On the Limits of Köhler Activation Theory: How do Collision and Coalescence Affect the Activation of Aerosols?

Activation is necessary to form a cloud droplet from an aerosol, and it occurs as soon as a wetted aerosol grows 1 beyond its critical radius. Traditional Köhler theory assumes that this growth is driven by the diffusion of water vapor. However, 2 if the wetted aerosols are large enough, the coalescence of two or more particles is an additional process for accumulating 3 sufficient water for activation. This transition from diffusional to collectional growth marks the limit of traditional Köhler 4 theory and it is studied using a Lagrangian cloud model in which aerosols and cloud droplets are represented by individually 5 simulated particles within large-eddy simulations of shallow cumuli. It is shown that the activation of aerosols larger than 6 0.1μm in dry radius can be affected by collision and coalescence, and its contribution increases with a power-law relation 7 toward larger radii and becomes the only process for the activation of aerosols larger than 0.4− 0.8μm depending on aerosol 8 concentration. Due to the natural scarcity of the affected aerosols, the amount of aerosols that are activated by collection is 9 small with a maximum of 1 in 10000 activations. The fraction increases as the aerosol concentration increases, but decreases 10 again as the number of aerosols becomes too high and the particles too small to cause collections. Moreover, activation by 11 collection is found to affect primarily aerosols that have been entrained above the cloud base. 12

We consider one particle which grows by coalescing with other particles. Accordingly, the particle's water mass after n 48 collections is given by 50 where m 0 terms the particle's initial water mass and m i (i > 0) the mass of water added by each collection. The second equals 51 sign introduces the assumption of a monodisperse ensemble of collected particles.

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Based on Köhler theory, it can be shown that the critical radius for activation is given by properties responsible for the solute effect are represented by b = 3ν s ρ s µ l /(4πρ l µ s ), with the van't Hoff factor ν s , the mass 57 density of the aerosol ρ s , and the molecular masses of water µ l and aerosol µ s , respectively. Accordingly, the critical mass for 58 activation after n collections yields 59 m crit,n = 4 3 πρ l · r 3 crit,n = 4 3 60 where m s,0 terms the initial aerosol mass and m s,i (i > 0) the aerosol mass added by each collection. Approximating the 61 summation in (3) demands further assumptions on the distribution of aerosol mass within the particle spectrum. Two scenarios 62 are defined. Scenario A: the collected particles contain a negligible amount of aerosols. Accordingly, the aerosol mass does 63 not change ( n i=1 m s,i = 0). Scenario B: each particle contains the same mass of aerosol. Correspondingly, the aerosol mass 64 increases proportionally to the number of collections ( n i=1 m s,i = n · m s ).
In Fig. 1, the evolving particle radius and critical radius are displayed as a function of the number of collections (details on 66 the particle properties are given in the figure's caption). The simultaneous examination of particle radius and critical radius 67 reveals if a particle is activated (particle radius larger than critical radius) or deactivated (particle radius smaller than critical 68 radius). For scenario A, the initially inactivated particle (black line) grows faster than the critical radius (blue line), and the aerosol activates after 3 collections. For scenario B, an initially inactivated particle (continuous red line) and an initially 70 activated particle (dashed red line) are examined. Since the critical radius for activation increases faster than the particle radius, 71 activation is inhibited or the deactivation of previously activated particle is caused.

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These considerations suggest that only the collection of particles with a large amount of water and a comparably small 73 amount of aerosol mass (i.e., highly dilute solution droplets) might lead to activation (as shown in scenario A). This, however,

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indicates that the collected particles are probably activated already. Therefore, the process of collectional activation will not in-75 crease the total number of activated aerosols since one ore more already activated aerosols need to be collected (or annihilated) 76 in the process of collectional activation. By contrast, the collection of particles with a comparably large amount of aerosol 77 (i.e., less dilute solutions, as shown in scenario B) might inhibit activation since the increase of the critical radius exceeds the 78 increase of the wet radius.

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The following part of the study is investigating how coalescence is able to cause aerosol activation in shallow cumulus clouds 80 using a detailed cloud model considering diffusional growth as well as detailed physics of collision and coalescence.  , 2008;Shima et al., 2009;Sölch and Kärcher, 2010;Riechelmann et al., 2012;Naumann and Seifert, 2015). A 86 summary of the governing equations and the extensions carried out for this study to treat aerosol mass change during collision 87 and coalescence is given in the Appendix A. The underlying dynamics model, the LES model PALM (Maronga et al., 2015), 88 solves the non-hydrostatic incompressible Boussinesq-approximated Navier-Stokes equations, and prognostic equations for 89 water vapor mixing ratio, potential temperature, and subgrid-scale turbulence kinetic energy. For scalars, a monotonic advec-90 tion scheme (Chlond, 1994) is applied to avoid spurious oscillations at the cloud edge (e.g., Grabowski and Smolarkiewicz, .

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The initial profiles and other forcings of the simulation follow the shallow trade wind cumuli intercomparison case by 93 Siebesma et al. (2003), which itself is based on the measurement campaign BOMEX (Holland and Rasmusson, 1973). A 94 cyclic model domain of 3.2 × 3.2 × 3.2 km 3 is simulated. (In comparison to Siebesma et al. (2003), the horizontal extent has 95 been halved in each direction due to limited computational resources.) The grid spacing is 20 m isotropically. Depending on 96 the prescribed aerosol concentration, a constant time step of ∆t = 0.2 − 0.5 s had to be used for the correct representation of 97 condensation and evaporation, but it is also applied to all other processes. The first 1.5 hours of simulated time are regarded as 98 model spin-up; only the following four hours are analyzed.

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The simulated particles, called super-droplets following the terminology of Shima et al. (2009), are released at the beginning the super-droplets is 4.3 m, yielding a total number of about 360 × 10 6 simulated particles and about 100 super-droplets per 102 grid box. Initial weighting factors, i.e., the number of real particles represented by each super-droplet, are 8 × 10 9 , 48 × 10 9 ,

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The dry aerosol radius is assigned to each super-droplet using a random generator which obeys a typical maritime aerosol 106 distribution represented by the sum of three lognormal distributions (Jaenicke, 1993) ( Fig. 2). However, only aerosols larger 107 than 0.005 µm are initialized since smaller aerosols do not activate in the current setup. The different aerosol concentrations 108 are created by scaling the weighting factor of each simulated particle to attain the desired concentration. The aerosols are 109 assumed to consist of sodium chloride (NaCl, mass density ρ s = 2165 kg m −3 , van't Hoff factor ν s = 2, molecular weight µ s = 110 58.44 g mol −1 ). The initial wet radius of each super-droplet is set to its approximate equilibrium radius depending on aerosol 111 mass and ambient supersaturation (Eq. (14) in Khvorostyanov and Curry, 2007). The applied collection kernel includes effects 112 of turbulence, which have been shown to increase the collection probability of small particles significantly (e.g., Devenish   depicts the ambient supersaturation experienced by that particle (black) and its critical supersaturation (red).
diffusional growth after activation, the activated particle is required to be located in a volume of air which exceeds the critical 119 supersaturation at the moment of activation (S > S crit at r = r crit ). This is always fulfilled in the case of diffusional growth, but 120 it is checked additionally in the case of collectional activation to ensure equivalence of collectional and diffusional activation.

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To decide if an activation is primarily driven by diffusion or collection, all simulated particles have been tracked throughout 122 the simulation and their mass growth has been integrated from their minimum mass before activation, min (m), to the critical 123 activation mass, m crit :

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where dm| diff and dm| coll are directly derived from the LCM's model equations (A2) and (A5) -(A6), respectively. Note the 127 following procedures for determining min (m), ∆m| diff , and ∆m| coll during the simulation: (i) If a particle shrinks below 128 min (m) before activation, ∆m| diff and ∆m| coll are set to zero and are re-calculated starting from this new minimum mass.

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(ii) If a particle becomes deactivated, i.e., evaporates smaller than its critical radius after being activated, the current mass is 130 considered the new min (m) and ∆m| diff and ∆m| coll are set to zero. (iii) If a collection does not result in an activation and 131 the particle evaporates back to its equilibrium radius afterwards, ∆m| diff will be negative and ∆m| coll positive. To avoid the 132 potentially incorrect classification of a following activation, ∆m| diff and ∆m| coll are set to zero if ∆m| diff becomes negative 133 and the current mass is considered as min (m).

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The following two processes are considered a collectional activation if the collectional mass growth exceeds the diffusional 135 (dm| coll > dm| diff ): first, the coalescence of two inactivated aerosols resulting directly or after some diffusional growth in an  activation; second, the coalescence of an inactivated aerosol with an activated aerosol resulting in an inactivated aerosol, which 137 activates after some diffusional growth. If the latter process results directly in an activated aerosol, this collection is only 138 considered a collectional activation if the wet radius of initially activated particle is smaller than the critical radius of the 139 newly formed activated particle. The latter restriction ensures that the coalescence of both particles is necessary to aggregate 140 the required amount of water for activation and excludes scavenging by large activated particles collecting smaller ones while 141 precipitating. Note that only collections of the first type are able to increase the number of activated aerosols, while the second 142 type might have no or a negative impact on the total number of activated aerosols as discussed in Section 2.

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To exemplify this methodology, Fig. 3 shows, for an aerosol selected from the LCM simulations discussed below, the time 144 series of its radius and critical radius (panel a) and the ambient supersaturation and critical supersaturation (panel b). Note that 145 this aerosol is actually one super-droplet, representing a larger ensemble of identical aerosols, which is, however, interpreted as 146 one aerosol here. The initial dry radius of the aerosol is 0.27 µm. On its way to activation, the particle experiences diffusional 147 growth, which can be easily identified by the continuous change of radius. One collection event, characterized by a distinct 148 increase in radius, is visible at 6220 s simulated time. At this point in time, the inactivated aerosol (wet radius 3.1 µm) coalesces 149 with an activated particle (wet radius 7.8 µm, aerosol dry radius 0.13 µm), but the product of coalescence (wet radius 7.9 µm, 150 aerosol dry radius 0.28 µm) remains inactivated. Due to the increased amount of aerosol mass, the critical radius (and to a lesser 151 extent the critical supersaturation) increases (decreases) after the coalescence. Afterwards, the particle grows by diffusion and 152 exceeds the critical radius at 6253 s simulated time, which can be identified as the time of activation. All in all, this activation 153 is considered a collectional activation since dm| coll = 1.9 × 10 −12 kg > dm| diff = 6.2 × 10 −13 kg.

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The last section showed that collection can contribute significantly to the mass growth leading to the activation of a single 156 aerosol. But how does collection contribute to the activation of aerosols in general? Figure 4 shows the vertical profiles of and vertical velocity above (e.g., Rogers and Yau, 1989, Chap. 7). Due to the larger number of water vapor absorbers, the 166 supersaturation as well as the maximum diffusion radius are generally smaller in the more aerosol-laden simulations.

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The collectional activation rate (Fig. 4 a) increases almost linearly with height. This increase can be related to the longer 168 lasting diffusional growth resulting in potentially larger particles at higher levels, which increases the collection kernel and 169 therefore the collection probability. The slope is larger in aerosol-laden environments, where more aerosols are available 170 for activation. Additionally, the height above cloud base where the collectional activation starts increases with the aerosol 171 concentration since the average particle radius is too small to enable collisions at lower levels. Accordingly, the collectional  Generally, the contribution of collectional activation to the number of activated aerosols is significantly smaller than the 177 contribution of diffusional activation (Fig. 5): only 1 activation in 10 000 to 35 000 is caused by collection, with a greater 178 contribution of collectional activation in moderately aerosol-laden environments up to 4000 cm −3 . As it will be outlined below, 179 this increase can be attributed to a shift of collectional activation to smaller, but more numerous aerosols. For 8000 cm −3 , 180 however, the fraction decreases again since the particles are too small to trigger a larger amount of collisions.  increases. This is primarily a result of the decreasing maximum radii that can be reached by diffusion alone (Fig. 4 b). Addi-202 tionally, the supersaturation decreases too (Fig. 4 c), which decelerates diffusional activation and therefore favors collectional 203 activation. Since small aerosols are significantly more abundant than larger ones (Fig. 2), the number of aerosols that are po- In Section 2, it has been argued that the collection of particles with a large fraction of liquid water (and accordingly less 207 aerosol) are more beneficial to collectional activation than particles with a large amount of aerosol mass. Figure 7 a displays the 208 average number of collisions that take place during a collectional activation, separated into collected activated and collected 209 inactivated particles. Accordingly, their sum yields the total number of collected particles necessary for a collectional activation. cloud well above the cloud base, which is located at 500 − 600 m. Accordingly, these particles miss the typical supersaturation 237 maximum located at cloud base (see Fig. 4 c), where a majority of these aerosols normally activates. Indeed, entrainment above 238 cloud base is generally favorable for collectional activation since these aerosols are mixed into an environment where larger 239 particles exist, triggering collisions among them more easily. For aerosols larger than 0.6 µm, the average entrainment height 240 is located closer to the cloud base. Since multiple collections are necessary for their activation (see Fig. 7 a), the lower average 241 entrainment height is more representative for the average entrainment height of all particles inside the cloud, which is the cloud 242 base (e.g., Hoffmann et al., 2015). the slope and the height, from which collectional activation starts, increase with the aerosol concentration.

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In conclusion, this study revealed collision and coalescence as an additional process for the activation of aerosols. This 262 process is not covered by commonly applied activation parameterizations (e.g., Twomey, 1959). But does this matter? First 263 the range between approximately 0.1 µm and 1.0 µm should be considered as a transition zone between (i) typical aerosols that 279 need to experience sufficiently strong supersaturations to grow beyond the critical radius and (ii) so-called giant and ultra-280 giant aerosols with sufficiently large wet radii to act like cloud droplets by triggering collision and coalescence without being 281 formally activated (e.g., Johnson, 1982).

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Finally, potential sources of uncertainty within this study shall be mentioned. First, the accuracy of the applied collection 283 kernel is limited. The widely-used collision efficiencies of Hall (1980) for small particles ( 20 µm) are slightly higher than 284 other estimates (e.g., Böhm, 1992). An effect of this uncertainty is the collectional activation of aerosols that are too small 285 to collide physically. Accordingly, collectional activation shall affect slightly larger radii than evaluated here. Further note i.e., the wet radius of these aerosols would be smaller and they would less likely cause collisions. Again, the range of aerosols 295 affected by collectional activation would be shifted to larger radii.

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In this section, the basic framework of the Lagrangian cloud model (LCM) applied in this study as well as the extensions 298 made to treat aerosol mass during collision and coalescence are described. One can refer to Riechelmann et al. (2012) for the 299 original description, Hoffmann et al. (2015) for the consideration of aerosols during diffusional growth, and Hoffmann et al.

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(2017, in review) for the most recent description of the LCM. This LCM, as all other available particle-based cloud physical 301 models (Andrejczuk et al., 2008;Shima et al., 2009;Sölch and Kärcher, 2010;Naumann and Seifert, 2015), are based on the 302 so-called super-droplet approach in which each simulated particle represents an ensemble of identical, real particles, growing 303 continuously from an aerosol to a cloud droplet. The number of particles within this ensemble, the so-called weighting factor, 304 is a unique feature of each particle, which is considered for a physical appropriate representation of cloud microphysics within 305 the super-droplet approach.

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The transport of a simulated particle is described by 308 where X i is the particle location and u i is the LES resolved-scale velocity at the particle location determined from interpolating 309 linearly between the 8 adjacent grid points of the LES. A turbulent velocity component u i is computed from a stochastic model 310 based on the LES sub-grid scale turbulence kinetic energy (Sölch and Kärcher, 2010). The sedimentation velocity w s is given 311 by an empirical relationship (Rogers et al., 1993). Equation (A1) is solved using a first-order Euler method.

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As described in Hoffmann et al. (2015), the diffusional growth of each simulated particle is calculated from 314 where r is the particle's radius and S terms the supersaturation within the grid box, in which the particle is located. Curvature 315 and solution effects are considered by the the terms −A/r and b · m s /r 3 , respectively. The factor f parameterizes the so-called 316 ventilation effect (Rogers and Yau, 1989). The coefficients F k = (L v /(R v T ) − 1) · L v ρ l /(T k) and F D = ρ l R v T /(D v e s ) repre-317 sent the effects of thermal conduction and diffusion of water vapor between the particle and the surrounding air, respectively.

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Here, k is the coefficient of thermal conductivity in air, D v is the molecular diffusivity of water vapor in air, L v is the latent 319 heat of vaporization, and e s is the saturation vapor pressure. Equation (A2) is solved using a fourth-order Rosenbrock method.

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Collision and coalescence are calculated from a statistical approach in which collections are calculated from the particle size 321 distribution resulting from all super-droplets currently located within a grid box (Riechelmann et al., 2012). These interactions 322 affect the weighting factor A n (i.e., the number of all particles represented by one super-droplet), the total water mass of a super-323 droplet M n = A n · m n (where m n is the mass of one particle represented by super-droplet n), and also the dry aerosol mass 324 M s,n = A n ·m s,n (where m s,n is the dry aerosol mass of one particle represented by super-droplet n), which has been introduced 325 for this study. The algorithm follows the all-or-nothing principle, which has been rigorously evaluated by Unterstrasser et al. 326 (2016, in review) and has been recently implemented into this LCM by Hoffmann et al. (2017, in review).

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It is assumed that the super-droplet with the smaller weighting factor (index n) collects A n particles from the super-droplet increase. Additionally, same-size collections of the particles belonging to the same super-droplet are considered. These inter-334 actions do not change M n and M s,n , but they decrease A n and accordingly increase r n and r s,n .

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These two processes yield in the following description for the temporal change of A n (assuming that the simulated particles 336 are sorted such that A n > A n+1 ): The first term on the right-hand-side denotes the loss of A n due to same-size collections; the second term the loss of A n due 339 to collisions with particles of a smaller weighting factor. The total water mass and the total aerosol mass of a super-droplet A m m s,n P mn , (A7) 344 respectively. In both equations, the first term on the right-hand-side denotes the increase of M n or M s,n by the collection of 345 water or dry aerosol mass from super-droplets with a larger weighting factor, while the second term describes the loss of these 346 quantities to super-droplets with a smaller weighting factor. The function P mn controls if a collection takes place: 347 P mn := P (ϕ mn ) =      0 for ϕ mn ≤ ξ,

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where ξ is a random number uniformly chosen from the interval [0, 1] and 349 ϕ mn = K(r m , r n , ) A n δt/∆V (A9) 350 is the probability that a particle with the radius r m collects one of A n particles with the radius r n within a volume ∆V during 351 the (collection) time step δt. The collection kernel K is calculated from the traditional collision efficiencies as given by Hall