Meteo 300 -- Introduction to the Atmospheric Sciences
Chapter 4. Atmospheric Aerosol and Cloud Microphysical Processes
 
 
Warm Cloud Processes.

 A warm cloud is one that has a temperature above 0oC.  These occur at low altitudes in the summertime midlatitudes and in the tropics.

In studying these clouds, we are particularly interested in a few variables:

Cloud condensation nuclei have a large effect on these characteristics of clouds.  For instance, marine clouds have a typical droplet radius of 30 microns, with a typical droplet concentration of less than 100 per cubic centimeter, while continental clouds have a droplet size of less than 20 microns, with a typical droplet concentration of hundreds per cubic centimeter.  Yet the liquid water content (LWC) of the two types of clouds can be similar.  Part of this difference can be attributed to the greater number of cloud condensation nuclei over the continent compared to the number over the ocean.
 

Cloud drop growth by vapor deposition.

We have already discussed the first steps leading to cloud droplet formation.  For supersaturated air, the CCN are activated (take up water), and when the drop radii exceed the critical drop radius, the cloud droplet forms.  The critical radius is typically a few hundredths to a few tenths of a micron.  The cloud drops grow at the expense of the haze drops, which shrink because the supersaturation is decreasing as more water vapor is deposited on the cloud drops.

So how do we end up with cloud droplets that are 10 to 30 microns in radius?  An obvious idea is that the cloud droplet continues to take up water vapor.  This seems like a good idea.  As long as supersaturated conditions remain, net condensation occurs.

However, we know that the typical lifetime of a cloud is 30 minutes to an hour.  Thus, for vapor deposition to be responsible for the cloud droplets, it must occur in a matter of tens of minutes.

To determine the rate of increase in cloud droplet size with time, we need to know the diffusion coefficient of water vapor in air (D), and the water density.  You can follow through the derivation of this process on page 169 - 170 in Wallace and Hobbs.  The important result of converting the expression for mass increase to radius increase is given by expression (4.10):

dr/dt = (1/r) [Drv(infinity)/(rl e(infinity))] [e(infinity) - e(r)]
But [e(infinity) - e(r)]/[e(infinity)] is approximately equal to the supersaturation (S), since e(r) must be quite close to es near the surface (for a drop greater than 1 micron in radius).  Thus,
r dr/dt = Gl S
where Gl is roughly constant in a given environment.

So, integrating this expression, we have that r(t) is proportional to the square root of time!  For typical conditions, we see that a typical drop can grow to 10 microns in about 2000 seconds and to 20 microns in about 7000 seconds.  Anything larger, like rain drops (see Figure 4.19), is even longer.

So another mechanism must also be working to produce cloud drops large enough to become precipitation.
 

Cloud drop growth by collision and coalescence.

Larger drops have greater fall speeds (settling velocities) than small drops and the terminal fall speed depends on the radius of the drop squared.  This result comes from the balance of the drag and gravitational forces on the drop.  As the larger drops begin to fall, they collide with smaller drops and collect them, thus making an even larger drop with an even larger fall speed.  Figure (4.20) shows this process.

However, very small drops will follow the air around the larger drop and not be collected.  We can define a collision efficiency, E, as:

E = y2/(r12 + r22)                                 (equation 4.12)

The collision efficiency depends on the size of the collector drop and on the size of the drops being collected.  We also need to know the probability that drops, once they collide, will actually join together (coalesce).  They can simply bounce off of each other.  So we must also determine a coalescence efficiency.

When we multiply these two efficiencies together, we end up with the collection efficiency, Ec.  This mechanism is called the collision-coalescence mechanism.

Models using these combined mechanisms can predict cloud growth reasonably well.

Seeing that smaller drops have a lower collection efficiency than larger drops, it is easy to see why marine cumulus clouds are more likely to rain than similar clouds over land.