Lecture 11: Growth of Cloud
Droplet in Warm Clouds
Wallace and Hobbs Section 6.4
Processes for Cloud Droplet
Growth
• How does this happen??
• By:
– condensation
– collision/coalescence
– ice-crystal process
today
Water Droplet Growth
Condensation & Collision
•
Condensational growth: diffusion of vapor to droplet
•
Collisional growth: collision and coalescence (accretion, coagulation)
between droplets
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
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 nonlinear 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 subvolumes, little change in others.
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
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.
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.
concept
• 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.
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
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??
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:
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
Factors promoting growth by
collision/coalescence
• Different drop sizes
• thicker clouds
• stronger updrafts
• consider a shallow
stratus deck....
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
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?
Water Droplet Growth - Collisional Growth
Continuum collection:
"capture" distance  VT R  VT rt
R
d(sweepout volume)
2
  R r VT R  VT r   R 2 VT R
dt

dm
collected mass :
  R 2 VT R LWC
dt
also :



VT(R)
VT(r)
dm
d 4 3 
2 dr

l
 r l  4r

dt
dt 3
dt
V R LWC
dR
substitution :
 T
dt
4 l
(increases w/R,
vs. condensation where
dR/dt ~ 1/R)
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - Collisional Growth
Integrating over size distribution of small droplets, r, and keeping R+r terms :
dR


dt
3 l
R  r 2
  R  VT R  VT r r 3 n(r) dr

PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - Collisional Growth
Accounting for collection efficiency, E(R,r):
dR


dt
3 l
R  r 2
  R  VT R  VT r E(R,r) r 3 n(r) dr
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 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - Collisional Growth
Collection Efficiency, E(R,r):
?
?
R&Y, p. 130
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - Collisional Growth
Terminal Velocity of Drops/Droplets:
• differences in fall speed lead to conditions for capture.
• terminal velocity condition: FG  FD
FD
VT(R)
VT(r)
constant fall velocity VT
4 3
FG   r  L g
3
C Re 
FD  6  m r u D
 24 
where
2u r
Re 
m
is the Reynolds’ number.
FG
r is the drop radius
L is the density of liquid water
g is the acceleration of gravity
m is the dynamic viscosity
u is the drop velocity (relative to air)
CD is the drag coefficient
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
Water Droplet Growth - Collisional Growth
Terminal Velocity Regimes:
Low Re; Stokes’ Law: r < 30 mm
CD Re
 1
24

2 r 2 gL
2
VT 
 k1 r ;
9m
k1 = 1.19 x 106 cm-1 s-1
High Re: 0.6 mm < r < 2 mm
CD ~ const.

0.5
VT
 
= k 2 r 0.5 ; k 2 = 2 x 10 3  o  cm 0.5 s-1
 
 o = 1.2 kg m-3
Intermediate Re: 40 mm < r < 0.6 mm
VT = k3 r ;
k3 = 8 x 10 3 s-1
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
air parcel
droplets
R&Y,
p. 132-133
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.
PHYS 622 - Clouds, spring ‘04, lect.4, Platnick
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
Collision Efficiency
•
•
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 622 - Clouds, spring ‘04, lect.4, Platnick