The Bern Model Puzzle

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.
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@ur momisugly 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.
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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.
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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
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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.