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.

Posts Evaluating Earth Energy Budget Problems

Here is a list of some of the posts in which I discuss some of the many problems in the Earth Energy Budgets put forth by the advocates of catastrophic man-made global warming:

Using Heat Transport Powers of the NASA Earth Energy Budget to Prove that Carbon Dioxide has an Insignificant Effect on Surface Temperatures, 15 June 2018

A Summary of Some of the Physics Errors of the NASA Earth Energy Budget, 8 June 2018

Water Vapor and Gravity Act Together to Warm the Earth, 19 June 2017

The Settled Science of Catastrophic Man-Made Global Warming Violates the Laws of Physics, 10 June 2017

mgh, Not Just Greenhouse Gases, Provides a Warm Earth, 23 May 2017

Greenhouse Gases Warmed the Earth Somewhat, but Additions Now Cool the Earth, 27 July 2015

Why Greenhouse Gas Theory is Wrong -- An Examination of the Theoretical Basis, 7 March 2015

Back-Radiation Insignificance for the Equilibrium Surface Temperature, 14 March 2014

The Stefan-Boltzmann Law at a Non-Vacuum Interface: Misuse by Global Warming Alarmists, 6 April 2013

The Unsettled Earth Energy Budget, 24 February 2013

Back-Radiation and the Highly Fallacious Kiehl-Trenberth Energy Budget, 10 July 2012

NASA Finally Produces A Realistic Energy Budget for the Earth, 26 May 2011

The Nested Black Body Shells Model and Extreme Greenhouse Warming

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 T_S and the power per unit area of surface it emits is P_S.  The first closely surrounding concentric shell has no power input except that radiated by the sphere.  Its own temperature is T_O1, 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 P_O1.  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 T_SI, given by the Stefan-Boltzmann Law, since the sphere is at that instant surrounded by T = 0K.

Q = P_SI = σT_SI⁴

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

Q = P_SE = P_O1E = P_O2E

Q = σ T_SE⁴ – σ T_O1E⁴ = σ T_O1E⁴ – σ T_O2E⁴ = σ T_O2E⁴

We can see that

T_SI = T_O2E

From the right side of the equilibrium equation we see that

T_O1E⁴ = 2 T_O2E⁴

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

T_SE⁴ – T_O1E⁴ = T_O2E⁴

T_SE⁴ – ( 2 T_O2E⁴ ) = T_O2E⁴

T_SE⁴ = 3 T_O2E⁴ or T_SE = 3^0.25 T_O2E = 3^0.25 T_SI  = 1.3161 T_SI, since T_O2E = T_SI

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

T_SE = 2^0.25 T_SI = 1.1892 T_SI

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

T_SE = (N+1)^0.25 T_SI

Thus,

For 10 shells: T_SE = 1.8212 T_SI

For 100 shells: T_SE = 3.1702 T_SI

For 1000 shells: T_SE = 5.6282 T_SI

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, ɛ_S, and an emissivity to shells representing absorptions by a greenhouse gas, ɛ_G.  The equations for a two-shell model then become:

Q = ɛ_S σ T_SI⁴

At equilibrium,

Q = ɛ_S σ T_SE⁴ - ɛ_G σ T_O1E⁴ = ɛ_G σ T_O1E⁴ - ɛ_G σ T_O2E⁴ = ɛ_G σ T_O2E⁴

ɛ_S σ T_SI⁴ =  ɛ_G σ T_O2E⁴, so T_O2E = (ɛ_S/ɛ_G)^0.25 T_SI

T_O1E⁴ = 2 T_O2E⁴

ɛ_S T_SE⁴ = 3 (ɛ_G T_O2E⁴) = 3 (ɛ_S T_SI⁴)

T_SE = 3^0.25 T_SI, 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:  T_SE = (N + 1)^0.25 T_SI 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

T_SE = 2^0.25 T_SI = 1.189 T_SI

At a thermal radiation fraction of 0.143, the greenhouse gas effect temperature rise would be about 0.143 (0.189) = 0.027 times T_SI.  If one takes T_SI = 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.

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.