I presented a summary of this preprint at the “Basic Science of a Changing Climate” conference in Porto, Portugal, on September 7, 2018. – Ed

Edwin X Berry, Ph.D., Physics

Climate Physics LLC, 439 Grand Dr #147, Bigfork, Montana 59911, USA

Copyright © 2018 by Edwin X Berry. This PREPRINT will be submitted to a journal for publication. Therefore the present copyright does not permit republication because journals allow only one PREPRINT for submitted papers.

Abstract

The United Nations Intergovernmental Panel on Climate Change (IPCC) climate theory assumes nature is constant. This assumption forces IPCC’s invalid claim that human emissions have caused all the increase in atmospheric CO2 above 280 ppm. IPCC’s argument to support its failed theory also fails logic because the argument itself assumes nature is constant.

IPCC’s theory cannot simulate the carbon-14 data from 1965 to 1995. The carbon-14 data prove human CO2 does not “reduce the buffer capacity of the carbonate system” as IPCC claims.

A Simple Model, based only on the continuity equation with CO2 outflow proportional to level, exactly replicates the carbon-14 data. The Model shows CO2 emissions do not accumulate in the atmosphere as IPCC theory claims but set balance levels for CO2.

Present human emissions increase the level by 18 ppm and present natural emissions increase the level by 392 ppm to produce today’s total level of 410 ppm.

The Simple Model requires us to think in a new paradigm about how CO2 flows into and out of our atmosphere. It changes entirely the dominant worldview of how human emissions change the level of CO2 in our atmosphere.

1. Introduction

The critical scientific questions about human-caused climate change are about cause-and-effect:

How much do human emissions increase atmospheric CO2? How much does increased atmospheric CO2 change climate?

This paper focuses on the first question.

The United Nations Intergovernmental Panel on Climate Change (IPCC, 2007) Executive Summary claims human emissions have caused ALL the increase in CO2 since 1750. They say the level was 280 ppm in 1750 and human emissions added 130 ppm to increase the level to today’s 410 ppm.

The IPCC claims “abundant published literature” shows, with “considerable certainty,” that nature has been a “net carbon sink” since 1750, so nature could not have caused the observed rise in atmospheric carbon dioxide.

IPCC (2007) claims human CO2 emissions have clogged the carbon cycle and thereby extended CO2 residence time in the atmosphere,

“The fraction of anthropogenic CO2 that is taken up by the ocean declines with increasing CO2 concentration, due to reduced buffer capacity of the carbonate system.”

The U.S. Global Change Research Program Climate Science Special Report (USGCRP, 2018) agrees with the IPCC and claims,

“This assessment concludes, based on extensive evidence, that it is extremely likely that human activities, especially emissions of greenhouse gases, are the dominant cause of the observed warming since the mid-20th century.”

The IPCC and the USGCRP claim there are “no convincing alternative explanations” other than their theory to explain “observational evidence.”

This paper proves all these IPCC and USGCRP claims are invalid.

“Abundant published literature” is irrelevant because votes don’t count in science. There is no “extensive evidence” except repetition of invalid claims, and evidence does not prove a theory is correct.

According to the scientific method, it is impossible to prove an idea is true, but if an idea makes only one false prediction then the idea is wrong. (Kemeny, 1959; Farnam Street, 2018a, 2018b; Feynman et al, 2011; ScienceNET, 2016; Science Today, 2017).

Many authors agree that human emissions have little effect on the level of atmospheric CO2, even though they used different methods to derive their conclusions.

Revelle and Suess (1957), Starr (1992), Segalstad (1992, 1996, 1998), Rorsch et al. (2005), Courtney (2008), Siddons and D’Aleo (2007), Quirk (2009), Spencer (2009), MacRae (2010, 2015), Essenhigh (2009), Glassman (2010), Wilde (2012), Caryl (2013), Humlum et al. (2013), Salby (2012, 2014, 2016), and Harde (2017a) concluded that human emissions cause only a minor change in the level of atmospheric CO2.

Segalstad (1998), Ball (2008, 2013, 2018), and Salby (2014) present evidence that the level in 1750 was higher than 280 ppm. Nevertheless, this paper uses IPCC data. The goal of this paper is to show how the IPCC climate theory fails even when using IPCC data.

Authors who argue for the IPCC view include Cawley (2011), Kern and Leuenberger (2013), Masters and Benestad (2013), Richardson (2013). Most notable is the Kohler et al. (2017) desperate attack on Harde (2017a) which concludes,

“Harde … uses a too simplistic approach, that is based on invalid assumptions, and which leads to flawed results for anthropogenic carbon in the atmosphere. We suggest that the paper be withdrawn by the author, editor or publisher due to fundamental errors in the understanding of the carbon cycle.”

There is no tolerance in Kohler’s world for a contradictory opinion. Like other promoters of Lysenkoism, Kohler wants Harde (2017a) withdrawn. In possible response, the Elsevier journal – Global and Planetary Change – refused to publish Harde’s (2017b) rebuttal to Kohler. Harde (2017c) replies to reviewer reports regarding the rejection of his rebuttal.

Kohler claims Harde (2017a) is wrong because Harde uses one reservoir (the atmosphere) and one equation. Kohler does not understand systems.

Science is replete with examples that use a simple system with a single equation for great benefit. The Carnot engine uses one equation to describe the maximum amount of work obtainable from the inflow and outflow of heat. The adiabatic process uses one equation to explain the physics inside one reservoir.

This paper supports Harde (2017a) and its key conclusions:

“Under present conditions, the natural emissions contribute 373 ppm and anthropogenic emissions 17 ppm to the total concentration of 390 ppm (2012).”

To keep the discussion simple, this paper converts all GtC (Gigatons of Carbon) units into the equivalent CO2 units of ppm (parts per million by volume in dry air), using:

1 ppm = 2.13 GtC

Munshi (2017) shows the “detrended correlation analysis of annual emissions and annual changes in atmospheric CO2” is zero. Therefore, IPCC’s claim of “considerable certainty” fails. Where there is no correlation, there is no cause and effect.

2. Two models take the carbon-14 test

2.1 The IPCC model

The IPCC theory assumes natural CO2 balances, but human CO2 does not. The IPCC inserted its theory into its climate models. The IPCC “Bern model” (Bern, 2002) is a seven-parameter curve fit to the output of IPCC’s climate models (Joos et al., 2013). The Bern model shows the effect of IPCC’s theory on climate models. It is described in Section 3.

The IPCC claims nature treats human-produced CO2 differently than it treats nature-produced CO2. However, that is impossible because the CO2 molecules from the two sources are identical.

Then the IPCC claims the Bern model applies only to human CO2 emissions. However, the Equivalence Principle requires the Bern model and all CO2 models to apply equally to human and natural emissions.

Einstein used the Equivalence Principle to derive his Theory of General Relativity. The Equivalence Principle says if data cannot tell the difference between two things then the two things are identical.

2.2 The Simple Model

The “Model” is the simplest possible model to describe how CO2 flows through the atmosphere. The Model is based on the continuity equation and the assumption that outflow is proportional to the level of CO2 in the atmosphere. The Model applies equally to human, natural, and carbon-14 CO2. It is described in Section 4.

Models, simple or complex, embody theories. Just as theories must be tested, models must be tested. The proper test is to make a prediction and compare the prediction with data. If the prediction fails, the model fails and maybe the theory fails as well.

Berry (1967) used a numerical model to predict how fast cloud-droplets collect to form raindrops. To test the model, Berry used predictions for three special cases of the core formula that had analytic solutions, and the three analytic solutions bounded the domain of known cloud-droplet collection. The successful simulation of the three analytic solutions gave credibility to the model’s calculation for cloud-droplet growth.

2.3 The carbon-14 data

The above-ground atomic bomb tests raised the level of carbon-14 in the atmosphere from the “normal” 100 percent to 180 percent until the tests were terminated in 1963. The carbon-14 atoms are in the form of CO2, so carbon-14 CO2 traces how carbon-12 CO2 flows out of the atmosphere.

Carbon-14 is a natural tracer of atmospheric CO2. The carbon-14 CO2 produced by the bomb tests follows the same carbon path as carbon-12 CO2.

A carbon-14 atom has 2 more neutrons that a carbon-12 atom. Carbon-14 CO2 is heavier than carbon-12 CO2. Therefore, it has a longer residence time.

Fig. 1 shows a plot of the carbon-14 data (Wikipedia, 2017). The natural concentration of carbon-14 CO2 is defined as 100 percent. The “pMC/ percent” is “percent of modern carbon” where “modern carbon” means the level in 1950 (Berger, 2014).

The half-life is the time taken for the level of carbon-14 CO2 to fall to one-half its initial level above its balance level. (Not to be confused with the radioactive half-life of carbon-14 of 5730 years.)

The carbon-14 level minus 100, loses half of its value every ten years. So, the half-life of carbon-14 CO2 is 10.0 years.

2.4 The carbon-14 test results

All valid CO2 models must replicate the decay of atmospheric carbon-14 data after 1963. According to the scientific method, if a prediction is wrong, the theory is wrong.

Fig. 2 shows how the IPCC Bern model and the Model simulate the outflow of CO2 from the atmosphere. The simulation subtracts 100 from the carbon-14 data to simulate the case where inflow is zero. This subtraction does not affect the shape of the carbon-14 decay curve. All models begin the simulation at 80 and assume inflow is zero. To download the Excel file with the calculations, see Berry (2018).

The blue curve is the carbon-14 data. This is “ground truth.”

The blue curve also shows the Model prediction using Eq. (8). The blue curve falls half-way to zero every 10 years. The simple Model exactly simulates the carbon-14 data.

The green curve shows the Model prediction with the residence time adjusted to simulate carbon-12 CO2. The green curve falls half-way to zero every 2.8 years.

The red curve shows the Bern model prediction calculated using Eq. (A.1). For the first year, it approximates the green curve of the simple model. Thereafter, the Bern model predicts progressively slower outflow. The level never goes below 12 on this plot because IPCC claims 15 percent of human emissions remain in the atmosphere forever.

The complex IPCC theory cannot simulate the carbon-14 data. Therefore, the IPCC theory is wrong.

The Bern model is unphysical. It changes its future with time. For example, the Simple Model predicts the level will fall one-half its value in a specific time interval no matter where you start on the curve. But the Bern model gives a different future if you begin at a different time on its curve.

The Model’s exact simulation of the carbon-14 data proves two things. First, the Model predicts the data. Second, the outflow of carbon-14 CO2 from the atmosphere is proportional to level. This implies the outflow of carbon-12 CO2 is similarly proportional to level.

3. Why the IPCC theory fails

3.1 IPCC assumes nature is constant

IPCC assumes nature is constant. This incorrect assumption forces the conclusion that human emissions caused all the increase in atmospheric CO2 above 280 ppm. If the IPCC would simply relax its constraints on nature, it would help the IPCC find the truth.

IPCC (2007) shows its basic assumption in two flow diagrams to represent the carbon cycle. The following discussion converts IPCC’s GtC units to ppm. Fig. 3 summarizes IPCC’s Figs. 3.1a and 3.1b. IPCC claims the “natural carbon cycle” is balanced and the “human perturbation” is unbalanced, leaving 1.5 ppm of human CO2 in the atmosphere each year.

The IPCC inserted its assumption that human CO2 causes all the increase in atmospheric CO2 into its climate models. IPCC is wrong because its assumption violates the Equivalence Principle and it cannot replicate the carbon-14 data.

3.2 IPCC argument fails logic

IPCC (2007) argues that during the time frame from 1750 to 2013:

Atmospheric CO2 increased 117 ppm. Total human CO2 emissions were 185 ppm. This total is 68 ppm more than the 117-ppm increase. Therefore, human emissions caused ALL the 117-ppm increase in atmospheric CO2, while nature absorbed the remaining human 68 ppm.

IPCC’s argument fails because it ignores natural CO2 inflow which totaled 26,000 ppm during the same period, and it ignores outflow. Simply put, IPCC’s core argument does not play with a full deck.

In Section 4, the Simple Model shows that present human CO2 emissions raise the level of atmospheric CO2 by only 18 ppm. If natural CO2 had not increased, the level of CO2 in the atmosphere would be IPCC’s 280 ppm plus 18 ppm for a total of 298 ppm. This result is consistent with IPCC’s first three points above.

3.3 IPCC Bern model

Appendix A shows how to remove the integral in the Bern (2002) model to reveal its level equation.

The Bern model Eq. (A.1) predicts human emissions of 4 ppm per year for 100 years will leave 60 ppm in the atmosphere forever.

The Equivalence Principle requires the Bern model to apply to natural as well as human CO2. The Bern model predicts natural emissions of 100 ppm per year for 100 years will leave 1500 ppm in the atmosphere forever. This clearly invalid prediction for natural emissions proves the Bern model is wrong. Therefore, IPCC’s basic assumption is wrong.

The creators of the original Bern model, Siegenthaler and Joos (1992), understood their model should reproduce the carbon-14 data and were disappointed that it did not do so.

The IPCC modified the original Bern model, described by Siegenthaler and Joos, that connected the atmosphere level to the upper ocean level, and the upper ocean level to the deep and interior ocean levels, as can be seen in their Fig. 1.

The IPCC removed the Bern model levels for the deep and interior ocean and connected their rates directly to the atmosphere level. That is why the Bern model has three residence times rather than one. Connecting flows that belong to the deep and interior ocean directly to the atmosphere violates the principles of systems (Forrester, 1968) and will give the wrong answer.

The Bern model assumes the flows with these three residence times act in series rather than in parallel. This is like having three holes of different sizes, in the bottom of a bucket of water, and claiming the smallest hole restricts the flow through the largest hole.

3.4 IPCC buffer capacity claim is wrong

IPCC’s Bern model is based on IPCC’s claim that human CO2 has overloaded natural carbon-dioxide sinks and therefore has slowed the outflow of CO2 from the atmosphere. IPCC claims,

“The fraction of anthropogenic CO2 that is taken up by the ocean declines with increasing CO2 concentration, due to reduced buffer capacity of the carbonate system.”

The IPCC put its unproven claim into its theory. Then the IPCC put its theory into its climate models. The Bern model simulates the climate models.

The IPCC theory predicts the half-life for carbon-14 would have increased from 1965 to 1995. This would have deviated the blue curve in Fig. 2 to look more like the Bern model red curve.

However, Fig. 1 shows the 10-year carbon-14 half-life did not change from 1965 to 1995. Therefore, human CO2 has not “reduced the buffer capacity of the carbonate system” as the IPCC claims.

3.5 IPCC adjustment time is invalid

The IPCC (1990) defines an “adjustment time” to support its claim that human emissions have a long residence time:

“The turnover time of CO2 in the atmosphere, measured as the ratio of the content to the fluxes through it, is about 4 years. This means that on average it takes only a few years before a CO2 molecule in the atmosphere is taken up by plants or dissolved in the ocean.

“This short time scale must not be confused with the time it takes for the atmospheric CO2 level to adjust to a new equilibrium if sources or sinks change.”

“This adjustment time… is of the order of 50 – 200 years, determined mainly by the slow exchange of carbon between surface waters and the deep ocean.

“The concentration will never return to its original value, but reach a new equilibrium level, about 15 percent of the total amount of CO2 emitted will remain in the atmosphere.”

IPCC’s short residence time of about 4 years is correct. The Bern model’s failure to simulate the carbon-14 data and the Equivalence Principle prove the IPCC’s “adjustment time” is invalid.

4. Physics Model for atmospheric CO2

4.1 A Simple Model

The simple model shows how inflow affects level, how level affects outflow, and how balance occurs when outflow equals inflow. Nothing in the world outside the atmosphere changes these conclusions. But these conclusions change entirely the dominant worldview of how human emissions change the level of CO2 in our atmosphere.

You have a bucket of water with a hole in the bottom. Water from your hose flows into your bucket while water leaks out of the hole. You adjust the inflow until the water level stays almost constant.

The water in your bucket represents the CO2 in the atmosphere. Water from your hose represents the flow of CO2 into the atmosphere. Water that flows out of the hole represents the flow of CO2 out of the atmosphere.

If inflow is greater than outflow, the level goes up, and vice-versa. Also, as the water level increases, outflow increases. The Model puts these two assumptions into a mathematical equation with more precision as applied to the atmosphere. The Model not only explains the simple flow of water in the bucket, the Model also explains the flow of CO2 in the atmosphere.

4.2 Model derivation

A model is a system used to describe a subset of nature. A model is composed of levels and flows between levels. Flows are rates. Levels set the flows and the flows set the new levels (Forrester, 1968).

The Model extends the model introduced by Salby (2016) and Harde (2017a).

Fig. 4 illustrates the atmosphere system. The Model includes the level (concentration) of CO2 in the atmosphere and the inflow and outflow of CO2.

The Model does not include processes outside the system but incorporates their effects if they modify inflow or outflow.

The mathematics used to describe the Model are analogous to the mathematics used to describe many engineering systems.

The continuity equation assures carbon atoms are conserved:

dL/dt = Inflow – Outflow (1)

Where

L = carbon dioxide level

dL/dt = the rate of change of L

t = time

Inflow = the rate carbon dioxide moves into the system

Outflow = the rate carbon dioxide moves out of the system

Outflow must be an increasing function of level, or there would be no natural balance. Assume outflow is proportional to level,

Outflow = L / Te (2)

where Te is the 1/e residence time.

(More generally, Outflow can equal f(L) / Te, where f(L) is a strictly increasing function of L. This allows the Simple Model to support analogies where f(L) does not equal L.)

Substitute Eq. (2) into the continuity equation (1),

dL/dt = Inflow – L / Te (3)

To find an equation for Inflow, let the level equal its balance level, Lb. Then the level is constant and Eq. (3) becomes

Inflow = Lb / Te (4)

where

Lb = balance level of L

Substitute Eq. (4) into Eq. (3) to get,

dL/dt = – (L – Lb) / Te (5)

Rearrange Eq. (5) to get

dL / (L – Lb) = – dt / Te (6)

Then integrate Eq. (6) from Lo to L on the left side, and from 0 to t on the right side, to get (Dwight, 1955),

Ln [(L – Lb) / (Lo – Lb)] = – t / Te (7)

where

Ln = natural logarithm, or logarithm to base e

Lo = Level at time zero (t = 0)

Lb = the balance level for a given inflow and Te

Te = Residence time for level to move (1 – 1/e) of the distance from Lo to Le

e = 2.7183

(The original integration of Eq. (7) contains two absolute functions, but they cancel each other because both L and Lo are always either above or below Le.)

Raise e to the power of each side of Eq. (7), to get the level as a function of time:

L = Lb + (Lo – Lb) exp(- t / Te) (8)

Equation (8) shows that Te is the 1/e residence time of CO2 in the atmosphere.

The Model applies independently to all forms and sources of CO2.

4.3 Balance level

Contrary to popular opinion, CO2 does not “accumulate” in the atmosphere. Constant inflows don’t add to the level of CO2.

The Model shows how inflows, human or natural, set independent balance levels. Constant inflows have constant balance levels. The sum of human and natural balance levels equals the total balance level. Fig. 4 shows how nature balances inflow by adjusting outflow until the level equals the balance level.

Solving Eq. (4) for Le gives

Lb = Inflow * Te (9)

Equation (9) shows how inflow and residence time set the balance level. Equation (2) shows how level and residence time set outflow.

Equation (5) shows how level always moves toward its balance level. If inflow is zero, Le is zero, and outflow will continue until the level goes to zero.

The level of CO2 in the atmosphere behaves like the level of water in a lake. If a river flows into a lake and lake water flows out over a dam, the inflow does not continue to increase the lake level. The inflow simply raises the level of the lake until the outflow over the dam equals the inflow from the river. Then the lake level remains constant so long as inflow remains constant.

4.4 Residence time

There are two definitions of residence times, half-life, Th, and 1/e residence time, Te. Both residence times are different measures of the same thing:

Residence time controls how level L approaches its balance level Le when inflow is constant.

When time t equals half-life Th, or

t = Th

then Eq. (7) becomes

Ln [(L – Lb) / (Lo – Lb)] = – Th / Te

Ln (1/2) = – Th / Te

Ln (2) = Th / Te

Te = Th / Ln (2)

Te = 1.4427 Th (10)

Equation (10) shows the relationship between residence half-life Th and 1/e residence time Te.

IPCC (2007) estimates today’s total natural carbon dioxide inflow is about 100 ppm per year. NOAA (2017) Mauna Loa data shows the 2015 level of atmospheric carbon dioxide is about 400 ppm.

Solve Eq. (4) for Te to get,

Te = Lb/ Inflow (11)

Insert the NOAA value for Le and the IPCC value for Inflow to get the residence time,

Te = 400 ppm / 100 ppm per year = 4 years (12)

IPCC (1990) agrees with Eq. (12) for residence time. This calculation of residence time applies to carbon dioxide levels from about 280 ppm to 1000 ppm.

4.5 Model replicates carbon-14 data

To reproduce the carbon-14 decay curve, use either rate Eq. (5) or analytic Eq. (8). Both equations give the same result. Use Eq. (10) to convert the carbon-14 half-life of 10.0 years,

Te = 1.4428 Th = 14.4 years (13)

Set,

Lo = 80 percent

Lb = 0 percent

4.6 Effect of human CO2

Data from Boden et al. (2017) show human carbon dioxide emissions from fossil-fuel burning, cement manufacturing, and gas flaring in 2014 was 4.6 ppm (9.855 GtC) per year.

Using Eq. (9) for the 2014 human emissions to get,

Lbh = (4.6 ppm/year) (4 years) = 18 ppm (14)

Using Eq. (9) for natural emissions to get,

Lbn = (98 ppm/year) (4 years) = 392 ppm (14)

Equation (14) shows human emissions create a balance level of 18 ppm. Equation (15) shows present natural emissions create a balance level of 392 ppm.

The total balance level for human and natural emissions, using the above data for 2014, is the total of Eq. (14) and Eq. (15), or 410 ppm.

If human and natural emissions stay constant after 2014, the carbon dioxide level would reach its balance level of 410 in about 2018. Mauna Loa data show 404 ppm for 2016. These calculations demonstrate the accuracy of the Model.

The ratio of Eq. (14) to Eq. (15) is independent of residence time,

Leh / Len = 18 / 392 = 4.6 percent (16)

Equation (16) shows the balance level ratio of human-produced to nature-produced carbon dioxide is the ratio of their inflows.

4.7 Effect of surface temperature

Rorsch et al. (2005), Courtney (2008), MacRae (2008, 2015), Humlum et al. (2013), Salby (2012, 2014, 2016), and Harde (2017a) show how changes in surface temperature precede changes in CO2.

Salby (2012) derives from data how the rate of change of CO2 level is a function of surface temperature Ts, or,

dL/dt = 3.5 (ppmv/year K) Ts(K) (17)

Salby shows how the increase in Ts since 1750 and since the Little Ice Age in 1650 explains the increase in the level of atmospheric CO2 from 1750 to the present.

Equation (2) shows outflow equals level divided by residence time. The carbon-14 data show outflow is proportional to level. Therefore, residence time is independent of level.

Equation (9) shows balance level equals inflow multiplied by residence time. Carbon-14 data show carbon sinks have not saturated and therefore have not changed residence time.

Therefore, in the absence of any other explanation, the increase of balance level after 1750 must be caused by increased inflow.

5. Conclusions

The IPCC theory is wrong because it cannot simulate the carbon-14 data, it claims nature treats human CO2 differently than natural CO2 which violates the Equivalence Principle, and its predictions with natural CO2 are extremely wrong.

Therefore, all climate models are wrong. All IPCC reports are wrong. Carbon sinks are not saturated. Human CO2 does not accumulate in the atmosphere. Human CO2 flows out of the atmosphere just like natural CO2.

The Simple Model is accurate. It exactly simulates the carbon-14 data.

All carbon dioxide emissions – human and natural, independently or in total – create an inflow that sets a balance level. Each level moves toward its balance level until its outflow equals its inflow. Then the level remains constant so long as its inflow remains constant.

Present human emissions create an inflow that adds 18 ppm to atmospheric CO2. Present natural emissions create an inflow that adds 392 ppm. Their total is 410 ppm.

If ALL human emissions stopped and natural emissions stayed constant, the level of CO2 would fall by only 18 ppm.

Nature’s CO2 emissions are 21 times human CO2 emissions. Therefore, nature changes climate. Human emissions do not.

Appendix A: Bern model math

The Bern (2002) model is an integral equation rather than a level or rate equation. The Bern model integrates the inflow of carbon dioxide from minus infinity to any time in the future.

To deconstruct the integral version of the Bern model, let inflow occur only in the year when “t-prime” equals zero (t’ = 0). Then the integral disappears, and the Bern model becomes a level equation.

The Bern level equation is,

L(t) = Lo [ A 0 + A 1 exp(- t/T 1 ) + A 2 exp(- t/T 2 ) + A 3 exp(- t/T 3 )] (A.1)

Where

t = time in years

Lo = the level of atmospheric carbon dioxide due to inflow in year t = 0

L(t) = the level of atmospheric carbon dioxide after year t = 0

where the Bern IPCC TAR standard values are,

A 0 = 0.152

A 1 = 0.253

A 2 = 0.279

A 3 = 0.319

T 1 = 173 years

T 2 = 18.5 years

T 3 = 1.19 years

The A-values merely weight the four terms on the right-hand side of Eq. (A.1):

A 0 + A 1 + A 2 + A 3 = 1.000

Here are two easy ways to show the Bern model contradicts real-world data.

Set t equal to 100 years. Then Eq. (A.1) becomes,

L = (A 0 + A 1 ) Lo = (0.152 + 0.253 * 0.56) Lo = 0.29 Lo (A.2)

Set t equal to infinity. Then Eq. (A.1) becomes,

L = Ao Lo = 0.152 Lo (A.3)

Equations (A.2) and (A.3) predicts a one-year inflow that sets Lo to 100 ppm, followed by zero inflow forever, will cause the level in 100 years to be 29 ppm and the future level will never fall below 15 ppm.

Competing Interests

The author declares he has no conflict of interest.

Acknowledgements

This research was funded by the personal funds of the author.

The author thanks Chuck Wiese, Laurence Gould, Tom Sheahen, and Charles Camenzuli, who provided scientific critique, and Daniel Nebert, Gordon Danielson, and Valerie Berry, who provided language and grammar improvements.

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