Guest Post by Willis Eschenbach
Although it sounds like the title of an adventure movie like the “Bourne Identity”, the Bern Model is actually a model of the sequestration (removal from the atmosphere) of carbon by natural processes. It allegedly measures how fast CO2 is removed from the atmosphere. The Bern Model is used by the IPCC in their “scenarios” of future CO2 levels. I got to thinking about the Bern Model again after the recent publication of a paper called “Carbon sequestration in wetland dominated coastal systems — a global sink of rapidly diminishing magnitude” (paywalled here ).
Figure 1. Tidal wetlands. Image Source
In the paper they claim that a) wetlands are a large and significant sink for carbon, and b) they are “rapidly diminishing”.
So what does the Bern model say about that?
Y’know, it’s hard to figure out what the Bern model says about anything. This is because, 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. The details of the model are given here.
For example, in the IPCC Second Assessment Report (SAR), the atmospheric CO2 was divided into six partitions, containing respectively 14%, 13%, 19%, 25%, 21%, and 8% of the atmospheric CO2.
Each of these partitions is said to decay at different rates given by a characteristic time constant “tau” in years. (See Appendix for definitions). The first partition is said to be sequestered immediately. For the SAR, the “tau” time constant values for the five other partitions were taken to be 371.6 years, 55.7 years, 17.01 years, 4.16 years, and 1.33 years respectively.
Now let me stop here to discuss, not the numbers, but the underlying concept. The part of the Bern model that I’ve never understood is, what is the physical mechanism that is partitioning the CO2 so that some of it is sequestered quickly, and some is sequestered slowly?
I don’t get how that is supposed to work. The reference given above says:
CO2 concentration approximation
The CO2 concentration is approximated by a sum of exponentially decaying functions, one for each fraction of the additional concentrations, which should reflect the time scales of different sinks.
So theoretically, the different time constants (ranging from 371.6 years down to 1.33 years) are supposed to represent the different sinks. Here’s a graphic showing those sinks, along with approximations of the storage in each of the sinks as well as the fluxes in and out of the sinks:
Figure 2. Carbon cycle.
Now, I understand that some of those sinks will operate quite quickly, and some will operate much more slowly.
But the Bern model reminds me of the old joke about the thermos bottle (Dewar flask), that poses this question:
The thermos bottle keeps cold things cold, and hot things hot … but how does it know the difference?
So my question is, how do the sinks know the difference? Why don’t the fast-acting sinks just soak up the excess CO2, leaving nothing for the long-term, slow-acting sinks? I mean, if some 13% of the CO2 excess is supposed to hang around in the atmosphere for 371.3 years … how do the fast-acting sinks know to not just absorb it before the slow sinks get to it?
Anyhow, that’s my problem with the Bern model—I can’t figure out how it is supposed to work physically.
Finally, note that 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. We simply don’t have enough years of accurate data to distinguish between the two.
Nor do we have any kind of evidence to distinguish between the various sets of parameters used in the Bern Model. As I mentioned above, in the IPCC SAR they used five time constants ranging from 1.33 years to 371.6 years (gotta love the accuracy, to six-tenths of a year).
But in the IPCC Third Assessment Report (TAR), they used only three constants, and those ranged from 2.57 years to 171 years.
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.
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?
All ideas welcome, I have no answers at all for this one. I’ll return to the observational evidence regarding the question of whether the global CO2 sinks are “rapidly diminishing”, and how I calculate the e-folding time of CO2 in a future post.
Best to all,
w.
APPENDIX: Many people confuse two ideas, the residence time of CO2, and the “e-folding time” of a pulse of CO2 emitted to the atmosphere.
The residence time is how long a typical CO2 molecule stays in the atmosphere. We can get an approximate answer from Figure 2. If the atmosphere contains 750 gigatonnes of carbon (GtC), and about 220 GtC are added each year (and removed each year), then the average residence time of a molecule of carbon is something on the order of four years. Of course those numbers are only approximations, but that’s the order of magnitude.
The “e-folding time” of a pulse, on the other hand, which they call “tau” or the time constant, is how long it would take for the atmospheric CO2 levels to drop to 1/e (37%) of the atmospheric CO2 level after the addition of a pulse of CO2. It’s like the “half-life”, the time it takes for something radioactive to decay to half its original value. The e-folding time is what the Bern Model is supposed to calculate. The IPCC, using the Bern Model, says that the e-folding time ranges from 50 to 200 years.
On the other hand, assuming normal exponential decay, I calculate the e-folding time to be about 35 years or so based on the evolution of the atmospheric concentration given the known rates of emission of CO2. Again, this is perforce an approximation because few of the numbers involved in the calculation are known to high accuracy. However, my calculations are generally confirmed by those of Mark Jacobson as published here in the Journal of Geophysical Research.
_Jim says:
May 6, 2012 at 1:12 pm
73’s back atcha …
H44WE
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.
ferd berple says:
May 6, 2012 at 12:39 pm
“Nonsense. The oceans cannot tell if that 1/2 comes from this year or last year.”
rgbatduke says:
May 6, 2012 at 2:35 pm
“Dearest Bart, Piffle.”
Guys… if you jump to conclusions and assume your opponents are completely witless without even bothering to understand their reasoning, you are never going to be effective. To get an idea of what they are thinking, consider a simple atmosphere/ocean coupled model of the form
dA/dt = a*O – b*A + H
dO/dt = b*A – a*O – k*(1+a/b)*O
A = atmospheric CO2 concentration
O = oceanic CO2 concentration
H = anthropogenic inputs
a,b,k = coupling constants
The “a” and “b” constants control how quickly CO2 from the atmosphere dissolves into the oceans. The “k” constant determines how quickly the oceans permanently (or, at least, semi-permanently, i.e., sufficiently long as to be of little consequence) sequester CO2.
If you are familiar with Laplace transforms, you can easily show that the transfer function from H to A is
A(s)/H(s)= (s + a + k*(1+(a/b)) / (s^2 + (a + b + k*(1+a/b))*s + k*(a+b) )
Under the assumption that “a” and “b” are much greater than k, this becomes approximately
A(s)/H(s) := (a / (a + b) ) / (s + k)
This approximate transfer function describes a system of the form
dA/dt := -k*A + (a / (a + b) )*H
A similar calculation for O will yield
dO/dt = -k*O + (b / (a+b) )*H
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.
The IPCC effectively says a is approximately equal to b, hence roughly 1/2 ends up in each reservoir in the short term. If this seems unreasonable to you, well, we aren’t done yet, so keep your powder dry. In actual fact, the processes involved are much more complicated than this. Obviously, for one thing, we haven’t included the dynamics of the land reservoir.
And, because the dynamics are governed by diffusion equations, partial differential equations (PDE) which can be cast as an infinite expansion of ordinary differential equations (ODE) (this is a key result of functional analysis), the scalar equations can be expanded into infinite dimensional vector equations, with the components of the vectors summing up to the total. Each component has its own gain and time constant associated with it, and can thereby be considered a partition of the total CO2. It is a mathematical construct, not a physical one, which approximates the physical reality only when it is all summed together.
That is how the Bern model is constructed. I am not saying it is constructed correctly, I am just telling you it is on firm theoretical grounds, and you guys are attacking the castle wall at its most fortified location, instead of just walking around to where they haven’t even laid the first stones.
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 look at the CO2 anual variations of the Hawaii observations due to just the seasonal temperature variations and it is 300% of the annual increase. The ocean’s breath is 3 times the annual increase of the supposed human footpring.
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
Seems to me plants will grab as much CO2 as they can get their grubby leaves on.
Since the CAGW claim is CO2 is “Well Mixed” then the reduction of CO2 in the atmosphere by plants should be governed only by how fast new CO2 can be transported via diffusion or wind into contact with the leaves.
Hydroponic Shop
Given the evidence from the wheat field (C3 plants) that plants will use all the CO2 in their vicinity, coupled with absorption of CO2 by water (rain is a weak acid due to dissolved CO2) I find the residence times over a couple of years very tough to swallow.
As Ian W says:
And of course as the CO2 is brought back to the surface the plants on land and in the ocean gobble it up.
I find myself agreeing with Bart here. And with Nullius, who I think is expressing the right idea, and also with the electrical circuit analogies.
To rephrase Bart (I think) you have a Laplace Transform representation, and you approximate the integrand by a set of poles. Or, if you want to think of it in the real domain, you have the idea that your response can be represented as a weighted average of a whole lot of exponentials (that’s just math), and then you choose a few to be representative.
But you can also see it in electrical terms with resistance-capacitance circuits. Each R-C pair has a time constant, reflecting the timescales in the Bern Model.
And yes, it’s also a multi-box model, and they have difficulties.
I’m glad to see Willis’ appendix – the two different time constants are indeed poorly understood.
The only thing that makes sense to me is the model they are using is assuming saturation of the various processes and assigning a % to that sink. That is, once the fastest sink saturates they assign a value to it, then they look at the 2nd fastest sink and so on. What’s eventually left goes into the slowest sink.
I didn’t read the link but I can’t think of any other way they could generate those percentages.
“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.
There seems to be confusion about the integral equation in the link. It is simply a “convolution integral” which says that the output is the input
convolved with the impulse response – very standard stuff. The impulse response, the term in [ ], does (in their formulation) contain a step, but IT is not inside the integral BY ITSELF. It is inside multiplied by E. E in turn can be thought of as a single impulse (or usually as many, possibly an infinite number) of impulses. Thus we may integrate TO a step. This simply means there will be a scaled version of the input in the output. A non-decaying exponential in the impulse response. Overall this corresponds to an exact (actual) circuit configuration.
It seems to me – we should think of CO2 here just like charge on a capacitor. In the circuit, individual (isolated) charges are restricted to their own capacitor (drained by an individual resistor). This is NOT what happens in the atmosphere – obviously. It’s all one capacitor and all the individual sinks are one resistor. Wrong model, and probably a faulty physical understanding.
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.
In the paper they claim that a) wetlands are a large and significant sink for carbon, and b) they are “rapidly diminishing”.
They’re also a large and significant source of methane. As for the “rapidly diminishing” part, I can only assume they believe that any (cue scary music) sea level rise (cut scary music) will cover existing wetlands (aka, “tidal swamps”) without creating new ones…
Bart says:
May 6, 2012 at 7:47 pm
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, and I didn’t see anything about that in your math, although I could have overlooked it. What am I missing?
w.
JFD says:
“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.”
You only mentioned sedimentary rocks, volcanic source rocks are not sedimentary rocks, they source material from the mantle. (as well as recycling material from the crust and from the ocean at plate boundaries). But until there is a mantle cycle ( eg mid ocean ridges) and a subduction cycle (eg largely at plate boundaries) in the carbon cycle diagram, the carbon cycle diagram as shown, is astonishingly incomplete. As I said before, carbonate and volcanoes in the oceans are involved in large scale exchanges, especially along mid ocean ridge systems, and in island arcs. These are not accounted for in the carbon cycle diagram of the IPCC. Carbonitites are another example, contianing >50% carbonate, although the origin of this carbonate is disputed.
As I also mentioned, this is important because I suspect the mid ocean ridges and other volcanoes play a role in e.g. buffering ocean acidity. This is not accounted for by marine biologists, of course.
Bart says:
May 6, 2012 at 7:47 pm
Bart, a question. Why is the rate at which the ocean sequesters CO2 a function of 1 + a/b ? Isn’t it a totally different physical process? Why should it depend on the rate of air/sea exchange?
w.
Nick Stokes says:
May 6, 2012 at 8:22 pm
“But you can also see it in electrical terms with resistance-capacitance circuits. Each R-C pair has a time constant, reflecting the timescales in the Bern Model.”
The characteristics of transmission lines fit this description, and transmission line models are often used to characterize so-called pink noise.
Willis Eschenbach says:
May 6, 2012 at 10:16 pm
“…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?”
At the part where I said: ” partial differential equations… can be cast as an infinite expansion of ordinary differential equations… Each component has its own gain and time constant… It is a mathematical construct, not a physical one, which approximates the physical reality only when it is all summed together.”
Bart says:
May 6, 2012 at 11:43 pm
Thanks, Bart, but I still don’t see where you get the partitioning. My understanding (which of course may be wrong) is that in an infinite expansion, each term applies to the entire thing, not e.g. 13% of the total for one term and 26% of the total for the second term and the like.
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 …”
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 …
Thanks for your patience in the explanations. As I said at the start, that’s what I’m looking for.
w.
Bart says:
May 6, 2012 at 7:47 pm
Erratum – This sentence should read: “The total a/(a+b) + b/(a+b) = 1, so all of it either ends up in the atmosphere or in the oceans.” The model I demonstrated did not include the land dynamics. Its main purpose was to show how roughly 1/2 of the CO2 could end up rapidly transported from the atmosphere into the oceans without becoming permanently sequestered from the overall system.
As I stated above, I believe this description is moot. That it is mathematically possible is not confirmation that it is the governing process, and the rather strong correlation between CO2 and temperature which I have pointed out indicates that to me that it is an unimportant question. Temperatures are driving CO2 concentration, and not the reverse.
Bart, let me try to explain it a different way. If I understand you, you are saying that the equation that they give in the linked file is an infinite expansion of a transfer function based on the underlying box model.
But if that were the case, why would there be a variety of numbers of terms, along with different coefficients for the terms? If it is an infinite expansion, wouldn’t it have defined coefficients with defined corresponding time constants? They use two forms, for example, one with six terms and the other with four terms. The terms have different coefficients, and apply to different fractions of the whole … how is it possible that that is “an infinite expansion of ordinary differential equations” as you say?
w.
“Another result from this assumption is that IPCC can invoke inappropriate chemical equilibrium equations to give the sequestering of sea water multiple simultaneous time constants, ranging from centuries to thousands in the IPCC reports, and up to 35,000 years in the papers of its key author, oceanographer David Archer, University of Chicago. The assumption is foolishness as shown by its consequences, but it tends to confirm oceanographer Wunsch’s 10,000 year memory claim. The science should have influenced Wunsch to distance himself from IPCC, neither joining with it in the lawsuit, nor identifying himself as a supporter of its conclusion, the existence of AGW”
http://www.rocketscientistsjournal.com/2007/06/on_why_co2_is_known_not_to_hav.html
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) + …
Willis Eschenbach says:
May 7, 2012 at 12:16 am
“If it is an infinite expansion, wouldn’t it have defined coefficients with defined corresponding time constants?”
The function is not known a priori, so the coefficients have to be estimated based on observables (my beef being that the observables are not enough to provide a complete description, and not very certain, either). Different estimation techniques and assumptions tend to yield different results.
In addition, practically speaking, you have to truncate the expansion at some point – there generally is not enough information rich data to estimate all the coefficients, and the more coefficients you try to estimate, the more uncertain each estimate becomes. Always, there is a tradeoff between bias and variance. The only question is whether the bias and variance can be small enough for the estimate to be useful.
So, to wrap it up for now, theoretically, the procedure is sound. But, practically speaking, there are plenty of good reasons to be wary, even skeptical (or, downright disbelieving, as I am), of the parameterization.
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.
Haven’t read all the replies, so this point might have been made.
Exponentials are not orthogonal functions like sine waves, and cannot be picked out of a mixture with any accuracy. Noisy data simply exacerbates the problem. Declaring exponential constants to four significant figures is a triumph of optimism.
Mike