Human CO2 has little effect on atmospheric CO2
American Meteorological Society 99th Annual Meeting, January 8, 2019, AMS website
(A journal is now reviewing this preprint. Copyright does not allow republishing of journal submissions. I improved Appendix A on March 19, 2019)
Edwin X Berry, Ph.D., CCM
Climate Physics LLC, Bigfork, Montana, USA
“The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advances in science.” – Albert Einstein
A simple Physics Model makes only one assumption: outflow is proportional to the level (or concentration) of CO2 in the atmosphere. This model replicates the decay of 14CO2 after 1970 using a constant e-time of 16.5 years. This replication has significant theoretical consequences.
Human and natural CO2 inflows set independent balance levels in proportion to their inflows, independent of e-time. The total balance level is the sum of the human and natural balance levels. The CO2 level moves with e-time to its balance level until outflow equals inflow. Then the level remains constant if inflow remains constant. Continued, constant human emissions do not add more CO2 to the atmosphere. Neither human nor natural CO2 accumulate in the atmosphere. Human CO2 has not caused all the increase in atmospheric CO2 since 1750, or above 280 ppm. Present human CO2 adds only 18 ppm to the balance level. Natural CO2 adds 392 ppm.
The complex United Nations Intergovernmental Panel on Climate Change (IPCC) model cannot reproduce the decay of 14CO2 after 1970. IPCC assumes human CO2 reduced the buffer capacity of the carbonate system. However, the 14C data show e-time is constant, which means the buffer capacity has not changed.
Keywords: carbon dioxide, CO2, climate change, anthropogenic, accumulation
The United Nations Intergovernmental Panel on Climate Change (IPCC)  Executive Summary claims human emissions caused atmospheric CO2 to increase from 280 ppm in 1750, to 410 ppm in 2018, for a total increase of 130 ppm.
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.
The U.S. Global Change Research Program Climate Science Special Report (USGCRP)  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.
IPCC and USGCRP claim there are “no convincing alternative explanations” other than their theory to explain the “observational evidence.”
This paper shows these IPCC and USGCRP claims are incorrect and presents a “convincing alternative explanation” that IPCC and USGCRP claim does not exist.
IPCC  bases all its climate conclusions on this flawed 3-step argument:
How do we know that in fact human activity has been responsible for the well documented 25% increase in atmospheric CO2 since the early 19th century? Couldn’t this rise instead be the result of some long-term natural fluctuation in the natural carbon cycle? Simple arguments allow us to dismiss this possibility.
First, the observational CO2 records from ice cores … show that the maximum range of natural variability about the mean of 280 ppm during the past 1000 years was small.
Segalstad , Jaworowski , and Salby  present evidence that the CO2 level before 1750 was much higher than 280 ppm. Nevertheless, this paper accepts IPCC’s assumption that the CO2 level was 280 ppm in 1750. However, this paper rejects the implication that the ice-core mean of 280 ppm implies nature’s CO2 emissions did not change after 1750. IPCC continues:
Second, the observed rate of CO2 increase closely parallels the accumulated emission trends from fossil fuel combustion and from land use changes.
Proper statistics show correlation does not mean causation and time-series correlations must be detrended. Munshi  shows the detrended correlation of annual human emissions with annual increase in atmospheric CO2 is zero. The zero correlation proves the human contribution to the increase in atmospheric CO2 is negligible. IPCC continues:
Third, the observed isotropic trends of 13C and 14C agree qualitatively with those expected due to the CO2 emissions from fossil fuels and the biosphere, and they are quantitatively consistent with results from carbon cycle modeling.
In fact, the observed isotropic trends of 14C and 13C support the Physics Model and reject the IPCC model.
For simplicity, this paper uses levels in units of ppm (parts per million by volume in dry air) and flows in units of ppm per year. GtC (Gigatons of Carbon) units are converted into CO2 units in ppm using:
1 ppm = 2.12 GtC
Authors who conclude human CO2 adds only a minor increase in atmospheric CO2 include Revelle and Suess , Starr , Segalstad , , , Rorsch , Courtney , Quirk , Essenhigh , Glassman , Humlum , Salby , , , Pettersson , Harde , , and Berry , .
Authors who support the IPCC conclusion include Archer , Cawley , Joos , Kern and Leuenberger , Richardson , and Kohler .
2. The Physics Model
2.1 Physics Model derivation
A system describes a subset of nature. A system includes levels and flows between levels. Levels set flows and flows set new levels .
Figure 1 illustrates the system for atmospheric CO2. The system includes the level (concentration) of CO2 in the atmosphere and the inflow and outflow of CO2.
Appendix A shows the mathematical derivation of the Physics Model. It begins with the continuity equation, Equation (A.1). Then it adds one hypothesis, Equation (A.2): Outflow equals Level divided by e-time (where e-time is the e-folding time).
All other Physics Model equations are deductions from the continuity equation and the one hypothesis. For example, the balance level equals inflow multiplied by e-time, Equation (A.4).
Equation (A.8) is the analytic solution to the Physics Model rate equation when inflow and e-time are constant. It calculates the level as a function of time for any starting level, balance level, and e-time.
The Physics Model shows inflow sets a “balance level.” The level always moves towards its balance level. When the level equals the balance level, outflow equals inflow, and the level remains constant with continuing inflow.
The level of CO2 in the atmosphere behaves like the level of water in a lake where water flows into the lake and then out over a dam. Inflow sets the balance level. The lake level changes until the level equals its balance level. Then outflow equals inflow. No water “accumulates” in the lake.
The level of CO2 in the atmosphere also behaves like water in a bucket where water flows into the bucket and flows out through a hole in the bottom. As the level increases, outflow increases. When outflow equals inflow, the level remains constant. No water “accumulates” in the bucket.
The bucket analogy provides insight into e-time. If the hole in the bucket gets smaller, e-time increases. If the hole in the bucket gets larger, e-time decreases.
The Physics Model is complete. Inflow and outflow include all the effects of outside processes. If the atmosphere level were connected to external carbon reservoirs in land and ocean, the Physics Model would behave exactly as derived in this paper.
The Physics Model applies independently and in total to all definitions of CO2, for example, 14CO2, 12CO2, human CO2, and natural CO2, and their sums. The mathematics used in the Physics Model are analogous to the mathematics used to describe many engineering systems.
Kohler  commented on Harde ,
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  wants Harde  withdrawn. In response, the journal refused to publish Harde’s  rebuttal to Kohler.
Kohler claims Harde’s  model and therefore the Physics Model is “too simplistic.” Kohler claims a valid CO2 system must contain at least two levels.
There is no such thing as a system being “too simplistic.” A system should be as simple as possible to solve a problem. Each level of a system is isolated and connected to other levels by inflows and outflows. So long as a level includes inflow and outflow, a system is complete.
The Physics system includes the effects of outside processes. The Physics system properly computes how inflow and outflow change the level of CO2. Its equations and conclusions for the atmosphere level would not change if the atmosphere level were connected to another level. Outside processes can change inflow and outflow. The Physics system shows how inflow and outflow change the level.
Kohler claims more complex models give more correct answers. It does not work that way. One must get the physics for each level correct independent of other levels. Nothing in physics says more complexity increases validity.
2.2 Physics Model consequences
IPCC ,  says nature emits into the atmosphere about 120 GtC from land and 90 GtC from ocean for a total of 210 GtC per year. This is equivalent to about 98 ppm per year. IPCC  admits its estimates of “gross fluxes generally have uncertainties of more than ±20%.”
Boden  shows human CO2 emissions in 2014 were 9.7 GTC per year, or 4.6 ppm per year.
Equation (A.4) shows the balance level equals the product of inflow and e-time. Using IPCC numbers, and subscripts “h” to mean human and “n” to mean natural, the balance levels of human and natural CO2 are 18.4 and 392 ppm:
Lbh = 4.6 (ppm/year) * 4 (years) = 18 ppm (1)
Lbn = 98 (ppm/year) * 4 (years) = 392 ppm (2)
Their ratio and percentage are independent of residence time,
Lbh / Lbn = 4.6 / 98 = 18 / 392 = 4.6 percent (3)
Lbh / (Lbn + Lbh ) = 4.6 / 102.6 = 18.4 / 410 = 4.5 percent (4)
Equation (1) shows present human emissions create a balance level of 18 ppm. This is independent of nature’s balance level. If nature’s balance level remained at 280 ppm after 1750, then present human emissions would have increased the CO2 level 18 ppm from 280 ppm to 298 ppm.
Equation (2) shows present natural emissions create a balance level of 392 ppm. The human contribution of 18 ppm brings the total balance level to 410 ppm, which is close to the level in 2018.
Equation (3) shows the ratio of human to natural CO2 in the atmosphere equals the ratio of their inflows, independent of e-time. The IPCC calls the ratio in Equation (3) the “airborne fraction.”
Equation (4) shows the percentage of human-produced CO2 in the atmosphere equals its percentage of its inflow, independent of e-time.
Equations (1) and (2) support Harde  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).
While the details are outside the scope of this paper, Appendix C, from Harde , shows how temperature can increase the balance level to account for the rise in atmospheric CO2 since 1750. Salby  and Pettersson  show how surface temperature determines the CO2 level. This paper shows the cause-effect path is (a) surface temperature sets CO2 inflow and (b) inflow sets the CO2 balance level.
3. Theories must replicate data
3.1 The Physics Model replicates the 14C data
The above-ground atomic bomb tests in the 1950s and 1960s almost doubled the concentration of 14C in the atmosphere. The 14C atoms were in the form of CO2, hereinafter called 14CO2.
The 14C data are in units of D14C per mil. In D14C units, the “natural” balance level is zero, as defined by the average measured level before 1950.
After the cessation of the bomb tests in 1963, the concentration of 14CO2 decreased toward its natural balance level. The decrease occurred because the bomb-caused 14C inflow became zero while the natural 14C inflow continued.
All valid CO2 models must replicate the 14C data after 1970. Otherwise, the model is invalid.
Hua  processed 14C data for both hemispheres from 1954 to 2010 using 61 mid-year data points. Turnbull  processed 14C data for Wellington, New Zealand, from 1954 to 2014 using 721 data points. After 1970, 14CO2 were well mixed between the hemispheres, and the 14C data from both sources are virtually identical after 1970.
Figure 2 shows the global average data for D14C . Figure 3 shows the New Zealand data for D14C .
Bern Equation (B.1) predicts 15 percent all human CO2 entering the atmosphere stays in the atmosphere forever, 25 percent stays in the atmosphere almost forever, and 28 percent stays in the atmosphere longer than 14CO2 stays in the atmosphere. Only 32 percent flows freely out of the atmosphere.
All the models reviewed by Archer  and by Joos  lack a proper Physics Model to represent the atmosphere. These models assume human CO2 caused all the rise in atmospheric CO2 since 1750.
IPCC  assumes the Bern model describes how CO2 flows through the atmosphere. This assumption excludes the Physics Model. Yet IPCC bases all its global warming claims on its Bern model that outrageously concludes that human CO2 sticks in the atmosphere.
The Physics Model shows how the CO2 level will rise until outflow equals inflow. It shows there is only one time constant, not four. No CO2 remains in the atmosphere forever.
The proper test is to see if the Bern model can replicate the 14CO2 data.
Figure 5 compares the Bern model with the Physics Model and with the 14CO2 data. All model calculations begin with the initial level set to 100 and the balance level set to zero.
Equation (A.8) of the Physics Model with e-time of 16.5 years replicates the 14CO2 data. Equation (A.8) with an e-time of 4 years simulates the curve for 12CO2. Equation (B.1) calculates the Bern model predictions.
The Bern model begins with a short e-time then its e-time increases. The Bern line crosses the 14C line and thus conflicts with the 14C data. The Bern model artificially constrains the outflow of CO2 from the atmosphere.
The inability of the Bern model to replicate the 14CO2 data proves the IPCC models reviewed by Joos  do not represent reality.
The Bern model is also an invalid model because it depends upon its history. If restarted at any point,
it cannot replicate its own prediction line. A valid model must continue its same prediction line if it is restarted at any point on its line. The Bern model is invalid because the CO2 in the atmosphere does not “know” its history. It only knows its present. The Bern model is a curve fit and not a real model.
The Bern model violates physics. IPCC  assumes its Bern model applies to human but not to natural CO2. That assumption is unphysical because CO2 molecules from human and natural sources are identical. All valid models must treat human and natural CO2 the same.
When applied to natural CO2, Bern Equation (B.1) predicts 100 ppm per year of natural CO2 will cause 15 ppm per year to stay in the atmosphere forever. In 1000 years, that is 15,000 ppm stuck in the atmosphere forever. This clearly invalid prediction proves the Bern model is invalid. That means IPCC’s theory and climate models produce wrong predictions.
3.3 The isotope 14CO2 follows 12CO2
Levin  concludes the C14 data provide “an invaluable tracer to gain insight into the carbon cycle dynamics.” The 12CO2 molecules participate in the same chemical reactions as 14CO2 except 12CO2 reacts faster because it is lighter than 14CO2.
RealClimate  agrees:
All isotopes of an element behave in a similar way chemically.
Kohler  claims 14CO2 does not trace 12CO2 because 12CO2 is restrained by the decreased the ocean’s buffer capacity while 14CO2 is not.
As an isotope, 14CO2 will undergo the same chemical reactions as 12CO2, except slower. Both isotopes will seek their balance level according to their e-time. The 14CO2 data show e-time is constant. Therefore, buffer capacity is constant.
3.4 The 14C data support the Physics Model
Figure 6 shows how the predictions of the Physics and IPCC models differ.
Human fossil-fuel CO2 is “14C-free.” Inflow of human CO2 lowers the 14C balance level. IPCC  and Kohler  claim this qualitative argument proves human CO2 caused all the rise in atmospheric CO2. The numbers show otherwise.
As shown in Equation (4), the Physics Model predicts human CO2 is 4.5 percent and natural CO2 is 95.5 percent of the total. This are the same percentages as in the inflow. Therefore, the Physics Model predicts human CO2 has lowered the balance level of 14C from zero to -4.5. Appendix D shows the calculations.
Figure 7 shows the Physics Model replicates the 14C data when the balance level is -4.5.
IPCC  says natural CO2 is 68 percent and human CO2 is 32 percent. Therefore, the IPCC model predicts human CO2 has lowered the 14C balance level to -32.
Figure 8 shows how a balance level of -32 does not fit the 14C data.
The 14C data support the Physics Model and reject the IPCC model.
Pettersson  shows how industrial emissions of 14C may have raised the 14C balance level and how the 12CO2 increase would lower the D14C balance level. However, Levin  used absolute values of 14C and still concluded the “ocean-atmosphere disequilibrium today is close to pre-industrial times.”
3.5 The 13C data support the Physics Model
Human CO2 has slightly less 13C than natural CO2. RealClimate  says the 13C/12C ratio for human CO2 is 2 percent less than for natural CO2. RealClimate says the 13C ratio in the atmosphere has declined about 0.15 percent since 1850. RealClimate says this qualitative argument proves human CO2 caused all the increase in atmospheric CO2 since 1850. The numbers show otherwise.
Using the numbers in Figure 6, the Physics Model predicts human CO2 has lowered the 13C ratio by 0.09. The IPCC model predicts human CO2 has lowered the 13C ratio by 0.64. Appendix E shows the calculations.
Figure 9 compares the Physics Model and IPCC model predictions to RealClimate’s numbers.
The 13C/12C data support the Physics Model and reject the IPCC model.
4. Other problems with the IPCC Model
4.1 IPCC’s core argument is illogical
In its core argument, IPCC  correctly notes that human emissions from 1750 to 2013 totaled 185 ppm while atmospheric CO2 increased by only 117 ppm. But IPCC incorrectly concludes that this proves human CO2 caused the increase.
IPCC argument omits natural CO2 which totaled about 26,000 ppm during the same period. So, the stronger logical counter-argument is that nature caused all the increase. IPCC conclusion assumes natural CO2 inflow remained exactly constant since 1750, an invalid assumption.
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.
Of course, nature has been a “net carbon sink” since 1750 because nature absorbs human CO2 emissions. But absorption of human CO2 does not prevent nature from increasing its own CO2 emissions. Inflow and outflow are two different physical processes. Nature will absorb the outflow of both human and natural CO2 because outflow is a function of level, independent of nature’s inflow.
Cawley  is a key paper for the IPCC theory. Cawley claims to prove that human CO2 caused all the increase of atmospheric CO2 above the 280 ppm in 1750. Cawley’s proof fails.
Cawley’s Equation (3) attempts to do the same job as Equation (A.2), namely, to represent how level sets outflow. But Cawley adds to his Equation (3) a term that represents a steady-state outflow that is independent of level. Cawley’s added term is fictitious because his first term on the right side of his Equation (3) is the true source of all outflow.
So, Cawley  adds outflow twice. First as a level-driven outflow. Second, as a fictitious steady-state outflow. As a result, Cawley’s Equations (3), (4), (5), and his equation after (5) are wrong, which makes his whole paper wrong.
Cawley’s Equation (7) should include his Fa for human inflow. His Equations (7) and (8) should omit his arbitrary Fe for outflow and set outflow equal to level (his C) divided by his residence time. His residence time is also inaccurate.
Cawley  argues the ratio of human to natural CO2 in the atmosphere is a function of residence-time, which is incorrect. The Physics Model Equation (3) and common sense show the ratio is independent of e-time. Cawley equations cannot replicate the 14C data.
Kohler  uses Cawley  to “prove” the IPCC case. But since Cawley fails physics, Kohler fails physics, as does the IPCC. The IPCC model is an attempt to derive a theory by pasting together observations without addressing the underlying physics.
4.2 IPCC’s time constants fail physics
The only hypothesis in the Physics Model is “outflow equals level divided by e-time” as shown in Equation (A.2). E-time is not a function of inflow.
The definition of e-time is precise. The Physics Model shows e-time is the time for the level L to move (1 – 1/e) of the distance from L to its balance level, Lb.
IPCC’s residence, adjustment, and turnover times have confusing definitions. IPCC  defines “turnover time (Tt)” as:
The ratio of the mass M of a reservoir (e.g., a gaseous compound in the atmosphere) and the total rate of removal S from the reservoir: Tt = M/S.
IPCC’s turnover time is not the same as e-time. Turnover time uses “total rate of removal” which is the negative difference between inflow and outflow.
IPCC  defines “adjustment time (Ta)” as:
The time-scale characterising the decay of an instantaneous pulse input into the reservoir.
Cawley  defines “adjustment time (Ta)” as:
The time taken for the atmospheric CO2 concentration to substantially recover towards its original concentration following a perturbation.
The word “substantially” indicates the definition is imprecise. IPCC’s fuzzy definition of adjustment time is necessary to allow for its fuzzy definition of residence time.
Cawley  follows IPCC  to define “residence time (Tr)” as:
The average length of time a molecule of CO2 remains in the atmosphere before being taken up by the oceans or terrestrial biosphere.
IPCC uses a different time constant depending upon whether the level is far from its balance level or close to its balance level:
- When the level is far from its balance level (which can be zero), IPCC thinks e-time is an adjustment time because the level is moving rapidly toward its balance level.
- When the level is close to its balance level, IPCC thinks e-time is a residence time because “molecules” are flowing in and out with little change in level.
IPCC’s time definitions do not properly model how CO2 flows through the atmosphere.
IPCC requires a decay to originate from a pulse. This is unphysical because a system does not know its history. Decay time depends only upon outflow which depends upon level. IPCC assumes decay time depends upon inflow.
Figure 10 illustrates how e-time relates to IPCC’s adjustment and residence times.
IPCC  claims:
In simple cases, where the global removal of the compound is directly proportional to the total mass of the reservoir, the adjustment time equals the turnover time: Ta = Tt.
The Physics Model’s replication of the 14C data shows the 14CO2 outflow is proportional to level. Therefore, by IPCC’s own definition, adjustment time equals residence time.
IPCC  says:
In more complicated cases, where several reservoirs are involved or where the removal is not proportional to the total mass, the equality T = Ta no longer holds.
Carbon dioxide is an extreme example. Its turnover time is only about 4 years because of the rapid exchange between atmosphere and the ocean and terrestrial biota.
Although an approximate value of 100 years may be given for the adjustment time of CO2 in the atmosphere, the actual adjustment is faster initially and slower later on.
IPCC concludes 12CO2 residence time is about 4 years but claims its adjustment time is much longer. IPCC claims adjustment time is “fast initially and slower later on.” IPCC’s confusion about residence time and adjustment time is why its Bern model cannot replicate the 14C data in Figure 4.
The 14C data (Figures 2 and 3) is the upper bound for CO2 e-time. The e-time for 14CO2 is 16.5 years, not hundreds of years.
Kohler  claims:
The IPCC summarizes the state of the art in peer-reviewed literature. Hence neither the residence time nor the adjustment time are assumptions or interpretations of the IPCC-AR5, but robust outcomes of the underlying science.
Kohler attempts to argue by authority. The implication of “Hence” is that IPCC summaries are so perfect that no one may disagree. That view violates the scientific method.
The IPCC theory makes wrong predictions. It contradicts physics. Its so-called “state of the art in peer-reviewed literature” is a repetition of inbred, invalid, protected claims.
4.3 IPCC’s buffer theory is invalid
IPCC theory says human but not natural emissions, reduce the “buffer capacity” of the carbonate system. There are three things wrong with this IPCC claim:
- It requires nature to treat human and natural CO2 differently, which is impossible.
- It assumes the much larger natural CO2 outflow does not reduce buffer capacity.
- It conflicts with the 14C data which show e-time is constant, which means buffer capacity has not changed.
IPCC  claims the reason the Bern model increases its residence time is because:
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.
IPCC’s claim requires the outflow of CO2 from the atmosphere to decrease as the level of CO2 in the atmosphere increases. That is the opposite of the Physics Model’s theory. The 14CO2 data prove the Physics Model theory is correct and the IPCC model theory is incorrect.
Kohler  claim human emissions reduced the “buffer capacity” of the carbonate system:
the rise in atmospheric and oceanic carbon content goes along with an increase in the Revelle factor, a phenomenon which is already measurable. This implies that the oceanic uptake of anthropogenic carbon will become slower if we continue to increase anthropogenic CO2 emissions. This is already seen in all CHIMP5 model simulations.
Kohler’s last sentence illustrates the absence of logic by Kohler and the IPCC. They claim a model proves what has been fed into the model.
The 14C data are the most accurate method to measure changes in the Revelle factor and “buffer capacity.” But the 14C data (Figures 2 and 3) prove e-time has been constant since 1970. Therefore, human CO2 has not changed the buffer capacity.
4.4 Human CO2 does not correlate
IPCC  claims annual human CO2 emissions cause annual increases in the level of CO2 in the atmosphere. Cawley  argues,
Lastly, the rise in atmospheric carbon dioxide closely parallels the rise in anthropogenic emissions, leading to an approximately constant airborne fraction, which would be somewhat of a coincidence if the rise were essentially natural in origin!
However, proper statistics requires a detrended analysis of a time series before concluding cause and effect. Munshi  shows the “detrended correlation of annual emissions with annual changes in atmospheric CO2” is zero. Where there is no correlation, there is no cause and effect.
In summary, statistics show human CO2 is insignificant to the increase in atmospheric CO2.
The exact replication of the 14C data by the Physics Model proves atmospheric CO2 flows out of the atmosphere in proportion to the level or concentration of CO2 in the atmosphere with a constant e-time. This fact overrides all the IPCC papers that claim outflow is not proportional to level and e-time increases with time. The consequences of this simple fact are significant.
The ratio of human to natural CO2 in the atmosphere equals the ratio of their inflows into the atmosphere, independent of e-time. Inflow sets the balance level. The level moves toward its balance level with an e-time. When the level equals its balance level, outflow equals inflow. Then, the level remains constant so long as inflow remains constant.
There are no cumulative carbon emissions. All CO2 emissions merely raise the level of CO2 in the atmosphere until outflow equals inflow. No CO2 accumulates in the atmosphere and all CO2 flows out of the atmosphere with one common e-time. Human-produced and nature-produced CO2 behave the same.
Present human CO2 inflow into the atmosphere raises the balance level by 18 ppm and present natural CO2 inflow raises the balance level by 392 ppm, for a total of 410 ppm. This conclusion is independent of e-time. Levels approach their balance levels according to the e-time.
If all human CO2 emissions stopped and natural inflow stayed constant, the CO2 level would fall to the natural balance level of 392 ppm with an e-time of about 4 years. Human CO2 does not increase atmospheric CO2 enough to change climate.
The IPCC model is pseudoscience. It cannot replicate the 14C data, it uses multiple, unphysical response times, its core arguments fail logic, and it treats human CO2 differently than it treats natural CO2.
The author thanks Chuck Wiese, Laurence Gould, Tom Sheahen, and Charles Camenzuli, who reviewed this paper and provided scientific critique, and Daniel Nebert, Gordon Danielson, and Valerie Berry, who provided language and grammar improvements. This research project was funded by the personal funds of Valerie and Edwin Berry.
Appendix A: Physics model math
We use the system definition of Section 3.1 to derive the physics model. We begin with the continuity equation:
dL/dt = Inflow – Outflow (A.1)
L = CO2 level (concentration in ppm)
t = time (years)
dL/dt = the rate of change of L (ppm/year)
Inflow = the rate CO2 moves into the system (ppm/year)
Outflow = the rate CO2 moves out of the system (ppm/year)
Assume outflow is proportional to level,
Outflow = L / Te (A.2)
where Te is the “e-folding time” or simply “e-time.”
Substitute Eq. (A.2) into the continuity Eq. (A.1) to get
dL/dt = Inflow – L / Te (A.3)
Define the balance level, Lb, as
Lb = Inflow * Te (A.4)
Equation (A.4) shows how Inflow sets the balance level. Substitute Equation (A.4) for Inflow into Equation (A.3) to get,
dL/dt = – (L – Lb) / Te (A.5)
Equation (A.5) shows how level always moves toward its balance level. At this point, both L and Lb are functions of time. Te can also be a function of time.
In the special case when Lb and Te are constant, there is an analytic solution to Equation (A.5). Rearrange Equation (A.5) to get
dL / (L – Lb) = – dt / Te (A.6)
Then integrate Eq. (A.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 (A.7)
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 = time for L to move (1 – 1/e) of the distance to Lb
e = 2.7183
(The original integration of Eq. (A.6) contains two absolute functions, but they cancel each other because both L and Lo are always either above or below Lb.)
Raise e to the power of each side of Eq. (A.7), to get the level as a function of time:
L(t) = Lb + (Lo – Lb) exp(- t / Te) (A.8)
Equation (A.8) is the analytic solution of Eq. (A.5).
The only assumption in the physics model is Eq. (A.2), namely, outflow equals level divided by residence time. All equations after Eq. (A.2) are deductions from this assumption.
Appendix B: 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 CO2 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)] (B.1)
t = time in years
Lo = the level of atmospheric CO2 due to inflow in year t = 0
L(t) = the level of atmospheric CO2 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. (B.1):
A0 + A1 + A2 + A3 = 1.000
Set t equal to infinity. Then Eq. (B.1) becomes,
L = Ao Lo = 0.152 Lo (B.2)
Equation (B.2) predicts a one-year inflow that sets Lo to 100 ppm, followed by zero inflow forever, will cause a permanent level of 15 ppm.
Appendix C: How temperature increases CO2
It is outside the scope of this paper to show how the balance level of CO2 changes with surface temperature. Here is reference information.
Harde (2017a) showed how both inflow and outflow depend on surface temperature, and how this causes the balance level to be a non-linear function of surface temperature. Harde used paleoclimate data as well as modern instrumental data to show how the natural balance level of CO2 in the atmosphere depends on surface temperature.
Kohler (2017) criticize Harde’s method. However, Harde (2017b) proves Kohler is wrong. Unfortunately, the journal did not publish the Harde (2017b) reply to Kohler.
Fig. C1 shows a plot using Harde’s Eq. (17).
Appendix D: How the models fit the 14C data
Table D.1. Row 1 shows the natural and human 14C ratios in units of D14C. Row 2 and Row 4 show the physics and IPCC natural and human fractions. Row 3 is the product of Row 1 and Row 2. Row 5 is the product of Row 1 and Row 4.
|1||14C Ratio||0||-100||Figs. 2 & 3|
Appendix E: How the models fit the 13C data
Table E.1. Row 1 shows the natural and human 13C ratios. Row 2 and Row 4 show the physics and IPCC natural and human fractions. Row 3 is the product of Row 1 and Row 2. Row 5 is the product of Row 1 and Row 4.
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