Stratospheric sulfate aerosol geoengineering relies on triggering an atmospheric perturbation through deliberate injections of sulfate aerosols or their precursors (often SO2) into the lower stratosphere to mimic the cooling effects seen after large volcanic eruptions (The Royal Society, 2009). Over the last several decades, this has been suggested as a possible means of reducing some of the impacts of climate change (e.g., Crutzen, 2006). There are, however, many open questions about the effects of radiative forcing from sulfate injections (Kravitz and MacMartin, 2020). The importance of choosing the altitude and latitudes of injection, and the distribution of injection rates across those, has been clearly demonstrated, as well as adjusting injection locations based on the season (Visioni et al., 2020). Additionally, even for sulfate aerosols, the method of dispersal will affect aerosol size distribution and, hence, the amount of material that needs to be injected. To date, many of these uncertainties are based on a climate response from fixed-injection locations (e.g., Robock et al., 2008; Heckendorn et al., 2009; Tilmes et al., 2017), which is a significant limitation for predicting dispersion in fully turbulent fluid flows. In fact, none of these studies consider the short-term variations in stratospheric winds or the organizing role of turbulent coherent structures in these time-varying flows. Driscoll et al. (2012) showed that it is impossible to correctly capture the impact of abrupt atmospheric perturbations on surface climate without a well-resolved stratospheric model. With the great significance of stratospheric dynamics for teleconnections and the state of the atmosphere (e.g., Jaiser et al., 2013; Domeisen et al., 2019), how can we optimize where to put aerosols or precursors so that we have greater influence on the mean climate and have better efficiency?

While benchmark studies have been quite successful at understanding the mean climatic response of geoengineering in sophisticated Earth system models (e.g., Kashimura et al., 2017; Kravitz et al., 2017), the injection protocols have all neglected presently available short-term predictive information useful for optimizing particle dispersion. An efficient dispersion of aerosol precursors is of crucial importance for aerosol coagulation (Kravitz and MacMartin, 2020); the particle size distribution is a critical and sensitive parameter for accurately determining surface cooling, stratospheric warming and changes in stratospheric dynamics (e.g., Rasch et al., 2008; Heckendorn et al., 2009; Tilmes et al., 2008; Niemeier et al., 2011). By only considering average flow behavior, one limits geoengineering evaluations to simple injection protocols that do not fully exploit turbulence, coherence and mixing in the stratosphere. This increases the likelihood of a heterogeneous spatial coverage and localized high concentrations of aerosols, leading to enhanced coagulation and sedimentation rates (e.g., Pierce et al., 2010). Without a more precise optimization of injection locations, we limit our ability to accurately model the full potential impacts of geoengineering.

Instead of standard fixed-locations, we propose a time-varying injection location protocol based on the identification and prediction of short-term Lagrangian stratospheric transport barriers. This method harnesses the theory of Lagrangian coherent structures (LCSs), a tool for highlighting the most influential material surfaces solely from fluid velocity fields without any further modeling of scalar transport (Haller, 2015). For a given unsteady fluid flow, any arbitrary surface of fluid particles, M, will block advective transport across these surface over time as the surface deforms with the flow. This is shown in a real 2D velocity field of geostrophic ocean surface currents in Fig. 1. Here, the blue line M separates regions of light and dark gray fluid particles. As the fluid flows from time t0 to t1, M is an advective transport barrier in that no dark gray fluid crosses M to mix with the light gray fluid. This result follows immediately from the continuity of the equations defining fluid motion.

Instead of looking for material barriers to advective transport, of which there are infinitely many, LCS theory identifies only exceptional distinguished material surfaces, such as those that are mathematically defined to be rotationally coherent, undergo minimal stretching over time or locally attract or repel nearby fluid particles at a significant rate. One example of the latter two structures, termed hyperbolic LCSs, and their time evolution in the same unsteady ocean flow is shown in Fig. 2. Over the time period t0 to t2, MA is the structure that is mathematically defined to most effectively attract nearby particles, and MR repels nearby particles. By identifying exceptional material barriers, such as the saddle feature in Fig. 2, LCS theory allows the organization of turbulent fluid flows into coherent patterns in a mathematically rigorous (nonempirical), physical and frame-independent manner (Haller, 2015).

Though using the mathematical definition of LCSs to define atmospheric flow structures is quite restrictive, LCSs have actually been identified throughout the atmosphere (Tang et al., 2010; Tallapragada et al., 2011; Rutherford et al., 2012; BozorgMagham and Ross, 2015; Knutson et al., 2015; Wang et al., 2017). Of particular relevance to the present research is the LCSs work of Beron-Vera et al. (2012), who demonstrated how zonal jets behave as meridional transport barriers at high latitudes. Olascoaga et al. (2012) analyzed LCSs in stratospheric winds to provide a rigorous definition of the transport barriers contributing to the loss of ozone from the Arctic ozone layer, and there was recent success in delineating LCSs along atmospheric rivers (Garaboa-Paz et al., 2015). Jupiter's Great Red Spot and zonal jets were identified as material transport barriers through video analysis and LCS theory (Hadjighasem and Haller, 2016). Using a null geodesic identification scheme, the northern polar vortex, a significant structure in high-latitude atmospheric mixing, was accurately identified as a transport-blocking LCS (Serra et al., 2017). Lastly, Wang et al. (2017) were able to use a related diagnostic strain tensor field to predict the location of space shuttle contaminant plumes in the thermosphere after 48 h of transport. These previous results indicate the potential for the most influential LCSs to be harnessed for geoengineering purposes. Specifically, hyperbolic LCSs that maximize or minimize dispersion may be used as time-varying injection locations that reduce coagulation of aerosols and increase their lifespan and utility.

Recently, Haller et al. (2018, 2020) derived an additional objective criterion that specifically identifies the strongest barriers and enhancers of diffusive particle transport. That is, one can identify the time-varying locations of material barriers in a fluid flow that either maximize or minimize the diffusive contribution in the advection–diffusion equations over a given time frame. They have obtained a diffusion barrier strength (DBS) field whose ridges highlight the strongest diffusive transport barriers in forward-time fluid flow analysis and the strongest diffusive transport enhancers by running a backward-time fluid flow analysis. Neither of these simulations actually require modeling the evolution of a diffusive scalar field but still rigorously define the structures that are most influential to diffusive transport. For atmospheric science, this significantly reduces the computational burden for predicting how scalar fields will evolve as it provides quantitative information about future attraction and dispersion patterns, without needing complex numerical machinery to model the advection–diffusion equations or making assumptions about their unknown initial and boundary conditions. In comparison, the effective diffusivity approach of Nakamura (2008) provides an a posteriori visualization of Eulerian barriers, but only after scalar transport simulations have been performed. DBS fields, however, give an a priori (predictive) characterization of material barriers to diffusion without ever running diffusive simulations. This new technique increases the rigor of Lagrangian atmospheric analysis and removes ambiguity arising from the lack of a universal definition of coherence in atmospheric LCSs work. As such, the DBS field is perfectly suited to optimizing aerosol dispersion and is computable solely from available wind field forecasts and hindcasts or reanalysis.

In this paper, we evaluate simulated stratospheric flows with the aim of identifying diffusive transport barriers and informing injection site selection for enhanced stratospheric geoengineering via aerosols. In doing so, we provide an initial demonstration of the benefits of incorporating short-term atmospheric dynamics into geoengineering analyses and provide suggestions to better assess its potential impacts. Our choice of dynamics-informed injections is evaluated against fixed-injection protocols via long-term metrics of pure advective transport and geoengineering scenarios simulated in a fully coupled climate-model. We find significant improvement in the ability of injected aerosols to both quickly surround the Earth and to be able to achieve similar coverage with fewer injection sites. We then introduce further practical and logistical restrictions on the DBS-based protocol and maintain our method's improved performance.

Here we have explored the use of diffusive transport barriers to guide strategic injection locations for stratospheric aerosol geoengineering. Compared to commonly used methods that rely on fixed-injection locations, this dynamic site selection allows for immediate improvements in particle dispersion and better global coverage, often with fewer injection sites. This has important implications for previous studies regarding the efficiency of aerosol optical depth versus injection rate. In particular, by focusing on only fixed-injection locations (e.g., Robock et al., 2008; Tilmes et al., 2017, among others) these studies neglected an influential variable, namely the time-varying locations of stratospheric diffusive transport barriers. It is safe to assume that, since these studies did not optimize injections for dispersion, they have thus far underestimated what is possible for geoengineered sulfate aerosols reflectance. Using a Lagrangian coherent-structure-informed approach via DBS fields shows promise for strengthening the role that geoengineered aerosols can play in altering climate dynamics, especially at short timescales or if logistical restrictions mean injection sites must be strategically chosen.

With dynamic injection locations, initial particle concentrations spread much more quickly, as indicated in our tracer experiments (Figs. 6 and 8) and in the full CESM2 simulations (run no. 2; Figs. 10 and 11). This reduces the probability of coagulation for each individual injection and likely influenced lower peak effective radii present in DI and ADI experiments (Fig. 12; e.g., Mills et al., 2017). With reduced coagulation, there will be a slower descent of sulfates from the stratosphere (and out of action) and increased scattering. Second, increased dispersion uniformity, as quantified by normalized entropy in Figs. 4 and 5, will reduce local stratospheric heating and result in more uniform radiative forcing. This is because heat transfer to aerosol particles in a given volume is proportional to the mass, which is reduced by lower concentrations. Lastly, in the fully coupled CESM2 (WACCM6) run no. 2 experiments, DBS-informed injections improved global coverage by SO4. This ability to more quickly achieve total coverage provides an opportunity to strategize geoengineering protocols with a shorter window for interference in chaotic flows.

The results here indicate a predictable enhancement of dispersion for geoengineering if influential hyperbolic structures are present in the stratosphere. When there are strong short-term DBSFW and DBSBW ridges, such as in the winter, spring and autumn CESM2 run no. 2 experiments, we show that we can exploit these ridges to optimize the immediate dispersion of aerosols. Additionally, the fine-scale behavior of aerosol dispersion can be explained by the presence of influential structures, such as the attraction and blocking that occurs in the Southern Hemisphere run no. 2 spring experiment. These fine-scale structures have not been actively considered in geoengineering research but may be exploited, as is clear in the Northern Hemisphere for the winter and spring experiments (Fig. 10). In our fully coupled microphysics and atmospheric chemistry climate simulations, we also verified that initial improvements in particle dispersion from simplified flow calculations can result in less coagulation and increased aerosol spread, as evidenced by more areal coverage by SO4 burden and reduced effective aerosol radii. The enhanced global distribution of SO4 for the two DBS-informed injection protocols after months of transport (e.g., Fig. 11) speaks to the utility of the strategic placement of aerosols or precursors near hyperbolic structures, as do long-term radiative forcing improvements (Table 2) and correlated, yet complex, relationships with reducing average effective radii.

The extent of this influence is dramatically portrayed in the autumn CESM2 run no. 2. At this time, a minimal modification of the injection site in the Southern Hemisphere near strong hyperbolic structures and a change of less than 650 km resulted in aerosols spreading over an additional 7.5 % of that hemisphere after 7 d. After 8 weeks, this immediate DI dispersion benefit was not as noticeable, but the ADI scheme still contributed to improved coverage over the FI aerosols. The enhanced coverage in the autumn experiment is, furthermore, coincident with a considerable improvement in RF for the DI experiments at 10, 30 and 365 d. Additionally, there was a minor but statistically significant reduction in average aerosol radii (p<1×10-5) 1 year after the SO2 injection. At longer timescales for several of the other CESM2 run no. 2 geoengineering experiments, the relationship between immediate enhanced dispersion, radiative forcing and aerosol radii was less clear. For example, in spring, the ability to achieve global coverage and reduce radiative forcing at long timescales was best for the DI protocol, but there was not a similar improvement in average aerosol effective radius. In the winter DI simulation, there was a significant reduction in aerosol radius and improved coverage compared to FI, but there was a weaker effect on radiative forcing 1 year after injection.

To manage uncertainties in atmospheric flow and climate response, several recent geoengineering climate modeling studies have employed a feedback algorithm that adjusts the SO2 injection rate at one or more latitudes (Jarvis and Leedal, 2012; MacMartin et al., 2014; Kravitz et al., 2017) in response to changes in surface climate; studies to date have updated the forcing once per year. In future studies, this slow feedback could be integrated with DBS-informed injection. Outlining this process, for every week of simulation, new injection locations would be determined based on wind fields from the previous week, using the DBS algorithm described previously to find locations within different latitude bands. The model would then be run forward for a week with the SO2 injected at those locations. This process, which essentially constitutes a form of model predictive control (García et al., 1989), could be carried out for a year, at which point the injection rates to use in each latitude range would be updated using the same slow climate-response-dependent feedback.

This research has shown that adapting aerosol geoengineering injections methods by considering 2D Lagrangian coherent structures provides an obvious advantage for dispersion of aerosols by enhancing longer-term dispersion dynamics from only short forecast data. All results in Sect. 3.1 suggest DBS-informed sites reliably outperform fixed locations when considering aerosol dispersion along isentropes as is rigorously guaranteed in the DBS metric derivation. The site-selection algorithm developed herein, however, does not consider the full three-dimensionality of stratospheric flows or any knowledge about common meteorological and climatic features. Thus, the user-independent injection protocol does not always result in enhanced radiative forcing or global coverage when strong attracting and repelling features are not present in all regions, such as the summer experiments in CESM2 run no. 2. With this is mind, we suggest that future injection experiments use DBS fields and diffusive transport barriers to constrain their choice of injection site but allow for user intervention in the absence of such strong dispersive ridges and consider other influential variables, such as seasonality of injection (e.g., Visioni et al., 2020), and aerosol microphysics, such as temperature and humidity.

In one injection season (winter), there is an appreciable reduction in the effective radius, and a more negligible effect in the others. This indicates that there is both the potential for dynamic injection to result in smaller aerosols, and it suggests there is room for improving our understanding of the role that dispersive stratospheric dynamics play in aerosol coagulation. Future work along these lines may further improve upon the findings indicated here and help us to understand the limits of what improvement in reducing aerosol size is still possible by considering time-varying, small-scale turbulent features.

Related to this study is the proposed idea of direct injection of H2SO4 droplets, instead of SO2 gas, which would ostensibly create a more monodisperse particle distribution and, thus, delay coagulation (Pierce et al., 2010). Further investigation is warranted to understand the relative effects of this method vs the SO2 injection simulated in our CESM2 (WACCM6) simulations, particularly if injection locations are chosen dynamically. This is especially important given the stratospheric chemistry involved in SO2 injection, including the approximately 1-month timescale of conversion from SO2 to sulfate aerosols, although, in principle, a transport barrier would apply to both gases and particles.

The results presented here are for a single model; different models will indicate different stratospheric features and, thus, different transport barrier locations and strengths. Of key importance is that the long-term dispersion analysis and structure identification methodology relied on 2D transport along isentropes. This method has proven to be successful for advancing the goals of optimizing sulfate precursor injections; a full 3D computation of DBS fields would further improve the results. On reviewing the results, it appears there was an overemphasis on the ability to separate unique attracting structures from the 2D isentrope data. With the automated algorithm defined in Table 1, pairs of injection sites were sometimes chosen to be close together as it appeared their injected aerosols would end up on separate structures. In fact, these features may have actually been connected along the third dimension. Additionally, long-term trends of the SO4 burden present in the stratosphere were mixed, suggesting further considerations of seasonality (e.g., Visioni et al., 2020) and consideration of what structure is likely being represented by a DBSBW ridge (e.g., a jet stream, polar vortex or something much less substantial).

Though not investigated in the present research, with the introduction of stratospheric heating, cross-isentropic flow is likely to occur (e.g., vertical uplift from the heating), potentially justifying a 3D analysis for the flows used here. Vertical transport of aerosol is likely inevitable, but a 3D DBS analysis would exponentially increase the complexity and the computational costs of finding injection locations. The currently proposed isentrope method is found to improve injection protocols at little to no increased operational cost as there are clear advantages in the short-time dynamics when using the DBS forecasts. One alternative improvement to 3D DBS fields would be a simultaneous 2D analysis of structures on a range of isentropes.

Several studies have found that the injection rate of SO2 is the limiting factor in geoengineering efficiency by increasing the coagulation (Heckendorn et al., 2009; Niemeier et al., 2011; Niemeier and Timmreck, 2015). These studies, however, did not optimize the dispersion of SO2 during the first days following injection and, therefore, did not maximize the potential of sulfate injections and consequent radiative forcing in model simulations. We conclude that the exploitation of readily available, short-range wind forecasts and reanalysis are a catalyst that will allow better understanding of what can be achieved with climate geoengineering. It is possible that one of the reasons the improvements seen here are not more drastic is the acute focus on the response to large individual injections, which is a method not commonly used. We ran simulations that included a single day of injection in an effort to demonstrate dispersion capabilities. As the ability of DBS ridges to predict dispersion dynamics has now been shown, a logical next step is to pursue more climate focused studies, such as injecting less mass over many successive injections using concurrent predictions. While the use of DBS-informed injections does not address many of the potential hazards of geoengineering (e.g., Robock et al., 2008; Heckendorn et al., 2009), it is an important step forward towards assessing the feasibility of geoengineering to prevent the climate from crossing a critical tipping point.