Stratospheric aerosol injection (SAI), as a possible supplement to emission reduction, has the potential to reduce some of the risks associated with climate change. Adding aerosols to the lower stratosphere would result in temporary global cooling. However, different choices for the aerosol injection latitude(s) and season(s) have been shown to lead to significant differences in regional surface climate, introducing a design aspect to SAI. Past research has shown that there are at least three independent degrees of freedom (DOFs) that can be used to simultaneously manage three different climate goals. Knowing how many more DOFs there are, and thus how many independent climate goals can be simultaneously managed, is essential to understanding fundamental limits of how well SAI might compensate for anthropogenic climate change, and evaluating any underlying trade-offs between different climate goals. Here, we quantify the number of meaningfully independent DOFs of the SAI design space. This number of meaningfully independent DOFs depends on both the amount of cooling and the climate variables used for quantifying the changes in surface climate. At low levels of global cooling, only a small set of injection choices yield detectably different surface climate responses. For a cooling level of 1–1.5

Reducing emissions of

Choosing where and when to inject aerosols can be thought of as a design problem

In this study, we estimate the number of DOFs of the design space for SAI. Knowing how many DOFs there are in the design space quantifies the number of independent climate goals that can be managed simultaneously by a SAI strategy. In order to be managed simultaneously, those independent climate goals cannot conflict. (For example,

The aerosols will primarily stay in the same hemisphere where they are injected and be transported mostly poleward by the stratospheric Brewer–Dobson circulation

There are two distinct steps in the analysis herein. The first step is to consider how different the spatiotemporal AOD patterns are for different injection choices. And second, to know whether the AOD patterns from two different injection choices are sufficiently similar to treat them as effectively equivalent, or sufficiently distinct to treat them as two separate DOFs, one needs to relate how similar or dissimilar the patterns of AOD are to how similar or dissimilar the resulting climate responses are. Identifying the number of DOFs only needs to consider injections that produce meaningfully different climates; herein, we define “meaningfully different” based on the ability to detect the difference in climate after 20 years, given natural variability – and this threshold clearly depends on the choice of climate variables to be considered and the amount of cooling desired. For example, injecting aerosols at 30

The next section describes the climate model and simulations used. Section

All simulations in this study were conducted using the Community Earth System Model version 1 with the Whole Atmosphere Community Climate Model as the atmospheric component, CESM1(WACCM). CESM1(WACCM) is a fully coupled Earth system model which includes atmosphere, ocean, land, and sea ice components

The 29 injection choices that we considered in our analysis for AOD patterns are shown in light green. The vertical axis shows the injection season of each injection choice, either injecting in only one season (DJF, MAM, JJA, or SON) or constantly throughout the year (ANN). The horizontal axis shows the injection latitude: from left to right, they are 60, 45, 30, 15

Injection design of the five existing SAI simulations analyzed in this study.

To assess the range of possible spatiotemporal patterns of AOD arising from different injection choices, we sample 29 possible choices in the AOD design space, including injections at low and middle latitudes as described by

Spatiotemporal AOD patterns of spring injections at

In addition to these shorter simulations that we use to assess the range of possible spatiotemporal AOD patterns from different injection choices, five sets of solar geoengineering simulations from existing studies

In this section, we consider 29 different injection choices, sampling from different latitudes and seasons of injection, as well as three additional cases that we use to verify that the set of 29 is sufficiently complete. The AOD pattern from a given injection choice (a given latitude and season of injection) is largely determined by the stratospheric circulation and aerosol lifetime, which constrains what spatiotemporal patterns are achievable

To describe any pattern of AOD, we consider the zonal-mean pattern as a function of both latitude and time of year. In order to treat these two dimensions consistently and yield AOD patterns independent of our sampling resolution in each dimension, we weight the monthly mean zonal mean AOD at each latitude and month by the corresponding incoming solar energy (petajoules) at the top of the atmosphere (TOA). We then represent the weighted spatiotemporal AOD pattern from each injection choice as a vector

With the vector representation explained above, our goal is to select a subset from the set of 29 injection choices such that any possible AOD pattern can be adequately represented by a linear combination of injection choices from this subset. Any given subset of linearly independent injection choices does not produce an orthogonal set but could be orthogonalized if needed. Determining the dimension of the set necessary to meet this goal is equivalent to determining the number of DOFs of SAI.

First, we need to verify that our set of 29 injection choices sufficiently describes all of the possible AOD patterns of other injection choices that we have not simulated. To do so, we choose three additional verification cases, which are annual injections at 7.5, 22.5, and 37.5

Mathematically, the linear combination that is most similar to the simulated pattern of AOD is the projection of its vector representation onto the space formed by the 29 injection choices. Solving the best approximation of the pattern of AOD can be formed as a constrained linear least-square problem of finding the projection onto the set of 29 injection choices:

By calculating the angle between the vector representation of the simulated AOD pattern and the vector representation of the approximated AOD pattern, we can assess how similar the simulated and approximated AOD patterns are. For annually constant injections at 7.5, 22.5, and 37.5

Comparison of simulated AOD patterns and the best approximation to these AOD patterns obtained from a linear combination of other injection choices within the set of 29 cases considered here. From top to bottom are the spatiotemporal AOD patterns for injections at 7.5, 22.5, and 37.5

With the set of 29 choices of injection locations and times, we can evaluate a wide range of possible selections of injection choices. Sets with different numbers of injection choices as well as different selections of the same number of injection choices will do a better or worse job at spanning the space of possible AOD patterns. One way to quantify how well the overall space of possible AOD patterns can be approximated by a particular subset of

In choosing subsets, we enforce hemispheric symmetry such that if an injection choice in one hemisphere is included, then the corresponding choice in the opposite hemisphere is also included (e.g., MAM in the Northern Hemisphere and SON in the Southern Hemisphere). While the seasonal circulation patterns are not exactly symmetric between the hemispheres, they are sufficiently similar that this is a reasonable simplification that reduces the number of sets to search over. For injections at the Equator, we similarly either include or do not include opposite seasons (e.g., DJF and JJA, or MAM and SON). With hemispheric symmetry, the only way to have a set with an odd number of DOFs is to include annually constant equatorial injection; we revisit this case in the discussion.

Mathematically, the steps above can be described as follows. First, for each subset

Taking

Angles (in degrees) between each unselected injection choice and a set of four injection choices (in orange): summer injection at 30, 15

Among all possible subsets of

Still using

A schematic diagram showing how to find the smallest maximum angle

For each possible value of

The maximum angle

Figure

To estimate this relationship, we consider the different strategies described in Table

The AOD pattern and corresponding surface climate responses are obtained by averaging over all available ensemble members for each strategy in Table

To estimate how large a change in the spatiotemporal pattern of AOD is needed to obtain a detectably different pattern of surface climate response, we consider detectability over a 20-year period. Therefore, we normalize the surface temperature and precipitation changes by the variability in 20-year averages, calculated from the across-ensemble variability from 2010–2029 in the RCP8.5 emissions scenario, where 21 ensemble members are available. If the variability were uncorrelated from year to year, this value would simply be a factor of

To analyze the differences in AOD and surface climate for different strategies, we define the AOD space, temperature space, and precipitation space. In Sect.

In the AOD space, temperature space, and precipitation space defined above, we evaluate the differences between each possible pair of the five SAI strategies described in Table

To estimate the relationships between

Data used here are from SAI designs that were considered in previous studies. Although they are not designed to span either the overall AOD design space or the surface climate design space, these available simulations do provide a useful set of data for analyzing the relationship between how similar or dissimilar the AOD patterns are and how similar or dissimilar the surface climate responses are.

Panels

To evaluate how different the surface climate responses are, we first perform Welch's

Panels

Here, we define that two strategies are considered to be detectably different if the difference in temperature or precipitation responses between them are detectable at a 95 % confidence level over a 20-year period on more than 5 % of the Earth's area. With the temperature and precipitation normalized by the standard deviation of 20-year means, temperature or precipitation responses at any grid point will be detectably different if the difference between the normalized data is more than 2 SD (standard deviations). To obtain a global aggregate metric, we note that roughly 5 % of the Earth's surface area has a temperature difference more than double the overall temperature distance

We compare the difference in temperature and precipitation responses between GLENS and iSpring and between GLENS and EQ, and show how the difference changes with levels of cooling using detectability plots, as shown in Figs.

From the detectability plots that compare these different SAI strategies, it is clear that the detectability of different injection strategies depends on both the level of cooling and the choice of climate variables. With the underlying assumption of linearity for surface climate responses, we estimate the difference in temperature and precipitation responses between GLENS and EQ at reduced levels of cooling (Fig.

Panels

As shown in Fig.

The cut-off AOD angle

In the previous sections, we estimate the relationship between the number of DOFs included and the maximum error in approximating AOD, and the relationship between the AOD angle and the level of cooling at which the resulting temperature or precipitation responses could be expected to be detectably different. In this section, we combine these two to estimate how many meaningfully independent DOFs there are as a function of the levels of cooling.

As changes in temperature are more detectable than those in precipitation, the extent to which two AOD patterns are sufficiently similar, and thus the number of DOFs in the SAI design space, is determined by the temperature response. Using Eq. (

The minimum number of DOFs,

The minimum number of DOFs corresponding to the worst-case error in approximating AOD,

As shown in Table

For a cooling level of 1

AOD patterns from the best set of six injections:

For a cooling level of 1.5

The SAI simulations analyzed in the previous sections are all high-altitude injections (6–7 km above the tropopause).

AOD patterns produced by low-altitude annually constant injection of 6 Tg yr

Angle between the AOD vector of each high-altitude injection and the set of all low-altitude injections.

Previous studies have shown that different choices of stratospheric aerosol injection latitudes and seasons lead to different surface climate responses. Choosing where and when to inject aerosols to the stratosphere to meet different climate goals can be considered a design problem. Previous studies have concluded that there are at least 3 DOFs; that is, at least three independent climate goals can be simultaneously met. These three – basically the global mean aerosol burden, the interhemispheric difference, and the Equator-to-pole difference – were motivated by physical intuition regarding stratospheric transport, which will ultimately constrain how many independent DOFs are achievable through different injection choices. A key observation is that the number of DOFs effectively depends on the amount of global cooling provided by SAI, because for a small amount of cooling, the difference in the climate response for different strategies may not be detectable. As the amount of cooling increases, the number of meaningfully independent DOFs increases. For a cooling level of 1–1.5

The choice of 20-year average, as well as the choice of 95 % confidence threshold over 5 % of the Earth's area, affects the number of DOFs that are “meaningfully different”. Considering the responses over a longer period of time might introduce additional DOFs. To date, current climate models have relatively high confidence in predicting temperature responses but have lower confidence in predicting circulation-related responses and much lower confidence in predicting regional-scale circulation-related extreme events

When evaluating the design space, we do not consider injections at different longitudes and additional altitudes beyond those evaluated in Sect.

Our estimation of the number of DOFs provides useful guidance to bound the number of injection choices that need to be considered when evaluating the range of possible different SAI strategies and the trade-offs among them. If only a small amount of cooling is needed from implementing SAI, a small set of selected injection choices would be sufficient to capture the range of possible resulting climate responses and evaluate how different those climate responses could be. As all possible injection choices form an extremely high-dimensional design space, only considering the meaningfully independent injection choices significantly reduces the dimension of the design space.

The number of meaningfully independent DOFs determines the number of independent climate metrics that SAI can manage simultaneously. Thus, for a cooling level of 1–1.5

It is important to note that all of these results are obtained from a single climate model. Other climate models may produce different numerical results. Nonetheless, the number of independent DOFs needed to span the range of possible different stratospheric AOD patterns can be reasonably expected to remain consistent as the transport of aerosols are constrained by stratospheric circulation. We make several simplifying approximations in order to make analysis tractable, particularly to estimate the relationship between how similar or dissimilar two patterns of AOD are and how similar or dissimilar the corresponding surface climate responses are; future research could explore the impact of these approximations. First, we only consider changes in temperature and precipitation, and we only consider changes in annual mean rather than shifts in seasonality, which could matter at high latitudes in particular

A key outcome of this study is that further research should be conducted to explore alternate SAI designs that can manage more than three and up to eight independent climate metrics simultaneously, and to compare the resulting climate responses and associated trade-offs. Research into more than eight is less policy-relevant, simply because any hypothetical deployment scenario would not reach more than 1.5

Data for the new simulations presented in this study (specifically, monthly AOD for spring injections at 60, 45

YZ conducted all analyses and wrote the paper with editing by DGM, BK, and DV. YZ and DGM conceived the study with input from all authors, and DV assisted with conducting simulations.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Resolving uncertainties in solar geoengineering through multi-model and large-ensemble simulations (ACP/ESD inter-journal SI)”. It is not associated with a conference.

The authors would like to acknowledge high-performance computing support from Cheyenne (

This research has been supported by the National Science Foundation (grant nos. CBET-1818759, CBET-2038246, and CBET-1931641).

This paper was edited by Roland Séférian and reviewed by two anonymous referees.