Is the Bern Model Non-Physical?

Guest essay by Joe Born

Is the Bern Model non-physical? Maybe, but not because it requires the atmosphere to partition its carbon content non-physically.

bern_irf[1]

A Bern Model for the response of atmospheric carbon dioxide concentration \rho_{CO_2} to anthropogenic emissions E_{CO_2} is arrived at by adopting the values of f_{CO_2}, \tau_{CO_2} (and maybe C_{CO_2}) that make the best fit of the following equation to the historical record:

\rho_{CO_2}(t)=C_{CO2}\int\limits_{-\infty}^{t}E_{CO_2}(t')\left[ f_{CO_2,0}+\sum\limits_{S=1}^{n} f_{CO_2,S}e^{-\frac{t-t'}{\tau_{CO_2,S}}} \right]dt' .

The “Bern TAR” parameters thus adopted state that the carbon-dioxide-concentration increment \rho initially caused by a slug of additional carbon dioxide will decay as follows:

l\rho(t)=\rho(0)(a_0+a_1e^{-t/\tau_1} +a_2e^{-t/\tau_2} +a_3e^{-t/\tau_3}),l

where the a‘s are coefficients that sum to unity, the \tau‘s are explicit time constants of 2.57, 18.0, and 171 years, and a time constant of infinity is implicitly assigned to a_0: a_0=0.152 of the concentration increase persists forever.

There are a lot of valid reasons not to like what that equation says, the principal one, in my view, being that the emissions and concentration record we have is too short to enable us to infer such a long time constant. What may be less valid is what I’ll call the “partitioning” version of the argument that the Bern model is non-physical.

That version of the argument was the subject of “The Bern Model Puzzle.” According to that post, the Bern Model “says that the CO2 in the air is somehow partitioned, and that the different partitions are sequestered at different rates. . . . 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?” (The 371.3 years came from another parameter set suggested for the Bern Model.)

The comments that followed that post included several by Robert Brown in which he advanced other grounds for considering the Bern Model non-physical. As to the partitioning argument, though, one of his comments actually came tantalizingly close to refuting it. Now, it’s not clear that doing so was his intention. And, in any event, he did not really lay out how the circuit he drew (almost) answered the partitioning argument.

So this post will flesh the answer out by observing that the response defined by the “Bern TAR” parameters is simply the solution to the following equation:

7910\frac{d^4\rho}{dt^3} + 3563\frac{d^3\rho}{dt^3} + 192\frac{d^2\rho}{dt^2} + \frac{d\rho}{dt} =7910\frac{d^3E}{dt^3} +2457\frac{d^2E}{dt^2} +78.2\frac{dE}{dt} +0.152E

where

E=E_{CO_2}C_{CO_2}\sum f_{CO_2} .

But that equation describes the system that the accompanying diagram depicts. And that system does not impose partitioning of the type that the above-cited post describes.

clip_image001

In the depicted system, four vessels of respective fixed volumes V contain respective variable quantities \rho of an ideal gas, which they keep at a constant temperature so that the pressure in each vessel is proportional to its respective value of \rho/V. The vessel on the left exchanges gas with each vessel on the right through membranes of respective permeabilities s, the net rate of gas exchange with a given vessel on the right being proportional to the difference between that vessel’s pressure and the left vessel’s pressure. For the ith vessel on the right, that is,

\frac{d\rho_i}{dt}= (\frac{\rho_0}{V_0}-\frac{\rho_i}{V_i})s_i .

Additionally, a gas source can add gas to the first vessel at a rate E(t), so the left vessel’s contents \rho_0 can be found by solving the following equation:

\frac{d\rho_0}{dt}= E(t)+\sum\limits_{i=1}^{3}(\frac{\rho_i}{V_i}-\frac{\rho_0}{V_0})s_i.

If appropriate selections are made for the V‘s and s‘s, then expressing the other \rho_i‘s in terms of \rho_0 converts that equation into the fourth-order equation above, i.e., into the system equation that the “Bern TAR” parameters dictate.

The gas represents carbon (typically as a constituent of carbon dioxide, cellulose, etc.), the first vessel represents the atmosphere, the other vessels represent other parts of the carbon cycle, the membranes represent processes such as photosynthesis, absorption, and respiration, and the stimulus E represents the rate at which carbon rejoins the carbon cycle after having been lost to it for eons.

I digress here to draw attention to the fact that I’ve just moved the pea. The flow from the source does not represent all emissions, or even all anthropogenic emissions. It represents the flow only of carbon that had previously been sequestered for geological periods as, e.g., coal, and that is now being returned to the cycle of life. Thus re-defining the model’s emissions quantity finesses the objection some have made that the Bern Model requires either that processes (implausibly) distinguish between anthropogenic and natural carbon-dioxide molecules or that atmospheric carbon dioxide increase without limit.

Now, there’s a lot to criticize about the Bern Model; many of the criticisms can be found in the reader comments that followed the partitioning-argument post. Notable among those were richardscourtney’s . Also persuasive to me was Dr. Brown’s observation that the atmosphere holds too small a portion of the total carbon-cycle content for the 0.152 value assigned to the infinite-time-constant component to be correct. And much in Ferdinand Engelbeen’s oeuvre is no doubt relevant to the issue.

As the diagram shows, though, the left, atmosphere-representing vessel receives all the emissions, and it permits all of the other vessels to compete freely for its contents according to their respective membranes’ permeabilities. So what is not wrong with the model is that it requires the atmosphere to partition its contents, i.e., to withhold some of its contents from the faster processes so that the slower ones get the share that the model dictates.

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rtj1211

Surely if you want to model carbon dioxide properly you should identify the possible fates of it and develop equations which mirror those possible fate processes?
So, you have to suggest the following fates of newly emitted carbon dioxide:
1. Absorption into the oceans (affected by the oceanic temperature and the steady state phytoplankton population, as well as an minerals rate-limiting for phytoplankton growth (if any are indeed limiting)).
2. Absorption by land-based plants and photosynthesising micro-organisms (affected by the air temperature, the sunshine hours and the overall density of photosynthetic factories within land-based plants etc).
3. Absorption into rain and snow with subsequent deposition into soil, rivers or snow.
4. No fate at all.
The other question to consider is where the carbon dioxide gets emitted (human/animal-related emissions will be pretty much at land level, wild fires may see heat driving the gas up to greater altitudes, whereas volcanic eruptions may emit it a few miles into the sky) and whether that affects the rate of that carbon dioxide’s fate into one of those four pathways (dependent on atmospheric/stratospheric/tropospheric mixing to equilibrium).
Fancy maths is all very well, but a model which is related to real earth processes is likely to produce a better prognostic model than one which happens to fit past data for no reason better than mathematical chance.
I don’t know if it is possible to monitor carbon dioxide fate safely using methods distinct from radioisotope labelling, but if people want to understand the kinetics of carbon dioxide fate, they are going to need to develop just such methods in order to succeed in their mission. I’m assuming of course that no-one will let you go and realise a load of carbon-14-labelled gas into the atmosphere right now……..

All models are wrong. Some are useful. The carbon cycle equation in the Bern model is a mathematical approximation of a very complex progress. This approximation was first published in 1988 by Ernst Maier-Reimer and Klaus Hasselmann. It is used in many simple climate models.
The question is not whether it is a physical representation. It is not, and nobody in their right mind ever claimed it was. The question is whether it is a good approximation. It is, unless you want to explore the very distant future or very extreme scenarios. The definitive work on this is by Georg Hooss, who tried his best to break the model but could not.

jhborn

Forgive my LaTeX blunder. The second equation should read \rho(t)=\rho(0)(a_0+a_1e^{-t/\tau_1} +a_2e^{-t/\tau_2} +a_3e^{-t/\tau_3}).

Professors Pettersson, supported to some extent by Professor Brown, maintains that since the atmosphere at 600 PgC represents only 1.5% of the active carbon sinks (38,000 PgC in the hydroisphere and 2000 PgC in the biosphere) only 1.5% of any excess CO2 we add to the atmosphere will remain indefinitely.
The Bern model’s value for alpha-zero, on that analysis, is overstated by an order of magnitude,
with the implication that the later points on the curve, in particular, show too slow a rate of decay, leaving more excess CO2 lingering in the atmosphere to cause warming than is correct.
As I explained in my posting on this, it is in the equilibrium constant – the excess remaining indefinitely resident in the atmosphere – that the chief difference between the Bern model and the bomb-test curve is to be found. The thread unfortunately became derailed by those who wanted to argue about semantic definitions of relaxation and adjustment times – not the main point..
The bomb-test curve shows a decay towards the equilibrium constant 0.015 derived by Professor Pettersson from values for the contents of the atmosphere and of the active sinks given by IPCC (2007, 2013). It would be interesting if, on this thread, the semantic quibblers were to exercise a self-denying ordinance, for I should be interested in comments on whether the Bern model’s value of 0.152 for the equilibrium constant is justifiably an order of magnitude greater than that which is derivable theoretically from the relative magnitudes of the contents of the atmosphere and of the active sinks, and empirically from the bomb-test curve.
I only ask because I want to know. As a curious layman, I do not know which position is correct. But the discrepancy is large, potentially influential in climate terms and, therefore, interesting.

TerryS

Richard,
What do you define as the very distant future? Is it 5 years, 50 years or 500 years or more?

db-uk

What most non-chemists are either forgetting or not mentioning for non-scientific reasons is the Dalton’s law going back to 1800s where the total pressure of air is:
P(air) = P(N2) + P(O2) + P(CO2) + …..
Since contribution from N2 and O2 is 99% and from CO2 only 0.04% it means that out of 2500 molecules that contribute towards any property of air, only 1 molecule comes from CO2. So, whenever is someone arguing what 1 molecule of CO2 is doing, one has to explain what are 2500 molecules of N2 and O2 surrounding that molecule of CO2 doing at the same time!

jhborn

Richard Tol: “The definitive work on this is by Georg Hooss, who tried his best to break the model but could not.”
Could you provide us a link and explain how his work supports, for instance, the high magnitudes for the infinite- and long-time constant components of the TAR impulse response?

jhborn

Monckton of Brenchley: “Professors Pettersson, supported to some extent by Professor Brown, maintains that since the atmosphere at 600 PgC represents only 1.5% of the active carbon sinks (38,000 PgC in the hydroisphere and 2000 PgC in the biosphere) only 1.5% of any excess CO2 we add to the atmosphere will remain indefinitely.”
To this layman that reasoning makes sense and militates against the Bern TAR parameters. It probably should be mentioned, though, that Professor Pettersson’s equation is equivalent to the first Bern equation above if n = 1 and f_CO2_0 is 0.015.

“So what is not wrong with the model is that it requires the atmosphere to partition its contents, i.e., to withhold some of its contents from the faster processes so that the slower ones get the share that the model dictates.”
I agree with that. I think a simpler way of putting it is that some sinks are layered. The slow ones fill not from the air, but from the sinks above.
The model is empirical. It’s true that we don’t have long enough observation to accurately measure a 171 year time number. But we can use that to describe the long term behaviour, acknowledging that a timestep of 160 years, with appropriate parameter, would probably also have done well. And the component said to remain indefinitely simply reflects time scales too long to estimate at all. As Richard Tol says, it’s a model that works within a prescribed range.
Mathematically, a sum of exponentials is very hard to fit uniquely, because they are not at all orthogonal. It is an ill-conditioned problem. But all that means is that the Bern model is just one of many that can describe the process.

tty

The empirical data show that Ao, the proportion of CO2 that remains in the atmosphere indefinitely must be very slightly less than zero. “Slugs” of CO2 are continuously being injected into the atmosphere by volcanoes, but the trend in CO2 in the atmosphere has been inexorably downward for the last 35 million years.

Roy Spencer

tty makes an interesting point I’d like to see comments on.

tty says: December 2, 2013 at 3:19 am
“The empirical data show that Ao, the proportion of CO2 that remains in the atmosphere indefinitely…”

It isn’t the proportion that remains in air indefinitely. It’s the proportion that remains so long that we can’t, in our limited observation span, measure the decay rate. The Bern model doesn’t claim to work for millennia. Here is an article from the originators in which they quote an expected range of 1765-2300 AD.

In answer to Roy Spencer, one should have regard to the various timescales to which the word “indefinitely” is applied. In the Neoproterozoic era, 750 Ma ago (Roy is too young to remember), there was at least 30% CO2 in the atmosphere. The CO2 was taken up in the oceans first as dolomitic limestone, then as amagnesic limestone, then as gypsum. During this phase, the equilibrium constant was demonstrably negative..
Now we are down to Henry’s Law, to the calcifying organisms, and to the growing net primary productivity of plants. In today’s geological conditions, therefore, the equilibrium constant may well be positive: and, if Professor Pettersson is right that it is the ratio of the carbon content of the atmosphere to that of the active sinks in the hydrosphere and biosphere, it is indeed slightly positive.
One problem with Professor Pettersson’s definition of the equilibrium constant is that, contrary to the geological evidence that it was for many ages negative, it can never be negative, for there cannot be a negative quantity of CO2 in the atmosphere. Another problem is that when the atmospheric partial pressure of CO2 doubles, the equilibrium “constant” also doubles by definition, and does so on a timescale of as little as a century.
I am beginning to wonder whether we have the slightest idea what the equilibrium constant is under today’s conditions. One cannot be sure that it remains negative. Equally, one cannot be sure that it is positive, still less that it is as strongly positive as the Bern model pretends. In this as in many other respects, the models are assuming that which cannot yet be known. What a relief it is, then, that The Science Is Settled, and we need not look any further into these matters.

jhborn

tty: “‘Slugs’ of CO2 are continuously being injected into the atmosphere by volcanoes, but the trend in CO2 in the atmosphere has been inexorably downward for the last 35 million years.”
According to the attempt I made above to make sense of the Bern Model, tty’s statement would be a correct characterization of volcanoes’ actions if they are indeed introducing into the carbon cycle some carbon that for eons has not been participating.
To include the decay of which tty speaks, one or more of the vessels in the above diagram would need to be provided a second membrane, through which the gas would be consigned to the exterior darkness. Those membranes’ permeabilities (flow conductances) would need to be exceedingly small, of course.

bobl

tty is right to a point.
tty, adding a slug of co2 all things being equal cant reduce CO2 but all things being equal, but you’d intuitively expect that after the slug, given the emission and uptake remain constant at the level before the slug, that the CO2 would return over time to some equilibrium about the same as before the slug were added.
Despite Lord Moncktons plea, this problem is stated wrongly and assumes that uptake is a fixed function of time – but it is not, it’s a variable function of time, temperature, and CO2 concentrations at the places of major sinking activity.
The Bern model may be correct but it represents only a fraction of the system, a very simple model with no dynamic reactivity. We know for a fact that slugs of CO2 do not leave a large residual because despite the biosphere emitting slugs of CO2 constantly, CO2 has been known to contract.
The bomb test shows the turnover of CO2 due to all causes dynamic and static is on average more than this, but still misses the point that the response to a slug of extra CO2 may be much faster than the average drawdown if photosynthesis grows quickly with CO2 concentration, that is, if the sinks are being rate limited by the low concentration of CO2, if the sinking of CO2 is rate limited, then it will be harder to push up concentration, since the negative feedback is large, and any excess is rapidly acted against by the system,
The bomb test curve gives the decay rate of sink operating at an average rate between maybe 320 PPM and 400 PPM but how have the sinks actually responded, to that increase, the difference between the Bern model and the Bomb test may well be that factor. So how does that extra sinking capacity play out if we were to stop producing extra CO2 IE keep CO2 emission constant. That would depend on the dynamics of the feedback system that is rate limiting the sinks, and I might add the random effects of temperature on the biosphere. One must also consider the possibility that random pertubations have a bias, since the atmosphere is clearly biassed to reducing CO2 then random temperature changes are likely to be biassed to reducing CO2 over the long term.
For example, temp rises, oceans outgas, photosynthesis takes up ocean emissions limiting final CO2 level, temps fall, CO2 absorbed by oceans, CO2 now lower than it began.

jhborn

Richard Tol: Thank you for the link. I assume you meant the dissertation to which that page in turn linked, i.e., to http://www.schoepfung-und-wandel.de/NICCS/docum/mpi_examen_83.pdf?
As to the software to which you did link, I’m flattered by the implication that I may be able to comprehend the nonlinear-system responses to which it is directed, but I must confess to skepticism that I will be enlightened.
Perhaps you could provide an executive summary of how you think that work supports the significant long-time-constant residues that the Bern TAR parameters dictate for the (linear) Bern Model.

cd

tty
If I understand your point then a0 term seems ludicrous. If true then this would suggest CO2 would rise indefinitely given “additional” CO2.
But this excludes fluxes in the carbon cycle. As Lord Monckton has alluded to. Geochemical sequestration happens all the time via biological activity => carbonate minerals (for clarity this excludes sulphates such as gypsum which often precipitate in the same environment). As part of the rock cycle volcanic input is balanced by subduction of carbonate minerals in rocks such as limestone at destructive continental plate margins; this produces the volcanoes. In short the volcanoes are returning the CO2 from the subducted rocks. Interestingly, the d13 signature may vary from volcanos depending on whether the subducted rocks are biogenic or chemical in origin.

bobl

Lord M
One more appeal, let me use an electrical analogy, the bomb test gives us a hint as to the rate of sinking of energy. For an amplifier it’s akin to the average power sunk in the load. But does the power sunk in the load due to the dc biasing of the amplifier, give us any information about the gain of the amplifier? A small change in the conditions, may lead to a great change in output or a small one depending on the feedback acting. What are the feedback assumptiins in the bern model?
Instead of focussing on averaging a rate of sinking, I think we also need to use the bomb test to infer how the sinking rate has changed between the 1960s and now.

jhborn

bobl: “We know for a fact that slugs of CO2 do not leave a large residual because despite the biosphere emitting slugs of CO2 constantly, CO2 has been known to contract”
In the diagram above, the CO2 emitted by the biosphere is represented by the leftward component of the net flow throw the membranes. Although this is something about which the model is silent, I would think of the net flow as a small difference between large leftward and rightward components. It is the sizes of those components–regarding which, again, the model is silent–that determines the rate of excess-carbon-14-concentration decay.

richardscourtney says: And the third model assumes that the carbon cycle is dominated by biological effects.
Looking at the annual changes in the amount of added CO2 to the atmosphere only this model makes sense.
It is worth noting that a large part of our emissions is absorbed by natural sinks. With this we agree everyone. From this it follows an important conclusion: natural sinks always react big increase on any new source. Natural sinks are not (almost) constants – there are (almost) in the balance with natural sources. The appearance of a new source always causes increase in productivity of sinks. Changes can not be linear.
How to react to organic of sinks – NPP, and decomposition (RH) on temperature changes?
Image 12 i 14 by M. Salby (http://wattsupwiththat.com/2013/11/22/excerpts-from-salbys-slide-show/) – here are the most important. They show – explained, that on the rapid changes in temperature RH reacts violently, and NPP (initially – and this is most important) responds slowly – much more slowly than RH. Land RH – respiration – responds rapidly to changes in temperature (rapidly – along with the temperature increase or decrease – much faster – more rapidly than NPP).
Therefore, I propose the following equation: “The Lotka–Volterra equations – models, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations”:
dx/dt = x (a -by)
dy/dt = – y (g-Sx)
where,
– x is the number of prey (for example, CO2 …);
– y is the number of some predator (for example, terrestrial plants …);
– dx/dt and dy/dt represent the growth rates of the two populations over time;
– t represents time; and
– a, b, g and S are parameters describing the interaction of the two, ie: where CO2 is the victim of a land photosynthesis is … predator.
Only by means of this equation we can prove that 4/5 – 5/6 (richardscourtney) added to the atmosphere of CO2 (presently) derived from natural sources. (http://climatechangescience.ornl.gov/content/historical-variations-terrestrial-biospheric-carbon-storage) : “Historical climatic variations which favored NPP over RH may have led to increased ecosystem carbon storage and might account for at least part of the “missing” sink required to balance the current century’s global carbon budget.”
When, however, the following: “NPP favored over RH”?
The higher density “prey” is easier, faster “hunting” …
Furthermore, according to the equation – model LV first grows RH, only when the NPP eg: produce seeds and produce new plants, it will be “favored NPP over RH” (http://en.wikipedia.org/wiki/File:Cheetah_Baboon_LV.jpg).
In the past it was. Sinks always coped with a much larger source of CO2. Never there have to saturation of sources. Always source increased.
Model Bern has a problem with sinks:
„It has been suggested that subtle and systematic changes in the net carbon fluxes as the result of climate variations or rising CO 2 concentrations over the past century may account for a substantial portion of the “missing” carbon sink of approximately 1 to 2 Gt C yr -1 needed to balance the contemporary atmospheric CO 2 budget.” (http://cdiac.esd.ornl.gov/pns/doers/doer34/doer34.htm)
NPP – in theory – be calculated as g C m-2 yr-1. Firstly, however, primarily calculated (or only) plant weight, and on this basis only estimated: g m-2 C-1 yr. In practice, not take into account the “nuances” such as: increase in CO2 = increase C concentration in plants: http://www.nature.com/scitable/knowledge/library/effects-of-rising-atmospheric-concentrations-of-carbon-13254108:
“The availability of additional photosynthate enables most plants to grow faster under elevated CO2, with dry matter production in FACE experiments being increased on average by 17% for the aboveground, and more than 30% for the belowground, portions of plants (Ainsworth & Long 2005; de Graaff et al. 2006). This increased growth is also reflected in the harvestable yield of crops, with wheat, rice and soybean all showing increases in yield of 12–14% under elevated CO2 in FACE experiments (Ainsworth 2008; Long et al. 2006).
Elevated CO2 also leads to changes in the chemical composition of plant tissues. Due to increased photosynthetic activity, leaf nonstructural carbohydrates (sugars and starches) per unit leaf area increase on average by 30–40% under FACE elevated CO2 (Ainsworth 2008; Ainsworth & Long 2005). Leaf nitrogen concentrations in plant tissues typically decrease in FACE under elevated CO2, with nitrogen per unit leaf mass decreasing on average by 13% (Ainsworth & Long 2005).”
Sorry by long quote…
Without a change in weight, the plants are able to accumulate many times more C than we think (and we are able to estimate by satellite). Tomatoes fertilized with CO2 – repeatedly increase the concentration of sugars in their juice and starch in the chloroplast – no mass change. In addition, NPP is not an important reservoir of C. This “tool” for the production of “remainders”. Them warmer (and more atm. CO2), but the more and faster (!), photosynthetic biosphere produces “remainders” (and these are the real “missing” sink) and .. decomposition “remainders” we are unable to (obviously good enough) estimated using satellites …. These are not “subtle” changes!
Conclusion: NPP can be removed from the cycle much more C than estimates CDIAC, the IPCC model Bern … “Missing” sink can be much larger than we expected.
Therefore: an increase in natural sources of C (XX century to today) can thus be large – larger than the size of our emissions.
… but of course we have to prove that Ferdinand Engelbeen is completely wrong here:
“… the 14C content of fossil fuel is zero: too old for 14C, which is below detection limit after ~60,000 years, while recent organics have recent levels of 14C incorporated. – the oxygen balance.”
That means that only humans are responsible for the δ13C decline , as the biosphere is not the cause and all other known sources are (too) high in δ13C.”
We must to prove that natural sources of old carbon increase during the twentieth century – supplemented a “small” carbon cycle. Trend of their increase was strongly positive. And I can try prove it.
… that if was not this increase (it according to L-V model) our C would not be added to the atmosphere …

cd

I must also add that most of the subducted rocks are the denser ocean crust but sedimentary rocks also get subducted.

DocMartyn

The input of CO2 into the system is about 0.3 GtC annualy, from volcanic sources.
This would replace the pre-industrial 535 GtC in only 1,800 years and completely turnover all the carbon in the biosphere in 135,000 years. The 800 Ky ice core data shows that CO2 varies between 240-300 ppm during this period, so it is reasonable to conclude that some process mineralizes carbon rapidly when atmospheric CO2 is high and mineralizes it more slowly when atmospheric CO2 is low.
Your three box model also ignores the fact that to interrogate the deep ocean, atmospheric CO2 must first interact with the surface layer; there is no direct route.
This was my simplistic three box model
http://i179.photobucket.com/albums/w318/DocMartyn/reseviours_zps4776b7df.png

jhborn

cd: “As Lord Monckton has alluded to. Geochemical sequestration happens all the time via biological activity => carbonate minerals (for clarity this excludes sulphates such as gypsum which often precipitate in the same environment).”
Quite right. And you thereby touch on something I had initially thought to point out in the post but omitted because it would serve to distract unnecessarily from the point of the post. Specifically, my use of “carbon cycle” above is squishy; in effect I use it to refer to cycling through the atmosphere that occurs on time scales of less than a few centuries.
Hardly a bright line, I know, and probably inconsistent with more-conventional uses of the phrase. But I don’t think that detracts from the post’s main point, which is that the Bern Model does not require atmospheric partitioning.

jhborn

DocMartyn: “Your three box model also ignores the fact that to interrogate the deep ocean, atmospheric CO2 must first interact with the surface layer; there is no direct route.”
Indeed. Moreover, the (actually, four-box) model shown there is not the only one that the Bern TAR parameters define; there no doubt are some that incorporate an indirect route. I have verified that for a system in which all four vessels are in series, for instance. And, although I don’t quite understand your model, but it likely can be characterized by the first Bern equation above.

Joe Born wrote in his guest essay,
“. . . a time constant of infinity is implicitly assigned [in Bern TAR] to ao=0.152 of the concentration increase persists forever.”

– – – – – – – –
And the BERN modelers explained that time constant how?
(a why question is pending)
John

jhborn

Monckton of Brenchley: “I should be interested in comments on whether the Bern model’s value of 0.152 for the equilibrium constant is justifiably an order of magnitude greater than that which is derivable theoretically from the relative magnitudes of the contents of the atmosphere and of the active sinks,”
I haven’t done the math. In fact, I can’t put my hands on the data I’d need for the attempt. But my guess is that some fitting of the n = 3 version of the first equation above to the data could indeed spit out parameters like the Bern TAR numbers–or not. But my understanding of Willis Eschenbach’s work is that the magnitude of the residuals thereby obtained would not be much less than that which results from fitting to a single exponential decay.
Mr. Tol seems to be in possession of knowledge that would refute the conclusion thereby to be drawn, but so far on this thread he has hidden his light under a bushel.

cd

Joe
I was really only answering the call by Dr Spencer to address tty’s comment.
The partitioning does seem a little peculiar and seems to assume some intelligence in the system. I think that the d13C signature may play a part though. If the atmosphere becomes enriched with 12C then biological uptake may be more rapid thus removing the extra CO2 (if depleted in 13C) more rapidly than say the inorganic processes. This then acts in a feedback loop until things stabilise? Perhaps this is the reason why they feel some partitioning is required.

chris y

Nick Stokes-
“Mathematically, a sum of exponentials is very hard to fit uniquely, because they are not at all orthogonal.”
Are you claiming that Fourier series are flawed? Surely not.

A. Ames

Perhaps the problem is not with a long term residual concentration, but the assumption that this residual has something to do with past history. The biosystem has its own central tendency whatever we or volcanoes do. To explain this requires an additional term in the model that is separate from the time dependent terms.

jhborn

John Whitman: “And the BERN modelers explained that time constant how?”
I’m putting words in their mouths here, but I think they’d say they’re not trying to explain anything but rather saying that, if you want to treat the system as a black box characterized by a linear differential equation, what equation of order n + 1 would you come up with if you fitted it to the data.
One criticism of their result is given in my last response above to Lord M. It is based on work that I thought I remembered Willis Eschenbach’s having done, but I haven’t been able to locate that work again. (His post I referred to above isn’t it.)

RC Saumarez

I agree with Nick Stokes’ comment on the difficulty of fitting multiple exponentials to decay curves. They are classically ill-conditioned and may result in very large errors.
Since there is a considerable biological component in CO2 uptake and release, is this likely to be a linear process? (I realse that with the available data this might be impossible to establish).

tty

cd says
” As part of the rock cycle volcanic input is balanced by subduction of carbonate minerals in rocks such as limestone at destructive continental plate margins; this produces the volcanoes. In short the volcanoes are returning the CO2 from the subducted rocks.”
Certainly, though some of the CO2 may also be from deep mantle sources, or entrained from carbonate rocks through which the magma rises. And the whole cycle: atmospheric CO2 -> oceanic CO2 -> marine organisms -> carbonate deposit on seabottom -> move to a subduction zone by plate tectonics -> subduct to a depth where magma forms -> magma rises to the surface -> magma degasses -> atmospheric CO2 probably takes a few hundred million years at the very least. And it is far from clear that it is a balanced process. Since atmospheric CO2 on the whole seems to have decreased during the Phanerozoic indications are that it is not.

tty

Doc Martyn says:
“The 800 Ky ice core data shows that CO2 varies between 240-300 ppm during this period, so it is reasonable to conclude that some process mineralizes carbon rapidly when atmospheric CO2 is high and mineralizes it more slowly when atmospheric CO2 is low.”
Its more like 170-300 ppm, and since the concentration tracks the glacial cycles fairly closely, with some lag (short during deglaciations, much longer during glacier growth phases) it would seem that the oceans are the only sink that could vary fast enough (on a time-scale of a few millenia). However I have no explanation why oceanic outgassing would lag rising temperature less than accumulation lags sinking temperatures.

Björn

The phrase “bern model” prodded my mmeory of having read an article by Jarl Ahlbeck a decade or so ago where he put forward a chemical mass balance model for the uptake of carbon dioxide by the oceans and biosphere, and then used it to predict the a size of around 17 GtC/year CO2 ocean/biosphere sink when the magic double of 560 ppmV concetration of CO2 in the atmosphere would happen instead of 8.5 GtC/yr (fro, The Bern Model I assume). He did this a little before the millennial shift and therefore had had only acess to data up to the year 1997 available estimate his parameters, but it should be check If his model has been tracking reality since then, or if we are really living inside the big bear bottles the author says the IPCC-(bern???)model thinks.
The article can be found at this URL.
http://www.john-daly.com/co2-conc/ahl-co2.htm

ZP

Joe,
Your equations for the pressure in each box are incorrect given your compartment box model. This is a system of coupled first order linear differential equations that must be solved by an approach such as that described here: http://web.ist.utl.pt/berberan/data/40.pdf.
For the given system, an acceptable system of equations can written as follows (where c represents the molar concentration of the species of interest in the ith compartment):
\frac{dc_{0}}{dt} = \frac{dE}{dt} - k_{1}c_{0} + k_{-1}c_{1} - k_{2}c_{0} - k_{-2}c_{2} - k_{3}c_{0} + k_{-3}c_{3}
\frac{dc_{1}}{dt} = k_{1}c_{0} - k_{-1}c_{1}
\frac{dc_{2}}{dt} = k_{2}c_{0} - k_{-2}c_{2}
\frac{dc_{3}}{dt} = k_{3}c_{0} - k_{-3}c_{3}
subject to the chemical thermodynamic equilibrium constraint (i.e. equilibrium constant or partition coefficient relation): K_{i} = \frac{k_{i}}{k_{-i}}.

Richard Tol:
George Box’s aphorism that “All models are wrong.” is obviously incorrect in reference to the natural laws. Until recently, it was correct in reference to models of complex systems such as the climate. Today, however, through the use of modern information theory, it is possible to build a model of a complex system that conforms to the principles of reasoning, thus not being wrong. All currently available climatological models are, however, wrong.

Fun stuff ladies and gentlemen.
Chaucer ably described the dominant factor in Earth`s carbon balance, as follows:
WHAN that Aprille with his shoures soote
The droghte of Marche hath perced to the roote,
And bathed every veyne in swich licour,
Of which vertu engendred is the flour…
We now have the benefit of the beautiful AIRS data animation of atmospheric CO2 at
http://svs.gsfc.nasa.gov/vis/a000000/a003500/a003562/carbonDioxideSequence2002_2008_at15fps.mp4
The CO2 seasonal sawtooth is dominated by the larger Northern Hemispheric (“NH“) landmass.
Atmospheric CO2 drops in NH Spring and Summer during Chaucer’s “shoures soote” as photosynthesis dominates, and then CO2 increases in NH Fall and Winter as oxidation becomes the dominating factor in this huge and wondrous equation.
The annual amplitude (from memory) of this magnificent seasonal CO2 sawtooth is about 16-18ppm in the far North (measured at Barrow AK), and as little as 1-2 ppm at the South Pole.
Nevertheless the average upward slope of the “global” (approx. equal to Mauna Loa) CO2 sawtooth is about 1-2 ppm per year. Aye, there`s the rub!
And some of us think we understand why CO2 increases (for example, the Mass Balance Argument attributes the CO2 increase to fossil fuel combustion), while Richard Courtney ably suggests that we don`t really know. I`m generally with Richard on this, although I wobble.
There are days when I think the Mass Balance Argument (`MBA`) has validity, although I suggest that other factors such as deforestation etc. may play a larger role, and it is not just fossil fuels.
Then there are other days when I think the MBA is overly simplistic. The limited data I have seen suggests that even in urban environments where fossil fuels are locally combusted, the daily CO2 signature is overwhelmingly natural. It appears that CO2 is sufficiently scarce that plants quickly gobble up excess CO2 close to the source. Yum!
In any case, the global CO2 Balance question is very interesting, but I suggest it is not that relevant to the oft-fractious global warming debate – because CO2 in Earth`s natural system is clearly driven by temperature and is at most an insignificant driver of temperature.
Some people insist that these matters must be quantified, so I will accommodate them:
The impact of the current increase in atmospheric CO2 on Earth temperature is less than one Standard Farticane*.
Regards to all, Allan
******
* 1 Standard Farticane = I Fart in a Hurricane, at Standard Temperature and Pressure.

mpainter

CD:
“As part of the rock cycle volcanic input is balanced by subduction of carbonate minerals in rocks such as limestone at destructive continental plate margins; this produces the volcanoes. In short the volcanoes are returning the CO2 from the subducted rocks.”
<<<<<<<<<<<<<<<<<<>>>>>>>>>>>>>>>>>>
Not all volcanoes return subducted material. Some occur at spreading centers, others at “hot spots”, i.e., Hawaian Islands, and such do not return subducted material but tap directly into the mantle. So carbon is not being recycled but added to the crust or atmosphere, insofar as such volcanoes produce CO2. However, subducted material is recycled by those volcanoes that occur on the margins of subduction zones, as you say.

jorgekafkazar

rtj1211 says: “I’m assuming of course that no-one will let you go and realise a load of carbon-14-labelled gas into the atmosphere right now……..”
Good news! Iran intends to go ahead with this experiment within the next ten years.

jorgekafkazar

Allan MacRae says: “The CO2 seasonal sawtooth is dominated by the larger Northern Hemispheric (“NH“) landmass.”
Or is it dominated by the larger Southern Hemisphere (“SH”) oceanic surface area?

jhborn

ZP: “Your equations for the pressure in each box are incorrect given your compartment box model. ”
Could you be a little more specific? The equations you wrote are not inconsistent with mine, except E in my system is a rate of mass flow, so I would have put E in your equations instead of dE/dt. Also, for parallelism I perhaps confusingly use rho to represent the mass (number of moles) in the vessels; it’s not a concentration. Other than that, your k’s are my S/V’s.
In short, you’ve told me the equations are wrong, but you haven’t pointed out where. I’d love to have someone vet my sums, but I’ll need a little more specificity if it is to do me any good.

Greg Goodman

DocMartyn: “Your three box model also ignores the fact that to interrogate the deep ocean, atmospheric CO2 must first interact with the surface layer; there is no direct route.”
That’s incorrect. There is significant up-welling of deep waters in certain ocean regions, esp. Indian Ocean and sinks at the poles. This is part of the thermohaline circulation. That provides a direct connection.
http://climategrog.wordpress.com/?attachment_id=715
The point you try to make may well apply to mid-ocean waters between the surface mixed layer and the thermocline.
I am writing up something that shows that the well-mixed layer has a time constant close to one year and equilibrates in under a decade with a change of the order of 10ppmv/K. The circa 15y ( Bern 18.6y ) time constant is presumably this mid level diffusion.
Lance Wallace fitted a time const of 1.17 years to Nordrap C14 data in an earlier discussion. This is close to what I’m finding by totally different methods that does not suffer from the lack of uniqueness described here for exponentials.
Gosta Pettersson is close to being correct in his conclusions but I think his working is wrong. He sees no need for the short period which IMO he should be applying to his El Nino changes.
He has recently pulled his papers 1 & 2 which he is rewriting them, so it will be interesting to see how he changes them.

Greg Goodman

me says: “with a change of the order of 10ppmv/K.”
That is atm 10ppmv. I view of the volume ratio, I would guess about 4x that for the mid-level reservoir giving total all the order of 50 ppmv/K. This is in rough agreement with Gosta’s figures by via a different (incompatible) calculation.
The resulting conclusion is the same w.r.t. solving the “missing sinks” issue: like the missing heat, it does not exist.

ZP

The k’s in the equations represent the specific rate constants governing the compartmental exchange rates. Your equations incorrectly assume that the forward rate constant (k_{i}) is equal to the reverse rate constant (k_{-i}), which allowed you to factor them out as a single constant S. There is no physical basis on which to assume that the forward rate constant should be equal to the reverse rate constant. And, the values will only be the same in the case where the equilibrium constant is unitary.

mpainter

regarding the comment by tty:
Since the late Eocene, the world has cooled considerably which must mean cooler oceans and a greater capacity for holding CO2, presumably. But, in terms of geologic time, the carbon equation must be incalculable.
For example, carbonates are precipitated directly from ocean waters at such places as the Bahamas or the Yucatan shelf, such areas being known as carbonate platforms. Thus carbon is removed on a semi-permanent basis at such places. An ocean richer in CO2 would positively affect the rate of precipitation, presumably. I doubt that all these factors can be untangled.

Greg Goodman

jorgekafkazar says:
Allan MacRae says: “The CO2 seasonal sawtooth is dominated by the larger Northern Hemispheric (“NH“) landmass.”
Or is it dominated by the larger Southern Hemisphere (“SH”) oceanic surface area?
Yes, indeed. Yet another urban climatology assumption I guess.
Annual ‘saw tooth’ works out rather nicely actually since it can be modelled by two ramps or 12moand 6mo cosines (the latter being more physically real, but the first is handy for getting flow rates).
http://climategrog.wordpress.com/?attachment_id=721

Bart

The Bern model can be thought of as a partition of reservoirs, but that is only an approximation. Generally, such “long tailed” models arise from partial differential diffusion equations. The sum of exponentials is an expansion of eigenfunctions of the partial differential equation.
So, the place to begin is the PDE model, the equations and their boundary conditions, assumed by the Bern model.

ATheoK

“tty says:
December 2, 2013 at 7:12 am
cd says
” As part of the rock cycle volcanic input is balanced by subduction of carbonate minerals in rocks such as limestone at destructive continental plate margins; this produces the volcanoes. In short the volcanoes are returning the CO2 from the subducted rocks.”
Certainly, though some of the CO2 may also be from deep mantle sources, or entrained from carbonate rocks through which the magma rises. And the whole cycle: atmospheric CO2 -> oceanic CO2 -> marine organisms -> carbonate deposit on seabottom -> move to a subduction zone by plate tectonics -> subduct to a depth where magma forms -> magma rises to the surface -> magma degasses -> atmospheric CO2 probably takes a few hundred million years at the very least. And it is far from clear that it is a balanced process. Since atmospheric CO2 on the whole seems to have decreased during the Phanerozoic indications are that it is not.”

Subduction is not required. Folding deposits deep enough for metamorphosis is sufficient to release CO2.
The CO2 sources for many of the ‘hot spot’ volcanoes, e.g., Hawaii, are unknown. Guesses about magma absorbing CO2 from sediments as magma passing through are just that, guesses.
Deep, really deep magmatic CO2 sources, are currently beyond our ken. CO2 out gassing from these hot spots may include primal CO2 still leaking from earth’ s core. Carbon content in metals often resides in carbides; oxygen in many forms, oxides to be simple. Bluntly speaking, carbon is abundant cosmically.

jhborn

ZP: “here is no physical basis on which to assume that the forward rate constant should be equal to the reverse rate constant.”
If you’re talking about the real world, in which various factors affect “natural” emission (leftward flow) and uptake (rightward flow), I agree, and to that extent the model does not reflect reality.
If you’re talking about the model world, in which a permeable membrane conducts net flow in accordance with the pressure difference–i.e., with the difference in moles/unit volume, then I’m free to assume that a pressure on one side causes the same flow as the same pressure on the other.
The concept of equilibrium constant may be causing the difficulty here. Note that the process so drives flows that vessel contents tend toward proportionality with their volumes V. That’s where the equilibrium constants come from.