**4.3 Ideal smooth pathways for CO2 emissions consistent with a prescribed UN climate target**

Whereas the model *E<sup>γ</sup>* ð Þ 0<*γ* < 1 is designed to fit a prescribed UN climate target <sup>L</sup><sup>∘</sup> C 1ð Þ *:*5≤L<2 *,* in the sense that the cumulative CO2 emissions will not raise the global average temperature by more than <sup>L</sup><sup>∘</sup> C above the pre-industrial level, its consistency turns out to be higher for more binding targets. Indeed, a numerical investigation (**Table 3**), based on specific criteria, shows that the lower the target the better the fit with the target. More precisely, there are (uncountable) many more *E<sup>γ</sup>* s, compatible with lower targets, that peak below 2.05 times (if not twice) the 2000 record (Criterion C1) project a reduction by at least 50% in 2050 (relative to 2000 level) (Criterion C2) and predict nearly-zero emission ð≤0*:*01 GtCO2) by 2100 (Criterion C3). From an analytical point of view, what is said about criterion (C1) is due to the decrease of the peak (coefficient *B* in the model *Eγ*) with decreasing remaining CO2 budget *R*, and therefore with decreasing target L. As for the comparison based on (C2) and (C3), this can be explained by the fact that the exponential decline will start more rapidly for a lower remaining budget *R* (due to a lower target), as it can be seen from the formula of the initial speed of the exponential reduction (at time *t* ¼ *v*):

$$|\frac{dE\_\gamma}{dt}| = aE\_0 = E\_0^2/(\chi R) \tag{22}$$

(using the formula of *α* given in (16)).

Consequently, the *E<sup>γ</sup>* s ð Þ *:*24 <*γ* <*:*27 appear to be the most consistent smooth pathways with the 1.5°C target, and among these, the *E:*<sup>26</sup> would be the ideal one as it satisfies the three criteria and predicts the lowest peak of emissions (by 2037), with a (constant) relative rate of exponential decline estimated at *α*≈14*:*2%*:* However, for more binding climate targets such as 1.4°C, which corresponding global mitigation needs to be implemented before 2034 (according to (5)), it turns out that half of the models *E<sup>γ</sup>* , namely those with 0*:*01<*γ* < 0*:*51*,* meet all criteria to ideally fit this target, and the ideal one of them would be the *E:*5*,* for the same reason as the *E:*<sup>26</sup> for the 1.5°C target, with a peak of emissions by 2028 and a (constant) relative rate of exponential decline estimated at *α*≈ 11*:*8%. See **Figure 3** for graphical illustration.

Nevertheless, if the 'zero' emission timing is prioritized over the peaking threshold, it is found that, among the *E<sup>γ</sup>* s, those with *γ* <0*:*00067 (resp. 0*:*00019) project the earliest 'zero' emission; by 2043 (resp. 2035) for the 1.5°C (resp. 1.4°C) target, with a peak estimated at 2.14 times (resp. twice) the 2000 level. But the interval between the peaking and the almost-zero moments seems to be extremely short; 2 months and 10 days respectively, making the curve look like a vertical line over this interval, which sounds rather unrealistic. However, feasible ideal *E<sup>γ</sup>* s for the earliest 'zero' emission could be found by considering higher values of parameter *γ* and time *z* (as close as possible to the unrealistic 'zero' emission moment, as found previously) that satisfy the following conditions:


As a result, the earliest feasible almost-zero emission will occur by 2061, 2050, and 2040 for the respective climate targets 1.6, 1.5, and 1.4°C, with the *E<sup>γ</sup>* s induced by *:*0179<*γ* <*:*0301*, :*0163 <*γ* <*:*033*,* and 0<*γ* <*:*038 respectively. **Figure 4** shows the most feasible of these pathways, with time periods between the peaking

*Ideal smooth pathways, for the earliest feasible 'zero'-CO2-emission; climate targets 1.6, 1.5 and 1.4°C*

*Ideal smooth pathways for CO2 emissions peaking below 2.05 times (if not twice) 2000 level and shrinking below 0.01 GtCO2 by 2100 (preferably); climate targets 1.6, 1.5 and 1.4°C (mitigation starting by 2020).*

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

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

An alternative ideal pathway could be made by juxtaposing the lowest restrictions of *E*<sup>1</sup> and one of the ideal *E<sup>γ</sup>* s consistent with the same climate target. But the resulting pattern would include two singularities (cusp shape); one in the beginning

Smooth pathways for CO2 emissions are designed taking into consideration not only their consistency with the UN climate targets but also the rapidity that has been urged by the UN for their reduction. Unlike the existing models, mostly produced by computer simulation such as the (IPCC) RCPs, a mathematical modelling, as an ideal host of interpolation and smoothing techniques, is presented

and the almost-zero moments estimated at 11, 8, and 6 years respectively.

of the mitigation and another at the junction between *E1* and *E<sup>γ</sup>* .

**5. Conclusions**

**73**

**Figure 3.**

**Figure 4.**

*(mitigation starting by 2020).*


*a Pathways E<sup>γ</sup> for CO2 emissions peaking below twice (2.05 times) 2000 level. <sup>b</sup>*

*Pathways E<sup>γ</sup> for CO2 emissions reduced by at least 50% in 2050 (rel. to 2000 level). <sup>c</sup>*

*Pathways E<sup>γ</sup> for CO2 emissions reduced to almost zero (below 0.01 GtCO2) by 2100.*

### **Table 3.**

*Ideal smooth pathways for CO2 emissions by UN climate target (UNCT) (global mitigation starting by 2020).*

**Figure 3.**

raise the global average temperature by more than <sup>L</sup><sup>∘</sup>

∣ *dE<sup>γ</sup>*

found previously) that satisfy the following conditions:

*Pathways E<sup>γ</sup> for CO2 emissions peaking below twice (2.05 times) 2000 level. <sup>b</sup> Pathways E<sup>γ</sup> for CO2 emissions reduced by at least 50% in 2050 (rel. to 2000 level). <sup>c</sup> Pathways E<sup>γ</sup> for CO2 emissions reduced to almost zero (below 0.01 GtCO2) by 2100.*

nential reduction (at time *t* ¼ *v*):

*Global Warming and Climate Change*

for graphical illustration.

*a*

**72**

**Table 3.**

(using the formula of *α* given in (16)).

level, its consistency turns out to be higher for more binding targets. Indeed, a numerical investigation (**Table 3**), based on specific criteria, shows that the lower the target the better the fit with the target. More precisely, there are (uncountable) many more *E<sup>γ</sup>* s, compatible with lower targets, that peak below 2.05 times (if not twice) the 2000 record (Criterion C1) project a reduction by at least 50% in 2050 (relative to 2000 level) (Criterion C2) and predict nearly-zero emission ð≤0*:*01 GtCO2) by 2100 (Criterion C3). From an analytical point of view, what is said about criterion (C1) is due to the decrease of the peak (coefficient *B* in the model *Eγ*) with decreasing remaining CO2 budget *R*, and therefore with decreasing target L. As for the comparison based on (C2) and (C3), this can be explained by the fact that the exponential decline will start more rapidly for a lower remaining budget *R* (due to a lower target), as it can be seen from the formula of the initial speed of the expo-

*dt* <sup>∣</sup> <sup>¼</sup> *<sup>α</sup>E*<sup>0</sup> <sup>¼</sup> *<sup>E</sup>*<sup>2</sup>

Nevertheless, if the 'zero' emission timing is prioritized over the peaking threshold, it is found that, among the *E<sup>γ</sup>* s, those with *γ* <0*:*00067 (resp. 0*:*00019) project the earliest 'zero' emission; by 2043 (resp. 2035) for the 1.5°C (resp. 1.4°C) target, with a peak estimated at 2.14 times (resp. twice) the 2000 level. But the interval between the peaking and the almost-zero moments seems to be extremely short; 2 months and 10 days respectively, making the curve look like a vertical line over this interval, which sounds rather unrealistic. However, feasible ideal *E<sup>γ</sup>* s for the earliest 'zero' emission could be found by considering higher values of parameter *γ* and time *z* (as close as possible to the unrealistic 'zero' emission moment, as

**UNCT (°C) (***C***1)-Pathways<sup>a</sup> (***C***2)-Pathways<sup>b</sup> (***C***3)-Pathways<sup>c</sup>** 1.6 *γ* >0*:*55 0ð Þ *:*46 None None 1.5 *γ* > 0*:*37 0ð Þ *:*24 *γ* < 0*:*34 *γ* < 0*:*27 1.4 *γ* >0*:*01 (All) All *γ* <0*:*51

*Ideal smooth pathways for CO2 emissions by UN climate target (UNCT) (global mitigation starting by 2020).*

*z* � *u*> max ð Þ *δ,* ϵ*=*3 *, E<sup>γ</sup>* ð Þ*z* <0*:*01 (23)

Consequently, the *E<sup>γ</sup>* s ð Þ *:*24 <*γ* <*:*27 appear to be the most consistent smooth pathways with the 1.5°C target, and among these, the *E:*<sup>26</sup> would be the ideal one as it satisfies the three criteria and predicts the lowest peak of emissions (by 2037), with a (constant) relative rate of exponential decline estimated at *α*≈14*:*2%*:* However, for more binding climate targets such as 1.4°C, which corresponding global mitigation needs to be implemented before 2034 (according to (5)), it turns out that half of the models *E<sup>γ</sup>* , namely those with 0*:*01<*γ* < 0*:*51*,* meet all criteria to ideally fit this target, and the ideal one of them would be the *E:*5*,* for the same reason as the *E:*<sup>26</sup> for the 1.5°C target, with a peak of emissions by 2028 and a (constant) relative rate of exponential decline estimated at *α*≈ 11*:*8%. See **Figure 3**

C above the pre-industrial

<sup>0</sup>*=*ð Þ *γR* (22)

*Ideal smooth pathways for CO2 emissions peaking below 2.05 times (if not twice) 2000 level and shrinking below 0.01 GtCO2 by 2100 (preferably); climate targets 1.6, 1.5 and 1.4°C (mitigation starting by 2020).*

### **Figure 4.**

*Ideal smooth pathways, for the earliest feasible 'zero'-CO2-emission; climate targets 1.6, 1.5 and 1.4°C (mitigation starting by 2020).*

As a result, the earliest feasible almost-zero emission will occur by 2061, 2050, and 2040 for the respective climate targets 1.6, 1.5, and 1.4°C, with the *E<sup>γ</sup>* s induced by *:*0179<*γ* <*:*0301*, :*0163 <*γ* <*:*033*,* and 0<*γ* <*:*038 respectively. **Figure 4** shows the most feasible of these pathways, with time periods between the peaking and the almost-zero moments estimated at 11, 8, and 6 years respectively.

An alternative ideal pathway could be made by juxtaposing the lowest restrictions of *E*<sup>1</sup> and one of the ideal *E<sup>γ</sup>* s consistent with the same climate target. But the resulting pattern would include two singularities (cusp shape); one in the beginning of the mitigation and another at the junction between *E1* and *E<sup>γ</sup>* .

## **5. Conclusions**

Smooth pathways for CO2 emissions are designed taking into consideration not only their consistency with the UN climate targets but also the rapidity that has been urged by the UN for their reduction. Unlike the existing models, mostly produced by computer simulation such as the (IPCC) RCPs, a mathematical modelling, as an ideal host of interpolation and smoothing techniques, is presented throughout this chapter. First, the global warming is quantified with time to determine the moment when a prescribed UN climate target will be hit (in case of no climate mitigation), which is then used to explicitly determine the remaining CO2 budget; crucial parameter in emissions modelling. Naturally, an exponential pattern is proposed at first for its rapid decline and long-term stabilization slightly above zero. Then, by means of quadratic interpolations, a parametrized collection of flexible pathways *E<sup>γ</sup>* ð Þ 0<*γ* <1 is derived to ensure more feasibility by including a smooth transition to the exponential trend, which will help compensate a certain lack of nocarbon energy. It turns out that the no-transition (exponential) and no-mitigation (linear) models correspond to the limit values of the involved parameter *γ* introduced as an arbitrary fraction of the remaining CO2 budget expected to be used during the exponential phase, which also gives an indication for the transition length.

GtCO2 gigatons of CO2

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

UN United Nations UNCT UN climate target

**Author details**

Department of Mathematics, College of Sciences and Human Studies,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Prince Mohammad Bin Fahd University, Al Khobar, KSA

\*Address all correspondence to: nizar.jaoua@gmail.com

provided the original work is properly cited.

Nizar Jaoua

**75**

IPCC Intergovernmental Panel on Climate Change NASA National Aeronautics and Space Administration

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

RCP representative concentration pathway

Graphically, the *E<sup>γ</sup>* s are comparable to the corresponding IPCC pathways; similar to the RCP4.5, for targets between 1.5 and 2°C, and to the RCP2.6 and no- and low-overshoot, for the 1.5°C target. However, they have the advantage of predicting the nearly-zero emission (<0.01 GtCO2), e.g., by 2090 for *γ* <0*:*22*,* or even as early as 2050 for *γ* <0*:*03*,* with no need for CO2 removal. Such similarities could be improved by using the IPCC estimation for the remaining CO2 budget (though determined with high uncertainties), which may lead to more representative pathways by involving further greenhouse gases.

Another virtue of the designed *E<sup>γ</sup>* s is their flexibility with regards to the constraints that would come with the climate target, which would provide climate policy makers with an uncountable set of ideal smooth pathways enlarging with decreasing target. For instance, whereas *E<sup>γ</sup>* s with 0*:*24<*γ* < 0*:*27 are recommended for the 1.5°C target, based on specific criteria including the peaking threshold, those with 0*:*01<*γ* <0*:*51 are recommended for a more binding target; the 1.4°C one. When it comes to the projection of the earliest feasible 'zero' emission, are recommended the *E<sup>γ</sup>* s with 0*:*017 <*γ* < 0*:*033 and 0 <*γ* <0*:*038*,* for the respective climate targets 1.5, and 1.4°C, which would result in the near extinction of CO2 emissions by 2050 and 2040 respectively.
