2.3. Transient response of CO₂ to a sinusoid change in temperature:

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):

Fig. 4: Response of bio-CO₂ on a continuous sinusoidal change in temperature [10].

It can be mathematically explained that the lag of the CO₂ response is maximum pi/2 or 90° after a sinusoidal temperature change [5]. 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 change in temperature:

To make the temperature changes and their result on CO₂ changes a little more realistic, we show here the result of a double sinusoid where the sinusoids have different periods. After all natural changes are not that smooth...:

Fig. 5: Response of bio-CO₂ on a continuous double sinusoidal change in temperature [10].

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...

2.5 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:

Fig. 6: Response of Natural CO₂ on a continuous double sinusoid plus slope change in temperature [10].

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 trend, 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 trend. 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.

A linear increase in temperature doesn't introduce a slope in the CO₂ rate of change at any level.

2.6 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.

Fig. 7: Response of CO₂ on a continuous double sinusoidal + slope change in temperature + emissions [10].

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...

The "match" between the slope in temperature and the CO2 slope in rate of change is entirely spurious.
 

3. The real world.

3.1 The cause of the variability:

Most of the variability in CO₂ rate of change is a response of (tropical) vegetation on (ocean) temperatures, mainly the Amazon.  That the main variability 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):

Fig. 8: 12 month averaged derivatives from temperature and CO2/ δ13C measurements at Mauna Loa [9].

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 [5]. 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 [6], [7].

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...

The response of CO₂ to the temperature variability is certainly from vegetation, but as vegetation is a proven small and increasing sink for CO₂, that is not the cause of the increase of CO₂ in the atmosphere or the slope in the CO₂ derivative.
 

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:

Fig. 9: Temperature, CO₂ emissions and increase in the atmosphere [9].

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 +/- 1.5 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 1985-2000, where several relative large temperature changes can be noticed like the 1991/2 Pinatubo eruption and the 1998 super El Niño:

Fig. 10: Influence of temperature variability on CO₂ variability around the CO2 trend [9]

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 22 ppmv measured...

The influence of temperature on the CO₂ variability is quite small: +/- 1.5 ppmv around the slope.

 

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:

 

3.4 The derivatives:

What does that show in the derivatives? First the transient response of the biosphere and oceans to temperature variability:

  

Fig. 13: RSS and HadCRUT-SH temperature changes compared to observed CO₂ rate of change and transient response of natural CO₂ (biosphere+oceans) rate of change [9][11].

It seems that the amplitude of the natural variability is overblown in the RSS plot, but not in the HadCRUT-SH plot. In 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 because the satellite temperature reacts much faster than the surface compilation, which is the average change of many stations, 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:

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 a very small offset to have a perfect match in the slopes. The calculated CO2 slope from the emissions and the observed CO2.dt slope have a small difference, but that is not measurable in the total increase of CO2.

A drawback of the artificial match of the slopes of temperature with the CO2 rate of change is that this also affects the amplitude of the variability as one factor is used to adjust the slopes and the variability. As both are caused by different processes (vegetation is dominant in the variability, but has a zero to negative slope over periods longer than 1-3 years), that leads to a too low amplitude of the variability if the difference in slope angles is large, as is the case for the HadCRUT-SH temperature...

As can be seen in these graphs, 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 graphs for the period 1987-2002, encompassing the largest temperature changes of the whole period, the 1991 Pinatubo eruption and the 1998 super El Niño:

4. Conclusion:

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 temperature induced variability.
The match between the slopes of temperature and CO₂ rate of change is entirely spurious.

All the above reflections of theoretical and observed temperature and CO2 changes were extensively discussed at Anthony Watts' WUWT blog with over 600 comments:
http://wattsupwiththat.com/2015/11/25/about-spurious-correlations-and-causation-of-the-co2-increase-2/