# Tradeoffs inherent in solar geoengineering peak-shaving strategies

###### Abstract

Some have suggested solar geoengineering, cooling the planet by reflecting
sunlight, could be deployed temporarily to “shave the peak” of global
temperatures. This would use limited deployment to keep global mean
temperatures below a given threshold, buying time for emissions reductions and
negative emissions to reduce greenhouse gas concentrations. However, we
suggest the trade-offs inherent in such scenarios are not widely appreciated.
In this study, we explore general features of such scenarios, using a
combination of simple analytical relationships between emissions and
temperatures and modelling to illustrate the strong trade-off between the
duration of geoengineering deployment and the temperature overshoot that is
suppressed by geoengineering, as well as the scale of negative emissions that
are required. Our results indicate that, unless climate sensitivity is at the
high end of estimates, shaving a substantial peak – say, of 2K to 1.5K – would
require at least a century of deployment of solar geoengineering and large-
scale negative emissions, under a broad range of emissions trajectories. If
negative emissions are not available at the scale of tens of gigatonnes of CO2
per year, substantial peak-shaving will require deployment of geoengineering
for multiple centuries. Our analysis emphasise the importance of factoring in
the scale, feasibility, and cost of the negative emissions required to avoid
indefinite commitment to solar geoengineering.

Earth’s Future

Department of Atmospheric, Oceanic and Planetary Physics, University of Oxford
Environmental Change Institute, University of Oxford

Thomas<EMAIL_ADDRESS>
Solar geoengineering “peak-shaving” strategies are considered in the abstract.

We find robust trade-offs between the suppressed temperature overshoot and the
required duration of SRM across a range of idealised and modelled scenarios.

The cost of, and committment to, long-term and large-scale negative emissions
must be considered in assessing the cost of any finite-duration geoengineering
scheme.

## Plain Language Summary

Scientists are confident that human emissions of greenhouse gases are warming
the planet. Some have suggested that solar geoengineering, where particles are
put into the atmosphere to reflect sunlight and cool the planet, could reduce
the negative effects of this climate change. Small amounts of geoengineering
may have more benefits and fewer risky side-effects than larger amounts. For
example, we could use geoengineering to keep to below a temperature threshold,
like those agreed upon in the Paris Climate Accord. In this paper, we use
mathematical equations and computer models to explore a range of ways that
this could be done. We show that there is a trade-off. In cases where you only
need geoengineering for a short time, this is unlikely to have much of an
effect on temperatures. Or, you can reduce temperatures by a larger amount,
but then you must keep geoengineering going for many decades, which could be
difficult politically. We also note that, because greenhouse gases remains in
the atmosphere for centuries, if you want to stop the geoengineering, you must
remove billions of tonnes of carbon dioxide. We argue that these trade-offs
should be carefully considered and made apparent in discussions of
geoengineering.

##  1 Introduction

Under what circumstances could we consider solar geoengineering? Many authors
have observed that solar geoengineering cannot be a substitute for mitigation
[Goes . (2011)] [Keith Irvine (2016)]. Solar radiation management (SRM)
imperfectly compensates for the effects of greenhouse gases, leading to
residual climatic changes. These residual climate changes are likely to be
worse as the amount of radiative forcing from SRM is increased. For example,
SRM reduces precipitation more quickly than temperature. This is because the
net result of cooling the surface with SRM, while the troposphere is warmed by
greenhouse gases, is an atmosphere that is more stable to convection.
Consequently, trade-offs exist between regions and between returning
temperature and precipitation to preindustrial baselines using SRM. [Moreno-
Cruz . (2012)] SRM would do little to address ocean acidification [Williamson
Turley (2012)], and may result in residual polar ocean warming or changes to
large-scale circulation [Fasullo . (2018)]. The greater the radiative forcing
that is offset by SRM, the more dangerous it would be to suddenly cease its
use due to the termination effect: a rapid rebound of temperatures [Jones .
(2013)]. For this reason, we suggest that the temperature change compensated
for by geoengineering is referred to as “suppressed” rather than “avoided”. In
addition, the magnitude and latitudinal distribution of the radiative forcing
that can be obtained with the most viable method, stratospheric aerosol
injection, is not yet clear, with model studies in disagreement [Niemeier
Timmreck (2015)] [Kleinschmitt . (2018)]. A scenario where humanity is
committed to ever-increasing deployment of SRM while emissions continue to
climb is evidently fraught with risk.

“Peak-shaving” scenarios, where SRM is used temporarily to keep temperatures
below a threshold – for example, the 1.5C or 2C targets of the Paris Agreement
– have been suggested as a more reasonable form of deployment, such as in the
IPCC SR1.5 report (Section 4.3.8 and Cross-Chapter Box 10) [de Coninck .
(2018)] [MacMartin . (2018)], which would buy time for mitigation to take
effect. Modelling suggests that limited deployment of SRM - for example, to
cancel half of the warming due to greenhouse gases - could reduce the negative
residual climatic changes in most regions. [Irvine . (2019)] Here, we do not
seek to evaluate whether the climate benefits of such peak-shaving outweigh
the risks, but instead to explore the parameter space of peak-shaving
scenarios.

First, we note that if SRM is to be deployed temporarily, CO2 emissions must
peak and decline to net zero. This follows from the approximate
proportionality of temperature changes to cumulative carbon emissions [Allen .
(2009)]. Once emissions reach net zero, carbon dioxide concentrations in the
atmosphere will gradually decline due to ocean uptake and weathering, but
substantial net-negative CO2 emissions are likely to be necessary to avoid
centuries of commitment to SRM in a peak-shaving scenario.

##  2 Peak-shaving in the abstract

###  2.1 A toy example: parabolic peak

As a simple example, consider only CO2 emissions. Assume that no mitigation
occurs until a particular threshold temperature - say, 1.5C - is crossed.
Thereafter, CO2 emissions begin falling at a constant rate, becoming net-
negative emissions at this same rate. Under these circumstances, the
temperature peak will be approximately parabolic, as the emissions in a given
year are approximately proportional to the rate of temperature change in that
year. Let D be the duration of peak-shaving above a given temperature
threshold, T be the temperature excess above threshold, and R be the rate of
warming when the threshold is first exceeded (noting that this is
approximately the Transient Climate Response to Cumulative Emissions, TCRE,
multiplied by the emissions when the threshold is exceeded.)

We can express this quadratic curve as $T(t)=\alpha t(t–D)$, where D is the
length of time for which the threshold is exceeded. Differentiating gives
$dT/dt=\alpha(2t–D)$, and $R=-\alpha D$ by definition. The maximum temperature
above the threshold, or the total overshoot that would be suppressed by SRM,
which we denote $T_{so}$, is equal to $–\alpha D^{2}/4.$ We can then express
the suppressed temperature overshoot in terms of the duration and the warming
rate when the threshold is first crossed: $T_{so}=RD/4$ or, equivalently,
$4T_{so}/R=D.$

In such an idealised scenario, the duration of the SRM commitment is twice the
time taken to reach net zero. The attainable mitigation rate therefore
determines the cumulative negative emissions required, and the duration of
commitment to SRM. When this is combined with the climate sensitivity - for
example, through the rate of warming - the temperature overshoot can be
inferred.

This allows us to see the tradeoff between the implied duration of deployment
and the suppressed temperature overshoot. For example, at the current rate of
warming of 0.25K per decade, shaving a peak of 2C to 1.5C implies a commitment
to deployment of around 80 years. (In reality, this is likely to be an
underestimate, as we will explain.)

Considered in the context of the Paris Agreement, current scenarios that limit
global warming to 1.5K (or 2K) include overshoot scenarios where temperature
temporarily rises above the target, before falling at the end of the century.
SRM could be deployed to suppress such an overshoot, as in SR1.5 Section
4.3.8. In the example considered in the IPCC report, overshoot of around 0.1K
is avoided. Assuming rates of warming and cooling are similar to today’s
magnitude of warming at 0.25K/decade, the rule-of-thumb implies a commitment
of 32 years, which is similar to the indicative time-scale shown (35 years).
This exemplifies the trade-off between the duration of the implied commitment
to SRM and the magnitude of the suppressed temperature overshoot.

By integrating, we can determine the area under the curve and hence the number
of degree–years avoided, as a rough metric for avoided damages [SJ. Smith
Rasch (2013)], although due to SRM’s imperfect compensation for increased GHG
concentrations, this should not be considered fungible with the equivalent
temperature increase avoided by mitigation. This gives $RD^{2}/6$ and
consequently would equate to around 267 degree–years avoided in the above
example.

Even this simple example illustrates the trade-off between duration of the
commitment and the suppressed temperature overshoot for a given rate of
warming. The more warming we use SRM to avoid – and hence, the more damaging
any potential termination – the longer our commitment must be.

###  2.2 Different rates of mitigation and net-negative emissions deployment

This analysis can be simply extended to consider the more general case for
overshoot trajectories where the rate of warming when the target is first
crossed and the rate of cooling are not the same, but parabolic trajectories
are assumed either side of the peak - in other words, mitigation and net-
negative emissions deployment occur at different rates.

In this case, we have (where warming and cooling rates are positive-definite
magnitudes):

$D_{warm}=\frac{2T_{so}}{R_{warm}}$ (1)

$D_{cool}=\frac{2T_{so}}{R_{cool}}$ (2)

$D_{overshoot}=\frac{2T_{so}}{R_{warm}}\left(1+\frac{R_{warm}}{R_{cool}}\right)$
(3)

If the rate of warming on the way up exceeds the rate of cooling on the way
down, then overshoot is longer. Given that rates of warming and cooling are
approximately proportional to the emissions of CO2, the ratio of $R_{warm}$ to
$R_{cool}$ is roughly the ratio of emissions for the year when the carbon
budget is exceeded and the negative emissions when the “carbon debt” has been
repaid.

The generalised formula for degree-years is similarly obtained:

$DY_{overshoot}=\frac{4T_{so}^{2}}{3R_{warm}}\left(1+\frac{R_{warm}}{R_{cool}}\right)$
(4)

If we denote the carbon budget overshoot by $\Delta C_{ex}$, and emissions as
a function of time as $E\left(t\right)$ then we have, with $\lambda$ as the
TCRE:

$D\approx 2\cdot\lambda\cdot\Delta C_{ex}\left(\frac{1}{\lambda\cdot
E\left(t_{warm}\right)}+\frac{1}{\lambda\cdot\left|E\left(t_{cool}\right)\right|}\right)$
(5)

$D\approx 2\cdot\Delta
C_{ex}\left(\frac{1}{E\left(t_{warm}\right)}+\frac{1}{\left|E\left(t_{cool}\right)\right|}\right)$
(6)

We see from this that the duration of the overshoot – and hence the implied
SRM commitment – depends on the amount the carbon budget is exceeded, the
timescale over which you continue to emit carbon dioxide and warm, and the
timescale to remove the full anthropogenic excursion of CO2. In this
approximation, the dependence on climate sensitivity arises from its
implications for the carbon budget for 1.5K or 2K. Higher climate sensitivity
reduces the cumulative emissions required for a given overshoot, which
shortens both the timescale of continued emissions and the timescale of
negative emissions required to remove them.

###  2.3 The implications of a negative emissions floor

Suppose that global net-negative emissions can only be increased up to a
certain “floor” (n). This could be due to geobiophysical limitations in the
case of large-scale BECCS, afforestation, or enhanced weathering, or societal
willingness to pay in the case of Direct Air Capture. We might also anticipate
that the average rate at which large-scale net-negative emissions are deployed
(m2) will differ from the average rate of mitigation (m1). If mitigation and
geoengineering start when the temperature threshold is crossed, and carbon
emissions fall linearly from a maximum of Cmax, then our carbon budget
formulation gives us:

$\Delta C_{ex}=\frac{1}{2}\frac{C^{2}_{max}}{m\textsubscript{1}}$ (7)

With tneg as the total time committment to any negative emissions, and tfloor
the time commitment to negative emissions at their lowest value, we have:

$t_{neg}=\frac{n}{m\textsubscript{2}}+t_{floor}$ (8)

Requiring that the total negative emissions cancel out the exceedance of the
carbon budget, we have:

$\Delta C_{ex}=\frac{1}{2}\frac{n^{2}}{m\textsubscript{2}}+t_{floor}n$ (9)

Solving for tfloor and with the total duration of the overshoot as D, we
obtain:

$D=\frac{C^{2}_{max}}{2m\textsubscript{1}n}+\frac{C_{max}}{m\textsubscript{1}}+\frac{n}{2m\textsubscript{2}}$
(10)

Which simplifies in the special case that m1 = m2 = m to:

$D=\frac{1}{2nm}(C_{max}+n)^{2}$ (11)

We also have that

$T_{so}=\Delta C_{ex}\cdot\lambda=\frac{C^{2}_{max}\cdot\lambda}{2\cdot
m\textsubscript{1}}$ (12)

Which allows us to re-express the duration in terms of the TCRE, the peak
shaved, the negative emissions floor, and the mitigation rate, if desired.

###  2.4 The double exponential emissions trajectory

An alternative model of emissions trajectories uses a double exponential
function. For peak-shaving emissions trajectories, subject to a negative
emissions floor, you can model the emissions after mitigation begins as:

$E(t)=C_{max}\exp(-m\textsubscript{1}t)-n(1-\exp(-m\textsubscript{2}t))$ (13)

where

$m\textsubscript{1,2}=\ln(2)/t\textsubscript{1,2}$ (14)

and t1,2 are the timescales for halving (gross) emissions and scaling up the
deployment of negative emissions to half of its maximum value, respectively.
Alternatively, to represent emissions which change by x% per year, one could
consider mi = ln(1+0.01x). In this idealised emissions scenario, rapid
mitigation is combined with expanding deployment of negative emissions at
scale to reach net zero, while both mitigation rate and negative emissions
deployment rate slow down over time. This corresponds to idealised
trajectories where emissions fall by a given average percentage each year.

This models well the trajectory of the Low Energy Demand scenario [Grubler .
(2018)] and the rapid decarbonisation pathway referred to as the “Carbon Law”,
which proposes halving emissions each decade while scaling up negative
emissions to a gigatonne scale. [Rockström . (2017)]

This gives a cumulative exceedance of the carbon budget as:

$\Delta
C_{ex}=\frac{C_{max}}{m\textsubscript{1}}(1-\exp(-m\textsubscript{1}t))+\frac{n}{m\textsubscript{2}}(1-\exp(-m\textsubscript{2}t))-nt$
(15)

This is a transcendental equation which can be solved numerically: the
duration of the implied overshoot occurs for the (non-trivial) root of $\Delta
C_{ex}=0$, while the maximum carbon budget overshoot - and, approximately, the
maximum temperature overshoot - occurs when emissions are zero. Therefore, in
a peak-shaving scenario, maximum SRM deployment is only reached when net zero
emissions are also reached. For a numerical and graphical exploration of the
double-exponential model, as well as a comparison to the quadratic
approximation, see Appendix A.

###  2.5 Path-dependence of the TCRE

The preceding analysis has assumed that the TCRE is constant for both negative
and positive emissions. However, more detailed modelling of the carbon-climate
cycle suggests that the TCRE is path-dependent, owing to the lagged thermal
response of the ocean [Zickfeld . (2016)], which means that negative emissions
are less effective at cooling than emissions are at warming, at least in the
short term.

This non-linearity is most prominent when the lagged ocean response is large,
which occurs when the carbon budget overshoot is large. For example, Zickfeld
et al. (2016) modelled an emissions trajectory where CO2 concentrations were
increased by 1% per annum to 4x preindustrial levels and then decreased in the
same way. They found that the TCRE was approximately 1.59K in the ascending
branch and 1.48K in the descending branch, i.e. 7% lower. In scenarios with
smaller total cumulative emissions, the difference between the ascending and
descending branch was smaller (or even of the opposite sign, implying negative
emissions appeared to be slightly more effective at cooling than positive
emissions were at warming.) This implies that the analytical estimates above
are likely to be slight underestimates for the real-world commitment to peak-
shaving; for more detailed analysis of this, see Appendix B.

##  3 Example: The SSP-Overshoot Scenario

We can validate these approximations by considering concrete peak-shaving
scenarios. Tilmes et al. (2016) [Tilmes . (2016)] used the Community Earth
System Model (CESM) to simulate the SSP5-3.4-OS scenario [Meinshausen .
(2019)]. In this scenario, emissions follow the RCP8.5 pathway until 2040.
Then, emissions follow an idealised pathway, declining to become net-negative
by the end of the century. The mitigation rate of -1.2Gt CO2/yr/yr, as well as
the floor of negative emissions at -18.5GtCO2/yr, are both drawn from the
Integrated Assessment Modelling literature: specifically, the Shared
Socioeconomic Pathway database [Riahi . (2017)], representing the maximum
decarbonisation and negative emissions rates respectively. This pathway
therefore represents a peak-shaving scenario at the upper end of what is
considered feasible in the current scenario literature: there is a significant
exceedance of temperature targets due to following RCP8.5 to 2040, then rapid
decarbonisation and large-scale negative emissions. Tilmes et al. simulate SRM
by using stratospheric aerosol injections to reduce temperature anomalies to
2K from this pathway.

In CESM, which has an equilibrium climate sensitivity of 4K, this pathway
results in temperature anomalies which peak at 3K above the preindustrial. The
warming rate in CESM approaches 0.05K/year at RCP8.5 2040 when its peak-
shaving begins (compare to 0.045K/year in the CMIP5 ensemble for RCP8.5 2040)
[Risbey . (2017)]. The cooling rate during the sustained period of negative
emissions is just over 0.01K/year.

The analytical approximation above therefore suggests a deployment time for
SRM to “peak-shave” from 3K to 2K as around 40 years while temperatures rise
and around 100 years as they decline. CESM’s explicit simulation of this
scenario found that geoengineering is deployed for 130 years to shave the peak
below 2C.

For a given emissions trajectory, uncertainties around the climate sensitivity
affect the overshoot trajectory. We use the FaIR simple climate model [CJ.
Smith . (2018)] to sample the IPCC AR5 “likely” range for climate sensitivity
[Stocker . (2013)]. Non-CO2 emissions are assumed to follow a 1.5C-compatible
mitigation pathway from 2020, stabilising at a constant radiative forcing of
$0.45W/m^{2}$ in the long term. In these versions of the scenario, the
negative emissions are switched off when temperatures reach 1.5C.

The results, for the standard overshoot parameters with mitigation beginning
in 2020 and 2040, are illustrated in Figure 1. The uncertainty in the
emissions and cumulative emissions pathways illustrated corresponds to the 67%
confidence interval for climate sensitivity sampled in FaIR, as the time for
which negative emissions are switched off and 1.5C is reached depends on the
TCRE. Note that for the low-end climate sensitivity in the AR5 range, no
negative emissions are required when rapid mitigation begins in 2020, as the
1.5C threshold is not exceeded.

Figure 1: Variants of the SSP5-34-OS scenario, which follows the RCP8.5
scenario with rapid mitigation beginning in 2020 and 2040 respectively. The
grey line corresponds to observed temperatures. The suppressed temperature
overshoot is shaded with a greater transparency, while the actual pathway
temperatures would be expected to follow with SRM is shaded with less
transparency. The necessary radiative forcing from SRM to keep temperatures to
below 1.5C in these overshoot scenarios is calculated and plotted in the lower
right-hand graph. Uncertainty in the temperature response and radiative
forcing required, as well as the emissions pathways, corresponds to the 66%
confidence interval for the TCRE from IPCC’s AR5 and is plotted with shading.
In some cases, this 66% interval intersects the axis. For example, in the case
where mitigation begins in 2020, and the TCRE is at the lower end of
estimates, no SRM or negative emissions are required to stay below 1.5C.

##  4 Variants on the overshoot scenario

To illustrate these points, we consider variants of the idealised SSP5-34-OS
case where emissions follow RCP8.5 until a given year (year_peak), then
decline and go negative with an average mitigation rate (mit_rate), eventually
reaching a negative emissions floor (neg_floor). We randomly sample 5000 sets
of year_peak, mit_rate, neg_floor and parameters at from “feasible” ranges,
and use these scenarios to drive FaIR in CO2-only mode. Note that FaIR
includes a sufficiently complex carbon cycle that its TCRE is path-dependent,
as discussed in Appendix B.

The mitigation rates are sampled from the interval [0.1, 2.3] GtCO2/yr/yr.
2.3GtCO2/yr/yr is the maximum mitigation rate achieved at any point in the
ambitious Low Energy Demand scenario [Grubler . (2018)]. The negative
emissions rate is sampled from the range [3.5, 30.0] GtCO2/yr. While the lower
end of this range assumes that sustained large-scale BECCS and DAC are
unavailable, and is determined by the land-use change negative emissions from
the Low Energy Demand scenario, the high end of this range is well beyond the
maximum considered feasible for BECCS by any 1.5C-compliant scenario, and
substantially higher than the largest negative emissions of 18.5GtCO2/yr in
the scenario database.

Figure 2: Trade-off between years of committment to solar radiation management
and the suppressed temperature overshoot in a range of idealised overshoot and
negative emissions scenarios, under the FaIR default ECS and TCR of 2.86K and
1.53K respectively. The idealised scenarios are simulated in FaIR, and are
generated by randomly selecting mitigation rate, maximum negative emissions
rate, and the year that mitigation begins from ranges of feasible values.
There is clearly a strong trade-off between the suppressed temperature
overshoot and the years of commitment to SRM in any of the scenarios. Even
with a very rapid mitigation rate and a high negative emissions floor,
scenarios that suppress more than 0.5K of temperature overshoot generally
require around a century of commitment to SRM.

Realmonte et al. (2019) compared the use of DAC across two integrated
assessment models, considering a maximum deployment by 2100 of 30GtCO2/yr and
a maximum scale-up rate of 1.5 GtCO2/yr/yr [Realmonte . (2019)]. This does not
imply that the most extreme scenarios sampled here are feasible - for example,
the rapid mitigation under the LED scenario may not be compatible with the
large-scale expansion in energy required for DAC on a 30GtCO2/yr scale. If
large-scale direct air capture is the predominant form of negative emissions,
then constraints are likely to be set by societal willingness-to-pay, energy
availability, and the speed that the technology can be scaled up and deployed.
Realmonte et al. explore some of the implications of deploying DAC at this
scale - which would also apply to corresponding peak-shaving scenarios.

The year that mitigation starts is sampled between [2020, 2060] – CO2
emissions follow RCP8.5 prior to this year. Scenarios that do not overshoot
1.5K, or which do not decline below 1.5K during the 750-year integration time
of the model, are filtered out. Under these scenarios, the trade-offs between
the suppressed temperature overshoot and the years of commitment to SRM are
illustrated in Figure 2.

This illustrates the trade-off between the duration of the commitment and the
magnitude of the suppressed temperature overshoot under a range of different
scenarios. Even if the most rapid mitigation and large-scale negative
emissions are assumed, shaving a peak of 2C to 1.5C requires a deployment in
excess of 80 years. If negative emissions are not available at sufficient
scale ($>2-3$GtCO2/yr), shaving a substantial ($>$0.5K peak) becomes
impossible on the 450-year timescale considered here.

Figure 3: As Figure 2, but with the UKESM1 ECS and TCR of 5.4K and 2.7K
respectively, at the higher end of any estimate for climate sensitivity. This
illustrates that the trade-off between geoengineering duration and the
suppressed temperature overshoot is arguably more favourable when climate
sensitivity is high. Because the TCRE is high, small CO2 excursions lead to
significant rises in temperature. This means that the CO2 corresponding to a
temperature rise of 0.5K could be removed in decades, rather than around a
century as in the low climate sensitivity case.

This is dependent on climate sensitivity, as illustrated by Figure 3. With the
UKESM climate sensitivity [Sellar . (2019)], temperature overshoots are larger
and occur more quickly for the same emissions trajectory. Higher climate
sensitivites mean smaller carbon budget overshoots correspond to larger
temperature overshoots. Given that the duration of SRM deployment is
equivalent to the time spent overshooting the carbon budget, added to the time
required to remove the carbon budget overshoot, higher climate sensitivity
implies shorter overshoot commitment times to avoid the same level of warming.

In other words, if climate sensitivity is low, then it is difficult to imagine
even a contrived scenario where SRM is both useful - suppressing a significant
amount of warming, say half a degree - and of short duration, over within a
few decades. Deployments of SRM that suppress a significant degree of warming
would need to be sustained for centuries, due to the significant additional
CO2 in the atmosphere for that degree of warming, which must be removed. This
tradeoff, however, is less dramatic when the climate sensitivity is higher.

The requirement to remove the entire overshoot of the carbon budget before
temperatures will decline to below the threshold results in a high sensitivity
of the duration of the commitment to any “floor” in achievable negative
emissions, especially for large overshoots. Table 1 shows how carbon budget
overshoots depend on the TCRE and the temperature overshoot. If geoengineering
is to be deployed as part of a “peak-shaving” strategy, then the cost of
removing carbon dioxide on this scale must be factored in to the cost of that
strategy.

TCRE $\downarrow$ /Overshoot $\rightarrow$ | 0.5K | 1K | 1.5K | 2K
---|---|---|---|---
0.92K (17%) | 543 | 1086 | 1629 | 2172
1.26K (33%) | 396 | 793 | 1189 | 1586
1.5K (50%) | 333 | 666 | 999 | 1332
1.71K (67%) | 292 | 584 | 876 | 1168
2.05K (83%) | 244 | 487 | 731 | 975
Table 1: Implied carbon budget overshoots (GtCO2) for various levels of
temperature overshoot and TCREs from the AR5 WG1 distribution. These
correspond to the cumulative net-negative emissions required to return to 1.5K
in each overshoot case.

##  5 Conclusion

This exercise is not intended to span the entire conceivable scenario space of
SRM deployment, nor to evaluate the circumstances under which SRM deployment
might result in improved climate outcomes, at least for some. Instead, we aim
to illustrate tradeoffs inherent in “peak-shaving” scenarios: between the
implied commitment time of SRM deployment and the suppressed temperature
overshoot, as well as to outline implications for the mitigation rate and
scale of negative emissions that are required to ensure the deployment is
limited to a particular timescale.

This analysis is arguably conservative, both for the timescales and negative
emissions commitment concerned. It assumes that the carbon budget is exceeded
rapidly, with high emissions, and that mitigation begins immediately after
emissions peak, which both reduce the carbon budget and temperature overshoot
compared to scenarios where changes in global emissions occur more slowly.
Here, we consider only the scale of global net-negative emissions. However,
additional negative emissions will likely be required even to reach net-zero
emissions [Anderson Peters (2016)], so the actual requirement for negative
emissions will be higher.

This analysis also allows for the largest mitigation and negative emissions
rates found in the ambitious 1.5C-compliant literature. In a world where SRM
is deployed, assuming it is effective in small doses at reducing climate risks
[Irvine . (2019)], the perceived urgency of reducing emissions and deploying
negative emissions at scale may be reduced, a form of “mitigation deterrence”.

Finally, as mentioned, the analytic section assumes that the TCRE is the same
for both positive and negative emissions, while more detailed carbon-cycle and
climate modelling suggests that the TCRE is path-dependent and that negative
emissions are less effective at reducing temperature than positive emissions
are at increasing temperature, especially when the cumulative emissions are
large. Where this is important, it is likely to increase the timescales
considered, unless the realised warming fraction is high.

Unless climate sensitivity is at the high end of estimates, deploying SRM to
shave a substantial peak over 1.5C – say, stabilising temperatures at 1.5C
when they might have peaked at 2C – is likely to require a commitment to SRM
on the order of a century, and to large-scale negative emissions, cumulatively
at least 350GtCO2. We find that, under median estimates of climate
sensitivity, deployments of SRM that are complete within a few decades are
unlikely to shift global mean temperatures a great deal beyond multidecadal
variability. Cost-benefit analyses that assume SRM deployment is not
indefinite must consider the cost and feasibility of cumulative negative
emissions equivalent to the exceedance of the carbon budget, as well as the
timescales required to draw these emissions down.

During the COVID-19 pandemic, to avoid loss of life, many of the world’s
governments have been implemented severe social distancing measures in order
to “flatten the curve” of infection. We see an analogy between these two
situations. The R-rate during the uncontrolled initial phase of the epidemic
is analagous to our emissions before net-zero is reached, and the rate of
negative emissions we can attain is analagous to the R-rate in lockdown. The
curves of new case numbers in countries that implemented lockdowns resembled
the asymmetric parabolas of section 2.2, as in general the R during the
uncontrolled epidemic phase was larger than the reciprocal of the R during
lockdown. Therefore, each additional week of delay prior to lockdown required
several weeks of commitment to lockdown to return infection numbers to their
previous levels, as well as entailing a greater overall burden in terms of
death and economic damages. Countries that were able to implement early
lockdowns, prior to substantial spread of the disease, were generally able to
adopt less stringent, less expensive, and shorter-duration social distancing
measures, while experiencing a lower rate of mortality. Similarly, the more
rapidly mitigation takes place, the shorter any long-term committment to
costly geoengineering measures would be.

In both cases, the inescapable conclusion is that the optimal scenario
involves early, swift and decisive action to tackle the root of the problem,
which obviates the need for a long-term commitment to a costly suppression
policy. When this becomes impossible, the trade-offs illustrated in this paper
must be confronted.

##  Appendix A Double-exponential model

Considering the double-exponential idealised model for emissions, with
negative emissions that increase towards a floor and emissions that decay
simply with mitigation, emissions after mitigation begins can be expressed as:

$E(t)=C_{max}\exp(-m\textsubscript{1}t)-n(1-\exp(-m\textsubscript{2}t))$ (16)

where

$m\textsubscript{1,2}={\ln}(2)/t\textsubscript{1,2}$ (17)

and t1,2 are the timescales for halving (gross) emissions and scaling up the
deployment of negative emissions to half of its maximum value, respectively.

Figure 4: Illustration of the tradeoff between duration and height of
overshoot in the double-exponential emissions model, taking TCRE to be
1.4K/GtC as in the FaIR model. Mitigation and negative emissions deployment
rates are both chosen in the interval [1, 7]% and three negative emissions
floors are sampled. The quadratic model is approximated as in Equation 6,
assuming that the emissions when the threshold is crossed are the peak of
emissions on the way up, and the negative emissions floor on the way down.

As an example, one could consider emissions peaking close to their present
value of 40GtCO2/year, then declining (through conventional mitigation) at a
given annual average percentage while negative emissions are scaled up to
their maximum value. Different examples illustrate the trade-off between the
implied duration of and height of the peak. For this example, we choose
mitigation rates between a maximum of 7% per annum, which corresponds to
halving emissions in just under a decade, and 0.5%, which corresponds to
halving emissions in around 140 years. This is illustrated in Figure 4. Note
the tradeoff between the duration of geoengineering and the height of the peak
shaved in all scenarios. The quadratic approximation, where mitigation occurs
at a constant rate in terms of gigatonnes of carbon, is also illustrated and
its gradient depends entirely on the rate of negative emissions, rather than
the mitigation and deployment rates.

It might be argued that the best case for peak-shaving is a scenario where
conventional mitigation is slow or expensive, but negative emissions can
eventually be deployed more rapidly - for example, if very cheap energy
becomes available at scale later in this century. These scenarios are sampled
here, but even in this extreme case shaving a peak of 0.5K involves 70+ years
of geoengineering.

##  Appendix B Path-dependence of the TCRE

Depending on the emissions scenario and the model used, the approximation that
global mean temperature is proportional to cumulative emissions - and hence
that TCRE is constant and well-defined - can be less accurate. Physically, the
concept of a constant TCRE depends on the approximate cancellation of two
factors: residual surface warming as the ocean comes into equilibrium with the
high-CO2 atmosphere, and uptake of carbon dioxide by the ocean sink and
biosphere [Allen . (2009)]. Consequently, factors that vary between models -
such as the rate of uptake of CO2 by the oceans and the biosphere, as well as
the timescale and magnitude of long-term warming as the deep ocean comes into
equilibrium with the atmosphere - can introduce a hysteresis effect in
temperature as cumulative emissions are reduced, and hence an apparent time-
dependence of the TCRE.

Figure 5: Ratio of TCRE during positive and negative emissions periods for
various Realised Warming Fractions (TCR/ECS ratios). This was calculated by
simulating simple emissions trajectory where emissions linearly decrease to
zero and then become negative at the same rate, until cumulative emissions are
zero. Differences both in the total anthropogenic CO2 excursion and the RWF
give rise to different levels of hysteresis. The approximation that the TCRE
is constant for removals and additions of CO2 \- which is relied on for the
simple algebraic models, but not the FaIR analysis - breaks down for RWF far
from the FaIR default of 0.53, or for large anthropogenic CO2 excursions.
Lower RWF or greater CO2 excursions would result in a greater implied duration
of geoengineering to maintain temperatures below a given threshold.

This behaviour can be modelled with FaIR by using idealised emissions
trajectories and varying the TCR/ECS ratio, known as the ”Realised Warming
Fraction” (RWF). RWFs for the CMIP5 models lie in the range 0.45-0.7, with an
average of 0.55: observationally constrained estimates tend to fall in the
upper half of this range. [Millar . (2017)] In these scenarios, emissions
linearly decrease from an initial maximum value E to -E over a period of 200
years, resulting in net cumulative emissions of zero. The TCRE is evaluated
for the ascending and descending branches by calculating the gradient of the
temperature-cumulative emissions curve, then averaging across the periods of
positive and negative emissions respectively. We plot cases for three peak
cumulative emissions trajectories. The ratio of the TCRE in the ascending and
descending branches is plotted against the RWF for several emissions scenarios
in Figure 5.

If the RWF is low, or the total carbon perturbation is high, the assumption of
linearity is less accurate. In these cases, global mean temperature can
continue to rise even as emissions go negative, as the large “unrealised”
warming dominates the effect of declining CO2 concentrations, reducing the
average TCRE across this descending branch. This will result in the analytical
formulae of Section 2 underestimating the commitment time to peak-shaving
geoengineering. By contrast, if the RWF is high, and the carbon perturbation
is small enough, the TCRE ratio can be less than 1: in other words, natural
removal of CO2 by sinks exceeds the effect of long-term warming, which results
in faster cooling during the negative emissions branch, as concentrations
decline more quickly.

The non-constant nature of the TCRE is an important caveat for the early,
analytical calculations of the trade-off between duration and the height of
the peak shaved - especially if the RWF is at the extreme ends of current
estimates, or the carbon perturbation is high. However, this effect is small
for RWF of around 0.5-0.6, similar to the multi-model CMIP5 average of 0.55
and the FaIR default of 0.53.

###### Acknowledgements.

T.H. was funded for his DPhil by the Natural Environment Research Council’s
Doctoral Training Partnership for Environmental Research.

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