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CO₂ change modelled with temperature

Temperature is strongly related to changes in the CO₂ concentration. If there is a causal relationship with CO₂, then temperature is the cause and CO₂ the effect. The increased CO₂ concentration, including the seasonal variability, can be accurately modelled based on the observed temperature changes.

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.

There is a clear correlation between temperature and changes in CO₂.
Figure 1: There is a clear correlation between temperature and changes in CO₂ on a monthly basis. Source: Hadcrut5, NOAA

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.

Changes in temperature are always followed by changes in CO2
Figure 2: Changes in temperature are always followed by changes in CO₂. Source: WoodForTrees

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.

Impulse Response Functions of ΔCO₂ and T
Figure 3: Impuls Response Functions for air temperature and changes in the CO₂ concentration. The impuls is at t = 0. Left: ∆T → ∆CO₂ is a potentially causal system as the response is later than the impuls. Right: ∆CO₂ → ∆T is a potentially anticausal system as the response precedes the impuls. Source: Koutsoyiannis, D. et al. (2023)

The conclusion of this analyses:

❝Changes in CO₂ concentration cannot be a cause of temperature changes. On the contrary, temperature change is a potential cause of CO₂ change on all time scales.❞

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.

Calculated CO2 concentration with temperature-dependent emission and absorption (Green).
Figure 4: Calculated CO2 concentration with temperature-dependent emission and absorption (Green). Compared against the observed record of CO2 from Mauna Loa (Blue Diamonds). Simulation without anthropogenic emissions (Magenta), and only human activities (Blue). Source: Harde, H. (2019)

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.

Change in CO₂ modeled with temperature data (R²=55%)
Figure 5: Change in CO₂ modeled with temperature data (R²=55%). Source: Koutsoyiannis, D. et al. (2023)
CO₂ concentration modeled with temperature data (R²=99.9%)
Figure 6: CO₂ concentration modeled with temperature data (R²=99.9%). Source: Koutsoyiannis, D. et al. (2023)

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.




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