developed a methodology to explore the meteorological and aerosol phase space of northeast Pacific (NEP) clouds by analyzing eight primary cloud-controlling factors from the European Center for Medium-Range Weather Forecasts (ECMWF) reanalysis version 5 (ERA5) along 2 d Lagrangian trajectories during summer 2018–2021 from six source locations (source locations shown in Fig. d). The factors were the trajectory mean 700 hPa water vapor mixing ratio, 700 hPa vertical velocity in pressure coordinates, estimated inversion strength (EIS), and 10 m wind speed, along with the corresponding changes (end minus beginning) in each variable over the trajectory. After performing principal component analysis on 1663 Lagrangian trajectories, showed that the first two principal components (PC1 and PC2) explain 43 % of the variance in the cloud controlling factors. At each of the six source locations, 9 trajectories spanning the PC1–PC2 phase space were also shown to reasonably capture the observed variability in surface pressure, liquid water path (LWP), droplet concentrations (Nd), background aerosol concentration (Na), boundary layer depth, and precipitation .

All 54 simulations were carried out with the System for Atmospheric Modeling , version 6.10.9, with the same model configuration outlined in that is enabled with a two-moment, prognostic aerosol scheme . Initial ERA5 temperature and moisture profiles required an inversion sharpening procedure to better align with microwave LWP observations and compensate for the coarse representation of inversion structure in ERA5 . During a 12 h, nocturnal spin-up period, temperature and moisture nudging within the boundary layer was applied on a 1 h timescale. In , boundary layer Na was initialized with the Modern Era Retrospective analysis for Research and Applications, version 2 (MERRA2) reanalysis product , but MERRA2 boundary layer Na is known to exhibit biases relative to aircraft measurements . For all 54 of the LES library members, the initial MERRA2 boundary layer Na values at each height level are corrected using the Clouds and the Earth's Radiant Energy System (CERES; ) Nd estimates using 1Nacorr=NaFc, where Nacorr is the CERES-corrected MERRA2 Na value and Fc is the correction factor defined as 2Fc=1AfNd‾Na‾, with Af being the assumed activation fraction, that is set to 80 % in accordance with previous studies , Nd‾ is the average of CERES-Nd during the first day of the trajectory, and Na‾ is the average of MERRA2-Na during the first day of the trajectory and over the entire boundary layer depth. Details on how CERES Nd is calculated based on measurements of cloud optical depth and effective radius are given in . MERRA2 Na is calculated from the mass mixing ratios of aerosol species, following the methodology described in Appendix A of .

After the 12 h initialization stage, the simulations are allowed to evolve freely within the boundary layer for 48 h, forced with time-varying ERA5 large-scale vertical velocity, large-scale temperature/moisture advective tendencies, geostrophic winds, and sea surface temperatures along each trajectory. Free tropospheric aerosols, temperature, moisture are nudged on an hourly timescale and winds are nudged on a 12 h timescale. The horizontal domain is 51.2 km × 51.2 km with a uniform horizontal grid spacing of 100 m. The vertical grid uses 7 m between 450 and 1200 m, and then gradually coarsens above and below this range, so that it is >100 m near the model top at 4.8 km and 20 m near the ocean surface.

When examining the shortwave cloud radiative effect (SWCRE), the bias between the LES library simulations and CERES-derived SWCRE indicates a clear dependence on cloud fraction (Fig. a and b). Since SWCRE is negative, a negative CERES-normalized bias indicates that the LES is underestimating the magnitude of SWCRE relative to the CERES-derived value. Cases with median cloud fractions above 50 % along the 2 d trajectory have substantially weaker normalized SWCRE biases than cases with median cloud fraction below 50 % (Fig. a and b), with biases generally smaller than 50 % for cloudy cases. Among the 54 LES simulations, 17 cases have median cloud fractions above 50 %, with 3 LES cases overestimating cloud reflectivity relative to CERES, 6 cases within 20 W m⁻², and 8 cases underestimating cloud reflectivity by 20–65 W m⁻² when examining the average over the two daytime periods (Fig. c).

Potential sources of persistent LES–observation biases are numerous, including uncertainties in the initialization of boundary-layer and free-tropospheric aerosol, as well as unresolved large-scale adjustment processes, domain-size and grid-spacing limitations. The goal of the present study is not to diagnose the root causes of these discrepancies, but instead to focus on the best performing cases and the most plausible candidates for MCB. Using the 17 cloudy cases, we build a representative summertime climatology of the Day 1 and Day 2 portions of the Lagrangian LES trajectories (Fig. d) spanning distinct aerosol, meteorological, and turbulence regimes, and use it to estimate the sensitivity of Twomey forcing to aerosol plume spreading rates. A manuscript describing the full ensemble's performance, behavior, and key characteristics is currently in preparation.

Given our objective of evaluating many different ship sprayer densities, aerosol lifetime assumptions, injection timings/durations, and sprayer motions, performing large-domain LES for all of these variations across the LES library is cost prohibitive. For this reason, estimating the horizontal spreading of ship tracks requires a more computationally efficient approach. As an alternative to explicitly modeling fluid motion and aerosol transport in physical space using the filtered Navier-Stokes equations solved on a numerical grid, as done in the LES, fluid flow can be described statistically by building probability distributions of individual stochastic particle trajectories that are governed by the following simplified Langevin equation described in and : 3dUt*=Ut‾-Ut*TLdt︸deterministicdriftterm+2σ2TLdWt︸Brownianmotionterm, where dUt* is the individual horizontal particle acceleration in m s⁻², Ut‾ is the boundary layer mean environmental velocity in m s⁻¹, σ2 is the boundary layer mean horizontal velocity variance in m² s⁻², and dWt is the random walk increment in s^1∕2. This study focuses on the horizontal spreading of ship tracks, meaning Eq. () is solved independently for the u and v components. The relaxation timescale (TL) is defined as 4TL=12σ234C0ε, where ε [m² s⁻³] is the boundary layer-averaged turbulence dissipation rate and C0 is a free parameter . found that C0 requires tuning to account for an anisotropic two-cell mesoscale circulation that develops within ship tracks in precipitating environments. The C0 parameter takes on one of two values depending on the surface precipitation rate (psfc) in the LES (Eq. ), with lower values of C0 increasing the relaxation timescale and spreading rates accordingly. Recent LES mechanism denial experiments suggest that precipitation suppression may not be the only mechanism that can initiate a spread-accelerating, two-cell mesoscale circulation, with cloud droplet sedimentation potentially being sufficient to drive the circulation in clean aerosol environments . Nonetheless, we choose to maintain the precipitation threshold, considering the uncertainty of a potential background aerosol threshold and the lack of observational data on two-cell mesoscale circulation occurrence in ship tracks. 5C0=0.15ifpsfc≥0.01mmd-1,0.5ifpsfc≤0.01mmd-1. In practice, higher order moments such as σ2 and ε are difficult to obtain or unavailable and are often parameterized in existing operational Langevin models such as FLEXPART . However, these turbulence-related quantities are accessible in the LES and can be used to directly force the Langevin particle model. In previous work, the Langevin particle model driven by domain- and boundary layer-averaged σ2 and ε was found to reliably reproduce horizontal spreading rates in the LES under a range of turbulence and aerosol conditions . As a result, the northeast Pacific library simulations, which do not include ship tracks, can be used in combination with the Langevin particle model to estimate the spreading rates that would occur if ship tracks were explicitly modeled in each LES simulation.

At each Langevin particle model time step (Δt=600 s) a fixed number of particles are emitted, controlled by the number of sprayers (Nsprayers) multiplied by the number of injected particles per sprayer (Sp=103), with the total number of particles in each batch given by Sbatch=NsprayersSp. Each modeled particle represents a multiplicity of particles (i.e., a superparticle). The number of real particles represented by each superparticle (Nrep) is given by 6Nrep=ΔtNinjSp, with Ninj being the injection rate, set to 10¹⁶ particles per second. Superparticle positions are converted to perturbed aerosol concentrations (Na′ in # mg⁻¹) with the following equation: 7Na′=ScNrepdxbindybinZinvρmg, with Sc representing the number of binned superparticles with a bin size of 103×103 m² (dxbin, dybin) and ρmg being the density in mg m⁻³. The vertical aerosol concentrations are assumed to be evenly distributed throughout the time-dependent boundary layer depth (Zinv) calculated from the LES library (Fig. e), implying that each bin volume remains in a well-mixed state.

All particle model simulations were performed on a 200×200 km² horizontal domain with doubly periodic boundary conditions. A 200×200 km² domain was used to minimize artificial plume self-interaction under periodic boundary conditions on diurnal timescales and to capture changes in plume geometry, such as elongation and changes in plume-axis orientation, associated with ship motion and time-varying mean winds. The ship plumes are initialized with a Gaussian particle position distribution that has a standard deviation (σ) of 1 km in both the x and y dimensions and no initial velocity perturbations. Ship motions are determined randomly from a sample of 8 possible compass directions (N, NE, E, SE, S, SW, W, NW) with a default ship velocity of 10 m s⁻¹. If a ship leaves the domain, its position and compass direction are randomly regenerated with the number of ships within the domain at any given time remaining fixed.

While the LES calculates wet scavenging within the aerosol microphysics, the impacts of scavenging processes (both wet and dry) are approximated in the particle model using a range of aerosol decay timescales (τdecay=0.25, 0.5, 1, 2 d), with the probability of a particle being removed (Premoval) from the population given by the exponential decay function 8Premoval=exp⁡-age*τdecay, where age^* is the particle age calculated as the time elapsed since injection. Particles are removed randomly from each batch of released particles if PremovalSbatch≥1. This age-dependent removal represents the accumulated probability of removal processes over the history of a particle batch, rather than implying that particle age itself is the physical cause of removal.

The Day 1 and Day 2 portions of each trajectory are run separately and represent distinct meteorological regimes with differing boundary layer depths, precipitation, mean winds, background aerosol, and turbulence (see Sect. 3.1). To approximate equilibrium conditions representative of multi-day, repeated injections as expected in an MCB scenario, we repeat the 12 h daytime aerosol injection until the domain-mean peak aerosol concentration changes by less than 2 % between consecutive days. Under this criterion, the number of repeated diurnal cycles necessary for periodic equilibrium for τdecay=0.25, 0.5, 1, 2 d is 1, 2, 4, and 8 cycles (days), respectively. Interpolated LES fields are smoothed with a 3 h averaging window to avoid sharp discontinuities from morning to night. Unless stated otherwise, aerosol injections begin at sunrise each day and continue for 12 h each day. To derive plume widths and spreading rates from the LPM, a single batch of particles is released and the 2σ width (single σ multiplied by 2) of the x and y particle positions is calculated as done in , neglecting particle decay/removal.

The Twomey forcing is diagnosed at 20 min intervals (the LES statistics output frequency) by merging the 2D LPM aerosol perturbation fields (Na′(xlpm,ylpm,t); Fig. a), defined on the LPM grid (xlpm, ylpm), with the radiative fields from each LES case, defined on the LES grid (xles, yles). The LES output provides the net shortwave radiative flux at TOA (SWnet(xles,yles,t)) and the total incoming shortwave insolation at TOA (Finc(t)), from which a scene reflectance (Rscene) can be computed using 9Rscene=-SWnet-FincFinc. Rscene contains the detailed cloud morphology information available from the LES (Fig. b).

The doubly periodic LES grid is tiled and clipped to match the size of the 200×200 km² LPM domain. The Rscene field is then coarse-grained to map from 100 m grid spacing in the LES to the 1 km bin size in the LPM by averaging the 10×10 chunk of grid boxes from the LES. We denote the tiled, clipped, and coarse-grained LES field as R‾(xlpm,ylpm,t). The Rscene values below 0.07 are set to zero to limit the inclusion of clear-sky/ocean-like reflectances. An example snapshot of the 2D Rscene field is shown in Fig. b.

Droplet concentration perturbations can be converted to changes in R‾ using Twomey's albedo susceptibility equation : 10ΔR‾=R‾(1-R‾)rn1/3-11+R‾rn1/3-1, where rn=Na′+Na,lesNa,les, with Na,les being the time-dependent, boundary layer-averaged background accumulation mode aerosol concentration from the LES. Twomey forcing depends on Nd, but we make the assumption that all aerosol is activated so that Nd=Na. As mentioned in , the primary sensitivity in Eq. () is the perturbation ratio term (rn) and not changes in R‾. The reflectance changes (ΔR‾(xlpm,ylpm,t)) are converted to the indirect shortwave radiative forcing (Twomey forcing) with 11ΔFgrid=-FincΔR‾, and an example snapshot of ΔFgrid is shown in Fig. c. The particle model ΔFgrid is converted to global mean Twomey forcing (ΔF) by averaging ΔFgrid over all LPM grid boxes and times during the final 24 h period of the equilibrium LPM simulations and multiplying that value by the fraction of global surface area that is assumed to be actively sprayed (taken to be 3.147 %) with a given sprayer density. This assumes that the sprayed ocean area is 4000×4000 km² (4.4 % of the ocean area), consistent with recent global modeling studies targeting the most susceptible cloud regions, primarily the northeast and southeast Pacific during the summer months . The primary ship densities explored in this paper represent low-end (Nsprayers=4000) and mid-range (Nsprayers=16000) MCB deployments , where we expect spreading rate impacts to potentially be most relevant. For Nsprayers=4000(16000) , there are 10 (40) sprayers active in the particle model domain during the injection period, corresponding to a ship density of 2.5×10-4 ships per km² (10⁻³ ships per km²) with a characteristic sprayer spacing of ≈63 km (32 km). For comparison with and global modeling studies, it is helpful to define the injected salt mass rate (Ms; Tg yr⁻¹), assuming a geometric mean dry diameter (Ds) of 100 nm, a geometric standard deviation (S) of 1.6, the density of sodium chloride (ρs=2160 kg m³), and Ninj, the following equation from can be solved 12Ms=NinjπρsDs36exp⁡4.5ln⁡(S)2, with Ms≈2 Tg yr⁻¹ for Nsprayers=4000 and ≈8 Tg yr⁻¹ for Nsprayers=16000.

There are two critical assumptions underlying the offline coupling of the LPM aerosol field with LES reflectances. First, we assume that cloud adjustments are negligible, meaning that the primary control on the radiative flux is the Twomey forcing and not LWP and/or cloud fraction changes. Since the aerosol concentrations from the LPM are not fully coupled to the LES, processes like precipitation suppression , entrainment enhancement related to smaller droplets and shifts in the radiative cooling profile , and potential cloud morphology changes associated with ship track/aerosol plume circulations are not captured. As outlined in the introduction, there is considerable uncertainty regarding cloud adjustments in high aerosol/MCB environments, but a growing body of observational studies suggests that radiative impacts of cloud adjustments may be minor compared to the Twomey forcing on both regional and global scales . In , the heuristic model used to estimate global Twomey forcing also assumed negligible cloud adjustments and was in broad agreement with existing LES of ship tracks.

Second, we assume that all injected aerosols are activated in cloudy regions (100 % activation fraction). This choice represents an upper-bound estimate of the Twomey forcing to highlight potential dependencies on plume spreading rates. In reality, activation will be sensitive to aerosol size, number, updraft velocity, and supersaturation, all of which vary in both space and time . As a result of these assumptions, the forcing should be interpreted as the cloud-state-prescribed sensitivity of reflectance to droplet number, rather than a fully coupled response of the cloud field to the aerosol.

Despite the simplifying assumptions of negligible cloud adjustments and 100 % activation, the LES-LPM merging process represents complex, non-linear interactions by capturing the spatial overlap between aerosol perturbations and cloud reflectance fields. Given the stochastic nature of both the Langevin particle model and ship motion, some degree of run-to-run variability is expected; however, even at low sprayer densities (Nsprayers=2000) the run-to-run variability in Twomey forcing is less than 3.5 %, suggesting LPM ensembles of each LES case are not necessary. The Twomey forcing is marginally sensitive to Sp, with Sp=1000 producing 0.7 %–1.8 % less forcing than simulations using Sp=2000.

The cloudy LES cases evolve consistently with theoretical, modeling, and observational understanding of climatological changes along equatorward-moving trajectories, which typically involve decreasing cloud fraction, LWP, and accumulation mode aerosol (NA_c), along with increasing surface precipitation occurrence, deepening boundary layers, and larger mesoscale cellular structures (Fig. ).

Cloud fraction varies considerably between individual LES cases (captured by R‾), experiencing abrupt changes at times, with collapses near midday and recoveries in the evening, in agreement with satellite observations of NEP clouds (Fig. a). The median cloud fraction on Day 2 of the trajectories is ∼25 % lower than Day 1 during the midday hours (Fig. a), representing a gently transitioning environment rather than the full stratocumulus-to-cumulus transition where cloud fractions would be expected to be much lower. The LWP evolution generally mirrors the cloud fraction behavior (Fig. b) and agrees well with satellite and aircraft observations, with peak LWP near the morning hours and minimum LWP in the late evening . The median boundary layer-averaged NA_c is on the order of 100 # mg⁻¹ on Day 1 as observed during the CSET campaign , which declines by nearly half on Day 2 (Fig. c). Surface precipitation is minimal on Day 1, but becomes much more common on Day 2, with several cases with surface rain rates greater than 1 mm d⁻¹ (Fig. d). Initial median boundary layer depth is near 1 km, growing to nearly 1.5 km on Day 2, with a corresponding increase in the characteristic mesoscale cell size , which is calculated by coarse-graining the horizontal velocity fields and counting the number of consecutive grid boxes of a given sign (Fig. e and d). Mesoscale cell sizes are generally smaller than reported sizes in , which may be related to domain-size limitations in the LES. Figure S1 in the Supplement provides an example of the counting method used to calculate mesoscale cell size.

The particle model forcing from the LES also has distinct differences in boundary layer-averaged mean winds, velocity variances, and relaxation timescales (TL) from Day 1 to Day 2 (Fig. ). Mean winds are consistent with regional climatological winds associated with persistent subtropical high pressure , as initially stronger north/northeasterly winds slacken and become more easterly on Day 2 (Fig. a and b).

Horizontal velocity variances show a clear diurnal cycle with maximum variances in the early morning hours and minimum variances in the late evening hours (Fig. c and d) related to daytime solar absorption that suppresses TKE production and reduces entrainment . The IQR of horizontal variances narrows considerably during the evening hours on both Day 1 and Day 2, despite large differences in boundary-layer depth, precipitation, LWP, and cloud fraction among cases. This behavior supports the idea that the cloud-radiation-turbulence-entrainment feedback acts to limit intercase spread . Larger LWP and deeper boundary layer cases have greater cloud-top negative buoyancy fluxes which drive stronger entrainment and may be more prone to drizzle, with both of these mechanisms promoting decoupling and limiting the ability of the cloud-top TKE production to be effectively mixed within the boundary layer. Thinner clouds have smaller cloud-top buoyancy fluxes, but can more easily mix TKE throughout the boundary layer depth. As a result, the IQR range of daytime LWP from 25–55 g m⁻² corresponds to a narrow range of boundary layer-mean horizontal variances , with higher case-to-case variability in the early morning hours. On average, Day 2 has ∼25 % larger velocity variances than Day 1; however, Day 2 variances near sunrise are ∼50 % larger than Day 1.

The relaxation timescale (TL) is almost a factor of 2 larger on Day 2 (Fig. e), suggesting that all else equal, particles would revert back to the mean more slowly, resulting in faster Day 2 spreading. It is useful to define a decoupling parameter (ϕ) to quantify the extent of decoupling , 13ϕ=qt,cl-qt,scqt,ft-qt,sc, where qt,cl is the total water mixing ratio in the cloud layer, subcloud layer (qt,sc), and free troposphere (qt,ft). As the boundary layer deepens on Day 2, ϕ increases by approximately 50 % across the ensemble (Fig. f), consistent with observational estimates of decoupling in the northeast Pacific . The TL evolution is strongly correlated with the mesoscale cell size (R2=0.59) and the degree of decoupling (ϕ) (R2=0.32), since the dissipation rate is sensitive to both flow geometry and stratification (Fig. S2).

In our merged LES-LPM modeling framework, cloud fraction, LWP, NA_c, precipitation, boundary layer depth, mean winds, vertical velocity variances, and relaxation timescales may all influence Twomey forcing in nonlinear ways. As shown in Fig. , the Day 1 and Day 2 states represent distinct meteorological, aerosol, and turbulence regimes along the trajectories and provide an opportunity to probe Twomey forcing sensitivity in each environment.

Horizontal aerosol plume widths vary substantially with the turbulence forcing used in the LPM, with the slowest- and fastest-spreading cases in the cloudy LES library differing by as much as 80 km after 48 h (Fig. a). Across the ensemble, plume width is well approximated by a linear growth rate of 1.5 km h⁻¹ (Fig. b), consistent with existing LES estimates of lateral plume spreading and with the observational estimate of 1.85 km h⁻¹ reported by . Although the ensemble-median plume width is well described by a linear spreading rate, the instantaneous spreading rate exhibits a pronounced diurnal cycle (Fig. c and d) associated with diurnal variations in the horizontal velocity variances and TL. There are several instances of plume spreading rates in excess of 3 km h⁻¹, predominantly occurring in the early morning hours (Fig. c). On average, Day 2 median spreading rates are 0.3 km h⁻¹ faster than Day 1 (∼20 % faster), but median spreading rates a few hours after sunrise vary by more than 1 km h⁻¹ (Fig. d).

In the Twomey forcing sensitivity analyses that follow, the Day 1 and Day 2 LPM simulations are treated as distinct climatological regimes, with injections initiated at sunrise and active injections for 12 h each day until equilibrium concentrations are reached. To better quantify plume spreading differences between these regimes, spreading rates were calculated from particle batches released at sunrise each day. During most daylight hours, median Day 2 spreading rates exceed those on Day 1 by approximately 1–2 km h⁻¹ (Fig. ). At night, spreading rates are slightly higher on Day 1 than on Day 2 (Fig. c). These results indicate that the two regimes differ not only in their background cloud and boundary layer properties, but also in the rate at which injected aerosol is dispersed, which has the potential to alter aerosol concentration probability distributions and the associated Twomey forcing. Day 2 has a higher incidence of surface precipitation, therefore, C0 values are adjusted accordingly. However, early morning spreading rates are controlled primarily by changes in variances and not C0 (Fig. S3). After 5 or so hours after the start of the injection, precipitation-related changes in C0 result in a 0.5–1 km h⁻¹ acceleration of the plume spreading rate during the midday period (Fig. S3).

Global mean forcing estimates (ΔF) from the merged LPM-LES library are of the same order of magnitude as implied by the heuristic framework of for the same total injected mass, although larger. For example, at 8 Tg yr⁻¹, τdecay=2 d, and Nsprayers=16000, our model suggests ΔF=1.8 W m⁻² (Fig. ), compared with about 1 W m⁻² in . The difference likely arises because our sprayers are limited to 4.4 % of the ocean surface rather than distributed across the entire eligible ocean area (54 %) and the injections are occurring preferentially in cloudy regions. As an additional point of comparison, found that by selectively spraying eastern Pacific cloudy regions (∼5 % of the ocean surface), nearly 1 °C of cooling was achieved. Assuming the mid-range injection scenario and τdecay=2 d, ΔF=-1.8 W m⁻² would impute a temperature sensitivity of 0.56 °C per W m⁻², which is broadly consistent with GCM estimates of temperature sensitivity for spatially inhomogeneous forcing . While the aim of this study is not to accurately determine ΔF, it is encouraging that the ΔF magnitudes are within a plausible range.

As shown in Fig. , and consistent with , Twomey forcing increases in strength with longer τdecay, with the largest fractional increases in -ΔF associated with the shorter aerosol lifetimes. Quadrupling Nsprayers only increases -ΔF by a factor of 2.1 at τdecay=2 d (Fig. ), indicating a saturation in forcing efficiency as sprayer density increases. Day 1 and Day 2 ΔF differ by less than 8 % for all values of τdecay and at both the low- and mid-range ship densities (Nsprayers=4000, 16 000) (Fig. ). This weak regime sensitivity suggests that for a fixed τdecay and negligible cloud adjustments, the deeper boundary layer and slightly more broken cloud conditions on Day 2 are approximately as susceptible to aerosol perturbations as the more overcast, shallower boundary layers on Day 1. The particle lifetime (set by τdecay) is the leading-order control on global mean Twomey forcing given a fixed sprayer density.

It is not immediately clear which of the meteorological, aerosol, and turbulence variables are most responsible for the weak regime sensitivity. To assess their relative roles, we perform a sensitivity analysis in which Day 2 values of boundary-layer depth, cloud fraction/morphology, particle model turbulence forcing, mean winds, and background aerosol concentration are substituted individually into the corresponding Day 1 simulations for each case, while all other variables retain their Day 1 evolution. We then quantify the resulting change in ΔF relative to the corresponding Day 1 control simulation.

Deeper boundary layers (which dilute the aerosol perturbation over a greater depth) and lower Day 2 cloud fractions both act to reduce Twomey forcing by 10 %–20 % relative to the Control, while the influence of mean winds is negligible (Fig. a). The predominant factor counteracting these reductions is the cleaner background aerosol environment on Day 2, resulting in a nearly 50 % increase in Day 1 Twomey forcing (Fig. a). Crucially, and of great relevance to this work, the Twomey forcing is insensitive to changes in the spreading rate between regimes (Fig. a). However, while the domain-averaged forcing remains stable when particle forcing is swapped, the underlying aerosol concentration distributions are impacted, with faster spreading rates resulting in reduced frequency of the smallest concentrations and more frequent extreme concentrations (Fig. b).

To our knowledge, the sensitivity of MCB forcing to sprayer velocity has received little direct attention in the literature. Sprayer velocity is potentially important as it constrains platform design and dictates total energy demand, which could increase substantially if high operating velocities are required. Given a theoretical understanding of the diminishing returns of Twomey forcing at high concentrations, it seems logical to assume that higher ship velocities would spread aerosol over larger regions, thus increasing Twomey forcing efficiency. However, the LPM results indicate that the Twomey forcing is largely insensitive to the ship velocity, with velocities ranging from 0 to 10 m s⁻¹ changing ΔF minimally (Fig. a). While the area-averaged forcing of ΔF remains nearly constant, the spatial distribution of the perturbed aerosol concentration does change, most notably at ship velocities of 10 m s⁻¹, with reduced occurrence of low concentrations and more frequent concentrations in the 20–80 # mg⁻¹ range (Fig. b). By running the LPM simulations with no mean wind (Ut‾=0 in Eq. 3), the sensitivity to ship velocity emerges with ship velocities below 5 m s⁻¹ resulting in dramatically reduced cooling efficiency (Fig. a). In the no mean wind scenario experiments, underlying probability distributions of aerosol concentrations begin to shift considerably, with the most susceptible Twomey forcing range experiencing reduced probabilities and a marked increase in concentrations above 200 # mg⁻¹ (Fig. b). This finding suggests that in environments like the northeast Pacific, the energy and mechanical requirements of MCB could be meaningfully reduced by utilizing stationary or slow-moving platforms without compromising radiative efficacy. Of course, it remains possible that this result could be impacted by processes that control the vertical dispersion of aerosols in different environments (i.e., evaporation, local buoyancy perturbations) . Simulations using uniformly-spaced sprayers moving in parallel, rather than randomly moving sprayers, indicate that Twomey forcing is also insensitive to ship organization (not shown).

The global mean Twomey forcing scales roughly linearly with increasing injection duration, with the slope of the increase depending on the assumed value of τdecay (Fig. a). Longer τdecay is associated with higher sensitivity to the injection duration, with τdecay=2 d resulting in 0.06 W m⁻² h⁻¹ of injection and τdecay=1 d resulting in 0.04 W m⁻² h⁻¹ of injection (Fig. a). Given 50 % increases in Twomey forcing from 6 to 12 h injection duration, it is likely worthwhile to continue injecting well after the “peak” susceptibility window in the morning hours . The sensitivity to injection timing is generally consistent with the modeling studies of , with 10 %–25 % enhancement in Twomey forcing achieved by injecting in the early mornings hours, versus a few hours after sunrise (Fig. b).

The particle model allows for the explicit tracking of particle age and the associated Twomey forcing from each particle age group. With τdecay=2 d, periodic equilibrium is reached after 8 d in the particle model, which means the maximum particle age is 8 d, despite the LES only having a run time of 2 d. Although not a strong function of Nsprayers or case-to-case variability, the percentage of Twomey forcing from a given particle age group is strongly dependent on τdecay, as expected, with 3+ day old particles contributing to 2 % of the forcing when τdecay=1 d and 41 % of the forcing when τdecay=4 d (Fig. a). Aerosols less than 1 d old are responsible for 25 %–65 % of the Twomey forcing, while aerosols 1–3 d old contribute close to 35 % across the τdecay range (Fig. a).

In addition to particle age, we quantified the contribution of each particle concentration bin to the Twomey forcing. For each bin, Twomey forcing was summed over all particle model grid cells falling within that concentration range. The contribution from each bin was then expressed as a percentage of the domain-total Twomey forcing (Fig. b). Our results reveal that for Nsprayers≤1600, only 10 %–15 % of the total Twomey forcing originates from concentration bins > 120 # mg⁻¹ assuming τdecay=2 d (Fig. b), emphasizing the difficulty of detecting the Twomey-forcing relevant aerosol perturbations , even in regions of relatively dense ship traffic. In the low-end MCB scenario (Nsprayers=4000), only 20 % of the forcing originates from bins below 120 # mg⁻¹ (Fig. b), making such a perturbation easily detectable.

We explore the maximum theoretical importance of spreading rate and quantify potential errors associated with assumptions made in GCMs (i.e., infinite spreading rate and uniform aerosol concentrations) using a changing particle model domain size with a single, fixed point source at the center of the domain. With doubly periodic boundary conditions, the changing domain size is a proxy for ship density. Cloud fraction is assumed to be unity, background aerosol is fixed at 20 # mg⁻¹, and boundary layer depth is constant at 1 km, to maximize the potential importance of spreading rate. First, we examine the fastest (FAST) and slowest (SLOW) spreading cases in the LES library, which have daily average spreading rates of 3.18 and 0.2 km h⁻¹, respectively. The difference between these two cases represents the upper bound on expected spreading-related Twomey forcing errors for naturally varying turbulence conditions, and the Twomey forcing magnitude and percentage differences are shown in Fig. a and b. For Nsprayers<5000 or a ship density < 3.125×10-4 ships per km², spreading rate-related Twomey forcing errors could potentially be in the 5 %–30 % range (Fig. b). Beyond Nsprayers=20000 (>1.25×10-3 ships per km²), errors related to spreading rates are likely to be small.

To estimate an upper bound on potential Twomey forcing errors in GCMs, we compare the SLOW case to the assumption made in current GCM simulations of MCB, which is that the aerosol is instantaneously spread evenly throughout the grid box. The working assumption in current GCM implementations is that the spreading rate is infinitely fast (INF), with Fig. c and d showing the Twomey forcing magnitude and percentage differences between the INF and SLOW cases. Errors become much more substantial when making the INF assumption, with ship densities below 2.5×10-4 ships per km² resulting in 10 %–200 % errors in global mean Twomey forcing, depending on τdecay (Fig. d). Even at sprayer densities representative of high-end MCB scenarios, spreading-related Twomey forcing errors could exceed 10 % in environments where τdecay is less than 1 d (Fig. d).

Additional idealized experiments indicate that concentration variability driven by random overlaps of ship tracks does not strongly impact the spreading rate and Twomey forcing relationship (Fig. S4). Using the median library spreading rate, instead of the extreme SLOW case, the expected spreading-related GCM errors for a mid-range MCB scenario (Nsprayer>10000) would generally be 10 % or less (Fig. S5). In the context of a mid- to high-end MCB scenario, Twomey forcing is unlikely to be impacted substantially by differences in naturally occurring horizontal spreading rates, but the instantaneous spreading rate assumption may still lead to overestimates of cooling in GCMs. These errors, especially in short aerosol lifetime environments, can be reduced by assuming a fixed, finite spreading rate (e.g., ∼1.5 km h⁻¹).

Assuming a finite spreading rate in GCMs could ameliorate some of the spreading-related radiative forcing errors, but in some global injection scenarios sprayer density might be on the lower end of Fig. , necessitating a more sophisticated approach to estimating the spreading rate within a GCM. One potential promising predictor of the particle model-derived spreading rate is the boundary layer-averaged horizontal TKE (TKE_hor) multiplied by TL, which has diffusivity units (m² s⁻¹) and explains nearly 50 % of the variance in spreading rate (Fig. ). When all times along the trajectories are included there is a clear time dependence of the spreading rate, with earlier times corresponding to faster spreading rates (Fig. a). Bearing in mind this time-dependence, the statistical spreading rate relationships may need to vary as a function of time since injection. When the first 10 h of spreading are neglected, the TKEhor×TL quantity explains ∼70 % of the variance. In future work, relationships such as these may be implemented within the GCM as an effective plume spreading parameterization. In models that employ CLUBB, such as E3SM , TL and TKE_hor are readily available quantities, but their agreement with LES requires further investigation.