Central to the model is Twomey's formulation for the susceptibility of cloud albedo αc to an increase in Nd assuming no cloud adjustments (Twomey, 1977), viz. 1dαcdNd=αc1-αc3Nd. Integrating Eq. (1) gives an expression for the increase in cloud albedo Δαc caused by an increase in Nd, as follows: 2Δαc=αc1-αcrN1/3-11+αcrN1/3-1, where rN=Nd′/Nd is the ratio of the droplet concentration in seeded vs. unperturbed clouds. It is worth noting that Eq. (2) is rather insensitive to αc, such that Δαc varies by only ∼ 10 % as αc changes from 0.3–0.7. Thus, the key sensitivity in Eq. (2) is to the value of rN.

To estimate the top-of-atmosphere (TOA) albedo for the same cloud requires a conversion to account for the absorption and scattering of solar radiation by the atmosphere above cloud. We follow the approach by Diamond et al. (2020; Eq. 17) and multiply the cloud albedo change by an atmospheric correction factor ϕatm as follows: 3ϕatm=Δαc,TOAΔαc=TFT21-αFTαc2, where TFT and αFT are the transmissivity and albedo of the free troposphere only. The more variable of these two parameters is TFT, which depends upon free-tropospheric water vapor. Here, we assume a value of TFT=0.8, consistent with values over dry regions of the Tropics and midlatitudes from the CERES-SYN product (Doelling et al., 2013). Free-tropospheric albedo is less variable, and we here assume a value of αFT=0.06 (also consistent with CERES-SYN). For typical cloud albedos αc in the range 0.25 to 0.75, ϕatm ranges from 0.66 to 0.70; for simplicity, we herein assume ϕatm=0.70. We estimate TOA indirect radiative forcing as -F⊙ϕatmΔαc, where F⊙ is the mean incoming solar irradiance averaged over day and night. Here, we assume a value of F⊙ equal to the global mean solar irradiance F⊙=342 W m-2. Geographical variation in insolation is not considered.

Marine cloud brightening, by definition, would only be deployed over the fraction of Earth covered by ocean. We further restrict this area to minimize the likelihood that plumes will intersect land areas. This is done by summing up the ocean area of those 10×10∘ latitude/longitude boxes that contain less than 10 % land area. The choice of boxes with 10∘ on a side is made because plumes are of the order of 1000 km in length (see Sect. 2.4). This limits the eligible fraction of Earth's surface for spraying, focean, to 0.54. We then assume that sprayers are confined to operate within some specified fraction fspray (0<fspray≤1) of this eligible area. If fspray is chosen to be less than unity, it is assumed that sprayed regions will be those with the highest climatological unobstructed low cloud cover. To determine the mean low cloud cover for the sprayed subregions flow, climatological monthly mean low cloud fractions are determined using MODIS Terra and Aqua level 3 liquid cloud fractions (years 2006–2010) for 10×10∘ boxes. As fspray decreases, the fraction of the ocean sprayed has a greater coverage of low clouds. If fspray is chosen to be very small, spraying would occur only in regions with the highest climatological monthly mean cloud cover (∼ 68 %). The MODIS data are well fitted with the following empirically determined expression: 4flow=0.32+0.36exp(-3.2fspray0.75).

Cloud condensate and coverage adjustments to injected aerosol are assumed to be zero, so MCB indirect radiative forcing arises only from the Twomey effect. In sprayed areas without low clouds, injected particles can exert a direct radiative forcing. The direct radiative forcing from injected aerosol in cloud-free regions between clouds is quantitatively estimated (see Sect. 2.7), but increasing direct radiative forcing is not a goal of the injection design. The global mean shortwave radiative forcing ΔF from MCB aerosol–cloud interactions is written as follows: 5ΔF=-F⊙foceanfsprayflowϕatmΔαc. To give a “back of the envelope” assessment of the potential for MCB, we take focean=0.54, assume fspray=1, and use Eq. (4) to set flow=0.33. If cloud albedo is increased by Δαc=0.01, then ΔF = -0.41 W m-2. Alternatively, it would take a cloud albedo increase of Δαc=0.09 to produce a radiative forcing of -3.7 W m-2, which would balance the longwave radiative forcing ΔF2×CO2 from doubling CO2. Figure 1 shows ΔF as a function of rN for different values of fspray and αc. Using Eq. (2), if we assume αc=0.56 (Bender et al., 2011, finds TOA cloud albedos of 0.35 to 0.42 for overcast stratocumulus in the major subtropical stratocumulus, Sc, decks, which must be corrected to cloud albedos with Eq. 3), then the ratio of seeded to unseeded cloud droplet concentration (rN=Nd′/Nd) would need to be 3.0 to produce a forcing with a magnitude equal to ΔF2×CO2. Assuming the entire ocean area could be seeded (focean=0.7; fspray=1), we find a value of rN=2.4, which is in the range of Nd increases over the ocean (2.10–2.85) that were needed to counter CO2 doubling in an analysis of three variants on a climate model (Slingo, 1990). If only half of the eligible ocean area is seeded (i.e., fspray=0.5), then rN would need to be at least 7 to counter CO2 doubling (Fig. 1). Stjern et al. (2018) analyzed an ensemble of different climate models in which Nd for all marine low clouds is increased by 50 % (rN=1.5) as a proxy for MCB and found an ensemble mean ΔF=-1.9 W m-2. Based on Fig. 1, and scaling the forcing to include the entire ocean, Eq. (5) produces a very similar forcing ΔF=-1.8 W m-2. This is also consistent with the models in Stjern et al. (2018) having small cloud adjustments overall so that the overwhelming bulk of the forcing is from the Twomey effect.

Any practical MCB deployment would be unable to produce uniform increases in Nd because seeding is necessarily discrete in nature rather than being distributed evenly. It is impractical to deploy sprayers at every point over the ocean; in practice, any deployment would likely consist of an array of floating particle injection systems distributed throughout regions where low clouds occur. To extend the heuristic model to account for this, assumptions are made about the spatiotemporal extent of the region affected by a single sprayer. Sprayers are assumed to be stationary so that air masses pass over them at the rate of the near-surface wind speed U0, which is taken as 7 m s-1, i.e., the mean value over oceans (Archer and Jacobson, 2005). Each sprayer injects sodium chloride particles continuously with a salt mass rate M˙s. Injected particles have a lognormal size distribution with geometric mean dry diameter (GMD) Ds and geometric standard deviation (GSD) S. The total number of particles sprayed per second from each sprayer N˙s is then as follows: 6N˙s=6M˙sπρsDs3e9(lnS)2/2, where ρs is the density of solid sodium chloride (2160 kg m-3). The volume into which particles are emitted increases with time as the plume expands to fill the depth of the MBL and widens horizontally. The timescale for vertical dispersion through the depth of the MBL is 10–20 min (Chosson et al., 2008), as evidenced by the fact that, in ship tracks, brightened clouds become evident typically 10–20 km downwind of the responsible ship. As satellite data readily show, ship tracks from commercial shipping are narrower close to the emitting ship and broaden downstream (Durkee et al., 2000). After rapid vertical dispersion through the MBL, dilution primarily occurs through lateral diffusion. Entrainment of lower-concentration free-tropospheric air also dilutes the plume but at a slower rate. The lateral track broadening rate is highly variable but is parameterized using the Heffter (1965) broadening rate K=1.85 km h-1 (see Fig. 7 in Durkee et al., 2000). This rate is broadly consistent with large eddy simulations of horizontal tracer spread in the cloudy MBL (Wang et al., 2011).

It has been proposed that a spray system to inject salt particles could derive the salt from sea water droplets (Salter et al., 2008; Cooper et al., 2014). Sea water is substantially more dilute than the equilibrium size of solution droplets at the surface (see, e.g., Hoffman and Feingold, 2021), and so there is some concern that cooling from the evaporation of water from equilibrating droplets may hinder or prevent the vertical mixing of injected particles. Reducing any negative buoyancy is an engineering challenge that may be addressed by increasing the turbulent mixing of the particle-laden plume with surrounding air and/or adding some thermal energy to the particle plume. This issue is beyond the scope of this study, and we herein assume the injected particles mix readily throughout the depth of the MBL. We also note that the additional water vapor introduced into the lower MBL from the evaporating sea water is negligible compared with the natural surface evaporative water flux and, thus, will have no impact on the MBL moisture budget.

Injected particles in the model have a characteristic residence e-folding timescale τres. This residence time incorporates several processes influencing particle lifetime, including removal by coalescence scavenging, scavenging by clouds and aerosol particles, and dry deposition. The value of τres varies with meteorological conditions, cloud, and precipitation properties and is also expected to be somewhat size dependent. In regions of marine stratocumulus values of τres of 2–3 d are consistent with estimates of precipitation scavenging (Wood et al., 2012), and τres=2 d is used as standard.

After a time t, the particles injected at time t=0 have moved a distance x=U0t. Considering both dilution and removal processes, given a plume width Wt=Kt and assuming dispersion through the entire MBL depth h, the injected particle concentration Ns(t) at time t is as follows: 7Ns(t)=N˙sU0hKte-t/τres. The wind speed and residence time define a length scale Lt=U0τres that effectively determines the streamwise length scale over which the particle concentration is affected by spraying. The area over the Earth's surface perturbed by each sprayer A is then determined by multiplying this length scale by a characteristic track width Wt, i.e., At=LtWt=U0τresWt. A linearly widening plume/track will expand to a width Kτres over the lifetime of the particles.

To estimate radiative forcing, the injected particle concentration Ns(t) is added to an assumed background aerosol over the entire track area and over the depth h of the MBL. Aerosol activation to form cloud droplets is carried out for the background aerosol and for the perturbed (background + injected) aerosol using an assumed updraft speed. Section 2.6 provides details of the activation scheme and the aerosol physical and chemical properties. The ratio of the perturbed to background cloud Nd from the activation scheme is used in the calculation of radiative forcing (Eqs. 2 and 5).

Figure 2 shows results from the model for a laterally spreading track, along with injected particle concentration and additional reflected shortwave from cloud brightening as a function of time/distance downstream of a point source sprayer. The heuristic model assumes an elongated cuboid plume (fixed width, height, and length), with the plume length Lt=U0τres and plume width taken to be the width of the linearly expanding plume at time τres/2, i.e., Wt=Kτres/2. The (time-independent) number concentration Ns1 of injected aerosol particles in the cuboid plume is as follows: 8Ns1=N˙shU0Wt=2N˙shU0Kτres. It is relatively straightforward to show that the overall injected particle concentration integrated over time is the same for the laterally spreading track (Eq. 7) and the cuboid track (Eq. 8). Although the reflected solar energy from the two tracks is not identical (Fig. 2c), the values are found to be close. A heuristic model track reflects slightly less than a spreading track for a given spray rate, with the difference growing as the magnitude of the Nd perturbation increases.

Experimentation with different spray, background aerosol, and cloud configurations shows that the reflected sunlight for the cuboid track is within 5 % of that for the spreading track for number spray rates N˙s<1016 s-1, with the cuboid model track being slightly less reflective. As N˙s increases, the ratio of the additional energy reflected by the spreading track to that from the cuboid track increases steadily, reaching 1.2 for N˙s=5×1016 s-1 and 1.5 for N˙s=1017 s-1, with the exact value dependent upon the background aerosol. As the magnitude of the aerosol number perturbation increases, an increasing fraction of the energy reflected occurs at times t>τres in the spreading plume. The albedo response in the cuboid track is relatively saturated due to the high aerosol/droplet concentrations (see Fig. 1), so the diluted but widespread aerosol in the spreading plume later on is more efficient at brightening. As we show in the discussion, coagulation losses during the high concentrations near to the sprayer are likely to be large for particle spray rates much greater than ∼ 1016 s-1. The cuboid tracks are, thus, a sufficiently accurate representation of reality for us to use them in the heuristic model, and this simplifies the treatment of overlapping tracks.

Given the plume dimensions for the heuristic model tracks, we estimate that the number of (nonoverlapping) tracks required to cover the 54 % of the ocean eligible for spraying (∼ 1.98 × 1014 m2), assuming Wt=44 km, τres=2 d, and U0=7 m s-1 (i.e., Lt≈1200 km; At=5.28 × 1010 m2) is 5300. If this number of sprayers was to be deployed either randomly or uniformly, then overlapping tracks would be unavoidable because air mass trajectories are not constant in time. Monte Carlo simulations were conducted, placing Nt randomly oriented or aligned rectangular tracks at random over a large domain of area A. The probability pn of n tracks overlapping in the domain is well predicted by a Poisson distribution as follows: 9pPn≈ζne-ζn!, where ζ=NtAt/A is the mean track density, i.e., the mean number of superimposed tracks (Fig. 3). Although not shown, pn is insensitive to both the track aspect ratio (Lt/Wt) and whether the tracks are aligned with their long sides in one direction or are randomly oriented.

The use of the Poisson distribution makes it straightforward to account for track overlap in the heuristic model; the injected particle concentration Ns at any given location is an integer multiple n(n≥0) of the single-track value Ns1 from Eq. (8) as follows: 10Ns=nNs1=2N˙snhU0Kτres, where the probability of n is given by Eq. (9). For low mean track densities ζ, the most likely value of n is zero (Fig. 3), and the ratio of the standard deviation to the mean value of n is high. A Poisson distribution has equal mean and variance, so the relative spatial heterogeneity of Ns, i.e., the ratio of the standard deviation to the mean track density, decreases as ζ-1/2. Because of the concave relationship between Δαc and Ns (see, e.g., Carslaw et al., 2013), a more homogeneous distribution of Ns over the seeded area will yield a radiative forcing with a larger magnitude for the same mean value of Ns.

Aerosol activation to form cloud droplets is treated using a five-dimensional look-up table derived from over 6000 numerical Lagrangian parcel model simulations (see the Appendix). In comparing with those, using the Abdul-Razzak and Ghan (2000) quasi-analytical activation scheme (henceforth ARG), we find significant differences that indicate a major underprediction of Nd with the ARG scheme when injected dry particle diameters are smaller than ∼ 200 nm (see the Appendix and also Sect. 4.2), and so we use the look-up table to treat activation in the heuristic model.

The aerosol size distributions used are the same for each activation approach. The background (unperturbed) aerosol particles are assumed to comprise lognormal accumulation and coarse modes. Accumulation-mode size values (D0,acc=175 nm; S0,acc=1.5) are taken from the synthesis of marine accumulation-mode measurements by Heintzenberg et al. (2000). Measured marine accumulation-mode number concentrations N0,acc vary considerably over the ocean, and the impacts of this on brightening are explored in Sect. 4.2. Although there is significant variability in the composition of marine cloud condensation nuclei (CCN), studies tend to find that the accumulation-mode aerosol in the unpolluted MBL consists of a mixture of sulfate, sea salt, and organic species. Different assessments of the hygroscopicity parameter (κ; from Petters and Kreidenweis, 2007) of CCN in the MBL provide a significant diversity of values, from values as low as 0.45 (Wex et al., 2010) to ∼ 0.7 (Andreae and Rosenfeld 2008). Here, we use the mean marine value of 0.7 from the model study of Pringle et al. (2010) for the unperturbed accumulation mode. The background coarse mode is lognormal, with GMD D0,coarse=615 nm and GSD S0,coarse=1.8 taken from summertime measurements at Graciosa island in the Azores (Zheng et al., 2018). The presence of the coarse mode suppresses the peak supersaturation in the updraft, increasing the minimum size of the particles that are activated, reducing the activated fraction (Ghan et al., 1998). This is explored further in Sect. 4.2. Injected aerosols are sodium chloride (κ=1.2; Petters and Kreidenweis, 2007), distributed lognormally with GMD Ds and GSD S, where Ds is allowed to vary and S=1.6. Table 1 provides a summary of the assumed aerosol properties used. A recent study suggests that the other inorganic species in sea salt render it slightly less hygroscopic than pure sodium chloride (κ=1.1; Zieger et al., 2017). Testing showed that the results of this study are largely insensitive to small variations in κ.

For most of the analysis presented in this study, a fixed updraft speed of w=0.4 m s-1 is assumed in the activation scheme. This is broadly representative of updrafts in the stratocumulus-topped MBL (Nicholls and Leighton, 1986; Wood, 2005; Bretherton et al., 2010; Zheng et al., 2016). Sensitivity to updraft speed is explored in Sect. 4.3. For simplicity, the temperature and pressure are set to be 280 K and 925 hPa, respectively, but the results are not highly sensitive to these values.

The heuristic model is also used to produce rough estimates of the aerosol direct radiative forcing from the injected aerosol. We assume direct forcing only in clear-sky regions. In an analogous formulation to Eq. (5), we estimate the global mean direct radiative forcing as follows: 11ΔFdirect=-F⊙foceanfsprayfclearE⋅AODspray, where fclear is the clear-sky fraction in the regions where sprayers operate, AODspray is the aerosol optical thickness (550 nm) of injected particles, and E is the clear-sky radiative forcing efficiency. We use E=-29 W m-2 AOD-1, which is the average over oceans for several models in the AeroCom study of Schulz et al. (2006). In this study, direct effects are only estimated for the case where fspray=1, i.e., sprayers operate in all eligible regions of the oceans, and so fclear=0.32 is taken as the complement of the total cloud cover during the daytime over the global oceans from Hahn and Warren (2007). To estimate AODspray, the injected aerosol lognormal size distribution (accounting for overlapping tracks as discussed in Sect. 2.5) is used to estimate extinction σspray at 550 nm, using the Mie code of Bohren and Huffman (1998). A mean relative humidity in clear-sky MBLs of 80 % is used to set a hygroscopic diameter growth factor for sodium chloride of 2.0 from Tang (1996). The assumed MBL depth h= 1 km (Table 1) is used to determine AODspray=hσspray. Direct forcing estimates are presented in Sect. 4.4.

Figure 5 presents results as a function of the number of sprayers Nsprayers and the salt mass injection rate per sprayer M˙s. For this case, injected particles have Ds=100 nm, and spraying occurs over all eligible ocean areas (focean=0.54 and fspray=1 in Eq. 5). A background accumulation-mode aerosol concentration N0,acc=100 cm-3 is assumed, representative of conditions over the open oceans (Heintzenberg et al., 2000), and a representative coarse mode is included (Sect. 2.6 and Table 1). Other parameters are set to the values provided in Table 1 and discussed in Sect. 2. The assumed MBL depth of 1000 m is representative of typical conditions in which stratiform marine low clouds occur. A global mean radiative forcing magnitude of 1–4 W m-2 can be achieved, with forcing generally increasing as total salt mass injection rates increase from ∼ 10 to ∼ 60 Tg yr-1 (Fig. 5a). As the total injection rate increases beyond 100 Tg yr-1, there are somewhat diminishing returns in terms of further brightening, and -ΔF reaches ∼ 5–8 W m-2 for an injection rate of 1000 Tg yr-1. The reduced sensitivity as more particles are injected is driven by increased competition for water vapor in the updraft, resulting in a decreasing fraction of injected aerosols activated to form droplets (Fig. 5b; dotted contours). When the injected aerosol concentration is less than a few hundred particles per cubic centimeter, such competition for vapor is relatively modest, and the activated fraction exceeds 70 %, but this reduces to only 30 %–40 % at injection rates of 300 Tg yr-1.

Given that -ΔF increases approximately as a function of total mass injection rate for mass injection rates of <50 Tg yr-1 (Fig. 5a), roughly the same forcing can be achieved either with a smaller number of high throughput sprayers, or a larger number of somewhat weaker sprayers. Scenario (1) in Fig. 5 has Nsprayers = 12 000, each injecting 6×1016 particles per second, for a total mass injection rate of 69 Tg yr-1; this achieves the same forcing (-3.7 W m-2) as scenario (2), which has Nsprayers=105, each injecting 6×1015 particles per second, for a total mass injection rate of 55 Tg yr-1. As we discuss in Sect. 5.1, particle spray rates approaching 1017 s-1 will likely result in significant particle losses due to high concentrations in the near field of the spray system, and so we consider scenario (1) to be close to the upper end of the injection rates that are likely to be feasible. From this, it may reasonably be concluded that, if MCB were ever to be used to achieve a radiative forcing close to that needed to offset a doubling of CO2, considerably more sprayers would be needed than are assumed in the estimate from Salter et al. (2008), where only ∼ 4500 spray vessels were assumed. The need for a greater number of sprayers in the heuristic model is primarily because of overlapping plumes which reduce effectiveness by introducing heterogeneity into the injected particle spatial distribution. Plume overlap is not accounted for in Salter et al. (2008), where each sprayer uniformly increases the particle concentration over an area of 7.7×1010 m2. Our assumed track area is At=5.28×1010 m2 (Sect. 2.5), which is quite similar to this, but our plumes overlap. The effect of plume overlap is demonstrated by noting that scenario (1), with fewer sprayers than scenario (2), requires ∼ 25 % more injected mass to achieve the same forcing. If we set Nsprayers to the number that would cover the ocean if there were no overlaps (5300; Sect. 2.5), the heuristic model would require over twice as much mass to produce a forcing sufficient to offset 2×CO2 as in the Nsprayers=105 case because the track coverage (fraction of the seeded area with at least one overlapping track) in seeded areas only marginally exceeds 50 % (Fig. 5b; dashed lines). Thus, almost half of the eligible ocean area remains unperturbed in this case, requiring increases in Nd to offset doubled CO2 in the perturbed clouds that are harder to achieve (see Fig. 1). Figure 6 (black curves) shows -ΔF for different values of Nsprayers plotted as a function of the total salt mass injection rate to further illustrate the need for a high number of sprayers to minimize the total mass of salt that needs to be injected to achieve a given forcing.

A key result from the heuristic model, for Ds ∼ 100 nm, is that the forcing to offset doubled CO2 should be achievable with a total salt mass injection rate of ∼ 50–70 Tg yr-1. This is much lower than the natural sea salt flux, which studies suggest ranges from 3000 to > 10 000 Tg yr-1 (Textor et al., 2006; Grythe et al., 2014). The residence time of natural sea spray particles is considerably shorter than the lifetime (τres=2 d) of the injected salt particles because sea spray particles have a much larger mean size. Thus, a perhaps more useful comparison is to examine the mass loading of the injected salt particles, which increases from 0.1 to 1 µg m-3 as the forcing magnitude increases from 1 to 4 W m-2 (Fig. 5c). The coarse-mode aerosol assumed in the model (Table 1) has a mass loading of 12.7 µg m-3, which is broadly representative of typical salt loadings in the MBL (5–20 µg m-3; Jaeglé et al., 2011). Thus, the mass loading of injected particles required to deliver a significant radiative forcing is a relatively small fraction (∼ 10 %) of the natural salt burden in the atmosphere. This is not the case with existing climate model studies of MCB, where much higher salt mass injection rates have been required in order to provide a globally significant radiative forcing. The reasons for this are explored the next section.

It is worth comparing the results from the heuristic model with estimates of radiative forcing from commercial shipping. Total SO2 emissions from shipping are ∼ 10 Tg yr-1 (see Lauer et al., 2007). Assuming this is all converted into sulfate, this equates to a mass of 15 Tg SO4 yr-1. Introducing injections of 15 Tg NaCl yr-1 would yield a Twomey radiative forcing of ∼ 1.5–2.0 W m-2 for Ds=100 nm (Fig. 6a). This represents a considerably higher efficacy (forcing per mass of solute) for MCB compared with commercial shipping. Although sea salt is slightly more hygroscopic than sulfate, it is unlikely that composition differences explain the greater efficacy. On the other hand, observations show that the modal diameter of accumulation-mode particles over the oceans, which consist mostly of sulfate, tend to be closer to 200 nm diameter than to 100 nm diameter (e.g., Heintzenberg et al., 2000). Although commercial ships do emit a considerable number of small Aitken particles (e.g., Hobbs et al., 2000), one would expect considerable growth into the accumulation mode over the lifetime of the emitted SO2. One hypothesis to explain the greater efficacy of MCB is that commercial shipping emissions result in larger accumulation-mode particles. Injection rates of 15 Tg yr-1 of NaCl particles with Ds=200 nm would yield a radiative forcing of only 0.5 W m-2 (Fig. 6a), a value within the range of estimates of forcing from marine shipping (0.06–0.6 W m-2; see the introduction). Additionally, shipping emissions are much more concentrated geographically than those from our heuristic model (assuming fspray=1), which reduces efficacy. A more thorough treatment of the effects of geographical heterogeneity of sulfate from commercial shipping is beyond the scope of this study.

A key unresolved question concerns what size of injected particles produces the most effective brightening. Prior studies using LES and climate models have used relatively large particles with modal dry diameters exceeding 200 nm. Although particles of this size serve as very effective CCN, it is important to take into consideration the overall mass injection rate, which determines the energetic requirements for particle generation and impacts on atmospheric chemistry (see the introduction). For the sprayer number and injection rate from scenario 2 (Sect. 4.1; Fig. 5) the optimal geometric mean diameter Ds of injected particles is 30–60 nm (Fig. 7a). This optimal size range is consistent with the parcel modeling of Connolly et al. (2014) and is relatively insensitive to the background accumulation-mode aerosol concentration N0,acc. For fixed mass injection rate, injected aerosol concentration increases as the inverse third power of Ds (Eqs. 5 and 7). For large injected particles (Ds ∼ 200 nm), most of the injected particles are activated (Fig. 7b), but each particle has a large mass, and so the overall mass injection rate required to produce a given forcing is roughly 5 times higher with Ds=200 nm than it is with Ds=100 nm (Fig. 6a), which is quantitatively consistent with the GCM (global climate model) sensitivity tests in Partanen et al. (2012). We find that 40 % more forcing can be achieved per mass injected if Ds is further decreased from 100 to 50 nm (Fig. 7a). With Ds in this optimal range in terms of forcing per mass injected, although the activated fraction is quite low, the gain in the added aerosol concentration counters this. This occurs up to a point where competition for vapor draws down supersaturation, and there is a reduction in the number of injected particles that have critical supersaturations sufficiently low to activate. When Ds is smaller than ∼ 40 nm, Nd begins to decrease again (Fig. 7b). The saturation effect of adding very small particles is also demonstrated by the gray lines in Fig. 6a, which show forcing as function of total salt mass injection rate for Ds=50 nm. For low mass injection rates (<10 Tg yr-1), these very small injected particles produce twice as much brightening as particles with Ds=100 nm. At higher rates, exceeding ∼ 50 Tg yr-1, brightening increases very modestly.

We find a major discrepancy between the parcel model activation used here and that estimated using the ARG parameterization. For Ds larger than 100 nm, droplet concentrations from ARG are in general agreement with those from the parcel model (Fig. 7b), but as the injected particle size decreases, ARG is unable to activate enough droplets. A significant tendency to underpredict activation fraction has been noted in several prior studies (Ghan et al., 2011; Connolly et al., 2014), and Simpson et al. (2014) found a systematic underprediction of peak supersaturations estimated with ARG, which we confirmed with the parcel model (Fig. 7c). This upshot of this issue is that, whereas we find that a maximum in brightening for Ds=40 nm, the competition for vapor in ARG is so strong that it prevents activation of almost all injected particles, and so the forcing is close to zero (Fig. 7a). It will, therefore, be very important to ensure that activation schemes used in climate modeling for MCB are sufficiently accurate to represent the unusual size distributions that would be needed for effective implementation of MCB.

Alterskjær and Kristjánsson (2013; henceforth AK13) used a climate model with ARG as its activation scheme and found that injected Aitken-mode particles (Ds=44 nm) produced a strong negative ΔF at the lowest injection rate used (48.2 Tg yr-1) but a positive ΔF for injection rates exceeding this. The behavior is consistent with our findings using the ARG scheme but is not consistent with the results from the parcel model, where brightening continues to increase with injected mass for particles of this size (Fig. 7a), and there is no cloud darkening (positive ΔF). AK13 also conducted a sensitivity study in which peak supersaturations were fixed at 0.2 % and found that the sign of the forcing changed from weakly positive to strongly negative. We find that the suppression of peak supersaturation as Ds decreases is similarly strong in the parcel model and the ARG parameterization for Ds>30 nm (Fig. 7c), implying that additional competition for vapor from the injected particles is not the main reason for the reduced activation fraction in ARG. Instead, it is the general underprediction of peak supersaturation in ARG occurring at all values of Ds that is the main reason for its inability to activate small Aitken particles. Indeed, the fixed supersaturation of 0.2 % in the AK13 sensitivity test is quite similar to that in the parcel model (Fig. 7c), and much higher than that for the ARG parameterization, showing that the use of ARG in AK13 is leading to misleading results regarding the efficacy of injecting Aitken-mode particles.

Although the optimal geometric mean diameter Ds of injected particles is 30–60 nm (Fig. 7a), there are other reasons against injecting very small Aitken-sized particles, including near-field coagulation and a higher loss rate to Brownian scavenging by cloud droplets. The former is explored in Sect. 5.1. The timescale for Brownian scavenging of injected aerosol in a cloud-topped MBL scales with Ds2 (Seinfeld and Pandis, 2003) and also decreases inversely with Nd2/3. For realistic liquid water contents, it can be shown that under MCB (i.e., Nd of a few hundred cubic centimeters) for Ds=60 nm, the timescale for particle losses to Brownian scavenging by cloud droplets is ∼ 4 d but falls to only ∼ 1 d for Ds=30 nm.

There is strong sensitivity of forcing to injected particle residence time τres (Fig. 6b). A longer τres increases the area of each sprayer track in proportion to τres2 (since both track width and length are proportional to τres; see Sect. 2.4), but the injected particle concentration over that area scales as τres-1 (Eq. 7). For the scenario of many overlapping tracks (Nsprayers=105), the total salt mass required to produce a given forcing scales with τres-1 (Fig. 6b). Thus, the lifetime of injected particles is a key determinant of MCB efficacy, warranting further study.

Higher background droplet concentration lowers a cloud's albedo susceptibility (dαc/dNd; Twomey, 1977; Platnick and Twomey, 1994). In addition, the increase in droplet concentration ΔNd with injection is also reduced when N0,acc is higher (Fig. 7b). This occurs because peak supersaturation is reduced when the background particles are more numerous, and so a lower fraction of the injected aerosol is activated. This results in a forcing that scales more weakly with N0,acc than would be expected based solely on the albedo susceptibility. As N0,acc increases from 50 to 150 cm-3, the albedo susceptibility decreases by a factor of 3, and yet the magnitude of the radiative forcing (e.g., for Ds=50 nm) decreases by less than a factor of 2.

The presence of a coarse mode in the unperturbed state imposes a relatively modest decrease in the effectiveness of aerosol injections in brightening clouds. For Ds in the range 50–100 nm, the realistic background coarse-mode concentration (N0,coarse=10 cm-3) used throughout this study results in a forcing that is less than 10 % smaller than in the absence of a coarse mode (Fig. 8), and the effect is weaker for larger Ds. Coarse-mode concentrations vary with wind speed and can reach values of ∼ 20–50 cm-3 at high wind speeds (Zheng et al., 2018). As Fig. 8 shows, it is only at concentrations well in excess of 10 cm-3 (mass loadings well more than 10 µg m-3) that there would be significant limits to brightening due to the coarse mode. It is important that MCB spray technology does not introduce a significant number of coarse-mode particles, as these will reduce brightening. However, it should be noted that the coarse-mode mass loadings required to produce a significant dampening of forcing are considerably more than those required to produce brightening using ∼ 100 nm particles.

Updraft speed w is a key determinant of the peak supersaturation during the activation process in updrafts (Sect. 2.6). A single value (w=0.4 m s-1) is assumed for the results shown in this study, but note that the sensitivity of ΔF to w is strongly dependent upon the injected particle size (Fig. 9a), with sensitivity decreasing strongly as Ds increases. The sensitivity is related to Nd (Fig. 9b), which itself depends upon the peak supersaturation in the updraft (Fig. 9c) and the size distribution of injected and unperturbed aerosol particles. Note that the suppression in peak supersaturation for the perturbed case compared with the unperturbed case is stronger for Ds=100 nm than it is for Ds=200 nm but falls no further for Ds=50 nm. Smaller and more numerous particles have a greater surface area and therefore remove vapor more rapidly, but kinetic limitations on growth rates restrict the continuation of this when Ds falls much below 100 nm. For the mass spray rates assumed here (scenario 2; see Sect. 4.1), almost all injected particles activate in updrafts exceeding 0.3 m s-1 when Ds = 200 nm (Fig. 9b). The suppression of peak supersaturation is relatively modest in this case (Fig. 9c). For Ds=100 nm, the forcing magnitude increase with w is stronger because the Nd′ increase with w is stronger. However, it should be noted that, for Ds=100 nm, the forcing magnitude only increases by 30 % over the range 0.2<w<0.6 m s-1, indicating relatively weak sensitivity to updraft speed overall. The greatest sensitivity to w occurs for the smallest injected particles (Ds=50 nm), where the forcing increases by 80 % over the range 0.2<w<0.6 m s-1. This reflects the fact that the greatest sensitivity of Nd′ to increasing peak supersaturation will occur when the critical supersaturation of the modal diameter (where the greatest number of particles lies) is close to the peak supersaturation. A 50 nm diameter salt particle has a critical supersaturation of 0.25 % (e.g., Petters and Kreidenweis, 2007), which is similar to the peak supersaturation at w=0.4 m s-1 (Fig. 9c). Given that activation in real MBL clouds occurs in a spectrum of updrafts (e.g., Snider et al., 2003), this result would caution against the use of injected particles that are too small to increase Nd in the majority of clouds seeded.

A recent study with three climate models found that direct forcing from injected salt aerosol may compete with or even exceed the indirect forcing in magnitude (Ahlm et al., 2017). We demonstrated (Sect. 4.2) that injecting particles with sizes Ds = 30–60 nm produces the greatest brightening for a given salt mass injected, and a forcing to offset doubled CO2 can be achieved with injection rates below 100 Tg yr-1 (dashed circle; Fig. 10a). These optimal particle sizes are much smaller than the dry modal diameters of 200, 260, and 880 nm for injected particles in the models used for the GeoMIP assessment of Ahlm et al. (2017). The magnitude of the heuristic model global direct forcing ΔFdirect (Sect. 2.7) is very small (<0.5 W m-2) for total salt mass injection rates of <100 Tg yr-1 (Fig. 10b). Indeed, generating ΔFdirect=-4 W m-2 requires an order of magnitude greater mass injection rate than it does to produce the same forcing from MCB (compare Fig. 10a and b). Partanen et al. (2012) found a very small contribution of direct radiative forcing with Ds=100 nm because the same indirect forcing in this case was achieved with ∼ 5 times less injected mass than their case (5 × GEO) with Ds=200 nm, wherein direct forcing constituted about 30 % of the forcing.

The dry size Ds to maximize the direct forcing, for a given mass injection rate, is ∼ 110 nm (Fig. 10b), which is around twice as large as the optimal size for MCB. As we have seen (Figs. 6 and 7), producing significant cloud brightening for the Ds values in the models in Ahlm et al. (2017) requires much higher mass injection, and this leads to significant direct radiative forcing. Indeed, for Ds=880 nm, the brightening efficiency (Fig. 7) is so low that we would expect very little brightening for spray rates of several hundred teragrams per year (hereafter Tg yr-1), which is consistent with the very small increases in Nd for the model that injected particles of this size, despite an injection rate of 590 Tg yr-1. In conclusion, the results here demonstrate that marine cloud brightening is not very effective without clouds when consideration is given to the injection rates/sizes required to produce a significant radiative forcing from aerosol–cloud interactions.

The results presented in Sect. 4.1 suggest that, to produce global radiative forcing from MCB that offsets a significant fraction of the forcing from doubling CO2, many sprayers will be required. To keep the number to below ∼ 105, particle number injection rates N˙s of ∼ 1016 s-1 will be needed. Similar forcing can be achieved with fewer sprayers, but this will necessitate higher N˙s. This implies very high particle concentrations in the near field of the spray system. Taking the spray system to be a collection of nozzles arranged over some area, A0, spraying into an airflow, vflow, then the initial particle concentration N0 in the immediate wake of the sprayer is N0=N˙s/A0vflow. The approach in Turco and Yu (1997) is used to model the downstream particle concentration, assuming a fixed coagulation kernel based on a particle diameter d=100 nm and vflow=10 m s-1. The coagulation kernel is not strongly dependent upon particle diameter for this size range (see Fig. 13.5 in Seinfeld and Pandis, 2003), so variations in injected dry size and the degree of hygroscopic swelling do not have major impacts. It is worth noting that seawater droplets ultimately yielding dry diameters in the effective range for MCB may be significantly larger and may coagulate somewhat more slowly. We consider a diluting slender plume based on the Gaussian plume dispersion, which yields a plume cross sectional area (and thus volume) that evolves with time t as V=V01+ttdila. Here a=ry+rz=1.49 and tdil=1.2 s, with tdil=vflow-1A0/πRyRz1/(ry+rz), where the plume widths at distance x downstream of the sprayer in the cross-wind and vertical directions are σyx=Ryxry and σzx=Rzxrz, respectively. The constants Ry, ry, Rz, and rz are those for neutral stability conditions from Klug (1969), as reproduced in Table 18.3 of Seinfeld and Pandis (2003). Numerical simulations had to be performed because the solution does not allow for analytical integration.

Results of the coagulation–dilution calculations (Fig. 11a) indicate that there are relatively weak particle losses from coagulation until particle injection rates N˙s exceed ∼ 1016 s-1, above which loss rates grow sharply. Without dilution, there are large losses within the first 100 s for rates exceeding 1015 s-1, and for rates approaching 1017 s-1, most of the losses occur within the first 10 s. Dilution immediately downstream of the spray system is therefore most important for particle survival. The volume profile for these simulations is shown in Fig. 11b. We assume that particle concentration within the expanding plume is uniform, which is somewhat unrealistic because the edges of the plume will become more diluted at a faster rate than those in the center. A multi-shelled Gaussian plume model was employed in Stuart et al. (2013), and this appears to result in somewhat weaker loss rates than what we find, but the same general dependencies were found. There is a somewhat weaker dependence of the fraction of particles remaining on A0 than might be imagined (Fig. 11c), given that A0 determines the initial concentration of particles, and loss rates scale with N2. This is because it takes longer for turbulent eddies to penetrate a wider plume and mix ambient air into the plume core, so that larger A0 is associated with a longer dilution timescale tdil.

The strong dependence of coagulation losses on N˙s (Fig. 11d) indicates that, as rates approach 1017 s-1, the fraction of particles remaining decreases as rapidly as N˙s increases, which essentially means no increase in the far-field concentration as injection rates are further increased. Because this is not strongly sensitive to A0 (Fig. 11c), this imposes a limit (∼ 5 × 1016 s-1) on the maximum rate of particles that a ship-deployable spray system can provide to the far-field environment. Recall that the spray scenario to offset doubled CO2 forcing with 15 000 ships discussed in Sect. 4.1 requires a number injection rate that is at this upper limit of feasibility. More rapid dilution than can be provided by boundary layer turbulence may be possible in the first few seconds downstream of the spray system if the initial flow rate can be increased, but more sophisticated fluid and aerosol dynamics modeling will be required to determine the maximum far-field injection rates that a sprayer can provide.

The results of this study have implications for both types of climate modeling MCB studies discussed in the introduction, namely those that fix Nd at some value in seeded regions and those that attempt to model aerosol–cloud interactions using salt aerosol injection.

The results presented in Sect. 4.1 demonstrate that there are diminishing returns on increasing Nd as spray rate increases (e.g., Fig. 5b). Producing cloud droplet concentrations of 1000 cm-3 is possible but requires mass injection rates approaching 1000 Tg yr-1 (compare Fig. 5a and b). Locally high mass injection rates would reduce the fractional area of the ocean required for spraying (see next paragraph), but it should be borne in mind that increasing Nd to 1000 cm-3, as has been done in some climate modeling studies (e.g., Rasch et al., 2009; Baughman et al., 2012), would increase the Brownian scavenging rate of interstitial injected aerosol and reduce the overall particle residence time τres, resulting in a reduced forcing (see Sect. 4.2; Fig. 6b). Increasing Nd from 300 to 1000 cm-3 reduces the Brownian scavenging timescale by a factor of over 2 (Seinfeld and Pandis 2003), suggesting that attempting to implement MCB with very high droplet concentrations may not be practical.

Several previous climate model studies have seeded a relatively small fraction of the ocean area. Jones et al. (2009) set Nd′ = 375 cm-3 in three regions that total only 4.7 % of the ocean and achieved ΔF=-0.97 W m-2. The unperturbed Nd is not known for this study, so it is not possible to predict the Twomey forcing for this case using the heuristic model, but for fspray=0.1, a peak forcing of -1.4 W m-2 is achievable (Fig. 12), but achieving a forcing magnitude in excess of 1 W m-2 requires a mean Nd of over 800 cm-3 (rN∼8) in the seeded area. One can only conclude that either the unperturbed Nd in Jones et al. (2009) was very low, or that the model produced significant positive cloud adjustments that augmented the Twomey effect. Figure 12 suggests that global forcing magnitudes greater than ∼ 4 W m-2 from the Twomey effect alone are only likely to be possible if ∼ 50 % of the eligible ocean area (∼ 40 % of the total ocean area) is seeded.

Climate modeling to investigate MCB by injecting surface salt sources has typically injected particles with Ds of 200 nm or greater (Alterskjær et al., 2012, 2013; Ahlm et al., 2017). The heuristic model sensitivity to injected particle size presented in Sect. 4.2 indicates that Ds=200 nm is inefficient (Fig. 7), requiring mass spray rates 5 times higher than for Ds=100 nm and over an order of magnitude higher than for Ds=50 nm (Fig. 6). This has led to mass spray rates in existing MCB studies of hundreds of Tg yr-1 (Ahlm et al., 2017), and these high mass spray rates produce significant direct radiative forcing (Ahlm et al., 2017). Our results suggest that smaller injected particles can yield global radiative forcing of -1 to -4 W m-2 with very little direct radiative forcing. Providing meaningful global radiative forcing using the aerosol direct effect is an extremely inefficient use of salt particles. We therefore suggest that future climate modeling should focus on smaller injected particles to build on the sensitivity study in Partanen et al. (2012). This may challenge some models because a dedicated injection particle mode independent of the model's accumulation mode will be required, and accurate treatment of activation for such cases is a major issue (see Sect. 4.2 and the Appendix).

One notable feature of existing LES studies designed to test the sensitivity of albedo to particle injections (Sect. 3) is that all existing simulation experiments show some degree of brightening, although some do show cloud cover or condensate adjustments that partly offset the Twomey effect. In most cases, especially very clean conditions, cloud adjustments significantly augment Twomey brightening. No LES studies in the literature show domain-wide cloud adjustments that completely offset Twomey brightening. However, predicting cloud adjustments accurately is a challenge, even for LES models; some models do not represent the physical processes needed to produce the correct adjustments. Such processes include size-dependent droplet evaporation, which requires an estimation of supersaturation (many LESs assume saturation adjustment; see Wang et al., 2003) and sedimentation–entrainment feedback (e.g., Bretherton et al., 2007). Tests to establish the importance of these processes for determining susceptibility to particle injections are incomplete. In addition, existing LES studies represent only a small subset of possible meteorological conditions, focusing primarily upon shallow MBLs that are probably more susceptible to aerosol injections. Studies examining the susceptibility to particle injections of deeper trade wind MBLs, including aggregated shallow convective systems, need to be conducted to establish the efficacy of MCB in these regions.

It should be noted that most LES cases to test MCB are of insufficient duration to examine the responses at timescales longer than the injected particle residence time τres, which suggests that longer simulations will be necessary to evaluate the true radiative forcing from injections. Furthermore, no LES studies to date have attempted to constrain the injected particle residence time τres, which is a key determinant of MCB forcing (Sect. 4.2; Fig. 6b). The dependence of τres on precipitation, entrainment, in-cloud Brownian scavenging, and other factors warrants exploration using LESs. In addition, there are several potentially important feedbacks involving aerosol residence time that are ideal for study using LESs. First, it is well understood that MBL precipitation is a major sink of CCN (Wood et al., 2012; Zheng et al., 2018; Wang et al., 2021). Residence time is expected to be shorter in precipitating MBLs, and suppression of precipitation by high Nd in seeded clouds will increase τres (Wood et al., 2012), but increased cloud surface area will increase Brownian scavenging of injected particles, reducing τres. Precipitation suppression will also impact the background aerosol properties, including potentially increasing the concentration of coarse mode and giant CCN that may counter some of the Nd-driven precipitation suppression (Feingold et al., 1999). These feedback processes may affect aerosol residence time and MCB forcing in ways not accounted for in current analyses. Finally, deficiencies in some activation schemes used in LESs are likely to be a significant issue hindering accurate representation of the effects of injected particles smaller than 100 nm (Sect. 4.2; Appendix; Connolly et al., 2014), so it will be important to ensure that LES models can handle the activation process faithfully.