Guest Post by Willis Eschenbach
A decade ago I wrote a post entitled “The Bern Model Puzzle”. It related to the following question.
Suppose we have a relatively steady-state condition, where the CO2 level in the atmosphere is neither rising or falling. Something like the situation around the year 1400 in the data below.
Figure 1. Historical airborne CO2 levels 1000AD to the present, from 10 ice cores and since 1959, from the Mauna Loa Observatory measurements (orange). Units are parts per million by volume (ppmv) of the atmosphere.
Now, suppose during that time, a volcano blows its top and dumps what we used to call a “metric buttload” of CO2 into the atmosphere. Over time, that pulse of CO2 will be absorbed by a variety of land and ocean sinks, and the status quo ante of atmospheric CO2 will be restored to the level it was before the eruption.
The “Bern Model” is a model used by the IPCC and various climate models. It purports to calculate how long it takes that pulse of CO2 to be reabsorbed by the natural sinks. And that’s where things get curious.
First off, the Bern Model says that 15.2% of that pulse of CO2 will stay airborne forever. Not 15% of the pulse, mind you … 15.2%.
I have never found anyone who can explain this to me. If this were true, it seems to me that every volcanic eruption would lead to a new and higher permanent level of airborne CO2 … but as you can see from Figure 1, that simply hasn’t happened.
For further evidence that the first claim of the Bern model is wrong, consider the annual swing of CO2 levels. From a low point around October to a high point around May of each year, there is a short sharp natural pulse of CO2 that leads to an increase in CO2 levels of about 6 parts per million by volume (ppmv). And this is matched by an equal sequestration of CO2 in natural sinks such that by the following October the previous CO2 level is restored. If that were not the case, CO2 levels would have been increasing every year since forever.
And during that same seven month period, at present we emit a pulse containing enough CO2 to result in an increase in CO2 levels of about 1.3 ppmv.
The Bern Model says that 15.2% of the 1.3 ppmv anthropogenic CO2 pulse will stay in the air forever … but the ~ 6 ppmv pulse is gone very quickly. So how does nature know the difference?
But that’s just the start of the oddity. It gets more curious. The Bern Model says that :
- 25.3% of the CO2 pulse decays back to the previous steady-state condition at a rate of 0.58% per year
- another 27.9% of the pulse decays at 5.4% per year, and
- a final 31.6% of the pulse decays back to the steady-state condition at 32.2% per year
This leads me to the same problem. How does nature know the difference? How is the CO2 partitioned in nature? What prevents the CO2 that’s still airborne from being sequestered by the fast-acting CO2 sinks?
There is, however, a more fundamental problem—the Bern Model simply doesn’t do a good job at representing reality. We have reasonably good information on CO2 emissions since 1850, available from Our World In Data. And we have reasonably good information on airborne CO2 concentrations since 1850 from ice cores and Mauna Loa, as shown in Figure 2.
Figure 1. Historical airborne CO2 levels from 1850AD to the present, from 10 ice cores and since 1959, from the Mauna Loa Observatory measurements (orange). Units are parts per million by volume (ppmv) of the atmosphere.
So I thought I’d take a look at the Bern Model, to see how well it could predict the airborne CO2 since 1850 from the emissions since 1850. The equation for the calculation is in the UNFCCC paper “Parameters for tuning a simple carbon cycle model“, and is also in the endnotes … bad news.
Figure 3. Actual atmospheric CO2 values, and values according to the Bern Model
No bueno … the fact that the Bern Model results are so much smaller indicates that it is incorrectly pushing much of the effect far out into the future.
So, is there a better way? Well, yes. The better way is to use the standard lagging formula:
CO2(t+1) = CO2(t)+ λ ∆E(t) * (1- exp( -1 / τ ) + CO2(t) exp( -1 / τ )
where:
- t = time
- E(t) = emissions at time t
- CO2(t) = CO2 concentration at time t
- λ = .47 (converts carbon emissions to ppmv)
- ∆ = difference from the previous value, so for example ∆CO2(t) = CO2(t) – CO2(t-1)
- τ = tau, the time constant for the decay
Using this formula, I find the time constant tau to be ~49 years. Here’s the result of that calculation.
Figure 4. Actual atmospheric CO2 values, and values according to a standard lagging model
This puts the halflife of a pulse of CO2 into the atmosphere at about 34 years …
Those are my questions and observations about the Bern Model. I’ve put the calculations and data into a spreadsheet here.
Now I need to go climb on the roof and pressure-wash the cedar-shingled walls in preparation for spraying FlameStop on them … dry times in California.
My very best wishes to all, comments and questions welcome.
w.
Of Course: As is my wont, I ask that you quote the exact words you are discussing. That way, nobody’s words get misconstrued. Well, fewer peoples’ words, at least.
The Equation: As promised …
Late News: Well, I’d just finished pressure-washing the upper part of the house when my pressure washer died … and while I know you may find this hard to believe, at that point I actually said very bad words …
Looks like Dr. W. is gonna have to engage in a forensic autopsy, to see if I can perform the Lazarus trick on the !@#$%^& pressure washer.
But not today … enough. And at least the pressure washing of the upper part of the house is done, done, done.
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Willis,
Several points:
The seasonal swings are temperature driven, not pressure driven: the fast growth of new leaves in spring + summer uptake gives the drop in CO2, which is larger than the simultaneous release from the warming oceans.
In fall/winter the fluxes are reverse. These processes are hardly influenced by any extra CO2 pressure in the atmosphere.
The Bern model expects that different reservoirs (oceans, vegetation, sediments,…) all have different uptake rates, which is true, but also do saturate (at different levels).
The latter is only true for the ocean surface, which saturates at about 10% of the change in the atmosphere: that is the Revelle/buffer factor.
That is not true for vegetation, where the optimum growth for all trees and most other plants which follow the C3-cycle is above 1000 ppmv.
That is also not true for the deep oceans, which are highly under saturated for CO2. The only problem is that these are largely isolated form the atmosphere. The only direct exchanges between the atmosphere and the deep oceans are via the sink places near poles and upwelling near the equator.
That is the main factor in the rather constant e-fold decay rate of ~49 years over the past 60+ years or the ~34 years half life time…
The problem with the Bern model is that they originally calculated that for 3000 PgC and 5000 PgC extra in the atmosphere, which is for resp. all available gas and oil used and more if also lots of coal are used. In that case, of course, even the deep oceans could get more and more saturated.
With the current total of around 400 GtC emitted by humans since about 1850, that is only 1% of what is already in the deep oceans. When everything is in equilibrium, that gives some 3 ppmv extra in the atmosphere, that is all…
The second problem is that they use the average of the ocean surface for the whole surface, while the sink places are extremely under saturated for CO2, thus bypassing the rest of the surface…
See: http://www.pmel.noaa.gov/pubs/outstand/feel2331/maps.shtm
That is the general problem of working with averages…
There were some interesting discussions in the past about the Bern model between Fortunat Joos (inventor of the model) and Ir. Peter Dietze:
http://www.john-daly.com/dietze/cmodcalc.htm
and
https://www.john-daly.com/dietze/cmodcalD.htm
I have never found anyone who can explain this to me. If this were true, it seems to me that every volcanic eruption would lead to a new and higher permanent level of airborne CO2
Well the settled science explanation is that anthropogenic CO2 is zombie apocalypse CO2. Other CO2 floats around in the air for a few years before getting sucked into plant stomata or silicate weathered into rock.
But not so man made zombie CO2. If this settles from the atmosphere, it waits for a suitable cinematic interval then zombie rises back into the air!
This is done by the magic power of the carbon Night King who periodically raises his arms slowly, just like Vladimir Furdik in Game of Thrones, at which all grounded zombie CO2 molecules then rise en masse into the night air.
I thought he was bad enough with the Wights.
As I wait for the last two books
He put the effort in at least – it was half a day’s work to get all his make-up sorted, even longer for the “children”.
“This puts the halflife of a pulse of CO2 into the atmosphere at about 34 years …”
I doubt that very much Willis. Look at plot #7 below, pulses of 12m∆ ML CO2 (green) that very closely follow the 12m∆ SST≥25.6°C (r=.84, lag=5mo), within months, not 34 years:
Your model is simple curve-fitting without ocean physics. The Bern model and your model are wrong because of the underlying assumption that there is no important and applicable temperature dependence for CO2 outgassing/sinking. Sure there is more CO2 because of emissions, but it is the ocean temperature which “decides” to partition the sources and sinks for all CO2 according to Henry’s Law of Solubility of Gases.
Most of the outgassing happens inside the yellow boundary within the tropics:
Bob, the response of CO2 to ocean surface temperatures is not more than 8 ppmv/K for Antarctic temperatures or 16 ppmv/K for global temperatures.
That is what Henry’s law says and is observed all over the oceans.
That means that the 0.8 K temperature increase since the LIA is good for about 13 ppmv CO2 increase in the atmosphere, not 120 ppmv as is observed.
Further, the 24 years is not the result of curve fitting, but of observations.
When there is a linear relationship between a disturbance and the resulting change (Le Chatelier’s principle), then the e-fold decay rate (where the residual disturbance is 1/e of the original disturbance) is as follows:
tau = cause / effect
tau = 120 ppmv (extra CO2 pressure) / 2.4 ppmv/year (net sink rate) = 50 years
Or a half life time of about 35 years.
The net sink rate is easy to calculate:
increase in the atmosphere = human emissions + natural emissions – natural sinks
2.1 ppmv = 4.5 ppmv + X – Y
X – Y = -2.4 ppmv
Whatever the exact height of X or Y, the net sink rate is known with sufficient accuracy.
That the net sink rate follows temperature changes with a small lag is true: but that is only the variability (+/- 1.5 ppmv) around the trend (90 ppmv) since Mauna Loa started its measurements. Nothing to do with the trend itself which is near zero in the derivatives. The temperature derivative is enhanced with a factor 3.5 to show similar amplitudes for its variability as for the CO2 variability:
I’ll leave the Berning to the experts while we lay folk and sundry deplorables take comfort that the worst dooming is now all under control-
‘Worst-case’ climate predictions are ‘no longer plausible,’ study (msn.com)
As to more serious problems like the longevity of domestic pressure washers thou shalt need to comprehend that commercial recirculating pressure pumps of various flow rates and pressures will consist of quality brass bodies with triple cylinder ceramic pistons like so-
Home page (interpump.com.au)
with appropriate fossil fuelled neddies to drive them but alas unlikely to frequent Home Depots Bunnings etc
How does the Bern model handle the pulse of C-14 from atmospheric nuclear weapons testing during the mid-20th century?
The Bern model does not matter. The 14CO2 pulse from atmospheric bombs was measured directly. The 1963 test ban marks the pulse beginning. 14CO2 peaked about a year later due to global mixing of the pulse. Then 14CO2 fell by 1/2 from 1965 to 1975.
This observation directly quantifies the Tau at 18 to 20 years, which occurred about 1985.
Almost none of the CO2 that existed in the Earth’s atmosphere in 1964 remains today.
I say again. People are conflating two very different matters. One is the “residence time”, how long an individual CO2 molecule stays airborne.
The other is the “pulse decay time”, how long it takes a pulse of CO2 into the atmosphere to be sequestered so that the atmospheric CO2 returns to the pre-pulse levels.
The two have NOTHING to do with each other. They have totally separate and different time constants “tau”.
w.
Willis,
The Tau of 18 to 20 years for the 14CO2 bomb pulse is not the residence time and indeed is much shorter than for a 12CO2 pulse for a different reason: there is an enormous lag (~1000 years) between what is going into the deep oceans and what returns.
Thus while the 14CO2 pulse was at it maximum around 1960, which was taken partly by the deep oceans, what returned from the deep oceans in the same year was the isotopic composition of long before the 14CO2 pulse.
That made that in 1960 some 97.5 of all 12CO2 returned, but only 45% of all 14CO2 and thus the decay rate of a 14CO2 pulse is much shorter than for a 12CO2 pulse…
If the 15.2% were correct, there would be more C14 in the atmosphere.
A good explanation of the non physical aspects of the Bern model is found here: https://youtu.be/rohF6K2avtY. At about 57 min. Salby shows part of the math error involved and more of it at about 1:05.
The Bern Model was falsified by OCO-2 data team papers published in 2017 in Sci Mag. The OCO-2 lead scientist just retired in January from the OCO-2 analysis group at NASA.
NASA no longer cares about global CO2 measured data. Only models.
As a UK resident I was going to ask why is it allowed to build houses with combustible cladding in a wildfire risk area.
But then I remembered we do that as well, as Grenfell Tower burnt down and 1000’s of other high rise buildings are having the combustible cladding removed.
Built 50 years ago when people didn’t think about such things.
w.
There is a very simple reason why 15% (sometimes 20%) is airborn. The same reason explains the slow decay of a pulse addition to the atmosphere.
The reason is the Revelle factor. It changes the solubility (residence time) of excess carbon in sea water.
The argumentation is like this:
In equilibrium (=pre-industrial level) a CO2 molecule on average stays about 10years in the surface sea layer. When the CO2 concentration in surface sea layer the residence time of the addition decrease drastically. Normal value of the Revelle factor is about 10. This means that the residence time for an addition is about 1 year, before it re-enter the atmosphere.
They just change the rate equations:
Outflow from surface sea to atmosphere at equilibrium = amount of carbon dioxide in surface layer / 10 years.
Outflow for addition= additional amount of carbon dioxide in surface layer / 1 year.
The effect is that the solubility of excess CO2 is 10 times lower than the equilibrium solubility.
The surface sea does not “want to” solve addition CO2, while it happily solves equilibrium CO2. They call this “buffering”.
All papers I have read about the Revelle factor are theoretical. Not measurments. Sometimes you can sea maps over the Revelle factor in the Oceans. That is calculated vales, but at first glance it looks as if it was measured.
It would be very straightforward to measure the Revelle factor. Take some sea water and add some CO2 to the atmosphere above. That will give the solubility.
I am pretty sure that experiments would show that Revelle factor is 1. If so, the IPCC carbon cycle model is totally wrong.
JonasW, the Revelle factor is measured at several seawater sampling stations over time and indeed shows a much smaller increase of CO2 derivatives in seawater than in the atmosphere, see my explanation at:
https://wattsupwiththat.com/2022/02/15/feeling-the-bern/#comment-3456103
The problem is that Henry’s law only applies to pure CO2 dissolved in water, not to bicarbonates and carbonates. If CO2 doubles in the atmosphere, pure CO2 in water doubles too, but in seawater that is only 1% of all inorganic carbon species, thus that doubles to 2%.
See the Bjerrum plot:
Are you sure it is measured ??
I think it is calculated. They measure salinity and PH and calculate the Revelle factor from that.
I strongly doubt the calculations.
The Revell factor changes the solubility. It says that excess CO2 has a lower solubility in sea water than “equilibrium” CO2 concentration.
A bucket with sea water with a hood above. Inject some extra CO2 and measure how much CO2 stays in the air -> will give a direct value of the Revell factor.
I’ve never understood the Bern model, although I’ll admit never trying very hard. The interesting thing about your example of major volcanic eruptions is that they lead to a decrease (not increase) in atmospheric CO2. I stumbled on that result in my model of yearly CO2 changes which assumes nature removes atmospheric CO2 at a rate proportional to the “excess” over some background level where sources and sinks are equal (see Fig. 5):
https://www.drroyspencer.com/2019/04/a-simple-model-of-the-atmospheric-co2-budget/
It turns out this effect of volcanoes on enhancing photosynthesis has been published before… the extra diffuse sky radiation after an eruption penetrates deeper into vegetation canopies;
https://www.researchgate.net/publication/10832080_Response_of_a_Deciduous_Forest_to_the_Mount_Pinatubo_Eruption_Enhanced_Photosynthesis
Fascinating, Dr. Roy. Now I’ll have to look at that, hang on … OK, here’s what I find.
The effect is not large, but it’s definitely visible. Well spotted.
w.
good stuff, Willis
The Bern Model says that 15.2% of the 1.3 ppmv anthropogenic CO2 pulse will stay in the air forever … but the ~ 6 ppmv pulse is gone very quickly. So how does nature know the difference?
The annual increase that is reversed by an annual decrease is from oscillatory source/sinks (mostly northern hemisphere vegetated areas). The Bern model is not about oscillatory source/sinks.
Sorry, but the Bern Model says that of any pulse of carbon into the atmosphere, 15% stays aloft. It says nothing about anything being off limits.
w.
Oscillating source-sinks of CO2 with an annual cycle are of a time period much shorter than the fastest-decaying component of all versions of the Bern model. The fastest-exponentially-decaying component that is properly in any version of the Bern model, no matter how many exponentially decaying components a Bern model is expanded/refined to have, is not shorter than the exponential decay (sinking into CO2 sinks) of individual CO2 molecules as indicated by exponential “decay”/sinking of individual carbon atoms of an isotope from nuclear bomb tests, with time constant (tau) mostly claimed 7-10 years, maybe as little as 6 years in a few claims although I also heard that short as being in the range of half-life as opposed to e-folding time. The annually-oscillating source-sink (or set of these) is not a model-disproving exception, but merely a cause of an annual cycle of ripple that gets added to the multi-year-scale exponential trends in both Bern models, and also non-Bern models that have a single exponential decay / sinking math component.
Donald,
Different processes at work: the Bern model is for the distribution and decay of any excess CO2 level (thus pressure), while the seasonal swings are temperature driven.
As the opposite δ13C and CO2 changes show, vegetation is the dominant effect of temperature changes over the seasons and the effect is a huge drop in CO2 when temperatures increase, no matter the CO2 pressure in the atmosphere.
http://www.ferdinand-engelbeen.be/klimaat/klim_img/seasonal_CO2_d13C_MLO_BRW.jpg
The same for year by year changes (Pinatubo, El Niño): again vegetation is the dominant effect, but then in opposite direction: more CO2 with higher temperatures (and drought in the Amazon) and less CO2 with lower temperatures and more light diffusion (Pinatubo).
For longer time periods (decades to multi-millennia), the oceans are the dominant effect…
All these processes work near independent of each other and are hardly influenced by any extra CO2 pressure in the atmosphere. The latter decay rate is mainly what is removed by the deep ocean cycle, which is highly undersaturated for CO2, but has a limited exchange with the atmosphere…
One way of coming up with the permanent component of addition of CO2 to the atmosphere, without effect of climate sensitivity, is ratio of amount of carbon in atmospheric CO2 (back when it was 280 PPMV or about 424 PPM by mass or about 595 gigatons of carbon) to the amount in the oceans (39,000 gigatons of carbon, not changed much in percent terms from what it was when atmospheric CO2 was 280 PPMV). In this oversimplification, about 1.5% of a pulse of CO2 emitted into the atmosphere remains in the atmosphere after the ocean water gains as much as it is ever going to, assuming the added CO2 does not warm the ocean. This neglects CO2 removal in ways other than being added to what’s dissolved in ocean water, such as being transferred to the lithosphere (which is on a time scale longer than the designers of the Bern model are concerned with). Also, the Bern model is oversimplified with a discrete number of exponential decay terms, and it gets more accurate as more exponential decay terms are added and the subset of the added ones that have decay rates longer than the longest decay rate already being used would take away some of the “permanent” component (which is 15.2% in the noted version of the Bern model that has only three exponential decay terms). Another thing: The permanent percentage of an addition of CO2 to the atmosphere varies with climate sensitivity. Demonstrating that a figure for this is excessive does not disprove the Bern model, but is merely evidence of using parameters for it that depend on a climate sensitivity figure (such as 3 degrees C/K per 2xCO2) that is greater than actual.
Regarding Figure 3: I just noticed it says that the particular Bern model being used says more CO2 should have been removed from the atmosphere by nature so far than has been the case, despite having an excessively large permanent component of CO2 remaining in the atmosphere. (Yellow “actual” looks like a little over 420 PPMV, red “Bern Model” looks like about 365 PPMV). I see this as not evidence of the Bern model in general being incorrect, but of the parameters of this particular version of Bern model (especially in one or both shortest term components) being incorrect. That is, assuming there are no errors whose correction would have modeled CO2 being higher than shown by the red curve.
Oops, I just looked again, I was a little incorrect when I said “a little over 420 PPMV”, it’s a little under.
Regarding Figure 4 showing atmospheric CO2 concentration being a fairly close match to a curve generated by modeling decay of each year’s emissions into the atmosphere being exponential with a time constant tau of 49 years: If atmospheric CO2 concentration in excess above the pre-industrial level and emissions are both growing exponentially at the same rate, then this can be modeled by a wide range of decay curve shapes for decay of a year’s emissions. That is a property of exponential growth of excess of atmospheric concentration and of emissions, if these two are growing at the same exponential rate. I expect the workable shapes of a decay curve for every year’s emissions includes a Bern one. Another workable decay curve shape here is decay happening in a year and being equal to some number around 50% of emissions of each year in question with atmospheric CO2 contribution not removed in a year being permanent, but ability of that curve to give a result that resembles Figure 4 does not prove correctness of such a decay curve. Because a variety of decay curves can give a result that resembles Figure 4, such a figure does not prove correctness of a decay curve, including of an exponential decay curve or of a Bern one that works (from choice of parameters for it) for such a purpose.
Regarding “I find the time constant tau to be ~49 years” just above Figure 4: How do you (or do you ?) reconcile this with “The calculation used best-fit values of 59 years as the time constant (tau)” in https://wattsupwiththat.com/2015/04/19/the-secret-life-of-half-life/ ?
I’m looking at a three times longer dataset in this post, 171 years (1850 -2020) versus 55 years (1959-2013).
w.
So, I suspect at this point that exponential growth rate of manmade emissions and of atmospheric excess above 283-285-whatever PPMV both reasonably support an exponential decay curve of annual emissions with only small errors, with exponential decay rate of “atmospheric excess” slowing a little as rate of exponential growth rate of emissions and of “atmospheric excess” slow a little. As I see this, the curve-fitting gets fairly good (so far) with modeling with choice of either 49 or 59 years of time constant tau, and the slower one works better for fewer past years and the faster one works better for more past years. Am I getting all of this correctly do far?
I see small slowdown of exponential-decay-rate model of nature removing CO2 from the atmosphere as exponent decreases by similarly small extent during this time of exponential growth of manmade CO2 emissions as being consistent with Bern models. I see consistency with more than one model, including a model I mentioned in a previous comment where nature’s removal of “excess CO2” from the atmosphere gets incompletely done (around halfway done) in about a year, with what added CO2 remains after that being permanent. Wide variety of models of “decay” of excess atmospheric CO2 that have fairly good fit to the data of growth of atmospheric CO2 and of human activities of transferring lithospheric carbon to atmosphere CO2 don’t disprove each other until they greatly disagree with ether.
Hi Willis,
I’ma little late to the party here, but I noted the smallish discrepancies between actual atmospheric concentration and modeled concentration might mostly disappear if your model included an ocean surface temperature effect. As Ferdinand has pointed out many times, there is an influence of average ocean surface temperature on atmospheric CO2, though it is not large… maybe 5 or 10 PPM per degree.
Dear Mr. Eschenbach
I 100% agree with you. I made my oun calculations concerning e-time and Bern-Model.
enclose an abstract out of it. Hope you enjoy it.
New basics assumptions to evaluate the e-time τ of CO₂ in the atmosphere
So we are pursuing the new approach:
· The historical CO₂ content based on natural CO₂ emissions/absorptions follows the same physical exchange principle as the anthropogenic emissions.
· There is no justification that pre-industrial emissions are constant.
· There is no reason to believe that the airborn fraction is constant.
· There is no reason to believe that Residence Time is constant.
· The Bern model served as a theoretical model for comprehension, but its use for forecasts and simulations for real conditions must be rejected.
· The increased CO₂ values from 280ppm to 411ppm are caused by several impacts: increase biomass by up to 30% see (34)C. Huntingford et al.This also increases the seasonal biomass cycle in the northern hemisphere.
· The increased biomass and the increased CO₂ partial pressure cause an additional increase in absorption and emission.
· We had a temperature increase of 0.8°C from 1975 to 2020. This results in an increased emission of CO₂ in the order of 2ppm/a. Takahashi et al (33)
· The results of the I⁴C study ((3)Skrable et al) were included in our thesis
· The residencel time τ can consist of different components.
· Since there was an equilibrium between Eland and Eocean before 1750 , we assume that the Ocean has an increase in absorption due to the increased CO₂ partial pressure, as assumed by the IPCC in Fig 6.1
· The increase of EDCNF was taken over tabulatally by (3) Skrable et all.
· Biomass combustion, which has risen sharply since the 1970s in particular, must be taken into account in the entire CO₂ budget. These are not included in ELUC and ENF (bp Statistic (4)).
Finaly we found an e-time of 3,4 years. If we calculate with these τ-values and data from Global Carbon Budget, MLO, EIA and EDGAR and formula (3) we receive a perfect match of measured and calculated CO2 concentrations between 1750 and 2020 .