text: Executive Summary One of the most controversial topics in understanding the build-up of carbon dioxide in the atmosphere is the question of timescales – the effect of the build-up depends not only on the amounts being released by human(-related) activities but also on how long the gas stays in the atmosphere. In fact much of the controversy/confusion stems from the fact that there are two relevant timescales, one which determines how the amount of carbon dioxide in the atmosphere equilibrates with other reservoirs (notably physical exchange with the oceans, and biological exchange via photosynthesis and respiration), and another which determines the exchange of carbon atoms. By analysing the amounts of a marker carbon isotope (carbon-13) it is possible to calculate these two timescales. The timescale for the amount of carbon dioxide is approximately twenty years, a significantly shorter timescale than often claimed (e.g. by the IPCC). From these figures, we can also deduce that the increased carbon dioxide in the atmosphere since the industrial revolution has led to a noticeable increase in the photosynthetic rate of the Earth’s plants and green algae (about 8%). This has clear implications for the on-going discussions on the costs, and indeed benefits, of increasing carbon dioxide levels. The reasons why the IPCC’s (and others’) estimates of carbon dioxide timescales in the atmosphere are overestimated are analysed – notably because no account is taken of changes in net respiration rates (ever more people, and domesticated animals, and animal pests that depend on them), because hydrocarbon usage by UN member states is underreported (quite possibly for reasons of political prestige), and finally because the models ignore the key empirical evidence (the carbon-13 isotope measurements). Bio – David Ellard David Ellard studied Natural Sciences at Kings College Cambridge with specialisations in mathematical and atmospheric chemistry. Since then he has worked over twenty years in the European Commission in Brussels in various science/technology/law-related areas, notably responsible for the Commission’s proposed directive on the patentability of computer-implemented inventions. Begins: Atmospheric carbon dioxide – a tale of two timescales Once upon a time, when the world (and this author) was young, students of atmospheric chemistry were taught about an entirely straightforward and uncontroversial concept. This was the residence timescale of a given (gaseous) component of the atmosphere such as nitrogen, oxygen and argon, or the trace gases such as carbon dioxide, sulphur dioxide and methane. The atmospheric timescale was easily calculated given the amount of the gas present and the known sources and sinks: tr = (atmospheric mass of component) / (average of sources and sinks) Strangely, amid all the current scientific controversy about atmospheric carbon dioxide, there is very little debate about the scale of the most important flows in the carbon cycle illustrated by the following diagram (other versions of the same diagram, including those of the IPCC, show numbers which do not differ significantly):
  [fig.1 Carbon cycle according to NASA] It is not a great mathematical achievement to plug in the flows (note that the sources are slightly greater than the sinks – carbon dioxide is increasing in the atmosphere of course) and calculate the residence timescale. The answer is:
  And yet, this simple result is now the subject of intense controversy. The IPCC has claimed that the relevant timescale for CO2 in the atmosphere is actually 50 years, figures of ‘hundreds of years’ are routinely quoted by “alarmist” websites. This blog itself has estimated the timescale at 33 years. So what gives? How do we reconcile these apparently contradictory claims? The simple answer is that there are two relevant timescales, one of which is the above-calculated atmospheric residence timescale, which really is of the order of 4 years. The other one, which is often – in fact almost invariably – also referred to as the atmospheric residence timescale, isn’t. And, furthermore, estimates of it – however it should be referred to – vary wildly. The purpose of this post is to try and explain the nature of the two timescales, and pin down using actual physical measurements (rather than computer games) the size of both. If you look at fig. 1, you will see that there are actually only two processes that count when it comes to calculating the atmospheric residence timescale of carbon dioxide. One is the flux between the atmosphere and the biota – the source being the respiration (and, indeed, combustion) of carbon-containing molecules by living creatures and the sink being photosynthesis. The second is the process of diffusion of CO2 across the boundary between the surface waters of the Earth (i.e. the oceans) and the atmosphere. Note that these fluxes overlap in space – an important part of photosynthesis takes place in the surface waters of the oceans by phytoplankton. Nonetheless they are quite distinct. I am going to start by looking at the latter – the physical (as opposed to biological) process of exchange of CO2 between the atmosphere and oceans. The key principle which determines the equilibrium between a substance in gaseous form and dissolved in a solvent is Henry’s Law. But I am going to illustrate the Law, which in fact derives from the deeper principles of thermodynamics, which are inherently statistical in nature, using a different example. The porter’s lodge of St. Henry’s College Oxbridge contains two sets of pigeon holes on opposite walls – the east wall and the west wall. One day the porter, who has a keen amateur interest in theoretical thermodynamics, releases an infinitesimal number of pigeons inside the lodge and observes what happens. Each pigeon alights in a given pigeon hole at random, on either wall of the lodge. The pigeon decides how much it likes the given pigeon hole it is in and, depending on that liking – which we shall shortly quantify as its utility function – spends a proportional amount of time in that hole before flying off and realighting in a random pigeon hole, just as before. These pigeons are a mite anti-social. If they sense another pigeon in a nearby pigeon hole, they will spend less time in that hole. All the pigeon holes on each wall are identical and therefore have an identical utility function as follows: U(either wall) = k x N + k2 x N2 where k and k2 are constants and N is the number of pigeons. If pigeons are anti-social, as gas molecules generally are, then k2 will be negative. The utility function is defined such that the ratio of time spent by pigeons on the east wall to the west wall will be given by the ratio of the utility of their respective pigeon holes. The porter immediately intuits that this ratio will, over the long term, be one, regardless of the number of pigeons, because the east wall utility function is identical to that of the west wall. One can well imagine the porter’s shock, on arriving for work one Sunday morning, to discover that some unruly students have come into the lodge during the previous night and painted all the pigeon holes of the east wall, and only the east wall, an unappealing shade of lime green. While the porter contemplates this new situation, (s)he is immediately struck by the consequences for the theoretical pigeons. Pigeons may or may not prefer the lime green pigeon holes, but there must undoubtedly now be a difference in the utility function for the two walls: U(east wall) = kE x N + k2 x N2 U(west wall) = kW x N + k2 x N2 So we now have two different first order utility constants kE and kW, but nonetheless the second order utility constant k2 remains the same (because this is based on dislike of pigeons of the proximity of other pigeons and the pigeon holes are the same size, and therefore have the same separation, on both walls). Disregarding the term in N2, we can see that the average ratio of pigeons on the east wall to the west wall is now given by the fixed expression kE/kW ≠ 1. Disregarding? What kind of science is that? The answer is, that is a reasonable approximation as long as N is small. It will indeed be an exact answer when N is infinitesimal – in other words when no pigeons are actually present (which is, one would hope, the normal state of the porter’s lodge). If N starts to grow though, the constant k2, the anti-sociability of pigeons, will become an increasingly important factor, and the ratio of east wall to west wall pigeons will change. And so we arrive at Henry’s Law which states that, ‘at infinite dilution’, the ratio of a given molecule in a gaseous phase which is in contact with a given solvent (so in a liquid phase) will be fixed, providing the two phases (the gas and the solvent) are in equilibrium. The gas phase and the solvent are like the (repainted) east and west walls. They have in principle a different attraction for the molecule (pigeon) in question. But as long as the assumption of ‘infinite dilution’ holds, the ratio of molecules in equilibrium between the two phases will be fixed. Hold on, infinite dilution? Henry’s Law only applies exactly when the molecule in question is not actually physically present! What use is that? The answer is, very useful, because Henry’s Law applies approximately to real situations where the molecule in question is present in significant, but non-infinite, dilution. What Henry’s Law is telling us, then, is that when we add molecules of carbon dioxide to the atmosphere, these molecules will ultimately partition themselves (leaving aside the effects of the biota) in an approximately fixed ratio between atmosphere and ocean (the solvent). Three questions arise: what is the dilution of carbon dioxide in the oceans? what does ‘ultimately’ mean? and what actually is the value of the fixed ratio? In order of asking: very dilute (the oceans are approximately 500 times undersaturated in molecular carbon dioxide), it depends on the mixing processes both within and between the atmosphere and ocean (discussed further on), and:
  This is, in my view, a rather startling result but, like the rather short atmospheric residence timescale, is extremely difficult to track down in current scientific literature. It is not actually absent so much as simply hidden in plain sight by only ever being quoted indirectly (as the product of other factors, see below). To rephrase then, for every six molecules of CO2 that are introduced into the atmosphere, five of the six (again ignoring biological processes) will end up in the oceans, only one of them will hang around in the air. Not only that but, as noted above, molecular CO2 is a very dilute solute in the oceans. At current rates, it would take tens of thousands of years for mankind to achieve saturation. The partition ratio 1:5 will continue to apply for the foreseeable future! The interested reader who attempts some research on this will be immediately confused. They will no doubt encounter the fact that the actual ratio of atmospheric carbon dioxide to that dissolved in the oceans (so-called Dissolved Inorganic Carbon, or DIC) is:
  The scientific literature will further confuse the unsuspecting amateur researcher by insisting that this ratio will change as further CO2 is added to the atmosphere, in apparent contradiction to Henry’s Law. Oh dear. Fortunately help is at hand. The answer lies in chemistry. Here’s a thought (or even real) experiment for you. Take a bucket of normal water, add some ordinary table salt (sodium chloride) until the water is distinctly salty. Sit back and watch. What doesn’t happen? Lots of things, obviously. One of the things that doesn’t happen is that a highly poisonous green gas, used as a weapon in the First World War, exits the solution in the bucket and drifts across to our unsuspecting (thought) experimenter. Why not? After all we just put pots of the element chlorine into the bucket in the table salt, and chlorine gas is made out of – the name gives it away really – chlorine. Why doesn’t the chlorine present in solution in the bucket equilibrate with that in the air around our intrepid experimenter, as Henry’s Law would seemingly predict? We can put this supposed equilibrium in a form beloved of chemistry teachers: [reaction 1: Air-gas equilibrium of chlorine/chloride: warning this is not a real chemical reaction!] 2 Cl– (aq) ⇌ Cl2 (g) where (aq) denotes aqueous phase i.e. dissolved in the bucket, (g) denotes gaseous phase i.e. in the air and Cl– denotes chloride, the ion present in table salt solution. Any chemists, however, reading this post will by now be experiencing severe heart palpitations, if they have not already undergone full cardiac arrest. In fact the chemical reaction given above is nonsense. No chlorine gas is emitted by table salt solution. Why not? The reason is that the reaction books don’t balance. It is Enron chemistry. The left hand side has two more electrons (the negative sign on the chloride ions) than the right hand side. Since (valence) electrons are more or less the entire point of chemistry, this is a major flaw. Chloride ions cannot equilibrate with chlorine molecules because they are fundamentally different things. Pigeons cannot equilibrate with badgers. On the other hand, if one visits a swimming pool, one is indeed very likely to smell the characteristic odour of chlorine molecules – which are also present, in accordance with Henry’s Law, in the water of the pool itself (which is the whole point, of course). The water contains chlorine molecules in solution, which are chemically different from chloride ions in solution. Swimming pool water does not normally taste salty. However there are also saltwater swimming pools out there. Chlorine molecules can also be added to the salty water to guard against infection. In principle there would be a Henry’s Law equilibrium between the chlorine molecules dissolved in the saltwater and those smellable in the gas phase above the pool. As good and careful chemists, we would differentiate between the chlorine molecules in solution in the water, participating in the Henry’s Law equilibrium, and the chloride ions in the same solution which don’t. If we were dealing with carbon dioxide, instead of chlorine, we would naturally take the same care. Sadly this is overwhelmingly not the case with the scientific literature on the question. There is a whole bunch of Enron carbon dioxide ocean chemistry out there which fails to make this crucial distinction. You have been warned. Like chlorine, carbon dioxide can also exist in ionic as well as molecular form in solution. These ions are referred to as carbonates. We will examine the exact chemistry in more detail shortly (it is more complicated than for chloride). The basic take home fact is that the ‘dissolved inorganic carbon’ or DIC in the world’s oceans is, in principle, a mixture of molecular carbon dioxide and dissolved carbonates. What is the ratio of molecular to ionic carbon dioxide? The smart among you will already have guessed: there is approximately 9 times as much ionic CO2 dissolved in the oceans as molecular. Only the latter is in Henry’s Law equilibrium with CO2 in the atmosphere. Hence the different ratios of 1:5 (atmospheric:molecular dissolved CO2) and 1:50 (atmospheric:molecular plus ionic dissolved CO2 i.e. DIC). In principle we can understand the difference by carrying out the thought experiment of boiling the world’s oceans dry (don’t do this for real please readers). After we have done this, 90% of the CO2 originally dissolved will end up in the form of carbonate salts precipitated out on the ocean floors. If you ever needed a top up of bath salts, this is the place to look. The other 10% – the fraction of dissolved molecular carbon dioxide – will have escaped into the atmosphere as CO2 gas (thus increasing the concentration there by a factor of 6, causing no doubt heart attacks to the folks at the IPCC, again please don’t do this at home readers). The ratio of DIC to dissolved molecular carbon dioxide (which is 10:1 since the former also includes the latter) is often referred to in the scientific literature as the Revelle factor. This factor actually varies with the surface temperature and salinity of the world’s oceans (as indeed does the Henry’s Law constant of carbon dioxide, but the implications of that would need to be covered by a whole other post). Thus, as we (both theoretically and actually) add carbon dioxide to the atmosphere, the ratio of atmospheric to dissolved molecular carbon dioxide (at equilibrium) will stay the same, in accordance with Henry’s law. The oft-repeated claim that the ratio of atmospheric to total dissolved CO2 or DIC (i.e. molecular plus ionic, the latter of which is fixed) will rise is therefore perfectly correct, and perfectly irrelevant. The ratio of atmospheric to dissolved molecular CO2 plus elephants will also rise. Most readers can readily see that elephants are not comparable to carbon dioxide molecules. Only the most chemically sophisticated, however, will appreciate that the comparison of molecular to ionic dissolved CO2 is, in the context of the Henry’s Law equilibrium, also specious. DIC is nonetheless important in one very significant respect. For at this point, it is time to look in more detail at what happens at the ocean-atmosphere interface. We are going to assume that someone (humankind you know who I am talking about) has added some carbon dioxide to an atmosphere that was previously in perfect Henry’s Law equilibrium with the oceans. Not only that, but they have added special CO2 molecules containing an atomic marker – a form, or isotope, of carbon which can be readily identified. In the following explanations, these molecules will be marked with a green dot. In a given time period, these ten surplus molecules from the atmosphere, all with atomic markers, will meet ten unremarkable, unmarked CO2-containing molecules from the ocean at the interface between the two – the surface. We now know that nine of these molecules will (in principle) be ionic – carbonates. Only one of the ten will be dissolved molecular carbon dioxide, capable of participating in the Henry’s Law equilibrium.
  [fig.2 Schematic of ocean-atmosphere physical exchange] As the atmosphere is now out of equilibrium, all ten surplus molecules will participate in forward reactions with their ocean counterparts. What reactions? When a gaseous neutral carbon dioxide meets a dissolved ionic counterpart, no Henry’s Law equilibration can take place. Nonetheless, the marker atom can be exchanged as follows: [reaction 2: Air-ocean carbon atom exchange]
  If you associate chemical reactions with things exploding in the front of chemistry labs at school, you are going to be sorely disappointed by this one. The reaction starts with one gaseous and one dissolved molecule. It ends exactly the same way. From the point of view of Henry’s Law, nothing has happened. But something has happened – something we can measure. Isotope exchange has taken place between the atmosphere and ocean. What about our lone molecule of dissolved neutral CO2? It meets its atmospheric counterpart and undergoes the following ‘reaction’: [reaction 3: Air-ocean carbon dioxide molecular exchange]
  This is the Henry’s Law equilibration. Our marked gaseous carbon dioxide molecule has dissolved in the oceans. So we can now recap. Before the exchange the atmosphere contained ten surplus marked molecules of carbon dioxide. After the exchange, there were still nine surplus molecules in the atmosphere, but none of them contained the marker! The ocean gained a single extra molecule of carbon dioxide but gained an extra nine atoms of marked carbon (and lost nine unmarked ones). At this point, I am hoping that you are experiencing an ‘a ha!’ moment. Do you begin to see why there are two relevant timescales to exchange of CO2 between atmosphere and ocean? We have to distinguish between the timescale for exchange of carbon isotopes (the marked molecules) which corresponds to the arrows in fig.1 showing diffusional exchange between atmosphere and ocean surface and the atmosphere. The flux is given as 92 Gt (gigatonnes = 109 tonnes) of carbon/year which corresponds to a timescale (relative to the atmospheric inventory) of about 8 years. But we now know that the timescale for exchange of carbon dioxide molecules is ten times slower! This timescale is thus 80 years. Having now explained the principle of what is known as ‘isotopic disequilibrium’, one of the most difficult and least intuitive of the principles of oceanic carbon chemistry, I must unfortunately now draw your attention to a further complication. When reading the scientific literature, you will doubtless come across a third ratio that will confuse you. You will read that:
  Nothing is simple in life, and neither is it in ocean carbon chemistry. The further complication for carbon dioxide and its aqueous ions, which is not the case for chlorine/chlorides, is that molecular carbon dioxide which has just reached the oceans from the atmosphere undergoes a further reaction. For each ten molecules which dissolve, nine of them react with a dissolved carbonate ion to produce an intermediate ion called bicarbonate (you may remember the name of the sodium salt which is used for baking – bicarbonate of soda): [reaction 4: Dissolution of gaseous CO2 in seawater] 10 CO2 (aq) + 9 H2O + 9 CO32- → 1 CO2 (aq) + 18 HCO3– This reaction is important from the point of view of ocean chemistry but it is, again, irrelevant to the Henry’s Law equilibrium which determines the ratio of atmospheric to dissolved molecular CO2 – only part of which (10%) consists of ‘free-floating’ aqueous CO2 molecules. Furthermore, if you consult the constants of equilibration between air and ocean, you will invariably be quoted the ratio between atmospheric and aqueous CO2 which is not the same as the Henry’s Law partition ratio! The ratio of atmospheric partial pressure of CO2 to aqueous CO2 in the oceans is treated as a Henry’s Law constant in the literature, but it isn’t because atmospheric CO2 is not in Henry’s Law equilibrium with aqueous CO2! The reason this is done is because, in practice, it is hard to measure the precise concentration of molecular carbon dioxide dissolved in water (not only because it exists in two forms, as aqueous CO2 and combined in a bicarbonate complex, but because bicarbonate ions themselves can be created as a result of interactions between water molecules and carbonate ions). We must nevertheless be extremely vigilant not to be confused by such ‘pseudo-Henry’s Law constants’ into thinking the partition ratio of CO2 between atmosphere and ocean is ten times greater than it actually is. A further confusion/deception in the scientific literature is that humanity’s tendency to pump CO2 into the atmosphere and hence into the oceans will change the ‘pseudo-Henry’s Law’ constant applying to gaseous versus aqueous CO2. The reason is because the ratio of aqueous to molecular dissolved CO2 will also change due to the complex chemical equilibrium in the oceans. But this is all irrelevant/deliberate misdirection because Henry’s Law must continue to apply! The partition ratio of 1:5 cannot change (or only very slightly as the CO2 in the oceans becomes less dilute). Having delved into ocean carbon chemistry in some depth, we are now ready to turn our attention back to the carbon cycle as a whole, have a look in particular at how the living world interacts with CO2 in the atmosphere, and introduce the four-box model.
  [fig.3 Ocean-surface-atmosphere-biosphere carbon exchange model] In principle we will need to make the same distinction for exchange of carbon isotopes vs. CO2 molecules when it comes to exchange between atmosphere and biota as we previously made with the ocean (the exact reasons for which we will examine later). In addition, there is a fifth timescale relevant which corresponds to how water is exchanged between the deep oceans and the surface waters (which is where the exchange with the atmosphere takes place, of course). In an effort to try to keep the mathematics as simple as possible, I am going to initially assume that this timescale is actually zero. This means that the ocean and surface are instantaneously well mixed and the concentrations of both carbon isotopes and CO2 molecules are always the same in both. Needless to say, this is highly unphysical. Do not fear, I will relax this assumption after completing the initial calculations! In the following calculations, we will use X to denote ‘concentration’, both for atoms/molecules in the atmosphere but also in the ocean and biota. In the atmosphere this will have a direct physical meaning (for CO2, I will generally quote in ppmv which means parts per million by volume), in the ocean it will denote a concentration that would be in Henry’s Law equilibrium with the corresponding concentration in the atmosphere. In the biota it means nothing physical at all, it is just a notional figure that allows us to calculate the flow of CO2 between biota and atmosphere. We now arrive at more or less the entire point of this article. We know three of the relevant four timescales from fig. 1 and our analysis of ocean carbon chemistry. The fifth timescale (the deep ocean-surface exchange) we are currently ignoring. These are then: toi – the timescale for exchange of carbon atoms/isotopes between atmosphere and oceans = 8 years (from fig. 1) toc – the timescale for exchange of carbon dioxide molecules between atmosphere and oceans = 80 years (multiply toi by the Revelle factor) tbi – the timescale for exchange of carbon atoms/isotopes between atmosphere and biota = 6 years (from fig.1) tbc – the timescale for exchange of carbon dioxide molecules between atmosphere and biota = ??? tos – the timescale for exchange of water between the deep oceans and the surface waters (which we are currently assuming is zero) We therefore now have two equations for the atmospheric residence timescale and a timescale that corresponds to how long it takes for additional CO2 molecules to dissipate from the atmosphere. This is the crucial missing timescale that I mentioned in the introduction. For want of a better name, I am going to call it the carbon dioxide atmospheric adjustment timescale or just ‘adjustment timescale’. These equations are (ignoring tos for the time being as discussed above): 1/tr (residence) = 1/toi + 1/tbi 1/ta (adjustment) = 1/toc + 1/tbc Whatever we do, nomenclature wise, does not alter the fact that we have an equation for the adjustment timescale with one unknown – the timescale for exchange of CO2 molecules (as opposed to carbon atoms) between atmosphere and biota. We badly need to constrain this equation if we want to solve it. And indeed we can, and we can. It is a simple consequence of the elementary laws of thermodynamics that increasing the amount of carbon dioxide in the air must, all things being equal, increase the rate of photosynthesis by the biota. This is known to chemists as Le Chetalier’s Principle and applies equally well to living/biochemical systems as it does to non-living/inorganic systems. Le Chetalier’s Principle tells us that the biota will try to reduce the ‘excess’ CO2 in the air by increasing their rate of photosynthesis, but it does not tell us by how much. In order to find out the rate order for photosynthesis, the increase in reaction rate with respect to the increase in concentration of the reactant CO2, we need to understand a bit of biochemistry. Plants fix carbon (convert it from gaseous to solid form) by absorbing CO2 via their leaves through microscopic openings called stomata. The process is entirely passive – CO2 enters plants because the effective concentration in plant tissues (yes this is another Henry’s Law equilibrium in principle) is lower than in the air – i.e. it takes place via molecular diffusion, through the stomata. This means that the rate order cannot be greater than one. Le Chetalier tells us it must be at least zero. The effective rate order will be given by the percentage of the biota (in terms of photosynthetic production) whose photosynthetic rate will be constrained by the rate of diffusion of CO2 into plant tissue. A detailed description of the various constraints on photosynthesis is beyond the scope of this post, but suffice to say that we can express the rate order as the ratio of the biotic equivalent of the two timescales we just explored for the ocean-atmosphere exchange i.e. tbi/tbc. So how to constrain the equation for the adjustment timescale of CO2? Remember that the timescales for elimination of carbon atoms from the atmosphere are much faster than for carbon dioxide molecules. Remember also, our discussion about marker atoms when we were looking at ocean-atmosphere exchange. I hope you can see where this is going? Carbon exists in three isotopes in nature. By far the most common isotope is carbon-12. A rare, but very useful, isotope is carbon-14 which is radioactive and decays with a half-life of about 6,000 years, and is hence especially prized by archaeologists to date ancient objects. But here we are going to look at the other carbon isotope – carbon-13. Unlike its heavier analogue, carbon-13 is stable (not radioactive). It constitutes approximately 1.1% of naturally occurring carbon on Earth. Approximately, because the ratio of carbon-13 varies very slightly depending on its origin. And therein lies the key. Carbon dioxide which is formed by the combustion of hydrocarbon fuels is depleted in carbon-13 relative to the carbon in the atmosphere. The measurement is referred to as d13C (pronounced ‘delta thirteen C’) and the typical signature of hydrocarbon-derived carbon is -25‰ (‘per mil’, so parts per thousand, relative to a fixed standard). This means that as mankind has been busy pumping carbon dioxide into the atmosphere, the carbon-13 ratio has been falling, from a pre-industrial estimate of -6.5‰ to a current figure of -8‰. But before we analyse these figures, we should calculate what we would expect to see based on the timescales discussed previously. If we assume that the pre-industrial concentration of carbon dioxide in the atmosphere is X0 and that the rate of addition of (anthropogenic) carbon dioxide is (dX/dt)anth, then it follows that the actual concentration of carbon dioxide will tend to: X0 + ta x (dX/dt)anth, where ta is the adjustment timescale for atmospheric CO2. But the concentration of marked CO2 (i.e. carbon dioxide with the isotope ratios of hydrocarbon fuels) will tend to a different concentration of: X0 + tr x (dX/dt)anth, where tr is the atmospheric residence timescale of CO2. We can now see that the ratio of total ‘excess’ carbon dioxide to ‘excess’ marked carbon dioxide is simply, lo and behold, the ratio of the two timescales ta / tr. In other words, the carbon-13 ratio in current atmospheric carbon dioxide allows us, since we started off this article stating the known atmospheric residence timescale, to calculate the adjustment timescale. Eureka! You are all, I hope, holding your breaths at this point. The proportion of atmospheric carbon dioxide which has a hydrocarbon fuel origin is given by: (d13C(atmosphere, present day) – d13C(atmosphere, pre-industrial)) / (d13C(hydrocarbon) – d13C(atmosphere, pre-industrial)) which gives:
  This is already very striking, because it is clearly much smaller than the proportion of ‘excess’ carbon dioxide in the atmosphere (i.e. the difference between current levels and those prevailing before the industrial revolution). What actually is (was) the pre-industrial concentration of carbon dioxide in the atmosphere? Conventional (including the IPCC’s) wisdom is 280 ppmv. However, accurate measurements have only been available since the Mauna Loa atmospheric observatory was set up in 1960. The level then was 315 ppmv, and that has been increasing since (at a steadily increasing rate) to just over 400 ppmv presently. If we take estimated hydrocarbon combustion figures from 1900-1960 and compare with 1960-present day, we arrive instead at a figure of 300 ppmv. I would need a whole other post to explain this in detail but I like the 300 ppmv figure not only because I think it is nearer the truth but also because it is a round number. Those who prefer to defer to scientific authority are of course welcome to carry out the calculation using the classic 280 ppmv figure. It will not make much difference. Of course, if you really defer to scientific authority, you probably shouldn’t be reading this post (or blog) in the first place. This gives us then the result that the ‘excess’ carbon dioxide is 100 ppmv and the ‘excess’ marked hydrocarbon-derived carbon is 30 ppmv (8% times 400 ppmv). This allows us our first estimate of the carbon dioxide adjustment timescale which is 12 years. This is already a surprising result! However, before we proceed further, there are two modifications to this number that we need to take into account. The first is the effect, as already briefly mentioned, of any delay in mixing between the deep ocean water and the surface. This will cause the surface waters to be more similar to the atmosphere in composition and will have the effect of making the equilibration timescales between atmosphere and ocean longer. Before we look into this, there is another implicit assumption I have made that needs to be examined more closely. I have assumed in the foregoing that all of the ‘excess’ (i.e. anthropogenic) carbon dioxide in the atmosphere derives from the combustion of hydrocarbon fuels, and hence has the characteristic carbon-13 profile. But does it? There is another (bio)chemical process, remarkably similar to combustion, which does the same job of converting solid (‘fixed’) carbon compounds to carbon dioxide gas. It is called respiration and the results of this (bio)chemical reaction come out of our noses and mouths every time we breathe. Since the industrial revolution, the human population of this planet has exploded. Not just humans though. We also have caused an explosion in the number of domestic animals, sheep, pigs, cows and chickens and the like. And not just the intended results of human food production. There are a myriad rats, cockroaches, potato blight funguses and the like out there which depend for their existence on our (unintended) generosity. They are also all busy respiring carbon dioxide into the atmosphere, thanks to us. We have to take this into account, as well as any changes in photosynthetic fluxes (which have the opposite tendency, to reduce atmospheric carbon dioxide). I would need a whole other post to discuss this in detail, but I am simply going to assume that one third of the ‘excess’ carbon dioxide is not of hydrocarbon origin. The crucial point is that this excess CO2 will not have the distinctive carbon-13 marking. Its carbon-13 profile will be almost identical to (well, pretty similar to, we will ignore the difference for simplicity) that already in the atmosphere. So we are going to calculate the carbon dioxide adjustment timescale as a function of the deep ocean-surface mixing timescale but reduce the result by a third to take into account non-hydrocarbon anthropogenic CO2 emissions. If you object to this piece of fudging, by all means feel free to do the calculation without it. Do we actually know what the deep ocean-surface mixing timescale is? There are estimates of the ocean overturning timescale of 300-1,000 years. This means in effect how long in the past the average drop of water in the oceans was last in contact with the atmosphere. To obtain the deep ocean-surface mixing timescale we need to multiply this timescale by the ratio of the mass of the well-mixed surface layer to that of the oceans as a whole. If we assume the well-mixed surface layer is 40 metres deep and the ocean is 4,000 metres deep (you can see how approximate all these estimates are) then a ballpark answer is 3-10 years, but basically no one really knows, as oceanographers (fairly) freely admit. At steady state, the concentration of isotopes and carbon dioxide molecules at the ocean surface will be given by: Xsi = (Xoitoi + Xaitos) / (toi + tos) Xsc = (Xoctoc + Xactos) / (toc + tos) We can now correct the equations linking the atmospheric residence and adjustment timescales (you may want to take my word for this): 1/ta = (1/tr + 1/(toi + tos) – 1/toi) x 30/100 x 3/2 + 1/toc – 1/(toc + tos) If you plot a graph of this using values of the deep ocean-surface mixing timescale of between, say, 0 and 100 years (which really should cover all eventualities), the value of the adjustment timescale varies between 16 and 23 years. Let’s take a happy median, thus:
  At this point, we can start to see why the adjustment timescale of CO2 is so important. We can derive a number of important results about the build-up (or not) of carbon dioxide in the atmosphere based on our best estimate (guess? a bit harsh if you don’t mind me saying so) above. To start with, we can now estimate the changes in global photosynthesis which have resulted from the increase in carbon dioxide in the atmosphere which is the rate order of photosynthesis as a function of atmospheric concentration, or tbi/tbc, multiplied by the proportionate increase in CO2 in the atmosphere, which gives:
  This tells us that the rate order of photosynthesis by the biota is in the region of 20-25%. To put it another way, it implies that 20-25% of photosynthetic production of the biota is drought constrained (why this is so would need another post). Considering that drought is not a problem for marine phytoplankton, nor for wet climes like northwestern Europe, that strikes me as a very reasonable result. We can now draw up a rough schematic of the excess flows of carbon dioxide into and out of the atmosphere i.e. the changes in the natural flows as a result of mankind’s activities. The current concentration of carbon dioxide in the atmosphere is 400 ppmv and is increasing by 2 ppmv/year. If the atmospheric adjustment timescale is 20 years then it means the oceans and biota are together absorbing 5 ppmv/year of the excess. Three quarters of this absorption is due to the increase in productivity of the biota and one quarter to the Henry’s Law re-equilibration in the oceans. So we can say that for every seven molecules of CO2 put into the air by mankind, of which just under five are from burning hydrocarbons, two accumulate there, one and a bit is dissolved into the oceans and just under four are reabsorbed by the biota via increased photosynthetic productivity. How does all this compare with the estimates of carbon dioxide timescales I introduced at the start of this piece? First off, I should point out that, if we take out my assumption that a third of anthropogenic CO2 comes from non-hydrocarbon sources (increase in respiration by the biota) then Euan’s timescale, bar a little terminological inexactitude, is spot on the money – the carbon-13 isotope results would then imply about 30 years. The IPCC’s estimate of 50 years is difficult to sustain. Even if we take the pre-industrial CO2 concentration as 280 ppmv instead of 300 ppmv and adjust the surface-deep ocean mixing timescale as generously as we dare, we cannot get a timescale larger than about 40 years. What is wrong with their estimate? Two things really. Firstly, the IPCC model ignores the carbon-13 isotope evidence and relies instead on submissions of hydrocarbon combustion rates by UN member states. To take a drastic analogy, it’s as if Sherlock Holmes stumbled on a man with a drawn blood-stained knife and a fresh corpse. Our hero has a choice of methods for deducing the identity of the murderer: he can test the blood on the knife blade for DNA matching the corpse, or he can ask the murderer if he is guilty i.e. rely on hearsay. The IPCC method is essentially the latter. The analogy is somewhat exact because, in considerable part because of political pressures brought to bear on countries as a result of the IPCC’s alarmist messages about CO2 emissions, there is a tendency of UN member states to underreport their emissions, figures that the IPCC then duly regards as gospel truth. Secondly, they ignore the obvious logic that increased human (and human-dependent) populations have led to increases in global respiration rates. Indeed the IPCC’s figures seem to assume that both photosynthetic production and net respiration by the biota have remained unchanged since the industrial revolution. In the light of facts such as the doubling of biological nitrogen throughput – thanks to the invention by humans of the Haber process – and the drastic alterations made by mankind to surface vegetation, those seem like heroic assumptions (Freeman Dyson has made a similar observation). The combination means that the IPCC has probably underestimated anthropogenic CO2 contributions. Hence it has overestimated the adjustment timescale. Models trump measurements indeed. And this, of course, has the effect of further exaggerating alarm at such CO2 contributions. But to my mind the most striking result, if we bring the carbon-13 isotope evidence fully to bear, is the increase in photosynthesis that must have taken place over the course of the twentieth century. The Henry’s Law equilibration between atmosphere and oceans is simply too slow to get rid of much of mankind’s excess CO2. The fact that there is not a lot more of this CO2 still lingering in the atmosphere (and therefore that the proportion which is hydrocarbon-derived is not even smaller) shows us that the donkey work of mopping up (most of) the excess has been carried out by the biota – all the phytoplankton, trees, grasses and algae that give wide areas of our planet’s surface its distinctive green colour. There is, then, some good news amid all the gloom and alarmism. There have been vast increases in human agricultural productivity throughout the twentieth and early twenty-first century. Most of this is of course due to improvements in plant strains, fertilisers, mechanisation etc; But nonetheless a (significant) part must also be down to the fact that it is easier for plants (including the ones we cultivate) to grow when there is more carbon dioxide in the air. This article appeared on the Energy Matters weblog at http://euanmearns.com/atmospheric-carbon-dioxide-a-tale-of-two-timescales/]]>	
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