by Edwin X Berry, PhD, Theoretical Physics, CCM
Ed Berry LLC, Bigfork, Montana
To read key referenced papers:
- CO2 Coalition paper
- Dia Ato paper
- Bernard Robbins paper
- Eike Roth paper
Click here
Responsiveness of Atmospheric CO2 to Fossil Fuel Emissins
by Jamal Munshi
Ferdinand Engelbeen says Jamal Munshi has not proved absence of correlation.
What do you think? Add your comment below.
A Thermal Acid Calcification Cause for Seasonal Oscillations in the Increasing Keeling Curve
Download this Excel file here: https://edberry.com/Excel-File
Here is table “Berry Carbon Flow Test” for discussion in our comments.
I request Ferdinand and anyone else who is contesting my calcuations to present your calculations for comparison.
We assume that the natural carbon cycle is at constant levels as shown in Figure 3.
With that information, we insert human carbon into the atmosphere at a constant rate of 10 PgC per year. Then we calculate annual time steps to see how much human carbon ends up in each reservoir each year.
This simple calculation is a way to compare our calculations because we keep human carbon inflow constant for each year.
The years run from zero to ten. All the L data are in PgC, and flow data are in PgC/Year.
Lg = land, La = atmosphere, Ls = surface ocean, Ld = deep ocean, L is the total PgC in the carbon cycle for each year. Ntice L increases by 10 PgC each year.
The CO2 ppm column simply converts the PgC in La to ppm.
Here’s how it works.
Year 0: 10 PgC is added to La, but you don’t see it until the beginning of Year 1.
Year 1: the 10 PgC in La produces outflows to Lg and Ls. We see the result in Year 2.
Year 2: the outflows from La have moved some carbon to Lg and Ls. Etc.
Notice that as La gets more PgC, its Outflow to Lg and Ls increase, etc.
While La increased by 7.14 PgC from Year 1 to Year 2, it increased by only 1.49 PgC from Year 9 to Year 10.
Also notice that as Lg and Ls get more carbon, they send carbon back to La.
Adjustment, Residence, E-time Compared
IPCC’s response times fail physics
Physics e-time has a precise definition. The IPCC times do not. In summary:
- Physics: e-time is the time for the level to move (1 – 1/e) of the distance to its balance level.
- IPCC: adjustment time is the time for the level to “substantially recover” from a perturbation.
- IPCC: residence time is the average time a CO2 molecule stays in the atmosphere.
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” is imprecise.
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.
In summary, IPCC uses two different response times where it should use only e-time:
- 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.
Figure A illustrates how e-time relates to IPCC’s adjustment and residence times.
Figure A. E-time covers the full range of movement of level to a balance level. IPCC adjustment and residence times apply to only each end of the range.
IPCC, 2001: Working Group 1: The scientific basis. Appendix 1 – Glossary.
Lifetime
Lifetime is a general term used for various time-scales characterising the rate of processes affecting the concentration of trace gases. The following lifetimes may be distinguished:
Turnover time (T) is 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: T = M/S. For each removal process separate turnover times can be defined.
Adjustment time or response time (Ta) is the time-scale characterising the decay of an instantaneous pulse input into the reservoir. The term adjustment time is also used to characterise the adjustment of the mass of a reservoir following a step change in the source strength.
Half-life or decay constant is used to quantify a first-order exponential decay process.
The term lifetime is sometimes used, for simplicity, as a surrogate for adjustment time.
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: T = Ta.
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 (CO2) 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.
However, a large part of that CO2 is returned to the atmosphere within a few years.
Thus, the adjustment time of CO2 in the atmosphere is actually determined by the rate of removal of carbon from the surface layer of the oceans into its deeper layers.
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.
It’s time the waffleator packed his bags for Oslo and spread his propaganda to a different audience. He said:
July 24, 2025 at 9:02 am
As I have a lot of other problems to solve for the moment and need to prepare for a (similar!) debate in Oslo, I stop here with my reactions
And
July 25, 2025 at 12:27 am
As a final comment
Yet here is he still waffling on. He just can’t help himself.
Brendan Godwin, July 26, 2025 at 4:41 am
If I have five minutes time in between the preparations for Oslo, it always is a joy to tease Brendan to see how he reacts…
BTW, what is the trend in ocean pCO2 in these Greys Reef? I didn’t find any trend data, only a lot of extreme (seasonal) noise…
Ferdinand Engelbeen
July 26, 2025 at 2:03 am
You claim assuming “that all outflows are only CO2 level/quantity/pressure dependent in the different reservoirs…is a fundamental error, as at least half of the total yearly outflow, the one into vegetation, is near completely independent of the actual CO2 level/pressure in the atmosphere.”
This contradicts CO2 Coalition’s own anecdotal evidence of four trees growing faster as a function of CO2. I say anecdotal, because you will claim it is the difference in CO2 today than 280 ppm then. But you would only be an assertion without scientific experimental evidence.
AI says, “Photosynthesis rate is the speed at which plants convert light energy into chemical energy, typically measured by the rate of carbon dioxide uptake or oxygen release. Factors like light intensity, carbon dioxide concentration, and temperature significantly influence this rate.”
Notice that it doesn’t say, “factors like carbon dioxide concentration difference between now and preindustrial time.” Plants don’t know what CO2 was centuries ago. If you want to make a convincing argument, then do or show an experiment that shows vegetation growth more proportional to the concentration difference than to the absolute CO2 concentration.
“[My gobbledygook] makes your back-calculation, which is based on the absolute CO2 pressure in the atmosphere, completely at odds with the real world.”
That speaks for itself, no further comment needed.
Dr. Spencer’s model, while applying the same physically wrong science you use, back calculates perfectly in the same manner as his forward calculation does. He used what you mistakenly call the “real net removal rate.” You have never shown how the difference between the actual CO2 level and the original equilibrium is the true rate determining mechanism. Other than the climate community, no other scientific community agrees with you. Examples to the contrary are welcome.
I wish you all the best in Oslo.
David Andrews
July 25, 2025 at 7:58 pm
In an earlier note to Ferdinand I said “I’m working on a way to demonstrate how the simple math you and David Andrews rely on can give ambiguous results based on somewhat arbitrary choices of variables.”
You responded July 8, 2025 at 7:44 pm with, “You seem to have been distracted from that goal or maybe have given up. My goal here is to keep you focused on it.”
Well, I put up the spreadsheets comparing Ferdinand and Spencer’s simple math models with my model demonstrating that one can arrive at the same results using different variables, i.e., removal rate constants and emission inputs.
Now you are saying I gave you no reason to look at time constants, “they are a distraction.” I really should stop beating my head against the wall, eh? [a reference to an old Bazooka Joe Gum joke]
You claim, “The net flow of ‘human carbon’ out of the atmosphere does not correspond to a net flow of carbon” should be self-explanatory. I’m sorry, but that seems like a denial of the equivalence principle.
Moving on, I’ll try to follow your logic. You categorize (separate) natural exchanges between atmosphere and land/sea reservoirs into two situations, (1) the preindustrial case where EXACTLY BALANCED exchanges homogenize the carbon composition of the various reservoirs without affecting AT ALL the total carbon levels in any of them, and (2) the present case where human emissions cause unbalanced exchanges resulting in net global uptake removing carbon from the atmosphere. Then you conclude, “[this] simple ‘net global uptake’ analysis shows that human emissions are responsible for the increase.” Voila!
That model is Magic Math, because it assumes no change in carbon levels, despite evidence to the contrary, and applies non-standard physical principles. My model needs to invoke “extra carbon” and changes in natural emissions to explain the known data and remain consistent with standard physical principles. Fortunately, the potential for finding evidence of extra carbon exists. Where is your evidence that carbon levels haven’t increased and how would you explain any evidence they have?
Jim Siverly, July 26, 2025 at 9:03 am
One need to make a differentiation between the seasonal and longer term fluxes between the atmosphere and the biosphere.
If the CO2 level in the atmosphere instantly doubles, the output in the biosphere hardly increases and needs decades to catch up with the extra CO2 in the atmosphere and even then doesn’t double.
With 50% extra in the atmosphere over the past 175 years, the increase of the fast cycle (thus the amount of extra living vegetation) only increased with 13%, that is all and only 25% of the extra FF of each year emitted in the atmosphere is absorbed into more permanent vegetation and soils as extra mass for longer periods.
“Dr. Spencer’s model, while applying the same physically wrong science you use, back calculates perfectly in the same manner as his forward calculation does.”
My calculations match these of Spencer within 1 PgC when you shift the 20 years start to 2022. It is your faulty interpretation that gets wrong, both in the future as in the past…
“Plants don’t know what CO2 was centuries ago.”
Plants adapt to temperature and the amount of CO2 that is in the atmosphere: when temperature and the CO2 level increased from an ice age to an interglacial, from about 190 to 290 ppmv, plant and animal life did expand and reversed when a new ice age arrived. Thus vegetation did “remember” its original CO2 level…
If one would stop all FF emissions, then the world will stop “greening” and instead will start “browning”, as the CO2 levels drop to the original atmospheric CO2 levels of around 280-300 ppmv which is the equilibrium with the ocean surface today. As they did over ice ages and interglacials: there is zero impact of plant life on the δ13C level over 800,000 years, while the CO2 levels increased and decreased with 90 ppmv, completely dominated by the slow changes in ocean surface temperatures and related pCO2.
“Other than the climate community, no other scientific community agrees with you.”
Dietze, already in 1997 and heavily against the IPCC’s Bern model, Spencer, Lindzen, and many others all use the same “real removal rate”, based on the difference in pCO2 between the two main reservoirs: atmosphere and oceans. The latter dominates the CO2 level in the atmosphere, except when humans add more CO2 than oceans and biosphere can remove.
As far as I know, that are not members of the “climate community”.
David Andrews
At July 25, 2025 at 6:27 pm I replied to your comment at July 25, 2025 at 8:55 am re: “Measured carbon levels in the ocean are increasing”. You failed to reply. Instead you let the waffleator respond for you. You did the same the previous time as well. You obviously don’t have the intellectual capacity or knowledge to reply on your own.
David Andrews.
At JULY 25, 2025 AT 7:58 PM you yet again cited a reference to consensus. Anyone who cedes to consensus as a reference is not a scientist.
On JULY 14, 2025 AT 5:43 PM you attacked Murry Salby’s science by reference to his employment. Whoever does this is not a scientist.
You continually taint your comments with these anti-science political statements. All this does is prove that you are not a scientist but a political activist. This renders your comments worthless. To be taken with a grain of salt. You are not in this discussion to find a scientific pathway to the truth, you are here to push your political agenda.
Brendan Godwin July 26, 2025 at 7:59 pm
Dear Brendan,
Well put. I agree with you.
Ed
Ferdinand Engelbeen
July 26, 2025 at 12:45 pm
No, one doesn’t need to do any differentiation with my model. I will make two modifications to it with 1) 50% pulse and 2) net zero with a continuing gradual natural emission increase. You can freeze the natural emissions at any point. In all cases, CO2 will return to a new equilibrium level according to a four-year e-time.
1) https://www.dropbox.com/scl/fi/g6zj3et0vs3o9pfk0jzim/Jim_fluxes1.xlsx?rlkey=kz95m7zhp7ohflh209x84niuy&dl=0
2) https://www.dropbox.com/scl/fi/603gxxzkdk3l5mw43slxo/Jim_fluxes2.xlsx?rlkey=b23hjfhdhpakttqeemy5wjtbc&dl=0
“…from about 190 to 290 ppmv, plant and animal life did expand and reversed when a new ice age arrived. Thus vegetation did “remember” its original CO2 level…”
That is speculation based on anecdotal data completely devoid of proof, Ferdinand.
Roy Spencer is the climatologist who publishes the monthly UAH temperatures chart. Richard Lindzen has been called “an expert on climate change for four decades.” In no way can they be considered outside the climate science community.
The reference to Peter Dietz is quite interesting. I previously ignored it (to my great regret), because I considered it an appeal to authority from another one of the 50-year Tau, not 4-year Te, proponents. I reviewed about a third of the discussions with fellow IPCC contributors Dietz documented in his open review comments relating to his Carbon Model Calculations paper. I will finish reading those comments before providing a more detailed analysis. Suffice for now to say Dietz assumes removal is proportional to the difference between pCO2 now and 280 ppm, the preindustrial pCO2, without any confirming evidence that it is based on sound physics. The 50-year Tau, 55-year in Dietz’s case, is based on the pCO2 now – 280 ppm difference. [(408-280)/50 = 2.56 ppm/year, the approximate average removal rate in 2018] That makes it a circular reasoning argument. The same one you make.
You referenced Dietz over a dozen times here. Many of those times Dr. Ed noted the circular reasoning. I tried asking about A B reactions hoping you would see the relevance to the rates being proportional to actual concentrations, not differences between now and some original level. The Feely equation is proportional to delta pCO2, meaning d(pCO2)/dt – k * dCO2aq/dt now, not a preindustrial difference. If you continue to promote the 50-year Tau, you really should come up with some scientific evidence of the underlying physics.
Jim,
Thanks for your excel work. Excellent. I guess David and Ferdinand can keep arguing but math is hard to argue with.
Ferdinand Engelbeen July 25, 2025 at 12:27 am
Dear Ferdinand,
You wrote:
So, if I understand you correctly, you are saying that these extra processes increase, not decrease, the flows between the reservoirs compared to my (2) which says Outflow = Level / Te.
If so, then you are simply arguing that the real Te are smaller than the Te I derived from IPCC’s own data. And it in that case, we agree.
That is because Delta14C data show the IPCC Te are too large because they would have reduced the balance level of Delta14C to near -300. (Ignore David Andrews’ objection to this because he is irrational.)
Your argument that these extra processes increase the flow argues for smaller Te to explain the Delta14C data.
This blows your argument that your Tau is 50 years or more.
Jim Siverly July 26, 2025 at 12:45 pm and July 26, 2025 at 11:57 pm
Dear Jim,
Thank you for your excellent Excel calculations.
I have one question that maybe indicates I do not follow your use of “back calculations.”
That is because we cannot start with any distribution of carbon among the carbon reservoirs and from that determine what a prior distribution may have been.
This is because the carbon flows between the reservoirs always move the carbon distribution closer to their equilibrium percentages. Since all distributions flow toward their equilibrium distributions, it is impossible to determine a prior distribution.
Or perhaps I misunderstand what you mean by back calculations.
Jim,
I advised you to understand disequilibrium isotope fluxes a few weeks ago. You flunked that assignment. I see you also have borrowed one of Ed’s favorite tricks. When you don’t have an answer to an argument, misrepresent that argument and argue against the misrepresentation. The balanced exchanges I am talking about have nothing to do with pre-industrial times and you know that. I fear you are purposely wasting my time, but I will give it another shot.
Remember: Natural Absorption(NA) = Natural Emissions(NE) + Net Global Uptake (NGU) by simple math, and NGU has been positive over the last century by data and carbon conservation. From the nominal numbers we have been using, NGU is only 2-3% of NA. Of course NGU mitigates the atmospheric carbon rise from human emissions by removing carbon from the atmosphere and adding it to land/sea reservoirs. NGU changes carbon levels in reservoirs. But the remaining 97-98% of NA is, call it NAb, is exactly equal to NE. Three questions for you.
1. If NE = NAb, and NE is the quantity of carbon moving from land/sea reservoirs to the atmosphere, and NAb is the quantity of carbon moving from the atmosphere to land/sea reservoirs, how much do these exchanges together effect carbon levels? (You may study this question with a spreadsheet if you need to.)
2. If the atmosphere is rich in “human carbon” compared to the land/sea reservoirs because that is where smokestacks and exhaust pipes put it, do the same balanced two-way exchanges move net “human carbon” in one direction or another? Which way?
3. Does your answer to 2. “violate the equivalence principle”?
You can vary assumptions about natural emissions all you want. To match the data you must then also vary natural absorption to fit the measured NGU. Have fun beating your head against the wall with that “extra carbon”.
I don’t know why you think I assume “no change in carbon levels”. They are indeed changing because previously sequestered FF carbon is being put in the atmosphere and from there spreads to other fast-cycle reservoirs. The mass balance argument is corroborated by an observed expansion of the biosphere and carbon increases in the ocean. (Note that I avoided saying “ocean acidification”; I don’t want to wake up Brendan.)
Which of my physical principles is nonstandard in your opinion? Note that I still haven’t linked my argument to time constants, not that they are unimportant.
I have not seen your spreadsheet, but I think I know what your problem is. You think ”human carbon” leaving the atmosphere implies level changes, because you didn’t do your disequilibrium isotope flux homework. So you add extra natural emissions to match the Mauna Loa data, then you need extra natural absorption … Pretty soon you have “exponentially” increasing natural emissions and a headache
Jim Siverly, July 26, 2025 at 11:57 pm
There is an error in your calculation of my return to the equilibrium, which makes part of the problem in the back-calculation: you calculate the net outflow from the FF emissions from the previous year, while for Spencer’s calculation you use the FF emissions for the same year. Makes some difference, but still not what it should be, if you use the “real” decay rate.
In both cases, you don’t reflect what Spencer, me and others have done, as we calculated the 50 years Te/Tau from the CO2 level in the atmosphere and the net outflow as calculated from known human inputs and observed increase in the atmosphere. While you calculated the one-way outflow only for the atmosphere out, based on the (roughly observed) total outflows, not the back-flows from the other reservoirs. These are implied in your calculation as increasing together with the overall outflow, as that is the difference with the
The main difference still is that you assume a constant ratio between output flow and total amount of CO2 in the atmosphere, no matter the change in Te/Tau with the CO2 increase.
Moreover, you use some “self fulfilling” prophecy’s by calculating the “natural” inflows from I don’t know where before 1958 and calculate them from the CO2 level in the atmosphere after Mauna Loa measurements started.
How do you know these inputs? Dr. Ed uses the levels in the oceans and biosphere to calculate the inputs to the atmosphere, but you calculate them with the same turnover time…
I don’t think that many will disagree with the fact that the biosphere expanded and did shrink together with temperature and CO2 levels over the ice ages and interglacials. The point is that this had no measurable effect on the δ13C levels, until humans started to emit fossil fuel CO2:
https://www.ferdinand-engelbeen.be/klimaat/klim_img/co2_d13C_lgm_cur.png
That is a definitive proof that the oceans are the dominant cause of the CO2 changes in pre-industrial times.
And as you may know, the oceans at the “edges” (poles and equator) have near-constant temperatures and thus pCO2(aq), where the ocean waters sink in the deep or are upwelling and that is the dynamic “memory” for the pCO2 in the atmosphere…
“In no way can they be considered outside the climate science community.”
Come on Jim, you know as good as I know, that these are not supporting the mainstream science about the “catastrophic” GH effect. Neither do they support the hundreds or thousands years “life time” of our extra CO2 in the atmosphere from the Bern model. But they all recognize the about 50 years “adjustment” time and reject the 4 years residence time or Te as completely irrelevant for the decay rate of any extra CO2 in the atmosphere, of whatever source.
“The 50-year Tau, 55-year in Dietz’s case, is based on the pCO2 now – 280 ppm difference. [(408-280)/50 = 2.56 ppm/year, the approximate average removal rate in 2018] That makes it a circular reasoning argument. The same one you make.”
That is where you are completely mistaken. Peter Dietze indeed used the “historical” CO2 level in the atmosphere as equilibrium, but the current equilibrium was calculated from the net removal rate over different years by Spencer, myself and David Burton, because we now have the net removal rates at different CO2 levels in the atmosphere. That makes that it is easy to back-calculate the “real”, current, equilibrium level where the net removal rate is zero.
https://sealevel.info/Global_Carbon_Budget_2023v1.1_with_removal_rate_plot2.png
Only the net removal rate is plotted against the CO2 level in the atmosphere, without any assumption of Te/Tau. That shows an equilibrium of between 280-300 ppmv and thus a Te/Tau of around 50 years as result. Not reverse.
Jim,
Half way you can see that my text doesn’t fit: that was when I saw that you used the output data to calculate the inputs to the atmosphere. Can you elaborate on that, where the input data to the atmosphere are based on?
David Andrews July 27, 2025 at 9:16 am
David, your so-called explanation is unnecessarily complicated, so much so that you can get pretty much any result you wish to promote.
The only way you can make a valid point is to use my equation (2) for human and natural carbon independently because they include all the flows we are talking about.
Then show how you get from these equations to your equations. We need that connection to evaluate your equations.
If you can’t do that, then you are wasting everyone’s time with pseudo-science.
Dr. Ed, July 27, 2025 at 8:26 am
“So, if I understand you correctly, you are saying that these extra processes increase, not decrease, the flows between the reservoirs compared to my (2) which says Outflow = Level / Te.”
Indeed the extra processes increase the flows between the different reservoirs in both directions: while the increase in the atmosphere is about 50% since 1750, the in/outflows with the oceans increased about 33% and with the biosphere about 13%. Thus not in ratio to the increase in the atmosphere.
Moreover, the absolute height of the flows is of zero interest for any changes in any reservoir, only the difference between inflows and outflows changes the CO2 mass in any reservoir.
That difference is directly proportional to the distance with the equilibrium, which for the current SST is between 280-300 ppmv, as can be back-calculated from the different CO2 levels over time and the observed net removal rate,
That makes that the calculated Te (or Tau) is around 50 years, not the 4 years of the turnover time.
The 14C and 13C/12C ratio’s are “thinned” by the return of deep ocean waters which contain relative more 13C and 14C than fossil fuel emissions. That makes the Tau of both 14C and the 13C/12C ratio are a lot shorter than for the extra 12C/13C removal out of the atmosphere.
Jim Siverly, July 26, 2025 at 11:57 pm
Some part where I didn’t react on:
“The Feely equation is proportional to delta pCO2, meaning d(pCO2)/dt – k * dCO2aq/dt now, not a preindustrial difference. If you continue to promote the 50-year Tau, you really should come up with some scientific evidence of the underlying physics.”
The Feely equation shows the different delta pCO2’s for different parts of the oceans. For 90% of the sea surface, that difference only leads to a very small uptake by the ocean surface layer of about 10% of the increase in the atmosphere. That is because of physical and chemical restrictions in uptake. That is known as the Revelle/buffer factor. And measured as increase of DIC in the ocean surface waters.
The IPCC uses the total ocean surface as limiting factor for the CO2 access to the deep oceans, with as result the long residual uptakes of part of the current increase of hundreds to thousands of years.
Far more important is where the ocean waters sink into the deep oceans.The temperature there is year-round around freezing near the floating ice, which makes that the pCO2 of the ocean waters can be as low as 150 μatm( ~ppmv). The same for the upwelling places near the equator, where year-round high SST is measured and as high as 750 μatm for pCO2(aq).
That makes that a lot of CO2 (and oxygen) is taken from the atmosphere and directly moved into the deep oceans, without the physical and chemical restrictions of the rest of the sea surface.
My own estimate, based on the “thinning” of the δ13C “fingerprint”, is around 40 PgC/year that goes directly into the deep oceans (mainly in the N.E. Atlantic) and comes back some 1,000 years later near the coast of Peru/Chile.
What does that make when CO2 increases in the atmosphere? The extra pressure will increase the output into the polar sink places and reduce the emissions at the upwelling sites. For the current pCO2 difference with the atmosphere, the estimated (to close the “gap” with atmosphere, ocean surface and vegetation) is about 20% of the human emissions as carbon mass. Again in complete ratio with the “old” equilibrium.
The carbon increase in vegetation also is calculated, based on the O2 balance since about 1990. That is about 25% of human emissions as carbon mass. Even with large year-by-year variability in uptake by vegetation, in average all three reservoirs do sink CO2 in ratio to the “old” equilibrium.
Dr. Ed
July 27, 2025 at 8:41 am
“My” idea of a back calculation is just reversing the forward calculation. Dr. Spencer’s model makes it easy because he has a simple formula Cn = Cn-1 + EHn – ( Cn-1 – C0 ) / 50 where EHn is the industrial emissions added in the current year. Notice the removal is NET removal of only 2%/year based on fitting the Cn results to the Mauna Loa data. So the 50-year figure is an artifact of the assumption that the removal process is proportional to difference between now and way back when. It never seemed right to me.
Now back to answering back-calculation question. One can solve Spencer’s simple formula for Cn-1 and get
Cn-1 = [ Cn – Ehn – C0/50 ] / ( 1 – 1/50 )
Since the Cn, EHn, and C0 are known, you just work back from there to C0 = 1750 or wherever you want to start. With my model, back calculating is only possible with the known CO2 values from 1959 on. I can estimate the natural emissions, ENn, from ENn = Cn – EHn – Cn-1 * ( 1 – 1/Te ). Before that I don’t know what Cn-1 is. That’s when I extrapolate back using exponentially-derived estimates of ENn from a fit to the Mauna Loa data. So both my model and the Spencer model fit data to a formula. But I argue my formula, dare I say your formula, makes more physical sense.
The bottom line is a back calculation is, at best, only as good as the forward one from which it is derived. While Spencer’s model is wrong, in my opinion, it is consistent. Ferdinand’s has a bug which may or may not be my error. His formula seems to be different than Spencer’s somehow.
The overall point I’m making and only one that really interests me is what the true science dictates. I’m OK being wrong, I just want to be wrong for the right reason.
“…we cannot start with any distribution of carbon among the carbon reservoirs and from that determine what a prior distribution may have been.”
I agree that we can’t know the exact past distribution. That’s why I always call my back calculation an estimate. The fact that it is based on a simple exponential function and nails the Mauna Loa data gives me confidence that the estimate is pretty darn close to accurate. I think I remember finding other ways to fit the data, but not both using constant Te and extrapolating back to around 280 in 1750.
David Andrews
July 27, 2025 at 9:16 am
“I fear you are purposely wasting my time…”
Sir, no one demands you comment here. If you feel compelled to respond, that’s on you.
Answer to 1. From year 2018, I estimate NE was 91.9 ppm and NAb was less than 89.5, therefore they were not the same. It was a typical year with natural contributing 2.5 ppm more than the year before.
2. In 2018, I estimate at least 4.5 ppm of human-sourced carbon was removed, almost as much as was emitted that year. I haven’t estimated how much human-sourced carbon returned to the atmosphere, but it’s a relatively small amount. That’s because human-sourced carbon in the ocean is diluted and the return is proportional to that diluted concentration.
3. My answer in 2 perfectly follows the equivalence principle. It’s what Dr. Ed’s article is all about.
I am pleased to find you are OK with a biosphere expansion. Your physical principles are based on Magic Math, not laws of nature. That may be why you avoid time-constants.
I think my problem is your problem accepting the consequences of biosphere expansion, because “Pretty soon you have “exponentially” increasing natural emissions.”
Ed,
Gosh, I thought the two points I have been making were pretty simple.
1. Since the atmosphere has been accumulating carbon at a rate slightly less than half the rate we have been adding it, natural proccesses must on balance have been removing carbon to the land/sea reservoirs, not adding it to the atmosphere.
2. The continual mixing between reservoirs homogenizes their content. That makes it incorrect to assume, as you do, that the small “human carbon” content of the present atmosphere implies only a small contribution to the rise from human emissions.
Which of these points do you find too complicated? You have had a few years to study them.
David Andrews July 27, 2025, at 2:34 pm
David, it is your duty show how your equations derive from my (2). Without that connection, all your claims and questions are without a foundation.
Is that too complicated for you to do?
Ferdinand Engelbeen July 27, 2025 at 11:38 am
Dear Ferdinand,
You wrote:
Your admission (1) is all that is needed. Thank you. Therefore, your model supports a shorter Te, which corresponds with the shorter Te revealed by the Delta14C data.
Your argument (2) is invalid because the Delta14C data show the Delta14C balance level was unaffected by the bomb tests, and the Delta14C balance level shows the proportion of the human carbon balance level to the natural carbon balance level.
For example, if human carbon caused all the CO2 increase, that would have lowered the Delta14C balance level to -333. Clearly, this has not happened.
The Delta14C balance level is between zero and -8. That means Te is much smaller than 4 years. This proves your Tau of 50 years is invalid.
Remember, the balance levels are determined by the present (not past) inflows of natural carbon and human carbon. There is no “replacement” argument that affects the balance level argument. Pure natural carbon has a Delta14C of zero.
Therefore, your arguments (1) and (2) converge to two simple conclusions: Te is smaller than 4 years and human carbon has a negligible effect on the level of atmospheric CO2.
Ed,
The topic is whether human emissions cause the CO2 increase, i.e, your H1. Carbon conservation and the measured positive net global uptake prove that they do.
Your C14 analysis is a joke.
David Andrews July 27, 2025 at 7:15 pm
David, you have just proved what I suspected.
You cannot connect your so-called argument with my equation (2).
Therefore, you have no argument. You are not even in the game.
You have been an imposter acting as if you knew something about this subject.
I asked you a very simple question related to the core of this discussion, which is equation (2).
This is simply physics, David.
Your unstated answer is that you don’t know how to connect your ramblings to equation (2).
Thanks for this revelation.
Ferdinand Engelbeen
July 27, 2025 at 9:25 am
Hopefully, something from my response to Ed at July 27, 2025 at 12:46 pm helps explain some of the confusion. It’s ironic that you cite self fulfilling prophecy while your whole model is based on self-fulfilling prophecy. At least I know why it is, as I’ve explained.
I’ll skip your real decay rate paragraph until I study the spreadsheet.
You claim, “you don’t reflect what Spencer, me and others have done….” I don’t understand how you can say that when I duplicated Spencer’s spreadsheet fully and exactly as he published it. I even downloaded an original version when he posted it on his website five years ago. His model is one-way outflows. If you have a problem with that, take it up with him. My model has both natural and human emission inflows, I just don’t assume a constant natural inflow as you do. I will add graphics for both our data and see which does a better job correlating Mauna Loa data.
My model doesn’t have a change in Te. Of course there’s a constant outflow/level. Nature is increasing its outflow as it strives to reach its new balance level, to put it in Dr. Ed’s vernacular.
“How do you know these inputs?” As explained to Ed, from fitting the Mauna Loa data assuming constant Te. It makes sense, because it goes to the likelihood that human activity, such as burning and growing stuff, increases CO2 output exponentially.
You may have “definitive proof that the oceans are the dominant cause of the CO2 changes in pre-industrial times,” but I don’t see how that rebuts anything I have proposed.
Not supporting not supporting the mainstream “catastrophic” science does not make one outside the climate community. Spencer complains about being boxed in on either side.
I’m not “completely mistaken.” Dietze calculated 55 years, because he didn’t have a quarter century more data that we have. But he got it the same way, a plot of CO2 versus deltaCO2 and assumed the same wrong formula you and the rest use. Again, I say wrong only happy to be shown otherwise. By the way, I have a scientific reason for assuming a constant Te: what physical processes have changed that warrant a variable Te?
Jim Siverly, July 27, 2025 at 8:19 pm
“His [Spencer’s] model is one-way outflows”
Spencer’s and many other’s and my model is about NET outflow, not one-way outflows…
“Cn = Cn-1 + EHn – ( Cn-1 – C0 ) / 50”
That is about the same formula that I used in my calculations:
La(n) = La(n-1) + Ff_start – (La(n-1) + Ff_start-La_base)/Te
I used the “In between” La(n-1) + Ff_start to calculate the net flow, Spencer used the “old” C(n-1).
Further, my calculations were asked to use a fixed 10 PgC/yr FF supply and not the real figures for Mauna Loa and the real supply.
Both Spencer’s and my approach doesn’t involve inflows or outflows at all, only net flows. Dr. Ed wanted to see what the in- and outflows do. That I did extra by looking at what the inflows and outflows could be.
So Spencer and I agree with each other. And you don’t agree:
“Notice the removal is NET removal of only 2%/year based on fitting the Cn results to the Mauna Loa data.”
Here we still agree.
“So the 50-year figure is an artifact of the assumption that the removal process is proportional to difference between now and way back when.”
Here again, we strongly disagree: The 2%/year figure is the observed (!) decay rate for every CO2 level increment in the atmosphere since Mauna Loa started its measurements. Thus not based on any past known or unknown “old” equilibrium.
“Observed” in that way that human emissions are quite accurately known from taxes on sales and burning efficiency and the increase in the atmosphere is very accurately known from measurements.
To the contrary, one can back-calculate the current equilibrium from the observed decay rates since 1958 as Spencer and many others and I and David Burton have done.
“As explained to Ed, from fitting the Mauna Loa data assuming constant Te. It makes sense, because it goes to the likelihood that human activity, such as burning and growing stuff, increases CO2 output exponentially.”
I didn’t talk about the outputs which are based on Te. I was talking about the inputs to the atmosphere, which have nothing to do with Te from the atmosphere, but with the Te from the oceans and the Te from the biosphere.
You used the Mauna Loa data to calculate the total input as a fit for the net removal of CO2 out of the atmosphere, while you don’t have any idea what the real inputs are. That is just curve fitting…
Moreover, for the mass balance: even in your calculations, the new inputs are less than the old outputs, thus nature NET removed CO2 mass out of the atmosphere, about half what humans as FF mass added in the previous year. Any bookkeeper will confirm you that the FF addition is the cause of the increase, no matter how much FF “labeled” CO2 returns to the atmosphere within the next season or next year(s).
Human activity increased the CO2 levels in the atmosphere with about 50%. The overall outputs into the oceans increased with 33% and in the biosphere with 13%. Not 50%. Thus Te, based on the 4 years La/F(out) was not constant and increased over time. The Te of 50 years, based on the NET outflow remained more or less constant.
Dr. Ed, July 27, 2025 at 7:03 pm
“Therefore, your model supports a shorter Te, which corresponds with the shorter Te revealed by the Delta14C data.”
I am not that familiar with the delta 14C data, but as far as I know the delta Te for 14C after the nuclear bomb tests is between 12 and 20 years, far beyond the 4 years Te that you use for the bulk CO2 output…
“Your argument (2) is invalid because the Delta14C data show the Delta14C balance level was unaffected by the bomb tests”
To the contrary, the 14C level before the bomb test was already affected by the zero-14C human FF supply, so that correction tables for radiocarbon dating were needed and after the bomb tests, the drop in 14C was way faster than calculated without FF emissions. In the coming decades, the 14C level will further drop to below the historical levels:
https://www.ferdinand-engelbeen.be/klimaat/klim_img/graven_14C.png
I have done calculations on the “thinning” of the δ13C human “fingerprint” in the atmosphere and that shows that about 2/3 of the human FF CO2 input directly into the atmosphere is replaced by CO2 from other reservoirs, mainly by deep ocean exchanges. Replaced, not removed as CO2 mass. One can calculate how much deep ocean water is needed to do that replacement: about 40 PgC exchange with the 1,000 year old deep ocean waters is sufficient.
https://www.ferdinand-engelbeen.be/klimaat/klim_img/deep_ocean_air_zero.jpg
0 PgC/year is the δ13C drop if all human FF emissions remained in the atmosphere…
And last but not least, as already explained to Jim, your Te of 4 years for the bulk CO2 in the atmosphere increased over time, because the total level of CO2 in the atmosphere increases with 50%, but the in/outflows with the oceans increased only 33% and with the biosphere only 13%…
Jim,
What you derisively call “Magic Math”, and Ed calls “too complicated”, I call “carbon conservation” and dead simple. Conservation principles are the bedrock of physics.
I looked for your spreadsheet on Ed’s site and didn’t find it, so I will continue to have to guess what you are up to. Your 7/27 9:16AM post, and talk of “extra carbon”, suggests that you are postulating a new category of carbon in addition to “human” and “natural”. (Demetris Koutsoyiannis tried the same thing. After I labeled his new category “super-natural carbon” he dropped it.) Perhaps you think some source like deep-sea volcanos is important and underestimated. Like FF, it would be an uncompensated transfer into the fast carbon cycle from a previously sequestered stock. But such a source is already included in the mass balance calculation. You might want to check your spreadsheet to see if human emissions are 2x atmospheric carbon growth as in the real world.
Your insistence that the equivalence principle does not allow balanced exchanges to move net “human carbon” from the atmosphere where it is dense to land/sea reservoirs where it is sparse is bonkers.
Ferdinand Engelbeen July 28, 2025 at 3:41 am
Dear Ferdinand,
I’m sorry but your explanation does not save your failed hypothesis.
You admit that you are not familiar with the Delta14C data. Therefore, you are not in a position to argue that it does not prove your hypothesis is wrong. You have not studied my papers and my rebuttal above to your CO2 Coalition paper.
When you understand the Delta14C level, you will see that we are not talking about the Te of Delta14C, which happens to be 16.5 years, and all your (and Dave Burton’s) calculations of this value are incorrect.
The key feature of the Delta14C data is that it proves its balance level has remained and still is less than about 8% below its historic balance level upon which all carbon dating is based.
You wrote:
“To the contrary, the 14C level before the bomb test was already affected by the zero-14C human FF supply, so that correction tables for radiocarbon dating were needed and after the bomb tests, the drop in 14C was way faster than calculated without FF emissions. In the coming decades, the 14C level will further drop to below the historical levels:
https://www.ferdinand-engelbeen.be/klimaat/klim_img/graven_14C.png.”
Indeed, human CO2 has lowered the Delta14C balance level a small amount below its long-term balance level. No problem with that. But that supports my calculations that show human CO2 has had only as small effect on the CO2 increase above the reference level of 280 ppm.
The key is that the present Delta14C balance level DOES NOT support your claim that human CO2 is a significant cause of the CO2 increase. It shows that natural CO2 is the dominant cause of the CO2 increase. There is no way you can deny this.
You wrote:
“I have done calculations on the “thinning” of the δ13C human “fingerprint” in the atmosphere and that shows that about 2/3 of the human FF CO2 input directly into the atmosphere is replaced by CO2 from other reservoirs, mainly by deep ocean exchanges. Replaced, not removed as CO2 mass.”
Sorry, but the Delta14C proves your calculations are wrong. Your hypothesis predicts an incorrect result. Therefore, your hypothesis is wrong.
You wrote:
“And last but not least, as already explained to Jim, your Te of 4 years for the bulk CO2 in the atmosphere increased over time, because the total level of CO2 in the atmosphere increases with 50%, but the in/outflows with the oceans increased only 33% and with the biosphere only 13%…”
Sorry, but the Delta14C data prove your above conclusions are wrong.
David Andrews July 28, 2025 at 9:03 am
David,
I can’t believe you are back with more of your math-deficient, hand-waving declarations that have no connection to real physics or to the issue under discussion.
You wrote to Jim:
“What you derisively call “Magic Math”, and Ed calls “too complicated”, I call “carbon conservation” and dead simple. Conservation principles are the bedrock of physics.”
But you have not shown anywhere that you are following conservative principles.
I asked you to derive your physics and math from my equation (2), which I connect with (1) to assure carbon mass conservation. So, all my work is based on carbon mass conservation because all my equations are derived from (1).
You can make no such claim legitimately. So, you resort to hand-waving statements. Maybe you should go into politics and forget physics.
You wrote toJim:
“You further claim that the climate equivalence principle does not allow balanced exchanges to move net “human carbon” from the atmosphere where it is dense to land/sea reservoirs where it is sparse is bonkers.”
Your argument is invalid, David, because you have no physics equations to even follow how carbon flows between the carbon reservoirs.
Please, next time you come back, please show us how you derive your imaginary model in terms of equations that derive from my (1) and (2).
Ferdinand Engelbeen
July 27, 2025 at 9:31 am
I will try to catch up and not duplicate too much. I was working on dropbox and Windows issues.
“Can you elaborate on that, where the input data to the atmosphere are based on?”
I think I answered this already in response to Dr. Ed. I’ll repeat with a better explanation, hopefully. I could see from Spencer’s original spreadsheet that he had no contribution from natural emissions. By default, his model assumes them to be whatever they are. Whether he started with a Dietz/Burton/Engelbeen-derived Tau and then found the Mauna Loa fit or first tweeked the fit and then found the same Tau as you and others, I don’t know. It’s a question for Dr. Spencer that I was too ignorant to ask at the time. What is clear, is that he assumed the removal rate was proportional to the difference between the pCO2 now and whatever it was in 1750. Just as Deitz, you, and others do.
Sensing this was wrong, I added natural emissions of about twenty times the human emissions and assumed the removal rate was directly proportional to whatever the pCO2 was in the current year. Then I had to figure out what the natural emissions would have to be to get a fit to the Mauna Loa data. That took a while, but was enabled by assuming the formula was exponential. I didn’t put the equation in the “Jim fluxes” spreadsheets, because it was originally in terms of ppm and I didn’t take time to convert it to PgC. So I just copied and pasted the ppm values and then converted to PgC. Call me lazy, but actually I was pressed for time. Now with all your questions, I’m even more pressed for time because I want to finish reading the Dietz comments.
With the appearance of your 20-year spreadsheet, I was able to attempt a back calculation, but failed. However, back calculation on Spencer worked perfectly. Next, I used the same approach to back calculate my spreadsheet. Because I had the extra variable, natural emissions, I could only do 1959 to 2018 where I had both Cn and Cn-1. So prior to 1959, I just used natural emissions from my original data fit. The formula to back calculate the Mauna Loa era natural emissions is
ENn = Cn – Ehn – Cn-1 * ( 1 – 1/Te )
I amended my spreadsheet to show the exponential equation and a graph of the fit. I noticed the exponential estimate at 2019 (cell AA274) was off by about 1 PgC. That’s not bad, considering my fit was an eye-ball one. Also, there was an error at cell AA214, which was corrected by replacing the wrong back-calculation formula with the exponential formula.
https://www.dropbox.com/scl/fi/ojyikq5ds5wgl6g5b0jsp/Jim_fluxes3.xlsx?rlkey=9lzxmuc33j6qs1hcgngy5mxtd&dl=0
July 27, 2025 at 12:19 pm
“…Again in complete ratio with the “old” equilibrium… all three reservoirs do sink CO2 in ratio to the “old” equilibrium.”
You continue to apply anecdotal data to validate your preconceived notions. Unscientific and unsatisfactory. This may fly at climate conferences, but I suspect you will continue to get pushback from anyone outside the Tau camp.
To be continued…
Dr. Ed, July 28, 2025 at 9:24 am
“You admit that you are not familiar with the Delta14C data.”
I didn’t say that I don’t know anything about the Δ14C data, but I haven’t looked at it with the same depth as for the δ13C data… David Burton did a lot of work on it and I am pretty sure that he has done a good job in our common work.
But here a graph I made already 20 years ago to show how the Δ14C data from the bomb tests are “diluted” by the deep ocean waters, making the bomb tests peak decay rate a lot faster than for an extra input of bulk 12/13CO2:
https://www.ferdinand-engelbeen.be/klimaat/klim_img/14co2_distri_1960.jpg
Of all CO2 as mass in 1960, about 97% returned back as mass from the deep oceans in the same year as absorbed. From the bomb tests, only 45% of 14C as “mass” returned in the same year, the rest did just begin their ~1000 year journey…
That makes that your calculation of the “fixed” 14C balance is completely at odds with reality, because you haven’t taken into account the ~1000 years delay between the extra 14C uptake and the current 14C “old” release.
Thus the decay rate of Δ14C from the bomb tests is much faster than the decay rate of about 50 years for an extra input of FF CO2 and moreover has absolutely nothing to do with the 4 years turnover time, which is the base for your Te…
The same happens with the 13C/12C ratio, which is “diluted” by 13C rich (compared to FF) deep ocean returns.
Thus looking only at 14C and 13C/12C ratio’s give you an underestimate of the real cause of the CO2 increase in the atmosphere, for the simple reason that an exchange in isotopes without a change in total mass has nothing to do with the decay rate of an extra CO2 mass (of whatever composition) in the atmosphere.
“Therefore, you are not in a position to argue that it does not prove your hypothesis is wrong. You have not studied my papers and my rebuttal above to your CO2 Coalition paper. ”
Wow. I have studied your papers in depth and that was a reason to react here. I still wonder why a clear explanation of the current scientific knowledge does stir so much controversy and hostile comments from some in the skeptic community… Maybe because one doesn’t like the result?
Take you stance against our graph nr 11 of the uptake/release of O2 as proof that nature is a net sink.
https://www.ferdinand-engelbeen.be/klimaat/klim_img/bolingraph.gif
You say:
“Figure 11 assumes H(1) is true. On that basis it plots O2 concentration as a function of CO2 concentration, assuming human CO2 causes all the CO2 increase”
Which is a complete false accusation. Nowhere is “assumed” that humans cause all the CO2 increase. All what was plotted is the calculated O2 use and CO2 increase from burning fossil fuels at the right side, based on sales (taxes) and burning efficiency, the observed decrease in O2 and increase in CO2 at the left side and the calculated O2 production of the biosphere to fill the gap between calculated and observed O2 use. When plants absorb CO2, they release O2 in equivalent amounts, thus plants also reduced the CO2 increase in the atmosphere. The oceans filled the remainder of the CO2 uptake.
Nowhere is there a NET natural release of CO2. None. All based on observations and direct calculations from observations.
Jim Siverly
July 28, 2025 at 11:47 am
“What is clear, is that he assumed the removal rate was proportional to the difference between the pCO2 now and whatever it was in 1750. Just as Deitz, you, and others do.”
Jim, again… Neither Spencer, or I or most other skeptics of the alarmist “catastrophic” global warming used the old equilibrium as base for the 50 years decay rate of extra CO2 in the atmosphere. Only Dietze did that, because the time period to calculate Te/Tau over a longer period was rather short.
As you have calculated yourself, the Te of the NET removal rate during the Mauna Loa time indeed is 2% of the increase, over any time period that you want (beyond the year-by-year noise). That points to a exponential decay rate, where the response is linear in ratio to the increase of CO2 in the atmosphere.
One then can calculate back to the zero net decay, where the true equilibrium is. Again: we didn’t use that in reverse.
Here the plot in real order:
https://www.ferdinand-engelbeen.be/klimaat/klim_img/acc_decay.png
A is the sum of all emissions, the observed increase in the atmosphere and the influence of T on the pCO2 of the oceans. That is a small change in ΔpCO2. If you don’t like it, just drop it. If you want the absolute sinks, be my guest and do the same calculations with a full ΔpCO2 from the atmosphere to zero pCO2. Gives you either some 16 PgC/year pressure dependent CO2 output flow (with the 50 year Tau plus 195 PgC/year non-pressure dependent outflows) or a lot longer Te/Tau…
B is the calculated net sink rate per year from A: increase minus emissions. Independent of the absolute height of CO2 in the atmosphere, only dependent of the difference between increase in the atmosphere and increase in emissions.
C is the calculated Te/Tau from A and B.
The order of calculation is from A to C. Not reverse…
No need to know or calculate or estimate any (uncertain) out- or inflow from or to the atmosphere, as we have much more accurate net outflow calculations…
Ferdinand Engelbeen
July 28, 2025 at 2:55 am
“Both Spencer’s and my approach doesn’t involve inflows or outflows at all, only net flows.”
I consider that a confession. Let’s examine net flows. I claim the 50-year figure is an artifact of the assumption that the removal process is proportional to difference between now and way back when. That doesn’t sound like net flow to me. Yet the removal (decay) rate both you and Spencer use is ( Cn-1 – Co ) / 50.
How is that a net flow? That basically says net flow is the amount of annual FF emissions that nature couldn’t remove, because she only removes a fiftieth of the difference of what’s in the atmosphere now and what she remembers it was in 1750. Unbelievable.
“The 2%/year figure is the observed (!) decay rate…Thus not based on any past known or unknown “old” equilibrium… “Observed” in that way that human emissions are quite accurately known…and the increase in the atmosphere is very accurately known from measurements.”
Your “not based on any past known or unknown “old” equilibrium” is contradicted by your use of ( Cn-1 – Co ) / 50. Can’t you see that? It’s circular logic. You observe the 2% rate from the slope of the plot of delta C vs C. Then you all use the intercept as the 1750 Co value Spencer, for example, used to produce his Mauna Loa data fit. To do that you have to assume no change in natural emissions and ignore e-time for the “no change in natural emissions” scenario changing from 3.14 to 4.86 years.
“That is just curve fitting…” Of course. What do think Spencer was doing using 0.02/year and 294 ppm to fit the Mauna Loa data? One makes certain assumptions to formulate a model. One can use the wrong assumptions and formulas and still get a good fit. Which assumptions are more physically correct is the question to be answered.
Finally, for your two last paragraphs, Magic Math bookkeeping is not scientific and even if your 33% and 13% numbers were accurate to the PgC, 46% is darn close to 50%.
Ferdinand Engelbeen
July 28, 2025 at 12:42 pm
Do you deny that your spreadsheet uses “the old equilibrium as base for the 50 years decay rate of extra CO2 in the atmosphere” after you just acknowledged both you and Dr. Spencer use ( Cn-1 – C0 ) / 50 as your decay rate in your formulas?
Please don’t use e-time Te when you mean adjustment time Tau, as in, “the Te of the NET removal rate.” The jargon is confusing enough as it is. While we are on the subject, “decay” is not appropriate for the current situation with increasing FF emissions and quite possibly increasing natural emissions, as well. As I have shown, there is no need to use anything but Te of about 4 years. Furthermore, there is no data that definitively proves that the atmosphere will decay to preindustrial times with an e-time of 50 years. That is a purely speculative model-based assertion of the Tau camp.
You have shown the acc_decay.png diagram with plots A, B, and C maybe a dozen times and I never really understood how you calculated the e-folding times. Would it be possible to show the formulas? Otherwise I have no idea how you “calculate back to the zero net decay, where the true equilibrium is.”
Ferdinand Engelbeen July 28, 2025 at 11:59 am
Dear Ferdinand,
1. Dave Burton does not correctly explain the Delta14C data.
2. Your old graph does not show the information that my plot shows.
3. The return of the Detla14C to its original zero-balance level after the bomb tests proves the Delta14C balance level remained near zero even as the bomb tests increased Delta14C.
4. The bomb tests had no effect on the natural inflow of Delta14C or its balance level of zero.
5. Since 1950, increased human 12C emissions caused an inflow of carbon with a Delta14C of -1000. This sets a balance level that reflects this inflow and this human carbon balance level mixes with the natural carbon balance level of zero.
6. Your hypothesis that “about 97% returned back as mass from the deep oceans in the same year as absorbed” is neither relevant nor backed by data.
7. Your hypothesis that “From the bomb tests, only 45% of 14C as “mass” returned in the same year, the rest did just begin their ~1000-year journey…” is imaginative.
8. The natural Delta14C inflow before the bomb tests continued during and after the bomb tests. The is no basis in data to claim that this natural inflow changed.
9. Your comment that my “calculation of the “fixed” 14C balance is completely at odds with reality” is incorrect. First, I do not consider the 14C balance level. Second, I use the Delta14C balance level that exactly matches excellent data.
10. Your comment that I “haven’t taken into account the ~1000 years delay between the extra 14C uptake and the current 14C “old” release” is irrelevant and ridiculous. I use the natural and human inflows since 1950, which is all that matters.
11. You wrote, “the decay rate of Delta14C from the bomb tests is much faster than the decay rate of about 50 years for an extra input of FF CO2.”
Indeed, it is. We can measure its Te at 16.5 years with a balance level of zero, proving your 50 years is imaginary.
12. The rate return of Delta14C back to its original balance level is set by its Te. A longer Te would have slowed the outflow of carbon from the atmosphere and extended the time for Delta14C to return to its balance level.
The Te of the return (16.5 years) rules out your hypothesis that Te (or Tau) is 50 years.
13. You commented that Delta14C “and 13C/12C ratio’s underestimate of the real cause of the CO2 increase in the atmosphere, for the simple reason that an exchange in isotopes without a change in total mass has nothing to do with the decay rate of an extra CO2 mass (of whatever composition) in the atmosphere.”
Your comment is not true. The Delta14C balance level is the combined balance level set by the Delta14C inflow with a balance level of zero and the human carbon inflow with a balance level of -1000. Delta14C data show the contribution of human carbon is very small.
You bring up an additional point separate from the above discussion:
This regards your Figure 1.4.2:O2-CO2trends 1990–2000, figure from the IPCC TAR (2003), that is also my draft Figure 11.
You wrote, “Nowhere is it “assumed” that humans cause all the CO2 increase.”
Well, this figure has an arrow titled “fossil fuel burning” that goes from 352 ppm to 382 ppm, and an arrow titled “atmospheric increase” that goes from 352 ppm to 367 ppm.
These arrows assume human CO2 emitted between 1990 to 2000 caused the CO2 level to increase from 352 ppm to 367 ppm.
Without this assumption, the figure should have named the arrows to, say, “total human CO2 inflow” and “net human CO2 inflow.”
The figure does not allow that natural CO2 might have caused some of the CO2 increase.
Jim Siverly, July 28, 2025 at 2:19 pm
“I claim the 50-year figure is an artifact of the assumption that the removal process is proportional to difference between now and way back when. ”
Sorry, my fault. Indeed figure C of the graph I sent is the calculation relative to the “old” equilibrium plus a small increase in equilibrium, due to the increased SST since 1850.
1. Let us begin with clearing some confusion about the differences in definitions…
Te (Berry) and τ/Tau (general) and Ta (IPCC) meanings are exactly the same: all assume an exponential decay rate for any change in one of the reactants of a process in equilibrium. That is the time needed to reduce the change to 1/e of the original disturbance in all three definitions. Only the exact figures are quite different.
In the case of Te, Berry and several others and you use the 4 years turnover/residence time, where a one-shot extra CO2 in the atmosphere follows the same decay speed as the overall output rate, which only is allowed if (and only if!) all outflows are unidirectional and no back flows are involved ánd if (and only if!) all outflows are driven by the CO2 pressure in the atmosphere. In that case Te = C/Fout and thus Fout = C/Te (equation 2 of Berry).
For any single CO2 molecule in the atmosphere, the turnover/residence time is always applicable, but only in that case the CO2 mass transfer is the same as the CO2 molecule transfer and both Te and Tau are equal:
https://www.ferdinand-engelbeen.be/klimaat/klim_img/mass_fluxes_classic.jpg
In the case of τ/Tau, that is the time needed to remove 1/e of the disturbance relative to the (“original”) equilibrium for a one-shot impulse.
If (and only if) the net (important: net!) output is in exact ratio to the change in disturbance, it doesn’t matter over what time frame Tau is calculated, even if the equilibrium is not known at all.
That is the case for both the oceans and vegetation (up to over 1,000) ppmv. Thus one doesn’t need the “original” equilibrium to know Tau.
Only in the case that all flows are unidirectional and all outflows are CO2 pressure related, then Tau = Te. In all other cases, Te and Tau are completely independent of each other:
https://www.ferdinand-engelbeen.be/klimaat/klim_img/mass_fluxes_real.jpg
The Ta of the IPCC’s Bern model is model based and assumes saturation of all reservoirs for CO2, which is only the case for a large part of the ocean surface. Not to over 1,000 ppmv for vegetation and not for the next thousands of years for the deep oceans, even if we burn lots of coal, oil and gas…
In all cases, it doesn’t matter that the input is a one-shot amount of FF CO2 or a continuous, increasing, supply of FF CO2. The (net) removal rate is a function of the total amount of CO2 in the atmosphere (for Te) or the difference between the total amount of CO2 in the atmosphere and the (old) equilibrium, thus the total increase over time, not only the extra input of one year.
2. The calculation of Tau.
If (and only if!) the net (!) output is directly proportional to the extra level in the atmosphere, then the formula for Tau is quite simple:
Tau = disturbance / effect
Or in the case of the atmosphere and what I used in graph C for each year:
Tau = [pCO2(atm) – pCO2(eq)] / net output
Where net output = outputs – inputs = FF emissions – increase in the atmosphere (= graph B).
As we know the FF emissions and the increase in the atmosphere quite exactly, there is no need to know any inputs or outputs or the sum of these, to know the net output with high accuracy.
Indeed I used the “old” equilibrium, but because the net outputs in this case are in exact ratio to the height of the change, one can use the current changes to obtain the current Tau.
That makes that one can calculate Tau from the accurate data from 1958 on.
Here the “old” equilibrium based Tau’s compared to the 10-year moving “modern” Tau’s.
Calculated with Tau(n) = [C(n) – C(n-9)] / [Fnetout(n) – Fnetout(n-9)]:
https://www.ferdinand-engelbeen.be/klimaat/klim_img/decay_10.png
The modern Tau seems to be somewhat higher (~55 years) in average than the “old” Tau, based on the old equilibrium. The old Tau was calculated on the polynomial through the net sink rates, that gets of course a lot less noise…
This calculation of Tau doesn’t involve any knowledge of CO2 in- or outfluxes or “old” equilibrium or natural/FF CO2 ratio…
Even if the atmospheric in- and outflows doubled over time (halving the turnover/residence time and thus your Te), that has zero influence on the net removal rate of CO2 out of the atmosphere, because only the difference between inflows and outflows is what changes the total mass of CO2 in the atmosphere, not how much comes in and goes out. The latter only determines the net outflow in the case that all outflows are CO2 pressure related, which for 95% is not the case in the real world…
As the 50-55 year Tau is clearly working over the full 170 years of human emissions, up to now, I don’t see any reason to assume that without human emissions the CO2 levels will not return back to the old equilibrium.
Except if you accept the hundreds to thousands years of the IPCC’s Bern model…
Dr. Ed, July 28, 2025 at 5:11 pm
1. “Dave Burton does not correctly explain the Delta14C data”.
There is some discussion with David Andrews, but David Andrews completely rejects your calculations. So, who is right? I am more confident in Andrews’ calculations than in yours, because of my past experiences…
2. “Your old graph does not show the information that my plot shows”.
It shows information that is not in your plot: that a large inflow of ~1,000 year old CO2 from the deep oceans returns today to the atmosphere. That CO2 is not contaminated by atomic bomb tests and fossil fuel emissions and thus is an important input that “dilutes” the δ13C and Δ14C input from the current FF input.
3.-5. “The return of the Detla14C to its original zero-balance level after the bomb tests proves the Delta14C balance level remained near zero even as the bomb tests increased Delta14C.”
The return is not yet ended and will go to below zero for Δ14C.
6.-7. “Your hypothesis that “about 97% returned back as mass from the deep oceans in the same year as absorbed” is neither relevant nor backed by data.”
The data for 1960 show a net uptake by oceans and vegetation, of which about 25% is absorbed by the biosphere, 5% by the ocean surface and 20% by the deep oceans.
The “dilution” of the δ13C data (and estimates from deep ocean sinks) shows a deep ocean-atmosphere flux of about 40 PgC/year. The level of Δ14C some 1,000 years ago was about 50% of the bomb tests and with a 10% radioactive decay that gives a 45% return of what did go into the deep oceans in 1960.
These were not exact worked-out figures, but give an indication of why the decay rates of δ13C and Δ14C are much faster than of the bulk mass removal of any extra CO2 injected in the atmosphere.
8.-10. Not relevant.
11.-12. “The rate return of Delta14C back to its original balance level is set by its Te. A longer Te would have slowed the outflow of carbon from the atmosphere and extended the time for Delta14C to return to its balance level.
The Te of the return (16.5 years) rules out your hypothesis that Te (or Tau) is 50 years.”
Again, you are mixing the decay of an isotopic ratio with the decay of the CO2 mass. The isotopic ratio is “thinned” by deep ocean returns, which you (and the IPCC) have not in your (or their) calculations. That makes that the Te for the isotopic ratio’s of δ13C and Δ14C are faster than for the total extra mass of CO2 in the atmosphere. The faster decay of the isotopic ratio’s in no way rules out the observed decay of 50-55 years of the extra CO2 mass in the atmosphere, but it rules out the 4 years Te, which has no bearing at all in the net (!) removal rate of any extra CO2 injected in the atmosphere.
13. “Your comment is not true. The Delta14C balance level is the combined balance level set by the Delta14C inflow with a balance level of zero and the human carbon inflow with a balance level of -1000”
And an input of ~1,000 year old pre-industrial CO2 with higher δ13C and lower Δ14C than in the current atmosphere…
Then we have figure 11 as example how your comments are at odds with what we did write:
“Well, this figure has an arrow titled “fossil fuel burning” that goes from 352 ppm to 382 ppm, and an arrow titled “atmospheric increase” that goes from 352 ppm to 367 ppm.
These arrows assume human CO2 emitted between 1990 to 2000 caused the CO2 level to increase from 352 ppm to 367 ppm.
Without this assumption, the figure should have named the arrows to, say, “total human CO2 inflow” and “net human CO2 inflow.”
The figure does not allow that natural CO2 might have caused some of the CO2 increase.”
“fossil fuel burning” and “total human CO2 inflow” are equivalent. So far so good.
“atmospheric increase” is what is measured at Mauna Loa and is “net human CO2 inflow.” as mass, as that is what remains as human “caused” CO2 mass in the atmosphere.
“The figure does not allow that natural CO2 might have caused some of the CO2 increase.”
That would be quite difficult: if FF CO2 is already larger than what is measured as increase in the atmosphere, then the natural flows only can net (!) remove CO2 out of the atmosphere.
What doesn’t imply that all human FF molecules still reside in the atmosphere. Even if all FF CO2 molecules were exchanged with “natural” CO2 molecules, that doesn’t change the fact that the increase in CO2 mass is fully caused by the FF emissions…
Thus we didn’t “assume” that FF emissions are the cause of the increase. The data show that the increase (as mass!) in the atmosphere is caused by the addition (as mass!) of more FF CO2 than is observed as increase in the atmosphere…
Ferdinand Engelbeen
July 29, 2025 at 4:23 am
I appreciate your interpretation of the e-time terminology. I would prefer and recommend we make a distinction between Te which is based on absolute concentrations, as I and Dr. Ed use, and Tau based on net flows proportional to century-long concentration differences. Can we agree on that for the purposes of these discussions?
My second request is that you explain where you find the rule that my 4-year e-time is only “allowed if (and only if!) all outflows are unidirectional and no back flows are involved?” A standard two-compartment model of a one-time pulse will always have bidirectional flows between the compartments. My model doesn’t show the second compartment, but that is where the “mostly” natural inflow to the atmosphere is coming from. If you like, I will link to a website demonstrating how the equations are derived.
Next, do you have an experimental evidence of “Classic view 2020” shown in your first link?
Can we agree that saturation of any reservoir is irrelevant because surface ocean CO2 goes to the deep and from there to the ocean floor? Nothing stops those processes continuing to infinity, if you apply realistic physical-chemical processes.
Also, can we agree that your hypothesis of net removal rate being a function of the difference between the total amount of CO2 in the atmosphere and the (old) equilibrium has never been experimentally verified?
Thank you for the calculation of Tau details. I will work on that and respond back soon.
Ferdinand Engelbeen
July 29, 2025 at 4:23 am
I previously and mistakenly assumed your smooth curve generating old Tau came from the linear fit you showed in graph B. Now you say it comes from a polynomial regression. Is that of Fnetout values? I was not able to duplicate your modern Tau data in your second link. What are the formulas for Fnetout(n) and Fnetout(n-9)?
I am trying to verify your claim, “This calculation of Tau doesn’t involve any knowledge of CO2 in- or outfluxes or “old” equilibrium or natural/FF CO2 ratio…” Until then, I prefer not to respond to the rest of your comment.
Ed,
Even the Trump Administration disagrees with you:
“The annual increase in concentration is only about half of the CO2 emitted because land and ocean processes currently absorb “excess” CO2 at a rate approximately 50 percent of the human emissions. Future concentrations, and hence future human influences on the climate, therefore depend upon two components: (1) future rates of global human CO2 emissions, and (2) how fast the land and ocean remove extra CO2 from the atmosphere. We discuss each of these in turn.”
Climate Working Group (2025) A Critical Review of Impacts of Greenhouse Gas Emissions on the U.S. Climate. Washington DC: Department of Energy, July 23, 2025
releasesd 7/29/25
Jim Siverly, July 29, 2025 at 10:42 am
Sorry, my fault: graph B indeed shows the linear fit, not the original polynomial that was used for the calculation. Doesn’t make much difference, except at the start and end of the calculations. Here the right one with the polynomial formula:
https://www.ferdinand-engelbeen.be/klimaat/klim_img/acc_decay.jpg
just replace the .png of the previous one with the .jpg of this one. I didn’t think about it, because the former one had the A/B/C marks and this one not…
That was used for the calculations of the “old” Tau’s, relative to the original equilibrium + the pCO2(eq) increase caused by the increase in SST since 1850, per formula of Takahashi.
The new Tau’s are based on the change in pCO2 of the atmosphere at Mauna Loa over a period of 9 years (not 10, was too much in a hurry, I suppose):
Tau(n) = [C(n) – C(n-9)] / [Fnetout(n) – Fnetout(n-9)]
Where C(n) and C(n-9) are the CO2 levels at Mauna Loa of the current year and 9 years back, thus the increment of CO2 in the atmosphere.
Fnetout(n) and Fnetout(n-9) are the calculated (raw) differences between increase in the atmosphere and human emissions for the current year and 9 years back in time, thus the net increase in absorption of CO2 mass, somewhere in other reservoirs, caused by the increase in concentration of CO2 in the atmosphere.
If the net removal of CO2 is directly proportional to the increase of CO2 in the atmosphere, then it doesn’t matter over what period one calculates Tau, even without any knowledge of the “old” equilibrium or the exact in- and outflows. One need only the observed increase of the pCO2 in the atmosphere and the calculated net uptake…
Jim Siverly, July 29, 2025 at 9:48 am
1. Agreed to use Te and Tau as separate things, to keep the discussion clear.
2. About the “no back-flow” point. That was used for a one compartment model: Many (Koutsoyiannis, Harde, Stallinga,… use the “lake/bath tube/container” model with one-directional inflows, container and one-directional outflows:
`Koutsoyiannis does that very detailed in his RRR approach, which indeed is superb for… lakes, which is his specialization:
https://www.itia.ntua.gr/en/getfile/2474/1/documents/water-16-02402-v2.pdf
Which is way over my head for his formula use, but when the result seems to prove that 2 + 2 = 3, something in his reasoning doesn’t fit reality…
I used a simple “river/lake/river” as example of such a process:
https://www.ferdinand-engelbeen.be/klimaat/klim_img/classic_view.png
Important points:
In that case Te = Tau = mass / output = RT (turnover/residence time)
In no way a “marker” in the inputs can exceed its input ratio in all following parts.
For a one compartment process, if you have back-flows, the situation is completely different:
I used the “fountain” model to make the difference as clear as possible:
https://www.ferdinand-engelbeen.be/klimaat/klim_img/real_view.png
Important points:
In that case, RT and Tau are completely decoupled and Te hardly exists as function of the pumping around, despite an enormous outflow of the basin into the cycling pump.
RT is a function of a process that is not or hardly influenced by the level/pressure in the basin
Tau is only influenced by the extra level in the basin and not at all by how much circulates, thus completely independent of the total “outflow” out of the basin.
Any “marker” in the input can asymptotically increase to 100% of the basin.
That was for the discussion in Athens, September last year…
In the case of multiple reservoirs, that makes it more complicated, but then the second problem: processes that move CO2, independent of its pressure in a reservoir, again plays a major role, compared to the small effect of CO2 pressure changes…
If we may assume that “earth” is a second reservoir where all outflows of the atmosphere go in and come back as similar inputs then we have:
Your 4 years Te is based on the calculated total sum of outputs out of the atmosphere into “earth”.
Your inputs are back-calculated to show the observed difference between the previous year and the new year CO2 concentration, based on the 4 years Te of the outflows.
Dr. Ed calculates the total inflow from the different Te’s of the different reservoirs, but also assumes that the outflow into “earth” is 4 years, based on the total outflows.
We don’t use inflows and outflows at all, only the extra pressure difference in the atmosphere with the “old” equilibrium and the observed net sink rate, or for modern times the ratio between the increments of CO2 pressure in the atmosphere and the observed increase in net sink rate. That gives the Tau of 50-55 years.
So which approach is the right one?
Te could be the right one, if all processes were 100% CO2 level/pressure dependent.
Because 95% of all processes that move huge quantities of CO2 (mainly diurnal and seasonal) between the reservoirs are temperature and biological dependent, Tau is the right one, as that shows the real, small, effect of the increase in CO2 pressure in the atmosphere on the change in net uptake.
3. “Can we agree that saturation of any reservoir is irrelevant because surface ocean CO2 goes to the deep and from there to the ocean floor? Nothing stops those processes continuing to infinity, if you apply realistic physical-chemical processes.”
Be careful, the ocean surface indeed gets saturated for any change in the atmosphere within a year and the IPCC uses that for the “isolation” of the deep oceans from the atmosphere, which leads to Tau’s of hundreds to thousands of years.
You do that indirectly: with a Te of 4 years, the “equilibrium” between atmosphere and “earth” also goes up with the increase in the atmosphere and when human emissions should stop completely, there is only a small change to a lower level…
4. “Also, can we agree that your hypothesis of net removal rate being a function of the difference between the total amount of CO2 in the atmosphere and the (old) equilibrium has never been experimentally verified?”
As the ratio between increase in the atmosphere and increase in net uptake remained and remains relative constant, the real world proof is given…
David Andrews July 30, 2025 at 5:49 am
David,
Thank you very much for this information.
Indeed, President Trump may lose the Lighthiser v Trump climate lawsuit on this very issue, which could overturn all his executive orders on climate and energy.
The courts properly make their decisions based on the data presented in a trial, as the court did in Held v Montana.
So, you and the Democrats could win a smashing big victory in Lighthiser v Trump because, at the moment, your side and the CO2 Coalition have Trump convinced that you are correct.
Dr. Ed, July 28, 2025 at 5:11 pm
A small addition about the O2/CO2 plot:
“The figure does not allow that natural CO2 might have caused some of the CO2 increase.”
The figure only shows the calculated release of CO2 and use of O2 from burning fossil fuels, the uptake of CO2 and release of O2 by the biosphere and the results of observations, where the oceans fill the gap of the remaining uptake.
There is no restriction at all that blocks the results of any natural release of CO2: if that was additional to the release by FF use, then the observations would be at the right side of the FF release of CO2.
If e.g. the biosphere was a net emitter of CO2 and not a net sink, then the oxygen use would have been larger than of FF only…
Ferdinand Engelbeen
July 30, 2025 at 6:41 AM
I see the equation on graph B is a polynomial fit and that the line just looks linear. That’s my fault, but it has nothing to do with my confusion over your calculation of new Tau from n and n-9 data. I generated a new spreadsheet to show my calculations. What am I doing wrong?
https://www.dropbox.com/scl/fi/6e7f6y10gukegc7vqk28v/New-Tau.xlsx?rlkey=8e1qnz65yi5at2ztdb24ir1fj&dl=0
I also tried plotting net removal of CO2 [delta CO2 – FF emissions] versus the increase of CO2 in the atmosphere [delta CO2] and found a very low correlation, not even close to being directly proportional. Please define the variables more clearly in terms of Mauna Loa data and known FF emissions.
Jim Siverly, July 30, 2025 at 11:23 am
Indeed you are right: my calculation (again) was based on the 10-year increments, compared to the “old” equilibrium of 285 ppmv for each year, not on the difference over 10 years in itself…
The problem is that there is an enormous – temperature induced – noise on the net sink rate, which is small: +/- 2 ppmv for the 100 ppmv trend, but very huge for a 5 ppmv/year increase.
So forget the 10-year increments, these remain too noisy.
What is left is the total increase since 1958 with accurate measurements and reasonable accurate FF emissions.
That is an increase of near 100 ppmv (414.24 – 315.97) between 1959 and 2020, with a net sink rate increase over the same time frame of 1.86 ppmv/year (2.29 – 0.44), based on the polynomial to avoid the influence of the huge noise.
Interesting to note: the increase of CO2 in the atmosphere since 1958 is already 80% of the total increase increase since 1850. That also is the case for about 80% of human emissions ánd also the case for 80% of the net sink rate.
Which points to a very linear ratio as response of the sinks to the extra CO2 in the atmosphere.
A small point of difference: I wondered about the difference between the 50 and 55 years Tau between the “old” and “new” calculations. That difference is in what is taken as human emissions:
Spencer and others and I in the past used the total emissions, including land use changes. In our common work for the CO2 Coalition we used only FF emissions, without far more uncertain land use changes. That gives a difference of about 10% in human emissions and Tau…
Ferdinand Engelbeen
July 31, 2025 at 2:29 am
I have been reviewing the evolution of Carbon Cycle Modeling starting with the comments of Peter Dietz on the John Daly website. The dialogue is fascinating considering some of our discussion here.
Tom Wigley in response to Peter Dietz: “It is, of course, quite possible to produce highly simplified (or ‘reduced-form’) versions of these models; but the original ‘first principles’ model must always be better and more credible than the simpler model. The real strength of first principles models is that they do not have free parameters that can be arbitrarily tuned to match observations.”
Peter responds to Tom Wigley, “The clue of my model is that I only consider the *change* in uptake flux which is proportional to the CO2 partial pressure increment against the equilibrium, e.g. 280 ppm. This reflects the acting main physical and chemical laws and thus is valid for the bulk of the sink fluxes….”
“It is remarkable that both Peter Dietze and Jarl Ahlbeck find that the present equilibrium value is the same as the pre-industrial. Shouldn’t the increase have started already? Best regards, Lars Kamél”
Peter responds, “Lars, you should not misinterpret the perturbation state as a shift of equilibrium. We consider the (final) equilibrium towards which any concentration increment will “decay” to 1/e within 55 years, as constant, i.e. 280 ppm. The equilibrium is *not* expected to shift within the geological short time of fossil fuel use – consider the huge oceanic sink capacity including shells and corals.”
Jarl Ahlbeck, “If the diffusion is controlled by a first order Fick’s equation to a greater extent than by the second order, the absorption rate is almost directly proportional to C(surf.carbon)-C(deep layer carbon) or to C(atm)-280 ppm. The rate constant may be calculated statistically from available data. The problem is the intercorrelation between emissions and concentration making an idealized batch describing recent data almost as well (constant airborne fraction) as when taking significant mass transfer to deeper layer into account (variable airborne fraction). I have walked around the problem by calculating the airborne fraction from data although I do not like this parameter, it feels a little unscientific.”
I found this interesting reference by Enting et al., 1994: https://unfccc.int/resource/brazil/documents/enting_2001a.pdf with the following quote,
“These fluxes are net fluxes superimposed on large natural gross fluxes of carbon. In many cases the models will consider the gross fluxes, particularly when calculating isotopic effects. The distinction is well-understood within the field of carbon cycle modelling, but sometimes causes confusion when results are reported.”
That last phrase is an understatement considering the exchanges you are having with Dr. Ed and some of us here.
From the Enting et al. paper comes the equation 2.123 dC/dt = Qfoss (t) + Dn (t) – Socean (t) – Sfert (t) – Sresid (t)
Except for Qfoss, the fossil emission rate, all other terms are net fluxes. One of the co-authors of that paper was Tom Wigley. He wrote an earlier paper with a similar equation, 2.123 d(deltaC)/dt = I + Dn – X – F, where I = Qfoss, X = Sfert, and F = Socean. He defines deltaC as C – Co. Since dCo/dt is zero, his equation is a form of Enting et al. Technically, each of the net flux terms should be written as k * S(r-a) – k’ * S(a-r) to reflect the true mass transfer process between the atmosphere and the reservoirs.
Wigley defined X as a terrestrial feedback (transfer from the atmosphere to the terrestrial biosphere). While Dn – X accounts for the “full” net terrestrial transfer, there is no similar appreciation for the effect of ocean outgassing which would moderate the value of F, the ocean sink. Without a +Y to offset F, analogous to X offsetting Dn, an important variable becomes embedded in the net variable F. I believe this leads to the misconception that flows are proportional to net concentration differences. The most egregious misuse of that concept is in assuming a net flux is proportional to the difference between a concentration now compared to what it was in 1750. That is the basis of the models of Dietz, Spencer, et al.
My contention is that flows are proportional to absolute concentrations which takes into account the probabilities affecting the mass transfer from one reservoir to another. Net fluxes erase that fundamental aspect. Now I just need to find the experimental evidence that proves it.
Sorry, the Enting et al. equation comes from https://unfccc.int/resource/brazil/documents/enting_2001a.pdf
Wigley’s paper can be found at https://b.tellusjournals.se/articles/10.3402/tellusb.v45i5.15739
Jim Siverly, July 31, 2025 at 5:43 am
Thanks for remaining me of the discussion between Peter Dietze and the others from the “mainstream” side…
There still are two main differences with the mainstream “model” and our “model” since that time:
1. The saturation of all reservoirs at different levels, no matter the emissions in the Bern and similar models:
https://gmd.copernicus.org/articles/11/1887/2018/
See Figure 5 for a pulse of 100 PgC (we have emitted 170 PgC over the past 65 years…): after 100 years still 40% remains in the atmosphere, because of the isolation of the deep oceans by the ocean surface layer and its physical and chemical restrictions for exchanges with the atmosphere.
Also in the Enting paper: formula 9.2 shows:
GO (t) = 0.131 + 0.201*e^-t/362.9 + 0.321*e^-t/73.6 + 0.249*e^-t/17.3 + 0.098*e^-t/1.9
The last term is the restricted uptake into the oceans surface at about 10% of the change in the atmosphere (the “Revelle”/buffer factor) and that restricts the speed of uptake for the other levels. The worst ones are the 363 years (for calcification?) and 13% that never gets away.
That is reflected in their Figure 9.2…
For a linear model (Dietze, Spencer, we) there is no such restriction (at the ocean “edges”) and in 100 years the airborne fraction will drop to 25%, after 150 years to 12.5%, etc…
Because we now have far more data than in 1997, it looks like that the Bern model is more and more at odds with reality, but still not (too) bad, as the last three terms are faster or similar than the constant t/50 of Dietze and others.
With a linear model, all human releases since 1750 will increase the deep ocean C content with only 1%, leading to an ultimate remaining increase of 1% in the atmosphere or 3 ppmv…
2. The difference between pressure related processes and other processes.
Almost all huge (seasonal) CO2 exchanges between the reservoirs are caused by temperature related processes and related biological processes, largely independent of the CO2 level in the different reservoirs.
Your (and Dr. Ed’s) approach implies that the total output depends of the total CO2 level/pressure in the atmosphere, and the inflows depend of the CO2 levels in the other reservoirs. Which, at least for the biosphere is for 95% not true for the CO2 flow from atmosphere into vegetation and for 100% not the case for the reverse flow…
Then: “Net fluxes erase that fundamental aspect.”.
The problem in this case is that the gross fluxes are only known with large margins of error, while the net flux is quite accurately known, be it with a huge, temperature related, noise…
Jim Siverly July 31, 2025 at 5:43 am
Dear Jim,
You wrote, “My contention is that flows are proportional to absolute concentrations which takes into account the probabilities affecting the mass transfer from one reservoir to another. Net fluxes erase that fundamental aspect. Now I just need to find the experimental evidence that proves it.”
There are two kinds of “evidence” that prove your contention for carbon cycle models.
First, the simple observation (that you have already made) that essential information contained in absolute concentrations is lost in the net fluxes. Net fluxes don’t control flows, but absolute concentrations do.
Second, it is well established in systems engineering that, to simulate natural processes, levels drive flows, and the flows reset the levels. This method of simulation retains the necessary information to simulate nature. Markov processes follow this systems methodology.
In addition, the fact that my carbon cycle equations can be reduced to proven electric circuit equations is further evidence that using absolute concentrations is the correct way to simulate reality.
By contrast, Ferdinand and company try to create a “model” based on net flows. This is impossible because net flows do not contain the information needed for a systems model.
It is OK to have observations for net flows assuming they are accurate. But such data must be used to check valid carbon cycle models that are based upon levels and resulting flows, but not to create a model.
Similarly, Ferdinand and company try to create a “model” based on uptakes. Uptakes don’t drive flows in a carbon cycle model. Natural processes effectively blow carbon through a straw rather than suck up carbon with a straw. Arguments based on uptakes are invalid.
Consider Dalton’s Law of Partial Pressures. The number of carbon atoms that impact a surface in a given time is proportional to their level (or partial pressure). This is best modelled by assuming these impacting atoms are absorbed but the absorber also sends back carbon according to the free carbon level in the absorber.
At equilibrium, the flows in each direction are equal. The flows continue but the level is constant. I think part of Ferdinand and company’s misuse of physics is that they do not understand this simple concept. They think they need a 50-year time constant to explain what happens when the flows are equal and opposite. They don’t get it that a 4-year time constant fully explains the data.
Dr. Ed, July 31, 2025 at 9:24 am
“Second, it is well established in systems engineering that, to simulate natural processes, levels drive flows, and the flows reset the levels.”
and
“Natural processes effectively blow carbon through a straw rather than suck up carbon with a straw. Arguments based on uptakes are invalid. ”
Which is not true for the biosphere: the CO2 flows are driven by temperature and sunlight, largely independent of the CO2 level in the atmosphere. Even reduces the level in the atmosphere when increasing from minimum in early spring to maximum in summer. Thus effectively “sucking” CO2 out of the atmosphere.
More than half (120 PgC/season on a total of 200 PgC in/outflows over a year) of the total outflow out of the atmosphere is driven by the by plant biochemistry of the biosphere. Hardly influenced by the CO2 level/pressure in the atmosphere.
Even so, half of that already returns the same day (!) at night, by soil bacteria and plant respiration, completely independent of what is already in the atmosphere. The other half returns when leaves fall down in fall/winter and are decaying or when vegetation is eaten by animals.
Only some 2.5 PgC/year is the net uptake, directly related to the CO2 level in the atmosphere…
And we fully understand what a dynamic process means. That means that any extra CO2 input in the atmosphere is removed by a Tau that depends of the difference between the disturbance and the equilibrium level. Not the absolute level. That is Le Chatelier’s principle.
Except if you know exactly what the inflows and outflows are, based on the absolute levels of all reservoirs. Which is quite problematic between the different reservoirs in the real world, while the net difference between the inflows and outflows is quite exactly known…
Dr. Ed
July 31, 2025 at 9:24 am
Dr. Ed,
Those decades-old conversations I quoted earlier indicate the carbon cycle modelling community have been group thinking without applying the Dalton’s Law principles. Even the recent Roy Spencer model employs those exact same assumptions. Seems as though not much has changed since 1994.
I should have pointed out that your bomb decay analysis is retrospective experimental evidence as is your model data which should have satisfied Ferdinand and company as sufficient proof. Are you aware of any research that carried out a prospective experiment to make the case that net fluxes don’t control flows? It may take more than a good argument to win court cases.
Jim Siverly, July 31, 2025 at 10:18 am
|
“I should have pointed out that your bomb decay analysis is retrospective experimental evidence”
Not at all… If you know that there is a huge difference between mass flows and isotopic ratio flows…
As mass, about half human FF emissions per year are removed by other processes.
As isotopic ratio, about 2/3 of human FF emissions per year are replaced by other processes.
Or the difference between removal and replacement (without removal)…
And I still am waiting for the explanation why the 14C decay rate is about 12 years and not yours 4 years. if all carbon/CO2 is equal…
Jim,
Also, there is all kinds of evidence for Te. There is zero evidence for Tau. Te is in keeping with how nature works, for instance, the gravitational constant, the equivalence principle, the gas laws, or the laws of conservation. Te includes Euler’s number in its derivation. Tau is a mathematical construct of a model that is continuously changing. God does not work that way. Tau is something Ferdinand and David believe, not something they’ve observed. They believe in the Science of Peter Pan.
Ferdinand,
Dr. Ed has already explained that in his paper. Because the 14C curve is 100% 14C. Nature is 98% 12C. Read his paper.
Ferdinand Engelbeen
July 31, 2025 at 8:26 am
The Bern model has been sufficiently discredited and needs no further trashing. What I have a problem with is your model, deltaC = Eh – (Cn – Co) / Tau, versus my model, deltaC = Eh + En – Cn / Te.
Please address the following:
You wrote, “Almost all huge (seasonal) CO2 exchanges between the reservoirs are caused by temperature related processes and related biological processes, largely independent of the CO2 level in the different reservoirs.”
And stated so many times, as yet without proof. Temperature affects transport rates but doesn’t drive them. They are driven by the magnitude of the reservoir concentrations.
Also, how can a net flux be quite accurately known, if the gross fluxes from which they are calculated are only known with large margins of error? Error propagates.
Stephen P Anderson
July 31, 2025 at 12:07 pm
Stephen,
People aren’t arguing against the Te, but rather that both Te and Tau are in play. As Dr. Ed wrote, “They think they need a 50-year time constant to explain what happens when the flows are equal and opposite. They don’t get it that a 4-year time constant fully explains the data.”
Still, explaining that, as Dr. Ed has done over and over, seemingly makes no difference to Ferdinand and David. As you wrote, it’s something they believe. But they believe that they HAVE observed Tau.
I won’t be going back to the lab bench, but what I think what we need is to find some data that proves Tau is the construct we think it is. Otherwise the issue won’t be going away, at least in my lifetime. As Tom Wigley wrote to Peter Dietz, “I’m sure you will realize that we cannot wait for 50 years to see whether you are right.” That was twenty-four years ago.
Dr Ed
July 31, 2025 at 9:24 am
“At equilibrium, the flows in each direction are equal. The flows continue but the level is constant. I think part of Ferdinand and company’s misuse of physics is that they do not understand this simple concept. They think they need a 50-year time constant to explain what happens when the flows are equal and opposite. They don’t get it that a 4-year time constant fully explains the data.”
Yes and when all else fails blame the deep oceans.
FE: ” after 100 years still 40% remains in the atmosphere, because of the isolation of the deep oceans by the ocean surface layer and its physical and chemical restrictions for exchanges with the atmosphere”
Stephen P Anderson, July 31, 2025 at 12:12 pm
“Dr. Ed has already explained that in his paper. Because the 14C curve is 100% 14C. Nature is 98% 12C. Read his paper.”
I have read his paper…
Te and Tau have the same scientific meaning: the e-fold decay rate for some extra injection of CO2 in the atmosphere.
Te for the bulk of 12/13CO2 is 4 years, according to Dr. Ed, Tau is 50 years, according to the rest of the scientific world, with a few exceptions.
Dr. Ed says that the Tau of 50 years is rejected, because the observed Tau of Δ14C from the atomic bomb tests is only 12 years. Which I explained by the facts that both the supply of 14C-free FF CO2 and the return of pre-bomb 14C (and pre-industrial 13C/12C ratio) via the deep oceans “dilute” the 14C (and 13C) “fingerprint”, making the decay of an extra impulse of 14CO2 faster than for the injection of some “bulk” 12/13CO2.
I didn’t receive any explanation why the decay rate of Δ14C is not 4 years as it should be for any CO2 molecule, whatever its isotopic composition…
Brendan Godwin, July 31, 2025 at 5:01 pm
“Yes and when all else fails blame the deep oceans.”
What you quoted is what the IPCC says, not what we said. To the contrary, we totally disagree with that concept and the resulting “hundreds to thousands” of years that the extra CO2 remains in the atmosphere.
“They think they need a 50-year time constant to explain what happens when the flows are equal and opposite.”
That is not what we said or even implied.
When the equilibrium is reached, of course the flows are equal and opposite, no matter if these flows are 10 or 100 or 1000 PgC/year.
Our difference in opinion is for when there is a disequilibrium: because humans inject nowadays some 10 PgC/year extra into the atmosphere. The question then is, how fast the 10 PgC/year is removed to get back to equilibrium.
According to Dr. Ed, that is 4 years. According to the increase in the atmosphere, that is 50 years…
Jim Siverly, July 31, 2025 at 12:41 pm
“The Bern model has been sufficiently discredited and needs no further trashing. What I have a problem with is your model, deltaC = Eh – (Cn – Co) / Tau, versus my model, deltaC = Eh + En – Cn / Te.”
Agreed, but you use the arguments of Wigley, Joos and others, while these “mainstream” scientists are arguing that the 50 years Tau for the decay of the extra CO2 is much too short (!) to “prove” that our 50 years Tau is much too long…
Then your formula.
deltaC = Eh + En – Cn / Te
In the real atmosphere as observed for 2020:
5 PgC/yr = 10 PgC/yr + En – 900 PgC / 4 yr = 10 PgC/yr + En – 225 PgC/yr
En = 220 PgC/yr
For 1960:
1 PgC/yr = 2 PgC/yr + En – 670 PgC / 4 yr = 168 PgC/yr
En = 167 PgC/yr
Because of the equivalency principle, one would expect that the removal of the extra 10 PgC would be 25% per year, not 50% per year… And the increase in inflow and outflow doesn’t match the observed increase in flows, according to the IPCC.
Our formula gives:
delta C = Eh – A(net)
where delta C is measured (Mauna Loa) and Eh is calculated from FF sales.
A(net) = 10 PgC/yr – 5 PgC/yr = 5 PgC/yr for 2020
No matter what En or Te or Tau is.
If A(net) is directly proportional to the extra CO2 above equilibrium, then delta A(net) is directly proportional to delta C (but be aware of the temperature induced huge noise in delta C), even without knowing or using that equilibrium:
An increase of near 100 ppmv (414.24 – 315.97) between 1959 and 2020, with a net sink rate increase over the same time frame of 1.86 ppmv/year (2.29 – 0.44), based on the polynomial to avoid the influence of the huge noise.
Tau = delta C / delta A(net)
Tau = 98.27 ppmv / 1.86 ppmv/yr = 52.8 yr
That is the calculated Tau, based on real life observations for the CO2 increase in the atmosphere, the calculated FF emissions, based on taxes on sales, and the change in difference between these two over a period of 60 years. Representing some 80% of all emissions, 80% of the increase in the atmosphere and 80% of the increase in both delta C and A(net).
The ~50 year Tau fits all the observed data…
To understand C14 dynamics and the bomb pulse, you all need to look at the trajectory of the atmospheric concentration of C14 as well as the trajectory of DeltaC14. See figures 2 and 4 of Presentation of Radiocarbon data
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/193CDF1F705B269BC975AF178CEF1AC3/S0033822224000274a.pdf/discussion-presentation-of-atmospheric-14co2-data.pdf
No, the C14 did not disappear from the atmosphere in 4 years, or even 12.
Jim Siverly, July 31, 2025 at 12:41 pm
“And stated so many times, as yet without proof. Temperature affects transport rates but doesn’t drive them. They are driven by the magnitude of the reservoir concentrations.”
I thought it was obvious from the observations that temperature and sunlight are the main drivers for the uptake of CO2 in spring/summer by the biosphere. Regardless of what is the CO2 level in the atmosphere, at least on short term.
But good, let us see what happens on different in only one time frame…
The diurnal cycle.
Plants take in CO2 during daylight and respire CO2 at night and so do soil bacteria from old debris all day round.
If there is an inversion layer in the atmosphere, CO2 levels near ground are not mixed with the bulk of the atmosphere and can show huge changes between day and night. Here for a few days at a modern station, Linden/Giessen in mid-west Germany:
https://www.ferdinand-engelbeen.be/klimaat/klim_img/giessen_background.png
The “background” stations from near the North Pole to the South Pole show very little change over days and little difference between each other of around 380 ppmv in average. These represent the CO2 levels of 95% of the CO2 mass in the atmosphere.
The levels in Giessen are taken at a few meters above ground in a semi-rural area, at a few km from forests.
Between the ground and up to the inversion level, a few hundred meters above ground, CO2 may be mixed during the day, but hardly at night. That makes that CO2 levels increase enormously at night and get lower that “background” during daylight, but less pronounced. Below 160 ppmv, for most plants, photosynthesis stops.
The overall global uptake by plants is about 120 PgC/year for the current level in the atmosphere, mainly in spring/summer and of which about half, 60 PgC/year, is already released at night.
The diurnal cycle is clearly completely independent of the “background” CO2 level in the atmosphere and heavily reacts on temperature and mainly sunlight.
That is already for 60 PgC/year in and out or about 30% of your outflows and thus your Te, which has near nothing to do with the CO2 pressure in the bulk of the atmosphere…
Similar conclusions can be taken for the rest of the biosphere, be it on a scale of months and for the seasonal ins and outs of the ocean surface.
Ferdinand Engelbeen August 1, 2025 at 4:27 am
Dear Ferdinand,
Please elaborate on your above comment to make it clear so I and maybe others can follow your argument.
Especially, please include your definitions of all the parameters you use in your calculations so I can follow your arguments. Trying to find your definitions in your previous comments is difficult.
Thanks, Ed
Ferdinand Engelbeen
August 1, 2025 at 4:27 am
I only used the arguments of Wigley, Joos and others to put your model in context. Dietz argued with them about the Revelle factor and supposedly other limitations of a saturated ocean to prolong an adjustment time. He was right. Nevertheless, the Dietz model still incorrectly bases the removal rate on proportionality to the difference between the current atmosphere concentration and a constant reference equilibrium value equal to what the atmosphere concentration was prior to the start of the industrial period.
Please don’t use anecdotal data in conjunction with Magic Math to try and discredit my model with the equivalence principle. In both 2020 and 1960, the 25% of the atmosphere removed applied equally to whatever amount of industrial carbon was in the atmosphere. That could have been from the current year or whatever remained from previous years, makes no difference.
“If A(net) is directly proportional to the extra CO2 above equilibrium, then delta A(net) is directly proportional to delta C….”
How is that not just a spurious correlation? I get an equally good correlation using delta C versus extra CO2 above equilibrium. Remember what Tom Wigley wrote, “It is, of course, quite possible to produce highly simplified (or ‘reduced-form’) versions of these models; but the original ‘first principles’ model must always be better and more credible than the simpler model. The real strength of first principles models is that they do not have free parameters that can be arbitrarily tuned to match observations.”
That means that just because your 50-year Tau fits all the observed data, it may be arriving at the correct fit with an incorrect model, i.e. one that uses nonphysical principles. I am working on a post that will show you that the Dietz/Engelbeen model is a unidirectional one-compartment model with a singular input from FF emissions. A bidirectional two-compartment model, while still inferior to multi-compartment models, will demonstrate with appropriate physical principles the sufficiency of a single 4-year Te and obviate a 50-year Tau.
August 1, 2025 at 8:31 am
“I thought it was obvious from the observations that temperature and sunlight are the main drivers for the uptake of CO2….”
You know what’s obvious? The sun comes up every day and goes down at night. The expected temperature swings follow. Meanwhile, concentration gradients continue to drive mass transfer 24/7.
Ferdinand,
“I didn’t receive any explanation why the decay rate of Δ14C is not 4 years as it should be for any CO2 molecule, whatever its isotopic composition…”
No, isotopic composition does affect reaction rates. Dr. Ed estimated Te for 14C at around16 years. That’s not 50 years. Why isn’t it 50 years? Oh, wait, I think your Tau has grown to 55 years. Why isn’t it 55 years?
Jim,
“I won’t be going back to the lab bench, but what I think what we need is to find some data that proves Tau is the construct we think it is.”
Why wouldn’t the C14 data be evidence against Tau?
Jim Siverly, August 1, 2025 at 12:03 pm
“You know what’s obvious? The sun comes up every day and goes down at night. The expected temperature swings follow. Meanwhile, concentration gradients continue to drive mass transfer 24/7.”
If you are not willing to accept that the upcoming sun and temperature drives 95% of the CO2 uptake and the concentration gradients only 5%, then you are just wasting everybody’s time…
Stephen Paul Anderson, August 1, 2025 at 1:44 pm
“No, isotopic composition does affect reaction rates. Dr. Ed estimated Te for 14C at around16 years. That’s not 50 years. Why isn’t it 50 years? Oh, wait, I think your Tau has grown to 55 years. Why isn’t it 55 years?”
Yes for “per mil” or tenths of a percent, as these ratio’s are expressed in that order of change.
Hardly for quantities (mass) of 14C or 13C, except when the ratio’s are “diluted” by other sources with a strong difference in isotopic ratio…
Stephen Paul Anderson August 1, 2025 at 1:44 pm
Dear Stephen,
The Delta14C data is excellent data. Since 1970, Delta14C has returned to within a few percent of its original long-term balance level of zero.
My curve fit of Delta14C is also very accurate. The return of Delta14C to its balance level of zero has a constant Te of 16.5 years. There is no way that Delta14C needs a Tau of 50 years to properly fit the data.
The Te of Delta14C is not to be confused with the Te of CO2. First, IPCC says this Te is “about 4 years.” Second, IPCC’s data for its natural carbon cycle at equilibrium at 280 ppm, shows Te for atmospheric CO2 is 3.5 years. There is no place for a Tau of 50 years.
Delta14C data show the balance level of human CO2 produced by its continuing inflow is very small compared to the balance level of natural CO2 produced by its continuing inflow.
Thus, the human CO2 balance level is less than 8 parts in 1000 and likely only 4 parts in 1000. For this to happen, the real Te for 12CO2 in the atmosphere must be LESS THAN 3.5 years.
Here’s one problem with Ferdinand’s claims that a long Tau occurs when a level is near its balance level:
In a medium of carbon or CO2, there can be only one Te for all carbon or CO2.
There cannot be a short Te for new carbon and a long Te for old carbon, because nature does not know how long any carbon atom or CO2 molecule has been in the atmosphere. There simply cannot be 50-year Tau CO2 floating around with 4-year Te CO2. Can’t happen.
Ferdinand Engelbeen August 1, 2025 at 3:00 pm
“If you are not willing to accept that the upcoming sun and temperature drives 95% of the CO2 uptake and the concentration gradients only 5%, then you are just wasting everybody’s time…”
Are you talking about how warmer CO2 can have more collisions per unit time with a surface than colder CO2?
Indeed, warmer temperature can decrease Te a small amount, but this is negligible because we must calculate such temperature changes in terms of absolute temperature in degrees K.
Ferdinand Engelbeen
August 1, 2025 at 3:00 pm
As far as I can tell, there is no natural emissions input to the Dietz model which I believe is basically Roy Spencer’s and your model, too. Natural emissions are assumed constant and equal to whatever share of the atmosphere they are considered to maintain. David Andrews has mentioned that they don’t have to remain constant as it makes no difference with his simple math and “disequilibrium isoflux.” I think you concur with that.
Before I stop wasting your time, I want to conclude my efforts to demonstrate how a one-compartment model embodies the basic concepts Dr. Ed has been trying to explain. Only one e-time is necessary to describe the evolution of CO2 in a physical-chemically meaningful way and perfectly match the Mauna Loa data. The one-compartment model is objectively bidirectional, not assumed to be automatically operating in the background like a fountain model.
The spreadsheet is a step-wise calculation of the CO2 accumulation based on the differential equation, dC/dt = Eh + En – C / Te, which simply states the increase in CO2 equals inputs minus outputs. The solution is
C(t) = [ C(0) – ( Eh + En ) * Te ] * exp( -t / Te ) + ( Eh + En ) * Te
I provided several scenarios in separate columns to illustrate the possibilities of a constant input rise to a balance level, a supplement of only FF emissions, and three scenarios with additional natural emissions. The last of the three includes an exponential addition of natural (or what could be any combination of other sources of emissions) that perfectly matches the Mauna Loa data. One can modify the starting balance level and input rates of the first four scenarios to get a feel for how starting balance levels and input rates affect the level. I used the same Te for all scenarios, so you would have to modify the formulas in each column to change Te. (Darn, I should have used a different cell for Te in each column).
https://www.dropbox.com/scl/fi/0fq76607u6onis8zopwgb/1-comp-CO2-model.xlsx?rlkey=kq01rp83cu9u50b2hkxpbvl5y&dl=0
Notice that the coefficient of time in the exponent (in cell J6) in the formula supplementing FF emissions with natural emissions equals 0.02. I don’t think that is a coincidence. I think it’s how Peter Dietz arrived at a Tau of 55 years.
In my previous comment, I left out a minus sign in front of exponential t / Te in the C(t) equation.
[ADMIN NOTE: I inserted the minus sign before t in Jim’s C(t) equation.]
I also wanted to reiterate that there is no logical reason to quantitate adjustment times. First, the clock could only start with net zero which isn’t going to happen. Second, what is the likelihood that any additional natural emissions will subside? They would be some combination of human-caused and strictly natural emissions that are not constrained by any adjustment time. Any reduction in atmospheric CO2 will be due to the removal processes contributing to the approximately four-year e-time.
Ed,
You insist on analyzing “human” and “natural” carbon cycles separately. But you also insist on analyzing C14 as a ratio to C12, via the DeltaC14 variable, rather than C14 and C12 concentrsations separately. Your strategy makes sense only if you don’t want to address what the concentration data shows.
Dr. Ed,
“There cannot be a short Te for new carbon and a long Te for old carbon, because nature does not know how long any carbon atom or CO2 molecule has been in the atmosphere. There simply cannot be 50-year Tau CO2 floating around with 4-year Te CO2. Can’t happen.”
Yes, the Equivalence Principle and the Law of Equipartition. Yes? Ferdinand’s Tau violates both of these.
Dr. Ed, August 1, 2025 at 3:32 pm
“The return of Delta14C to its balance level of zero has a constant Te of 16.5 years. There is no way that Delta14C needs a Tau of 50 years to properly fit the data.”
If Te is 4 years for any isotopic composition of CO2, then the bomb spike should have returned to its balance level faster than with the 4 years e-fold decay, because of its dilution with 14C-free fossil fuel CO2 ánd the return of the “old” 14C ratio from the deep oceans, long before the bomb tests.
The 16.5 years Te/Tau definitely rejects the 4 years Te, but is faster than for a “bulk” CO2 injection.
“First, IPCC says this Te is “about 4 years.”
That is NOT what the IPCC says. The IPCC says that the turnover time or residence time is 4 years. That is the formula:
(R)T = mass/output
That is the residence time of any single CO2 molecule in the atmosphere, NOT the exponential decay time for an extra CO2 mass injection in the atmosphere.
If (and only if!) that is a one-way process, then (and only then!):
Te = (R)T
In all other cases of direct return flows and/or mass/pressure independent processes, Te and (R)T are largely to completely independent of each other.
“Thus, the human CO2 balance level is less than 8 parts in 1000 and likely only 4 parts in 1000.”
The observed FF level in the atmosphere is over 10%, or 100 parts in 1000. In the ocean surface over 6% and increasing both in vegetation and the deep oceans.
Again, the 4 years Te widely rejected.
“Here’s one problem with Ferdinand’s claims that a long Tau occurs when a level is near its balance level:
In a medium of carbon or CO2, there can be only one Te for all carbon or CO2.”
Besides that different reservoir exchanges have different Te/Tau’s (and saturation for the sea surface), the overall Te/Tau can be calculated from the increase in the atmosphere and the increase in net uptake over the same time span, if the uptake is in exact ratio to the increase in the atmosphere. That calculation shows a Te/Tau of 50 years, not of 4 years.
Dr. Ed, August 1, 2025 at 3:53 pm
“Are you talking about how warmer CO2 can have more collisions per unit time with a surface than colder CO2?”
Absolutely not: I am talking about the amount of CO2 that is released by warming ocean surfaces in spring/summer, releasing some 25 ppmv CO2 and absorbing that again in fall/winter, largely independent of how much CO2 is in the atmosphere. Another 20 ppmv CO2 is just circulating between the equator and poles to get there in the deep oceans and returns some 1,000 years later.
And vegetation, absorbing some 60 ppmv CO2 in spring/summer by warmer temperatures and photosynthesis, of which half is already released again at night by plant respiration and soil bacteria, the other half in fall/winter, again largely independent of how much CO2 is in the atmosphere for the uptake side and completely independent for the release side by all bio-life.
These quantities form the turnover/residence time of 4 years, not a decay rate Te of 4 years.
The Te/Tau caused by any extra CO2 pressure in the atmosphere is 50 years, not 4 years.
Stephen P Anderson, August 3, 2025 at 11:28 am
“Yes, the Equivalence Principle and the Law of Equipartition. Yes? Ferdinand’s Tau violates both of these.”
Both a Te/Tau of 4 years or of 50 years are theoretically possible, depending of how you see the processes at work in the atmosphere.
The defenders of the 4 years Te/Tau, see the atmosphere as a single process, where only the CO2 amount/pressure in the atmosphere causes the overall output, for the full 100%.
The defenders of the 50 years Te/Tau see a lot of different processes at work, where the influence of the CO2 amount/pressure in the atmosphere only plays a very small role in the CO2 flows (less that 5%).
Jim Siverly, August 2, 2025 at 8:13 pm
“I also wanted to reiterate that there is no logical reason to quantitate adjustment times. First, the clock could only start with net zero which isn’t going to happen. Second, what is the likelihood that any additional natural emissions will subside?”
That is based on the idea that there are “additional” natural emissions, while nature is more sink that source for every year that we have accurate data…
As said now several times: there is not the slightest need to know any natural input or output, no matter if the natural cycles doubled or halved over time, no matter if these were 10 or 100 or 1000 PgC/year. Only the difference between inputs and outputs matters and that was negative over the past 67 years, with a few borderline (El Niño) exceptions.
Moreover, a Te of 4 years doesn’t exist. There is a turnover/residence time of 4 years, but nobody did prove that any extra CO2 injected in the atmosphere decays with a Te/Tau of 4 years. The calculated Te/Tau, based on the real, observed decay rate is around 50 years, not 4 years. That is for any bulk injection of CO2, including human FF CO2.
For the δ13C and Δ14C ratio decays: these are faster than for the bulk CO2 decay, because of the return of pre-industrial CO2 with the “old” δ13C and Δ14C levels and the addition of 14C-free and low-13C FF CO2.
Then again your formula:
deltaC = Eh + En – Cn / Te
As observed for 2020:
En = 220 PgC/yr; An = 225 PgC/yr; net removal rate in nature: 5 PgC/yr
For 1960:
En = 167 PgC/yr; An = 168 PgC/yr; net removal rate in nature: 1 PgC/yr
No matter if the removal is 100% natural or 100% FF, it is about quantities, not isotopic ratio’s.
There is no way that a negative contribution adds to an increase…
Jim Siverly, July 31, 2025 at 12:41 pm
“Also, how can a net flux be quite accurately known, if the gross fluxes from which they are calculated are only known with large margins of error? Error propagates.”
That is the beauty of alternative measurements…
dC/dt = F(g-a) – F(a-g) + F(s-a) – F(a-s) + F(d-a) – F(a-d) + E(h)
That is the problem of Dr. Ed’s approach: all these flows are only know with +/- 30% margin, which makes the calculation of F(net) from these flows quite problematic.
The alternative. without knowing even one natural input or output:
dC/dt = F(net) + E(h)
or F(net) = dC/dt – E(h)
Where dC/dt is accurate to 0.2 ppmv/415 ppmv and E(h) accurate to +/- 10% on 10 PgC/year or +/- 1 PgC/year.
“Ed,
You insist on analyzing “human” and “natural” carbon cycles separately. But you also insist on analyzing C14 as a ratio to C12, via the DeltaC14 variable, rather than C14 and C12 concentrsations separately. Your strategy makes sense only if you don’t want to address what the concentration data shows.”
David,
Your left hand doesn’t know what your right hand is doing, does it? It is the CO2 Coalition that believes the 14C:12C ratio is important.
“CO2C concludes,
The decrease of the14C/C ratio in the atmosphere supports the fact that fossil fuels are the cause of the CO2 increase in the atmosphere……”
Dr. Ed. is only pointing out that your logic is flawed. He’s mathematically consistent. You’re not.
Ferdinand,
“Both a Te/Tau of 4 years or of 50 years are theoretically possible, depending of how you see the processes at work in the atmosphere.
The defenders of the 4 years Te/Tau, see the atmosphere as a single process, where only the CO2 amount/pressure in the atmosphere causes the overall output, for the full 100%.
The defenders of the 50 years Te/Tau see a lot of different processes at work, where the influence of the CO2 amount/pressure in the atmosphere only plays a very small role in the CO2 flows (less that 5%).”
Tau doesn’t exist. It is a figment of your imagination. If human emissions went away, the new balance level would still be set by Te, not by Tau. It would be (Total Emission)Te=Lb. Nothing would change.
Stephen Paul Anderson, August 4, 2025 at 8:52 am
“If human emissions went away, the new balance level would still be set by Te, not by Tau.”
A balance level is not set by Te (or Tau), Tau (or Te) is the result of the distance to the balance level… And a Te of 4 years even doesn’t exist. The turnover/residence time is 4 years, but Te/Tau is not equal to the turnover/residence time, if several processes exits that cycle CO2 independent of the quantity of CO2 in the atmosphere…
And the global balance level is ultimately set by the temperature of the sea surface, per Henry’s law. For the current average SST, the balance level would be 295 ppmv, not over 400 ppmv.
Ferdinand,
Per the continuity equation, the balance level is set by Inflow(Te)= Balance Level. Henry’s Law is that the solubility of a gas in a liquid is proportional to the partial pressure of the gas above the liquid. Henry’s Law is a basis for Dr. Ed’s theorem along with the Ideal Gas Law. You need to re-take Chemistry 101.
Stephen Paul Anderson, August 4, 2025 at 8:47 am
“CO2C concludes:
The decrease of the14C/C ratio in the atmosphere supports the fact that fossil fuels are the cause of the CO2 increase in the atmosphere……”
That is one of the many points that supports the conclusion that humans are the cause of the recent increase of CO2 in the atmosphere.
The most important one is the carbon mass balance:
If humans add 10 PgC/year CO2 from burning fossil fuels and the increase in the atmosphere is only 5 PgC/year, then the rest of the world absorbs 5 PgC/year CO2 and there is no way that nature can be the cause of the CO2 increase, or you violate the carbon mass balance.
For the rest, the 14C decay rate is faster than for a bulk CO2 decay, as that is “diluted” by fossil 14C-free CO2 and the return of pre-industrial CO2 from other reservoirs.
Stephen Paul Anderson, August 4, 2025 at 10:19 am
“Per the continuity equation, the balance level is set by Inflow(Te)= Balance Level.”
Not in the slightest way: you are talking about a one-way reactor or lake or bath tube,… Not about the real world.
The balance level of CO2 in the atmosphere is set by the solubility of CO2 in seawater and that changes with the temperature of seawater: about 4.3%/°C, about 16 ppmv/°C for glacial-interglacial transitions.
Some 13 ppmv increase at equilibrium since the Little Ice Age. That is all.
Not over 100 ppmv for less than 1°C warming of the oceans…
“Henry’s Law is that the solubility of a gas in a liquid is proportional to the partial pressure of the gas above the liquid.”
Yes, but that works in both directions… If you open a bottle of Coke, it will get its 6 atm CO2 out into the atmosphere, until equilibrium with the atmosphere and that is 0.000425 bar CO2 nowadays.
The ocean surface adds and subtracts CO2 to/from the atmosphere over the seasons, depending of its temperature.
“Yes, but that works in both directions… If you open a bottle of Coke, it will get its 6 atm CO2 out into the atmosphere, until equilibrium with the atmosphere and that is 0.000425 bar CO2 nowadays.
The ocean surface adds and subtracts CO2 to/from the atmosphere over the seasons, depending of its temperature.”
I don’t know of any textbook that recognizes your definition of Henry’s Law.
Ferdinand Engelbeen August 4, 2025 at 3:48 am
Dear Ferdinand,
I wrote: “The return of Delta14C to its balance level of zero has a constant Te of 16.5 years. There is no way that Delta14C needs a Tau of 50 years to properly fit the data.”
You wrote: “If Te is 4 years for any isotopic composition of CO2, then the bomb spike should have returned to its balance level faster than with the 4 years e-fold decay, because of its dilution with 14C-free fossil fuel CO2 ánd the return of the “old” 14C ratio from the deep oceans, long before the bomb tests.”
The Te is NOT 4 years for 14CO2. Do you see the “14” in Delta14C?
The Te of 12CO2 is about 4 years. The Te of 14CO2 is longer than 4 years. That’s why the Te of Delta14C is longer than 4 years.
You wrote, in summary, The IPCC does NOT say Te of 12CO2 is “about 4 years.”
You are wrong because I define Te = Level / Outflow …… (2)
You wrote, this Te “is the residence time of any single CO2 molecule in the atmosphere, NOT the exponential decay time for an extra CO2 mass injection in the atmosphere.
The definition (2) does not say anything about a single CO2 molecule. It says “level” divided by “outflow.” Which obviously does not refer to a single molecule.
You use your tactic of redefining what others, like me, say and then you attack your own invalid definition.
Furthermore, your implication that (2) is a “one-way process” is further BS. In (2), we need to know only two of the variables to calculate the third variable.
You write: “In all other cases of direct return flows and/or mass/pressure independent processes, Te and (R)T are largely to completely independent of each other.”
Your claim is good science fiction. If you can formulate your claim as a hypothesis, data will show it makes invalid predictions.
I wrote that Delta14C data show “the human CO2 balance level is less than 8 parts in 1000 and likely only 4 parts in 1000.”
You claim: “The observed FF level in the atmosphere is over 10%, or 100 parts in 1000.”
Where’s your data? There are no measurements that can tell the difference between human-caused 12CO2 and natural-caused 12CO2.
I wrote: “In a medium of carbon or CO2, there can be only one Te for all carbon or CO2.”
You replied: “different reservoir exchanges have different Te/Tau’s (and saturation for the sea surface), the overall Te/Tau can be calculated from the increase in the atmosphere and the increase in net uptake over the same time span, if the uptake is in exact ratio to the increase in the atmosphere. That calculation shows a Te/Tau of 50 years, not of 4 years.”
I reply, your calculation that “shows Te/Tau of 50 years” is not a valid calculation. Your calculation is not based on physics. You have not presented anywhere any valid calculation that Te/Tau is 50 years.
We are talking about constructing a simple equation that explains how carbon flows between the reservoirs. (2) is that simple equation. It defines “balance levels.”
(2) explains the return of Delta14C to its original level of zero and the small drop below zero caused by human 12CO2 emissions.
(2) explains how there can exist an equilibrium carbon distribution between the reservoirs.
(2) can replicate IPCC’s equilibrium at 280 ppm without any need for a Tau of 50 years.
You have not shown any data that proves (2) is invalid.
You wrote: “The Te/Tau caused by any extra CO2 pressure in the atmosphere is 50 years, not 4 years.”
I reply, your conclusion is not based on physics. You have not even described a testable hypothesis that describes your claim.
Ferdinand Engelbeen August 4, 2025, at 11:07 am
… in your reply to Stephen Paul Anderson, August 4, 2025, at 10:19 am
You wrote: “The balance level of CO2 in the atmosphere is set by the solubility of CO2 in seawater and that changes with the temperature of seawater: about 4.3%/°C, about 16 ppmv/°C for glacial-interglacial transitions.”
No, these parameters do not directly set the balance level.
The parameters can influence the value of Te that determines Outflow = Level / Te for an outflow of a reservoir, e.g., the Air reservoir.
This outflow becomes an inflow to a receiving carbon reservoir, e.g., Land or Surface Ocean, wherein this inflow sets a “balance level” equal to Inflow * Te of the receiving reservoir.
So FE says that Dr. Ed’s model is to simple and the processes involved are too complicated to model but the fact that there is more each year than the last year proves that sinks are less than sources. Further the gain is less than human sources so the increase is caused by the human sources. Somehow the natural sinks know how much humans will put in and adjust to get half of it out even when the human sources are growing. Further the complicated processes and poor measurements make statistical analysis of the data that show no correlation of rate of change of human emissions to rate of change of atmospheric content for periods up to 5 years inaccurate even though the author of these statistics doesn’t think that is the case. Further Mr. Andrews states that if human emissions stop the increase will end but we can’t tell how long or by how much it will decline. Do we know if the sinks will decline as well or will they continue to grow without human sources?
That’s my summary of this fascinating dialog. I am certainly thankful for this site promoting it and all those that have contributed.
It would be interesting to know what Te has been through the millennia, but I suspect it hasn’t changed much in the last billion years or so since the planet has cooled. It is probably about what it was pre-Cambrian.
Ferdinand Engelbeen
August 4, 2025 at 6:45 am
“Moreover, a Te of 4 years doesn’t exist.” That is wishful thinking and argument by assertion.
“There is a turnover/residence time of 4 years,” is a concession that a Te of 4 years exists.
“nobody did prove that any extra CO2 injected in the atmosphere decays with a Te/Tau of 4 years.” That’s impossible to do, because one cannot end FF emissions and make nature stay constant long enough to prove any decay rate of an injected pulse of CO2.
“The calculated Te/Tau, based on the real, observed decay rate is around 50 years, not 4 years.” That is a logical fallacy, because there has been no real observed decay of injected CO2.
“No matter if the removal is 100% natural or 100% FF, it is about quantities, not isotopic ratio’s.” ???? Proof-read before you post.
“There is no way that a negative contribution adds to an increase…” Another false premise. When have natural emissions ever been negative? I know what you are implying, but it’s unfairly camouflaged in Magic Math.
August 4, 2025 at 7:02 am
Thank you for explaining correctly how a “net flux” can be accurately known. But what does that actually tell us?
Look at the models:
Te group: F(net) = dC/dt – Eh = En – C / Te
Tau group: F(net) = dC/dt – Eh = (Cn – Co) / Tau
I contend my quite accurately known F(net) is the difference between En and C / Te , a real physically meaningful difference between annual natural emissions and amounts removed based on an average observed e-time/turnaround/residence time. In contrast, your quite accurately known F(net) is calculated from two assumed and unknown values, Co and Tau, both never derived from any scientifically-verified physical model.
ADMIN NOTE:
With almost 600 comments, it is difficult to scroll through the comments and to find a comment, and even to make and edit a comment.
Therefore, I activated the WordPress option to separate comments into pages. I set it for 100 comments per page.
You will find a link at the top and bottom of each page to move to a previous or subsequent page of 100 comments.
Ferdinand, Jim, and Ed,
Don’t you think that if you wanted to study 14C movements between the atmosphere and other reservoirs you would start by plotting its concentration over time? It is true that the dominant dating application of radiocarbon requires knowing the 14C/12C ratio, measured by “Delta14C”, and that means it is easier to find plots of this quantity. Nevertheless, with a very small amount of digging you can find plots of the quantity you need: 14C ATMOSPHERIC CONCENTRATION, measured as a molar fraction. See for example the article linked below;
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/193CDF1F705B269BC975AF178CEF1AC3/S0033822224000274a.pdf/discussion-presentation-of-atmospheric-14co2-data.pdf
Please, stop arguing about what Delta14C is doing and look at what 14C concentration is doing. Perhaps, Ferdinand, you should ask Jim how the concentration curve shown in Figure 2, quite different from the Delta14C curve, can be represented simply by a Te. Perhaps you can ask Ed why 14C concentration didn’t return anywhere near to its pre-bomb test “balance level”. Look also at Figures 1 and 4.
I know this is mostly a repeat of an earlier post, but ignorance of the difference between Delta14C and concentration continues.
Dear Ferdinand,
On July 30, 2025 at 8:30 am, you addressed four requests I made on July 29, 2025 at 9:48 am.
1) You agreed on making a distinction between Te and Tau and then conflated Te/Tau on August 4, 2025 at 6:45 am.
2) You explained the Unidirectional Rule that a four-year e-time is only “allowed if (and only if!) all outflows are unidirectional and no back flows are involved” by claiming everybody does it, comparing it to your fountain model, and asking which approach is the right one. “Te could be the right one, if all processes were 100% CO2 level/pressure dependent.” That’s an assertion without evidence. First, inflows don’t need to be level/pressure dependent. Outflows are level/pressure dependent, whether temperature is constant or not. I posted a spreadsheet that shows a four-year e-time one-compartment model with bidirectional flows perfectly explains Mauna Loa data.
The Unidirectional Rule has a corollary [July 20, 2025 at 1:04 pm]: “If (and only if!) all inflows, container and outflows are unidirectional from inputs to outputs, without any direct or indirect recycling from outputs back to inputs, then (and only then!) Te = Tau.”
I claim that both your Unidirectional Rule and its corollary have no physical basis and have been proven false by my one-dimensional model.
3) You asserted without proof and by authority of the IPCC “the ocean surface indeed gets saturated for any change in the atmosphere within a year.” I argue that saturation only means absorption of CO2 by the ocean increases carbonate which dissolves CaCO3 without preventing absorption of more CO2. https://www.dropbox.com/scl/fi/r6uclpqpuyybtpdnv571b/Aqueous-CaCO3-equilibria-vs-p-CO2.png?rlkey=35ao5f191azncby40pw2ppgy2&dl=0
4) You claim, “the ratio between increase in the atmosphere and increase in net uptake remained and remains relatively constant.” However, that is a correlation, not real-world proof of any adjustment time. It’s another logical fallacy. Have you heard of it? Correlation is not causation. Yet, you have argued it axiomatically. Let’s call it the Fnet Axiom.
The Fnet Axiom also has a corollary, [July 29, 2025 at 4:23 am and elsewhere]: “If (and only if) the net output is in exact ratio to the change in disturbance, it doesn’t matter over what time frame Tau is calculated, even if the equilibrium is not known at all.”
The corollary to Fnet Axiom also follows from the same correlation from which the Fnet Axiom is postulated without a sound physical basis. So far, no applicable physical principles, such as Dalton’s Law, have been invoked.
Finally, we have the Natural Flux Theorem [postulated July 6, 2025 at 10:23 am]: “There is indeed one and only one possibility that the natural fluxes are the cause of the increase in the atmosphere: if, and only if, they increased in exact the same ratio as the human emissions.” This also remains unproven.
Stephen Paul Anderson, August 4, 2025 at 11:19 am
“I don’t know of any textbook that recognizes your definition of Henry’s Law.”
The definition of Henry’s law is that there is a fixed ratio between a gas in a liquid and the atmosphere above it for a fixed temperature of the liquid.
That is known for engineering, for 1 bar CO2 pressure, the solubility of CO2 in fresh water is known for different temperatures:
https://www.engineeringtoolbox.com/gases-solubility-water-d_1148.html
That doesn’t imply that the transfer is one-way from atmosphere to liquid.
If the amount of (free) gas in the liquid is higher than the fixed ratio, then gas will leave the liquid and add to the atmosphere, just as what happens with a bottle of Coke if you open it and leave it open.
The same for the oceans: if the temperature of the surface increases in spring/summer, then CO2 will leave the ocean surface with an increase of about 25 ppmv into the atmosphere and when it cools it will absorb about the same amount of CO2…
Jim Siverly, August 4, 2025 at 3:05 pm
“Moreover, a Te of 4 years doesn’t exist.” That is wishful thinking and argument by assertion.
and
“There is a turnover/residence time of 4 years,” is a concession that a Te of 4 years exists.
That are not equivalent sentences. Dr. Ed, to my surprise, made it clear that his “Te” is the simple calculation of mass/outflow, which is the formula for the turnover/residence time. NOT the exponential decay rate for any extra injection of CO2 (of whatever source) in the atmosphere.
Te “may” be equal to the turnover time, if (and only if!) all inflows are unidirectional into the reservoir and to the outflows. Not when e.g. over 50% of the CO2 outputs is sucked out by vegetation in spring/summer and recycled the same day/year into the atmosphere, largely independent of the amounts in the atmosphere. Even increasing from minimum to maximum output while the levels in the atmosphere drop.
That makes no difference at all for the residence time. neither how long a single molecule CO2 (of whatever origin) remains (“resides”) in the atmosphere, but it makes a tremendous difference for the speed that removes any extra CO2 above equilibrium, wherever that may be, even if that is zero CO2.
In the case of circulating flows and processes that are independent of the CO2 pressure in the atmosphere, the decay rate for any extra CO2 injection is NOT equal to the turnover/residence time and need to be calculated out of the observed change in decay rate in ratio to the change in level in the atmosphere. no matter if these changes are positive or negative.
“That’s impossible to do, because one cannot end FF emissions and make nature stay constant long enough to prove any decay rate of an injected pulse of CO2.”
and
“That is a logical fallacy, because there has been no real observed decay of injected CO2.”
Sorry, but one can calculate Tau independent of the FF emissions: the change of total CO2 in the atmosphere is what leads to the change in net absorption of CO2, no matter if FF emissions are one-shot, fixed or in/decreasing over time.
That has nothing to do with the decay of the CO2 injection of one year, but the decay rate of the total CO2 amount in the atmosphere. Both the total CO2 increase and the resulting increase in net CO2 uptake are quite exactly known.
“Another false premise. When have natural emissions ever been negative?”
Any bookkeeper worth his/hers money in this world will tell you that, if you have more expenses than income, you are getting into trouble. No matter if your income is $1,000 or $10,000 or $100,000 per month…
“Te group: F(net) = dC/dt – Eh = En – C / Te
Tau group: F(net) = dC/dt – Eh = (Cn – Co) / Tau”
Te does NOT equal the residence time, thus the calculation of F(net) by a Te of 4 years has no physical basis.
And one can calculate Tau out of the observed change in C and the observed change in F(net), without knowing Co:
Tau = (Cn – Cn-x) / (F(net)n – F(net)n-x)
Co then can be back-calculated from F(net) over the years to where F(net) is zero.
DMA, August 4, 2025 at 2:36 pm
“So FE says that Dr. Ed’s model is to simple and the processes involved are too complicated to model”
Indeed the model that Dr. Ed uses is a simple “lake/bath tube” model where inflows are all unidirectional flowing into the reservoir and unidirectional going to the outflows.
The IPCC has done some work on it and Dr. Ed did show that in his work as Figure 2.
I am trying to translate that scheme into monthly data, as these are quite important to show that within a year enormous CO2 flows are going in and out of the atmosphere, near completely independent of how much CO2 is in the atmosphere and that only a very small part is CO2 pressure related at the end of the year…
Here a first overview (but still far from exact):
https://www.ferdinand-engelbeen.be/klimaat/klim_img/seasonal_flows.png
The net result at Mauna Loa for the same year (2021):
https://www.ferdinand-engelbeen.be/klimaat/klim_img/seasonal_flows_mlo.png
“Somehow the natural sinks know how much humans will put in and adjust to get half of it out even when the human sources are growing.”
The natural sinks only react on the total extra CO2 in the atmosphere, not on the injection of one year. That it is around half human emissions is the result of the linear increasing FF emissions: that leads to a slightly quadratic increase in the atmosphere, a resulting slightly quadratic increase in net uptake and thus a quite constant ratio between emissions and uptake.
If human emissions halved, there wouldn’t be any increase in the atmosphere anymore…
“Further the complicated processes and poor measurements make statistical analysis of the data that show no correlation of rate of change of human emissions to rate of change of atmospheric content for periods up to 5 years inaccurate even though the author of these statistics doesn’t think that is the case.”
For that point, the problem is not in the processes or measurements, but in the fact that two separate independent variables both influence the increase of CO2 in the atmosphere:
One has a huge trend and hardly any variability.
The other has a huge variability and hardly any trend.
That caused a huge trend with a lot of variability in the dependent variable.
Any type of program that uses variability as a method to separate the influence of both independent variables on the dependent variable is doomed to fail… Especially of one de-trend the variables, thus effectively removing the cause of the trend of the dependent variable…
Ferdinand Engelbeen
August 5, 2025 at 3:05 am
“Te “may” be equal to the turnover time, if (and only if!) all inflows are unidirectional into the reservoir and to the outflows.”
That’s the Unidirectional Rule corollary that remains to be proven and has already been discredited by my one-compartment model with inputs originating from the same reservoirs which constitute the outputs. Those outputs are not largely independent of the amounts in the atmosphere as has been demonstrated experimentally by the CO2 Coalition. Please identify the cases of circulating flows and processes and verify they are independent of the CO2 pressure in the atmosphere. Notice that inputs are not necessarily concentration dependent. And consider this: which generates more natural emissions, a biosphere with a given amount of decomposing vegetation or a biosphere with 50 % more decomposing vegetation?
“Sorry, but one can calculate Tau independent of the FF emissions: the change of total CO2 in the atmosphere is what leads to the change in net absorption of CO2, no matter if FF emissions are one-shot, fixed or in/decreasing over time.”
That is a variation of the Fnet Axiom which emanates from the correlation between net uptake and growth in atmospheric CO2 concentration. The correlation gives rise to a model that produces a Tau associated with a presumed decay that has never happened and likely never will.
“Any bookkeeper” will also admit that taxes put some people in the red despite their incomes exceeding their expenses.
“Te does NOT equal the residence time, thus the calculation of F(net) by a Te of 4 years has no physical basis.” I thought we agreed not to conflate Te and Tau. Te is residence time which equals turnover time. Tau is an adjustment time for a hypothetical scenario of a decay in CO2 which likely will never happen. Claiming otherwise is argument by assertion. Where is the logical derivation of a model that supports your assertion?
“And one can calculate Tau out of the observed change in C and the observed change in F(net), without knowing Co.” That is the corollary to the Fnet Axiom, “If (and only if) the net output is in exact ratio to the change in disturbance, it doesn’t matter over what time frame Tau is calculated, even if the equilibrium is not known at all.”
Once again from Tom Wigley, “It is, of course, quite possible to produce highly simplified (or ‘reduced-form’) versions of these models; but the original ‘first principles’ model must always be better and more credible than the simpler model. The real strength of first principles models is that they do not have free parameters that can be arbitrarily tuned to match observations.”
The first principles one-compartment bidirectional four-year e-time model obviates Unidirectional Rules, Fnet Axioms, and the Natural Flux Theorem to account for the observed increase in atmospheric CO2.