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 has been submitted to a journal for publication. Therefore the present copyright does not permit republication because journals allow only one PREPRINT for submitted papers.
The United Nations Intergovernmental Panel on Climate Change (IPCC) assumes nature treats human-produced and nature-produced carbon dioxide differently. This assumption is wrong because it violates the Equivalence Principle.
IPCC’s basic assumption infects climate models. IPCC’s Bern model, a 7-parameter curve-fit to climate model output, predicts human carbon dioxide stays in the atmosphere for a long time, some of it forever. That conclusion is a result of IPCC’s basic assumption and it is wrong.
Applying the Equivalence Principle, the Bern model predicts natural emissions will cause a runaway carbon dioxide level that contradicts data. Therefore, IPCC climate models are wrong. IPCC’s model cannot simulate the carbon-14 data.
A Model, derived from the continuity equation with outflow proportional to level, accurately simulates the carbon-14 data with no arbitrary curve-fitting parameters.
The Model shows constant carbon dioxide emissions, human or natural, do not add more carbon dioxide to the atmosphere. Rather their inflows set equilibrium levels for atmospheric carbon dioxide.
Using IPCC data, the Model shows 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.
Any climate change caused by increased CO2 is 96 percent from natural CO2 and only 4 percent from human CO2.
The effect of human emissions is the same as if natural emissions had increased by the same amount and human emissions had remained zero.
The critical scientific questions about climate change are about cause-and-effect:
- How much do human emissions increase atmospheric carbon dioxide?
- How much does increased atmospheric carbon dioxide change climate?
This paper focuses on the first question.
The United Nations Intergovernmental Panel on Climate Change (IPCC, 2007) answers the first question with the following two claims:
- Human emissions have caused ALL the increase in carbon dioxide since 1750.
- 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.
“Abundant published literature” is irrelevant because votes are irrelevant to science. It is impossible to prove an idea is true, but it is possible to prove an idea is wrong. As Richard Feynman explained so well, 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 carbon dioxide, 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 carbon dioxide.
Authors who argue for the IPCC view include Cawley (2011), Kern and Leuenberger (2013), Masters and Benestad (2013), Richardson (2013), and many others. 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.”
Kohler’s world cannot tolerate a contradictory position, so Kohler wants to censor Harde (2017a) withdrawn. That is a new low in science, but it may have had an effect. Possibly unique in scientific publishing, 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, because
“the most simple model of the carbon cycle must at minimum contain equations of at least two reservoirs (the atmosphere and the surface ocean), which are solved simultaneously.”
Kohler is wrong because Kohler does not understand systems.
The history of 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.
Kohler argues that Harde is wrong because Harde makes science too simple. This paper shows Kohler and the IPCC are wrong because they think complexity proves accuracy. The ultimate scientific test is a prediction. The IPCC model cannot replicate the carbon-14 data.
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).”
Munshi (2017) shows the “detrended correlation analysis of annual emissions and annual changes in atmospheric carbon dioxide” is zero. Therefore, IPCC’s claim of “considerable certainty” fails. Where there is no correlation, there is no cause and effect.
2. IPCC model contradicts science
2.1 IPCC basic assumption violates Equivalence Principle
IPCC’s (2007) basic assumption treats human carbon dioxide differently than it treats natural carbon dioxide in the atmosphere. IPCC shows its basic assumption in two figures. Each figure has arrows to represent the carbon dioxide flow of an external source to and from the atmosphere.
The IPCC figures use GtC units. To keep the discussion simple, this paper converts all GtC (Gigatons of Carbon) units into the equivalent carbon dioxide units of ppm (parts per million by volume in dry air), using:
1 ppm = 2.13 GtC
Fig. 1 shows IPCC’s Fig. 3.1a that represents the “natural carbon cycle” as balanced with average annual inflow and outflow equal to 98 ppm per year.
Fig. 2 shows IPCC’s Fig. 3.1b that represents the “human perturbation” as an average annual inflow of 3.3 ppm per year with an outflow of only 1.8 ppm per year.
Table 1 summarizes the relevant numbers from IPCC’s Fig. 3.1a and 3.1b, after converting GtC to ppm.
Table 1. IPCC assumes natural carbon dioxide inflow and outflow balance, but human carbon dioxide inflow and outflow do not balance. This converts IPCC GtC units to ppm.
IPCC’s basic assumption violates the Equivalence Principle. Therefore, it is wrong.
The Equivalence Principle says if data cannot tell the difference between two things then the two things are identical. The Equivalence Principle applies to climate physics because nature cannot distinguish between human-produced and nature-produced carbon dioxide in the atmosphere. Einstein used the Equivalence Principle to derive his Theory of General Relativity.
It does not matter how much complexity the IPCC added outside the atmosphere system. The IPCC got the atmosphere system wrong. The IPCC constraint on the outflow of human-produced carbon dioxide is artificial and incorrect.
Processes outside the atmosphere system can change inflow and outflow to the atmosphere but they cannot cause the atmosphere to artificially restrict the outflow of human carbon dioxide. The atmosphere will balance human carbon dioxide emissions just like it balances natural carbon dioxide emissions.
2.2 IPCC basic argument fails logic
IPCC (2007) claims in the third paragraph of its Executive Summary,
“The present atmospheric carbon dioxide increase is caused by anthropogenic emissions of carbon dioxide.”
The IPCC assumes the level of atmospheric carbon dioxide in 1750 was 280 ppm. Segalstad (1998), Ball (2008), and Salby (2014) present evidence that the level in 1750 was much higher than 280 ppm. Nevertheless, this paper will use this IPCC assumption because the IPCC basic argument still fails.
IPCC’s basic argument has six steps. This paper agrees with the first 4 steps that are the premise:
- In 1750, the atmospheric carbon dioxide level was 280 ppm (MacFarling Meure et al. (2006)).
- In 2013, the atmospheric carbon dioxide level was 397 ppm (NOAA, 2017; Olivier, 2015).
- So, the atmospheric carbon dioxide level increased 117 ppm from 1750 to 2013.
- The sum of all human carbon dioxide emissions from 1750 to 2013 (Boden, 2017) was 185 ppm. This is 68 ppm more than the 117-ppm increase in the atmospheric carbon dioxide level from 1750 to 2013.
- Therefore, human emissions caused ALL the 117-ppm increase in atmospheric carbon dioxide, while nature absorbed the remaining human 68 ppm.
- Since nature was a net absorber of carbon dioxide since 1750, nature cannot have caused the increase in atmospheric carbon dioxide.
Step #5 is wrong because it ignores natural carbon dioxide emissions which are 21 times greater than human emissions. It assumes nature will not absorb most human emissions, which is an invalid assumption that forces IPCC’s conclusion. It violates the Equivalence Principle.
For example, alternatives to Steps #4 and #5 are:
4a. The sum of natural carbon dioxide emissions from 1750 to 2013 is about 100 ppm per year for 260 years or 26,000 ppm. This is 25,883 ppm more than the increase in atmospheric carbon dioxide since 1750.
5a. Therefore, nature caused all the 117-ppm increase in atmospheric carbon dioxide, while nature absorbed the remaining 25,883 ppm.
Table 2 shows how Step #5 assumes natural CO2 is balanced while human CO2 drives all the increase in atmospheric CO2.
Table 2. Step #5 neglects effects of natural CO2. Units are total ppm from 1750 to 2013.
Table 3 shows one of the infinite number of scenarios that satisfy Steps #1 through #4. This shows Step #5 is arbitrary.
Table 3. A scenario that satisfies Steps #1 through #4 of IPCC’s basic argument. Units are total ppm from 1750 to 2013.
Table 4 shows an alternative explanation to the IPCC basic argument that satisfies IPCC’s first four points of its argument. Let natural emissions be 96 percent of the total inflow while human emissions contribute the remaining inflow of 4 percent. Then at equilibrium, the atmosphere will contain the same percentages as the inflows, and the outflows will be the same percentages as the atmosphere.
Table 4. Physics shows an alternative explanation to the IPCC basic argument that satisfies Steps #1 through #4. Units are ppm per year and percent.
Step #6 is invalid because it is based on the invalid Step #5.
In conclusion, IPCC’s basic argument fails to support its basic assumption.
2.3 Its Bern model fails prediction
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 basic assumption on climate models.
Appendix A shows how to remove the integral in the Bern (2002) model to reveal its level equation.
Table 5 shows the Bern model predictions for human and natural CO2 emissions. The Bern model predicts human emissions of 4 ppm per year for 100 years will leave 60 ppm in the atmosphere forever.
Table 5. Example Bern model predictions for human and natural CO2 emissions
The Equivalence Principle requires the Bern model to also apply to natural emissions. The Bern model predicts natural emissions of 100 ppm per year for 100 years will leave 1500 ppm forever. This clearly invalid prediction for natural emissions proves the Bern model is wrong.
The Bern model cannot reproduce the carbon-14 data. (Section 3.4 shows the carbon-14 data and a plot of Bern model output.) A valid model must be able to reproduce the carbon-14 data.
The creators of the original Bern model, Siegenthaler and Joos (1992), understood that 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, where the smallest hole restricts the flow through the largest hole.
2.4 IPCC adjustment time fails Equivalence Principle
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.
“For example, the first reduction by 50 percent occurs within some 50 years, whereas the reduction by another 50 percent (to 25 percent of the initial value) requires approximately another 250 years.
“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. IPCC’s “adjustment time” for human emissions is incorrect.
The Equivalence Principle requires IPCC’s “adjustment time” also apply to natural emissions. As the previous section shows, inserting natural emissions into the Bern model produces a clearly invalid prediction. Therefore, “adjustment time” is wrong.
2.5 IPCC basic assumption fails simple physics
The effect of human emissions is the same as if natural emissions increased by the same amount. So, the IPCC basic assumption is equivalent to the absurd notion that nature was so precipitously balanced in 1750 that even a one-percent increase in natural emissions would initiate a continuous rise in atmospheric carbon dioxide.
Imagine water from two sources flowing into a beaker with a hole in the bottom. Let 96 percent of the total inflow come from the first source and 4 percent from a second source. When outflow equals inflow, 96 percent of the water in the beaker will be from the first source and 4 percent from the second source.
Yet the IPCC basic assumption says nature can restrain the outflow of water from the second source and cause it to accumulate in the beaker, in violation of the Equivalence Principle.
3. An explanation of carbon dioxide flow
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).
Fig. 3 illustrates the atmosphere system. The system Model includes the level (concentration) of carbon dioxide in the atmosphere and the inflow and outflow of carbon dioxide.
The Model does not include processes outside the system but incorporates their effects if they modify inflow or outflow.
The Model shows how the inflows from any source, human or natural, set independent equilibrium levels, and the sum of these equilibrium levels equals the total equilibrium level.
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)
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 equilibrium. Following the Ideal Gas Law, this paper assumes outflow is proportional to level,
Outflow = L / Te (2)
where Te is a constant that has the dimension of time.
Salby (2016) and Harde (2017) use a similar form of Eq. (2). Equation (2) causes the level to always move toward its equilibrium level.
Substituting Eq. (2) into the continuity equation (1), gives,
dL/dt = Inflow – L / Te (3)
To find an equation for Inflow, let the level equal its equilibrium level, Le. Then the level is constant and Eq. (3) becomes
Inflow = Le / Te (4)
Le = equilibrium level of L
Substituting Eq. (4) into Eq. (3), gives Model Eq. (5),
dL/dt = – (L – Le) / Te (5)
Rearrange Eq. (5) to get
dL / (L – Le) = – 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 – Le) / (Lo – Le)] = – t / Te (7)
Ln = natural logarithm, or logarithm to base e
Lo = Level at time zero (t = 0)
Le = the equilibrium 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 = Le + (Lo – Le) exp(- t / Te) (8)
Equation (8) reveals that Te is the 1/e residence time of carbon dioxide in the atmosphere.
3.2 Balance level
Inflows don’t add to the level of carbon dioxide. Inflows set balance levels.
Fig. 4 shows how nature balances inflow by adjusting outflow until the level equals the equilibrium level. This applies to the inflows of each partial pressure component and to their totals.
Solving Eq. (4) for Le gives
Le = Inflow * Te (9)
Equation (9) shows how inflow and residence time set the equilibrium level. Equation (2) shows how level and residence time set outflow.
Equation (5) shows how level always moves toward its equilibrium level. If inflow is zero, Le is zero, and outflow will continue until the level goes to zero.
With only minor modifications, the Model describes these three analogies to atmospheric carbon dioxide.
- Pump air into a balloon (or inner tube) that has a hole in it. The greater the air pressure inside the balloon, the faster the air leaks out of the hole. If the inflow is constant, the balloon will expand or contract until outflow equals inflow.
- A river flows into a lake and the lake water flows out over a fixed dam. The lake level rises or falls until outflow equals inflow. If inflow increases, the lake level will rise until, once again, outflow equals inflow.
- Physics students let water flow into the top of the beaker with a hole in the bottom. As level increases, outflow increases. For every inflow, there is an equilibrium level where outflow equals inflow.
3.3 Residence times
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 is a measure of how level L approaches its equilibrium level Le when inflow is constant.
When time t equals half-life Th, or
t = Th
then Eq. (7) becomes
Ln [(L – Le) / (Lo – Le)] = – 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 the 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 = Le / 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)
Equation (12 for residence time agrees with IPCC (1990). This calculation of residence time applies to carbon dioxide levels from about 280 ppm to 1000 ppm.
3.4 The Model replicates carbon-14 data
Every valid theory must make valid predictions. Therefore, the Model must replicate the atmospheric carbon-14 data after 1963.
The atomic bomb tests in the 1960’s increased atmospheric carbon-14 by more than 80 percent. After the halt of the tests in 1963, the concentration of carbon-14 decreased exponentially toward its previous equilibrium level of 100 percent.
Fig. 5 shows a plot of the carbon-14 data (Wikipedia, 2017). The natural concentration of carbon-14 carbon dioxide 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 for the level of carbon-14 carbon dioxide to fall to one-half its initial level above its equilibrium level. (Not to be confused with the radioactive half-life of carbon-14 of 5730 years.)
The data show carbon-14 lost one-half of its level above its equilibrium level every 10 years. The decrease in the level of carbon-14 carbon dioxide follows an exponential curve to its equilibrium value.
Table 5 shows data taken from Fig. 5 every ten years.
Table 5. The carbon-14 level minus 100, loses half of its value every ten years.
The data show the residence half-life for carbon-14 carbon dioxide is 10.0 years,
Th = 10.0 years
Therefore, using Eq. (11),
Te = 1.4428 Th = 14.4 years (13)
To test the Model, use the carbon-14 data,
Lo = 180 percent
Le = 100 percent
Te = 14.4 years
Use Eq. (5) to calculate the natural inflow of carbon-14 carbon dioxide,
Inflow = Le / Te (4)
Inflow = 100 / 14.4 = 6.9 percent per year (14)
Then use either Eq. (5) or Eq. (5) to calculate the level as a function of time in years. Insert the residence time from Eq. (13) and the natural inflow from Eq. (14).
Fig. 6, calculated in Excel using Eq. (5), shows a perfect fit to the carbon-14 data in Fig. 3. Level L moves half the distance to its equilibrium level Le every ten years.
Fig. 6 also shows the Eq. (5) calculation for carbon-12 carbon dioxide and the Bern model prediction using Eq. (A.1).
The Model accurately predicts how the level of carbon-14 carbon dioxide approaches its equilibrium level. Therefore, the Model will also correctly predict how carbon-12 carbon dioxide will approach its equilibrium level.
3.5 The effect of human carbon dioxide
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 gives,
Leh = (4.6 ppm/year) (4 years) = 18 ppm (15)
Using Eq. (9) for natural emissions gives,
Len = (98 ppm/year) (4 years) = 392 ppm (16)
Equation (15) shows human emissions create an equilibrium level of 18 ppm. Equation (16) shows present natural emissions create an equilibrium level of 392 ppm.
The total equilibrium level for human and natural emissions, using the above data for 2014, is the total of Eq. (15) and Eq. (16), or 410 ppm. If human and natural emissions stay constant after 2014, the carbon dioxide level would reach its equilibrium 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. (15) to Eq. (16) is independent of residence time,
Leh / Len = 18 / 392 = 4.6 percent (17)
Equation (17) shows the equilibrium level ratio of human-produced to nature-produced carbon dioxide is the ratio of their inflows.
3.6 Level can follow temperature
Rorsch et al. (2005), Courtney (2008), MacRae (2008, 2015), Humlum et al. (2013), and Salby (2012, 2014, 2016) show how changes in surface temperature precede changes in carbon dioxide. More specifically, they use data to show how the rate of change of carbon dioxide level, dL/dt, is a function of surface temperature.
For dL/dt to follow surface temperature, Le must follow surface temperature, or
Le = c Ts (18)
where c is a constant. Salby (2014) uses data to evaluate c.
IPCC’s basic assumption, that nature treats human carbon dioxide emissions differently than it treats nature’s carbon dioxide emissions, is wrong because it violates the Equivalence Principle.
IPCC’s claim, that human carbon dioxide emissions will linger in the atmosphere for hundreds of years and 15 percent will remain forever, is invalid.
IPCC’s claim that human emissions have caused all the rise in the level of carbon dioxide in the atmosphere since 1750, is invalid.
The Model has no arbitrary curve-fit parameters. Yet, it accurately simulates the carbon-14 data.
The Model shows human emissions add only 18 ppm and nature adds 392 ppm to produce today’s 410 ppm level of carbon dioxide.
If all human CO2 emissions were stopped, the level of carbon dioxide in our atmosphere would fall by only 18 ppm.
The effect of human emissions is the same as if natural emissions had increased by the same amount and human emissions had remained zero.
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 [ A0 + A1 exp(- t/T1) + A2 exp(- t/T2) + A3 exp(- t/T3)] (A.1)
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,
A0 = 0.152
A1 = 0.253
A2 = 0.279
A3 = 0.319
T1 = 173 years
T2 = 18.5 years
T3 = 1.19 years
The A-values merely weight the four terms on the right-hand side of Eq. (A.1):
A0 + A1 + A2 + A3 = 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 = (A0 + A1) 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.
The author declares he has no conflict of interest.
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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