# CO2 Residence Times, Take Two

Guest essay by Joe Born

In a recent post Christopher Monckton identified me as a proponent of the following proposition: The observed decay of bomb-generated atmospheric-carbon-14 concentration does not tell us how fast elevated atmospheric carbon-dioxide levels would subside if we discontinued the elevated emissions that are causing them. He was entirely justified in doing so; I had gone out of my way to bring that argument to his attention.

But I was merely passing along an argument to which a previous WUWT post had alerted me, and the truth is that I’m not at all sure what the answer is. Moreover, semantic issues diverted the ensuing discussion from what Lord Monckton probably intended to elicit. So, at least in my view, we failed to join issue.

In this post I will attempt to remedy that failure by explaining the weakness that afflicts the position attributed (again, understandably) to me. I hasten to add that I don’t profess to have the answer, so be forewarned that no conclusion lies at the end of this post. But I do hope to make clearer where at least this layman thinks the real questions lie.

To start off, let’s review the argument I made, which is that the atmospheric-carbon-dioxide turnover time is what determined how long the post-bomb-test excess-carbon-14 level took to decay. That argument was based on the “bathtub” model, which Fig. 1 depicts. The rate at which the quantity <i>m</i> of water the tub contains changes is equal to the difference between the respective rates <i>e</i> (emissions) and <i>u</i> (uptake) at which water enters from a faucet and leaves through a drain:

$\frac{dm}{dt}= e- u$

The same thing can, <i>mutatis mutandis</i>, be said of contaminants (read carbon-14) in the water. But in the case of well-mixed contaminants one of the <i>mutanda</i> is that the rate at which the contaminants leave is dictated by the rate at which water leaves:

$\frac{dc}{dt}= e_c-\frac{c}{m} u,$

where $c$ is the contaminant quantity and $e_c$ is the rate at which contaminants are added.

Consequently, if the water quantity increases for an interval during which <i>e</i> exceeds <i>u</i>, it will thereafter remain elevated if the emissions rate <i>e</i> then falls no lower than the drain rate <i>u</i>. If a dose of contaminants is added to the water, though, the resultant contaminant amount falls, even when there’s no difference between <i>u</i> and <i>e</i>, in accordance with the <i>turnover</i> rate, i.e., with the ratio of <i>u</i> to <i>m</i>. So, to the extent that this model reflects reality’s relevant aspects, we can conclude that the rate at which the carbon-14 concentration decays tells us nothing about what happens when total-CO2 emissions return to a “normal” level.

But among the foregoing model’s deficiencies is that it says nothing about a possible dependence of overall drain rate <i>u</i> on the water quantity <i>m</i>, whereas we may expect biosphere uptake (and emissions) to respond to the atmosphere’s carbon-dioxide content. Nor does it deal with the possibility that after contamination has flowed out the drain it will be recycled through the faucet. In contrast, the biosphere no doubt returns to the atmosphere at least some of the carbon-14 it has previously taken from it.

A model that takes such factors into account could support a conclusion different from the one to which the bathtub led us. Consistently with my last post’s approach, Fig. 2 uses interconnected pressure vessels to represent one such model. In this case there are only two vessels, the left one representing the atmosphere and the right one representing carbon sinks such as the ocean and the biosphere.

The vessels contain respective quantities $m_1$ and $m_2$ of an ideal gas, which represents carbon. This model is based on the assumption that those quantities’ ratio is naturally driven to a preferred equilibrium value. The vessels are assigned “volumes” $V_1$ and $V_2$ to represent that equilibrium ratio. We assume that the vessels keep their contents at a constant temperature so that the resultant pressures can represent the emission and uptake processes’ tendencies to proceed, and a volume flow rate $S$ expresses the flow rate’s proportionality to those pressures:

$\frac{dm_1}{dt}= -\frac{S}{V_1}m_1 + \frac{S}{V_2}m_2 + e(t)$

$\frac{dm_2}{dt}= \frac{S}{V_1}m_1 - \frac{S}{V_2}m_2.$

Those equations tell us that the carbon quantity $m_1$ responds as follows to a magnitude-$m_0$ injection of new carbon:

$m_1(t) = \left[ \frac{V_1}{V_1+V_2}+\frac{V_2}{V_1+V_2} \exp\left(-\frac{V_1+V_2}{V_1V_2}St \right)\right] m_0 ,$

which the substitutions $k = \frac{V_1}{V_2}$ and $\mu = \frac{V_1+V_2}{V_1V_2}S$ transform into the Gösta Pettersson equation set forth in Lord Monckton’s post.

Note that in the Fig. 2 system any constituent of the gas would be exchanged between vessels in accordance with the partial-pressure difference of that constituent alone, as if it were the only component the vessels contained. By thus constraining the flow from (and to) the first, atmosphere-representing vessel, this model supports the conclusion that the overall-carbon-dioxide quantity would, contrary to my previous argument, decay just as the excess, bomb-caused quantity of atmospheric carbon-14 did.

Could providing more than one sink enable us to escape that conclusion? Not necessarily. Consider the more-complex system that Fig. 3 depicts. Just as the system that my previous post described, this one can embody the TAR Bern-model parameters. As that post indicated, describing such a system requires a fourth-order linear differential equation. So that system does have more degrees of freedom in its initial conditions and can therefore exhibit a wider range of responses.

But it still constrains the flow among its four vessels linearly in accordance with partial pressures, just as the Fig. 2 system does. From complete equilibrium, therefore, its behavior for any constituent is the same as for any other constituent as well as for the contents as a whole. In other words, this model, too, seems to support the notion that the bomb-test results do indeed tell us how long excess carbon dioxide will remain if we stop taking advantage of fossil fuels.

In a sense, though, the models of Figs. 2 and 3 beg the question; they use the same uptake- and emissions-process-representing $S$ parameters for every constituent. In contrast, although carbon’s different isotopes are often called “chemically identical,” they differ in their chemical kinetics. Whereas the inter-vessel flow in those models makes no distinction based on type of constituent, the real-world processes it represents do.

So one question is how significant that difference is in the present context. I don’t have the answer, although my guess is, not very. But readers attempting to answer that question could do worse than start by referring to a previous WUWT post by Ferdinand Engelbeen.

Another way in which carbon-14 differs from the other two carbon isotopes is that it’s unstable. So, if the Fig. 3 model is adequate for carbon-12 or -13, a different model, which Fig. 4 depicts, would have to be used for carbon-14 if its radioactive decay is significant. That diagram differs from Fig. 3 in that it includes a flow $S_5$to represent beta-decay “leakage” from the deep oceans.

To the extent that those different models produce different responses, using bomb-test data to predict the total carbon content’s behavior is problematic. But the Engelbeen post mentioned above seems to say that even deep-ocean residence times tend to be only a minor fraction of carbon-14’s half-life: this factor’s impact may be small.

A possibly more-significant factor is that the carbon cycle is undoubtedly non-linear, whereas the conclusions we tentatively drew from the models above depend greatly on their linearity. Before I reach that issue, though, I should point out an aspect of the Bern model that was not relevant to my previous post. The Bern equations I set forth in my last post were indeed linear. But that does not mean that their authors meant to say that the carbon cycle itself is. Although for the sake of simplicity I’ve discussed the models’ physical quantities as though they represented, e.g., the entire mass of carbon in a reservoir, their authors no doubt intended their (linear) models’ quantities to represent only the differences from some base, pre-industrial values. Presumably the purpose was to limit their range enough that the corresponding real-world behavior would approximate linearity.

But such linearization compromises the conclusions to which the models of Figs. 2 and 3 led us. A linear system is distinguished by the fact that its response to a composite stimulus always equals the sum of its individual responses to the stimulus’s various constituents; if the stimulus equals the sum of a step and a sine wave, for example, the system’s response to that stimulus will equal the sum of what its respective responses would have been to separate applications of the step and the sine wave. And this “superposition” property was central to drawing the conclusions we did from those models: the response to a large stimulus is proportionately the same as the response to a small one.

To appreciate this, consider Fig. 5, which depicts scaled values of the Fig. 2 model’s left-vessel total-carbon and carbon-14 contents. Initially, the system is at equilibrium, with zero outside emissions $e_a$, and with balanced emissions $e_n$ and uptake $u$ recycling contents fast enough to turn that vessel’s contents over in seven years.

At time t = 5, a bolus of carbon-14 appears in the (atmosphere-representing) left vessel. Compared with the total carbon content, the added quantity is tiny, but it is large enough to double the small existing carbon-14 content. As the distance between the red dotted vertical lines shows, the resultant increase in carbon-14 content decays toward its new equilibrium value with a time constant of just about seven years. (I’ve assumed that the processes greatly favor the sink-representing right vessel—i.e., that its “volume” is much greater than the left vessel’s—so that the new equilibrium value is not much greater than the original.)

Now consider what happens at t = 45, when the left vessel’s total-carbon quantity suddenly increases. Although the two quantities are scaled to their respective initial values, this total-carbon increase is orders of magnitude greater than the t = 5 carbon-14 increase. Yet, as the black dotted vertical lines show, the decay of the left vessel’s total-carbon content proceeds just as fast proportionately as the much-smaller carbon-14 content did. As was observed above, this could tempt one to conclude that the carbon-14 decay we’ve observed in the real world tells us how fast the atmosphere would respond to our discontinuing fossil-fuel use.

But now consider what can happen if we relax the linearity assumption. Specifically, let’s say that the Fig. 2 model’s proportionality “constant” $S$ varies with the system’s carbon content. Maybe various sinks saturate or become less efficient with increased concentration. Whatever the reason, the resultant non-linearity could cause behavior like that shown in Fig. 6.

In that plot, the red lines show that the carbon-14 decay occurs just as fast as in the previous plot, the carbon-14 content falling to exp(-1) above its new equilibrium value in around seven years. But the much-larger total-carbon increase brings the system into a lower-efficiency range, so that quantity subsides at a more-leisurely pace, taking over forty years. If such results are any indication, bomb-test results are a poor predictor of how long total-carbon content will settle.

Now permit me a digression in which I attempt to forestall pointless discussion of precisely what the quantities are that the graphs show. I believe the exposition is clearest if it is directed, as in Figs. 5 and 6, to ratios that carbon 14 and total carbon bear to their own initial values. But it appears customary to express the carbon-14 content instead in terms of its ratio to total carbon content. This means that, since total carbon has been increasing, the numbers commonly used in carbon-14 discussions could fall below the pre-bomb values, even though total carbon-14 has in fact increased.

For the sake of those to whom that issue looms large, I have attached Fig. 7 to illustrate how the values for carbon-14 itself could differ from those of its ratio to total carbon in a situation in which new (carbon-14-depleted) carbon is continually injected into the atmosphere.

But that’s a detail. More important is the issue that Fig. 6 raises.

Now, I “cooked” Fig. 6’s numbers to emphasize the point that nonlinearity can undermine conclusions based on linear models. Specifically, Fig. 6 depicts the results of making the flows proportional only to the fifth root of the carbon content.

But non-linearity must have some effect. How much? I don’t know. Together with the differences in behavior between carbon-14 and its stable siblings, though, it is among the considerations to take into account in assessing how informative the bomb-test data are.

As I warned at the top of the post, this post draws no conclusions from these considerations. But maybe the foregoing analysis will prompt knowledgeable readers’ comments that help narrow the issues.

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bushbunny

Having studied archaeology, C14 is a radioactive isotope that is absorbed by all organic matter. Then it decays slowly. It has fluctuated over the years because of sun spot activity that deflect galactic sub atomic particles bombarding the earth. The same sub atomic particles also when they meld with water vapor molecules form clouds. I don’t know if it is correct, but before they banned atmospheric Atom and Hydrogen bombs the C.14 increased. We had terrible weather too lots of rain and cold temps in UK. That is why any carbon 14 dating usually gets a + or minus, and anything younger than say 2,000 is harder to date accurately and other dating methods have to be used.

OT but very funny and scary at the same time.
Stephen Colbert Tells David Keith Government is Already Spraying Us « GeoEngineering Exposed

Bernie Hutchins

Connecting CO2 sources/sinks in the manner shown here is analogous to the way an electrical engineer might connect multiple resistors and capacitors in a network without using any active elements (amplifiers or buffers). Because you can never achieve complex poles with this restriction, the resulting low-pass filters are not very useful (frequency characteristics are way too wimpy). We don’t do it that way.
This is NOT to suggest that the electrical equivalents could not be completely correct and very useful as models for CO2 migrations. The mathematics (that is – the physics) is very similar if not identical.
While NOT very useful for usual signal processing (analog filtering here) we do use these schemes for “envelope generators” for electronic music generation as I have recently reviewed at
http://electronotes.netfirms.com/EN210.pdf
[ In synthesizer design, “envelopes” are slowly-varying contours (way sub-audio) which are the parameters or controls for single tones (single notes).] My summary referenced here shows four approaches to solutions: the differential equation approach (eigen-analysis), but also the corresponding Laplace transform method, numerical simulation solutions, and finally a (gasp!) experiment. This comparison smorgasbord may be of interest here.

mobihci

seeing as temperature over the history of the earth has changed dramatically, and the CO2 has changed with that temperature with a lag, then who cares. it seems rather obvious that the earth was & still is capable of absorbing CO2 in cooling conditions, so really it must be linear or something close to it or earth would not have survived as we know it.

Hoser

Hopelessly over complicated. From Fig 5 and discussion, I question whether the writer understands the system we are talking about. 14C is effectively a tracer. There is a steady state level maintained by adding the same amount to the atmosphere each year from transmutation via cosmic ray interactions with nitrogen atoms in the atmosphere. The total amount of carbon is essentially unchanged. The atomic bombs did not increase the amount of CO2 in the atmosphere, but did almost double the tiny amount of 14C.
A basic reasonable assumption of a single turnover kinetics is the tracer leaves the reservoir and does not come back. 14C leaving the atmosphere can safely be assumed to not return in any significant fraction. Hence, when we see it leave, that is the off rate cleanly determined. Now when we have an equilibrium level, we can calculate the on rate as well. It doesn’t matter how complicated the reservoir system you create. The only part that matters is the off rate constant. We don’t know how the material partitions into other reservoirs, but that issue is beyond the scope of this experiment.
I showed previously [1] the math works out nicely, and essentially you can slice off the steady-state amount of 14C and look at the decay of the excess alone. The t1/2 is about 5 years. Even accounting for the increase in ppmv CO2 diluting the 14C from 1963 to 1993 and beyond, the t1/2 is still about 5 years (at least it was in my hands). Once you know the t1/2 is 5 years, a number of interesting calculations follow.
For one thing, it becomes pretty clear the increase in atmospheric CO2 cannot possibly be due to anthropogenic CO2. The CO2 quantity changes simply don’t match what humans have produced given the amount of CO2 we have produced each year from the late 1700s and how much would be left Y years after emission.
1) http://wattsupwiththat.com/2013/11/21/on-co2-residence-times-the-chicken-or-the-egg/#comment-1481426

Dear Joe,
By thus constraining the flow from (and to) the first, atmosphere-representing vessel, this model supports the conclusion that the overall-carbon-dioxide quantity would, contrary to my previous argument, decay just as the excess, bomb-caused quantity of atmospheric carbon-14 did.
The main problem with that model is that there is no lag included between the outflow (from the atmosphere in the deep oceans) and the inflow (from the deep oceans to the atmosphere). That makes that there is not only a huge difference in mass, but also a huge difference in concentrations of 14C.
What goes into the deep oceans is in direct ratio to the partial pressure difference of CO2 between atmosphere and ocean surface. What comes out is in direct ratio to the partial pressure difference of CO2 between ocean surface and atmosphere. That is the same for 12CO2 and 14CO2, with a small difference due to kinetics.
What goes into and comes from the deep oceans is thus directly related to their relative partial pressure differences.
For 12CO2 that is about 97% (in mass) * 99% (in concentration) that returns of what is going into the deep oceans, but for 14CO2, that is only 97% (in mass) * 45% (in concentration) at the time of the peak bomb test (1960):
http://www.ferdinand-engelbeen.be/klimaat/klim_img/14co2_distri_1960.jpg
The mass difference in this scheme is only at the inputs (to make the difference clear), but in reality it is equally distributed between deep ocean input and output to/from the atmosphere.
In pre-industrial times about 90% (45% of the bomb spike) of 14CO2 returned from the deep oceans for a ~1000 years lag, but that was about compensated by fresh production from cosmic rays.
Thus the 14CO2 decay is a mix of the mass spike decay (about equal to a 12CO2 mass spike decay) which depends of the atmospheric partial pressure and the “contamination” spike decay, which depends of the residence time.
That makes that the 14CO2 bomb spike decay rate of ~14 years is longer that the residence time, but shorter than the decay rate of a 12CO2 mass spike.
Something similar happens to the 13CO2 rate decay caused by the use of low-13C fossil fuels. If we look at the decrease of the 13C/12C ratio in the atmosphere, we see a decay rate which is only 1/3rd of what can be expected from the use of fossil fuels. That too is caused by the “thinning” from 13C-rich upwelling waters from the deep oceans. That can be used to estimate the deep ocean – atmosphere exchanges over a year:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/deep_ocean_air_zero.jpg
or about 40 GtC/year in and out between atmosphere and deep oceans.
With that in mind, the residence times of a “human” CO2 spike can be estimated if we give a pulse of 13c-depleted CO2 of 100 GtC to the pre-industrial CO2 level of 580 Gtc:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/fract_level_pulse.jpg
where FA is the fraction of “human” CO2 in the atmosphere, FL in surface waters, tCA total carbon in the atmosphere and nCA natural carbon in the atmosphere.
After ~60 years all low-13C from fossil fuels is replaced by “natural” CO2 from the deep oceans, while still 45% of the original total CO2 spike is present…

Hoser says:
December 11, 2013 at 11:32 pm
Your calcualtion is for the thinning of 14CO2 by the total CO2 turnover, which gives you the residence time (which is mainly temperature difference dependent), but that has nothing to do with the decay time for a mass pulse (which is mainly pressure difference dependent).

bushbunny

It is nothing humans can control anyway.

I’m grateful to Mr. Born for his detailed consideration of the CO2-lifetime question..The literature from Revelle onwards supports Hoser’s contention that the turnover time is 5-7 years: but that, as Mr. Engelbeen reminds us, is not the whole story. Dick Lindzen estimates that most CO2 molecules added to the atmosphere will have found their way out of it again in about 40 years. That is below the IPCC’s 50-200 years.
And the missing sink remains missing. Why is half of all the CO2 we emit disappearing instantaneously from the atmosphere? I’d be most grateful if wiser heads than mine were able to explain that.

phlogiston

Ferdinand Engelbeen says:
December 12, 2013 at 12:27 am
Hoser says:
December 11, 2013 at 11:32 pm
Your calcualtion is for the thinning of 14CO2 by the total CO2 turnover, which gives you the residence time (which is mainly temperature difference dependent), but that has nothing to do with the decay time for a mass pulse (which is mainly pressure difference dependent).
NO NO NO NO NO NO NO NOOOOOO! (adopts facial expression of The scream by Munch)
Ferdninand we all respect your erudition in regard to atmospheric CO2 but I fully agree here with Hoser that all this discussion is completely missing what a radio tracer measurement really is. And vastly over-complicating the discussion as a result.
A radiotracer measures a single removal term. PERIOD. A pulse of CO2 enters the atmosphere different from the other CO2 due to 14C. So it can be tracked in exclusion of any other CO2.
It is COMPLETELY IRRELEVANT all the other cycling and dilution and dynamics, pressure, temperature etc. of CO2 that are going on, the 14 tracer simply tells us the removal term for CO2. That is the whole point of a radiotracer measurement.
From the bomb test data we know that:
CO2 removal half life = 5 years
CO2 residence time = half life / ln2 = 5 / 0.693 = 7.7 years
That is the WHAT. Everything else is the WHY.

bushbunny

Cl4 can not be absorbed once the organism dies, or is buried. So that is how they can calculate by the amount of half life remaining how old the organism is. Anything organic including bones, trees etc., that have been unearthed in an archaeological site. I can’t see the problem really, we are bombarded all the time by galactic sub atomic particles. When sun pot activity is large then we don’t receive as much. Inactive and we receive a lot, including more clouds.

OldEngland

As a layman reading this it seems to over complicate the issue but you will see my ignorance shine through from my comment ! C14 as a tracer should provide a reasonable means of tracking the removal or sequestration rate of CO2.
We know from NASA that higher levels of CO2 have resulted in an increase of ~15% in vegetative matter globally which has ‘captured’ and removed CO2, although I don’t know if any calculations or studies show the likely amount of CO2 by weight and percentage that is. I assume there must be and that forward projections of increased rates of capture in higher CO2 levels are there somewhere. As an aside – I wonder if the increase in tree growth and number of leaves increases the release rate of volatile organics which in turn serve to increase forest related cloud formation and increased rainfall – or maybe with less water requirement (elevated CO2 and fewer stoma) they respond by producing less VO ?
Bacteria (some anyway) capture carbon and there is a strong likelihood that the populations of these may explode or implode depending on ambient CO2 levels which suit them best, again I don’t know if this has been studied or evaluated although it potentially may have very dramatic effects on the rate of carbon sequestration. Ditto bacteria which emit CO2.
I cannot see how the ‘bath tub’ simile has any value at all without including a means to show some form of active sequestration within the first bath tub. As you can see I am filled with curious ignorance.

Coldish

Joe Born: “The observed decay of bomb-generated atmospheric-carbon-14 concentration does not tell us how fast elevated atmospheric carbon-dioxide levels would subside if we discontinued the elevated emissions that are causing them.”
Yes, but as Lubos Motl has pointed out, we can easily calculate this rate without bothering with 14C.
Lubos Motl says:
November 22, 2013 at 1:42 am
“It’s trivial to see that the residence time of CO2 is of order 30 years or longer. We emit 4 ppm worth of CO2 a year; the CO2 concentration increases by 2 ppm per year. So it’s clear that the “excess uptake” (which is natural and depends on the elevated CO2 relatively to the equilibrium) is also 2 ppm pear year. The excess CO2 above the equilibrium value for our temperature- which is still around 280 ppm – is about 120 ppm so one needs about 30 years to halve the excess CO2 and 50 years to divide it by e.”
Residence time τ can been defined (Barth, 1952, quoted by Henderson, 1982§) as τ = A/(dA/dt), where A is the amount of a substance dissolved or contained in a reservoir and dA/dt is the rate of efflux of that substance from the reservoir. In the case here considered the ‘substance’ can be considered as the excess atmospheric CO2 above the equilibrium value for today’s temperatures (around 280ppm). So if today’s CO2 concentration is about 400ppm, then A = 400 – 280 = 120ppm.
If (as Lubos states) the equivalent of 4ppm excess (i.e. anthropogenic) CO2 is being emitted to the atmosphere annually, while the concentration in the atmosphere is increasing by 2ppm/year, then dA/dt = 2ppm/year and τ (the residence time in the atmosphere of the excess CO2) is 120/2 years, so 60 years. Of course that assumes that anthropogenic omissions cease now, which they won’t…but any talk of residence time being less than 10 years is unrealistic.
§ Henderson, P. 1982. Inorganic geochemistry. p. 287.

jhborn

Ferdinand Engelbeen:
Thank you very much for the response. I’m going to have to get some coffee in me before I digest it all, but it seems to me that your third graph is actually your answer to the ultimate question. I think I’ll have questions about how you get there, but I won’t impose upon you until I’ve prayed and fasted a bit over what you’ve already said. Except for one question now:
I didn’t quite understand that passage:
“For 12CO2 that is about 97% (in mass) * 99% (in concentration) that returns of what is going into the deep oceans, but for 14CO2, that is only 97% (in mass) * 45% (in concentration) at the time of the peak bomb test (1960).”
. The 45% figure, I take it, comes from the fact that the deep oceans absorbed half the bomb-peak concentration and so are returning that, minus a 10% loss from beta decay: 0.5 * (1.0 – 0.1) = 0.45. So the 14C02 partial pressure from the deep oceans is 45% that of the bomb-peak atmosphere’s? And what is the “97% (mass)” by which you multiply the concentrations?

Nick

There are other ways of getting an estimate on the withdrawal rate of CO2.
http://wattsupwiththat.com/2008/04/06/co2-monthly-mean-at-mauna-loa-leveling-off/
Has graphs of the CO2 measured in Hawaii.
You can see that at particular times of the year, there are drops. That drop shows the rate at which C02 can be withdrawn by the system. It’s large.
There are other examples with temperature lags in the system. Now if only we could experiment with the earth we could find out those lags. Right? Hmmm, how about turning the sun off for 12 hours and seeing how quickly things cool. Small drops in temperature mean large lags. Large drops small lags. Low an behold the night day temperature difference shows very small lags in the system to changes in forcings.

Berényi Péter

Do we have data on the response to 14C bomb spike of carbon reservoirs other than the atmosphere?

phlogiston says:
December 12, 2013 at 12:53 am
A radiotracer measures a single removal term. PERIOD. A pulse of CO2 enters the atmosphere different from the other CO2 due to 14C. So it can be tracked in exclusion of any other CO2.
In this case, the radiotracer meausures not only the removal term (as mass), but also the “thinning” of the concentration, because what returns from the deep is only halve the concentration (at the height of the bomb spike) of what goes into the deep oceans. Two distinct removal rates without much connection with each other.
The decay rate of a 12CO2 pulse only depends of the mass balance between ins and outs, the decay rate of a 14CO2 pulse mainly depends of the concentration balance and hardly the (total) mass balance between ins and outs.

TFN Johnson

An important problem is emerging. The Kyoto Annexe II countries are gearing up to impose financial penalties on Annexe I (‘developed’) countries proportional to their cumulative CO2 emissions. But if the half-life of CO2 in the air is (say)20 years, instead of the thousands that some claim, then the ‘legacy’ CO2 of Annexe I countries is much lower. Indeed, as China now emits more COS then the US, then it will soon have a larger legacy concentration. China is of course an Annexe II country, and determined to remain so. But their share of the COs now in the atmosphere is rapidly heading into second place, given that mush of the US CO2 was emitted long ago.

John Marshall

studying the decay times for C14 does not help calculate the CO2 residence time. C12 is preferentially used by plants in photosynthesis as opposed to other isotopes. They remain in the atmosphere so would give a false reading.

I loved your discussion but it caused a flashback to my nightmarish college calculus classes.

Joe Born says:
December 12, 2013 at 1:49 am
Some background:
The 1950-1960 bomb spike almost doubled the existing 14CO2 “background” levels. If we take the maximum bomb spike level of 1960 as 100%, the pre-industrial 14CO2 level in the atmosphere thus was 50% of the bomb spike.
– in pre-industrial times there was a balance between decay and production of 14CO2:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/14co2_distri_1850.jpg
Some 14CO2 is distributed and removed via vegetation and the ocean surface, but the bulk is removed by the deep oceans: 50% bomb spike level goes into the deep oceans, 45% comes out after ~1000 years, that is 90% coming back, the difference is the radioactive decay of 14CO2 over that time span + mixing in of older 14CO2 free carbon from deep sources (CH4, carbonate dissolution).
The continuous 14CO2 production in the atmosphere from cosmic rays compensated for the losses.
– since ~1850 there is an increase of 14CO2-free fossil emissions, which had some impact on the 14C/12C ratio, which made it necessary to adjust the radiocarbon dating tables.
– about 1960, at the height of the 14CO2 bomb spike, there was already a measurable increase of (mainly) 12CO2 in the atmosphere:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/14co2_distri_1960.jpg
That caused an imbalance between atmosphere and deep oceans where inputs and outputs aren’t equal anymore. While that mainly influenced the bulk of CO2 (which is ~99% 12CO2), that also influenced the 14CO2 spike to a lesser extent: 40.5 GtC out of the atmosphere into the deep oceans, 39.5 GtC into the atmosphere, caused by the extra pressure from the 100 GtC increase of CO2 in the atmosphere. That is for 100% CO2 out into the deep, 97.5% is coming back from the deep oceans. The difference in 12CO2 concentration between ins and outs (both ~99%) is negligible.
Not so for 13CO2 and 14CO2.
From the 100% 14CO2 spike going into the deep (1960), 97.5% (as mass) * 45% (as concentration) is coming back, that is only 42.8% is coming back from the deep oceans.
The difference between a 12CO2 spike and a 14CO2 spike is caused by the long delay between what goes into the oceans and what comes out, which makes that the 14CO2 (and 13CO2) input is effectively decoupled from the output, while that is hardly the case for 12CO2.
For the year 2000, things changed for both 14CO2 and 12CO2:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/14co2_distri_2000.jpg

Berényi Péter says:
December 12, 2013 at 1:53 am
Do we have data on the response to 14C bomb spike of carbon reservoirs other than the atmosphere?
Here an interesting reference for the 14CO2 distribution in general and specific for the bomb spike:

Mike Jowsey

Isn’t open science fascinating?

Chuck Nolan

“….. if we stop taking advantage of fossil fuels.”
Damn! I like that term.
That’s exactly how it should be looked at and expressed every time – fossil fuel is not pollution…it’s an advantage.
cn

Speed

Ahhh. My first day of pharmacology. This is a multi-compartment model — atmosphere, oceans, plants, soil etc.
A multi-compartment model is a type of mathematical model used for describing the way materials or energies are transmitted among the compartments of a system. Each compartment is assumed to be a homogenous entity within which the entities being modelled are equivalent. For instance, in a pharmacokinetic model, the compartments may represent different sections of a body within which the concentration of a drug is assumed to be uniformly equal.
Hence a multi-compartment model is a lumped parameters model.
Multi-compartment models are used in many fields including pharmacokinetics, epidemiology, biomedicine, systems theory, complexity theory, engineering, physics, information science and social science. The circuits systems can be viewed as a multi-compartment model as well.

http://en.wikipedia.org/wiki/Multi-compartment_model

Quinn the Eskimo

I have seen the point made hundreds of times that the residence time of an average molecule is different than the time for a pulse to decay back to equilibrium. Usually this point is made in pedantic fashion, as if it were a tiresome chore to explain such a simple point to such simpletons. But I still don’t get it. Are the molecules in the pulse exempt from the processes giving rise to the properties of the residence time of average molecules? Are they not average molecules? How so? What qualifies a molecule, or group of molecules, to be treated as part of a slug or pulse on the one hand that is governed by pulse decay rules or merely part of the background sources of emissions governed by residence time of average molecule rules? How do it know? Where human emissions are a single digit percentage of natural emissions, which source is the slug? What is the threshold at which or the rules by which emissions from one or the other, human or natural, transition from being subject to average molecule processes to slug molecule processes? How do it know?

jhborn

Ferdinand Engelbeen: “40.5 GtC out of the atmosphere into the deep oceans, 39.5 GtC into the atmosphere, caused by the extra pressure from the 100 GtC increase of CO2 in the atmosphere. That is for 100% CO2 out into the deep, 97.5% is coming back from the deep oceans.”
Thanks a lot for that response; it answers my question nicely.
If I can again abuse your patience, what are K and k in your diagrams? I couldn’t readily find the accompanying text on your site.

Ferdinand Englebeen: “From the 100% 14CO2 spike going into the deep (1960), 97.5% (as mass) * 45% (as concentration) is coming back, that is only 42.8% is coming back from the deep oceans.” Is this the loss of that spike since 1960 or is it a suggestion that the kinetic difference between 14C and 12C is that large? I couldn’t quickly find kinetic differences for carbon isotopes but would be surprised if they are that large. I’d also expect the kinetic differences to be process dependent.

jhborn

Bernie Hutchins “This is NOT to suggest that the electrical equivalents could not be completely correct and very useful as models for CO2 migrations. The mathematics (that is – the physics) is very similar if not identical.”
For the linear models above, RC-circuit mathematics is indeed identical. But using a signed quantity (charge) to represent an unsigned quantity (mass) can lead to some conceptual confusion. Although the left vessel is connected in series between the source and the right vessel, for example, the analog’s current source would have to be wired in parallel with the capacitor representing the left vessel and with the RC-series combination representing the right vessel and its flow restriction.

I assume all the discussion of CO2 mass balance is based on the assumption that the CO2 distribution in the atmosphere is homogeneous and can be represented by one measuring location. Other things, liquid or gas, require relatively great amounts of energy to get homogeneous mixtures at much lower volumes. Is the atmospheric CO2 that homogeneous and if not, how much does that affect these estimates?

jhborn

Bob Greene: “Is the atmospheric CO2 that homogeneous and if not, how much does that affect these estimates?”
Someone has probably already answered this question somewhere, but I haven’t found it yet myself. I’ve wondered whether the “missing sink” may be a reflection of greatly enhanced uptake at areas of locally intense near-power-plant (or -highway) concentrations not reflected in the more-general CO2 numbers we see–uptake that is not offset substantially by locally enhanced natural emissions.
No doubt a naive notion that has already been laid to rest elsewhere, but it’s one of which I haven’t yet been disabused.

Quinn the Eskimo says:
December 12, 2013 at 4:07 am
I have seen the point made hundreds of times that the residence time of an average molecule is different than the time for a pulse to decay back to equilibrium.
I know, this seems to be one of the most difficult points to be explained…
The simplest way is to describe it as the difference between the turnover of capital (thus goods) in a factory and the gain or loss that that capital makes (over a year).
The turnover gives how much times a capital is going through the factory: from the purchase of raw materials to the sales of the endproduct.
The yield of the factory is what is gained or lost of its capital after the turnover(s).
Both are about the same money, but they are largely independent of each other: you can double the turnover, but that may or may not increase the gain, because you have to pay higher wages for overtime, or you may get from a loss to a gain…
The same for 14CO2 vs. 12CO2:
With the 14CO2 bomb spike, you are mostly measuring the turnover of CO2 through the atmosphere, without influencing the mass (total capital).
With a 12CO2 spike, you are increasing the total mass (capital), whithout influencing the turnover that much: the turnover will be somewhat diluted (more in the atmosphere, about the same througput).
The decay rate of an excess mass of 12CO2 depends of how fast the deep oceans (less for other reservoirs) take an extra mass of CO2 away from the atmosphere (partial pressure difference related), while the decay rate of an excess concentration of 14CO2 depends of the turnover (which is temperature difference related, mostly between equator and poles)…

Monckton of Brenchley says:
December 12, 2013 at 12:51 am
Why is half of all the CO2 we emit disappearing instantaneously from the atmosphere?
============
This does appear to be the nub of the problem. Why does this number remain at 1/2 of human emissions, rather than a fraction of cumulative CO2 excess?
If we consider the bathtub model, then the amount flowing out remains at 1/2 the amount flowing in, regardless of the height of water in the tub. Which makes no sense. The amount flowing out should vary as the height of water in the tub, not as the amount flowing in.
Which argues that the bathtub model does not describe CO2 reality.

Joe Born says:
December 12, 2013 at 4:16 am
If I can again abuse your patience, what are K and k in your diagrams? I couldn’t readily find the accompanying text on your site.
You are welcome…
K and k should all be k and are the rate constants for all transfers, where I should switch the +’s and -‘s, as normally one looks at the atmosphere as starting and end place…
Still to be worked out, as good as the whole page I need to devote to the explanation of all these diagrams and a lot more…

aaron

The methane residency time is also likely much lower than currently believed. As we discover more and more sources, we must realize that there more sinks or that sinks respond to emissions to keep methane levels from growing much.

ferd berple says:
December 12, 2013 at 4:47 am
This does appear to be the nub of the problem. Why does this number remain at 1/2 of human emissions, rather than a fraction of cumulative CO2 excess?
In fact it is pure coincidence: human emissions increased slightly quadratic over time, which gives a slightly quadratic increase of the residuals in the atmosphere (= pressure) and which causes a slightly quadratic increase in sink rate. The result is an astonishing fixed ratio between emissions and increase in the atmosphere (the “airborne fraction”):
http://www.ferdinand-engelbeen.be/klimaat/klim_img/temp_co2_acc_1960_cur.jpg
or in ratio since 1900:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/acc_co2_1900_cur.jpg
or since 1960:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/acc_co2_1960_cur.jpg
where the South Pole (SPO) lags the Mauna Loa (MLO) data.
If the human emissions would stay the same, the “airborne fraction” would decrease and CO2 levels would go assymptotically to a new steady state equilibrium CO2 level…

glenncz

Hoser says:
December 11, 2013 at 11:32 pm
Hopelessly over complicated.
—————————
Indeed. All this analysis of 1 part of 20,000 parts of the atmosphere changing from “something” to CO2. A change of .005%, while the other 99.995% purportedly has been expected to remain stable and have influence. Oh the effort that has been spent on solving a puzzle that is simply a figment of someone’s overactive imagination.

Bob Greene says:
December 12, 2013 at 4:29 am
I assume all the discussion of CO2 mass balance is based on the assumption that the CO2 distribution in the atmosphere is homogeneous and can be represented by one measuring location.
For most of the atmosphere, the CO2 levels are within +/- 2% of full scale, despite that some 20% of all CO2 in the atmosphere is exchanged with CO2 of other reservoirs each year. Seems to be quite nicely and fast mixed. There are lags between near-ground and height and between the NH and the SH. The largest differences are in the first few hundred meters over land, where the largest/fastest sources and sinks are at work.
But have a look at what different stations find:
Thus while there are huge CO2 level differences around vegetation (day/night difference of several hundred ppmv!), that doesn’t exist for the air over the oceans, where the 14C/12C decay rates have the highest difference in decay rate…

aaron

It also means there’s a good bit of fossil carbon in the atmosphere from natural sources.

Jquip

I take no joy in this, but the fallacies are so deep that it warrants no less than a full fisking of Ferdinand:
” 50% bomb spike level goes into the deep oceans, 45% comes out after ~1000 years,
Maxwell called and said he wanted his demon back. Putting aside gedankens of nonsense, it is wholly irrelevant to the topic at hand: The bomb spikes that happened ~50 years ago. Perhaps the argument can be salvaged with clarity, and perhaps it would be worthwhile to do so if we were discussing some other subject.
“That caused an imbalance between atmosphere and deep oceans where inputs and outputs aren’t equal anymore. ”
The ice core records show that ‘aren’t equal anymore‘ places ‘anymore’ prior to the ice core records themselves. There has never been a proper equality. This is confused and nonresponsive at best, and an attempt at sophistical obfuscation at worst.
“40.5 GtC out of the atmosphere into the deep oceans, 39.5 GtC into the atmosphere, caused by the extra pressure from the 100 GtC increase of CO2 in the atmosphere. ”
Based on what measurements? This is first-order circular nonsense. As it presumes the rates to exist — without any measurements taken — on the basis of global average temperatures as compared to salinity and etc. The 14CO2 discussion is precisely about discarding this vulgar fallacy and making use of actual observations — for good or ill. One cannot ‘prove’ your conclusion by assuming it as a premise. Again, a thoroughly confused point at best, or a sophistry at worst.
“The difference in 12CO2 concentration between ins and outs (both ~99%) is negligible.
Not so for 13CO2 and 14CO2.”
With respect to 14C there is the issue of the radioactive decay that would occur with or without Maxwell’s lesser demon of oceanic circulation. But it was already stated in the same post that: in pre-industrial times there was a balance between decay and production of 14CO2:
Whatever the case may be with radioactive decay is meaningless here. For there was, and ppresumptively is, the same process extant. To the degree there is not that can be haggled over. But tagging a herd of Carbon Dioxide with radioactive tracers isn’t going to upset this long running process that has been stated to be balanced. Again, deep confusion or sophistry.
With respect to isotopic fixation in organisms, it only needs noted that 12C is preferred. So we would see
more 12C sequestered — versus exchanged — than 13C and 14C. That is, the decay curve in the bomb spike represents an upper bound on time for the process in question. But this has precisely nothing to do with the thrust of the ‘ins and outs’ statement or the statements about 14C and CH4.
“The difference between a 12CO2 spike and a 14CO2 spike is caused by the long delay between what goes into the oceans and what comes out, which makes that the 14CO2 (and 13CO2) input is effectively decoupled from the output, while that is hardly the case for 12CO2.”
This is again an instance of Maxwell’s demon. But the difference is between a bolus of CO2 — of any and all isotopes — and radioactively tagging a set of the existing bolus. And to be sure, the idea that there is an increasing ppm of atmospheric CO2 is the entire argument of AGW. That there is a constant excess above equilibrium injected into the atmospheric reservoir. So yes, they are different creatures. One is a bolus without remark, as a pure gedanken, and the other is the actual physical process the gedanken represents — just with extra tagging of the molecules. No more nor less.
So the 13 and 14 CO2 are only decoupled from the output in the same manner that the pedestrian evil of 12CO2 is decoupled. Which is: They aren’t. Perhaps in the purely fictional land of idealized models that have no application outside of validation with measurement. But, once more, this entire discussion is about the rare sighting of a wild observation. A practically mythical beast long thought to be extinct.
Lastly, and out of order from the rest: “From the 100% 14CO2 spike going into the deep (1960), 97.5% (as mass) * 45% (as concentration) is coming back, that is only 42.8% is coming back from the deep oceans.”
Sadly, this one is full of confusion and/or sophistry again. There is again the circular argument based on the models proving the models. There is again, the idea that theory trumps measurement. There is again the special pleading about the deep oceans. The only relevant part to this is that the exchange does matter. And is the entire point behind looking at the bomb test curve in the first place. As given the increasing ppm of CO2, it is not a question that more CO2 is entering the atmosphere than leaving it. And the 14CO2 decay curve gives us an idea of the net flow on the basis of that decay.
There is a lot of heel digging on this issue for no good reason, as the math is entirely straight forward for it. We know the 14CO2 production rate, the (Bomb14)CO2 decay rate, and the historical levels — and so the partial pressure of the atmospheric reservoir of 14CO2; if you’re into bathtub models. Everything is right there to gain the answer.
It is, of course, wholly optional. As we can also get the same sanity check on CO2 cycling by looking at the unsmoothed data from Mauna Loa and other measuring sites. As every summer season the CO2 levels fair plummet. To the degree we add man’s culpability, and have a good measure of how much CO2 that dire beast is putting in the atmostphere, then the rest can be figured out independently from there as well.
That’s two — count them, two — independent sources of actual measurement that can be used to ground the entire discussion in science. The utter allergy to approaching the simple math for either of these is wholly incomprehensible. But Ferdinand deserves a round of applause for suffocating his straw man under a heap of errors.

Ferdinand Engelbeen says:
December 12, 2013 at 5:07 am
In fact it is pure coincidence: …. The result is an astonishing fixed ratio between emissions and increase in the atmosphere
==============
As I explained to the police, it was pure co-incidence my gun was smoking when the victim dropped dead.
What if we make the other assumption, that it is not co-incidence? Does this have implications for the choice of model? If the ratio is not due to co-incidence, but rather the nature of the system, how would this change the model?

philincalifornia

phlogiston says:
December 12, 2013 at 12:53 am
It is COMPLETELY IRRELEVANT all the other cycling and dilution and dynamics, pressure, temperature etc. of CO2 that are going on, the 14 tracer simply tells us the removal term for CO2. That is the whole point of a radiotracer measurement.
———————————————————
Amen to that, and no amount of mental masturbation can change it.
This math(s) is virtually identical to that used by pharmacologists/pharmacokineticists for calculating drug dosage in animal models and human studies and treatment. They get it wrong, people die.
Typically, the drug is considered to be, for all intents and purposes, gone at six half-lives, explaining approximately, by analogy, Lindzen’s number (40 years) mentioned in Lord Monckton’s post above.
What reemerges from the sinks, and why is a separate discussion from this. This is the experimentally observed removal term which anyone can see from eyeballing the bomb data is a half-life of 5 or so years.

stevefitzpatrick

Ferdinand,
You have the patience of a saint, but are you really making progress? There are none so blind as those who refuse to see.

Gary Pearse

An elegant mathematical exploration of the travels of our most talked about substance. However indeed,
“Maybe various sinks saturate or become less efficient with increased concentration.”
So far, only the gas guys have responded. CO2 is also a complex chemical, forming carbonic acid in the atmosphere with rain and in its solution into the ocean. Once in the water the acid dissociates into three species:
CO2 +;H2O => H2CO3 => H^+ + HCO3^- => 2nd H^+ + CO3^-2
Now in seawater we have Ca^++ (and other cations which reacts which forms both inorganic limestone precipitate and biologically produced CaCO3 in shells (possibly through the bicarbonate stage), etc. This is sequestered CO2. Also, probably C14 is probably taken up by algae, plankton and the like, similar to the absorption by land plants. Go for the model where the sinks INCREASE their intake by sequestration.

ferd berple says:
December 12, 2013 at 5:43 am
If the ratio is not due to co-incidence, but rather the nature of the system, how would this change the model?
The “model” in this case is the most simple form of reaction: the sink rate of CO2 is directly proportional to the increase in the atmosphere above the (temperature controlled) equilibrium…

Oh my! I remember the hours spent learning to produce qualitatively effective curves of individual isotope activities in a closely empirically engineered system, a nuclear reactor. To attempt the same in an open system is amazing and beyond my ken. I knew that my curves were merely CARTOONS of reality.

Gary Pearse says:
December 12, 2013 at 5:47 am
Go for the model where the sinks INCREASE their intake by sequestration.
==========
In such a bathtub, the drain gets bigger as you increase the input flow. The output is no longer coupled to the height of the bath, rather to the inflow rate.
So, in such a model we have a reverse of the co-incidence proposed above. It is the cumulative excess that is the co-incidence, and the inflow rate that is the determining factor in the outflow.
Such a model may in fact be more likely than the standard bathtub model because the water in a bathtub is not in short supply. But in nature there is ample evidence that CO2 is barely above the minimum required by plants to maintain photosynthesis. As we add CO2 the drain (life) is increasing in size.
In this case it would not be partial pressure that is steering the good ship CO2, rather it is life.

DocMartyn

Your figure 4 is essentially right from a simplest box model approach; with V1 atmosphere, V2 terrestrial biomass, V3 the ocean SURFACE and V4 the depths.
Please do not use the word SINK when you mean reservoir, a sink in local or process that is infinite, a kinetic black hole where there is, on the time scale analyzed, zero back rate. In your figure 4 you include a true sink out of V4, the mineralization of carbon.
Now mechanistically we know that the atmosphere. V1, can ‘talk’ to the ocean surface, V3, but cannot ‘talk’ to the deep ocean, V4.
Let us take what we know we know:-
The disappearance of 14CO2 from the atmosphere is first order all the way from the 70’s with a t1/2 of approx 12.5 years.
The disappearance of 14CO2 from the atmosphere has a projected endpoint very near to the pre-1945 levels.
What we can logically conclude from these two observations.
As atmospheric 14CO2 is being diluted into a reservoir that >40 times bigger, and we know the approximate sizes of all the reservoir’s, we can conclude that the only reservoir big enough to dilute 14CO2 is the deep ocean.
We know that atmospheric 14CO2 MUST interact with the surface layer, before it can interrogate the depths.
The rate of 14CO2 in V1, into V3 and then into V4 has an overall t1/2 of 12.5 years, which is the rate of V1 to V3 or V3 to V4.
As some estimates of residency time of atmospheric CO2 suggest half-lives of 4-7 years, it follows that the simplest mechanism is that
V1 to V3 has a half-life of 6 years, a first order rate of 0.11 y-1
V3 to V4 has a half-life of 12.5 years, a first order rate of 0.055 y-1
The sink rate, mineralization, should match the rate that volcanic CO2 is added to the system, over geological time.
Over geological time, 800,000 years, the volcanic injection of CO2 into the atmosphere is not stable, indicated by ice-core sulphate levels.
Over geological time, 800,000 years, the levels of atmospheric CO2 are pretty stable (230-290 ppm), indicated by ice-core CO2 levels.
It follows that the sink rate is dynamic with respect to atmospheric CO2, and when CO2 levels are low, atmospheric CO2 is not mineralized, but when large amounts of CO2 is injected into the atmospheric by large scale volcanic activity, atmospheric CO2 levels fall quickly back to the ‘normal’ steady state levels.
Typically, chemical processes do not have such concentration threshold effects, but crucially, biological processes like carbon fixation do.
We can therefore speculate that the mineralization sink rate is governed by biological, and not chemical, activity and kinetics.
PS Carbon dioxide molecules do no know what they are doing in a reservoir and do not calculate if it is the thermodynamical appropriate thing to do to move from one reservoir to another, they do not know what a delta[CO2] is and don’t care a damn about entropy; they just move, without thought, plan or knowledge of the system they are in.

catweazle666

Thinking back many years to my chemical engineering phase, it seems to me this process may bear some resemblance to the concepts I studied and applied then, the ‘purge stream’, for example.
So this document might throw some light on the subject (or not, of course!)
Just a thought…

Jquip says:
December 12, 2013 at 5:40 am
Putting aside gedankens of nonsense, it is wholly irrelevant to the topic at hand: The bomb spikes that happened ~50 years ago.
Jquip, what goes into the deep oceans is the isotopic composition of today (minus some fractionation), what comes out of the deep oceans is the isotopic composition of ~1000 years ago (minus some fractionation and some radioactive dexay). That is irrelevant for what happens with a 12CO2 mass spike, but that is highly relevant for the 14CO2 concentration spike.
That gives completely different decay rates for a 12CO2 spike, which is only mass/pressure dependent and for a 14CO2 spike which is (total CO2) mass dependent and concentration dependent. Which makes that the 14CO2 concentration decay is much faster than for a 12CO2 excess mass decay.
The ice core records show that ‘aren’t equal anymore‘ places ‘anymore’ prior to the ice core records themselves. There has never been a proper equality.
The ice core records show a thight ratio between CO2 levels and T levels of ~8 ppmv/K ranging over periods from a few decades (MWP-LIA) to multi-millennia. The rate of change was at maximum 0.8 K for 6 ppmv over 5o years (MWP-LIA) or 0.12 ppmv/yr. The current rate of change in the past 50 years is near 20 times faster for an increase of 0.6 K and increasing even in the past 17 years without T increase. Just by coincidence, while humans are emitting twice the increase of CO2 in the atmosphere?
As it presumes the rates to exist — without any measurements taken —
The 40 GtC/yr exchange rate is my own estimate based on the difference between the theoretical decrease of the 13C/12C ratio in the atmosphere and the observed decrease. Thus based on measurements:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/deep_ocean_air_zero.jpg
It may be 40 GtC/yr or more or less. That is important for the decay rate of the 14CO2 concentration spike, but is completely unimportant for a 12CO2 mass spike: the latter only decays from a differencein mass flow between inputs and outputs, not from a difference in concentration between inputs and outputs, neither from the absolute height of the carbon exchange.
Any uptake of CO2 by the oceans (and reverse) is directly proportional to the partial pressure difference between atmosphere and oceans. If the levels in the atmosphere increase, more is going into the ocean sinks and less is coming out of the upwelling sites. It is the unbalance between the uptake and release that makes that a part of the excess CO2 (currently ~2 ppmv for an excess level of 110 ppmv) is absorbed by the oceans and other reservoirs. If there was no unbalance, then any 12CO2 spike would remain forever in the atmosphere, whatever the 14CO2 spike decay. But there is hardly any connection between the height of the throughput (the 40 GtC/yr) and the unbalance.
But, once more, this entire discussion is about the rare sighting of a wild observation.
Except that the observation does observe the exchange rate of 14CO2 and 12CO2 between the different reservoirs alike, but doesn’t show the decay rate of an excess amount of CO2 in the atmosphere, as that is independent of the exchange rate…
There is again the circular argument based on the models proving the models.
Nothing to do with models: what goes into the deep is the measured isotopic composition of the day. What comes out is the measured composition of ~1000 years ago minus the radioactive decay. 14CO2: 100% in, 45% out; 12CO2: 99% in, 99% out (1960) * the ins/outs as total mass.