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4. Initiation of Raindrops by Collision and Coalescence. 4.1 Introduction to precipitation physics. 4.2 Setting the stage for coalescence. 4.3 Droplet growth by collision and coalescence. 4.4 Growth models and discussion. 4.1 Precipitation Physics.

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4. Initiation of Raindrops by

Collision and Coalescence

4.1 Introduction to precipitation physics

4.2 Setting the stage for coalescence

4.3 Droplet growth by collision and coalescence

4.4 Growth models and discussion


4.1 Precipitation Physics

  • Central task: Explain how raindrops can be created by condensation and coalescence in times as short as 20 minutes.
    • Observed interval between initial drop development of a cumulus cloud and the first appearance of rain.
  • Collisions and coalescence
    • Theory still uncertain
  • Drops need to grow to 20m before collision and coalescence processes occur in significant numbers.

Drops size

  • Small drops
    • small collision cross-section and slow settling speeds.
    • Little chance of colliding.
  • Coalescence
    • Drops spectrum has a spread of sizes and fall velocities
    • Some drops of 20m must exist.


  • Rate of collision  r4.
    • Once coalescence begins, it proceeds at an accelerating pace.
    • By time a few drops reach 30m, coalescence is the dominating processes.
  • Typical 1mm diameter raindrop may be the result of order of 105 collisions.
  • Before this can begin, some other processes must account for production of a few droplets as large as 20m in radius
    • 1 in 105 drops, or 1 per liter of cloud volume!

4.2 Setting the stage

  • A cloud drop can grow by condensation to a radius of 20m in 10 minutes under constant S if super-saturation is 0.5%.
    • This would require a sustained updraft of 5 m/s or greater.
  • Something else is causing this spectral broadening toward larger sizes.
  • Giant sea ice nuclei?
    • Not observed
  • Mixing between the cloud and its environment
    • Most likely explanation.

Homogeneous mixing

  • Sub-saturated (RH < 100%) cloud-free air is entrained into a cloud.
  • Mixing occurs quickly and completely
    • all droplets at a given level are exposed to the same sub-saturation.
  • Drops evaporate until saturation is once again reached.

Homogeneous mixing

  • Effect:
    • Reduces all droplet sizes by evaporation
    • Reduces the concentration by dilution in proportion to the amount of outside air introduced.
    • Possibly introduce newly activated droplets.
  • Time required for mixing is short compared to the time for drops to evaporate and re-establish vapor equilibrium.
  • Explains broadening of drops to smaller sizes, but not the creation of larger drops.

Inhomogeneous mixing

  • Time scale of droplet evaporation is short compared to that of turbulent mixing.
  • Evaporation proceeds rapidly in region just exposed to entrained air.
    • Creating volumes of air that are drop free, but saturated.
  • Volumes mix with unaffected cloud, reducing concentration by dilution without changing their size.
  • Cloud consists of many of these volumes with different sizes and mixing histories.
  • Again, can be used to explain drop broadening to smaller sizes, but not broadening to large sizes.

Cloud top entrainment

  • Mixing of dry air and environmental air at cloud top.
  • Contributes to broadening of cloud drop spectra to larger drops sizes.
  • “Understanding the significance of cloud-top entrainment may eventually explain many of the observed microphysical characteristics of clouds.” – 1989

4.3 Droplet growth by collision and coalescence


  • Gravitational force
    • Dominates in clouds.
    • Large drops fall faster than small drops.
    • Large drops overtake and capture a fraction of these small drops.
  • Electrical force
    • Enhance the collection of small droplets
    • Usually strong local effect.
  • Aerodynamic force
    • Some drops are swept aside in the air-stream around the drops.

Collision efficiency

  • Ratio of actual number of collisions to number of collisions geometrically possible.
  • Factors:
    • Size of collector drop.
    • Size of the collected droplets.
  • Collisions don’t guarantee coalescence.


  • Bounce apart.
  • Coalesce, and remain permanently together.
  • Coalesce temporarily, separate, and retain their identities.
  • Coalesce temporarily, separate, and break into a number of small drops.
  • For r < 100 m, 1 and 2 are important.

More efficiency

  • Coalescence efficiency
    • Ratio of Number of coalescences/ Number of collisions.
  • Collection efficiency
    • Collision efficiency x Coalescence efficiency
  • Charged drops or electrical fields present?
    • Coalescence efficiency  1
    • Clouds, collection efficiency = collision efficiency
  • Task: Determine the collision efficiency or collision rates among a population of droplets.


  • Three steps
  • Determine droplet terminal fall speed.
  • Determine collision frequency.
  • Growth equations.


A drop with a radius of 40 micrometers is

at the cloud base (z=0). The cloud has a

liquid water content of 1.5g/m and a steady

updraft of 2m/s. The terminal velocity of the

drop is given by u=(8X10 s )R. R is in mm.

Assume a collection efficiency of unity.




  • What is the size of the drop when it
  • begins to fall?
  • 2. What is the maximum height that the drop
  • will reach?

Another Example

The liquid water content of a cloud 2 km in depth varies linearly from 1

at the base to 3

at the top. A drop of 100 diameter starts to fall from

the top of the cloud. What will be its size when it leaves the cloud base?

Assume that the collection efficiency is 0.8 and that there is no updraft.


4.4 Growth models and discussion

* Statistical-discrete growth

The Telford and Robertson Models

  • Statistical fluctuation in droplet concentration

- Initial bimodal size distribution of droplets

- A drop grows by discrete collision and capture

events, not by continuous growth processes

- Some drops have more collisions than others


- Important in early stage growth to get a few

larger drops

- Rain is produced when one drop in 10 gets an

initial head start and then grows by gravitational


- Require shorter time for a droplet to reach raindrop

than continuous growth

- Collision efficiencies are not unity


* Stochastic growth

  • Consider few larger drops which have
  • made a coalescence collision after a
  • rather short time

- The next collisions are more favorable

giving a further widening of the drop

size spectrum


- To describe how a size distribution of droplets

changes with collection

  • Every possible combination of droplets
  • that can coalesce
  • The probability of each coalescence
  • The change in these probabilities after
  • each coalescence


- 10 of 100 large droplets will

collect a small droplet during

a given time

- Then 1 in 10 of each large size

will collect a smaller droplet

- Large droplets then grow at

different rates

- The distribution spreads


Physical Processes responsible

for broadening size distributions

- Autoconversion

- Accretion

- Large hydrometeor

self collection


* Effect of condensation on coalescence


Narrowing Spectrum

Accelerating coalescence


* Effects of turbulence on collisions

and coalescence

Three possible mechanisms:

i. Drops of different sizes respond differently to

a fluctuating velocity

ii. The overlapping of turbulent eddies

iii. Abrupt inhomogenieties – a few intense turbulence

surrounded by areas of weak turbulence


* Remarks

- The collision-coalescence process is how precipitation

forms in warm clouds (those clouds that remain above


- For the process to work efficiently, the droplet spectrum

cannot be too narrow. Otherwise, the droplets will have

similar terminal velocities and collisions will be infrequent.

- As the large drops get larger than 4 or 5 mm, they

become unstable and break apart. This creates some

additional large drops which can themselves start to



* There are still some unknown features of the

collision-coalescence process

i. Rain has been observed to occur in warm clouds within 15 minutes. Yet, our current understanding of collision-coalescence suggests a much longer time period is needed.

  • The mechanism by which a few, large drops form is unknown, but once they do form they can grow.

iii. The answer probably lies in a better understanding

of how turbulence affects droplet populations.

Statistical methods are probably also needed to

better understand and model the process by which

warm cloud precipitation develops.


Meteorology 342

Homework (4)

1. Problem 8.1

2. A drop with an initial radius of 100 micrometers falls through

a cloud containing 100 droplets per cubic centimeter, which

it collects in a continuous manner with a collection efficiency

of 0.8. If all the cloud droplets have a radius of 10 micrometers,

how long will it take for the drop to reach a radius of 1 millimeter?

Assume a drop fall speed similar to that in problem 1. Also assume

the cloud droplets are stationary and that the updraft velocity in

the cloud is negligible.