**2. Background materials**

using a mathematical approach based on the law of conservation of energy (e.g., [7–10]). When it comes to the climate mitigation, the existing models were mostly produced by computer simulation, which involved rather climatologists. Among these are the representative concentration pathways (RCPs; [5, 11–13]), which were adopted by the Intergovernmental Panel on Climate Change (IPCC) to predict future annual CO2 emissions by simulating representative mitigation scenarios of

mitigation. In the same setting, the C4MIP as part of the CMIP (Coupled Model Inter-comparison Project) provided a set of earth system models, involving the carbon cycle [14], also adopted by the IPCC (AR5, WG I). Among these were included models that infer CO2 emissions based on atmospheric CO2 concentrations targets. More recently, mixed models were developed using a combination of simulation climate and socio-economic models [15] to limit the radiative forcing to

The purpose of this chapter is to provide climate policy makers with smooth patterns of global CO2 emissions consistent with a prescribed UN climate target, i.e., a limit L (°C) to the rise in global average temperature above the pre-industrial level. Unlike in literature where modelling is often based on computer simulation, an accessible mathematical analysis is used to design such models. Basically, two parameters are required; an estimation of the emissions level in the beginning of their mitigation (fixed parameter) and the remaining CO2 budget (dependent parameter) which, by definition, consists of the cumulative CO2 emissions (from the starting time) that will raise the global average temperature up to the given climate target. These parameters will be determined using a very strong and highly significant linear regression involving recent data on the gas emissions [16], which also provides a time model for these emissions in case of no climate policy. Based on this model, the second parameter will be explicitly determined in terms of the climate target. Modelling future emissions to make them fit the given UN target would be nothing else but connecting their initial state (predicted level in the beginning of the mitigation) to their desired final state (zero or almost-zero emission). Naturally, an exponential interpolation would provide such a connecting way with a rapid reduction of the emissions over the first 50–60 years, their stabilization slightly above zero in the long term, along with their extinction in far future due to the asymptotic behavior of the exponential model. Another source of mathematical modelling with regards to climate mitigation is the transition to this exponential trend, which can provide more feasible patterns for CO2 emissions. Indeed, an independent parameter is introduced as an arbitrary fraction of the remaining CO2 budget expected to be used exponentially, which also gives an indication for the transition length. Then suitable quadratic interpolations are performed to smoothly

connect the current linear trend to the exponential decline. As a result, an uncountable range of exponential pathways is designed with smooth and flexible transition, which will not only overcome a global shortage of no-carbon energy but also lead to the nearly-zero emission as soon as desired depending on the climate target. The graphical representation of the designed models will help to explore their similarities to the (IPCC) RCPs and no- and low-overshoot 1.5°C pathways [17]. The rest of the chapter is organized as follows. The required materials are presented in Section 2, including recent annual data on CO2 (concentration in the air as well as emissions level) and the correlation between the global warming and the atmospheric CO2. In Section 3, a time model for global warming is presented along with a formulation of the hitting time for a given UN climate target. Section 4 is devoted to the elaboration and discussion of smooth mathematical models for global CO2 emissions consistent with the UN climate targets. The results are sum-

, going from the highest to the lowest

radiative forcing; 2.6, 4.5, 6, and 8.5 W m�<sup>2</sup>

*Global Warming and Climate Change*

, and hence to meet the 1.5°C target.

1.9 W m�<sup>2</sup>

marized in Section 5.

**64**

It is well-known that the global warming is due to the growing concentration of the greenhouse gases in the atmosphere, particularly the anthropogenic CO2. Its quantification with time would therefore require the consideration of both; its correlation with and annual data of the atmospheric CO2. On the other hand, recent data on global CO2 emissions will be necessary to design appropriate pathways for these emissions in order to limit their warming effect to a prescribed UN target. Additionally, non-linear interpolations are inevitable to ensure a smooth transition from the current trend to the rapid decline of the emissions as urged by the UN.
