The Bern Model Puzzle

There’s a massive CO2 sink that resides over Siberia during winter, it is rapidly ‘taken up’ by foliage during Spring and Summer.

excellent thought post Willis……..
I still can’t figure out how CO2 levels rose to the thousands ppm….
….and crashed to limiting levels
Without man’s help……….

Their graph is made in a manner which would make readers not realize how much biomass growth occurs from human CO2 emissions.
Human emissions averaged around 27 billion tons a year of CO2 during the decade of 1999-2009 (on average 7 billion tons annually of carbon), which amounted to about 270 billion tons of CO2 added to the atmosphere. Meanwhile there was a measured increase in atmospheric CO2 levels of 19.4 ppm by volume, 155 billion tons by mass, an amount about 57% of the preceding but only 57% of it.
If one looks at where the other 115 billion tons went, it was a mix of uptake by the oceans and it going into increased growth of biomass (carbon fertilization from higher CO2 levels) / soil.
Approximately 18% (49 billion tons CO2, 13 billion tons carbon) went into accelerated growth of biomass / soil, and about 25% went into the oceans.
To quote TsuBiMo: a biosphere model of the CO2-fertilization effect:
“The observed increase in the CO2 concentration in the atmosphere is lower than the difference between CO2 emission and CO2 dissolution in the ocean. This imbalance, earlier named the ‘missing sink’, comprises up to 1.1 Pg C yr–1, after taking land-use changes into account.” “The simplest explanation for the ‘missing sink’ is CO2 fertilization.”
http://www.int-res.com/articles/cr2002/19/c019p265.pdf
In fact, global net primary productivity as measured by satellites increased by 5% over the past three decades. And, for example, estimated carbon in global vegetation increased from approximately 740 billion tons in 1910 to 780 billion tons in 1990:
http://cdiac.esd.ornl.gov/pns/doers/doer34/doer34.htm
Other observations include those discussed at http://www.co2science.org/subject/f/summaries/forests.php

Ah! A warmist recently told me that the residence time of CO2 is 2 to 500 years. I replied that that is quite the error bar. He probably had looked up that Bern model and got his information from there. But it really doesn’t make any sense and I would file it under make-work schemes or epicycles. A product of the Warmist Works Progress Administration.

“Calculating” to 4.16 or 1.33 years indicates a remarkable of “accuracy” 3.65 days residence time for “some” CO2. The oceans do NOT absorb CO2 from the atmosphere. The oceans have “vast pools of liquid CO2 on the ocean floors” which keep constant supply of dissolved CO2 and other elemental gases in the water column ready to disperse. See Timothy Casey’s excellent “Volcanic CO2” posted at http://geologist-1011.net While there visit the “Tyndall 1861” actual experiment with Casey’s endnotes and the original translation of Fourier on greenhouse effect. Reality is different than the forced orthodoxy.

I’m way out of my depth here, but The Hockey Schtick wrote about The Bern Model with advice from “Dr Tom V. Segalstad, PhD, Associate Professor of Resource and Environmental Geology, The University of Oslo, Norway, who is a world expert on this matter.”
http://hockeyschtick.blogspot.co.uk/2010/03/co2-lifetime-which-do-you-believe.html

CO2 is determined by climatic factors. Temperature-independent CO2 fluxes into/out of the atmosphere (especially minor ones like the human input) are compensated by the system (oceans).
If we could magically remove 100 ppm of the CO2 from the atmosphere in one day, what would be the transient system response (after 10 days, 1 month, a year…)?

The idea CO2 is partitioned in any way sounds like complete bull to me – CO2 is quickly well mixed in the atmosphere. But its possible some CO2 sinks could be diminishing, that might help explain some of the increase in atmospheric concentration (plus as others have pointed out higher temperatures mean less absorption by the oceans).
I don’t think any discussion of the carbon cycle should be without mention of Salby’s work, if you haven’t seen it:
http://youtu.be/YrI03ts–9I
[REPLY: It was discussed here on April 19. -REP]

FTA: “Anyhow, that’s my problem with the Bern model—I can’t figure out how it is supposed to work physically.”
It is because the process of CO2 sequestration is not solved by an ordinary differential equation in time, but by a partial derivative diffusion equation. It has to do with the frequency of CO2 molecules coming into contact with absorbing reservoirs (a.k.a. sinks). If the atmospheric concentration is large, then molecules are snatched from the air frequently. If it is smaller, then it is more likely for an individual molecule to just bob and weave around in the atmosphere for a long time without coming into contact with the surface.
This gives rise to a so-called “fat-tail” response. Such a fat-tail response can be approximated as a sum of exponential responses with discrete time constants.
I am not, of course, advocating the Bern model parameters. The modeling process is reasonable and justifiable, but the parameterization is basically pulled out of a hat.
What we actually see in the data is that CO2 rate of change is effectively modulated by the difference in global temperatures relative to a particular baseline. What is more likely? That CO2 rate of accumulation responds to temperatures, or that temperatures respond to the rate of change of CO2? The latter would require that temperatures be independent of the actual level of CO2, which is clearly not correct. Hence, we must conclude that CO2 is responding to temperature, and not the other way around.
In case anyone misses the point, let me spell the implications out clearly: fat tail or no, the response time for sequestering the majority of anthropogenic CO2 emissions is relatively short, and the system is having no trouble handling it. CO2 levels are being dictated by temperatures, by nature, not by humans.

Thanks. I thought I was the only one that didn’t believe this fallacy.
The fast processes will finish with it’s CO2 then go after the next batch !
The alarmists just want an excuse to say CO2 remains in the atmosphere for 100 years which is can’t possibly do.

CO2 works like resistors in parallel.
1/RT=1/R1 + 1/R2+1/R3
So the total resistance can’t be more than the smallest resistor.
for CO2 the 1/2 life of the total can’t be greater than the 1/2 life which is shortest.

Bart says:
May 6, 2012 at 11:51 am
I want to repeat this part of my post, because people may miss it in with the other stuff, and I think it is important.
What we actually see in the data is that CO2 rate of change is effectively modulated by the difference in global temperatures relative to a particular baseline.
What is more likely? That CO2 rate of accumulation responds to temperatures, or that temperatures respond to the rate of change of CO2? The latter would require that temperatures be independent of the actual level of CO2, which is clearly not correct. Hence, we must conclude that CO2 is responding to temperature, and not the other way around.
In case anyone misses the point, let me spell the implications out clearly: fat tail or no, the response time for sequestering the majority of anthropogenic CO2 emissions is relatively short, and the system is having no trouble handling it. CO2 levels are being dictated by temperatures, by nature, not by humans.

Surely they don’t serious use the sum of five or six exponentials, Willis. Nobody could be that dumb. The correct ordinary differential equation for CO_2 concentration , one that assumes no sources and that the sinks are simple linear sinks that will continue to scavenge CO_2 until it is all gone (so that the “equilibrium concentration” in the absence of sources is zero (neither is true, but it is pretty easy to write a better ODE) is:

Interpretation: Since CO_2 doesn’t come with a label, EACH process of removal is independent and stochastic and depends only on the net atmospheric CO_2 concentration. Suppose is the rate at which the ocean takes up CO_2. Left to its own devices and with only an oceanic sink, we would have:






where is the constant of integration. I mean, this is first year calculus. I do this in my sleep. The inverse of $R_1$ is the exponential decay constant, the time required for the original CO_2 level to decay to of its original value (for any original value ). If there are two processes running in parallel, the rate for each is independent — if (say) trees remove CO_2 at rate $R_2$, that process doesn’t know anything about the existence of oceans and vice versa, and both remove CO_2 at a rate proportional to the concentration in the actual atmosphere that runs over the sea surface or leaf surface respectively. The same diffusion that causes CO_2 to have the same concentration from the top of the atmosphere to the bottom causes it to have the same concentration over the oceans or over the forests, certainly to within a hair. So both running together result in:

If (say) trees and the ocean both remove CO_2 at the same independent rate, the two together remove it at twice the rate of either alone, so that the exponential time constant is 1/2 what it would have been for either alone. If there are five such independent sinks (where by independent I mean independent chemical processes), all with equal rate constants , the exponential time constant is 1/5 of what it would be for one of them alone. This is not rocket science.
This is completely, horribly different from what you describe above. To put it bluntly:

Compare this when , :

(correct) versus

(incorrect). The latter has exactly twice the correct decay time, and makes no physical sense whatsoever given a global pool of CO_2 without a label. The person that put together such a model for CO_2 — if your description is correct — is a complete and total idiot.
Note that this would not be the case if one were looking at two different processes that operated on two different molecular species. If one had one process that removed CO_2 and one that removed O_3, then the rate at which one lowered the “total concentration of CO_2 + O_3” would be a sum of independent exponentials, because each would act only on the partial pressure/concentration of the one species. However, using a sum of exponentials for independent chemical pathways depleting a shared common resource is simply wrong. Wrong in a way that makes me very seriously doubt the mathematical competence of whoever wrote it. Really, really wrong. Failing introductory calculus wrong. Wrong, wrong, wrong.
(Dear Anthony or moderator — I PRAY that I got all of the latex above right, but it is impossible to change if I didn’t. Please try to fix it for me if it looks bizarre.)
rgb

Willis Eschenbach says:
May 6, 2012 at 12:07 pm
“What I don’t get is what causes the fast sequestration processes to stop sequestering, and to not sequester anything for the majority of the 371.6 years … and your explanation doesn’t explain that.”
The best I can tell you is what I stated:”It has to do with the frequency of CO2 molecules coming into contact with absorbing reservoirs (a.k.a. sinks). If the atmospheric concentration is large, then molecules are snatched from the air frequently. If it is smaller, then it is more likely for an individual molecule to just bob and weave around in the atmosphere for a long time without coming into contact with the surface.” The link I gave explains it from a mathematical viewpoint.

I’m not sure of the significance of the e-folding time. I presume it must be related to the rate at which a particular sink absorbs CO2, in which case why not use the absorption time? As for the partitions, I just don’t get it. Surely there must be a logical explanation in the text for the various percentages listed.

Cos the other sinks are fully saturated, or already absorbing all they can, so they leave the rest behind for the other longer sinks – then they can claim the natural sinks are saturated and we are evil despite it making no sense. Its all based on the assumption that before 1850 co2 levels were constant and everything lived in a perfect state of equilibirum just on the very the systems absorbtion capacity. Ahhh the world of climate science!

Willis Eschenbach says:
May 6, 2012 at 12:33 pm
“No, the link you gave explains simple exponential decay from a mathematical viewpoint, which tells us nothing about the Bern model.”
No, that’s not what it explains at all. It is a statistical model in which the probability distribution is exponential, to be used in finding a solution of the Fokker-Planck equation. The “decay” he shows is actually 1/(1+a*sqrt(t)), the reciprocal of 1 plus a constant time the square root of time.
Sorry I cannot explain it better right now. Must go.

I am particularly intrigued by the 17.01 year figure, I had no idea climate science was so precise.

Willis Eschenbach – “However, there is nothing that I know of that allows us to establish any of those numbers. Once again, it seems to me that the authors are just picking parameters.”
I would point out a very important part of the link you referenced (http://unfccc.int/resource/brazil/carbon.html):
“All IRFs are obtained by running the Bern model (HILDA and 4-box biosphere) as used in SAR or the Bern CC model (HILDA and LPJ-DGVM) as used in the TAR.” – (IRF’s -> impulse response functions, time factors, and final percentages)
The percentages you quoted are the resulting partial absorptions of various climate compartments resulting from running the Bern model, which is described in Siegenthaler and Joos 1992 (http://tellusb.net/index.php/tellusb/article/viewFile/15441/17291). In short, those percentages are results, not the inputs, of running the Bern model – presented by Joos et al for use by other investigators if they wish to apply the Bern model to their calculations.
Also note the statement that “Parties are free to use a more elaborate carbon cycle model if they choose.” Again – the results of the Bern model were offered as an available computational tool for further work.
I hate to say this, but you give the impression you did not fully read the UN reference (with percentages) that you opened the discussion with…

I just went through that post with Salby video (didn’t have time before). Amazing that people still misunderstand the natural CO2-rise argument (like by Salby). Again:
The rise in the atmospheric CO2 is caused by warming climatic factors. The source is anthropogenic CO2, because it’s available in the atmosphere, but the cause is the warmth. Without anthropogenic CO2, oceans would have to release the necessary CO2 to achieve the climatically driven atmospheric CO2.

To attempt to clarify what I wrote in my previous post (http://wattsupwiththat.com/2012/05/06/the-bern-model-puzzle/#comment-978032):
The exponentials, percentages, and time factors in the link Willis Eschenbach provided are approximations that reproduce the results of running the Bern model – much as 3.7 W/m^2 direct forcing per doubling of CO2 is the approximation of running radiative code such as MODTRAN, allowing quick calculations without having to run the model over and over again. I.e., the percentages and time factors are shorthand for the model.
As Joos stated in that link (http://unfccc.int/resource/brazil/carbon.html), the Bern model approximations were offered as a tool for use by others, and “Parties are free to use a more elaborate carbon cycle model if they choose.”

In relation to sources and sinks, can Willis, or anyone else explain this image of global CO2 concentrations ?
Why don’t warm tropical oceans give high CO2 ?
Why is there a band of high CO2 around the 35S ?
How is the distribution over Africa and S America explained ?
Why does Antarctic ice appear to be such a strong absorber in parts and why such strong striation?
http://www.seos-project.eu/modules/world-of-images/world-of-images-c01-p05.html

Well, whatever the Bern model does, it must be correct. After all, once you match the results of 9 GCM models (except one outlier), you have matched them all. CO2 sensitivity was assumed to be 2.5 K to 4.5 K in the models.
” After 80 years, the increase in global average surface temperature is 1.6 K and 2.4 K, respectively. This compares well with the results of nine A/OGCMs (excluding one outlier) which are in the range of 1.5 to 2.7 K”

Fascinating stuff, cheers Willis. So instead of trying to capture CO2 underground why not just dump into a high speed sink?

So … does anyone understand how 13% of the atmospheric CO2 is supposed to hang around for 371.6 years without being sequestered by the faster sinks?
Perhaps you need to look at this the other way around. We have often heard about the 800 year time lag between high temperatures and CO2 concentrations. Is it a coincidence that 371.6 is about half of 800? Is it possible that if the CO2 concentration were to suddenly drop, then various processes would act to raise the CO2? And that the part of the CO2 that is in the deep oceans may take 371.6 years to reach the atmosphere and add 13% to the overall increase in CO2 concentration?

They may be discussing on-rate constants only. The concept might be equilibrium and saturation in different reservoirs. The on-rate for a given reservoir is the atmospheric concentration times the rate constant. The off rate is the reservoir concentration times the off-rate constant. If the on-rate equals the off-rate, then the reservoir is at equilibrium. No net uptake would occur.
rate(on) = k(on) * [conc(air)]
rate(of) = k(off) * [conc(res)]
No net uptake occurs when rate(on) = rate(off), that’s equilibrium.
In the real case, you also have a loss rate for CO2 via other routes, e.g. diatom skeletons in the ocean, and leaves or grasses on land. Which means the reservoirs don’t necessarily saturate, and they can continue to take up more CO2. In some cases, the on-rate may depend on the rate of dropout loss in that reservoir since near equilibrium limits the net uptake.
d(CO2 air)/dt = k(off) * [CO2(res)] – k(on) * [CO2(air)]
d(CO2 res)/dt = k(on) * [CO2(air)] – k(off) * [CO2(res)] – k(dropout) * [CO2(res)]
If the reservoir is in dynamic equilibrium, and to a reasonable approximation the reservoir concentration doesn’t change. Then
-k(on) * [CO2(air)] – k(off) * [CO2(res)] = k(dropout) * [CO2(res)] ,
which means both sides of the equation are constant, since CO2(res) doesn’t change.
Since the dropout material doesn’t readily cycle back to the air via the reservoir, it only makes sense when the dropout material can be returned to the atmosphere via another route, e.g. biological digestion, fire, volcanoes, or burning fossil fuel (coal). Returning the dropout material to the atmosphere creates the Carbon cycle we think about.
Partitions in the atmosphere itself make no sense, unless you are silly enough to count flying birds (%P) <- that's an emoticon.
From 1976 to 1997, atmospheric 14CO2 was measured. The levels spiked after due to atmospheric testing of nuclear weapons that ended in the 1960s. These data show the 14CO2 half-life is about 11 years (see raw data here: http://cdiac.ornl.gov/trends/co2/cent-scha.html). In the first approximation, uptake mechanisms won't know the difference between carbon isotopes. It is quite safe to say half of the CO2 in the atmosphere is turned over in about 11.4 years. Five half-lives is about 57 years. That means only about 3% of the CO2 present in the atmosphere 57 years ago is still in the air.
It seems we have an idea of what the dropout rate for CO2 must be, and thus what the replenishment rate must be to keep the atmosphere in roughly steady state. In this simple model, a sudden change in atmospheric CO2 concentration could shift the equilibrium concentration in the reservoirs, and then establish a new constant uptake rate somewhat higher than the old one. If you think about it, perhaps that does make some sense in some cases, such as faster plant growth, or more alkalinty in the ocean (sorry catastrophists, biological action converts carbonic acid to bicarbonate, e.g. by nitrogen fixation).
Yes, models can be misused. It's up to you to decide when they are appropriate.

Since you like differential equations…
Start with three variables A, B, and C. The volume of flow from A to B is k_AB (A-B), and the volume of flow from A to C is k_AC (A-C).
So
dA/dt = k_AB (B-A) + k_AC (C-A) + E
dB/dt = k_AB (A-B)
dC/dt = k_AC (A-C)
where E is the rate of emission.
Treat L = (A, B, C) as a vector, ignore E for the moment to get dL/dt = ML where M is a matrix of constants. Diagonalise the matrix to get two independent differential equations (the rows of M are not linearly independent), each giving a separate exponential decay with a different time constant. Transforming back to the original variables gives a sum of exponentials.
(I think the two time constants are -k_AB-k_AC +/- Sqrt(k_AB^2-k_AB k_AC +k_AC^2) but I did it quickly.)

Following on from an earlier poster’s comments, the science of decompression diving uses a similar approach to that of the Bern model, to understand the movement of nitrogen into and out of a divers’ body tissues. The approach was pioneered by JS Haldane in 1907. He came up with the idea of using tissue compartments which exchanged nitrogen at different rates. Tissues like blood and nerves were fast, and were able to quickly equilibrate with any changes in the partial pressure of nitrogen. Tissues like muscle and fat were of intermediate speed, while bone was extremely slow. What Haldane did was to come up with a crude multi-compartment (tissue) model, and then carry out very extensive tests to tweak the model so that divers (goats in his initial experiments) did not get decompression sickness (gas bubbles forming in tissue). Over a hundred years on, Haldane-type tissue models are used in most divers’ decompression computers, and they work extremely well.
Some important points about diving science: 1. Haldane, and those who followed, constantly tested and refined their models against experimental data – a process which continues today – over a hundred years on. 2. The models are a crude approximation of a complex system (the human body), but at least the physics is reasonably well understood e.g. the local temperature and pressure gradients and the kinetics of the physical processes – solubility, perfusion and diffusion. 3. In decompression models, only one of the tissue compartments usually controls the behaviour of the model (rate of ascent/decompression stop timing). Therefore, small errors or uncertainties in each compartment are not a major problem.
The application of this approach to climate science is, in my view, highly problematic because: 1. There simply has not been enough time/effort to refine these models against experimental data – a process which can take many decades. 2. The models are a profoundly crude approximation of a bewilderingly complex system (global carbon cycle), about which most of physics/biology/geology/chemistry/vulcanology etc, etc are not well understood. 3 In climate models all of the compartments contribute to the behaviour of the model (CO2 sequestration rate) and so errors and uncertainties in each compartment are cumulative.

I have been wondering, if there is any reason to expect the rate of carbon exchange between the athmosphere and the ocean ( and other sinks) to be diffrent for diffrent isotopes of C ( CO2 for that matter ) . In other word might it be possible to infere something about the uptake rate of the sinks from the data (its accessible at the CDIAC website ) for the athmospheric dC14 content and the spike caused in it by open air nuclear bomb testing in the last century. I belive the i have somwhere seen a statment claming the “athmospheric half life” calculated from this “experiment” is 5.5 – 6 for the c14 isotope.

It is because the process of CO2 sequestration is not solved by an ordinary differential equation in time, but by a partial derivative diffusion equation. It has to do with the frequency of CO2 molecules coming into contact with absorbing reservoirs (a.k.a. sinks). If the atmospheric concentration is large, then molecules are snatched from the air frequently. If it is smaller, then it is more likely for an individual molecule to just bob and weave around in the atmosphere for a long time without coming into contact with the surface.
Dearest Bart,
Piffle. I am talking about the integral in the document Willis linked, which is an integral over time only. If you set all of the , effectively making the entire sum under the integral zero — this is what you would get if you made carbon dioxide sequestration in these imagined modes instantaneous — then the remainder of the function under the integral is . This will cause CO_2 concentration to grow without bound as long as we are emitting CO_2 at all. Nor will it ever diminish, even if . Worse, all of the terms in the integral I forced to zero by making their time constants absurdly small are themselves non-negative. None of them cause a reduction of . They only make CO_2 concentration grow faster as one makes their time constants longer as long as .
Now, as to your actual assertion that the rate that CO_2 molecules are “snatched from the air” is proportional to the concentration of molecules in the air — absolutely. However, for each mode of removal it is proportional to the total concentration in the air, not the “first fraction, second fraction, third fraction”. The CO_2 molecules don’t come with labels, so that some of them hang out anomalously long because they are “tree removal” molecules instead of “ocean removal” molecules. The ocean and the trees remove molecules at independent rates proportional to the total number of molecules. That is precisely the point of the simple linear response model(s) I wrote down. They actually “do the right thing” and remove CO_2 faster when there is more of it in total, not broken up into fractions that are somehow removed at different rates, as if some CO_2 is “fast decay” CO__2 and comes prelabelled that way and is removed in 2.57 years, but once that fraction is removed none of the rest of the CO_2 can use that fast removal process.
Except that the integral equation for the concentration is absurd — it doesn’t even do that. There is no open hole through which CO_2 can ever drain in this formula — it can do nothing but cause CO_2 to inexorably and monotonicall increase, for any value of the parameters, and CO_2 can never equilibrate unless you set to zero.
As I said, piffle. I do teach this sort of thing, and would be happy to expound in much greater depth, but note well that all of this is true before one considers any sort of PDE or multivariate dependence. Those things all modulate the time constants themselves, or make the time deriviative a nonlinear function of the concentration. In general, one can easily handle those things these days by integrating a set of coupled ODEs with a simple e.g. runge-kutta ODE solver in a package like the Gnu Scientific Library or matlab or octave. I tend to use octave or matlab for quick and dirty solutions and the GSL routines (some of which are very slick and very fast) if I need to control error more tightly or solve a “big” problem that needs the speed more than the convenience of programming and plotting.
But one thing one learns when actually working with meaningful equations over a few decades is how to read them and infer meaning or estimate their asymptotics. The “simple carbon cycle model” Willis linked, wherever it came from, is a travesty that quite literally never permits CO_2 concentration to diminish and that purports to break a well-mixed atmosphere into sub-concentrations with different decay rates, which is absurd at the outset because it violates precisely the principle you stated at the top of this response, where removal/sequestration by any reasonable process is proportional to the concentration, not the sub-concentration of “red” versus “blue”, “ocean” vs “tree” CO_2.
rgb

@ Latitude, May 6, 2012 at 10:38 am
“I still can’t figure out how CO2 levels rose to the thousands ppm….
….and crashed to limiting levels
Without man’s help……….”
Perhaps it is because your ( and your teacher’s ) view of the subject is incorrect.
The real question is: “Why is there any CO2 in the atmosphere at all?”
Answer that question and you will have the puzzle solved. CO2 is constantly and irreversibly being sequestered into the formation of insoluble carbonates (organically e.g. Foraminifera and chemically e.g. Calcium carbonate) over millennia.
One possible answer is the concept of a Hydritic Earth. With never ending up dwelling methane being oxidized to CO2.
Dan Kurt

And by the way “The Bern Model” triggered a memory in my head about having read an article by Jarl Ahlbeck some years ago on the John Daly website, where he among other things maintains that the future athmospheric carbon dioxide concentrations the bern model predicts are just a simple minded parabolic fit to to some unrealistic assumptions (not data). I never could really make up my mind if he was right or wrong in that, but for what it is worth the link to the paper is on the line below:
http://www.john-daly.com/ahlbeck/ahlbeck.htm

Final comment and then time to do some actual “work”. I do, actually respect the notion that CO_2 concentration should be modelled by a set of coupled ODEs. I also am perfectly happy to believe that some of the absorption mechanisms — e.g. the ocean — are both sources and sinks, or rather are net one or the other but which they are at any given time may well depend on some very complicated, nonlinear, non-Markovian dynamics indeed. In this case trying to write a single trivial integral equation solution for CO_2 concentration (one with a visibly absurd asymptotic behavior) is counterindicated, is it not? In fact, in this case on has to just plain “do the math”.
The point is that one may, actually, be able to write an integrodifferential equation that represents the CO_2 concentration as a function of time. It appears to be the kind of problem for which a master equation can be derived (somebody mentioned Fokker-Planck, although I prefer Langevin, but whatever, a semideterministic set of coupled ODEs with stochastic noise). That is not what the equation given in the link Willis posted is. That equation is just a mistake — all gain terms and no loss terms. Perhaps there is a simple sign error in it, but as it stands it is impossible.
rgb

E.M.Smith says:
May 6, 2012 at 2:38 pm
Plankton are a huge consumer of CO2, and they are rate limited by iron in nutrient rich waters and by silicon for diatom shells in others. NOT by CO2. So a significant modulator of CO2 will be volcanism that puts iron and silicon into the biosphere.
=========================
Saharan/African dust……….

Dan Kurt says:
May 6, 2012 at 2:48 pm
CO2 is constantly and irreversibly being sequestered into the formation of insoluble carbonates
==================
Gosh Dan, you just explained how denitrification is possible without carbon………….

Willis,
I have not yet plowed through all the comments so if this is redundant please forgive me. What you have described as the Bern model sounds a lot like a multi-compartment first-order elimination model similar to that of some drugs. A simple one-compartment, first-order elimination model is concentration dependent. That is, you put Drug X into the body and it will be eliminated primarily through one route (usually the kidneys). You have t1/2 elimination constant and about five half-lives later the body has essentially cleared the drug. Some drugs like aminoglycoside antimicrobials have a simple and rather restricted distribution in the human body. Their apparent volume of distribution (a purely theoretical metric derived for the purposes of calculation) is roughly that of the blood volume. Other drugs have very unusual volumes of distribution. An ancient (but good) example is that of digoxin. You can give a few daily doses of 250 µg of digoxin and end up with an observed serum concentration in the ng range much lower than one might anticipate. That drug has gone somewhere else other than the apparent blood volume.
This is where we get into multi-compartment models. A single drug may occupy the blood volume, the serum proteins, adipose tissue, muscle tissue, lung tissue, kidney tissue, lung tissue and brain tissue. Each tissue “compartment” is associated with its own in-and-out elimination constants so a steady state, single elimination constant is virtually impossible to quantify.
I know nothing about the the Bern Model so I can only surmise that maybe this is an elaborate model built on multi-compartment, first-order elimination kinetics. Then again, you have to consider the possibility of zero-order (non-concentration dependent) kinetics. Drugs like ethanol follow this model. If one keep ingesting alcohol at a certain point the liver’s capacity to metabolize alcohol is overwhelmed. We see a real-life “tipping point.” Once those metabolic pathways in the liver become saturated, every additional gram of alcohol ingestion produces a geometrically higher EtOH serum concentration (i.e. blackouts).
I have no idea if any of this is relevant. But what you described bore an amazing resemblance to multi-compartment, first-order kinetics. Still…on a personal level I think it’s BS. Different drugs behave differently in the human body. I have a hard time believing CO2 (a “single drug”) behaves differently in the atmosphere as a whole.

Oddly enough I read your post whilst in a “Tidal wetland” that I had visited to observe the spring *super moon” tides. Been doing this for more years than I care to remember but the tide was no higher than I’ve seen before (nowhere near), the estuary just as vibrant as ever but the first time I have ever seen a bird surface before me with a wriggling fish in its beak and gobble it down.

Wet lands are usually replaced by pasture. Another sink.
Secondly, every previous run up of CO2 has been followed by significant drop. So obviously some other factor9s) may come into play.

the first thing I noticed is “tau” time calculated to 3 (three!!!) significant figures. A good sign that those involved have NO clue what they are doing. Now let me finish reading…..

right or wrong in that, but for what it is worth the link to the paper is on the line below:
http://www.john-daly.com/ahlbeck/ahlbeck.htm

Good paper. Agree or disagree, he is very clear about what he models and the assumptions in it and how he sets his parameters.
You are kidding right? Of course there is no actual partition it is a model so you can think through how carbon moves in and out of the atmosphere. You do get that, right? You do understand that to change sinks effects you change what is in each bucket (partition) to model how quickly that sink removes it from the atmosphere, right?
Please tell me you are not this rigid in your thought process – where is your degree from?
I’m not rigid in my thought process at all. I am looking at the equation Willis linked! Are you? Is there something in that equation that makes you think that it it could possibly be correct? The point I’ve been making is that even if you remove the exponential decaying parts from the kernel entirely, you are left with an integral of from to the present. I can do this integral in my head for any non-compact function — it is infinite. Ignoring the and integrating from “a long time ago but not infinity” in such a way that you get the right baseline behavior is obviously wrong in so many ways, if that is what they do.
In any event, this integral basically says that 15% of what is added every year is never going to be removed, and in fact is still around from every belch or fart of CO_2 gas since the planet began. The decay kernel then strictly increases this cumulative concentration, it does not decrease it, so that even if we all vanished from the planet tomorrow would remain constant for eternity.
This is clearly absurd, as I’ve tried to say so many times now.
We could then go on and address the rest of the kernel, the part that actually might make physical sense, depending on how it is derived. But it is then difficult, actually, to have it make a LOT of sense because the result would almost certainly have a completely incorrect form if $latexE(t)$ were suddenly set to zero. I could be convinced otherwise, but it would certainly take some effort, because then those “buckets” that are basically fairly arbitrary terms in an approximation to a very odd decay function, one that describes a very highly nonexponential process, not a sum of mixed differential processes.
In physics mixed exponential processes are far from unknown. For example, if one activates silver with slow neutrons, two radioactive isotopes are produced with different half-lives. If you try to determine the half lives from the raw count rate, you find that one of the two isotopes decays much more quickly than the other, so that after a suitable time the observed rate is almost all the slow process. One can fit that, then subtract the back-projected result and fit the faster time constant. Or nowadays, of couse, you could use a nonlinear least squares routine to fit the two at the same time and maybe even be able to get the result from a much shorter observation time if you have enough signal.
But note well, two different isotopes. I’m having a very hard time visualizing how, if CO_2 sources all turned off tomorrow, 1/e of 32% of it would have disappeared from the atmosphere within 2.56 years via one channel, but 28% of it will have only gone down by a factor of , while 25% of the rest will have diminished by and 15% of it will not have changed at all. It might even be correct, but what does this mean? All of the CO_2 molecules in the atmosphere are identical. What one is really describing is some sort of saturation (as you might have noted) of some process that can never take up more than 32% of the atmospheric CO_2, no matter how long you wait, with absolutely no sources at all.
At that point I have to say that I become very dubious indeed. First of all, this implies a complete lack of coupling across the “buckets”, which is itself impossible. By the time the fast process has removed 32% of the atmospheric CO_2 — call it ten or fifteen years, depending on how many powers of 1/e you want to call zero, the concentration exposed to the intermediate process has had its baseline concentration dropped by a third or more. This, in turn, destroys the assumptions made in writing out sums of exponentials in the first place, and so its time constant is now meaningless because the CO_2, unlike the silver atoms, has no label! It is quite possible that whatever process was involved in the 18 year exponential decay constant removal has switched sign and become a CO_2 source, because the reason given for not just summing the exponential decay rates of the independent processes is that they are not independent.
Finally, one then has to question the uniqueness of the decomposition of the decay kernel. Why three terms (plus the impossible fourth term)? How “linearized” were the assumptions that went into constructing it, and how far does have to change before the assumptions break down? This is a pretty complex model — wouldn’t simpler models work just as well, or even better? Why write the solution as an integral equation at all instead of as a set of coupled ODEs?
The latter is the big question. If were constant or slowly varying or there was some kernel of meaning to be extracted from converting the ODEs into an integral equation, there might be some point. But when one looks at the leading constant term, presumably added because without it the model is just wrong, it leads to instantly incorrect asymptotic behavior. Surely that is a signal that the rest of the terms cannot be trusted!. The evidence is straightforward — there are times in the past when CO_2 concentration has been much higher. Obviously the monotonic term is fudged over the real historical record or CO_2 now would not be less. But nothing in this equation predicts the asymptotic equilibrium CO_2 concentration if is zero. In fact, it creates a completely artificial baseline CO_2 that the decay kernel parts will regress to, one that varies with time to be ever higher now in spite of the fact that one simply didn’t do the integral over all past times and in fact imposed an arbitrary cut-off or something so that it didn’t diverge.
Am I somehow mistaken in this analysis? Is there some way that the baseline CO_2 concentration produced by this model is not strictly increasing from an absolutely arbitrary amount that is whatever value you choose to assign the integral before you really start to do it, say 1710 years in the past (ten of the slowest decay times)?
I’ve done my share of fitting nonlinear multiple exponentials, and you can get all kinds of interesting things if you have three of them and a constant to play with, but there is no good reason to think that the resulting fit is meaningful or extensible.
rgb
P.S. My degree in physics is from Duke. And I’ve published papers on Langevin models in quantum electrodynamics, and spent a decade doing Monte Carlo and finite size scaling analysis that involved fitting exponentially divergent quantities (and made my share of mistakes, and could easily be mistaken here — this is the first few hours I have looked at the equation, after all). But still, wrong/completely nonphysical asymptotic form is not a good sign when looking at a model, as I point out just as emphatically when it is CAGW doubters (like Nikolov and Zeller who propose a model for explaining atmospheric heating that contains utterly nonphysical dimensioned parameters) that come up with it.
And yeah, it disturbs me a lot to talk about “buckets” in a three term exponential decomposition of an integral equation kernel supposed to describe a systems of great underlying complexity with many feedback channels and mechanisms. It’s too many, or too few. Too few to be a good approximation to a laplace transform of the actual integral kernel. Too many to be physically meaningful in a simple linearized model. If you want to write I’m all for it, but be aware that the you end up with from an empirical fit are, well, shall we say open to debate in any discussion of physical relevance or meaning, especially one where the mechanisms they supposedly represent can themselves have nontrivial functional dependences.
And in the end, if you have a believable model, why not just integrate the coupled ODEs? That’s what I’d do, every time. If nothing else it can reveal places where your linearization hypotheses are terrible, as you add or tweak detail and the model predictions diverge.
Do you disagree?
[Formatting fixed … I think … -w.]

Freeman Dyson proposed “growing” topsoil (Biomass) as a means to fighting climate change – which he is sceptical of – as this would be more cost effective than reducing emissions and have a positive benefit for agriculture whilst resolving excess CO2.
Why haven’t the greens supported this innovative idea ?
I think it clearly shows the “climate change” debate is about political powerand not much else !

Willis Eschenbach
You asked: “what is the physical mechanism that is partitioning the CO2 so that some of it is sequestered quickly, and some is sequestered slowly?”
Perhaps the writeup you linked to was just not clear enough, or you interpreted the approximations presented as the model itself – if so, my sincere apologies. It seems quite clear to me what the page (http://unfccc.int/resource/brazil/carbon.html) presented: approximations of the Bern model (http://tellusb.net/index.php/tellusb/article/viewFile/15441/17291) results for how much and how fast CO2 ends up in different partitions, based upon that model of the carbon cycle, as those percentages. Not parameters, not the model itself, but an approximation of the model results. Results presented so that other researchers could use that approximation in their own work, with the stated caveat that “Parties are free to use a more elaborate carbon cycle model if they choose.”
I’m therefore finding quite difficult to see how you arrived at the interpretation you posed when writing the original post – that there is somehow an initial “partitioning”. That’s neither a correct description of the Bern model results nor of the UN page you linked to…

jimboW,
Thanks. It’s appreciated.
rgbatduke,
“However, for each mode of removal it is proportional to the total concentration in the air, not the “first fraction, second fraction, third fraction”. The CO_2 molecules don’t come with labels,”
As I mentioned above, the fractions are not a separation of the atmosphere into labelled portions, but a consequence of the relative sizes of the reservoirs. If water flows from tank A to tank B until levels equalise, and the tanks have equal surface area, they converge on the midpoint and half the water added to tank A stays there. If you add another bucketload to A, they equalise again and half the new bucket goes to B. It’s not some magic form of CO2 that hangs around for longer, it’s just the effect of the level increasing in the destination reservoir.
“In any event, this integral basically says that 15% of what is added every year is never going to be removed, and in fact is still around from every belch or fart of CO_2 gas since the planet began. The decay kernel then strictly increases this cumulative concentration, it does not decrease it, so that even if we all vanished from the planet tomorrow \rho_{CO_2} would remain constant for eternity.”
Ah, right. I’ve figured out what you’re talking about, now.
Yes, that is what the equations says, because it doesn’t include the very long term geological sequestration processes that take place on the order of thousands of years. Those components would have no significant effect – they’d look effectively constant – over the time intervals they ran the simulations for. The first fraction represents all those time constants too big to measure.
CO2 is conserved. If you chuck it into a system with no exits, it will stay there forever.
The integral is a convolution of the emission history with an impulse response function. If you get a pulse of emissions and then nothing, the emission falls further and further behind, t-t’ gets larger and larger, and the impact of the emissions is given a large negative weighting, shrinking it. It does decay.

All modelers are looking towards the next funding round and sensationalism wins the day every time. “Nature”, despite its unwarranted kudos, is not a scientific journal, it is a magazine.

Let’s back away from the problem just a bit and look at some data on carbon. Carbon in the world is located in the following locations/situations:
99.9% is in the sedimentary rocks in the form of limestone and dolomite
.002% is the fossils in the form of crude oil, natural gas, lignite and coal
.06% is in water bodies, primarily the oceans, in the form of CaCO3 and HCO3
.001% is in the atmosphere in the form of CH4, CO2, CO, VOCs and halogens
.005% is in the mineral soil in the form of humus, forest litter, bottom of mires and bogs
.001% is in living organisms, mainly vegetation
The question is then how does carbon dioxide convert -get into- the various forms of carbon storage or sinks? The times for each sink are obviously highly variable with the water bodies being the shortest, vegetation being second and the sedimentary rocks probably being the longest. With data perhaps one could develop relative time intervals. To me, using different times is acceptable for a model, I just don’t like the times and percentages used by the authors. They have made it too simple and precise of a problem.
One has to be careful of pinch points when dealing with something that is only .1% of the whole. In human time frames only water bodies and vegetation are of probable interest. Winds and currents are of the most interest with clearing and replanting of forests and jungles being of important interest as well.
With the CO2 in/out ratio being so constant, I am suspicious of oxidation of limestones and dolomites being undercounted in any material balance calculations. Ninety Nine point nine percent doesn’t have to change very much to sway the other times considerably.

To electrical engineers, the impulse response model (the equation) in the link is quite unremarkable – just an ordinary linear system. On our benches, it would look like a bunch of R-C low-pass circuits in parallel (six of them I guess). The partitions (gain constants) and time constants are parameters of the model, and could be derived from rho(t) by deconvolution, if we knew E(t). This assumes the system really is linear (the one on the bench is), and is exactly the sum of six real poles. The theory is straightforward and manipulations such as “Prony’s method” and “Kautz function analysis” are long-established (and quite beautiful ).
That noted, attempting to apply this mathematical procedure to a true CO2 concentration curve is of course, utter nonsense. Likely the CO2 situation is not even linear in the first place, and the measured curves are subject to large systematic and random errors. For a circuit on the bench, we could at least cheat and peek at some of the component values. But for the atmosphere, there are no actual partitions, or separable processes with-defined characteristic times. There are NO discrete components – let alone ones we could identify and measure! It’s just a silly over-beefy model.
For CO2, It is doubtful there would be any usable physical reality to even a single pole model. It’s very very far from being a circuit on a bench.

Willis, you are one of the great ones. I very much appreciate your keen mind and the quickness and width of your knowledge and interests. I read your treatises first, always.
I just see that it doesn’t take much exposure to the air of near surface carbonates, arising from landslides, floods, hurricanes, earthquakes; you name it, to introduce enough additional CO2 to the atmosphere to offset the removal by the other sinks. Thus, in human time, there will always be CO2 in the atmosphere no matter what the faster processes do in removing CO2.
I have 99.9% versus .1% in my favor, grin.
JFD

Bill Illis says:
May 6, 2012 at 1:16 pm
“It will take about 150 years to draw down CO2 to the equilibrium of 275 ppm if we stop adding to the atmosphere each year. Alternatively, we can stabilize the level just by cutting our emissions by 50%”
——————————————–
Why would we want to do that?

Sure, I’ve heard of carbonitite volcanoes. They have a high percentage of limestone and dolomite (calcium/magnesium carbonates) in them. They are in the 99.9% of carbon listed first in my post.

There was an extensive set of meaurement published in several peer reveiwed papers by by teams of scientists working at Princeton University at the beginning of the 21Century.
Unlike the bovine pasture patties “precise” half-entry, non-debit only, “bookeeping accounting” produced by the EPA, these scientyists measured the CO2 content on the Air blowing in from the Pacific prevailing winds, into North America and they measured what happened to it as it traversed over the continent, and then measured CO2 as it exited on the prevailing winds blowing out over the Atlantic. They discovered it rose over the industrialized Coasts, and the again in industrial midwest, but decreased as it traversed the forests, ranchlands of the West, the breadbaskets of the grasslands and the eastern and southern forests. They also reported the North America continent absorbs much more than it emits by both Man and Nature. The Air blowing out over the Atlantic has much less CO2 the air entering the continent, despite all that the most industrialized country adds.
North America is the biggest Carbon Sink on the Planet and proves there is absolutely no need for America to have any concerns about CO2, even if you concede that CO2 is of any concern at all except to a botanist. If Eurasia produces net CO2, let them remove it. We have already done all we need to do and much more.
“A Large Terrestrial Carbon Sink in North America Implied by Atmospheric and Oceanic Carbon Dioxide Data and Models
S. Fan, M. Gloor, J. Mahlman, S. Pacala, J. Sarmiento, T. Takahashi and P. Tans” Science 16 October 1998:
Vol. 282 no. 5388 pp. 442-446
DOI: 10.1126/science.282.5388.442
, is just typically one of many such papers, that the CAGW Eco-Druids, managed to suppress and/or ignore.
Meanwhile the EPA eco-druids total the reports which estimate the kilo pounds that human industries report emiting; and in their half-assed accounting haven’t found a way to intimidate the mighty Oak and Pine to fill out their bureaucratic forms and to report how many megatons they and their saplings absorb, so don’t bother to include any considerations of that.

I’ll acknowledge a personal lack of mathematical virtuosity, but there seem to be two ways of applying mathematics to physical phenomena.
The first is to come up with some ‘cocktail-shaker’ combination of numbers and functions that somehow fits observations and accurately models the past, thus inspiring confidence that it may be useful for predictions. This is essentially ‘hit-and-miss’ with little genuine understanding required, but can be useful for complex, analysis-defying systems.
The second is to accurately quantify and incorporate ALL relevant factors with their correct relationships. This requires a high degree of understanding and becomes progressively more difficult as system complexity increases. Indeed, with something like the weather or CAGW I have to wonder at the claimed reliability of ANY model.
I like to think that mathematicians are gainfully employed but, surely, some phenomena are not readily amenable to mathematical modelling. Will a significant increase in CO2 lead to evolution of more CO2-hungry organisms? Exactly how big a role does the sun play and how do we know what it’s going to do next? What about cosmic rays? I’ve barely scratched the surface here and we’re arguing about the application of high mathematics to poorly-understood phenomena. What’s the weather going to be like next month?
Frankly, I believe that the study of crystal orbs or chicken entrails is just as likely to deliver an understanding of climate as the Bern Model or its rivals.
Surely, fans of this website have noticed that it is a lot easier to poke holes in the asinine pronouncements of the doomsayers than it is to come up with a robust alternative. Think about the reasons for this.
(Wet blanket now hung out to dry.)

I think David Archer is describing the “Bern” model clearly here.
http://geosci.uchicago.edu/~archer/reprints/archer.2008.tail_implications.pdf
I think they’re arguing that CO2 quickly reaches a balance with sea surface and plants, but is slow to reach balance with the ocean depths. Forr a simple example, if the CO2 ratios in
the oceans’ top layer, in plants,,in the atmosphere, and ocean depths was 2, 1, 1, and 48,
and an additional unit of CO2 was dumped into the system, the balance would quickly reach
2.5, 1.5, and 1.5, but the 48 ocean depth is acting a lot slower, on a scale of about 800 years.
I think the drop in C14 since the nuclear testing spike in the early 1960s is a counter example to the Bern- David Archer model.
http://en.wikipedia.org/wiki/File:Radiocarbon_bomb_spike.svg

“CO2 evolves according to a higher-order linear equation (or a system of first-order linear equations that is the same). Very reasonable. That is where the “partitioning” comes,”
NO, NO, NO, NO, NO.
CO2 does not do anything “according to” any equation.
We humans use equations as fair, simlar MODELS of reality. A molecule of ANYTHING never checks some equation to see how to behave. Never.
That is our human imagination that a falling object’s speed “follows” some formula, etc.
This may not seem like a big point, but it makes all the difference in the world. The natural world does not behave according to formulas, with us discovering the formula. The natural world behaves. We develop models that APPROXIIMATE this behavior. If we are lucky.

First thing is, the expression is not the Bern model, it is approximation (regression) of the Bern model. You can understand individual factors as regression coeficients which usually have very limited connection to reality.
Second thing, the categorization of CO2 sinks in SAR has probably nothing to do with categorization of CO2 sinks in TAR – SAR works with five “main sinks”, TAR works with three completely different “main sinks”. In each case real considered CO2 sinks are assigned to five or three groups for simplicity in a way so that differences in each group more or less cancel out to provide consistent behavior of the group.
In order to visualise the expression, divide earth surface to six (SAR) or four (TAR) parts, proportional to coefficients a(0) to a(n). a(0) is part of earth surface which does not act as carbon sink, the rest are carbon sinks with “sinking” effectivity given as tau(n). Also understand that proportion of Earth surface also corresponds to proportion of atmospheric volume above that surface.
The coefficients tau(n) specify effectivity of individual sinks – if tau(1) is 171 and tau(2) is 18 it only means that sink 1 would do with the atmosphere in 171 years what sink 2 would do with it in 18 years, if there was only sink 1, respective sink 2 all over the world.

Willis Eschenbach says:
May 7, 2012 at 12:06 am
‘For example, the expansion of Cos(x)/x = 1/x – x/2 + x3/24 – x5/720 + …
But nowhere in there do I find x broken into “13% x / 720 +26% x / 720 …”’
I assume you mean, if “y = Cos(x)/x”, nowhere do you find y broken into “13% x / 720 +26% x / 720 …”’
But, nothing is stopping you from doing so.
“Also, normally an infinite expansion has alternating positive and negative terms which decrease in size. This is not true of their expression, where all terms are positive and are of different sizes …”
Not generally. For example, exp(x) = 1 + x + x^2/2 + x^3/6 + … The coefficients have to decrease in size or the expansion will not converge. But, the decrease does not have to be monotonic. For example, (1 + x^2)*exp(x) = 1 + x + 1.5*x^2 + 7/6*x^3 + …
Of course, exp(x) is unbounded as a function of x. But, the polynomial base functions are, themselves, unbounded. In the Bern model, the basis functions are decaying exponentials, so this is not a concern. For example, I can expand
(1 + exp(-x)^2)*exp(exp(-x)) = 1 + exp(-x) + 1.5*exp(-2*x) + 7/6*exp(-3*x) + …

Dr Burns says:
May 6, 2012 at 1:30 pm
In relation to sources and sinks, can Willis, or anyone else explain this image of global CO2 concentrations ?
Why does Antarctic ice appear to be such a strong absorber in parts and why such strong striation?
http://www.seos-project.eu/modules/world-of-images/world-of-images-c01-p05.html

I think the striations are due to mid-troposphere southward flows of air with relatively high CO2 feeding surface northward katabatic winds.
But I was unable to find a study that supports this, so just a guess on my part.

“Thanks, Bart. So … where in your derivation do we find the part about the division of the atmosphere into partitions, each of which has a different time constant? That’s the part that seems problematic to me”
Where he says:
“Thus, the fraction a/(a+b) of H accumulates in the atmosphere, and b/(a+b) accumulates in the oceans. The total a/(a+b) + b/(a+b) = 1, so all of it either ends up in the land or in the oceans.”
If a fraction a/(a+b) stays in the air, the time constant for the transfer only applies to the CO2 transferred, which is a proportion b/(a+b) of the total atmosphere.
Some have said that the rate has to be proportional to the total CO2 content of the atmosphere, but this isn’t true. The rate is proportional to the difference in concentrations of the source and destination reservoir, like conductive heat flow is proportional to the difference in temperatures, electrical current is proportional to the difference in voltages, water flow is proportional to the difference in heights, etc.
And the difference between the atmosphere and one sink may be different to the difference between the atmosphere and another sink.
The other thing being assumed is that the CO2 transferred has no effect on the level of CO2 in the sink, that it acts like an infinite void, or an idealised thermodynamic cold sink that can absorb any amount of heat without changing temperature. While there’s no doubt the ocean is a lot bigger than the atmosphere, it’s not infinitely bigger, and to start with only the top layer of the ocean is affected. The level in the sink increases as it absorbs CO2, and the rate of flow is proportional to the difference in levels. So the flow pushes the source to decrease and the sink to rise until they converge on an intermediate level, when flow stops. It’s not because the reservoir is full, or saturated. It’s simply because the levels are equal. If you add more CO2, it will go up again. If you keep adding CO2 it will go up continuously. Stop and the flow will slow and stop, but it decays away exponentially to the intermediate level, not the original level. (A fraction a/(a+b) stays in the air.) This is at some fraction of the difference between them, and that’s where the apparent partitioning comes from.
I’ll try the tanks example with some numbers, in case that helps. Tanks A and B have 1 square metre horizontal cross sections, but are very tall. Tank C has a 98 square metre cross section, and is again tall. The level of water in all three tanks is 3 metres. We now dump a pulse of 3 cubic metres into tank A. Instantaneously the level in that tank doubles to 6 metres. Very quickly, because of the wide pipe connecting them, A drops to 4.5 metres and B rises to 4.5 metres as 1.5 cubic metres of water flows between them. Note that the 3 cubic metres added has been partitioned – half of it has transferred and half stayed. Only 1.5 metres is affected by the rapid exponential decay. Then over a far longer period, the 4.5 metres in A and B equalises with the 3 metres in C. 3 cubic metres (2x(4.5-3)) is spread evenly over 100 square metres, for an equilibrium depth of 3 centimetres. Tanks A and B slowly drop to 3.03 metres, and C even more slowly rises to 3.03 metres. There is 3 cm of permanent rise.
None of the tanks are full. If you dump another three cubic metres into tank A, half of it will still drain into tank B just as rapidly as before. The sink is not saturated, its capacity for absorption not reduced. You could dump 300 cubic metres into it and half would still transfer to B with the same rapid rate constant. But only half.

Thanks Willis for responding to my comment, which was based on a misunderstanding of the link I gave. But what an interesting post, question and comments. I learn something new here every day.
The question, ” what is the physical mechanism that is partitioning the CO2 so that some of it is sequestered quickly, and some is sequestered slowly?” seems to be unanswerable, which perhaps makes the Bern Model wrong and therefore its use by the IPCC wrong.
Particularly as “there is no experimental evidence that will allow us to distinguish between plain old exponential decay (which is what I would expect) and the complexities of the Bern model”.
The Svensmark paper mentioned carbon dioxide being scarce when supernovas were high based on the idea that plants dislike carbon dioxide molecules containing carbon-13, which were then absorbed by the ocean. But this doesn’t seem relevant so apologies if I’m talking through my proverbial.
I look forward to your further posts on sinks and the e-folding time. You have that rare knack for getting straight to the heart of a theory.

Allan McRae,
Nice analysis! Temperature variations cause a lagged CO2 response because of the solubility pump’s dependence on temperature. But CO2 change is contributed to by many sources and sinks, and just because one component is caused by temperature doesn’t mean all the others are.
You might find it useful to plot a graph of the correlation coefficient between temperature and lagged CO2 as a function of the lag. That’s the usual way to make lagged relationships clear.

I notice that the little cartoon diagram fails to include the sequestration as concrete sets.

I should also add about this to the “rapidly diminishing” part.
Running water performs three closely interrelated forms of geologic work:
1. Erosion
2. Transportation
3. Deposition
As rivers erode the river bed the angle of incline becomes lower. In “old age” or where a river/stream dumps into a lake, pond or ocean you get flood plains with low relief across which the water flows meander causing braided channels, deltas and marshes. As the eroded sediment is dumped into lakes and ponds they fill and the later stages are swamps and marshes.
Mankind using dredging can to some extent modify this geologic progression but with “wetlands protection” in many advanced western countries “rapidly diminishing” stopped thirty years ago and has more likely changed to “rapidly advancing” ever since.
In 1971, the international Convention on Wetlands was adopted…The Ramsar List of Wetlands of International Importance includes nearly 2,000 sites, covering all regions of the planet.
In other words this is a recycling of the old “Boogy Man” from the 1960’s for a new generation of ignorant bleeding hearts who are unfamiliar with the complex tangle of international treaties, conventions and accords.
(BTW bleeding hearts is not a derogatory term, it is the con-men using people’s compassion that I have the problem with. Civilization requires EDUCATED bleeding hearts to keep the predators in check. Most people on WUWT fit the category of educated bleeding hearts, or we would not give a darn what happens to future generations.)

Some have said that the rate has to be proportional to the total CO2 content of the atmosphere, but this isn’t true. The rate is proportional to the difference in concentrations of the source and destination reservoir, like conductive heat flow is proportional to the difference in temperatures, electrical current is proportional to the difference in voltages, water flow is proportional to the difference in heights, etc.
Yes, but in order for the model to be correct, none of the reservoirs can be in contact with absolute zero, and if the model doesn’t permit the attainment of equilibrium on a reasonable time scale and in a reasonable way, it is just wrong. So why, exactly, will the biosphere adjust its equilibrium upwards? Why won’t the ocean follow the temperature instead of some imagined “fractional difference” leading it to an every higher base equilibrium concentration? Noting well that the total atmospheric CO_2 is just a perturbation of oceanic CO_2 so that for all practical purposes it is an infinite reservoir.
rgb

MikeG says:
May 7, 2012 at 1:31 am
“Declaring exponential constants to four significant figures is a triumph of optimism.”
Or, something. Agree completely.
richardscourtney says:
May 7, 2012 at 2:09 am
“This indicates the ‘adjusting carbon cycle’ assumption is plausible but, of course, it does not show it is ‘true’.”
Absolutely. The system is underdetermined and not fully observable. Thus, to get an answer, the analysts have to insert their own biases. The likelihood that they would hit on a faithful model, out of all the possibilities, is vanishingly small.
Nullius in Verba says:
May 7, 2012 at 2:45 am
“The matrix is of rank n-1, since we lose one degree of freedom from the conservation of mass. We therefore ought to get n-1 decaying exponentials for n reservoirs.”
The reservoirs themselves are a continuum. For example, what is the land reservoir? It is trees, grasslands, soil bacteria, rock and sediment formation, mammals, reptiles, amphibians, insects, etc… And, the dynamics of the atmosphere are diffusive. Areas of high and low concentration appear randomly, and the divergence increases near the surface sink. The eigevalues of the Laplacian operator are limitless. So, basically, you are correct, but n tends to infinity.

rgbatduke says:
May 7, 2012 at 8:06 am
“There is no rate of addition of CO_2 that can lead to equilibrium with this solution but zero.”
Yes. As I have been saying, the model is mathematically, theoretically sound. But, they have parameterized it in such a way that it gives the answer they wanted, vastly underestimating the power of the sinks to draw out a substantial fraction of the atmospheric constituents in the near term. There is no data available to establish that the model is correct.
In fact, we know it is incorrect, for the very simple reason that temperature is driving the rate of CO2, and not the other way around. It is obvious in this plot that anthropogenic inputs have, at best, a minor role in establishing the overall concentration. Temperature variation accounts for almost all of it.

Legatus says:
May 6, 2012 at 11:45 am
The basic assumption of this model seems to be that there is some “perfect” amount of CO2 that the earth tries to return to. Otherwise, if adding CO2 causes it to slowly go away, we should have no CO2 now, right? Thus, they must believe that it does down to this “perfect” amount and just stays there. Why does it stop diminishing? What mechanism could cause it to do so? For that matter, what mechanism would cause it to try and return to some “perfect” amount?

My Occamized Hypothesis is:
Geology and geogenesis put a sh**-load of CO2 into the early atmosphere. Eventually, life forms (flora) arose able to build themselves and proliferate by using photons to combine H2O and CO2. Shortly, eaters (fauna) evolved to munch on said flora. The flora expanded with little restraint and consumed CO2 until it began to run short. They then evolved to survive on less, but kept going. They are now approaching a lower limit, variously guesstimated to be in the region of 130-260 ppm — famine time. Fauna mass and volume and numbers track this, more or less.
So the “ideal number” is the lower limit, as that’s what the dominant life forms (flora) keep trying to achieve. Warmies are mental vegetables, so they also want to flirt forever with suicide by famine.
Simples!
So

“a2videodude says:
The bottom line is that simultaneously deducing the distribution of amounts AND half-lives from decay data (either radioactive decay or CO2 concentration decay) is incredibly difficult and the uncertainties are enormous because the functions you are using to model the decay (a series of exponentials) are far, far from being orthogonal. Any negative exponential can, to excellent accuracy, be approximated by a sum of other exponentials with different decay rates. You can either deduce decay rate if you know you have a single (or at least very simple but known) combination of reservoirs, or you can deduce the amounts in different reservoirs if you know their decay rates independently. You just can’t to both things simultaneously to any useful degree.”
Absolutely correct. In a chemical reaction the measured first order rate constant IS the sum of all the first order rate constants, which are individual collisions of molecules of differing energy and colliding on different vectors.

I am just amazed that nobody has commented on the relationship I have pointed out numerous times, which clearly shows that temperature has been driving CO2 concentration. It is obvious here that there simply is no need to consider human emissions to any significant level. This very simple observation kicks the very foundation out from under the Climate Change imbroglio.

Willis Eschenbach – “”…as far as I can see, the Bern model proposes an impossibility. It says that the CO2 in the air is somehow partitioned, and that the different partitions are sequestered at different rates”
I really don’t understand this statement. There are multiple parallel and serial processes occurring in the fairly simple Bern model, and the approximations (constants for weighting and time factor) are just approximations for the various and multiple (parallel and serial) inter-compartment transfer rates. The percentages allow considering different CO2 pulses, and the time factors show how the Bern model inter-compartment movements show up.
There certainly is no initial partitioning of a CO2 input in the Bern model. The complexities of the model, its behavior, are just curve-fitted. Much in the way nuclear reactor fuel decay can be fit with a decaying exponential, regardless of the internal physics – a behavioral description. The exponentials are purely descriptive, behavioral analogs to the Bern model, which was (if I interpret that initial page correctly) provided as one potential resource. The issues raised with compartmentalization are really irrelevant.
I (IMO) don’t believe it’s appropriate to criticize the Bern (or any other model) from that standpoint – two steps back, arguing about the curve-fit to the model behavior. Rather, if you wish to truly critique a model, you need to show where the model itself breaks down. And since there is as far as I can see no discussion here of the assumptions, parameters (fit to among other things the carbon-14 data), and compartmentalization of that model, what is being discussed really isn’t the model at all.
That is not to say that the Bern model is a thing of perfection. It’s 20 years old, does not include long term sequestration such as CaCO3 or rock weathering, a simplistic biosphere compartment, and (as Joos et al note in their paper) has some latitude dependent inaccuracies with replicating C-14 measurements. But to quote George Box, “Essentially, all models are wrong, but some are useful”. A critique of this model needs to show how the model fails to meet observations – something that simply hasn’t been done on this thread.
There has been no demonstration that the model itself isn’t useful – that requires an evaluation against the data. If it fits the data, it’s useful. If it doesn’t, it’s not. I have seen no discussion of the model behavior against observations.

Bart
The Bern model is a carbon cycle model, not a temperature model, and the observations used to calibrate the Bern model are carbon-14 distributions in the oceans, also checked against (IIRC) CFC-11 distributions. In regards to the mid-term carbon cycle, the model being discussed is reasonably accurate – it matches those observations.
It would be an error to assume that CO2 is the only forcing WRT temperature, however – methane, CFC’s, aerosols, solar, and the ENSO variations are also in play. All of those affect climate forcings (and hence temperatures) as well. And all of those need to be (and are, in the literature) considered when looking at forcings and climate responses – issues beyond the realm of the carbon-cycle model discussed in this thread.

KR says:
May 7, 2012 at 8:26 pm
“It would be an error to assume that CO2 is the only forcing WRT temperature…”
Indeed it would. CO2 is not forcing temperature. Temperature is forcing CO2. The fact that the derivative of CO2 is highly correlated with temperature anomaly establishes it. As I related above, the forcing cannot be the other way around without producing absurd consequences.
“In regards to the mid-term carbon cycle, the model being discussed is reasonably accurate – it matches those observations. “
A subjective exercise in curve fitting which cannot gainsay the above.

For those who have not been following along, the way in which CO2 concentration can be insensitive to human inputs while its derivative is effectively proportional to delta-temperature is explained here.