1. Lecture 11: Growth of Cloud Droplet in Warm Clouds
Wallace and Hobbs Section 6.4

2. Processes for Cloud Droplet Growth
How does this happen??
By:
condensation
collision/coalescence
ice-crystal process
today

3. Condensation & Collision
Water Droplet Growth
Condensation & Collision
Condensational growth: diffusion of vapor to droplet
Collisional growth: collision and coalescence (accretion, coagulation) between droplets
PHYS Clouds, spring ‘04, lect.4, Platnick

4. Water Droplet Growth - Collisions
Droplets collide and coalesce (accrete, merge, coagulate) with other droplets. Collisions require different fall velocities between small and large droplets (ignoring turbulence and other non-gravitational forcing).
Diffusional growth gives narrow size distribution. Turns out that it’s a highly non-linear process, only need 1 in 105 drops with r ~ 20 µm to get process rolling.
How to get size differences? One possibility - mixing.
Homogeneous Mixing: time scale of drop evaporation/equilibrium much longer relative to mixing process. All drops quickly exposed to “entrained” dry air, and evaporate and reach a new equilibrium together Dilution broadens small droplet spectrum, but can’t create large droplets.
Inhomogeneous Mixing: time scale of drop evaporation/equilibrium much shorter than relative to turbulent mixing process. Small sub-volumes of cloud air have different levels of dilution Reduction of droplet sizes in some sub-volumes, little change in others.
Fig. from Bill Olsen
PHYS Clouds, spring ‘04, lect.4, Platnick

5. After a water droplet forms, it continues to grow by added condensation or sublimation
directly onto the particle. This is the slower of the two methods and usually results in
drizzle or very light rain or snow.
Cloud particles collide and merge into a larger drop in the more rapid growth process.

7. Water Droplet Growth - Collisions
Droplets collide and coalesce (accrete, merge, coagulate) with other droplets.
Collisions governed primarily by different fall velocities between small and large droplets (ignoring turbulence and other non-gravitational forcing).
Collisions enhanced as droplets grow and differential fall velocities increase.
Not necessarily a very efficient process (requires relatively long times for large precipitation size drops to form).
Rain drops are those large enough to fall out and survive trip to the ground without evaporating in lower/dryer layers of the atmosphere.
concept
Fig. from Bill Olsen
PHYS Clouds, spring ‘04, lect.4, Platnick

8. Collision/Coalescence
Collision/Coalescence - cloud droplet growth by collision
this is a dominant process for precipitation formation in warm clouds (tops warmer than about -15°C)
some cloud droplets will grow large enough and will start to fall in the cloud -->>
since the bigger drops fall faster than the smaller drops, they will "collect" the smaller drops - the bigger drop grows
droplet fall speed is called its terminal velocity
need droplets of different sizes for this process to really work
Q: what determines the droplets fall speed relative to the ground??

9. Droplet Fall Speeds and Droplet Growth
Q: what determines the droplets fall speed relative to the ground??
A: droplet size and updraft strength -->
given a growing cu with an updraft strength of 4 ms-1:
if the particle terminal velocity is -2 ms-1, the particles fall speed is: ANSWER
if the particle terminal velocity is -4 ms-1, the particles fall speed is: ANSWER
if the particle terminal velocity is -6 ms-1, the particles fall speed is:
2ms-2
0ms-1
2ms-1 downward

10. Life cycle of a droplet Growth by collision
the drop initially forms in the updraft of the cloud near cloud base
it grows in size by collisions
since Vg = w + Vt
Vg = ground relative fall speed of the drop
w = updraft velocity
Vt = drop's terminal velocity
then the drop will begin to fall when Vt > w

11. Factors promoting growth by collision/coalescence
Different drop sizes
thicker clouds
stronger updrafts
consider a shallow stratus deck....

12. Droplet Growth in a Shallow Stratus Deck
Often, drops will evaporate from shallow stratus before reaching the ground
or you may get drizzle if they are large enough

14. QUESTION FOR THOUGHT:
1.  Why is a warm, tropical cumulus cloud more likely to produce precipitation than a cold, stratus cloud?
2.  Clouds that form over water are usually more efficient in producing precipitation than clouds that form over land. Why?

15. Water Droplet Growth - Collisional Growth
Continuum collection:
VT(R) R
VT(r)
(increases w/R,
vs. condensation where
dR/dt ~ 1/R)
PHYS Clouds, spring ‘04, lect.4, Platnick
PHYS Clouds, spring ‘04, lect.4, Platnick

16. Water Droplet Growth - Collisional Growth
Integrating over size distribution of small droplets, r, and keeping R+r terms :
PHYS Clouds, spring ‘04, lect.4, Platnick
PHYS Clouds, spring ‘04, lect.4, Platnick

17. Water Droplet Growth - Collisional Growth
Accounting for collection efficiency, E(R,r):
If small droplet too small or too far center of collector drop, then capture won’t occur.
• E is small for very small r/R, independent of R.
• E increases with r/R up to r/R ~ 0.6
• For r/R > 0.6, difference is drop terminal velocities is very small.
–drop interaction takes a long time, flow fields interact strongly and
droplet can be deflected.
–droplet falling behind collector drop can get drawn into the wake
of the collector; “wake capture” can lead to E > 1 for r/R ≈ 1.
PHYS Clouds, spring ‘04, lect.4, Platnick
PHYS Clouds, spring ‘04, lect.4, Platnick

18. Water Droplet Growth - Collisional Growth
Collection Efficiency, E(R,r): ? ?
R&Y, p. 130
PHYS Clouds, spring ‘04, lect.4, Platnick
PHYS Clouds, spring ‘04, lect.4, Platnick

19. Water Droplet Growth - Collisional Growth
VT(R)
VT(r)
Terminal Velocity of Drops/Droplets:
• differences in fall speed lead to conditions for capture.
• terminal velocity condition:
constant fall velocity VT
where r is the drop radius
rL is the density of liquid water
g is the acceleration of gravity
m is the dynamic viscosity
is the Reynolds’ number u is the drop velocity (relative to air)
CD is the drag coefficient FD FG
B. Olsen
µ = dynamic viscosity of fluid
PHYS Clouds, spring ‘04, lect.4, Platnick

20. Water Droplet Growth - Collisional Growth
Terminal Velocity Regimes:
Low Re; Stokes’ Law: r < 30 mm
High Re: 0.6 mm < r < 2 mm
Intermediate Re: 40 mm < r < 0.6 mm
k1 = 1.19 x 106 cm-1 s-1
B. Olsen
PHYS Clouds, spring ‘04, lect.4, Platnick

21. PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
air parcel
droplets
R&Y, p
collector
drop
• Fig. 8.4: collision/coalescence process starts out slowly, but VT and E increase rapidly with drop size, and soon collision/coalescence outpaces condensation growth.
• Fig. 8.6:
– with increasing updraft speed, collector ascends to higher altitudes, and emerges as a larger raindrop.
– see at higher altitudes, smaller drops; lower altitudes, larger drops.
From Bill Olsen.
PHYS Clouds, spring ‘04, lect.4, Platnick

22. Precipitation Growth in Cold Clouds - Warm versus Cold Clouds
Our previous discussion regarding droplet growth by condensation and collisions is valid for warm clouds:
warm clouds - have tops warmer than about 0°C
comprised entirely of water

24. Collision Efficiency

32. Water Droplet Growth - Collisional Growth
Approach:
We begin with a continuum approach (small droplets are uniformly distributed, such that any volume of air - no matter how small - has a proportional amount of liquid water.
A full stochastic equation is necessary for proper modeling (accounts for probabilities associated with the “fortunate few” large drops that dominate growth).
Neither approach accounts for cloud inhomogeneities (regions of larger LWC) that appear important in “warm cloud” rain formation.
PHYS Clouds, spring ‘04, lect.4, Platnick
PHYS Clouds, spring ‘04, lect.4, Platnick