Transcript Meteorology 342 - Iowa State University

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.
Coalescence
• 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
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.
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 is
at the cloud base (z=0). The cloud has a
liquid water content of 1.5g/m3 and a steady
updraft of 2m/s. The terminal velocity of the
3 -1
drop is given by u=(8X10 s )R. R is in mm.
Assume a collection efficiency of unity.
1.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 g / m
at the base to 3
g / m3
3
at the top. A drop of 100 m 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.
Continuous-Growth equation
The 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
- 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
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 responsible
for 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
* Remarks
- 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.
* 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.