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The Nested Black Body Shells Model and Extreme Greenhouse Warming

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And Lessons from this model that show us how limited the Greenhouse Effect actually is

In a previous post, Critique of The Steel Greenhouse by Willis Eschenbach, I wrote about the Willis Eschenbach thought experiment model of a perfectly conducting sphere closely surrounded by a concentric perfectly conducting shell in which these bodies behave like black body radiators and the only energy loss mechanism is thermal radiation.  That black body thought experiment was not presented as a good model of the greenhouse gas effect for the Earth.  It was presented to illustrate that there is a warming effect due to thermal radiation absorption in the atmosphere, which many call the greenhouse gas effect.  I pointed out that Eschenbach is right that the surrounding shell causes the inner sphere to have a higher temperature than it would if it were only surrounded by vacuum at T = 0 K because it loses radiant energy more slowly.

But Eschenbach makes a very serious error in common with almost all scientists because he believes the surrounding shell radiates the surface of the inner sphere with the same radiation that the outer shell radiates from its outer surface toward T = 0 K surroundings.  I have explained why this is an error in my post Solving the Parallel Plane Black Body Radiator Problem and Why the Consensus Science is Wrong.  By making this error, he multiplies the photon energy density by a factor of 3 in the space between the sphere and the shell, making this one of many ways that the catastrophic man-made global warming advocates greatly strengthen the greenhouse gas effect.  I pointed out that the current NASA Earth Energy Budget amplifies the photon energy density in the atmosphere even more by a factor of 12.8!  This is critical because the warming effect due to greenhouse gases is proportional to the photon energy density at a wavelength times the absorption cross section at that wavelength.  Consequently, increasing the photon energy density by a factor of 12.8 increases the warming energy by a factor of 12.8 at all wavelengths at which absorption occurs for the gas molecule.

In this post, I will provide the radiative equilibrium solution for a black body sphere with 2 surrounding black body shells.  This might seem important because the mean free path length for absorption of longwave infrared photons that can be absorbed by water vapor and carbon dioxide is very short at the primary absorption wavelengths, though it can be much longer in the wings of those primary absorption wavelengths.  The solution will then be generalized to N surrounding black body shells since the absorption of photons of different wavelengths by a greenhouse gas in the atmosphere can take different numbers of spheres in a model.  In fact, for a given infrared-active molecule, the number of shells is a function of the absorption cross-section as a function of the wavelength because the mean free path length varies over orders of magnitude from the main absorption wavelengths out into the tails or wings of those principal absorption peaks.

Having developed this nested shell model for black body absorbers/radiators, I will then generalize this model to account for a sphere surface which has an emissivity different from the emissivity of the shell surfaces.  Because the absorption of greenhouse gases is over a substantially smaller range of frequencies than is that of the black body material, one might expect a major change in the result for the inner sphere equilibrium temperature based on the effective emissivity/absorptivity of the inner sphere surface and of the nested shell surfaces that might be an analog to greenhouse gases.  This result proves to be very interesting.

The results will inform us of interesting properties of an energy loss problem dominated by radiation loss and absorption.  However, this model is not a good model of the Earth’s greenhouse gas effect.  Energy loss and transport from the Earth’s surface is not dominated by radiant energy loss and absorption.  It is dominated instead by the effects of the Earth’s gravitational field moderated by convection and the evaporation-condensation cycle of water.  The temperature of the Earth’s surface and of each successively higher layer of air in which an infrared-active molecule will absorb the longwave radiation from the Earth’s surface or a lower layer of air is not determined primarily by radiation transport of energy, but by gravity, convection, and the water cycle.

One has to remember that there are many competing effects in determining the Earth’s climate, including many cooling effects by both water vapor and carbon dioxide that are too often underestimated.  Radiative cooling of the surface is much less than NASA and the UN IPCC claim it is.  The alarmist rendition of greenhouse gas warming does everything it can to amplify that effect, to ignore the many cooling effects, and to over-emphasize the role to carbon dioxide relative to that of water vapor.  Finally, I will adapt a portion of the shell model to make an estimate of the total greenhouse gas effect in the real world of the Earth’s climate.  I will then proceed to make an estimate for the size of the greenhouse effect for the first 400 ppm of carbon dioxide in the atmosphere and discuss briefly what one can expect as one adds higher concentrations of carbon dioxide to the atmosphere.  These estimates are not precise, but they are of the proper scale and will inform us that the warmer Earth surface compared to its overall radiative temperature as seen from space is in very little part due to the absorption of longwave radiation by carbon dioxide.  It also makes it clear that the infrared-absorbing effect due to water vapor is also a small fraction of the 33K effect normally attributed to greenhouse gases, though that effect is many times the effect due to carbon dioxide.

The inner core sphere section of unit area has a power input of Q, the temperature of the sphere is TS and the power per unit area of surface it emits is PS.  The first closely surrounding concentric shell has no power input except that radiated by the sphere.  Its own temperature is TO1, which when the power to the sphere is turned on is 0 K and the shell is only then warmed by radiation from the sphere which travels at the speed of light to it.  It radiates no photons toward the sphere, but does radiate photons as its temperature rises toward the second shell with power PO1.  The second concentric shell has the same parameters but with a 2 in the subscript, rather than a 1.  It also starts from T=0K.  Only vacuum exists between the sphere and the planes so that there are no heat losses except by means of thermal radiation.

When the power Q to the sphere is first turned on, the sphere has an initial temperature of TSI, given by the Stefan-Boltzmann Law, since the sphere is at that instant surrounded by T = 0K.

Q = PSI= σTSI4

Let the thermal equilibrium values of each parameter be denoted with the addition of an E in the subscript, then at thermal equilibrium:

Q = PSE= PO1E = PO2E

Q = σ TSE4σ TO1E4 = σ TO1E4σ TO2E4 = σ TO2E4

We can see that


From the right side of the equilibrium equation we see that

TO1E4= 2 TO2E4

Then plugging this value in the part of the equilibrium equation involving TSE, we have

TSE4– TO1E4 = TO2E4

TSE4– ( 2 TO2E4 ) = TO2E4

TSE4= 3 TO2E4 or TSE = 30.25 TO2E= 30.25 TSI  = 1.3161 TSI, since TO2E = TSI

With one surrounding shell, the equilibrium temperature of the enclosed sphere was given by

TSE= 20.25 TSI = 1.1892 TSI

Thus the second shell causes a sufficient reduction in the cooling rate that the equilibrium temperature of the sphere is 1.067 times higher than it would be with one surrounding shell.  The rise in the sphere temperature is less with each added shell.

The Nth surrounding shell results in a radiative equilibrium sphere temperature of

TSE= (N+1)0.25 TSI


For 10 shells: TSE = 1.8212 TSI

For 100 shells: TSE = 3.1702 TSI

For 1000 shells: TSE = 5.6282 TSI

Imagine modeling the absorption of surface radiation by water vapor using such a model.  Of course, water vapor is not a black body absorber or radiator of longwave infrared radiation.  For most of the radiation that it absorbs, it has a mean free path for absorption which is short, though in the wings of an absorption maximum, the absorption cross section can be much lower and the corresponding mean free path is much longer.  But for a crude model, one might say that the absorption mean free path is something like 10 meters.  One might then say that to account for water vapor absorption out to 8000 meters altitude, one needs 800 shells.  Of course, one would think each shell would absorb only a fraction of the power that a black body would and it would emit only a fraction of that energy also.  Carbon dioxide has a longer mean free path and it absorbs a smaller fraction of the power that a black body would compared to water vapor, but it also does not have a relatively sharp cut-off in its density at higher altitudes in the atmosphere.  One might model it with a much smaller fraction of absorption compared to water with a shell every 40 meters but with 300 shells to get to an altitude of 12000 meters.

Let us see what happens in this simple model if we assign an emissivity to the sphere surface, ɛS, and an emissivity to shells representing absorptions by a greenhouse gas, ɛG.  The equations for a two-shell model then become:

Q = ɛSσ TSI4

At equilibrium,

Q = ɛSσ TSE4ɛG σ TO1E4 = ɛGσ TO1E4ɛG σ TO2E4 = ɛGσ TO2E4

ɛS σ TSI4 =  ɛG σ TO2E4, so TO2E= (ɛS/ɛG)0.25 TSI

TO1E4= 2 TO2E4

ɛS TSE4 = 3 (ɛGTO2E4) = 3 (ɛS TSI4)

TSE= 30.25 TSI, the same solution for the equilibrium sphere temperature we had for the black body emitters and absorbers.

Consequently, for N greenhouse gas shells we have:  TSE= (N + 1)0.25 TSI just as with N black body shells.

So, the greenhouse gas hypothesis is looking as though it could indeed cause a disastrous increase in the Earth’s surface temperature even though greenhouse gases are far less efficient absorbers and emitters than are black body absorbers, right?

Wrong.  There is a fatal flaw in the model.  That fatal flaw is the assumption that the only way that heat is transported from the surface of the Earth is by means of thermal radiation.  In reality, much more heat is transported up through the atmosphere by means of the water evaporation and condensation cycle and by convection currents.  Let us look once again at the NASA Earth Energy Budget:

There is no back radiation of 100% and the surface radiation given as 117% in terms of the top of the atmosphere solar insolation is hugely exaggerated by NASA.  The real thermal radiation from the surface is the difference between these values or 17%.  This is perhaps a decent value for the loss from the surface itself and of this 12% is lost immediately to space.  This leaves only 5% of thermal radiation from the surface that is absorbed by the atmosphere along with 5% attributed to convection and 25% lost by water evaporation.  So, in this energy balance only 5% / (5% + 5% + 25%) = 0.143 of the surface energy is absorbed by the atmosphere as radiation from the surface.

Even this is a huge overstatement of the fraction of the surface energy carried off by means of transport by radiation from shell to shell.  This is the fraction that is absorbed by the first shell of the many-shell model.  Once that first absorption occurs, the absorbing greenhouse gas molecule passes off the absorbed energy to the nitrogen and oxygen molecules and the argon atoms that collide with it 6.9 billion times a second at sea level and 2.1 billion times a second at 10 Km altitude in the U.S. Standard Atmosphere of 1976.  Yes, the greenhouse gas molecule very quickly comes into temperature equilibrium with the molecules in the same layer of air with it.  These greenhouse gas molecules can radiate thermal energy to the layer of air just above which is slightly cooler, but the time between such radiation events is extremely long compared to the gas collision frequency.  Water molecules radiate only about every 0.2 seconds and carbon dioxide molecules radiate only about once a second.

Consequently, almost all of the 5% of surface energy that is radiated to the first shell and absorbed is thereafter transported by convection.  The assumption in the model above that all energy is transported through the atmosphere as radiation and in stages could not be more wrong.  In fact, while the little bit of further radiation transport from one layer of air to a cooler layer of air a short distance above it would be handled in the nested shell model as a further warming of the surface.  But, this should actually be viewed as a cooling effect, because any energy transported from air layer to air layer is transported to higher altitude faster compared to the alternative transport mechanism of convection.  These realizations constitute a good lesson in the need to check your premises!

Based on almost all of the 5% of surface radiation being absorbed by the first shell and then thereafter being transported by convection, about what surface temperature might we expect?  Recall that in the one shell case, the surface equilibrium temperature is given by

TSE= 20.25 TSI = 1.189 TSI

At a thermal radiation fraction of 0.143, the greenhouse gas effect temperature rise would be about 0.143 (0.189) = 0.027 times TSI.  If one takes TSI = 255 K, the radiative temperature of the Earth system as a whole with respect to space, then the change of temperature attributable to the greenhouse gas effect for our present atmosphere is 
ΔT = 6.9 K

Note that 6.9 K is only 21% of the 33K difference of the surface temperature with respect to the Earth’s effective radiative temperature as seen from space.  Consequently, the claim that this temperature difference is due to the absorption of longwave infrared radiation emitted from the Earth’s surface is exaggerated by nearly a factor of 5.

Almost all of this 6.9K warming effect is due to water vapor, not the 400 ppm of carbon dioxide in the atmosphere.  The absorption by carbon dioxide compared to that by water vapor is about 8 times less.  Thus the portion of the total infrared warming effect due to carbon dioxide is one ninth of 6.9 K, which is 0.8 K.  My recent post Using Heat Transport Powers of the NASA Earth Energy Budget to Prove that Carbon Dioxide has an Insignificant Effect on Surface Temperatures, estimated that the present 400 ppm of CO2 in the atmosphere causes a temperature increase of only 0.2 K, but this is the result of a different approach to the issue and it took into account the increased absorption of solar insolation caused by carbon dioxide in the atmosphere.  The fraction of the 6.9 K net greenhouse gas temperature increase that can be attributed to carbon dioxide overall is then about 3%.

Given that the 0.2 K net warming effect of carbon dioxide is already mostly saturated, additions of further carbon dioxide to the atmosphere will have very little effect on the surface temperature of the Earth.  What is more, the feedback effect of water vapor at present common levels of water vapor is more likely negative than positive and is certainly very small.

The reason that the surface of the Earth is about 33 K warmer than the effective radiative temperature of the Earth system as a whole as seen from space is because most of the energy dissipated by the Earth’s surface and through the troposphere (the lower atmosphere) is transported upward by a combination of water evaporation and convection.  It is very important that transport by thermal radiation is a much more minor actor in the transport of heat in the troposphere.  It is critically important that most of the Earth’s heat is radiated to space from the upper part of the troposphere which is at temperatures that are close to those of the Earth system as observed from space.  This actually forces the air at the altitudes where most of the radiation to space occurs to be near and slightly lower than this low effective radiative temperature of the Earth system.  Once that is a given, then the temperature gradient in our troposphere due to the Earth’s gravitational field dictates the surface temperature with adjustments due to such cooling mechanisms as convection and the effects of the water cycle and clouds.


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