by Edwin X Berry, Ph.D., Physics
October 11, 2019: I posted the first draft of this paper for your comments.
November 7, 2019: I updated this post. Thank you all for your comments.
January 23, 2020: I am updating this preprint to correct an error I made in interpreting the IPCC data. The correction will not affect the conclusions of this preprint but it may reduce the human effect below the present 31 ppm. It will take me about a week to compute and post the new results.
Copyright (c) 2019 by Edwin X Berry
The scientific basis for the effect of human carbon dioxide on atmospheric carbon dioxide rests upon correctly calculating the human carbon cycle. This paper uses the United Nations Intergovernmental Panel on Climate Change (IPCC) carbon-cycle data and allows IPCC’s assumption that the CO2 level in 1750 was 280 ppm. It derives a framework to calculate carbon cycles. It makes minor corrections to IPCC’s time constants for the natural carbon cycle to make IPCC’s flows consistent with its levels. It shows IPCC’s human carbon cycle contains significant, obvious errors. It uses IPCC’s time constants for natural carbon to recalculate the human carbon cycle. The human and natural time constants must be the same because nature must treat human and natural carbon the same. The results show human emissions have added a negligible one percent to the carbon in the carbon cycle while nature has added 3 percent, likely due to natural warming since the Little Ice Age. Human emissions through 2019 have added only 31 ppm to atmospheric CO2 while nature has added 100 ppm. If human emissions were stopped in 2020, then by 2100 only 8 ppm of human CO2 would remain in the atmosphere.
Keywords: carbon dioxide, CO2, climate change, carbon cycle; climate politics; global warming
1.1 The problem
The United Nations Intergovernmental Panel on Climate Change (IPCC)  incorrectly claims,
With a very high level of confidence, the increase in CO2 emissions from fossil fuel burning and those arising from land use change are the dominant cause of the observed increase in atmospheric CO2 concentration.
The primary source of the increased atmospheric concentration of carbon dioxide since the pre-industrial period results from fossil fuel use.
The United Nations World Meteorological Organization (WMO) Global Carbon Project  incorrectly claims,
With solid justification, one can describe the annual carbon budgets as products of high scientific quality with strong political relevance.
The problem is to calculate the total effect of all human CO2 emitted since 1750 through 2019 on the carbon cycle and atmospheric CO2.
Authors who conclude that human CO2 increases atmospheric CO2 by only a small amount include Revelle and Suess , Starr , Segalstad , Jaworoski [6, 7], Beck , Rorsch, Courtney, and Thoenes , Courtney , Quirk , Essenhigh , Glassman , Salby [14-17], Humlum , Harde [19, 20], Berry [21-23], and Munshi [24-28].
Authors who support the IPCC  position – that human CO2 has caused all the increase in atmospheric CO2 above about 280 ppm – include Archer et al. , Cawley , Kohler , and their many references.
Courtney  (pp. 6-7) concluded in 2008,
“… the relatively large increase of CO2concentration in the atmosphere in the twentieth century (some 30%) is likely to have been caused by the increased mean temperature that preceded it. The main cause may be desorption from the oceans. … Assessment of this conclusion requires a quantitative model of the carbon cycle, but – as previously explained – such a model cannot be constructed because the rate constants are not known for mechanisms operating in the carbon cycle.”
IPCC  used different rules to calculate the human carbon cycle than it used to calculate the natural carbon cycle. The correct calculation of the human carbon cycle must use the same rules as the natural carbon cycle because nature cannot distinguish between human and natural carbon atoms.
1.2 The solution
Courtney  commented in his review of this paper’s Preprint, that this paper,
“quantifies the anthropogenic and natural contributions to changes in atmospheric CO2 concentration without need for knowledge of rate constants for individual mechanisms. This is a breakthrough in understanding which (other authors) including myself all failed to make.”
This paper uses the Physics model  with IPCC  data to determine the “rate constants” or “e-times” for IPCC’s natural carbon cycle. Then, this paper uses the e-times for IPCC’s natural carbon cycle to calculate the human carbon cycle. IPCC did not do this which is why IPCC got the wrong answer for the human carbon cycle. And the wrong answer has led to the incorrect public perception of the influence of human carbon emissions.
The correct calculation, described herein, shows that all human carbon emissions through 2019 have increased atmospheric CO2 by only 31 ppm (parts per millions by volume).
This paper converts carbon units of GtC (Gigatons of Carbon) and PgC (Petagrams of Carbon) into CO2 units of ppm (parts per million by volume in dry air) using:
1 ppm = 2.12 GtC = 2.12 PgC
These new carbon-cycle calculations are not complicated. Anyone competent in basic physics and in simple numerical calculations should be able to reproduce the results shown in this paper.
(Some readers may wish to read the summary in Section 5.3 first.)
2. The Physics Model
2.1 Physics Model description
There are four key carbon reservoirs: land, atmosphere, surface ocean, and deep ocean. The Physics model (Berry, 2019b) applies independently to each carbon reservoir. The “level” of each reservoir is the mass of carbon in each reservoir.
Each reservoir has an e-time defined as the time for the level to move (1 – 1/e) of the distance from its present level to its balance level. The balance level is defined below.
Figure 1 shows the Physics model system for carbon in a reservoir. The carbon in the atmosphere is in the form of CO2.
The Physics model shows how inflow, outflow, and e-time control the level of carbon in each reservoir.
The only way external processes can change a reservoir’s level is by changing the reservoir’s inflow, outflow, or e-time. Therefore, the Physics model INCLUDES ALL EFFECTS OF EXTERNAL PROCESSES (chemical, biological, etc.) on the level of carbon in a reservoir.
The Physics model rides above chemical processes. Chemical processes can change reservoir levels only by changing inflow, outflow, or e-time, which the Physics model includes.
2.2 Physics Model derivation
The calculation of the carbon cycle requires a theoretical base. The Physics model (Berry, 2019b) provides the base that is reviewed here.
A system describes a subset of nature. A system includes levels and flows between levels. Levels set flows and flows set new levels. The mathematics used in the Physics model are analogous to the mathematics used to describe many engineering systems.
Following , the Physics model derivation begins with the continuity equation (1) which says the rate of change of level is the difference between inflow and outflow:
dL/dt = Inflow – Outflow (1)
- L = CO2 level (concentration in ppm)
- t = time (years)
- dL/dt = rate of change of L (ppm/year)
- Inflow = rate CO2 moves into the system (ppm/year)
- Outflow = rate CO2 moves out of the system (ppm/year)
The Physics model has only one hypothesis, which is outflow is proportional to level:
Outflow = L / Te (2)
where Te is the “e-folding time” or simply “e-time.” E-time is the time for the level to move (1 – 1/e) of the distance from its present level to its balance level.
Substitute (2) into (1) to get,
dL/dt = Inflow – L / Te (3)
When dL/dt is zero, the level will be at its balance level. Define the balance level, Lb, as
Lb = Inflow * Te (4)
Substitute (4) for Inflow into (3) to get,
dL/dt = – (L – Lb) / Te (5)
Equation (5) shows the level always moves toward its balance level. 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, which means Inflow is constant, there is an analytic solution to (5). Rearrange (5) to get
dL / (L – Lb) = – dt / Te (6)
Then integrate (6) from Lo to L on the left side, and from 0 to t on the right side to get
Ln [(L – Lb) / (Lo – Lb)] = – t / Te (7)
- 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) from L to Lb
- e = 2.7183
The original integration of (6) contains two absolute values, 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 (7), to get the level as a function of time:
L(t) = Lb + (Lo – Lb) exp(– t/Te) (8)
Equation (8) is the analytic solution of (5) when Lb and Te are constant.
All equations after (2) are deductions from hypothesis (2) and the continuity equation (1).
2.3 Physics Model properties
Hypothesis (2) is a linear function of level. This means the Physics model applies independently and in total to human and natural carbon. The balance levels of human and natural carbon are independent.
The Physics model also applies independently and in total to all definitions of carbon or CO2. For example, it applies independently to human CO2, natural CO2, and their sums, and to 12CO2, 13CO2, and 14CO2, and their sums.
However, if outflow were a “strictly increasing function” of level other than level to the power of one, then the Physics model would not apply independently and in total to human CO2 and natural CO2.
Because of (2), it is not necessary (or desirable) to compute the carbon cycle for human and natural carbon simultaneously. It is better (and simpler) to compute their effects separately. Just ADD ANOTHER INSTANCE of the Physics model for each carbon definition. The separate results can be summed to produce the total result.
Equation (4) shows how inflow sets a balance level. Equation (5) shows how the level moves toward the balance level with a speed determined by e-time. When the level equals the balance level, outflow will equal inflow. At the balance level, continuing constant inflow will maintain a constant level of carbon in the reservoir.
Equation (4) shows CO2 does not accumulate in the atmosphere. If inflow decreases, the balance level decreases, and the level follows the balance level. The response is immediate. When inflow to a reservoir increases the level of the reservoir, that reservoir immediately increases its outflow.
2.4 Physics Model verification
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, called 14CO2.
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.
Hua  processed 14C data for both hemispheres from 1954 to 2010. Turnbull  processed 14C data for Wellington, New Zealand, from 1954 to 2014. The 14C data from both sources are virtually identical after 1970. After 1970, 14CO2 molecules were well mixed between the hemispheres and 14CO2 in the stratosphere moved to the troposphere.
The 14C data are in units of D14C per mil [33, 34]. The lower bound in D14C units is -1000 which corresponds to zero 14C in the atmosphere. The “natural” balance level, defined by the average measured level before 1950, is zero.
A carbon atom has three isotopes, 12C, 13C, and 14C. Isotopes have the same number of protons and electrons but different numbers of neutrons. Isotopes undergo the same chemical reactions but the rates that isotopes react can differ.
Lighter isotopes form weaker chemical bonds and react faster than heavier isotopes (Wikipedia, 2019). Because 12CO2 is a lighter molecule than 14CO2, it reacts faster than 14CO2. Therefore, the 12CO2 e-time will be shorter than the 14CO2 e-time.
Levin et al.  conclude the 14C data provide “an invaluable tracer to gain insight into the carbon cycle dynamics.” The 14C data trace how CO2 flows out of the atmosphere. All valid models of atmospheric CO2 must replicate the 14C data.
The Physics Model, (5) and (8), accurately replicates the 14CO2 data from 1970 to 2014 with e-time set to 16.5 years, balance level set to zero, and starting level set to the D14C level in 1970.
Figure 2 shows how the Physics Model replicates the 14C data.
The Physics model is not a curve fit equation. The Physics model uses hypothesis (2) and allows only 2 parameters to be adjusted: balance level and e-time. Both are physical parameters.
The replication of the 14C data by the Physics Model has significant consequences. It shows hypothesis (2) is correct. It shows the 14C natural balance level has remained close to zero and e-time has remained constant since 1970. If the e-time had changed since 1970, it would have required a variable e-time to make the Physics Model fit the data
The Physics model’s replication of the 14C data may be the most elegant and important fit of a hypothesis to data in climate change literature.
2.5 Physics Carbon-Cycle Formulation
“The formulation of a problem is often more essential than its solution…” – Albert Einstein
There can be no solution for the carbon cycle until there has been a formulation of the problem. IPCC does not provide a formulation to calculate the carbon cycle. The following may be the first time the fundamental equations for the carbon cycle have been derived and presented.
This paper uses IPCC  numbers for the carbon cycle. These numbers will change as new data becomes available. This formulation will likely endure. It provides an easy way to update these carbon cycle calculations as new data become available.
The carbon-cycle question for climate change is:
HOW MUCH does human CO2 increase atmospheric CO2 after we account for the recycling of human carbon from the land and ocean back into the atmosphere?
There are two different ways to view the carbon-cycle system. Figure 3 shows individual outflows where the arrows are all positive numbers.
Figure 4 shows net flows where the arrows can be positive or negative numbers.
The IPCC model uses individual flows. The Physics model uses net flows because they simplify the following derivations. .
Define the Levels:
- Lg = level of carbon in the land
- La = level of carbon in the atmosphere
- Ls = level of carbon in the surface ocean
- Ld = level of carbon in the deep ocean
Define flow e-times:
- Tga = e-time for flow from land to atmosphere
- Tag = e-time for flow from atmosphere to land
- Tas = e-time for carbon to go from atmosphere to surface ocean
- Tsa = e-time for flow from surface ocean to atmosphere
- Tsd = e-time for flow from surface ocean to deep ocean
- Tds = e-time for flow from deep ocean to surface ocean
Define reservoir e-times:
- Ta = e-time for flow from atmosphere to land and surface ocean
- Ts = e-time for flow from surface ocean to atmosphere and deep ocean
Notice these relationships:
1/Ta = 1/Tag + 1/Tas (9)
1/Ts = 1/Tsa + 1/Tsd (10)
Define other variables:
- t = time in years
- Hin = Inflow of human carbon
The Physics model (2) defines the net flows in Figure 4:
Fga = Lg/Tga – La/Tag (11)
Fas = La/Tas – Ls/Tsa (12)
Fsd = Ls/Tsd – Ld/Tds (13)
The rate equations for these flows are:
dLa/dt = Fga – Fas + Hin (15)
dLs/dt = Fas – Fsd (16)
dLd/dt = Fsd (17)
Now, insert the flows (11-13) into the rate equations (14-17) to get the Physics rate equations:
dLa/dt = Ls/Tsa + Lg/Tga – La/Tag – La/Tas + Hin (19)
dLs/dt = La/Tas + Ld/Tds – Ls/Tsa – Ls/Tsd (20)
dLd/dt = Ls/Tsd – Ld/Tds (21)
Rather than use different e-times, the IPCC model specifies the “splits” to each connected reservoir.
- Kag = fraction of carbon flow from atmosphere to land = 0.64
- Kas = fraction of carbon flow from atmosphere to surface ocean = 0.36
- Ksa = fraction of carbon flow from surface ocean to atmosphere = 0.4
- Ksd = fraction of carbon flow from surface ocean to deep ocean = 0.6
Kag + Kas = 1 (22)
Ksa + Ksd = 1 (23)
IPCC’s splits are related to the Physics e-times as follows:
Kag = Ta / Tag (24)
Kas = Ta / Tas (25)
Ksa = Ts / Tsa (26)
Ksd = Ts / Tsd (27)
Substitute (24-27) into (18-21) and use (22-23) to get IPCC’s rate equations:
dLg/dt = Kag*La/Ta – Lg/Tg (28)
dLa/dt = Ksa*Ls/Ts + Lg/Tg – La/Ta + Hin (29)
dLs/dt = Kas*La/Ta + Ld/Td – Ls/Ts (30)
dLd/dt = Ksd*Ls/Ts – Ld/Td (31)
With the above formalities, we tested and found that IPCC’s splits different from 0.5 do not give significantly different results than IPCC’s splits. Using this simplification, (24-27) become:
Tag = 2 Ta (32)
Tas = 2 Ta (33)
Tsa = 2 Ts (34)
Tsd = 2 Ts (35)
Tga = Tg (36)
Tds = Td (37)
Equations (32-37) simplify the Physics rate equations (18-21) to:
dLg/dt = La/2Ta – Lg/Tg (38)
dLa/dt = Ls/2Ts + Lg/Tg – La/Ta + Hin (39)
dLs/dt = La/2Ta + Ld/Td – Ls/Ts (40)
dLd/dt = Ls/2Ts – Ld/Td (41)
This paper uses both the IPCC rate equations (28-31) and the Physics simplified rate equations (38-41). The Physics and IPCC results are identical when IPCC splits are set to 0.5. The Excel file used for the calculations is included in Supplemental Materials.
3. The Natural Carbon Cycle
3.1 IPCC natural carbon cycle
IPCC assumes (a) the level of atmospheric CO2 in 1750 was 280 ppm and (b) human emissions caused all the increase in atmospheric CO2 above 280 ppm.
Regarding (a), Segalstad  and Jaworowski [6, 7] present evidence that the CO2 level before 1750 was much higher than 280 ppm. Nevertheless, this paper uses assumption (a) to make the Physics carbon-cycle calculations consistent with IPCC’s natural and human carbon cycles. This paper proves assumption (b) is not compatible with IPCC data.
This paper uses IPCC  definitions for natural carbon and human carbon. Human carbon is the result of human emissions. All carbon inflow that does not result from human emissions is defined as natural carbon.
IPCC calculations of the human carbon cycle attempt to “tag” every human-produced carbon atom and follow them through the whole calculation. The Physics model shows why it is best to calculate the natural and human carbon cycles independently. After the independent calculations, the human and natural carbon-cycle results can be summed to get the total result. Independent calculations automatically keep track of human carbon as it flows through the carbon cycle.
IPCC  missed this important simplification and calculates human and natural effects together. As a result, IPCC made obvious and significant errors in its calculations.
IPCC  Fig. 6.1 shows IPCC’s version of the carbon cycle. Its legend says,
Black numbers and arrows indicate reservoir mass and exchange fluxes estimated for the time prior to the Industrial Era, about 1750.
Figure 5 shows the IPCC Figure 6.1 carbon cycle values for natural carbon.
IPCC’s marine biota level of 3 PgC is negligible because it is 0.3 percent of IPCC’s surface ocean level of 900. IPCC’s dissolved organic carbon level of 700 PgC is negligible because it is 1.9 percent of IPCC’s deep ocean level. This paper adds IPCC’s carbon flow through marine biota of 11 PgC per year to IPCC’s flow from surface ocean to deep ocean of 90 PgC per year to get 101 PgC per year.
IPCC’s levels and flows produce these e-times, using (2), for the natural carbon cycle:
- Tg = 2300 / 107 = 21.5 years
- Ta = 590 / 170 = 3.5 years
- Ts = 900 / 161 = 5.6 years
- Td = 37100 / 100 = 371 years
Table 1 shows selected years of the Physics carbon-cycle calculation for IPCC’s natural carbon levels for 1750. The Physics model shows IPCC’s flows do not maintain IPCC’s constant levels.
- Table 1. IPCC’s e-times and splits increase the level of atmospheric CO2 to 302 ppm rather than keep IPCC’s claimed 280 ppm. Values for levels are in PgC except for the ppm column.
IPCC’s natural flows support a natural level of CO2 in the atmosphere of about 302 ppm rather than IPCC’s claimed 280 ppm after 1750. The difference is not significant, but it shows it is possible to correct IPCC’s natural carbon cycle calculations.
3.2 Corrected IPCC natural carbon cycle model
To correct the IPCC data to be internally consistent, we use IPCC’s natural carbon levels and find e-times, and therefore flows, that make the levels constant over time.
Table 2 shows corrected e-times for IPCC splits. These e-times produce flows that maintain the atmosphere level at 280 ppm, as IPCC claims, and other levels constant.
- Table 2. Corrected e-times for IPCC splits that maintain IPCC’s levels for IPCC’s natural carbon cycle. Values for levels are in PgC except for the ppm.
Table 3 shows the corrected e-times for 0.5 splits. These e-times maintain the atmosphere level at 280 ppm and other levels constant.
- Table 3. Corrected e-times for 0.5 splits to maintain IPCC’s levels for IPCC’s natural carbon cycle. Values for levels are in PgC except for the ppm.
The flows in Tables 2 are unequal flows. The flows in Table 3 are equal.
The End % values are the same in Table 3 and Table 2. So, the End % values are independent of IPCC’s splits. These End % values represent the long-term equilibrium percentages for the natural carbon cycle.
Figure 6 shows the IPCC natural carbon cycle corrected as in Table 3.
The Physics model found e-times that properly model the IPCC natural carbon cycle. The corrected e-times slightly changed the End % in the reservoirs. IPCC’s splits produce the same result as 50-50 splits.
4. The Human Carbon Cycle
4.1 IPCC’s invalid human carbon cycle
Because human carbon atoms are identical to nature’s carbon atoms, nature will treat human carbon the same as it treats natural carbon. This is an extension of the Equivalence Principle that Einstein used to derive his theory of relativity.
According to this extended Equivalence Principle, the human carbon cycle must have the same e-times as the natural carbon cycle. Also, the human carbon long-term percentages will equal the natural long-term percentages.
IPCC  AR5 Fig. 6.1 shows IPCC’s version of the carbon cycle. Its legend says,
Red arrows and numbers indicate annual ‘anthropogenic’ fluxes averaged over the 2000–2009 time period. These fluxes are a perturbation of the carbon cycle during Industrial Era post 1750.
Figure 7 shows IPCC’s Figure 6.1 data for the human carbon cycle.
IPCC shows 9 PgC per year (from fossil fuels, cement production, and land use change) flows into the atmosphere. IPCC shows a net 2.6 PgC per year flows from atmosphere to land, and a net 2.3 PgC per year flows from atmosphere to surface ocean. The leaves 4 PgC per year added to the atmosphere.
There are five obvious errors in IPCC’s human carbon cycle:
- The surface ocean level remains at 0 PgC, unaffected by the net 2.3 PgC inflow. That cannot happen because outflow is proportional to level, which means a level cannot go to zero so long as there is an inflow.
- The surface ocean, with zero outflow to the deep ocean, magically adds 155 PgC to the deep ocean. That cannot happen because no level can increase if its inflow is zero. And once the deep ocean level is greater than zero, carbon will flow back to the surface ocean.
- The net flow of 2.6 PgC per year from atmosphere to land does not add carbon to the land as it should. Rather it sucks carbon out of the land. This makes the land level decrease from 0 to -30 PgC. A negative level is impossible when there is only positive human carbon to fill the reservoirs. It is like having a glass filled with negative water.
- The carbon level in the atmosphere is 66%. That just happens to be the level IPCC needs to justify its assumption that human carbon caused ALL the increase in atmospheric CO2 above 280 ppm. IPCC used circular reasoning, not science, to achieve its desired result.
- IPCC’s human carbon level (Figure 7) for the atmosphere is 66% while its natural carbon level is 1.5% (Figure 6). This very significant difference shows IPCC treats human carbon differently than it treats natural carbon.
IPCC says human carbon is a “perturbation” on the natural carbon cycle. That is not the correct way to model the effect of human carbon on the carbon cycle. The correct way is to model the human carbon cycle independently as this paper does.
4.2 Human carbon added to the carbon cycle
This paper calculates the human carbon cycle independently from the natural carbon cycle. That eliminates the need to keep separate track of carbon from human and natural sources. These independent calculations add up to produce the same result as calculating human and natural carbon together, which is much more complicated.
Boden et al.  provides human CO2 emissions data from 1750 to 2014. The calculations in this paper add estimates of human emissions from 2014 through 2019.
Initially, the human carbon level in all reservoirs is zero. Then the calculations insert annual human carbon emissions into the atmosphere from 1750 through 2019. Each year, the calculations allow carbon to flow from the atmosphere to land and surface ocean, and from surface ocean to deep ocean.
All levels must have non-negative numbers. The addition of human carbon cannot remove natural carbon from any reservoir. The surface ocean level will rise before it can flow carbon to the deep ocean.
Table 4 shows the result of these calculations. All human emissions since 1750 have added 452 PgC of carbon to the natural carbon cycle. This human-carbon addition is about one percent of natural carbon. Human carbon has increased the 2020 level of atmospheric CO2 by 31 ppm.
- Table 4. The carbon-cycle model for IPCC splits shows all human CO2 emissions from 1750 to January 1, 2020, increase atmospheric CO2 by 35 ppm. The calculation sets inflow to zero on January 2, 2020, to see how fast human CO2 exits the atmosphere.
Table 4 shows, at the beginning of 2020, only 14.7% of all human carbon remains in the atmosphere, 36.7% is in the land, 10.7% is in the surface ocean, and 37.9% is in the deep ocean.
If human emissions were to stop in 2020, then by 2100, 3.9% of human carbon would remain in the atmosphere, 19.3% would be in the land, 3.9% would be in the surface ocean, and 72.8% would be in the deep ocean.
Figure 8 shows the calculated increase in atmospheric CO2 caused by human emissions through 2019 and how this would decay if all human CO2 emissions were stopped in 2020.
Figure 9 shows the combined effects of human and natural CO2 on the level of atmospheric CO2.
Figure 10 shows how the reservoir levels change with time. Most human carbon finds its way to the deep ocean just as natural carbon finds its way to the deep ocean. The smallest amount ends up in the atmosphere.
The fall of human carbon in the atmosphere after 2020, when the calculation stops human emissions, shows human carbon has little long-term effect.
4.3 Human carbon for constant emissions
Rather than set human CO2 inflow to zero in 2020, this section sets human inflow to its 2019 value from 2020 to 2100.
Table 5 shows the calculated values using the Physics carbon-cycle model.
- Table 5. Results of Physics carbon-cycle model when human emissions are held constant beginning in 2020.
Figure 11 shows the effect of continued constant human CO2 emissions after 2019. The human-caused increase is still much smaller than the increase caused by natural emissions.
Figure 11 shows the continuation of constant human emissions after 2020 would cause a total increase in atmospheric CO2 of 52 ppm by 2100.
4.4 Pulse decay: Physics versus IPCC Bern
The IPCC Bern model, Joos , represents IPCC’s claim that human carbon sticks in the atmosphere much longer than natural carbon. Berry  shows how to deconstruct  to get an equation to represent the results of one pulse of human CO2.
The Physics carbon-cycle model uses IPCC data for the natural carbon-cycle.
Figure 12 shows how the Physics carbon-cycle model and the IPCC Bern model predict the decay of a 100-ppm pulse.
The Physics model shows the pulse decays to 15 ppm in 10 years and to 4 ppm in 100 years. By contrast, the IPCC Bern model predicts the pulse decays to 55 ppm in 10 years and to 30 ppm in 100 years. The incorrect Bern model says it is impossible for a pulse of human CO2 to ever decay below 15 percent.
Figure 13 shows how the carbon moves from the atmosphere to the other reservoirs.
Human carbon in the atmosphere moves rapidly to the land and the deep ocean because it flows between the reservoirs exactly like natural carbon flows. The IPCC human carbon cycle does not allow human carbon to flow like natural carbon.
Table 6 shows a summary of the pulse calculations.
- Table 6. Human carbon moves from atmosphere to land and deep ocean.
After 200 years, only 2.2% of the human pulse remains in the atmosphere and 85% is in the deep ocean. Initially, the carbon moved to the land but after 30 years, carbon from the land moved to the deep ocean.
The Bern model contradicts the IPCC  data. The IPCC Bern model is a curve fit to the calculations of IPCC’s climate models. Threfore, IPCC’s climate models do not represent the data that the IPCC puts into its own reports.
4.5 Physics carbon cycle entropy
One might ask,
Why does the carbon in a system flow to other reservoirs? What makes the system seek an equilibrium? What defines equilibrium?
In physics, entropy drives a system toward equilibrium. Left alone, the entropy of a system always increases. Equilibrium occurs when the entropy of a system is at its maximum value within the system’s constraints.
We might further ask,
What parameter of the system represents the entropy?
Equations (38-41) define the simplified Physics carbon-cycle system. Equilibrium occurs when the flows are zero. When the flows are zero, the levels are constant, and the L/Te are equal:
Lg/Tg = La/2Ta = Ls/2Ts = Ld/Td (42)
Equation (42) defines equilibrium. The sum of the L/Te’s are an inverse measure of the system’s entropy. The inverse of entropy is negentropy:
Negentropy = Lg/Tg + La/2Ta + Ls/2Ts + Ld/Td (43)
Think of negentropy as the ability to do work. Negentropy is maximum when all the carbon is in the reservoir with the smallest e-time. In year zero, all the carbon is in the atmosphere which is the reservoir with the smallest e-time. When the carbon flows to the other reservoirs, the negentropy decreases. Negentropy is at its minimum when there is no more flow which is when (42) is true.
Figure 14 illustrates the system in year zero when all the carbon is in the atmosphere. Carbon flow from A to G is defined as a negative flow for mathematical purposes.
An analogy is four water buckets connected by tubes. If all the water is in A then the system can do work, say, if turbines were in the tubes.
Figure 15 illustrates the system when the L/Te are distributed evenly between the reservoirs. At that point, the net flows between the reservoirs are zero. The entropy is maximum. If this were the analogy of four buckets, the system cannot do work.
Figure 16 shows how the L/Te levels decrease as carbon flows from the atmosphere to the other reservoirs. The total L/Te begins near 33 and decreases uniformly with time.
Table 7 shows how the L/Te values for each reservoir change with time. Some values go up but only to speed the decrease of the total L/Te that represents negentropy of the system.
- Table 7. The L/Te values as a function of time. The total always decreases.
The system seeks equilibrium because system entropy will increase as required by the Second Law of Thermodynamics.
4.6 The Principle of Least Action
The Principle of Least Action says a system will take the path from Start to Finish that requires the least “action.”
The formal definition of “action” is the time integral of the difference between kinetic energy and potential energy. OK, that is a bit heavy for non-physicists. So, let’s make it simpler.
Action is how something moves from state A to state B. Action is the path you take to get from your home to the grocery store. The quickest or least costly way to get there is the path of least action.
The top curve in Figure 16 represents the total negentropy of the system. It trends downward smoothly because the flows between the reservoirs find the fastest way to lower negentropy and move the system to equilibrium.
In Figure 16, carbon flows into the land and surface ocean in the first 10 years because that is the fastest path to reduce negentropy of the system.
The very definitions (11-13) of the flows in the Physics carbon-cycle model are in terms of entropy levels, not of carbon levels. This definition for the flows is a result of the hypothesis of the Physics model , namely,
Outflow = L / Te (2)
The Principle of Least Action says human carbon will flow from the atmosphere to the other reservoirs in a way that reduces the system negentropy the fastest. The calculation presented in this paper may not be the least action scenario, but it is likely close.
The calculations presented here assume IPCC’s estimated values for natural levels and approximate flows are accurate. Should better estimates become available these calculations can be quickly updated using the Excel file (Supplemental Materials).
5.1 Why the IPCC carbon-cycle models are wrong
Archer et al.  tests all IPCC carbon-cycle models and finds that all these models
“agree that 25-35% of the CO2 remains in the atmosphere after equilibrium with the ocean (2-20 centuries).”
However, the agreement among models does not means they are correct. ALL  models can be equally wrong. All  models use different rules for human carbon than for natural carbon.
Archer et al.  conclude,
“Some CO2 from the release would remain in the atmosphere thousands of years into the future, and the atmosphere lifetime calculated at that time would be thousands of years.”
The only difference in the human carbon cycle and the equilibrium natural carbon cycle is that human carbon adds new carbon to the carbon cycle. Human carbon still flows from reservoir to reservoir as described by the Physics model.
Contrary to , the decrease in atmospheric carbon dioxide has no long tail. What  calls a long tail is caused by the increase in the balance level of human carbon dioxide.
The  models assume natural carbon stays balanced while human carbon, which adds only 1% to nature, throws nature out of balance. That can’t happen because nature cannot tell the difference between human and natural carbon atoms.
There is no physics basis to assume what nature does is “good” and what humans do is “bad.” Maybe it is “good” to restore locked carbon to the atmosphere by burning carbon fuels.
There is no evidence that 1% more carbon in the carbon cycle changed the rules for the carbon cycle. If an added 1% did change the rules, it would change the rules for natural as well as for human carbon, and the effect would be very large. No such effect has been documented.
The Physics carbon-cycle model first simulates the natural carbon cycle using IPCC . Then the Physics carbon cycle model uses the rules for natural carbon to calculate the human carbon cycle. The  models have no physics model.
Table 4 shows the human carbon level in the atmosphere never gets to 25% of total human carbon. The calculations show only 15% of all human carbon is in the atmosphere by 2020. That is because human carbon flows to the other reservoirs fast enough to keep the human carbon in the atmosphere below 25%. If human emissions were to stop in 2020, then by 2100 only 4% of all human carbon would remain in the atmosphere. There is no significant long-term effect of human carbon emissions.
Figure 12 compares a simulated pulse of human carbon with the Bern model. The Bern model, which simulates the Archer et al. models rather than IPCC data, incorrectly claims 15% of human carbon will remain in the atmosphere forever. The simulated pulse of human carbon will decrease to 15% in 10 years. Table 6 shows only 4.2% will remain after 100 years and 2.2% will remain after 200 years.
5.2 The effect of temperature on CO2
Salby [14-17] shows how changes in surface temperature precede CO2 changes. Harde [19, 20] shows how atmospheric carbon dioxide increases with surface temperature.
There are only two ways to increase atmospheric CO2: (a) add new carbon to the carbon cycle or (b) increase the e-time of the atmosphere.
Tests using the Excel file (Supplemental Materials) show the only permanent way to increase atmospheric CO2 is to add new carbon to the carbon cycle. Also, the 14C data show the e-time for the atmosphere has not changed since 1970 .
This is a summary of the fundamental information presented in this paper. Four charts show the percent distribution of natural or human carbon among the four key carbon reservoirs. These reservoirs are land, atmosphere, surface ocean, and deep ocean, in that order because that is how they connect. Carbon-cycle data is from IPCC  Fig. 6.1.
Figure 17 shows IPCC’s distribution for natural carbon in 1750 when, according to IPCC, the level of natural CO2 in the atmosphere was 280 ppm. Only 1.45 percent is in the atmosphere and 90 percent is in the deep ocean.
The Figure 17 data represent the long-term equilibrium distribution for natural carbon. It is also the long-term equilibrium distribution for human carbon because natural and human carbon atoms are identical, so nature treats them the same.
Figure 18 shows IPCC’s human carbon distribution as of about 2013. An overwhelming 66 percent of human carbon is in the atmosphere.
Question: In Figure 18, how did IPCC get the 66 percent of human carbon in the atmosphere?
Answer: IPCC assigned 66 percent of human carbon emissions to the atmosphere because that is the percent necessary to support IPCC’s invalid claim that human carbon emissions have caused all the increase in atmospheric CO2 above 280 ppm. IPCC simply used its assumed result rather than calculate the human carbon cycle.
IPCC’s errors are obvious. IPCC shows human carbon in land is negative 8 percent. That is like having a glass of negative water. It is impossible. Also, no carbon can get to the deep ocean if there is no carbon in the surface ocean. IPCC’s human carbon cycle is clearly wrong. Yet, these IPCC errors are the basis of all climate alarmism.
Figure 19 shows the physics model calculation for human carbon. The physics model simply applies IPCC’s rules for natural carbon to the human carbon cycle.
Figure 19 shows less than 15 percent of all human carbon emissions from 1750 through 2019 remain in the atmosphere in 2019. About 25 percent is in the atmosphere and surface ocean. About 75 percent of human carbon has flowed to land and deep ocean. As human carbon enters the atmosphere, it simultaneously flows to land and surface ocean, and from the surface ocean to the deep ocean.
Figure 20 shows the physics model distribution of human carbon in 2100 under the assumption that human carbon emissions are terminated in 2020. In 2100, less than 4 percent, or 8 ppm, of human carbon remains in the atmosphere.
In the absence of human carbon inflow, the carbon level in the atmosphere falls as carbon flows into the deep ocean. The distribution of human carbon moves toward the natural carbon equilibrium shown in Figure 17. Given enough time, the distribution of human carbon will be the same as the equilibrium distribution of natural carbon.
This summary shows the result of using proper physics to calculate the natural and human carbon cycles. The result shows that human carbon is not a danger to the planet.
This paper shows, possibly for the first time, a formulation of the carbon-cycle model and separate calculations of human and natural carbon cycles using the same time constants.
IPCC’s own data show human emissions through 2019 have added only one percent to the carbon in the carbon cycle. During the same period, nature has added 3 percent to the carbon in the carbon cycle, likely due to natural warming since the Little Ice Age.
In terms of quantity, human emissions through 2019 have added 31 ppm to atmospheric CO2 while nature has added 100 ppm. So, natural emissions have increased atmospheric CO2 from 280 ppm to 380 ppm and human emissions have increased atmospheric CO2 from 380 ppm to 412 ppm.
If human emissions were to stop in 2020, then by 2100, 8 ppm of human carbon would remain in the atmosphere. If nature remained constant, then stopping all human carbon emissions could not reduce atmospheric CO2 below 388 ppm. But any small changes by nature would overwhelm the decrease of 23 ppm achieved by eliminating all human carbon emissions.
Human carbon emissions cause no significant long-term change to atmospheric carbon dioxide and are not the cause of climate change because human carbon flows from the atmosphere to the land and oceans as fast as natural carbon does.
The author thanks those who reviewed and commented on the draft of this paper: Richard Courtney, Nils-Axel Morner, Chuck Wiese, Gordon Fulks, Gordon Danielsen, Larry Lazarides, John Knipe, Ron Pritchett, Alan Falk, Leif Asbrink, Mark Harvey, Case Smit, Stephen Anderson, and Chic Bowdrie. This research project was funded by the personal funds of Valerie and Edwin Berry.
The author declares he is the only contributor to the research in this paper.
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