# 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 accelerates, 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. For instance, the terminal velocity of a 10 μm radius cloud drop is about 1 mm s–1, while the terminal velocity for a 100 μm drop is about 1 m s–1. Air flow around a falling particle. The shaded area is the cross sectional area of the particle. Note the movement of air around the particle. Only the air in innermost streamline collides with the particle; the rest goes around it. Credit: W. Brune (after Lamb and Verlinde)

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

$\frac{d m_{L}}{d t}=$ Area swept out*efficiency of collection*velocity difference*liquid water content

$\frac{d m_{L}}{d t}=A_{g} \cdot E_{c} \cdot\left(v_{L}-v_{s}\right) \cdot L W C$

$\frac{d m_{L}}{d t}=\pi\left(r_{L}+r_{s}\right)^{2} \cdot E_{c} \cdot\left(v_{L}-v_{s}\right) \cdot L W C$

• mL is the mass of the large drop that is falling,
• Ag is the geometric cross-sectional area for which collisions between the falling large drop and the many drops below is possible,
• c 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),
• vL is the velocity of the large drop and vs is the velocity of the smaller, slower falling drops below,
• 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-section becomes less than the actual cross-section, 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. Schematic of the maximum possible geometric cross-section of a large and small drop and the actual cross-section due to particles following air streamlines around the big particle. Credit: W. Brune (after Lamb and Verlinde)

Ec 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). Ecincreases 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. Collision–collection efficiencies for two drops, with the percent collision efficiency on the vertical axis; the ratio of the radius of the small drop, rs, to the radius of the large drop, rL, on the horizontal axis; and lines for individual large drop radii. Credit: W. Brune (after Rogers and Yau)

Once a collecting drop has reached a radius of a few hundred μm, it is falling fast and its collision–coalescence efficiency is close to 100%. The growth of its radius then has the approximate form:

$r_{d} \propto \exp (\text {time})$

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 they 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 particles. 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 break-up.

How Raindrops are Formed