The enhancement of droplet collision by electric charges and atmospheric electric fields

The effect of electric charges and atmospheric electric fields on droplet spectrum evolution is studied numerically. Collision efficiencies for droplet pair with radii from 2 to 1024 μm and charges from -32 r to +32 r (in unit of elementary charge, droplet radius r in unit of μm) in different strength of downwards electric fields (0, 200 and 400 V cm) is computed. It is seen that collision efficiency is increased by electric charges and fields, especially for a pair of small droplets. 10 The evolution of cloud droplet spectrum with different initial sizes is simulated using the stochastic collection equation. Results show that the electric effect is not notable for the cloud with the initial mean droplet radius ?̅? =15 μm or larger. For the cloud with the initial ?̅? = 9 μm, the electric charge without field could evidently accelerate large-drop formation compared to the uncharged condition, and the existence of electric fields further accelerates it. For the cloud with the initial ?̅? = 6.5 μm, it is difficult for gravitational collision to occur, and the electric field could significantly enhance the collision process. Results 15 of this study indicate that electric charges and fields could accelerate large-drop formation in natural conditions, particularly for clouds with small droplet size.

enhanced by an order of magnitude in thunderstorm condition, while collision between large droplets is hardly affected. Note that Schlamp et al. (1976) didn't simulate the spectrum evolution process.
As for the electric effect on droplet spectrum evolution, few researches have been conducted. Khain et al. (2004), focused on weather modification, showed that droplet electric charges could enhance precipitation. They considered interaction of 35 droplet pair by image charge, and use Stokes Flow to calculate hydrodynamic interaction. The charge limit is set up to the airbreakdown limit. It is found that a small fraction of extremely charged particles could trigger the collision process, and thus accelerate raindrop formation or fog elimination significantly.
Previous studies about Albrecht (1989) effect show that increase of aerosol number decreases cloud droplet size, and thus extending cloud lifetime and suppress precipitation. But with the existence of electric charges, the Albrecht effect might be 40 partially weakened. As mentioned above, Schlamp et al. (1976) had already shown that smaller droplets are more sensitive to electric effect. So, the coupling of electric effect and Albrecht effect needs to be considered.
This study investigates the effect of electric charges and fields on droplet collision efficiency and the evolution of droplet spectrum. Different initial droplet size spectra and different electric conditions are considered. Section 2 describes the theory of droplet collision and stochastic collection equation. Section 3 and 4 present the numerical methods. Section 5 shows the 45 numerical results of electric effects on collision efficiency, and on cloud spectrum evolution.

Stochastic Collection Equation
The evolution of droplet size spectrum due to collision-coalescence is described by the stochastic collection equation (SCE), 50 which was first proposed by Telford (1955), and is shown as (Lamb and Verlinde, 2011, p.442)  (1) where ( ) is the spectrum density of droplets, and is the collection kernel between the two classes of droplets. The collection kernel describes the rate that droplets of mass collected by − and form new droplets of mass . In Eq. experiments.
And is the coalescence efficiency, namely the coalescence probability when two droplets collide. In fact, two droplets do not always coalesce when they collide with each. Instead, observations show that the pair can possibly rebound away in some 70 cases, because of an air film temporally trapped between the two surfaces. Especially for droplets with radius both larger than 100 μm, the "coalescence efficiency" is remarkably less than 1.0 (Beard and Ochs, 1984). The formula of coalescence efficiency is used from Beard and Ochs (1984).
In this study, electric charges and fields are taken into consideration. Cloud spectrum has two parameters-droplets mass (or radius ) and electric charge . So, SCE with the two parameters ( , ) is 75 Bott (2000) proposed a method to solve SCE on two-parameter spectrum, which took droplet mass and also interior aerosol mass into consideration. In this work, however, the problem is more complicated, since the electric charge could 80 affect the collection kernel, just like Khain et al. (2004).

Droplet motion equation 85
In order to get the collision efficiency, the motion equation of droplets is integrated to get the trajectories of droplets. Droplet motion not only depends on gravity and flow drag, but also depends on the electric force between droplets.
The motion equations for a pair of droplets are shown below, where is the gravitational acceleration, is the velocity vector of each droplet relative to the earth, is the flow velocity field induced by each droplet (also relative to earth), is the fluid viscosity, and is the drag coefficient, which is a function https://doi.org/10.5194/acp-2019-1140 Preprint. Discussion started: 31 January 2020 c Author(s) 2020. CC BY 4.0 License.
of Reynolds number. Droplet mass m = 4 3 /3, and is the electrostatic force caused by droplet electric charge and the external vertical electric field. The fluid property is treated as air with temperature T = 283 K and pressure p = 900 hPa. The 95 three terms on the rhs is gravity, flow drag force, and electric force respectively.

The drag force term
The superposition method is used to solve the motion equation of droplets in a hydrodynamic flow, assuming that each droplet 100 moves in the flow field induced by the other one moving alone. This method has been successfully used in many researches of collision efficiency calculation (Pruppacher and Klett, 1997).
Considering a sphere moving in a viscous fluid, the exact solution of the induced flow velocity field is to solve the Navier-Stokes equations. However, the computation is too complicated in this study. An appropriate idea is that the stream function depends on Reynolds number . Johnson (1962, 1963) gave the stream function induced by a moving 105 rigid sphere: which converges to stokes flow when → 0. Then the induced flow field is derived, The drag coefficient is function of , 115 where = ln ( ), and fitting constants 0 , 1, 2 are from table 2 of Beard (1976 where is the free path of air molecules, and is the droplet radius.
Then, electric force on droplet 1 could be derived immediately by 140 The comparison between inverse-square law and conductor model are shown in Fig. 2, where the electric force of opposite charges (dashed lines) and of same charges (solid lines) varies with distance. It is shown that in remote distance, two models are basically identical. But when the spheres approach closely, the conductor interaction (blue lines) turns into strong attraction, because of electrostatic induction. It is significant that interaction must turn to attraction as long as the distance is small enough, 145 regardless of sign of charge.
If there is only inverse-square law without electrostatic induction, it is obvious to say that same-sign charges must decrease collision efficiency. However, after taking electrostatic induction into account, the effects of same-sign and opposite-sign charges need to be reconsidered.

Droplet trajectory and the effective cross section
Eq. (4) is integrated to get the trajectories for the two droplets of any possible droplet pair ( 1 , 1 and 2 , 2 ) in various strengths of downwards electric fields (0, 200 and 400 V cm -1 ). The 2-order Runge-Kutta method is used for the integration.
Following their trajectories, the two droplets can either collide or not. In order to get the collision cross section = 2 (or say the collision efficiency E), we vary the initial horizontal distance between the two droplets using the bisection method, until we find a threshold distance that makes the two droplets follow the grazing trajectories and just exactly collide. The threshold distance is found with a precision of 0.1%.
After the collision efficiency is derived for droplet pair with ( 1 , 1 ) and ( 2 , 2 ) , the collection kernel K( 1 , 1 , 2 , 2 ) is derived, where the coalescence efficiency is restricted in the range from 0.3 to 1.0, using the formula of Beard and Ochs 160 (1984). With the collection kernel ( 1 , 1 , 2 , 2 ), the effect of electric charges and fields on droplet collision is determined by the SCE.

165
The evolution of the droplet spectrum is described by the 2-parameter SCE, i.e. Eq. (1). To solve the equation numerically, droplet radii and charges are divided into discrete logarithmically equidistant bins. Droplet radius ranging from 2 to 1024 μm is divided into 37 bins. The radius is increased by a factor of 2 1/4 from one bin to the next. For droplet radius > 1024 μm, they are assumed to precipitate out and not included in droplet spectrum.
Usually the droplet mass and charge after coalescence do not fall in any existing bins. A simple method is to linearly redistribute the droplets to the two neighbouring bins (Khain et al, 2004). We first redistribute droplets to the certain mass 175 bins. The ratio of redistribution is based on total-mass conservation and droplet-number conservation simultaneously. For example, to redistribute droplets with mass ( < < +1 ) and number , a proportion of is added to the (i+1)th bin. After mass redistribution to the ith and (i+1)th mass bins, the charge is redistributed within each of the mass bins, satisfying total-charge conservation and droplet-number conservation. For example, to redistribute droplets with charge ( , < < , +1 ) within the ith mass bin, a proportion of 180 , = , +1 −q , +1 − , is added to the bin of (i, j), and a proportion of , +1 = q− , , +1 − , is added to the bin of (i, j+1).
As shown in Fig. 3, the coalescence between bin ( 1 , 1 ) and bin ( 2 , 2 ), shown with black dots, generates a droplet show with a red dot. This newly generated droplet is then redistributed into 4 bins shown with blue dots. Note that the numbers of each blue dot in Fig. 3 are the percentages in the redistribution of droplets to the bins. In fact, this method only reaches the first-order precision. Although Bott (1998) compared several methods to redistribute droplets with high-order correction, the 185 two-parameter spectrum is too complicated to do the high-order correction in this study.
where =1 g m -3 is liquid water content, and ̅ is the mean droplet mass. The number density spectrum ( ) can also multiply droplet mass, thus producing the mass spectrum, 190 and can also be written as: where ̅ is mean droplet radius (with ̅ = 4 ̅ 3 /3), which is an important variant to describe the cloud droplet size.
12 cases with different initial conditions are considered to study the spectrum evolution. The mean droplet radius ̅ is set 195 by 3 different sizes: 15 μm, 9 μm and 6.5 μm, where ̅ = 6.5 μm represents polluted conditions, and 15 μm case represents the clean conditions. For each ̅ , comparisons are made among 4 different electric conditions, including the uncharged cloud, charged cloud, charged cloud with a field of 200 V cm -1 , and charged cloud with a field of 400 V cm -1 (This study considers the downward electric fields as positive). For the uncharged cloud, the initial distribution is shown in Fig. 4a, where all droplets are put in the bins with no charge. For charged clouds, the initial charge is distributed symmetrically, as shown in 200 Fig. 4b: 14% with charge +1 2 , 14% with charge -1 2 , 22% with charge +0.5 2 , 22% with charge -0.5 2 , and 28% with no charge (charge in unit of elementary charge, and r in μm), which represent the electric state in a normal precipitation process.
During the computation of spectrum evolution, each bin could coalesce with any other bins in each step time Δ , which require the collection kernel between the two bins. Thus, a large matrix of kernel ( 1 , 2 , 1 , 2 ) is computed in advance.
The initial electric charges, and electric field strength are set according to the conditions in the early stage of thunderstorms 205 or warm clouds. In fact, in some extreme thunderstorm cases, both the electric charge and field could be one order of magnitude larger (Takahashi, 1973) than the values used in this study. Furthermore, in natural clouds, the electric charge on a droplets leaks away gradually. In this study, the charge leakage is assumed as a process of exponential decay (Pruppacher and Klett, 1997), and the relaxation time is set to =120 min. Namely, all the bins lose Δ of electric charge in each step time Δ = 1 . 210

Collision efficiency 215
The collision efficiencies for droplets without electric charge or field are shown in Fig. 5. The radius of the larger droplet 1 ranges from 30 to 305 μm. The coloured lines are computation results in this study, and the dots are from previous experiment results. It is clear that our results are basically consistent with those from previous studies. Collision efficiencies https://doi.org/10.5194/acp-2019-1140 Preprint. Discussion started: 31 January 2020 c Author(s) 2020. CC BY 4.0 License. increase with 2 from 2 to 14 μm, and also increase with 1 from 30 to 305 μm. For droplet pair that are both large enough, 220 collision efficiencies are close to 1.
With the collision efficiencies of droplets with different radii and charges in different strength of electric fields all computed, it is found that the electric effect is sensitive to droplet radii. Results are only discussed for 1 = 30 μm and shown in Fig. 6. Totally 6 combinations of electric conditions are selected to be shown here, and the details are summarized in table 1. The droplet pair is set to have no charge, same-sign charges, or opposite charges. The electric field is set to be 0 or 400 V 225 m -1 . Compared to the no-charge pair (curve 1), the same-sign charges without electric field (curve 2) slightly decreases collision efficiency, because of the repulsive force. The results of both positively charged pair and negatively charged pair are identical, since there is no electric field. In a downward electric field, the collision efficiency of the two situations is changed. For a positively charged pair (curve 3), the collision efficiency is very close to the no-charge pair, which implies that enhancement of electric field offset the repulsive effect. For a negatively charged pair in a downward field (curve 4), the 230 collision efficiency with small 2 is significantly enhanced. This could be easily explained by electrostatic induction: the strong downward electric field induces positive charge on the lower part of the larger droplet (even though it is overall negatively-charged), so the smaller negative-charged droplet below feels attraction.
As for a pair with opposite charge, curve 5 shows that the collision efficiency is higher than the pair with no charge. For 2 < 5 μm, the collision efficiency is nearly an order of magnitude higher; which for larger droplet the increase is not so strong. 235 This means that the electric effect is sensitive to the radius of droplets, and mainly affects small droplets. Curve 6 shows that with an electric field of 400 V cm -1 , the electric effect becomes significantly stronger. Collision efficiency is increased by more than one order of magnitude compared to no-charge condition when 2 < 5 μm. Even if 2 is large, the collision efficiency could still be increased by about 2 times. 240

Evolution of cloud spectrum
This part shows the electric effect on spectrum evolution with different initial size distributions, i.e., ̅ = 15 μm, 9 μm and 6.5 μm. For each initial size distribution, comparisons are made among four different electric conditions, including 245 uncharged, charged without field, charged with 200 V cm -1 and charged with 400 V cm -1 . Note that "charged" here refers to initial distribution shown in Fig. 4. The magnitude of 400 V cm -1 corresponds to the early stage of a thunderstorm. Figure 7 shows the evolution of the spectrum with initial ̅ = 15 μm. The 4 rows show different times (t = 7.5, 15, 22.5, and 30 min) during spectrum evolution. The left side denotes the spectrum mass density, and the right side shows the droplet number concentration. In each panel, the dotted line denotes initial spectrum distribution (t = 0 min) for reference. It is seen 250 that droplet spectra under 4 electric conditions have similar behaviour. All the spectra evolve to a double-peak form, regardless of electric charge or field. At 30 min, the 4 cases all have a modal radius of about 200 μm (Fig. 7d). The electric https://doi.org/10.5194/acp-2019-1140 Preprint. Discussion started: 31 January 2020 c Author(s) 2020. CC BY 4.0 License. effect is not notable for large droplets, and the initial radius is large enough to start gravitational collision quickly.
Consequently, the electric effect is negligible in this case. Figure 8 shows the evolution of the spectrum with initial ̅ = 9 μm. For the uncharged spectrum, it takes 60 min to have 255 the second peak grow to about 200 μm. Therefore, the 4 panels of Fig. 8 show the spectrum evolution for t = 15, 30, 45, and 60 min. The charges and the electric fields have more significant effect in the ̅ = 9 μm case than in the ̅ = 15 μm case. It is seen that, at 15 and 30 min, the spectra with different electric conditions evidently differ from each other, but the second mode is not obvious. At 45 min, the effect of charges and electric fields on the second peak is evident. The small-droplet peak on the left is lower, and the second peak on the right is higher, indicating that the charged cloud (red line) evolves more 260 quickly than the uncharged cloud. Moreover, vertical electric fields further boost the collision-coalescence process of charged droplets (green and purple lines). Under the electric field of 200 V cm -1 , the second peak is two times higher than the no-field case (red line) at 45 min. Under the electric field of 400 V cm -1 , the second peak is even higher. At 60 min, modal radius of second peak is about 200 μm for uncharge situation, 300 μm for charged without field situation, 500 μm for charged with 200 V cm -1 situation, and 700 μm for charged with 400 V cm -1 situation, respectively. 265 The 2-dimensional spectrum for ̅ = 9 μm at 60 min is shown in Fig. 9. Figure 9a is for the uncharged situation. Figures   9b, 9c, and 9d are for the situations with charges and with electric fields of 0, 200, 400 V cm -1 , respectively. After 60 min of evolution, these charge distributions are still nearly symmetric. These clearly show the process that charges transport to large droplets during coalescence growth. Note that the integration of this 2-dimensional spectrum along the charge axis gives the size distribution at 60 min shown in Fig. 8d. 270 Figure 10 shows the evolution of the spectrum with initial ̅ = 6.5 μm. For the uncharged cloud, it takes 120 min to have the second peak grow to about 200 um. Therefore, the 4 panels of Fig. 10 show the spectrum evolution for t = 30, 60, 90 and 120 min. The enhancement of the electric field on collision-coalescence process is much more obvious than ̅ = 9 μm.. After 90 min of evolution, the spectra of the uncharged cloud (blue line) and charged cloud without field (red line) are almost the same as the initial spectrum. This is because the droplets are too small to initiate gravitational collision. At 120 min, a second 275 peak has formed for the situations with no charge and with charge but no field. In comparison, under the external electric field of 200 and 400 V cm -1 (green and purple lines), the cloud droplets grow much more quickly than the no-field situations.
Some droplets even have evolved to larger than 1024 μm, which are supposed to precipitate out from the clouds. These results show that, the electric field would remarkably trigger the collision-coalescence process for the small droplets.
As for the initial mean droplet radius ̅ < 6 μm (figure not shown), similar to Fig. 10, the spectra of uncharged and charged 280 cloud without electric field would nearly have no difference, while the effect of electric fields is much stronger. This means that charge effect is relatively small compared to electric fields when the initial droplet radius of the cloud is small enough.
According to Eq. (2), collection kernel K is composed of the collision efficiency E, relative terminal velocity, and coalescence efficiency ε. It is found that the total electric effect on K is mainly contributed by E. The electric enhancement of collision efficiency E is particularly significant for small droplets, as shown in Sect. 6.1. The relative terminal velocity term also contributes to the collection kernel, and the electric field can affect terminal velocity of small charged droplets significantly. As shown in Fig. 11, in downwards electric field 400 V cm -1 , terminal velocity of a large droplet is nearly not 290 affected. The difference of velocity at = 1000 μm does not exceed 1%, and difference at 100 μm does not exceed 5%. On the contrary, electric fields strongly affect the sedimentation of charged small droplets. For r < 5 μm, the terminal velocity of negative-charged droplet even turns "upwards". This is due to the fact that droplet mass ∝ 3 , while droplet charge ∝ 2 according to observation. So, ∝ 2/3 means that acceleration of electric force decreases with increasing droplet mass, which indicates that small droplets are more sensitive to electric charges and fields. 295 This study neglects some possible electric effects in collision-coalescence process. Electric effect on coalescence efficiency ε is neglected. Rebound (collide but not coalesce) happens because of an air film temporally trapped between the two surfaces, which is a barrier to coalescence. This barrier may be overcome by strong electric attraction occurring at small distance. Many experiments show that electric charges and fields would enhance coalescence efficiency, such as Jayaratne and Mason (1964). But there is no proper numerical model to evaluate the effect. So, this study may underestimate the 300 electric effect on droplet collision-coalescence process.
Induced charge redistribution is also neglected when rebound happens. For instance, let us consider a rebound event in a positive (downwards) electric field. The larger droplet is often above the smaller droplet, and the smaller one will carry positive charge instantaneously according to electrostatic induction, then move apart. So, the rebound would cause charge redistribution between the pair. This may lead to some change in spectrum evolution. 305

Conclusion
The effect of electric charges and atmospheric electric fields on the evolution of cloud droplet spectrum is studied numerically.
The motion equation of droplets in the atmosphere is solved to get the trajectories of droplet pair of any radii (2 to 1024 μm) 310 and charges (-32 to +32 2 , in unit of elementary charge, droplet radius r in unit of μm) in different strength of downwards electric fields (0, 200 and 400 V cm -1 ). Based on trajectories, we determine whether a droplet pair collide or not. Thus, collision efficiencies for the droplet pairs are derived. It is seen that collision efficiency is increased by electric charges and fields, especially when the droplet pair are oppositely charged or both negatively charged in a downward electric field. The increase is particularly significant for a pair of small droplets. 315 With collision efficiencies derived in this study, SCE is solved to simulate the evolution of cloud droplet spectrum. The initial droplet size conditions include ̅ = 15 μm, 9 μm, and 6.5 μm, and the initial electric conditions include uncharged and charged (with charge amount proportional to droplet surface area) in different strength of electric fields (0, 200 and 400 V cm - (1) 0 0 0 (2) +32 r1 2 +32 r2 2 0 (3) +32 r1 2 +32 r2 2 +400 (4) -32 r1 2 -32 r2 2 +400 (5) +32 r1 2 -32 r2 2 0 (6) +32 r1 2 -32 r2 2 +400

FIG.
3. An example of redistribution of coalescence between two bins. Black dots denote the two bins of droplets before coalescence. The red dot refers to the droplet after coalescence but not on the bin grids. The blue dots show the redistribution method and proportion of each redistributed bin, which is constrained by particle-number conservation, mass conservation and charge conservation. conditions. It is significant that terminal velocity of negatively charged droplets smaller than 5 μm would turn upwards, which leads to the discontinuity of the lower curve in the figure.