Guest essay by Ferdinand Engelbeen
Both Bart Bartemis and Dr. Murray Salby are confident that temperature is the only/main cause of the CO2 increase in the atmosphere. I am pretty sure that human emissions are to blame. With this contribution I hope to give a definitive answer…
Some of you may remember the lively discussions of already 5 years ago about the reasons why I am pretty sure that the CO2 increase in the atmosphere over the past 57 years (direct atmospheric measurements) and 165 years (ice cores and proxies) is manmade. That did provoke hundreds of reactions from a lot of people pro and anti.
Since then I have made a comprehensive overview of all the points made in that series of discussions at:
There still is one unresolved recurring discussion between mainly Bart/Bartemis and me about one – and only one – alternative natural explanation: if the natural carbon cycle is extremely huge and the sinks are extremely fast, it is -theoretically- possible that the natural cycle dwarfs the human input. That is only possible if the natural cycle increased a fourfold in the same time frame as human emissions (for which is not the slightest indication) and it violates about all known observations. Nevertheless, Bart’s (and Dr. Salby’s) reasoning is based on a remarkable correlation between temperature variability and the CO2 rate of change variability with similar slopes:
Capture: Fig.1: Bart’s combination of T and dCO2/dt from WoodForTrees.org
Bart (and Dr. Salby) thinks that the match between variability and slopes (thanks to an arbitrary factor and offset) proves beyond doubt that temperature causes both the variability and slope of the CO2 rate of change. The following will show that variability and slope have nothing in common and temperature is not the cause of the slope in the CO2 rate of change.
2. The theory.
2.1 Transient response of CO2 to a step change in temperature.
To make it clear we need to show what happens with CO2 if one varies temperature in different ways. CO2 fluxes react immediately on a temperature change, but the reaction on CO2 levels needs time, no matter if that is by rotting vegetation or the ocean surfaces. Moreover, increasing CO2 levels in the atmosphere reduce the CO2 pressure difference between ocean surface and the atmosphere, thereby reducing the average in/out flux, until a certain CO2 level in the atmosphere is reached where in and out fluxes again are equal.
In algebraic form:
dCO2/dt = k2*(k*(T-T0) – ΔpCO2)
Where T0 is the temperature at the start of the change and ΔpCO2 the change in CO2 partial pressure in the atmosphere since the start of the temperature change, where pCO2(atm) was in equilibrium with pCO2(aq) at T0. The transient response in rate of change is directly proportional to the CO2 pressure difference between the pCO2 change in water (caused by a change in temperature) and the CO2 pressure in the atmosphere.
When the new equilibrium is reached, dCO2/dt = 0 and:
k*(T-T0) = ΔpCO2
Where k = ~16 ppmv/°C which is the value that Henry’s law gives for the equilibrium between seawater and the atmosphere.
In the next plot we assume the response is from vegetation, mainly in the tropics, as that is a short living response as will be clear from measurements in the real world in chapter 3:
Caption: Fig. 2: Response of bio-CO2 on a step change of temperature
As one can see, a step response in temperature gives an initial peak in dCO2/dt rate of change which goes back to zero when CO2 is again in equilibrium with temperature. That equilibrium can be static (for an open bottle of Coke) or dynamic (for the oceans). In the latter case one speaks of a “steady state” equilibrium or a “dynamic equilibrium”: still huge exchanges are going on, but the net result is that no CO2 changes are measurable in the atmosphere, as the incoming CO2 fluxes equal the outgoing CO2 fluxes.
Taking into account Henry’s law for the solubility of CO2 in seawater, any in/decrease of 1°C has the same effect if you take a closed sample of seawater and let it equilibrate with the above air (static) or have the same in/decrease in (weighted) average global ocean temperature with global air at steady state (dynamic): about 16 ppmv/°C.
2.2 Transient response of CO2 to an increasing temperature trend.
If the temperature has a slope, CO2 will follow the slope with some delay.
Caption: Fig. 3: Response of bio-CO2 on a continuous increase of temperature
A continuous increase of temperature will induce a continuous increase of CO2 with an increasing dCO2/dt until both increases parallel each other and dCO2/dt remains constant. This ends when the “fuel” (like vegetation debris) gets exhausted or the temperature slope ends. In fact, this type of reaction is more applicable to the oceans than on vegetation, but this all is more about the form of the reaction than what causes it…
A typical example is the warming from the depth of a glacial period to an interglacial: it takes about 5,000 years to reach the new maximum temperature and CO2 lags the temperature increase with some 800 +/- 600 years.
2.3 Transient response of CO2 to a sinusoid.
Many changes in nature are random up and down, besides step changes and slopes. Let’s first see what happens if the temperature changes with a nice sinus change (a sinusoid):
Caption: Fig. 4: Response of bio-CO2 on a continuous sinusoidal change in temperature
It can be mathematically explained that the lag of the CO2 response is maximum pi/2 or 90° after a sinusoidal temperature change . Another mathematical law is that by taking the derivatives, one shifts the sinusoid forms 90° back in time. The remarkable result in that case is that changes in T synchronize with changes in dCO2/dt, that will be clear if we plot T and dCO2/dt together in next item.
2.4 Transient response of CO2 to a double sinusoid.
To make the temperature changes and their result on CO2 changes a little more realistic, we show here the result of a double sinusoid for sinusoids with different periods. After all natural changes are not that smooth…:
Caption: Fig. 5: Response of bio-CO2 on a continuous double sinusoidal change in temperature
As one can see, the change in CO2 still follows the same form of the double sinusoid in temperature with a lag. Plotting temperature and dCO2/dt together shows a near 100% fit without lag, which implies that T changes directly cause immediate dCO2/dt changes, but that still says nothing about any influence on a trend. In fact still T changes lead CO2 changes and dT/dt changes lead dCO2/dt changes, but that will be clear in next plot…
2.4 Transient response of CO2 to a double sinusoid plus a slope.
Now we are getting even more realistic, not only we introduced a lot of variability, we also have added a slight linear increase in temperature. The influence of the latter is not on CO2 from the biosphere (that is an increasing sink with temperature over longer term), but from the oceans with its own amplitude:
Caption: Fig. 6: Response of Natural CO2 on a continuous double sinusoidal plus slope change in temperature
As one can see, again CO2 follows temperature as well for the sinusoids as for the slope. So does dCO2/dt with a lag after dT/dt, but with a zero slope, as the derivative of a linear trend is a flat line with only some offset from zero.
This proves that the trend in T is not the cause of any trend in dCO2/dt, as the latter is a flat line without a slope. No arbitrary factor can match these two lines, except (near) zero for the temperature trend to match the dCO2/dt trend, but then you erase the amplitudes of the variability…
Thus while the variability in temperature matches the variability in CO2 rate of change, there is no influence at all from the slope in temperature on the slope in CO2 rate of change.
Conclusion: A linear increase in temperature doesn’t introduce a slope in the CO2 rate of change at any level.
2.4 Transient response of CO2 to a double sinusoid, a slope and emissions.
All previous plots were about the effect of temperature on the CO2 levels in the atmosphere. Volcanoes and human emissions are additions which are independent of temperature and introduce an extra amount of CO2 in the atmosphere above the temperature dictated dynamic equilibrium. That has its own decay rate. If that is slow enough, CO2 builds up above the equilibrium and if the increase is slightly quadratic, as the human emissions are, that introduces a linear slope in the derivatives.
Caption: Fig. 7: Response of CO2 on a continuous double sinusoidal + slope change in temperature + emissions
Several important points to be noticed:
– The variability of CO2 in the atmosphere still lags the temperature changes, no matter if taken alone or together with the result of the emissions. No distortion of amplitudes or lag times. Only simple addition of independent results of temperature and emissions.
– The slope of the natural CO2 rate of change still is zero.
– The relative amplitude of the variability is a small factor compared to the huge effect of the emissions.
– The slope and variability of temperature and CO2 rate of change is a near perfect match, despite the fact that the slope is entirely from the slightly quadratic increase of the emissions and the effect of temperature on the slope of the CO2 rate of change is zero…
Conclusion: The “match” between the slopes in temperature and CO2 rate of change is entirely spurious.
3. The real world.
3.1 The variability.
Most of the variability in CO2 rate of change is a response of (tropical) vegetation on (ocean) temperatures, mainly the Amazon. That it is from vegetation is easily distinguished from the ocean influences, as a change in CO2 releases from the oceans gives a small increase in 13C/12C ratio (δ13C) in atmospheric CO2, while a similar change of CO2 release from vegetation gives a huge, opposite change in δ13C. Here for the period 1991-2012 (regular δ13C measurements at Mauna Loa and other stations started later than CO2 measurements):
Caption: Fig. 8: 12 month averaged derivatives from temperature and CO2/ δ13C measurements at Mauna Loa .
Almost all the year by year variability in CO2 rate of change is a response of (tropical) vegetation on the variability of temperature (and rain patterns). That levels off in 1-3 years either by lack of fuel (organic debris) or by an opposite temperature/moisture change . Over periods longer than 3 years, it is proven from the oxygen balance that the overall biosphere is a net, increasing sink of CO2, the earth is greening , .
Not only is the net effect of the biological CO2 rate of change completely flat as result of a linear increasing temperature, it is even slightly negative in offset…
The oceans show a CO2 increase in ratio to the temperature increase: per Henry’s law about 16 ppmv/°C. That means that the ~0.6°C increase over the past 57 years is good for ~10 ppmv CO2 increase in the atmosphere that is a flat line with an offset of 0.18 ppmv/year or 0.015 ppmv/month in the above graph.
There is a non-linear component in the ocean surface equilibrium with the atmosphere for a temperature increase, but that gives not more than a 3% error on a change of 1°C at the end of the flat trend or a maximum “trend” of 0.00045 ppmv/month after 57 years. That is the only “slope” you get from the influence of temperature on CO2 levels. Almost all of the slope in CO2 rate of change is from the emissions…
3.2 The slopes.
Human emissions show a slightly quadratic increase over the past 115 years. In the early days more guessed than calculated, in recent decades more and more accurate, based on standardized inventories of fossil fuel sales and burning efficiency. Maybe more underestimated than overestimated, because of the human nature to avoid paying taxes, but rather accurate +/- 0.5 GtC/year or +/- 0.25 ppmv/year.
The increase in the atmosphere was measured in ice cores with an accuracy of 0.12 ppmv (1 sigma) and a resolution (smoothing) of less than a decade over the period 1850-1980 (Law Dome DE-08 cores). CO2 measurements in the atmosphere are better than 0.1 ppmv since 1958 and there is a ~20 year overlap (1960 – 1980) between the ice cores and the atmospheric measurements at Mauna Loa. That gives the following graph for the temperature – emissions – increase in the atmosphere:
Caption: Fig. 9: Temperature, CO2 emissions and increase in the atmosphere .
While the variability in temperature is high, that is hardly visible in the CO2 variability around the trend, as the amplitudes are not more than 4-5 ppmv/°C (maximum +/- 1 ppmv) around the trend of more than 90 ppmv. To give a better impression, here a plot of the effect of temperature on the CO2 variability in the period 1990-2002, where two large temperature and CO2 changes can be noticed: the 1991/2 Pinatubo eruption and the 1998 super El Niño:
Caption: Fig. 10: Influence of temperature variability on CO2 variability around the CO2 trend .
It is easy to recognize the 90° lag after temperature changes, but the influence of temperature on the variability is small, here calculated with 4 ppmv/°C. For the trend, the CO2 increase caused by the 0.2°C ocean surface temperature increase in that period is around 3 ppmv of the 17 ppmv measured…
3.3 The response to temperature variability and human emissions:
With the theoretical transient response of CO2 to temperature in mind, we can calculate the response of vegetation and oceans to the increased temperature and its variability:
Caption: Fig. 11: Transient response of bio and ocean CO2 to temperature .
The bio-response to temperature changes is very fast and zeroes out after a few years , the response to the temperature amplitude is about 4-5 ppmv/°C, based on the response to the 1991 Pinatubo eruption and the 1998 El Niño.
The response of the ocean surface is slower, but stronger in effect. The 16 ppmv /°C is based on the long-term response in ice cores and Henry’s law for the solubility of CO2 in ocean waters (4-17 ppmv /°C in the literature).
In reality, both oceans and the biosphere are net sinks for CO2, due to the increased CO2 pressure in the atmosphere and the biosphere also a net sink due to increased temperature on periods of more than 3 years. That is not taken into account here, but is used in the calculation of the net increase of CO2 in the atmosphere with the introduction of human emissions.
If we introduce human emissions , that gives a quite different picture of the relative dimensions involved:
Caption: Fig. 12: Human emissions + calculated and measured CO2 increase + transient response of bio and ocean CO2 to temperature .
The influence of temperature both in variability and increase rate is minimal, compared to the effect of human emissions, based on the transient response of oceans and biosphere and the calculated decay rate of human emissions.
The long tau (e-fold decay rate) of human emissions is based on the calculated sink rate (human emissions – increase in the atmosphere) and the increased CO2 pressure in the atmosphere above dynamic equilibrium (“steady state”), which is ~290 ppmv for the current weighted average ocean surface temperature. That is thus ~110 ppmv above steady state and that gives ~2.15 ppmv net sink rate per year. For a linear response, the e-fold decay rate can be calculated:
disturbance / response = decay rate
or for 2012:
110 ppmv / 2.15 ppmv/year = 51.2 years or 614 months.
That the sink process is quite linear can be seen in the similar calculation by Peter Dietze with the figures of 27 years ago :
1988: 60 ppmv, 1.13 ppmv/year, 53 years
Or from earliest accurate CO2 measurements:
1959: 25 ppmv, 0.5 ppmv/year, 50 years
Conclusion: Within the accuracy of the CO2 emission inventories and the natural variability, the decay rate of any extra CO2 above the dynamic equilibrium (whatever the cause) behaves like a linear process…
3.4 The derivatives.
What does that show in the derivatives? First the transient response of the biosphere and oceans to temperature variability:
Caption: Fig. 13: RSS temperature compared to CO2 increase and transient response of natural CO2 (biosphere+oceans) rate of change .
It seems that the amplitude of the natural variability is overblown, but for the rest both the temperature and the transient response of CO2 are equally synchronized with the observed CO2 rate of change with hardly any slope in the transient response. Thus while all the variability is from the transient response, there is hardly any contribution of oceans or biosphere to the slope in CO2 rate of change.
The overdone amplitude of the natural variability may be a matter of CO2/temperature ratio or a too short transient response time, but that is not that important. The form and timing are the important parts.
Now we can add human emissions into the rate of change:
Fig. 14: RSS temperature compared to CO2 increase and transient response of natural CO2 + emissions rate of change .
For an exact match of the slopes of RSS temperature and CO2 rate of change one need to multiply the temperature curve with a factor and add an offset. The match of the slopes of the observed CO2 rate of change and the calculated rate of change from the emissions plus the small slope of the natural transient response needed no offset at all: it was a perfect match. Only the amplitude of the variability was reduced, but that has no effect on the small natural CO2 rate of change slope.
As can be seen in that graph, both temperature rate of change and CO2 rate of change from humans + natural transient response show the same variability in timing and form. That is clear if we enlarge the graph for the period 1987-2002, encompassing the largest temperature changes of the whole period:
Fig. 15: RSS temperature compared to CO2 increase and transient response of natural CO2 + emissions rate of change in the period 1987-2002 .
As is very clear in this graph, there is an exact match in timing and form between temperature and the transient response of the CO2 rate of change, as was the case in the theoretical calculations. Where there is a discrepancy between the observed and calculated rates of change of CO2 , temperature shows the same discrepancy, like the 1991 Pinatubo eruption which increased photosynthesis by scattering incoming sunlight.
Conclusion: it is entirely possible to match the slopes and variability by temperature only or by the effect of human emissions + natural variability.
Which of the two possible solutions is right is quite easy to know, by looking which of the two matches the observations.
The straight forward result:
– The temperature-only match violates all known observations, not at least Henry’s law for the solubility of CO2 in seawater, the oxygen balance – the greening of the earth, the 13C/12C ratio, the 14C decline,… Together with the lack of a slope in the derivatives for a transient response from oceans and vegetation to a linear increase in temperature.
– The emissions + natural variability matches all observations. See: http://www.ferdinand-engelbeen.be/klimaat/co2_origin.html
Most of the variability in the rate of change of CO2 is caused by the influence of temperature on vegetation. While the influence on the rate of change seems huge, the net effect is not more than about +/- 1.5 ppmv around the trend and zeroes out after 1-3 years.
Most of the slope in the rate of change of CO2 is caused by human emissions. That is about 110 ppmv from the 120 ppmv over the full 165 years (about 70 from the 80 ppmv over the past 57 years). The remainder is from warming oceans which changes CO2 in the atmosphere with about 16 ppmv/°C, per Henry’s law, no matter if the exchanges are static or dynamic.
Yearly human emissions quadrupled from over 1 ppmv/year in 1958 to 4.5 ppmv/year in 2013. The same quadrupling happened in the increase rate of the atmospheric CO2 (at average around 50% of human emissions) and in the difference, the net sink rate.
There is not the slightest indication in any direct measurements or proxy that the natural carbon cycle or any part thereof increased to give a similar fourfold increase in exactly the same time span, which was capable to dwarf human emissions…
Conclusion: Most of the CO2 increase is caused by human emissions. Most of the variability is natural variability. The match between temperature and CO2 rate of change is entirely spurious.
Fourth comment by Paul_K, and further on in that discussion, gives a nice overview of the effect of a transient response of CO2 to temperature. Ignore the warning about the “dangerous” website if you open the referenced image.
 Lecture of Pieter Tans at the festivities of 50 years of Mauna Loa measurements, from slide 11 on:
 http://www.sciencemag.org/content/287/5462/2467.short full text free after registration.
 temperature trend of HadCRUT4 and CO2 trend and derivatives from Wood for trees.
CO2 and δ13C trends from the carbon tracker of NOAA: http://www.esrl.noaa.gov/gmd/dv/iadv/
CO2 emissions until 2008 from: http://cdiac.ornl.gov/trends/emis/tre_glob.html
CO2 emissions from 2009 on from: http://www.eia.gov/cfapps/ipdbproject/IEDIndex3.cfm?tid=90&pid=44&aid=8
 The spreadsheet can be downloaded from: http://www.ferdinand-engelbeen.be/klimaat/CO2_lags.xlsx