### 5.8 Did you know most precipitation comes from collision–coalescence?

There are two types of processes for growth into precipitation drops: **warm cloud processes** and **cold cloud processes**. In warm clouds, the processes all involve only liquid drops. In cold clouds, the processes can involve only solid particles, as well as mixed phases (both supercooled liquid and ice). Some of the most important processes involve collisions between drops, whether they be liquid or solid.

### Collisions

Collisions occur in both cold and warm clouds and can involve either liquid drops or solid particles or both.

**Collision–Coalescence:**Large liquid drop scavenges smaller liquid drops as it falls.**Riming:**Falling ice collects liquid water, which freezes on its surface.**Capture Nucleation:**Large liquid drop captures small ice particle, which acts as an ice nucleus and causes the large drop to freeze. The particle that is collected can be either an ice nucleus (IN) or a piece of ice, which also is a good ice nucleus. In either case, the supercooled liquid drop freezes on contact with the IN.**Aggregation:**Falling snowflake scavenges other snowflakes that aggregate to make a larger snowflake bundle.

For a cloud drop at rest, gravity is the only external force. Once the cloud drop starts to fall, then the air resistance forms another force called *drag*, which is a function of the velocity.

In less than a second, the particle reaches a fall speed such that the drag force exactly balances the gravitational force and the velocity becomes constant. This velocity is called the *terminal velocity*. Because the gravitational force depends on the volume of the drop, it goes as the cube of the drop radius. In contrast, drag acts on the surface of the drop, and so it depends on the drop area and goes as the square of the drop radius (times the velocity). Setting the gravitational and drag forces equal to other and then solving for the terminal velocity, it is easy to show that the terminal velocity should vary linearly with drop radius. Measurements bear this linear relationship out. For instance, the terminal velocity of a 50 μm radius drop is about 0.3 m s^{–1}, while the terminal velocity for a drop 10 times larger (500 μm radius) is about 4 m s^{–1}, which just a little more than a factor-of-10 increase.

The growth of a cloud drop into a precipitation drop by collision–coalescence is given by the equation:

$$\begin{array}{l}\frac{d{m}_{L}}{dt}=Area\text{\hspace{0.17em}}swept\text{\hspace{0.17em}}out\cdot efficiency\text{\hspace{0.17em}}of\text{\hspace{0.17em}}collection\cdot velocity\text{\hspace{0.17em}}difference\cdot liquid\text{\hspace{0.17em}}water\text{\hspace{0.17em}}content\\ \frac{d{m}_{L}}{dt}={A}_{g}\cdot {E}_{c}\cdot ({v}_{L}-{v}_{s})\cdot LWC\\ \frac{d{m}_{L}}{dt}=\pi {\left({r}_{L}+{r}_{s}\right)}^{2}\cdot {E}_{c}\cdot ({v}_{L}-{v}_{s})\cdot LWC\end{array}$$

*m*is the mass of the large drop that is falling,_{L}*A*is the geometric cross-sectional area for which collisions between the falling large drop and the many drops below are possible,_{g}*E*is the collision–coalescence efficiency (i.e., a collection efficiency), which is the fraction of the actual cross-sectional area that is swept out compared to the cross-sectional area that is geometrically possible (smaller drops can follow air streamlines and go around the big drop) (see the figure below),_{c}*v*is the velocity of the large drop and_{L}*v*is the velocity of the smaller, slower falling drops below,_{s}- and
*LWC*is the liquid water content.

The figure below provides a good conceptual picture of collision–coalescence. The collector drop must be falling faster than the smaller collected drop so that the two of them can collide. As the air streamlines bow out around the drop, they carry the smaller drops with them around the drop, and the effective cross-sectional area becomes less than the actual cross-sectional area, which is simply the cross-sectional area of a disk with a radius that is the sum of the radii of the large collector drop and the smaller collected drops. As drops get bigger, they have too much inertia to follow the air streamlines, thus making the collision more likely.

*E _{c}* is small for 10 μm drops, so by a random process, some drops become bigger than others and begin collecting smaller drops (see figure below).

*E*increases as the radius of the falling drop increases. When the larger falling drop gains a radius of more than 100 μm, its collision–coalescence efficiency is very good for all smaller drops down to sizes of about 10–20 μm.

_{c}Once a collecting drop has reached a radius of a few hundred μm, it is falling fast (*v _{L}* >>

*v*) and its collision–coalescence efficiency is close to 100%. Now take the following steps to rewrite Equation 5.16: (1)

_{s}*E*equal to 1, (2)

_{c}*v*>>

_{L}*v*, (3)

_{s}*v*= constant x

_{L}*r*, (4)

_{L}*m*= 4

_{L}*ρ*π

_{l}*r*

_{L}^{3}/3 , and (5) solve for

*dr*/

_{L}*dt*. Once you take these steps, you can show that

*dr*/

_{L}*dt*is proportional to

*r*. That is, the bigger the drop gets the faster it grows. Separating variables (

_{L}*r*and

_{L}*t*) and integrating from

*r*= 0 at

_{L}*t*= 0 to arbitrary values of

*r*and

_{L}*t*reveals that

*r*increases exponentially with time:

_{L}$${r}_{d}\propto \mathrm{exp}(time)$$

With the constant of proportionality between the terminal velocity and drop radius set at 0.8 x 10^{–3} s^{–1} and LWC = 1 g m^{–3}, it can be shown that a drop can grow from 50 μm to 1000 μm by collision–coalescence in only 25 minutes. So, the activated cloud drops grow to 10–20 μm by the slow growth of vapor deposition (square root of time). Then when collision–coalescence starts and produces a few big drops, they can grow exponentially with time.

Smaller drops are typically spherical. Once these drops get to be above a mm in radius, they become increasingly distorted, with a flattened bottom due to drag forces, and they look a little like the top half of a hamburger bun. They can be further distorted so that the middle of the bun-shape gets pushed up by the drag forces so that the drop takes on a shape resembling an upside down bowl.

Eventually the drops break up, either by getting thin enough in the middle that they break into pieces or by colliding with other drops so hard that filaments or sheets of liquid break off to form other drops. These processes create a whole range of sizes of drops. Thus rain consists of drops that have a wide spectrum of sizes. The following video (2:50) entitled "How Raindrops are Formed" starts with a simplified view of the atmosphere's water cycle, but then shows examples of a falling drop, collision–coalescence, and cloud-drop breakup.

For riming, capture nucleation, and aggregation, there are similar equations with terms similar to those in Equation 5.16—an area swept out, a collection efficiency, the relative velocity, and the liquid or solid mass concentration of the smaller drops or ice. These are typically a bit more complicated if the ice is not spherical, but the concepts are the same. These ice collision–coalescence processes are able to produce ice particles big enough to fall, and if these particles warm as they pass through the warm part of the cloud, they can turn into liquid rain. A significant fraction of rain in the summer can come from ice collision–coalescence processes above the freezing line in the clouds.