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 20m 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 20m must exist.

Coalescence

• Rate of collision r4.– Once coalescence begins, it proceeds at an

accelerating pace.– By time a few drops reach 30m, 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 20m 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 20m 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

Collisions• 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.

4.3 Droplet growth by collision and coalescence

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.

Options

1. Bounce apart.

2. Coalesce, and remain permanently together.

3. Coalesce temporarily, separate, and retain their identities.

4. 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.

Procedure

Three steps

• Determine droplet terminal fall speed.

• Determine collision frequency.

• Growth equations.

Terminal Velocity

Collision efficiencies

Collision efficiencies

Example:

A drop with a radius of 40 micrometers isat the cloud base (z=0). The cloud has a liquid water content of 1.5g/m and a steadyupdraft of 2m/s. The terminal velocity of thedrop is given by u=(8X10 s )R. R is in mm. Assume a collection efficiency of unity.

3

3 -1

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

The liquid water content of a cloud 2 km in depth varies linearly from 1 3/mg

at the base to 3 3/mg at the top. A drop of 100 diameter starts to fall fromm

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.

Another Example

Continuous-Growth equationThe Bowen Model

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 coalescence

- 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

• Every possible combination of droplets that can coalesce

- To describe how a size distribution of droplets changes with collection

• The probability of each coalescence

• The change in these probabilities after each coalescence

Example:

- 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 responsiblefor broadening size distributions

- Autoconversion

- Accretion

- Large hydrometeor self collection

* Effect of condensation on coalescence

Condensation

Narrowing Spectrum

Accelerating coalescence

East (1957)

* 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

- The collision-coalescence process is how precipitation forms in warm clouds (those clouds that remain above freezing).

- 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 grow.

* Remarks

* 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.

ii. 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.