CO₂ change modelled with temperature
Correlation CO₂ change vs. temperature
An important indication that temperature has a strong influence on the CO₂ rise is the clear correlation between the temperature and changes in the CO₂ concentration over the past 65 years. The following chart in Figure 1 shows the temperature anomaly versus the monthly changes of the CO₂ concentration. To compensate for seasonal variations, the change in every month is compared to the same month in the previous year. The CO₂ follows the temperature: if the temperature is high the CO₂ level goes up, if the temperature is low, the change is much less.
Causality
The remarkable thing is that changes in temperature are always followed by changes in CO₂. We can see it from the phase difference in Figure 1, but it is also illustrated in the following chart from WoodForTrees, which represents the average temperature and CO₂ concentration on a yearly basis. Changes in temperature precede the changes in carbon dioxide.
This indicates that if there is a causal relationship, then temperature is the cause and the CO₂ change is the effect. Koutsoyiannis, D. et al. (2023) have done detailed research into the causal relationship. They used a stochastic analysis in which the causality is investigated with the help of an impulse response function. An impulse response is the reaction of a dynamic system in response to a brief input signal, called an impulse. You can compare it to a sound engineer clapping his hands in front of a microphone to hear how the sound behaves in a concert hall. In the following graphs the IRFs are given, left the response for T ➜ ΔCO₂, right for the response ΔCO₂ ➜ T. The first response (green) is positive, which indicates a potential causal relationship. The second response (red) is negative, which indicates an anti-causal response. Anti-causal means that the actual causality direction is opposite to that assumed.
The conclusion of this analyses:
Models
Based on the physical explanation, the strong correlation and the causal relationship, it is possible to describe the CO₂ concentration in a model, with only the observed changes in temperature and human emissions as input. In his 2019 research (Harde, H. (2019)), Hermann Harde showed that the changes in the CO₂ concentration can be well described in a simple model that is based on two equations:
eN(t) = eN0 + ße . ΔT(t)
τR(t) = τR0 + ßT . ΔT(t)
where eN is the natural emission, τR is the residence time, ΔT is the temperature anomaly, and βe and βτ are the temperature coefficients of the natural emission and the residence time. In the chart of Figure 3 shows that this simple model can well describe the observed CO₂ rise. Also the relatively small impact of human emissions is visible.
The following two charts come from a model of Koutsoyiannis that is based on the derived Impulse Response Function from T ➜ ΔCO₂. Also this calculation is straightforward and results in a very accurate quantification of atmospheric carbon exchanges, based on measured temperature fluctuations.


The framework is in good agreement, not just with the increase in atmospheric CO₂ concentration, but also the seasonal variations and the expansion of the biosphere.