## Sunday, October 9, 2016

### Falsification of Anthropogenic Global Warming

The theory of Anthropogenic Global Warming, so far as we understand it, consists of the following two assertions.

(1) If we increase the concentration of CO2 in the atmosphere to 600 ppmv, we will cause the world to warm up by at least 2°C. (The concentration in pre-industrial times was 300 ppmv and is currently 400 ppmv, where ppmv is parts per million by volume.)

(2) If we continue burning fossil fuels at our current rate, emitting 10 petagrams of carbon into the atmosphere every year, we will raise the concentration of CO2 in the atmosphere to 600 ppmv within the next one hundred years.

We can falsify the second assertion using our observations of the carbon-14. We present a detailed analysis of atmospheric carbon-14 in a series of posts starting with Carbon-14: Origins and Reservoir. Here we present a summary, with approximate numerical values that are easy to remember.

Each year, cosmic rays create 8 kg of carbon-14 in the upper atmosphere. If carbon-14 were a stable atom, all carbon in the Earth's atmosphere would be carbon-14. But carbon-14 is not stable. One in eight thousand carbon-14 atoms decays each year. The rate at which the Earth's inventory of carbon-14 decays must be equal to the rate at which it is created. There must be 64,000 kg of carbon-14 on Earth.

The Earth's atmosphere contains 800 Pg of carbon (1 Pg = 1 Petagram = 1012 kg) bound up in gaseous CO2. One part per trillion of this carbon is carbon-14 (1 ppt = 1 part in 1012). There are 800 kg of carbon-14 in the atmosphere. That leaves 63,200 kg of the total inventory somewhere else. We'll call this "somewhere else" the carbon-14 reservoir.

Each year, 8 kg of carbon-14 is created in the atmosphere by cosmic rays, and each year the atmosphere loses 8 kg of carbon-14 to the reservoir. (Here we are ignoring the 0.1 kg of atmospheric carbon-14 that decays each year.) There is no chemical reaction that can separate carbon-14 from normal carbon. Every 1 kg of carbon-14 that leaves the atmosphere for the reservoir will be accompanied by 1 Pg of normal carbon.

Consider the atmosphere before we began to add 10 Pg of carbon to it each year. The mass of carbon in the atmosphere is constant. If 1 Pg of carbon leaves the atmosphere and enters the reservoir, 1 Pg of carbon must go in the opposite direction, leaving the reservoir and entering the atmosphere.

The only way for there to be a net loss of carbon-14 from the atmosphere to the reservoir is if the concentration of carbon-14 in the reservoir is lower than in the atmosphere. The only place on Earth that is capable of acting as the reservoir is the deep ocean, in which the concentration of carbon-14 is 80% of the concentration in the atmosphere. Each year 40 Pg of carbon leaves the atmosphere and enters the deep ocean, carrying with it 40 kg of carbon-14, while 40 Pg of carbon leaves the ocean and enters the atmosphere, carrying with it 32 kg of carbon-14. The result is a net flow of 8 kg/yr of carbon-14 into the ocean. Furthermore, the ocean contains 63,200 kg of carbon-14 in concentration 0.8 ppt, so the total mass of carbon in the oceans is roughly 80,000 Pg.

With the ocean and the atmosphere in equilibrium, 40 Pg of carbon is absorbed by the ocean each year, and 40 kg is released by the ocean. If we were to double the quantity of carbon in the atmosphere, we would double the amount absorbed by the ocean each year. Instead of 40 Pg being absorbed each year, 80 Pg would be absorbed. We could double the concentration of carbon in the atmosphere by emitting 40 Pg/yr. But we emit only 10 Pg/yr. Our emissions are sufficient to increase the mass of carbon in the atmosphere by 25%, after which everything we emit will be absorbed by the oceans. The oceans contain 80,000 Pg of carbon. If we add 10 Pg/yr, it will take roughly eight thousand years to double the carbon concentration in the oceans, after which the concentration in the atmosphere will double also.

Back in the 1960s, atmospheric nuclear bomb tests doubled the concentration of carbon-14 in the atmosphere. Such tests stopped in 1967. In our more precise calculation we predict that the concentration of carbon-14 must relax after 1967 with a time constant of 17 years, so that it would be 1.37 ppt in 1984 and 1.05 ppt in 2018. The concentration did relax afterwards, with a time constant of roughly 15 years, and in 2016, the carbon-14 concentration in the atmosphere is indistinguishable from its value before the bomb tests. During that time, almost every CO2 molecule that existed in the atmosphere in 1967 passed into the ocean and was replaced by another from the ocean. Anyone claiming that our carbon emissions will remain in the atmosphere for thousands of years, such as the author of this article, is wrong. If we stopped burning fossil fuels tomorrow, the CO2 concentration of the atmosphere would return to its pre-industrial value within fifty years.

When carbon is absorbed or emitted by the ocean, it does so as a molecule of CO2. Statistical mechanics dictates that the rate of absorption is weakly dependent upon temperature, but the rate of emission is strongly dependent upon temperature. When we calculate the effect of temperature upon the equilibrium between the ocean and the atmosphere, we conclude that a 1°C warming of the oceans will cause a 10 ppmv increase in the concentration of CO2 in the atmosphere. When we look back at the record of CO2 concentration and temperature over the past 400,000 years, we see the correlation we expect, with the magnitude of the changes in good agreement with our prediction. For a 12°C increase in temperature, for example, the concentration of CO2 increases by 110 ppmv.

If we consider the atmosphere of the Earth in pre-industrial times, its atmospheric CO2 concentration was roughly 300 ppmv. A more exact value for the creation of carbon-14 is 7.5 kg/yr and we conclude that 37 Pg/yr or carbon was being absorbed and emitted by the ocean. When we add 10 Pg/yr human emissions from burning fossil fuels, we expect the concentration of CO2 in the atmosphere to rise by 27% to 380 ppmv, which is close to the 400 ppmv we observe.

Our analysis of the carbon cycle makes three independent and unambiguous predictions all of which turn out to be correct to within ±10%. Our analysis is reliable, and it tells us that it will take roughly eight thousand years to double the CO2 concentration of the atmosphere if we continue burning fossil fuels at our current rate. Assertion (2) above is wrong by two orders of magnitude. The theory of Anthropogenic Global Warming, as stated above, is untrue.

POST SCRIPT: Assertion (1) is harder to falsify, and we do not claim to have done so in a manner convincing to all readers. Nevertheless, we did conclude that assertion (1) had to be wrong in our series of posts on the greenhouse effect, which we summarize in Anthropogenic Global Warming. We calculated that doubling the CO2 concentration of the atmosphere will cause the Earth to warm up by 1.5°C, provided we ignore changes in water vapor and cloud cover. As the world warms up, however, water evaporates more quickly from the oceans, and we get more clouds. Clouds reflect sunlight. The warming effect of doubling CO2 concentration is reduced by an increase in cloud cover. Clouds stabilize the Earth's temperature because they become more frequent as the Earth warms up, and less frequent as it cools down. Our simulation of the atmosphere with clouds suggests that the actual warming caused by a doubling of CO2 will be 0.9°C. So far as we can tell, the climate models used by the majority of climate scientists do not account for the increase in cloud cover that occurs as the world warms up. But they do account for the increase in water vapor in the atmosphere. Clouds cool the world, but water vapor is another greenhouse gas, and warms the world. By including water vapor but excluding the increasing cloud cover, these climate models conclude that the effect of doubling CO2 concentration will be 2°C or larger.

POST POST SCRIPT: Some readers suggest that the atmosphere-ocean system cannot be modeled with linear diffusion because the dissolved CO2 does not increase in proportion to atmospheric CO2 concentration. We address and reject their claim in an update to Probability of Exchange. They claim that Henry's Law does not apply to CO2 and seawater, despite many measurements to the contrary, such as Tsui et al., and general acceptance of Henry's Law in academic texts such as this.

1. I think your analysis is confusing the residence time of CO2 molecules with the adjustment time of an enhancement in atmospheric CO2. The residence time of individual molecules is indeed only years. The adjustment time of an enhancement in atmospheric CO2, however, is more like centuries. This paper by Gavin Cawley does a nice analysis that illustrates this. This paper is also worth looking at.

2. Dear TTP,

I am sorry you found my presentation confusing. In a two-reservoir model, the adjustment time is equal to the residence time.

http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-analytic-solution-to.html

In the long term, absorption by the ocean is proportional to atmospheric concentration, no matter how complex the model. For more complex models, in which we divide the ocean into layers, the time it takes for the system to adjust to a new equilibrium could be 20% larger or smaller than for the simpler single-layer system. In other words: there is no significant difference between the multi-layer model of the ocean and the single-layer model. I guess I could set up some spreadsheets to prove that, or a few simulations, but I'm short of time. The Arnold et al paper proves the point.

http://www.hashemifamily.com/Kevan/Climate/Dist_C14_Nature.pdf

Given that the single-layer model predicts the bomb test aftermath, the CO2/temperature correlation in ice cores, and the current rise in CO2 concentration, all to within 10%, it seems to me there is no need to introduce further layers in the ocean.

I am debating with Gavin Cawley here:

http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-origins-and-reservoir.html

Thanks for visiting.

Yours, Kevan

3. In a two-reservoir model, the adjustment time is equal to the residence time.
Well, I'm disputing this and I think it's remarkable, if true. We know that, in equilibrium, the ocean/biosphere would take up as much CO2 per year as it emits per year. The atmospheric concentration would therefore not change, but a typical CO2 molecule in the atmosphere would not stay there indefinitely; it would probably be taken up by one of the sinks (ocean/biosphere) within 15 years, or so. You're suggesting that if the atmospheric concentration were to be enhanced, this enhancement would decay on the same timescale as the typical residence time of an atmospheric CO2 molecule. This seems a remarkable coincidence.

I had a look at your equations, but I can't quite work out the terms. Maybe you could define them.

4. Dear TTP,

Yes, that's exactly what I'm saying. It's not a coincidence. It's a feature of systems in which the rate at which a quantity changes is proportional to the quantity itself. The residence time and the adjustment time are two sides of the same phenomenon.

Maybe I can be less confusing, with a one-reservoir example.

Suppose we are adding 8 kg/yr of carbon-14 to a pile of carbon-14. That's what cosmic rays are doing to the Earth. The rate of change in the total mass of carbon-14 is equal to +8 kg/yr minus what decays, which is M/8000, where M is the mass that exists in the pile. Let "dM/dt" be "the change per year in the mass M".

dM/dt = + 8 kg - M/8000

If we start with 0 kg, M will do this (this is the solution to the above differential equation for the boundary condition M=0 at t=0).

M = 64,000 kg * (1 - exp(-t/8000))

When t=8000, we have added a total of 64,000 kg, which is happens to be the value M will reach eventually, but after 8000 years the pile is only (1-exp(-8000/8000)) = 63% of its eventually value. So the adjustment time is 8000 years (time it takes to get to 63% of the final equilibrium), but that's also the decay time of carbon-14, which is the same as the residence time of carbon-14 (the time in which the chance of a carbon-14 atom decaying is 63%).

Does that make sense? Now, the two-reservoir model we have derived for the atmosphere (based on that of Arnold et. al.), contains one small reservoir (800 Pg of carbon in the atmosphere) and one large reservoir (80,000 Pg of carbon in the ocean). When we double the CO2 to the atmosphere, it is the behavior of the atmosphere that dominates the response for the first ten centuries, because the larger oceanic reservoir is so large it is unaffected by the CO2 it absorbs from the atmosphere.

So, we we have a one-reservoir system like that of the carbon-14 pile, and the residence time is the adjustment time. Call it a coincidence if you like, but that's the way it is.

Yours, Kevan

5. I think the issue is that this is not correct.
When we double the CO2 to the atmosphere, it is the behavior of the atmosphere that dominates the response for the first ten centuries, because the larger oceanic reservoir is so large it is unaffected by the CO2 it absorbs from the atmosphere.
The addition of CO2 to the atmosphere changes both the flux of CO2 from the atmosphere into the oceans AND the flux of CO2 from the oceans back into the atmosphere. Hence, the timescale over which an enhancement of atmospheric CO2 will decay is not the same as the timescale over which a typical CO2 molecule will move between the atmosphere and the ocean.

6. Dear TTP,

The probability that a given CO2 atom will be absorbed by the ocean each year is independent of the number of CO2 molecules in the atmosphere. If that's not clear to you, we can talk about it. The mass of carbon absorbed by the oceanic reservoir depends only upon the concentration of CO2 in the atmosphere.

The probability of a CO2 molecule being emitted by the ocean is dependent upon temperature, but not upon the number of CO2 molecules in the ocean. At a certain temperature, the rate at which CO2 is emitted by the oceanic reservoir depends only upon the concentration of CO2 in the oceanic reservoir.

The equilibrium concentrations of CO2 in the atmosphere and the ocean are the ones at which these two rates are equal and the total amount of carbon is constant.

Right now, these two rates are roughly 40 Pg/yr. If we double the concentration in the atmosphere, CO2 starts moving into the ocean at 80 Pg/yr rather than 40 Pg/yr. The ocean continues to emit 40 Pg/yr until its concentration increases. There is a net 40 Pg/yr entering the ocean. After 100 years, the mass of carbon in the ocean will have increased from 80,000 Pg to 84,000 Pg, which is only a 5%, so the rate at which it releasees CO2 will increase by only 5%, which is small.

Yours, Kevan

7. PS. If you look at the equations for the two-reservoir CO2 model, you'll see two time constants: 17 years (residence and adjustment in the short-term) and 8200 years. The second one is the adjustment time and residence time of the oceanic system. Kevan

8. You're making a number of claims that are - as I understand it - simply wrong. The ocean may contain 80000PgC (I think it's more like 40000PgC, but - again - that's not really relevant) but the atmosphere does not exchange with the entire ocean; it interacts initially with the upper ocean, which contains something more like 1000PgC. So, your assumption that the ocean is a massive reservoir that is unaffected by an atmospheric is wrong. The transfer from the upper ocean to the deep ocean is much slower and is the main reason why the atmospheric CO2 concentration is expected to remain enhanced for thousands of years.

9. Daer TTP,

The Earth holds 64,000 kg of carbon-14. Only 800 kg is in the atmosphere. But 8 kg is created in the atmosphere every year. These are severely constraining observations for any model of the carbon cycle.

You claim the ocean contains only 40,000 Pg of carbon. Let us consider the implications of your claim. The concentration of carbon-14 in the deep ocean is 80% of atmospheric, or 0.8 ppt. Your ocean will contain only 32,000 kg of carbon-14. But there absolutely must be 64,000 kg of carbon-14 on Earth, and only 800 kg are in the atmosphere. Where is the missing 31,200 kg? Do you think it's in the biomass? The biomass concentration of carbon-14 is 99% of that in the atmosphere. So there must be 32,000 Pg of carbon in the biomass. But the residence time of carbon in the biomass is also constrained by the concentration of carbon-14 in the biomass: at 99% the atmospheric concentration, the residence time in the biomass is close to 80 years. That means 32,000 Pg of biomass carbon must be exchanged with the atmosphere every 80 years, or 400 Pg/yr, which would put the residence time of carbon in the atmosphere at 2 years, which is contradicted directly by the bomb test aftermath. So we have reached a contradiction.

The mass of carbon in the ocean must be 80,000 Pg.

Draw a line around the ocean and look at it as one big lump. If the one-lump model does not work, we'll throw it away. But it does work. Carbon from the atmosphere obviously must go through the top 1 km of water before it gets to the deep ocean. But that first 1 km of water transfers carbon to the layer below in proportion to its own concentration. It does not act as some kind of barrier. It's water that is mixing around. What goes on inside the ocean may be complex, but that won't make any significant difference to its outward behavior.

Yours, Kevan

10. Can I ask if you're familiar with the Revelle factor?

11. PS. Lets suppose you were correct when you said, "The transfer from the upper ocean to the deep ocean is much slower." We know that there must be a net flow of 8 kg of carbon-14 out of the atmosphere and into the ocean each year. You are saying this transfer takes place into the top 1 km of the ocean, but does not proceed into the deep ocean at the same rate. The top layer contains 10,000 Pg. The concentration of carbon-14 in the top 1 km of the ocean is 96% of the concentration in the atmosphere, so the top 1 km contains 10,000 kg of carbon-14. Each year 1.25 kg decays and another 8 kg of carbon-14 arrives from the atmosphere. The quantity of carbon-14 in your top layer is growing at 6.75 kg/yr. But that's not the case: the concentration of the top 1 km will be constant at equilibrium.

Thus the claim that "atmospheric CO2 concentration is expected to remain enhanced for thousands of years" cannot be true. The 8 kg of carbon-14 moves freely through the top layers of the ocean to the deep ocean.

Yours, Kevan

12. I'm still interested in whether or not you're familiar with the Revelle factor.

Also, your claim that carbon-14 in the top layer would grow at 6.75kg/yr is assuming that the flux from the ocean to the atmosphere never contains a carbon-14 rich CO2 molecule, which is clearly wrong.

13. Dear TTP,

Revelle Factor: Never heard of it before. Just looked it up. Very interesting. The PH of the ocean affects how likely CO2 is to be absorbed, and the concentration of CO2 in the ocean affects the PH. Thank you for that.

I should have said: to the first approximation, the CO2 absorption rate is independent of ocean concentration and temperature and weather.

The Revelle Factor responds to the log of the dissolved carbon concentration. The molar fraction of dissolved inorganic carbon changes by up to 50% for a -1 change in PH, or x10 increase in H+, or x10 increase in dissolved carbon. So it looks like a doubling of dissolved carbon concentration will cause no more than a 15% drop in the absorption rate into the ocean, and probably less than that. So, let's ignore it.

Yours, Kevan

14. Revelle Factor: Never heard of it before.
Yes, I guessed as much and thanks for being honest. If you're going to rewrite our scientific understanding of a topic such as this, can I suggest that you first familiarise yourself with our current understanding. You can always then prove it all wrong, but not actually knowing what it is, is not a particularly good way to do so.

15. Dear TTP,

"Also, your claim that carbon-14 in the top layer would grow at 6.75kg/yr is assuming that the flux from the ocean to the atmosphere never contains a carbon-14 rich CO2 molecule, which is clearly wrong."

You make a very good point. But I made no such assumption. Yes, carbon-14 is moving into the top layer of your ocean and out of your top layer. But the net flow into the top layer from the atmosphere must be 8 kg/yr.

Let G be the mass of carbon that moves into the top layer of the ocean each year, for the Earth in equilibrium. The rate at which it leaves the top layer is also G. The rate at which carbon-14 enters the top layer is G * 1 ppt and the rate at which it leaves is G * 0.96 ppt. We have:

G * 1 ppt - G * 0.96 ppt = 8 kg/yr

From which we find that G = 200 Pg/yr. In your 10,000 Pg per year model, we would have to have 25% of the Earth's carbon being exchanged with the top layer of the ocean each year and we would still have a build-up of 6.75 kg/yr of carbon-14 in the top layer. So that's not going to work, so far as I can see.

Yours, Kevan

16. Dear TTP,

Forgot to say: I'm getting the carbon-14 concentrations from a table here:

http://www.hashemifamily.com/Kevan/Climate/Dist_C14_Nature.pdf

As to your comment, "I suggest that you first familiarise yourself with our current understanding. You can always then prove it all wrong, but not actually knowing what it is, is not a particularly good way to do so."

Thank you. I know you mean well. But until you can actually point out a flaw in my calculations, you don't have standing to suggest changes to the manner in which I conduct myself as a scientist. As soon as you show that my model of the carbon cycle fails to fit the evidence, I'll be a keen student.

Right now, mine is the only carbon cycle model that fits the evidence. No other model is consistent with the carbon-14 observations. Please go ahead and show me I'm wrong. I would really appreciate it.

Yours, Kevan

17. Personally, I think that understanding the existing models is an important part of doing science. Science isn't done in a vacuum in which each scientist works without becoming aware of what others are doing. If you don't understand these other models, it's hard for you to claim that your does best.

Also, can your model explain the changes in ocean pH? My guess is that it can't, given that you're assuming that atmosphere exchanged instantly with the entire ocean, rather than just with the surface ocean.

18. Dear TTP,

I will explain to you the problems with the Gavin Cawley paper you mentioned. After that, I'm not going to waste my time dissecting any more of the existing non-functioning models.

Gavin assumes that the ocean concentration remains constant, and so assumes that the emission by the ocean, Fe, remains constant. He then asserts that the absorption into the ocean will not be proportional to the concentration of CO2 in the atmosphere, C, but rather it will be linear with C. That violates the fundamental principles of molecular chemistry: the probability that a gaseous CO2 molecule will be absorbed is a independent of the concentration. So his model begins with a parameter that violates the laws of chemistry.

Even if we set aside the fact that the model is based upon a violation of the laws of chemistry, he then proceeds to differentiates the atmospheric CO2 concentration as measured by an instrument on a mountain top and claims that the slope of this derivative, which is the second derivative of the concentration, is a fundamental property of the Earth's CO2 system.

All measurements have non-linear, systematic errors. When you take the derivative like that, you amplify those errors. Thus, if Cm is the measured C and Cc is the correct C at the ocean surface, we'll have Cm = Cc + a*Cc^2. Suppose:

Cm = C + C^2/100000

Then Cm measures 310 ppmv when it should measure 300 ppmv. We would never know the difference. But we can hope that Cm is correct to 10%. Now suppose that after 1940, Cc = 300 + (t - 1940) That is, Cc is increasing at 1 ppmv per year since 1940, when it was 300 ppmv. So now we have

Cm = 300 + (t-1940) + (300+(t-1940))^2/100000

And when Gavin differentiates he'll get:

dCm/dt = 1 + 600/100000 + 2*(t-1940)/100000

He fits his inverse a=adjustment time to 2*(t-1940)/100000, which is entirely a function of the small parabolic error in our measurement of Cm, and claims that is a fundamental property of the carbon cycle.

Yours, Kevan

19. Dear TTP,

Also, you say, "Science isn't done in a vacuum in which each scientist works without becoming aware of what others are doing."

Now that's an interesting statement. It's certainly true that it takes many people to do effective experiments. So measuring CO2 and temperature for the past 400k years takes a team of people, and so on. There's no doubt that we all work together to acquire observations of nature.

But when it comes to building an understanding of those observations, one person is perfectly adequate, and indeed it is usual for one person to do the theorizing and others around her to gather data.

You are claiming that I can't build an accurate model of the CO2 system because I have not looked at other people's models. That's not true. All I have to do is look at other people's measurements. And I have done that. My model fits the data.

Your argument against my model is not one based in calculation or observation, but one based upon a presumed authority held by a group of people writing climate models. I understand that you do recognize that authority, but please be assured that I do not, so it is a waste of your precious time to try to convince me that I'm wrong because someone else says I am. I will be convinced only by calculations or evidence.

Yours, Kevan

20. so it is a waste of your precious time to try to convince me
At least we agree about something.

I'm not suggesting you're wrong because others say you're wrong. I'm suggesting that you should at least understand the other models and I'm pointing out that your assumption that the atmosphere exchanges instantly with the entire ocean is probably incorrect; it initially exchanges with the upper ocean which contains far less CO2 than you assume is in that reservoir.

Also, you still haven't answered my question about whether or not you model can explain the changes in pH.

1. Dear TTP, What changes in PH? Please be specific. I would like numbers and calculations. Yours, Kevan

21. How often in science does an outsider (i.e. a researcher from another field) disprove the basics of some other field of research without first taking the time to properly understand the foundational work in that field? Practically never, I can't offhand think of a single example. Examples of people who think they have proven Einstein wrong and can't be convinced otherwise are much more common.

Of course you can make a model from first principles that fits SOME of the observations, but how do you know it fits the data for the right reason? How do you know there isn't some hidden factor that explains why Archer's model is correct? Or the model of Enting and Pearman that I mentioned (that does model the various isotopes independently and apparently does explain 14C reasonably well)? You don't, and you probably never will if you continue with this sort of attitude towards people who are trying to help you by discussing your assumptions.

Feynman said "The first principle is that you must not fool yourself — and you are the easiest person to fool." Do you think taking the time to properly understand the existing carbon cycle models makes it more difficult to fool yourself or less?

(Gavin Cawley)

22. Dear Dikran Marsupial,

May I say, that's an awesome alias. I'm going to try to respond only to calculations and measurements from now on. I was offensive to you once today already, and I don't intend to be rude a second time.

Yours, Kevan

23. BTW the idea that the CO2 measurements from Mauna Loa are so inaccurate that the annual growth rate is meaningless is absurd (and no evidence whatsoever was provided to support that assertion). This casual dismissal of an argument that you disagree with is exactly the sort of thing that Feynman was talking about.

24. BTW an apology that goes on to accuse someone of incompetence is a non-apology apology, and did you little credit. You have been offensive more than once today!

BTW regarding "bombastic and arrogant", I would say that

"Modern models of the CO2 cycle are presented in papers such as Archer et al., but these models are contradicted by carbon-14 observations, so they cannot be correct."

is hardly meek or modest, especially when it isn't actually true (e.g. the model of Enting and Pearman).

Suppose we have the Earth's atmosphere in equilibrium. We know such a state does not really exist, because the atmosphere is always changing and has always changed. So let's allow ourselves to assume that the rates of exchange are fluctuating by no more than 10% about a hypothetical equilibrium value. Now we can use carbon-14 observations, or whatever we have on hand, to estimate the quantities that are stable to 10%.

If we take the derivative of one of these quantities, the derivative will not be stable to 10%, but could fluctuate by 100%. If we want the derivative to be stable to 10%, we must have the other quantities stable to 1%. If we want to use the second derivative, our original rates of exchange will have to be stable to 0.1%.

In Gavin's paper, he considers what happens if we start adding CO2 to the equilibrium atmosphere. He takes the derivative of CO2 concentration and fits a straight line to this derivative. The slope of that straight line is the second derivative of the CO2 concentration, and Gavin uses that second derivative to obtain his adjustment time. (Please correct me if I have misread your paper, Gavin.)

A non-linearity in the Mauna Loa measurement of order 10 ppmv during the interval 300-400 ppmv will give you an error in that slope of order 0.01 which is about the size of the slope that Gavin finds in the data. A change of a few percent in the emission of CO2 from the ocean, which arises from a change of temperature during that period of time, or any other cause, could also give rise to that second derivative.

Therefore, the adjustment time Gavin derives has no physical significance. It is a model parameter whose meaning he has pre-supposed, but which he could fit to the data under any circumstances, whether the second derivative of CO2 concentration was due to the phenomena he describes or any other phenomenon.

In the case of the other models in the Archer et al. review, they are all wrong because it is impossible for a 1000 Pg pulse to disturb the CO2 concentration for more than 200 years, not without violating either the carbon-14 constraints, or diffusion constraints, or the assumption of constant temperature.

In Gavin's model, he argued that the rate of absorption into the ocean was not proportional to the concentration of CO2. It could be that the way these models are generating these 200-year time constants is by violating the laws of diffusion when they implement their ocean layers. From where I look at it, the rate of absorption is proportional to the concentration, and the rate of movement through the ocean also.

Another way to get the Earth's atmosphere to persist in elevated CO2 would be to assume that doubling CO2 warms the Earth by 2C, in which case CO2 will come out of the ocean and until the world cools down again, the CO2 concentration in the atmosphere will be higher. But I'm talking about a model of the carbon cycle at constant temperature, which we can then perturb with temperature to later.

40 Pg/yr into the deep ocean to cary 40 kg of carbon-14.
40 Pg/yr out of the deep ocean to cary 32 kg of carbon-14
Net flow 8 kg into the deep ocean.
Double the concentration in atmosphere, double the flow into the deep ocean.

Well, after a while, when the ocean concentration changes, there will be a second-order effect where acidification slows down absorption. But I estimate these effects to be minimal. (Please correct me with a calculation if you have time.)

At constant temperature, and staying within the laws of diffusion, and conforming to the carbon-14 observations, none of those models can be right.

Yours, Kevan

26. Dear Gavin,

I was working on a derivation of the Lorentz transform for some friends of mine, and it got me thinking. Observer A sees Observer B's clocks going slow, but Observer B says she sees the same thing the other way around: his clocks are going slow. They might argue about it, because it seems impossible that they could both be right. Happily, the Lorentz transform shows that they are both right.

It occurs to me that something like that might be going on between you and I. No matter whether I disagree with your paper or not, I greatly appreciate the fact that you also are interested in this subject, and have done me the honor of looking at my home-made model.

Yours, Kevan

27. "A non-linearity in the Mauna Loa measurement of order 10 ppmv during the interval 300-400 ppmv will give you an error in that slope of order 0.01 which is about the size of the slope that Gavin finds in the data"

But have you provided any evidence that there actually is a non-linearity of 10pmmv in the interval between 300 and 400 ppmv in the measurements taken at Mauna Loa? No, hence the objection remains specious. The idea that there was a calibration problem of this magnitude in the measurements seems rather absurd to me, given that the magnitude of the annual cycle is less than 10ppmv even at Mauna Loa.

Incidentally,

"Yes, that's [In a two-reservoir model, the adjustment time is equal to the residence time] exactly what I'm saying. It's not a coincidence. It's a feature of systems in which the rate at which a quantity changes is proportional to the quantity itself."

I think this is only true for a system where the flux is proportional to the quantity, but not for a system where the NET flux is proportional to the quantity (which appears to be the case for the ocean-atmosphere flux - Box 4.2 of "The Global Carbon Cycle" by David Archer, Princeton Primers in Climate).

28. Dear TTP, What changes in PH? Please be specific.
You're the expert here. Surely you know these numbers?

1. I do not know the numbers, and even if I did, I would still ask you to state them as you see them.

29. "But have you provided any evidence that there actually is a non-linearity of 10pmmv in the interval between 300 and 400 ppmv in the measurements taken at Mauna Loa?"

No, I have not. It's an example of the kind of error that is insignificant when you are looking at the measurement itself, but becomes significant when you take derivatives. I don't know what errors, if any, there are in the Mauna Loa measurements. It is for you to prove that all those other sources are negligible, not for me to prove that they are significant. At least: that's how it is in my business. My job is designing instruments that measure things.

"I think this is only true for a system where the flux is proportional to the quantity"

Yes.

"but not for a system where the NET flux is proportional to the quantity"

When you say NET flux, do you mean the difference between two fluxes, each of which is proportional to concentration in two parts of the system? I think that's what you mean, in which case, I agree. In the two-part system the solution is the sum of two exponentials. If the time constant of the two exponentials are similar then they interact.

In my two-part model, one time constant is 8200 years and the other is 17 years, so in the short term, it's one time constant. See below for derivation if you want proof.

http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-analytic-solution-to.html

Yours, Kevan

30. "No, I have not."

Then it is a specious criticism of the model, which you shouldn't have made without at least pointing out that you had no evidence of such a non-linearity. Do you think making specious arguments against models that disagree with yours makes it easier to fool yourself or more difficult?

Suggesting that there is an 10% non-linearity in the data that the research groups that collect the data are not aware of and haven't calibrated out is basically a suggestion that those research groups are incompetent. Rather like suggesting that carbon cycle models are falsified by observations that have been extensively studied is basically suggesting that the worlds carbon cycle modelling groups are also incompetent. I don't think either is very likely.

"It is for you to prove that all those other sources are negligible"

I'm sorry, but that really is an exemplification of the Feynman quote. It is for you to prove that the adjustment time doesn't depend on the stratification of the ocean, but you haven't, you have just shown it fits the 14C data over the last 60 years or so, but that doesn't mean you get a fit for the right reason. Again you are putting higher requirements on my evidence than you are on yours, which is a recipe for fooling yourself.

"When you say NET flux, do you mean the difference between two fluxes, each of which is proportional to concentration in two parts of the system? "

No, what I mean is that there are fluxes from the oceans to the atmosphere and from the atmosphere to the ocean that go on all the time and it is the difference between those fluxes that are proportional (in a "spherical cow" model) to the difference in partial pressures. As explained in the Archer reference I gave. Even when the difference in partial pressures is zero, there are still (very large) fluxes in either direction (which is why the residence time is short), it is just that they cancel out (so atmospheric CO2 levels remain constant). This is explained in my paper.

You say

"My job is designing instruments that measure things."

However you earlier said

"From your side: please be assured that it is extraordinarily rude in my field to argue from personal authority or authority of the published authors as you are doing. We don't do that. It's considered an admission of incompetence. "

no inconsistency there then? ;o)

31. I do not know the numbers, and even if I did, I would still ask you to state them as you see them.
Again, IMO, if you're going to make the kinds of claims that you are, then you should know these numbers. They're a pretty fundamental part of understanding the ocean uptake of CO2. Similarly, it's remarkable that you've made these claims without ever coming across the Revelle factor.

FWIW, here are some numbers. The Ocean pH has dropped from about 8.2 to around 8.1. That's a 25% increase in acidity.

If you do the full oceanic chemistry calculation (which I have done and have just gone and checked my code) a change in pH from 8.2 to 8.1 is associated with an increased in Dissolved Inorganic Carbon (DIC) from about 2000 micro-mol/kg to about 2060 micro-mol/kg. If you also include that we've also warmed by about 1K, then it is more like an increase from 2000 micro-mol/kg to about 2050 micro-mol/kg. In other words, the pH change is associated with a 2.5% increase in DIC.

In your calculation, you're assuming that the ocean contains 80000 PgC. About 25% of our emissions are in the ocean, which would be around 150 PgC. If this is all well-mixed within the whole ocean, that would be a 0.2% increase in DIC. That's 10 times smaller than is required to explained the observed change in pH. The reason is - as I've pointed out before - that the atmosphere initially exchanges with the upper ocean which contains for less carbon that you're assuming is in the reservoir.

32. Dear TTP,

Thank you. Yes, the atmosphere exchanges with the top layer of the ocean first, so a one-lump model of the ocean cannot predict changes in the top of the ocean. My two-part model shows how carbon moves between the atmosphere and the deep ocean. It does not tell us anything about the details of the route between the two. It assumes that this route carries carbon a a rate proportional to concentration, and it predicts how that rate will vary with temperature. If you would like a model with ten ocean layers, I suppose I could generate a spreadsheet with ten layers, all bound to fit the carbon-14 flow. Do you agree that a ten-layer model will have produce no significant difference in the exchange between the atmosphere and the deep ocean?

Yours, Kevan

33. Dear Gavin,

"no inconsistency there then? ;o)"

Good call. I did not mean it as an argument by authority, but I see it certainly looks that way, and I apologize.

What I meant was: it's perfectly reasonable for you to ask me to prove your measurements have errors, and it's perfectly reasonable for me to ask you to prove your measurements don't have errors. It's a question of protocol, which depends upon your working culture. In my working culture, it's up to you to prove you don't have errors. But I see nothing wrong with the culture being the other way round. After all, in a court of law, you are innocent until proven guilty. In my business, it's as if you are guilty until proven innocent.

So, if it's the other way around in this field, then fine: I will take a look at the annual sinusoidal component of the Mauna Loa measurement and see if it can produce a large enough error when you run it through a twelve-month running average filter.

"Even when the difference in partial pressures is zero, there are still (very large) fluxes in either direction (which is why the residence time is short), it is just that they cancel out (so atmospheric CO2 levels remain constant)."

I see. In that case: the adjustment time is the same as the residence time, because each flux is proportional to a concentration.

In the case of the ocean, there are various layers exchanging at different rates. The top layer may exchange faster than the bottom layer. The residence time of a carbon atom could be only 2 years in the atmosphere if we look at the exchange between the atmosphere and the top layer of the ocean. But the adjustment time of the atmosphere to a large disturbance could be seventeen years, because the 17-year time constant of the atmospheric exchange with the deep ocean dominates the 2-year time constant of its exchange with the top of the ocean.

In short: there can be many residence times and many adjustment times. It is perfectly possible for the physical residence time of a carbon atom to be 2 years in the atmosphere, while the longer-term adjustment time is 74 years, as in your paper. But if it's 74 years, then 74 years would have to be the residence time of carbon outside the deep ocean. Which it's not.

Yours, Kevan

34. Except, you're claiming that the residence time and the adjustment time are the same. As I understand it, this implies that it essentially mixes with the entire ocean instantly (i.e., there is no difference between the upper and deep ocean). Hence, I don't think your model can explain the observed pH change because the change in DIC in your model would be far too low.

35. "Good call. I did not mean it as an argument by authority, but I see it certainly looks that way, and I apologize."

No problem.

"What I meant was: it's perfectly reasonable for you to ask me to prove your measurements have errors, and it's perfectly reasonable for me to ask you to prove your measurements don't have errors."

That isn't what you did though, you didn't ask me if the measurements had errors, you raised the issue of a huge un-noticed non-linearity in the measurements to conclude:

"Therefore, the adjustment time Gavin derives has no physical significance."

"So, if it's the other way around in this field, then fine: I will take a look at the annual sinusoidal component of the Mauna Loa measurement and see if it can produce a large enough error when you run it through a twelve-month running average filter."

This is a goal-post shift, we were talking about a non-linearity in the measurement process, now you are talking about seasonal and inter-annual variability. If you read my paper, you will find that it demonstrates that there is considerably year-to-year variability in the annual growth rate (Figure 2), which is why the data don't all end up as a straight line, which is why the regression (Figure 6) must assume a noise process.

"I see. In that case: the adjustment time is the same as the residence time, because each flux is proportional to a concentration."

No, you don't see. The residence time depends on the magnitude of the flux out of the atmosphere, the adjustment time depends on the net flux out of the atmosphere. If you have non-zero steady state fluxes in and out of the atmosphere, the two things are not the same. This is explained in my paper.

36. Dear Gavin,

"The residence time depends on the magnitude of the flux out of the atmosphere, the adjustment time depends on the net flux out of the atmosphere."

Equations, please. Clearly, I am not understanding what you are talking about, and no, I am not going to go and read again about coins in a jar, I have a full teaching load today.

Yours, Kevan

37. Dear TTP,

"Except, you're claiming that the residence time and the adjustment time are the same. As I understand it, this implies that it essentially mixes with the entire ocean instantly"

Yes. In a two-part model, one assumes that both parts are homogenous, which is to say: perfectly mixed. If you want to explore how one part of the ocean behaves compared to another, we need more than one layer for the ocean. So I could build another model with three, four, or ten layers. It won't take me more than half an hour, but I have a full teaching load today, so can't do it today. But the two-part model shows us how the atmosphere responds to a pulse of carbon-14, to changes in temperature over the past 400k years, and to anthropogenic emissions. That was its purpose.

Yours, Kevan

38. The equation for the residence time is the definition given in my paper (on page three, before the coin analogy), I'll repeat it here:

"The exchange fluxes are very substantial, exchanging approximately 20% of the total atmospheric reservoir each year; hence the residence time, which can be calculated as the ratio of the mass of the atmospheric reservoir and the volume of the flux out of the atmosphere, is short 762/(92.2+122.6) ≈ 3.5 years. "

The equation for the adjustment time depends on the model used to estimate it. However given that it is obvious that the rate of change in atmospheric CO2 depends only on the difference between total emissions into the atmosphere and total uptake from the atmosphere and that is independent of the steady state exchange fluxes (which dominate the residence time), whatever the exact functional form of the dependence.

It is a shame that you disparage the coin analogy, analogies are often useful when someone has a conceptual block that leaves them unable to see their error in the actual system.

I also have a full teaching load, but I find it easier if the students are willing to do their homework (as I was when I learned the basics of the carbon cycle). Trying to prove the world's scientists wrong without learning the basics of their field is not a good idea.

1. Dear Gavin,

Do you agree with the following:

The solution to any linear differential equation is a sum of terms of the form A*exp(bt) where A is real, b can be complex, and t is the variable that we are differentiating with respect to.

In my model there are two terms, two values of "b". One is 17 years. The other is 8200 years. The time constant of 17 years is the time it takes for 63% of carbon atoms in the atmosphere to move into the ocean and be replaced from the ocean. It's also the time it takes the system to go 63% of the way to a new equilibrium point.

There is no need for us to talk about "residence time" and "adjustment time" because those terms are ambiguous. There will be one or more "time constants", b.

Yours, Kevan

2. "There is no need for us to talk about "residence time" and "adjustment time" because those terms are ambiguous"

No, they aren't, they are well defined concepts and well understood by the worlds climatology (although the terminology is often inconsistent, the concepts are not). There is no point in trying to write equations if you don't understand the system you are trying to model. The reason 14C observations are misleading is to do with the distinction between residence time and adjustment time. Until you grasp these concepts you will not be able to understand the source of your error.

39. I look forward to your model explaining the change in ocean pH.

All that Gavin is saying is that the residence time depends only on the rate at which CO2 leaves the atmosphere. The adjustment time, however, depends on both the rate at which it leaves the atmosphere AND the rate at which it enters the atmosphere (i.e., net rather than gross).

1. Indeed, which is why one (the residence time) depends on the "steady-state" fluxes and the other (the adjustment time) does not. Thus if you have a model where the residence time and the adjustment time are equal, that probably means the steady state fluxes are zero, which is not representative of the real world carbon cycle where the steady state exchange fluxes are very large.

2. I think that Kevan's model assumes that the flux out of the ocean remains constant (because it is assumed to have such a high C mass compared to the atmosphere that it is unaffected by an small increase in C mass) and that the flux out of the atmosphere scales with the total C mass in the atmosphere. Hence if you double atmospheric CO2 you double the flux out of the atmosphere, without any increase in the flux back into the atmosphere.

In your model you assumed that the net flux scaled with the enhancement in atmosphere concentration, which - I think - is because it should decay exponentially back towards its steady state concentration (which is also not quite correct, but not really relevant for this discussion). Maybe you can clarify something about this. My understanding of this is that this is because in reality an enhancement in atmospheric CO2 leads to an enhancement of flux out of the atmosphere, but also leads to an enhancement of the flux back into the atmosphere. Ultimately, we therefore expect the net change in flux to scale with the enhancement in atmospheric CO2, not with the total atmospheric concentration. Is that about right?

3. "I think that Kevan's model assumes that the flux out of the ocean remains constant"

I have a full analytical solution to the two-reservoir system, solved from the differential equations that express the dependence of carbon flow upon concentration in the source.

"My understanding of this is that this is because in reality an enhancement in atmospheric CO2 leads to an enhancement of flux out of the atmosphere"

Mathematical equations are the way to answer this question. I have the mathematical equations for my model here:

http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-analytic-solution-to.html

There is no need for you to speculate on what I'm doing in my model: it is expressed unambiguously in those equations.

Yours, Kevan

4. Firstly, there is nothing fundamentally wrong with describing something in words. In fact, it is regarded as quite an important part of describing what you're doing. Also, mathematical equations typically require that you define all your terms very clearly. Could you do so?

5. Dear TTP,

"I look forward to your model explaining the change in ocean pH."

Oh no, I'm not going to do anything with PH. I know nothing about it, while you know a lot.

I will, however, help you out in your investigation by building a spreadsheet that implements a 10-layer atmospheric and ocean system, and you can insert into the model the sizes of the reservoirs that you think are sensible, and the spreadsheet will make sure that all your choices fit the carbon-14 restrictions.

Then you can see if the model predicts PH concentrations. I think what you will end up having to do is adjust the top ocean layer size until it matches your PH observations, which would not be satisfying. So, before you play with the model, I suggest you write down here, for the record, how much carbon you think there is in the top-layer of the ocean. You don't have to decide the rate of exchange with the atmosphere, because that will be dictated by the carbon-14 flow, and I think we can use 0.96 ppt as the concentration of the top-layer carbon-14, because we have measured that.

Yours, Kevan

6. Yes, I think so (if I understand you correctly). The enhancement in atmospheric CO2 is relative to the level that is set by the dynamic equilibrium between all natural sources and sinks, so it is basically Le Chatelier's principle

"When any system at equilibrium is subjected to change in concentration, temperature, volume, or pressure, then the system readjusts itself to (partially) counteract the effect of the applied change and a new equilibrium is established."

so the change in the net flux will indeed scale with the enhancement in atmospheric CO2, rather than total atmospheric CO2.

This suggests that if we add CO2 to the atmosphere the natural sources and sinks will alter to oppose the increase to some extent. However this might be from natural sources weakening or sinks strengthening, or even sources strengthening but sinks strengthening even more (which is what seems to be happening, according to the IPCC reports).

I think it is worth pointing out that a simple proportionality is unrealistic and the true relationship is non-linear to some extent, so my model (actually it is an improvement on Essenhigh's model rather than strictly speaking mine) is best thought of as a local linearisation around current conditions, rather than a model of the carbon cycle under all conditions.

7. "This suggests that if we add CO2 to the atmosphere the natural sources and sinks will alter to oppose the increase to some extent."

Yes. The other sources will fill up, and so the rate at which they emit carbon back into the atmosphere will increase.

"I think it is worth pointing out that a simple proportionality is unrealistic and the true relationship is non-linear to some extent,"

Every relationship is non-linear to some extent, I'll grant you that. Just as the carbon dioxide concentration measurement will be non-linear to some extent. So we start with the linear relationship and, so long as we are strict about not taking the derivatives of the non-linear parts, we can hope to get a good model.

If the model makes several independent predictions that are correct, then we can trust it. Which is why I trust my model. It is impossibly unlikely that my model could predict the bomb-test aftermath, the CO2/temperature correlation in the ice cores, and the recent rise in CO2 concentration, and be wrong about the size of the ocean reservoir and the way it sucks carbon out of the atmosphere.

Yours, Kevan

8. "Every relationship is non-linear to some extent, I'll grant you that. "

in which case you will see why the intercept term in the regression model was necessary. If it were a simple linear system, the intercept would be zero, but that is inconsistent with the observations.

9. "If the model makes several independent predictions that are correct, then we can trust it. Which is why I trust my model. "

The Enting and Pearman model makes even more independent predictions that are correct. Should I trust it more than your model?

40. Dear TTP,

"The Ocean pH has dropped from about 8.2 to around 8.1. That's a 25% increase in acidity. "

To show you how little I know about ocean PH: I did not even realize the ocean was alkaline. All the talk of "ocean acidification" and I assumed the PH was 5 or 6. At 8 and dropping to 7, can you show me how to calculate the effect of acidification upon the probability of a carbon atom being captured by the ocean?

Yours, Kevan

1. What do you mean "can I show you"? Are you actually asking me to explain the various chemical processes associated with the uptake of CO2 by the oceans?

2. Dear TTP,

How do I know what I'm asking for? I don't know anything about ocean acidification. I don't even know enough to know what to ask for.

But you said up above that the Revelle thingy-bob would cause the absorption of CO2 from the atmosphere to be affected by the concentration of carbon in the ocean. So I assumed you knew what you were talking about, and could therefore educate me on the matter.

I did a rough calculation, but that was after fifteen minutes of reading. You have just revealed that you have spent many hours on this subject, so please enlighten me.

Yours, Kevan

3. I spent some time working through some of the material and I think I have a reasonable understanding of the basics. I don't mind spending some time explaining this to others, but you appear to be presenting a result that contradicts our current understanding, so I had assumed that you to would be familiar with this material. The basics involves Henry's Law, which relates the amount of dissolved CO2 in the ocean to the partial pressure of CO2 in the atmosphere. On top of that, there are various chemical reactions, such as the reaction of CO2 with water to produce bicarbonate and H+, and the dissociation of bicarbonate to form carbonic acid and H+. The various reaction constants have temperature dependencies, but one can combine all of this to show the various relationships between the partial pressure of CO2, pH, DIC, etc. I did write this up once, but it involved 3 pages of text and 18 equations, so I don't have the time to repeat it all here in a blog comment. There was a book that explained it all quite well, but I've forgotten the reference. If I remember it, I will post it.

41. ATTP wrote "Firstly, there is nothing fundamentally wrong with describing something in words."

More than that, it is very important, without describing the model in words, how do we know whether the model really represents reality? Equations can be correct in the sense of being internally consistent and giving the right result, but that doesn't mean they are a meaningful representation of reality, or that you got the right answer for the right reason. We can't determine the validity of the model assumptions just by looking at equations.

1. Only this morning, I said to my wife, "May the derivative of your happiness be positive." And she said to me, "Yes dear." And I said to my three sons as they departed for school, "Let C be the quantity of knowledge in your brains, and let t be time in hours, I hope that dC/dt is positive for you today, and look forward to hearing about the boundary conditions of your class experience." And they said, "Whatever Dad." So, you can see that it clearly works for me in my family, but I suppose all families are different. Kevan

42. Dear TTP,

"Also, mathematical equations typically require that you define all your terms very clearly. Could you do so?"

Yes, you'd have to start reading here:

http://homeclimateanalysis.blogspot.com/2015/09/carbon-14-origins-and-reservoir.html

Yours, Kevan

1. Surely you have somewhere where you clearly describe the terms in your equations?

2. http://homeclimateanalysis.blogspot.com/2015/10/carbon-14-carbon-cycle.html

Breaking off now. I will build the 10-layer spreadsheet model after TTP tells me how much carbon he thinks is in the top-layer of the ocean that controls acidification, but not before.

I will look into the demodulation effect of a 12-month averaging filter upon the change in the amplitude of the annual fluctuation in the carbon dioxide measurements after Gavin accepts that demonstrating that such an effect is significant to his 74-year result will constitute proof that there are many significant sources of error he has ignored. Right now, he's saying I'm changing the goal posts by looking into that particular error.

Yours, Kevan

1. I wrote "If you read my paper, you will find that it demonstrates that there is considerably year-to-year variability in the annual growth rate (Figure 2), which is why the data don't all end up as a straight line, which is why the regression (Figure 6) must assume a noise process."

Kevan writes

"Gavin accepts that demonstrating that such an effect is significant to his 74-year result will constitute proof that there are many significant sources of error he has ignored."

Clearly I haven't ignored it. Kevan clearly has been ignoring my responses to his comments though.

It is a goal post shift as the original contention was that there was an undiagnosed non-linearity in the measurements. Sorry, I no longer believe this is a good-faith discussion, so I'll leave it there.

2. I will build the 10-layer spreadsheet model after TTP tells me how much carbon he thinks is in the top-layer of the ocean that controls acidification, but not before.
I'm not that bothered one way or the other; I'm not doing this for my benefit. I have already told you a rough answer. To explain the observed change in pH the amount of DIC must increase by around 2.5%. This might be a slight underestimate, but I think it's not far off.

3. Dear TTP, What carbon mass should we use for the top layer of the ocean? Yours, Kevan

4. Dear Gavin,

I said IF there were a 10 ppmv non-linearity in the CO2 measurement THEN this would account for the slope of your dC/dt graph, and I showed that this was true. I did not show, nor did I claim, that such an error actually existed. I merely gave TTF and example of the kinds of error that are amplified by taking the derivative of a measurement.

Yours, Kevan

5. I don't know why you can't simply see if your model could produce a 2.5%, or so, increase in DIC, but if you consider the Revelle factor being about 10, then

10 delta_DIC/DIC = delta_pCO2/pCO2

The RHS is about 0.4, and I've already said that delta_DIC is about 50 micro-mol/kg, so that would mean that the DIC of the upper layer would be around 2000 GtC. Of course, we aren't actually in equilibrium, so this is approximate.

6. Dear TTM,

I can make a model that produces 2.5% increase in dissolved inorganic carbon, by choosing the parameters to make sure it produces that increase. The parameter required is the mass of the surface layer. So, if I pick that after I build the model, the model has no significance, because I could always make it work. Thus I was hoping you could say what the ocean top-layer is first, and then put that in the model and see if we get the right output.

Thanks for the equation, in which I'm guessing pCO2 is partial pressure of CO2. I will think about this, because I prefer to express the exchanges in terms of two flows, so I need to understand how to make the absorption by the ocean change correctly with concentration in the ocean.

Yours, Kevan

7. Actually, I didn't get that right. The Revelle factor tells you that if delta_pCO2/pCO2, then delta_DIC/DIC would be about 0.04 (i.e., DIC would be 25 times bigger than delta_DIC). If we assume that delta_DIC is all of the CO2 taken up by the oceans (150 PgC) then the total C mass of the upper layer would be 3750 PgC.

There are some caveats with this, though. Not all of the CO2 taken up by the oceans will still be in the upper layer. Also, the above assumes that it has reached equilibrium with an atmospheric CO2 concentration of 400ppm, which it has not. However, it's probably a reasonable rough estimate if you're looking for something to work with. I still don't quite get why you can't simply see if your model can produce an upper layer with a 3%, or so, increase in C.

Here's an additional question for you. Why don't you run your model for the period 1880 - 2016. You know the emission profile and you claim to know the adjustment time. If you did that, would you get an atmospheric CO2 concentration of 400ppm today?

8. "if delta_pCO2/pCO2 is 0.4"

44. Dear Gavin,

"The Enting and Pearman model makes even more independent predictions that are correct. Should I trust it more than your model?"

Thank you very much for prodding me again about that paper. I assumed the Enting and Pearman model was among those of the Archer et al. list, and therefore had to be wrong. But it's not, so far as I can see. And I did not bother looking the paper up until now. Furthermore, I failed to find it the first time I looked through the literature. But it's a great paper.

The Enting and Pearman model is a multi-layer version of the Arnold et. al. model (and my model is the same as the Arnold et. al. model). It has much more detail, and pays full attention to the bomb-test curve, which was of course unavailable to Arnold et al. in 1956. Their model has an empirical equation for the air-sea interface (equation 2.11) that models the temperature-dependence of the equilibrium, it has concentration-dependent flow into the deep ocean (equations 2.18 and 2.20) and and the deep ocean reservoir is vast (2.4 moles per m3 carbon is about 38,000 Pg) and communicates with the atmosphere at about 50 Pg/yr to get the carbon-14 down into the region where the concentration is low.

So, if they were to turn on 10 Pg/yr and see how long it took to double the CO2 concentration of the atmosphere, they would come up with around 3000 years, while my answer is 6000 years.

They get the temperature-dependence of the atmospheric concentration from equation 2.11, which is an empirical relationship not consistent with either the Van t'Hoff equation or the solubility of CO2 versus temperature. Their model has 31 free parameters, and they don't test the temperature-dependence. That was not their objective. Their model would predict half the temperature-induced CO2 concentration changes we see in the ice-core data. My model predicts those changes within 10% with temperature dependence derived from first principles, so I would trust my model more for the terperature dependence.

But that's a minor point. The major point is: this paper uses the same calculations as my model and Arnold et. al. It will come up with the same long-term predictions. Are you saying the Enting and Pearlman model is flawed, because it contradicts all the models in the Archer et al. summary? Or are you saying that you agree with Enting and Pearlman, in which case you agree with me?

Yours, Kevan

1. one last try:

"But that's a minor point. The major point is: this paper uses the same calculations as my model and Arnold et. al. It will come up with the same long-term predictions. "

But it doesn't. Check out figure 1.7 of the first IPCC WG1 report (the 1990 one), it is on page 15. Note the reference for the model used is Enting and Pearman. Curve (a) shows the decrease in atmospheric CO2 if all anthropogenic emissions ceased in 1990, and it shows that only about half (judging by eye) has been removed after 100 years. From the text (any typos mine):

"In case a (all emissions stopped; Figure 1.7), the atmospheric concentration declines, but only slowly (from 351ppmv in 1990 to 331ppmv in 2050 and 324ppmv in 2100 [DM: the baseline being just over 300 in 1950]), because the penetration of man-made CO2 to deeper ocean layers takes a long time. [DM: which is why you need a stratified model to match the stratified ocean]" [DM: comments mine]

Clearly the calculations in Enting and Pearman are not the same as yours, as they give different long term predictions. I suspect you can't see the difference because you don't yet understand the difference between residence time and adjustment time.

Note also that their model shows a doubling of CO2 could be achieved with emissions fixed at something like 2010 levels within 100 years (curve c'), which again contradicts your model.

BTW note that one of the contributing authors to the chapter is one G Pearman, so I rather doubt that Enting and Pearman's paper is significantly inconsistent with what is in that chapter of the IPCC report.

"Are you saying the Enting and Pearlman model is flawed, because it contradicts all the models in the Archer et al. summary?"

It doesn't contradict them, you only think it does because you don't understand the difference between residence time and adjustment time. The models used by Archer et al are similar to the models used by Enting and Pearman (because AFAICS most carbon cycle modellers use that sort of model), so I'd be very surprised if they greatly disagreed.

45. Dear Gavin,

I see the figure now. That's intriguing. I don't understand how they arrived at that response from the same model they wrote up in the 1986 paper, which followed the first thirty years of the bomb-test curve successfully with a time constant of around 15 years.

The Enting et al. model appears to be based upon linear differential equations, where each rate of transfer is proportional to the concentration of the source. The analytic solution to their multi-layer system will be a sum of exponential terms, some added and some subtracted. The equations governing the response to doubling the carbon-14 concentration in the atmosphere are identical to those that govern a doubling of total carbon concentration, because all the equations for carbon-14 movement are exactly the same as for carbon, and when we double the carbon-14 in the atmosphere, we are doing exactly the same thing, mathematically, as when we double the total carbon concentration.

Any model of linear differential equations that follows the bomb-test curve all the way down from 2 ppt to today's 1 ppt must, if it is implemented correctly, also give the same response for a pulse of 800 Pg of carbon, and in the case of stopping emissions at time zero, will show no significant residue after sixty years. That's why I am so confident that the papers in Archer et. al. are wrong.

I see five possible explanations for Figure 1.7:

(1) My argument above is wrong: somehow the solution for carbon-12 is different than for carbon-14.

(2) The IPCC simulation was run with an insufficiently small time step, and so failed to run correctly. I had lots of problems with my numerical model of just a two-reservoir system. I had to solve the analytical equations in order to make sure it was running correctly. Enting et al. do not present an analytic solution, and I can't say I blame them, with six layers. But they would have been unable to check their simulation against an analytic solution.

(3) The IPCC implementation of the Enting model uses different parameters from the Enting paper of 1986.

(4) The Enting model, despite my presumption, would not follow the entire 60-year bomb test curve if we ran it now, but instead would show carbon-14 lingering after the first 30 years, in the same way that the Figure 1.7 graph shows carbon lingering after 100 years. If so, the model is wrong, which is also surprising, because it appears to have carbon-14 going at the correct rate all the way down into the deep ocean.

(5) Despite my presumption, the Enting paper is not a set of linear equations, but has exchange rates that are not proportional to source concentration. In that case: I was too quick to celebrate the paper as realistic. But I can't see where they have done any such thing.

I'm guessing you're going to pick (1). The thought of being this dramatically wrong about math is rather exciting: I will learn something. But I just can't see it myself. So please help me out, if you think it's (1).

Yours, Kevan

1. The problem is that you haven't done your homework and don't have a sufficient understanding of the system you are trying to model (e.g. Revelle). However as you are so resistant to advice, I don't have the time or energy to help you any further. The best place to start is to acknowledge that you haven't done your homework and learn about the carbon cycle by reading the papers and understanding them before going back to the modelling. What you are doing on this blog is great, but as ATTP was advising, science doesn't happen in a vacuum and you need to understand other peoples models to understand why they don't agree with yours.

46. Dear TTP,

"Here's an additional question for you. Why don't you run your model for the period 1880 - 2016."

I did that, but I did not post it. I put in human emissions and the model says CO2 should be 70 ppmv above where it was in 1900. And I think we are now at 90 ppmv above 1900, but I'm not sure, because we don't have a continuous CO2 measurement from 1900 to now, and I figured if I posted about it I would have to get into the issue of how to estimate CO2 before Moana Loa. So instead I did a step increase of 10 Pg/yr from human emissions and plotted the atmospheric concentration afterwards, starting from 300 ppm, which you see here:

http://homeclimateanalysis.blogspot.com/2015/12/carbon-cycle-with-ten-petagrams-per-year.html

So there, it goes up to 380 ppmv after about thirty years in my model. That's pretty close to 400 ppmv we see today, compared to 300 ppmv (maybe) in 1900. I know there are some cogent arguments being made that the 100 ppmv increase is natural, but my model says otherwise. It looks like its almost all man-made.

Yours, Kevan

1. Your model leads to a change from 300ppm to 400ppm if we emit 10PgC per year for 100 years. That's a total of 1000 PgC. Since pre-industrial times, we've emitted a total of about 600PgC and atmospheric concentrations have increased from 280ppm to 400ppm. Would seem your model underestimates the rise in atmospheric CO2. Conversely, if you used the known emission profile, you wouldn't end up at 400ppm if you started at 280ppm.

2. You are absolutely right: when I use the known emissions profile and keep the temperature of the system constant, and start at 300 ppmv, the concentration rises only to 380 ppmv. If I add in a 1C rise in temperature, the model adds another 12 ppmv.

I'm not at all confident in the pre-industrial value of 280 ppmv. I would give that a +-30 ppmv margin of error, which is why I did not want to post about it. And when I picked the value I thought best, I looked at a bunch of measurements and picked 300 ppmv, not 280 ppmv. I am not an expert on the actual measurement process, so have no way to determine which measurements are reliable and which are not.

In any case: my model does not account for natural changes other than the temperature effect, and I assume CO2 can vary by +-10% in a century, as it appears to have done leading up to 1950.

So, I'm not claiming the model can do any better than 10% (+-30 ppmv). And I'm certainly not going to tweak its parameters to make it fit the data better. Thank you for checking the numbers.

47. Dear TTP,

Putting your comment and correction together we have.
If delta_pCO2/pCO2 = 0.4, then delta_DIC/DIC = 0.04. And carbon mass of top layer around 2000 Pg. I have to go through your comments in order and make sure I understand what you have said, but thank you very much for your explanation.

Yours, Kevan

48. Dear Gavin,

Another possibility:

(6) They allow the temperature of the Earth to change in response to CO2, and when emissions stop, we are at a new equilibrium point, which takes a while to go away. That would be combining a CO2 cycle model and a CO2-causes-warming model.

Do you think that's likely? Surely they would have said something about it.

Yours, Kevan

49. Dear Gavin,

A linear system obeys the principle of superposition, and the carbon-14 equations are simply the carbon behavior multiplied by 10^-12, and with the decay rate added in, and some minor changes in concentration from 100% to 80% going down into the ocean. But over 200 years, the decay is only 2.5%, so to within 2.5%, the bomb-test is the impulse response of any carbon-cycle model that consists of linear differential equations and predicts the bomb-test curve. No matter how many layers it contains, no matter how complicated, the model is, if it's linear, it's impulse response defines its behavior entirely and completely.

Therefore, the Figure 1.7 is not the response of a correctly-implemented linear model that fits the bomb-test curve. Nor are any of the models in Archer et. al.

If the Figure 1.7 model is non-linear, this should be described in the Enting et al. paper, and I'm just not seeing it. In any case, if it's non-linear, then I'm confident it's wrong: over a factor of two increase in concentration, there is no way that the non-linearity in a diffusion system could grow to 50% of the response. Diffusion is one of the most linear processes in the natural world.

So my best guess is that whoever ran the simulation of the Enting et al. model for Figure 1.7 did it wrong.

Yours, Kevan

October 19, 2016 at 11:01 PM

1. Sorry, if you keep going back to the idea that it is the experts who have got it all wrong and not you, you will never learn anything. I suspect you will find that the person that did the model run was either Enting or Pearman or both (note that Pearman was a contributing author of the chapter, so if the model had been run incorrectly then I am sure he would have noticed). Sadly I think you are beyond help at this point, which is a shame as you are obviously a bright guy.

50. Dear Gary,

It would really great for you and your career if it turned out that I was right and you were wrong, because you could then write a paper with me and it would a big deal (you would do all the work, of course). And if you prove me wrong, well, you'd have my thanks, but that's about it. You wasted your time on another upstart.

Your faith in the competence of your colleagues gives you confidence to say I'm wrong. But I have the principle of superposition, which gives me absolute certainty that your experts are wrong, incredible though it may seem. When you eliminate the impossible, the improbable must be true.

The behavior of any system governed by linear differential equations is entirely described by its impulse response. This may seem incredible, but it is true. It is the basis of all fields of engineering. Your field of study runs models of linear systems without checking that they obey the principle of superposition. They are the ones who are flagrantly ignoring the work done by a century of scientists. It is your field that is in the minority. It is you and your colleagues who are the upstarts who think you can re-write the mathematics of physics and engineering.

But that's fine with me: go for it. But please try to see the opportunity that stands before you. You have forced me, with your splendid debating skills, to think about these things more clearly. This realization that the superposition theory is the answer to these faulty models comes from the debate with you and TTF. It is not something I came up with on my own. So, you can take ownership of your share of it, or you can leave it on the table. Either way: thank you.

Yours, Kevan

1. You don't need me to write the paper, I encouraged you on the other thread to write it up and send it to the journal - go for it. The problem isn't with the maths, it is with the physics, you don't understand all of the processes involved (and you have ignored my attempts to point them out to you) so the maths doesn't match all of reality, just bits of it. Sadly it looks like your hubris is getting in the way of you seeing this.

2. BTW this isn't a debate, it is discussion of the science where ATTP and I have tried to help you see where the flaws in your model lie, but you have been too unreceptive to criticism to take it on board.

"This realization that the superposition theory is the answer to these faulty models comes from the debate with you"

no, that is all your invention, and sadly, I don't think it is true (note an approximation of the full Bern model is often used, which is a linear DE, which obviously conforms to the principle of superposition, you would know that if you did your homework).

51. Correction: "Dear Gavin," not "Gary". Yours, Kevan

52. Dear Gavin,

The Bern model does not predict the bomb-test aftermath, so it's wrong. No linear model can predict the bomb-test aftermath and have a pulse response that lasts more than fifty years. This is a consequence of the superposition principle.

No scientist uses argument by authority. Science is based upon observations of nature, not worship of the word of experts. And yet argument by authority is all you have. Any time you try to debate me on actual numbers and calculations, you lose.

You could be a scientist, if you could swallow your pride, stop using argument by authority, and actually answer my point about the superposition principle. You will lose the debate, of course, but at least you will lose it with honor, like a scientist, not like a religious zealot shouting, "Heretic!"

Your field has struck out on its own, inventing its own math, its own physics, making predictions fifty years in the future that nobody can prove wrong. When one expert runs a faulty model, none of you know how to test it. None of you have any will to question authority. Once you make a mistake it is impossible for your field to recover, because any result that conflicts with your mistake you dismiss with your blind faith in your own infallibility.

The climate-modeling field has done nothing whatsoever for humanity, other than make vague predictions about the future that scare people. There is nothing you do that could not be done by someone walking in of the street. Anyone can make vague predictions about the future that will forgotten by the time they get proved wrong. All it takes is convincing yourself you are right, and shouting it out loud enough, and some people will be fooled by your confidence. You use the jargon of science as a costume to give you authority, in the same way that hermits don't bathe to make themselves look holy.

For every one of you, there are a thousand people like me. I can design and build a radio out of transistors. No layman has any chance of doing that. Physics, medicine, biology, chemistry, physics, these are all fields that have build the reputation for science that you exploit by masquerading as members of our community.

Your field has been able to deceive the public by exploiting the reputation science has earned through the efforts of people like me. You are not the first pseudo-scientific field to do this, nor will you be the last. Fields like yours are like tumors that grow inside the wonderful creature that is the scientific community. I am an antibody from the scientific community, doing my part to try to shrink you until you are harmless. I am here because I know that your experts are wrong, and that your field is cancerous.

Yours, Kevan

1. "For every one of you, there are a thousand people like me. I can design and build a radio out of transistors. No layman has any chance of doing that."

So what makes you think that you [in this context a layman, just as I am] can build a better model of the carbon cycle than those who have had the specialist training, without learning even the basics of the carbon cycle actually works (e.g. Revelle)?

As it happens, I am not a carbon cycle modeller or environmental scientist. I am an electronic engineer (I can also design a radio, although a computer would be easier in my case), that now teaches in a computer science department. However, I have self-skepticism, so if I think some research in another field is incorrect, I assume that is because I am probably missing something (or I don't understand it), so I take the trouble to understand why they do what they do the way they do. I do my homework. If I still see a problem, I ask questions to make sure. That is how to avoid the Dunning-Kruger effect. So when I read about Prof. Essenhigh's argument about atmospheric CO2, I made sure I understood his model, made sure I could get the same results as he did, discussed it with him via email, and sought help from carbon cycle specialists before publishing my paper on why Prof. Essenhigh was wrong.

Kevan writes "All it takes is convincing yourself you are right, and shouting it out loud enough, and some people will be fooled by your confidence."

Kevan earlier wrote "But I have the principle of superposition, which gives me absolute certainty that your experts are wrong," [emphasis mine]

http://rs848.pbsrc.com/albums/ab49/norbrookc/Fail/irony-meter_zps6a643b0b.jpg~c200

"Your field has been able to deceive the public by exploiting the reputation science has earned through the efforts of people like me. You are not the first pseudo-scientific field to do this, nor will you be the last. Fields like yours are like tumors that grow inside the wonderful creature that is the scientific community. I am an antibody from the scientific community, doing my part to try to shrink you until you are harmless. I am here because I know that your experts are wrong, and that your field is cancerous."

This suggests some "motivated reasoning" going on, rather than dispassionate, rational scientific investigation.

2. Kevan,
You really should read your comment back to yourself while also reminding yourself of what you're suggesting for others.

53. "The Bern model does not predict the bomb-test aftermath, so it's wrong."

Again, this is because you don't understand the distinction between residence time and adjustment time. The Bern model is predicting the response of the climate to anthropogenic emissions, which depends on the adjustment time. The aftermath of the Bomb 14C depends on residence time.

The Second IPCC WG1 report even has a section on this (section 2.1.4 Bomb Lifetime vs. Perturbation Lifetime, starting on page 85). There has been lots of work on understanding this over the years, the section ends:

"Because of the different dynamic behaviour 14C can not be taken as a simple analogue tracer for the excess anthropogenic CO2, a fact that has long been known to the carbon cycle modelling community (e.g. Revelle and Suess, 1957; Oeschger et al. 1975; Broecker et al.,1980)."

Note Revelle cropping up again. Of course, had you done your homework, it would have been common knowledge for you as well. If you we able to take criticism on board you would have learned it from this discussion, but you haven't. There is no better way to convince yourself you are right than by ignoring criticism.

1. I should add that through this discussion I have discovered that the calculation of the true residence time from the 14C observations isn't quite as straightforward as I had thought. It is still mostly telling you about residence time though.

2. residence time = adjustment time in a linear model with a single dominant time constant. In order for the adjustment time to be different, the model has to be non-linear.

3. TTF: Your comment contains argument from incredulity, also not valid in scientific debate. Please restrain yourself.

4. Are you referring to me? If so, which comment?

5. Here, "You really should read your comment back to yourself while also reminding yourself of what you're suggesting for others." You claim that, because my actions do not make sense to you, they are nonsensical, and that if I were to think about them, I would change my mind.

6. No, I'm suggesting that you are doing what you suggest the rest of us avoid doing. You're appealing to some kind of authority (yourself) as an argument against what is being said. If you're going to continue to give advice about how scientists should behave, while not following your own advice, this is all going to be rather pointless.

54. Dear Gavin,

"As it happens, I am not a carbon cycle modeller or environmental scientist. I am an electronic engineer"

My mistake, I assumed you were in the climate modeler field. I retract my statements about you as a professional, and please accept my apologies. Writing a paper with me would in no way benefit your career, just as it would not benefit mine.

In so far as you use argument by authority, you are not acting like a scientist in this debate. You are welcome to keep telling me I don't understand things, but I'm not convinced.

To you and TTF: Stop using argument by authority on my blog and you will never hear it from me again. So far you have been unable to do so. You revert to it whenever you cannot answer my questions.

How about this: if you say anything on my blog it is strictly about observations and calculations, and never an expression of incredulity that I question the orthodoxy of climate science.

Yours, Kevan

55. Actually Fig 2.8 of the SAR shows that the Bern model actually does predict the bomb 14C aftermath, it is just the predictions of the removal of pulse of excess carbon into the atmosphere (what Archer et al are discussing) isn't it. Again this shows you just don't know enough about the carbon cycle to see where your misunderstanding lies. Unfortunately this seems a classic example the Dunning-Kruger effect.

For anyone else still reading: The difference between the two is that there is nothing to distinguish anthropogenic carbon in the atmosphere from "natural" carbon. So if the exchange fluxes between the atmosphere and oceans/biosphere swap a molecule of anthropogenic CO2 in the atmosphere with a "natural" molecule from the oceans/biosphere, that has no effect whatsoever on the atmospheric concentration. This is why the characteristic timescale on which excess carbon is removed from the atmosphere (the adjustment time) depends on only the difference between total uptake and total emissions and not on the magnitude of the exchange.

14C on the other hand is distinguishable from 13C or 12C, it is isotropically labelled. Thus if a molecule of 14C02 in the atmosphere is swapped with a molecule of 12CO2 or 13CO2 from the oceans/biota, the concentration of CO2 in the atmosphere remains the same, but the isotopic ratio falls. The rate at which this happens depends on the magnitude of the exchange between the atmosphere and the oceans/biosphere, and is essentially the residence time. If the oceanic/biospheric reservoirs are large by comparison with the atmosphere (they are) then they will stay isotropically the same for a long time, which means more 14C is replaced by 12/13C than vice versa. This isn't rocket science, it is carbon cycle 101 material.

1. See below for clarification, the phenomena you are describing are possible only in non-linear models.

56. Regarding argument from authority, you have used ad-hominems, e.g. "ields like yours are like tumors that grow inside the wonderful creature that is the scientific community. ", which is just the flip-side of argument from authority, so perhaps you need to look a bit more careful at your own arguments.

As it happens, we are not using argument from authority. We are not saying that the science is necessarily right because that is what the experts say. The point is that the experts may be wrong, but (and this is a bit but) if you want to try and prove them wrong, you need to first understand why they hold the position that they do, because it is probably that a layman (as both of us are) have gaps in our knowledge that if filled would resolve the disagreement. This is common sense, and why self-skepticism is a skill that scientists really cannot do without.

Also both ATTP and myself have pointed out the particular flaws in your method (the maths is fine, its just the maths doesn't describe reality), but you have ignored that as well.

1. Please see effort to clarify below.

57. "How about this: if you say anything on my blog it is strictly about observations and calculations, and never an expression of incredulity that I question the orthodoxy of climate science."

O.K. Do you accept that Fig 2.8 of the IPCC SAR shows that your assertion "The Bern model does not predict the bomb-test aftermath," is factually incorrect, and that the Bern model does actually predict a rapid decline in 14C over the course of a few decades?

1. You told me the BERN model was linear and that it had carbon lingering in the atmosphere past 200 years. Those two facts imply that the BERN model does not follow the bomb-test. So, either the BERN model is non-linear, or it has no carbon lingering after 200 years. I don't know.

2. Do you agree that the fact Figure 2.8 shows different rates for the dissipation of an excess of CO2 and the isotopic ratio suggests there mat be some physical process that affects one but not the other that you don't know about?

3. Just to be clear:

"You told me the BERN model was linear and that it had carbon lingering in the atmosphere past 200 years. Those two facts imply that the BERN model does not follow the bomb-test."

This is a non-sequitur because an excess of carbon lingering in the atmosphere for 200 years does not mean that the isotopic ratio perturbation from the bomb tests will linger for that long. Making such an inference is an indication that your understanding of the carbon cycle is substantially incomplete.

4. "This is a non-sequitur because an excess of carbon lingering in the atmosphere for 200 years does not mean that the isotopic ratio perturbation from the bomb tests will linger for that long."

Yes it does, provided the BERN model is linear and correctly implemented. I can prove this using the superposition principle, or you can prove it yourself, see my post (1)-(7) below.

5. Gavin: I see you deleted one of your comments. I permanently deleted it without looking at it, and will do so for all other corrections you want to make in the future. Yours, Kevan

6. "Yes it does, provided the BERN model is linear and correctly implemented."

... unless there is some physical process you are unaware of that means the two rates are different. There is, it was pointed out to you by ATTP in the very first comment. It is a non-sequitur, as clearly demonstrated by Figure 2.8 of the IPCC SAR, which show the two things are expected occur at different rates and the Bern model's prediction of the isotopic ratio is "in accord with the observations".

7. At equilibrium (i.e. the concentrations of CO2 in the surface ocean and atmosphere at the same - we can ignore the biosphere for the moment for the sake of clarity), what are the fluxes from the ocean to the atmosphere and from the atmosphere into the ocean?

8. I assume you mean the concentration of CO2 in the surface ocean and the atmosphere are "constant with time", right? The concentration is not "the same". Well, maybe it is, I did not have any need to calculate the absolute concentration in the ocean in units of CO2 mass per ocean mass.

Assuming you mean: constant with time, then the top layer of the ocean has 96% carbon-14 concentration of the atmosphere. We need to move 8 kg of carbon-14, atmospheric concentration is roughly 1.0 ppt, so we have:

Flux into top ocean = 8 kg / 1 ppt / (1.00 - 0.96)
= 200 Pg/yr

So that's in both directions. There are 800 Pg of carbon in the atmosphere (very roughly) so 63% of carbon atoms in the atmosphere will move into the top layer of the ocean in 4 years.

If we start adding 10 Pg/yr to the atmosphere, and if we could assume that the top layer of the ocean were infinitely large, it will take 4 years for the 200 Pg/yr to increase to 210 Pg/yr, which is brought about by an increase in atmospheric CO2 concentration of 5%.

Here we have a contradiction because the bomb-test curve shows the impulse response time constant is not 4 years. It's 15 years. The top layer of the ocean is not infinite.

I am confused. Let's clear the slate and start again, so I can try to understand where exactly we disagree. Which of the following statements do you dispute?

(1) A linear model of the carbon cycle obeys the principle of superposition.

(2) If the numerical implementation of a linear model does not obey the principle of superposition, the implementation is wrong.

(3) Suppose A is a linear model of the carbon cycle. Multiply all concentrations by 10^-12, and in each reservoir multiply by the relative carbon-14 concentration, and add terms to each reservoir to show the decay of carbon-14 with time constant 8000 years. Now you have a linear model, B, of the carbon-14 cycle. The impulse responses of A and B will each be a sum of exponentials. The first fifty years of the impulse responses of A and B will be dominated by one or more exponentials with time constants less than 200 years, and these time constants will be the same to within 20% of those in B. (Carbon-14 decay is 1% in 50 years and maximum concentration difference atmosphere to ocean is 20%.)

(4) The linear carbon-14 model must predict the bomb-test curve, which means its dominant behavior in 50 years is a total relaxation with time constant roughly 15 years.

(5) The dominant behavior of the carbon model in the first 50 years must be a relaxation with time constant roughly 15 years +-20%.

(6) No linear model of the carbon cycle can have atmospheric carbon concentration elevated by more than 1% fifty years after an 800 Pg pulse.

(7) The only way for a carbon cycle model to have a pulse response that endures more than 50 years is for it to have non-linear diffusion that dominates the linear diffusion.

Which of those statements do you require me to prove to you?

Yours, Kevan

1. As I keep saying, the error isn't in the maths but in your lack of understanding of the physics of the carbon cycle. I have already explained this several times on this thread. The paper I wrote is precisely concerned with the issue that you are missing. The problem is orthogonal to your points (1)-(7).

2. That's argument by authority. Please restrain yourself.

3. I clearly do not understand what you mean by residence time and adjustment time, you are using different definitions from me, which is fine, but irrelevant.

If you accept all of (1)-(7) above, then (7) stands, and were done here, right? No linear model can behave like the model in your Figure 1.7 from IPCC and also match the bomb-test results, so either the Enting paper is non-linear, or it was improperly implemented for IPCC. QED.

4. "That's argument by authority. Please restrain yourself."

Rubbish. To which authority did I appeal? Certainly not mine as I have already explicitly pointed out that on this topic I am a layman, just like you are.

"I clearly do not understand what you mean by residence time and adjustment time, you are using different definitions from me, which is fine, but irrelevant."

If you don't understand what I mean by these terms then perhaps you shouldn't be dismissing them as irrelevant until you do.

"No linear model can behave like the model in your Figure 1.7 from IPCC and also match the bomb-test results"

Demonstrably incorrect, see Fig 2.8 from SAR.

"either the Enting paper is non-linear, or it was improperly implemented for IPCC"

or you don't understand the difference between residence time and adjustment time, which explains the apparent inconsistency, and are unwilling to find out, perhaps because of your views of the carbon cycle modellers:

"Fields like yours are like tumors that grow inside the wonderful creature that is the scientific community. I am an antibody from the scientific community, doing my part to try to shrink you until you are harmless. I am here because I know that your experts are wrong, and that your field is cancerous."

Perhaps it is you that needs to restrain yourself and accept that maybe you don't know all you need to know about the carbon cycle to model it meaningfully, but others do.

5. When you say "Your lack of understanding of the carbon cycle" you are claiming someone else understands it better than me. That is argument by authority. You attempt to dismiss my mathematical argument by saying it can't be true because I don't understand the carbon cycle. Please restrain yourself.

Which of the above (1)-(7) do you dispute? If you dispute none of them, the debate is over. The following statements are all consequences of (1)-(7).

(a) The impulse response of any linear climate model that predicts the bomb-test curve will be dominated by a single exponential term with time constant close to 15 years.

(b) The step response of any linear climate model, being the integral of the impulse response, will have a time constant of close to 15 years also. All adjustments will be compete to within 1% within 50 years.

No definition of "residence time" or "adjustment time" affects the above two conclusions. The only way to have a model predict an impulse response that takes longer than 50 years to go away is to give the model non-linear terms that dominate the linear terms. In that case, you you have twice as many parameters to choose, and agreement with the bomb-test curve tells you nothing at all about the validity of the non-linear terms responsible for predicting century-long impulse response.

6. "You attempt to dismiss my mathematical argument by saying it can't be true because I don't understand the carbon cycle. "

No, I am saying that your maths doesn't represent the real world because it doesn't make the correct distinction between residence time and adjustment time. That is an identifiable technical flaw with your model. The fact that you don't understand the difference is the reason you have difficulty seeing the flaw in your model, even after it has been pointed out repeatedly. That is not an appeal to authority any more than you saying the IPCC have implemented the Enting and Pearman model incorrectly (despite Pearman being a corresponding author of the chapter!) is an appeal to your authority.

Lets go through this step by step "At equilibrium (i.e. the concentrations of CO2 in the surface ocean and atmosphere at the same - we can ignore the biosphere for the moment for the sake of clarity), what are the fluxes from the ocean to the atmosphere and from the atmosphere into the ocean?"

59. I see you asked this above also, reposting here for clarity.

I assume you mean the concentration of CO2 in the surface ocean and the atmosphere are "constant with time", right? The concentration is not "the same". Well, maybe it is, I did not have any need to calculate the absolute concentration in the ocean in units of CO2 mass per ocean mass.

Assuming you mean: constant with time, then the top layer of the ocean has 96% carbon-14 concentration of the atmosphere. We need to move 8 kg of carbon-14, atmospheric concentration is roughly 1.0 ppt, so we have:

Flux into top ocean = 8 kg / 1 ppt / (1.00 - 0.96)
= 200 Pg/yr

So that's in both directions. There are 800 Pg of carbon in the atmosphere (very roughly) so 63% of carbon atoms in the atmosphere will move into the top layer of the ocean in 4 years.

If we start adding 10 Pg/yr to the atmosphere, and if we could assume that the top layer of the ocean were infinitely large, it will take 4 years for the 200 Pg/yr to increase to 210 Pg/yr, which is brought about by an increase in atmospheric CO2 concentration of 5%.

Here we have a contradiction because the bomb-test curve shows the impulse response time constant is not 4 years. It's 15 years. The top layer of the ocean is not infinite. It starts to fill up quickly. But you will note that the time for 63% of atmospheric carbon atoms to move into the ocean remains 4 years, even though the impulse response is 15 years.

The two-reservoir model lumps the top and deep oceans together, and we get a time constant of 15 years. In 15 years, 63% of atmospheric carbon atoms will make it into the deep ocean. In this two-part model we don't have a parameter to represent how long a carbon atoms spends in the atmosphere before entering the top of the ocean. The two-part model treats the path between the atmosphere and the deep ocean as a diffusion path with flow in each direction proportional to the concentration at one end.

60. "We need to move 8 kg of carbon-14, atmospheric concentration is roughly 1.0 ppt, so we have:"

"If we start adding 10 Pg/yr to the atmosphere, "

please don't obfuscate by digressing and give a straight answer to the question. At equilibrium (for example in the pre-industrial carbon cycle), what are the fluxes of CO2 from the atmosphere into the oceans and from the oceans into the atmosphere (Pg/yr).

61. You are using argument by authority by suggesting that I am obfuscating and digressing. Please restrain yourself.

I answered your question. The flux is 200 Pg/yr from the atmosphere into the top layer of the ocean, and 200 Pg/yr back again, see calculations above.

1. I did ask if we could go through this step by step, which is easier if you just answer the question and do not digress.

Right, so you would agree that this flux of 200 Pg/yr in either direction has no effect whatsoever on the total amount of CO2 in the atmosphere, but it does govern the rate at which the bomb 14C pulse decays?

62. I'm not sure I understand what you mean by "has no effect whatsoever". By assumption, we are at equilibrium, so nothing is changing the CO2 concentration. If the 200 Pg/yr remains the same in both directions, then it cancels, and so does not cause any change in the CO2 concentration. But that can't be what you are asking. Please clarify.

No, the bomb 14C pulse curve is not determined by the 200 Pg/yr. It is determined by the rate at which carbon flows into the deep ocean, which is only 40 Pg/yr.

1. "If the 200 Pg/yr remains the same in both directions, then it cancels, and so does not cause any change in the CO2 concentration. But that can't be what you are asking."

Yes, of course that is what I was asking. Not all questions are difficult, but the best way forward is often to get explicit agreement on the basics, so we have a foundation on which to build.

"No, the bomb 14C pulse curve is not determined by the 200 Pg/yr. It is determined by the rate at which carbon flows into the deep ocean, which is only 40 Pg/yr."

No, this is incorrect. Imagine that the surface ocean reservoir was vast compared to the atmospheric reservoir, that it initially contains no 14C and the rate of exchange with the deep ocean was infinitely slow (none of which is true, but extreme case thought experiments are often useful). Assume the atmosphere initially contains no 14C, but at t=0 a pulse of 14C is injected which is too small to affect the fluxes into and out of the atmosphere (e.g. 1kg). What happens to the isotopic ratio of the atmosphere (or the 14C concentration, if you prefer)?

2. You start with no carbon-14 in the system. The top layer of the ocean is infinite. Let the pulse be 800 kg of carbon-14 delivered into an atmosphere of 800 Pg, so we jump from 0 ppt in the atmosphere to 1 ppt in the atmosphere.

We now have 200 Pg/yr entering the infinite ocean, which carries with it 200 kg/yr of carbon-14. Because the ocean is infinite, the carbon-14 concentration in the ocean remains 0 ppt. No carbon-14 returns into the atmosphere with the 200 Pg/yr of carbon that leaves the ocean and enters the atmosphere. Thus the 800 kg pulse of carbon-14 is removed from the atmosphere with a time constant of 4 years.

3. yes and is completely independent of the rate rate which carbon flows into the deep ocean (which in this case is 0 Pg/yr), do you agree?

4. In your example, the top layer of the ocean is infinitely deep. It is the deep ocean. The flow into that deep ocean is, in your example, 200 Pg/yr, and that flow into the deep ocean dominates the carbon-14 removal.

5. No it isn't infinitely deep, there is another layer below it. I also didn't say it was infinite, just vast (e.g. 10 times the size?). The top layer is still connected to the deep layer, just the transport between them is so slow as to be negligible. As I said we are investigating a limiting case of a two-box ocean, but it is still a two box ocean.

6. To avoid this, lets assume the surface ocean is 100 times the capacity of the atmosphere, but it sits on top of an infinite deep ocean, and the exchange between them is 1KgC/yr. What happens to the 14C now? (pretty much the same as before, but this time it can't be avoided by claiming that the surface ocean is the deep ocean).

7. If it's ten times the size as the atmospheric reservoir, then it qualifies as a deep ocean, doesn't it? How much bigger does it have to be for you to call it "deep"?

Anyway, what happens if the top layer is 2000 Pg and the atmosphere is 800 Pg? We release 800 kg of carbon into the atmosphere. Please show me.

8. It seems to me what you are saying is: the carbon-14 removal is dominated by the nearest reservoir that is ten or a hundred times the size of the atmosphere. Yes, that's right. On Earth, that reservoir is the deep ocean, not the top layer, which is only a few times larger than the atmosphere. So: please calculate for me.

9. Oh, let's do this too: calculate what would happen if we doubled the carbon concentration in the atmosphere in your example. We double the atmospheric carbon to 1600 kg from 800 kg. Now we have 400 Pg/yr going into the top layer that is 100x as big as the atmosphere, and only 200 Pg/yr coming out, so atmospheric carbon concentration relaxes with time constant 4 years.

In every example, let's do both, so we can see that they are always the same, or very close to the same.

10. Correction: double to 1600 Pg from 800 Pg

If the top layer is 2000 Pg, the same thing happens for carbon and carbon-14, as I am sure you can confirm for yourself when you present me with your calculation.

11. "If it's ten times the size as the atmospheric reservoir, then it qualifies as a deep ocean, doesn't it? "

No, it doesn't, you are just trying to evade the line of argument. The surface ocean is the part that is in direct exchange with the atmosphere (hence "surface"). The deep ocean is the bit below the thermocline, where the waters stratify and don't mix very much.

"It seems to me what you are saying is: the carbon-14 removal is dominated by the nearest reservoir that is ten or a hundred times the size of the atmosphere. Yes, that's right. On Earth, that reservoir is the deep ocean, not the top layer, which is only a few times larger than the atmosphere. "

No it isn't, it is dominated by the nearest reservoir (note withput the qualification of size). This (for a waterworld) is the surface ocean. They couldn't actually be any nearer given that they share a boundary!

"calculate what would happen if we doubled the carbon concentration in the atmosphere in your example. We double the atmospheric carbon to 1600 kg from 800 kg. Now we have 400 Pg/yr going into the top layer that is 100x as big as the atmosphere, and only 200 Pg/yr "

No, there has been no substantial change in the total amount of carbon in the atmosphere (1600kg is negligible compared to the 750Pg it currently contains), so the fluxes would remain the same at 200 Pg/yr in either direction. But I'm glad we agree that the relaxation time is still 4 years.

We have made some progress here as 4 years is the mainstream estimate of the residence time (the average time a particular molecule of CO2 remains in the atmosphere), and the 200 Pg/yr is a reasonable estimate for the pre-industrial exchange flux between the atmosphere and ocean/biosphere.

So we can see that the rate uptake into the deep ocean doesn't affect the rate at which the isotopic ratio falls. What happens if we make the exchange between the surface ocean and the deep ocean 1000 times bigger (i.e. 1000Kg rather than 1Kg)?

As to equations, there is no point in writing equations until we reach an understanding of what the physical processes are that we are representing with the equations. Patience please.

12. I should have said "Pg", not "kg". So let me clarify.

What happens if we double the carbon content of the atmosphere from 800 Pg to 1600 Pg? The rate into the top layer increases from 200 Pg/yr to 400 Pg/yr, while the rate coming out remains the same, at 200 Pg/yr. So the carbon mass in the atmosphere drops at 200 Pg/yr, relaxing back to 800 Pg with a time constant of four years.

The carbon-14 does the same thing when we add it into the atmosphere in your example: it disappears with a time constant of 4 years.

Even that small amount of carbon I entered by mistake would disappear with the same time constant. For example, 1600 kg added to 800 Pg is an increase in concentration of 2x10^-9, which, when we multiply by 200 Pg/yr increases the flow into the top-layer of the ocean by 400 kg/yr, and so the 1600 kg disappears with a time constant of 4 years.

I don't understand your question, "So we can see that the rate uptake into the deep ocean doesn't affect the rate..." In your model, the rate was 0 kg. Now you want to increase it to 1000 kg of carbon-14 each year? Please clarify.

In any case: the system you have described has the same relaxation time for carbon and carbon-14, provided we use a linear model for diffusion. The same is true for any system of reservoirs you care to propose, and I'm happy to do the calculations to prove this to you, within limits of time, of course.

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15. Ah, since you mentioned Kg I assumed we were still talking about 14C (which we haven't finished discussing), but it appears that you have gone off on another digression again (despite me asking you not to). Do you want to go through this step by step or not?

If you keep reiterating your conception of the carbon cycle, rather than paying attention to what I am trying to explain to you, we won't get anywhere.

Now, if we take our thought experiment waterworld, and run the same simulation, but this time the transfer rate between the surface waters and the deep ocean is larger by a factor of 1000, (i.e. 1000 Kg/yr), what happens to the atmospheric 14C?

16. Dear Gavin,

When you say "gone off on another digression" you are arguing from authority. You presume to know where this discussion is going, and so you presume to know what is a "digression" and what is not. Please restrain yourself.

You do not have the authority to forbid me to perform calculations on my own web site. Nor do you have the authority to decide which questions can be asked and which cannot be asked.

The behavior of the carbon in the carbon cycle you proposed is pertinent to the discussion, because we are debating my claim that the behavior of carbon-14 and carbon-12 are almost identical in any linear carbon cycle model. As you propose new models, therefore, it seems to me our top priority should be to seek out any differences between the carbon-12 and carbon-14 behavior.

You proposed a two-reservoir system. You asked me to calculate what would happen if we put a pulse of 800 kg of carbon-14 in the atmosphere. I did so. I also calculated what would happen if we put 800 Pg of carbon in the atmosphere. In both cases, the disturbance relaxes with time constant 4 years.

Before we proceed with a new model, I suggest that we agree upon how the first one behaves. Do you agree that both the carbon-12 and the carbon-14 have the same behavior in your first model? If you do not agree, please show me where my calculation was wrong.

Yours, Kevan

17. "When you say "gone off on another digression" you are arguing from authority."

You seem to like that phrase, but again your are using it in a specious manner. I suggested we go through this step by step and asked politely for there to be no digressions (so that I could set out the argument clearly). There is no authority there, just a request that you have repeatedly ignored.

Now I am not forbidding you to do calculations on your blog, I just asked you to let me set out my argument step by step so that I could do so clearly and make sure no confusions arose. Being able to identify the things we agree on as we go is an excellent way of narrowing down on the exact point where we disagree.

Now of course if you don't want to help me explain my position and would rather hamper my efforts then that is fine, but it isn't very polite, rational or scientific behaviour. Your blog, your choice.

"Before we proceed with a new model, I suggest that we agree upon how the first one behaves. Do you agree that both the carbon-12 and the carbon-14 have the same behavior in your first model?"

again by jumping ahead (we have not discussed 12C injections yet), you are obfuscating attempts to discuss this correctly. We haven't even fully uncovered the 14C response yet.

But to answer your question, the response to a perturbation of the isotopic ratio is not the same as the response to a perturbation of the mass of the atmospheric reservoir, because one causes a change in the carbonate buffering system in the oceans and the other (the former) does not (as it doesn't change the amount of carbon in the oceans). Hence the rate constants are different. This is the Revelle effect, which has been pointed out to you several times already on this thread.

As I said, the equations you have are fine, but some of the physics are missing, so it represents a "spherical cow" carbon cycle, not the one we actually have.

Now are you willing to go through my argument step by step? If not that is fine, I'll leave it there.

18. Dear Gavin,

Splendid. We are now in agreement.

You agree that only non-linear effects (like the Revelle effect) can cause the carbon-14 behavior to be different from the carbon-12 behavior.

If we build a linear model of the carbon cycle, its carbon-12 and carbon-14 predictions will have the same impulse and step responses. No linear model of the carbon cycle can predict the bomb-test curve and have a carbon pulse response that lingers for more than fifty years.

The model in Enting and Pearman is linear. If it fits the bomb-test curve, it cannot follow path (1) in Figure 1.7 of the IPCC 1990 document, in which our carbon emissions linger in the atmosphere for more than fifty years. That figure, which was used to advise government policy, is wrong.

All the models in the Archer et. al. paper, and the BERN model, predict our carbon lingering in the atmosphere for more than fifty years. They must all be either non-linear, or they don't fit the bomb-test curve.

Now, we can talk about the absurdity of diffusion models dominated by non-linear effects at a later date. Until then, thank you for sticking with this, I look forward to debating with you in the future.

Yours, Kevan

19. O.K. it is obvious that Kevan is going to continue obstructing my attempts to explain the physics to him. Given his earlier meltdown it is pretty clear he is unable to accept that he is wrong for reasons that have nothing to do with the science.

This is a great example:

"The model in Enting and Pearman is linear. If it fits the bomb-test curve, it cannot follow path (1) in Figure 1.7 of the IPCC 1990 document, in which our carbon emissions linger in the atmosphere for more than fifty years. That figure, which was used to advise government policy, is wrong."

It can't possibly be that Kevan has made a mistake, it must be the IPCC, when one of the authors of that chapter was Pearman himself, who would know whether his model could produce Figure 1.7 or not. Pure hubris.

63. What happens if the top layer of the ocean has a mass of 2000 Pg and an initial concentration of 0.96 ppt, while the atmosphere has a mass of 800 Pg and an initial concentration of 2.00 ppt?

1. one step at a time please, if we keep digressing away from the central point, we will never get there (the above thought experiment was dealing with one digression already, but it looks to have been perhaps a profitable one).

64. Dear TTP,

I doubt you are still listening, but if so, I would like some clarification on the Revelle Factor. You say:

10 * delta_DIC/DIC = delta_pCO2/pCO2

The solution to this equation is:

pCO2 = constant * DIC^10

That is: the partial pressure in the atmosphere is proportional to the dissolved inorganic carbon content raised to the tenth power. In order to double the dissolved inorganic carbon content, we must increase the atmospheric CO2 concentration by a factor of 1024. I can find no evidence of this unusual tenth-power relationship in the literature, nor any empirical evidence. So far as I can tell, it would be rather hard to make a carbonated drink if this law were the case.

Meanwhile, Henry's Law, which is the go-to partial pressure law in Chemistry, states that:

pCO2 = constant * dissolved_carbon_concentration

Henry's law is in good agreement with all physical measurements I can find tabulated or graphed in the literature.

Did you type the equation incorrectly? Did I miss some a correction?

Yours, Kevan

1. Henry's Law applies to the relationship between atmospheric CO2 and dissolved aqueous CO2, not to total Dissolved Inorganic Carbon. As you say below, there are various reactions that convert aqueous CO2 into H+ and bicarbonate and then convert bicarbonate into carbonic acid. So not all of the Dissolved Inorganic Carbon in the ocean in the ocean is in the form of aqueous CO2.

The Revelle factor really tells you the relationship between a fractional change in atmospheric CO2 and a fractional change in total Dissolved Inorganic Carbon. Essentially, a 10% change in pCO2 is associated with a about 1% change in DIC.

To be clear, this is not my expertise, so I've only got this from reading up on it and asking others. I don't claim absolute accuracy.

2. TTP: Thank you for your continued assistance. I'm looking at Figure 5.7 here:

http://lawr.ucdavis.edu/classes/ssc102/Section5.pdf

We have dissolved CaCO3 to complicate the problem. For Log(p_CO2) from -13 to +2 the dissolved CaCO3 remains constant. The sum of the three other carbon species appears to be proportional to p_CO2, to the extent that I can add log concentrations visually. There is a peak in the CO3(-2) concentration, for example, that corresponds to an dencrease in slope of the HCO2(-1) concentration as this latter species takes over being the majority species. At p_CO2 400 ppmv, or log(p_CO2) = -3.4, the slope appears to be close to 7/7 = 1.0. So it seems to me that Henry's Law applies to the partial pressure of CO2 and the amount of carbon dissolved minus the amount that we put in with CaCO3.

What is interesting, and I'm guessing what you are interested in, is the fact that the pH of the solution changes with p_CO2.

65. CO2 Dissolved Concentration versus Partial Pressure
--------------------------------------------

When CO2 dissolves in water, it takes several forms.

http://lawr.ucdavis.edu/classes/ssc102/Section5.pdf

"Dissolved carbon is distributed among three species H2CO3, HCO3 and CO3 as a function of pH"

Figure 5.1 shows H2CO3 is dominant for pH below 6, HCO3 from 6 to 10, and CO3 for 10 and up.

Figure 5.4 is a log-log plot of the concentration of the three species versus CO2 partial pressure above pure water, where the CO2 above the water is mixed with gases that do not affect the pH of the water. The pressure is in atmospheres. At 400 ppmv the partial pressure of CO2 in the atmosphere is of order 0.0004 atmospheres, the log of which is -3.4. The pH of this solution will be about 6 (the log of the H+ concentration, slightly acidic). The dominant dissolved species is H2CO3.

The total dissolved carbon concentration we can obtain by adding the three concentrations together, which is pretty much the same as taking the largest one on a log plot. The total dissolved carbon concentration is increasing linearly with partial pressure.

If we add something alkaline to the water, the pH will rise. In the ocean, the pH is around 8.1, slightly alkaline. At pH = 8.1, the dominant species of dissolved CO2 is no longer H2CO3, it is HCO3.

According to the ionic equilibria derived in the above chapter, if were were to remove from a solution all the members of one species, the concentration of that species would soon be restored. The three species must exist in a certain ratio at any given pH.

At pH 7.0, for very low concentrations of dissolved CO2, the concentration increases linearly with partial pressure of CO2 above the water.

There is no mention in the above chapter of any deviation from Henry's Law when the pH of the solution reaches 8.1, even though their graphs of activity versus pH span pH from 4 to 12.

A "Revelle factor of 10" means the slope of the log-log plot of total dissolved concentration versus partial pressure would go stop rising with slope 1.0 in the region of log(concentration) = -3.4 and instead rise only with slope 0.1, then start rising again at higher concentrations to return to Henry's Law after its brief deviation.

After a few hours searching this evening, I have been unable to find any such plot of concentration versus pressure for pH = 8.1, nor any empirical source of the assumption that the slope of the graph will be 0.1.

1. There's an online app that does this

http://biocycle.atmos.colostate.edu/shiny/carbonate/

I can't seem to get it to actually work, but the tabs include a description and the code.

66. I could not get the app to work either. The rabett post is about the temperature-dependence of Henry's constant. I calculated that from first principles here:

http://homeclimateanalysis.blogspot.com/2016/01/carbon-cycle-effect-of-temperature.html

And used it to show how, according to my linear, two-part carbon system model, temperature drives the changes in CO2 concentration over the past 400k years:

http://homeclimateanalysis.blogspot.com/2016/01/carbon-cycle-correlation-between.html

As to the concentration of dissolved carbon increasing with partial pressure of carbon, I don't feel confident that I really understand the graphs in that UC Davis chapter. It could be that one of the constraints they put on their fourth-order ionic balance equation is that the dissolved CO2 gas (as distinct from the dissolved CaCO3 carbon) concentration increases in accordance with Henry's law at a fixed temperature, in which case the graphs themselves do not prove that this assumption is correct.

67. Another variable in regards to the increase in CO2 is the increase in growth rates of algae and terrestrial plants.

CO2Science.org catalogs experiments involving the growth rates of plant under increased CO2 conditions. The effects of raising CO2 by an extra 300 ppm can cause a dramatic increase in plant growth rate.

Per an article at CO2Science, an increase in sea algae growth would increase the emission of cloud forming sulfur compounds.

I am still reading through your site (when I can spare the time). There is a lot there to absorb.

1. I'd like to see measurements of algae growth rate and CO2 concentration in air and water.

68. If you're interested, I wrote these posts. The first presents the basic chemistry associated with dissolving CO2 in seawater, and the second presents some basic analysis and includes some python scripts that you can download, if you wish.

https://andthentheresphysics.wordpress.com/2016/10/30/ocean-co2-uptake/

https://andthentheresphysics.wordpress.com/2016/11/02/ocean-co2-uptake-part-2/

1. Thank you. I am studying your two posts.

I have a three-layer linear carbon cycle model for you, I will play with it some more before I upload it. It turns out that the carbon mass of the top layer of the ocean (carbon-14 concentration 96% of atmospheric) is already constrained by the shape of the bomb-test response: about 350 Pg so far as I can figure, otherwise the clean-out of carbon-14 would be sharper in the first ten years. Anyway: top layer of the ocean carbon concentration rises in proportion to atmospheric, with a response time of a few years.

69. ATTP: Left the following comment on your blog, but don't see it there, maybe something went wrong.

Thanks for your posts. I note that your calculation of the effect of temperature upon atmospheric CO2 is consistent with mine: you have 9C rise causing 280 ppmv to 400 ppmv, I have 12C causing a 40% increase, and the ice-core CO2-temperature measurements show 12C causing 190 ppmv to 300 ppmv.

In your first post, you appear to be assuming that Henry's Law applies only to the CO2(aq) species, but I could be mis-reading you. To clarify, for pure water and CO2 we have:

pCO2 = k * ([CO2(aq)] + [H2CO3] + [HCO3-] + [CO3--])

where k is a constant and pCO2 is the partial pressure of CO2 gas above the water, and I'm using square brackets for concentration of each species. If we dissolve CaCO2 in the water, we have have:

pCO2 = k * ([CO2(aq)] + [H2CO3] + [HCO3-] + [CO3--] - [Ca++])

I would call the concentration on the right side of each equation the "dissolved CO2 concentration", but you may be using that phrase in a different way, to refer only to the H2CO3 concentration, in the graph at the top of this post. I can see why you would interchange the two names: I have read a dozen chemistry chapters on this subject in the past week, and they interchange the terms H2CO3*, H2CO3, CO2(aq), Dissolved CO2, and Total Carbon Concentration. I think are lax about these terms because CO2(aq) is the dominant species in systems of pure water and CO2.

The Revelle Factor appears to express the fact that most of the carbon in the ocean is not from atmospheric CO2, but from dissolved CaCO3 and other sources. You have the Revelle Factor around 10 for the ocean today. If the total carbon mass in the top layer of the ocean is 2000 GT, then only 200 GT of that carbon is dissolved CO2 taking part in the CO2 cycle. Is that right?

I note that dissolved CaCO3 does not affect the rate at which CO2 molecules strike the ocean surface and are dissolved (an exothermic reaction 19 kJ/mol), nor does it change the rate at which CO2 emerges from the ocean by thermal excitation (endothermic 19 kJ/mol). The concentration of CO2 in the atmosphere will be in proportion to the mass of dissolved CO2 in the ocean. If we double the CO2 pressure in the atmosphere, the rate at which CO2 dissolves will double. Thus I see no means by which the CaCO3, nor any change in pH, can affect the linearity of the carbon cycle's diffusion equations. Do you agree? If not, can you show me some equations that express the non-linearity generated by the ocean chemistry?

70. I am certainly no expert on this subject, but here are a few quotes from by blog regarding the Revelle Factor and why I think it's incorrect:

"Take note there is no time-variable in the Revelle Factor formula (ΔPCO2ml/PCO2ml)/(ΔDIC/DIC) and the total amount of CO2 water can absorb based on that formula remains eternally constant over time until the relative concentrations of DIC change. Hence if the deep-ocean had the same DIC ratio as the surface-ocean the total amount of anthropogenic CO2 the whole ocean would absorb at equilibrium would only be 10%, in violation of Henry’s law. Once again, Henry’s law governs the solubility of gases in water and states that at a given temperature the amount of a gas dissolved in water is directly proportional to its partial pressure in the air adjacent to the solvent at equilibrium. The law can be described mathematically as: p = kHc. Where p is the partial pressure of the gas above the solute, kH is the proportionality constant (i.e. Henry’s constant) and c is the concentration of dissolved gas in the liquid. The constant of proportionality for CO2 at the average surface temperature of 15°C gives us a partitioning ratio between the atmosphere and the oceans of 1:50 respectively. If the Revelle Factor were correct and the solubility of CO2 changed as the relative concentrations of DIC shifted (which occurs when the partial pressure of CO2 changes) then kH in Henry’s law (and thus CO2’s partitioning ratio) would not be a constant for a given temperature. Note that Henry’s constant (in the equilibrium state of the law) is the ratio of the partial pressure of a gas at the liquid interface with the concentration of that gas dissolved in the liquid. Hence the constant does not change with concentration. It is a linear law. This means that the partitioning ratio of a gas (including that of CO2) is unchanged by changes to the atmospheric mass and can be multiplied up proportionally for any specified concentration in ppmv. Obviously this is in conflict with the Revelle Factor which suggests that the solubility of CO2 is affected by the relative concentrations of DIC as the partial pressure of CO2 changes".

Regarding the Revelle Factor Bolin et al (1959) states: “Less than 10% of the excess fossil CO2 in the atmosphere should have been taken up by the mixed layer. It is therefore obvious the mixed layer acts as a bottleneck in the transport of fossil fuel CO2 into the deep sea”. This bottleneck inhibiting the transport of anthropogenic CO2 to the deep-ocean would appear to be at odds with the removal of anthropogenic 14CO2 from the atmosphere after the 1963 nuclear test-ban treaty. These tests doubled the concentration of 14CO2 in the atmosphere above its natural equilibrium level. The observations show a half-life for 14CO2 of 10-12 years (Figure 16). Equilibriation would therefore essentially be complete (by 94%) after 4 half-lifes = 40-48 years. Considering that the combined amount of carbon in the soil, vegetation and surface-ocean is ∼3,400Gts (according to the IPCC in AR5) and the total amount of carbon in the atmosphere is ∼800Gts then if such a bottleneck existed in the surface-ocean the concentration of 14CO2 in the atmosphere would have stabilized at around 23% (i.e. 800/3400). With only about 4% of 14CO2 remaining in the atmosphere today it implies that the 14CO2 has become mixed with a reservoir 25 times larger than the amount of CO2 in the atmosphere and the only place that much CO2 is known to exist (and be in exchange with the atmosphere) is in the deep-oceans. Interestingly the residence time of 14CO2 from the nuclear bomb-tests has been measured longer than that for 12CO2. This may be due to the fact that the extreme heat from the nuclear explosions ejected a large portion of 14CO2 into the stratosphere where it has a 5-8 year delay for its transfer to the troposphere".

Fantastic posts by the way Kevan.

71. This comment has been removed by the author.

72. Segalstad has a good paper that explains some of the issues with the Revelle Factor. I may as well just link it instead of quoting large chunks out of it: http://www.co2web.info/ESEF3VO2.htm

73. CHIP,

I have a lot of respect for Henry's Law. It is widely used in chemical engineering. But consider this: suppose a gas dissolves into two species in a liquid. One species can convert back into the original gas, but the other cannot. So long as the two species remain in the same proportion, the rate at which the gas is emitted by the liquid will be proportional to the amount of gas dissolved in the liquid. There is a Henry's Law constant and Henry's Law is obeyed.

But suppose we add acid to the solution, or make some other chemical change, and as a result, the two species no longer occur in the same proportion, but one is now rarer than the other. The rate at which the gas is emitted will change, because the concentration of the species that can emit gas has changed, even though the combined concentration of the two species remains the same. The Henry's Law constant changes, which means the solution is not obeying Henry's Law.

The promotors of the Revelle Factor are claiming that the atmosphere-ocean system is in such a transitional state: as we increase the partial pressure of CO2, the ocean pH changes, and the distribution of carbonate ions changes rapidly, resulting in a deviation from Henry's Law.

The question is: why do they think the ocean is in this state? It has taken me a while to figure out how they came to this conclusion, but I think I have now figured it out. I have added an update here to explain:

http://homeclimateanalysis.blogspot.com/2015/12/carbon-14-probability-of-exchange.html

But I'll summarize here. If you take the CO2-water system at CO2 partial pressure 0.0003 atmospheres, and nitrogen for the rest of the gas above the water, its pH is 5.8. But it follows Henry's Law. This system is not a good model of the ocean, because the ocean pH is 8.2. If, however, you add OH- to this system (by dissolving NaOH, for example) until the pH = 8.2, then hey presto, you have something with CO2 and water and the right pH and that exhibits an extraordinary transition of ionic concentrations as you increase the partial pressure of CO2, so that, far less CO2 is dissolved than you would expect from Henry's Law.

Alternatively, you can model the top layer of the ocean as a saturated solution of CaCO3 with CO2 above at 0.0003 atmospheres, and this system, without any modification, has pH = 8.5, so it's very close to the real ocean. It also has the advantage of being realistic: the top layer of the ocean is indeed saturated with CaCO3 most of the time. This system does not undergo any dramatic changes in carbonate concentrations from 100 ppmv to 10,000 ppmv CO2 in the atmosphere, and therefore does follow Henry's Law.

Now, why Climate Science would choose the OH-CO2-water model rather than the CaCO3-CO2-water model is anyone's guess, but I'm going with the CaCO3-CO2-water model for my carbon cycle.

As a result, I claim the carbon cycle is linear, and by the principle of superposition, whatever carbon-14 does in the system, carbon-12 must do also.

Yours, Kevan

1. “I claim the carbon cycle is linear, and by the principle of superposition, whatever carbon-14 does in the system, carbon-12 must do also”.

That follows logically and I agree. It is plausible that some small absorption preference would arise on the basis of different isotope frequencies, but not to the dramatic extent that the IPCC would have us believe. Regarding the Revelle Factor, there is so much confusion surrounding it. Whenever anyone tries to get to grips with the actual ‘science’ behind the Revelle Factor it always turns eventually into a kaleidoscope of technical images that have no coherent logical relationships and which leave the enquirer no clearer or better-informed. Different papers have different interpretations of how the Revelle Factor operates. The argument by Segalstad is similar to yours. He states that: “Current global carbon cycle models are made to fit the assumption that the level of CO2 in the pre-industrial atmosphere was about 280ppmv and that due to a ‘Buffer Factor’ the ocean can remove only about 10% of the atmospheric CO2 added by man’s activities (e.g. Siegenthaler and Oeschger, 1987). This ‘Buffer Factor’ was calculated by assuming that the chemical interaction of atmospheric CO2 is limited only to the reactions CO2 <-> HCO3 <-> CO32 in the 75 meters thick upper ocean layer and by neglecting other seawater species and buffer systems, and by assuming that CO2 removal will be limited to this upper layer”. He points out that: “In the current global carbon cycle models the last partial chemical reaction is neglected: CO2(g) + H2O + Ca2+(aq) <-> CaCO3(s) + 2 H+. Any additional CO2 entering the ocean from the atmosphere will have the potential of precipitating calcium carbonate according to the Principle of Le Châtelier (average ocean depth 3.8 km; average calcite saturation depth 4 km)”. The paper goes into more detail and is available to read here if you’re interested: http://www.co2web.info/np-m-119.pdf

2. Kevan,

I agree that the response of any increase of CO2, be it 14CO2 or 12CO2 is (more or less) the same and that the response of the (deep) ocean sinks is surprisingly linear in the past 57 years of accurate measurements.

However, there is one problem with 14C as tracer in the atomic bomb tests: what did go into the deep oceans in 1960, at the peak of the 14C concentration, was the isotopic composition of that year, what got out of the oceans in the same year was the isotopic composition of the deep oceans, with much lower 14C content.
That makes that in that year about 97.5% of 12CO2 sinking in the deep oceans returned out of the upwelling zones, while only some 44% of the sinking 14CO2 did return. That makes that the decay rate for any excess 12CO2 (that is the bulk) above the steady state is much slower than any excess 14CO2...
Similar, but reverse, problem for 13CO2 (from human emissions).

The calculated e-fold decay rate for 12CO2 is ~51 years or a half life time of ~35 years, much shorter than the IPCC expects, but much longer than for 14CO2.

The IPCC uses the Bern model, where each compartiment has its own decay rate for an excess CO2 in the atmosphere (which is true), but also a saturation level, which is only true for 10% of the atmospheric change in the ocean surface, questionable for the deep oceans (zero indication until now) and non-existing for the biosphere.

Here the fate of 12CO2/14CO2 in the different fluxes and compartiments for 1960:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/14co2_distri_1960.jpg

About the Revelle factor, that is empirically confirmed: The increase of CO2 in the atmosphere is known, the increase of DIC (CO2 + bi + carbonate) is measured in the ocean surface, with longer series at a few places. Here for Bermuda:
http://www.biogeosciences.net/9/2509/2012/bg-9-2509-2012.pdf
In fig.5, one can see the increase in DIC. If you compare that with the increase in the atmosphere over the same period, it is about 10%.
That is only the case for the ocean surface layer, which contains ~1000 GtC (the atmosphere currently ~800 GtC).

That doesn't affect the deep ocean saturation, as these are largely isolated from the atmosphere and the main exchanges with the atmosphere are in the largest temperature differences: equator and poles, where temperature is the main driver, not chemistry...

3. Dear Ferdinand,

What a delight for me to see your 14CO2 distribution schematic with the cosmic ray generation included. I think you'll find that my two-reservoir model of the carbon cycle plays out the bomb-test curve rather well. See here:

http://homeclimateanalysis.blogspot.com/2015/11/carbon-14-bomb-tests.html

Is the Bern model linear? That is: does it follow linear differential equations for the exchange between the atmosphere and the ocean? If so, it cannot follow the bomb test carbon-14 observations and at the same time predict that our CO2 emissions will remain in the atmosphere for centuries. If it implements the Revelle Factor, then it's using non-linear differential equations.

The top layer of the ocean is exchanging dissolved carbon with the deep ocean. In your schematic you appear to have direct exchange with the deep ocean, bypassing the top layer. I don't see how that's possible, by definition of the top layer. The carbon that dissolves in the top layer diffuses downwards. Thus I don't expect the top layer to show the same increase in dissolved carbon concentration as the atmosphere.

Consequently, I'm puzzled by your statement that the small increase in top-layer carbon concentration implies that the Revelle factor has been proven. The way to prove the Revelle factor is to take a sealed barrel of seawater with a lump of chalk in the bottom, to keep it saturated with calcium carbonate, and pump CO2 into the air above the water. Then measure how much dissolves at equilibrium. So far as I know, this experiment has not been performed by climate scientists recently, although I did find some experiments done in the early twentieth century, and they showed the CO2 dissolving according to Henry's Law.

Furthermore, the ionic equations used to calculate the Revelle factor are for CO2 dissolved in pure water with sodium hydroxide dissolved in it to give it an alkaline pH. If we use the ionic equations for a saturated solution of calcium carbonate, the liquid does obey Henry's law for CO2.

Thus the Revelle factor appears to have no basis, either in ionic theory or in experiment.

I have built a three-reservoir model of the carbon cycle, with a top-layer for the ocean. The total behavior is almost identical to the two-reservoir model, but there is a 4-year time constant for initial absorption of CO2 (and carbon 14), combined with 15-year for the deep ocean absorption. There is a rapid drop in carbon-14 down to something like 80% of initial concentration over the first 4 years, then the 15-year time constant after that.

I could present the three-part model in another post, but I'm not sure it's worth it. I have come to the same conclusion as Arnold et al. in the paper linked below: the behavior of a two-layer ocean model does not differ significantly from the three or four or five-layer model when it comes to predicting the atmospheric concentration of carbon-14 or CO2.

http://www.hashemifamily.com/Kevan/Climate/Dist_C14_Nature.pdf

4. Dear Kevan,

Indeed the deep oceans - atmosphere exchange is largely bypassing the surface: upwelling occurs mainly near land with wind from the land side (like trade winds near the equator) and downwelling is near near the poles where colder, densier waters sink in the deep. All together that is about 5% of the ocean surface each. The rest of the deep oceans is largely isolated from the surface for temperature and minerals exchanges. One exception: biolife which rests drop out of the surface and enrich the deep with organic and inorganic carbon.
Thus one can split the ocean-atmosphere exchanges between surface which is easily saturated and the deep oceans which need much more time to equilibrate with the atmosphere, both because of the huge mass and the relative small exchange rate of ~40 GtC/year.
The Bern model still is rather linear, despite the Revelle factor for the ocean surface only. That factor is only the limit (~10%) where the exchange rate is going to end. More info about the Bern model is here:
http://unfccc.int/resource/brazil/carbon.html
Where each compartment has its own decay rate for an excess caused by emissions and its own fraction of maximum uptake and some 15% remains in the atmosphere for centuries to milennia...
The fastest decay rate is for the ocean surface, the second for the deep oceans and the third for vegetation.

Anyway, it is too soon to decide if the Bern model doesn't fit the real world, as the multi-decay model with limits still shows the same decay speed as a single decay rate without limits...

The smaller increase of CO2 in the mixed layer of the oceans is in fact the real life experiment, even if there is not much sold carbonate present, with the exception of coccoliths like e-hux in all surfaces and in shallow oceans near land.
As the ocean surface is largely isolated from the deep oceans, there you have your oversized barrel...

More in a second part (due to size limits?)

5. One need to take into account that Henry's law is for pure dissolved CO2 in water alone, not for bicarbonates and carbonates.
If you double the CO2 content in the atmosphere, that doubles free CO2 in fresh and seawater alike. For fresh water there it ends. For seawater, the chain reaction moves to more bicarbonate and more H+, but less carbonate, until a new equilibrium is reached. That is at an increase in all inorganic carbon species (DIC) about 10 times the total increase of free dissolved CO2. Thus while limited to 10% of the change in the atmosphere, that change is 10 times the change in CO2 mass compared to the change in fresh water...

More background about the Revelle factor from chipster07 (the same person as CHIP here?) at:
https://chipstero7.blogspot.be/2016/12/the-revelle-factor-vs-henrys-law.html
Where also the Bjerrum plot is shown. That gives the relative C species at different pH. In the case of a CO2 doubling, the pH shifts a few tenths to the left (less basic), CO2(aq) from ~1 to ~2%, bicarbonates get somewhat higher and carbonates somewhat lower.

Then what the "consensus" says:
"The Revelle Factor applies to the whole oceans (and not just the surface), as Archer (2005) says"
That is largely wrong. the ratio factor in Henry's law depends of the temperature of the liquid. In the case of CO2 in seawater that changes the solubility with 16 ppmv/K. As most of the deep oceans are isolated from direct contact with the atmosphere and simple diffusion of CO2 in calm water is extremely small, the main sinks into the deep oceans are near the poles, where the seawater temperature is around -2°C. That makes that the pressure difference between CO2 in the atmosphere (at ~400 μatm ~= ppmv) and the ocean waters (at ~150 μatm) is over 250 μatm, that pushes some 40 GtC CO2 in highly undersaturated waters into the deep oceans.

That means that the Revelle factor plays no role in the deep coean water - atmosphere exchanges. Not now and not in the far future and the real buffer factor by far exceeds the expectations in the Bern model.

Where Chipster07 is wrong is that he applies Henry's law to DIC, while it is only to dissolved CO2 not the other carbon species. Thus both Henry's law and the Revelle factor are at work without conflict, but at different levels: Henry's law for free CO2 alone and the Revelle factor for total carbon species. Thus the partitioning ratio of free CO2 in the oceans is not affected by the relative concentrations of DIC as the partial pressure of CO2 in the atmosphere changes (or reverse), but the relative concentrations of the other C species is affected...
His last chapter needs a lot of comment too, but that is not to be discussed here...

6. Ferdinand, Thank you for your explanation. I have read it once and you have given me a lot to think about. I am busy this weekend, but look forward to taking the time to check the links and consider ocean up-welling. Yours, Kevan

7. Ferdinand,

I must accept that up-welling and down-welling currents could dominate the exchange of CO2 between the deep ocean and the atmosphere. I accept that such exchanges take place because they motivate the growth of algae and krill that feed whales, but that's water coming up near the poles. Given that water at atmospheric pressure has greatest density at 4C, I don't see how temperature can drive the convection. But I accept that it takes place. So, there is not much use in me implementing a three-part exchange model because the exchange may not be taking place through the middle part. And in any case, the two-part model works very well.

When you say the "surface which is easily saturated" I am not sure what you mean by "saturated". My understanding of the word "saturated" is "no longer responds to further increases", so that the surface being "saturated" with CO2 would mean "a doubling of CO2 does not increase the dissolved CO2". But that's not what you mean, judging from study of the rest of your comment.

On the Revelle Factor, "that change is 10 times the change in CO2 mass compared to the change in fresh water" I see how the change in carbon mass in the ocean will be double the net amount of carbon absorbed from the atmosphere, because one more CO3 group is dissolved from CaCO3 for every CO2 dissolved. But I am unaware of the reactions that lead to nine CO3 being dissolved. So far as I can tell, the Revelle Factor comes from a calculation of how pure CO2 plus water would behave if you added NaOH to give it a pH of 8.2, and not from any realistic consideration of ocean chemistry. At the site by CHIP, he reproduces the CO2 and pure water ionic balance calculations, and appears to be unaware of the fact that these calculations do not apply to a saturated solution of CaCO3, nor does he seem to have realized that a solution of CO2 and water is always acidic, and to make it alkaline like the ocean, we must add something to it. What we add must not contain CO3 or else the ionic equations change dramatically and the Revelle Factor disappears.

So I'd like to see ionic equations for the ocean that show a Revelle Factor. The equations for a saturated solution of CaCO3 and CO2 are well worked out, see below.

http://lawr.ucdavis.edu/classes/ssc102/Section5.pdf

That is: the Revelle Factor is a confusion generated by applying an ionic balance to the wrong solution, nothing more.

Yours, Kevan

8. Dear Kevan,

Your link is for the CO2/carbonate system in soils, which is mostly the same as for the oceans, with as main difference the direct availability of solid carbonates in soils (by addition or naturally present), while for the oceans that is only true coastal or with shallow seas. However, here a similar work for the oceans, where the dissolved carbonates are part of the buffer capacity of the oceans:
http://www.soest.hawaii.edu/oceanography/courses/OCN623/Spring2012/CO2pH.pdf

Anyway, all what the Revelle/buffer factor says is that a 100% change of CO2 in the atmosphere gives only a ~10% change in total C species, still 100% of dissolved "pure" CO2, but less from the other C species. That is all. As free CO2 is less than 1% of all C species in seawater, a doubling of free CO2 as in fresh water is peanuts, compared to a 10% increase of all C species...

Some more detail:
http://onlinelibrary.wiley.com/doi/10.1029/2008GB003407/full

Upwelling of deep ocean waters is mostly near land, where (trade) wind from the landside pulls deep ocean waters to the surface and sinks are near the poles as freezing water expels salts which increase the density of remaining waters... So it is a mix of winds and temperature. Not at all an important point, but if you use a multi-compartment model (including the biosphere), one can differentiate between all the different fluxes and their result...

I used "saturated" as the time factor to reach the CO2 equilibrium between the ocean surface and the atmosphere, not the absolute end of the ocean buffer capacity: thanks to wind and waves, any change in the atmosphere (or reverse) is fast redistributed between these two (exchange rate less than a year). As both have comparable quantities of C species and the buffer factor of the oceans, the total change in the atmosphere was from ~590 to ~800 PgC (as CO2) since 1850 and in the ocean surface from ~1000 PgC to ~1030 PgC (for all inorganic C species). The latter shows up as a small increase in DIC in the ocean surface where longer time series were taken.

As only the ocean surface is in direct, fast contact with the atmosphere, it seems to me more interesting to separate that from the deep oceans, as the latter are much larger in capacity but have a much slower exchange rate...

9. I'm looking at this:

http://www.soest.hawaii.edu/oceanography/courses/OCN623/Spring2012/CO2pH.pdf

The author states that CO2 "controls the pH of the ocean". Given that a solution of CO2 and water is always acidic, it seems to me a strange thing to claim that CO2 controls the pH when the pH is alkaline because of dissolved CaCO3. Reading the presentation, it sounds as if the HCO3- and CO2-2 concentrations in the ocean are dominated by CO2 gas, when they are dominated by dissolved CaCO3, which is how the acidity of the CO2 solution was overcome and reversed. So I am still confused as to why such a presentation would focus on CO2 when the ocean pH and ionic balances are controlled by CaCO3. Furthermore, the author's statement that it takes ten thousand years for the CaCO3 system to respond to changes in ocean pH seems unlikely to me. What is the author's evidence for such a statement?

I'm looking at:

http://onlinelibrary.wiley.com/doi/10.1029/2008GB003407/full

and here the authors make sense to me. They are not talking about the Revelle Factor as a deviation from Henry's Law, but rather looking at how increased CO2 partial pressure changes the equilibrium for dissolved CaCO3.

I now understand your use of the word "saturated".

Why do you believe that the mixing of the ocean surface and the deep ocean is negligible?

10. Dear Kevan,

"controls" is big word in this case. All depends of what is actually changing. If some 100 volcanoes at the ocean bottom all together start to spew a lot of SO2, the pH will go down and CO2 will be lost to the atmosphere and the total carbon content of the oceans (DIC) would go down. At such a moment the volcanoes "control" the pH and DIC levels in the oceans. Nowadays we see the opposite happening: CO2 levels in the atmosphere increase with as result that CO2 levels in the ocean surface increase while the pH gets (slightly) lower. That is one of the reasons to conclude that the average CO2 flux is from the atmosphere into the oceans, not reverse.

That it takes a lot of time to get everything in equilibrium is a matter of exchange/mass ratio. Most oceans have little solid CaCO3 compared to the total mass and most is at the ocean bottom (above the saturation level). As we push more CO2 into the surface, that has no direct contact with solid CaCO3, except in shallow seas and coastal. The difference is in the exchange ratio: ~50 GtC/year in/out the ocean surface for ~1000 GtC in that part of the oceans, mainly seasonal, while for the deep oceans it is ~40 GtC/year in and out between upwelling and sink places for a ~36,000 GtC reservoir.
There are some exchanges between the surface and the deep oceans, mostly biochemical, as are the dropouts from dead plants and animal material. But most CO2 exchanges between the atmosphere and the deep oceans bypasses the ocean surface. Thus an equilibrium change in the surface is fast, but lacks access to solid carbonates...

There are a few stations in several oceans which measured DIC over longer periods. If you compare the CO2 change in the atmosphere over the same period with the DIC change in the ocean surface, you will see that the change in the surface is about 10% of the change in the atmosphere (fig.5):
http://www.biogeosciences.net/9/2509/2012/bg-9-2509-2012.pdf

11. Dear Ferdinand, Thank you for your answer. I'm trying to make sure I understand what you mean by "DIC change in ocean surface" and "change in the atmosphere". I'm looking at the paper you linked to. I'm embarrassed to say I don't know what pCO2 means in seawater, only what it means for a mixture of gases. The DIC change 1983-2011 was around 2020 to 2060 umol/kg, an increase of around 2%. The pCO2 change was 320 ppmv to 370 ppmv, which I guess is +50 uAtm, or 15%. If the partial pressure of CO2 in the atmosphere were 0 uAtm, what would DIC be? Kevan

74. CHIP, I looked up Le Chatelier's principle (and I don't know how you got the letter with the hat on it into your comment). I looked at the paper. It's from 1992. I did not realize that the climate models were non-linear at such an early date. Thank you for the reference. Kevan

75. Kevan,

I recently discovered your blog after engaging a friend in climate change discussions. I've been studying the issues for some time now using my chemistry background to understand the nuances involved in this multidisciplinary field. I came to the same conclusion as you that the two main issues are how much CO2 affects global temperature and how much human emissions have increased CO2. I gained an additional insight from you and that is the possibility that the CO2 ocean reservoir is so vast that atmospheric CO2 may never double. Let me know if I am misstating your position.

Has there been any further discussion about the appropriate use of your CaCO3-CO2-water system over the OH-CO2-water model which you attribute to mainstream climate scientists? The correctness of CO2 ocean models seems to revolve around this question. I plan to look more deeply into this without having to reinvent the wheel.

Are you aware of papers by Ed Berry [https://edberry.com/blog/climate/climate-physics/human-co2-has-little-effect-on-the-carbon-cycle/] and Hermann Harde? [http://www.sciencepublishinggroup.com/journal/paperinfo?journalid=161&doi=10.11648/j.earth.20190803.13]