because of their isotropic emission model in comparison to the time it would take to go directly from the surface to space if there were no infrared-active molecules.  So what enhancement of radiative energy dwell time are they getting?  With this two atmospheric absorption layer model it would be that the sum of radiation back to the surface from L1, Su, is

This means that the energy dwell time in the lower troposphere has been doubled by this two-layer 100% radiative energy loss model with isotropic emission.  But keeping about 76% of this energy around for about 3.7 x 10^-5 s is not such a big deal.

What if we put three atmospheric absorbing layers into the model?  Then I find that the series of emissions to space from the top layer L3 is

Sp = 1/8 + 1/8 + 7/64 + 3/32 + 41/512 + 35/512 + 239/4096 + 577/16384 + ….

These first eight terms sum to 0.695129, so with eight emissions from L3 to space 69.5% of the unit of energy emitted from the surface has been lost to space.  Consequently, 69.5% of the energy is lost in about 8 (11,000 m) / 3 x 10⁸ m/s or 2.9 x 10^-4 s.

Su = ½ + 3/8 + 5/16 + 17/64 + 29/128 + 99/512 + 169/1024 + 577/4096 + 1731/16384 + …..

The first eight terms of Su sum to 2.01898.  So when Sp is a bit over 2/3 after eight terms and must sum to one, Su is a bit over 2 and seems most likely to sum to about 3.

So, let us make a leap here and assume that with an atmospheric absorption model of n layers with isotropic emission, Su will sum to about n.  [If someone has the time to work this series out to more terms or can find a way to solve it exactly, I would enjoy seeing the result.]

Alright now, let us assume that we have 100 absorption layers in our atmosphere, corresponding to an absorption distance of about 110 m.  Most of the radiation energy emitted from the surface will find its way to space in less time than 100 (11,000) / 3 x 10⁸ = 3.7 x 10^-3 s.  If you have 10,000 atmospheric absorption layers (an absorption distance of about 1.1 m), the time is then 0.37 s.  But the alternative means of removing that energy to space is convection and that takes hours to do the job.

In fact, radiation between the layers only occurs long after an absorption event during which time many, many collisions with other molecules would occur and the absorbing molecule would give up virtually all the energy it had absorbed from radiation from another layer to the 2500 times as plentiful non-greenhouse gas molecules in the air.  It is a comparatively very long time before the infrared-active molecule emits radiation again.  In the meantime, it is only 1/2500 of the molecules moving energy as part of a convection current.  Yet insofar as these infrared-active molecules do emit radiation, they are acting to speed up the emission of surface energy to space.  They are therefore acting to cool the atmosphere from top to bottom of the troposphere compared to the convection energy transport mechanism.

The lesson here is that the very first absorption event of thermal radiation from the surface in the atmosphere is very important because it puts the transfer of that energy into the hands of a much slower convection cooling process than is that of radiation.  However, whatever further thermal radiation events occur in the atmosphere simply speed up the loss rate of energy to space compared to the rate due to convection.  The addition of further infrared-active gases to the atmosphere causes there to be more absorption layers in the model.  If the atmospheric load of infrared absorption gases was so high that the mean free path length for absorption of their emissions was as short as 1 meter, then the time to dissipate most of the energy to space would still be less than a second, while the time it takes for the alternative energy transport by convection is still hours.

Do additional greenhouse gases warm or cool the Earth?  The addition of a gas in just enough concentration that there is absorption by that gas in the lower troposphere once which would not otherwise have occurred at a given wavelength slows the rate of radiative heat loss and may be regarded as effectively warming the Earth.  It does this by converting the cooling mechanism from rapid radiative cooling to that of slow convection cooling.  However, once that threshold concentration is exceeded for a given wavelength, additions of that gas simply cool the atmosphere more quickly than would the alternative of convection currents.  At such an above threshold concentration, that gas can be regarded as cooling the Earth faster compared to the rate it would cool without its addition.

The lapse rate is the temperature gradient with altitude in the troposphere.  At normal levels of humidity, the adiabatic lapse rate is less than the dry lapse rate is.  This tells us that at normal water vapor concentrations, the water vapor concentration is already high enough to produce a cooling effect on the surface and lower troposphere temperatures.  Water vapor does this with the cooling effect at the surface as liquid water becomes water vapor and then the water vapor rises with convection until it reaches an altitude at which it condenses and releases energy.  At the warm surface it cools with evaporation and at the cooler altitudes it warms by condensing.  The evaporation process increases the water molecule’s kinetic energy, including its vibrational modes, and the molecule carries that energy upward by convection and then releases the kinetic energy of evaporation as it condenses to liquid or solid form.   Each water molecule carries more energy per molecule at a given temperature than can a nitrogen or oxygen molecule.  Thus, as they rise with convection, they are transporting more energy upward per molecule than are the dry air molecules in the same convection current.  In addition, the water molecule is radiating energy to the layer of air above it, which is usually cooler and able to absorb that radiated energy if it also has water vapor molecules in it or sometimes if it has carbon dioxide in it. 

Unlike water vapor molecules, a carbon dioxide molecule carries less energy at a given temperature than do the nitrogen and oxygen molecules with which it shares a convection current.  This means an added CO₂ molecule causes a convection current to become a less effective cooling mechanism.  However, it retains the ability to warm the air layer above it throughout the troposphere as it cools its local surroundings by radiation to the layer above it.  In addition, more CO₂ in the upper troposphere and in the stratosphere means more molecules radiating energy directly to space.  There is good evidence that the addition of more hot molecules of CO₂ in the stratosphere has resulted in a measured  cooling of the stratosphere, as would be expected because these hot molecules are effective radiators.  I believe the concentration of carbon dioxide in the atmosphere is already high enough that additions of CO₂ are cooling the troposphere as well or at least counterbalancing the mild warming effect of additional carbon dioxide molecules to a great degree.

Whether I am right or not about this, the claim that a very small warming effect by additional CO₂ will be amplified by a greater warming effect by increased water vapor (a positive feedback) is surely false.  The fact that the wet adiabatic lapse rate is less than the dry adiabatic lapse rate makes it clear that the claim of a positive feedback is wrong.  The IPCC and other alarmists depend upon this false claim of a positive feedback by water vapor to make it appear possible that additional carbon dioxide will cause significant harm.  The reality is that additional carbon dioxide has no net significant effect on temperatures at the Earth’s surface or in the lower troposphere.

Meanwhile, additional carbon dioxide in the atmosphere provides plants with the means for easier growth.  With a growing human population, this is very helpful in producing the additional food we need to produce.  This should be a factor in reducing human anxiety for the future.  Of course, I understand that some people just have to have something to worry about.  I suggest you worry, if you must, about another ice age which additional carbon dioxide in our atmosphere cannot prevent.  Or, you might worry about an asteroid  striking the Earth.  But it is even more foolish to worry about problems created by more CO₂ in the atmosphere.

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.