Climate sensitivity and feedback

**Specific heat capacity** The specific heat capacity *c* of a substance is the heat capacity *C* of the substance divided by the mass of the substance:

$$c = \frac{C}{M} = \frac{1}{M} \frac{\Delta \boldsymbol{\varepsilon}}{\Delta \boldsymbol{T}} \qquad (8.7)$$

where $\Delta \boldsymbol{\varepsilon}$ is the amount of heat required to raise the temperature of the substance by $\Delta \boldsymbol{T}$. Here, *M* is the mass of the substance in kilograms (kg), $\Delta \boldsymbol{\varepsilon}$ is the change in energy in joules (J), $\Delta \boldsymbol{T}$ is the change in temperature in Kelvin (K) and *c* is the specific heat capacity in joules per kilogram per Kelvin (J kg⁻¹ K⁻¹). For example, the specific heat capacity of water is 4 186 J kg⁻¹ K⁻¹. From Equation (8.7), we deduce that:

$$\Delta \boldsymbol{\varepsilon} = M c \Delta \boldsymbol{T} \qquad (8.8)$$

In this case, the amount of energy required to raise the temperature of 1 m³ of water by 10°C is equal to:

$$\Delta \boldsymbol{\varepsilon} = 10^3 \times 4 186 \times 10 = 4 186 000 \, \text{J}$$

The specific heat capacity of air is about 1 000 J kg⁻¹ K⁻¹, while the mass of the atmosphere is 5.148 × 10¹⁸ kg. Therefore, the amount of energy required to raise the temperature of the atmosphere by 1°C is:

$$\begin{aligned} \Delta \boldsymbol{\varepsilon} &= (5.148 \times 10^{18} \, \text{kg}) \times (1 000 \, \text{J kg}^{-1} \, \text{K}^{-1}) \times 1 \, \text{K} \\ &= 5.148 \times 10^{21} \, \text{J} \end{aligned}$$

We can also write Equation (8.7) as follows:

$$\Delta \boldsymbol{T} = \frac{\Delta \boldsymbol{\varepsilon}}{M c} \qquad (8.9)$$

This equation gives the change in temperature for a change in energy. For example, adding 1 000 joules of energy to one liter of water will increase its temperature by about 0.239°C:

$$\Delta \boldsymbol{T} = \frac{1000}{1 \times 4 186} = 0.239$$

Equations (8.7)–(8.9) can be modified by scaling the mass *M* of the substance to standardize the required energy $\Delta \boldsymbol{\varepsilon}$. A possible scaling factor can be the surface area:

$$m = \frac{M}{\text{Area}}$$

and we get:

$$\Delta \boldsymbol{\varepsilon} = m c \Delta \boldsymbol{T} \qquad (8.10)$$

For the atmosphere, we have:

$$m = \frac{M}{\text{Area}} = \frac{5.148 \times 10^{18} \, \text{kg}}{510.0645 \times 10^6 \times 10^6 \, \text{m}^2} = 1.0093 \times 10^4 \, \text{kg m}^{-2}$$

because the radius *r* of the Earth is 6 371 km and the surface of the Earth is approximately *Area* = 4π*r*² = 510.0645 million km². In some climate modeling textbooks, Equation (8.10) is expressed using other formulas for the mass of the atmosphere per unit area. For example, *m* can be replaced
by the product of height h and density ρ, or by the ratio of pressure p to gravitational acceleration g. However, all these quantities are equivalent because we have:

$$m = h\rho = (8.2 \times 10^3 \text{ m}) \times (1.225 \text{ kg m}^{-3}) = 1.0045 \times 10^3 \text{ kg m}^{-2}$$

and:

$$m = \frac{p}{g} = \frac{101\,325 \text{ Pa}}{9.81 \text{ m s}^{-2}} = \frac{101\,325 \text{ m}^{-1} \text{ kg s}^{-2}}{9.81 \text{ m s}^{-2}} = 1.0329 \times 10^4 \text{ kg m}^{-2}$$

where h = 8.2 km is the height of the atmosphere<fnref n="45" />, ρ = 1.225 kg m⁻³ is the density of the atmosphere, p = 101 325 Pa is the standard atmospheric pressure at sea level on Earth, and g = 9.81 m s⁻² is the acceleration due to gravity at the Earth's surface. Therefore, we obtain the following equivalent formulas:

$$mc \Delta T = h \rho c \Delta T = \frac{p}{g} c \Delta T = \Delta \varepsilon$$

**Radiative relaxation timescale** We transform Equation (8.10) into a differential equation:

$$mc \frac{d\mathcal{T}}{dt} = \frac{d\varepsilon}{dt}$$

We consider a black body. At the equilibrium, we have:

$$F_{\text{solar}} - \sigma \mathcal{T}^4 = 0$$

We deduce that:

$$\begin{aligned} \varepsilon &= F_{\text{solar}} - \sigma \mathcal{T}^4 \\ &= \sigma \mathcal{T}_e^4 - \sigma \mathcal{T}^4 \end{aligned}$$

Let us assume that $\mathcal{T} = \mathcal{T}_e + \Delta\mathcal{T}$. It follows that:

$$mc \frac{d\Delta\mathcal{T}}{dt} = -4\sigma\mathcal{T}_e^3 \Delta\mathcal{T}$$

because:

$$\frac{\partial}{\partial \Delta \mathcal{T}} (\sigma \mathcal{T}_e^4 - \sigma \mathcal{T}^4) = -4\sigma \mathcal{T}_e^3$$

Let $\tau_e$ be the radiative relaxation timescale defined as:

$$\tau_e = \frac{mc}{4\sigma\mathcal{T}_e^3}$$

We have:

$$\left\{ \begin{aligned} \frac{d\Delta\mathcal{T}}{dt} &= -\frac{1}{\tau_e}\Delta\mathcal{T} \\ \Delta\mathcal{T}(0) &= \Delta\mathcal{T}_0 \end{aligned} \right.$$

The solution of this ordinary differential equation is well-known and we get:

$$\Delta\mathcal{T}(t) = \exp\left(-\frac{t}{\tau_e}\right) \Delta\mathcal{T}_0$$

<footnote n="45">⁴⁵The height of the atmosphere depends on the choice of layer. Here we consider the troposphere, which is the lowest layer of the atmosphere at the polar regions.</footnote>
$\Delta T(t)$ gives the impulse response of an initial temperature shock of $\Delta T_0$. Because $\tau_e > 0$, we conclude that the system is stable:

$$ \lim_{t \to \infty} \Delta T(t) = 0 $$

We note that the equation for $\Delta T(t)$ describes an exponential survival function with parameter $\tau_e^{-1}$. We deduce that the radiative relaxation timescale $\tau_e$ is the mean lifetime.

Using the previously obtained values for the atmosphere ($m = 1.0093 \times 10^4 \text{ kg m}^{-2}$, $c = 1000 \text{ J kg}^{-1} \text{ K}^{-1}$ and $T_e = 255.81 \text{ K}$), the radiative relaxation timescale is equal to 31 days$^{46}$:

$$ \begin{aligned} \tau_e &= \frac{(1.0093 \times 10^4 \text{ kg m}^{-2}) \times (1000 \text{ J kg}^{-1} \text{ K}^{-1})}{4 \times (5.67 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}) \times (255.81 \text{ K})^3} \\ &= 2658427 \text{ J W}^{-1} \\ &= \frac{2658427 \text{ s}}{24 \times 3600 \text{ s}} \\ &= 30.77 \text{ days} \end{aligned} $$

Figure 8.43 shows the impulse response function for $\Delta T_0 = +1^\circ\text{C}$. We check that the effect of the initial shock disappears after a period of a few weeks. However, the memory of the temperature perturbation depends strongly on the effective temperature $T_e$, as illustrated in the second panel of Figure 8.43. This explains why a temperature shock does not have the same effect all over the terrestrial globe, e.g., in the polar regions or in the equatorial regions.

<figure><img src="image_323.png" /><figcaption>Figure 8.43: Impulse response function for $\Delta T_0 = +1^\circ\text{C}$ and a black body</figcaption></figure>

<footnote>$^{46}$We use the fact that $\text{J W}^{-1} = \text{s}$.</footnote>
<figure><img src="image_324.png" /><figcaption>Figure 8.44: Impulse response function for $\Delta T_0 = +1^\circ C$ and a gray body</figcaption></figure>

Now consider the case of a gray body and the one-layer model with emissivity. On page 668, we found that:

$$ \sigma T_e^4 = (2 - \varepsilon) \sigma T_a^4 = \left(\frac{2 - \varepsilon}{2}\right) \sigma T_s^4 $$

We deduce that:

$$ mc \frac{dT_a}{dt} = \frac{d}{dt} (\sigma T_e^4 - (2 - \varepsilon) \sigma T_a^4) $$

and:

$$ mc \frac{dT_s}{dt} = \frac{d}{dt} \left( \sigma T_e^4 - \left( \frac{2 - \varepsilon}{2} \right) \sigma T_s^4 \right) $$

Using the same reasoning as above, we obtain:

$$ \left\{ \begin{array}{l} \Delta T_a(t) = \exp\left(-\frac{t}{\tau_a}\right) \Delta T_0 \\ \Delta T_s(t) = \exp\left(-\frac{t}{\tau_s}\right) \Delta T_0 \end{array} \right. $$

where:

$$ \tau_a = \frac{mc}{4(2 - \varepsilon) \sigma T_e^3} \quad \text{and} \quad \tau_s = \frac{mc}{2(2 - \varepsilon) \sigma T_e^3} $$

We consider an emissivity value of 65%, and find that $\tau_a = 22.79$ and $\tau_s = 45.58$ days. The corresponding impulse response functions are given in Figure 8.44. Although these calculations are rough and approximate$^{47}$, they clearly show that the response at the Earth's surface is not the same as in the different layers of the atmosphere.

<footnote>$^{47}$In fact, it is not realistic to use the same value for m, c and $T_e$.</footnote>
**Mathematical definition of climate sensitivity and feedback** Before giving the mathematical expression of these two concepts, here is how Hartman introduces them in his book:

"[...] Climate forcing is a change to the climate system that can be expected to change the climate. Examples would be doubling the CO₂, increasing the total solar irradiance (TSI) by 2%, introducing volcanic aerosols into the stratosphere, etc. Climate forcings are usually quantified in terms of how many W/m² they change the energy balance when imposed. For example, instantaneously doubling the CO₂ changes the energy balance at the top of the atmosphere by about 4 W/m². A feedback process is a response of the climate system to surface warming that then alters the energy balance in such a way as to change the temperature response to the forcing. A positive feedback makes the forced response bigger, and a negative feedback makes it smaller. Classic examples of positive feedbacks are ice-albedo feedback and water-vapor feedback. When it warms, ice melts, and this reduces Earth's albedo and causes further warming. When it cools, ice grows, and this increases Earth's albedo, causing further cooling [...]." (Hartmann, 2016, page 294).

Mathematically, we assume that some extra energy $\Delta F$ is added to the system. $\Delta F$ is called the radiative forcing and is measured in W/m². The climate response is generally measured as the change in the surface temperature $\Delta T_s$. The climate sensitivity is defined as:

$$ \phi := \frac{\Delta T_s}{\Delta F} $$

Using differential notation, we have:

$$ \phi := \frac{dT_s}{dF} $$

implying that:

$$ dT_s = \phi dF $$

Following Hartmann (2016, page 295), we assume that the perturbation $dF$ depends on the temperature $T_s$ and some exogenous factors $x_i$:

$$ dF = \frac{\partial F}{\partial T_s} dT_s + \sum_{i=1}^{n} \frac{\partial F}{\partial x_i} dx_i $$

We conclude that:

$$ \left(1 - \phi \frac{\partial F}{\partial T_s}\right) dT_s = \phi \sum_{i=1}^{n} \frac{\partial F}{\partial x_i} dx_i $$

We then get a feedback mechanism, because the temperature dynamics depend on the factors $x_i$, but also on the temperature response.

Let us apply this framework to the one-layer model with emissivity. At the equilibrium, we have $\mathcal{E} = 0$ where:

$$ \begin{aligned} \mathcal{E} &= F_{\text{solar}} - \left(\frac{2 - \varepsilon}{2}\right) \sigma T_s^4 \\ &= \frac{1}{4} (1 - \alpha_p) S_0 - \left(\frac{2 - \varepsilon}{2}\right) \sigma T_s^4 \end{aligned} $$

The first-order Taylor Series expansion of $\mathcal{E} = 0$ gives:

$$ \Delta \mathcal{E} = \frac{1}{4} (1 - \alpha_p) \Delta S_0 - \frac{1}{4} S_0 \Delta \alpha_p + \frac{1}{2} \sigma T_s^4 \Delta \varepsilon - 4 \left(\frac{2 - \varepsilon}{2}\right) \sigma T_s^3 \Delta T_s $$
Remember that each perturbation $\Delta y$ depends on the temperature $T_s$ and some exogenous factors $x_i$:

$$ \Delta y = \frac{\partial y}{\partial T_s} \Delta T_s + \sum_{i=1}^{n} \frac{\partial y}{\partial x_i} \Delta x_i $$

We have:

$$ \Delta \boldsymbol{\epsilon} = \frac{1}{4}(1 - \alpha_p) \Delta S_0 - \frac{1}{4} S_0 \left( \frac{\partial \alpha_p}{\partial T_s} \Delta T_s + \sum_{i=1}^{n} \frac{\partial \alpha_p}{\partial x_i} \Delta x_i \right) + \frac{1}{2} \sigma T_s^4 \left( \frac{\partial \varepsilon}{\partial T_s} \Delta T_s + \sum_{i=1}^{n} \frac{\partial \varepsilon}{\partial x_i} \Delta x_i \right) - 4 \left( \frac{2 - \varepsilon}{2} \right) \sigma T_s^3 \Delta T_s $$

We deduce that$^{48}$:

$$ \Delta \boldsymbol{\epsilon} = \lambda \Delta T_s + \sum_{i=0}^{n} \Delta F_i \qquad (8.11) $$

where $\Delta F_0 := \Delta F_{solar} = \frac{1}{4}(1 - \alpha_p) \Delta S_0$,

$$ \Delta F_i = \left( -\frac{1}{4} S_0 \frac{\partial \alpha_p}{\partial x_i} + \frac{1}{2} \sigma T_s^4 \frac{\partial \varepsilon}{\partial x_i} \right) \Delta x_i \quad \text{for } i \ge 1 $$

and:

$$ \lambda = -\frac{1}{4} S_0 \frac{\partial \alpha_p}{\partial T_s} + \frac{1}{2} \sigma T_s^4 \frac{\partial \varepsilon}{\partial T_s} - 4 \left( \frac{2 - \varepsilon}{2} \right) \sigma T_s^3 $$

Since $\Delta \boldsymbol{\epsilon}$ and $\Delta F_i$ are measured in W/m$^2$ and $\Delta T_s$ is measured in Kelvin, we deduce that $\lambda$ is measured in W m$^{-2}$ K$^{-1}$. As before, we transform $\Delta \boldsymbol{\epsilon}$ into $\Delta T_s$ by considering the heat capacity $c$ expressed in W m$^{-2}$ K$^{-1}$ s. Then Equation (8.11) becomes:

$$ c \frac{d\Delta T_s}{dt} = \lambda \Delta T_s + \Delta F \qquad (8.12) $$

where $\Delta F = \sum_{i=0}^{n} F_i$. The climate feedback parameter $\lambda$ can be positive or negative, and we have the following mathematical properties:

* If $\lambda > 0$, the system is unstable;

* If $\lambda < 0$, the system is stable and the equilibrium is reached when:

$$ \Delta T_s = \Delta T_s^* = -\frac{\Delta F}{\lambda} = -\phi \Delta F \qquad (8.13) $$

We see that the climate feedback parameter can be decomposed into three components:

$$ \lambda = \lambda_0 + \lambda_{\alpha_p} + \lambda_{\varepsilon} $$

where:

1. $\lambda_0$ is the Planck feedback or the black body response:

$$ \lambda_0 = -4 \left( \frac{2 - \varepsilon}{2} \right) \sigma T_s^3 $$

<footnote>$^{48}$ $\lambda = \phi^{-1}$ is the inverse of the climate sensitivity parameter $\phi$ defined by Hartmann (2016).</footnote>
2. $\lambda_{\alpha_p}$ is the surface albedo feedback:

$$ \lambda_{\alpha_p} = -\frac{1}{4} S_0 \frac{\partial \alpha_p}{\partial T_s} $$

3. $\lambda_{\epsilon}$ is the emissivity feedback:

$$ \lambda_{\epsilon} = \frac{1}{2} \sigma T_s^4 \frac{\partial \epsilon}{\partial T_s} $$

Since $\epsilon < 1$, the Planck feedback is negative, meaning that it stabilizes the climate and counter-acts global warming. In fact, as the Earth warms, it emits more thermal radiation into space, and this increased longwave radiation acts as a natural cooling mechanism. Conversely, as the Earth cools, it emits less thermal radiation into space, and this decreased longwave radiation acts as a natural warming mechanism. Earlier we found that $\epsilon = 0.78$. So the estimate of $\lambda_0$ is$^{49}$:

$$ \begin{aligned} \lambda_0 &= -4 \left( \frac{2 - 0.78}{2} \right) \times (5.67 \times 10^{-8}) \times (273.15 + 15)^3 \\ &= -3.310 \, \mathrm{W} \, \mathrm{m}^{-2} \, \mathrm{K}^{-1} \end{aligned} $$

The sign of the surface albedo feedback depends on the sign of $\frac{\partial \alpha_p}{\partial T_s}$. Two main factors play a role in determining $\alpha_p$:

> "The albedo of the planet for solar radiation is primarily determined by the clouds and surface, with the main variable component of the latter being the ice/snow cover." (Hansen et al., 1984, page 166).

The feedback is positive for the ice-albedo mechanism because $\frac{\partial \alpha_p}{\partial T_s} < 0$. In fact, decreased temperature leads to increased sea ice and snow cover. This increases the albedo and decreases the absorption of shortwave radiation. This mechanism is satisfied globally, but we observe a lot of regional and seasonal variation (Stephens et al., 2015). If the Earth were completely covered in ice, its albedo $\alpha_p$ would be about 84%, meaning it would reflect most of the sunlight that hits it. On the other hand, if the Earth were covered by a dark green forest canopy, the albedo would be about 14% and most of the sunlight would be absorbed$^{50}$. This explains why the ice-albedo feedback is greatest in the polar regions (Goosse et al., 2018). Cloud albedo feedback occurs because changes in cloud cover, cloud altitude or cloud properties affect the amount of reflected shortwave radiation. This feedback can be either positive or negative, depending on the specific change. The overall effect is a positive feedback (IPCC, 2021, Chapter 7, page 926). According to IPCC AR6, the value of the global surface albedo feedback is estimated to be $0.35 \, \mathrm{W} \, \mathrm{m}^{-2} \, \mathrm{K}^{-1}$, with a 90% confidence interval from 0.10 to $0.60 \, \mathrm{W} \, \mathrm{m}^{-2} \, \mathrm{K}^{-1}$ (IPCC, 2021, Chapter 7, page 971). This means that:

$$ \frac{\partial \alpha_p}{\partial T_s} = -\frac{4\lambda_{\alpha_p}}{S_0} = -\frac{4 \times 0.35}{1368} = -1.023 \times 10^{-3} $$

<footnote>$^{49}$For comparison, the most recent estimate is $-3.22 \, \mathrm{W} \, \mathrm{m}^{-2} \, \mathrm{K}^{-1}$, while the 90% confidence interval is from $-3.4$ to $-3.0 \, \mathrm{W} \, \mathrm{m}^{-2} \, \mathrm{K}^{-1}$ (IPCC, 2021, Chapter 7, page 968).</footnote>

<footnote>$^{50}$Source: https://earthobservatory.nasa.gov/images/84499/measuring-earths-albedo.</footnote>
Box 8.9: Ice-albedo feedback modeling

We have seen that the surface albedo feedback is equal to $\lambda_{\alpha_p} = -\frac{1}{4}S_0 \frac{\partial \alpha_p}{\partial T_s}$. So we need to parameterize the temperature dependence of the albedo. If we assume that $\alpha_p = f_{\text{albedo}}(T_s)$, then we get $\lambda_{\alpha_p} = -\frac{1}{4}S_0 f'_{\text{albedo}}(T_s)$. For example, Sellers (1969) suggested:

$$ \alpha_p = \begin{cases} b(\phi) - 0.009T_s & \text{if } T_s \le 283.16 \text{ K} \\ b(\phi) - 2.548 & \text{if } T_s \ge 283.16 \text{ K} \end{cases} $$

where $b(\phi)$ is an estimated coefficient that depends on the latitude $\phi$. Sellers (1969) found $b(\phi) \in [2.798, 2.992]$. Below, we show the function $b(\phi)$ with respect to the spherical coordinate $\phi$. On average we have $\bar{b}(\phi) = 2.8811$. We deduce that:

$$ \lambda_{\alpha_p} = \frac{1}{4} \times 1368 \times 0.009 = 3.078 \text{ W m}^{-2} \text{ K}^{-1} $$

It is obvious that this positive feedback has been overestimated. The reason is that snow and sea ice cover about 10% of the Earth's surface. Therefore, we get $\lambda_{\alpha_p} \approx 0.3 \text{ W m}^{-2} \text{ K}^{-1}$.

<figure><img src="image_325.png" /></figure>

Budyko (1969) proposed another famous ice-albedo model, assuming that the land and oceans are completely covered by snow and sea ice when the temperature is below a threshold $T_{\text{cold}}$, and that the planet is free of ice and snow when the temperature is above another threshold $T_{\text{warm}}$:

$$ \alpha_p = \begin{cases} \alpha_{\text{cold}} & \text{if } T_s \le T_{\text{cold}} \\ \alpha_{\text{warm}} + (\alpha_{\text{cold}} - \alpha_{\text{warm}}) \left( \frac{T_{\text{warm}} - T_s}{T_{\text{warm}} - T_{\text{cold}}} \right)^{\eta} & \text{if } T_{\text{cold}} \le T_s \le T_{\text{warm}} \\ \alpha_{\text{warm}} & \text{if } T_s \ge T_{\text{warm}} \end{cases} $$

where $\eta \ge 1$. It follows that:

$$ \lambda_{\alpha_p}(T_s) = \frac{1}{4} \eta S_0 (\alpha_{\text{cold}} - \alpha_{\text{warm}}) \frac{(T_{\text{warm}} - T_s)^{\eta-1}}{(T_{\text{warm}} - T_{\text{cold}})^{\eta}} \cdot 1 \{T_{\text{cold}} \le T_s \le T_{\text{warm}}\} $$

Using the values $\alpha_{\text{cold}} = 0.7$, $T_{\text{cold}} = 260 \text{ K}$, $\alpha_{\text{warm}} = 0.3$, $T_{\text{warm}} = 295 \text{ K}$ and $\eta = 2$, we get $\lambda_{\alpha_p} (282 \text{ K}) = 2.90 \text{ W m}^{-2} \text{ K}^{-1}$, $\lambda_{\alpha_p} (288 \text{ K}) = 1.56 \text{ W m}^{-2} \text{ K}^{-1}$ and $\lambda_{\alpha_p} (293 \text{ K}) = 0.45 \text{ W m}^{-2} \text{ K}^{-1}$.
The third feedback is the change in emissivity, which is the sum of several components:

$$ \lambda_{\varepsilon} = \lambda_{\text{water vapor}} + \lambda_{\text{lapse rate}} + \lambda_{\text{cloud longwave}} + \dots $$

The water vapor feedback, also known as the specific humidity feedback, is the most important positive and destabilizing feedback. It can be described as follows:

> "[...] As the temperature increases, the amount of water vapor in saturated air increases. Since water vapor is the principal greenhouse gas, increasing water vapor content will increase the greenhouse effect of the atmosphere and raise the surface temperature even further." (Hartmann, 2016, page 297).

The mechanism behind water vapor feedback is related to the thermodynamics of moist (or humid) air in the atmosphere. Water molecules move freely between the liquid and vapor phases, and the equilibrium depends on the water vapor saturation $e_s$. The relationship between $e_s$ and temperature is given by the Clausius-Clapeyron equation:

$$ \frac{\mathrm{d}e_s}{\mathrm{d}\mathcal{T}_s} = \frac{1}{(\alpha_{\text{vapor}} - \alpha_{\text{liquid}}) \mathcal{T}_s} = \frac{L_{\text{vapor}} e_s}{R_{\text{vapor}} \mathcal{T}_s^2} $$

where $\alpha_{\text{vapor}}$ is the specific volume of the vapor phase, $\alpha_{\text{liquid}}$ is the specific volume of the liquid phase, $R_{\text{vapor}}$ is the gas constant of water vapor and $L_{\text{vapor}}$ is the specific latent heat of evaporation of water. If we rearrange this equation, we obtain the following relationship:

$$ \frac{\mathrm{d}e_s}{e_s} = \left( \frac{L_{\text{vapor}}}{R_{\text{vapor}} \mathcal{T}_s} \right) \frac{\mathrm{d}\mathcal{T}_s}{\mathcal{T}_s} = \varphi \frac{\mathrm{d}\mathcal{T}_s}{\mathcal{T}_s} $$

According to (Hartmann, 2016, page 298), $\varphi \approx 20$, which means that a 1% change in temperature is associated with a 20% change in saturation humidity. This means that a 20% change in saturation humidity causes a change in temperature of $1\% \times (273.15 + 15) \approx 2.9$ K. According to IPCC AR6, the value $\lambda_{\text{water vapor}}$ of the water vapor feedback is assessed to be $1.85 \text{ W m}^{-2} \text{ K}^{-1}$ (IPCC, 2021, Chapter 7, page 969). The lapse rate mechanism describes the relationship between temperature and altitude in the atmosphere (Colman and Soden, 2021, page 13):

$$ \Gamma = -\frac{\partial \mathcal{T}}{\partial z} \in [4 \text{ K km}^{-1}, 10 \text{ K km}^{-1}] $$

where $z$ is the altitude in kilometers. On average, the lapse rate is about 6.5°C per kilometer. This means that the temperature is not uniform throughout the atmosphere. According to IPCC (2021, Chapter 7, page 969), “the warming is larger in the upper troposphere than in the lower troposphere [...] leading to a larger radiative emission to space and therefore a negative feedback” and the average value of $\lambda_{\text{lapse rate}}$ is $-0.50 \text{ W m}^{-2} \text{ K}^{-1}$. The case of cloud feedbacks is more complicated because it involves several mechanisms: (1) high cloud altitude, (2) tropical high cloud amount, (3) subtropical marine low cloud, (4) land cloud, (5) midlatitude cloud amount, (6) extratropical cloud optical depth, and (7) Arctic cloud. According to IPCC AR6, the value $\lambda_{\text{cloud}}$ of the net cloud feedback is estimated to be $0.42 \text{ W m}^{-2} \text{ K}^{-1}$ (IPCC, 2021, Chapter 7, page 974). One of the difficulties is to decompose the cloud feedback into shortwave and longwave feedbacks, since the global surface albedo feedback already includes shortwave cloud mechanisms. Assuming that 2/3 of the cloud feedback is longwave radiation, we get:

$$ \lambda_{\varepsilon} \approx 1.85 - 0.50 + \frac{2}{3} \times 0.42 = 1.63 \text{ W m}^{-2} \text{ K}^{-1} $$
Finally, we can compute the total feedback parameter by summing the Planck feedback, the surface albedo feedback, and the emissivity feedback:

$$
\begin{aligned}
\lambda &= \lambda_0 + \lambda_{\alpha_p} + \lambda_{\varepsilon} \\
&= -3.31 + 0.35 + 1.63 \\
&= -1.33 \, \mathrm{W} \, \mathrm{m}^{-2} \, \mathrm{K}^{-1}
\end{aligned}
$$

This value is obtained with a simple one-layer model with emissivity. It is relatively close to the average climate feedback parameter calculated with more complex models. However, using global climate models, we can assess the uncertainty in the climate feedback parameter. In this case, we consider the parameter $\lambda$ as a Gaussian random variable $\tilde{\lambda} \sim N(\mu_{\lambda}, \sigma_{\lambda}^2)$. As before, we can decompose the feedback as a sum of individual feedbacks:

$$ \tilde{\lambda} = \sum_{i=1}^{n} \tilde{\lambda}_{i} $$

where $\tilde{\lambda}_i \sim N(\mu_i, \sigma_i^2)$. Assuming that the individual feedbacks are independent, we have:

$$ \left\{ \begin{array}{l} \mu_{\lambda} = \sum_{i=1}^{n} \mu_i \\ \sigma_{\lambda} = \sqrt{\sum_{i=1}^{n} \sigma_i^2} \end{array} \right. $$

<table><caption>Table 8.10: Parameters $\mu_i$ and $\sigma_i$ of feedback parameters</caption><thead><tr><th>Feedback mechanism</th><th>μ<sub>i</sub></th><th>σ<sub>i</sub></th><th>Shortwave</th></tr></thead><tbody><tr><td>High-cloud altitude</td><td>+0.20</td><td>0.10</td><td></td></tr><tr><td>Tropical marine low cloud</td><td>+0.25</td><td>0.16</td><td></td></tr><tr><td>Tropical anvil cloud area</td><td>-0.20</td><td>0.20</td><td></td></tr><tr><td>Land cloud amount</td><td>+0.08</td><td>0.08</td><td></td></tr><tr><td>Middle-latitude marine low-cloud amount</td><td>+0.12</td><td>0.12</td><td></td></tr><tr><td>High-latitude low-cloud optical depth</td><td>+0.00</td><td>0.10</td><td>✓</td></tr><tr><td>Planck feedback</td><td>-3.20</td><td>0.10</td><td>✓</td></tr><tr><td>Water vapor + lapse rate</td><td>+1.15</td><td>0.15</td><td></td></tr><tr><td>Surface albedo</td><td>+0.30</td><td>0.15</td><td>✓</td></tr><tr><td>Total cloud</td><td>+0.45</td><td>0.33</td><td></td></tr><tr><td>Stratospheric</td><td>+0.00</td><td>0.10</td><td></td></tr><tr><td>Atmospheric composition changes</td><td>+0.00</td><td>0.15</td><td></td></tr><tr><td>Climate feedback parameter</td><td>-1.30</td><td>0.44</td><td></td></tr></tbody></table>

Source: (Sherwood et al., 2020, Table 1, page 18).

In Table 8.10, we report the estimated value of $\mu_i$ and $\sigma_i$ for individual climate feedbacks calculated by Sherwood et al. (2020, Table 1, page 18). The estimation process is based on multiple lines of evidence, including historical observations, general circulation models, and theory. The first part of the table lists the parameters for the individual cloud feedbacks. Their corresponding probability density functions are shown in Figure 8.45. Using this modeling framework, we can compute the probability that the feedback is positive:

$$ \mathrm{Pr}\{\tilde{\lambda}_i \geq 0\} = 1 - \Phi\left(-\frac{\mu_i}{\sigma_i}\right) $$
<figure><img src="image_326.png" /><figcaption>Figure 8.46: Probability density function of positive feedbacks</figcaption></figure>

<figure><img src="image_327.png" /></figure>
<figure><img src="image_328.png" /><figcaption>Figure 8.47: Comparison of $\tilde{\lambda}_{\text{Planck}}$ and $\tilde{\lambda}_{\text{positive}}$</figcaption></figure>

or the confidence interval at the $\alpha$ confidence level: $[\mu_i - c_\alpha\sigma_i, \mu_i + c_\alpha\sigma_i]$ where $c_\alpha = \Phi^{-1}((1+\alpha)/2)$. In IPCC AR6, the confidence intervals of $\tilde{\lambda}_i$ are called likely and very likely ranges and correspond to $\alpha = 66\%$ and $\alpha = 90\%$, respectively$^{51}$. Using the parameters of the individual cloud feedbacks, we can infer the distribution of the total cloud feedback. We have:

$$ \mu_{\text{total cloud}} = 0.20 + 0.25 - 0.20 + 0.08 + 0.12 + 0.00 = 0.45 \, \text{W m}^{-2} \text{K}^{-1} $$

and:

$$ \sigma_{\text{total cloud}} = \sqrt{0.10^2 + 0.16^2 + 0.20^2 + 0.08^2 + 0.12^2 + 0.10^2} = 0.3262 \, \text{W m}^{-2} \text{K}^{-1} $$

These are the numbers obtained by Sherwood et al. (2020). We now plot the probability density functions of the five main positive feedback components in Figure 8.46 and aggregate them into a single feedback. We have:

$$ \mu_{\text{positive}} = 1.15 + 0.30 + 0.45 + 0.00 + 0.00 = 1.90 \, \text{W m}^{-2} \text{K}^{-1} $$

and:

$$ \sigma_{\text{positive}} = \sqrt{0.15^2 + 0.15^2 + 0.33^2 + 0.10^2 + 0.15^2} = 0.4317 \, \text{W m}^{-2} \text{K}^{-1} $$

The aggregated positive feedback is shown in Figure 8.47 and can be compared to the Planck feedback. Finally, Sherwood et al. (2020) aggregated all the positive and negative feedbacks to obtain the climate feedback parameter: $\tilde{\lambda} \sim \mathcal{N}(\mu_\lambda, \sigma_\lambda^2)$ where $\mu_\lambda = -1.30 \, \text{W m}^{-2} \text{K}^{-1}$ and $\sigma_\lambda = 0.44 \, \text{W m}^{-2} \text{K}^{-1}$. The probability density function of $\tilde{\lambda}$ is given in Figure 8.48. The probability that the climate feedback is positive is small$^{52}$, less than 1%.

$$ \text{Pr}\{\tilde{\lambda} \ge 0\} = \text{Pr}\left\{\frac{\tilde{\lambda} - \mu_{\lambda}}{\sigma_{\lambda}} \ge -\frac{\mu_{\lambda}}{\sigma_{\lambda}}\right\} = 1 - \Phi\left(-\frac{\mu_{\lambda}}{\sigma_{\lambda}}\right) = 1 - \Phi\left(\frac{1.30}{0.44}\right) = 0.16\% $$

<footnote>$^{51}$ $c_\alpha$ is equal to 0.95 and 1.65.</footnote>

<footnote>$^{52}$ We have:</footnote>
<figure><img src="image_329.png" /><figcaption>Figure 8.48: Probability density function of the climate feedback parameter</figcaption></figure>

**Equilibrium climate sensitivity (ECS)** We recall that the definition of equilibrium is given by Equation (8.13):

$$ \Delta T_s^* = -\frac{\Delta F}{\lambda} = -\phi \Delta F $$

Therefore, we can calculate the effect of a specific value of radiative forcing $\Delta F$ on temperature change. In particular, Earth's equilibrium climate sensitivity (ECS) is the long-term global mean surface temperature change due to a specific value of radiative forcing corresponding to a doubling of CO₂ in the atmosphere (Jeevanjee, 2023). We have:

$$ \text{ECS} := \Delta T_{2 \times \text{CO}_2} = -\frac{\Delta F_{2 \times \text{CO}_2}}{\lambda} $$

where ECS (or $\Delta T_{2 \times \text{CO}_2}$) is the equilibrium climate sensitivity, $\lambda$ is the climate sensitivity parameter and $\Delta F_{2 \times \text{CO}_2}$ is the radiative forcing resulting from a doubling of the atmospheric carbon dioxide concentration.

**Stochastic equilibrium temperature modeling** Let us assume that $\Delta F = 4 \text{ W/m}^2$ and $\lambda = -1.33 \text{ W m}^{-2} \text{ K}^{-1}$. The equilibrium warming is then equal to 3°C:

$$ \Delta T_s^* = -\frac{4 \text{ W/m}^2}{-1.33 \text{ W m}^{-2} \text{ K}^{-1}} = 3.0075 \text{ K} $$

Remember that the dynamics of the temperature change is given by the ordinary differential equation:

$$ c \frac{d\Delta T_s}{dt} = \lambda \Delta T_s + \Delta F $$
We assume that the solution has the following form:

$$ \Delta T_s(t) = e^{At}B + C $$

We deduce that:

$$ \frac{d\Delta T_s}{dt} = Ae^{At}B = A(\Delta T_s(t) - C) $$

The identification of the parameters results in:

$$ \left\{ \begin{array}{l} A = \frac{\lambda}{c} \\ -AC = \frac{\Delta F}{c} \\ B + C = \Delta T_s(0) \end{array} \right. $$

The solutions are then $A = c^{-1}\lambda$,

$$ C = -\frac{\Delta F}{Ac} = -\frac{\Delta F}{\lambda} = \Delta T_s^* $$

and:

$$ B = \Delta T_s(0) - C = \Delta T_s(0) - \Delta T_s^* $$

Finally, we conclude that:

$$ \Delta T_s(t) = \exp\left(-\frac{t}{\tau}\right) (\Delta T_s(0) - \Delta T_s^*) + \Delta T_s^* \quad (8.14) $$

where:

$$ \tau = -\frac{c}{\lambda} \qquad (8.15) $$

Again, we obtain that the equation for $\Delta T_s(t)$ describes an exponential survival function with parameter $\tau^{-1}$. Using a specific heat capacity of $c = 4 \times 10^8 \text{ J m}^{-2} \text{K}^{-1}$, we get$^{53}$:

$$ \tau = -\frac{c}{\lambda} = -\frac{4 \times 10^8 \text{ J m}^{-2} \text{K}^{-1}}{-1.33 \text{ W m}^{-2} \text{K}^{-1}} = \frac{4 \times 10^8 \text{ W m}^{-2} \text{K}^{-1} \text{ s}}{1.33 \text{ W m}^{-2} \text{K}^{-1}} = 3.0075 \times 10^8 \text{ s} $$

The relaxation time $\tau$ of the climate system is then equal to 3480.92 days or 9.53 years$^{54}$. Due to the exponential distribution, $\tau$ is also the mean lifetime. The computation of the half-life$^{55}$ gives $t_{1/2} = \tau \ln(2) = 6.61$ years. Considering that the initial state is $\Delta T_s(0) = 0$, the dynamics of $\Delta T_s(t)$ when $\Delta F = 4 \text{ W/m}^2$ and $\lambda = -1.33 \text{ W m}^{-2} \text{K}^{-1}$ is shown in Figure 8.49. We check that the system converges to the equilibrium: $\lim_{t \to \infty} \Delta T_s(t) = \Delta T_s^* = 3.0075^\circ\text{C}$.

The previous analysis assumes that the climate feedback parameter is certain. In fact, it is stochastic, which means that the equilibrium temperature and the temperature dynamics are stochastic. If $\tilde{\lambda} \sim \mathcal{N}(\mu_{\lambda}, \sigma_{\lambda}^2)$, the equilibrium temperature is equal to:

$$ \Delta \tilde{T}_s^* = -\frac{\Delta F}{\tilde{\lambda}} = \frac{1}{\xi} $$

$$ \Delta T_s(t) - \Delta T_s^* = \exp\left(-\frac{t}{\tau}\right) (\Delta T_s(0) - \Delta T_s^*) $$

the half-life $t_{1/2}$ is defined as the solution of the equation $\exp(-\frac{t_{1/2}}{\tau}) = \frac{1}{2}$.

<footnote>$^{53}$We have $\text{J m}^{-2} \text{K}^{-1} = \text{J s}^{-1} \text{m}^{-2} \text{K}^{-1} = \text{W m}^{-2} \text{K}^{-1}$ s.</footnote>

<footnote>$^{54}$We scale $\tau$ by $24 \times 3600$ seconds to get the relaxation time expressed in days and assume that one year is 362.25 days.</footnote>

<footnote>$^{55}$Since we have:</footnote>
<figure><img src="image_330.png" /><figcaption>Figure 8.50: Probability density function of the equilibrium temperature (ΔF = 4 W/m², µλ = -1.30 W m⁻² K⁻¹ and σλ = 0.44 W m⁻² K⁻¹)</figcaption></figure>

<figure><img src="image_331.png" /></figure>
<figure><img src="image_332.png" /><figcaption>Figure 8.52: Monte Carlo simulation of the surface temperature dynamics after a radiative forcing of $4 \text{ W/m}^2$ ($\mu_{\lambda} = -1.30 \text{ W m}^{-2} \text{ K}^{-1}$ and $\sigma_{\lambda} = 0.44 \text{ W m}^{-2} \text{ K}^{-1}$)</figcaption></figure>

<figure><img src="image_333.png" /></figure>
where:

$$ \xi = -\frac{\tilde{\lambda}}{\Delta F} \sim \mathcal{N}(\mu_{\xi}, \sigma_{\xi}^2) \equiv \mathcal{N}\left(-\frac{\mu_{\lambda}}{\Delta F}, \frac{\sigma_{\lambda}^2}{\Delta F^2}\right) $$

We deduce that $\Delta\tilde{T}_s^*$ follows a reciprocal normal distribution, and its probability density function is$^{56}$:

$$ f(x) = \frac{1}{\sigma_{\xi} x^2 \sqrt{2\pi}} e^{-\frac{1}{2} \left( \frac{x^{-1} - \mu_{\xi}}{\sigma_{\xi}} \right)^2} $$

Using the previous calibration of the climate feedback parameter $\tilde{\lambda} \sim \mathcal{N}(-1.30, 0.44^2)$, Figure 8.50 shows the probability density function of the equilibrium temperature when the radiative forcing is equal to 4 W/m$^2$. The distribution of $\Delta\tilde{T}_s^*$ is right skewed and has an excess of kurtosis. Using numerical integration, we compute the cumulative distribution function and the exceedance probability:

<table><thead><tr><th>Temperature θ</th><th>2°C</th><th>3°C</th><th>4°C</th><th>5°C</th><th>7°C</th><th>10°C</th></tr></thead><tbody><tr><td>Pr{ΔT̃s* ≥ θ}</td><td>94.26%</td><td>52.86%</td><td>24.61%</td><td>12.63%</td><td>4.73%</td><td>1.88%</td></tr></tbody></table>

The probability of observing an equilibrium temperature greater than 4°C and 7°C is close to 25% and 5%, respectively. Therefore, there is a high uncertainty in $\Delta\tilde{T}_s^*$ when we introduce the random nature of the climate feedback. The uncertainty of $\tilde{\lambda}$ also concerns the dynamics of $\Delta T_s(t)$. Indeed, the mean lifetime becomes random and follows a reciprocal normal distribution:

$$ \tilde{\tau} = -\frac{c}{\tilde{\lambda}} \sim \mathcal{RN}\left(-\frac{\mu_{\lambda}}{c}, \frac{\sigma_{\lambda}^2}{c}\right) $$

Figure 8.51 shows the probability distribution of $\tilde{\tau}$ when $\tilde{\lambda} \sim \mathcal{N}(-1.33, 0.44^2)$. Finally, we consider a Monte Carlo exercise and simulate 50 random paths of the surface temperature in Figure 8.51.

**Equilibrium climate sensitivity estimation** According to IPCC (2021, Chapter 7, page 926), the total anthropogenic ERF$^{57}$ over the industrial era (1750–2019) is 2.72 W/m$^2$, the contribution of anthropogenic greenhouse gas emissions is 3.84 W/m$^2$ and the combined effect of all radiative feedbacks (including Planck feedback) is estimated to be -1.16 W/m$^2$. Thus, the best estimate of the ECS is 3°C. The likely range is 2.5°C to 4°C (with a probability of 66%), and the very likely range is 2°C to 5°C (with a probability of 90%).

Figure 8.53 reproduces the IPCC graph showing the change in effective radiative forcing (ERF) from 1750 to 2019 by contributing forcing agents. The total anthropogenic effective radiative forcing is divided into six categories: carbon dioxide, other well-mixed greenhouse gases (methane, nitrous oxide, and halogens), ozone, stratospheric water vapor, albedo, contrails & aviation-induced cirrus, and aerosols (aerosol-cloud and aerosol-radiation). The contributions of CO$_2$, CH$_4$ and N$_2$O are 79.41%, 19.9% and 17.3% of the total ERF, respectively. They are partially compensated by three negative contributors (aerosol-cloud, aerosol-radiation and land use). In Table 8.11, we report the evolution of these contributing forcings. In 2019, the total anthropogenic effective radiative forcing is 2.72 W/m$^2$ and has increased by 29.3% over the last decade. The natural ERF due to volcanic activity and solar variations is 0.118 W/m$^2$. We deduce that the total ERF, which is the sum of anthropogenic and natural ERF, is equal to 2.838 W/m$^2$ in 2019. It has been multiplied by a factor of 4.85 in the last 70 years.

<footnote>$^{56}$We apply the theorem of variable change, which is explained on page 1156.</footnote>

<footnote>$^{57}$Effective radiative forcing (ERF) is a measure of the change in radiative flux at the top of the atmosphere and does not include the Planck feedback.</footnote>
<figure><img src="image_334.png" /><figcaption>Figure 8.53: Anthropogenic effective radiative forcing (ERF) from 1750 to 2019 by contributing forcing agents</figcaption></figure>

Source: (IPCC, 2021, Figure 7.6, Chapter 7, page 959) & https://www.ipcc.ch/report/ar6/wg1/figures/chapter-7/figure-7-6.

<table><caption>Table 8.11: Effective radiative forcing from 1750 to 2019</caption><thead><tr><th>Forcing agent</th><th>1800</th><th>1850</th><th>1900</th><th>1950</th><th>2000</th><th>2010</th><th>2019</th></tr></thead><tbody><tr><td>CO<sub>2</sub></td><td>0.070</td><td>0.140</td><td>0.346</td><td>0.648</td><td>1.561</td><td>1.854</td><td>2.156</td></tr><tr><td>CH<sub>4</sub></td><td>0.025</td><td>0.049</td><td>0.119</td><td>0.245</td><td>0.509</td><td>0.518</td><td>0.544</td></tr><tr><td>N<sub>2</sub>O</td><td>0.004</td><td>0.007</td><td>0.032</td><td>0.069</td><td>0.157</td><td>0.181</td><td>0.208</td></tr><tr><td>Other GHG</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.010</td><td>0.375</td><td>0.392</td><td>0.408</td></tr><tr><td>O<sub>3</sub></td><td>0.015</td><td>0.030</td><td>0.081</td><td>0.167</td><td>0.399</td><td>0.443</td><td>0.474</td></tr><tr><td>H<sub>2</sub>O (stratospheric)</td><td>0.002</td><td>0.005</td><td>0.011</td><td>0.022</td><td>0.047</td><td>0.048</td><td>0.050</td></tr><tr><td>Contrails</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.005</td><td>0.039</td><td>0.044</td><td>0.058</td></tr><tr><td>Aerosol</td><td>-0.018</td><td>-0.078</td><td>-0.346</td><td>-0.708</td><td>-1.221</td><td>-1.266</td><td>-1.058</td></tr><tr><td>Black carbon on snow</td><td>0.002</td><td>0.006</td><td>0.020</td><td>0.032</td><td>0.069</td><td>0.085</td><td>0.080</td></tr><tr><td>Land use</td><td>-0.011</td><td>-0.031</td><td>-0.084</td><td>-0.144</td><td>-0.194</td><td>-0.197</td><td>-0.200</td></tr><tr><td>Total (anthropogenic)</td><td>0.089</td><td>0.128</td><td>0.179</td><td>0.346</td><td>1.739</td><td>2.103</td><td>2.720</td></tr><tr><td>Volcanic</td><td>0.183</td><td>0.194</td><td>0.198</td><td>0.182</td><td>0.175</td><td>0.137</td><td>0.140</td></tr><tr><td>Solar</td><td>-0.043</td><td>0.008</td><td>-0.037</td><td>0.057</td><td>0.110</td><td>-0.008</td><td>-0.022</td></tr><tr><td>Total (natural)</td><td>0.140</td><td>0.202</td><td>0.160</td><td>0.239</td><td>0.285</td><td>0.129</td><td>0.118</td></tr><tr><td>Total</td><td>0.229</td><td>0.330</td><td>0.339</td><td>0.585</td><td>2.025</td><td>2.232</td><td>2.838</td></tr></tbody></table>

Source: (IPCC, 2021, Figure 7.6, Chapter 7, page 959) & https://www.ipcc.ch/report/ar6/wg1/figures/chapter-7/figure-7-6.
**Remark 78** The previous decomposition based on radiative sources can be complemented by an attribution based on GHG emissions. For example, Panel (a) in Figure 8.54 shows the global and annual mean ERF attributed to emitted compounds over the period 1750–2019 based on AerChemMIP simulations (Thornhill et al., 2021), while the emissions-based contributions to GSAT change are estimated in Panel (b).

<figure><img src="image_335.png" /><figcaption>Figure 8.54: Contribution to effective radiative forcing (ERF) (a) and global mean surface air temperature (GSAT) change (b) from component emissions between 1750 to 2019 based on CMIP6 models</figcaption></figure>

Source: (IPCC, 2021, Figure 6.12, Chapter 6, page 854) & https://www.ipcc.ch/report/ar6/wg1/figures/chapter-6/figure-6-12.

We now have all the information we need to calculate the equilibrium climate sensitivity. Using the parameters from Sherwood et al. (2020) $\lambda \sim N(-1.30, 0.44^2)$ and $\Delta F_{2 \times CO_2} \sim N(4.00, 0.30^2)$, we get:

$$ 
\begin{aligned}
\text{ECS} &= -\frac{\Delta F_{2 \times CO_2}}{\lambda} \\
&= -\frac{N(4.00, 0.30^2)}{N(-1.30, 0.44^2)} \\
&= NRD(-4.00, 0.30^2, -1.30, 0.44^2)
\end{aligned}
$$

where $NRD$ is the normal ratio distribution defined on page 1157. The probability density function of ECS is shown in Figure 8.55 and the exceedance probability is given below:

<table><thead><tr><th>Temperature θ</th><th>2°C</th><th>3°C</th><th>4°C</th><th>5°C</th><th>7°C</th><th>10°C</th></tr></thead><tbody><tr><td>Pr {ECS ≥ θ}</td><td>93.24%</td><td>52.79%</td><td>24.92%</td><td>12.85%</td><td>4.81%</td><td>1.91%</td></tr></tbody></table>

Therefore, we have a probability about 25% to observe a global warming greater than 4°C. In Figure 8.55, we compare the probability distribution when $\Delta F_{2 \times CO_2}$ is constant and the probability distribution when $\Delta F_{2 \times CO_2}$ is stochastic. This means comparing $RN(1.30, 0.44^2, 4.00, 4.00^2)$ and $NRD(-4.00, 0.30^2, -1.30, 0.44^2)$. We conclude that the feedback uncertainty is the main contributor to the ECS uncertainty.
<figure><img src="image_336.png" /><figcaption>Figure 8.55: Probability density function of the equilibrium climate sensitivity ($\Delta F = +4.00$ W/m$^2$, $\sigma (\Delta F) = 0.30$ W/m$^2$, $\mu_{\lambda} = -1.30$ W m$^{-2}$ K$^{-1}$ and $\sigma_{\lambda} = 0.44$ W m$^{-2}$ K$^{-1}$)</figcaption></figure>

Instability, tipping point and chaos

**An example of unstable equilibrium** We combine the Budyko model presented on page 677 with the one-layer model with emissivity. At equilibrium, we have:

$$ \varepsilon = \frac{1}{4} (1 - \alpha_p(T_s)) S_0 - \left(\frac{2 - \varepsilon}{2}\right) \sigma T_s^4 = 0 $$

where$^{58}$:

$$ \alpha_p(\mathcal{T}_s) = \begin{cases} \alpha_{\text{cold}} & \text{if } \mathcal{T}_s \le \mathcal{T}_{\text{cold}} \\ \alpha_{\text{warm}} + (\alpha_{\text{cold}} - \alpha_{\text{warm}}) \left( \frac{\mathcal{T}_{\text{warm}} - \mathcal{T}_s}{\mathcal{T}_{\text{warm}} - \mathcal{T}_{\text{cold}}} \right)^{\eta} & \text{if } \mathcal{T}_{\text{cold}} \le \mathcal{T}_s \le \mathcal{T}_{\text{warm}} \\ \alpha_{\text{warm}} & \text{if } \mathcal{T}_s \ge \mathcal{T}_{\text{warm}} \end{cases} $$

In the first panel of Figure 8.56 we plot the albedo function $\alpha_p(\mathcal{T}_s)$ when $\eta$ is set to 3. The second panel shows the two forcing functions:

$$ \left\{ \begin{array}{l} F_{\text{solar}}(\mathcal{T}_s) = \frac{1}{4}(1 - \alpha_p((\mathcal{T}_s))) S_0 \\ F_{\text{blackbody}}(\mathcal{T}_s) = \left(\frac{2 - \varepsilon}{2}\right) \sigma \mathcal{T}_s^4 \end{array} \right. $$

We then compare the net forcing function $F_{\text{solar}}(\mathcal{T}_s) - F_{\text{blackbody}}(\mathcal{T}_s)$ to the equation $\varepsilon(\mathcal{T}_s) = 0$

<footnote>$^{58}$We use the values $\alpha_{\text{cold}} = 0.7$, $\mathcal{T}_{\text{cold}} = 260$ K, $\alpha_{\text{warm}} = 0.3$ and $\mathcal{T}_{\text{warm}} = 295$ K.</footnote>
<figure><img src="image_337.png" /><figcaption>Figure 8.56: Equilibrium states of the Budyko ice-albedo model</figcaption></figure>

in the third panel. We can see graphically that there are three equilibrium states. By solving the equation $F_{\text{solar}}(\mathcal{T}_s) - F_{\text{blackbody}}(\mathcal{T}_s) = 0$ with a bisection algorithm, we find that the three equilibria are $\mathcal{T}_1^* = 233.38$ K, $\mathcal{T}_2^* = 268.43$ K, and $\mathcal{T}_3^* = 288.13$ K. To see if the equilibria are stable, we compute the total feedback:

$$ \lambda(\mathcal{T}_s) = \lambda_0(\mathcal{T}_s) + \lambda_{\alpha_p}(\mathcal{T}_s) $$

where:

$$ \lambda_0(\mathcal{T}_s) = -4 \left( \frac{2 - \varepsilon}{2} \right) \sigma \mathcal{T}_s^3 $$

and:

$$ \lambda_{\alpha_p}(\mathcal{T}_s) = \frac{1}{4}\eta S_0 (\alpha_{\text{cold}} - \alpha_{\text{warm}}) \frac{(\mathcal{T}_{\text{warm}} - \mathcal{T}_s)^{\eta-1}}{(\mathcal{T}_{\text{warm}} - \mathcal{T}_{\text{cold}})^{\eta}} \cdot 1 \{\mathcal{T}_{\text{cold}} \le \mathcal{T}_s \le \mathcal{T}_{\text{warm}}\} $$

The results are shown below:

<table><thead><tr><th>State</th><th>Temperature</th><th>λ<sub>0</sub>(T<sub>s</sub>)</th><th>λ<sub>α<sub>p</sub></sub>(T<sub>s</sub>)</th><th>λ(T<sub>s</sub>)</th></tr></thead><tbody><tr><td>T<sub>1</sub><sup>*</sup></td><td>233.38 K -39.77°C</td><td>-1.76</td><td>0.00</td><td>-1.76</td></tr><tr><td>T<sub>2</sub><sup>*</sup></td><td>268.43 K -4.72°C</td><td>-2.68</td><td>6.75</td><td>+4.08</td></tr><tr><td>T<sub>3</sub><sup>*</sup></td><td>288.13 K 14.98°C</td><td>-3.31</td><td>0.45</td><td>-2.86</td></tr></tbody></table>

We conclude that $\mathcal{T}_1^*$ and $\mathcal{T}_3^*$ are two stable equilibria because $\lambda$ is negative, but $\mathcal{T}_2^*$ is an unstable equilibrium. This means that if the climate system is in equilibrium $\mathcal{T}_2^*$, any perturbation will cause instability and the temperature will not return to that equilibrium. To illustrate this instability, we consider the dynamics of the temperature:

$$ c \frac{d\mathcal{T}_s}{dt} = F_{\text{solar}}(\mathcal{T}_s) - F_{\text{blackbody}}(\mathcal{T}_s) + \Delta F $$
where $c = 4 \times 10^8 \text{ J m}^{-2} \text{ K}^{-1}$ is the heat capacity and $\Delta F$ is the perturbation. We assume that the system is in equilibrium $T_s(0) = T_2^*$. We introduce a small perturbation $\Delta F = +0.1 \text{ W/m}^2$ and solve the above ordinary differential equation numerically. The solution is reported in the fourth panel of Figure 8.56. In this case, we have:

$$ \lim_{t \to \infty} T_s(t) = T_3^* $$

If the perturbation $\Delta F = -0.1 \text{ W/m}^2$, we get:

$$ \lim_{t \to \infty} T_s(t) = T_1^* $$

We check that $T_2^*$ is an unstable equilibrium. The previous analysis concerns the sensitivity to the initial conditions. We can also consider the sensitivity to the parameters. For example, we can show that there is only one solution if $\eta < 1.9235$, while there are three solutions if $\eta > 1.9236$. Furthermore, there exists a parameter $\eta \in [1.9235, 1.9236]$ such that we get two solutions. This type of analysis is called bifurcation analysis.

**Tipping point definition** The concept of a tipping point emerged from stability analysis and bifurcation. The term was popularized in 2000 by Malcolm Gladwell in his book “*The Tipping Point: How Little Things Can Make a Big Difference*”, which explored the concept in sociological change. In climate change, it was popularized in the late 2000s by Lenton et al (2008). In common parlance, a tipping point is a change. When applied to climate change, the term usually has a negative connotation and is generally associated with a change that could have devastating consequences for human society. However, when we look at scientific publications on climate change, a tipping point is a neutral mathematical concept. For example, here are the definitions adopted by Timothy Lenton and IPCC:

*“The term tipping point commonly refers to a critical threshold at which a tiny perturbation can qualitatively alter the state or development of a system.” (Lenton et al, 2008, page 1786).*

*“A climate tipping point occurs when a small change in forcing triggers a strongly non-linear response in the internal dynamics of part of the climate system, qualitatively changing its future state.” (Lenton, 2011, page 201).*

*“Tipping points refer to critical thresholds in a system that, when exceeded, can lead to a significant change in the state of the system, often with an understanding that the change is irreversible.” (IPCC, 2018, page 262).*

In climate, a tipping point is “a hypothesized critical threshold when global or regional climate changes from one stable state to another stable state. The tipping point event may be irreversible.” (IPCC, 2021, page 1463).

In fact, these scientific definitions contrast with the content of these scientific publications, because a tipping point generally announces catastrophic events (Kemp et al., 2022). For example, the term “climate endgame” refers to a hypothetical scenario in which climate change becomes catastrophic for human civilization and seems to be the Ten Plagues of Egypt with a series of super disasters: collapse of ecosystems, mass extinctions, global food shortages, water scarcity, etc.
**Bifurcation theory** Lenton (2011, Figure 2, page 203) gives a heuristic interpretation of the global warning when approaching a bifurcation point with a ball in a landscape of valleys and mountains. In Figures 8.57-8.59 we use an analogous story where the valleys represent stable attractors and the ball represents the state of the system. Figure 8.57 shows a stable equilibrium corresponding to the green ball. If the ball moves away from the equilibrium, but not too far, it will return to the equilibrium. This situation corresponds to the two red balls. This illustrates a reversible behavior where the system can retrace its path and return to its initial state. Figure 8.58 shows a red ball representing a tipping point. In fact, this equilibrium is unstable and corresponds to a bifurcation point, since the red ball cannot remain in this position indefinitely. If the system is initially at the red ball and is then slightly perturbed by a small disturbance, it will move toward the green balls and settle there. However, it is unpredictable which direction the red ball will take. It depends on many parameters. The red arrows represent the transitions between equilibria and indicate the direction between the tipping point and a stable equilibrium. Figure 8.59 shows the new position of the ball after it has moved to the left valley. Again the ball is in equilibrium. However, when it leaves this equilibrium, it cannot return to its initial position as shown in 8.59. In fact, the system has changed because part of the mountain has filled the valley. Instead, there is a new valley, but it's higher than the old one. Therefore, when the green ball leaves its equilibrium, it can move to the position represented by the yellow ball, but it cannot reach the position represented by the grey ball because it no longer exists.

To present the bifurcation theory of one-dimensional dynamical systems, we use chapters 2 and 3 of the classic textbook on bifurcation theory and chaos by Strogatz (2015). We consider the dynamical system of the form:

$$ \frac{dx}{dt} = f(x, \mu) $$

where $\mu$ is the parameter set. We look at the mathematical properties of the system by analyzing the trajectories (or flows<fnref n="59" />) of $x(t)$. If $f(x, \mu) > 0$, then $x$ increases with time, while if $f(x, \mu) < 0$, then $x$ decreases with time. A fixed point $x^*$ is then a value of $x$ such that the system does not change with time, i.e., $f(x^*, \mu) = 0$. Let $x = x^* + \Delta x$ where $\Delta x$ is small. We have:

$$ \begin{aligned} \frac{dx}{dt} &= f(x^* + \Delta x, \mu) \\ &= f(x^*, \mu) + \frac{\partial f(x^*, \mu)}{\partial x} \Delta x + O(\Delta x^2) \\ &= f(x^*, \mu) + \lambda \Delta x + O(\Delta x^2) \end{aligned} $$

where $\lambda$ is the feedback of the system. Since we have:

$$ \frac{dx}{dt} = \frac{d(x^* + \Delta x)}{dt} = \frac{d\Delta x}{dt} $$

and:

$$ f(x^* + \Delta x, \mu) \approx f(x^*, \mu) + \lambda \Delta x = \lambda \Delta x $$

We deduce that:

$$ \frac{d\Delta x}{dt} = \lambda \Delta x $$

If $\lambda > 0$, then any small perturbation of the fixed point grows exponentially, while if $\lambda < 0$, then any small perturbation of the fixed point decays exponentially. In the first case the fixed point is

<footnote n="59">⁵⁹A flow is used here as a synonym for a trajectory, and corresponds to a vector field that gives the direction and magnitude of the trajectory.</footnote>
<figure><img src="image_338.png" /><figcaption>Figure 8.58: Unstable equilibrium</figcaption></figure>

<figure><img src="image_339.png" /><figcaption>Figure 8.59: Irreversible tipping point</figcaption></figure>

<figure><img src="image_340.png" /></figure>
unstable, in the second case it is stable. When $\lambda = 0$, we need to study the second derivative of $f(x, \mu)$ to conclude. We also note that the relaxation timescale is equal to:

$$ \tau = \frac{1}{|\partial_x f(x^*, \mu)|} $$

**Example 27** Consider the dynamical system $\frac{dx}{dt} = x^2 + \mu$. We have $\partial_x f(x, \mu) = 2x$. The fixed points are solutions of the equation $x^2 = -\mu$. If $\mu > 0$, there are no fixed points. If $\mu = 0$, the fixed point is $x^* = 0$ and is not stable. If $\mu < 0$, there are two fixed points. $x_1^* = -\sqrt{-\mu}$ is stable while $x_2^* = \sqrt{-\mu}$ is unstable.

The stability analysis evaluates the behavior of $f(x^* + \varepsilon, \mu)$ and checks if the point $x^* + \Delta x$ converges to the point $x^*$. We can consider a second approach to stability where we evaluate the behavior of $f(x^*, \mu + \varepsilon)$, i.e., we apply the perturbation not directly to $x$ but to the control parameter $\mu$. In particular, we say that the value $\mu^*$ is a bifurcation value if $f(x, \mu^*)$ is not structurally stable. In the previous example, the dynamical system exhibits a bifurcation that occurs at $\mu = 0$. This type of bifurcation is called a saddle-node or fold bifurcation, because fixed points are created or destroyed. The first panel of Figure 8.60 shows the bifurcation diagram of the system. As $\mu$ increases, the distance between the stable and unstable fixed points decreases, and the equilibria are destroyed when $\mu$ is strictly positive. Following Strogatz (2015), other types of bifurcation are also common. A transcritical bifurcation occurs when fixed points exchange stability for a critical value of $\mu$. For example, the system $\frac{dx(t)}{dt} = \mu x - x^2$ has a transcritical bifurcation at $\mu = 0$ (second panel of Figure 8.60). A pitchfork bifurcation occurs when the system is unchanged and exhibits symmetry when $x = -x$. There are two forms of pitchfork bifurcation:

1. In a supercritical pitchfork bifurcation, a stable fixed point becomes unstable at a critical value of $\mu$. A canonical example is:

$$ \frac{dx}{dt} = \mu x - x^3 $$

The corresponding bifurcation diagram is shown in the third panel of Figure 8.60. We observe that $\mu = 0$ is a supercritical pitchfork bifurcation.

2. In a subcritical pitchfork bifurcation, an unstable fixed point becomes stable at a critical value of $\mu$. A canonical example is:

$$ \frac{dx}{dt} = \mu x + x^3 - x^5 $$

The corresponding bifurcation diagram is shown in the fourth panel of Figure 8.60. Let $\mu^* = -0.25$ be the first value of $\mu$ that exhibits fixed points. When $\mu < \mu^*$, there is a stable fixed point at $x^* = 0$ because $\partial_x f(x^*, \mu) < 0$. When $\mu = \mu^*$, two new equilibria are created. When $\mu^* < \mu \le 0$, there are five fixed points. Three of these are stable equilibria. When $\mu > 0$, the stable equilibrium $x^* = 0$ is destroyed and jumps to the stable branches or becomes unstable.

In bifurcation theory, hysteresis refers to a phenomenon where the behavior of the system depends on the history of its past states. For example, consider the previous subcritical pitchfork bifurcation and suppose the system is at equilibrium $x^* = 0$. If we increase $\mu$, the system jumps to one of the stable branches. If we decrease $\mu$, the system does not return to its past equilibrium, but stays on the stable branch. However, if we decrease $\mu$ even more, the system jumps to its past equilibrium $x^* = 0$ when $\mu$ reaches $\mu^* = -0.25$. Therefore, the path of the system has formed a hysteresis loop.
<figure><img src="image_341.png" /><figcaption>Figure 8.60: Bifurcation diagram</figcaption></figure>

**Remark 79** Chaos theory is closely related to bifurcation theory because chaos theory is primarily concerned with the behavior of dynamical systems that are highly sensitive to initial conditions. Since chaos is synonymous with disorder, a deterministic chaotic system exhibits aperiodic long-term behavior that appears to be random. These include strange attractors, feedback loops, limit cycles, self-similarity, fractals, butterfly effects, irregular oscillations, etc. The most famous chaotic models are the Hénon map, the Lorenz system, the logistic map, and the Rössler attractor described in Box 8.10.

Let us return to the Budyko ice-albedo model described on page 689. Using the default values $S_0 = 1368 \text{ W/m}^2$, $\varepsilon = 78\%$, $\eta = 3$, $\alpha_{\text{cold}} = 0.7$, $T_{\text{cold}} = 260 \text{ K}$, $\alpha_{\text{warm}} = 0.3$ and $T_{\text{warm}} = 295 \text{ K}$, we have obtained three fixed points: $T_1^* = 233.38 \text{ K}$, $T_2^* = 268.43 \text{ K}$, and $T_3^* = 288.13 \text{ K}$. $T_1^*$ and $T_3^*$ were stable, while $T_2^*$ was unstable. To perform the bifurcation analysis of the Budyko ice-albedo model, we consider the range $T_s \in [200 \text{ K}, 300 \text{ K}]$ and divide the range into ten intervals. Using the bisection algorithm, we solve the equation for each interval of $T_s$ and each value of the parameter of interest:

$$ \frac{1}{4} (1 - \alpha_p ((T_s))) S_0 - \left(\frac{2 - \varepsilon}{2}\right) \sigma T_s^4 = 0 $$

We collect all fixed points $\{T_1^*, T_2^*, \dots\}$. For each fixed point we calculate the feedback:

$$ \lambda(T_s) = \frac{1}{4}\eta S_0 (\alpha_{\text{cold}} - \alpha_{\text{warm}}) \frac{(T_{\text{warm}} - T_s)^{\eta-1}}{(T_{\text{warm}} - T_{\text{cold}})^{\eta}} \cdot 1 \{T_{\text{cold}} \le T_s \le T_{\text{warm}}\} - 4 \left(\frac{2 - \varepsilon}{2}\right) \sigma T_s^3 $$

We perform this analysis for a given range of the parameter of interest and plot the fixed points $\{T_1^*, T_2^*, \dots\}$ against the parameter of interest. We also use a solid green line if the fixed point is stable or $\lambda(T_s) < 0$, and a dashed red line otherwise. Results are shown in Figure 8.61, when the
Box 8.10: Rössler chaotic model

The Rössler model is a system of ordinary differential equations that exhibits chaotic dynamics (Rössler, 1976). It has become one of the most studied and widely used examples of a chaotic system. The Rössler equations are given by:

$$ 
\left\{ 
\begin{aligned}
\frac{dx(t)}{dt} &= -(y(t)+z(t)) \\
\frac{dy(t)}{dt} &= x(t)+ay(t) \\
\frac{dz(t)}{dt} &= b+x(t)z(t)-cz(t)
\end{aligned}
\right. 
$$

where a, b and c are three parameters that control the behavior of the system. The Rössler attractor is a three-dimensional surface described by $(x(t), y(t), z(t))$. The attractor is very sensitive to the initial conditions $(x(0), y(0), z(0))$ and the set of parameters $(a, b, c)$. Below we plot the attractor in the first panel and the bifurcation diagram of $x(t)$ with respect to each parameter in the other three panels. We assume that the initial condition is $(x(0), y(0), z(0)) = (1, 1, 0)$ and the parameters$^a$ are $a = 0.2$, $b = 0.2$ and $c = 5.7$. We check that the dynamic system $(x(t), y(t), z(t))$ does not necessarily converge to a fixed point and is highly sensitive to the parameters. While the Rössler model is deterministic, its behavior is unpredictable like a stochastic model.

<figure><img src="image_342.png" /></figure>

parameters of interest are $\eta$, $\varepsilon$, $S_0$ and $\alpha_{cold}$. We can make several remarks. First, we note that we can have one, two, or three fixed points, meaning that there is at least one equilibrium and no more than three equilibria for a set of parameters. Second, one of these fixed points is stable and it always generates a saddle-node bifurcation since two new equilibria are created. Third, the unstable fixed point is always between the two stable fixed points. Figure 8.62 shows the relaxation timescale of the different equilibria:

$$ \tau = \frac{c}{|\lambda(T_s)|} $$

<footnote>$^a$For the bifurcation diagram of a, we set $b = 2$.</footnote>
<figure><img src="image_343.png" /><figcaption>Figure 8.62: Relaxation timescale of the Budyko ice-albedo model (in years)</figcaption></figure>

<figure><img src="image_344.png" /></figure>
where $c = 4 \times 10^8 \text{ J m}^{-2} \text{ K}^{-1}$ is the heat capacity. $\tau$ is expressed in years. Note that $\tau$ is less than 50 years, which is not realistic. In fact, the framework we have used to model the ice-albedo feedback is very simplistic and far from reality.

<figure><img src="image_345.png" /><figcaption>Figure 8.63: Bifurcation and overshooting tipping points</figcaption></figure>

Source: Ritchie et al. (2021, Figures 1c & 3c, pages 518 & 520) & https://www.nature.com/articles/s41586-021-03263-2.

The Budyko ice-albedo model produces a relatively classical bifurcation where equilibria are created or destroyed at a bifurcation point. To obtain a more complex bifurcation diagram, we need to use a higher dimensional model or nonlinear feedback relationships. In Figure 8.63 we show two examples calculated by Ritchie et al. (2021). The first panel compares slow and fast overshooting tipping points and the importance of the radiative timescale<fnref n="60" />. The blue curve represents a rapidly overshooting tipping point system, such as the dieback of the Amazon forest, while the orange curve shows that the transition can take a long time even after we have passed the tipping point, such as the collapse of the Atlantic meridional overturning circulation (AMOC). This means that some tipping points may have already been reached, but we will not see their effects until the next century or millennium. The second panel shows the behavior of the AMOC when implementing an overshoot<fnref n="61" />. Two different trajectories are considered, one with a small overshoot and long recovery time (blue curve) and one with a large overshoot and short recovery time (orange curve). Contrary to expectations, the small overshoot leads to a persistent collapse of the AMOC, while the large overshoot allows a recovery. The resilience of the AMOC in this second scenario can be explained by the rapid reversal of global warming.

<footnote n="60">⁶⁰Here is the description of the figure provided by the authors: "System state vs. global warming for the fast onset tipping element (blue) and slow onset tipping element (orange). Both systems have a desired state, which is the upper solid black curve and represent contemporary conditions. An undesired stable state, given by the lower solid black curve, coexists with the desired state for warming levels below the 2°C threshold, separated by an unstable state (black dashed curve). Above the threshold, only the undesired equilibrium state remains." (Ritchie et al., 2021, page 518).</footnote>

<footnote n="61">⁶¹This figure shows the flow strength of the AMOC against global warming for short long and large fast overshoots: "The blue curve is a small and long overshoot, while the orange curve is a much larger yet quick overshoot. An AMOC on state (upper solid black curve) and an AMOC off state (lower solid black curve) both coexist for warming levels below the threshold of 4°C and are separated by an unstable state (black dashed curve). Above the threshold only the AMOC off state remains." (Ritchie et al., 2021, page 520).</footnote>
**Climate tipping elements** Lenton et al (2008) introduced the term “tipping element” to describe large-scale components of the Earth system that may pass a tipping point. They identified 9 climate tipping elements: (1) Arctic sea ice, (2) Greenland ice sheet, (3) West Antarctic ice sheet, (4) Atlantic thermohaline circulation, (5) El Niño Southern Oscillation (ENSO), (6) Indian summer monsoon, (7) Sahara/Sahel and West African monsoons, (8) Amazon rainforest, and (9) boreal forest. They added to this list other candidate tipping elements that did not meet all the criteria for a tipping point: (10) Antarctic bottom water, (11) tundra, (12) permafrost, (13) marine methane hydrates, (14) ocean anoxia, and (15) Arctic ozone. The above short list of climate tipping elements has evolved over time (Lenton, 2011; Steffen et al., 2018; Lenton et al., 2019). Today, there is a consensus on the most important tipping points, which has been published by Armstrong McKay et al. (2022). We report this short list in Table 8.12, while Figure 8.64 shows their location on the Earth.

<figure><img src="image_346.png" /><figcaption>Figure 8.64: Geographical distribution of global and regional tipping elements</figcaption></figure>

Source: Armstrong McKay et al. (2022) & https://www.pik-potsdam.de/en/output/infodesk/tipping-elements.

Armstrong McKay et al. (2022) distinguish 9 global core and 7 regional impact tipping elements that affect the cryosphere, ocean-atmosphere, and biosphere. For each climate tipping point, they estimate the threshold $\theta$, the timescale $\tau$ and the maximum impact $\delta$. This means that the tipping point is reached when the temperature anomaly is greater than $\theta$. The result is a new equilibrium that induces a temperature change of $\delta$ after $\tau$ years:

$$ \Delta T_s (0) \geq \theta \Rightarrow \text{tipping point} \Rightarrow \Delta T_s (\tau) = \delta $$

For example, if $\theta = +1^{\circ}\text{C}$, $\tau = 50$ years and $\delta = 0.2^{\circ}\text{C}$, the tipping point is crossed when the temperature increase is $1^{\circ}\text{C}$. The new equilibrium will be reached after 50 years with a supplementary temperature increase of $0.2^{\circ}\text{C}$. The plausible range of these estimates is given in parentheses.
<table><caption>Table 8.12: Threshold, timescale, and impact estimates for the global and regional tipping elements</caption><thead><tr><th>Category</th><th>#</th><th>Climate tipping element</th><th>Tipping point</th><th>Threshold</th><th>Timescale</th><th>Maximum impact</th></tr></thead><tbody><tr><td rowspan="15">Global</td><td>1</td><td>Greenland ice sheet</td><td>collapse</td><td>1.5°C (0.8-3.0)</td><td>10 kyr (1-15)</td><td>0.13°C (0.5-3.0)</td></tr><tr><td>2</td><td>West Antarctic ice sheet</td><td>collapse</td><td>1.5°C (1.0-3.0)</td><td>2 kyr (0.5-13)</td><td>0.05°C (1.0)</td></tr><tr><td>3</td><td>Labrador-Irminger seas</td><td>collapse</td><td>1.8°C (1.1-3.8)</td><td>10 yr (5-50)</td><td>-0.50°C (-3.0)</td></tr><tr><td>4</td><td>East Antarctic subglacial basins</td><td>collapse</td><td>3.0°C (2.0-6.0)</td><td>2 kyr (0.5-10)</td><td>0.05°C</td></tr><tr><td>5</td><td>Amazon rainforest</td><td>dieback</td><td>3.5°C (2.0-6.0)</td><td>100 yr (50-200)</td><td>0.20°C (0.4-2.0)</td></tr><tr><td>6</td><td>Boreal permafrost</td><td>collapse</td><td>4.0°C (3.0-6.0)</td><td>50 yr (10-300)</td><td>0.40°C</td></tr><tr><td>7</td><td>AMOC</td><td>collapse</td><td>4.0°C (1.4-8.0)</td><td>50 yr (15-300)</td><td>-0.50°C (-4/-10)</td></tr><tr><td>8</td><td>Arctic winter sea ice</td><td>collapse</td><td>6.3°C (4.5-8.7)</td><td>20 yr (10-100)</td><td>0.60°C (0.6-1.2)</td></tr><tr><td>9</td><td>East Antarctic ice sheet</td><td>collapse</td><td>7.5°C (5.0-10.0)</td><td>&gt; 10 kyr</td><td>0.60°C (2.0)</td></tr><tr><td rowspan="16">Regional</td><td>10</td><td>Low-latitude coral reefs</td><td>die-off</td><td>1.5°C (1.0-2.0)</td><td>10 yr</td><td></td></tr><tr><td>11</td><td>Boreal permafrost</td><td>abrupt thaw</td><td>1.5°C (1.0-2.3)</td><td>200 yr (100-300)</td><td>0.04°C</td></tr><tr><td>12</td><td>Barents sea ice</td><td>abrupt loss</td><td>1.6°C (1.5-1.7)</td><td>25 yr</td><td></td></tr><tr><td>13</td><td>Mountain glaciers</td><td>loss</td><td>2.0°C (1.5-3.0)</td><td>200 yr (50-1000)</td><td>0.08°C</td></tr><tr><td>14</td><td>Sahel and West African monsoon</td><td>greening</td><td>2.8°C (2.0-3.5)</td><td>50 yr (10-500)</td><td></td></tr><tr><td>15</td><td>Boreal forest (southern)</td><td>die-off</td><td>4.0°C (1.4-5.0)</td><td>100 yr (50+)</td><td>-0.18°C (-0.5/-2)</td></tr><tr><td>16</td><td>Boreal forest (northern)</td><td>expansion</td><td>4.0°C (1.5-7.2)</td><td>100 yr (40+)</td><td>0.14°C (0.5-1.0)</td></tr></tbody></table>

Source: Armstrong McKay et al. (2022, Table 1, page 3).
Below is a list of the different tipping elements and some explanations of the mechanism behind them:

1. The Greenland ice sheet is the second largest ice sheet in the world. It covers 80% of the surface of Greenland. Its melting would increase sea level rise, possibly up to 7.4 meters, accelerate ocean acidification, and have a potentially positive feedback effect on climate change (Shepherd et al., 2020).

2. The West Antarctic ice sheet is a large ice sheet in Antarctica. It sits on a bedrock that is mostly below sea level and has formed a deep subglacial basin due to the weight of the ice sheet, which can be up to 4 kilometers thick in places. Its collapse could raise global sea levels, possibly up to 3 meters (Alley et al., 2015).

3. The Labrador and Irminger seas are located in the subpolar North Atlantic, between Canada, Greenland and Iceland. These seas are characterized by cold and salty waters that generate deep convection. This deep convection and the regulation of ocean salinity influence the circulation of the Atlantic meridional overturning circulation (AMOC). The collapse of the deep convection system in the Labrador and Irminger seas would affect the overall circulation in the North Subpolar Gyre (Chafik et al., 2022).

4. East Antarctic subglacial basins are large, ice-filled depressions in the bedrock adjacent to the East Antarctic ice sheet. They also serve as reservoirs for meltwater. Certain subglacial basins, such as Wilkes, Aurora, and Recovery, are more susceptible to a situation called marine ice sheet instability (MISI). As the ice shelf at the edge of the ice sheet retreats, warm ocean water can flow into the deeper basin, further destabilizing the ice shelf. This process can create a self-perpetuating cycle of ice melt and ice shelf retreat (Stokes et al., 2022).

5. The Amazon rainforest, also known as the Amazon jungle or Amazonia, is the largest rainforest in the world. It spans nine countries: Brazil, Peru, Colombia, Ecuador, Bolivia, Venezuela, Guyana, Suriname, and French Guiana. It contains the largest and most biodiverse area of tropical rainforest in the world. The Amazon rainforest acts as a massive carbon sink, absorbing and storing vast amounts of carbon dioxide through the process of photosynthesis. If the forest were to be subject to widespread deforestation or degradation, the stored carbon could be released back into the atmosphere, contributing to increased greenhouse gas concentrations. The Amazon also plays a critical role in the Earth's water cycle, influencing regional and global weather patterns. Its dense vegetation effectively captures rainwater and slowly releases it into streams and rivers, helping to maintain stable water levels, prevent flooding, and provide a steady source of fresh water (Gatti et al., 2021).

6. Boreal permafrost is a permanently frozen layer of soil and rock that underlies much of the world's boreal forest. It is found in Siberia, Alaska, Northern Canada, and the Tibetan Plateau. Permafrost forms when the soil temperature remains below 0°C for at least two consecutive years. Boreal permafrost contains large amounts of organic carbon stored in the form of dead plant material that could not decompose due to the cold temperatures. It is also one of the largest reservoirs of methane. Rising temperatures may cause the boreal permafrost to thaw, releasing large amounts of carbon and methane into the atmosphere (Schuur et al., 2015).

7. The Atlantic meridional overturning circulation (AMOC) is a large, complex system of ocean currents that transports warm water from the tropics to the North Atlantic and cold water from the North Atlantic to the subtropics. It is also known as the Gulf Stream system. A weakening
of the AMOC could have complex and regionally specific effects on temperatures. On a global scale, it could result in less warm water reaching higher latitudes, leading to cooler sea surface temperatures in the North Atlantic and warmer temperatures in the Southern Hemisphere (Liu et al., 2017).

8. Arctic winter sea ice is the maximum extent of sea ice that forms in the Arctic Ocean during the winter months. It helps to regulate global temperatures by reflecting sunlight back into space. This albedo reflection helps to cool the Arctic. As sea ice melts, more sunlight is absorbed by the ocean, causing a further warming trend<fnref n="62" />. However, the impact of the albedo effect remains controversial (Bathiany et al., 2016).

9. The East Antarctic ice sheet is the largest and thickest ice sheet on Earth. A complete collapse would raise the global sea levels by 50 meters. However, the East Antarctic ice sheet is generally considered to be more stable than the West Antarctic ice sheet, due to its higher elevation and more remote location (Stokes et al., 2022).

10. Low-latitude coral reefs occur in the Atlantic, Indian, and Pacific Oceans, most notably in the Philippines, Indonesia, and Australia. They require warm, sunny weather and unpolluted water. Therefore, coral reefs can be affected by climate change, although their impact on climate change is more limited (Descombes et al., 2015).

11. We have already seen that the boreal permafrost is a global tipping element, but it is also a regional tipping element. In fact, an abrupt thaw of the boreal permafrost would have devastating consequences for the region, affecting infrastructure (roads, buildings, transportation), the environment (flooding, forests, vegetation), and living conditions and health (Hjort et al., 2022).

12. Barents sea ice is found in the Barents sea, an arm of the Arctic Ocean between Norway and Russia. The sea ice forms during the winter months and melts during the summer months. This regional tipping element is strongly related to two global tipping elements: Labrador-Irminger seas and AMOC (Moore et al., 2022).

13. Mountain glaciers are large masses of ice that form on mountains at high altitudes. They are formed from compacted snow that has accumulated over many years. The melting of mountain glaciers would have a major regional impact on human life (Carey, 2010).

14. The West African monsoon is a seasonal wind pattern that affects the Sahel, bringing moisture from the Atlantic Ocean during the rainy season and drying out the region during the dry season. It is responsible for the region's agriculture and supports the livelihoods of millions of people. Changes in rainfall can affect vegetation, agriculture and people (Pausata et al., 2020).

15. The boreal forest, also known as the taiga, is a biome that surrounds the Arctic region. Countries with significant areas of boreal forest include Canada, Russia, Sweden, Norway and Finland. The southern edge of the boreal forest is the boundary between the boreal forest and temperate forests or grasslands. The risk could be an abrupt die-off (Venäläinen et al., 2020).

16. The northern edge of the boreal forest is typically found at higher latitudes, closer to the Arctic Circle. The change could be an abrupt expansion into a tundra forest characterised by treeless landscapes and permafrost (Girona et al., 2023).

<footnote n="62">⁶²Moreover, the warming Arctic has the potential to release methane from permafrost.</footnote>
Almost all the climate tipping elements induce a temperature increase, except Labrador-Irminger seas, AMOC and the southern edge of the boreal forest. We also find some large differences in timescale (between 10 years and more than 10 000 years) and impact. According to Armstrong McKay et al. (2022), the Greenland and West Antarctic ice sheets have certainly passed their tipping points, and other tipping points such as the die-off of tropical coral reefs and boreal permafrost are close to crossing their threshold.

Cascading tipping points and climate domino effects

**Mathematical framework of coupling tipping points** Based on an original idea by Brum-
mitt et al. (2015), Klose et al. (2020) assumes that the climate tipping elements follow an ordinary
differential equation with a double fold bifurcation:

$$
\frac{dx_i}{dt} = f_i(x_i, \mu_i) = \frac{\alpha_i x_i - \beta_i x_i^3 + \mu_i}{\tau_i} \quad (8.16)
$$

where $x_i$ is the $i^{th}$ tipping element, $\tau_i$ is the timescale, $\alpha_i > 0$ and $\beta_i > 0$ are two fixed parameters$^{63}$ and $\mu_i$ is the bifurcation parameter. Note that the cubic equation $f_i(x_i, \mu_i) = 0$ has three roots (real and complex). The feedback is equal to:

$$
\lambda = \frac{\partial f_i(x_i, \mu_i)}{\partial x_i} = \frac{\alpha_i - 3\beta_i x_i^2}{\tau_i} \quad (8.17)
$$

Since our system is a special case of an ODE with a RHS term which is a cubic polynomial, we can
use the results given in Box 8.11. Comparing with the form of $f_i(x_i, \mu_i)$, we have $a = -\beta_i/\tau_i$, $b = 0$,
$c = \alpha_i/\tau_i$, and $d = \mu_i/\tau_i$. Substituting these values into the expression for the discriminant, we get:

$$
\Delta_i = \frac{4\beta_i\alpha_i^3 - 27\beta_i^2\mu_i^2}{\tau_i^4}
$$

The number of stable equilibria depends on the discriminant $\Delta_i \propto 4\alpha_i^3 - 27\beta_i\mu_i^2$. If $\Delta_i < 0$ or $4\alpha_i^3 < 27\beta_i\mu_i^2$ there is one stable equilibrium, while if $\Delta > 0$ or $4\alpha_i^3 > 27\beta_i\mu_i^2$ there two stable equilibria$^{64}$. Therefore, the bifurcation occurs when $\Delta_i = 0$ or:

$$
\mu_i = \pm \sqrt{\frac{4\alpha_i^3}{27\beta_i}}
$$

Figure 8.65 shows the bifurcation diagram of the system when<fnref n="65" /> αᵢ = βᵢ = 1. If μᵢ is strongly negative, there is only one stable equilibrium. A first bifurcation occurs at μᵢ = -√(4/27) and two more equilibria are created, one stable and another one unstable. Then two equilibria are destroyed when μᵢ reaches the value √(4/27) and x*ᵢ = -√(1/3). They are the lower stable equilibrium and the middle unstable equilibrium. Only the upper equilibrium remains when μᵢ is greater than √(4/27).

<footnote n="63">⁶³According to Klose et al. (2020), αᵢ corresponds to the distance between the upper and lower layers of stable equilibria of the cusp, while βᵢ controls the strength of the nonlinearity in the system.</footnote>

<footnote n="64">⁶⁴Since −βᵢ < 0, the graph of the cubic function fᵢ(xᵢ, μᵢ) shows that the function is decreasing, then increasing and then decreasing. If we note that x*ᵢ,₁, x*ᵢ,₂, and x*ᵢ,₃ are the roots of fᵢ(xᵢ, μᵢ) = 0 with x*ᵢ,₁ ≤ x*ᵢ,₂ ≤ x*ᵢ,₃, we deduce that f'ᵢ(x*ᵢ,₁, μᵢ) ≤ 0, f'ᵢ(x*ᵢ,₂, μᵢ) ≥ 0, and f'ᵢ(x*ᵢ,₃, μᵢ) ≤ 0.</footnote>

<footnote n="65">⁶⁵Note that the bifurcation diagram does not depend on the timescale $\tau_i$.</footnote>
Box 8.11: First-order ODE with a cubic polynomial

Consider the following ordinary differential equation (ODE):

$$ \frac{dx}{dt} = f(x) = ax^3 + bx^2 + cx + d $$

where $f(x)$ is a general cubic polynomial. The discriminant of the general cubic equation $f(x) = 0$ is:

$$ \Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2 $$

and the first derivative is:

$$ f'(x) = 3ax^2 + 2bx + c $$

The stability of the equilibria depends on the sign of the discriminant $\Delta$ and the sign of the derivative $f'(x)$:

1. Stable equilibria

If $\Delta > 0$, the ODE can have three distinct real roots. In this case, the equilibrium points can be classified based on the sign of $f'(x)$ at each root. If $f'(x)$ is negative for all three roots, then all equilibrium points are stable.

2. Saddle equilibria

If $\Delta < 0$, the ODE can have one real root and two complex conjugate roots. The real root corresponds to a saddle equilibrium point.

3. Unstable equilibria

If $\Delta = 0$, the ODE can have a repeated real root. Stability depends on higher order derivatives.

Let $\alpha_i = \beta_i = 1$ and $\mu_i = 0.5$. We assume that the current state of the system is $x_i(0) = -1$. Figure 8.13 shows the convergence of $x_i(t)$ to equilibrium for three values of the timescale. Whatever the value of $\tau_i$, we have $\lim_{t \to \infty} x_i(t) = 1.1915$. The equilibrium is reached in seven years when $\tau_i = 1$, while we need more than 70 years when $\tau_i = 10$. In the case $\tau_i = 10$, convergence to equilibrium takes between 300 and 400 years.

To introduce the cascading effects between the tipping elements, Brummitt et al. (2015) and Klose et al. (2020) assume that $x_i$ depends on the other tipping elements $x_j$ with $j \neq i$:

$$ \frac{dx_i}{dt} = f_i(x_i, \mu_i) + c_i(x_{-i}) = g_i(x_1, \dots, x_n, \mu_i) $$

where $c_i(x_{-i})$ is the coupling function. In particular, they study the linear case where $c_i(x_{-i}) = \tau_i^{-1} \sum_{j \neq i} \gamma_{i,j} x_j$ and $\gamma_{i,j}$ measures the influence of subsystem $x_j$ on subsystem $x_i$. Therefore, Equation (8.16) becomes:

$$ \frac{dx_i}{dt} = \frac{\alpha_i x_i - \beta_i x_i^3 + \mu_i + \sum_{j \neq i} \gamma_{i,j} x_j}{\tau_i} \quad (8.18) $$

We may wonder how the coupling function affects the tipping points. For example, in the two-dimensional case we have:

$$ \left\{ \begin{aligned} \frac{dx_1}{dt} &= \frac{\alpha_1 x_1 - \beta_1 x_1^3 + \mu_1 + \gamma_{1,2} x_2}{\tau_1} \\ \frac{dx_2}{dt} &= \frac{\alpha_2 x_2 - \beta_2 x_2^3 + \mu_2 + \gamma_{2,1} x_1}{\tau_2} \end{aligned} \right. $$

Several coupling models can be distinguished:
<figure><img src="image_347.png" /><figcaption>Figure 8.65: Double fold bifurcation</figcaption></figure>

Source: Klose et al. (2020, Figure 1, page 3).

<figure><img src="image_348.png" /><figcaption>Figure 8.66: Convergence to the equilibrium</figcaption></figure>
* If $\gamma_{1,2} = \gamma_{2,1} = 0$, then the two subsystems are uncoupled and the previous analysis remains valid.

* If $\gamma_{1,2} = 0$ and $\gamma_{2,1} \neq 0$, then the first subsystem drives the second subsystem ($x_1 \rightarrow x_2$). We obtain a master-slave system with linear coupling (Brummitt et al., 2015, Section 2.2, pages 3-5). For the first subsystem, we have $\Delta_1 = \frac{4\beta_1\alpha_1^3 - 27\beta_1^2\mu_1^2}{\tau_1^4}$, which means that we obtain the same results as in the uncoupled case. Assuming that $x_1$ is at the equilibrium $x_1^*$, the discriminant for the second subsystem becomes:

$$ \Delta_2 = \frac{4\beta_2\alpha_2^3 - 27\beta_2^2(\mu_2 + \gamma_{2,1}x_1^*)^2}{\tau_2^4} \\ \propto 4\alpha_2^3 - 27\beta_2(\mu_2 + \gamma_{2,1}x_1^*)^2 $$

$\Delta_2$ then depends on the bifurcation parameter $\mu_2$, the coupling strength $\gamma_{2,1}$ and the equilibrium $x_1^*$. If $\Delta_2 < 0$ or $4\alpha_2^3 < 27\beta_2(\mu_2 + \gamma_{2,1}x_1^*)^2$ there is one stable equilibrium, while if $\Delta_2 > 0$ or $4\alpha_2^3 > 27\beta_2(\mu_2 + \gamma_{2,1}x_1^*)^2$ there two stable equilibria. Therefore, the bifurcation occurs when:

$$ \Delta_2 = 0 \Leftrightarrow 4\alpha_2^3 - 27\beta_2(\mu_2 + \gamma_{2,1}x_1^*)^2 = 0 \\ \Leftrightarrow \mu_2 = -\gamma_{2,1}x_1^* \pm \sqrt{\frac{4\alpha_2^3}{27\beta_2}} $$

As noted by Brummitt et al. (2015), the saddle-node bifurcation of the slave subsystem now depends on the equilibrium $x_1^*$ of the master subsystem and the coupling strength $\gamma_{2,1}$. Therefore, we have to distinguish three cases: $\mu_1 \le \mu_1^- = -\sqrt{\frac{4\alpha_1^3}{27\beta_1}}$, $\mu_1 \ge \mu_1^+ = \sqrt{\frac{4\alpha_1^3}{27\beta_1}}$, and $\mu_1^- \le \mu_1 \le \mu_1^+$.

* If $\gamma_{1,2} \neq 0$ and $\gamma_{2,1} \neq 0$, then the two subsystems are dependent. We can get complex equilibria with the possibility of loops and hysteresis effects.

We consider the previous example $\alpha_i = \beta_i = \tau_i = 1$ and assume that $\gamma_{1,2} = 0$ and $\gamma_{2,1} = 0.10$. We have a master-slave system ($x_1 \rightarrow x_2$). In the case of uncoupling, the stable equilibrium is equal to $x_2^* = -0.8376$ if we consider the lower branch and $\mu_2 = 0.25$. Let us assume that the equilibrium for the first subsystem is $x_1^* = 5$. Therefore, the second system moves to the new equilibrium$^{66}$ $x_2^* = 1.2626$. Previously, the equilibrium $x_2^*$ was a function of the bifurcation parameter $\mu_2$: $x_2^* = h(\mu_2 | \alpha_2, \beta_2)$. Now it also depends on the equilibrium $x_1^*$:

$$ x_2^* = h(\mu_2, x_1^* | \alpha_2, \beta_2) $$

For example, we have $x_2^* = h(0.25, -10 | 1, 1) = -1.2626$. If $x_1^* = 6$, we get three solutions: $-1.1429, 0.4289$ and $0.7140$. All these solutions are shown in Figure 8.67. This implies that one subsystem can jump from one (stable) equilibrium to another equilibrium because the state of the other subsystem changes. In Figure 8.68, we represent all the solutions of the slave subsystem $x_2^* = h(\mu_2, x_1^* | \alpha_2, \beta_2)$ with respect to $\mu_2$ and $x_1^*$. The first and third panels correspond to the stable lower and upper branches, while the second panel shows the unstable solutions.

<footnote>$^{66}$We solve the equation $x_2 - x_2^3 + (0.25 + 0.10 \times 5) = 0$.</footnote>
<figure><img src="image_349.png" /><figcaption>Figure 8.67: Master-slave bifurcation</figcaption></figure>

Source: Klose et al. (2020, Figure 2, page 6).

<figure><img src="image_350.png" /><figcaption>Figure 8.68: Equilibria of the slave subsystem</figcaption></figure>
We can extend the previous analysis to the three-dimensional case to illustrate the concept of cascading tipping points. Brummitt et al. (2015) analyzed the following master-slave-slave system ($x_1 \rightarrow x_2 \rightarrow x_3$):

$$ 
\left\{ 
\begin{aligned}
\frac{dx_1}{dt} &= \frac{\alpha_1 x_1 - \beta_1 x_1^3 + \mu_1}{\tau_1} \\
\frac{dx_2}{dt} &= \frac{\alpha_2 x_2 - \beta_2 x_2^3 + \mu_2 + \gamma_{2,1} x_1}{\tau_2} \\
\frac{dx_3}{dt} &= \frac{\alpha_3 x_3 - \beta_3 x_3^3 + \mu_3 + \gamma_{3,2} x_2}{\tau_3}
\end{aligned}
\right. 
$$

In particular, they showed that:

> "Regime shifts can spread in two ways in this system $x_1 \rightarrow x_2 \rightarrow x_3$: First, if all three systems are sufficiently close to their tipping points, then a cascade of regime shifts can occur, one causing the next. The second way is more novel: if the intermediate system $x_2$ is relatively far from its tipping point whereas $x_1$ and $x_3$ are close to their tipping points, then the sequence of regime shifts can hop over the intermediate system $x_2$. That is, a regime shift in the master subsystem can nudge the intermediate system $x_2$ enough to trigger a regime shift in the third system $x_3$ but not so much that $x_2$ undergoes a regime shift." (Brummitt et al., 2015, page 5).

### An application with Greenland and West Antarctic ice sheets, AMOC and Amazon rainforest
Using the previous framework developed by Brummitt et al. (2015) and Klose et al. (2020), Wunderling et al. (2021) analyze the dynamics of four tipping climate elements, namely (1) Greenland ice sheet, (2) West Antarctic ice sheet (WAIS), (3) the AMOC and (4) the Amazon rainforest (Figure 8.69). The dynamics of the tipping elements are defined as:

$$ \frac{dx_i}{dt} = \frac{1}{\tau_i} \left( x_i - x_i^3 + \sqrt{\frac{4}{27} \frac{\Delta T_s}{\Delta T_i^*}} + \frac{\gamma}{10} \sum_{j \neq i} s_{i,j} (x_j + 1) \right) $$

where $\Delta T_s$ is the variation of the global mean surface temperature, $\Delta T_i^*$ is the critical temperature anomaly to reach a tipping point, $\gamma$ is the overall interaction strength and $s_{i,j}$ is the coupling strength between $x_i$ and $x_j$. Using the previous notations, we have $\alpha_i = 1$, $\beta_i = 1$, $\mu_i = \sqrt{\frac{4}{27} \frac{\Delta T_s}{\Delta T_i^*}} + \frac{\gamma}{10} \sum_{j \neq i} s_{i,j}$ and $\gamma_{i,j} = \frac{\gamma}{10} s_{i,j}$.

<table><caption>Table 8.13: Parameters of the cascading model</caption><thead><tr><th>Tipping element</th><th>Greenland</th><th>WAIS</th><th>AMOC</th><th>Amazon rainforest</th></tr></thead><tbody><tr><td>ΔT<sup>*</sup><sub>i</sub> (in °C)</td><td>0.8 – 3.2</td><td>0.8 – 5.5</td><td>3.5 – 6.0</td><td>3.5 – 4.5</td></tr><tr><td>Greenland</td><td></td><td>+10</td><td>+10</td><td></td></tr><tr><td>WAIS</td><td>+2</td><td></td><td>±3</td><td></td></tr><tr><td>AMOC</td><td>-10</td><td>+1.5</td><td></td><td>±2 – ±4</td></tr><tr><td>Amazon rainforest</td><td></td><td></td><td></td><td></td></tr></tbody></table>

Source: Wunderling et al. (2021, pages 608 and 610).

To better understand the model, we report the magnitude of $\Delta T_i^*$ and the values of $s_{i,j}$ used by Wunderling et al. (2021). For example, the critical temperature $\Delta T_1^*$ for the Greenland ice sheet is between 0.8°C and 3.2°C, which is close to the range estimated by Armstrong McKay et al. (2022)
<figure><img src="image_351.png" /><figcaption>Figure 8.69: Interactions between climate tipping elements and their roles in tipping cascades</figcaption></figure>

Source: Wunderling et al. (2021, Figure 1, page 603) & https://esd.copernicus.org/articles/12/601/2021.

(see Table 8.12 on page 700). The results of the Monte Carlo simulation exercise are presented in Table 8.14. Wunderling et al. (2021) found that tipping occurs in 61% of all simulations under the assumption of 2°C global warming. The split between single tipping and cascading tipping differs with respect to the global strength $\gamma$. In particular, when $\gamma$ is equal to one, the authors observed two-element cascades in 21% of all MC simulations, three-element cascades in 15% of all MC simulations, and there is a 4% chance of observing a full cascade phenomenon. They concluded that “*the interactions tend to destabilize the network of tipping elements*”, even under the assumption that the Paris Agreement is respected.

<table><caption>Table 8.14: Share of tipping events</caption><thead><tr><th rowspan="2">γ</th><th colspan="4">Cascade size</th></tr><tr><th>1</th><th>2</th><th>3</th><th>4</th></tr></thead><tbody><tr><td>1.00</td><td>22%</td><td>21%</td><td>15%</td><td>3%</td></tr><tr><td>0.75</td><td>26%</td><td>18%</td><td>14%</td><td>2%</td></tr><tr><td>0.50</td><td>31%</td><td>15%</td><td>14%</td><td>1%</td></tr><tr><td>0.25</td><td>42%</td><td>13%</td><td>6%</td><td></td></tr><tr><td>0.10</td><td>56%</td><td>5%</td><td></td><td></td></tr><tr><td>0.00</td><td>61%</td><td></td><td></td><td></td></tr></tbody></table>

Source: Wunderling et al. (2021, Table 3, page 611).

**Remark 80** This section on the physics of climate change provides a basic overview of the mechanisms behind climate change. It may be supplemented by Section 12.1 on Chapter 12 of Physical Risk Modeling, which is a comprehensive presentation of general circulation models.
## 8.2 The ecosystem of climate change

The climate change ecosystem is relatively complex and involves many third parties. In this section, we focus on three types of actors: the scientific community, the Conference of the Parties, and regulators.

### 8.2.1 Scientists

#### Intergovernmental Panel on Climate Change

As mentioned above, the United Nations Environment Programme (UNEP) and the World Meteorological Organization (WMO) established the Intergovernmental Panel on Climate Change (IPCC) in 1988. The members of the IPCC are then governments (95 as of January 2024), not scientists, academic institutions, or NGOs. Its original mandate was to “prepare a comprehensive review and recommendations on the state of knowledge of the science of climate change, the social and economic impacts of climate change, and possible response strategies and elements for inclusion in a possible future international climate convention.” The first IPCC Assessment Report (AR) was published in 1990. It consists of a synthesis report (March 1990, 180 pages), followed by three working group reports: Scientific Assessment of Climate Change (June 1990, 414 pages), Impacts Assessment of Climate Change (July 1990, 296 pages), and the IPCC Response Strategies (October 1990, 332 pages). This first report presented scientific evidence that underscored the urgency of addressing climate change through international cooperation and was a catalyst for the creation of the United Nations Framework Convention on Climate Change (UNFCCC), the international treaty to reduce global warming. Among its findings, the report concludes that under a business-as-usual emissions scenario, the global mean temperature is likely to increase by about 0.3°C per decade.

<table><caption>Table 8.15: List of IPCC Chairs</caption><thead><tr><td>Date</td><td>Name</td><td>Country</td><td>Occupation</td></tr></thead><tbody><tr><td>1988–1997</td><td>Bert Bolin</td><td>Sweden</td><td>Meteorologist</td></tr><tr><td>1997–2002</td><td>Robert T. Watson</td><td>UK</td><td>Chemist</td></tr><tr><td>2002–2015</td><td>Rajendra Kumar Pachauri</td><td>India</td><td>Engineer</td></tr><tr><td>2015</td><td>Ismail A.R. Elgizouli</td><td>Sudan</td><td>Civil servant</td></tr><tr><td>2015–2024</td><td>Hoesung Lee</td><td>South Korea</td><td>Economist</td></tr></tbody></table>

This first report already reflected the structure of the IPCC, which is organized around three working groups led by two co-chairs. These three IPCC Working Groups are:

* **Working Group I (WGI)**
  Assesses the scientific aspects of the climate system that underpin past, present and future climate change.

* **Working Group II (WGII)**
  Assesses climate change impacts, adaptation options and vulnerabilities on human and natural systems.

* **Working Group III (WGIII)**
  Focuses on mitigation of climate change, assessment of methods to limit and reduce greenhouse gas emissions, and removal of greenhouse gases from the atmosphere.
<table><caption>Table 8.16: List of IPCC Assessment Reports</caption><thead><tr><td>Date</td><td>Symbol</td><td>Title of the report</td><td>website</td></tr></thead><tbody><tr><td>1990</td><td>FAR/AR1</td><td>First IPCC Assessment Report</td><td>www.ipcc.ch/reports/?rp=ar1</td></tr><tr><td>1995</td><td>SAR/AR2</td><td>Second Assessment Report</td><td>www.ipcc.ch/reports/?rp=ar2</td></tr><tr><td>2001</td><td>TAR/AR3</td><td>Third Assessment Report</td><td>www.ipcc.ch/reports/?rp=ar3</td></tr><tr><td>2007</td><td>AR4</td><td>Fourth Assessment Report</td><td>www.ipcc.ch/reports/?rp=ar4</td></tr><tr><td>2014</td><td>AR5</td><td>Fifth Assessment Report</td><td>www.ipcc.ch/reports/?rp=ar5</td></tr><tr><td>2023</td><td>AR6</td><td>Sixth Assessment Report</td><td>www.ipcc.ch/reports/?rp=ar6</td></tr></tbody></table>

The composition of the working groups reflects the task assigned, e.g., the first working group has a majority of physicists, while we find more economists in the third working group. The three working groups are more or less independent, but the results of the second working group must be coordinated with the results of the first working group, while the analysis of the third working group must be consistent with the stylized facts found by the first and second working groups. This explains why the WGI report is published first, followed by the WGII report, and then the WGIII report. Finally, the synthesis is published later. For example, the WGI contribution to the Sixth Assessment Report (Climate Change 2021: The Physical Science Basis, 2409 pages) was published on 9 August 2021, the WGII contribution (Climate Change 2022: Impacts, Adaptation and Vulnerability, 3068 pages) on 28 February 2022, and the WGIII contribution (Climate Change 2022: Mitigation, 2042 pages) on 4 April 2022. The synthesis report (Climate Change 2023: Synthesis Report, 186 pages) was released on 20 March 2023 to inform the 2023 Global Stocktake under the United Nations Framework Convention on Climate Change (UNFCCC).

<figure><img src="image_352.png" /><figcaption>Figure 8.70: Climate Change 2023: Synthesis Report (AR6)</figcaption></figure>

The process for publishing IPCC reports<fnref n="67" /> is relatively complex and well standardized (IPCC, 1999). Authors are selected through an extensive process in which candidates submit detailed CVs. Selection is a collaborative effort involving governments, observer organizations and the scientific community. The aim is to form author teams that represent different scientific, technical and socio-

<footnote n="67">⁶⁷In addition to publishing assessment reports, the IPCC also publishes special reports, the most prominent of which is the Special Report on Global Warming of 1.5°C (IPCC, 2018).</footnote>
economic perspectives, and to ensure a balanced mix of experts from different regions and from both developed and developing countries. The idea is to avoid bias towards any particular nation or group and ensures that issues relevant to different regions receive appropriate attention in the reports. Chapter teams consist of coordinating lead authors, lead authors, and review editors. The Bureau of each IPCC Working Group selects scientists for these roles from the pool of nominated experts. For the Sixth Assessment Report, 782 experts were selected as chapter authors from some 2827 nominations (Table 8.17). Once teams of authors are formed, each team must synthesize the scientific evidence on a given topic. This means that the IPCC does not conduct its own research, run its own models, or make its own measurements of climate or weather phenomena. It uses existing research that has been published in scientific journals and other relevant publications. For example, the Sixth Assessment Report contains more than 65 000 citations (Table 8.17). Once the first-order draft is prepared by the authors based on the existing literature in scientific journals and other relevant publications, it is reviewed by experts. After expert review of the first-order draft, the author teams prepare a second-order draft of the report, taking into account the review comments received. This second draft and an initial policy summary are then reviewed again by experts and governments. The review comments are incorporated into a final draft, which is then reviewed only by governments. For example, there were 61 513, 124 235, and 13 889 review comments for the first-order, second-order, and final drafts of the Sixth Assessment Report, respectively (Table 8.17). A plenary session with governments is then organized to accept the underlying final report and approve the policy summary line by line.

<table><caption>Table 8.17: Statistics of the Sixth Assessment Report</caption><thead><tr><th>Working Group</th><th>WGI</th><th>WGII</th><th>WGIII</th></tr></thead><tbody><tr><td>Author team (Lead Authors, Review Editors)</td><td>234</td><td>270</td><td>278</td></tr><tr><td>Review comments</td><td></td><td></td><td></td></tr><tr><td>First order draft (experts)</td><td>23 462</td><td>16 348</td><td>21 703</td></tr><tr><td>Second order draft (experts and governments)</td><td>51 387</td><td>40 293</td><td>32 555</td></tr><tr><td>Final draft (governments)</td><td>3 158</td><td>5 777</td><td>4 954</td></tr><tr><td>Number of citations</td><td>14 000</td><td>34 000</td><td>18 000</td></tr></tbody></table>

The AR6 Synthesis Report (Summary for Policymakers) is relatively long (more than 40 pages), but the three main conclusions are as follows:

1. Unequivocal human influence

The report reaffirms that human activities are unequivocally causing climate change. The evidence is now overwhelming, and it is highly likely that human influence has been the dominant cause of the observed warming since the mid-20th century. The report states that global surface temperatures have reached 1.1°C above pre-industrial levels.

2. Widespread and intensifying impacts

The report emphasizes that climate change is already having a profound impact on the planet, affecting ecosystems, human health, and infrastructure. These impacts are projected to worsen in the future, even with moderate warming.

3. Urgency of mitigation and adaptation

The report underscores the urgency of taking action to address climate change. Limiting global warming to 1.5°C will require rapid and far-reaching changes in energy systems, food production and land management. These changes are possible and would bring significant environmental and human health benefits.
## Climate research institutions

We reiterate that the IPCC does not conduct research on climate change. It uses scientists (academic professors, experts, etc.) working in climate research centers, universities and non-academic institutions<sup>68</sup> (e.g., international bodies, intergovernmental organizations). Climate research institutes and university research centers are then an important component, as they provide most of the climate experts. Table 8.18 contains a list of some well-known climate research institutions, but the list is unofficial, non-exhaustive, far from complete, and expresses the personal views and biases of the author. In this list, we have excluded climate research centers that are part of universities. However, the distinction between an independent research center and a university research center is not always clear. For example, the Scripps Institution of Oceanography (SIO) is considered independent, while it is affiliated with the University of California San Diego (UCSD). These climate research institutions can then be a governmental organization (GO), a non-governmental organization (NGO), an international non-governmental organization (INGO), foundations, etc. Moreover, the scope of climate research institutions is not always easy to define. For example, do we include the Chinese Academy of Sciences or just the Institute of Atmospheric Physics of the Chinese Academy of Sciences? Do we include the Institut Pierre-Simon Laplace or the most famous of the ten research centers of the IPSL, the Laboratoire des Sciences du Climat et de l'Environnement (LSCE)? Once this selection is made, we find that the US largely dominates climate research. The most prominent institutions are GISS, NCAR, and NOAA in the United States, and IAP, IIASA, IPSL, MOHC, NIES, and PIK outside the United States.

### Box 8.12: National Center for Atmospheric Research (NCAR)

The National Center for Atmospheric Research is a federally funded research and development center managed by the University Corporation for Atmospheric Research (UCAR) and funded by the National Science Foundation (NSF). NCAR was established in 1960, and its founding director was Walter Orr Roberts. Its headquarters are located in Boulder, Colorado. NCAR's annual budget was $173 million in 2017. The center is organized into eight laboratories with approximately 1500 members. Notable scientists include or were Guy Brasseur, Clara Deser, Brant Foote, Marika M. Holland, Paul R. Julian, Jean-François Lamarque, David M. Lawrence, Gerald A. Meehl, Joanne Simpson, Kevin E. Trenberth, Warren M. Washington, and Tom M. L. Wigley.

### Box 8.13: Potsdam Institute for Climate Impact Research (PIK)

The Potsdam Institute for Climate Impact Research is a government-funded research institute in Potsdam, Germany. It was founded in 1991 by Hans Joachim Schellnhuber, who is now Director of IIASA. With a network of about 400 researchers, PIK's mission is to address scientific issues in the fields of climate risks and sustainable development. The current directors of the institute are Ottmar Edenhofer, who also serves as chief economist, and Johan Rockström, former director of the Stockholm Resilience Center. In 2022, the Institute received about 13.3 million euros in institutional funding. Additional project funding from external sources amounted to approximately 18.2 million euros. The institute boasts many renowned researchers, including Elmar Kriegler, Christoph Müller, Ottmar Edenhofer, Alexander Popp, Stefan Rahmstorf and Johan Rockström.

<footnote n="68"><sup>68</sup>Examples of non-academic institutions include the International Energy Agency (IEA), the Energy Transitions Commission (ETC), the World Meteorological Organization (WMO), and the Organization for Economic Cooperation and Development (OECD).</footnote>
<table><caption>Table 8.18: Some well-known climate research institutions</caption><thead><tr><th>Symbol</th><th>Climate research institutions</th><th>Country</th><th>Website</th></tr></thead><tbody><tr><td>AARI</td><td>Arctic and Antarctic Research Institute</td><td>Russia</td><td>www.aari.ru</td></tr><tr><td>ARC</td><td>Australian Research Council Centre of Excellence for Climate Extremes</td><td>Australia</td><td>climateextremes.org.au</td></tr><tr><td>AWI</td><td>Alfred Wegener Institute for Polar and Marine Research</td><td>Germany</td><td>www.awi.de</td></tr><tr><td>CCCma</td><td>Canadian Centre for Climate Modelling and Analysis</td><td>Canada</td><td>www.canada.ca</td></tr><tr><td>CICERO</td><td>Center for International Climate and Environmental Research</td><td>Norway</td><td>www.cicero.oslo.no</td></tr><tr><td>CIMA</td><td>Centro de Investigaciones del Mar y la Atmosfera</td><td>Argentina</td><td>www.cima.fcen.uba.ar</td></tr><tr><td>CIRED</td><td>Centre international de recherche sur l'environnement et le développement</td><td>France</td><td>www centre-cired.fr</td></tr><tr><td>CSIC</td><td>Spanish National Research Council</td><td>Spain</td><td>www.csic.es</td></tr><tr><td>CSIRO</td><td>Commonwealth Scientific and Industrial Research Organisation</td><td>Australia</td><td>www.csiro.au</td></tr><tr><td>CMCC</td><td>Euro-Mediterranean Center on Climate Change</td><td>Italy</td><td>www.cmcc.it</td></tr><tr><td>CNRS</td><td>French National Centre for Scientific Research</td><td>France</td><td>www.cnrs.fr</td></tr><tr><td>GISS</td><td>NASA Goddard Institute for Space Studies</td><td>US</td><td>www.giss.nasa.gov</td></tr><tr><td>IAP</td><td>Institute of Atmospheric Physics Chinese Academy of Sciences</td><td>China</td><td>english.iap.cas.cn</td></tr><tr><td>IIASA</td><td>International Institute for Applied Systems Analysis</td><td>Austria</td><td>www.iiasa.ac.at</td></tr><tr><td>IITM</td><td>Indian Institute of Tropical Meteorology</td><td>India</td><td>www.tropmet.res.in</td></tr><tr><td>IPSL</td><td>Institut Pierre-Simon Laplace (LSCE)</td><td>France</td><td>www.ipsl.fr</td></tr><tr><td>INECC</td><td>National Institute of Ecology and Climate Change</td><td>Mexico</td><td>www.gob.mx/inecc</td></tr><tr><td>JAMSTEC</td><td>Japan Agency for Marine-Earth Science and Technology</td><td>Japan</td><td>www.jamstec.go.jp</td></tr><tr><td>MOHC</td><td>Met Office Hadley Centre for Climate Science and Services</td><td>UK</td><td>www.metoffice.gov.uk</td></tr><tr><td>MPIM</td><td>Max Planck Institute for Meteorology</td><td>Germany</td><td>www.mpmet.mpg.de</td></tr><tr><td>NCAR</td><td>National Center for Atmospheric Research</td><td>US</td><td>ncar.ucar.edu</td></tr><tr><td>NERC</td><td>Natural Environment Research Council</td><td>UK</td><td>www.ukri.org/councils/nerc</td></tr><tr><td>NIES</td><td>National Institute for Environmental Studies</td><td>Japan</td><td>www.nies.go.jp</td></tr><tr><td>NOAA</td><td>National Oceanic and Atmospheric Administration</td><td>US</td><td>www.noaa.gov</td></tr><tr><td>NREL</td><td>National Renewable Energy Laboratory</td><td>US</td><td>www.nrel.gov</td></tr><tr><td>PBL</td><td>PBL Netherlands Environmental Assessment Agency</td><td>Netherlands</td><td>www.pbl.nl</td></tr><tr><td>PCMDI</td><td>Program for Climate Model Diagnosis &amp; Intercomparison (LLNL)</td><td>US</td><td>www.pcmdi.llnl.gov</td></tr><tr><td>PIK</td><td>Potsdam Institute for Climate Impact Research</td><td>Germany</td><td>www.pik-potsdam.de</td></tr><tr><td>PNNL</td><td>Pacific Northwest National Laboratory</td><td>US</td><td>www.pnnl.gov</td></tr><tr><td>REKLIM</td><td>Helmholtz Climate Initiative Regional Climate</td><td>Germany</td><td>www.reklim.de</td></tr><tr><td>SEI</td><td>Stockholm Environment Institute</td><td>Sweden</td><td>www.sei.org</td></tr><tr><td>SIO</td><td>Scripps Institution of Oceanography</td><td>US</td><td>www.scripps.ucsd.edu</td></tr><tr><td>SMHI</td><td>Rossby Centre of the Swedish Meteorological and Hydrological Institute</td><td>Sweden</td><td>www.smhi.se</td></tr><tr><td>SPREP</td><td>Secretariat of the Pacific Regional Environment Programme (PCCC)</td><td>Samoa</td><td>www.sprep.org</td></tr><tr><td>WHOI</td><td>Woods Hole Oceanographic Institution</td><td>US</td><td>www.whoi.edu</td></tr></tbody></table>

Source: Author's research.
Box 8.14: International Institute for Applied Systems Analysis (IIASA)

The International Institute for Applied Systems Analysis is an independent international research institute located in Laxenburg near Vienna, Austria. IIASA was founded by a charter signed on October 4, 1972 by representatives of the Soviet Union, the United States, and ten other countries from the Eastern and Western blocs. IIASA brings together experts from various fields to study complex issues such as climate change, energy, and sustainable development. In 2022, IIASA's annual budget was 24.4 million euros, of which just under half came from the Institute's national and regional member organizations (Austria, Brazil, China, Egypt, Finland, Germany, India, Iran, Israel, Japan, South Korea, Norway, Russia, Slovakia, Sub-Saharan Africa, Sweden, Ukraine, United Kingdom, United States, Vietnam). IIASA has about 500 researchers from 50 countries. Since December 2023, the Director General is Hans Joachim Schellnhuber. Among the researchers who work, have worked or have visited the research center, we can mention George Dantzig, Shinichiro Fujimori, Petr Havlík, Leonid Kantorovich, Tjalling Koopmans, Nebojsa Nakicenovic, William D. Nordhaus, Michael Obersteiner, Howard Raiffa, Keywan Riahi, Joeri Rogelj and Thomas Schelling.

Box 8.15: National Oceanic and Atmospheric Administration (NOAA)

The National Oceanic and Atmospheric Administration is an agency of the United States Department of Commerce (DOC) responsible for monitoring and managing the nation's weather, climate, and oceans. NOAA was established in 1970. The scope of NOAA is vast: weather forecasting, oceanography, fisheries management, satellite operations. As a result, it employs 12000 people worldwide, while the number of NOAA scientists and engineers is about 6500. The Office of Oceanic and Atmospheric Research (OAR) is the primary research arm of NOAA. It is responsible for conducting a wide range of research on the Earth's atmosphere, oceans, and coasts. One of OAR's goals is to understand the causes and effects of climate change. The most prominent affiliated research center is the Geophysical Fluid Dynamics Laboratory (GFDL), a joint program of Princeton University and NOAA. GFDL researchers include Thomas L. Delworth, Larry W. Horowitz, Thomas R. Knutson, Vaishali Naik, and Venkatachalam Ramaswamy.

Box 8.16: Institut Pierre-Simon Laplace (IPSL)

The Institut Pierre-Simon Laplace is a French university research institute that brings together 10 laboratories with about 1500 members. The institute was founded in 1991 by Gérard Mégie, and one of its directors was Jean Jouzel from 2001 to 2008. The 10 laboratories are (1) the Centre d'Enseignement et de Recherche en Environnement Atmosphérique (CEREA), (2) Géosciences Paris-Sud (GEOPS), (3) the Laboratoire Atmosphères, Milieux, Observations spatiales (LATMOS), (4) the team TASQ of the Laboratoire d'Études du Rayonnement et de la Matière en Astrophysique et Atmosphères (LERMA), (5) the Laboratoire Inter-universitaire des Systèmes Atmosphériques (LISA), (6) the Laboratoire de Météorologie Dynamique (LMD), (7) the Laboratoire d'Océanographie et du Climat Expérimentation et Approches Numériques (LOCEAN), (8) the Laboratoire des Sciences du Climat et de l'Environnement (LSCE), (9) the research center Milieux Environnementaux, Transferts et Interactions dans les hydrosystèmes et les Sols (METIS), and (10) the team Surface & Réservoirs of the Laboratoire de Géologie de l'ENS. IPSL is placed under the supervision of Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU), Université Versailles Saint-Quentin (UVSQ), École Polytechnique, Commissariat à l'Énergie Atomique et aux Énergies Alternatives (CEA), Institut de Recherche pour le Développement (IRD) and École Nationale des Ponts et Chaussées (ENPC). Researchers include Sandrine Bony, Laurent Bopp, Olivier Boucher, Pascale Braconnot, Philippe Ciais, Jean Jouzel, Pierre Friedlingstein, Valérie Masson-Delmotte, Robert Vautard, and Nicolas Viovy.
The picture of climate research centers would not be complete without mentioning university research, which does most of the work on the subject. However, compiling a list of top universities is a complex task for many reasons: the number of universities is very large; sometimes there are several research centers working on climate risks within the same university; the prominence of a research center may depend on the leadership of an emblematic individual or on the strength of a collaborative team, etc. Nevertheless, we can agree on a restrictive list of academic institutions.

To get a better idea of the leading universities and scientists influencing the climate change debate, we use Reuters 2021 Hot List ranking of top climate scientists. The list can be found at www.reuters.com/investigates/section/climate-change-scientists. The Hot List identifies the 1000 most influential scientists using three rankings based on climate change research. The rankings take into account the number of published papers, citations by peers, and public reach through social and mainstream media. All data comes from Dimensions<fnref n="69" />, Digital Science's academic research portal, which has been tracking academic research since 1988. The first ranking focuses on the number of climate-related papers published by each scientist, requiring at least one citation. The second ranking uses Dimensions' Field Citation Ratio to measure influence among peers in different scientific fields. The third ranking uses Dimensions' Altmetric Attention Score to assess a paper's public reach and influence outside the academic community. The final score for each scientist is the sum of these rankings, with lower scores indicating greater overall influence. In Table 8.19, we report the top 30 climate scientists on the Reuters Hot List. While this list is not perfect and has some limitations, including a potential bias toward prolific scientists and a reliance on titles and abstracts that may miss some relevant studies, it provides very consistent and informative results about the top climate scientists. The top five names are:

* Keywan Riahi, who is the Director of the Energy, Climate and Environment (ECE) Program at the International Institute for Applied Systems Analysis (IIASA) and one of the principal developers of the Representative Concentration Pathway (RCP) and Shared Socio-economic Pathway (SSP) concepts.

* Anthony A. Leiserowitz, Professor at Yale University, who studies public perceptions of climate change.

* Pierre Friedlingstein, who holds the Chair in Mathematical Modeling of the Climate System at the University of Exeter and coordinates the annual publication of the Global Carbon Budget.

* Detlef P. Van Vuuren, who is Professor at Utrecht University, Project Leader of the IMAGE Integrated Assessment Team at PBL Netherlands Environmental Assessment Agency, and one of the main developers of the Representative Concentration Pathway (RCP) and Shared Socio-economic Pathway (SSP) concepts.

* James E. Hansen, who is considered one of the world's most influential climate scientists and was the director of NASA Goddard Institute for Space Studies from 1981 to 2013.

All of these scientists (except Anthony A. Leiserowitz) have more than 100 000 citations according to Google Scholar. As mentioned above, the list has several biases. The most important is that it favors younger authors. In fact, there are more researchers working on climate change today than 20 or 30 years ago. And newer publications tend to cite more recent research because of the scientific standards of publication. This explains why Nobel laureates Klaus Hasselmann and Syukuro Manabe are ranked 639 and 755 respectively, and a leading researcher like Raymond Pierrehumbert is missing

<footnote n="69">⁶⁹The website is www.dimensions.ai.</footnote>
<table><caption>Table 8.19: Top 30 climate scientists on the Reuters Hot List</caption><thead><tr><th>Rank</th><th>Name</th><th>Gender</th><th>Institution</th><th>Location</th></tr></thead><tbody><tr><td>1</td><td>Keywan Rialli</td><td>M</td><td>International Institute for Applied Systems Analysis</td><td>Austria</td></tr><tr><td>2</td><td>Anthony A. Leiserowitz</td><td>M</td><td>Yale University</td><td>United States</td></tr><tr><td>3</td><td>Pierre Friedlingstein</td><td>M</td><td>University of Exeter</td><td>United Kingdom</td></tr><tr><td>4</td><td>Detlef Peter Van Vuuren</td><td>M</td><td>Utrecht University</td><td>Netherlands</td></tr><tr><td>5</td><td>James E. Hansen</td><td>M</td><td>Columbia University</td><td>United States</td></tr><tr><td>6</td><td>Petr Havlik</td><td>M</td><td>International Institute for Applied Systems Analysis</td><td>Austria</td></tr><tr><td>7</td><td>Edward Wile Maibach</td><td>M</td><td>George Mason University</td><td>United States</td></tr><tr><td>8</td><td>Josep G. Canadell</td><td>M</td><td>Commonwealth Scientific and Industrial Research Organisation</td><td>Australia</td></tr><tr><td>9</td><td>Sonia Isabelle Seneviratne</td><td>F</td><td>ETH Zurich</td><td>Switzerland</td></tr><tr><td>10</td><td>Mario Herrero</td><td>M</td><td>Commonwealth Scientific and Industrial Research Organisation</td><td>Australia</td></tr><tr><td>11</td><td>David B. Lobell</td><td>M</td><td>Stanford University</td><td>United States</td></tr><tr><td>12</td><td>Carlos Manuel Duarte</td><td>M</td><td>King Abdullah University of Science and Technology</td><td>Saudi Arabia</td></tr><tr><td>13</td><td>Kevin E. Trenberth</td><td>M</td><td>National Center for Atmospheric Research</td><td>United States</td></tr><tr><td>14</td><td>Stephen A. Sitch</td><td>M</td><td>University of Exeter</td><td>United Kingdom</td></tr><tr><td>14</td><td>Glen P. Peters</td><td>M</td><td>Center for International Climate and Environmental Research</td><td>Norway</td></tr><tr><td>16</td><td>Ove I. Hoegh-Guldberg</td><td>M</td><td>University of Queensland</td><td>Australia</td></tr><tr><td>17</td><td>Richard Arthur Bets</td><td>M</td><td>Met Office</td><td>United Kingdom</td></tr><tr><td>18</td><td>Michael G. Oppenheimer</td><td>M</td><td>Princeton University</td><td>United States</td></tr><tr><td>18</td><td>William Neil Adger</td><td>M</td><td>University of Exeter</td><td>United Kingdom</td></tr><tr><td>20</td><td>William Wai Lung Cheung</td><td>M</td><td>University of British Columbia</td><td>Canada</td></tr><tr><td>21</td><td>Christopher B. Field</td><td>M</td><td>Stanford University</td><td>United States</td></tr><tr><td>23</td><td>Shinichiro Fujimori</td><td>M</td><td>Kyoto University</td><td>Japan</td></tr><tr><td>23</td><td>Elmar Kriegler</td><td>M</td><td>Potsdam Institute for Climate Impact Research</td><td>Germany</td></tr><tr><td>25</td><td>Yadvinder Singh Malhi</td><td>M</td><td>University of Oxford</td><td>United Kingdom</td></tr><tr><td>26</td><td>Ken Caldeira</td><td>M</td><td>Carnegie Institution for Science's Department of Global Ecology</td><td>United States</td></tr><tr><td>27</td><td>Chris D. Thomas</td><td>M</td><td>University of York</td><td>United Kingdom</td></tr><tr><td>28</td><td>Stéphane Hallegatte</td><td>M</td><td>World Bank</td><td>United States</td></tr><tr><td>28</td><td>Andy P. Haines</td><td>M</td><td>London School of Hygiene &amp; Tropical Medicine</td><td>United Kingdom</td></tr><tr><td>30</td><td>Michael Obersteiner</td><td>M</td><td>International Institute for Applied Systems Analysis</td><td>Austria</td></tr><tr><td>40</td><td>Philippe Clais</td><td>M</td><td>Laboratoire des Sciences du Climat et de l'Environnement</td><td>France</td></tr><tr><td>75</td><td>Pete Smith</td><td>M</td><td>University of Aberdeen</td><td>United Kingdom</td></tr><tr><td>164</td><td>Richard S. J. Tol</td><td>M</td><td>VU Amsterdam</td><td>Netherlands</td></tr><tr><td>173</td><td>William D. Nordhaus</td><td>M</td><td>Yale University</td><td>United States</td></tr><tr><td>240</td><td>Phil D. Jones</td><td>M</td><td>University of East Anglia</td><td>United Kingdom</td></tr><tr><td>338</td><td>Filippo Giorgi</td><td>M</td><td>International Centre for Theoretical Physics</td><td>Italy</td></tr><tr><td>639</td><td>Klaus Hasselmann</td><td>M</td><td>Max Planck Institute for Meteorology</td><td>Germany</td></tr><tr><td>755</td><td>Syukuro Manabe</td><td>M</td><td>Princeton University</td><td>United States</td></tr></tbody></table>

Source: Reuters (2021), www.reuters.com/investigates/special-report/climate-change-scientists-list.
from the list<fnref n="70" />. The list is also biased towards researchers from large teams involved in collaborative projects. There is no distinction between lead and secondary authors. In fact, writing a theoretical research paper alone is not on the same level as signing a research article with 100 other researchers. A third bias is that the number of citations and the market impact of a researcher increases with the number of peers involved in the research field. This explains why economist Richard S. J. Tol and Nobel laureate William D. Nordhaus are ranked only 164th and 173rd, respectively, despite their massive impact on several climate change issues (integrated assessment model, social cost of carbon, carbon tax, economic cost of climate change).

<table><caption>Table 8.20: Top 20 climate research institutions on the Reuters Hot List</caption><thead><tr><th>Institution</th><th>Count</th><th>Location</th></tr></thead><tbody><tr><td>Potsdam Institute for Climate Impact Research (PIK)</td><td>14</td><td>Germany</td></tr><tr><td>University of Reading</td><td>13</td><td>United Kingdom</td></tr><tr><td>Institute of Atmospheric Physics (CAS)</td><td>13</td><td>China</td></tr><tr><td>Utrecht University</td><td>12</td><td>Netherlands</td></tr><tr><td>Met Office (MOHC)</td><td>12</td><td>United Kingdom</td></tr><tr><td>National Center for Atmospheric Research (NCAR)</td><td>11</td><td>United States</td></tr><tr><td>Columbia University</td><td>10</td><td>United States</td></tr><tr><td>ETH Zurich</td><td>10</td><td>Switzerland</td></tr><tr><td>Laboratoire des Sciences du Climat et de l'Environnement (IPSL)</td><td>10</td><td>France</td></tr><tr><td>International Institute for Applied Systems Analysis (IIAS)</td><td>9</td><td>Austria</td></tr><tr><td>University of Melbourne</td><td>9</td><td>Australia</td></tr><tr><td>University of Leeds</td><td>9</td><td>United Kingdom</td></tr><tr><td>Geophysical Fluid Dynamics Laboratory (NOAA)</td><td>9</td><td>United States</td></tr><tr><td>Max Planck Institute for Meteorology (MPIM)</td><td>9</td><td>Germany</td></tr><tr><td>Pacific Northwest National Laboratory (PNNL)</td><td>8</td><td>United States</td></tr><tr><td>Lamont-Doherty Earth Observatory (Columbia University)</td><td>8</td><td>United States</td></tr><tr><td>Massachusetts Institute of Technology</td><td>8</td><td>United States</td></tr><tr><td>University of Washington</td><td>8</td><td>United States</td></tr><tr><td>Wageningen University &amp; Research</td><td>8</td><td>Netherlands</td></tr><tr><td>University of Tokyo</td><td>8</td><td>Japan</td></tr><tr><td>University of Bremen</td><td>8</td><td>Germany</td></tr></tbody></table>

Source: Reuters (2021), www.reuters.com/investigates/special-report/climate-change-scientists-list.

Despite these biases in individual rankings, the Reuters Hot List can be used to assess global rankings and the importance of research institutions on an aggregate basis. In Table 8.20 we have counted the number of scientists present in the top 1000 by research center. Not surprisingly, PIK, IAP, MOHC, NCAR, IPSL, IIASA, MPIM and PNNL are among the top 20 climate research institutions. When we group the scientists by location, the United States largely dominates the

<footnote n="70">⁷⁰In 2015, Carbon Brief conducted a survey to nominate the most influential climate change papers of all time. To do this, they asked all coordinating lead authors, lead authors, and review editors of the IPCC's AR5 report to nominate three papers from any time in history. The question asked was: "What do you consider to be the three most influential papers in the field of climate change?". The IPCC scientists' top choice for the most influential climate change paper of all time was the seminal paper *Thermal Equilibrium of the Atmosphere with a Given Distribution of Relative Humidity* by Manabe and Wetherald (1967), followed by ex aequo by *Atmospheric Carbon Dioxide Variations at Mauna Loa Observatory, Hawaii* by Keeling et al. (1976) and *Robust Responses of the Hydrological Cycle to Global Warming* by Held and Soden (2006). However, despite the innovative and pioneering nature of these research, these papers are only cited 2212, 1157, and 4763 times, respectively, according to Google Scholar as of January 1, 2024.</footnote>
<table><caption>Table 8.21: Top 15 country on the Reuters Hot List</caption><thead><tr><th>Country</th><th>Count</th><th>Frequency</th><th>Cumulative</th></tr></thead><tbody><tr><td>United States</td><td>292</td><td>29.3%</td><td>29.3%</td></tr><tr><td>United Kingdom</td><td>112</td><td>11.3%</td><td>40.6%</td></tr><tr><td>Germany</td><td>91</td><td>9.1%</td><td>49.7%</td></tr><tr><td>China</td><td>87</td><td>8.7%</td><td>58.5%</td></tr><tr><td>Australia</td><td>74</td><td>7.4%</td><td>65.9%</td></tr><tr><td>France</td><td>40</td><td>4.0%</td><td>69.9%</td></tr><tr><td>Netherlands</td><td>39</td><td>3.9%</td><td>73.9%</td></tr><tr><td>Canada</td><td>37</td><td>3.7%</td><td>77.6%</td></tr><tr><td>Japan</td><td>30</td><td>3.0%</td><td>80.6%</td></tr><tr><td>Switzerland</td><td>28</td><td>2.8%</td><td>83.4%</td></tr><tr><td>Spain</td><td>22</td><td>2.2%</td><td>85.6%</td></tr><tr><td>Italy</td><td>20</td><td>2.0%</td><td>87.6%</td></tr><tr><td>Norway</td><td>13</td><td>1.3%</td><td>88.9%</td></tr><tr><td>Denmark</td><td>12</td><td>1.2%</td><td>90.2%</td></tr><tr><td>South Korea</td><td>12</td><td>1.2%</td><td>91.4%</td></tr><tr><td>Austria</td><td>10</td><td>1.0%</td><td>92.4%</td></tr></tbody></table>

Source: Reuters (2021), www.reuters.com/investigates/special-report/climate-change-scientists-list.

<figure><img src="image_353.png" /><figcaption>Figure 8.71: Location of the top 1000 climate scientists</figcaption></figure>

Source: Source: Reuters (2021), www.reuters.com/investigates/special-report/climate-change-scientists-list & Author's calculations (created by Datawrapper).
world of climate change science with 292 scientists in the top 1000 (Table 8.21). This is followed by the United Kingdom (112 scientists) and Germany (91 scientists). These 3 countries account for nearly 50% of the scientists on the Reuters Hot List<sup>71</sup>. In fact, the top 1000 scientists on the Reuters Hot List are located in a small number of countries, as shown in Figure 8.71.

The Reuters Hot List already provides some insight into the most important universities involved in the climate change debate. The University of Reading, Utrecht University, Columbia University and ETH Zurich are among the top 10. For a more complete picture, we look at the annual tables compiled by *Nature*. We cannot filter on climate change because it is too restrictive. The closer category is Earth & Environmental Sciences, and the results are shown in Tables 8.22 and 8.23. These rankings reflect the size of the university or the institution. The larger the university, the better the ranking. These rankings are dominated by American and Chinese institutions. Outside these two countries, the top five institutions<sup>72</sup> are Helmholtz Association of German Research Centers (Germany), CNRS (France), ETH Zurich (Switzerland), University of Cambridge (UK) and University of Queensland (Australia).

## Scientific journals

We cannot end this section without mentioning scientific journals, which play an important role in the dissemination of research and in the reputation of researchers. In addition to general scientific journals (Nature, Nature Communications, Science, Science Advances, American Economic Journal, Proceedings of the National Academy of Sciences, etc.), we find more specialized journals. Below is a non-exhaustive list of the most prestigious journals, classified by research topic:

* Atmospheric and Environmental Sciences

Atmospheric Chemistry and Physics; Bulletin of the American Meteorological Society; Climate Dynamics; Climate in the Past; Earth System Dynamics; Earth System Science Data; Earth's Future; Environmental Research Letters; Journal of Advances in Modeling Earth Systems; Journal of Climate; Journal of Geophysical Research: Atmospheres; Geophysical Research Letters; Geoscientific Model Development; Global Change Biology; Journal of the Atmospheric Sciences; npj Climate and Atmospheric Science; Quarterly Journal of the Royal Meteorological Society; Reviews of Geophysics; Tellus.

* Climate change

WIREs Climate Change; Climatic Change; Current Climate Change Reports; Global Environmental Change; Global and Planetary Change; Nature Climate Change; Nature Sustainability.

* Economics

Climate Change Economics; Climate Policy; Ecological Economics; Environmental and Resource Economics; Environmental Modeling & Assessment; Journal of Environmental Economics and Management; Resource and Energy Economics; Review of Environmental Economics and Policy.

<footnote n="71"><sup>71</sup>Note that five scientists were not affiliated with an institution at the time of the Reuters Hot List, so the frequency is calculated as the number of scientists divided by 995, not 1000.</footnote>

<footnote n="72"><sup>72</sup>The other non-Chinese and non-US institutions ranked in the 2023 annual tables are (in descending order): University of Toronto (Canada), University of Oxford (UK), University of Tokyo (Japan), Utrecht University (Netherlands), Imperial College London (ICL), United Kingdom (UK), Max Planck Society (Germany), JAMSTEC (Japan), Leibniz Association (Germany), Spanish National Research Council (Spain), National Institute of Geophysics and Volcanology (Italy), Tohoku University (Japan), Monash University (Australia), Institute of Research for Development (France), University of Oslo (Norway), University of Leeds, (UK), University of Alberta (Canada), Curtin University (Australia), University of British Columbia (Canada), Technical University of Denmark (Denmark), McGill University (Canada), University of Bergen (Norway), and Stockholm University (Sweden).</footnote>
<table><caption>Table 8.22: Top 30 leading institutions in Earth & Environmental Sciences (Nature index, 2016 ranking)</caption><thead><tr><th>Institution</th><th>2016</th><th>2017</th><th>2018</th><th>2019</th><th>2020</th><th>2021</th><th>2022</th><th>2023</th></tr></thead><tbody><tr><td>Chinese Academy of Sciences (CAS), China</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Helmholtz Association of German Research Centres, Germany</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>3</td></tr><tr><td>French National Centre for Scientific Research (CNRS), France</td><td>3</td><td>4</td><td>4</td><td>3</td><td>3</td><td>3</td><td>3</td><td>6</td></tr><tr><td>National Aeronautics and Space Administration (NASA), United States of America (USA)</td><td>4</td><td>3</td><td>3</td><td>5</td><td>4</td><td>6</td><td>6</td><td>9</td></tr><tr><td>Swiss Federal Institute of Technology Zurich (ETH Zurich), Switzerland</td><td>5</td><td>5</td><td>6</td><td>4</td><td>5</td><td>5</td><td>5</td><td>10</td></tr><tr><td>U.S. Geological Survey (USGS), United States of America (USA)</td><td>6</td><td>9</td><td>8</td><td>12</td><td>7</td><td>9</td><td>17</td><td>31</td></tr><tr><td>University of Colorado Boulder (CU-Boulder), United States of America (USA)</td><td>7</td><td>10</td><td>9</td><td>7</td><td>14</td><td>20</td><td>22</td><td>26</td></tr><tr><td>University of California, Berkeley (UC Berkeley), United States of America (USA)</td><td>8</td><td>19</td><td>12</td><td>9</td><td>17</td><td>18</td><td>15</td><td>24</td></tr><tr><td>Stanford University, United States of America (USA)</td><td>9</td><td>8</td><td>13</td><td>22</td><td>9</td><td>15</td><td>14</td><td>20</td></tr><tr><td>California Institute of Technology (Caltech), United States of America (USA)</td><td>10</td><td>6</td><td>5</td><td>8</td><td>12</td><td>7</td><td>8</td><td>15</td></tr><tr><td>National Oceanic and Atmospheric Administration (NOAA), United States of America (USA)</td><td>11</td><td>7</td><td>7</td><td>14</td><td>8</td><td>8</td><td>18</td><td>17</td></tr><tr><td>Columbia University in the City of New York (CU), United States of America (USA)</td><td>12</td><td>12</td><td>16</td><td>11</td><td>11</td><td>13</td><td>23</td><td>22</td></tr><tr><td>University of Washington (UW), United States of America (USA)</td><td>13</td><td>11</td><td>10</td><td>10</td><td>10</td><td>17</td><td>21</td><td>21</td></tr><tr><td>University of California, San Diego (UC San Diego), United States of America (USA)</td><td>14</td><td>14</td><td>18</td><td>26</td><td>32</td><td>24</td><td>10</td><td>29</td></tr><tr><td>The University of Tokyo (UTokyo), Japan</td><td>15</td><td>15</td><td>20</td><td>23</td><td>20</td><td>31</td><td>29</td><td>50</td></tr><tr><td>Peking University (PKU), China</td><td>16</td><td>16</td><td>19</td><td>13</td><td>19</td><td>11</td><td>7</td><td>4</td></tr><tr><td>The University of Texas at Austin (UT Austin), United States of America (USA)</td><td>17</td><td>13</td><td>14</td><td>15</td><td>21</td><td>27</td><td>32</td><td>27</td></tr><tr><td>The University of Queensland (UQ), Australia</td><td>18</td><td>57</td><td>62</td><td>37</td><td>43</td><td>26</td><td>35</td><td>36</td></tr><tr><td>Nanjing University (NJU), China</td><td>19</td><td>18</td><td>11</td><td>6</td><td>6</td><td>4</td><td>4</td><td>2</td></tr><tr><td>Massachusetts Institute of Technology (MIT), United States of America (USA)</td><td>20</td><td>21</td><td>17</td><td>20</td><td>22</td><td>23</td><td>24</td><td>32</td></tr><tr><td>University of Minnesota (UMN), United States of America (USA)</td><td>21</td><td>45</td><td>22</td><td>47</td><td>39</td><td>41</td><td>68</td><td>52</td></tr><tr><td>University of California, Los Angeles (UCLA), United States of America (USA)</td><td>22</td><td>17</td><td>15</td><td>21</td><td>29</td><td>28</td><td>26</td><td>23</td></tr><tr><td>University of Maryland, College Park (UMCP), United States of America (USA)</td><td>23</td><td>32</td><td>55</td><td>40</td><td>53</td><td>66</td><td>77</td><td>80</td></tr><tr><td>Woods Hole Oceanographic Institution (WHOI), United States of America (USA)</td><td>24</td><td>34</td><td>50</td><td>30</td><td>37</td><td>51</td><td>56</td><td>67</td></tr><tr><td>The Pennsylvania State University (Penn State), United States of America (USA)</td><td>25</td><td>27</td><td>29</td><td>25</td><td>25</td><td>30</td><td>28</td><td>56</td></tr><tr><td>University of Michigan (U-M), United States of America (USA)</td><td>26</td><td>20</td><td>23</td><td>35</td><td>27</td><td>35</td><td>50</td><td>73</td></tr><tr><td>University of Toronto (U of T), Canada</td><td>27</td><td>38</td><td>34</td><td>29</td><td>34</td><td>34</td><td>40</td><td>40</td></tr><tr><td>Georgia Institute of Technology (Georgia Tech), United States of America (USA)</td><td>28</td><td>50</td><td>27</td><td>72</td><td>55</td><td>76</td><td>36</td><td></td></tr><tr><td>National Institute of Geophysics and Volcanology (INGV), Italy</td><td>29</td><td>35</td><td>45</td><td>69</td><td>97</td><td>88</td><td>77</td><td></td></tr><tr><td>Princeton University, United States of America (USA)</td><td>30</td><td>67</td><td>42</td><td>44</td><td>46</td><td>47</td><td>52</td><td>43</td></tr></tbody></table>

Source: Nature (2023), www.nature.com/nature-index/annual-tables.
<table><caption>Table 8.23: Top 30 leading institutions in Earth & Environmental Sciences (Nature index, 2023 ranking)</caption><thead><tr><th>Institution</th><th>2016</th><th>2017</th><th>2018</th><th>2019</th><th>2020</th><th>2021</th><th>2022</th><th>2023</th></tr></thead><tbody><tr><td>Chinese Academy of Sciences (CAS), China</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Nanjing University (NJU), China</td><td>19</td><td>18</td><td>11</td><td>6</td><td>6</td><td>4</td><td>4</td><td>2</td></tr><tr><td>Helmholtz Association of German Research Centres, Germany</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>3</td></tr><tr><td>Peking University (PKU), China</td><td>16</td><td>16</td><td>19</td><td>13</td><td>19</td><td>11</td><td>7</td><td>4</td></tr><tr><td>University of Chinese Academy of Sciences (UCAS), China</td><td>77</td><td>24</td><td>18</td><td>13</td><td>10</td><td>9</td><td>5</td><td>5</td></tr><tr><td>French National Centre for Scientific Research (CNRS), France</td><td>3</td><td>4</td><td>4</td><td>3</td><td>3</td><td>3</td><td>3</td><td>6</td></tr><tr><td>Sun Yat-sen University (SYSU), China</td><td></td><td></td><td></td><td>65</td><td>42</td><td>19</td><td>13</td><td>7</td></tr><tr><td>Tongji University, China</td><td>60</td><td>43</td><td>43</td><td>17</td><td>15</td><td>12</td><td>16</td><td>8</td></tr><tr><td>National Aeronautics and Space Administration (NASA), United States of America (USA)</td><td>4</td><td>3</td><td>3</td><td>5</td><td>4</td><td>6</td><td>6</td><td>9</td></tr><tr><td>Swiss Federal Institute of Technology Zurich (ETH Zurich), Switzerland</td><td>5</td><td>5</td><td>6</td><td>4</td><td>5</td><td>5</td><td>5</td><td>10</td></tr><tr><td>Zhejiang University (ZJU), China</td><td>75</td><td>73</td><td>53</td><td>36</td><td>40</td><td>25</td><td>11</td><td>11</td></tr><tr><td>University of Science and Technology of China (USTC), China</td><td>74</td><td>39</td><td>31</td><td>16</td><td>16</td><td>16</td><td>20</td><td>12</td></tr><tr><td>China University of Geosciences (CUG), China</td><td>44</td><td>37</td><td>31</td><td>18</td><td>14</td><td>19</td><td>13</td><td>13</td></tr><tr><td>China Meteorological Administration (CMA), China</td><td>100</td><td>95</td><td>70</td><td>54</td><td>23</td><td>25</td><td>27</td><td>14</td></tr><tr><td>California Institute of Technology (Caltech), United States of America (USA)</td><td>10</td><td>6</td><td>5</td><td>8</td><td>12</td><td>7</td><td>8</td><td>15</td></tr><tr><td>Ministry of Natural Resources (MNR), China</td><td></td><td></td><td></td><td>75</td><td>59</td><td>44</td><td>38</td><td>16</td></tr><tr><td>National Oceanic and Atmospheric Administration (NOAA), United States of America (USA)</td><td>11</td><td>7</td><td>7</td><td>14</td><td>8</td><td>8</td><td>18</td><td>17</td></tr><tr><td>Tsinghua University, China</td><td>53</td><td>36</td><td>30</td><td>28</td><td>26</td><td>21</td><td>12</td><td>18</td></tr><tr><td>Southern University of Science and Technology (SUSTech), China</td><td></td><td></td><td></td><td></td><td></td><td>72</td><td>47</td><td>19</td></tr><tr><td>Stanford University, United States of America (USA)</td><td>9</td><td>8</td><td>13</td><td>22</td><td>9</td><td>15</td><td>14</td><td>20</td></tr><tr><td>University of Washington (UW), United States of America (USA)</td><td>13</td><td>11</td><td>10</td><td>10</td><td>10</td><td>17</td><td>21</td><td>21</td></tr><tr><td>Columbia University in the City of New York (CU), United States of America (USA)</td><td>12</td><td>12</td><td>16</td><td>11</td><td>11</td><td>13</td><td>23</td><td>22</td></tr><tr><td>University of California, Los Angeles (UCLA), United States of America (USA)</td><td>22</td><td>17</td><td>15</td><td>21</td><td>29</td><td>28</td><td>26</td><td>23</td></tr><tr><td>University of California, Berkeley (UC Berkeley), United States of America (USA)</td><td>8</td><td>19</td><td>12</td><td>9</td><td>17</td><td>18</td><td>15</td><td>24</td></tr><tr><td>Wuhan University (WHU), China</td><td></td><td></td><td>95</td><td>55</td><td>58</td><td>67</td><td>42</td><td>25</td></tr><tr><td>University of Colorado Boulder (CU-Boulder), United States of America (USA)</td><td>7</td><td>10</td><td>9</td><td>7</td><td>14</td><td>20</td><td>22</td><td>26</td></tr><tr><td>The University of Texas at Austin (UT Austin), United States of America (USA)</td><td>17</td><td>13</td><td>14</td><td>15</td><td>21</td><td>27</td><td>32</td><td>27</td></tr><tr><td>Harbin Institute of Technology (HIT), China</td><td></td><td></td><td>67</td><td>48</td><td>54</td><td>45</td><td>49</td><td>28</td></tr><tr><td>University of California, San Diego (UC San Diego), United States of America (USA)</td><td>14</td><td>14</td><td>18</td><td>26</td><td>32</td><td>24</td><td>10</td><td>29</td></tr><tr><td>Nanjing University of Information Science and Technology (NUIST), China</td><td></td><td></td><td></td><td>100</td><td>77</td><td>49</td><td>46</td><td>30</td></tr></tbody></table>

Source: Nature (2023), www.nature.com/nature-index/annual-tables.
* Energy
  Energy and Environmental Science; Energy Economics; Energy Journal; Energy Policy; Energy Studies Review; Journal of Cleaner Production.

* Specialized topics
  Arctic, Antarctic, and Alpine Research; Artic Ice Journal; Cryosphere; Frontiers in Earth Science: Cryospheric Science; Polar Science; Quaternary Science Reviews; Water Research.

**Remark 81** In this section, we do not examine some organizations that can have a major impact on the scientific debate on climate change and transition. Among them, the most important and influential is certainly the International Energy Agency (IEA), an autonomous intergovernmental organization based in Paris, founded in 1974, that provides policy advice, analysis and data on the entire global energy sector (www.iea.org). Its famous reports (World Energy Outlook, Net Zero by 2050) have a considerable impact and a large audience in the general media. Another example is the Energy Transitions Commission (ETC), an international think tank focused on economic growth and climate change mitigation (www.energy-transitions.org). More regional organizations can also have a local impact. For example, the Institute for Climate Economics (I4CE) is a registered non-profit organization founded by the French National Bank, Caisse des Dépôts, and the French Development Agency (AFD). It provides policy analysis on climate change mitigation and adaptation (www.i4ce.org). All of these organizations participate in the public debate and provide accessible research, surveys and databases that are of great interest in understanding current climate risk issues.

## 8.2.2 Conferences of the Parties

### Earth Summit

In June 1972, the United Nations held its first conference on the global environment (United Nations Conference on the Human Environment) in Stockholm and established the United Nations Environment Programme (UNEP) to coordinate responses to environmental issues within the United Nations system. Over the next 20 years, UNEP organized several conferences and participated in several initiatives, including the establishment of the Intergovernmental Panel on Climate Change (IPCC). In June 1992, a major achievement of UNEP was the organization of a conference in Rio de Janeiro, Brazil, to address the challenges of environment and development. Known as the United Nations Conference on Environment and Development (UNCED) or the Earth Summit, it was one of the largest UN conferences ever held, with the participation of 175 governments and more than 1000 non-governmental organizations. It is considered a great success with several outcomes:

* The Rio Declaration on Environment and Development was signed by more than 170 countries and consists of 27 principles to guide countries in future sustainable development. The first principle declares that “human beings are at the centre of concerns for sustainable development and are entitled to a healthy and productive life in harmony with nature”, while the eighth principle states that in order “to achieve sustainable development and a higher quality of life for all people, States should reduce and eliminate unsustainable patterns of production and consumption and promote appropriate demographic policies” (United Nations, 1992a).

* The Forest Principles (also known as the Rio Forest Principles) is a non-legally binding document that makes several recommendations for the conservation and sustainable development of forests. The second principle declares that “States have the sovereign and inalienable right to utilize, manage and develop their forests in accordance with their development needs and level of socio-economic development [...] including the conversion of such areas for other uses
within the overall socio-economic development plan and based on rational land-use policies. Forest resources and forest lands should be sustainably managed to meet the social, economic, ecological, cultural and spiritual needs of present and future generations" (United Nations, 1992b).

* Agenda 21, the third document drafted since 1989 and adopted at the 1992 Earth Summit, is a non-binding, comprehensive plan of action for sustainable development. Its original goal was to achieve numerous sustainable development goals by the year 2000 or earlier<fnref n="73" />. Agenda 21 is divided into four main sections:

1. Social and economic dimensions

This section focuses on promoting sustainable development through poverty eradication, sustainable consumption and production patterns, sustainable demography, promoting human health conditions, and integrating environment and development in decision-making.

2. Conservation and management of resources for development

This section focuses on protecting the atmosphere, managing land resources, combating deforestation, managing fragile ecosystems, conserving biodiversity, protecting oceans, managing freshwater resources, and controlling toxic chemicals and wastes (hazardous, solid, and radioactive).

3. Strengthening the role of major groups

This section emphasizes the importance of involving all sectors of society in sustainable development, including women, children, indigenous peoples, non-governmental organizations, local authorities, workers and their trade unions, business and industry, the scientific and technological community and farmers.

4. Means of implementation

This section outlines the financial resources, technology transfer and capacity building needed to implement Agenda 21.

In fact, Agenda 21 anticipated the Sustainable Development Goals (SDGs) and laid the ground-work for addressing environmental and sustainable development. In a sense, the SDGs are a reaffirmation of Agenda 21, reiterating the commitment to sustainable development made in Agenda 21.

* The so-called Rio Conventions

- The United Nations Framework Convention on Climate Change (UNFCCC) was adopted in 1992 and entered into force in 1994. The UNFCCC is the first legally binding international agreement on climate change. The main objective of the UNFCCC is to stabilize greenhouse gas concentrations at a level that would prevent dangerous anthropogenic interference with the climate system. It recognizes the importance of sustainable development and acknowledges the different responsibilities and capabilities of nations in addressing climate change.

- The Convention on Biological Diversity (CBD) was adopted in 1992 and entered into force in 1993. The main objective of the CBD is to promote the conservation of biological diversity, the sustainable use of its components and the fair and equitable sharing of the benefits derived from genetic resources. It recognizes the intrinsic value of biodiversity and its critical role in maintaining ecosystem stability and supporting human well-being.

<footnote n="73">⁷³The number 21 in Agenda 21 refers to the 21st century.</footnote>
- The United Nations Convention to Combat Desertification (UNCCD) was adopted in 1994 and entered into force in 1996. The main objective of the UNCCD is to combat desertification and land degradation in arid, semi-arid and dry sub-humid areas, particularly in Africa. It recognizes the importance of promoting sustainable land management practices, restoring degraded ecosystems, and encouraging the active participation of local communities in combating desertification in order to achieve land degradation neutrality.

* The launch of the UNEP Finance Initiative (UNEP FI), which works with banks, insurers and investors to create a sustainable finance sector.

## United Nations Framework Convention on Climate Change (UNFCCC)

The UNFCCC is the international treaty (or a convention<fnref n="74" />) adopted at the Earth Summit. It consists of 26 articles and two annexes. The objective of the UNFCCC is defined in Article 2:

> "The ultimate objective of this Convention and any related legal instruments that the Conference of the Parties may adopt is to achieve, in accordance with the relevant provisions of the Convention, stabilization of greenhouse gas concentrations in the atmosphere at a level that would prevent dangerous anthropogenic interference with the climate system. Such a level should be achieved within a time-frame sufficient to allow ecosystems to adapt naturally to climate change, to ensure that food production is not threatened and to enable economic development to proceed in a sustainable manner." (United Nations, 1992d, page 9).

The commitments are set out in Article 4. The principle of common but differentiated responsibilities recognizes that all countries have a common responsibility to address climate change, but acknowledges that developed countries should take the lead because of their historical contributions to greenhouse gas emissions and their greater financial and technological capabilities. The Convention therefore distinguishes between three types of Parties: Annex I Parties, Annex II Parties, and Non-Annex I Parties. All Parties (including Non-Annex I Parties) must publish national inventories of anthropogenic emissions and implement climate change mitigation programs (Article 4.1). Annex I Parties are subject to more stringent obligations than Non-Annex I Parties. In particular, Annex I Parties are required to adopt national policies and take appropriate measures to mitigate climate change by limiting their anthropogenic emissions of greenhouse gases and to report on the steps taken to return to their 1990 emission levels (Article 4.2). Annex I Parties are industrialized countries that were members of the OECD in 1992 and countries with economies in transition<fnref n="75" />. Annex II Parties are a subset of Annex I Parties and correspond to developed countries. In addition to the commitments in Articles 4.1 and 4.2, they have specific financial responsibilities as described in Article 4.3. They must provide additional financial resources, including for technology transfer, to meet the costs of developing countries and some specific vulnerable countries. The full list of this last category is given in Article 4.3.8, including small island countries, countries with low-lying coastal areas, etc. These countries are either located in areas that are particularly vulnerable to the effects of climate change or have economies that are heavily dependent on climate-sensitive sectors.

<footnote n="74">⁷⁴ A United Nations convention is a formal, legally binding international agreement that is negotiated and approved by representatives of the participating countries.</footnote>

<footnote n="75">⁷⁵ In 1992, Annex I Parties were Australia, Austria, Belarus*, Belgium, Bulgaria*, Canada, Czechoslovakia*, Denmark, European Economic Community, Estonia*, Finland, France, Germany, Greece, Hungary*, Iceland, Ireland, Italy, Japan, Latvia*, Lithuania*, Luxembourg, Netherlands, New Zealand, Norway, Poland*, Portugal, Romania*, Russian Federation*, Spain, Sweden, Switzerland, Turkey, Ukraine*, United Kingdom of Great Britain and Northern Ireland, United States of America. The symbol * indicated countries in transition.</footnote>
<table><caption>Table 8.24: Chronological list of the meetings of the Conference of the Parties</caption><thead><tr><th>Year</th><th>COP</th><th>CMP</th><th>CMA</th><th>City</th><th>Country</th><th>Treaty</th></tr></thead><tbody><tr><td>1992</td><td></td><td></td><td></td><td>Rio de Janeiro</td><td>Brazil</td><td>Convention</td></tr><tr><td>1995</td><td>1</td><td></td><td></td><td>Berlin</td><td>Germany</td><td></td></tr><tr><td>1996</td><td>2</td><td></td><td></td><td>Geneva</td><td>Switzerland</td><td></td></tr><tr><td>1997</td><td>3</td><td></td><td></td><td>Kyoto</td><td>Japan</td><td>Kyoto Protocol</td></tr><tr><td>1998</td><td>4</td><td></td><td></td><td>Buenos Aires</td><td>Argentina</td><td></td></tr><tr><td>1999</td><td>5</td><td></td><td></td><td>Bonn</td><td>Germany</td><td></td></tr><tr><td>2000</td><td>6-1</td><td></td><td></td><td>The Hague</td><td>Netherlands</td><td></td></tr><tr><td>2001</td><td>6-2</td><td></td><td></td><td>Bonn</td><td>Germany</td><td></td></tr><tr><td>2001</td><td>7</td><td></td><td></td><td>Marrakech</td><td>Morocco</td><td></td></tr><tr><td>2002</td><td>8</td><td></td><td></td><td>New Delhi</td><td>India</td><td></td></tr><tr><td>2003</td><td>9</td><td></td><td></td><td>Milan</td><td>Italy</td><td></td></tr><tr><td>2004</td><td>10</td><td></td><td></td><td>Buenos Aires</td><td>Argentina</td><td></td></tr><tr><td>2005</td><td>11</td><td>1</td><td></td><td>Montreal</td><td>Canada</td><td></td></tr><tr><td>2006</td><td>12</td><td>2</td><td></td><td>Nairobi</td><td>Kenya</td><td></td></tr><tr><td>2007</td><td>13</td><td>3</td><td></td><td>Bali</td><td>Indonesia</td><td></td></tr><tr><td>2008</td><td>14</td><td>4</td><td></td><td>Poznań</td><td>Poland</td><td></td></tr><tr><td>2009</td><td>15</td><td>5</td><td></td><td>Copenhagen</td><td>Denmark</td><td></td></tr><tr><td>2010</td><td>16</td><td>6</td><td></td><td>Cancún</td><td>Mexico</td><td></td></tr><tr><td>2011</td><td>17</td><td>7</td><td></td><td>Durban</td><td>South Africa</td><td></td></tr><tr><td>2012</td><td>18</td><td>8</td><td></td><td>Doha</td><td>Qatar</td><td></td></tr><tr><td>2013</td><td>19</td><td>9</td><td></td><td>Warsaw</td><td>Poland</td><td></td></tr><tr><td>2014</td><td>20</td><td>10</td><td></td><td>Lima</td><td>Peru</td><td></td></tr><tr><td>2015</td><td>21</td><td>11</td><td></td><td>Paris</td><td>France</td><td>Paris Agreement</td></tr><tr><td>2016</td><td>22</td><td>12</td><td>1-1</td><td>Marrakech</td><td>Morocco</td><td></td></tr><tr><td>2017</td><td>23</td><td>13</td><td>1-2</td><td>Bonn</td><td>Germany</td><td></td></tr><tr><td>2018</td><td>24</td><td>14</td><td>1-3</td><td>Katowice</td><td>Poland</td><td></td></tr><tr><td>2019</td><td>25</td><td>15</td><td>2</td><td>Madrid</td><td>Spain</td><td></td></tr><tr><td>2021</td><td>26</td><td>16</td><td>3</td><td>Glasgow</td><td>United Kingdom</td><td></td></tr><tr><td>2022</td><td>27</td><td>17</td><td>4</td><td>Sharm El Sheikh</td><td>Egypt</td><td></td></tr><tr><td>2023</td><td>28</td><td>18</td><td>5</td><td>Dubai</td><td>United Arab Emirates</td><td></td></tr><tr><td>2024</td><td>29</td><td>19</td><td>6</td><td>Baku</td><td>Azerbaijan</td><td></td></tr></tbody></table>

To assist in the implementation of the UNFCCC and to give it life, a Conference of the Parties is established (Article 7):

"The Conference of the Parties, as the supreme body of this Convention, shall keep under regular review the implementation of the Convention and any related legal instruments that the Conference of the Parties may adopt, and shall make, within its mandate, the decisions necessary to promote the effective implementation of the Convention." (United Nations, 1992d, Article 7.2, page 17).

The list of Conferences of the Parties is given in Table 8.24. These annual conferences take place every year, except for 2020 due to the COVID-19 pandemic, and focus primarily on negotiations and debates to assess progress toward the goal of limiting climate change. Occasionally, COPs result in new agreements and treaties, often aimed at refining targets, establishing guidelines, or creating legally binding treaties such as the Kyoto Protocol or the Paris Agreement. A critical aspect of these
meetings is the assessment of each party's contributions to climate change mitigation and adaptation efforts. COPs are attended by representatives of governments and observer organizations, such as non-governmental organizations (NGOs), charities, and advocacy groups. The first COP was held in Berlin in April 1995, and the last in Dubai in December 2023. According to Oberthür and Ott (1995), 4000 delegates, including more than 2000 journalists, 200 observer organizations, 117 UNFCCC member countries and 50 observer states, followed the proceedings in Berlin, while there were more than 100 000 delegates in Dubai<sup>76</sup> (97 372 on-site and 3 074 virtual-only attendees).

## Kyoto Protocol

At the first COP in Berlin in April 1995, Parties recognised that the voluntary approach of the UNFCCC was a shortcoming and that there was a need for a more formal, comprehensive and binding approach to implementing the Convention. The Berlin Mandate sets the stage for new negotiations to establish quantitative targets and specific deadlines. In addition, the IPCC published the Synthesis Report of the Second Assessment Report (SAR) in October 1995 and the Revised 1996 IPCC Guidelines for National Greenhouse Gas Inventories. The negotiations started at COP1 continued at COP2 in Geneva, Switzerland, in 1996, but it was the intensive negotiations at COP3 in Kyoto, Japan, that led to the adoption of the Kyoto Protocol on 11 December 1997 (Oberthür and Ott, 1999). After a complex ratification process, it entered into force on 16 February 2005. As of 1 January 2024, there are currently 192 Parties to the Kyoto Protocol. The Protocol's first commitment period started in 2008 and ended in 2012. At COP18 in Doha, Qatar, a second commitment extended the first one from 2013 to 2020. In fact, there is no third commitment period for the Kyoto Protocol. In order to follow up the specific commitments of the Kyoto Protocol, a new Conference of the Parties was created: the Conference of the Parties serving as the Meeting of the Parties to the Kyoto Protocol or CMP. In fact, the CMP takes place during the same period as the COP and can be seen as parallel sessions of the COP dedicated to the Kyoto Protocol. The same mechanism will be created for the Paris Agreement in 1995. The CMA, or the Conference of the Parties serving as the Meeting of the Parties to the Paris Agreement, oversees the implementation of the Paris Agreement and corresponds to other parallel sessions of the COP. Table 8.24 summarises the chronology of events and the history of the COP, CMP and CMA.

Unlike the original UNFCCC international treaty, the Kyoto Protocol introduces quantified commitments to limit or reduce greenhouse gas (GHG) emissions from 1990 levels<sup>77</sup> for the five-year period 2008–2012. These targets are listed in Table 8.25. The Kyoto Protocol also defines the list of greenhouse gases to be measured (Table 8.26). New commitments and a new scope of greenhouse gases will be adopted at COP18 in 2012 (Tables 8.26 and 8.27). To meet their targets, countries must first implement national measures as part of a coherent climate policy. However, the Kyoto Protocol also provides them with an additional tool in the form of three market-based mechanisms:

* International Emissions Trading (EIT)

* Clean Development Mechanism (CDM)

* Joint Implementation (JI)

The EIT is a form of carbon emissions trading that allows countries to reduce their emissions by buying or selling emission allowances. Each Annex B Party<sup>78</sup> is allocated a number of Assigned Amount

<footnote n="76"><sup>76</sup>The list is available at https://unfccc.int/documents/634503.</footnote>

<footnote n="77"><sup>77</sup>Some countries have a base year earlier than 1990.</footnote>

<footnote n="78"><sup>78</sup>Annex B Parties to the Kyoto Protocol are Annex I Parties to the UNFCCC minus Belarus and Turkey, plus Liechtenstein and Monaco.</footnote>
<table><caption>Table 8.25: Quantified first commitment under the Kyoto Protocol to limit or reduce emissions (% of base year)</caption><thead><tr><th>R</th><th>Country</th></tr></thead><tbody><tr><td>+10%</td><td>Iceland</td></tr><tr><td>+8%</td><td>Australia</td></tr><tr><td>+1%</td><td>Norway</td></tr><tr><td>0%</td><td>New Zealand, Russian Federation, Ukraine</td></tr><tr><td>-5%</td><td>Croatia</td></tr><tr><td>-6%</td><td>Canada, Hungary, Japan, Poland</td></tr><tr><td>-7%</td><td>United States of America</td></tr><tr><td>-8%</td><td>Austria, Belgium, Bulgaria, Czech Republic, Denmark, Estonia, European Community, Finland, France, Germany, Greece, Ireland, Italy, Latvia, Liechtenstein, Lithuania, Luxembourg, Monaco, Netherlands, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland, United Kingdom of Great Britain and Northern Ireland</td></tr></tbody></table>

Source: UNFCCC (1997, Annex B, page 24).

<table><caption>Table 8.26: List of greenhouse gases under the Kyoto Protocol and the Doha Amendment</caption><thead><tr><th>Greenhouse gas</th><th>Symbol</th><th>Kyoto Protocol</th><th>Doha Amendment</th></tr></thead><tbody><tr><td>Carbon dioxide</td><td>CO<sub>2</sub></td><td>✓</td><td>✓</td></tr><tr><td>Methane</td><td>CH<sub>4</sub></td><td>✓</td><td>✓</td></tr><tr><td>Nitrous oxide</td><td>N<sub>2</sub>O</td><td>✓</td><td>✓</td></tr><tr><td>Hydrofluorocarbons</td><td>HFCs</td><td>✓</td><td>✓</td></tr><tr><td>Perfluorocarbons</td><td>PFCs</td><td>✓</td><td>✓</td></tr><tr><td>Sulphur hexafluoride</td><td>SF<sub>6</sub></td><td>✓</td><td>✓</td></tr><tr><td>Nitrogen trifluoride</td><td>NF<sub>3</sub></td><td></td><td>✓</td></tr></tbody></table>

Source: UNFCCC (1997, Annex A, page 22) & UNFCCC (2012, Article 1.B, page 4).

<table><caption>Table 8.27: Quantified second commitment under the Kyoto Protocol (Doha Amendment) to limit or reduce emissions (% of base year)</caption><thead><tr><th>R</th><th>Country</th></tr></thead><tbody><tr><td>-0.5%</td><td>Australia</td></tr><tr><td>-5%</td><td>Kazakhstan</td></tr><tr><td>-12%</td><td>Belarus</td></tr><tr><td>-16%</td><td>Liechtenstein, Norway, Switzerland</td></tr><tr><td>-20%</td><td>Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia, European Union, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, United Kingdom of Great Britain and Northern Ireland</td></tr><tr><td>-22%</td><td>Monaco</td></tr><tr><td>-24%</td><td>Ukraine</td></tr></tbody></table>

Source: UNFCCC (2012, Article 1.A, pages 1-2).
Units (AAUs), which represent its allowed emissions for the Kyoto Protocol's commitment period. Parties that exceed their emission targets can sell their surplus AAUs to Parties that have difficulty meeting their targets. A central registry tracks the transfer of AAUs between parties. The European Union Emissions Trading Scheme (EU ETS) is the largest EIT program. The Clean Development Mechanism (CDM) allows Annex B Parties to implement emission reduction projects in developing countries. If certified, such projects can generate marketable Certified Emission Reduction (CER) credits that can be used to calculate Kyoto targets. The third market-based mechanism, known as Joint Implementation, allows Annex B Parties to obtain Emission Reduction Units (ERUs) from an emission reduction project in another Annex B Party, which can be used to calculate Kyoto targets.

The Kyoto Protocol has been extensively studied by economists (Böhringer, 2003). For example, in June 1999, the Energy Journal published a special and complete issue on the Kyoto Protocol (Weyant and Hill, 1999). Some famous economists such as William Nordhaus and Richard Tol participated in the issue (Nordhaus and Boyer, 1999; Tol, 1999). An overview of the economic literature up to the year 2002 can be found in Böhringer (2003). The economic questions concern the efficiency of quotas versus taxes, the rationality of a partially cooperative solution, the credibility of sanctions, the uncertainty of abatement costs, the cost-benefit analysis, and the negative externalities of the Kyoto Protocol. Most of the academic conclusions are negative about the Kyoto Protocol. For example, Nordhaus and Boyer (1999) wrote that “[...] the benefit-cost ratio of the Kyoto Protocol is 1/7. Additionally, the emissions strategy is highly cost-ineffective, with the global temperature reduction achieved at a cost almost 8 times the cost of a strategy which is cost-effective in terms of where and when efficiency.” Tol (1999) concluded that “[...] the agreements of the Kyoto Protocol are not readily reconciled with economic rationality.” In the introduction and overview to the special issue of the Energy Journal, John Weyant and Jennifer Hill summarized the findings of the 13 articles and 390 pages as follows:

> "Despite these considerable uncertainties, a number of common results and insights emerge from the set of model results considered here. First, meeting the requirements of the Kyoto Protocol will not stop economic growth anywhere in the world, but it will not be free either. In most Annex I countries, significant adjustments will need to be undertaken and costs will need to be paid. Second, unless care is taken to prevent it, the sellers of international emissions rights (dominantly the Russian Federation in the case of Annex I trading, and China and India in the case of global trading) may be able to exercise market power raising the cost of the Protocol to the other Annex I countries. Third, meaningful global trading probably requires that the non-Annex I countries take on emissions targets; without them accounting and monitoring (even Annex I monitoring and enforcement may be quite difficult) becomes almost impossible. Finally, it appears that the emissions trajectory prescribed in the Kyoto Protocol is neither optimal in balancing the costs and benefits of climate change mitigation, nor cost effective in leading to stabilization of the concentration of carbon dioxide at any level above about 500 ppmv." (Weyant and Hill, 1999, page xliv)

As economists have pointed out, one of the issues is the monitoring of the Kyoto Protocol. Another issue is the actual impact of the Kyoto Protocol. The Kyoto Protocol required each country to ratify the agreement. The EU and its member states ratified the Protocol in 2002, Russia ratified the Protocol in 2004, but the US never ratified the Protocol. Canada withdrew from the Kyoto Protocol in 2011. In addition, Canada, Japan, New Zealand, Russia, and the United States did not participate in the second commitment of the Kyoto Protocol. Therefore, we have to question the success of the Kyoto Protocol because it has mainly affected the European Union and Australia. On
the European Union's website<sup>79</sup>, we can read that total EU emissions, excluding Cyprus and Malta, which have no target, were 23.5 GtCO<sub>2</sub>e for the period 2008–2012. This represents a reduction of about 19% below base year levels, excluding additional reductions from carbon sinks (LULUCF) and international credits. For the second commitment (2013–2020), we can read in January 2024 that the EU countries are on track...

## Paris Climate Agreement

The Paris Climate Agreement is another international UNFCCC treaty on climate change, adopted at COP21 in Paris in December 2015 and entered into force in November 2016. It can be seen as a successor to the Kyoto Protocol, particularly its second commitment to go one step further. Indeed, as we have already seen, while the Kyoto Protocol is generally seen as a major political achievement on climate change, it has had little impact because it only applies to Annex I Parties and most non-European major emitting countries, with the exception of Australia, did not participate in the Doha Amendment. However, there are a number of important differences between the Kyoto Protocol and the Paris Agreement:

* The ultimate goal of the Paris Agreement is more precise than the Kyoto Protocol. In fact, it aims to “strengthen the global response to the threat of climate change, in the context of sustainable development and efforts to eradicate poverty, including by: (a) Holding the increase in the global average temperature to well below 2°C above pre-industrial levels and pursuing efforts to limit the temperature increase to 1.5°C above pre-industrial levels, recognizing that this would significantly reduce the risks and impacts of climate change; (b) Increasing the ability to adapt to the adverse impacts of climate change and foster climate resilience and low greenhouse gas emissions development, in a manner that does not threaten food production; and (c) Making finance flows consistent with a pathway towards low greenhouse gas emissions and climate-resilient development.” (UNFCCC, 2015, Article 2.1, page 2).

* In the Kyoto Protocol, targets are set top-down and are mandatory. The Paris Agreement takes a bottom-up approach, with each Party deciding on its own Nationally Determined Contribution (NDC), meaning that a country can change its contribution to the global objective. Furthermore, the definition of a NDC is left to the discretion of each country, but countries are expected to make them increasingly ambitious over time.

* The Paris Agreement is global and applies to all Parties to the UNFCCC, while the Kyoto Protocol only covers industrialized countries (or Annex I Parties). As of January 2024, 195 of the 198 members of the UNFCCC are Parties to the Agreement. Three UNFCCC members have not ratified the agreement: Iran, Libia and Yemen.

* The timeframe also varies, with countries submitting their NDCs every few years and generally every five years. For example, some countries have already submitted their updates twice.

In summary, the Kyoto Protocol was a more prescriptive and legally binding treaty than the Paris Agreement, which is more flexible, global, and based on voluntary action by countries.

As noted in Schleussner et al. (2016), the Paris Agreement is the first international treaty to set a global long-term temperature goal. Although the Paris Agreement has its roots in the COP15 in Copenhagen, Denmark, which in turn is influenced by the IPCC AR4, it is clear that the Paris Agreement is strongly influenced by the IPCC AR5, whose synthesis report is completed in November

<footnote n="79">⁷⁹The website is https://climate.ec.europa.eu/eu-action/international-action-climate-change.</footnote>
2014 (IPCC, 2014b). But the full rationale of the Paris Agreement will come later, with the famous IPCC publication of Global Warming of 1.5°C in 2018:

> [...] Five years ago, the IPCC's Fifth Assessment Report provided the scientific input into the Paris Agreement, which aims to strengthen the global response to the threat of climate change by holding the increase in the global average temperature to well below 2°C above pre-industrial levels and to pursue efforts to limit the temperature increase to 1.5°C above pre-industrial levels. [...] This Report responds to the invitation for IPCC to provide a Special Report in 2018 on the impacts of global warming of 1.5°C above pre-industrial levels and related global greenhouse gas emission pathways' contained in the Decision of the 21st Conference of Parties of the United Nations Framework Convention on Climate Change to adopt the Paris Agreement. [...] The IPCC accepted the invitation in April 2016, deciding to prepare this Special Report on the impacts of global warming of 1.5°C above pre-industrial levels and related global greenhouse gas emission pathways, in the context of strengthening the global response to the threat of climate change, sustainable development, and efforts to eradicate poverty. (IPCC, 2018, pages v and 4).

SR15 is the IPCC publication that had the greatest impact on politicians and media. It strengthened the legitimacy of the Paris Agreement, because after this publication, politicians could not say they did not know. But it also illustrates the ambiguous relationship between science and politics, because SR15 can be seen as serving the Paris Agreement (Livingston and Rummukainen, 2020). As noted by Jewell and Cherp (2020) and Cointe and Guillemot (2023), the initial skepticism of the 1.5°C target has given way to ownership and acceptance<fnref n="80" />, but the political and scientific feasibility of the 1.5°C target remains an issue. In fact, it's rare to find politicians or scientists who actually believe that the 1.5 degree target is achievable. The front page of *The Economist* at COP27 is not so ironic. Still, having an ambitious but surely unattainable goal ultimately helps us move forward.

<figure><img src="image_354.png" /><figcaption>Figure 8.72: The Economist, Say goodbye to 1.5°C, November 5, 2022</figcaption></figure>

<footnote n="80">⁸⁰For example, Cointe and Guillemot (2023) interviewed one of the SR15 authors who said “honestly, as a scientist, I thought it was a bad idea to produce a report on 1.5°C, but in fact, after having done so, I think it was a good idea. So, even though it was very much driven by political motivation, I think that there was still an intuition behind it that was quite powerful.”</footnote>
To understand what a Nationally Determined Contribution is, we can consult the NDC Registry at https://unfccc.int/NDCREG. The first thing to note is that each NDC has its own format and may contain different information. For example, if we look at the first Party, Afghanistan, the NDC is an 8-page document with the following executive summary:

<table><tr><td>Base year</td><td>2005</td></tr><tr><td>Target years</td><td>2020 to 2030</td></tr><tr><td>Contribution type</td><td>Conditional</td></tr><tr><td>Sectors</td><td>Energy, natural resource management, agriculture, waste management and mining</td></tr><tr><td>Gases covered</td><td>Carbon dioxide (CO<sub>2</sub>), methane (CH<sub>4</sub>), and nitrous oxide (N<sub>2</sub>O)</td></tr><tr><td>Target</td><td>There will be a 13.6% reduction in GHG emissions by 2030 compared to a business-as-usual (BAU) 2030 scenario, conditional on external support</td></tr><tr><td>Financial needs</td><td>Total: $17.405 bn<br>Adaptation: $10.785 bn + Mitigation: $6.62 bn (2020-2030)</td></tr></table>

If we look at the second Party, Albania, the NDC is documented in a 145-page report with no executive summary. The last Party is Zimbabwe, and the updated submission is a 56-page report. It contains a table summarizing the changes from the intended (or original) NDC. We report the excerpt on the emissions reduction target below:

## 2015 INDC

- Business-as-usual (BAU) scenario

> "The INDC BAU baseline focused solely on per capita energy emissions. Zimbabwean per capita energy emissions were projected to be 1.06 tCO₂e in 2020, 2.57 tCO₂e in 2025 and 3.31 tCO₂e in 2030 under business-as-usual."

- Emission reduction target

> "The INDC emission reduction target was a 33% reduction in energy-related emissions per capita compared to BAU by 2030, conditional on international support. In the mitigation scenario, energy-related emissions per capita were projected to be 2.21 tCO₂e in 2030."

## 2021 Revised NDC

- Business-as-usual (BAU) scenario

> "Updated to include all IPCC sectors. National total emissions in the base data period ranged between 25.24 MtCO₂e in 2011 and 41.66 MtCO₂e in 2015. Emissions in 2017 were 35.84 MtCO₂e. National total emissions per capita in the base data period ranged between 2.03 tCO₂e in 2011 and 2.98 tCO₂e in 2015. Emissions per capita in 2017 were 2.45 tCO₂e. The NDC revision process incorporated impacts of COVID-19 on emissions trends and macroeconomic parameters, including GDP, which fed into the updated baseline."

- Emission reduction target

> "The updated target is a 40% reduction in economy-wide GHG emissions per capita compared to BAU by 2030, conditional on international support. In the mitigation scenario, economy-wide emissions per capita are projected to be 2.3 tCO₂e in 2030."

There is no official NDC database that can be used to compare different targets across countries. The most comprehensive unofficial database is the IGES NDC Database (IGES, 2022). Unfortunately, the last updated version was October 2022. The alternative is to use the databases of
Climate Watch (www.climatewatchdata.org/ndcs-explore) or the German Development Insti-
tute (https://klimalog.idos-research.de/ndc). The different NDCs can be classified into the
following categories: mitigation, scope (GHG & sectors), implementation period, conditionality, fi-
nancial needs, technology needs, market mechanisms, adaptation, climate policy, transparency and
information.

In Table 8.28 we report the NDC reduction rate for the 45 largest GHG emitters in the world. For each NDC, we give the value of the reduction rate, the base year, the target year and the type of reduction. For example, China’s climate commitment is to reduce carbon dioxide emissions per GDP by 65% from 2005 levels by 2030, as this is a carbon intensity reduction or CIR type. In April 2021, when the United States rejoined the Paris Agreement after withdrawing from it under the Trump administration, the Biden administration commits to reducing greenhouse gas emissions by 50% from 2005 levels by 2030, as this is an absolute emissions reduction or AER type. Indonesia submits an updated NDC in 2021, committing to reduce its GHG emissions by 31.89% unconditionally and 43.20% conditionally by 2030 compared to a business as usual (BAU) pathway. This means that Indonesia will implement climate policies that would reduce GHG emissions by 32% in 2030 compared to a situation where it does not implement these climate policies. It is therefore a relative emissions reduction or RER type. Several observations can be made. First, as mentioned above, climate pledges are not homogeneous. Most developed countries have an absolute emissions reduction or AER commitment, while developing countries express their commitments in terms of intensity reduction (per GDP) or in relation to a business-as-usual scenario (RER type). In the case of the European Union countries, their climate commitment is aligned with the Fit for 55 package. Therefore, all EU countries commit to reducing their GHG emissions by 55% in 2030 compared to 1990 levels (AER type). The second observation concerns the conditionality of the climate pledges for developing countries. In Table 8.28 we only report unconditional commitments, as conditional commitments depend on financial support from developed countries. The third observation is the definition of the RER type when it does not assume a business-as-usual scenario. For example, Saudi Arabia aims to avoid 278 MtCO₂e of GHG emissions per year by 2030, with 2019 as the base year for this NDC, while South Africa’s annual GHG emissions would be in the range of 350-420 MtCO₂e in 2030. We have transformed these NDCs using a proxy reduction rate relative to a base year of 2020.

Will these NDCs be enough to meet the Paris Agreement target? Academics agree that the
answer is no. In fact, academics have identified two main reasons:

1. The first reason is that the NDCs are not ambitious enough, which means that even if the climate pledges are met, the temperature anomaly will not stay below 2°C, much less if the target is 1.5°C;

2. The second reason is that some NDCs are not credible compared to the recent carbon emission trend of some countries and the lack of means to implement these NDCs.

For example, here is the abstract of Liu and Raftery (2021):

"[...] we find that the probabilities of meeting their nationally determined contribu-
tions for the largest emitters are low, e.g., 2% for the USA and 16% for China. On current
trends, the probability of staying below 2°C of warming is only 5%, but if all countries
meet their nationally determined contributions and continue to reduce emissions at the
same rate after 2030, it rises to 26%. If the USA alone does not meet its nationally
determined contribution, it declines to 18%. To have an even chance of staying below
2°C, the average rate of decline in emissions would need to increase from the 1% per
<table><caption>Table 8.28: 2022 GHG emissions and NDCs of top emitting countries</caption><thead><tr><th>Rank</th><th>Country</th><th>(1)</th><th>(2)</th><th>(3)</th><th>(4)</th><th>(5)</th><th>(6)</th><th>(7)</th><th>(8)</th><th>(9)</th><th>(10)</th></tr></thead><tbody><tr><td>1</td><td>China</td><td>15 685</td><td>29.16</td><td>29.16</td><td>10.95</td><td>0.61</td><td>35.62</td><td>65</td><td>2005</td><td>2030</td><td>CIR</td></tr><tr><td>2</td><td>United States</td><td>6 017</td><td>11.19</td><td>40.35</td><td>17.90</td><td>0.28</td><td>-10.01</td><td>50</td><td>2005</td><td>2030</td><td>AER</td></tr><tr><td>3</td><td>India</td><td>3 943</td><td>7.33</td><td>47.68</td><td>2.79</td><td>0.39</td><td>38.85</td><td>45</td><td>2005</td><td>2030</td><td>CIR</td></tr><tr><td>4</td><td>Russia</td><td>2 580</td><td>4.80</td><td>52.48</td><td>17.99</td><td>0.64</td><td>14.90</td><td>30</td><td>1990</td><td>2030</td><td>AER</td></tr><tr><td>5</td><td>Brazil</td><td>1 310</td><td>2.44</td><td>54.91</td><td>6.05</td><td>0.40</td><td>11.27</td><td>50</td><td>2005</td><td>2030</td><td>AER</td></tr><tr><td>6</td><td>Indonesia</td><td>1 241</td><td>2.31</td><td>57.22</td><td>4.47</td><td>0.36</td><td>50.00</td><td>32</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>7</td><td>Japan</td><td>1 183</td><td>2.20</td><td>59.42</td><td>9.41</td><td>0.23</td><td>-10.85</td><td>46</td><td>2013</td><td>2030</td><td>AER</td></tr><tr><td>8</td><td>Iran</td><td>952</td><td>1.77</td><td>61.19</td><td>11.20</td><td>0.70</td><td>17.94</td><td></td><td></td><td></td><td></td></tr><tr><td>9</td><td>Mexico</td><td>820</td><td>1.52</td><td>62.71</td><td>5.99</td><td>0.33</td><td>6.64</td><td>22</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>10</td><td>Saudi Arabia</td><td>811</td><td>1.51</td><td>64.22</td><td>22.64</td><td>0.45</td><td>30.84</td><td>34</td><td>2020</td><td>2030</td><td>RER</td></tr><tr><td>11</td><td>Germany</td><td>784</td><td>1.46</td><td>65.68</td><td>9.49</td><td>0.17</td><td>-16.68</td><td>55</td><td>1990</td><td>2030</td><td>AER</td></tr><tr><td>12</td><td>Canada</td><td>757</td><td>1.41</td><td>67.09</td><td>19.79</td><td>0.40</td><td>4.22</td><td>40</td><td>2005</td><td>2030</td><td>AER</td></tr><tr><td>13</td><td>International Shipping</td><td>751</td><td>1.40</td><td>68.48</td><td></td><td></td><td>7.17</td><td></td><td></td><td></td><td></td></tr><tr><td>14</td><td>South Korea</td><td>726</td><td>1.35</td><td>69.83</td><td>14.01</td><td>0.31</td><td>7.79</td><td>40</td><td>2018</td><td>2030</td><td>AER</td></tr><tr><td>15</td><td>Turkey</td><td>688</td><td>1.28</td><td>71.11</td><td>8.09</td><td>0.24</td><td>61.73</td><td>21</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>16</td><td>Australia</td><td>571</td><td>1.06</td><td>72.17</td><td>21.98</td><td>0.43</td><td>-5.06</td><td>43</td><td>2005</td><td>2030</td><td>AER</td></tr><tr><td>17</td><td>Pakistan</td><td>546</td><td>1.02</td><td>73.19</td><td>2.53</td><td>0.42</td><td>36.49</td><td>15</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>18</td><td>South Africa</td><td>535</td><td>0.99</td><td>74.18</td><td>8.91</td><td>0.66</td><td>-9.11</td><td>40</td><td>2020</td><td>2030</td><td>RER</td></tr><tr><td>19</td><td>Vietnam</td><td>489</td><td>0.91</td><td>75.09</td><td>4.88</td><td>0.44</td><td>58.37</td><td>9</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>20</td><td>Thailand</td><td>464</td><td>0.86</td><td>75.95</td><td>6.67</td><td>0.37</td><td>8.76</td><td>20</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>21</td><td>France</td><td>430</td><td>0.80</td><td>76.75</td><td>6.50</td><td>0.14</td><td>-16.37</td><td>55</td><td>1990</td><td>2030</td><td>AER</td></tr><tr><td>22</td><td>United Kingdom</td><td>427</td><td>0.79</td><td>77.54</td><td>6.27</td><td>0.14</td><td>-28.29</td><td>68</td><td>1990</td><td>2030</td><td>AER</td></tr><tr><td>23</td><td>International Aviation</td><td>426</td><td>0.79</td><td>78.34</td><td></td><td></td><td>-8.38</td><td></td><td></td><td></td><td></td></tr><tr><td>24</td><td>Nigeria</td><td>408</td><td>0.76</td><td>79.09</td><td>1.88</td><td>0.38</td><td>11.08</td><td>20</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>25</td><td>Poland</td><td>401</td><td>0.75</td><td>79.84</td><td>10.62</td><td>0.29</td><td>-4.13</td><td>55</td><td>1990</td><td>2030</td><td>AER</td></tr><tr><td>26</td><td>Italy</td><td>395</td><td>0.73</td><td>80.57</td><td>6.70</td><td>0.15</td><td>-21.62</td><td>55</td><td>1990</td><td>2030</td><td>AER</td></tr><tr><td>27</td><td>Argentina</td><td>383</td><td>0.71</td><td>81.29</td><td>8.27</td><td>0.37</td><td>10.60</td><td>10</td><td>2020</td><td>2030</td><td>RER</td></tr><tr><td>28</td><td>Egypt</td><td>378</td><td>0.70</td><td>81.99</td><td>3.55</td><td>0.27</td><td>19.16</td><td>33</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>29</td><td>Iraq</td><td>368</td><td>0.68</td><td>82.67</td><td>8.41</td><td>0.90</td><td>75.42</td><td></td><td></td><td></td><td></td></tr><tr><td>30</td><td>Malaysia</td><td>354</td><td>0.66</td><td>83.33</td><td>10.50</td><td>0.37</td><td>26.48</td><td>45</td><td>2005</td><td>2030</td><td>CIR</td></tr><tr><td>31</td><td>Kazakhstan</td><td>332</td><td>0.62</td><td>83.95</td><td>17.33</td><td>0.65</td><td>3.50</td><td>15</td><td>1990</td><td>2030</td><td>AER</td></tr><tr><td>32</td><td>Spain</td><td>329</td><td>0.61</td><td>84.56</td><td>7.08</td><td>0.17</td><td>-13.07</td><td>55</td><td>1990</td><td>2030</td><td>AER</td></tr><tr><td>33</td><td>Taiwan</td><td>308</td><td>0.57</td><td>85.13</td><td>12.86</td><td>0.19</td><td>1.57</td><td></td><td></td><td></td><td></td></tr><tr><td>34</td><td>United Arab Emirates</td><td>295</td><td>0.55</td><td>85.68</td><td>29.33</td><td>0.42</td><td>31.21</td><td>31</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>35</td><td>Algeria</td><td>284</td><td>0.53</td><td>86.21</td><td>6.38</td><td>0.57</td><td>36.01</td><td>7</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>36</td><td>Bangladesh</td><td>281</td><td>0.52</td><td>86.73</td><td>1.62</td><td>0.26</td><td>29.87</td><td>7</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>37</td><td>Philippines</td><td>265</td><td>0.49</td><td>87.22</td><td>2.35</td><td>0.27</td><td>50.56</td><td>3</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>38</td><td>Uzbekistan</td><td>227</td><td>0.42</td><td>87.65</td><td>6.67</td><td>0.79</td><td>9.25</td><td>35</td><td>2010</td><td>2030</td><td>CIR</td></tr><tr><td>39</td><td>Colombia</td><td>216</td><td>0.40</td><td>88.05</td><td>4.23</td><td>0.27</td><td>18.88</td><td>51</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>40</td><td>Ukraine</td><td>209</td><td>0.39</td><td>88.43</td><td>4.84</td><td>0.55</td><td>-47.32</td><td>65</td><td>1990</td><td>2030</td><td>AER</td></tr><tr><td>41</td><td>Qatar</td><td>195</td><td>0.36</td><td>88.80</td><td>67.38</td><td>0.74</td><td>41.86</td><td>25</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>42</td><td>Ethiopia</td><td>192</td><td>0.36</td><td>89.15</td><td>1.63</td><td>0.66</td><td>53.40</td><td>14</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>43</td><td>Venezuela</td><td>170</td><td>0.32</td><td>89.47</td><td>4.99</td><td>1.02</td><td>-35.03</td><td>20</td><td>BAU</td><td>2030</td><td>RER</td></tr><tr><td>44</td><td>Myanmar</td><td>169</td><td>0.31</td><td>89.78</td><td>3.04</td><td>0.76</td><td>42.10</td><td></td><td></td><td></td><td></td></tr><tr><td>45</td><td>Kuwait</td><td>168</td><td>0.31</td><td>90.10</td><td>37.96</td><td>0.80</td><td>35.24</td><td>7</td><td>BAU</td><td>2035</td><td>RER</td></tr><tr><td></td><td>Total</td><td>53 786</td><td></td><td></td><td>6.76</td><td>0.39</td><td>14.46</td><td></td><td></td><td></td><td></td></tr></tbody></table>

(1) GHG emissions in 2022 in MtCO₂e, (2) share in %, (3) cumulative share in %, (4) GHG emissions per capita in 2022 in tCO₂e, (5) GHG emissions per GDP in 2022 in kgCO₂e, (6) GHG emissions growth between 2010 and 2022 in %, (7) NDC reduction rate, (8) base year (BAU = business-as-usual), (9) target year, (10) mitigation type (AER = absolute emissions reduction, CIR = carbon intensity reduction, RER = relative emissions reduction).

Source: Crippa et al. (2023), https://edgar.jrc.ec.europa.eu/report_2023, IGES (2022), https://unfccc.int/NDCREG & Author's calculations.
year needed to meet the nationally determined contributions, to 1.8% per year.” (Liu and Raftery, 2021, page 1).

More recently, Meinshausen et al. (2022) and Wang et al. (2023) are more optimistic about the 2°C pathway when all conditional climate pledges from developing countries are included in the analysis. However, the 1.5°C target still appears unrealistic, with Wang et al. (2023) estimating that there is still an emissions gap of 10-15 GtCO₂e by 2030 to follow the 1.5°C pathway. Ahead of COP28 in Dubai, the UNFCCC Secretariat has produced two reports to inform Parties to the CMA that national climate action plans remain insufficient to limit global temperature rise to 1.5°C and meet the goals of the Paris Agreement (UNFCCC, 2023a,b). The second report states:

> “Full implementation of all latest NDCs is estimated to lead to a 5.3% (2.3%-8.2%) emission reduction by 2030 relative to the 2019 level; while implementation of all latest NDCs excluding any conditional elements is estimated to result in 1.4% higher emissions in 2030 than in 2019 (ranging from 1.5% lower to 4.2% higher).” (UNFCCC, 2023b, Article I.11, page 5).

In Table 8.28 we have reported for each country the GHG emissions in 2022 in MtCO₂e and also the corresponding share in % of total emissions. It can be seen that GHG emissions are concentrated in a few countries and regions. In 2022, China accounts for 29.16% of total GHG emissions, followed by the US, India, Russia and Brazil with contributions of 11.19%, 7.33%, 4.80% and 2.44% respectively. The cumulative share of the top five countries then amounts to almost 55% of total GHG emissions, while more than 75% of GHG emissions are concentrated in 18 countries and the international shipping. The European Union (27 countries) has a share of 6.67% in 2022. In this context, it is clear that the 1.5°C target is not in the hands of the Europeans, who were the backbone of the Kyoto Protocol and made a significant contribution to the signing of the Paris Agreement. Using the EDGAR database<sup>81</sup>, we calculate the growth of GHG emissions since 1990 and plot its evolution for some developed countries (Figure 8.73) and developing countries (Figure 8.74). It is interesting to see the heterogeneity of emissions trajectories between the US and the EU, between China and Russia, etc. For example, the European Union has dramatically reduced its emissions since 1990 and Japan more recently, while the United States is close to 1990 levels and Canada has sharply increased its GHG emissions during this period. Nevertheless, all these countries have experienced a trajectory below the global trajectory of GHG emissions. If we look at the top developing countries, they generally have a trajectory above the global trajectory, with the exception of Russia. This is particularly true for China and India.

To understand the drivers of the emissions growth, we can again use a Kaya decomposition<sup>82</sup>. For example, we can consider a simplified approach where emissions are driven by population:

$$ \text{GHG emissions} = \text{Population} \times \frac{\text{GHG emissions}}{\text{Population}} $$

or by GDP:

$$ \text{GHG emissions} = \text{GDP} \times \frac{\text{GHG emissions}}{\text{GDP}} $$

Figure 8.75 shows the scatter plot of per capita GHG emissions between 1990 and 2002. On a global basis, this ratio is relatively stable between 6 and 7 tCO₂e per capita (black circle). On average, the number of countries that have increased this intensity measure is roughly equal to the number

<footnote n="81"><sup>81</sup>The link is https://edgar.jrc.ec.europa.eu.</footnote>

<footnote n="82"><sup>82</sup>The Kaya identity is defined in Box 8.6 on page 651.</footnote>
<figure><img src="image_355.png" /><figcaption>Figure 8.74: Growth of greenhouse gas emissions in % for developing countries (base year = 1990)</figcaption></figure>

<figure><img src="image_356.png" /></figure>
<figure><img src="image_357.png" /><figcaption>Figure 8.76: GHG emissions per GDP (1990 vs. 2022)</figcaption></figure>

<figure><img src="image_358.png" /></figure>
of countries that have decreased it. In fact, this ratio is not interesting for projecting the evolution of GHG emissions because it is difficult to control. A better decomposition is to use the per capita approach, as shown in Figure 8.76. On average, it was equal to 0.69 kgCO₂e per $ of GDP in 1990. It is now equal to 0.39 kgCO₂e per $ of GDP in 2022. The countries with the highest reduction rate are Bosnia and Herzegovina (86%), Armenia (84%), Estonia (83%), Ireland (82%) and Malta (80%). China reduced its emissions per GDP by 76% to a level of 0.61. Peters et al. (2017) proposed to use another Kaya identity to track the Paris Agreement NDCs:

$$ \text{GHG emissions} = \text{GDP} \times \frac{\text{Energy}}{\text{GDP}} \times \frac{\text{GHG emissions}}{\text{Energy}} $$

This simplified Kaya decomposition shows that the achievement of NDCs depends on three factors: the GDP growth of the economy, its energy intensity and its emission intensity. As explained by Peters et al. (2017), energy intensity is difficult to manage because it is highly dependent on the economy of the country. Therefore, the only way to curb GHG emissions is to improve emission intensity, which needs to be reduced by a factor greater than GDP growth.

**Remark 82** To track government climate action and progress towards the Paris Agreement goals, we can use the Climate Action Tracker, an independent scientific project that collects and measures this information (Figure 8.77). As of 1 January 2024, CAT tracks 39 countries and the EU covering around 85% of global emissions.

<figure><img src="image_359.png" /><figcaption>Figure 8.77: The CAT thermometer</figcaption></figure>

Source: Climate Action Tracker (2023), https://climateactiontracker.org.
### 8.2.3 Regulation policies

A comprehensive presentation of national climate change regulations around the world would take too much time, as the number of regulations today is huge and has increased dramatically since 2015 and the signing of the Paris Agreement. In this section, we only illustrate the impact of the UNFCCC and COPs at the country or regional level. In addition, we do not list all regulations affecting the financial sector, as these have already been presented in chapters 1 and 4. For example, it is clear that SFDR, CSRD and the EU Taxonomy are European regulations related to COP21.

#### United States

The US Environmental Protection Agency (EPA) is an independent agency of the US government, established in 1970. It is responsible for protecting the environment and enforcing standards under a variety of environmental laws. Here are some examples of regulations implemented or proposed by the EPA since 2020:

* An 85% reduction in the production and consumption of hydrofluorocarbons over the next 15 years, beginning in 2021;

* Vehicle emissions standards, including stricter GHG emissions standards for cars and light trucks beginning in 2023;

* Aviation GHG standards for commercial aircraft and large business jets;

* Renewable Fuel Standard (RFS) program, a federal program that requires a certain amount of renewable fuel to be blended into transportation fuel each year^83^;

* Regulation of power plant emissions through the Acid Rain Program (ARP) and Mercury and Air Toxics Standards (MATS).

These new regulations complement existing regulations such as the Clean Air Act (CAA) of 1970, the federal law that regulates air emissions from stationary and mobile sources, including power plants, factories, and vehicles. In August 2022, the United States passed the Inflation Reduction Act (IRA). One of its goals is to support clean energy development, including investments in solar, wind, and geothermal power, as well as energy efficiency and grid modernization. In December 2023, the EPA issued a new methane rule to reduce methane emissions from oil and natural gas operations. In 2022 and 2023, some states also increased their renewable portfolio standards (RPS), which are mandates to increase energy production from renewable sources such as wind, solar, biomass, and geothermal^84^.

**Remark 83** At the financial level, the Securities and Exchange Commission (SEC) has taken steps to improve climate-related disclosures for public companies. In March 2022, the SEC proposed a new climate disclosure rule that is aligned with the TCFD framework and is similar in approach to the European CSRD/ESRS and the IFRS sustainability disclosure standards (ISSB). As of January 2024, the SEC has not finalized the proposed rule and no timeline has been provided. In fact, the proposed rule is controversial, with some critics arguing that it would be too costly to comply with.

<footnote>^83^Under the RFS, EPA sets annual renewable fuel volume targets, measured in Renewable Volume Obligations (RVOs), for four categories of renewable fuel: cellulosic biofuel, biomass-based diesel, advanced biofuel, and total renewable fuel.</footnote>

<footnote>^84^In the United States, this is known as the Renewable Electricity Standard (RES).</footnote>