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:
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 PS, 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 TO will increase. What
will happen to TS?
The
power transferred from the inner core to the outer shell is PS. The
power radiated from the outer surface of the shell section will be PO
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
= PS = σ TS4
- σ TO4
PO
= σ TO4
At
equilibrium, PS = PO, so
TSE4
- TOE4 = TOE4, where the added E in
the subscripts designates the equilibrium values.
Therefore,
TSE4 = 2 TOE4 or TSE =
1.189 TOE and
PS
= σ ( 2 TOE4 - TOE4 ) = σ TOE4.
But
Q = PS always, so when the shell was still at T=0 K, Q = PS
= σ TSI4 , where TSI 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,
TOE
= TSI
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
TSE
= 1.189 TOE = 1.189 TSI
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 TSI
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:
PS
= 470 W/m2 = σ TSE4
TSE
= 301.74 K
In
my case,
PS
= σ TSI4 = 235 W/m2
TSI
= 253.73 K
TSE
= 1.189 TSI = 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.
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