Let us examine the net effect of infra-red active (so-called
greenhouse) gases on the Earth’s surface temperature under present conditions
and then the effect of a perturbation of that condition. First, the net effect of the greenhouse
gases presently on the surface temperature is usually found as the presently
measured surface temperature minus the temperature predicted by a simple black
body radiation calculation. The average
power flux of energy from solar insolation at the top of the atmosphere on the
Earth system is usually given as
S (1-A)/ 4,
where S is the total solar insolation or radiation at the top of the atmosphere, A is the
albedo or the fraction of the solar radiation reflected without absorption by
the Earth system, and the factor of 4 is the average reduction of solar flux
due to the projection of a rotating sphere onto a disk in the daily cycle.
However, the Earth has a tilt angle of its daily rotational axis with respect to the axis of its annual rotation in orbit about the sun of ψ
= 23.44° or 0.4094 radians. The paper mistakenly calls the this the precession angle. The tilt angle does precess over tens of thousands of years, but the angle of its precession is not important for this present calculation. [Thanks to Tom Anderson for pointing out the proper identity of this angle. See the comments below.] According to Sorokhtin, Chilingar, Khilyuk
and Gorfunkel in Evolution of the Earth’s Global Climate, Energy Sources, Part
A, (2007), 1-19 and Sorokhtin, Chilingar and Khilyuk, Global Warming and Global
Cooling: Evolution of Climate on Earth, Elsevier, Amsterdam (2007),
p.313, the correction factor for the rotational tilt effect, ø, according to Chilingar, replaces the factor
4 in the divisor above with 4ø, where ø is
[π/2 – ψ]/π/2 +
(ψ/π/2) [1/(1+cos ψ)] = 0.8754 for the
Earth
So 4ø = 3.5016 for the Earth.
However, to calculate surface temperature of the Earth without
any infra-red active gases such as water vapor, carbon dioxide, or methane, one
has to delete the losses of reflected solar insolation due to reflections from
clouds. If there is no water vapor,
there are no clouds. Let us examine the
2013 NASA Earth Energy Budget of Fig. 1 or a means to estimate the fraction of
the solar insolation incident upon the surface, the only location where
absorption occurs, which is reflected.
The albedo A of an Earth without infra-red active gases is 0.127 from
this NASA Earth Energy Budget, rather than the 0.3 value for our present Earth
with infra-red active gases. The Earth’s
surface temperature without infra-red active gases, TS, is then
TS =
[S(1-A)/(3.5016)σ]0.25 = [1367 W/m2 (1 - 0.127)/
(3.5016)(5.6697 x 10-8 W/m2 K4)]0.25
TS = 278.4
K
So if the present average temperature of the Earth is taken
to be 288.2 K, the net warming effect of all of the present infra-red active
gases is 9.8 K. This is a far cry from
the 33 K warming effect which is often claimed as the result of the so-called
greenhouse gas effect. But it is true
that without the so-called greenhouse gases, the Earth’s surface would be
cooler than it is now because the surface itself would be in radiative
equilibrium with space instead of a combination of the surface, a more heavily
weighted altitude at the top of the troposphere, and a much lighter weighting
of the stratosphere. The
movement of the altitude of the final emission to space of infra-red radiation
upward gives slower energy transport mechanisms in the troposphere the primary
task of cooling the surface and warming the lower troposphere.
Fig. 1. The NASA
Earth Energy Budget of 2013 is shown.
There is a great deal of nonsense in this energy budget, but the one
thing we are taking from it is the fraction of solar radiation incident upon
the surface which is reflected, which is 7% / (48% + 7%) = 0.127.
There are many effects that are caused by the infra-red
active gases. The first molecules of
these gases added to the atmosphere were able to absorb energy that would
otherwise have been radiated directly from the surface straight out into
space. That absorbed energy was then
most often transferred to non-active infra-red molecules of nitrogen, oxygen,
and argon gas which then mostly transported the energy upward by convection
processes until the energy was deposited in the atmosphere where the molecular
collision rate was lower and the mean free path for infra-red energy absorption
was longer. This absorption effect is
large at first, but becomes rapidly smaller as the number of infra-red
molecules becomes larger. Other effects
do not shrink as rapidly or at all as the number of infra-active molecules
increases, however. For instance, water
vapor and CO2 also absorb incoming solar insolation in the
atmosphere and that absorption is less saturated at the present concentrations
of water vapor and CO2 in the atmosphere. This is a surface cooling effect in that the radiation never arrives at the surface to warm it. The differential effects of water vapor and
CO2 compared to N2 and O2 on the heat
transported by convection scale linearly with the increase in water vapor and
CO2, so they do not diminish as their concentrations are
increased. Water vapor condensation in
the atmosphere also increases linearly with the amount of water vapor.
So, it is not a foregone conclusion that adding CO2
to the present mix of gases in the Earth atmosphere will cause further warming,
just because the additions of the first molecules did cause warming. We do not immediately know whether the
so-called greenhouse effect is increasing or decreasing with further additions
of greenhouse gases. This is a question
I have been discussing for years on this blog and since I wrote a book chapter
called Do IR-Absorbing Gases Warm or Cool the Earth’s Surface?,
in Slaying the Sky Dragon -- Death of the Greenhouse Gas Theory, Stairway Press, published in January 2011. Of course,
the presence of water on the Earth’s surface and water vapor in the atmosphere
causes the Earth’s surface to be warmer than it would be without water, but
unlike the common assumption, this does not tell us that further additions of
the so-called greenhouse gases will cause further warming. I have many times explained why the physics commonly
and vaguely offered as the reason why such gases would continue to warm the
Earth’s surface is wrong.
Tγ p1-γ
= constant, where γ = cP/cV, or
T = (constant) pα,
with α = (γ – 1)/γ
They note that for atmospheres with a pressure greater than
0.2 atm,
Th = bα
[S(1-A)/(4 ø σ)]0.25 (ph / p0)α,
Where Th is the temperature in K at altitude h, ph
is the pressure at altitude h, σ is the Stefan-Boltzmann constant, and b is a
constant. For Earth, S = 1367 W/m2,
the albedo A = 0.3, and 4 ø = 3.5016.
Taking the surface temperature TS = 288.2 K, one can
calculate the value of bα to be 1.094. For the Earth’s present atmosphere, α =
0.1905.
The adiabatic exponent α is known to be
α = R / µ (cP
+ cW + cR),
where R is the gas constant or 1.987 cal/K mole, µ is the
air molecular weight, cW is the heat capacity per gram due to water
vapor, cR is the additional specific heat capacity per gram due to
infra-red radiation, and µ cP is the partial pressure weighted
average of the cP per gram of each gas molecule given as
µ cP = [µN2
pN2 cP (N2) + µO2 pO2 cP
(O2) + µCO2 pCO2 cP (CO2)
+ µAr pAr cP (Ar)]/p,
which is not the way this is expressed in the paper. Note that µN2 cP (N2)
is the heat capacity per mole of nitrogen gas and each atmospheric gas
component should be handled similarly. cW
+ cR is the effective heat capacity of the sum of the water
condensation processes and the absorption by infra-red active gases of the
incoming solar insolation in the atmosphere.
A decrease in the value of α will cause a temperature decrease at any
given altitude in the troposphere and a temperature decrease at the surface.
The value of µ should also be adjusted for additions with a
weighted average based on component gas partial pressures as I showed above,
though the paper does not present the issue in this way. Additions of carbon dioxide with a mass of 44
amu increase the overall air µ since N2 has a mass of 28 amu and O2
has a mass of 32 amu, with normal air being about 28.96 amu on average. So additions of carbon dioxide will decrease
α by increasing the average molecular mass.
On the other hand, additions of water vapor (18 amu) or methane (16 amu),
both reduce the average air molecular weight, which acts to increase α. To find the overall effect of a gas component
in convection, however, one needs to examine the heat capacity of each gas in
terms of its µ cP or its constant pressure heat capacity per mole.
Unfortunately, the paper incorrectly equates specific heat
with heat capacity in the discussion.
Specific heats are given in relation to that of water. While they misuse the term, the results are
handled correctly.
Because the infra-red active gases have internal modes of
vibration which are excited and hence carry energy in addition to the
translational kinetic energy of these molecules, they have larger heat
capacities per mole than do the non-infra-red active gases such as N2
and O2. For instance, at
atmospheric pressure N2 has a heat capacity at constant pressure of
6.96 cal/K mol, while H2O vapor has a heat capacity of 8.02 cal/K
mol, CO2 has a heat capacity of 8.87 cal/K mol, and methane, CH4,
has a heat capacity of 8.44 cal/K mol.
The constant pressure heat capacities per mole of water vapor, carbon
dioxide, and methane are all greater than those of nitrogen gas, so they reduce
the value of α by increasing the convective heat capacity in the denominator of
α. A reduced α means a reduced
temperature. The paper confuses this
issue in the discussion because it gives the heat capacities for each molecule
as the heat capacity per gram, which is lower for CO2 than it is for
N2 and O2 due to its substantially greater molecular
weight. They state the right conclusion,
but the reasoning is hard to follow.
More water vapor increases both cW and cR,
while an increase in carbon dioxide or methane increases cR. So α and the temperature are still further
reduced by the increased net heat capacity.
The effective temperature of radiative equilibrium with
space, Te, is not precisely defined in the paper, but is this:
Te =
[S(1-A)/(3.5016)σ]0.25 = [1367 W/m2 (1 - 0.3)/
(3.5016)(5.6697 x 10-8 W/m2 K4)]0.25
Te = 263.5
K
In addition, the heat in the atmosphere per gram, Q is given
as
Q = cR Te
But we also have
Q = (cP +
cW) (TS – Te)
Consequently,
CR = (cP
+ cW) (TS – Te)/ Te
Note that Equation 5 in the paper is in error, though 5’,
which is derived from equation 5, is correct.
Using the fact that α = R / µ (cP + cW + cR),
we find that
cR =
(R/µα) (TS – Te)/ TS
Also,
CW =
(R/µα) (Te/TS) - cP
Calculating these values for Earth with α = 0.1905, µ = 29, the
dry air heat capacity cP = 0.2394 cal/g K, TS = 288 K, Te
= 263.5 K, one finds that
cR = 0.306
cal/g K
cW =
0.0897 cal/g K
The heat energy transport by convection, water condensation,
and radiation of infra-red active gases is proportional to the cP, cW,
and cR values. Convection is
responsible for 66.56% of the heat transfer, water condensation for 24.94%, and
radiation by infra-red active gases accounts for 8.51% of the energy transport
in the troposphere.
The paper uses this methodology to show an excellent match
with the surface temperatures and the lower atmosphere temperature gradients
for both Earth and Venus. It points out
that an all methane Earth atmosphere would have almost exactly the same surface
temperature, while an all CO2 Earth atmosphere would have a surface
temperature of about 281K, instead of 288K.
These are under the assumption that the total weight of the atmosphere is
preserved in the comparisons.
So, as I have often said, the net warming of the Earth’s
surface by infra-red gases is much less than it is claimed to be. It is about 9.8 K, not about 33 K. Also, as I have said by other empirical approaches,
the effect of adding water vapor to the atmosphere is now a cooling effect,
though water vapor is responsible for most of the prior warming due to its role
in preventing a direct radiative equilibrium between the surface and space for
most of the heat at the surface. I have
also said that adding CO2 has a very small effect on the surface
temperature, which is borne out by this paper where CO2 is only
responsible for a small portion of the small cR effect and a very small
increase of cP. I have long
said that it was not clear that adding CO2 would not decrease the
temperature a wee bit. It now appears
clear that just as adding water vapor now decreases the surface temperature, so
too does adding either CO2 or methane gas. This paper I have just discussed shows why additions
of the infra-red active (greenhouse) gases now have a net cooling effect upon
our troposphere and upon surface temperatures.
There is a warming of the surface by infra-red active gases,
the so-called greenhouse gases, but that effect was maximized at lower
concentrations of those gases than we now have.
Increases in those gases now cause small decreases in surface and
general tropospheric temperatures.
This
is because the mean free length for infra-red absorption by these gases is now
too short for them to move the upper troposphere radiative equilibrium altitude
to higher altitudes in the dense troposphere.
With that space radiation shell at the top of the troposphere relatively
stabilized, the increased role of the gases in transporting heat energy upward
from the surface means they are stronger coolants than they are “greenhouse”
heaters.
In addition, the less saturated effect of the infra-red active gases in absorbing solar insolation prior to its reaching the Earth's surface is in effect a cooling of the surface. This solar insolation absorbing cooling effect has gained significance with respect to the surface temperature warming effect that was due to the absorption of thermal energy emitted from the Earth's surface and which broke the surface radiative equilibrium with space. The disruption of the radiative equilibrium of the Earth's surface with space is the only important means by which infra-red active molecules warm the surface. This has nothing to do with back-radiation from the atmosphere as I have discussed
here,
here, and
here. A small amount of radiation from the atmosphere is absorbed by the Earth's surface, but only in those local conditions when the absorbing molecule in the Earth's surface has a lower temperature than does the emitting infra-red active molecule in the atmosphere. An example is when a warm wind blows over a cool surface or when photons emitted from a molecule in the air in a sunny area are absorbed by a surface area which is shaded from the sun.
This article was updated on 6 August 2015.