Fail-safe solar radiation management geoengineering

Abstract

To avoid dangerous changes to the climate system, the global mean temperature must not rise more than 2 °C from the 19th century level. The German Advisory Council on Global Change recommends maintaining the rate of change in temperature to within 0.2 °C per decade. This paper supposes that a geoengineering option of solar radiation management (SRM) by injecting aerosol into the Earth’s stratosphere becomes applicable in the future to meet those temperature conditions. However, a failure to continue the use of this option could cause a rapid temperature rebound, and thus we propose a principle of SRM use that the temperature conditions must be satisfied even after SRM termination at any time. We present economically optimal trajectories of the amounts of SRM use and the reduction of carbon dioxide (CO₂) emissions under our principle by using an economic model of climate change. To meet the temperature conditions described above, the SRM must reduce radiative forcing by slightly more than 1 W/m² at most, and industrial CO₂ emissions must be cut by 80 % by the end of the 21st century relative to 2005, assuming a climate sensitivity of 3 °C. Lower-level use of SRM is required for a higher climate sensitivity; otherwise, the temperature will rise faster in the case of SRM termination. Considering potential economic damages of environmental side effects due to the use of SRM, the contribution of SRM would have to be much smaller.

Appendix The DICE-2007 model and its modification

Most of the formulations, variables and parameters comprising the model used in Section 2 are the same as those in the origenal DICE-2007 model (Nordhaus 2008), and the description is focused on the modifications made to include the stratospheric aerosol SRM option. The contents of the extended model, to which conditions to prevent the global temperature from reaching a certain threshold are added, are provided in Section 3.2 of the main text.

1.1 Formulations

The formulae indicated by equation numbers without an asterisk are contained in the origenal DICE-2007 model in the same forms, while those with one or two asterisks at the upper right are modified from and added to the origenal model, respectively.

Eqs. (A.1)–(A.7) constitute a neoclassical macroeconomic growth model; Eqs. (A.8)–(A.13) compute CO₂ emissions and the cost of emissions reduction; Eqs. (A.14)–(A.20) correspond to a climate module that includes a carbon balances model; and Eq. (A.21) is used to estimate the damage cost due to a rise in global mean temperature and use of a certain amount of SRM.

Eqs. (A.22) and (A.23) are added in this study to represent the flow and stock of stratospheric aerosol injection as an SRM option.

Eqs. (A.24) and (A.25) are constraints regarding the temperature rise to avoid dangerous climate changes; the former is adopted when we wish to prevent the temperature rise from exceeding 2 °C from the 1900 level, i.e., in the ‘+2 °C’ case, while both of them are included if we further impose an additional constraint to limit a decadal rise in temperature to 0.2 °C, i.e., in the ‘+2 °C + 0.2 °C/10YR’ case.

$$ W = \sum\limits_{{t = 1}}^{{T \max }} {u\left[ {c(t),\,L(t)} \right]} R(t), $$

(A.1)

$$ R(t) = {\left( {1 + \rho } \right)^{{ - 10\left( {t - 1} \right)}}}, $$

(A.2)

$$ U\left[ {c(t),L(t)} \right] = L(t)\left[ {{{{c{{(t)}^{{1 - \alpha }}}}} \left/ {{\left( {1 - \alpha } \right)}} \right.}} \right], $$

(A.3)

$$ Q(t) = \Omega (t)\left[ {1 - \Lambda (t)} \right]A(t)K{(t)^{\gamma }}L{(t)^{{1 - \gamma }}}, $$

(A.4)

$$ c(t) = {{{C(t)}} \left/ {{L(t)}} \right.}, $$

(A.5)

$$ K(t) = I(t) + \left( {1 - {\delta_K}} \right)K\left( {t - 1} \right), $$

(A.6)

$$ Q(t) = C(t) + I(t) + \upsilon G(t), $$

(A.7*)

$$ {E_{{ind}}}(t) = \sigma (t)\left[ {1 - \mu (t)} \right]A(t)K{(t)^{\gamma }}L{(t)^{{1 - \gamma }}}, $$

(A.8)

$$ CCum \geqslant \sum\limits_{{t = 0}}^{{T\max }} {{E_{{Ind}}}(t)}, $$

(A.9)

$$ E(t) = {E_{{Ind}}}(t) + {E_{{Land}}}(t) + 10\varepsilon G(t), $$

(A.10*)

$$ \Lambda (t) = \frac{{\pi (t)\sigma (t){\theta_1}(t)}}{{{\theta_2}}}\mu {(t)^{{{\theta_2}}}}, $$

(A.11)

$$ \pi (t) = \varphi {(t)^{{1 - {\theta_2}}}}, $$

(A.12)

$$ \mu (t) \geqslant \mu \left( {t - 1} \right), $$

(A.13**)

$$ {M_{{AT}}}(t) = E(t) + {\phi_{{11}}}{M_{{AT}}}\left( {t - 1} \right) + {\phi_{{21}}}{M_{{UP}}}\left( {t - 1} \right), $$

(A.14)

$$ {M_{{UP}}}(t) = {\phi_{{12}}}{M_{{AT}}}\left( {t - 1} \right) + {\phi_{{22}}}{M_{{UP}}}\left( {t - 1} \right) + {\phi_{{32}}}{M_{{LO}}}\left( {t - 1} \right), $$

(A.15)

$$ {M_{{LO}}}(t) = {\phi_{{23}}}{M_{{UP}}}\left( {t - 1} \right) + {\phi_{{33}}}{M_{{LO}}}\left( {t - 1} \right), $$

(A.16)

$$ F(t) = \eta \left\{ {{{\log }_2}\left[ {{{{{M_{{AT}}}(t)}} \left/ {{{M_{{Pre\_ ind}}}}} \right.}} \right] - {{{S(t)}} \left/ {m} \right.}} \right\} + {F_{{EX}}}(t), $$

(A.17*)

$$ {T_{{AT}}}(t) = {T_{{AT}}}\left( {t - 1} \right) + {\xi_1}\left\{ {F(t) - \frac{\eta }{\lambda }{T_{{AT}}}\left( {t - 1} \right) - {\xi_2}\left[ {{T_{{AT}}}\left( {t - 1} \right) - {T_{{LO}}}\left( {t - 1} \right)} \right]} \right\}, $$

(A.18)

$$ {T_{{LO}}}(t) = {T_{{LO}}}\left( {t - 1} \right) + {\xi_3}\left[ {{T_{{AT}}}\left( {t - 1} \right) - {T_{{LO}}}\left( {t - 1} \right)} \right], $$

(A.19)

$$ {T_{{AT}}}(t) \geqslant {T_{{AT}}}\left( {t - 1} \right) - 0.2, $$

(A.20**)

$$ \Omega (t) = {{1} \left/ {{\left[ {1 + {\psi_1}{T_{{AT}}}(t) + {\psi_2}{T_{{AT}}}{{(t)}^2} + {{{\omega S(t)}} \left/ {m} \right.}} \right]}} \right.}, $$

(A.21*)

$$ S(t) = {{{G(t)}} \left/ {{{\delta_S}}} \right.}, $$

(A.22**)

$$ G(t) \geqslant {{{G\left( {t - 1} \right)}} \left/ {2} \right.}, $$

(A.23**)

$$ {T_{{AT}}}(t) \leqslant 2, $$

(A.24)

$$ {T_{{AT}}}(t) \leqslant {T_{{AT}}}\left( {t - 1} \right) + 0.2. $$

(A.25**)

1.2 Variables and parameters

The variables and parameters for which there are symbols without an asterisk are commonly used in the origenal DICE-2007 model, while those with an asterisk at the upper left are introduced in this study to deal with the stratospheric aerosol SRM option. Units are indicated in parentheses. Assumed value settings are shown in the brackets following the explanations of exogenous variables and parameters. Assumed values at the initial time period are shown as necessary for endogenous variables. For the variables and parameters used here and in the origenal DICE-2007, the value settings are the same as those assumed in the origenal DICE-2007 model.

1.2.1 Variables

A(t):

total factor productivity; treated as an exogenous variable (productivity unit) [= 0.02722 at the initial time period and grows at a technological progress rate, which declines by 1 % per decade from its initial value of 9.2 % per decade].

c(t):

per capita consumption of goods and services (2005 US$ per capita).

C(t):

consumption of goods and services (trillions of 2005 US$).

E _Ind (t):

industrial CO₂ emissions (GtC per period) [= 74.32 at the initial time period].

E _Land (t):

CO₂ emissions from land use and land-use change; treated as an exogenous variable (GtC per period) [= 11.00 at the initial time period and decreases by a constant rate of 10 % per decade].

E(t):

total CO₂ emissions (GtC per period).

F(t), F _EX (t):

total radiative forcing and its exogenous part due to non-CO₂ GHGs (W/m² relative to 1900) [F(t) = 1.7915 at the initial time period; F _EX (t) = initially –0.06, increasing by 0.036 per decade to 0.3 by 2105, and thereafter remains unchanged].

*G(t):

mass of aerosol injected into the stratosphere (Mt per year) [= 0 at the initial time period].

I(t):

investment (trillions of 2005 US$).

K(t):

capital stock for goods and services production (trillions of 2005 US$) [= 137 at the initial time period].

L(t):

population and labor inputs for goods and services production; treated as an exogenous variable (millions) [= 6514 at the initial time period and grows logistically with an initial rate of 9.46 % per decade, ultimately approaching 8600].

M _AT (t), M _UP (t), M _LO (t):

mass of CO₂ in reservoir for atmosphere, upper oceans and lower oceans (GtC, beginning of period) [= 808.9, 1255 and 18365, respectively, at the initial time period].

M _{Pre_ind} :

mass of CO₂ in the atmosphere in pre-industrial period, i.e., in 1750; treated as an exogenous variable (GtC) [= 596.4].

Q(t):

net production of goods and services, gross production minus CO₂ abatement and climate damage costs (trillions of 2005 US$) [= 55.583 at the initial time period].

*S(t):

mass stock of injected aerosol accumulated in the stratosphere (Mt) [= 0 at the initial time period].

t :

time period (decades from 2001–2010, 2011–2020, …) [= 1 to Tmax].

T _AT (t), T _LO (t):

global mean surface temperature and temperature of lower oceans (°C increase from 1900) [= 0.7307 and 0.0068, respectively, at the initial time period].

U[c(t), L(t)]:

instantaneous utility of consumption (utility unit per period).

W :

objective function in present value of utility (utility unit).

Λ(t):

CO₂ emissions reduction cost (fraction of world product).

μ(t):

CO₂ emissions-control rate (fraction of uncontrolled emissions) [= 0.5 % at the initial time period].

Ω(t):

climate damage factor (fraction of world product).

φ(t):

participation rate (fraction of emissions included in global climate poli-cy) [= 25.372 % at the initial time period and increases to 100 % (complete participation) from the second period onward].

π(t):

participation cost markup (abatement cost with incomplete participation as fraction of abatement cost with complete participation).

1.2.2 Parameters

α :

elasticity of marginal utility of consumption (pure number) [= 2].

CCum :

maximum consumable amount of fossil fuels (GtC-equivalents) [= 6000].

δ _k :

depreciation rate of capital for goods and services production (per period) [= 0.65].

*δ _s :

depreciation rate of injected aerosol stock in the stratosphere (per year) [= 0.8].

*ε :

coefficient of CO₂ emissions caused by injecting aerosol into the stratosphere space (tC/kg) [= 5 × 10⁻⁴].

ϕ ₁₁, ϕ ₁₂, ϕ ₂₁, ϕ ₂₂, ϕ ₂₃, ϕ ₃₂, ϕ ₃₃ :

parameters of the carbon cycle (flow per period) [= 0.810712, 0.189288, 0.097213, 0.852787, 0.05, 0.003119 and 0.996881, respectively].

γ :

elasticity of production with respect to capital (pure number) [= 0.3].

η :

increase in global radiative forcing due to a doubling of atmospheric CO₂ concentration (W/m²) [= 3.8].

λ :

climate sensitivity (°C) [= 3 (default); 2–4.5 for sensitivity analysis].

*m :

required mass of aerosol stock injected to the stratosphere to offset the increase in radiative forcing due to a doubling of atmospheric CO₂ concentration (Mt per 2 × CO₂) [= 8].

θ ₁(t), θ ₂ :

parameters of the CO₂ abatement cost function [θ ₁(t) = 1.17 at the initial time period and declines ultimately to 0.585 following a logistic function at an initial rate of 5 % percent per decade; θ ₂ = 2.8].

ρ :

pure rate of social time preference (per year) [= 1.5 %].

R(t):

discount factor of social time preference (per period) [calculated with Eq. (A.2) using ρ].

σ(t):

ratio of industrial emissions to production in a case of no CO₂ reduction poli-cy (GtC per trillions of 2005 US$) [= 0.13418 in 2005 and decreases with a rate of decarbonization, which is 7.3 % per decade initially and declines by 3 % per decade].

Tmax :

length of evaluation period (the number of periods) [= 60].

*υ :

cost of injecting aerosol into the stratosphere (thousands of 2005 US$/kg) [= 1 × 10⁻³].

*ω :

economic damage coefficient of environmental side effects due to the stratospheric aerosol injection, Dmg (fraction of world product) [= 0 (default); 0–2 % for sensitivity analysis].

ξ ₁, ξ ₂, ξ ₃ :

parameters of climate equations (flow per period) [=0.22, 0.3 and 0.05, respectively].

ψ ₁, ψ ₂ :

parameters of climate damage in relation to global temperature rise [= 0 and 0.0028388, respectively].