**4.2 CO2 emissions pathways consistent with the UN climate targets**

One obvious way to regularly reduce the CO2 emissions would suggest a constant rate of reduction, which will definitely put an end to them at time *t* ¼ *z*

(limiting therefore the rise in global temperature to the UN target L), as described by the following piece-wise linear model, applicable from time *t* ¼ *t*<sup>0</sup> < *h*:

$$L(t) \approx \begin{cases} \frac{E\_0}{t\_0 - \mathbf{z}}(t - \mathbf{z}), & t\_0 \le t < \mathbf{z} \\\\ 0, & t \ge \mathbf{z} \end{cases} \tag{8}$$

it follows that *E*0*=α*<sup>1</sup> ¼ *R,* which gives *α*<sup>1</sup> as in (12).

*Eγ*ð Þ*t* ≈

*u*¼ *t*<sup>0</sup> þ ϵ*, v* ¼ *u* þ *δ,* ϵ ¼ *αδE*0*=α*<sup>0</sup>

*a*¼ *αE*0ð Þ *αE*<sup>0</sup> þ *α*<sup>0</sup> *, b* ¼ 3*αE*0ð Þ *E*<sup>0</sup> þ 1

Δ

� � <sup>¼</sup> *v, E*0Þ*, m* <sup>¼</sup> *dE<sup>γ</sup>*

8 >>><

>>>:

� � <sup>p</sup> *<sup>=</sup>*2*a,* <sup>Δ</sup> <sup>¼</sup> *<sup>b</sup>*<sup>2</sup> <sup>þ</sup> <sup>12</sup>*aα*0ð Þ <sup>1</sup> � *<sup>γ</sup> <sup>R</sup>*

*A*1*, A*2*,* and *B* follow immediately from (2) applied with *x*0*, y*<sup>0</sup>

exponential decline) to get *A*<sup>2</sup> as in (16) and another formulation of

� � *t*¼*v*

*B B*ð Þ ¼ *E*<sup>0</sup> þ *αδE*0*=*2 . Then, by equating the two expressions of *B*, one gets the announced formula for *ϵ*. As for *δ,* it is found to be the unique positive solution of

which discriminant is given by: <sup>Δ</sup> <sup>¼</sup> *<sup>b</sup>*<sup>2</sup> <sup>þ</sup> <sup>12</sup>*aα*0ð Þ <sup>1</sup> � *<sup>γ</sup> R,* where *<sup>a</sup>* and *<sup>b</sup>* are as announced with (16). Indeed, by evaluating the integrals in the remaining CO2

> ð*v u*

Then by plugging the expressions of *A*1*, A*2*, B,* and *ϵ* into (19) then simplify-

*E<sup>γ</sup>* ð Þ*t dt* þ

ð Þ *<sup>A</sup>*1*=*<sup>3</sup> <sup>ϵ</sup><sup>3</sup> <sup>þ</sup> *<sup>B</sup>*<sup>ϵ</sup> � � <sup>þ</sup> ð Þ *<sup>A</sup>*2*=*<sup>3</sup> *<sup>δ</sup>*<sup>3</sup> <sup>þ</sup> *<sup>B</sup><sup>δ</sup>* � � <sup>þ</sup> ð Þ¼ *<sup>γ</sup><sup>R</sup> <sup>R</sup>* (19)

*dt* � �

and the exponential decline as follows:

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

*α*¼ *E*0*=γR, R* ¼ *R t*ð Þ<sup>0</sup>

*<sup>δ</sup>*¼ �*<sup>b</sup>* <sup>þ</sup> ffiffiffiffi

(12) using Ð <sup>∞</sup>

*dt <sup>t</sup>*¼*t*<sup>0</sup> <sup>¼</sup> *dE*<sup>0</sup>

� �

then with *x*0*, y*<sup>0</sup>

budget equation:

one gets:

**69**

*dt*

�

the following quadratic equation:

ð<sup>∞</sup> *t*0

*E<sup>γ</sup>* ð Þ*t dt* ¼

ing, one gets the following quadratic equation in *δ*:

ð*u t*0

*E<sup>γ</sup>* ð Þ*t dt* þ

*dE<sup>γ</sup>*

�

Although the model *E*<sup>1</sup> appears to better fit the UN climate goal, in comparison with the linear and power pathways, the abrupt reduction of CO2 emissions may threaten fundamental industries such as food production. To overcome this risk, an alternate parametrized model *E<sup>γ</sup>* ð Þ 0<*γ* < 1 would start with a succession of two smooth parabolic junctions (ascending then descending) between the linear growth

**<sup>2</sup>** <sup>þ</sup> *<sup>B</sup>, <sup>t</sup>***<sup>0</sup>** <sup>≤</sup>*<sup>t</sup>* <sup>&</sup>lt;*<sup>u</sup>*

**<sup>2</sup>** <sup>þ</sup> *<sup>B</sup>, <sup>u</sup>*≤*<sup>t</sup>* <sup>&</sup>lt;*<sup>v</sup>*

(15)

(16)

� � <sup>¼</sup> ð Þ *<sup>t</sup>*0*, E*<sup>0</sup> *, m* <sup>¼</sup>

*E<sup>γ</sup>* ð Þ*t dt* ¼ *R* (18)

¼ �*αE*<sup>0</sup> (for a smooth transition to the

*ax*<sup>2</sup> <sup>þ</sup> *bx* � <sup>3</sup>*α*0ð Þ <sup>1</sup> � *<sup>γ</sup> <sup>R</sup>* <sup>¼</sup> <sup>0</sup> (17)

ð<sup>∞</sup> *v*

*E***<sup>0</sup>** *e*�*α*ð Þ *<sup>t</sup>*�*<sup>v</sup> , t* ≥*v*

*A***1**ð Þ *t* � *u*

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

*A***2**ð Þ *t* � *u*

*A*1¼ �*α*0*=*2ϵ*, A*<sup>2</sup> ¼ �*αE*0*=*2*δ, B* ¼ *E*<sup>0</sup> þ *α*0ϵ*=*2*, α*<sup>0</sup> as in 6ð Þ

The free parameter *γ* ð Þ 0< *γ* <1 is introduced to split the remaining budget into two parts; one ð Þ *γR* will go for the exponential reduction and the other 1 ð Þ ð Þ � *γ R* for the quadratic transition requiring two consecutive time periods; *ε* to slow the emissions, then *δ* for their initial reduction. Let *u* ¼ *t*<sup>0</sup> þ ϵ*, v* ¼ *u* þ *δ* be the ending times of these periods. The coefficient *α* can be determined the same way as *α*<sup>1</sup> in

*<sup>v</sup> E<sup>γ</sup>* ð Þ*t dt* ¼ *γR* (instead of *R*). On the other hand, the coefficients

*<sup>t</sup>*¼*t*<sup>0</sup> <sup>¼</sup> *<sup>α</sup>*<sup>0</sup> (for a smooth slowdown) to get *<sup>A</sup>*<sup>1</sup> and *<sup>B</sup>*, as given in (16),

where *z* ¼ *t*<sup>0</sup> þ 2*R=E*<sup>0</sup> ðwith *E*<sup>0</sup> ¼ *E*0ð Þ *t*<sup>0</sup> *, R* ¼ *R t*ð ÞÞ <sup>0</sup> is determined by solving the remaining CO2 budget equation:

$$\int\_{t\_0}^{x} L(t)dt = (x - t\_0)E\_0/2 = R. \tag{9}$$

Nevertheless, the zero-emission ensured by the linear pattern will probably cause an environmental issue, as it could not be hit before the 2070s, for UN targets as low as 1.5°C, or late 2150s for even medium targets such as 1.8°C. In addition, a constant rate of reduction (annually 31% for 1.8°C and 79% for 1.5°C) will presumably not be compatible with a struggling switch to no-/low-carbon energy.

To avoid the issue regarding the zero-emission, one could simply bring these emissions as close as possible to zero by considering a smooth non-linear pathway with an asymptotic behavior, such as the following power model:

$$P(\mathbf{t}) \approx \mathbf{E}\_{\mathbf{0}} \cdot (\mathbf{t}\_{\mathbf{0}}/\mathbf{t})^p, \quad \mathbf{t} \ge \mathbf{t}\_{\mathbf{0}}, \quad p = \mathbf{1} + \mathbf{t}\_{\mathbf{0}} \mathbf{E}\_{\mathbf{0}}/\mathbf{R}, \quad \mathbf{E}\_{\mathbf{0}} = \mathbf{E}\_{\mathbf{0}}(\mathbf{t}\_{\mathbf{0}}), \quad \mathbf{R} = \mathbf{R}(\mathbf{t}\_{\mathbf{0}}) \tag{10}$$

where the suitable power *p*, with *p* > 1 for integrability over ½ � *t*0*,* ∞ *,* that makes the model fit the given UN target, is determined by solving (for *p*) the associated remaining CO2 budget equation:

$$\int\_{t\_0}^{\infty} P(t)dt = E\_0 \left. t\_0^p \right|\_{t\_0}^{\infty} t^{-p}dt = t\_0 E\_0 / (p - \mathbf{1}) = R. \tag{11}$$

Unfortunately, the emissions could not be made as low as 0.1 (GtCO2) even for the 1.5°C target and after six centuries of reduction.

However, an exponential decrease of CO2 emissions would not only ensure their rapid reduction (as recommended in the 2015 Paris Agreement, Art. 4), but will also tolerate very low emissions, relatively earlier (compared with the power model *P*), that will disappear in far future, which could maintain food production, especially in the regions where transition to no-carbon energy might be extremely challenging. This leads to the following 1-phase model for CO2 emissions consistent with a prescribed UN climate target, applicable from time *t* ¼ *t*<sup>0</sup> <*h*:

$$E\_1(\mathbf{t}) \approx E\_0 e^{-a\_1(t - t\_0)}, \quad \mathbf{t} \ge \mathbf{t}\_0, \quad \mathbf{a}\_1 = \mathbf{E}\_0/\mathbf{R}, \quad \mathbf{E}\_\mathbf{0} = \mathbf{E}\_0(\mathbf{t}\_0), \mathbf{R} = \mathbf{R}(\mathbf{t}\_0) \tag{12}$$

Indeed, to ensure an exponential decrease of the annual amount of CO2 emissions from the initial level *E*0, the model *E*<sup>1</sup> must satisfy the initial-value problem:

$$dE\_1/dt = -a\_1E\_1(a\_1 > 0), \quad E\_1(t\_0) = E\_0 \tag{13}$$

which unique solution is in the form given in (10). Now, from the remaining CO2 budget equation:

$$\int\_{t\_0}^{\infty} E\_1(t)dt = R,\tag{14}$$

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

it follows that *E*0*=α*<sup>1</sup> ¼ *R,* which gives *α*<sup>1</sup> as in (12).

(limiting therefore the rise in global temperature to the UN target L), as described

0*, t* ≥*z*

where *z* ¼ *t*<sup>0</sup> þ 2*R=E*<sup>0</sup> ðwith *E*<sup>0</sup> ¼ *E*0ð Þ *t*<sup>0</sup> *, R* ¼ *R t*ð ÞÞ <sup>0</sup> is determined by solving

Nevertheless, the zero-emission ensured by the linear pattern will probably cause an environmental issue, as it could not be hit before the 2070s, for UN targets as low as 1.5°C, or late 2150s for even medium targets such as 1.8°C. In addition, a constant rate of reduction (annually 31% for 1.8°C and 79% for 1.5°C) will presum-

To avoid the issue regarding the zero-emission, one could simply bring these emissions as close as possible to zero by considering a smooth non-linear pathway

where the suitable power *p*, with *p* > 1 for integrability over ½ � *t*0*,* ∞ *,* that makes the model fit the given UN target, is determined by solving (for *p*) the associated

Unfortunately, the emissions could not be made as low as 0.1 (GtCO2) even for

However, an exponential decrease of CO2 emissions would not only ensure their rapid reduction (as recommended in the 2015 Paris Agreement, Art. 4), but will also tolerate very low emissions, relatively earlier (compared with the power model *P*), that will disappear in far future, which could maintain food production, especially in the regions where transition to no-carbon energy might be extremely challenging. This leads to the following 1-phase model for CO2 emissions consistent with a

*<sup>E</sup>***1**ð Þ*<sup>t</sup>* <sup>≈</sup>*E***0***e*�*α***1**ð Þ *<sup>t</sup>*�*t***<sup>0</sup>** *, <sup>t</sup>* <sup>≥</sup>*t***0***, <sup>α</sup>***<sup>1</sup>** <sup>¼</sup> *<sup>E</sup>***0***=R, <sup>E</sup>***<sup>0</sup>** <sup>¼</sup> *<sup>E</sup>***0**ð Þ *<sup>t</sup>***<sup>0</sup>** *,<sup>R</sup>* <sup>¼</sup> *R t*ð Þ**<sup>0</sup>** (12)

Indeed, to ensure an exponential decrease of the annual amount of CO2 emissions from the initial level *E*0, the model *E*<sup>1</sup> must satisfy the initial-value problem:

which unique solution is in the form given in (10). Now, from the remaining

ð<sup>∞</sup> *t*0

*dE*1*=dt* ¼ �*α*1*E*1ð Þ *α*<sup>1</sup> >0 *, E*1ð Þ¼ *t*<sup>0</sup> *E*<sup>0</sup> (13)

*E*1ð Þ*t dt* ¼ *R,* (14)

*, t* ≥ *t***0***, p* ¼ **1** þ *t***0***E***0***=R, E***<sup>0</sup>** ¼ *E***0**ð Þ *t***<sup>0</sup>** *, R* ¼ *R t*ð Þ**<sup>0</sup>** (10)

�*pdt* <sup>¼</sup> *<sup>t</sup>*0*E*0*=*ð Þ¼ *<sup>p</sup>* � <sup>1</sup> *<sup>R</sup>:* (11)

ably not be compatible with a struggling switch to no-/low-carbon energy.

with an asymptotic behavior, such as the following power model:

*p* 0 ð<sup>∞</sup> *t*0 *t* ð Þ *t* � *z , t***<sup>0</sup>** ≤ *t* <*z*

*L t*ð Þ*dt* ¼ ð Þ *z* � *t*<sup>0</sup> *E*0*=*2 ¼ *R:* (9)

(8)

by the following piece-wise linear model, applicable from time *t* ¼ *t*<sup>0</sup> < *h*:

*E***0** *t***<sup>0</sup>** � *z*

8 < :

*L t*ð Þ≈

ð*z t*0

the remaining CO2 budget equation:

*Global Warming and Climate Change*

*P t*ð Þ<sup>≈</sup> *<sup>E</sup>***0***:*ð Þ *<sup>t</sup>***0***=<sup>t</sup> <sup>p</sup>*

CO2 budget equation:

**68**

remaining CO2 budget equation:

ð<sup>∞</sup> *t*0

*P t*ð Þ*dt* ¼ *E*<sup>0</sup> *t*

prescribed UN climate target, applicable from time *t* ¼ *t*<sup>0</sup> <*h*:

the 1.5°C target and after six centuries of reduction.

Although the model *E*<sup>1</sup> appears to better fit the UN climate goal, in comparison with the linear and power pathways, the abrupt reduction of CO2 emissions may threaten fundamental industries such as food production. To overcome this risk, an alternate parametrized model *E<sup>γ</sup>* ð Þ 0<*γ* < 1 would start with a succession of two smooth parabolic junctions (ascending then descending) between the linear growth and the exponential decline as follows:

$$\mathbf{E}\_{\mathbf{y}}(t) \approx \begin{cases} \mathbf{A}\_{1}(t-u)^{2} + \mathbf{B}, & \mathbf{t}\_{0} \le t < u \\\\ \mathbf{A}\_{2}(t-u)^{2} + \mathbf{B}, & u \le t < v \\\\ \mathbf{E}\_{0} \ e^{-a(t-v)}, & t \ge v \end{cases} \tag{15}$$
 
$$a = E\_{0}/\gamma \mathbf{R}, \quad \mathbf{R} = \mathbf{R}(t\_{0})$$
 
$$A\_{1} = -a\_{0}/2\epsilon, \quad A\_{2} = -a\mathbf{E}\_{0}/2\delta, \quad \mathbf{B} = E\_{0} + a\_{0}\epsilon/2, \quad a\_{0} \quad \text{as in} \quad (6)$$
 
$$u = t\_{0} + \epsilon, \quad v = u + \delta, \quad \epsilon = a\delta \mathbf{E}\_{0}/a\_{0} \tag{16}$$
 
$$\delta = \left(-b + \sqrt{\Delta}\right)/2\mathfrak{a}, \quad \Delta = b^{2} + 12aa\_{0}(1-\gamma)\mathbf{R}$$
 
$$a = a\mathbf{E}\_{0}(a\mathbf{E}\_{0} + a\_{0}), \quad b = 3a\mathbf{E}\_{0}(\mathbf{E}\_{0} + 1)$$

The free parameter *γ* ð Þ 0< *γ* <1 is introduced to split the remaining budget into two parts; one ð Þ *γR* will go for the exponential reduction and the other 1 ð Þ ð Þ � *γ R* for the quadratic transition requiring two consecutive time periods; *ε* to slow the emissions, then *δ* for their initial reduction. Let *u* ¼ *t*<sup>0</sup> þ ϵ*, v* ¼ *u* þ *δ* be the ending times of these periods. The coefficient *α* can be determined the same way as *α*<sup>1</sup> in (12) using Ð <sup>∞</sup> *<sup>v</sup> E<sup>γ</sup>* ð Þ*t dt* ¼ *γR* (instead of *R*). On the other hand, the coefficients *A*1*, A*2*,* and *B* follow immediately from (2) applied with *x*0*, y*<sup>0</sup> � � <sup>¼</sup> ð Þ *<sup>t</sup>*0*, E*<sup>0</sup> *, m* <sup>¼</sup> *dE<sup>γ</sup> dt <sup>t</sup>*¼*t*<sup>0</sup> <sup>¼</sup> *dE*<sup>0</sup> *dt* � � � � *<sup>t</sup>*¼*t*<sup>0</sup> <sup>¼</sup> *<sup>α</sup>*<sup>0</sup> (for a smooth slowdown) to get *<sup>A</sup>*<sup>1</sup> and *<sup>B</sup>*, as given in (16), then with *x*0*, y*<sup>0</sup> � � <sup>¼</sup> *v, E*0Þ*, m* <sup>¼</sup> *dE<sup>γ</sup> dt* � � � � *t*¼*v* ¼ �*αE*<sup>0</sup> (for a smooth transition to the exponential decline) to get *A*<sup>2</sup> as in (16) and another formulation of *B B*ð Þ ¼ *E*<sup>0</sup> þ *αδE*0*=*2 . Then, by equating the two expressions of *B*, one gets the announced formula for *ϵ*. As for *δ,* it is found to be the unique positive solution of the following quadratic equation:

$$a\mathbf{x}^2 + b\mathbf{x} - \mathbf{3}a\_0(\mathbf{1} - \boldsymbol{\gamma})\mathbf{R} = \mathbf{0} \tag{17}$$

which discriminant is given by: <sup>Δ</sup> <sup>¼</sup> *<sup>b</sup>*<sup>2</sup> <sup>þ</sup> <sup>12</sup>*aα*0ð Þ <sup>1</sup> � *<sup>γ</sup> R,* where *<sup>a</sup>* and *<sup>b</sup>* are as announced with (16). Indeed, by evaluating the integrals in the remaining CO2 budget equation:

$$\int\_{t\_0}^{\infty} E\_\gamma(t)dt = \int\_{t\_0}^u E\_\gamma(t)dt + \int\_u^v E\_\gamma(t)dt + \int\_v^{\infty} E\_\gamma(t)dt = R \tag{18}$$

one gets:

$$\left( (A\_1/3)\mathbf{e}^3 + B\mathbf{e} \right) + \left( (A\_2/3)\delta^3 + B\delta \right) + (\chi R) = R \tag{19}$$

Then by plugging the expressions of *A*1*, A*2*, B,* and *ϵ* into (19) then simplifying, one gets the following quadratic equation in *δ*:

$$\left(a^2 E\_0^2/(\Im a\_0) + a E\_0/\Im\right) \delta^2 + \left(E\_0 + a E\_0^2/a\_0\right) \delta - (1 - \chi) \mathcal{R} = \mathbf{0} \tag{20}$$

or equivalently,

$$aE\_0(aE\_0 + a\_0)\delta^2 + \mathfrak{Z}aE\_0(E\_0 + \mathfrak{z})\delta - \mathfrak{Z}a\_0(\mathfrak{1} - \chi)\mathbb{R} = \mathbf{0} \tag{21}$$

which gives (17) with *x* ¼ *δ:*

On the other hand, Δ >0*,* and more precisely Δ >*b*<sup>2</sup> *,* and hence *δ,* as given in (16), is the unique positive solution of Eq. (17).

Based on the model formulated in (15), **Table 1** provides an estimation of the expected reduction (in % below the 2000 level) of CO2 emissions, due to a smoothly-implemented exponential mitigation starting by 2020, considering two different years (2050 and 2100), two climate targets (1.5 and 1.8°C), and two *γ* values (0.4 and 0.6). For example, whereas the emissions consistent with the 1.8°C target (for both *γ*s) will be still above the 2000 record in 2050, the 1.5°C-pathway projects, for the same year, their reduction by 39% for *γ* ¼ 0*:*6 and by 46% for *γ* ¼ 0*:*4*:* However, the latter predicts for 2050 a similar reduction as half of the IPCC-1.5°C scenarios (70–90% below 2010 record, [5]) for 0*:*11≤*γ* < 0*:*59 (long to medium transition), and as the other half (95% or more below 2010 record, [17, 21]), for *γ* <0*:*11 (long transition). More generally, as it can be seen from the rate of decline ð Þ �*αE*<sup>0</sup> *,* with *α* as in (16), a lower target or a longer transition will require a faster reduction. See also **Figure 2** for the effect of transition length.

### *4.2.1 Notes*

i. The parameter *γ* represents the fraction of the remaining budget to be used during the exponential reduction of CO2 emissions. Consequently, the remaining fraction ð Þ *γ* � 1 will go for the transition period. Therefore, the closer to 1*γ* is, the less CO2 will be emitted during the transition, and the shorter the transition will be; about 14 years (resp. 4 years) with *γ* ¼ 0*:*9*,* compared to about 25 years (resp. 7 years) with *γ* ¼ 0*:*8*,* for the climate target 1.8°C (resp. 1.5°C). In the limit case where *γ* ¼ 0 (no reduction because of no climate mitigation), the model *E<sup>γ</sup>* degenerates into the linear pathway *E*<sup>0</sup> (given in (6)). However, in the other limit case where *γ* ¼ 1 (no transition), the model is simply reduced to the exponential pathway *E*<sup>1</sup> (given in (12)).

But this would be in contradiction with the case when L≤1*:*65*,* for which

*Projected level of CO2 emissions (GtCO2) consistent with the 1.5°C target by the end of the current century*

*CO2 emissions pathways consistent with the UN climate target 1.5°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*

**Pathway** *E<sup>γ</sup>* **Slowdown (years) 2080 2090 2100** *γ* ¼ 1 0 4.7 3.2 2.2 *:*7 ≤*γ* ≤ *:*9 3–7 3.1–4.3 1.8–2.8 1*:*1 � 1*:*9 *:*4≤ *γ* ≤*:*6 10–14 0.8–2.4 *:*3 � 1*:*3 *:*1 � *:*7 *:*1<sup>≤</sup> *<sup>γ</sup>* <sup>≤</sup>*:*<sup>3</sup> <sup>16</sup>–<sup>21</sup> 3 10�<sup>5</sup> � *:*9 8 10�<sup>7</sup> � *:*08 2 10�<sup>8</sup> � *:*<sup>04</sup>

two transition phases cannot be proportional. This is due to the fact that their ratio *δ=*ϵ is a non-constant function of the climate target L*,* as it can be seen from

transition 0ð Þ < *γ* <0*:*3 *,* the first phase (quadratic slowdown) will be much longer than the second (quadratic reduction). However, for a short transition ð Þ 0*:*7 <*γ* <1 *,* the first phase will be longer for low targets (e.g., 5 vs. 2.3 years, for 1.5°C and *γ* = 0.8) but shorter for higher targets (e.g., 11.5 vs. 13.4 years, for 1.8°C and *γ* = 0.8). The case of the 1.5°C target is illustrated in **Table 2** and **Figure 2**, where one can see that the longer the transition (decreasing *γ*) the longer the slowdown (increasing ϵ) and the higher the peak of emissions (coefficient B).

iii. Whereas themodel *E<sup>γ</sup>* for the 1.8°C target seems to be close to the (IPCC) RCP4.5, the 1.5°C version appears to be similar to the (IPCC) RCP2.6 and no- and lowovershoot over the first 30 years (see **Figure 2**), with low emissions (<1 GtCO2), e.g., by 2080 for *γ* < 0*:*43 (see **Table 2**), the same way as the no-overshoot and half of the RCP2.6 models [17, 21], or earlier, e.g., by 2050 for *γ* < 0.08) with the advantage of predicting the nearly-zero emission (<0.01 GtCO2), e.g., by 2090

for *γ* < 0*:*22 (see **Table 2**), or even as early as 2050 for *γ* < 0.03.

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

Whereas the model *E<sup>γ</sup>* ð Þ 0<*γ* < 1 is designed to fit a prescribed UN climate

C 1ð Þ *:*5≤L<2 *,* in the sense that the cumulative CO2 emissions will not

<sup>0</sup>*=α*0. More generally, for any pathway *E<sup>γ</sup> ,* the times it will take the

<sup>0</sup>*=*ð Þ *α*0*γR* . This same formula shows that, for a long

*R*<2001<*E*<sup>2</sup>

*(global mitigation starting by 2020).*

**Figure 2.**

**Table 2.**

**climate target**

target <sup>L</sup><sup>∘</sup>

**71**

the formula: <sup>ϵ</sup>*=<sup>δ</sup>* <sup>¼</sup> *<sup>E</sup>*<sup>2</sup>

ii. For all pathways *E<sup>γ</sup> ,* the two transition phases cannot have the same duration, i.e., there is no parameter *γ* for which ϵ ¼ *δ*. Indeed, if this were the case, one would necessary have *<sup>γ</sup>* <sup>¼</sup> *<sup>E</sup>*<sup>2</sup> <sup>0</sup>*=*ð Þ *α*0*R ,* and this would imply that, for any climate target L*,* the corresponding remaining budget *R* (which is an increasing function of <sup>L</sup>) would be bounded below (by the constant *<sup>E</sup>*<sup>2</sup> <sup>0</sup>*=α*0).


*a Reduction (in %) of CO2 emissions below 2000 level. <sup>b</sup>*

*40% of remaining budget to be used for transition to exponential decline* ð Þ *<sup>γ</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>6</sup> *. <sup>c</sup>*

*60% of remaining budget to be used for transition to exponential decline* ð Þ *γ* ¼ 0*:*4 *.*

### **Table 1.**

*Remaining CO2 budget (*R*) from 2020 and projected reduction of CO2 emissions for 2050 and 2100 due to a global mitigation (starting by 2020) consistent with the UN climate target (UNCT).*

### **Figure 2.**

*α*2 *E*2

or equivalently,

*4.2.1 Notes*

*a*

**70**

**Table 1.**

which gives (17) with *x* ¼ *δ:*

*Global Warming and Climate Change*

would necessary have *<sup>γ</sup>* <sup>¼</sup> *<sup>E</sup>*<sup>2</sup>

*Reduction (in %) of CO2 emissions below 2000 level. <sup>b</sup>*

*40% of remaining budget to be used for transition to exponential decline* ð Þ *<sup>γ</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>6</sup> *. <sup>c</sup> 60% of remaining budget to be used for transition to exponential decline* ð Þ *γ* ¼ 0*:*4 *.*

*global mitigation (starting by 2020) consistent with the UN climate target (UNCT).*

<sup>0</sup>*=*ð Þþ <sup>3</sup>*α*<sup>0</sup> *<sup>α</sup>E*0*=*<sup>3</sup> *<sup>δ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>E</sup>*<sup>0</sup> <sup>þ</sup> *<sup>α</sup>E*<sup>2</sup>

On the other hand, Δ >0*,* and more precisely Δ >*b*<sup>2</sup>

in (16), is the unique positive solution of Eq. (17).

<sup>0</sup>*=α*<sup>0</sup>

*<sup>α</sup>E*0ð Þ *<sup>α</sup>E*<sup>0</sup> <sup>þ</sup> *<sup>α</sup>*<sup>0</sup> *<sup>δ</sup>*<sup>2</sup> <sup>þ</sup> <sup>3</sup>*αE*0ð Þ *<sup>E</sup>*<sup>0</sup> <sup>þ</sup> <sup>1</sup> *<sup>δ</sup>* � <sup>3</sup>*α*0ð Þ <sup>1</sup> � *<sup>γ</sup> <sup>R</sup>* <sup>¼</sup> 0 (21)

Based on the model formulated in (15), **Table 1** provides an estimation of the

i. The parameter *γ* represents the fraction of the remaining budget to be used during the exponential reduction of CO2 emissions. Consequently, the remaining fraction ð Þ *γ* � 1 will go for the transition period. Therefore, the closer to 1*γ* is, the less CO2 will be emitted during the transition, and the shorter the transition will be; about 14 years (resp. 4 years) with *γ* ¼ 0*:*9*,* compared to about 25 years (resp. 7 years) with *γ* ¼ 0*:*8*,* for the climate target 1.8°C (resp. 1.5°C). In the limit case where *γ* ¼ 0 (no reduction because of no climate mitigation), the model *E<sup>γ</sup>* degenerates into the linear pathway *E*<sup>0</sup> (given in (6)). However, in the other limit case where *γ* ¼ 1 (no transition), the model is simply reduced to the exponential pathway *E*<sup>1</sup> (given in (12)).

ii. For all pathways *E<sup>γ</sup> ,* the two transition phases cannot have the same duration, i.e., there is no parameter *γ* for which ϵ ¼ *δ*. Indeed, if this were the case, one

climate target L*,* the corresponding remaining budget *R* (which is an increasing function of <sup>L</sup>) would be bounded below (by the constant *<sup>E</sup>*<sup>2</sup>

**UNCT (°C)** *R* **(GtCO2) Reduction by 2050 (%)***<sup>a</sup>* **Reduction by 2100 (%)***<sup>a</sup>* 1.8 2929 None*<sup>b</sup>* (None)*<sup>c</sup>* 35.2*<sup>b</sup>* (46.2)*<sup>c</sup>* 1.5 1155 39.0*<sup>b</sup>* (45.8)*<sup>c</sup>* 97.2*<sup>b</sup>* (99.5)*<sup>c</sup>*

*Remaining CO2 budget (*R*) from 2020 and projected reduction of CO2 emissions for 2050 and 2100 due to a*

<sup>0</sup>*=*ð Þ *α*0*R ,* and this would imply that, for any

<sup>0</sup>*=α*0).

expected reduction (in % below the 2000 level) of CO2 emissions, due to a smoothly-implemented exponential mitigation starting by 2020, considering two different years (2050 and 2100), two climate targets (1.5 and 1.8°C), and two *γ* values (0.4 and 0.6). For example, whereas the emissions consistent with the 1.8°C target (for both *γ*s) will be still above the 2000 record in 2050, the 1.5°C-pathway projects, for the same year, their reduction by 39% for *γ* ¼ 0*:*6 and by 46% for *γ* ¼ 0*:*4*:* However, the latter predicts for 2050 a similar reduction as half of the IPCC-1.5°C scenarios (70–90% below 2010 record, [5]) for 0*:*11≤*γ* < 0*:*59 (long to medium transition), and as the other half (95% or more below 2010 record, [17, 21]), for *γ* <0*:*11 (long transition). More generally, as it can be seen from the rate of decline ð Þ �*αE*<sup>0</sup> *,* with *α* as in (16), a lower target or a longer transition will require a faster reduction. See also **Figure 2** for the effect of transition length.

*<sup>δ</sup>* � ð Þ <sup>1</sup> � *<sup>γ</sup> <sup>R</sup>* <sup>¼</sup> 0 (20)

*,* and hence *δ,* as given



### **Table 2.**

*Projected level of CO2 emissions (GtCO2) consistent with the 1.5°C target by the end of the current century (global mitigation starting by 2020).*

But this would be in contradiction with the case when L≤1*:*65*,* for which *R*<2001<*E*<sup>2</sup> <sup>0</sup>*=α*0. More generally, for any pathway *E<sup>γ</sup> ,* the times it will take the two transition phases cannot be proportional. This is due to the fact that their ratio *δ=*ϵ is a non-constant function of the climate target L*,* as it can be seen from the formula: <sup>ϵ</sup>*=<sup>δ</sup>* <sup>¼</sup> *<sup>E</sup>*<sup>2</sup> <sup>0</sup>*=*ð Þ *α*0*γR* . This same formula shows that, for a long transition 0ð Þ < *γ* <0*:*3 *,* the first phase (quadratic slowdown) will be much longer than the second (quadratic reduction). However, for a short transition ð Þ 0*:*7 <*γ* <1 *,* the first phase will be longer for low targets (e.g., 5 vs. 2.3 years, for 1.5°C and *γ* = 0.8) but shorter for higher targets (e.g., 11.5 vs. 13.4 years, for 1.8°C and *γ* = 0.8). The case of the 1.5°C target is illustrated in **Table 2** and **Figure 2**, where one can see that the longer the transition (decreasing *γ*) the longer the slowdown (increasing ϵ) and the higher the peak of emissions (coefficient B).

iii. Whereas themodel *E<sup>γ</sup>* for the 1.8°C target seems to be close to the (IPCC) RCP4.5, the 1.5°C version appears to be similar to the (IPCC) RCP2.6 and no- and lowovershoot over the first 30 years (see **Figure 2**), with low emissions (<1 GtCO2), e.g., by 2080 for *γ* < 0*:*43 (see **Table 2**), the same way as the no-overshoot and half of the RCP2.6 models [17, 21], or earlier, e.g., by 2050 for *γ* < 0.08) with the advantage of predicting the nearly-zero emission (<0.01 GtCO2), e.g., by 2090 for *γ* < 0*:*22 (see **Table 2**), or even as early as 2050 for *γ* < 0.03.
