**Abstract**

This chapter aims to provide climate policy makers with smooth patterns of global carbon dioxide (CO2) emissions consistent with the UN climate targets. An accessible mathematical approach is used to design such models. First, the global warming is quantified with time to determine when the climate targets will be hit in case of no climate mitigation. Then, the remaining budget for CO2 emissions is derived based on recent data. Considering this for future emissions, first proposed is an exponential model for their rapid reduction and long-term stabilization slightly above zero. Then, suitable interpolations are performed to ensure a smooth and flexible transition to the exponential decline. Compared to UN climate simulation models, the designed smooth pathways would, in the short term, overcome a global lack of no-carbon energy and, in the long term, tolerate low emissions that will almost disappear as soon as desired from the 2040s with no need for direct removal of CO2.

**Keywords:** atmospheric carbon dioxide (CO2), global CO2 emissions, global warming, remaining CO2 budget, time model, UN climate target

## **1. Introduction**

The climate change has been declared as an urgent global threat [1] since the spread of devastating floods, severe droughts, and ravaging wildfires, due to rising temperatures especially in the past three decades [2–4]. In response to this threat, the UN parties adopted the *2015 Paris Agreement on Climate Change* along with its implementation by 2020. Was included 'holding the increase in global annual average temperature above the pre-industrial level well below 2°C and pursuing efforts to limit it to 1.5°C' ([1], Art.2). Was also comprised 'projecting global peaking of greenhouse gases emissions as soon as possible along with their rapid reduction' ([1], Art.4).

Since the last century, the atmospheric carbon dioxide (CO2) has been largely dominating the other greenhouse gases [5, 6] due to increasing anthropogenic CO2 emissions as a consequence of a growing global demand for fossil-fuel-based products. Subsequently, climate policies would include a massive reduction of these emissions by shifting to no-carbon energy and introducing gas capture/removal technologies.

Climate mathematical modelling has so far focused on the physics behind the global warming, and has therefore described the rise in global average temperature 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 radiative forcing; 2.6, 4.5, 6, and 8.5 W m�<sup>2</sup> , going from the highest to the lowest 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 1.9 W m�<sup>2</sup> , and hence to meet the 1.5°C target.

**2. Background materials**

*DOI: http://dx.doi.org/10.5772/intechopen.88961*

**2.1 Global warming vs. atmospheric CO2**

doubling CO2 concentration [5, 18, 19].

**2.2 Atmospheric CO2 data (2000–2017)**

**2.3 Global CO2 emissions data (2000–2013)**

Analysis Center (CDIAC) up to 2013 [16].

ments from 2000 to 2017 [20].

**2.4 Quadratic interpolation**

emissions.

*parabola is*:

**65**

<sup>Δ</sup>*T r*ð Þ<sup>≈</sup> *<sup>β</sup> <sup>r</sup>* � <sup>1</sup>

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.

*Mathematical Model for CO2 Emissions Reduction to Slow and Reverse Global Warming*

One of the key results in [10], reminded below, will be of great use in modelling with time the global warming. Based on the physics law of conservation of energy, this result states that the rise in global average temperature above the pre-industrial record is growing with the ratio *r* of CO2 concentration to the pre-industrial level. It can be seen as a generalization of the well-investigated climate response to

The warming effect of the atmospheric CO2 as quantified in (1), along with the

It is necessary to determine the trend of the global CO2 emissions over the past two decades prior to modelling with time the desired effect of any projected mitigation in line with the UN climate goal. This trend will be estimated by linear regression of the annual gas emissions recorded by Carbon Dioxide Information

Classically, a quadratic interpolation consists of determining a quadratic function using the values that it takes on at exactly three particular values of its variable. The following result provides an original quadratic interpolation using also three given data on the parabola representing the function: its symmetry axis, one of its points (other than the vertex), and the slope of the tangent line at that point. This technique will be used to add a smooth transition to an exponential model for CO2

*If a parabola is symmetric about the line: x* ¼ *u, passes through a point x*0*, y*<sup>0</sup>

*with x*<sup>0</sup> 6¼ *u, and is tangent at this point to the line of slope m, then an equation of this*

*,*

trend of its concentration over the past two decades, will allow to describe the global warming through time. This trend will be estimated by linear regression of the annual average concentration of the gas based on the NASA monthly measure-

*<sup>r</sup>* � *<sup>k</sup>* ð Þ *<sup>β</sup>* <sup>≈</sup> <sup>5</sup>*:*84*, k*<sup>≈</sup> � <sup>0</sup>*:*<sup>85</sup> (1)

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 summarized in Section 5.

*Mathematical Model for CO2 Emissions Reduction to Slow and Reverse Global Warming DOI: http://dx.doi.org/10.5772/intechopen.88961*
