Watts Up With That? December 11, 2016
As a factor in Global Warming, increases in the atmospheric concentration of CO2 have been, and will continue to be, largely irrelevant
Guest essay submitted by William Van Brunt
Copyright © William Van Brunt, 2016. All rights reserved.
The following are the basic principles and assumptions underlying the calculations set out in this paper:
1. The heating provided by CO2 is radiant heating and for purposes of this paper, when calculating the increase in heating that is a result of the buildup of CO2 in the atmosphere the only source of any increase in heating in these calculations is CO2 and the Water Vapor Feedback Effect it creates.
2. In order to maintain a given temperature, the power of the radiant heating absorbed by the Earth’s surface must at least equal the power of the thermal radiation emitted by the surface.
3.The total heating power, ΔF, required to drive a given increase in the temperature of the surface of Land can be determined as,
ΔF = [(TLo + ΔTL) / TLo)4 – 1] × RULo / Eff
Where: TLo is the initial average temperature of Land
ΔTL is the change in the average surface temperature of Land,
RULo is the initial Up Radiation at TLo
Eff is the percentage of an increase in Total Heating that heats the Earth’s Land surface.
The increase in heating power, ΔRadCO2, caused solely by an increase in the concentration of CO2 from the initial concentration, C0 to C, in ppmv, is determined by this formula –
ΔRadCO2 = 5.35 × ln (C / C0) (w/m2),
which means that there is but one result for ΔRadCO2 for a given change in concentration.
4. The increase in heating from the Water Vapor Feedback Effect provided by an increase in average temperature, ΔTCO2, resulting solely from the increase in heating from a buildup in CO2 is determined by this formula:
ΔWV = 1.6 ×ΔTCO2 (w/m2)
5. The Maximum increase in heating power received at the surface cannot exceed the sum of the results of the calculations set out in statements 4 & 5.
6. The Maximum average increase in Land temperature in degrees Fahrenheit, ΔTL resulting from of the calculations set out in statements 4 & 5 of ΔRadCO2 and ΔWV is determined by this formula as –
ΔTL = TLo × [(1 + Eff × (ΔWV + ΔF) / NHLo)1/4– 1]
NHLo is the initial Net Heating of the Land surface in watts per square meter.
7. One cannot determine the increase in Average Global Temperature based upon a change in heating because the surface temperature change response of Land and the Oceans to an increase in heating is significantly different. However, it is possible to determine the increase in Average Global Land Temperature based upon a change in heating and then estimate the change in Average Global Temperature.
8. The Maximum increase in average temperature cannot exceed the increase in temperature caused by the result of the calculations set out in statement 6.
The following are the results of the applications of these principles:
A. The change in the Average Global Temperature for Land between 1880 and 2002 was 2.6oF. To effect such an increase requires an increase of 13.8 w/m2 in total Average Heating Power. The Maximum total increase in total Average Heating Power that the buildup of CO2 over this period could have effected is 1.6 w/m2 and the Maximum increase in the Average Global Temperature for Land that the buildup of CO2 over this period could drive cannot exceed 0.3 oF.
B. CO2 is not THE cause nor is it the primary cause of Global Warming
C. The Maximum increase in Average Global Temperature that a doubling of the concentration of CO2 from 400 to 800 ppmv can effect is 0.8oF. The IPCC’s predictions of 3.4oF to 7.9oF are 325% to 900% too high and this would require an increase in heating of 800% to 900% greater than that determined in accordance with the calculation set out in statement 3 above.
D. The IPCC is simply wrong.
I have no direct, or indirect, links or ties to any business or investment that has any interest, whatsoever, in this matter. I have neither sought, been offered or received any funding, benefit or any form of consideration or promises to prepare this work – none. This has all been an independent pursuit of truth.
At the time, of the award of the 2007 Nobel Peace Prize to former Vice President Albert Gore and the Intergovernmental Panel on Climate Change (the IPCC) which was accepted on behalf of the IPCC by Dr. Pachauri, then Chair of the IPCC, I was honored to accept an invitation from a colleague to attend a gathering to celebrate the granting of this award, in Oslo.
I should also note that my academic training is not in meteorology or climate studies but this is also true, not only for Al Gore, but Dr. Pachauri as well. And, unlike both, for several years I was part of a team of scientists designing vehicles for the vacuum of space and calculating the extreme rates of heating to which they are exposed as they slammed into the atmosphere of Earth or Venus. In the case of the probes into the planet Venus this work took into account radiative heating.
In terms of absorbing and emitting radiative heat, our planet is just another object in space, with sufficient mass to maintain an atmosphere that contains a small percentage of gases that both absorb and reradiate infrared (IR) radiation, the Greenhouse Gases (GHG).
With a basic grasp of physics, radiative heating and thermodynamic principles, a determination of the Maximum increase in the Average Global Temperature (the Upper Bound) that a buildup in the concentration of CO2 can effect is possible. Otherwise, the only option is to rely on the purported “experts” which I did for a couple of decades.
I sat there the night of this celebration listening to the speakers with the belief that Global Warming had occurred and hoping that at this celebration there would be an explanation as to why there was this exclusive focus on an atmospheric increase of ~ one part per million per year or one part per ten thousand over a century, of CO2, the effect of which is merely logarithmically proportional to increases in concentration over 290 ppmv, (at this level, a 10% increase in concentration results in a 1.7% increase in heating power) and on a molecule-for-molecule is less effective as a Greenhouse Gas than the primary Greenhouse Gas, Water Vapor, which, on average, is present in the atmosphere at levels, and varies by factors, that are an order of magnitude greater than that of CO2 and, …..what this had to do with peace?
There was no presentation that demonstrated how an inconsequential change in such a minor component of the atmosphere could be responsible for Global Warming. Instead, what I heard were assumption based conclusions, summaries of the results of unexplained computer models, political speak and predictions of a parade of horribles, which may or may not be realistic, but could be the result of warming, irrespective of the cause.
That night, as the advocates for this belief played on our fears of Global Warming including a totally irrelevant and nonsensical analogy to horrible conditions on planet Venus, something I knew a little about, at the same time they appeared to be seeking to impute an unquestioning sense of guilt for all of Humankind stemming from having so benefitted from the massive consumption of fossil fuels along with a need to make amends by paying whatever it takes to stem the tide of Carbon buildup and minimize the effects of various potential doomsday scenarios, (reminded me of some preachers, “Atone for your sins or suffer hellfire and brimstone for eternity.”). For the first time, I began to wonder, based on the lack of scientific proof offered at a celebration of a Nobel Prize on the work of the role of CO2 in Global Warming, whether, and if so, why, the world was being taken in, misdirected into thinking that CO2 was THE or the primary cause of Global Warming.
Since then, my question – why the exclusive focus on such an inconsequential component of the atmosphere – went unanswered. Having read many justifications from those who make claims that Global Warming was/is caused solely by increases in the concentration of CO2. They basically boil down to:
1. Correlations of temperature increases with increases in the concentration of CO2;
2. Formulations/approximations that do not comply with the basic laws of physics, ignore the actual effects of heating and, at times, either alone or together with a theoretical, inflated and incorrect Water Vapor Feedback Effect formulation, substantially overstate the increases in temperature that the buildup in the concentration of CO2 can effect; and,
3. Determining that CO2 must be THE cause, because, if one does not include subsequent increases in the concentration of CO2 since the 1800s in the climate change computer models, these models do not show global warming, Lindzen (2007), but only do when subsequent increases in the concentration of CO2 are included (and then they overstate the increase in temperature, suggesting they are premised on the above formulations) which, of course, assumes that these models are correct; they are not; See Gray (2012);
concluding, therefore, that Global Warming has been driven by the buildup of CO2 since the advent of industrialization.
These responses are all based upon the assumption that the buildup of CO2, alone was responsible for Global Warming.
When it comes to CO2, I wondered, rather than make assumptions, why not simply calculate the Maximum incremental heating that an increase in the concentration of atmospheric Carbon Dioxide can provide and the resulting MAXIMUM temperature increase? It is not difficult.
The average temperature of the surface cannot exceed the MAXIMUM average temperature that the Net Heating can effect. Therefore, if one knows the additional net-heating that a buildup in CO2 can cause, including the Water Vapor Feedback Effect, one can calculate the theoretical MAXIMUM increase in the Average Global Land temperature that the buildup in the concentration of CO2, alone, can effect.
While I could find many papers that calculated the increase in heating, radiative forcing, that increases in the concentration of CO2 could drive and then draw conclusions about the relationship to net surface temperatures based on the assumption that these temperature changes were caused by increases in the concentration of CO2, I could find very few analyses that went beyond the calculation of incremental heating.
There were only a few that purported to explain how to calculate the increase in the Average Global Temperature resulting from increases in the concentration of CO2. Of these there were only a few that calculated the historical increase and then only at the conclusion of the time period in question. (e.g. “Between 1880 and 2002 the temperature increase caused by the prior buildup of CO2 was equal to X.”) I found no studies for the changes in the temperature of Land caused by CO2, which for the reasons set out below, enables the most straightforward comparison.
My back of the envelope calculations for the heating power required from increases in the concentration of CO2 to effect the actual temperature increases over time called all of the IPCC’s conclusions about the role played by CO2 in Global Warming, into question. Therefore, I looked into this issue in greater detail, which resulted in this paper, in which, will calculate the MAXIMUM (not the precise) increase in average temperature that the buildup of CO2 can effect.
How to Calculate the Power and Maximum Temperature Increase Caused by an Increase in the Concentration of CO2
The Earth constantly emits thermal infrared radiation (IR) which I will term “Up Radiation”, RU.
The sole source of heating of the Earth’s surface is the net radiant heating absorbed from the Sun and the “Back Radiation” from GHG, the Net Heating.
If the average surface temperature is constant for a period of time, this means that the average power per square meter of the Net Heating, NH, is at least equal to the power per square meter of the average Up Radiation. Therefore,
Net Heating, NH = RU
Comparing Land to Ocean, the temperature of the surface of Land is far more responsive to the same changes in Net Heating. See Figure 1, below.
Figure 1. Average, Ocean and Land Temperature Anomalies (NOAA 2010)
Due to the percentage that goes into subsurface heating as a result of the thermal diffusivity of the Oceans, the surface temperature of the Oceans is not as responsive to the same radiant heating as Land.
Thus, changes in Average Global Land Temperature is a far better gauge of the changes in Net Heating than changes in the Average Global Ocean Surface Temperature or Average Global Temperatures (Land & Ocean, above), which includes the Oceans comprising 70.57% of the Earth’s surface. Therefore, I will use changes in Land temperature as a gauge.
The Up Radiation per square meter of the Land surface, RUL is equal to εσTL4 (Luciuk) where, ε is emissivity, a dimensionless constant between 0 and 1 that determines the efficiency of a body to radiate and absorb energy, which in this paper, for the surface of Land is assumed to be 1; σ is the Stefan-Boltzmann constant, 5.40×10-9 w/m2 T-4 and TL is the Global Average Land temperature in degrees Rankine.
RUL = εσTL4
At a constant average surface temperature, Net Heating, NHL = RUL, and, initially, NHLo = RULo
To maintain a given temperature, the Net Heating, NHL must equal the Up Radiation
NHL = RUL = εσTL4
NHLN / NHLo = NHLN / RuLo = εσTLN4 / εσTLo4 = TLN4 / TLo4
TLN = TLo + ΔTL
NHLN = NHLo + ΔNHL
The increase in Net Heating power, ΔNHL, required to support this increase in temperature is,
ΔNHL = RULo × [(TLo + ΔTL) / TLo)4 – 1]
Where ΔTL is the change in the average surface temperature of Land, and
RULo is the initial Up Radiation at TLo
The minimum change in Total Heating power, ΔF, required to drive a given increase in the temperature of the surface of Land can be determined as, ΔNHL / Eff
ΔF = ΔNHL / Eff = [(TLo + ΔTL) / TLo)4 – 1] × RULo / Eff
So for an increase of 2.6o from an initial temperature of 507.9oR and an initial Up Radiation of 360 w/m2, for this change in temperature, the minimum change in Total Heating, ΔF, required to effect this is,
ΔF = [(507.9 + 2.6) / 507.9)4 – 1] × 360 / 0.55 = 9.5 w/m2
If there is a change in Net Heating, ΔNHL
This will result in a change in temperature, ΔT and the new temperature, TLN
TLN = TLo + ΔTL
The new Up Radiation, RULN, is equal to the initial Up Radiation, RULo plus the change in Up Radiation, ΔRuL.
RULN = RULo + ΔRuL
and, as noted above, where NHn is the New Net Heating,
RULN = NHn
NHn is equal to the initial Net Heating, NHo, plus the change in Net Heating, ΔNHL. Therefore,
RULN = NHn = NHo + ΔNHL = RULo + ΔRuL
Since, NHLo= RULo
∴ ΔNHL = ΔRuL
Further, given that
RULN = εσTLN4
Therefore, the ratio RULN / RULo
RULN / RULo = εσTLN4/ εσTLo4 = TLN4/TLo4
Since, RULN = RULo + ΔRuL
This ratio can then be written as,
(RULo + ΔRuL) / RULo = TLN4/TLo4
Given that ΔRuL = ΔNHL, then,
(RULo + ΔNHL) / RULo = TLN4/TLo4
And given that TLN = TLo + ΔTL, then,
(TLo + ΔT)4 /TLo4 = (RULo + ΔNHL) / RULo
Taking the fourth root of each side
(TLo + ΔTL)/TLo = [(RULo + ΔNHL) / RULo]1/4
Then solving for ΔTL
ΔTL = TLo × [(RULo + ΔNHL) / RULo]1/4– TLo
ΔTL = TLo × [(RULo + ΔNHL) / RULo)1/4– 1]
The next step is to determine the increase in Net Heating as a result of an increase in the concentration of CO2.
The IR frequency band within which atmospheric CO2 can absorb IR radiation is nearly saturated, meaning that, today, the pre-existing concentration of CO2 effectively absorbs almost all of the Up IR Radiation that fall within this narrow band. In addition, this band overlaps with absorption band for Water Vapor. The consequence, there is very little IR radiation remaining that falls within this band that added CO2 can absorb. Therefore, the absorption within this band is not directly proportional to increases in the concentration of CO2.
The effect of this IR band saturation can be accurately modeled on the University of Chicago’s Modtran computer model, climatemodels.uchicago.edu/modtran/modtran.doc.html, for simulating the absorption and emission of infrared radiation in the atmosphere, including the effect of variations in the concentration of CO2. This computer model was first developed for the U.S. Air Force and has been verified by satellite measurements. It is a very accurate way of determining the effects of band saturation on the ability of changes in the concentration of CO2 to change IR Back Radiation. However, this model is both change in concentration and geographically specific. In order to gauge the heating effect of changes in the concentration of CO2, each change in the concentration requires a separate computer run.
Instead, in this paper, the increase in heating from an increase in the concentration of CO2 in watts per square meter, ΔRadCO2, is calculated, in accordance with the IPCC’s formula as:
where, C is the CO2 concentration in parts per million by volume at the later date, ppmv and, C0 is the concentration at the date from which the change is being measured, in ppmv,
not because it is correct (it overstates the heating power from the increase in concentration) but because it is the only consensus model I have found and will clearly result in the calculation of the MAXIMUM temperature increase a buildup of CO2 can cause.
Knowing that the increase in heating from the buildup of CO2, alone, ΔNHLCO2 is equal to the percentage of ΔRadCO2 that goes into heating the Land, Eff, and substituting Eff × ΔRadCO2 for ΔNHLCO2, the change in temperature caused solely by an increase in heating from the buildup in the concentration of CO2, can be expressed as,
ΔTLCO2 = TLo × [(RULo + Eff × ΔRadCO2) / RULo)1/4– 1]
ΔTLCO2 = TLo × [(1 + Eff × ΔRadCO2 / RULo)1/4– 1]
Set out in Table 1, below, are my estimates of the key components of the Earth’s energy budget in 1880 and 2002 for Land.
Earth’s Average Global Land Heating Budget for 1880 and 2002, (w/m2)
|Up Radiation Land, RUL or Net Heating Land, NHL||360||367|
|Back Radiation from GHG||312||324|
|Evaporative Power, Land||13||13|
|Thermal Convection Land||99||105|
This heating budget for Land for 1880 and 2002 together with the Average Global Temperature for Land in these respective years sets a base from which one can calculate the MAXIMUM temperature changes increases in the concentration of CO2 can effect.
As both the Sun and the GHG heat the surface of the Earth they simultaneously drive evaporation, subsurface warming and convection. The power that goes into evaporation, subsurface warming and convection cannot go into heating of the surface. In this paper, Net Heating is defined as the percentage of Total Heating that does not go into the evaporation, sub surface warming and convection. The Effective heating percentage (“Eff”) is defined as the percentage of Total Heating that heats the Earth’s Land surface. Referring to Table 1, for Land, about 53% of the Total Heating of the Earth results in the Net Heating of the surface.
To be conservative, Eff is set at 55%. Therefore, to determine the Net Heating Power,
Net Heating Power = Eff × Total Heating = 0.55 × Total Heating
This increase in heating and temperature will gives rise to an increase in evaporation, which will in turn increase the GHG and give rise to an additional increase in temperature, determined as follows:
The increase in Average Global Temperature can be determined from the increase in Land Temperature. It is approximately equal to the increase in Average Global Land Temperature multiplied by the ratio of the increase in Average Global Temperature between 1880 and 2002,1.4oF to the increase in Average Global Land Temperature over this period 2.6oF = 1.4oF / 2.6 = 0.56
The Maximum measured and estimated long term Water Vapor Feedback is 1.6 w/m2 per degree Fahrenheit change in Average Global Temperature Dessler (2014).
Thus, the heating caused by the Water Vapor Feedback Effect, ΔWVCO2, as a result of an increase in Average Global Land Temperature, ΔTLCO2, in degrees Fahrenheit, can be expressed as:
ΔWVCO2 = 0.56 × 1.6 × ΔTLCO2
Taking into account the Water Vapor Feedback Effect, WVCO2, the MAXIMUM increase in net heating of the Land, ΔNHL, that can be caused by an increase in the concentration of CO2 from a given date can be determined as follows:
The Net Heating Increase, ΔNHL = Eff × (ΔRadCO2 + ΔWVCO2)
Thus, this is how the MAXIMUM Average Global Land temperature increase can be calculated for a buildup of CO2.
ΔTL = TLo × [(1 + Eff × (ΔWVCO2 + ΔRadCO2) / RULo)1/4– 1]
So, for 1880, which is a starting point commonly used,
C0 is 291 ppmv,
ToL for Land is 507.9oR
Ruo is 360 w/m2
Eff is = 0.55
ΔRadCO2 = 5.35 × ln (C / C0)
In 2002, C is 373 ppmv,
∴ ΔRadCO2 = 5.35 × ln (C / C0) = 5.35 × ln (373/ 291) = 1.33 w/m2
Then the increase in temperature from the increase in CO2, alone.
ΔTLCO2 = TLo × [(1 + Eff × ΔRadCO2 / RULo)1/4– 1]
ΔTLCO2 = 507.9 × [(1 + .55 × 1.3 / 360)1/4– 1] = 0.14oF
The Water Vapor Feedback Effect is:
ΔWVCO2= 0.56 × 1.6 × Δ TLCO = 0.56 × 1.6 ×0.36 = 0.22 w/m2
The increase in total heating from this increase in the concentration of CO2, ΔRadCO2 + ΔWVCO2 = 1.33 w/m2 + 0.22 w/m2 = 1.6 w/m2 is consistent with the IPCC estimates of total increase in heating from all man made sources between 1750 and 2007.)
Then the temperature increase on Land with Eff = 0.55, resulting from the buildup of CO2 between 1880 and 2002, including the Water Vapor Feedback Effect, is:
ΔTL = TLo × [(1 + Eff × (ΔWVCO2 + ΔRadCO2) / RULo)1/4– 1]
ΔTL = 507.9 × [(1 + 0.55 × (0.32 + 1.32) / 360)1/4– 1] = 0.3 oF
Compare this Maximum increase in the Average Global Land Temperature effected by the buildup in CO2, 0.3 oF, to the actual increase in Average Global Land Temperature of 2.6 oF.
Using the ratio of Average Temperature to Land Temperature, 0.56, the increase in Average Global Temperature effected by the buildup in CO2 over this period is 0.2oF compared to the actual increase in Average Global Temperature over this period of 1.4oF.
Clearly the buildup of CO2 over this period, 1880 – 2002, is not the cause of this temperature increase.
The IPCC Formulations for Determining the Temperature Increase from the Buildup of CO2 are Incorrect and Substantially Overstate the Resulting Temperature Increase
The IPCC uses different formulae for calculating the increase in average global temperature from a buildup of CO2, which appear to be based upon the formulation of Arrhenius (1896) who set out his formula for a change in Average Global Temperature in degrees Celsius, as
ΔTArr = S × log2 (C/Co)
S, is the doubling sensitivity and it is normally given in degrees Celsius.
In Arrhenius’ paper, S can be determined as equaling 5.8o C. However, in his subsequent book, he suggests a smaller climate sensitivity, S = 4. Arrhenius & Borns (1906)
The IPCC’s most recent report (2013) states: “equilibrium climate sensitivity (the doubling sensitivity) is likely in the range 1.5 K [S] to 4.5 K [S] (high confidence).” IPCC (2013)
Since the IPCC is focused on the effects of doubling the concentration of CO2 from 400 ppmv to 800 ppmv, I will focus on this as well.
Such a doubling would result in an increase of 3.7 w/m2 in total heating power from the buildup of CO2, after applying the applying the IPCC formula for increases in heating of, ΔRadCO2 = 5.35 × ln (C / C0), increasing this for the Water Vapor Feedback Effect and with Eff = 0.55, this would give rise to an increase in Average Global Land Temperature, using the above formulas, of 0.8oF.
Referring to Figure 1, above, a 0.8 degree increase in Average Global Land Temperature corresponds to ~ a 0.4 degree, increase in Average Global Temperature.
Set out below in Table 2, below, is a comparison of the temperature results based on using the Arrhenius formulation for such a doubling, for values of S ranging from 1.5 to 4.5 and comparing the required increase in heating to effect such a change to the 4.3 w/m2 determined as set out above.
Temperature and power required using Arrhenius Formulation for various values of S Proposed by the IPCC
|% Increase over actual temperature increase of 0.4oF||% Increase in Power required to effect this temperature increase compared to actual power increase of 4.3 w/m2|
The IPCC formulation for determining an increase in heating, ΔRadCO2, is dependent solely on the change in concentration, ΔRadCO2 is proportional to ln (C / C0). There is no “S” variable in this formulation. Therefore, the increase in heating is 3.7 w/m2, regardless of the value of S.
An increase in heating of 4.3 w/m2 can cause a 0.8oF increase in Average Global Temperature – no more; much less a range of temperature increases as high as 8.1oF.
To publish a range of the Maximum increases in temperature for the same increase in concentration and, therefore, the same heating is nothing short of scientifically absurd. If the Maximum temperature increase that the rate of heating can cause, is 0.8oF, that is it. This is best illustrated by column 4 which sets out the percentage increase in heating power required to cause the corresponding increase in temperature.
While some propose far greater increase in power from the Water Feedback Effect based on some theoretical concepts, the fact is the Water Feedback Effect has been measured. Any theoretical calculation or computer model that predicts a greater heating from this effect is wrong.
Moreover, the basic and fundamental law that energy is always conserved, stands as a complete and total bar to any increase in temperature greater than 0.8oF.
Further, that the Arrhenius formulation, ΔTArr = S × log2 (C/Co) is simply wrong can be shown as follows:
Converting this expression to natural log function, then
ΔTArr = S × 1.44 × ln (C/Co)
As noted above, according to the IPCC, the increase in radiative power per square meter, ΔRadCO2, from an increase in the concentration of CO2, can be determined as:
ΔRadCO2 = 5.35 × ln (C/Co)
ln (C/Co) = ΔRadCO2 / 5.35
Substituting ΔRadCO2 / 5.35 for ln (C/Co) in the Arrhenius formulation for calculation for change of temperature results in,
ΔTArr = 1.8 × S × 1.44 × ln (C/Co) = S × 1.44 × ΔRadCO2 / 5.35
which means that ΔTArr is directly proportional to changes heating, ΔRadCO2.
As noted above, based upon the basic principles of radiative heating,
ΔTCO2 = To × [(1 + ΔRadCO2 / RUo)1/4– 1]
which means that instead of being directly proportional to changes heating, ΔRadCO2, as Arrhenius assumes, ΔTCO2 is proportional to the fourth root of changes in heating, ΔRadCO21/4. Arrhenius’ conjecture is clearly not founded on the principles of physics.
The Arrhenius formulation and IPCC approach cannot possibly be correct.
Another writer, Ellis (2013) derives the equation for the increase in temperature, ΔTEll, in degrees Fahrenheit, resulting from an increase in heating, ΔRadCO2, which can be expressed as:
ΔTEll = 1.8 × 0.31× ΔRadCO2 = 0.56 × ΔRadCO2
Comparing this to Arrhenius, effectively in Ellis’ formulation, S is ~ 2.
These and similar calculations, Jacob (1999: § 7.4.3), in which the change in temperature is also directly proportional to changes in ΔRadCO2, instead of being proportional to the fourth root of the change in ΔRadCO2 as (1+ΔRadCO2 / Ru).25, do not comply with the radiative heating laws of thermodynamics and are simply wrong.
Given how straightforward the correct formulation is, one wonders why this is not employed by the IPCC and why “The IPCC’s range of uncertainty in the value of k[S] extends from 1.5 C to 4.5 C, with a central value of 3.0 C.”
The IPCC’s determinations overstate, significantly, the role of CO2 in Global Warming and are wrong.
The change in the Average Global Temperature for Land between 1880 and 2002 was 2.6oF. To effect such an increase requires an increase of 13.8 w/m2 in Total Average Heating Power. The Maximum total increase in total Average Heating Power that the buildup of CO2 over this period could have effected is 1.6 w/m2. The Maximum increase in the Average Global Temperature for Land that the buildup of CO2 over this period could drive cannot exceed 0.3 oF. Comparing 1) the Maximum increase in heating power of 1.6 w/m2 to the required increase in power to effect a temperature change of Land of 2.6oF, 13.8 w/m2,2) the Maximum increase in temperature that can be effected by this increased heating of 1.6 w/m2, 0.3oF in the Average Global Temperature of Land, resulting from the actual increase in the concentration of CO2 between 1880 and 2002, to the actual temperature change of Land of 2.6oF and 3) comparing the correct prediction for a doubling of the concentration of CO2 of a Maximum increase of 0.8oF increase in Average Global Temperature compared to the IPCC’s range of 2.7 to 8.1oF, demonstrates, conclusively, that the IPCC is wrong. As a factor in Global Warming, increases in the atmospheric concentration of CO2 have been, and will continue to be, largely irrelevant.
This is not merely a scientific debate.
Governments across the globe are in the process of implementing and planning to implement, laws regulations, changes in taxing and offering direct and indirect subsidies and credits that in the future could result in costs that, in the aggregate, could equal the Annual Gross Domestic Product of the economies of all the countries in the World, based upon the determinations of and pronouncements from the IPCC. While potentially devastating to the economies and peoples of all nations, these efforts may not result in any meaningful reduction in the buildup of CO2, but even if they succeed in achieving this goal, this almost certainly will not result in a reduction of the Average Global Temperatures, because as a factor in Global Warming, the buildup of CO2 is largely irrelevant.
There will be no return on these economically damaging and tremendously costly investments.
Let me conclude with a few questions:
With all of the data possessed by the IPCC and all of the experts it has mustered, why is it that I have not seen any publications in which the IPCC, and its affiliates have:
1. Shown or discussed the increase in total heating power required to have caused the 2.6oF global average increase in land temperature since 1880?
2. Applied the computer models it uses for predictions to the period 1880 to today and compared the results to the actual average annual global temperature trends from 1880 to today?
3. Used the straightforward formulation, based on classical physics, to calculate the Maximum temperature increase a buildup in Carbon Dioxide can cause or explained why they view this as inapplicable?
Surely, the IPCC has considered these questions. If not, it should.
Looking back, it is now clear. The 2007 the Nobel Peace Prize was awarde because the work of the recipients would not qualify for an award of the Nobel Prize for Physics or Economics.
Arrhenius, S (1896) “On the influence of carbonic acid in the air upon the temperature of the ground” Philosophical Magazine Series 5 Vol. 41
Arrhenius, S. & Borns, H. (1908) “Worlds in the Making; the Evolution of the Universe” New York, Harper” pp. 53 & 56
Cox, J.D. “Understanding the Weather’s Water Cycle” Weather For Dummies (www.dummies.com/how-to/content/understanding-the-weathers-water-cycle.html).
Dessler, A., (2014) “Measuring the effect of Water Vapor on climate warming.” (phys.org/news/2014-03-effect-vapor-climate.html).
Ellis, R. (2013b) (www.globalwarmingequation.info/global%20warming%20eqn.pdf).
Gray, W.M. (2012) “The Physical Flaws of the Global Warming Theory and Deep Ocean Circulation Changes as the Primary Climate Driver” (https://tropical.atmos.colostate.edu)
IPCC (2013) Intergovernmental Panel on Climate Change, Fifth Assessment Report (AR5) WG1, https://www.climatechange2013.org/images/report/WG1AR5_SPM_FINAL.pdf
Jacob, D.J. (1999) “§ 7.4.3 Radiative forcing and surface temperature.”, Introduction to Atmospheric Chemistry”, Princeton University Press, (acmg.seas.harvard.edu/people/faculty/djj/book/bookchap7.html)
Lindzen, R.S. (2007) “Taking Greenhouse Warming Seriously” Energy & Environment, Vol. 18 No. 7+8
Luciuk, M. “Temperature and Radiation” (https://www.asterism.org/tutorials/tut40RadiationTutorial.pdf)
NOAA (2010) “Global Land and Ocean Temperature Anomalies January – December.”, NOAA’s National Climatic Data Center
Trenberth, K.E. (2011) “Tracking Earth’s energy: A key to climate variability and change.” (www.skepticalscience.com/print.php?n=865).
 Imagine a football stadium filled with 10,000 people (representing the atmosphere), with 100 to 400 people close to the field hollering all of the time (representing the initial level of GHG). The noise increase at field level from the addition of one more hollering person assigned to the highest seating level is similar in effect to the heating increase of one part per ten thousand of CO2.
 “…general circulation models (GCMs) can be used to estimate the surface warming associated with an increase in Greenhouse Gas concentrations. The GCMs are 3-dimensional meteorological models that attempt to capture the ensemble of radiative, dynamical, and hydrological factors controlling the Earth’s climate through the solution of fundamental equations describing the physics of the system. In these models, a radiative perturbation associated with increase in a Greenhouse Gas (radiative forcing) triggers an initial warming; complex responses follow…… There is still considerable doubt regarding the ability of GCMs to simulate perturbations to climate, and indeed different GCMs show large disagreements in the predicted surface warmings resulting from a given increase in Greenhouse Gases. …. Despite these problems, all GCMs tend to show a linear relationship between the initial radiative forcing and the ultimate perturbation to the surface temperature, the difference between models lying in the slope of that relationship.” (Jacob §7.4) (Emphasis added)
As noted below, the relationship between temperature and radiative heating is that temperature increases as heating to the ¼ power. (∆T µ ∆F1/4). It is not linear, which would greatly overstate the increase in temperature by hundreds of a percent. A “linear relationship between the initial radiative forcing and the ultimate perturbation to the surface temperature” is contrary to correct “fundamental equations describing the physics of the system”. Basic thermodynamics also teaches that the rate of heat transfer to the Earth’s surface cannot exceed the sum of the net radiative heating from current solar and back radiation.
 The MODTRAN algorithm solves the Line By Line radiative transfer equations at very fine spectral resolution.
 This equation is based on a determination for the optical (IR) opacity of CO2 and the assumption that the most significant and variable GHG, Water Vapor, was constant. This is not a valid assumption. More importantly this calculation ignores the very real and complex effects of CO2 band saturation, which can only be determined accurately using a very sophisticated computer model. Based on the simulations I have performed; the IPCC model produces results that are consistently higher than the output of the Modtran computer calculations.
 Knowing that Land covers 29.4% of the Earth’s surface, the Oceans account for 84% of total evaporation (Cox), in 1880 the average Land temperature was 2.6oF lower, using the energy budget data from Trenberth (2011), measured changes in solar heating and the Water Vapor Feedback Effect for changes in temperature, with –
1. Up Radiation adjusted for relative changes in Average Global Land Temperature to the fourth power,
2. Back Radiation adjusted to take these changes in Up Radiation into account after accounting for the Water Vapor Feedback Effects, and
3. Thermal Convection calculated as Total Heating less Up Radiation and Evaporative Power Land for the respective year.
one can estimate the Earth’s average energy budget.
 “From 2002 to 2009, an infrared sounder aboard NASA’s Aqua satellite measured the atmospheric concentration of Water Vapor. Combined with a radiative transfer model, Gordon et al. used these observations to determine the strength of the Water Vapor Feedback. According to their calculations, atmospheric Water Vapor amplifies warming by 2.2 plus or minus 0.4 watts per square meter per degree Celsius. This value, however, is only the “short-term” feedback—the strength of the feedback as measured during the observational period. This value is subject to short-term climate variability. The true value of the feedback, the “long-term” value, is what the short-term observed values should trend towards when given enough time.”
Using a series of climate models, the authors estimate the strength of the long-term Water Vapor Feedback. Extrapolating from their short-term observations they calculate a long-term feedback strength of 1.9 to 2.8 watts per square meter per degree Celsius.” 2.8 watts per square meter is the Water Vapor Feedback measure employed in this paper for temperature measured in degrees Celsius which is converted to 1.6 for temperature measured in degrees Fahrenheit.
While this measurement relates this amplification in heating to linear changes in Average Global Temperature, not to changes in temperature to the fourth power, this is likely so because this is a measure of changes in concentration which, over time, are driven by evaporation which changes linearly with temperature, the effects of which are orders of magnitude greater than changes in heat flux from changes in temperature.
 Of course, temperatures vary across the globe. If one performs this calculation for a range of initial temperature changes, ± 30oF, for example, and adjusts the Up Radiation accordingly, the average temperature change of this range is within one percent of the average temperature change calculated using this formula. Therefore, the Average Global Land temperature increase is calculated as set forth above.
 Referring to Figure 1, it is evident that Global Warming did not commence until the late 1970’s and ceased prior to 2002.
If one does the same calculations for the 38 ppmv increase in CO2 over this period; the Maximum increase in the Average Global Land Temperature effected by this buildup in CO2 is 0.2 oF. The actual increase in Average Global Land Temperature over this period is 1.8 oF.
The increase in Average Global Temperature effected by the buildup in CO2 over this period is 0.1oF compared to the actual increase in Average Global Temperature over this period of 1oF, or 10% of the actual increase.
 Global Warming nonetheless occurred between the late 1970’s and 2002. I show in another paper what the likely causes of this were.
 There are those who believe a range is appropriate due to the inability to precisely predict the impact of delays in reaching an equilibrium temperature and the difficulties associated with modeling the thermal diffusivity and responses of the Oceans and the manner in which the atmosphere responds to increases in heating.
Given an increase of 1 -2 ppmv per year in the concentration of CO2, reaching an equilibrium temperature on Land should occur far faster than the rate of change. But, whether or not this is correct, this paper assumes that the equilibrium temperature, which is the Maximum temperature, is reached and while all of these oceanic and atmospheric factors make it difficult to predict the precise effects of increases in GHG heating, these ranges must all be less than the Maximum increase in the Average Global Temperature that the heating can effect. They cannot exceed the Maximum number.
Copyright © William Van Brunt, 2016. All rights reserved.
William Van Brunt is a practicing lawyer and President and CEO of JFA, LLC. Before attending law school, he was a senior scientist and part of a highly successful design team engaged in state of the art research and development for, and writing the complex software necessary to determine the aerodynamics and heating of hypersonic vehicles for the U.S. Air Force and Navy and probes into the planet Venus, for NASA. Relevant to this topic are the degrees he holds from the Pennsylvania State University, B.S. (Aeronautical Engineering) and the Massachusetts Institute of Technology, M.S. (Aeronautics and Astronautics), where he was elected to the Society of Sigma Xi. Fascinated by the claims made about the role of Carbon Dioxide in Global Warming and causes therefor, his is a novel, in depth and totally independent assessment of this topic.