## Monday, September 30, 2013

### Mathematical & observational proof that CO2 has no significant effect on climate

A recent comment at WUWT sets out the mathematical and observational proof that the effect of CO2 on climate is effectively nil.

The comment is a succinct summary of many recent posts here on the Hockey Schtick.

Bart says:
‘As Monk would say, “Here’s what happened”.’
That Monk is a smart guy. It’s been pretty obvious, actually, for a long time that the climate modelers had conflated the natural cyclical upswing with a sudden anthropogenic rise. But, the fact that the rise from approximately 1970-2000 was almost precisely the same as the rise from 1910-1940 gave the game away.
“This analysis shows that the real AGW effect is benign and much more likely to be less than 1 °C/century than the 3+ °C/century given as the IPCC’s best guess for the business-as-usual scenario.”
It’s actually pretty obvious from other data that it must be even less than that, and effectively zero. If we look at the relationship between CO2 and temperatures, it is apparent that to a very high degree of fidelity that
dCO2/dt = k*(T – Teq)
CO2 = atmospheric concentration
k = sensitivity factor
T = global temperature anomaly
Teq = equilibrium temperature
k and Teq are parameters for a 1st order fit. They may change over time, but are well represented by constants for the modern era since 1958 when precise measurements of CO2 became available.
This is a positive gain system – an increase in temperatures produces an increase in CO2 concentration. If we now presume that there is a positive feedback between CO2 and temperature, we get a positive feedback loop, which would be unstable.
There are other negative feedbacks, e.g., the T^4 radiation of heat. But, to maintain stability, these would have to be dominant, in which case the overall effect of CO2 on temperature would be negligible anyway. All roads lead to Rome –whatever the overall system response is, it must be such that the effect of CO2 on temperatures is effectively nil.
Now, a note on how the relationship above comes about. Atmospheric CO2 obeys a partial differential diffusion equation. The interface with the oceans sets boundary conditions. The boundary condition can be considered to obey something akin to Henry’s law (buffering processes complicate the actual relationship)
CO2(boundary) = Kh*CO2_Oceans(boundary)
The derivative of this is
dCO2(boundary)/dt = dKh/dt*CO2_Oceans(boundary) + Kh*dCO2_Oceans(boundary)/dt
Kh is a function of temperature, and thus can be expanded to first order as
Kh = Kh_eq + Kh_partial*(T – Teq)
where Kh_partial is the partial derivative of Kh to temperature. The oceans have been a net source of CO2 to the atmosphere. Assuming these are dominant, then
dCO2(boundary)/dt := (Kh_partial*dCO2_Oceans(boundary)/dt) * (T – Teq)
which is the form of the equation above with
k = Kh_partial*dCO2_Oceans(boundary)
In words, the influx of CO2 from the oceans produces a temperature dependent pumping action into the atmosphere.
The full dynamics are an atmospheric diffusion equation, with ocean boundary conditions as above, as well as a boundary condition with the land, which establishes a flow from the atmosphere into the miinerals and biota of the land, and an outflow from anthropogenic release of latent CO2. This is vastly simplified, of course, as the oceans contain their own biota and other CO2 absorbing processes. So, rather than division strictly into oceans and land, there is some overlap between the two reservoirs. In any case, though I have not yet worked out the details, it is clear where all this is heading. A very simplified ODE system model is
dCO2/dt = (CO2eq – CO2)/tau + H
dCO2_eq/dt = k*(T – Teq)
CO2 = atmospheric CO2
CO2eq = equilibrium CO2 established by the oceanic boundary condition
H = human inputs
tau = a time “constant”
The equilibrium CO2 is established by the interface with the oceans, and is relentlessly driven upward by temperatures above the equilibrium level. These feed into the atmospheric diffusion equation, which is being driven by human inputs, but is also being depleted by natural sinks which react in proportion to the CO2 level above equilibrium.
If “tau” is short, then H will be dramatically attenuated, and have little overall effect, and CO2 will track CO2eq. The actual dynamics are undoubtedly much more complicated, and “tau” would be more precisely modeled as an operator theoretic value which smooths the CO2 differential, leading to a “long tail” response, though not too long in the most significant components, as the data show that human inputs are being fairly rapidly sequestered.
But, this is effectively what the data show is happening. There really is no doubt about it. And, because of the positive feedback effect noted above, CO2 concentration cannot have a significant effect on temperature, because otherwise, we already would have maxxed out at some enormous level of CO2 and exceedingly high temperatures eons ago.

#### 31 comments:

1. Here my (long) reaction at WUWT:

Bart, you are essentially wrong on several points:

If we now presume that there is a positive feedback between CO2 and temperature, we get a positive feedback loop, which would be unstable.

If the positive feedback is modest, then the system is not unstable, only gives an extra increase of temperature and CO2 levels with (fb) and [error: must be vs.] without (nofb) feedback of CO2 on temperature:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/feedback.jpg

If we look at the relationship between CO2 and temperatures

The strong relationship is between the variability of (d)temperature(/dt) and dCO2/dt, not with the slope of dCO2/dt. By fitting the trends with an arbitrary factor and bias, you attribute the whole slope of dCO2/dt to temperature, but the slope is the result of all contributions to the increase, including human emissions.

The oceans have been a net source of CO2 to the atmosphere

Vegetation is a net sink for CO2 (~1 GtC/yr, humans emit ~9 GtC/yr), based on the oxygen balance. Besides vegetation and oceans, all other known natural sinks are either to small or too slow. The atmospheric increase is ~4 GtC/yr. Some 4 GtC/yr human emissions (as mass) + the extra release from the oceans goes where?

In words, the influx of CO2 from the oceans produces a temperature dependent pumping action into the atmosphere.

According to Henry's Law a temperature increase gives an increase in equilibrium setpoint of ~16 µatm CO2 with the atmosphere. Thus an increase of ~16 ppmv in the atmosphere will bring the in- and outfluxes of the ocean-atmosphere system back to what they were previous to the temperature increase. Starting from a system in dynamic equilibrium:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/upwelling_temp.jpg
The increase of CO2 in the atmosphere both reduces the increased upwelling and increased the downwelling

If the increase of CO2 was caused by a sudden extra upwelling of extra CO2 from the deep oceans (the "Coke effect"), that would have a similar effect on the balance, as in the case of a temperature increase:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/upwelling_incr.jpg

Temperature changes and upwelling changes act independent of each other and are simply additive:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/upwelling_incr_temp.jpg

Thus the dynamics of the ocean processes prove that a continuous release of the oceans from a sustained difference in temperature is impossible, as the increased CO2 level in the atmosphere influences both the release and uptake of CO2 from/into the oceans.

2. There is no evidence in the paleoclimate record that CO2 has any effect on climate. For example during the pre industrial revolution Holocene, there have been warming and cooling cycles with temperatures both cooler and warmer than today but CO2 levels were relatively constant at about .028% The climate change that we are experiencing today is very similar to what has been going on during the Holocene for the past 10,000 years. The primary causes appear to be changes in solar activity modified by ocean current osculations. CO2 warming is not required to explain what has been happening.

The primary greenhouse gas in our atmosphere is H2O, if you believe in greenhouse effect theory. H2O provides negative feedbacks to the addition of other greenhouse gases so as keep our climate relatively stable to changes in greenhouse gases.

Adding CO2 to the atmosphere adds to the atmosphere's radiant thermal insulation properties that restricts energy flow so as to cause warming in the lower atmosphere and cooling in the upper atmosphere where earth radiates to space in the LWIR. The warming in the lower atmosphere causes more H2O to enter the atmosphere which adds to the greenhouse insulation effect which is a positive feedback.

Besides being a greenhouse gas, H2O is a major coolant, moving heat from the earth's surface to where clouds form via the heat of vaporization. More heat is moved this way then by both convection and LWIR non window band radiation.. More H2O means that more heat is transfered which a negative feedback to the addition of CO2.

More H2O means more clouds form. Clouds reflect solar energy and they radiate LWIR energy to space much more efficiently then the clear air that they replace. More clouds are another negative feedback.

As the lower atmosphere warms the upper atmosphere where earth radiates to space in the LWIR cools . That is how insulation works. The cooling causes a rediction in H2O in the upper atmosphere which counteracts the addition of CO2. This upper atmosphere reduction on H2O is another negative feedback.

If it were not for these negative feedbacks our oceans would have boiled away millions of years ago. It is these negative feedbacks that mitigate any effect that added CO2 might have on climate.

3. "If the positive feedback is modest..."

Positive feedback is always unstable, unless a particular positive feedback is overcome by an overpowering negative feedback. Many people get confused by this. They will say, for example, that the small gain theorem says the plant will be stable if the feedback gain is less than unity. However, that only holds if the original plant is already stabilized, i.e., is dominated by negative feedback.

This was the point of my "all roads lead to Rome" comment. No matter how you slice it, the dominant feedback is overall negative, and CO2 cannot significantly produce warming.

"...but the slope is the result of all contributions to the increase, including human emissions."

Human emissions have negligible effect on atmospheric concentration. It is evident in the fact that the temperature relationship explains everything to a high degree of fidelity. We have argued this endlessly. You do not see it. In time, you will.

"Some 4 GtC/yr human emissions (as mass) + the extra release from the oceans goes where?"

Minerals and biota, would be my guess. I do not have to know where it goes to know what the data are telling me.

"The increase of CO2 in the atmosphere both reduces the increased upwelling and increased the downwelling."

If ocean upwelling at least equivalent to human inputs cannot drive atmospheric CO2, then neither can human inputs. As this contradicts your claim that it is human inputs driving things, you are in a rather difficult position.

"Temperature changes and upwelling changes act independent of each other and are simply additive:"

No, they are, to a valid degree of approximation, multi-linear. They modulate one another. The derivation is shown above.

- Bart

4. Bart,

The small gain theorem is for non-linear systems. In the case of linear systems (which seems the case for the CO2-temperature system), you can go up to unity for the combined feedback before the system gets unstable:
http://en.wikipedia.org/wiki/Positive_feedback
If the functions A and B are linear and AB is smaller than unity, then the overall system gain from the input to output is finite, but can be very large as AB approaches unity. In that case, it can be shown that the overall or "closed loop" gain from input to output is:
G = A/(1-AB)
When AB > 1, the system is unstable, so does not have a well-defined gain; the gain may be called infinite.

Further:

It is evident in the fact that the temperature relationship explains everything to a high degree of fidelity.

The temperature variation explains the short term variability, but the arbitrary offset and factor is only curve fitting, attributing the combination of all contributing factors to temperature alone, even if the increase of dCO2/dt may be 90% human emissions and 10% temperature...

Biota (all combined: ocean and land, microbes to animals) is a net sink of ~1 GtC/yr. That is all. Rock weathering is a much too slow process to give a short-term response to the changing CO2 levels...

If ocean upwelling at least equivalent to human inputs cannot drive atmospheric CO2, then neither can human inputs.

What you forget is that at the other end of the world, the oceans are sinks, absorbing about the remainder of the CO2 balance. Both CO2 releases and sinks of the oceans are heavily influenced by the CO2 level in the atmosphere, that is a dynamic equilibrium reaction. Human emissions are not influenced by the levels in the atmosphere.

The CO2 fluxes are only influenced by the pCO2 difference ocean-atmosphere (and wind speed), influenced both by temperature and concentration. The increase in the atmosphere balances the fluxes again with a e-fold time of ~15 years, not the extremely fast response time you need to dwarf the human emissions...

The main problem is that you (wrongly) attribute near all increase in the atmosphere to the temperature increase, which implies a short response time, but if the increase is (mainly) caused by the human emissions and only a small part by temperature, then the response is modest: an e-fold time of ~52 years.

The underlying point is that you think that the CO2 response is caused by one process, but there are a lot of different processes at work, each with their own response time: from fast (ocean surface) but limited (10% of the CO2 change in the atmosphere), slower (deep oceans) to very slow (rock weathering)... The fast processes give the short-term variability around the slope, the slower processes are the cause of the slope.

No, they are, to a valid degree of approximation, multi-linear. They modulate one another. The derivation is shown above.

The solubility curves of CO2 in seawater are near linear for several % in concentration or 1 K temperature increase. There is negligible influence on each other beyond simple addition. Again you derive a non-existing relationship from a wrong attribution...

5. "The small gain theorem is for non-linear systems. In the case of linear systems (which seems the case for the CO2-temperature system), you can go up to unity for the combined feedback before the system gets unstable:"

The small gain theorem is general, and applies to all cases. You are simply repeating it. And, you are making the mistake I alluded to above. The systems AB are already L2, i.e., stable, i.e., dominated by negative feedback.

"The temperature variation explains the short term variability, but the arbitrary offset and factor is only curve fitting..."

No. The slope of dCO2/dt matches the slope in T. Human inputs also have a slope. Therefore, there is no room to fit them in.

"What you forget is that at the other end of the world, the oceans are sinks, absorbing about the remainder of the CO2 balance."

The oceans are a net source.

"Human emissions are not influenced by the levels in the atmosphere."

Upwelling ocean CO2 content is what it is, regardless of atmospheric levels.

"The fast processes give the short-term variability around the slope, the slower processes are the cause of the slope."

There is no natural system response which can high pass the temperature related CO2 pumping, and low pass the human contributions, blending them together with no observable phase distortion. This is an unphysical paradigm.

"There is negligible influence on each other beyond simple addition."

I have demonstrated mathematically how they modulate one another. My math beats your assertion.

If this discussion progresses true to previous form, you will just find ways to re-assert what you have already stated, and I will repeat to you why you are wrong. How about we bypass that unproductive activity, and just leave things off at this point, agreeing to disagree, and letting any observers make up their own minds?

-Bart

6. Bart,

One step further:

The whole base of your thesis is the fact that the arbitrary offset and factor of Tanom matches the variability and slope of dCO2/dt

So let us see how that covers the opposite thesis in a simulated emissions / temperature setup.

We simulate an increase of CO2, caused by a combination of two processes:
The first process is a straight addition of CO2 into the atmosphere, where the yearly addition is increasing over time (slightly quadratic). The second process is a temperature process that is composed of a simple linear increase superimposed with a sinusoid of a few years length, which combined cause a small (8 ppmv/K) effect on CO2 with a small delay behind the temperature fluctuations/trend. In graph form:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/sim_co2_temp.jpg

Over 90% of the total CO2 increase over time is caused by the external injection, less than 10% by the temperature increase.

Let us have a look at the derivatives of the above simulation:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/sim_dco2_dT_Tanom.jpg

Something remarkably happened: the temperature anomaly perfectly fits the variability ánd the trend of the CO2 increase, while in the simulation over 90% of the increase is not from the temperature increase. What happened is that the factor and offset used to fit the curves are in fact including the result of the increasing emissions and falsely attribute these to temperature. In reality, there is zero trend in the derivative of the CO2 increase caused by temperature, only the full variability. All of the trend in the year-by-year CO2 increase in the above simulation is caused by the slight curvature of the emissions (and therefore in the totals of the atmosphere), not by the linear increase of temperature.

Thus the base of your thesis is completely wrong, one can't use an arbitrary offset and factor to attribute the full slope in rate of change to one of the variables.

For the real situation in the atmosphere we have:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/base_co2_temp_1960-cur.jpg
The derivatives:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/base_dco2_Tanom_1960-cur.jpg

The trend of the temperature derivative is zero. Thus whatever the effect of temperature on any CO2 production process, that probably has zero effect on the rate-of-change trend. The trend derived from the emissions fully fits the trend and only the variability around the trend is mainly caused by the temperature variability.

This definitively proves that the temperature anomaly has not the slightest connection with the origin of the CO2 increase over time and that one need to look at which process has a non-linear curvature which may cause the increasing rate of change of CO2 in the atmosphere.
The simplest answer that fits all observations is the human emissions...

With thanks to jimmi_the_dalek who inspired me to make the problem with Bart's curve fitting clear in graphs.

7. Another one:

The oceans are a net source.

I will comment there...

8. "The whole base of your thesis is the fact that the arbitrary offset and factor of Tanom matches the variability and slope of dCO2/dt"
Yes and, to match emissions to concentration, you also have to use an arbitrary offset and scale factor. No advantage for you there.

"Let us have a look at the derivatives of the above simulation:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/sim_dco2_dT_Tanom.jpg"

Your temperature variations are 90 degrees out of phase with your dCO2(temp)/dt. In the real world, these are perfectly in phase. Your model fails.

"Something remarkably happened: the temperature anomaly perfectly fits the variability ánd the trend of the CO2 increase, while in the simulation over 90% of the increase is not from the temperature increase."

Well, of course. You set it up to do that. GIGO.

"For the real situation in the atmosphere we have..."

Poor resolution, and a trivially superficial resemblance. In the real world, the rates of emissions and atmospheric concentration are diverging.

"The trend of the temperature derivative is zero. Thus whatever the effect of temperature on any CO2 production process, that probably has zero effect on the rate-of-change trend."

Honestly, it appears you do not understand the model at all. The trend in the temperature record integrates into the quadratic curvature in the CO2 record. The accumulated emissions also have pronounced curvature. But, that curvature is already accounted for by temperature dependent term.

Global temperatures are beginning to fall. That accelerating divergence between emissions and concentration will become very stark in the not-too-distant future. Keep your eye on it. At some point, you will realize I am right.

-Bart

9. Bart,

Yes and, to match emissions to concentration, you also have to use an arbitrary offset and scale factor. No advantage for you there.

No offset needed at all and without any scale factor, the slope of the rate of change in emissions is already twice the observed slope...

Your temperature variations are 90 degrees out of phase with your dCO2(temp)/dt. In the real world, these are perfectly in phase. Your model fails.

Have a better look: CO2(temp) lags T, dCO2(temp) lags dT, but dCO2(temp) and T are perfectly in phase, as good as in the simulation as in the real world.

Well, of course. You set it up to do that. GIGO.

Of course I have set it up, but with quite realistic figures for emissions and temperature influence on CO2 (4.5 ppmv/K short term, 8 ppmv/K long term). It simply shows that you can match the rate of change of CO2 by the temperature anomaly, even if temperature has hardly any influence on CO2 levels.

The trend in the temperature record integrates into the quadratic curvature in the CO2 record. The accumulated emissions also have pronounced curvature. But, that curvature is already accounted for by temperature dependent term.

The trend in the rate of change of CO2 is the result of both the temperature influence AND the influence of the emissions. By matching the temperature record with the rate of change of CO2 you attribute both influences to the temperature influence alone. That is circular reasoning.

Further, integrating the temperature record in this case gives a quadratic temperature function, what kind of physical process is that?

That accelerating divergence between emissions and concentration will become very stark in the not-too-distant future.

The emissions were more constant in the past decade or so, thus the derivative isn't increasing, but the sinks still are, as the real driver of the sinks is the CO2 level in the atmosphere which still is increasing. That has little influence on the integrals up to now:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/acc_co2.jpg
But it will give a change if the emissions remain constant over longer periods. Then we will see a leveling off of CO2 levels in the atmosphere. But that is hardly influenced by a sustained temperature difference.

10. "No offset needed at all..."

The offset is in your data. The proxy data which you rely upon for your starting point has been adjusted to flow into the direct measurements.

... and without any scale factor, the slope of the rate of change in emissions is already twice the observed slope...

Which is meaningless.

"...dCO2(temp) and T are perfectly in phase..."

At e.g., about 19 years, dCO2(temp)/dt is at a peak, while T(anom) is at maximum upswing. 90 degrees out of phase. If I didn't know you better, I'd suggest you were trying to fool people. From what I know of you, though, I think you are trying to fool yourself.

"Of course I have set it up, but..."

Sorry, the model fails. You are 90 deg out of phase.

"By matching the temperature record with the rate of change of CO2 you attribute both influences to the temperature influence alone."

Not at all. You are misinterpreting what I am doing. Once again, I am proposing a temperature dependent process, not a temperature dominant process. There are two primary actions occurring: outgassing of CO2 from CO2-rich upwelling waters and a temperature modulation of that process.

"Further, integrating the temperature record in this case gives a quadratic temperature function, what kind of physical process is that?"

It is the very straightforward integration of a temperature modulated process.

"The emissions were more constant in the past decade or so..."

No, they weren't. If anything, they accelerated after 2000. Look at the plot.

-Bart

11. Bart,

Starting point is 1960 from direct measurements. Any increase in CO2 is from that point on, either from temperature or emissions. No offset needed.
It is easier to explain an increase in the atmosphere from a known source that shows twice the increase in rate of change and fits all observations than from an unknown source which needs ignoring a lot of observations...

At e.g., about 19 years, dCO2(temp)/dt is at a peak, while T(anom) is at maximum upswing. 90 degrees out of phase.

Sorry, my fault:
CO2 lags T, dCO2 lags dT and dCO2 matches T, as good in my simulation as in reality. dCO2(temp) in the graph was in fact dT/dt * 4.5 to show the impact of dT on the variability of dCO2 (which lags dT). I have changed the graph accordingly.

Anyway, if you look at Tanom, that matches dCO2/dt perfectly. By your thesis sufficient to declare that temperature is the only variable that controls dCO2, while we used 90% emissions and 10% temperature in the simulation. That works for any simulation of a mix of "emissions" and temperature, as long as you include a lag between T and CO2 and choose the right parameters for offset and slope of Tanom.

There are two primary actions occurring: outgassing of CO2 from CO2-rich upwelling waters and a temperature modulation of that process.

Again, you are mixing in an extra variable which is only partly influenced by temperature.
- increased CO2 upwelling (concentration and/or quantity) causes an increase in CO2 influx independent of temperature.
- increased temperature causes an increase in CO2 influx and a decrease in CO2 outflux independent of upwelling.
- an increased temperature has little influence on an increase in upwelling caused by concentration and non if caused by quantity, besides its own effect. These two influences are practically independent of each other.
- the increase in upwelling and/or temperature increases the CO2 level in the atmosphere.

But more important: you completely ignore that:
- the increase of CO2 in the atmosphere decreases the upwelling fluxes and increases the downwelling fluxes.
Which makes that after a reasonable amount of time, the influence of temperature and/or extra upwelling is balanced again at a higher CO2 level.

No, they weren't. If anything, they accelerated after 2000.

Agreed, was wrong there. We will see what happens with the sink rate in the near future...

12. "Starting point is 1960 from direct measurements. Any increase in CO2 is from that point on, either from temperature or emissions. No offset needed."

But, this is a trivial fit. You have two series dominated by linear trends starting at zero, so of course they are scale similar, and integrate into a scale similar total quantity. It's tautological.

Moreover, the accumulations start to diverge in the 2000s, right at the time the rate of change of CO2 flattens in lock-step with the flattening of temperatures.

Temperature anomaly requires an offset in this model. It already has one built in because it is an anomaly, but that offset is arbitrary. So, there is no basis to object to offsetting it with an appropriate value. Moreover, the offset has no effect on the trend, which is the quantity upon which I am basing my contention that CO2 is not dependent on human inputs.

"By your thesis sufficient to declare that temperature is the only variable that controls dCO2..."

No, again, the model is of CO2 pumping from the upwelling waters which is modulated by the temperature anomaly from a particular baseline.

"...while we used 90% emissions and 10% temperature in the simulation."

I am really not interested in contrived simulations. I am interested in the real world, where dCO2/dt and T are in phase, leading to a 90 deg phase lag in accumulated CO2 relative to temperature. You cannot get this phase response without having the relationship of CO2 being dependent on the integral of temperature anomaly. And, when you integrate the temperature anomaly, scaled to fit the variational components, you fit the curvature. As you must, because the temperature anomaly has a slope which matches the slope of dCO2/dt when it is scaled so that the variability matches. The offset to the temperature anomaly has no effect on that slope.

Let me repeat that last: The offset to the temperature anomaly has no effect on that slope. The offset does not create a term upon which I am basing my argument.

"- an increased temperature has little influence on an increase in upwelling caused by concentration and non if caused by quantity, besides its own effect."

Again, I have shown the math. My math beats your assertion.

"Agreed, was wrong there. We will see what happens with the sink rate in the near future..."

That is probably a good point at which to leave off. We will wait and see.

-Bart

13. I didn't notice this before, but kudos to Roy Spencer for pointing out in 2009 that much of the CO2 rise may be natural

http://wattsupwiththat.com/2009/05/12/spencer-on-an-alternate-view-of-co2-increases/

14. MS,

If you look at the comments on Dr. Spencer's essay, you will find mine too...

Bart,

The point is not that you need a scale factor and offset to match T and dCO2.
The point is that you can match any mix of human emissions and CO2 increase caused (indirectly) by temperature in the atmosphere in the derivative with a different set of offset and factor of T vs. dCO2. Thus your perfect match between T and dCO2 doesn't say anything about the mix that caused the increase in rate of change.
But I will work that out tomorrow...

Another problem is that the temperature is more or less linearly increasing and thus completely flat in the derivative (including a lag of dCO2 vs. dT), which has no direct contribution to the slope of dCO2. That needs an unknown process which responds non-linearly to a linear temperature increase. That is quite certainly not ocean upwelling.

That while the trend of emissions is slightly quadratic itself and largely explains the slope of the rate of change of CO2 (be it that the real slope is from the total increase in the atmosphere above equilibrium, not from the rate of change of CO2).

1. These items have all been addressed. There is nothing new. You're just not getting the argument, and there appears to be no means to bridge the gap, given how many pages we have dedicated to it here and elsewhere.

We will wait and see. -Bart

15. Bart,

The mix and match of different processes by the T-anomaly graph has never been addressed by you. You simply assume that the perfect match is caused by some natural, temperature dependent process and that it is. But any mix of natural and human releases can be matched with its own set of offset and factor. Here some thoughts...

First a comparison between dT and Tanom and CO2 increase in the atmosphere as observed.
The dCO2 variability near perfectly matches dT variability with a lag of about 90°. Indeed Tanom matches dCO2 at exact the same timing, but that is simply because CO2 lags T and all what happened is that by comparing the derivative of CO2 with T, you bring the shift between these two back to zero.

In the case of a direct influence of temperature (on e.g. the solubility of CO2 in the oceans), T increase causes (a part of) CO2 increase and dT variability causes dCO2 variability. But dT shows zero contribution to the slope in dCO2 (even slightly negative). Thus the slope in dCO2 is entirely from a different process (temperature dependent or not).

WFT doesn't have human emissions in its database, or it would be possible to show that the emissions increase over time with about twice the slope seen in dCO2. But we can simulate that (with realistic figures for emissions and influence of T on CO2 levels, each of which about halve is remaining in the atmosphere:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/sim_co2_temp_95.jpg
with its derivative:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/sim_dco2_dT_Tanom.jpg

If there is a non-linear temperature dependent natural process involved, the increase in the atmosphere may be caused by a 50:50 mix of natural and human emissions:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/sim_co2_temp_50.jpg
in the derivative:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/sim_dco2_dT_Tanom_50.jpg

No problem to match the slope and timing of dCO2 with Tanom, only a problem with the amplitude of the variability. But if the underlying natural process shows much more reaction to fast changes in temperature than to slower changes, that may be solved. If the fast and slow processes are independent of each other, then dT still shows the full amplitude and d(emissions) still show the full slope of dCO2.

One step further: 90% natural, 10% human:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/sim_co2_temp_10.jpg
and its derivative:
http://www.ferdinand-engelbeen.be/klimaat/klim_img/sim_dco2_dT_Tanom_10.jpg

Interesting observations:
- The more the natural contribution increases, the more difficult it is to match the amplitude of the variability with Tanom, while that is no problem at all for dT.
- The slope and the amplitude of dCO2 may increase with Tanom, depending of the reaction of CO2 from some natural process, but in all cases temperature alone doesn't do the job, except for the case that temperature hardly plays a role...

Main result:
Temperature anomaly matching has zero predictive power for the attribution of the cause of the increase of CO2 in the atmosphere...

1. Nothing new. Already addressed. - Bart

16. Arctic Oscillation seems strongly related to global rate of change of atm. CO2 but with considerable lag.

This may explain why the short segment from Barrow has greater variability.

The Arctic seems to play a key role in how much of our emissions get absorbed.

The IPCC seems intent on avoiding any boarder analysis and is still trying to reduce everything to CO2 plus random stochastic “noise”.

http://climategrog.wordpress.com/?attachment_id=259

17. http://www.ipa.org.au/library/publication/1339463007_document_break_paper_apjas_ipa.pdf

18. http://notrickszone.com/2013/10/08/carbon-dioxide-and-the-ocean-temperature-is-driving-co2-and-not-vice-versa/

1. I am reacting there...

19. http://www.scribd.com/doc/129802522/Natural-or-Not

20. http://icecap.us/index.php/go/joes-blog/carbon_dioxide_in_not_the_primary_cause_of_global_warming_the_future_can_no/

21. Satellite movie showing how nature controls CO2 levels

http://svs.gsfc.nasa.gov/vis/a000000/a003500/a003562/carbonDioxideSequence2002_2008_at15fps.mp4

22. http://climategrog.wordpress.com/?attachment_id=259

http://wattsupwiththat.com/2013/12/02/is-the-bern-model-non-physical/#comment-1490103

http://climategrog.wordpress.com/page/2/

23. Related comments at WUWT:

Hoser says:
November 21, 2013 at 7:08 pm
The 14C (as CO2) bomb test is pretty close to a single-turnover experiment, essentially measuring the off rate, that is the rate of exit from the atmosphere. There is a steady state level of production from cosmic rays, that sets the minimum level of 14C. So there is an on rate as expressed in the equations above.

Lets do math.

dC/dt = N – kC (1),
where C is 14C as CO2, N is a fixed rate of 14C formation from cosmic rays, and k is the off rate, that is the rate 14C leaves the atmosphere, not to return. If you can’t get past this part, give up.

This is a non-homogeneous differential equation. Let assume C can be expressed as the product of U and V where U is the solution to the homogeneous equation.

C = U*V (2)
dU/dt + kU = 0 (3).

With solutions
U = A*e^(-kt) (4)

Because dC/dt = dU/dt*V + U*dV/dt, we can substitute (2,3) into (1).

dU/dt*V + U*dV/dt + U*V = N (5)

Rearranging

V*(dU/dt + U) + U*dV/dt = N (6), and because of (3)

U*dV/dt = N (7) and

dV/dt = N / U (8).

Substitute (4) into (8) and

dV/dt = N*A*e^(kt) (9).

Integrating from 0 to t

V = N/Ak * (e^(kt) – 1) (10).

Solutions of C are A1 * U*V + A2 * U, so after some multiplication

C = A1*N/k *(1-e^(-kt) ) + A2 * e ^(-kt) (11)

At t=0, C = A2, and at t = inf, C = A1*N/k. Let A1 = k and A2 = N + Xo, where Xo is the excess 14C we start with at t = 0, then

C = N * (1-e^(-kt)) + (N + Xo)* e^(-kt) (12).

Rearranging we get

C = N*(1 – e^(-kt) + e^(-kt) ) + X * e^(-kt) (13), or finally

C = N + Xo e^(-kt) (14), and since C = N + X, after rearranging

X = Xo e^(-kt) (15).

Clearly, we see the fixed amount N and the decay of the excess X with rate constant k. When I fit the data using this equation, I get a half-life of about 5 years for 14C using ORNL data. My previous attempt failed to subtract N from C first. I just fit the fall, which was a mistake.

24. Hoser says:
December 11, 2013 at 11:32 pm
Hopelessly over complicated. From Fig 5 and discussion, I question whether the writer understands the system we are talking about. 14C is effectively a tracer. There is a steady state level maintained by adding the same amount to the atmosphere each year from transmutation via cosmic ray interactions with nitrogen atoms in the atmosphere. The total amount of carbon is essentially unchanged. The atomic bombs did not increase the amount of CO2 in the atmosphere, but did almost double the tiny amount of 14C.

A basic reasonable assumption of a single turnover kinetics is the tracer leaves the reservoir and does not come back. 14C leaving the atmosphere can safely be assumed to not return in any significant fraction. Hence, when we see it leave, that is the off rate cleanly determined. Now when we have an equilibrium level, we can calculate the on rate as well. It doesn’t matter how complicated the reservoir system you create. The only part that matters is the off rate constant. We don’t know how the material partitions into other reservoirs, but that issue is beyond the scope of this experiment.

I showed previously [1] the math works out nicely, and essentially you can slice off the steady-state amount of 14C and look at the decay of the excess alone. The t1/2 is about 5 years. Even accounting for the increase in ppmv CO2 diluting the 14C from 1963 to 1993 and beyond, the t1/2 is still about 5 years (at least it was in my hands). Once you know the t1/2 is 5 years, a number of interesting calculations follow.

For one thing, it becomes pretty clear the increase in atmospheric CO2 cannot possibly be due to anthropogenic CO2. The CO2 quantity changes simply don’t match what humans have produced given the amount of CO2 we have produced each year from the late 1700s and how much would be left Y years after emission.

1) http://wattsupwiththat.com/2013/11/21/on-co2-residence-times-the-chicken-or-the-egg/#comment-1481426

http://wattsupwiththat.com/2013/11/21/on-co2-residence-times-the-chicken-or-the-egg/#comment-1481426

25. phlogiston says:
December 12, 2013 at 12:53 am
Ferdinand Engelbeen says:
December 12, 2013 at 12:27 am

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

NO NO NO NO NO NO NO NOOOOOO! (adopts facial expression of The scream by Munch)

Ferdninand we all respect your erudition in regard to atmospheric CO2 but I fully agree here with Hoser that all this discussion is completely missing what a radio tracer measurement really is. And vastly over-complicating the discussion as a result.

A radiotracer measures a single removal term. PERIOD. A pulse of CO2 enters the atmosphere different from the other CO2 due to 14C. So it can be tracked in exclusion of any other CO2.

It is COMPLETELY IRRELEVANT all the other cycling and dilution and dynamics, pressure, temperature etc. of CO2 that are going on, the 14 tracer simply tells us the removal term for CO2. That is the whole point of a radiotracer measurement.

From the bomb test data we know that:
CO2 removal half life = 5 years
CO2 residence time = half life / ln2 = 5 / 0.693 = 7.7 years

That is the WHAT. Everything else is the WHY.

http://wattsupwiththat.com/2013/12/11/co2-residence-times-take-two/#comment-1498559

26. http://wattsupwiththat.com/2013/12/11/co2-residence-times-take-two/#comment-1499508

27. Hoser says:
December 12, 2013 at 6:07 pm
Ferdinand Engelbeen says:
December 12, 2013 at 2:38 pm
Regarding how long to remove the extra mass.

If the off rate measured for 14C does scale to the whole atmosphere, then equilibrium is roughly maintained by an on rate. A pulse of excess CO2 should be taken up by one or more reservoirs unless there is saturation of these reservoirs or another factor alters the on rate. I suppose we are talking about feedbacks now. But just for fun, let’s say the simple model is correct. What would we expect for an off rate (that is how much CO2 should exit the atmosphere per year)? With 3264 Gton CO2 in atmosphere, a roughly 5 year t1/2 gives us 413 Gton/yr flux. Approximately 450 Gton/yr is estimated from http://www.ipcc.ch/publications_and_data/ar4/syr/en/contents.html and http://en.wikipedia.org/wiki/Carbon_dioxide_in_Earth's_atmosphere.

Not bad for a simple model.
Hoser says:
December 12, 2013 at 6:29 pm
To summarize, there are two conclusions I tentatively come to. 1) Humans cannot be the cause of the rise in CO2 because the rise is much greater than the amount of CO2a that should be present given the simple model and t1/2 of 5 years. 2) There is another natural process at work shifting the equilibrium between CO2 reservoirs such that atmospheric CO2 is rising, that is, the on rate has increased.

Using Oak Ridge Natl. Lab global CO2 emission data (1751-2010), I estimate anthropogenic CO2 is now about 200 Gton of the total CO2 in the atmosphere. If we removed it all, atmospheric CO2 would be about 375 ppmv. This view seems relatively balanced with what we know.

http://wattsupwiththat.com/2013/12/11/co2-residence-times-take-two/#comment-1499714

28. The Bart/Ferdinand debate continues:

http://wattsupwiththat.com/2015/07/28/carbon-sink-detected-underneath-worlds-deserts/#comment-1999370