The Community Earth System Model, version 2 (CESM2), is a state-of-the-art Earth system model developed by the U.S. National Science Foundation's National Center for Atmospheric Research. We run the model with fully-coupled atmosphere, land, ocean, sea ice, land ice, and river runoff components. For the atmosphere component, we use the Whole Atmosphere Community Climate Model WACCM6,; this configuration, CESM2(WACCM6), contributed to Phase 6 of CMIP and Phase 6 of GeoMIP and has been used extensively to model SAI e.g.,. We run the model with a horizontal resolution of 0.9° latitude by 1.25° longitude, and WACCM6 uses 70 vertical layers with a model top at approximately 140 km (4.5×10-6 hPa). This configuration includes comprehensive tropospheric, stratospheric, mesospheric and lower thermospheric (“TSMLT”) chemistry and prognostic aerosol physics and chemistry using the Modal Aerosol Module MAM4,, which includes Aitken, accumulation, and coarse mode representation for sulfate aerosols, with some modifications to modal size distributions introduced by . For the ocean component, we use the Parallel Ocean Program version 2 POP2,, and for the land component, we use the Community Land Model version 5 CLM5,.

All of the simulations described in the next section (novel, and previously published) use this same configuration. For simulations with SAI, the SAI is implemented by placing SO₂ directly into a gridbox at pre-defined latitudes, approximately 5 km above the tropopause.

In this portion of the study, we consider nine CESM2 experiments: four simulations of scenarios without SAI (all previously published elsewhere), and five simulations of SAI (two previously published, and three novel). These simulations are summarized in Table  below.

The four no-SAI simulations are PI control, 1 % CO₂, Historical, and SSP2-4.5. PI control refers to the 500-year CMIP6 preindustrial control simulation. 1 % CO₂ is a CMIP6 GHG forcing scenario, branching from year 70 of the PI control simulation, in which CO₂ concentrations increase by 1 % annually. Historical refers to CMIP6 simulations of the 1850-2014 historical period. SSP2-4.5, part of the Shared Socioeconomic Pathway framework used in CMIP6 , is a moderate warming, “middle-of-the-road” projection of future climate change in which emissions do not deviate substantially from historical trends. The SSP2-4.5 simulations considered here begin in model year 2015, branching from Historical, with five ensemble members running until 2100 and the other five running until 2070.

The two previously-published SAI simulations we analyze are ARISE-SAI-1.5 and G6-1.5K-SAI ; both branch from the SSP2-4.5 emissions scenario in model year 2035 and use SAI to maintain reference period temperatures corresponding to the 2020–2039 SSP2-4.5 average. G6-1.5K-SAI injects at 30° N and 30° S in equal quantities, with injection amounts chosen using a feedback algorithm (described below) to maintain global mean temperature (T0) only; ARISE-SAI-1.5 injects at four latitudes (30° N, 15° N, 15° S, and 30° S) in different quantities, using a more complex feedback algorithm to manage not only (T0) but also the interhemispheric temperature gradient (T1) and the equator-to-pole temperature gradient (T2). ARISE-SAI-1.5 runs through model year 2069 (35 years), and G6-1.5K-SAI runs through model year 2084 (50 years).

The three novel SAI experiments we present are the G2-SAI experiments, individually named G2-SAI-1DOF, G2-SAI-3DOF, and G2-SAI-hybrid; the first two are named after their respective injection strategies, while the “hybrid” experiment combines elements from each of the other two (described in detail in the next section). Each branches from the PI control simulation in the 70th year (the same year that 1 % CO₂ begins) and runs for 150 years with 1 % CO₂ forcing, using SAI to maintain PI control temperatures averaged over 51 years centered on the branch year (years 45 to 95 of the PI control simulation, inclusive). The G2-SAI simulations are designed to mirror the contemporary SAI strategies used by the other experiments considered here, and others described in Sect. 1: the 1DOF simulation injects in equal amounts at 30° N and 30° S latitude, with the total amount chosen to maintain global mean temperature (T0) only, the same strategy as G6-1.5K-SAI; and the 3DOF and hybrid simulations inject in different amounts at 30° N, 15° N, 15° S, and 30° S to maintain T0, T1, and T2 simultaneously. The 3DOF and hybrid simulations have the same temperature targets, but there are differences in how the injection rates are chosen to meet these targets, described in the next section. Temperature targets for all SAI experiments are documented in Table .

As described in the previous section and in Table , The G2-SAI-3DOF and G2-SAI-hybrid experiments use the same scenario, set of injection locations, and temperature objectives. However, as we will show in the next section, the injection rates used to meet these targets – and subsequent impacts on the Earth system – are very different. In this section, we explain how injection rates are chosen via algorithm, and why and how the algorithms used for the 3DOF and hybrid experiments are different. This section may not be of interest to the general reader, and those mainly interested in the impacts of different injection strategies on the Earth system may proceed to Sect. 3.

G6-1.5K-SAI and ARISE-SAI-1.5 both use feedforward-feedback proportional-integral control algorithms (colloquially, “feedback algorithms”) to choose injection rates to maintain desired objectives. This approach was first developed to maintain desired T0, T1, and T2 temperature targets with 15 and 30° N/S injections in CESM1 , and the same framework has since been commonly used in simulation design, as discussed in Sect. 1. The full structure and design of the algorithm is described in the aforementioned studies, but to briefly summarize, the feedback controller consists of constants of proportionality, called gains, which determine how much SO₂ to inject each year. These gains consist of feedforward gains and feedback gains. Feedforward gains prescribe linearly increasing injection amounts based on the expected temperature change (see Fig. 1) and the known sensitivity of temperature to injection in the model, i.e., a “best guess” of how much injection will be needed over time to meet the targets. Feedback gains then adjust the injection rates each year based on the error (the difference between the actual and desired model behavior) over the course of the simulation.

For such a controller, the number of outputs equals the number of inputs, also called the number of degrees of freedom (DOF). In other words, each pair of one feedforward + one feedback gain allows for the control of one climate objective. G6-1.5K-SAI controls for T0 only (one objective), and therefore requires a “1-DOF” controller with one feedforward gain and one feedback gain. The gains determine the total SO₂ injection rate each year, which is divided evenly between the 30° N and 30° S injection sites. ARISE-SAI-1.5 manages T0, T1, and T2 simultaneously and therefore uses a more complicated “3-DOF” controller with three sets of feedforward and feedback gains, which are named “ℓ0”, “ℓ1”, and “ℓ2” (corresponding to the respective temperature metrics they manage). These gains are handled by the controller in sequence: first, the ℓ0 gains determine how much SO₂ is placed at 15° N + 15° S to manage T0; second, the ℓ1 gains determine whether any of this SO₂ should be diverted to either 30° N + 15° N or 30° S + 15° S to preferentially cool one hemisphere and manage T1; lastly, the ℓ2 gains determine whether any 15° N + 15° S injection should be shifted to 30° N + 30° S injection to preferentially cool the high latitudes and manage T2. The result is some combination of injection across the four latitudes that attempts to meet all three goals simultaneously, but prioritizing T0 first, T1 second, and T2 last. In ARISE-SAI-1.5, this combination largely converges to 15° S injection; this happens because 15° S injection tends to cool the planet relatively evenly in CESM2, but that behavior is model-specific see.

All three G2-SAI simulations also use feedforward-feedback algorithms to choose injection rates: G2-SAI-1DOF uses a 1-DOF algorithm as in G6-1.5K-SAI, and G2-SAI-3DOF uses a 3-DOF algorithm as in ARISE-SAI-1.5. G2-SAI-hybrid also uses a 3-DOF algorithm, but with some changes relative to G2-SAI-3DOF, with the following rationale. T0 and T2 changes in the 1 % CO₂ and SSP2-4.5 scenarios are similar for the first 50 years of injection, and the trends continue similarly thereafter for 1 % CO₂ (Fig. 1); however, while T1 changes are similar initially, T1 behavior is very different in the two scenarios after the first ∼20 years. T1 is highly variable in the no-SAI scenarios, likely due to long-term ocean processes in the model. Because of the nonlinear nature of long-term T1 evolution in the 1 % CO₂ scenario, we use two distinct sets of controller gains in our two simulations with 3-DOF algorithms to more fully explore the design space. In G2-SAI-3DOF, all three pairs of ℓ0, ℓ1, and ℓ2 feedforward and feedback gains are used. In G2-SAI-hybrid, all three feedback gains are used, but the ℓ1 and ℓ2 feedforward terms are set to 0. As a result, the 3DOF controller prioritizes 15° S injection, while the hybrid controller “defaults” to 15° N + 15° S injection and has more freedom to adjust the injection strategy as the simulation evolves (this simulation eventually converges to mostly 30° N + 30° S injection, as shown in Fig. 4 below; the term “hybrid” is chosen here to reflect the combined aspects of both the 3DOF and 1DOF strategies).

In Fig. , we present the total SO₂ injection rates required to maintain the temperature targets (Fig. a) and cooling per unit SO₂ injection (Fig. b) for the five SAI scenarios. As seen in Fig. a, the rates of increase in global mean temperature (T0) to be offset are similar in the first 50 years of the SSP2-4.5 and 1 % CO₂ scenarios; however, G2-SAI scenarios begin injecting at the same time that temperatures begin increasing (year 70 of PI control, or year 1 of 1 % CO₂) whereas the G6-1.5K-SAI and ARISE-SAI-1.5 scenarios are slightly offset (i.e., begin injecting in 2035 to control to ∼2030 conditions). SO₂ injection rates increase more than linearly for G2-SAI, following the warming trend in the 1 % CO₂ scenario (Fig. a). The cooling efficiency per unit injection per year is similar for SAI in the two scenarios for regimes under 10 Tg yr⁻¹ (approximately 1 °C per 10 Tg yr⁻¹ injected); above 10 Tg yr⁻¹, G6-1.5K-SAI cools slightly more efficiently than the average of the G2-SAI scenarios, but this could be variability given the small sample size. For G2-SAI, the injections cool less efficiently above 10 Tg yr⁻¹ (first 50 years, 9.5 Tg yr⁻¹ per 1 °C; years 51-100, 11.6 Tg yr⁻¹ per 1 °C; last 50 years, 13.4 Tg yr⁻¹ per 1 °C). Additional diagnostics concerning injection rates and cooling efficiency are presented in Sect. S1, Table S1 in the Supplement.

In Fig. , we present timeseries of T0, T1, and T2 for all SAI and no-SAI scenarios. All 3-DOF SAI scenarios (ARISE-SAI-1.5, G2-SAI-3DOF, and G2-SAI-hybrid) manage T0 and T2 well relative to the amount of warming in their respective scenarios, but interannual variability in T1 is higher relative to long-term change. The hemispherically symmetrical 1-DOF injection strategies (G2-SAI-1DOF and G6-1.5K-SAI) overcool the Northern Hemisphere initially, resulting in negative T1 and T2 error (i.e., NH and poles too cold), but G2-SAI-1DOF exhibits both positive T1 and T2 error (i.e., NH and poles too warm) by the end of the 150-year simulation.

Figure  shows differences in how the different SAI simulations meet their respective targets, and the resultant surface temperature distributions. The 1-DOF strategies (G6-1.5K-SAI and G2-SAI-1DOF) inject equal amounts in both hemispheres to control for T0 only (Fig. a–b), with similar distributions of stratospheric 550 nm aerosol optical depth (henceforth “AOD”; Fig. f). The three 3-DOF simulations (ARISE-SAI-1.5, G2-SAI-3DOF, G2-SAI-hybrid) distribute injections as needed to maintain T0, T1, and T2 simultaneously to the greatest extent possible. ARISE-SAI-1.5 converges to mostly 15° S injection, supplemented by some 30° S and NH injections (Fig. e). G2-SAI-3DOF, designed similarly, converges to a similar distribution, with some long-term variation visible in the extent of 15° N + 15° S versus 30° N + 30° S injection (Fig. d); AOD distributions in years 16–35 of injection for G2-SAI-3DOF and ARISE-SAI-1.5 likewise have similar shapes (Fig. g). The G2-SAI-hybrid simulation injects less into the SH and more into the NH compared to G2-SAI-3DOF, both initially and in the long-term. This results in slightly less overcooling in the SH and somewhat more initial overcooling in the NH (similar to 1DOF) for G2-SAI-hybrid compared to G2-SAI-3DOF. The larger SH injections in G2-SAI-3DOF may be critical for initiating stronger feedback in the cloud and corresponding warming responses, which required continued larger SH injections in following years. A more detailed investigation of this feedback warrants further study and may have contributed to changes in the AMOC (see below). Over time, the G2-SAI-hybrid injection strategy shifts towards symmetrical injection, and by the end of the experiment, the injection distribution and AOD more closely resemble that of the 1DOF strategy (Fig. c, h–j). The 3DOF and hybrid strategies meet the same set of temperature targets, but the zonal mean temperature distributions are different (Fig. n–o): the 3DOF strategy has warmer NH subtropics and colder higher latitudes, while the hybrid strategy has colder subtropics and warmer high latitudes.

The differences in the NH extra-tropical temperature responses between the G2-SAI-3DOF and G2-SAI-hybrid simulations are partly driven by the corresponding differences in the AMOC response (Fig. ). The AMOC – Atlantic Meridional Overturning Circulation – is a major feature of the Earth’s circulation: the upper branch carries warm water poleward from the tropics, and the lower branch carries cooler water towards the tropics. Observations and simulations find that the strength of the AMOC is declining under global warming e.g., due to reductions in surface heat fluxes and salinity in the North Atlantic, which reduces the rate of overturning. Studies have found that SAI can mitigate, prevent, or reverse the trend of AMOC decline; attribute the impacts of GLENS and ARISE-SAI-1.5 on AMOC to changes in surface heat fluxes, and attribute changes to AMOC in G6sulfur to changes in surface ocean-air temperatures, while also finding that freshening from summer sea ice melt may also play a role in its weakening. tested SAI at separate latitudes individually in CESM2(WACCM6), and found that the effect of SAI on AMOC strength was strongly dependent on the latitude of injection; while any of the considered injection latitudes (ranging from 45° N to 45° S) increased AMOC strength relative to the SSP2-4.5 baseline, injections in the Northern Hemisphere had a much stronger impact on AMOC strength and associated predictors, such as North Atlantic sea surface temperatures (SSTs), surface salinity and density.

In agreement with the aforementioned studies, we observe a substantial decrease in the AMOC strength under 1 % CO₂ (Fig. c). Such AMOC weakening is consistent with the reduction in the North Atlantic mixed layer depth (Fig. d) and surface density (Fig. f), driven both by the reductions in the North Atlantic surface heat flux (Fig. e; negative values indicate energy loss from the ocean) and salinity (Fig. h). Ocean temperatures in the North Atlantic (Fig. g) increase initially (∼0–50 years) under global warming but then decrease again as the AMOC weakens and less warm tropical water is transported poleward.

The 3DOF strategy injects largely in the SH throughout the experiment (Fig. d). Under this intervention, the decline of the AMOC is slowed down relative to 1 % CO₂ forcing alone, but not prevented entirely; similar trends are seen in mixed layer depth and North Atlantic surface heat flux, density, and salinity, and in the absence of net global warming, North Atlantic ocean temperatures only decrease as AMOC strength declines. Changes in AMOC strength further impact NH surface temperature changes and modulate the SAI injection rates needed to maintain the temperature targets, one of the feedbacks identified by : the weakening AMOC results in a slower transfer of heat from the tropics to the NH mid and high latitudes, decreasing NH vs. SH temperature gradient (T1) and encouraging more injection in the tropics and SH and less at 30° N (Fig. a). In contrast, the 1DOF strategy and hybrid strategies, which inject more in the NH (30° N specifically), maintain the strength of the AMOC (and associated metrics) relative to PI control. The AMOC feedback operates in the opposite direction as under the 3DOF strategy, with the stronger AMOC carrying more heat to the NH mid and high latitudes, increasing T1 and encouraging more NH injection (Fig. b). The result is two SAI strategies which control for the same T0–T1–T2 temperature targets, but result in different temperature distributions which satisfy those targets: G2-SAI-3DOF injects mostly in the SH and maintains the temperature targets with a weak AMOC, warmer NH tropics and subtropics, and cooler NH midlatitudes and pole; and G2-SAI-hybrid injects in the midlatitudes and maintains the same temperature targets in with a strong AMOC, cooler NH tropics and subtropics, and warmer NH midlatitudes and pole (Fig. n–o).

While T1 does not depend solely on AMOC, as other circulation changes that happen as a result of SAI will also affect the distribution of future injections, our results strongly support the conclusions of both , in that the injection latitude can have a first order impact on determining the AMOC response to SAI, and , in that the AMOC response itself can further influence the distribution of injection rates needed to reach specific temperature targets. Importantly, and regardless of the cause, our results demonstrate that G2-SAI-3DOF and G2-SAI-hybrid both successfully maintain the same temperature targets by converging to two distinct climate states with different injection strategies. The results thus demonstrate that over longer periods of time, the same T0–T1–T2 combinations can correspond to multiple substantially different climate states. We discuss the implications for SAI experiment design in Sect. 5.

In this section, we analyze annual mean temperature and precipitation for the G2-SAI simulations, with select comparisons to the experiments on which they are based (G6-1.5K-SAI and ARISE-SAI-1.5). To do this, we compare maps averaged over selected periods for each experiment: the reference period, the warmed world, and the new climate state reached after offsetting GHG warming with SAI. This is the same framework used by , and the periods for each experiment are defined in Fig. . These definitions for periods “1”, “2”, and “3” are used throughout the remainder of the manuscript, as well as in the Supplement.

In Fig. , we present maps of temperature changes. In order to compare patterns of warming under GHGs, cooling under SAI, and the combination of this warming and cooling, the three topmost rows are normalized by the amount of warming in the background warming scenario relative to the reference period: 5.13 °C for 1 % CO₂ relative to PI control, and 0.91 °C for late SSP2-4.5 relative to early SSP2-4.5. The first row shows the pattern of the GHG warming at the end of the experimental period (i.e., the last 20 years of injection) relative to the period from which temperature targets are derived, and the second row shows the pattern of cooling due to SAI in the same period. The last two rows plot the residual temperature difference between the last 20 years of SAI and the temperature target period, showing how the imperfect cancellation of the GHG warming and SAI cooling affect regional surface temperatures. Because the lengths of the target periods, amounts of warming, and ensemble sizes are different across the sets of simulations, distributions of the shading denoting statistically insignificant changes are not directly comparable.

Warming patterns across the two background GHG scenarios (Fig. a–b) share several broad characteristics, including polar amplification in both hemispheres; increased warming over land relative to the ocean; a “wedge” of increased warming in the Eastern tropical Pacific indicative of an El-Niño-like response; and a warming hole in the North Atlantic indicative of a weakening of the AMOC. All five SAI scenarios also cool the land more than the ocean. This is a common response in SAI modeling experiments; while this has been attributed to the higher heat capacity of water relative to land , the enhanced warming over land has also been attributed to a number of lapse rate and hydrological feedbacks, which SAI could be preventing . The relative cooling in the North Atlantic is much stronger in the SSP2-4.5 scenario than in the 1 % CO₂ scenario, as the warming period considered in this study (i.e., the 2020–2039 period to the 2050–2069 period) takes place entirely during the fastest decline of the AMOC, whereas the warming in the 1 % CO₂ scenario averages over the entire AMOC decline, including the plateau at the end (compare Figs. c and d). The equatorial eastern Pacific warming is visible in the residual temperature maps of all five SAI scenarios, indicating relatively few SAI-induced changes in this mode of climate variability in this model. While absolute temperature residuals are, in general, larger for the G2-SAI scenarios than for the SSP2-4.5-branching SAI scenarios (fourth row), they tend to be smaller per unit of warming being offset (third row). For symmetrical injection strategies, while G6-1.5K-SAI overcools most of the Northern Hemisphere and undercools most of the Southern Hemisphere, G2-SAI-1DOF does not. All three G2-SAI scenarios have substantial residual warming over northern Asia and the Southern Ocean. However, clear differences are visible between G2-SAI-hybrid and G2-SAI-3DOF, showing a substantial cooling over the North Atlantic in G2-SAI-3DOF, which is much weaker for G2-SAI-hybrid, while a slight but significant cooling exists over the continental US, Southern Europe, and Asia, more similar to G2-SAI-1DOF. Determining whether these responses are scenario-specific and/or model-specific will require further study and intermodel comparison; however, the CESM2 simulations of with higher cooling (e.g., the PI +0.5 °C target simulations) do show similar patterns (see Supplement, Sect. S2, Fig. S1), suggesting that this could be a robust response when large injections are used to offset large amounts of warming in CESM2.

Figure  plots precipitation changes. Global mean precipitation is expected to increase under global warming, and it is common result that cooling the planet via the reflection of sunlight decreases precipitation by a greater amount than it increased under GHG forcing; this result has been observed in, among others, the GLENS , ARISE-SAI-1.5 , and G6sulfur and G6-1.5K-SAI experiments . In the PI control baseline period (45–95), global average precipitation is 2.91±0.01 mm d⁻¹ (± denoting the standard deviation of annual means); by the last 20 years of 1 % CO₂, it has increased to 3.14±0.02 mm d⁻¹, while under G2-SAI, it instead decreases to 2.76±0.02 mm d⁻¹ (1DOF), 2.72±0.02 mm d⁻¹ (3DOF), and 2.75±0.02 mm d⁻¹ (hybrid), offsetting approximately 175 % of the increase under global warming. In comparison, ARISE-SAI-1.5 and G6-1.5K-SAI offset 119 % and 106 %, respectively, of the precipitation increase under SSP2-4.5 between the 2020–2039 and 2050–2069 periods. More study is needed to understand whether this pattern is robust across models, or CESM specific. Detailed investigation of regional and seasonal rainfall changes is beyond the scope of this paper. Additionally, the G2-SAI simulations have only one ensemble member each, and due to the high variability of precipitation, analyses of specific impacts would benefit from multiple ensemble members.

Both warming scenarios result in southward shifts and net increases in tropical precipitation; increased precipitation in the midlatitudes and polar regions in both hemispheres; and smaller changes in many parts of the subtropics, though not everywhere. All five SAI interventions reduce precipitation in the midlatitudes and poles in both hemispheres and in the tropics, but the cancelation of the GHG effect is largely imperfect, with residual drying or wetting in one or both hemispheres and a chevron-shaped pattern of residual wetting and drying in the tropical Pacific. G2-SAI-3DOF and ARISE-SAI-1.5 both show increased drying over land in the tropics relative to the 1-DOF strategies. G2-SAI-3DOF also has stronger residual drying in the North Atlantic and southwest Europe. Plots comparing zonal mean precipitation among the simulations considered in this study are provided in the Supplement (Sect. S3, Fig. S2).