There are two parts of the climate discussion:

- The sensitivity of the temperature w.r.t. the atmospheric content
- The amount of in the atmosphere

While the sensitivity dominates the scientific climate discussion, the political decisions are dominated by “carbon budget” criteria on the basis of numbers, which are hardly publicly discussed.

It has been claimed that more that 20% of the emitted will remain in the atmosphere for more than 1000 years.

This article will investigate the functional relation between emissions and the actual content in the atmosphere.

Atfter finding this relation several future emission scenarios and their effect on the atmospheric content are investigated.

#### Carbon dioxid emissions in the past

Starting point are the actual emissions during the last 170 years

It is very informative to look at the relative changes of this time series (here the mathematical derivation). This is the equivalent of economic growth for emissions.

The largest increase in emissions was between 1945 and 1980, the period of great growth in wealth and quality of life primarily in the industrialized countries, with the absolute peak of global emissions growth passed in 1970, interestingly 3 years before the first oil crisis. At the turn of the millennium, there was another increase in emissions, this time caused by the economic boom of the emerging economies. **Since 2003, the growth of emissions has been steadily declining, and has de facto already fallen below the zero line**, i.e. **from now on, emissions are not expected to grow, despite the growth in China, India and other emerging and developing countries**.

This is convincingly illustrated in the time-series graph of the Global Carbon Project:

The long-standing decline in emissions in industrialized countries is currently balancing out with the slowing rise in emerging economies China and India since 2010. **Accordingly, it is realistic to call constant emissions from 2019 onward “business as usual” (2020 was a Covid-19 driven emissions decline).**

#### CO_{2} content of the atmosphere with simple emission models

It is assumed that before 1850 the level was approximately constant and that the measured content is the sum of the pre-industrial constant level and a function of the emissions. The aim of this chapter is to find a simple function, which explains the atmospheric content.

Three different models are tested:

- The first model assumes that all emissions will remain in the atmosphere forever. This means that the additional – on top of the pre-industrial leval – content would be the cumulative sum of all -emissions.
- The second model assumes an exponential decay of emitted into the oceans or biosphere with a half life time of 70 years, i.e. half of all emitted is absorbed after 70 years. This is achieved by a convolution with an exponential decay kernel and a time constant years
- The third model assumes an exponential decay of emitted into the oceans or biosphere with a half life time of 35 years, i.e. half of all emitted is absorbed after 35 years. This is achieved by a convolution with an exponential decay kernel and a time constant years.

In order to make the numbers comparable, the emissions, that are measured in Gt have to be converted to ppm. This is done with the equivalence of 3210 Gt = 410 ppm .

The yellow graph are the measured actual emissions from the diagram above, and the blue graph is the measured actual content.

The first “cumulative” model approximates the measured content quite well from 1850 to 1910, but heavily overpredicts the content after 1950. This falsifies the hypothesis that stays in the atmosphere for “thousands of years”.

Also the second model model with a half life time of 70 years of emitted overshoots considerably after 1950, it approximates the time between 1925 and 1945. The third model with a half life time for emissions of 35 year fits the actual content from 1975 till now.

This confirms, what has very recently been published in Nature , that the rate of absorption into the oceans increases with increasing atmospheric content.

The same relation, in particular the increasing “carbon sink” of oceans and biosphere, is reported from the Global Carbon Project in this graphics:

Although we can expect a further increase of the flux into the ocean in the future, **we can therefore safely use the third model with a half life time of 35 years for conservative, i.e. non-optimistic predictions**.

#### ARX model for estimating the precise CO_{2} half life time

The estimate of 35 the years half life time in the previous chapter was found by guessing (which is a legitimate scientific approach). Now we take an approach based of first principles. Assuming that the absorption by the oceans and the biosphere can be represented by the diffusion equation, after integration over the surface of the earth and sampling over discrete time intervals this can be mapped into an extended autoregressive process ARX(n), where the carbon content at year i is assumed to be a sum of

- a linear function of the emissions with offset and slope b,
- a weighted sum of the carbon content of previous years,
- random (Gaussian) noise

In the most simple case ARX(1) we get

##### Model selection and scaling

There is an apparent subjective choice on how n should be chosen. Large n includes also effects over a longer time span but increases the complexity and can introduce meaningless parameters. One way of introducing a more long term system response is to scale and subsample the data, e.g. by a factor of 2, taking the mean of neighboring data points and then dropping each second data point. Details of this procedure are described in more detail here.

It turned out in this case that the most robust statistical behaviour was achieved by sub-sampling with a factor 2 and then using n=1, i.e. applying only a single AR parameter.

##### Applying the model to real data

This model enforces a homogeneous absorption over the whole investigated time span. Using our previous knowledge that the absorption changed over time, and that it appeared not to have changed a lot since 1950, we use the time range 1950-2014 for estimating the ARX-parameters. After converting the emission data from Gt/a to ppm and sub-sampling the data by 2, we get these least squares estimation results (with the Python module OLS from package statsmodels.regression.linear_model):

The Ljung-Box test results in Q = 12 with p = 0.72, which means that the residue of the least squares estimate is sufficiently uncorrelated.

The -parameter is an indicater of the diffusion, it measures the percentage of the remaining after 2 years. The pulse response (only a single non-zero emission signal) results in this exponential decay curve, with a half life time of 28 years.

This is a remarkable result. It says, that half of todays emitted will be absorbed by the oceans and biosphere after 28 years only. As we know that the absorption is increasing, it can be expected to be even less than that.

With these parameterns the carbon content training data set is reproduced entirely by the 3 ARX(1) parameters and the emissions data set:

#### Future scenarios

In order to evaluate policy decisions, I will apply this model to predict the future content with 3 different emission scenarios:

- The first scenario (red) I’d like to call the “Business as usual” scenario, in the sense that China already now increases emissions only marginally and has committed itself to stop increasing emissions after 2030. Today emissions are not growing any more. This scenario means, that we keep global emission on the 2019 level.
- The second scenario (green) is the widely proclaimed decarbonisation by 2050
- The third scenario (blue) is a compromise proposal, reducing emissions to 50% of the 2019 value (37 Gt) by 2100. This scenario reflects the facts that
**fossil fuels are finite, and that research and development of sound new technologies takes time**:

The consequences for the -content based on the simple model with 35 years half life time are these:

- The first scenario (red) increases content, but not beyond 510ppm in the long distant future, which is less double the amount of the pre-industrial era. Depending on the sensitivity this means a hypothetical induced temperature increase of 0,16° to 0,8° from current temperatures, resp. 0,45° to 1,3° since pre-industrial times, depending on the sensitivity.
- The second scenario — worldwide fast decarbonisation — (green) hardly increases the content any more, and eventually reduces the atmospheric content to pre-industrial levels.

Do we really want this???**This would mean deprivation of all plants, which thrive best at -levels larger that 400 ppm. Not even the IPCC ever formulated this as a desirable goal.** - The compromise scenario (blue) will slightly raise content but keep it below 460 ppm, and then gradually reduce it to the 1990 level. The atmospheric levels will begin to fall after 2065.

The precise ARX(1) model, corresponding to a 28 years half life time, gives these predictions. Even the worst case scenario, the “business as usual” will remain under 500 ppm content, and the economically viable reduction to 50% by 2100 eventually reduces the content to less than 400 ppm, which is below the current level.

#### Conclusion

Not even the most pessimistic of the scenarios described above reaches a “catastrophic” content in the atmosphere. **The complete decarbonisation scenario by 2050 can only be judged as utter nonsense.** **No one can wish to go back to pre-industrial levels. **

On the other hand, the limited fossile resources motivate to replace them in a feasible and humane way. This is reflected in the “compromise” scenario, which gradually reduces long-term emissions to the level of approximately the year 1990.