Radiation physics constraints on global
warming: C02 increase has little effect
By Denis G. Rancourt
Former physics professor, University of Ottawa, Ottawa, Canada.
Posted on June 3, 201 1 :
http://climateguy.blogspot.com/201 1/06/radiation-physics-constraints-on-global.html
(Small clarifications added and several typos corrected here in this December 3, 2011, pdf version.)
Abstract — I describe the basic physics of planetary radiation balance and
surface temperature, using the simplest model possible that is sufficiently realistic
for correct evaluations of predicted surface temperature response sensitivities to
the key Earth parameters. The model is constrained by satellite absolute integrated
intensity and spectroscopic measurements and the known longwave absorption
cross section of C02 gas. I show the predicted Earth temperature for zero
atmospheric resonant absorption of longwave radiation (no greenhouse effect in
the otherwise identical atmosphere) to be -46°C, not -19°C as often wrongly
stated. Also, the net warming effect from the atmosphere, including all
atmospheric processes (not just greenhouse forcing), without changing anything
else (except to add the removed atmosphere) is +18°C, not the incorrect textbook
value of +33°C. The double-layer atmosphere model with no free parameters
provides: (a) a mean Earth surface temperature of +17°C, (b) a post-industrial
warming due only to C02 increase of 5T = 0.4°C, (c) a temperature increase from
doubling the present C02 concentration alone (to 780 ppmv C02; without water
vapour feedback) equal to 8T = 1 .4°C, and (d) surface temperature response
sensitivities that are approximately two orders of magnitude greater for solar
irradiance and for planetary shortwave albedo and longwave emissivity than for
the atmospheric greenhouse effect from C02. All the model predictions robustly
follow from the straightforward underlying assumptions without any need for
elaborate global circulation models. The same longwave optical saturation that
provides such a large radiative warming of the planet surface also ensures that the
warming effect from increasing C02 concentration is minimal. I conclude with
suggested implications regarding warming alarmism, errors by sceptics, research
funding, and scientific ignorance regarding climate feedbacks.
Rancourt on radiation physics - C02 little effect
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Introduction
Historically, the greatest ability of the physicist has been to perform simple calculations
that capture the essential features of a physical phenomenon in order to correctly
elucidate the underlying principal causes. This is the ultimate "What is going on?"
challenge.
Too many practicing physicists, in the many areas where physics is applied, have lost or
never had this ability. Instead, they have been incorporated into the enterprise of using
computers to simulate reality using questionable selections of approximate or invalid
algorithms in large simulation programs.
These programs develop lives of their own, as careers and reputations are invested in
their incremental development and in their predictions. A pathological optimism envelops
the practitioners with the illusion that their algorithms capture complex features in some
"average" or "effective" way and efforts are made to demonstrate this in so-called
"validation" exercises rather than perform simple calculations that would demonstrate the
algorithms themselves to be wrong for the intended application.
Physicists have largely abandoned their gadfly role of fundamentally challenging broad
interpretive schemes in order to serve and benefit from career-enhancing collective
enterprises, often dressed in elaborate conceptual edifices and often supported by
computer simulations.
I believe this situation is playing itself out today in climate modelling science. As a
physicist, if on close examination I can't understand what the C02 warming alarmism is
about and I can't get any of my colleagues to explain it - without computer-black-box
magic, in published papers or elsewhere - then I am not going to believe it.
At its core, planetary surface temperature is a macroscopic radiation balance phenomenon
that has been understood for one hundred years or so. If global warming alarmism is
justified then it must be possible to explain why it is justified in simple terms and without
appealing to faith or authority for any essential point in the argument.
I've tried to do this, as honestly and openly as possible, and I have asked my peers to find
any errors. I believe the present article to be error-free and to conclusively show that we
should not be focussed on C02 if we are concerned about the planet's surface
temperature. I am additionally of the opinion that we should not be concerned about the
planet's surface temperature.
Regarding the sceptic-warmist debate, my conclusion is: The sceptics say many incorrect
things but they are right whereas the warmists say many correct things but they are
wrong. The skeptics appear to be motivated by skepticism whereas the warmists appear
to be motivated by conformism. The skeptics' incorrect things have been used to discredit
the skeptics and the warmists' correct things have been used to mask a lie.
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Simplest models with essential features
My goal is to construct the simplest possible models of planetary radiation balance,
realistic enough to capture the essential global average features using different physical
assumptions about the type of atmosphere.
I take the planet to be a perfect sphere with a smooth and homogeneous surface and to
have a thin (compared to the planet radius, but longwave optically thick) atmosphere. The
planet is uniformly irradiated by a distant sun.
The incident intensity (in Watts per square -meter, W/m ) of "shortwave" radiation
(largely visible light) from the sun at the planet is the so-called solar constant, I s , where
for Earth I s = 1366 W/m 2 (having a real seasonal variation in magnitude from 1412 to
1321 W/m , or 6.7% of its average value).
Different parts of the planet's surface receive different intensities of incident shortwave
radiation. This is because the surfaces at different latitudes receive the incident rays at
different angles and because half of the planet's surface is shielded from all incident rays
(only one hemisphere is exposed to the sun at any given time).
Rather than deal with the latter complexity of non-uniform irradiation, instead, as is
commonly done, we take the entire planet's surface to be uniformly irradiated with an
intensity equal to the corresponding average solar constant. The correct average solar
constant is * = (1/4)I S = 341.5 W/m 2 , as is well known and easy to calculate.
In my models, therefore, every part of the planet's surface is identical in terms of the
radiation balance conditions. Each part of the planet's surface represents what is
happening on average, in terms of radiation balance, and of the planet properties which
we take to be the Earth's average properties.
Basic concepts and Earth with no atmosphere
Of all the incident shortwave solar radiation that strikes the planet a fraction is reflected
back into outer space without being absorbed by any part of the planet (surface or
atmosphere). This fraction (from zero to one) of the incident shortwave solar radiation
energy that is reflected out from the planet is called the planet's (Bond) albedo.
The reflected outgoing shortwave radiation need not have the same spectral distribution
(radiation intensity versus radiation frequency or wavelength) as the incoming incident
solar shortwave radiation because the amount of absorption/reflection can be (and
generally is) dependent on wavelength. The albedo is the net energy fraction that is
reflected.
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Modern satellite spectroscopic measurements can quantify the solar constant and the
amount of out-reflected shortwave radiation, can resolve these radiations from longwave
thermal radiation, and can measure continuously in orbit to obtain planet- wide averages.
Satellite measurements allow us to conclude that the average Earth albedo is = 0.30
[1]. Arguably-more-direct and reliable Earth-based so-called "Earthshine observation"
measurements give = 0.297(5) where, using scientific error notation, the latter means
0.297 ± 0.005 [2]. There are daily changes in Earth's albedo (from large scale weather
changes) of -5% and seasonal variations of -15% (from snow and ice cover, vegetation,
and weather and cloud cover) [2].
The main source of heat on the planet is the planet's surface that absorbs shortwave solar
radiation. The physical absorption process transforms the electromagnetic energy of the
incident solar radiation into heat energy (vibrational energy of the surface's molecules).
In the case with an atmosphere, the atmosphere also directly absorbs a fraction of the
incident shortwave solar radiation.
Any opaque body at any temperature above 0 K (i.e., having vibrating rather than
motionless molecules) in turn emits electromagnetic radiation. The latter so-called
"thermal" or "black-body" radiation has characteristics that depend on the body's
"emitting surface" temperature. The spectral distribution of such emitted thermal
radiation follows Planck's Law (modified to allow a wavelength-dependent emissivity).
For the temperatures of interest the surface thermal radiation is longwave (or infra-red)
radiation.
The intensity I e (in W/m ) of the emitted thermal (here longwave) electromagnetic
radiation coming from the effective emitting surface of a somewhat opaque body is given
by the Stefan-Boltzmann law:
I e = s a T 4 (eq.l)
where T is the temperature of the emitting surface in K, o is the Stefan-Boltzmann
8 2 4
constant a = 5.6704 x 10" W/m K , and s is the "emissivity" of the emitting surface valid
for the relevant emitted frequencies.
The emissivity has a dimensionless value between zero and one. It is the fractional
energy emission from the surface compared to the surface's emission if it were an ideal
black body emitter, s = 1 for an ideal black body surface and s = 0 for an ideally
reflective surface (i.e., a surface having an albedo of exactly 1).
The global average emissivity (for the relevant longwave radiation), *~~, of Earth's
surface is difficult to evaluate. It can be reasonably estimated by considering the known
measured longwave emissivities of typical Earth surface materials, such as liquid water,
vegetation, and sand.
Let us next describe how the planet's mean surface temperature is established.
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If the net radiant energy into the planet is larger than the net radiant energy escaping from
the planet then the net received energy will heat the planet and increase its temperature.
Likewise, if the net radiant energy out from the planet is larger than the net energy into
the planet then the net loss of energy of the planet will cause the planet to loose heat and
decrease its temperature.
Therefore, in a "steady state" situation, after a certain planetary response time following
any change affecting radiation balance, the temperature of the planet's longwave
radiation emitting surface will stabilize at some value corresponding to the rate of
energy-in being equal to the rate of energy-out and there will be planetary "radiation
balance" at a stable planetary surface temperature.
The net energy-in is the incident solar radiation minus the albedo loss. With no
atmosphere, the net energy-out is the longwave emission energy from the planet's surface
escaping into outer space. By setting in = out we can solve for the resulting radiation-
balancing planet surface temperature.
The corresponding radiation balance equation, therefore, can be written as:
~~* (1 - **) = a T 4 (eq.2)
Solving for the planet surface temperature T 0 (in K) for no atmosphere, eq.2 gives:
T 0 = [ (1 - )** / o ] 1/4 . (eq.3)
At this point, virtually all researchers and authors have used *~~ = 1 , usually without
providing a stated justification. That is, they have assumed that the Earth's surface is an
ideal black body emitter for longwave radiation.
Using the latter assumption for ~~~~ and (for now, wrongly) assuming that the Earth's
mean albedo ~~~~ is the same with and without its atmosphere ( = 0.30) eq.3 gives T 0
= 254.8 K or minus (-) 18.3°C. Compared to the accepted actual mean global surface
temperature of 14.0°C this would imply a total global atmosphere (greenhouse) effect
warming on Earth of 32.3°C - corresponding to the repeatedly stated textbook nominal
value of 33°C of greenhouse effect warming [3].
This surface temperature (nominally -19°C) is also the surface temperature (with ~~~~ =
1) of the bare planet (no atmosphere but preserving the same albedo) that would give the
same total emission of longwave radiation presently escaping from Earth into outer
space.
Many authors have stated that this thus calculated nominal -19°C temperature is "the
Earth's temperature as seen from outer space". The latter statement is incorrect because
although the actual present integrated emission intensity would, via eq. 1 , give this
temperature, the actual longwave emission spectrum of Earth is not a black-body
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emission spectrum (i.e., does not follow Planck's Law, due to significant atmospheric
absorption) and only a black-body-radiation spectrum can be interpreted as corresponding
to an emitter's "temperature".
Wrong textbook views of Earth warming
The above described repeatedly stated textbook [3] value of 33°C of Earth greenhouse
effect warming is wrong because this is not the correct predicted value of planet surface
warming (or cooling) that occurs on turning on (or off) the greenhouse effect in an
otherwise unchanged Earth atmosphere and otherwise unchanged Earth.
I also taught the incorrect 33°C value in my university physics courses and repeated it in
my 2007 critique of global warming [4]. Wikipedia is no exceptions [5]. American
Geophysical Union (AGU) press releases typically announce [6]:
"Overall, the greenhouse effect warms the planet by about 33 °C, turning it from
a frigid ice-covered ball with a global average temperature of about -1 7 °C, to
the climate we have today. Heat-absorbing components contribute directly to that
warmth by intercepting and absorbing energy passing through the atmosphere as
electromagnetic waves. "
In describing the "physical science basis" the Intergovernmental Panel on Climate
Change (IPCC) in its 2007 "Contribution of Working Group I to the Fourth Assessment
Report" (incorrectly, see below) put it this way [7]:
"The energy that is not reflected back to space is absorbed by the Earth 's surface
and atmosphere. This amount is approximately 240 Watts per square metre (W
m—2). To balance the incoming energy, the Earth itself must radiate, on average,
the same amount of energy back to space. The Earth does this by emitting
outgoing longwave radiation. Everything on Earth emits longwave radiation
continuously. That is the heat energy one feels radiating out from a fire; the
warmer an object, the more heat energy it radiates. To emit 240 W m—2, a surface
would have to have a temperature of around -19°C. This is much colder than the
conditions that actually exist at the Earth 's surface (the global mean surface
temperature is about 14°C). Instead, the necessary -19°C is found at an altitude
about 5 km above the surface.
The reason the Earth 's surface is this warm is the presence of greenhouse gases,
which act as a partial blanket for the longwave radiation coming from the
surface. This blanketing is known as the natural greenhouse effect. "
The scientists at RealClimate.org also (incorrectly, see below) use this 33°C number in
their interpretations [8]:
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"Since we are looking at the whole of the present-day greenhouse effect (around
33 C), it is not surprising that the radiative forcings are very large compared to
those calculated for the changes in the forcing. The factor of~2 greater
importance for water vapour compared to C02 is consistent with the first
calculation. "
Virtually all mainstream science and teaching has accepted and parrots this idea that the
planetary greenhouse effect on Earth causes a warming of approximately 33°C.
In all of these sources the assumption ~~~~ = 1 is virtually never explicitly justified. It is
important to provide a justification because, at first glance, the assumption appears to
violate Kirchoff s Law of radiation physics.
Kirchoff s Law of radiation physics says generally that the larger the reflectivity the
smaller the emissivity. More precisely, Kirchoff s Law is expressed for a given
wavelength X as:
1 - a(X) = s(k). (eq.4)
It is essential to note that the law holds at each wavelength (and direction) of radiation
but that albedo at one wavelength need not be related to emissivity at a different
wavelength.
On Earth, the relevant mean (Bond) albedo is for shortwave radiation (solar radiation,
largely visible) and has a value ~~~~ = 0.30 whereas the needed emissivity is for
longwave radiation (infra-red or thermal Earth-emission radiation) such that ~~~~ can
have a value significantly different from the value 0.70 incorrectly predicted by eq.4.
We must therefore appeal to measurements of s for representative Earth surface
materials. A main Earth surface material is water. The longwave emissivity of water is
almost 1 . This is understandable because water is almost perfectly absorbing in the
infrared. Dry rocks and sand also have near-one values of their longwave emissivities,
that is values of -0.91-0. 92. Any vegetation coverage of dry soil significantly increases
the value of the emissivity, given the water content of vegetation. For example, green
grass has emissivity in the range -0.97-0. 99 [9].
This is why it is not unreasonable to use ~~~~ ~ 1 for our ocean, lake and vegetation-
covered Earth. Therefore, the assumed value of unity for emissivity is not significantly in
error for our purposes.
Similarly, note that the above calculation leading to the nominal -19°C surface
temperature is not for an Earth without its atmosphere but that is otherwise unchanged;
because such an instantaneously bared Earth would not retain its albedo of 0.30. Indeed, a
large contribution to Earth's albedo is from clouds and the atmosphere itself - it is not a
purely planet-surface albedo.
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With no atmosphere we should use the albedo of the Earth's present solid surface, in its
present state. The latter shortwave albedo ~~~~ has been measured by satellite and is
23/(23+161) = 0.125 ([1]: Fig.l). This gives (eq.3) the significantly higher no-atmosphere
mean surface temperature of T 0 = 269.4 K (or -3.7°C), for a total atmosphere warming
effect without changing anything else on the present Earth of +18°C, not +33°C. The
correct predicted surface temperature of an Earth with no atmosphere but otherwise
unchanged is -4°C.
Of course without an atmosphere there would be no vegetation, etc., and significantly
more snow and ice cover, thereby increasing the surface albedo. The latter changes are
difficult to predict and obviously have not been measured by satellite. In any case, the
relevant question for the present discussion is "What is the net warming effect from the
atmosphere, including all its processes, without changing anything else?" The answer is
+18°C,not+33°C.
It should be clear therefore that the oft-repeated nominal -19°C surface temperature both
is not the Earth temperature without an atmosphere (but otherwise unchanged) and is not
"the temperature of Earth as seen from outer space". It is a nonsense number arising from
an incorrect application of eq. 1 .
But error in most warming establishment spin is even larger than that because the
presence or absence of the atmosphere is not equivalent (in terms of global radiation
balance) to the presence or absence of an atmospheric greenhouse forcing, by any means.
In other words, in terms of the planetary radiation balance, removing the atmosphere and
removing the greenhouse action of the greenhouse gases in the atmosphere have
dramatically different effects.
This is shown below and is because the atmosphere impacts surface temperature by much
more than only via greenhouse forcing. And the other impacts cause surface cooling
rather than warming. These other impacts are: (1) direct absorption by the atmosphere of
incident shortwave solar radiation (78 W/m 2 ; [1]), (2) increased albedo from clouds and
atmosphere (0.30 vs. 0.125), (3) surface cooling via atmospheric thermals (17 W/m 2 ; [1]),
and (4) surface cooling via the evaporation/condensation via the water cycle (80 W/m 2 ;
[1]).
I show below that, as a result, the correct Earth surface temperature in the absence of
greenhouse forcing but with an otherwise unchanged atmosphere is a freezing -46°C.
This means that "global warming" from atmospheric resonant scattering of infrared
radiation on Earth is +60°C, from -46°C to +14°C, not +33°C.
Note that despite the large (~60°C) predicted atmospheric greenhouse warming on Earth
this includes the total planet greenhouse effect whereas C02 absorption is presently
saturated (see below), such that a large C02 change impact is not implied. It is essential
to recognize that a large overall planetary greenhouse warming does not imply a
significant warming from increasing the concentration of C02. Indeed only a much
smaller overall planetary greenhouse warming could give rise to a large warming effect
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Page 8 of 22
from increasing the C02 concentration in the atmosphere. The two factors (total warming
versus sensitivity to C02 increase) are anti-correlated via the phenomenon of optical
saturation (see below).
Earth with homogeneous atmosphere
Consider first the simplest atmosphere; an atmosphere which is uniform in its
temperature, composition and density and that attains its own steady state temperature by
balancing its own net in and out radiative and other fluxes of energy. Let the temperature
of the atmosphere (in K) be denoted Tp and the temperature of the planet surface be
denoted T a .
In this case, the longwave emission of the atmosphere (eq.l) up and out is equal to its
longwave emission down and in (which is fully absorbed by the planet surface, ~~~~ ~ 1).
In addition, since the atmosphere layer emits both up and down, it has a thermal emission
surface of 2~~~~ a T„ 4 is the average longwave emission intensity from the Earth surface
and ~~~~ is the longwave emissivity of Earth's surface, as defined above
• "C" is the (mean global) upward energy flux intensity from thermals, taking heat
from the surface and delivering it into the atmosphere, 17 W/m [1]
• "D" is the (mean global) upward energy flux intensity from the water cycle
(evaporation and latent heat condensation/freezing), taking heat from the surface
and delivering it into the atmosphere, 80 W/m 2 [1]
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Eq.5 represents all the radiation and other energies absorbed by and emitted by the planet
surface. If the atmospheric temperature (of the assumed homogeneous atmosphere) was
known, eq.5 would lead to a predicted surface temperature.
Likewise, the following energy flux balance equation must hold for the planet's
atmosphere:
B + (1 - a at )a- 2(3 + C + D = 0 ... (eq.6)
where
• "B" is the intensity of incident shortwave solar radiation directly absorbed by the
atmosphere, 78 W/m 2 [1]
• a at is the longwave "albedo" of the atmosphere, the fraction (from zero to one) of
longwave radiation intensity incident on the atmosphere that is not absorbed by
the atmosphere, referred to elsewhere as the atmosphere's transmission coefficient
[10]
The global average value of a at is known from satellite measurements to be a at = 40 W/m
/ 396 W/m 2 = 0.10 [1]. In addition, eq.4 implies s at = 1 — a at = 0.90 (both are longwave
values for the atmosphere).
This system of two equations (eqs.5 and 6) is immediately solved for the two unknowns
(a and (3) such as:
a = (1 + a at )-'[2A + B - C - D] ... (eq.7)
And gives:
T a = 264 K (or -9°C) and ... (eq.8a)
T p = 254 K (or -19°C). ... (eq.8b)
It is also possible to "turn off all greenhouse effect forcing by setting a at = 1 (i.e., letting
a at approach a value of one). This gives a zero-greenhouse-warming Earth surface
temperature of
T a = 227 K (or -46°C) ... in the absence of greenhouse effect warming ... (eq.9)
The latter value for the surface temperature of a non-greenhouse Earth is maintained for
all multi-layer atmosphere models, with any number of atmosphere layers and with any
distributions of energy deliveries to the different layers (energies B, C, and D partitioned
to the different atmosphere layers). This is shown below, for all models using equally
opaque (sufficiently optically thick) layers sharing the same value of a at .
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Interpreting the homogeneous atmosphere prediction
The above predicted Earth surface temperature T a = 264 K (or -9°C) is low compared to
the accepted measured average of 14°C [11] and the predicted difference with the
homogeneous atmosphere temperature Tp = 254 K (or -19°C) is only 10°C.
This occurs because a single-layer homogeneous atmosphere does not allow the air closer
to the surface to be warmer than air of higher altitude. Energy transport from the surface
via thermals and the water cycle is considered to occur towards the full atmosphere rather
than be concentrated near the surface.
We can consider that the near-surface air is part of an effective surface and that the
thermal and water cycle energy deliveries occur near the surface (e.g., fog, wind-mixing
of thermal transport, etc.) and are cyclically confined to a near-surface region by simply
setting C = D = 0. This implies that no thermal and water cycle energy exchanges occur
with the atmosphere -proper that is taken to be distinct from a near-surface atmosphere
layer considered to be part of the effective surface.
When, in this way, only radiative heating and cooling are considered we obtain: T a = 283
K (or +10°C) and Tp = 239 K (or -34°C). These values are close to actual values for
Earth. This suggests that multi-layer atmosphere models are needed for sufficient realism.
Earth with double-layer atmosphere
Let the planet surface be denoted a, the inner atmosphere layer be denoted [3 and the outer
atmosphere layer be denoted y. For simplicity, let the direct incident solar, thermals, and
water cycle energy inputs to the atmosphere be divided between and delivered equally to
the two atmosphere layers.
Therefore, the energy flux balance condition for a is:
A - C - D - a + p + a at y = 0 ... (eq.10)
where
• Y = s at cj T y 4 is the longwave emission intensity from the y-layer of the
atmosphere to the Earth and T 7 is the uniform temperature of the y-layer
Each atmosphere layer emits equal longwave emission intensities both up (towards
space) and down (towards Earth). Each atmosphere layer is equally optically opaque,
with the same values of a at . The latter arises because of the high degree of longwave
optical opaqueness (high degree of resonant absorption over saturation, ~4 orders of
magnitude or so) of the total atmosphere.
And, the energy flux balance equations for [3 and y are, respectively:
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B/2 + C/2 + D/2 + (1 - a at )a -2p + (1 - a at )y = 0 ... (eq. 1 1)
and
B/2 + C/2 + D/2 + a at (l - a at )a + (1 - a at )p - 2y = 0 ... (eq. 12)
These equations (eqs.9, 10, 11) give:
a = (l/2)(3 + 2a at - a at 2 )(l + a at )" 2 [2A + B - C - D] ... (eq.13)
and
T a = 290 K (or +17°C), ... (eq.l4a)
T p = 280 K (or +7°C), and ... (eq.l4b)
T Y = 251 K(or-22°C). ... (eq.l4c)
These temperature values are close to actual values for Earth despite the remarkable
simplicity of the model with no free parameters. This suggests that a radiation balance
approach is correct despite ancillary complications related surface roughness, lapse rate
constraints, Earth's rotation, non-uniform irradiation, thermal response times, an
inhomogeneous atmosphere, and an inhomogeneous surface. It also suggests that the
model is sufficiently realistic to calculate response sensitivities to changing its key
parameters.
We note that the only difference between eq.13 and eq.7 is in the pre factor to the
intensity terms and that eq.13 gives the same no-greenhouse-effect (a at =1) prediction as
eq.7:
a = [A + B/2 - C/2 - D/2] ... (eq.15)
giving T a (a at = 1) = 227 K (or -46°C) (same as eq.8). Indeed, the structure of the
equations show that this is a robust result that would be the same for any number of
optically opaque layers and using any division or distribution of direct incident solar,
thermals, and water cycle energy inputs to the atmosphere layers.
Eq.15 shows that under no-greenhouse-effect conditions half of B and of C and of D are
longwave re-radiated back to Earth by the emitting atmosphere, irrespective of its
vertically inhomogeneous structure. The other halves of B, C, and D are radiated out to
space.
Temperature change scenarios and sensitivity predictions
In this section we use our double-layer atmosphere model to predict surface temperature
responses to various C02 scenarios and to other changes in key physical parameters.
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First we consider the model (eq.13) predictions for doubling the present atmospheric
C02 concentration and for the post-industrial increase in atmospheric C02 concentration,
without changing anything else. That is, in the absence of any water vapour feedback or
any other such positive or negative feedback.
For these calculations we must develop an equation that relates changes in atmospheric
C02 concentration to corresponding changes in atmosphere albedo a at , equavalent to net
longwave forward transmission through the atmosphere (denoted in a previous paper
[10]). A given atmospheric C02 concentration (in ppmv, parts per million per volume) is
denoted C co2 .
The present known value of a at (-0.10) is significantly smaller than 1 and C02 longwave
absorption predominantly occurs in a limited wavelength range (from -600 to -800
wavenumber, 1/cm) centered on -15 jam (micro-meter wavelength), such that absorption
saturation occurs in this main relevant C02 absorption band [12].
This implies that the induced change in a at is not simply (anti-)proportional to the
considered change in C02 concentration (change in C C0 2) but instead is highly attenuated.
Indeed, the decrease in a a t from an increase in C C0 2 arises not from an increased
absorption at resonance but instead from increased absorption on the outer edges of the
absorption band, thereby increasing the wavenumber- width of the absorption region
corresponding to saturation absorption conditions (e.g., [12]: Fig. 2).
Here, we derive the needed relation between a at and C C0 2 as follows.
We take the main relevant C02 longwave absorption band to be mathematically
represented by a Gaussian function having a height and width equal to the height and
width of the actual (non-saturated) absorption cross section for the C02 band centered at
the radiation frequency (v 0 ) corresponding to 15 [am wavelength.
This choice is mathematically convenient, is motivated by the fact that a single motion-
broadened resonance line in a gas atmosphere has a near-Gaussian shape, and gives a fair
though approximate representation of the actual resonant absorption cross section for
C02 in the atmosphere.
The Gaussian cross section is written:
2 2
G(v) = c m exp[-(v-v 0 ) / 2ca ] ... (eq.16)
where a m is the (maximum) absorption cross section at resonance (at v 0 ) and co is the
Gaussian width of the cross section function. Note: I am using total sample (atmosphere)
intrinsic cross section, not specific cross section on a per-molecule or per-mass of gas
basis. The Gaussian function is such that the half width at half maximum (HWHM) of the
cross section function (intrinsic absorption band) is related to a> as:
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HWHM = (2 ln(2)Y u co ... (eq.17)
Next, we find the needed frequency-width of the region of absorption saturation by
setting G(v) equal to the cross section a e defined to be the cross section at which the C02
longwave absorption becomes effectively saturated in the atmosphere. That is, we set
G(v) = c e and we solve for the two absorption band edge positions in frequency v, on
either side of the central resonance frequency v 0 .
This gives a saturation band full width as:
Av = 2(0 [2 /«(o m /c e )] 1/2 ... (eq.18)
Here, a e is a constant property of a C02-bearing Earth atmosphere and c m , by definition,
is directly proportional to the atmospheric concentration of C02. Also c m /c e ~ 10 4 for
C02 at Earth concentrations ([12]: Fig. 2, using intrinsic specific not total cross section).
The precise value of the latter ratio does not significantly impact our calculations because
it appears as the argument of the logarithmic function.
We then examine the variation (8(Av)) of Av (eq. 1 8) with a m and obtain:
8(Av) / Av = [ 2 ln(oJo e ) J" 1 5(a m )/o m ... (eq.19)
where 8(a m ) is the considered variation or change in c m . Next, we note that:
8(Av) / Av = -5(a at ) / (1 - a at ) ... (eq.20)
since the saturation band width, by definition, negatively and proportionally affects the
relevant C02 longwave absorption through the atmosphere (nothing within the saturation
width escapes through any thick layer of atmosphere in our simplified approach), and
m F co 2 5(C co2 )/C co2 ... (eq.21)
where F co2 is the present fraction (from 0 to 1) of all greenhouse effects that arise from
C02. Eq.21 follows from the linear proportionality of cross section with greenhouse
effect gas concentration for a given gas.
Therefore we need F co2 . It can most reliably be obtained from satellite spectral
measurements. This was done in [13] where F co2 ~ 0.26 (for clear sky conditions).
Given equations such as eq.7 and eq.13 for a where a is defined as a = ~~~~ a T a 4 , it
follows that changes in Earth surface temperature T a are related to changes in a (arising
from changes in a a t) as:
8(T„) = (l/4)T„8(a)/~~