1. Drop growth by condensation: Quick review
Drop growth by condensation is modeled as a process of diffusion of water vapor to the surface
The “driving force” is the difference between the vapor pressure of water over the droplet, and the vapor pressure of the environment
A gradient (profile of concentration changes with distance) exists between the drop surface and the far-field environment
Sign can be positive or negative – same equation is used for evaporation / condensation
The rate of mass addition is largest for larger drops, BUT the rate of change of radius is largest for smaller drops (condensation narrows a drop size distribution)
What is the water vapor pressure at the drop surface?
May depend on curvature and on solutes in the drop
If the drop is very dilute, and large enough to approximate a plane surface, it may simply equal the water saturation vapor pressure … BUT
The drop temperature is not the same as the environmental T if the drop is gaining or losing mass, because of the release / absorption of latent heat during the phase change!
So we need to find out what the T of the drop is, in order to look up the saturation vapor pressure at that temperature
The latent heating effect will tend to REDUCE the driving force (slows evaporation or condensation)

2. Review, continued
The drop T rises until the T gradient between the drop and the environment is large enough to make the rate of heat conduction from the drop, equal to the rate of heat released by condensation
Can see from above the equations are coupled (can’t get Ta until you know growth rate, which in turn depends on Ta)
Invoke Clausius-Clapeyron: plug into drop growth equation and use approx that δ << 1
End up with single drop growth equation with Ta not appearing explicitly anywhere – can solve numerically

3. How does nature create a raindrop this big??
Wallace & Hobbs (1977)
How does nature create a raindrop this big??
We only need about 1 per liter…
Growth mechanisms:
Water vapor condensation
Droplet coalescence
Ice processes
Simple calculation shows that for constant supersaturation of 1%, it takes an hour for a drop to grow to ~100 µm
 Only 0.1% of the mass of an average raindrop!
“indirect effects”

4. Collision-Coalescence Growth
Since we know that condensation growth alone cannot produce precipitation-sized drops (> 100 µm) in reasonable cloud lifetimes, we must invoke other mechanisms to explain the production of precipitation in ‘warm’ clouds. Warm clouds contain no ice and their summit temperatures are greater than 0°C.
Invoke the collision-coalescence mechanism
Collisions between cloud droplets can occur to produce a larger droplet by coalescence with neighboring droplets
These initial collisions are motivated by the presence of different-sized cloud droplets -- this size distribution can be caused by nucleation on both small (hygroscopic) and large (wettable) CCN. (Broadening of the cloud droplet size distribution is discussed later)
Once a larger cloud droplet is formed, it will collide with other droplets and thus grow rapidly.
CCN spectra (giant nuclei)
Broadening via turbulence
Inhomogeneous mixing
Three mechanisms that act to broaden the droplet size distribution.

5. Droplet fall speeds
As a droplet falls, the streamlines around it bend. The terminal fallspeed is determined by the balance between acceleration due to gravity (which depends on droplet MASS) and the drag on the particle (which depends on an “effective” CROSS SECTIONAL AREA – size and shape, and speed)

6. Regimes for droplets

7. Over certain ranges, power laws apply:
vp a Dpn

8. Conceptual model A bimodal spectrum produced by some means r1
r1=radius of collector (drop)
r2=radius of collected droplet
Collision efficiency E
drops making collisions
not all drops in path of collector are expected to collide with the collector E=
drops in path of collector
Coalescence efficiency E’
sticking collisions
E’=
not all drops that collide are expected to coalesce
total collisions
Collection efficiency

9. Geometry (from Lamb & Verlinde)
Imagine we have a bimodal spectrum, produced by some means:
So drops are falling with DIFFERENT fallspeeds
rL=radius of collector (drop)
rs=radius of collected droplets
Volume of cylinder that is swept out, per unit time:
Area of cylinder that is swept out

10. Some of the small drops will follow the air streamlines around the drop, as shown. This means that not all the drops located within the cylinder that is swept out, will actually hit the collecting drop.
To account for the “limited success” of potential collisions, we introduce an efficiency factor, E
Here yc is the limiting impact parameter
SO the remaining problems are:
- Defining the limiting impact parameters for pairs of drop sizes
- Deciding if each collision actually “sticks” the drops together (coalescence)

11. Collection efficiencies near unity only for
Points to note:
– E is small if rs < 5 µm, independent of rL
(so, for E > 0.1, rs > 5 µm – ~minimum threshold in cloud)
– E increases rapidly for rs > 5 µm (don’t follow streamlines as well)
– but also need rL > 20 µm for E > 0.1
– Decrease in E after maximum is due to increasing similarity in fallspeeds
– when E increases (upturn) near end of curves, this is due to wake effect (lower drop distorts flow field, lets larger drop fall faster)
Displayed as family of constant-rL curves
Collection efficiencies near unity only for
rL > 50 µm and rs > 10 µm

12. For fixed , E increases with a1. Larger drop has more inertia.
Pruppacher & Klett (1978)
For fixed , E increases with a1. Larger drop has more inertia.
E is small for , small. Small droplets tend to move around collector in streamline flow.
When , increases, E increases. Collected droplets tend to move more in straight line paths and do not flow around collection as readily.
When , E decreases. Relative velocity between the droplets of similar size is small allowing deflective forces to inhibit collection
For large a, and WAKE CAPTURE

13. As drop sizes approach each other, coll eff increases (but prob in atmosphere is small…)
Drops with diameters < 50 µm have small collision efficiency
 growth to sizes larger than this is required to accelerate accretional growth
Need to multiply these numbers by coalescence efficiencies

14. COALESCENCE Ec = E × ε water surface water surface
Grazing Collision
Ec = E × ε
water surface
More direct
collision
water surface
results in coalescence
event
Critical angle is
but relative drop sizes is also an important factor.

15. The collision Weber number expresses the ratio of external pressure force to surface tension force. As Weber number increases, drop deforms more easily.
The figure, from experimental data, shows that the ability of the large drop to deform as the small drop approaches, results in cushion of air that impedes coalescence.

16. If collision occurs, coalescence will result.
Not all collisions will result in coalescence.
For drops rebound due to cushion of air that exists between them.
Air cushion most likely squeezed out for dissimilar sized drops.

17. Net effects Ec ~ 50% over a broad range Rapid increase as rs > 5 µm
Strong decrease as rs increases
Net effect: Ec maximizes at intermediate small-drop sizes
Ec ~ 50% over a broad range

18. Note about breakup (large drops)

19. Continuous Collection Equation
GOAL: Derive an expression for the growth rate of a droplet by collection of small droplets
terminal fall speed
r2 with fall speed V2 (depends on r)
Assume small collected droplets are uniformly distributed in the cloud.
Collector drops of the same size collect droplets at the same rate, that is, collector drops of a given size all grow at the same rate.
Drops of size R have the same probability for undergoing collisions with small droplets in given time
We need to consider the portion of the cloud liquid water content that is collected by the drop. The mass to be collected resides within an imaginary cylinder swept out by the collector drop.

20. wl Ec Liquid water content Assuming cloud drafts are small
Volume swept out
per unit time
wl Ec
collection efficiency
Mass of water in volume collected
This is the Continuous Collection Equation

21. For spherical water droplets,
Since
Equation (1) becomes
For spherical water droplets,
Therefore,
Since increases with , and increases with ,
Coalescence growth is an accelerating process
condensation
collection
time

22. Consider growth of a droplet in a modest cumulus cloud -
collisional
growth
Drop fall speed exceeds updraft speed –
drop falls back through the cloud
w, updraft speed
CCN activated
Derive equation for drop size as a function of position in the cloud
Motion of collector drop
updraft speed
drop fall speed
(function of )
From chain rule,
from continuous collection equation

23. drop size as a function of drop position .or
Consider drop having size at [cloud base].
Let at some height in the cloud. Drop
grows as it moves upward in the cloud to height
Now integrate above equation;
For constant with height,
On drops downward path, assume
Therefore
(- sign needed since increases with decreasing )

24. Continuous growth through a layer with liquid water content 1
EXAMPLE
Continuous growth through a layer with liquid water content 1
Uniform liquid water
content . Assume
Then
Therefore,

25. Calculations with this equation indicate the following, as an example,
1; 45mins, ascent height 2.2km, r=.15
2; 15mins, cloud base, r=0.7mm 2 1
Examples:
Produce small drops since drops begin to fall
back to cloud base with relatively small sizes
weak
updraft
case
Drops carried to greater heights since path length through liquid water content is much larger compared to clouds with weak updrafts – however clouds with strong updrafts must also be deeper – which they normally are – and they contain larger
strong
updraft
case

26. Updraft velocity (m s-1) Maximum supersaturation (%)
Some Useful (Ballpark) Values (Table 15.3, Seinfeld & Pandis)
Cloud Type
Updraft velocity (m s-1)
Maximum supersaturation (%)
Continental cumulus
~1 – 17
0.25 – 0.7
Maritime cumulus
~1 – 2.5
0.3 – 0.8
Stratiform
~0 – 1
~0.05
Fog --
~0.1