The analysis in this paper shows that the fundamental theory of the CO₂ level in the atmosphere, under the influence of changing CO₂ emissions, can be modeled as a first order linear differential equation with a forcing function, describing industrial emissions. 

 

Observations of the CO₂ level at the Mauna Loa CO₂ observatory and official statistics of global CO₂ emissions, from Edgar, the Joint Research Centre at the European Commission, are used to estimate all parameters of the forced CO₂ differential equation.

 

The estimated differential equation has a logical theoretical foundation and convincing statistical properties. It is used to reproduce the time path of the CO₂ data from Mauna Loa, from year 1990 to 2018, with very small errors. Furthermore, the differential equation shows that the global CO₂ level, without emissions, has a stable equilibrium at 280 ppm. This value has earlier been reported by IPCC as the pre-industrial CO₂ level.

 

The differential function is applied to derive four dynamic cases of the global CO₂ level, from the year 2020 until 2100, conditional on four different strategies concerning the development of global CO₂ emissions: 

i. Emissions continue to increase according to the trend during 1990–2018

ii. Emissions stay for ever at the 2020 level

iii. Emissions are reduced with a linear trend to become zero year 2100

iv. Emissions are reduced with a linear trend to become zero year 2050

In case i., the CO₂ level year 2100 will be 688 ppm. In cases ii. and iii., the CO₂ levels in 2100 will be 517 ppm and 389 respectively. In case iv., the CO₂ level in 2050 is 408 ppm and then rapidly falls.