The effects of electric charges and fields on droplet collision–coalescence and the evolution of cloud droplet size distribution are studied numerically. Collision efficiencies for droplet pairs with radii from 2 to 1024

The evolution of the cloud droplet size distribution with the electrostatic
effects is simulated using the stochastic collection equation. Results show
that the electrostatic effect is not notable for clouds with the initial
mean droplet radius of

Clouds are usually electrified (Pruppacher and Klett, 1997). For thunderstorms, several theories of electrification have been proposed in the
past decades. The proposed theories assume that the electrification involves
the collision of graupel or hailstones with ice crystals or supercooled
cloud droplets, based on radar observational results indicating that the onset of strong electrification follows the formation of graupel or hailstones within the cloud (Wallace and Hobbs, 2006). However, the exact conditions and mechanisms are still under debate. One charging process could be due to the thermoelectric effect between the relatively warm, rimed graupel or hailstones and the relatively cold ice crystals or supercooled cloud droplets. Another charging process could be due to the polarization of
particles by the downward atmospheric electric field. The thunderstorm
electrification can increase the electric fields to several thousand volts per centimeter, while the magnitude of electric fields in fair weather air is only about 1 V cm

Liquid stratiform clouds do not have such strong charge generation as those in thunderstorms. But the charging of droplets can indeed occur at the upper and lower cloud boundaries as the fair weather current passes through the clouds (Harrison et al., 2015; Baumgaertner et al., 2014). The global fair weather current and the electric field are in the downward direction. Given the electric potential of 250 kV for the ionosphere, the exact value of the fair weather current density over a location depends on the electric resistance of the atmospheric column, but its typical value is about 2

In general, the charging of droplets can lead to the following effects on warm cloud microphysics. First, for charged haze droplets, the charges can lower the saturation vapor pressure over the droplets and enhance cloud droplet activation (Harrison and Carslaw, 2003; Harrison et al., 2015). Second, the electrostatic induction effect between charged droplets can lead to a strong attraction at a very small distance (Davis, 1964) and higher collision–coalescence efficiencies (Beard et al., 2002). However, Harrison et al. (2015) showed that charging is more likely to affect the collision processes than activation for small droplets.

The electrostatic induction effect can be explained by regarding the charged cloud droplets as spherical conductors. The electrostatic force between two conductors is different from the well-known Coulomb force between two point charges. When the distance between a pair of charged droplets approaches infinity, the electrostatic force converges to a Coulomb force between two point charges. But, when the distance between the surfaces of two droplets is small (e.g., much smaller than their radii), their interaction shows extremely strong attraction. Even when the pair of droplets carry the same sign of charges, the electrostatic force can still change from repulsion to attraction at small distances. Although there is no explicit analytical expression for the electrostatic interaction between two charged droplets, a model with high accuracy has been developed (Davis, 1964) for the interaction of charged droplets in a uniform electric field. Many different approximate methods are also proposed for the convenience of computation in cloud physics (e.g., Khain et al., 2004).

Based on this induction concept, the electrostatic effects on the droplet collision–coalescence process have been studied in the past decades. A few experiments show that electric charges and fields can enhance coalescence between droplets. Beard et al. (2002) conducted experiments in cloud chambers and showed that even a minimal electric charge can significantly increase the probability of coalescence when the two droplets collide. Eow et al. (2001) examined several different electrostatic effects in a water-in-oil emulsion, indicating that electric fields can enhance coalescence by using several mechanisms, such as film drainage.

Model simulations indicate that charges and fields can increase droplet
collision efficiencies because of the electrostatic forces. Schlamp et al.
(1976) used the model of Davis (1964) to study the effect of electric
charges and atmospheric electric fields on collision efficiencies. They
demonstrated that the collision efficiencies between small droplets (about
1–10

As for the electrostatic effect on the evolution of droplet size distribution and the cloud system, few studies have been conducted. Focusing on weather modification, Khain et al. (2004) showed that a small fraction of highly charged particles can trigger the collision process and, thus, accelerate raindrop formation in warm clouds or fog dissipation significantly. In their study, the electrostatic force between the droplet pair is represented by an approximate formula. The charge limit is set to the electrical breakdown limit of air. Stokes flow is adopted to represent the hydrodynamic interaction that can be used to derive the trajectories of droplet pairs. Harrison et al. (2015) calculated droplet collision efficiencies affected by electric charges in warm clouds. When simulating the evolution of droplet size distribution in their study, the enhanced collision efficiencies were not used. Instead, the collection cross sections were multiplied by a factor of no more than 120 % to approximately represent the electric enhancement of the collision efficiency. This approximation can roughly show the enhancement of droplet collision and raindrop formation from the charges in warm clouds. Further studies are still needed to evaluate the electrostatic effect more accurately and for various aerosol conditions that are typical in warm clouds.

The increased aerosol loading by anthropogenic activities can lead to an increase in cloud droplet number concentration, a reduction in droplet size, and, therefore, an increase in cloud albedo (Twomey, 1974). This imposes a cooling effect on the climate. It is further recognized that the aerosol-induced reduction in droplet size can slow droplet collision–coalescence and cause precipitation suppression. This leads to increased cloud fraction and liquid water amount and imposes an additional cooling effect on climate the (Albrecht, 1989). As the charging of cloud droplets can enhance droplet collision–coalescence, especially for small droplets, it is worth studying to what extent the electrostatic effect can mitigate the aerosol effect on the evolution of droplet size distribution and precipitation formation.

This study investigates the effect of electric charges and fields on droplet collision efficiency and the evolution of the droplet size distribution. The electric charges on droplets are set as large as in typical warm clouds, and the electric fields are set as the early stage of thunderstorms. A more accurate method for calculating the electric forces is adopted (Davis, 1964). The correction of the flow field for large Reynolds numbers is also considered. Section 2 describes the theory of droplet collision–coalescence and the stochastic collection equation. Section 3 presents the equations of motion for charged droplets in an electric field. A method for obtaining the terminal velocities and collision efficiencies for charged droplets is also presented. Section 4 describes the model setup for solving the stochastic collection equation. Different initial droplet size distributions and different electric conditions are considered. Section 5 shows the numerical results of the electrostatic effects on collision efficiency and on the evolution of droplet size distribution. We intend to find out to what extent the electric charges and fields, as in the observed atmospheric conditions, can accelerate the warm rain process and how sensitive these electrostatic effects are to aerosol-induced changes in droplet sizes.

The evolution of droplet size distribution due to collision–coalescence is described by the stochastic collection equation (SCE), which was first proposed by Telford (1955) and is expressed as follows (Lamb and Verlinde, 2011, p. 442):

The collection kernel between droplets with mass

For a pair of droplets, each one induces a flow field that interacts
with the other. As the collector falls and sweeps the air volume, the
droplets in the volume tend to follow the streamlines of the flow field
induced by the collector. Droplets collide with the collector only when they
have enough inertia to cross the streamlines. Collision efficiency is then
defined as the ratio of the actual collisions over all possible collisions
in the swept volume. It can be much smaller than 1.0 when the sizes of the
two droplets are significantly different. The physical meaning of the collision efficiency is shown in Fig. 1 for a droplet pair. The collector droplet falls faster and induces a flow field that interacts with the small droplet. The small droplet follows a grazing trajectory (as shown in Fig. 1), when the centers of the two droplets have an initial horizontal distance

A schematic diagram for a droplet pair collision. The initial
vertical distance between the center of the two droplets is set to be 30 (

Two droplets may not coalesce even when they collide with each other.
Observations show that the droplet pair can rebound in some cases because of
an air film temporarily trapped between the two surfaces. Especially for
droplets with radii both larger than 100

In this study, electric charges and external electric fields are taken into
consideration for droplet collision–coalescence. The droplet distribution
function has two variables, namely droplet mass

The collection kernel for charged droplets in an external electric field has
the same form as Eq. (2). However, terminal velocity, collision efficiency,
and coalescence efficiency in the kernel may all be affected by the electric
charge and field. We consider these as electrostatic effects. In a vertical
electric field, the terminal velocity of a charged droplet may be increased
or decreased, depending on the charge sign and the direction of the field.
The threshold horizontal distance

As will be seen in this study, the electrostatic effect on collision
efficiency is much stronger than on terminal velocity. Therefore, the
electrostatic effect on terminal velocity is presented in Sect. 6 as the
discussion, and we focus on the electrostatic effects on collision
efficiency in this paper. The method for obtaining droplet terminal velocity
and collision efficiency with the electrostatic effects will be presented in
Sect. 3. The electrostatic effect on coalescence efficiency is not
considered here. The coalescence efficiency used in this study is the same
as that for uncharged droplets, based on the results of Beard and Ochs
(1984). In their study, coalescence efficiency is a function of

In order to calculate the terminal velocity and collision efficiency, the
equations of motion need to be solved. Droplet motion depends on the
following three forces: gravity, the flow drag force, and the electrostatic
force due to droplet charge and the external electric field. The equations
of motion for a pair of droplets are as follows:

The flow drag force is described by the second term on the right hand side
of Eq. (4), which assumes a simple hydrodynamic interaction of the two
droplets. That is, each droplet moves in the flow field induced by the other
one moving alone, and it is called the superposition method in cloud physics. This method has been successfully used in many studies for the calculation of collision efficiencies (Pruppacher and Klett, 1997). The superposition method can also ensure that the stream function satisfies the no-slip boundary condition (i.e., Wang et al., 2005). To calculate the flow drag force, the induced flow field

Considering a rigid sphere moving in a viscous fluid with a velocity

Both the Stokes and Hamielec stream functions satisfy the no-slip boundary condition, i.e., the fluid velocity on the surface of the droplet is equal
to the velocity of the droplet. The Hamielec stream function is no-slip because those functions

According to an empirical equation of Beard (1976), the drag coefficient

For droplets with

The electric force is described by the third term on the right side of Eq. (4). The electric force includes the interactive force between the two
charged droplets, and it is also an external electric force if there is an external electric field. For two point particles, we apply Coulomb's law as follows:

The interaction between charged conductors is a complex mathematical problem
in physics. Davis (1964) demonstrated an appropriate computational method
for an electric force between two spherical conductors in a uniform external
field. The electric force depends on droplet radius (

The electric force directly from the external field is shown as two
terms in Eq. (13) and can be simply written as

Similarly, the resultant electric force

A schematic diagram of all the forces acting on two charged droplets and droplet velocities and the induced flow velocities. The electric field

Comparison of the electric force from the conductor model (Davis,
1964; Eq. 15 in this study) and the inverse-square law (Eq. 12 in this
study). Positive force represents repulsion and negative force represents
attraction. Radius of the pair is set to

If there is no external electric field but only a charge effect, Eq. (13)
is reduced to the following:

The equations of motion (Eq. 4), along with the other equations in this section, are used to calculate the terminal velocities of charged droplets.
Note that the terminal velocity refers to the steady-state velocity of a droplet relative to the flow when there are no other droplets present, as
mentioned earlier. Therefore, by setting the induced flow

Equation (4), along with other equations, is also integrated to obtain the
trajectories for the two droplets in any possible droplet pair (

After computing the collision efficiencies

To solve the stochastic collection equation (Eq. 3) numerically, droplet
radius and charge are both divided into discrete bins that are logarithmically equidistant. Droplet radius, ranging from 2 to 1024

In each radius bin, droplets may have different amounts and different signs of charges. For the bin of radius

Droplet size and charge after collision–coalescence usually do not fall into
any existing bins. A simple method is to linearly redistribute the droplets
to the two neighboring bins (Khain et al., 2004). We first redistribute the
droplets to the size bins. The ratio of redistribution is simultaneously based on total mass conservation and droplet number conservation. For example, to redistribute droplets with mass

An example of droplet redistribution to new size and charge bins after collision–coalescence. Black dots denote the two bins of droplets before collision–coalescence. The red dot denotes the droplets after collision–coalescence but not on the bin grids. Blue dots denote the droplets that are redistributed to the new bins. Numbers close to the blue dots are the percentage of droplets that are redistributed into that bin. The redistribution method is constrained by particle number conservation, mass conservation, and charge conservation.

As shown in Fig. 4, the collision–coalescence between bin (

The initial droplet size distribution used in this study is derived based on
an exponential function in Bott (1998) as follows:

The initial droplet mass distributed over the size and charge bins.
Colors represent water mass content in the bins (in units of g m

A total of 12 cases with different initial conditions are considered for studying the evolution of droplet distribution. The mean droplet radius

Total number concentration and charge content for all initial droplet distributions.

For each

The initial electric charges and electric field strength are set according
to the conditions in warm clouds or in the early stage of thunderstorms. In
fact, in some extreme thunderstorm cases, both the electric charge and field
could be 1 order of magnitude larger (Takahashi, 1973) than the values
used in this study. Furthermore, in natural clouds, the electric charge on a
droplet leaks away gradually. In this study, the charge leakage is assumed
to be a process of exponential decay (Pruppacher and Klett, 1997), and the
relaxation time is set to

Here we present collision efficiencies for typical droplet pairs to illustrate the electrostatic effects. During the evolution of droplet size distribution, the radius and charge amount of colliding droplets have large variability. In addition, the charge sign of the colliding droplets may be the same or the opposite. Therefore, only some examples are shown.

Collision efficiency for droplets with no electric charge or field.
Lines are the results computed in this study. Different lines represent the
different collector radius

Collision efficiency for droplets with electric charge and field.
The radius of the collector droplet

The collision efficiencies for droplet pairs with no electric charge and field are presented in Fig. 6 as a reference. Collector droplets with radii
larger than 30

Figure 7 shows the collision efficiencies for droplet pairs with electric
charge and field. Basically, droplet pairs that have no charge, same-sign charges, and opposite-sign charges are selected here and
under the 0 and 400 V m

Collision efficiency for droplets with electric charge and field.
The radius of the collector droplet

For the collector droplet with a radius of 30

As for a pair with opposite-sign charges, line 5 in Fig. 7a shows that the
collision efficiency is enhanced by the electrostatic effect even when there
is no electric field. The collision efficiency is nearly 1 order of magnitude higher, with

Figure 8 shows the collision efficiencies for droplet pairs with charge and
field and with smaller collectors. The collector droplet has a radius of 10

It is evident that droplet charge and field can significantly affect the collision efficiency, especially for smaller collectors. This means that the electrostatic effects depend on the radius of collector droplets, and it mainly affects small droplets. The section below provides a detailed description on how these electrostatic effects can influence droplet size distributions.

This section shows the electrostatic effects on the evolution of different
droplet size distributions. As discussed in Sect. 4, this study uses three
initial size distributions, where

The evolution of droplet size distribution with initial

Temporal changes in droplet total number concentration and total
charge content for

Figure 9 shows the evolution of the droplet size distribution with initial

The evolution of the droplet size distribution with initial

The evolution of droplet total number concentration and total positive
charge concentration (also equal to the total negative charge concentration)
is shown in Fig. 10. It is evident that droplet total number concentration
decreases from 71 to less than 5 cm

Temporal changes in droplet total number concentration and total
charge content for

Comparison of evolutions of the 2D distribution of droplet
mass concentration with different electric conditions at 60 min (initial

The evolution of the droplet size distribution with initial

Figure 11 shows the evolution of the droplet size distribution with initial

Temporal changes in droplet total number concentration and total
charge content for

As for the evolution of the droplet total number concentration and charge concentration, Fig. 12 shows that they are distinctly affected by the four
different electric conditions. The charged cloud with a field of 400 V cm

Figure 14 shows the evolution of the droplet size distribution with initial

As for the initial mean droplet radius

Now we compare the electrostatic effects shown above with the aerosol effects. Let us take the cases with

According to Eq. (2), the collection kernel

Terminal velocities of droplets in an external electric field 400 V cm

This study still neglects some possible electrostatic effects in the
collision–coalescence process. The electrostatic effect on coalescence efficiency

Induced charge redistribution is also neglected when rebound happens. For instance, let us consider a rebound event in a positive (downward) 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. The rebound would cause a charge redistribution between the pair. This may lead to some change in the evolution of clouds.

The effect of electric charges and atmospheric electric fields on cloud
droplet collision–coalescence and on the evolution of cloud droplet size
distribution is studied numerically. The equations of motion for cloud
droplets are solved to obtain the trajectories of droplet pair of any radii (2 to 1024

With the collision efficiencies derived in this study, the SCE is solved to
simulate the evolution of cloud droplet size distribution under the
influence of electrostatic effects. The initial droplet size distributions
include

It is known that the increase in aerosol number and, therefore, the decrease
in cloud droplet size lead to suppressed precipitation and a longer cloud
lifetime. But, with the electrostatic effect, the aerosol effect can be
mitigated to a certain extent. The three initial droplet size distributions
used in this study, with

Data and programs are available from Shian Guo (guoshian@pku.edu.cn) upon request.

SG developed the model, wrote the codes of the program, and performed the simulation. HX advised on the case settings of the numerical simulation. HX and SG worked together to prepare the paper.

The authors declare that they have no conflict of interest.

We are grateful to Jost Heintzenberg and Shizuo Fu for their constructive discussions on this study.

This study has been supported by the National Innovation and Entrepreneurship Training Program for College Students and the Chinese Natural Science Foundation (grant no. 41675134).

This paper was edited by Patrick Chuang and reviewed by three anonymous referees.