# Simple Model for rebalancing stable CO2 emissions

As requested by @richardbetts and @edhawkins here is a simple model to back up the arguments from the previous post – Stabilising Climate.

Simple Model

CO2 levels rise when the rate of change of the sources – S exceeds the rate of change of sinks – K. Without human emissions  then S = K, when averaged over one year. However with ever increasing human emissions the situation becomes dynamic

If C is the yearly value of CO2,  S  the net sources of CO2 and K the net sinks, then at time t.

$C(t) = C_0 +\int_{t_0}^{t} \frac{dS}{dt} - \frac{dK}{dt} dt$

However it has been measured for at least the last 60 years that

If   $\frac{d^2S}{dt^2} > 0$   then   $\frac{dK}{dt} \simeq 0.5 \times \frac{dS}{dt}$

Now let’s assume that the world manages to stabilise annual emissions at current rates of 34 Gtons CO2/year  indefinitely.  CO2 sinks currently absorb roughly half of that figure – 17 Gtons and have been increasing proportional to the increase in partial pressure of CO2 in the atmosphere – currently that of 400ppm. Stabilising emissions now results in a decreasing fractional uptake by carbon sinks as the partial pressure imbalance between the surface and atmosphere begins to fall. The simplest assumption is that the sink increase depends only on the partial pressure difference for a given year. Therefore  if this pressure difference is reduced by half in one year then the next year it will be reduced by one quarter, then one eighth  and so on. The same argument applies for the case that it takes longer to reduce pressure difference by a half.

Year 1: 50%  Year 2: 25% Year 3: 12.5% Year 4: 6.25% etc. which is simply equal to the infinite sum

$\sum_{n=1}^{\infty} {\frac{1}{2}}^{-n} = 1$

So in this simplest of models, CO2 levels in the atmosphere will  taper off after just ~10 years to reach a new long term value equivalent to adding an additional one year of emissions 34 Gtons of CO2 to the atmosphere. The atmosphere currently contains 3.13 x 10^12 tons of CO2 so the net increase at equilibrium would in this simple model be just   1%. Therefore for the years following 2016 the resultant CO2 curve would look like the red curve below. If instead it takes say 4 years for the sinks to increase  by $\frac{1}{2}^n$ then we get the blue curve. In this case it would take 30 years for CO2 levels to to stabilise and the increase would be 5 times larger.

CO2 stabilisation curves for different time constants. The red curve assumes sinks match half the imbalance in 1 year while the blue matches it in 4 years.

PhD High Energy Physics Worked at CERN, Rutherford Lab, JET, JRC, OSVision
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### 44 Responses to Simple Model for rebalancing stable CO2 emissions

1. edhoskins says:

Hi Clive

In my book there is no significant temperature effect of any added man-made CO2

So why worry unless you dance to the alarmist tune

https://edmhdotme.wordpress.com/2014/09/13/the-diminishing-influence-of-increasing-carbon-dioxide-co2-on-temperature/

very best Ed

• Clive Best says:

Only time will tell Ed,

The climate is so complicated that no-one really knows what ‘climate sensitivity’ is. However if we can fairly easily stop CO2 levels rising anyway, it would at least be an insurance policy against being continuously hectored by doom mongers.

cheers

P.S. I think I may have found a source for good Rose de Provence !

2. Lance Wallace says:

The simple model is based on the belief that anthropogenic CO2 emissions are split about half between adding to the atmospheric CO2 concentration and half disappearing into a sink. But this is true only if the sum of all the far greater sources and sinks in the carbon budget are precisely constant. Where are the data showing this?

Not to mention that our estimates of fossil fuel emissions may not be very accurate–witness the recent Chinese admission that they were underestimating emissions by I think 17%. Moreover, Jamal Munshi at ResearchGate has a paper showing no relation of yearly emissions with yearly increases in CO2.

RESPONSIVENESS OF ATMOSPHERIC CO2 TO FOSSIL FUEL EMISSIONS: PART 2
JAMAL MUNSHI
ABSTRACT: This short note is a validation of a previous work which found no correlation between changes in atmospheric CO2
and fossil fuel emissions at an annual time scale. In this work, this result is tested for robustness with respect to sample period
selection within a range of data availability. A resampling procedure similar to bootstrap is used. Resampling ensures that the
failure to find a correlation is not an artifact of the sample period chosen. The results validate the robustness of the previous
finding and imply that here is no evidence that atmospheric CO2 is responsive to fossil fuel emissions at an annual time scale
net of long term trends. This result is robust. It holds for all possible combination of years in the study period 1958-20151.

• Clive Best says:

You are right that there are huge annual 2-way fluxes from air to ocean and from biosphere to air. However on average these average to zero because otherwise CO2 levels would have been changing forever in the past. What have never seen explained though is quite why 280ppm is the correct level. Why not 500ppm ? Only when IPCC scientists can explain this number can they also claim to understand the carbon cycle.

Here is another analogy of what I think happens, with the proviso that all analogies are not fully true.

I have a very large bath of water with the plug out and the tap on. For thousands of years the depth of water has remained 28 inches because the water pressure induced flow out of the plug exactly balance the flow in from the tap. A little boy turns up and starts peeing in the bath and the level rises ever so slightly, but then he gets a hose pipe and keeps increasing the flow into the bath and the water level keeps rising. Next he gets a firehose and increases that to full throttle then he gets another, then three firehoses forever increasing the flow rate. Finally he gives up the game and goes away leaving all the hoses still running. By that time the level has risen to 40 inches and is still rising. The water pressure finally equalises the drain rate to the in-flow rate and the level stops rising.

The giant’s house doesn’t get flooded!

3. Clive,
Does the figure below look as though it is correctly representing your model for a pulse of 280ppm at t = 0 and with a decay profile so that it halves every year (black) or every 4 years (red)?

• Clive Best says:

No you have misunderstood completely. It is not about how sinks react to a pulse of CO2.

It is about how sinks relax back to a steady state once $\frac{d^2c}{dt^2} = 0$

I should have written that explicitly.

• Okay, can you then provide a graph showing what would happen if you injected a pulse of 280ppm and then let the system relax?

• Clive Best says:

That is a mini – PETM event

So decays back in 100,000 years

The model describes how sinks relax into to a new stable situation with fixed sources unchanging in time.

If you throw suddenly vast quantities of CO2 into the atmosphere it will cause massive instability. $\frac{d^2c}{dt^2}$ is huge.

• As I understand it, we’re emitting CO2 about 10 times faster than during the PETM, so not quite sure why you would expect the response to be different. However, the pulse size doesn’t really matter. How would a smaller pulse decay?

• Clive Best says:

Look this is not about changes in carbon sources. It is about reaching a new balance with a now fixed source (us). Think out of the box for a change

4. I’m trying to do what Ed suggested. That requires a model for how each pulse/emission of CO2 decays with time. This shouldn’t depend on whether the emissions are constant, or not. This isn’t really about thinking inside, or outside, the box. It’s simply a pretty reasonable test of what you’re suggesting.

Write eqn for how you think sinks change as function of emissions & time. Then use past emissions to predict CO2 in atmosphere.

• Clive Best says:

It is more or less the physics of a bathtub with a drain. We have been increasing the rate of low into the tub at an exponential rate for the last 100 years, and the level has been rising, but only at half the rate expected because the outgoing drain flow has simultaneously been increasing. Now we keep the flow constant for the next 50 years. The drain now begins to start losing more than half (1/2) of the now fixed amount of water than is coming in. First it loses 3/4 of the tap flow. Then it loses 7/8th, 15/16th 31/32nd etc. of the incoming flow until a constant level is reached.

So my model is simply based on past clear evidence that the drain increases at half the rate that the tap increases. The equation is shown above.

If you want to continue to nit pick then it is probably better if you do it on twitter instead.

• I’m not nit-picking. Just to be clear, using your model you cannot (given the known historical emissions) produce an estimate for the atmospheric CO2 concentration as function of time (i.e., from whenever we started emitting CO2 to now)? I’m genuinely interested in doing this, but you seem to be suggesting that your model can’t do it.

• Clive Best says:

OK I can do it but not tonight !

5. kap55 says:

“Stabilising emissions now results in a decreasing fractional uptake by carbon sinks as the partial pressure imbalance between the surface and atmosphere begins to fall.”

There is no such imbalance. Ocean and air are always in balance, and always will be. (Henry’s Law.) It is impossible for it to be otherwise: the flux between air and ocean is roughly 50x annual human emissions. They MUST be in equilibrium, at all times.

To a first approximation, a year’s worth of emissions adds 55% of that emission to the air, and 45% into oceans and soils. That 55% stays in the air indefinitely.. And next year, another 55% stays in the air indefinitely. And the year after that, and the year after that. Oceans, soils, and air all rise together, and all fall together.

All you’ve done is invent an imaginary unfilled infinite sink, and computed what would happen to the air if it were brought into contact with that sink. But there is no such infinite sink in contact with the air. It’s all fairy dust and unicorns.

• Clive Best says:

Ocean and air are not always in balance and Henri’s law concerns the solubility of CO2 with temperature.

a year’s worth of emissions adds 55% of that emission to the air, and 45% into oceans and soils. That 55% stays in the air indefinitely.

If you include land change the figure is more like 50%. However your figure is an observation which has never been derived from first principals. Indeed it was a surprise to climate scientists.

The 50% figure means that 3 times as much carbon is absorbed today than was emitted in 1960.

The 50% airborne fraction depends on $\frac{d^2c}{dt^2} > 0$

• kap55 says:

Sorry, that’s just not correct. The primary driver of solubility of a gas in water is the partial pressure of the gas above the surface. The exact amount of dissolution is determined by a constant that is different for every gas. There is a temperature dependence of the constant with water temperature, but pressure is the biggest factor. Dissolution in water CANNOT go up unless gas pressure goes up (or temperature declines, which ain’t gonna happen). The only way the oceans can double uptake is for double the gas pressure above. But if gas pressure is doubled above, then oceans have to double uptake AGAIN. And so on.

Here’s a graph of oceanic pCO2 (Aloha Station, Hawaii) vs. atmospheric pCO2 (Mauna Loa, Hawaii) for 1988-2015. Seawater is a noisier dataset (because ocean mixing is much slower than air), but the trend lines are identical. As they have to be.

• Clive Best says:

Yes I am agreeing with you. Increasing CO2 Partial Pressure drives dissolution in the ocean surface and they march in step. That is a beautiful plot. However Hawaii is in the tropics so would be interesting to see the same thing for Alaska.

6. I think the notation used here is rather confusing, the first equation (presuming this works in the comments:

$\int_{t_0}^{t} \frac{dS}{dt} - \frac{dK}{dt}$

Is just a restatement of the familiar mass balance equation. As integration is linear

$C(t) = C_0 + \int_{t_0}^t dS/dt dt - \int_{t_0}^t dK/dt dt$

which implies

$C(t) = C_0 + S(t) - K(t)$

in other words, the current atmospheric CO2 mass is just its initial mass plus cumulative emissions since t_0 minus cumulative emissions since t_0. There doesn’t seem any value in introducing dS/dt and dK/dt as far as I can see.

Note the mass balance also woks perfectly well as a discrete time difference equation, i,e,

$\Delta C(t) = S(t) - K(t)$

Where $\Delta C(t)$ is the change in atmospheric CO2 in year t, and S(t) and K(t) are the total emission and total uptake in year t.

The second statement

$\frac{dK}{dt} \simeq 0.5 \times \frac{dS}{dt}$

is simply incorrect. Firstly an airborne fraction of 0.5 refers to the fraction of cumulative anthropogenic emissions that remain in the atmosphere each year. The “cumulative” part means that it can’t be used to infer that instantaneous source/sink rates in this way, and the “anthropogenic” part means that we can’t use it to make statements about all “net sources” and “net sinks”, as implied by the first equation. Note the article doesn’t really explain what you mean by “net sinks” and “net sources”.

A model that separated out anthopogenic sources (there are no anthropogenic sinks to speak of), total (rather than net) natural sources and total natural sinks would be less ambiguous. My attempt at doing so is here (preiprint), which might be a useful basis for making your argument, however take note of the caveats regarding such simplistic models given in section on “Limitations of the One Box Model” (which implies they are not useful for quantative predictions such as this, but can be useful for qualitative explanations).

“The simplest assumption is that the sink increase depends only on the partial pressure difference for a given year. Therefore if this pressure difference is reduced by half in one year then the next year it will be reduced by one quarter, then one eighth and so on.”

A conclusion is only as valid as the assumptions on which it is based. What is your evidence that this assumption is correct (and models, such as the Berne model, constructed by those who have studied the carbon cycle for many years are by implication incorrect)?

• Incidentally, the approximately constant airborne fraction is largely the consequence of exponentially rising anthropogenic emissions (see my paper); I don’t think it a very reliable means of quantifying the behaviour of the carbon cycle for the purposes of predicting the response of the carbon cycle to changes in anthropogenic emissions, and there are more reliable means (still bearing in mind that simple models are of questionable value for quantative predictions).

• Clive Best says:

Yes I looked at your ‘proof’ and it looks good. However you don’t explain why the value is 0.5 So in the absence of anything better I will stick to the 1/2 life model. I would like to have a physical reason why the value is 0.5.

Complex models are also of dubious value. The Berne model being a prime example.

• ” However you don’t explain why the value is 0.5″

It is a consequence of the parameters estimated from the observations, not a property of the carbon cycle.

“Complex models are also of dubious value. The Berne model being a prime example.”

Einstein said that “everything should be made as simple as possible”, but no simpler (or words to that effect). You can’t just dismiss years of research because the model is complex, the complexity is required for the model to be useful, as simple models don’t accurately reflect reality.

• Clive Best says:

I think I fixed the latex. Let me know if not.

Fair points. The basic point though is that I am just trying to model the transition from exponential growth to stability.

I don’t like the Berne model. I think it gives unphysical results namely that 25% of our emissions remain in the atmosphere forever. Their fit is based on a complex ocean modelling code whose assumptions are opaque. I think there is much more rapid mixing of the mixed layer with deep oceans than assumed. For example one ignored possibility is tidal mixing in cold arctic/antarctic waters where most CO2 is absorbed.

• Thanks.

“Fair points. The basic point though is that I am just trying to model the transition from exponential growth to stability.”

That is fine, but the assumptions need to be valid and justifiable for the conclusions to be valid/persuasive, and you haven’t provided that justification/evidence.

“Their fit is based on a complex ocean modelling code whose assumptions are opaque.”

This is not the case. The modelling techniques used in the Berne model have been published in journals and several edited volumes, but it takes a long time to read and understand them.

“For example one ignored possibility is tidal mixing in cold arctic/antarctic waters where most CO2 is absorbed.”

To know whether it is ignored, or inappropriately modelled, you first need to understand the model (and the science behind it). I certainly am not in a position to do that, just summarise some of the very basics for a lay audience as I did in my paper. Having said which, I would have thought the transport of carbon from the mixed layer to the deep ocean is mediated by the biological pump, rather than direct mixing of the layers of the ocean, so I suspect that is a minor issue that is already accommodated in the model (the layered ocean models do have transport between layers).

• Clive Best says:

“Firstly an airborne fraction of 0.5 refers to the fraction of cumulative anthropogenic emissions that remain in the atmosphere each year”
In your case it does but not in the AR4 figure which shows the annual emissions. There is a huge natural seasonal change in sources and sinks which swamps out the smaller adjustment of sinks within one year, but we know it works on a yearly interval. So it is OK – It’s just that you don’t like it.

$\frac{dK}{dt} \simeq 0.5 \times \frac{dS}{dt}$

• You are missing the point. The 0.5 is the result of the carbon cycle being driven by approximately exponentially increasing anthropogenic emissions, which means that the response is dominated by the first order behaviour of the carbon cycle. That does not mean that the carbon cycle will behave in the same way when we are no longer driving the system exponentially. To do that you need to model the physics, which is what the Berne model does, rather than a simple terminal analogue fit. Terminal analogue models are often reliable under the conditions under which they are calibrated, but give bad predictions outside those conditions. Physical models on the other hand, because they are modelling the causal relationships, tend to do rather better.

You still have provided no evidence or justification for your assumption.

7. Clive,
I was going to write a post about this, but before I do that, maybe we can see if we clear something up. As I understand it, $dS/dt$ is simply emissions per unit time (GtC/yr, for example). The integral over time (as Dikran says) is then simply cumulative emissions at time $t$. Your model is therefore simply saying that the concentration at time $t$ is the initial concentration plus half of the cumulative emissions at time $t$.

Essentially

$C(t) = C_o + 0.5 S(t).$

You’re suggesting that if we fix emissions, then concentrations will quickly stabilise. For that to be true, S(t) has to become constant. However, fixing emissions means $dS/dt = constant$, not $S(t) = constant$. The integral of $dS/dt$ continues to increase with time (i.e., cumulative emissions increase even if emissions are fixed). Therefore, unless I’m misunderstanding the terms in your own model, $C(t)$ should continue to increase.

8. Clive Best says:

Sorry It should have read If $\frac{d^2S}{dt^2} > 0$

In other words while the rate of change of annual emissions keeps rising the airborne fraction remains ~ 0.5. Only when it has reduced to zero does the half life model starts to apply after an unknown lag time.

Look – I am being bombarded with comments (on your favourite WUWT website) so have had no time to write the software I wanted to do to properly describe it.

Since I know in advance that your intention is to rubbish my proposal – why don’t you just go ahead anyway!

• Okay, your model is airborne fraction is 0.5 as long as emissions are increasing, but if emissions become constant then sinks – after some relatively short time period – start to take up all our emissions so that atmospheric concentrations stabilise? Can you write an equations for $C(t)$ that also incorporates this latter process?

Since I know in advance that your intention is to rubbish my proposal – why don’t you just go ahead anyway!

No, that’s not really my intent. It might, however, be the outcome.

• Clive Best says:

Obviously there is a transition between the two phases so that afterwards sinks rise to one (airborne fraction falls to zero). So model is for the airborne fraction to decay by a half-life function $e^{-at}$ with decay time unknown but short – 1y,2y, 4y, 8y, 12 y max.

• Don’t you mean the other way around? The airborne fraction can’t go to 1 in your model because it is the fraction of our cumulative emissions. If emissions are constant, then cumulative emissions increase and an increasing airborne fraction would mean an increasing atmospheric CO2 concentration.

• Clive Best says:

Yes of course I am tired – sorry. I just corrected it.

• “Since I know in advance that your intention is to rubbish my proposal”

In my case, and I suspect ATTP, the intention is to point out potential flaws and weaknesses in your argument, so you can either strengthen it, or decide that the proposal is unsupportable. That is how science works, and what we do when we review papers. Your attitude to criticism is not doing you any favours.

• Clive Best says:

Of course. I welcome criticism so long as it is fair. I have written many papers in the past and reviewed those written by others. The basic question independent of whether the details of my model makes sense is: Will CO2 levels stabilise once we keep emissions constant. Yes/No

I think the answer must be yes, and It seems that there are several other eminent scientists who agree. If you or ATTP or anyone else can prove me wrong about this then I will retract it.

As far as my friend ATTP goes I just know him from the past, and if you look above at the comment he made on December 15, 2016 at 8:21 pm, you’ll see what I mean. I suspect he wanted to put that up in a blog post as part of a put-down.

Of course the decay of a doubling of CO2 doesn’t decay with a half life of 1 year, and nothing I wrote implied that! I actually calculated the decay here (using yet another simple model ;-0 ) and compared it to the Berne Model. There is a long tail out to geological time scales left for rock weathering.

• ” I welcome criticism so long as it is fair.”

is obviously inconsistent with

“Since I know in advance that your intention is to rubbish my proposal”

ATTP and I have offered technical criticisms of your argument, which you have largely ignored, for instance you have not provided evidence or justification for your central assumption “The simplest assumption is that the sink increase depends only on the partial pressure difference for a given year. “. Assuming that the airborne fraction will remain at a half is also very unlikely to be true as it is largely the result of exponentially rising emissions expanding the partial pressure difference each year, a point that you don’t seem to have adequately responded to either. Nor the point that you need to at least model the total (rather than net) fluxes into and out of the atmosphere to model the physics (even very approximately).

If I was an peer-reviewer, faced with such a response, I would not recommend the editor accept a revised manuscript. Would you?

• Clive Best says:

Don’t be so pompous. This is a blog post for Christ sake not a paper for Nature!

So you are now saying, if I understand you correctly, that the airborne fraction will increase if emissions stall !

That is the exact opposite of what you wrote on your own blog !

• “Don’t be so pompous. This is a blog post for Christ sake not a paper for Nature!”

LOL, it was you that brought up the topic of reviewing ;o)

“So you are now saying, if I understand you correctly, that the airborne fraction will increase if emissions stall !”

No, you don’t understand me correctly.

“That is the exact opposite of what you wrote on your own blog !”

which should have told you that.

You STILL have not addressed any of the criticisms I made. Perhaps you should ask yourself why you are indulging in name calling, but not actually answering my questions?

I think I’ll leave the discussion there. I have tried to engage with your argument with constructive criticism, and you obviously are not interested in engaging with them, so there seems little point in continuing,

• if you look above at the comment he made on December 15, 2016 at 8:21 pm, you’ll see what I mean. I suspect he wanted to put that up in a blog post as part of a put-down.

No, I just wanted to check if I was correctly representing your model. I wasn’t, but that’s mainly because your model appears incapable of modelling the decay of a pulse of emission.

• Clive Best says:

OK sorry. When I saw that 560 -> 280 drop I thought you were taking the p**s

No it only describes the assumption that sinks will increase to offset a new steady source of anthropogenic CO2. Anyway the post you have put up is fine by me and yes the Revelle Factor could be a killer as it chokes uptake.

• OK sorry. When I saw that 560 -> 280 drop I thought you were taking the p**s

Okay, fair enough.

Anyway the post you have put up is fine by me and yes the Revelle Factor could be a killer as it chokes uptake.

Yes, that is essentially the point.

• It is not correct to say that we only need emissions to be rising in order for the airborne fraction to be a half (or more accurately for the net flux of CO2 out of the atmosphere that year to be half of anthropogenic emissions). For this to be a constant requires anthropogenic emissions to be rising exponentially (at least using a naive model of the carbon cycle, see my paper).

It is hard to see how arguments made for d^2S/dt^2 > 0 tell you what will happen when d^2S/dt^2 = 0.

At the end of the day, the condition for a stabilisation of atmospheric CO2 is that total uptake equals total emissions. If we carry on emitting, that can’t happen when the difference in partial pressures becomes zero, because at that point natural emissions will balance natural uptake, so there is nothing to offset anthropogenic emissions. Eitherway the calculation cannot be based on uptake of a proportion of emissions as the uptake depends not on the emissions but on the difference in partial pressures, the constant fraction is just a coincidental consequence of approximately exponentially rising emissions. It is not a fundamental property of the carbon cycle.

9. “The simplest assumption is that the sink increase depends only on the partial pressure difference for a given year. ”

It is the magnitude of the sink that depends on the partial pressure difference, not the sink increase.

• Clive Best says:

OK but M(P) – M(P-DP) = DM(P)

10. The climate enforcers are here. Strangely they do not realise that their enforcement activities strengthen scepticism