3. Droplet Growth by Condensation

3.1 Growth of an individual droplet by condensation

3.2 Evaporation of droplets

3.3 Growth of droplet populations

3.4 Factors affecting growth theory

• CCN – once a drop is activated.– Cloud condensation nuclei

• How fast does a drop grow?– Rate of diffusional growth of water molecules

from the vapor onto its surface.

– Occurs before and after r*.

3.1 Growth of an individual droplet by condensation

Vapor Diffusion

• Drop is located in a vapor field with the concentration of vapor molecules a distance R from the droplet center, n(R). Drop has radius r.

• Isotropic conditions assumed, so n(R) or (R) does not depend on direction from the droplet.

• Concentration of vapor field molecules is assumed to satisfy the diffusion equation.

Heat Diffusion

• Temperature is usually not the same as the ambient temperature.

• Must be determined by considering the heat transfer equation between the droplet and its surroundings.

• Condensation Latent heat release Drop temperature rises above the ambient value.

Rate of Drop Growth

• Rain drops form in around 30 minutes.

• Condensation

41,000s = 11.4 hours

• Something else must be happening.

3.2 Evaporation of droplets

• Once a droplet falls from a cloud it enters an unsaturated environment (S<1). The droplets will begin to evaporate.

• The evaporation of the droplet is also governed

by the same equation as condensation.

• A small droplet will evaporate very quickly and disappear, while larger droplets will last longer.

• Rapid increase of distance with the increase of radius

• Cloud drops

r < 0.1mm

• Drizzle drops

r is near 0.1mm

• Raindrops

r > 0.1mm (100m)

3.3 Growth of droplet populations

• Goal of cloud physics:– To understand the processes that shape the droplet

distribution.

• Facts– A droplet that forms on a small

condensation nuclei is seen to grow initially at a rate faster than droplets with large nuclei.

– Once small nuclei drops reach a comparable radius, the growth rate is about the same.

– Droplet size distribution is narrowing.

• If environment has an excess of vapor over the equilibrium value…….– Growth by condensation

• If dry environmental air is mixed with cloudy air……– Evaporation

• Sedimentation (gravitational settling) Coagulation (coalescence)

– Important for growth of larger droplets

Early Development

• Cloud drops are too small for sedimentation or coalescence.

• Condensation is the dominant growth process.

• Controlled by the ambient saturation ratio.

• Therefore, we need to know the details of a developing cloud.

Variations

Model

• It is possible to start with an assumed condensation nuclei distribution and updraft velocity, and calculate the evolution of the droplet spectrum

– 15 cm/s updraft

– Moderate concentration of CN.

Results

More Results

Observations

• Drop distributions within a short distance from the cloud base are narrow (5-10m).

– Consistent with drop growth by condensation.

• Higher in cloud, droplet spectra are often broader than predicted by drop growth by condensation.

3.4 Factors affecting growth theory

• Kinetic effects

• Ventilation effects

• Non-stationary growth

• Unsteady updraft

• Statistical effects

Kinetic Effects

• Rate of heat mass and momentum transfer between drop and its surroundings depends on Knudsen number.

• Knudsen number = l/rl = Mean free path in air

r = radius of drop

l 0.06m for normal conditions at sea level.

l/r << 1

• Fields of vapor and temperature are continua.

• Exchange described by Maxwell continuum approximation.

• Everything we learned before is valid.

l/r >> 1

• Vapor and temperature fields are not considered continua.

• Exchange described by Molecular Collision theory.

• Appropriate for particles in the air smaller than 0.06m.

• Drop has to travel a distance before encountering vapor in the air.

l/r ~ 1

Neither the free molecular nor the Maxwell continuum approximation is valid.

For small cloud droplets having radiusbetween 0.1 and 1 micron,

Kinetic Effects• Tends to reduce drop radius

for all times.

• As drop grows, solution tends toward continuum solution.

• Rate of growth after sufficient time becomes the same.

• Growth of smaller drops are slowed more than large drops.

• Broader cloud drop spectrum.

Kinetic Effects

Ventilation Effects

• Vapor field is spherically symmetric (isotropic)– Good for a drop at rest. What about falling drops?

• Rate of heat and mass transfer increase and are greatest on upstream side.

• For large drops, negligible compared to growth by collision or coalescence.

• Very significant for evaporation of rain drops and other precipitation particles..

Non-Stationary growth

• Fields of vapor and temperature about a drop are not exactly steady.– Surface of drop is always expanding or contracting.

– Drops is moving around requiring adjustment of the diffusion field.

• Temperature and vapor fields adjust very quickly to the presence of a drop– molecular diffusion

– ~10s to reach steady state conditions.

– Good approximation to neglect this effect..

Unsteady Updraft

• Accelerating and decelerating regions of updraft.

• Capable of broadening the spectrum by continual activation of fresh nuclei.

• Except for the creation of newly activated droplets in regions of upward acceleration, this effect had little impact on drop evolution.

Statistical effects

• Drops have different histories– Experience different super-saturations.

• Broadening of the drop spectrum.

• Currently regarded as a small effect.

• Drop spectral broadening is still an active area of research.

Meteorology 342 Homework (3)

1. Problem 7.2

2. For radii greater than a few microns, the curvature and solution effects become negligible. Find how long it would take to grow a droplet from 5 to 50 microns for the following conditions, a saturation ratio of 1.0005, a temperature of 273K, and a pressure of 1000 hPa.

3. Consider the diffusional growth stage which will dominate the growth process until the drop grows to sufficient size so that collisions become important. Briefly discuss each physical process that must be considered if you were to accurately model the growth of an initial population of cloud droplets by diffusional processes.