I’ve been getting a lot of requests to cover this story, probably 20 or so now with wonderings about “why haven’t you covered this yet?”
How quickly you all forget. WUWT was the very first to cover this story back on November 10th, 2009.
Everybody else in the media today is playing catch-up. So if you’d like to read the original press release and participate in the already ripe comments left then, see this WUWT story:
Willis Eschenbach (22:16:13) :
Our comments crossed. See above.
philincalifornia (22:02:26) :
Thanks for your support. It is nice to know at least one person is listening in who knows where I am coming from, and to whom the things I am stating are common experience, rather than unfathomable gibberish.
A note: the “gains” S1, S2, … in the model above would be, in general, 1 X 2 matrices, and dT1, dT2, .. 2 X 1 vectors consisting of the sine and cosine components of the Fourier series. Or, S1, S2,… could be complex, with dT1, dT2… also complex, and C would be the “real” output of the model. Put in other terms, S1, S2,… would have both gain and phase characteristics.
Eric (skeptic) (18:50:35) :
Ferdinand has ice core data that he showed me once a while back (somewhere in Greenland, forgot where) but it had something like 10’s of years resolution thanks to lots of snow causing the air pockets to lock up a lot faster. Maybe he can dredge up a temperature sensitivity estimate from that data. Also would be interesting to see a C13/C12 ratio sensitivity to temperature from the ice cores.
Hi Eric, as far as I remember, the ice core measurements were from Antarctica, as CO2 levels from Greenland are unreliable due to acids/carbonate deposits from the Iceland volcanoes, which causes in situ CO2 production… The highest accumulation cores are from Law Dome, more towards the Antarctic coast. That makes that they show a very good resolution (2 cores – 8 years average for the past 150 years) to moderate (1 core – 40 years average for the past 1,000 years). The combined graph for the past 1,000 years is here:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/antarctic_cores_001kyr_large.jpg
There is a some dip in the period 1600-1800 of about 6 ppmv in the finest resolution cores, which points to the colder LIA (CO2 lagging temperature with 50-100 years in this case). Depending of which temperature reconstruction you use, the largest difference between MWP and LIA was about 0.8 K (Esper, Moberg), thus that gives a CO2 sensitivity of about 8 ppmv/K in line with the ice core of Vostok for much longer time periods. Other reconstructions show smaller temperature variations MWP-LIA, but that are e.g. Mann’s hockeystick, I don’t think that one is very reliable…
The d13C/12C ratio is interesting too, as that shows a similar trend. Here the trend for the past 400 years, both in the oceans as in the ice cores + firn + atmosphere:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/sponges.gif
Over the past 23,000 years, that is from the coldest part of the previous glacial until now:
http://epic.awi.de/Publications/Khl2004e.pdf
The change in temperature LGM-Holocene (5 K?) gives a change of about + 0.4 per mil, probably due to increasing vegetation growth at higher temperatures.
The d13C values in the atmosphere didn’t change much in the past 10,000 years (around -6.4 per mil), except for some temperature (vegetation growth) influence of about +/- 0.2 per mil. Since about 1800, levels in the atmosphere dropped to -8 per mil and the upper level oceans from near +5 to less than +4 per mil nowadays.
Thus the general conclusion from both CO2 levels and 13C/12C ratio’s is that the current increase in CO2 and the decrease of d13C is not induced by temperature variations.
philincalifornia (16:35:04) :
If you start from first principles, Bart has shown that you can achieve the same result (the one that is observed) as you can by assuming that all of the increase is due to anthropogenic emissions then back-filling the rate constants to fit the conclusion. The latter is a circular argument.
Indeed it is circular reasoning to use the rate constants, but what Bart shows doesn’t fit any of the observations, not even the increase in the atmosphere in constant ratio with the human emissions, without invoking some impossible feedback. That is a practical problem for a theoretical solution. Thus the theoretical solution is wrong…
Bart, I haven’t read every single comment in this thread, but has anyone mentioned the slight decrease in pH, which could (would) also affect the equilibrium towards the atmospheric compartment, and could be chalked up as an anthropogenic component since it would presumably be a direct positive feedback (for CO2 levels in the atmosphere that is, just to be clear
In the latest discussions I had with other non-human CO2 increase enthousiasts, this seems the new argument to show that the increase of CO2 in the atmosphere is due to the lowering pH in the oceans. That may have been caused by some unknown undersea volcanic release of a lot of acids… Besides that it should be highly synchronised to human CO2 releases, that is not very likely: a decrease in pH due to adding a non-CO2 acid shifts the equilibrium in seawater from carbonate to bicarbonate to CO2. As CO2 pressure increases, more is released to the atmosphere. As result, the total sum of CO2 + bi + carbonate (= dissolved inorganic carbon, DIC) reduces, because less bi/carbonate is left when more CO2 escapes to the atmosphere. But we see the reverse: DIC increased over the past decades. Thus more CO2 entered the oceans, and that is the cause of the (very slight) decrease of pH… See:
http://www.bios.edu/Labs/co2lab/research/IntDecVar_OCC.html
Bart (15:12:10) :
Ferdinand Engelbeen (14:22:04) :
Let me put the temperature dependent part back into my equation
Cdot = (Co-C)/tau + (1+Ko)*adot + S*dT
You will recall from earlier I left this out because, I said, this does not affect the sensitivity to adot. It does, however, affect the calculation of the magnitude of the time constant, because of the cyclical temperature dependence you have brought up.
I think you have misunderstand this, the temperature dependency is already in the factor adot, where you presume that adot is (currently) only 3% of the total inflow. The natural inflow of about 150 GtC is what is temperature dependent and mainly occurs as result of upper ocean temperatures in one direction and vegetation growth in the other direction, that is simply pumping CO2 from/to other compartiments, without draining anything (as Joel already said). In the bathtube example, that is the effect of a seasonal heating and cooling of the bathtube in separate compartiments. That doesn’t affect the average real, natural release/drain speed, which is much lower, unknown, and largely what is going into the deep oceans near the poles and comes out again near the equator.
That is what we (Joel and I) try to show you. Thus your adot is not 0.03 but may be 0.5 or 2.0 or… That makes that you need to recalculate the rest of the equations too (and you may assume Ko = 0 for my part…).
Further, far more important, my impression still is that you reverse cause and effect: if you add CO2 to the atmosphere, that isn’t added in ratio to the other inputs. It simply is added 100% to the atmosphere. As that increases the C – Co difference, the sinks increase in ratio to the increase/previous difference. Not the reverse, that the inflow rate dictates the increase in the atmosphere… Thus in this case, after one year:
Cdot = (Co – (C + Cem))/tau = (580 – (808))/55 = -4.15 GtC which means that of the original 8 GtC added, some 3.85 GtC remains in the atmosphere. For adot (assuming Ko = 0), that means that adot is about halve the original equilibrium flows. But that is mainly of academical interest, as we do know tau, C, Co and Cem with reasonable accuracy. Thus we don’t need to know or estimate the real natural in or outflows.
The rise in global temperature since 1970 has been about 0.6 degC. This suggests, according to the equation above, that atmospheric CO2 concentration should have increased about 0.6*189 = 113 ppmv. Anthropogenic forcing should have contributed about 0.015*280 = 4.2 ppmv (I used 1.5% instead of 3% because anthropogenic production has actually been ramping up to 3%, and has not been 3% all the time). So, we have a total expected delta of about 117 ppmv from a base of 280 ppmv, or 397 ppmv.
A little overblown and it doesn’t fit reality. The 189 ppmv/K is far too high. The actual global change over the seasons is about 3 ppmv for 1 K increase, mainly as result of the NH (land) temperature swings. See:
http://www.esrl.noaa.gov/gmd/ccgg/trends/
and
http://data.giss.nasa.gov/gistemp/seas_cycle.html (just push on show map), thus 3 ppmv/K.
Further, the direct influence of temperature on the increase/sink rate of CO2 around the trend is about 4 ppmv/K, see:
http://icecap.us/images/uploads/CO2vsTMacRae.pdf
And from the past 1,000 years to the 800,000 years old ice cores, the ratio between temperature and CO2 level changes is about 8 ppmv/K, if the temperature difference is maintained for hundreds to thousands of years. See:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/Vostok_trends.gif
Moreover, there are two periods in the recent past where temperatures were flat/decreased: 1945-1975 and 2000-now. In both periods CO2 increased and still increases in ratio with the emissions, even while temperature/CO2 correlations went negative… See:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/temp_emiss_increase.jpg
Not too shabby, given all the approximations I have made. It follows that, if temperatures genuinely start to decline, after 2-6 years lag time (3 time constants is commonly referred to as “settling time” in systems theory), you should expect to see the CO2 levels decline as well.
I shouldn’t hold my breath until that happens… ever.
In Joel Shore (19:32:05) we read “It will certainly help them correctly decide how seriously to weigh your views and it will help to re-enforce the conception that AGW “skeptics” are merely people who will deny reality no matter how strong the scientific evidence is.”
As you know Joel, sometimes reality is simple and sometimes it is not. Sometimes we interpret evidence in such a way as to make reality look simpler than it really is.
Then you said “I am in fact happy to see some skeptics like Willis and Ferdinand take on those with more extreme views.” Not only that, you show up on these threads to take on those extreme views yourself. So you are not just happy to see solid science, but you are contributing to it. In this particular case the reality of manmade emission in the multi-hundred Gt range (carbon) and corresponding CO2 and corresponding rise in atmospheric CO2.
But allow me a tiny tangent now. The CAGW believers point to models of aerosols as delaying warming. They point to warming of deep ocean as delaying atmospheric warming. They point out positive feedback in models, and also in measurements, some of which are adjusted using models. They make conclusions about sensitivity and then put forth theories of rapid ice melt and catastrophe.
How much of this is based on reality? Strip away the models and speculation and their own denial of reality (e.g. recent slowing of Greenland’s glaciers) and there are few direct measurements of reality. Yet there are many claims that people who deny CAGW are denying reality. You imply that you would hope that the bad science (the clearly denying reality kind) get mixed and conflated with the legitimate scientific disputes over models or complex measurements (e.g. UTWV). But I don’t think that will advance the overall cause of science. I believe it has harmed science and one of the results is reactionary bad science.
OTOH, what you are doing here, at least in this thread is useful and good for science, so please keep it up.
Bart, I don’t get it.
You first claim that the sensitivity of CO2 to temperature is 189 ppmv per degree Celsius. On this basis, you say that as soon as temperature starts to fall, so will CO2 … a reasonable claim based on your numbers, if they were correct..
Joel and Ferdinand and I point out that your numbers are in error by a couple of orders of magnitude, and that the actual sensitivity to temperature is only 1 ppmv per degree Celsius or less.
You claim this makes no difference to your thesis, saying:
If you truly think that an error of two orders of magnitude in your main variable doesn’t invalidate your thesis, I fear I can’t help you. This is especially true since in addition to the incorrect magnitude, your error entirely removes the annual variation which is the basis of your claims.
Here’s the best response in your situation. Say ‘Ooops, I was looking at the wrong data. My bad, my conclusions were wrong. Nothing to see here, folks, move along, better luck next time.’ If you do that, you look like a scientist.
Any other response, like ‘I made a major foolish error but I’m still right, it doesn’t invalidate my thesis’, merely marks you as a fanatic rather than someone seeking the truth.
My best wishes to you,
w.
Bart came in here professing to be a systems theorist with the mathematical model of the CO2 cycle which experimental data must agree with or be wrong! Despite the fact that he doesn’t understand the physical chemistry of the system (and having it explained to him, notably by Ferdinand) he bashed on regardless!
I’ll try one more time, the ocean/atmosphere exchange of CO2 can reasonably be approximated by Henry’s Law, p=k*c, where p is the partial pressure of the gas, c the concentration of the dissolved gas in the liquid phase and k the coefficient (depends on temperature).
So for a fixed volume of gas in equilibrium with a fixed volume of water the mass of the solute in the gas phase will have a constant ratio with the mass of solute in the liquid phase. A consequence of this is that new solute is added to the gas phase it will distribute itself between the two phases in the same ratio. Knorr’s findings illustrate that this is indeed true for CO2 in the earth’s atmosphere/ocean system. Note that because the total mass of solute in the system has increased the equilibrium concentration in each phase has increased too, this is the fatal flaw of Bart’s model, he assumes that the equilibrium concentration must stay the same and that the system is ‘trying’ to return to that concentration whereas it is actually ‘trying’ to maintain the same ratio of CO2 between the phases. Since k is a function of T if temperature varies then the setpoint also varies, this sensitivity must be small because the annual fluctuation of CO2 is small as Ferdinand has pointed out (189ppm/K would be totally impossible).
The conclusion to be drawn from Knorr’s paper is that the system is consistent with a dominant exchange between the ocean and atmosphere which can be well described by a Henry’s Law model over a multi-decadal time scale between the atmosphere and an oceanic mixed layer to which CO2 is steadily being added with a small scale temperature fluctuation.
This does not precluded longer term exchange with the deep ocean or the solid state (CaCO3).
Ferdinand Engelbeen (07:37:45) :
““I think you have misunderstand this, the temperature dependency is already in the factor adot, where you presume that adot is (currently) only 3% of the total inflow.”
I think there may have been some confusion in my nomenclature, “adot” refers solely to anthropogenic input. I presume adot is currently only 3% of the total inflow because that is what the IPCC says. The question, which you have brought to the fore, is: what is the definition of that total inflow within my model?
“The natural inflow of about 150 GtC is what is temperature dependent and mainly occurs as result of upper ocean temperatures in one direction and vegetation growth in the other direction, that is simply pumping CO2 from/to other compartiments, without draining anything (as Joel already said).”
My problem with this thesis has always been that you have never suggested a mechanism by which such an independent cycle could be achieved and maintained. “Equilibrium is never achieved by luck” has been my mantra. You may not have noticed, but at Bart (23:07:49), I gave you a mechanism by which such a cycle could be established and maintained with minimal regulatory input.
It is a rather advanced mechanism, and one I doubt has been considered explicitly and in all of its ramifications. There are fundamental limits, also, on how such a mechanism can function while maintaining overall stability. I still do not believe that it dominates to the degree needed to attribute the full 30% rise in CO2 of the last 50 years to anthropogenic causes, but I do believe it could allow substantially more of a contribution than 3%.
I have not sorted out all of the ramifications, myself, yet. Nor do I expect I will for some time – I do have a job and a living to make. I would advise withholding final judgment until I can get back to you with something more meaty to sink your teeth into ;-).
Willis Eschenbach (08:41:28) :
“Say ‘Ooops, I was looking at the wrong data. My bad, my conclusions were wrong. Nothing to see here, folks, move along, better luck next time.’ If you do that, you look like a scientist.”
Quite the opposite. I have put forward a plausible mechanism by which the sensitivity to temperature cycles can be substantially attenuated, and my thinking is evolving as I digest the ramifications of that mechanism. Indeed, it must be significantly attenuated based on the empirical evidence. We have CO2 variation on the order of +/- 3 ppmv, and we have absolute temperature variation on the order of +/- 2 degC.
A scientist does not throw up his hands and say “my initial hypothesis was not entirely on the mark, so I give up – it cannot be explained”. A scientist looks for new possibilities, within the realm of the possible and the plausible, by which he can explain the discrepancy.
Phil. (09:20:29) :
The model hasn’t changed. It is fundamental. The interpretation of the model, and the assignation of parameter values, is all that will, or can, change.
I do not understand this impulse to shut down debate before full understanding has been achieved. For a child, it would not be unexpected, but you would expect one’s outlook to evolve as one reaches adulthood.
Bart (11:44:30) :
Phil. (09:20:29) :
The model hasn’t changed. It is fundamental. The interpretation of the model, and the assignation of parameter values, is all that will, or can, change.
It’s fundamentally wrong, that’s your problem which you’re apparently incapable of realizing.
I do not understand this impulse to shut down debate before full understanding has been achieved. For a child, it would not be unexpected, but you would expect one’s outlook to evolve as one reaches adulthood.
No one’s trying to shut down debate, just trying to avoid wasting time down a blind alley. Your resort to childish insults when the flaws in your model are pointed out doesn’t help your case.
Phil. (12:51:52) :
It was a mistake to recognize your input. I won’t make it again. You have nothing of value to add to the conversation. Go fling your feces over at Tamino, or wherever you like to hang out with other small minded folk.
I hope more thoughtful others recognize that I am actually flinging a lifeline to rescue Joel and Ferdinand from their own folly.
The only way to satisfy their demand for an uncoupled, steady state CO2 cycle is via an internal model feedback, which effectively notches out the response to the full temperature variation in the CO2 response. You can doubt the existence of such feedback, or support Joel and Ferdinand’s interpretation, but not both. They are mutually exclusive. A steady state independent cycle cannot happen by luck. Something has to get it going and keep it going.
“They are mutually exclusive.”
Which is to say, Joel and Ferdinand’s conception of a steady state cycle is equivalent to saying there is an internal model feedback.
“…which effectively notches out the response to the full temperature variation in the CO2 response.”
This is a necessary consequence of such a feedback.
Bart (14:21:10) :
A steady state independent cycle cannot happen by luck. Something has to get it going and keep it going.
They call that the THC, the thermohaline circulation. Kept going by the fact that the sun shines and the earth rotates…
Ferdinand Engelbeen (15:30:09) :
“They call that the THC, the thermohaline circulation. Kept going by the fact that the sun shines and the earth rotates…”
In that case, the dynamics are not decoupled, and you do not get more than 3.1% (with appropriate error bars) rise due to anthropogenic forcing.
I hope there is no perception that I have somehow conceded that my model results were wrong. The question of using absolute temperature variation versus temperature anomaly variation only affects the calculation of the sensitivity factor S1. The actual value of that parameter was immaterial to the conclusion drawn – the variation in CO2 was still +/- 3 ppmv regardless. At that point, it was just a number.
The part where I got into trouble was saying that, if you used the value I had derived of S1 for S0, then you could also explain the observed rise in atmospheric CO2 concentration as a consequence of the rise in temperature. But, if S0 = S1, and S1 is the sensitivity to absolute temperature variation, then S0 is too small to do that. If, however, S1 = S0*N, where N is the gain of a notch filter at the yearly cycle, then S0 still could be large enough.
That led me to realize that a notch filter would be the natural consequence of an internal model feedback, and I could still get the value of S0 I desire. But, the wrench in this is that it would allow a decoupling of the anthropogenic and natural dynamics such as Ferdinand and Joel have been insisting upon without physical justification, and the upshot there would likely be a stronger role for anthropogenic CO2 in the observed rise.
I think it is likely that the ultimate fallout will be both: there will plausibly be a stronger than 3% role for anthropogenic forcing in the observed rise, but this will also necessarily dictate a greater role for temperature forcing of the CO2 level than has previously been accepted.
Bart says:
We have given you lots of physical justification for the accepted picture of what is going on, along with lots of analogies, and discussion of the underlying mathematics. You have just chosen to ignore them.
At any rate, I am glad that you seem to be coming around at least somewhat in your thinking. Perhaps you’ve realized the error of making arrogant statements like “I guess mathematics doesn’t reflect reality after all” when people challenge the reality of your model? It is always good to approach science with a healthy dose of humility and a realization that you have thought about the problem a lot less than those who have been working on it for a long time…and that many of them are at least as intelligent and knowledgeable as you are.
Bart (11:44:30)
Not entirely on the mark???
Your calculations were out by two freakin’ orders of magnitude, and on my planet that is a total miss. Yet your model was in complete harmony with that totally incorrect calculation.
Now you say it makes no difference … m’kay. But if your model is untouched and unaffected by your error of two orders of magnitude, you have more problems than I can help you with.
I’m not asking you to throw up your hands and quit. I am saying that the mechanism that you were so sure of, the mechanism that you were willing to insult people to defend, was wrong by two orders of magnitude. In order for you to rescue a shred of credibility, you need to start by acknowledging that you were emphatically proclaiming that you were right, right, right, when in fact you were ludicrously wrong, wrong, wrong. “Not entirely on the mark” doesn’t do it. The reality is you were two orders of magnitude wide of the mark from a newbie error. Trying to peanut butter over that crack and then rushing to impress us with your new you-beaut theory won’t get you any traction.
Willis Eschenbach (20:18:19) :
Suppose we have two numbers, x and y, and I say they are constrained by the equation
x + y = 5
Let’s say I suggest “one solution could be x = 1 and y = 4, then you come back and say “we know x is greater than 2, so your equation is bunk. That is tantamount to what you are claiming here.
Joel Shore (19:20:02) :
We have given you lots of physical justification for the accepted picture of what is going on, along with lots of analogies, and discussion of the underlying mathematics. You have just chosen to ignore them.”
That is because you have not proposed any rigorous mathematics. You are just hand-waving and guessing at how you think the system mighta’ oughta’ kinda’ work.
For the rest, you have misinterpreted my comments, but I don’t really see the point of pursuing it right now.
Let me try to lay this out for you all one more time:
1) Ferdinand’s suggestion was that the 3% figure for adot was for 3% of more than just Co/tau, but also had to take into account natural flows
2) I showed that, even taking these into account, the figure barely budged. This conclusion was based on the observed CO2 yearly variation, and was in no way dependent on the sensitivity parameter
3) Entirely apart from this exercise, I erroneously noted that the sensitivity parameter was quite large, and hastily proclaimed that this, then, could also account for the secular variation in CO2 from temperature forcing
4) Willis noted that my calculation of the sensitivity parameter was way too large, so I could no longer claim the secular variation in CO2 was from temperature forcing from that calculation – that is all it meant
5) I found, however, a striking (to me) correlation between the temperature anomaly and the first and second harmonic amplitudes of CO2 variation. I then cast about for a way to rescue the notion that the sensitivity was to temperature anomaly, rather than absolute temperature variation
6) Based upon my experience in designing actual control systems for working real-world systems, I realized that this was possible if I inserted an internal model into the feedback loop. I then considered that deciduous plants and other biota do, in fact, internalize the yearly clock and might behave in a manner representative of such a feedback mechanism. I then realized that this did give Joel and Ferdinand a potential out – that such a feedback mechanism could perhaps set up a decoupled exchange system which could be considered independent of the anthropogenic cycle.
It goes without saying that, even though such feedback mechanisms are routinely designed into working feedback systems, the likelihood of a similar and substantial climate feedback mechanism is remote. But, it still allows the possibility, however slim, that such feedback action could exist, and that could tie everything up into a neat, tidy ball: the otherwise mysterious rise in CO2 concentrations, the observable variations in temperature and CO2 fluctuations… everything.
7) I now wish to delve into this subject, and determine whether there is any plausibility in the concept at all. But, my previous conclusions still hold:
A) the sensitivity of the CO2 dynamics to anthropogenic forcing can be determined by the equation
Cdot = (Co-C)/tau + (1+Ko)*adot
We do not need to have the temperature forcing in this equation because it is a linear system, and superposition holds. Temperature dynamics can affect the values of tau and Ko, but that is already here.
B) Including temperature dynamics leads to an equation of the form
Cdot = (Co-C)/tau + (1+Ko)*tau + S0*dT0 + S1*dT1 + S2*dT2 + …
Any linear dynamics, including the potential internal model feedback of which I have spoken, can be included by the appropriate evaluation of the sensitivities Ko, S0, S1, S2, …
C) Any real world dynamics must be able to be expressed in this form in a local operating regime. Any hypothetical dynamics which do not, in some representation, conform to this model are invalid.
“I then realized that this did give Joel and Ferdinand a potential out – that such a feedback mechanism could perhaps set up a decoupled exchange system which could be considered independent of the anthropogenic cycle.”
However, since the feedback, if there is any, depends on a small subset of links in the carbon cycle chain, it is almost certain that it is not sufficient to entirely, or even mostly, decouple the anthropogenic and natural forcing.
That is my position as of now. If anyone still has gripes with it, I apologize for not being able to make it clear enough for you. And now, I really must say au revoir to this thread.
Bart (23:26:29) :
Ferdinand’s suggestion was that the 3% figure for adot was for 3% of more than just Co/tau, but also had to take into account natural flows
Last time now, until the moment you are able to give us the results of your “model” which fits all known observations (including measured increase and d13C decrease in the atmosphere and oceans in ratio with the emissions…) here my take on what is reality based.
Starting with your formula:
Cdot = (Co-C)/tau + (1+Ko)*adot
adot is near independent from temperature. That is the THC: at the upwelling side temperature hardly change the deep ocean CO2 surplus release and at the downwelling side, there is no temperature change, only a shift of where the downwelling will take place (at the edge of the sea ice). The main change is from the overall pressure increase, which is directly related to the emissions. The original downwelling flows are largely unknown and probably small. Thus again (seems very hard to bring into your head): the human emissions are not 3% of the pressure dependent source/sink flows, they may be 50% or 200%. The measured pressure effect (over the past 160 years!) is an increase of 50% of the human addition.
Besides that, we have the temperature dependent seasonal and multi-year flows. These are massive flows with little effect on pressure. In general 3 ppmv/K for seasonal and 4-8 ppmv/K for (multi-year to multi-millennia) temperature changes.
Thus we have two equilibrium reactions: the temperature influence which mainly changes Co with a small pressure shift and huge flow changes between carbon compartments (biosphere and upper oceans) for a small temperature change and a pressure related equilibrium which has a small effect on temperature and flows (between atmosphere and deep oceans), but a huge effect on pressure. These two equilibrium reactions and their related flows are largely (but not completely) decoupled.
Thus the combined formula is (assuming that Ko is low, compared to the influence of dT):
Cdot = (Co-C)/tau + adot + F(dT)
Further, as the emissions are not a constant addition over time, but a (near) constant increasing addition, adot isn’t applicable but instead of that, one may assume a rise in constant ratio to the addition. In fact what was found by Knorr… Thus the realistic, empirical formula that covers the increase over the past 160 years to over 800,000 years is:
Cdot = 0.55*Cem + 4*dT
Where 4*dT must be expanded to 8*dT for (very) long time periods of changed temperature.
Thus please come back when you have figured out that your model fits reality (quantities and isotope ratio’s). Only then we can have a fruitfull further discussion…
Ferdinand I agree with what you said above with one further point about the equation Cdot = (Co-C)/tau + (1+Ko)*adot.
The real problem with this is the Co term which takes no account of the solution/dissolution process, this term should be Ctotal/(1+k) where Ctotal is the total CO2 present in the atmosphere/mixed ocean layer and k is a Henry’s Law coeff. Assuming a starting equilibrium with Co in the atmosphere the initial Ctotal = Co(1+k) but this term grows with time as follows:
Ctotal = Co(1+k)+∫adot
The temperature dependence enters via k and there is no need for the Ko term.
With that change that formula might come reasonably close to reality.