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.